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Investments T

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ZVI BODIE Boston University

ALEX KANE University of California, San Diego

ALAN J. MARCUS Boston College

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INVESTMENTS, TENTH EDITION Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright © 2014 by McGraw-Hill Education. All rights reserved. Printed in the United States of America. Previous editions © 2011, 2009, and 2008. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 0 DOW/DOW 1 0 9 8 7 6 5 4 3 ISBN 978-0-07-786167-4 MHID 0-07-786167-1 Senior Vice President, Products & Markets: Kurt L. Strand Vice President, Content Production & Technology Services: Kimberly Meriwether David Managing Director: Douglas Reiner Executive Brand Manager: Chuck Synovec Executive Director of Development: Ann Torbert Development Editor: Noelle Bathurst Director of Digital Content: Doug Ruby Digital Development Editor: Meg B. Maloney Digital Development Editor: Kevin Shanahan Executive Marketing Manager: Melissa S. Caughlin Content Project Manager: Bruce Gin Senior Buyer: Michael R. McCormick Design: Debra Kubiak Cover Image: Aleksandar Velasevic/Getty Images Typeface: 10/12 Times Roman Compositor: Laserwords Private Limited Printer: R. R. Donnelley All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. Library of Congress Cataloging-in-Publication Data Bodie, Zvi. Investments / Zvi Bodie, Boston University, Alex Kane, University of California, San Diego, Alan J. Marcus, Boston College.—10th Edition. pages cm.—(The McGraw-Hill/Irwin series in finance, insurance and real estate) Includes index. ISBN-13: 978-0-07-786167-4 (alk. paper) ISBN-10: 0-07-786167-1 (alk. paper) 1. Investments. 2. Portfolio management. I. Kane, Alex. II. Marcus, Alan J. III. Title. HG4521.B564 2014 332.6—dc23 2013016066 The Internet addresses listed in the text were accurate at the time of publication. The inclusion of a website does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites. www.mhhe.com

About the Authors ZVI BODIE

ALEX KANE

ALAN J. MARCUS

Boston University

University of California, San Diego

Boston College

Zvi Bodie is the Norman and Adele Barron Professor of Management at Boston University. He holds a PhD from the Massachusetts Institute of Technology and has served on the finance faculty at the Harvard Business School and MIT’s Sloan School of Management. Professor Bodie has published widely on pension finance and investment strategy in leading professional journals. In cooperation with the Research Foundation of the CFA Institute, he has recently produced a series of Webcasts and a monograph entitled The Future of Life Cycle Saving and Investing.

Alex Kane is professor of finance and economics at the Graduate School of International Relations and Pacific Studies at the University of California, San Diego. He has been visiting professor at the Faculty of Economics, University of Tokyo; Graduate School of Business, Harvard; Kennedy School of Government, Harvard; and research associate, National Bureau of Economic Research. An author of many articles in finance and management journals, Professor Kane’s research is mainly in corporate finance, portfolio management, and capital markets, most recently in the measurement of market volatility and pricing of options.

v

Alan Marcus is the Mario J. Gabelli Professor of Finance in the Carroll School of Management at Boston College. He received his PhD in economics from MIT. Professor Marcus has been a visiting professor at the Athens Laboratory of Business Administration and at MIT’s Sloan School of Management and has served as a research associate at the National Bureau of Economic Research. Professor Marcus has published widely in the fields of capital markets and portfolio management. His consulting work has ranged from new-product development to provision of expert testimony in utility rate proceedings. He also spent 2 years at the Federal Home Loan Mortgage Corporation (Freddie Mac), where he developed models of mortgage pricing and credit risk. He currently serves on the Research Foundation Advisory Board of the CFA Institute.

Brief Contents Preface

xvi

PART III

PART I

Equilibrium in Capital Markets 291

Introduction 1

9

1

The Capital Asset Pricing Model

The Investment Environment 1

291

2

10

Asset Classes and Financial Instruments 28

Arbitrage Pricing Theory and Multifactor Models of Risk and Return 324

11

3 How Securities Are Traded

The Efficient Market Hypothesis

59

349

4

12

Mutual Funds and Other Investment Companies 92

Behavioral Finance and Technical Analysis 388

13

PART II

Empirical Evidence on Security Returns

Portfolio Theory and Practice 117

PART IV

5

Fixed-Income Securities 445

Risk, Return, and the Historical Record 117

14

6 Capital Allocation to Risky Assets

Bond Prices and Yields

168

445

15

7 Optimal Risky Portfolios

The Term Structure of Interest Rates

205

16

8 Index Models

414

Managing Bond Portfolios

256

vi

515

487

Brief Contents PART V

PART VII

Security Analysis 557

Applied Portfolio Management 835

17 Macroeconomic and Industry Analysis

557

24

18 Equity Valuation Models

Portfolio Performance Evaluation

591

25

19 Financial Statement Analysis

International Diversification

635

882

26 Hedge Funds

PART VI

926

27

Options, Futures, and Other Derivatives 678

The Theory of Active Portfolio Management 951

28

20 Options Markets: Introduction

835

Investment Policy and the Framework of the CFA Institute 977

678

21 Option Valuation

722

REFERENCES TO CFA PROBLEMS

22 Futures Markets

GLOSSARY

770

NAME INDEX

23 Futures, Swaps, and Risk Management

G-1

799

SUBJECT INDEX

vii

I-1 I-4

1015

Contents Preface

Reverses / Federal Funds / Brokers’ Calls / The LIBOR Market / Yields on Money Market Instruments

xvi

PART I

2.2

CHAPTER 1 1.1

Real Assets versus Financial Assets 2

1.2

Financial Assets 3

1.3

Financial Markets and the Economy

34

Treasury Notes and Bonds / Inflation-Protected Treasury Bonds / Federal Agency Debt / International Bonds / Municipal Bonds / Corporate Bonds / Mortgages and Mortgage-Backed Securities

Introduction 1 The Investment Environment

The Bond Market

2.3

1

Equity Securities 41 Common Stock as Ownership Shares / Characteristics of Common Stock / Stock Market Listings / Preferred Stock / Depository Receipts

5

2.4

The Informational Role of Financial Markets / Consumption Timing / Allocation of Risk / Separation of Ownership and Management / Corporate Governance and Corporate Ethics

Stock and Bond Market Indexes

Stock Market Indexes / Dow Jones Averages / Standard & Poor’s Indexes / Other U.S. Market-Value Indexes / Equally Weighted Indexes / Foreign and International Stock Market Indexes / Bond Market Indicators

1.4

The Investment Process 8

1.5

Markets Are Competitive 9

Options / Futures Contracts

The Risk–Return Trade-Off / Efficient Markets

End of Chapter Material

1.6

2.5

Derivative Markets 51

The Financial Crisis of 2008

CHAPTER 3

How Securities Are Traded

15

Antecedents of the Crisis / Changes in Housing Finance / Mortgage Derivatives / Credit Default Swaps / The Rise of Systemic Risk / The Shoe Drops / The Dodd-Frank Reform Act 1.8

3.1

3.2

59

How Securities Are Traded 63 Types of Markets

24–27

Direct Search Markets / Brokered Markets / Dealer Markets / Auction Markets

CHAPTER 2

Asset Classes and Financial Instruments The Money Market

How Firms Issue Securities

59

Privately Held Firms / Publicly Traded Companies / Shelf Registration / Initial Public Offerings

Outline of the Text 23 End of Chapter Material

2.1

54–58

The Players 11 Financial Intermediaries / Investment Bankers / Venture Capital and Private Equity

1.7

44

Types of Orders

28

Market Orders / Price-Contingent Orders Trading Mechanisms

29

Dealer Markets / Electronic Communication Networks (ECNs) / Specialist Markets

Treasury Bills / Certificates of Deposit / Commercial Paper / Bankers’ Acceptances / Eurodollars / Repos and

viii

Contents 3.3

The Rise of Electronic Trading

3.4

U.S. Markets 69

68

5.1

New Trading Strategies 71 5.2

Algorithmic Trading / High-Frequency Trading / Dark Pools / Bond Trading 3.6

Comparing Rates of Return for Different Holding Periods 122 Annual Percentage Rates / Continuous Compounding

Globalization of Stock Markets 74 Trading Costs 76

5.3

Bills and Inflation, 1926–2012

3.8

Buying on Margin

5.4

Risk and Risk Premiums

3.9

Short Sales 80

3.7

76

3.10 Regulation of Securities Markets

83

5.5

87–91

Mutual Funds and Other Investment Companies 92 4.2

Investment Companies 92 Types of Investment Companies 93

The Normal Distribution

5.7

Deviations from Normality and Risk Measures

135 137

Value at Risk / Expected Shortfall / Lower Partial Standard Deviation and the Sortino Ratio / Relative Frequency of Large, Negative 3-Sigma Returns

Commingled Funds / Real Estate Investment Trusts (REITs) / Hedge Funds

5.8

Historic Returns on Risky Portfolios

141

Portfolio Returns / A Global View of the Historical Record

Mutual Funds 96 Investment Policies

5.9

Money Market Funds / Equity Funds / Sector Funds / Bond Funds / International Funds / Balanced Funds / Asset Allocation and Flexible Funds / Index Funds Costs of Investing in Mutual Funds

Long-Term Investments 152 Normal and Lognormal Returns / Simulation of LongTerm Future Rates of Return / The Risk-Free Rate Revisited / Where Is Research on Rates of Return Headed? / Forecasts for the Long Haul

How Funds Are Sold 4.4

130

5.6

Unit Investment Trusts / Managed Investment Companies / Other Investment Organizations

4.3

Time Series Analysis of Past Rates of Return

Time Series versus Scenario Analysis / Expected Returns and the Arithmetic Average / The Geometric (TimeWeighted) Average Return / Variance and Standard Deviation / Mean and Standard Deviation Estimates from Higher-Frequency Observations / The Reward-toVolatility (Sharpe) Ratio

CHAPTER 4

4.1

125

127

Holding-Period Returns / Expected Return and Standard Deviation / Excess Returns and Risk Premiums

Self-Regulation / The Sarbanes-Oxley Act / Insider Trading End of Chapter Material

118

Real and Nominal Rates of Interest / The Equilibrium Real Rate of Interest / The Equilibrium Nominal Rate of Interest / Taxes and the Real Rate of Interest

NASDAQ / The New York Stock Exchange / ECNs 3.5

Determinants of the Level of Interest Rates

99

End of Chapter Material

161–167

Fee Structure Operating Expenses / Front-End Load / Back-End Load / 12b-1 Charges

CHAPTER 6

Capital Allocation to Risky Assets

Fees and Mutual Fund Returns 6.1

4.5

Taxation of Mutual Fund Income

103

4.6

Exchange-Traded Funds 103

4.7

Mutual Fund Investment Performance: A First Look 107

4.8

Information on Mutual Funds

Risk and Risk Aversion

168

168

Risk, Speculation, and Gambling / Risk Aversion and Utility Values / Estimating Risk Aversion 6.2

Capital Allocation across Risky and Risk-Free Portfolios 175

6.3

The Risk-Free Asset 177

PART II

6.4

Portfolios of One Risky Asset and a Risk-Free Asset 178

Portfolio Theory and Practice 117

6.5

End of Chapter Material

110

112–116

6.6

182

Passive Strategies: The Capital Market Line End of Chapter Material

CHAPTER 5

Risk, Return, and the Historical Record

Risk Tolerance and Asset Allocation Nonnormal Returns

187

190–199

Appendix A:Risk Aversion, Expected Utility, and the St. Petersburg Paradox 199

117 ix

Contents Appendix B:Utility Functions and Equilibrium Prices of Insurance Contracts 203 Appendix C:The Kelly Criterion

8.5

Is the Index Model Inferior to the Full-Covariance Model? / The Industry Version of the Index Model / Predicting Betas / Index Models and Tracking Portfolios

203

CHAPTER 7

Optimal Risky Portfolios

End of Chapter Material

205

7.1

Diversification and Portfolio Risk

7.2

Portfolios of Two Risky Assets 208

7.3

Asset Allocation with Stocks, Bonds, and Bills

206 215

Equilibrium in Capital Markets 291

The Markowitz Portfolio Optimization Model

220

Security Selection / Capital Allocation and the Separation Property / The Power of Diversification / Asset Allocation and Security Selection / Optimal Portfolios and Nonnormal Returns 7.5

CHAPTER 9

The Capital Asset Pricing Model 291

Risk Pooling, Risk Sharing, and the Risk of LongTerm Investments 230

9.1

234–244

Appendix A:A Spreadsheet Model for Efficient Diversification 244 Appendix B:Review of Portfolio Statistics

9.2

249

8.1

256

A Single-Factor Security Market

257

The Input List of the Markowitz Model / Normality of Returns and Systematic Risk 8.2

The Single-Index Model

259

Estimating the Single-Index Model

302

9.3

The CAPM and the Academic World

313

9.4

The CAPM and the Investment Industry

315

316–323

CHAPTER 10

Arbitrage Pricing Theory and Multifactor Models of Risk and Return 324

264

10.1 Multifactor Models: An Overview

The Security Characteristic Line for Hewlett-Packard / The Explanatory Power of the SCL for HP / Analysis of Variance / The Estimate of Alpha / The Estimate of Beta / Firm-Specific Risk / Correlation and Covariance Matrix 8.4

Assumptions and Extensions of the CAPM

End of Chapter Material

The Regression Equation of the Single-Index Model / The Expected Return–Beta Relationship / Risk and Covariance in the Single-Index Model / The Set of Estimates Needed for the Single-Index Model / The Index Model and Diversification 8.3

291

Assumptions of the CAPM / Challenges and Extensions to the CAPM / The Zero-Beta Model / Labor Income and Nontraded Assets / A Multiperiod Model and Hedge Portfolios / A Consumption-Based CAPM / Liquidity and the CAPM

CHAPTER 8

Index Models

The Capital Asset Pricing Model

Why Do All Investors Hold the Market Portfolio? / The Passive Strategy Is Efficient / The Risk Premium of the Market Portfolio / Expected Returns on Individual Securities / The Security Market Line / The CAPM and the Single-Index Market

Risk Pooling and the Insurance Principle / Risk Sharing / Investment for the Long Run End of Chapter Material

284–290

PART III

Asset Allocation with Two Risky Asset Classes 7.4

Practical Aspects of Portfolio Management with the Index Model 278

325

Factor Models of Security Returns 10.2 Arbitrage Pricing Theory 327 Arbitrage, Risk Arbitrage, and Equilibrium / WellDiversified Portfolios / Diversification and Residual Risk in Practice / Executing Arbitrage / The No-Arbitrage Equation of the APT

Portfolio Construction and the Single-Index Model 271

10.3 The APT, the CAPM, and the Index Model

Alpha and Security Analysis / The Index Portfolio as an Investment Asset / The Single-Index-Model Input List / The Optimal Risky Portfolio in the Single-Index Model / The Information Ratio / Summary of Optimization Procedure / An Example

334

The APT and the CAPM / The APT and Portfolio Optimization in a Single-Index Market 10.4 A Multifactor APT 338 10.5 The Fama-French (FF) Three-Factor Model

Risk Premium Forecasts / The Optimal Risky Portfolio

End of Chapter Material

x

342–348

340

Contents Limits to Arbitrage and the Law of One Price

CHAPTER 11

The Efficient Market Hypothesis

“Siamese Twin” Companies / Equity Carve-Outs / Closed-End Funds

349

11.1 Random Walks and the Efficient Market Hypothesis 350

Bubbles and Behavioral Economics / Evaluating the Behavioral Critique

Competition as the Source of Efficiency / Versions of the Efficient Market Hypothesis 11.2 Implications of the EMH

12.2 Technical Analysis and Behavioral Finance

354

Momentum and Moving Averages / Relative Strength / Breadth

Technical Analysis / Fundamental Analysis / Active versus Passive Portfolio Management / The Role of Portfolio Management in an Efficient Market / Resource Allocation

Sentiment Indicators Trin Statistic / Confidence Index / Put/Call Ratio A Warning

11.3 Event Studies 359 11.4 Are Markets Efficient?

End of Chapter Material

362

The Issues

Empirical Evidence on Security Returns 414

Weak-Form Tests: Patterns in Stock Returns Returns over Short Horizons / Returns over Long Horizons

13.1 The Index Model and the Single-Factor APT 415 The Expected Return–Beta Relationship

Predictors of Broad Market Returns / Semistrong Tests: Market Anomalies

Setting Up the Sample Data / Estimating the SCL / Estimating the SML

The Small-Firm-in-January Effect / The NeglectedFirm Effect and Liquidity Effects / Book-to-Market Ratios / Post–Earnings-Announcement Price Drift

Tests of the CAPM / The Market Index / Measurement Error in Beta 13.2 Tests of the Multifactor CAPM and APT 421

Strong-Form Tests: Inside Information / Interpreting the Anomalies

Labor Income / Private (Nontraded) Business / Early Versions of the Multifactor CAPM and APT / A Macro Factor Model

Risk Premiums or Inefficiencies? / Anomalies or Data Mining? / Anomalies over Time

13.3 Fama-French-Type Factor Models

Bubbles and Market Efficiency 11.5 Mutual Fund and Analyst Performance

426

Size and B/M as Risk Factors / Behavioral Explanations / Momentum: A Fourth Factor

375

Stock Market Analysts / Mutual Fund Managers / So, Are Markets Efficient?

13.4 Liquidity and Asset Pricing

433

13.5 Consumption-Based Asset Pricing and the Equity Premium Puzzle 435

380–387

Consumption Growth and Market Rates of Return / Expected versus Realized Returns / Survivorship Bias / Extensions to the CAPM May Resolve the Equity Premium Puzzle / Liquidity and the Equity Premium Puzzle / Behavioral Explanations of the Equity Premium Puzzle /

CHAPTER 12

Behavioral Finance and Technical Analysis 388 12.1 The Behavioral Critique

407–413

CHAPTER 13

The Magnitude Issue / The Selection Bias Issue / The Lucky Event Issue

End of Chapter Material

400

Trends and Corrections

389

End of Chapter Material

442–444

Information Processing Forecasting Errors / Overconfidence / Conservatism / Sample Size Neglect and Representativeness

PART IV

Behavioral Biases

Fixed-Income Securities 445

Framing / Mental Accounting / Regret Avoidance Affect

CHAPTER 14

Prospect Theory

Bond Prices and Yields

Limits to Arbitrage

14.1 Bond Characteristics 446

Fundamental Risk / Implementation Costs / Model Risk

Treasury Bonds and Notes

xi

445

Contents Accrued Interest and Quoted Bond Prices

16.2 Convexity 525

Corporate Bonds

Why Do Investors Like Convexity? / Duration and Convexity of Callable Bonds / Duration and Convexity of Mortgage-Backed Securities

Call Provisions on Corporate Bonds / Convertible Bonds / Puttable Bonds / Floating-Rate Bonds

16.3 Passive Bond Management

Preferred Stock / Other Domestic Issuers / International Bonds / Innovation in the Bond Market

533

Bond-Index Funds / Immunization / Cash Flow Matching and Dedication / Other Problems with Conventional Immunization

Inverse Floaters / Asset-Backed Bonds / Catastrophe Bonds / Indexed Bonds

16.4 Active Bond Management

14.2 Bond Pricing 452 Bond Pricing between Coupon Dates

End of Chapter Material

14.3 Bond Yields 458 Yield to Maturity / Yield to Call / Realized Compound Return versus Yield to Maturity 14.4 Bond Prices over Time

Security Analysis 557

Yield to Maturity versus Holding-Period Return / ZeroCoupon Bonds and Treasury Strips / After-Tax Returns

CHAPTER 17

468

Macroeconomic and Industry Analysis 557

Junk Bonds / Determinants of Bond Safety / Bond Indentures Sinking Funds / Subordination of Further Debt / Dividend Restrictions / Collateral

17.1 The Global Economy

558

17.2 The Domestic Macroeconomy

Yield to Maturity and Default Risk / Credit Default Swaps / Credit Risk and Collateralized Debt Obligations End of Chapter Material

545–556

PART V

463

14.5 Default Risk and Bond Pricing

543

Sources of Potential Profit / Horizon Analysis

479–486

560

17.3 Demand and Supply Shocks

562

17.4 Federal Government Policy

563

Fiscal Policy / Monetary Policy / Supply-Side Policies 17.5 Business Cycles 566

CHAPTER 15

The Term Structure of Interest Rates

The Business Cycle / Economic Indicators / Other Indicators

487

17.6 Industry Analysis 571

15.1 The Yield Curve 487

Defining an Industry / Sensitivity to the Business Cycle /

Bond Pricing 15.2 The Yield Curve and Future Interest Rates

Sector Rotation / Industry Life Cycles

490

Start-Up Stage / Consolidation Stage / Maturity Stage / Relative Decline

The Yield Curve under Certainty / Holding-Period Returns / Forward Rates 15.3 Interest Rate Uncertainty and Forward Rates 15.4 Theories of the Term Structure

Industry Structure and Performance

495

Threat of Entry / Rivalry between Existing Competitors / Pressure from Substitute Products / Bargaining Power of Buyers / Bargaining Power of Suppliers

497

The Expectations Hypothesis / Liquidity Preference 15.5 Interpreting the Term Structure

501

End of Chapter Material

582–590

15.6 Forward Rates as Forward Contracts 504 End of Chapter Material

506–514

CHAPTER 18

Equity Valuation Models CHAPTER 16

Managing Bond Portfolios 16.1 Interest Rate Risk

18.1 Valuation by Comparables

515

591

591

Limitations of Book Value 18.2 Intrinsic Value versus Market Price

516

Interest Rate Sensitivity / Duration / What Determines Duration?

18.3 Dividend Discount Models

593

595

The Constant-Growth DDM / Convergence of Price to Intrinsic Value / Stock Prices and Investment Opportunities / Life Cycles and Multistage Growth Models / Multistage Growth Models

Rule 1 for Duration / Rule 2 for Duration / Rule 3 for Duration / Rule 4 for Duration / Rule 5 for Duration

xii

Contents Index Options / Futures Options / Foreign Currency Options / Interest Rate Options

18.4 Price–Earnings Ratio 609 The Price–Earnings Ratio and Growth Opportunities / P/E Ratios and Stock Risk / Pitfalls in P/E Analysis / Combining P/E Analysis and the DDM / Other Comparative Valuation Ratios

20.2 Values of Options at Expiration

Price-to-Book Ratio / Price-to-Cash-Flow Ratio / Price-to-Sales Ratio 18.5 Free Cash Flow Valuation Approaches

20.3 Option Strategies 689 Protective Put / Covered Calls / Straddle / Spreads / Collars

617

Comparing the Valuation Models / The Problem with DCF Models 18.6 The Aggregate Stock Market End of Chapter Material

20.4 The Put-Call Parity Relationship

Callable Bonds / Convertible Securities / Warrants / Collateralized Loans / Levered Equity and Risky Debt

623–634

20.6 Financial Engineering 707

CHAPTER 19

20.7 Exotic Options 709

635

Asian Options / Barrier Options / Lookback Options / Currency-Translated Options / Digital Options

19.1 The Major Financial Statements 635

End of Chapter Material

The Income Statement / The Balance Sheet / The Statement of Cash Flows 19.2 Measuring Firm Performance

Option Valuation

19.3 Profitability Measures 641 Return on Assets, ROA / Return on Capital, ROC / Return on Equity, ROE / Financial Leverage and ROE / Economic Value Added

722

21.1 Option Valuation: Introduction 722 Intrinsic and Time Values / Determinants of Option Values 21.2 Restrictions on Option Values

19.4 Ratio Analysis 645

725

Restrictions on the Value of a Call Option / Early Exercise and Dividends / Early Exercise of American Puts

Decomposition of ROE / Turnover and Other Asset Utilization Ratios / Liquidity Ratios / Market Price Ratios: Growth versus Value / Choosing a Benchmark

21.3 Binomial Option Pricing

729

Two-State Option Pricing / Generalizing the Two-State Approach / Making the Valuation Model Practical

19.5 An Illustration of Financial Statement Analysis 655

21.4 Black-Scholes Option Valuation 737

19.6 Comparability Problems 658

The Black-Scholes Formula / Dividends and Call Option Valuation / Put Option Valuation / Dividends and Put Option Valuation

Inventory Valuation / Depreciation / Inflation and Interest Expense / Fair Value Accounting / Quality of Earnings and Accounting Practices / International Accounting Conventions End of Chapter Material

710–721

CHAPTER 21

640

19.7 Value Investing: The Graham Technique

698

20.5 Option-Like Securities 701

622

Financial Statement Analysis

685

Call Options / Put Options / Option versus Stock Investments

21.5 Using the Black-Scholes Formula

746

Hedge Ratios and the Black-Scholes Formula / Portfolio Insurance / Option Pricing and the Crisis of 2008–2009 / Option Pricing and Portfolio Theory / Hedging Bets on Mispriced Options

665

665–677

PART VI

21.6 Empirical Evidence on Option Pricing End of Chapter Material

Options, Futures, and Other Derivatives 678

758

759–769

CHAPTER 22

Futures Markets CHAPTER 20

22.1 The Futures Contract

Options Markets: Introduction 678

770

771

The Basics of Futures Contracts / Existing Contracts 22.2 Trading Mechanics 775

20.1 The Option Contract 679

The Clearinghouse and Open Interest / The Margin Account and Marking to Market / Cash versus Actual Delivery / Regulations / Taxation

Options Trading / American and European Options / Adjustments in Option Contract Terms / The Options Clearing Corporation / Other Listed Options

xiii

Contents 22.3 Futures Markets Strategies

The Role of Alpha in Performance Measures / Actual Performance Measurement: An Example / Performance Manipulation and the Morningstar Risk-Adjusted Rating / Realized Returns versus Expected Returns

781

Hedging and Speculation / Basis Risk and Hedging 22.4 Futures Prices 785 The Spot-Futures Parity Theorem / Spreads / Forward versus Futures Pricing 22.5 Futures Prices versus Expected Spot Prices

24.2 Performance Measurement for Hedge Funds

24.3 Performance Measurement with Changing Portfolio Composition 854

791

Expectations Hypothesis / Normal Backwardation / Contango / Modern Portfolio Theory End of Chapter Material

24.4 Market Timing 855 The Potential Value of Market Timing / Valuing Market Timing as a Call Option / The Value of Imperfect Forecasting

793–798

24.5 Style Analysis 861

CHAPTER 23

Futures, Swaps, and Risk Management

Style Analysis and Multifactor Benchmarks / Style Analysis in Excel

799

23.1 Foreign Exchange Futures 799

24.6 Performance Attribution Procedures 864

The Markets / Interest Rate Parity / Direct versus Indirect Quotes / Using Futures to Manage Exchange Rate Risk

Asset Allocation Decisions / Sector and Security Selection Decisions / Summing Up Component Contributions

23.2 Stock-Index Futures 806

End of Chapter Material

The Contracts / Creating Synthetic Stock Positions: An Asset Allocation Tool / Index Arbitrage / Using Index Futures to Hedge Market Risk

International Diversification 25.1 Global Markets for Equities

Hedging Interest Rate Risk Swaps and Balance Sheet Restructuring / The Swap Dealer / Other Interest Rate Contracts / Swap Pricing / Credit Risk in the Swap Market / Credit Default Swaps

883

25.2 Risk Factors in International Investing

887

Exchange Rate Risk / Political Risk 25.3 International Investing: Risk, Return, and Benefits from Diversification 895

822

Pricing with Storage Costs / Discounted Cash Flow Analysis for Commodity Futures

Risk and Return: Summary Statistics / Are Investments in Emerging Markets Riskier? / Are Average Returns Higher in Emerging Markets? / Is Exchange Rate Risk Important in International Portfolios? / Benefits from International Diversification / Misleading Representation of Diversification Benefits / Realistic Benefits from International Diversification / Are Benefits from International Diversification Preserved in Bear Markets?

825–834

PART VII

Applied Portfolio Management 835

25.4 Assessing the Potential of International Diversification 911

CHAPTER 24

Portfolio Performance Evaluation

882

Developed Countries / Emerging Markets / Market Capitalization and GDP / Home-Country Bias

23.4 Swaps 815

End of Chapter Material

870–881

CHAPTER 25

23.3 Interest Rate Futures 813

23.5 Commodity Futures Pricing

851

25.5 International Investing and Performance Attribution 916

835

Constructing a Benchmark Portfolio of Foreign Assets / Performance Attribution

24.1 The Conventional Theory of Performance Evaluation 835

End of Chapter Material

Average Rates of Return / Time-Weighted Returns versus Dollar-Weighted Returns / Dollar-Weighted Return and Investment Performance / Adjusting Returns for Risk / The M2 Measure of Performance / Sharpe’s Ratio Is the Criterion for Overall Portfolios / Appropriate Performance Measures in Two Scenarios

920–925

CHAPTER 26

Hedge Funds

926

26.1 Hedge Funds versus Mutual Funds 26.2 Hedge Fund Strategies

Jane’s Portfolio Represents Her Entire Risky Investment Fund / Jane’s Choice Portfolio Is One of Many Portfolios Combined into a Large Investment Fund

927

928

Directional and Nondirectional Strategies / Statistical Arbitrage

xiv

Contents 26.3 Portable Alpha 931

CHAPTER 28

An Example of a Pure Play

Investment Policy and the Framework of the CFA Institute 977

26.4 Style Analysis for Hedge Funds 933 26.5 Performance Measurement for Hedge Funds

935

28.1 The Investment Management Process

Liquidity and Hedge Fund Performance / Hedge Fund Performance and Survivorship Bias / Hedge Fund Performance and Changing Factor Loadings / Tail Events and Hedge Fund Performance 26.6 Fee Structure in Hedge Funds End of Chapter Material

Objectives / Individual Investors / Personal Trusts / Mutual Funds / Pension Funds / Endowment Funds / Life Insurance Companies / Non–Life Insurance Companies / Banks

943

28.2 Constraints 983

946–950

Liquidity / Investment Horizon / Regulations / Tax Considerations / Unique Needs

CHAPTER 27

28.3 Policy Statements 985

The Theory of Active Portfolio Management 951 27.1 Optimal Portfolios and Alpha Values

Sample Policy Statements for Individual Investors 28.4 Asset Allocation 992

951

Taxes and Asset Allocation

Forecasts of Alpha Values and Extreme Portfolio Weights / Restriction of Benchmark Risk

28.5 Managing Portfolios of Individual Investors

Adjusting Forecasts for the Precision of Alpha / Distribution of Alpha Values / Organizational Structure and Performance

The Tax-Deferral Option / Tax-Deferred Retirement Plans / Deferred Annuities / Variable and Universal Life Insurance

962

Black-Litterman Asset Allocation Decision / Step 1: The Covariance Matrix from Historical Data / Step 2: Determination of a Baseline Forecast / Step 3: Integrating the Manager’s Private Views / Step 4: Revised (Posterior) Expectations / Step 5: Portfolio Optimization

28.6 Pension Funds 1000 Defined Contribution Plans / Defined Benefit Plans / Pension Investment Strategies Investing in Equities / Wrong Reasons to Invest in Equities

27.4 Treynor-Black versus Black-Litterman: Complements, Not Substitutes 968

28.7 Investments for the Long Run

The BL Model as Icing on the TB Cake / Why Not Replace the Entire TB Cake with the BL Icing? 27.5 The Value of Active Management

1003

Target Investing and the Term Structure of Bonds / Making Simple Investment Choices / Inflation Risk and Long-Term Investors

970

A Model for the Estimation of Potential Fees / Results from the Distribution of Actual Information Ratios / Results from Distribution of Actual Forecasts / Results with Reasonable Forecasting Records

End of Chapter Material

1004–1014

REFERENCES TO CFA PROBLEMS 1015 GLOSSARY

27.6 Concluding Remarks on Active Management 972 End of Chapter Material

994

Human Capital and Insurance / Investment in Residence / Saving for Retirement and the Assumption of Risk / Retirement Planning Models / Manage Your Own Portfolio or Rely on Others? / Tax Sheltering

27.2 The Treynor-Black Model and Forecast Precision 958

27.3 The Black-Litterman Model

978

G-1

NAME INDEX

973–974

I-1

SUBJECT INDEX

Appendix A:Forecasts and Realizations of Alpha 974 Appendix B:The General Black-Litterman Model 975

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I-4

Preface

W

e’ve just ended three decades of rapid and profound change in the investments industry as well as a financial crisis of historic magnitude. The vast expansion of financial markets during this period was due in part to innovations in securitization and credit enhancement that gave birth to new trading strategies. These strategies were in turn made feasible by developments in communication and information technology, as well as by advances in the theory of investments. Yet the financial crisis also was rooted in the cracks of these developments. Many of the innovations in security design facilitated high leverage and an exaggerated notion of the efficacy of risk transfer strategies. This engendered complacency about risk that was coupled with relaxation of regulation as well as reduced transparency, masking the precarious condition of many big players in the system. Of necessity, our text has evolved along with financial markets and their influence on world events. Investments, Tenth Edition, is intended primarily as a textbook for courses in investment analysis. Our guiding principle has been to present the material in a framework that is organized by a central core of consistent fundamental principles. We attempt to strip away unnecessary mathematical and technical detail, and we have concentrated on providing the intuition that may guide students and practitioners as they confront new ideas and challenges in their professional lives. This text will introduce you to major issues currently of concern to all investors. It can give you the skills to conduct a sophisticated assessment of watershed current issues and debates covered by the popular media as well as more-specialized finance journals. Whether you plan to

become an investment professional, or simply a sophisticated individual investor, you will find these skills essential, especially in today’s rapidly evolving environment. Our primary goal is to present material of practical value, but all three of us are active researchers in financial economics and find virtually all of the material in this book to be of great intellectual interest. Fortunately, we think, there is no contradiction in the field of investments between the pursuit of truth and the pursuit of money. Quite the opposite. The capital asset pricing model, the arbitrage pricing model, the efficient markets hypothesis, the option-pricing model, and the other centerpieces of modern financial research are as much intellectually satisfying subjects of scientific inquiry as they are of immense practical importance for the sophisticated investor. In our effort to link theory to practice, we also have attempted to make our approach consistent with that of the CFA Institute. In addition to fostering research in finance, the CFA Institute administers an education and certification program to candidates seeking designation as a Chartered Financial Analyst (CFA). The CFA curriculum represents the consensus of a committee of distinguished scholars and practitioners regarding the core of knowledge required by the investment professional. Many features of this text make it consistent with and relevant to the CFA curriculum. Questions from past CFA exams appear at the end of nearly every chapter, and, for students who will be taking the exam, those same questions and the exam from which they’ve been taken are listed at the end of the book. Chapter 3 includes excerpts from the “Code of Ethics and Standards of Professional Conduct” of the CFA Institute. Chapter 28, which discusses investors and the investment process, presents the

xvi

Preface A second theme is the risk–return trade-off. This too is a no-free-lunch notion, holding that in competitive security markets, higher expected returns come only at a price: the need to bear greater investment risk. However, this notion leaves several questions unanswered. How should one measure the risk of an asset? What should be the quantitative tradeoff between risk (properly measured) and expected return? The approach we present to these issues is known as modern portfolio theory, which is another organizing principle of this book. Modern portfolio theory focuses on the techniques and implications of efficient diversification, and we devote considerable attention to the effect of diversification on portfolio risk as well as the implications of efficient diversification for the proper measurement of risk and the risk–return relationship. 2. This text places greater emphasis on asset allocation than most of its competitors. We prefer this emphasis for two important reasons. First, it corresponds to the procedure that most individuals actually follow. Typically, you start with all of your money in a bank account, only then considering how much to invest in something riskier that might offer a higher expected return. The logical step at this point is to consider risky asset classes, such as stocks, bonds, or real estate. This is an asset allocation decision. Second, in most cases, the asset allocation choice is far more important in determining overall investment performance than is the set of security selection decisions. Asset allocation is the primary determinant of the risk–return profile of the investment portfolio, and so it deserves primary attention in a study of investment policy. 3. This text offers a much broader and deeper treatment of futures, options, and other derivative security markets than most investments texts. These markets have become both crucial and integral to the financial universe. Your only choice is to become conversant in these markets—whether you are to be a finance professional or simply a sophisticated individual investor.

CFA Institute’s framework for systematically relating investor objectives and constraints to ultimate investment policy. End-of-chapter problems also include questions from test-prep leader Kaplan Schweser. In the Tenth Edition, we have continued our systematic collection of Excel spreadsheets that give tools to explore concepts more deeply than was previously possible. These spreadsheets, available on the Web site for this text (www. mhhe.com/bkm), provide a taste of the sophisticated analytic tools available to professional investors.

UNDERLYING PHILOSOPHY In the Tenth Edition, we address many of the changes in the investment environment, including the unprecedented events surrounding the financial crisis. At the same time, many basic principles remain important. We believe that attention to these few important principles can simplify the study of otherwise difficult material and that fundamental principles should organize and motivate all study. These principles are crucial to understanding the securities traded in financial markets and in understanding new securities that will be introduced in the future, as well as their effects on global markets. For this reason, we have made this book thematic, meaning we never offer rules of thumb without reference to the central tenets of the modern approach to finance. The common theme unifying this book is that security markets are nearly efficient, meaning most securities are usually priced appropriately given their risk and return attributes. Free lunches are rarely found in markets as competitive as the financial market. This simple observation is, nevertheless, remarkably powerful in its implications for the design of investment strategies; as a result, our discussions of strategy are always guided by the implications of the efficient markets hypothesis. While the degree of market efficiency is, and always will be, a matter of debate (in fact we devote a full chapter to the behavioral challenge to the efficient market hypothesis), we hope our discussions throughout the book convey a good dose of healthy criticism concerning much conventional wisdom.

Distinctive Themes Investments is organized around several important themes:

NEW IN THE TENTH EDITION

1. The central theme is the near-informational-efficiency of well-developed security markets, such as those in the United States, and the general awareness that competitive markets do not offer “free lunches” to participants.

The following is a guide to changes in the Tenth Edition. This is not an exhaustive road map, but instead is meant to provide an overview of substantial additions and changes to coverage from the last edition of the text.

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Preface Chapter 1 The Investment Environment This chapter contains updated coverage of the consequences of the financial crisis as well as the Dodd-Frank act.

Chapter 2 Asset Classes and Financial Instruments We devote additional attention to money markets, including recent controversies concerning the regulation of money market mutual funds as well as the LIBOR scandal.

concerning the use of financial ratios as tools to evaluate firm performance.

Chapter 21 Option Valuation We have added substantial new sections on risk-neutral valuation methods and their implementation in the binomial option-pricing model, as well as the implications of the option pricing model for tail risk and financial instability.

Chapter 3 How Securities Are Traded

Chapter 24 Portfolio Performance Evaluation

We have extensively rewritten this chapter and included new sections that detail the rise of electronic markets, algorithmic and high-speed trading, and changes in market structure.

New sections on the vulnerability of standard performance measures to manipulation, manipulation-free measures, and the Morningstar Risk-Adjusted Return have been added.

Chapter 5 Risk, Return, and the Historical Record

ORGANIZATION AND CONTENT

This chapter has been updated with considerable attention paid to evidence on tail risk and extreme stock returns.

Chapter 9 The Capital Asset Pricing Model We have streamlined the explanation of the simple CAPM and updated and integrated the sections dealing with extensions of the CAPM, tying together extra-market hedging demands and factor risk premia.

Chapter 10 Arbitrage Pricing Theory The chapter contains new material on the practical feasibility of creating well-diversified portfolios and the implications for asset pricing.

Chapter 11 The Efficient Market Hypothesis We have added new material documenting the behavior of market anomalies over time, suggesting how market inefficiencies seem to be corrected.

Chapter 13 Empirical Evidence on Security Returns Increased attention is given to tests of multifactor models of risk and return and the implications of these tests for the importance of extra-market hedging demands.

Chapter 14 Bond Prices and Yields This chapter includes new material on sovereign credit default swaps.

Chapter 18 Equity Valuation Models This chapter includes a new section on the practical problems entailed in using DCF security valuation models and the response of value investors to these problems.

Chapter 19 Financial Statement Analysis We have added a new introduction to the discussion of ratio analysis, providing greater structure and rationale

The text is composed of seven sections that are fairly independent and may be studied in a variety of sequences. Because there is enough material in the book for a twosemester course, clearly a one-semester course will require the instructor to decide which parts to include. Part One is introductory and contains important institutional material focusing on the financial environment. We discuss the major players in the financial markets, provide an overview of the types of securities traded in those markets, and explain how and where securities are traded. We also discuss in depth mutual funds and other investment companies, which have become an increasingly important means of investing for individual investors. Perhaps most important, we address how financial markets can influence all aspects of the global economy, as in 2008. The material presented in Part One should make it possible for instructors to assign term projects early in the course. These projects might require the student to analyze in detail a particular group of securities. Many instructors like to involve their students in some sort of investment game, and the material in these chapters will facilitate this process. Parts Two and Three contain the core of modern portfolio theory. Chapter 5 is a general discussion of risk and return, making the general point that historical returns on broad asset classes are consistent with a risk–return trade-off, and examining the distribution of stock returns. We focus more closely in Chapter 6 on how to describe investors’ risk preferences and how they bear on asset allocation. In the next two chapters, we turn to portfolio optimization (Chapter 7) and its implementation using index models (Chapter 8).

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Preface Part Four is the first of three parts on security valuation. This part treats fixed-income securities—bond pricing (Chapter 14), term structure relationships (Chapter 15), and interest-rate risk management (Chapter 16). Parts Five and Six deal with equity securities and derivative securities. For a course emphasizing security analysis and excluding portfolio theory, one may proceed directly from Part One to Part Four with no loss in continuity. Finally, Part Seven considers several topics important for portfolio managers, including performance evaluation, international diversification, active management, and practical issues in the process of portfolio management. This part also contains a chapter on hedge funds.

After our treatment of modern portfolio theory in Part Two, we investigate in Part Three the implications of that theory for the equilibrium structure of expected rates of return on risky assets. Chapter 9 treats the capital asset pricing model and Chapter 10 covers multifactor descriptions of risk and the arbitrage pricing theory. Chapter 11 covers the efficient market hypothesis, including its rationale as well as evidence that supports the hypothesis and challenges it. Chapter 12 is devoted to the behavioral critique of market rationality. Finally, we conclude Part Three with Chapter 13 on empirical evidence on security pricing. This chapter contains evidence concerning the risk–return relationship, as well as liquidity effects on asset pricing.

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A Guided Tour This book contains several features designed to make it easy for students to understand, absorb, and apply the concepts and techniques presented.

CHAPTER OPENING VIGNETTES SERVE TO OUTLINE the upcoming material in the chapter and provide students with a road map of what they will learn.

1

CHAPTER ONE

The Investment Environment

AN INVESTMENT ISthe current commitment of money or other resources in the expectation of reaping future benefits. For example, an individual might purchase shares of stock anticipating that the future proceeds from the shares will justify both the time that her money is tied up as well as the risk of the investment. The time you will spend studying this text (not to mention its cost) also is an investment. You are forgoing either current leisure or the income you could be earning at a job in the expectation that your future career will be sufficiently enhanced to justify this commitment of time and effort. While these two investments differ in many ways, they share one key attribute that is central to all investments: You

CONCEPT CHECKS A UNIQUE FEATURE of this book! These self-test questions and problems found in the body of the text enable the students to determine whether they’ve understood the preceding material. Detailed solutions are provided at the end of each chapter.

Broadly speaking, this chapter addresses three topics that will provide a useful perspective for the material that is to come later. First, before delving into the topic of “investments,” we consider the role of financial assets in the economy. We discuss the relationship between securities and the “real” assets that actually produce goods and services for consumers, and we consider why financial assets are important to the functioning of a developed economy. Given this background, we then take a first look at the types of decisions that confront investors as they assemble a portfolio of assets. These investment decisions are made in an environment where higher returns usually can be obtained only at the price of

claim and limited liability features. Residual claim means that stockholders are the last in line of all those who have a claim on the assets and income of the corporation. In a liquidation of the firm’s assets the shareholders have a claim to what is left after all other claimants such as the tax authorities, employees, suppliers, bondholders, and other creditors have been paid. For a firm not in liquidation, shareholders have claim to the part of operating income left over after interest and taxes have been paid. Management can either pay this residual as cash dividends to shareholders or reinvest CONCEPT CHECK 2.3 it in the business to increase the value of the shares. Limited liability means that the most shareholders a. If you buy 100 shares of IBM stock, to what can lose in the event of failure of the corporation is their are you entitled? original investment. Unlike owners of unincorporated b. What is the most money you can make on businesses, whose creditors can lay claim to the personal this investment over the next year? assets of the owner (house, car, furniture), corporate c. If you pay $180 per share, what is the most bod61671_ch01_001-027.indd 1 03/05/13 12:08 AM shareholders may at worst have worthless stock. They are money you could lose over the year? not personally liable for the firm’s obligations.

Stock Market Listings

Example 4.2

NUMBERED EXAMPLES NUMBERED AND TITLED examples are integrated throughout chapters. Using the worked-out solutions to these examples as models, students can learn how to solve specific problems step-by-step as well as gain insight into general principles by seeing how they are applied to answer concrete questions.

Fees for Various Classes

Here are fees for different classes of the Dreyfus High Yield Fund in 2012. Notice the trade-off between the front-end loads versus 12b-1 charges in the choice between Class A and Class C shares. Class I shares are sold only to institutional investors and carry lower fees.

Front-end load Back-end load 12b-1 feesc Expense ratio

Class A

Class C

Class I

0–4.5%a 0 .25% .70%

0 0–1%b 1.0% .70%

0 0%b 0% .70%

a

Depending on size of investment. Depending on years until holdings are sold. Including service fee.

b

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c

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Investors are jumping out of mutual funds managed by professional stock pickers and shifting massive amounts of money into lower-cost funds that echo the broader market. Through November 2012, investors pulled $119.3 billion from so-called actively managed U.S. stock funds according to the latest data from research firm Morningstar Inc. At the same time, they poured $30.4 billion into U.S. stock exchange-traded funds. The move reflects the fact that many money managers of stock funds, which charge fees but also dangle the prospect of higher returns, have underperformed the benchmark stock indexes. As a result, more investors are choosing simply to invest in funds tracking the indexes, which carry lower fees and are perceived as having less risk. The mission of stock pickers in a managed mutual fund is to outperform the overall market by actively trading individual stocks or bonds, with fund managers receiving higher fees for their effort. In an ETF (or indexed mutual fund), managers balance the share makeup of the fund so it accurately reflects the performance of its underlying index, charging lower fees.

WORDS FROM THE STREET

Investors Sour on Pro Stock Pickers

WORDS FROM THE STREET BOXES

Morningstar says that when investors have put money in stock funds, they have chosen low-cost index funds and ETFs. Some index ETFs cost less than 0.1% of assets a year, while many actively managed stock funds charge 1% a year or more. While the trend has put increasing pressure lately on stock pickers, it is shifting the fortunes of some of the biggest players in the $14 trillion mutual-fund industry. Fidelity Investments and American Funds, among the largest in the category, saw redemptions or weak investor interest compared with competitors, according to an analysis of mutual-fund flows done for The Wall Street Journal by research firm Strategic Insight, a unit of New York-based Asset International. At the other end of the spectrum, Vanguard, the world’s largest provider of index mutual funds, pulled in a net $141 billion last year through December, according to the company. Many investors say they are looking for a way to invest cheaply, with less risk.

SHORT ARTICLES FROM business periodicals, such as The Wall Street Journal, are included in boxes throughout the text. The articles are chosen for real-world relevance and clarity of presentation.

Source: Adapted from Kirsten Grind, “Investors Sour on Pro Stock Pickers” The Wall Street Journal, January 3, 2013.

or a mutual fund company that operates a market index fund. Vanguard, for example, operates the Index 500 Portfolio that mimics the S&P 500 index fund. It purchases shares of the firms constituting the S&P 500 in proportion to the market values of the outstanding equity of each firm, and therefore essentially replicates the S&P 500 index. The fund thus duplicates the performance of this market index. It has one of the lowest operating expenses (as a percentage of assets) of all mutual stock funds precisely because it requires minimal managerial effort. A second reason to pursue a passive strategy is the free-rider benefit. If there are many active, knowledgeable investors who quickly bid up prices of undervalued assets and force

eXcel APPLICATIONS: Two–Security Model

EXCEL APPLICATIONS

he accompanying spreadsheet can be used to measure the return and risk of a portfolio of two risky assets. The model calculates the return and risk for varying weights of each security along with the optimal risky and minimum-variance portfolio. Graphs are automatically generated for various model inputs. The model allows you to specify a target rate of return and solves for optimal combinations using the risk-free asset and the optimal risky portfolio. The spreadsheet is constructed with the

T

THE TENTH EDITION features Excel Spreadsheet Applications with new Excel questions. A sample spreadsheet is presented in the text with an interactive version available on the book’s Web site at www.mhhe.com/bkm. bod61671_ch06_168-204.indd 189

two-security return data from Table7.1. This spreadsheet is available at www.mhhe.com/bkm.

Excel Question 1. Suppose your target expected rate of return is 11%. a. What is the lowest-volatility portfolio that provides that expected return? b. What is the standard deviation of that portfolio? c. What is the composition of that portfolio? Expected

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B

C

D

E

F

Return (%)

Asset Allocation Analysis: Risk and Return

2 3

Expected

Standard

Correlation

Return

Deviation

Coefﬁcient

4

Security 1

0.08

0.12

5

Security 2

0.13

0.2

6

T-Bill

0.05

0.3

Covariance 0.0072 11

7 8

Weight

Weight

Expected

Standard

Reward to

9

Security 1

Security 2

Return

Deviation

Volatility

1

0.08000

0.12000

0.25000

11

0.9

0.1

0.08500

0.11559

0.30281

12

0.8

0.2

0.09000

0.11454

0.34922

13

0.7

0.3

0.09500

0.11696

0.38474

14

0.6

0.4

0.10000

0.12264

0.40771

10

A B C D E F 1 2 Gross HPR = Wealth Implicitly Assumed Squared 3 Period HPR (decimal) 1 + HPR Index* Probability = 1/5 Deviation 4 0.8811 0.8811 2001 .2 −0.1189 0.0196 5 0.0586 2002 .2 0.7790 0.6864 −0.2210 6 0.0707 0.8833 2003 .2 1.2869 0.2869 7 0.1088 2004 1.1088 .2 0.0077 0.9794 8 .2 1.0491 1.0275 0.0491 2005 0.0008 9 0.0210 10 Arithmetic average AVERAGE(C5:C9) = SUMPRODUCT(B5:B9, C5:C9) = 0.0210 11 Expected HPR Standard deviation SUMPRODUCT(B5:B9, D5:D9)^.5 = 0.1774 Check: 12 STDEV(C5:C9) = 0.1983 1.0054^5= 13 GEOMEAN(E5:E9) − 1 = 0.0054 Geometric average return 1.0275 14 15 *The value of $1 invested at the beginning of the sample period (1/1/2001).

Spreadsheet 5.2

eXcel

Time series of HPR for the S&P 500

Please visit us at www.mhhe.com/bkm

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5

0 0

5

10

15

20

25

30

35

Standard Deviation (%)

EXCEL EXHIBITS SELECTED EXHIBITS ARE set as Excel spreadsheets and are denoted by an icon. They are also available on the book’s Web site at www.mhhe.com/bkm.

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End-of-Chapter Features 1. Unit investment trusts, closed-end management companies, and open-end management companies are all classified and regulated as investment companies. Unit investment trusts are essentially unmanaged in the sense that the portfolio, once established, is fixed. Managed investment companies, in contrast, may change the composition of the portfolio as deemed fit by the portfolio manager. Closed-end funds are traded like other securities; they do not redeem shares for their investors. Open-end funds will redeem shares for net asset value at the request of the investor.

SUMMARY

2. Net asset value equals the market value of assets held by a fund minus the liabilities of the fund divided by the shares outstanding.

SUMMARY

3. Mutual funds free the individual from many of the administrative burdens of owning individual securities and offer professional management of the portfolio. They also offer advantages that are available only to large-scale investors, such as discounted trading costs. On the other hand, funds are assessed management fees and incur other expenses, which reduce the investor’s rate of return. Funds also eliminate some of the individual’s control over the timing of capital gains realizations.

AT THE END of each chapter, a detailed summary outlines the most important concepts presented. A listing of related Web sites for each chapter can also be found on the book’s Web site at www. mhhe.com/bkm. These sites make it easy for students to research topics further and retrieve financial data and information.

4. Mutual funds are often categorized by investment policy. Major policy groups include money market funds; equity funds, which are further grouped according to emphasis on income versus growth; fixed-income funds; balanced and income funds; asset allocation funds; index funds; and specialized sector funds. 5. Costs of investing in mutual funds include front-end loads, which are sales charges; back-end loads, which are redemption fees or, more formally, contingent-deferred sales charges; fund operating expenses; and 12b-1 charges, which are recurring fees used to pay for the expenses of marketing the fund to the public. 6. Income earned on mutual fund portfolios is not taxed at the level of the fund. Instead, as long as the fund meets certain requirements for pass-through status, the income is treated as being earned by the investors in the fund.

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CHAPTER 5

PROBLEM SETS WE STRONGLY BELIEVE that practice in solving problems is critical to understanding investments, so a good variety of problems is provided. For ease of assignment we separated the questions by level of difficulty Basic, Intermediate, and Challenge.

Risk, Return, and the Historical Record

163

1. The Fisher equation tells us that the real interest rate approximately equals the nominal rate minus the inflation rate. Suppose the inflation rate increases from 3% to 5%. Does the Fisher equation imply that this increase will result in a fall in the real rate of interest? Explain.

PROBLEM SETS

2. You’ve just stumbled on a new dataset that enables you to compute historical rates of return on U.S. stocks all the way back to 1880. What are the advantages and disadvantages in using these data to help estimate the expected rate of return on U.S. stocks over the coming year?

Basic

3. You are considering two alternative 2-year investments: You can invest in a risky asset with a positive risk premium and returns in each of the 2 years that will be identically distributed and uncorrelated, or you can invest in the risky asset for only 1 year and then invest the proceeds in a risk-free asset. Which of the following statements about the first investment alternative (compared with the second) are true? Its 2-year risk premium is the same as the second alternative. The standard deviation of its 2-year return is the same. Its annualized standard deviation is lower. Its Sharpe ratio is higher. It is relatively more attractive to investors who have lower degrees of risk aversion.

4. You have $5,000 to invest for the next year and are considering three alternatives:

Intermediate

a. A money market fund with an average maturity of 30 days offering a current yield of 6% per year. b. A 1-year savings deposit at a bank offering an interest rate of 7.5%. c. A 20-year U.S. Treasury bond offering a yield to maturity of 9% per year. What role does your forecast of future interest rates play in your decisions? 5. Use Figure5.1 in the text to analyze the effect of the following on the level of real interest rates: a. Businesses become more pessimistic about future demand for their products and decide to reduce their capital spending. b. Households are induced to save more because of increased uncertainty about their future Social Security benefits. c. The Federal Reserve Board undertakes open-market purchases of U.S. Treasury securities in order to increase the supply of money.

us at www.mhhe.com/bkm

a. b. c. d. e.

5. Characterize each company in the previous problem as underpriced, overpriced, or properly priced.

EXAM PREP QUESTIONS

6. What is the expected rate of return for a stock that has a beta of 1.0 if the expected return on the market is 15%?

PRACTICE QUESTIONS for the CFA® exams provided by Kaplan Schweser, A Global Leader in CFA® Education, are available in selected chapters for additional test practice. Look for the Kaplan Schweser logo. Learn more at www.schweser.com.

a. 15%. b. More than 15%. c. Cannot be determined without the risk-free rate.

hhe.com/bkm

7. Kaskin, Inc., stock has a beta of 1.2 and Quinn, Inc., stock has a beta of .6. Which of the following statements is most accurate? a. The expected rate of return will be higher for the stock of Kaskin, Inc., than that of Quinn, Inc. b. The stock of Kaskin, Inc., has more total risk than Quinn, Inc. c. The stock of Quinn, Inc., has more systematic risk than that of Kaskin, Inc.

Intermediate

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8. You are a consultant to a large manufacturing corporation that is considering a project with the following net after-tax cash flows (in millions of dollars): Years from Now

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CFA PROBLEMS WE PROVIDE SEVERAL questions from past CFA examinations in applicable chapters. These questions represent the kinds of questions that professionals in the field believe are relevant to the “real world.” Located at the back of the book is a listing of each CFA question and the level and year of the CFA exam it was included in for easy reference when studying for the exam.

a. McCracken was correct and Stiles was wrong. b. Both were correct. c. Stiles was correct and McCracken was wrong.

Challenge

17. Assume a universe of n (large) securities for which the largest residual variance is not larger than ns2M. Construct as many different weighting schemes as you can that generate well-diversified portfolios. 18. Derive a more general (than the numerical example in the chapter) demonstration of the APT security market line: a. For a single-factor market. b. For a multifactor market. 19. Small firms will have relatively high loadings (high betas) on the SMB (small minus big) factor. a. Explain why. b. Now suppose two unrelated small firms merge. Each will be operated as an independent unit of the merged company. Would you expect the stock market behavior of the merged firm to differ from that of a portfolio of the two previously independent firms? How does the merger affect market capitalization? What is the prediction of the Fama-French model for the risk premium on the combined firm? Do we see here a flaw in the FF model?

1. Jeffrey Bruner, CFA, uses the capital asset pricing model (CAPM) to help identify mispriced securities. A consultant suggests Bruner use arbitrage pricing theory (APT) instead. In comparing CAPM and APT, the consultant made the following arguments: a. Both the CAPM and APT require a mean-variance efficient market portfolio. b. Neither the CAPM nor APT assumes normally distributed security returns. c. The CAPM assumes that one specific factor explains security returns but APT does not.

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51.50

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49.25

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54.75

300

49.00

200

58.25

100

48.50

600

EXCEL PROBLEMS

a. If a market buy order for 100 shares comes in, at what price will it be filled? b. At what price would the next market buy order be filled? c. If you were a security dealer, would you want to increase or decrease your inventory of thisstock?

eXcel Please visit us at www.mhhe.com/bkm

SELECTED CHAPTERS CONTAIN problems, denoted by an icon, specifically linked to Excel templates that are available on the book’s Web site at www.mhhe.com/bkm.

9. You are bullish on Telecom stock. The current market price is $50 per share, and you have $5,000 of your own to invest. You borrow an additional $5,000 from your broker at an interest rate of 8% per year and invest $10,000 in the stock. a. What will be your rate of return if the price of Telecom stock goes up by 10% during the next year? The stock currently pays no dividends. b. How far does the price of Telecom stock have to fall for you to get a margin call if the maintenance margin is 30%? Assume the price fall happens immediately.

eXcel Please visit us at www.mhhe.com/bkm

10. You are bearish on Telecom and decide to sell short 100 shares at the current market price of $50 per share. a. How much in cash or securities must you put into your brokerage account if the broker’s initial margin requirement is 50% of the value of the short position? b. How high can the price of the stock go before you get a margin call if the maintenance margin is 30% of the value of the short position?

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E-INVESTMENTS BOXES

E-INVESTMENTS EXERCISES The Federal Reserve Bank of St. Louis has information available on interest rates and economic conditions. A publication called Monetary Trends contains graphs and tables with information about current conditions in the capital markets. Go to the Web site www. stls.frb.org and click on Economic Research on the menu at the top of the page. Find the most recent issue of Monetary Trends in the Recent Data Publications section and answer these questions.

THESE EXERCISES PROVIDE students with simple activities to enhance their experience using the Internet. Easy-to-follow instructions and questions are presented so students can utilize what they have learned in class and apply it to today’s Web-driven world.

1. What is the professionals’ consensus forecast for inflation for the next 2 years? (Use the Federal Reserve Bank of Philadelphia line on the graph to answer this.) 2. What do consumers expect to happen to inflation over the next 2 years? (Use the University of Michigan line on the graph to answer this.) 3. Have real interest rates increased, decreased, or remained the same over the last 2 years? 4. What has happened to short-term nominal interest rates over the last 2 years? What about long-term nominal interest rates? 5. How do recent U.S. inflation and long-term interest rates compare with those of the other countries listed? 6. What are the most recently available levels of 3-month and 10-year yields on Treasury securities?

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Supplements and feedback is provided and EZ Test’s grade book is designed to export to your grade book. • PowerPoint Presentation These presentation slides, also prepared by Anna Kovalenko, contain figures and tables from the text, key points, and summaries in a visually stimulating collection of slides that you can customize to fit your lecture. • Solutions Manual Updated by Marc-Anthony Isaacs, this Manual provides detailed solutions to the end-ofchapter problem sets. This supplement is also available for purchase by your students or can be packaged with your text at a discount.

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Supplements improve student engagement in and out of class. You can select and use any asset that enhances your lecture. The Connect Finance Instructor Library includes all of the instructor supplements for this text.

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Supplements connecting selected questions in the text and the test bank to the six general knowledge and skill guidelines in the AACSB standards. The statements contained in Investments Tenth Edition are provided only as a guide for the users of this textbook. The AACSB leaves content coverage and assessment within the purview of individual schools, the mission of the school, and the faculty. While Investments Tenth Edition and the teaching package make no claim of any specific AACSB qualification or evaluation, within this edition we have labeled selected questions according to the six general knowledge and skills areas.

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AACSB STATEMENT The McGraw-Hill Companies is a proud corporate member of AACSB International. Understanding the importance and value of AACSB accreditation, Investments Tenth Edition recognizes the curricula guidelines detailed in the AACSB standards for business accreditation by

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Acknowledgments Throughout the development of this text, experienced instructors have provided critical feedback and suggestions for improvement. These individuals deserve a special thanks for their valuable insights and contributions. The following instructors played a vital role in the development of this and previous editions of Investments:

Anna Craig Emory University

Mahmoud Haddad Wayne State University

Elton Daal University of New Orleans

Greg Hallman University of Texas at Austin

David C. Distad University of California at Berkeley

Robert G. Hansen Dartmouth College

Craig Dunbar University of Western Ontario

Joel Hasbrouck New York University

J. Amanda Adkisson Texas A&M University

David Durr Murray State University

Andrea Heuson University of Miami

Sandro Andrade University of Miami at Coral Gables

Bjorn Eaker Duke University

Eric Higgins Drexel University

Tor-Erik Bakke University of Wisconsin

John Earl University of Richmond

Shalom J. Hochman University of Houston

Richard J. Bauer Jr. St. Mary’s University

Michael C. Ehrhardt University of Tennessee at Knoxville

Stephen Huffman University of Wisconsin at Oshkosh

Scott Besley University of Florida

Venkat Eleswarapu Southern Methodist University

Eric Hughson University of Colorado

John Binder University of Illinois at Chicago

David Ellis Babson College

Delroy Hunter University of South Florida

Paul Bolster Northwestern University

Andrew Ellul Indiana University

A. James Ifflander A. James Ifflander and Associates

Phillip Braun University of Chicago

John Fay Santa Clara University

Robert Jennings Indiana University

Leo Chan Delaware State University

Greg Filbeck University of Toledo

George Jiang University of Arizona

Charles Chang Cornell University

James Forjan York College of Pennsylvania

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Kee Chaung SUNY Buffalo

David Gallagher University of Technology, Sydney

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Anna Kovalenko Virginia Tech University

L. Michael Couvillion Plymouth State University

Weiyu Guo University of Nebraska at Omaha

Josef Lakonishok University of Illinois at Champaign/Urbana

xxvii

Acknowledgments Malek Lashgari University of Hartford

Andrew Prevost Ohio University

Charles A. Trzincka SUNY Buffalo

Dennis Lasser Binghamton SUNY

Herbert Quigley University of the District of Columbia Murli Rajan University of Scranton Speima Rao University of Southwestern Louisiana Rathin Rathinasamy Ball State University William Reese Tulane University Craig Rennie University of Arkansas Maurico Rodriquez Texas Christian University Leonard Rosenthal Bentley College Anthony Sanders Ohio State University Gary Sanger Louisiana State University Don Seeley University of Arizona John Settle Portland State University Edward C. Sims Western Illinois University Robert Skena Carnegie Mellon University Steve L. Slezak University of North Carolina at Chapel Hill Keith V. Smith Purdue University

Yiuman Tse Binghamton SUNY

Hongbok Lee Western Illinois University Bruce Lehmann University of California at San Diego Jack Li Northeastern University Larry Lockwood Texas Christian University Christopher K. Ma Texas Tech University Anil K. Makhija University of Pittsburgh Davinder Malhotra Philadelphia University Steven Mann University of South Carolina Deryl W. Martin Tennessee Technical University Jean Masson University of Ottawa Ronald May St. John’s University William McDonald University of Notre Dame Rick Meyer University of South Florida Bruce Mizrach Rutgers University at New Brunswick Mbodja Mougoue Wayne State University Kyung-Chun (Andrew) Mun Truman State University Carol Osler Brandeis University Gurupdesh Pandner DePaul University

Patricia B. Smith University of New Hampshire Ahmad Sohrabian California State Polytechnic University– Pomova

Robert Pavlik Southwest Texas State

Eileen St. Pierre University of Northern Colorado Laura T. Starks University of Texas Mick Swartz University of Southern California Manuel Tarrazo University of San Francisco Steve Thorley Brigham Young University

Marianne Plunkert University of Colorado at Denver

Ashish Tiwari University of Iowa

Jeffrey Pontiff Boston College

Jack Treynor Treynor Capital Management

Don B. Panton University of Texas at Arlington Dimitris Papanikolaou Northwestern University Dilip Patro Rutgers University

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Joe Ueng University of St. Thomas Gopala Vasuderan Suffolk University Joseph Vu DePaul University Qinghai Wang Georgia Institute of Technology Richard Warr North Carolina State University Simon Wheatley University of Chicago Marilyn K. Wiley Florida Atlantic University James Williams California State University at Northridge Michael Williams University of Denver Tony R. Wingler University of North Carolina at Greensboro Guojun Wu University of Michigan Hsiu-Kwang Wu University of Alabama Geungu Yu Jackson State University Thomas J. Zwirlein University of Colorado at Colorado Springs Edward Zychowicz Hofstra University For granting us permission to include many of their examination questions in the text, we are grateful to the CFA Institute. Much credit is due to the development and production team at McGraw-Hill/ Irwin: our special thanks go to Noelle Bathurst, Development Editor; Chuck Synovec, Executive Brand Manager; Bruce Gin, Content Project Manager; Melissa Caughlin, Senior Marketing Manager; Jennifer Jelinski, Marketing Specialist; Michael McCormick, Senior Buyer; and Debra Kubiak, Designer. Finally, we thank Judy, Hava, and Sheryl, who contribute to the book with their support and understanding. Zvi Bodie Alex Kane Alan J. Marcus

CHAPTER ONE

The Investment Environment

Broadly speaking, this chapter addresses three topics that will provide a useful perspective for the material that is to come later. First, before delving into the topic of “investments,” we consider the role of financial assets in the economy. We discuss the relationship between securities and the “real” assets that actually produce goods and services for consumers, and we consider why financial assets are important to the functioning of a developed economy. Given this background, we then take a first look at the types of decisions that confront investors as they assemble a portfolio of assets. These investment decisions are made in an environment where higher returns usually can be obtained only at the price of greater risk and in which it is rare to find assets that are so mispriced as to be obvious bargains. These themes—the risk–return trade-off and the efficient pricing of financial assets—are central to the investment process, so it is worth pausing for a brief discussion of their implications as we begin the text. These implications will be fleshed out in much greater detail in later chapters. We provide an overview of the organization of security markets as well as the various players that participate in those markets. Together, these introductions should give you a feel for who the major participants are in

PART I

AN INVESTMENT ISthe current commitment of money or other resources in the expectation of reaping future benefits. For example, an individual might purchase shares of stock anticipating that the future proceeds from the shares will justify both the time that her money is tied up as well as the risk of the investment. The time you will spend studying this text (not to mention its cost) also is an investment. You are forgoing either current leisure or the income you could be earning at a job in the expectation that your future career will be sufficiently enhanced to justify this commitment of time and effort. While these two investments differ in many ways, they share one key attribute that is central to all investments: You sacrifice something of value now, expecting to benefit from that sacrifice later. This text can help you become an informed practitioner of investments. We will focus on investments in securities such as stocks, bonds, or options and futures contracts, but much of what we discuss will be useful in the analysis of any type of investment. The text will provide you with background in the organization of various securities markets; will survey the valuation and risk-management principles useful in particular markets, such as those for bonds or stocks; and will introduce you to the principles of portfolio construction.

1

(concluded) the securities markets as well as the setting in which they act. Finally, we discuss the financial crisis that began playing out in 2007 and peaked in 2008. The crisis dramatically illustrated the

1.1

connections between the financial system and the “real” side of the economy. We look at the origins of the crisis and the lessons that may be drawn about systemic risk. We close the chapter with an overview of the remainder of the text.

Real Assets versus Financial Assets

The material wealth of a society is ultimately determined by the productive capacity of its economy, that is, the goods and services its members can create. This capacity is a function of the real assets of the economy: the land, buildings, machines, and knowledge that can be used to produce goods and services. In contrast to real assets are financial assets such as stocks and bonds. Such securities are no more than sheets of paper or, more likely, computer entries, and they do not contribute directly to the productive capacity of the economy. Instead, these assets are the means by which individuals in well-developed economies hold their claims on real assets. Financial assets are claims to the income generated by real assets (or claims on income from the government). If we cannot own our own auto plant (a real asset), we can still buy shares in Ford or Toyota (financial assets) and thereby share in the income derived from the production of automobiles. While real assets generate net income to the economy, financial assets simply define the allocation of income or wealth among investors. Individuals can choose between consuming their wealth today or investing for the future. If they choose to invest, they may place their wealth in financial assets by purchasing various securities. When investors buy these securities from companies, the firms use the money so raised to pay for real assets, such as plant, equipment, technology, or inventory. So investors’ returns on securities ultimately come from the income produced by the real assets that were financed by the issuance of those securities. The distinction between real and financial assets is apparent when we compare the balance sheet of U.S. households, shown in Table1.1, with the composition of national wealth in the United States, shown in Table1.2. Household wealth includes financial assets such as bank accounts, corporate stock, or bonds. However, these CONCEPT CHECK 1.1 securities, which are financial assets of households, are liabilities of the issuers of the securities. For example, Are the following assets real or financial? a bond that you treat as an asset because it gives you a claim on interest income and repayment of principal from a. Patents Toyota is a liability of Toyota, which is obligated to make b. Lease obligations these payments to you. Your asset is Toyota’s liability. c. Customer goodwill Therefore, when we aggregate over all balance sheets, d. A college education these claims cancel out, leaving only real assets as the net wealth of the economy. National wealth consists of struce. A $5 bill tures, equipment, inventories of goods, and land.1 1

You might wonder why real assets held by households in Table1.1 amount to $23,774 billion, while total real assets in the domestic economy (Table1.2) are far larger, at $48,616 billion. A big part of the difference reflects the fact that real assets held by firms, for example, property, plant, and equipment, are included as financial assets of the household sector, specifically through the value of corporate equity and other stock market investments. Also, Table1.2 includes assets of noncorporate businesses. Finally, there are some differences in valuation methods. For example, equity and stock investments in Table1.1 are measured by market value, whereas plant and equipment in Table1.2 are valued at replacement cost.

2

CHAPTER 1

Assets

$ Billion

% Total

Real assets Real estate Consumer durables Other

$18,608 4,821 345

24.4% 6.3 0.5

$23,774

31.2%

Total real assets

Liabilities and Net Worth

$ Billion

% Total

Liabilities Mortgages Consumer credit Bank and other loans

$ 9,907 2,495 195

13.0% 3.3 0.3

Security credit Other Total liabilities

Financial assets Deposits Life insurance reserves Pension reserves Corporate equity Equity in noncorp. business Mutual fund shares Debt securities Other

$ 8,688 1,203 13,950 9,288 7,443 5,191 5,120 1,641

Total financial assets

52,524

68.8

76,298

100.0%

Total

268 568 $13,433

Net worth

17.6%

62,866

82.4

$76,298

100.0%

Balance sheet of U.S. households Note: Column sums may differ from total because of rounding error. Source: Flow of Funds Accounts of the United States, Board of Governors of the Federal Reserve System, June 2012.

Assets

$ Billion

Commercial real estate Residential real estate Equipment and software Inventories Consumer durables

$12,781 23,460 5,261 2,293 4,821

Table 1.2 Domestic net worth

$48,616

Note: Column sums may differ from total because of rounding error. Source: Flow of Funds Accounts of the United States, Board of Governors of the Federal Reserve System, June 2012.

We will focus almost exclusively on financial assets. But you shouldn’t lose sight of the fact that the successes or failures of the financial assets we choose to purchase ultimately depend on the performance of the underlying real assets.

1.2

0.4 0.7

11.4% 1.6 18.3 12.2 9.8 6.8 6.7 2.2

Table 1.1

Total

3

The Investment Environment

Financial Assets

It is common to distinguish among three broad types of financial assets: fixed income, equity, and derivatives. Fixed-income or debt securities promise either a fixed stream of income or a stream of income determined by a specified formula. For example, a corporate

4

PART I

Introduction

bond typically would promise that the bondholder will receive a fixed amount of interest each year. Other so-called floating-rate bonds promise payments that depend on current interest rates. For example, a bond may pay an interest rate that is fixed at 2 percentage points above the rate paid on U.S. Treasury bills. Unless the borrower is declared bankrupt, the payments on these securities are either fixed or determined by formula. For this reason, the investment performance of debt securities typically is least closely tied to the financial condition of the issuer. Nevertheless, fixed-income securities come in a tremendous variety of maturities and payment provisions. At one extreme, the money market refers to debt securities that are short term, highly marketable, and generally of very low risk. Examples of money market securities are U.S. Treasury bills or bank certificates of deposit (CDs). In contrast, the fixed-income capital market includes long-term securities such as Treasury bonds, as well as bonds issued by federal agencies, state and local municipalities, and corporations. These bonds range from very safe in terms of default risk (for example, Treasury securities) to relatively risky (for example, high-yield or “junk” bonds). They also are designed with extremely diverse provisions regarding payments provided to the investor and protection against the bankruptcy of the issuer. We will take a first look at these securities in Chapter2 and undertake a more detailed analysis of the debt market in Part Four. Unlike debt securities, common stock, or equity, in a firm represents an ownership share in the corporation. Equityholders are not promised any particular payment. They receive any dividends the firm may pay and have prorated ownership in the real assets of the firm. If the firm is successful, the value of equity will increase; if not, it will decrease. The performance of equity investments, therefore, is tied directly to the success of the firm and its real assets. For this reason, equity investments tend to be riskier than investments in debt securities. Equity markets and equity valuation are the topics of Part Five. Finally, derivative securities such as options and futures contracts provide payoffs that are determined by the prices of other assets such as bond or stock prices. For example, a call option on a share of Intel stock might turn out to be worthless if Intel’s share price remains below a threshold or “exercise” price such as $20 a share, but it can be quite valuable if the stock price rises above that level.2 Derivative securities are so named because their values derive from the prices of other assets. For example, the value of the call option will depend on the price of Intel stock. Other important derivative securities are futures and swap contracts. We will treat these in Part Six. Derivatives have become an integral part of the investment environment. One use of derivatives, perhaps the primary use, is to hedge risks or transfer them to other parties. This is done successfully every day, and the use of these securities for risk management is so commonplace that the multitrillion-dollar market in derivative assets is routinely taken for granted. Derivatives also can be used to take highly speculative positions, however. Every so often, one of these positions blows up, resulting in well-publicized losses of hundreds of millions of dollars. While these losses attract considerable attention, they are in fact the exception to the more common use of such securities as risk management tools. Derivatives will continue to play an important role in portfolio construction and the financial system. We will return to this topic later in the text. Investors and corporations regularly encounter other financial markets as well. Firms engaged in international trade regularly transfer money back and forth between dollars and 2

A call option is the right to buy a share of stock at a given exercise price on or before the option’s expiration date. If the market price of Intel remains below $20 a share, the right to buy for $20 will turn out to be valueless. If the share price rises above $20 before the option expires, however, the option can be exercised to obtain the share for only $20.

CHAPTER 1

The Investment Environment

other currencies. Well more than a trillion dollars of currency is traded each day in the market for foreign exchange, primarily through a network of the largest international banks. Investors also might invest directly in some real assets. For example, dozens of commodities are traded on exchanges such as the New York Mercantile Exchange or the Chicago Board of Trade. You can buy or sell corn, wheat, natural gas, gold, silver, and so on. Commodity and derivative markets allow firms to adjust their exposure to various business risks. For example, a construction firm may lock in the price of copper by buying copper futures contracts, thus eliminating the risk of a sudden jump in the price of its raw materials. Wherever there is uncertainty, investors may be interested in trading, either to speculate or to lay off their risks, and a market may arise to meet that demand.

1.3

Financial Markets and the Economy

We stated earlier that real assets determine the wealth of an economy, while financial assets merely represent claims on real assets. Nevertheless, financial assets and the markets in which they trade play several crucial roles in developed economies. Financial assets allow us to make the most of the economy’s real assets.

The Informational Role of Financial Markets Stock prices reflect investors’ collective assessment of a firm’s current performance and future prospects. When the market is more optimistic about the firm, its share price will rise. That higher price makes it easier for the firm to raise capital and therefore encourages investment. In this manner, stock prices play a major role in the allocation of capital in market economies, directing capital to the firms and applications with the greatest perceived potential. Do capital markets actually channel resources to the most efficient use? At times, they appear to fail miserably. Companies or whole industries can be “hot” for a period of time (think about the dot-com bubble that peaked in 2000), attract a large flow of investor capital, and then fail after only a few years. The process seems highly wasteful. But we need to be careful about our standard of efficiency. No one knows with certainty which ventures will succeed and which will fail. It is therefore unreasonable to expect that markets will never make mistakes. The stock market encourages allocation of capital to those firms that appear at the time to have the best prospects. Many smart, well-trained, and well-paid professionals analyze the prospects of firms whose shares trade on the stock market. Stock prices reflect their collective judgment. You may well be skeptical about resource allocation through markets. But if you are, then take a moment to think about the alternatives. Would a central planner make fewer mistakes? Would you prefer that Congress make these decisions? To paraphrase Winston Churchill’s comment about democracy, markets may be the worst way to allocate capital except for all the others that have been tried.

Consumption Timing Some individuals in an economy are earning more than they currently wish to spend. Others, for example, retirees, spend more than they currently earn. How can you shift your purchasing power from high-earnings periods to low-earnings periods of life? One way is to “store” your wealth in financial assets. In high-earnings periods, you can invest your savings in financial assets such as stocks and bonds. In low-earnings periods, you can sell these assets to provide funds for your consumption needs. By so doing, you can “shift” your consumption over the course of your lifetime, thereby allocating your consumption to

5

6

PART I

Introduction

periods that provide the greatest satisfaction. Thus, financial markets allow individuals to separate decisions concerning current consumption from constraints that otherwise would be imposed by current earnings.

Allocation of Risk Virtually all real assets involve some risk. When Ford builds its auto plants, for example, it cannot know for sure what cash flows those plants will generate. Financial markets and the diverse financial instruments traded in those markets allow investors with the greatest taste for risk to bear that risk, while other, less risk-tolerant individuals can, to a greater extent, stay on the sidelines. For example, if Ford raises the funds to build its auto plant by selling both stocks and bonds to the public, the more optimistic or risk-tolerant investors can buy shares of its stock, while the more conservative ones can buy its bonds. Because the bonds promise to provide a fixed payment, the stockholders bear most of the business risk but reap potentially higher rewards. Thus, capital markets allow the risk that is inherent to all investments to be borne by the investors most willing to bear that risk. This allocation of risk also benefits the firms that need to raise capital to finance their investments. When investors are able to select security types with the risk-return characteristics that best suit their preferences, each security can be sold for the best possible price. This facilitates the process of building the economy’s stock of real assets.

Separation of Ownership and Management Many businesses are owned and managed by the same individual. This simple organization is well suited to small businesses and, in fact, was the most common form of business organization before the Industrial Revolution. Today, however, with global markets and large-scale production, the size and capital requirements of firms have skyrocketed. For example, in 2012 General Electric listed on its balance sheet about $70 billion of property, plant, and equipment, and total assets of $685 billion. Corporations of such size simply cannot exist as owner-operated firms. GE actually has more than half a million stockholders with an ownership stake in the firm proportional to their holdings of shares. Such a large group of individuals obviously cannot actively participate in the day-today management of the firm. Instead, they elect a board of directors that in turn hires and supervises the management of the firm. This structure means that the owners and managers of the firm are different parties. This gives the firm a stability that the owner-managed firm cannot achieve. For example, if some stockholders decide they no longer wish to hold shares in the firm, they can sell their shares to other investors, with no impact on the management of the firm. Thus, financial assets and the ability to buy and sell those assets in the financial markets allow for easy separation of ownership and management. How can all of the disparate owners of the firm, ranging from large pension funds holding hundreds of thousands of shares to small investors who may hold only a single share, agree on the objectives of the firm? Again, the financial markets provide some guidance. All may agree that the firm’s management should pursue strategies that enhance the value of their shares. Such policies will make all shareholders wealthier and allow them all to better pursue their personal goals, whatever those goals might be. Do managers really attempt to maximize firm value? It is easy to see how they might be tempted to engage in activities not in the best interest of shareholders. For example, they might engage in empire building or avoid risky projects to protect their own jobs or overconsume luxuries such as corporate jets, reasoning that the cost of such perquisites is largely borne by the shareholders. These potential conflicts of interest are called agency problems because managers, who are hired as agents of the shareholders, may pursue their own interests instead.

CHAPTER 1

The Investment Environment

Several mechanisms have evolved to mitigate potential agency problems. First, compensation plans tie the income of managers to the success of the firm. A major part of the total compensation of top executives is often in the form of stock or stock options, which means that the managers will not do well unless the stock price increases, benefiting shareholders. (Of course, we’ve learned more recently that overuse of options can create its own agency problem. Options can create an incentive for managers to manipulate information to prop up a stock price temporarily, giving them a chance to cash out before the price returns to a level reflective of the firm’s true prospects. More on this shortly.) Second, while boards of directors have sometimes been portrayed as defenders of top management, they can, and in recent years, increasingly have, forced out management teams that are underperforming. The average tenure of CEOs fell from 8.1 years in 2006 to 6.6 years in 2011, and the percentage of incoming CEOs who also serve as chairman of the board of directors fell from 48% in 2002 to less than 12% in 2009.3 Third, outsiders such as security analysts and large institutional investors such as mutual funds or pension funds monitor the firm closely and make the life of poor performers at the least uncomfortable. Such large investors today hold about half of the stock in publicly listed firms in the U.S. Finally, bad performers are subject to the threat of takeover. If the board of directors is lax in monitoring management, unhappy shareholders in principle can elect a different board. They can do this by launching a proxy contest in which they seek to obtain enough proxies (i.e., rights to vote the shares of other shareholders) to take control of the firm and vote in another board. However, this threat is usually minimal. Shareholders who attempt such a fight have to use their own funds, while management can defend itself using corporate coffers. Most proxy fights fail. The real takeover threat is from other firms. If one firm observes another underperforming, it can acquire the underperforming business and replace management with its own team. The stock price should rise to reflect the prospects of improved performance, which provides incentive for firms to engage in such takeover activity.

Example 1.1

Carl Icahn’s Proxy Fight with Yahoo!

In February 2008, Microsoft offered to buy Yahoo! by paying its current shareholders $31 for each of their shares, a considerable premium to its closing price of $19.18 on the day before the offer. Yahoo’s management rejected that offer and a better one at $33 a share; Yahoo’s CEO Jerry Yang held out for $37 per share, a price that Yahoo! had not reached in more than 2 years. Billionaire investor Carl Icahn was outraged, arguing that management was protecting its own position at the expense of shareholder value. Icahn notified Yahoo! that he had been asked to “lead a proxy fight to attempt to remove the current board and to establish a new board which would attempt to negotiate a successful merger with Microsoft.” To that end, he had purchased approximately 59 million shares of Yahoo! and formed a 10-person slate to stand for election against the current board. Despite this challenge, Yahoo’s management held firm in its refusal of Microsoft’s offer, and with the support of the board, Yang managed to fend off both Microsoft and Icahn. In July, Icahn agreed to end the proxy fight in return for three seats on the board to be held by his allies. But the 11-person board was still dominated by current Yahoo management. Yahoo’s share price, which had risen to $29 a share during the Microsoft negotiations, fell back to around $21 a share. Given the difficulty that a well-known billionaire faced in defeating a determined and entrenched management, it is no wonder that proxy contests are rare. Historically, about three of four proxy fights go down to defeat.

3

“Corporate Bosses Are Much Less Powerful than They Used To Be,” The Economist, January 21, 2012.

7

8

PART I

Introduction

Corporate Governance and Corporate Ethics We’ve argued that securities markets can play an important role in facilitating the deployment of capital resources to their most productive uses. But market signals will help to allocate capital efficiently only if investors are acting on accurate information. We say that markets need to be transparent for investors to make informed decisions. If firms can mislead the public about their prospects, then much can go wrong. Despite the many mechanisms to align incentives of shareholders and managers, the three years from 2000 through 2002 were filled with a seemingly unending series of scandals that collectively signaled a crisis in corporate governance and ethics. For example, the telecom firm WorldCom overstated its profits by at least $3.8 billion by improperly classifying expenses as investments. When the true picture emerged, it resulted in the largest bankruptcy in U.S. history, at least until Lehman Brothers smashed that record in 2008. The next-largest U.S. bankruptcy was Enron, which used its now-notorious “specialpurpose entities” to move debt off its own books and similarly present a misleading picture of its financial status. Unfortunately, these firms had plenty of company. Other firms such as Rite Aid, HealthSouth, Global Crossing, and Qwest Communications also manipulated and misstated their accounts to the tune of billions of dollars. And the scandals were hardly limited to the United States. Parmalat, the Italian dairy firm, claimed to have a $4.8 billion bank account that turned out not to exist. These episodes suggest that agency and incentive problems are far from solved. Other scandals of that period included systematically misleading and overly optimistic research reports put out by stock market analysts. (Their favorable analysis was traded for the promise of future investment banking business, and analysts were commonly compensated not for their accuracy or insight, but for their role in garnering investment banking business for their firms.) Additionally, initial public offerings were allocated to corporate executives as a quid pro quo for personal favors or the promise to direct future business back to the manager of the IPO. What about the auditors who were supposed to be the watchdogs of the firms? Here too, incentives were skewed. Recent changes in business practice had made the consulting businesses of these firms more lucrative than the auditing function. For example, Enron’s (now-defunct) auditor Arthur Andersen earned more money consulting for Enron than by auditing it; given Arthur Andersen’s incentive to protect its consulting profits, we should not be surprised that it, and other auditors, were overly lenient in their auditing work. In 2002, in response to the spate of ethics scandals, Congress passed the Sarbanes-Oxley Act to tighten the rules of corporate governance. For example, the act requires corporations to have more independent directors, that is, more directors who are not themselves managers (or affiliated with managers). The act also requires each CFO to personally vouch for the corporation’s accounting statements, created an oversight board to oversee the auditing of public companies, and prohibits auditors from providing various other services to clients.

1.4

The Investment Process An investor’s portfolio is simply his collection of investment assets. Once the portfolio is established, it is updated or “rebalanced” by selling existing securities and using the proceeds to buy new securities, by investing additional funds to increase the overall size of the portfolio, or by selling securities to decrease the size of the portfolio. Investment assets can be categorized into broad asset classes, such as stocks, bonds, real estate, commodities, and so on. Investors make two types of decisions in constructing their

CHAPTER 1

The Investment Environment

portfolios. The asset allocation decision is the choice among these broad asset classes, while the security selection decision is the choice of which particular securities to hold within each asset class. Asset allocation also includes the decision of how much of one’s portfolio to place in safe assets such as bank accounts or money market securities versus in risky assets. Unfortunately, many observers, even those providing financial advice, appear to incorrectly equate saving with safe investing.4 “Saving” means that you do not spend all of your current income, and therefore can add to your portfolio. You may choose to invest your savings in safe assets, risky assets, or a combination of both. “Top-down” portfolio construction starts with asset allocation. For example, an individual who currently holds all of his money in a bank account would first decide what proportion of the overall portfolio ought to be moved into stocks, bonds, and so on. In this way, the broad features of the portfolio are established. For example, while the average annual return on the common stock of large firms since 1926 has been better than 11% per year, the average return on U.S. Treasury bills has been less than 4%. On the other hand, stocks are far riskier, with annual returns (as measured by the Standard & Poor’s 500 index) that have ranged as low as –46% and as high as 55%. In contrast, T-bills are effectively riskfree: You know what interest rate you will earn when you buy them. Therefore, the decision to allocate your investments to the stock market or to the money market where Treasury bills are traded will have great ramifications for both the risk and the return of your portfolio. A top-down investor first makes this and other crucial asset allocation decisions before turning to the decision of the particular securities to be held in each asset class. Security analysis involves the valuation of particular securities that might be included in the portfolio. For example, an investor might ask whether Merck or Pfizer is more attractively priced. Both bonds and stocks must be evaluated for investment attractiveness, but valuation is far more difficult for stocks because a stock’s performance usually is far more sensitive to the condition of the issuing firm. In contrast to top-down portfolio management is the “bottom-up” strategy. In this process, the portfolio is constructed from the securities that seem attractively priced without as much concern for the resultant asset allocation. Such a technique can result in unintended bets on one or another sector of the economy. For example, it might turn out that the portfolio ends up with a very heavy representation of firms in one industry, from one part of the country, or with exposure to one source of uncertainty. However, a bottom-up strategy does focus the portfolio on the assets that seem to offer the most attractive investment opportunities.

1.5

Markets Are Competitive

Financial markets are highly competitive. Thousands of intelligent and well-backed analysts constantly scour securities markets searching for the best buys. This competition means that we should expect to find few, if any, “free lunches,” securities that are so underpriced that they represent obvious bargains. This no-free-lunch proposition has several implications. Let’s examine two. 4

For example, here is a brief excerpt from the Web site of the Securities and Exchange Commission. “Your ‘savings’ are usually put into the safest places or products ... When you ‘invest,’ you have a greater chance of losing your money than when you ‘save.’” This statement is incorrect: Your investment portfolio can be invested in either safe or risky assets, and your savings in any period is simply the difference between your income and consumption.

9

10

PART I

Introduction

The Risk–Return Trade-Off Investors invest for anticipated future returns, but those returns rarely can be predicted precisely. There will almost always be risk associated with investments. Actual or realized returns will almost always deviate from the expected return anticipated at the start of the investment period. For example, in 1931 (the worst calendar year for the market since 1926), the S&P 500 index fell by 46%. In 1933 (the best year), the index gained 55%. You can be sure that investors did not anticipate such extreme performance at the start of either of these years. Naturally, if all else could be held equal, investors would prefer investments with the highest expected return.5 However, the no-free-lunch rule tells us that all else cannot be held equal. If you want higher expected returns, you will have to pay a price in terms of accepting higher investment risk. If higher expected return can be achieved without bearing extra risk, there will be a rush to buy the high-return assets, with the result that their prices will be driven up. Individuals considering investing in the asset at the now-higher price will find the investment less attractive: If you buy at a higher price, your expected rate of return (that is, profit per dollar invested) is lower. The asset will be considered attractive and its price will continue to rise until its expected return is no more than commensurate with risk. At this point, investors can anticipate a “fair” return relative to the asset’s risk, but no more. Similarly, if returns were independent of risk, there would be a rush to sell high-risk assets. Their prices would fall (and their expected future rates of return rise) until they eventually were attractive enough to be included again in investor portfolios. We conclude that there should be a risk–return trade-off in the securities markets, with higher-risk assets priced to offer higher expected returns than lower-risk assets. Of course, this discussion leaves several important questions unanswered. How should one measure the risk of an asset? What should be the quantitative trade-off between risk (properly measured) and expected return? One would think that risk would have something to do with the volatility of an asset’s returns, but this guess turns out to be only partly correct. When we mix assets into diversified portfolios, we need to consider the interplay among assets and the effect of diversification on the risk of the entire portfolio. Diversification means that many assets are held in the portfolio so that the exposure to any particular asset is limited. The effect of diversification on portfolio risk, the implications for the proper measurement of risk, and the risk–return relationship are the topics of Part Two. These topics are the subject of what has come to be known as modern portfolio theory. The development of this theory brought two of its pioneers, Harry Markowitz and William Sharpe, Nobel Prizes.

Efficient Markets Another implication of the no-free-lunch proposition is that we should rarely expect to find bargains in the security markets. We will spend all of Chapter 11 examining the theory and evidence concerning the hypothesis that financial markets process all available information about securities quickly and efficiently, that is, that the security price usually reflects all the information available to investors concerning its value. According to this hypothesis, as new information about a security becomes available, its price quickly

5

The “expected” return is not the return investors believe they necessarily will earn, or even their most likely return. It is instead the result of averaging across all possible outcomes, recognizing that some outcomes are more likely than others. It is the average rate of return across possible economic scenarios.

CHAPTER 1

The Investment Environment

adjusts so that at any time, the security price equals the market consensus estimate of the value of the security. If this were so, there would be neither underpriced nor overpriced securities. One interesting implication of this “efficient market hypothesis” concerns the choice between active and passive investment-management strategies. Passive management calls for holding highly diversified portfolios without spending effort or other resources attempting to improve investment performance through security analysis. Active management is the attempt to improve performance either by identifying mispriced securities or by timing the performance of broad asset classes—for example, increasing one’s commitment to stocks when one is bullish on the stock market. If markets are efficient and prices reflect all relevant information, perhaps it is better to follow passive strategies instead of spending resources in a futile attempt to outguess your competitors in the financial markets. If the efficient market hypothesis were taken to the extreme, there would be no point in active security analysis; only fools would commit resources to actively analyze securities. Without ongoing security analysis, however, prices eventually would depart from “correct” values, creating new incentives for experts to move in. Therefore, even in environments as competitive as the financial markets, we may observe only near-efficiency, and profit opportunities may exist for especially diligent and creative investors. In Chapter 12, we examine such challenges to the efficient market hypothesis, and this motivates our discussion of active portfolio management in Part Seven. More important, our discussions of security analysis and portfolio construction generally must account for the likelihood of nearly efficient markets.

1.6

The Players

From a bird’s-eye view, there would appear to be three major players in the financial markets: 1. Firms are net demanders of capital. They raise capital now to pay for investments in plant and equipment. The income generated by those real assets provides the returns to investors who purchase the securities issued by the firm. 2. Households typically are net suppliers of capital. They purchase the securities issued by firms that need to raise funds. 3. Governments can be borrowers or lenders, depending on the relationship between tax revenue and government expenditures. Since World War II, the U.S. government typically has run budget deficits, meaning that its tax receipts have been less than its expenditures. The government, therefore, has had to borrow funds to cover its budget deficit. Issuance of Treasury bills, notes, and bonds is the major way that the government borrows funds from the public. In contrast, in the latter part of the 1990s, the government enjoyed a budget surplus and was able to retire some outstanding debt. Corporations and governments do not sell all or even most of their securities directly to individuals. For example, about half of all stock is held by large financial institutions such as pension funds, mutual funds, insurance companies, and banks. These financial institutions stand between the security issuer (the firm) and the ultimate owner of the security (the individual investor). For this reason, they are called financial intermediaries. Similarly, corporations do not market their own securities to the public. Instead, they hire agents, called investment bankers, to represent them to the investing public. Let’s examine the roles of these intermediaries.

11

12

PART I

Introduction

Financial Intermediaries Households want desirable investments for their savings, yet the small (financial) size of most households makes direct investment difficult. A small investor seeking to lend money to businesses that need to finance investments doesn’t advertise in the local newspaper to find a willing and desirable borrower. Moreover, an individual lender would not be able to diversify across borrowers to reduce risk. Finally, an individual lender is not equipped to assess and monitor the credit risk of borrowers. For these reasons, financial intermediaries have evolved to bring the suppliers of capital (investors) together with the demanders of capital (primarily corporations and the federal government). These financial intermediaries include banks, investment companies, insurance companies, and credit unions. Financial intermediaries issue their own securities to raise funds to purchase the securities of other corporations. For example, a bank raises funds by borrowing (taking deposits) and lending that money to other borrowers. The spread between the interest rates paid to depositors and the rates charged to borrowers is the source of the bank’s profit. In this way, lenders and borrowers do not need to contact each other directly. Instead, each goes to the bank, which acts as an intermediary between the two. The problem of matching lenders with borrowers is solved when each comes independently to the common intermediary. Financial intermediaries are distinguished from other businesses in that both their assets and their liabilities are overwhelmingly financial. Table1.3 presents the aggregated balance sheet of commercial banks, one of the largest sectors of financial intermediaries. Notice that the balance sheet includes only very small amounts of real assets. Compare

Assets

$ Billion

Real assets Equipment and premises Other real estate

$

121.3 44.8

0.9% 0.3

$

166.1

1.2%

Total real assets

% Total

Liabilities and Net Worth Liabilities Deposits Debt and other borrowed funds Federal funds and repurchase agreements Other Total liabilities

Financial assets Cash Investment securities Loans and leases Other financial assets Total financial assets Other assets Intangible assets Other Total other assets Total

$ 1,335.9 2,930.6 7,227.7 1,161.5

9.6% 21.0 51.9 8.3

$12,655.7

90.9%

$

371.4 732.8

$ Billion $10,260.3 743.5 478.8 855.8 $12,338.4

% Total 73.7% 5.3 3.4 6.1 88.6%

2.7% 5.3

$ 1,104.2

7.9%

$13,926.0

100.0%

Net worth

Table 1.3 Balance sheet of FDIC-insured commercial banks and savings institutions Note: Column sums may differ from total because of rounding error. Source: Federal Deposit Insurance Corporation, www.fdic.gov, July 2012.

$ 1,587.6

11.4%

$13,926.0

100.0%

CHAPTER 1

Assets

$ Billion

% Total

Real assets Equipment and software Real estate Inventories Total real assets

$ 4,259 9,051 2,010 $15,320

13.9% 29.5 6.6 50.0%

Financial assets Deposits and cash Marketable securities Trade and consumer credit Direct investment abroad Other Total financial assets Total

$

967 769 2,555 4,055 6,983 $15,329 $30,649

3.2% 2.5 8.3 13.2 22.8 50.0% 100.0%

13

The Investment Environment

Liabilities and Net Worth

$ Billion

% Total

Liabilities Bonds and mortgages Bank loans Other loans Trade debt Other Total liabilities

$ 5,935 612 1,105 1,969 4,267 $13,887

19.4% 2.0 3.6 6.4 13.9 45.3%

$16,762 $30,649

54.7% 100.0%

Net worth

Table 1.4 Balance sheet of U.S. nonfinancial corporations Note: Column sums may differ from total because of rounding error. Source: Flow of Funds Accounts of the United States, Board of Governors of the Federal Reserve System, June 2012.

Table1.3 to the aggregated balance sheet of the nonfinancial corporate sector in Table1.4 for which real assets are about half of all assets. The contrast arises because intermediaries simply move funds from one sector to another. In fact, the primary social function of such intermediaries is to channel household savings to the business sector. Other examples of financial intermediaries are investment companies, insurance companies, and credit unions. All these firms offer similar advantages in their intermediary role. First, by pooling the resources of many small investors, they are able to lend considerable sums to large borrowers. Second, by lending to many borrowers, intermediaries achieve significant diversification, so they can accept loans that individually might be too risky. Third, intermediaries build expertise through the volume of business they do and can use economies of scale and scope to assess and monitor risk. Investment companies, which pool and manage the money of many investors, also arise out of economies of scale. Here, the problem is that most household portfolios are not large enough to be spread across a wide variety of securities. In terms of brokerage fees and research costs, purchasing one or two shares of many different firms is very expensive. Mutual funds have the advantage of large-scale trading and portfolio management, while participating investors are assigned a prorated share of the total funds according to the size of their investment. This system gives small investors advantages they are willing to pay for via a management fee to the mutual fund operator. Investment companies also can design portfolios specifically for large investors with particular goals. In contrast, mutual funds are sold in the retail market, and their investment philosophies are differentiated mainly by strategies that are likely to attract a large number of clients. Like mutual funds, hedge funds also pool and invest the money of many clients. But they are open only to institutional investors such as pension funds, endowment funds, or wealthy individuals. They are more likely to pursue complex and higher-risk strategies. They typically keep a portion of trading profits as part of their fees, whereas mutual funds charge a fixed percentage of assets under management.

WORDS FROM THE STREET

Separating Commercial Banking from Investment Banking Until 1999, the Glass-Steagall Act had prohibited banks in the United States from both accepting deposits and underwriting securities. In other words, it forced a separation of the investment and commercial banking industries. But when Glass-Steagall was repealed, many large commercial banks began to transform themselves into “universal banks” that could offer a full range of commercial and investment banking services. In some cases, commercial banks started their own investment banking divisions from scratch, but more frequently they expanded through merger. For example, Chase Manhattan acquired J.P. Morgan to form JPMorgan Chase. Similarly, Citigroup acquired Salomon Smith Barney to offer wealth management, brokerage, investment banking, and asset management services to its clients. Most of Europe had never forced the separation of commercial and investment banking, so their giant banks such as Credit Suisse, Deutsche Bank, HSBC, and UBS had long been universal banks. Until 2008, however, the stand-alone investment banking sector in the U.S. remained large and apparently vibrant, including such storied names as Goldman Sachs, Morgan-Stanley, Merrill Lynch, and Lehman Brothers. But the industry was shaken to its core in 2008, when several investment banks were beset by enormous losses on their holdings of mortgage-backed securities. In March, on the verge of insolvency, Bear Stearns was merged into JPMorgan Chase. On September 14, 2008, Merrill Lynch, also suffering steep mortgage-related losses, negotiated an agreement to be acquired by Bank of America. The next day, Lehman Brothers entered into the largest bankruptcy in U.S. history, having failed to find an acquirer able and willing to rescue it from its steep losses. The next week, the

only two remaining major independent investment banks, Goldman Sachs and Morgan Stanley, decided to convert from investment banks to traditional bank holding companies. In doing so, they became subject to the supervision of national bank regulators such as the Federal Reserve and the far tighter rules for capital adequacy that govern commercial banks. The firms decided that the greater stability they would enjoy as commercial banks, particularly the ability to fund their operations through bank deposits and access to emergency borrowing from the Fed, justified the conversion. These mergers and conversions marked the effective end of the independent investment banking industry—but not of investment banking. Those services now will be supplied by the large universal banks. Today, the debate about the separation between commercial and investment banking that seemed to have ended with the repeal of Glass-Steagall has come back to life. The Dodd-Frank Wall Street Reform and Consumer Protection Act places new restrictions on bank activities. For example, the Volcker Rule, named after former chairman of the Federal Reserve Paul Volcker, prohibits banks from “proprietary trading,” that is, trading securities for their own accounts, and restricts their investments in hedge funds or private equity funds. The rule is meant to limit the risk that banks can take on. While the Volcker Rule is far less restrictive than Glass-Steagall had been, they both are motivated by the belief that banks enjoying Federal guarantees should be subject to limits on the sorts of activities in which they can engage. Proprietary trading is a core activity for investment banks, so limitations on this activity for commercial banks would reintroduce a separation between their business models.

Economies of scale also explain the proliferation of analytic services available to investors. Newsletters, databases, and brokerage house research services all engage in research to be sold to a large client base. This setup arises naturally. Investors clearly want information, but with small portfolios to manage, they do not find it economical to personally gather all of it. Hence, a profit opportunity emerges: A firm can perform this service for many clients and charge for it.

Investment Bankers Just as economies of scale and specialization create profit opportunities for financial intermediaries, so do these economies create niches for firms that perform specialized services for businesses. Firms raise much of their capital by selling securities such as stocks and bonds to the public. Because these firms do not do so frequently, however, investment bankers that specialize in such activities can offer their services at a cost below that of maintaining an in-house security issuance division. In this role, they are called underwriters. Investment bankers advise the issuing corporation on the prices it can charge for the securities issued, appropriate interest rates, and so forth. Ultimately, the investment banking firm handles the marketing of the security in the primary market, where new issues 14

CHAPTER 1

The Investment Environment

of securities are offered to the public. Later, investors can trade previously issued securities among themselves in the so-called secondary market. For most of the last century, investment banks and commercial banks in the U.S. were separated by law. While those regulations were effectively eliminated in 1999, the industry known as “Wall Street” was until 2008 still comprised of large, independent investment banks such as Goldman Sachs, Merrill Lynch, and Lehman Brothers. But that stand-alone model came to an abrupt end in September 2008, when all the remaining major U.S. investment banks were absorbed into commercial banks, declared bankruptcy, or reorganized as commercial banks. The nearby box presents a brief introduction to these events.

Venture Capital and Private Equity While large firms can raise funds directly from the stock and bond markets with help from their investment bankers, smaller and younger firms that have not yet issued securities to the public do not have that option. Start-up companies rely instead on bank loans and investors who are willing to invest in them in return for an ownership stake in the firm. The equity investment in these young companies is called venture capital (VC). Sources of venture capital are dedicated venture capital funds, wealthy individuals known as angel investors, and institutions such as pension funds. Most venture capital funds are set up as limited partnerships. A management company starts with its own money and raises additional capital from limited partners such as pension funds. That capital may then be invested in a variety of start-up companies. The management company usually sits on the start-up company’s board of directors, helps recruit senior managers, and provides business advice. It charges a fee to the VC fund for overseeing the investments. After some period of time, for example, 10 years, the fund is liquidated and proceeds are distributed to the investors. Venture capital investors commonly take an active role in the management of a start-up firm. Other active investors may engage in similar hands-on management but focus instead on firms that are in distress or firms that may be bought up, “improved,” and sold for a profit. Collectively, these investments in firms that do not trade on public stock exchanges are known as private equity investments.

1.7

The Financial Crisis of 2008

This chapter has laid out the broad outlines of the financial system, as well as some of the links between the financial side of the economy and the “real” side in which goods and services are produced. The financial crisis of 2008 illustrated in a painful way the intimate ties between these two sectors. We present in this section a capsule summary of the crisis, attempting to draw some lessons about the role of the financial system as well as the causes and consequences of what has become known as systemic risk. Some of these issues are complicated; we consider them briefly here but will return to them in greater detail later in the text once we have more context for analysis.

Antecedents of the Crisis In early 2007, most observers thought it inconceivable that within two years, the world financial system would be facing its worst crisis since the Great Depression. At the time, the economy seemed to be marching from strength to strength. The last significant macroeconomic threat had been from the implosion of the high-tech bubble in 2000–2002. But the Federal Reserve responded to an emerging recession by aggressively reducing interest

15

16

PART I

Introduction

8

3-month LIBOR 3-month T-Bill TED spread

7 6

Percent

5 4 3 2 1

Jul-12

Jul-11

Jan-12

Jul-10

Jan-11

Jul-09

Jan-10

Jan-09

Jul-08

Jul-07

Jan-08

Jul-06

Jan-07

Jul-05

Jan-06

Jul-04

Jan-05

Jul-03

Jan-04

Jul-02

Jan-03

Jul-01

Jan-02

Jul-00

Jan-01

Jan-00

Figure 1.1 Short-term LIBOR and Treasury-bill rates and the TED spread

rates. Figure1.1 shows that Treasury bill rates dropped drastically between 2001 and 2004, and the LIBOR rate, which is the interest rate at which major money-center banks lend to each other, fell in tandem.6 These actions appeared to have been successful, and the recession was short-lived and mild. By mid-decade the economy was apparently healthy once again. Although the stock market had declined substantially between 2001 and 2002, Figure 1.2 shows that it reversed direction just as dramatically beginning in 2003, fully recovering all of its posttech-meltdown losses within a few years. Of equal importance, the banking sector seemed healthy. The spread between the LIBOR rate (at which banks borrow from each other) and the Treasury-bill rate (at which the U.S. government borrows), a common measure of credit risk in the banking sector (often referred to as the TED spread 7), was only around .25% in early 2007 (see the bottom curve in Figure1.1), suggesting that fears of default or “counterparty” risk in the banking sector were extremely low. Indeed, the apparent success of monetary policy in this recession, as well as in the last 30 years more generally, had engendered a new term, the “Great Moderation,” to describe the fact that recent business cycles—and recessions in particular—seemed so mild compared to past experience. Some observers wondered whether we had entered a golden age for macroeconomic policy in which the business cycle had been tamed. The combination of dramatically reduced interest rates and an apparently stable economy fed a historic boom in the housing market. Figure1.3 shows that U.S. housing prices 6

LIBOR stands for London Interbank Offer Rate. It is a rate charged on dollar-denominated loans in an interbank lending market outside of the U.S. (largely centered in London). The rate is typically quoted for 3-month loans. The LIBOR rate is closely related to the Federal Funds rate in the U.S. The Fed Funds rate is the rate charged on loans between U.S. banks, usually on an overnight basis. 7 TED stands for Treasury–Eurodollar spread. The Eurodollar rate in this spread is in fact LIBOR.

CHAPTER 1

The Investment Environment

Cumulative Value of a $1 Investment

40 35 30 25 20 15 10 5 0 1980

1983

1986

1989

1992

1995

1998

2001

2004

2007

2010

2013

Figure 1.2 Cumulative returns on the S&P 500 index

Index (January 2000 = 100)

250 200 150 100 50 0 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007 2009 2011 2013

Figure 1.3 The Case-Shiller index of U.S. housing prices

began rising noticeably in the late 1990s and accelerated dramatically after 2001 as interest rates plummeted. In the 10 years beginning 1997, average prices in the U.S. approximately tripled. But the newfound confidence in the power of macroeconomic policy to reduce risk, the impressive recovery of the economy from the high-tech implosion, and particularly the housing price boom following the aggressive reduction in interest rates may have sown the seeds for the debacle that played out in 2008. On the one hand, the Fed’s policy of reducing interest rates had resulted in low yields on a wide variety of investments, and investors were hungry for higher-yielding alternatives. On the other hand, low volatility and growing complacency about risk encouraged greater tolerance for risk in the search

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Introduction

for these higher-yielding investments. Nowhere was this more evident than in the exploding market for securitized mortgages. The U.S. housing and mortgage finance markets were at the center of a gathering storm.

Changes in Housing Finance Prior to 1970, most mortgage loans would come from a local lender such as a neighborhood savings bank or credit union. A homeowner would borrow funds for a home purchase and repay the loan over a long period, commonly 30 years. A typical thrift institution would have as its major asset a portfolio of these long-term home loans, while its major liability would be the accounts of its depositors. This landscape began to change when Fannie Mae (FNMA, or Federal National Mortgage Association) and Freddie Mac (FHLMC, or Federal Home Loan Mortgage Corporation) began buying mortgage loans from originators and bundling them into large pools that could be traded like any other financial asset. These pools, which were essentially claims on the underlying mortgages, were soon dubbed mortgage-backed securities, and the process was called securitization. Fannie and Freddie quickly became the behemoths of the mortgage market, between them buying around half of all mortgages originated by the private sector. Figure 1.4 illustrates how cash flows passed from the original borrower to the ultimate investor in a mortgage-backed security. The loan originator, for example, the savings and loan, might make a $100,000 home loan to a homeowner. The homeowner would repay principal and interest (P&I) on the loan over 30 years. But then the originator would sell the mortgage to Freddie Mac or Fannie Mae and recover the cost of the loan. The originator could continue to service the loan (collect monthly payments from the homeowner) for a small servicing fee, but the loan payments net of that fee would be passed along to the agency. In turn, Freddie or Fannie would pool the loans into mortgage-backed securities and sell the securities to investors such as pension funds or mutual funds. The agency (Fannie or Freddie) typically would guarantee the credit or default risk of the loans included in each pool, for which it would retain a guarantee fee before passing along the rest of the cash flow to the ultimate investor. Because the mortgage cash flows were passed along from the homeowner to the lender to Fannie or Freddie to the investor, the mortgagebacked securities were also called pass-throughs. Until the last decade, the vast majority of securitized mortgages were held or guaranteed by Freddie Mac or Fannie Mae. These were low-risk conforming mortgages, meaning that eligible loans for agency securitization couldn’t be too big, and homeowners had to meet underwriting criteria establishing their ability to repay the loan. For example, the ratio of loan amount to house value could be no more than 80%. But securitization gave rise to a new market niche for mortgage lenders: the “originate to distribute” (versus originate to hold) business model.

$100 K Homeowner

$100 K Agency

Originator P&I

$100 K

P&I – servicing fee

Figure 1.4 Cash flows in a mortgage pass-through security

Investor P&I – servicing – guarantee fee

CHAPTER 1

The Investment Environment

Whereas conforming loans were pooled almost entirely through Freddie Mac and Fannie Mae, once the securitization model took hold, it created an opening for a new product: securitization by private firms of nonconforming “subprime” loans with higher default risk. One important difference between the government agency pass-throughs and these so-called private-label pass-throughs was that the investor in the private-label pool would bear the risk that homeowners might default on their loans. Thus, originating mortgage brokers had little incentive to perform due diligence on the loan as long as the loans could be sold to an investor. These investors, of course, had no direct contact with the borrowers, and they could not perform detailed underwriting concerning loan quality. Instead, they relied on borrowers’ credit scores, which steadily came to replace conventional underwriting. A strong trend toward low-documentation and then no-documentation loans, entailing little verification of a borrower’s ability to carry a loan, soon emerged. Other subprime underwriting standards quickly deteriorated. For example, allowed leverage on home loans (as measured by the loan-to-value ratio) rose dramatically. By 2006, the majority of subprime borrowers purchased houses by borrowing the entire purchase price! When housing prices began falling, these loans were quickly “underwater,” meaning that the house was worth less than the loan balance, and many homeowners decided to walk away from their loans. Adjustable-rate mortgages (ARMs) also grew in popularity. These loans offered borrowers low initial or “teaser” interest rates, but these rates eventually would reset to current market interest yields, for example, the Treasury bill rate plus 3%. Many of these borrowers “maxed out” their borrowing capacity at the teaser rate, yet, as soon as the loan rate was reset, their monthly payments would soar, especially if market interest rates had increased. Despite these obvious risks, the ongoing increase in housing prices over the last decade seemed to lull many investors into complacency, with a widespread belief that continually rising home prices would bail out poorly performing loans. But starting in 2004, the ability of refinancing to save a loan began to diminish. First, higher interest rates put payment pressure on homeowners who had taken out adjustable-rate mortgages. Second, as Figure1.3 shows, housing prices peaked by 2006, so homeowners’ ability to refinance a loan using built-up equity in the house declined. Housing default rates began to surge in 2007, as did losses on mortgage-backed securities. The crisis was ready to shift into high gear.

Mortgage Derivatives One might ask: Who was willing to buy all of these risky subprime mortgages? Securitization, restructuring, and credit enhancement provide a big part of the answer. New risk-shifting tools enabled investment banks to carve out AAA-rated securities from original-issue “junk” loans. Collateralized debt obligations, or CDOs, were among the most important and eventually damaging of these innovations. CDOs were designed to concentrate the credit (i.e., default) risk of a bundle of loans on one class of investors, leaving the other investors in the pool relatively protected from that risk. The idea was to prioritize claims on loan repayments by dividing the pool into senior versus junior slices, called tranches. The senior tranches had first claim on repayments from the entire pool. Junior tranches would be paid only after the senior ones had received their cut.8 For example, if a pool were divided into two tranches, with 70% of the pool allocated to the senior tranche and 30% allocated to the junior one, the senior investors 8

CDOs and related securities are sometimes called structured products. “Structured” means that original cash flows are sliced up and reapportioned across tranches according to some stipulated rule.

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PART I

Introduction

would be repaid in full as long as 70% or more of the loans in the pool performed, that is, as long as the default rate on the pool remained below 30%. Even with pools composed of risky subprime loans, default rates above 30% seemed extremely unlikely, and thus senior tranches were frequently granted the highest (i.e., AAA) rating by the major credit rating agencies, Moody’s, Standard & Poor’s, and Fitch. Large amounts of AAA-rated securities were thus carved out of pools of low-rated mortgages. (We will describe CDOs in more detail in Chapter 14.) Of course, we know now that these ratings were wrong. The senior-subordinated structure of CDOs provided far less protection to senior tranches than investors anticipated. When housing prices across the entire country began to fall in unison, defaults in all regions increased, and the hoped-for benefits from spreading the risks geographically never materialized. Why had the rating agencies so dramatically underestimated credit risk in these subprime securities? First, default probabilities had been estimated using historical data from an unrepresentative period characterized by a housing boom and an uncommonly prosperous and recession-free macroeconomy. Moreover, the ratings analysts had extrapolated historical default experience to a new sort of borrower pool—one without down payments, with exploding-payment loans, and with low- or no-documentation loans (often called liar loans). Past default experience was largely irrelevant given these profound changes in the market. Moreover, the power of cross-regional diversification to minimize risk engendered excessive optimism. Finally, agency problems became apparent. The ratings agencies were paid to provide ratings by the issuers of the securities—not the purchasers. They faced pressure from the issuers, who could shop around for the most favorable treatment, to provide generous ratings.

CONCEPT CHECK

1.2

When Freddie Mac and Fannie Mae pooled mortgages into securities, they guaranteed the underlying mortgage loans against homeowner defaults. In contrast, there were no guarantees on the mortgages pooled into subprime mortgage-backed securities, so investors would bear credit risk. Was either of these arrangements necessarily a better way to manage and allocate default risk?

Credit Default Swaps In parallel to the CDO market, the market in credit default swaps also exploded in this period. A credit default swap, or CDS, is in essence an insurance contract against the default of one or more borrowers. (We will describe these in more detail in Chapter 14.) The purchaser of the swap pays an annual premium (like an insurance premium) for protection from credit risk. Credit default swaps became an alternative method of credit enhancement, seemingly allowing investors to buy subprime loans and insure their safety. But in practice, some swap issuers ramped up their exposure to credit risk to unsupportable levels, without sufficient capital to back those obligations. For example, the large insurance company AIG alone sold more than $400 billion of CDS contracts on subprime mortgages.

The Rise of Systemic Risk By 2007, the financial system displayed several troubling features. Many large banks and related financial institutions had adopted an apparently profitable financing scheme: borrowing short term at low interest rates to finance holdings in higher-yielding, long-term

CHAPTER 1

The Investment Environment

illiquid assets,9 and treating the interest rate differential between their assets and liabilities as economic profit. But this business model was precarious: By relying primarily on shortterm loans for their funding, these firms needed to constantly refinance their positions (i.e., borrow additional funds as the loans matured), or else face the necessity of quickly selling off their less-liquid asset portfolios, which would be difficult in times of financial stress. Moreover, these institutions were highly leveraged and had little capital as a buffer against losses. Large investment banks on Wall Street in particular had sharply increased leverage, which added to an underappreciated vulnerability to refunding requirements—especially if the value of their asset portfolios came into question. Even small portfolio losses could drive their net worth negative, at which point no one would be willing to renew outstanding loans or extend new ones. Another source of fragility was widespread investor reliance on “credit enhancement” through products like CDOs. Many of the assets underlying these pools were illiquid, hard to value, and highly dependent on forecasts of future performance of other loans. In a widespread downturn, with rating downgrades, these assets would prove difficult to sell. The steady displacement of formal exchange trading by informal “over-the-counter” markets created other problems. In formal exchanges such as futures or options markets, participants must put up collateral called margin to guarantee their ability to make good on their obligations. Prices are computed each day, and gains or losses are continually added to or subtracted from each trader’s margin account. If a margin account runs low after a series of losses, the investor can be required to either contribute more collateral or to close out the position before actual insolvency ensues. Positions, and thus exposures to losses, are transparent to other traders. In contrast, the over-the-counter markets where CDS contracts trade are effectively private contracts between two parties with less public disclosure of positions, less standardization of products (which makes the fair value of a contract hard to discover), and consequently less opportunity to recognize either cumulative gains or losses over time or the resultant credit exposure of each trading partner. This new financial model was brimming with systemic risk, a potential breakdown of the financial system when problems in one market spill over and disrupt others. When lenders such as banks have limited capital and are afraid of further losses, they may rationally choose to hoard their capital instead of lending it to customers such as small firms, thereby exacerbating funding problems for their customary borrowers.

The Shoe Drops By fall 2007, housing price declines were widespread (Figure1.3), mortgage delinquencies increased, and the stock market entered its own free fall (Figure1.2). Many investment banks, which had large investments in mortgages, also began to totter. The crisis peaked in September 2008. On September 7, the giant federal mortgage agencies Fannie Mae and Freddie Mac, both of which had taken large positions in subprime mortgage–backed securities, were put into conservatorship. (We will have more to say on their travails in Chapter 2.) The failure of these two mainstays of the U.S. housing and mortgage finance industries threw financial markets into a panic. By the second week of September, it was clear that both Lehman Brothers and Merrill Lynch were on the verge of bankruptcy. On September 14, Merrill Lynch was sold to Bank of America, again with the benefit of government brokering and protection against losses. The next day, Lehman Brothers, which was denied equivalent treatment, filed for bankruptcy protection. Two 9

Liquidity refers to the speed and the ease with which investors can realize the cash value of an investment. Illiquid assets, for example, real estate, can be hard to sell quickly, and a quick sale may require a substantial discount from the price at which the asset could be sold in an unrushed situation.

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Introduction

days later, on September 17, the government reluctantly lent $85 billion to AIG, reasoning that its failure would have been highly destabilizing to the banking industry, which was holding massive amounts of its credit guarantees (i.e., CDS contacts). The next day, the Treasury unveiled its first proposal to spend $700 billion to purchase “toxic” mortgagebacked securities. A particularly devastating fallout of the Lehman bankruptcy was on the “money market” for short-term lending. Lehman had borrowed considerable funds by issuing very short-term debt, called commercial paper. Among the major customers in commercial paper were money market mutual funds, which invest in short-term, high-quality debt of commercial borrowers. When Lehman faltered, the Reserve Primary Money Market Fund, which was holding large amounts of (AAA-rated!) Lehman commercial paper, suffered investment losses that drove the value of its assets below $1 per share.10 Fears spread that other funds were similarly exposed, and money market fund customers across the country rushed to withdraw their funds. The funds in turn rushed out of commercial paper into safer and more liquid Treasury bills, essentially shutting down short-term financing markets. The freezing up of credit markets was the end of any dwindling possibility that the financial crisis could be contained to Wall Street. Larger companies that had relied on the commercial paper market were now unable to raise short-term funds. Banks similarly found it difficult to raise funds. (Look back to Figure1.1, where you will see that the TED spread, a measure of bank insolvency fears, skyrocketed in 2008.) With banks unwilling or unable to extend credit to their customers, thousands of small businesses that relied on bank lines of credit also became unable to finance their normal business operations. Capital-starved companies were forced to scale back their own operations precipitously. The unemployment rate rose rapidly, and the economy was in its worst recession in decades. The turmoil in the financial markets had spilled over into the real economy, and Main Street had joined Wall Street in a bout of protracted misery.

The Dodd-Frank Reform Act The crisis engendered many calls for reform of Wall Street. These eventually led to the passage in 2010 of the Dodd-Frank Wall Street Reform and Consumer Protection Act, which proposes several mechanisms to mitigate systemic risk. The act calls for stricter rules for bank capital, liquidity, and risk management practices, especially as banks become larger and their potential failure would be more threatening to other institutions. With more capital supporting banks, the potential for one insolvency to trigger another could be contained. In addition, when banks have more capital, they have less incentive to ramp up risk, as potential losses will come at their own expense and not the FDIC’s. Dodd-Frank also mandates increased transparency, especially in derivatives markets. For example, one suggestion is to standardize CDS contracts so they can trade in centralized exchanges where prices can be determined in a deep market and gains or losses can be settled on a daily basis. Margin requirements, enforced daily, would prevent CDS participants from taking on greater positions than they can handle, and exchange trading would facilitate analysis of the exposure of firms to losses in these markets. The act also attempts to limit the risky activities in which banks can engage. The socalled Volcker Rule, named after former chairman of the Federal Reserve Paul Volcker, 10

Money market funds typically bear very little investment risk and can maintain their asset values at $1 per share. Investors view them as near substitutes for checking accounts. Until this episode, no other retail fund had “broken the buck.”

CHAPTER 1

The Investment Environment

prohibits banks from trading for their own accounts and limits total investments in hedge funds or private equity funds. The law also addresses shortcomings of the regulatory system that became apparent in 2008. The U.S. has several financial regulators with overlapping responsibility, and some institutions were accused of “regulator shopping,” seeking to be supervised by the most lenient regulator. Dodd-Frank seeks to unify and clarify lines of regulatory authority and responsibility in one or a smaller number of government agencies. The act addresses incentive issues. Among these are proposals to force employee compensation to reflect longer-term performance. The act requires public companies to set “claw-back provisions” to take back executive compensation if it was based on inaccurate financial statements. The motivation is to discourage excessive risk-taking by large financial institutions in which big bets can be wagered with the attitude that a successful outcome will result in a big bonus while a bad outcome will be borne by the company, or worse, the taxpayer. The incentives of the bond rating agencies are also a sore point. Few are happy with a system that has the ratings agencies paid by the firms they rate. The act creates an Office of Credit Ratings within the Securities and Exchange Commission to oversee the credit rating agencies. It is still too early to know which, if any, of these reforms will stick. The implementation of Dodd-Frank is still subject to considerable interpretation by regulators, and the act is still under attack by some members of Congress. But the crisis surely has made clear the essential role of the financial system in the functioning of the real economy.

1.8

Outline of the Text

The text has seven parts, which are fairly independent and may be studied in a variety of sequences. Part One is an introduction to financial markets, instruments, and trading of securities. This part also describes the mutual fund industry. Parts Two and Three contain the core of what has come to be known as “modern portfolio theory.” We start in Part Two with a general discussion of risk and return and the lessons of capital market history. We then focus more closely on how to describe investors’ risk preferences and progress to asset allocation, efficient diversification, and portfolio optimization. In Part Three, we investigate the implications of portfolio theory for the equilibrium relationship between risk and return. We introduce the capital asset pricing model, its implementation using index models, and more advanced models of risk and return. This part also treats the efficient market hypothesis as well as behavioral critiques of theories based on investor rationality and closes with a chapter on empirical evidence concerning security returns. Parts Four through Six cover security analysis and valuation. Part Four is devoted to debt markets and Part Five to equity markets. Part Six covers derivative assets, such as options and futures contracts. Part Seven is an introduction to active investment management. It shows how different investors’ objectives and constraints can lead to a variety of investment policies. This part discusses the role of active management in nearly efficient markets and considers how one should evaluate the performance of managers who pursue active strategies. It also shows how the principles of portfolio construction can be extended to the international setting and examines the hedge fund industry.

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PART I

SUMMARY

Introduction

1. Real assets create wealth. Financial assets represent claims to parts or all of that wealth. Financial assets determine how the ownership of real assets is distributed among investors. 2. Financial assets can be categorized as fixed income, equity, or derivative instruments. Topdown portfolio construction techniques start with the asset allocation decision—the allocation of funds across broad asset classes—and then progress to more specific security-selection decisions.

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3. Competition in financial markets leads to a risk–return trade-off, in which securities that offer higher expected rates of return also impose greater risks on investors. The presence of risk, however, implies that actual returns can differ considerably from expected returns at the beginning of the investment period. Competition among security analysts also promotes financial markets that are nearly informationally efficient, meaning that prices reflect all available information concerning the value of the security. Passive investment strategies may make sense in nearly efficient markets. 4. Financial intermediaries pool investor funds and invest them. Their services are in demand because small investors cannot efficiently gather information, diversify, and monitor portfolios. The financial intermediary sells its own securities to the small investors. The intermediary invests the funds thus raised, uses the proceeds to pay back the small investors, and profits from the difference (the spread). 5. Investment banking brings efficiency to corporate fund-raising. Investment bankers develop expertise in pricing new issues and in marketing them to investors. By the end of 2008, all the major stand-alone U.S. investment banks had been absorbed into commercial banks or had reorganized themselves into bank holding companies. In Europe, where universal banking had never been prohibited, large banks had long maintained both commercial and investment banking divisions.

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6. The financial crisis of 2008 showed the importance of systemic risk. Systemic risk can be limited by transparency that allows traders and investors to assess the risk of their counterparties, capital requirements to prevent trading participants from being brought down by potential losses, frequent settlement of gains or losses to prevent losses from accumulating beyond an institution’s ability to bear them, incentives to discourage excessive risk taking, and accurate and unbiased analysis by those charged with evaluating security risk.

security selection security analysis risk–return trade-off passive management active management financial intermediaries investment companies investment bankers

primary market secondary market venture capital private equity securitization systemic risk

KEY TERMS

investment real assets financial assets fixed-income (debt) securities equity derivative securities agency problem asset allocation

PROBLEM SETS

1. Financial engineering has been disparaged as nothing more than paper shuffling. Critics argue that resources used for rearranging wealth (that is, bundling and unbundling financial assets) might be better spent on creating wealth (that is, creating real assets). Evaluate this criticism. Are any benefits realized by creating an array of derivative securities from various primary securities?

Basic

2. Why would you expect securitization to take place only in highly developed capital markets? 3. What is the relationship between securitization and the role of financial intermediaries in the economy? What happens to financial intermediaries as securitization progresses?

CHAPTER 1

The Investment Environment

25

4. Although we stated that real assets constitute the true productive capacity of an economy, it is hard to conceive of a modern economy without well-developed financial markets and security types. How would the productive capacity of the U.S. economy be affected if there were no markets in which to trade financial assets? 5. Firms raise capital from investors by issuing shares in the primary markets. Does this imply that corporate financial managers can ignore trading of previously issued shares in the secondary market?

Intermediate

6. Suppose housing prices across the world double. a. Is society any richer for the change? b. Are homeowners wealthier? c. Can you reconcile your answers to (a) and (b)? Is anyone worse off as a result of the change?

a. Lanni takes out a bank loan. It receives $50,000 in cash and signs a note promising to pay back the loan over 3 years. b. Lanni uses the cash from the bank plus $20,000 of its own funds to finance the development of new financial planning software. c. Lanni sells the software product to Microsoft, which will market it to the public under the Microsoft name. Lanni accepts payment in the form of 1,500 shares of Microsoft stock. d. Lanni sells the shares of stock for $80 per share and uses part of the proceeds to pay off the bank loan. 8. Reconsider Lanni Products from the previous problem. a. Prepare its balance sheet just after it gets the bank loan. What is the ratio of real assets to total assets? b. Prepare the balance sheet after Lanni spends the $70,000 to develop its software product. What is the ratio of real assets to total assets? c. Prepare the balance sheet after Lanni accepts the payment of shares from Microsoft. What is the ratio of real assets to total assets? 9. Examine the balance sheet of commercial banks in Table 1.3. What is the ratio of real assets to total assets? What is that ratio for nonfinancial firms (Table 1.4)? Why should this difference be expected? 10. Consider Figure 1.5, which describes an issue of American gold certificates. a. Is this issue a primary or secondary market transaction? b. Are the certificates primitive or derivative assets? c. What market niche is filled by this offering? 11. Discuss the advantages and disadvantages of the following forms of managerial compensation in terms of mitigating agency problems, that is, potential conflicts of interest between managers and shareholders. a. A fixed salary. b. Stock in the firm that must be held for five years. c. A salary linked to the firm’s profits.

Figure 1.5 A gold-backed security

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7. Lanni Products is a start-up computer software development firm. It currently owns computer equipment worth $30,000 and has cash on hand of $20,000 contributed by Lanni’s owners. For each of the following transactions, identify the real and/or financial assets that trade hands. Are any financial assets created or destroyed in the transaction?

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PART I

Introduction 12. We noted that oversight by large institutional investors or creditors is one mechanism to reduce agency problems. Why don’t individual investors in the firm have the same incentive to keep an eye on management? 13. Give an example of three financial intermediaries and explain how they act as a bridge between small investors and large capital markets or corporations. 14. The average rate of return on investments in large stocks has outpaced that on investments in Treasury bills by about 7% since 1926. Why, then, does anyone invest in Treasury bills? 15. What are some advantages and disadvantages of top-down versus bottom-up investing styles? 16. You see an advertisement for a book that claims to show how you can make $1 million with no risk and with no money down. Will you buy the book? 17. Why do financial assets show up as a component of household wealth, but not of national wealth? Why do financial assets still matter for the material well-being of an economy? 18. Wall Street firms have traditionally compensated their traders with a share of the trading profits that they generated. How might this practice have affected traders’ willingness to assume risk? What is the agency problem this practice engendered?

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19. What reforms to the financial system might reduce its exposure to systemic risk?

E-INVESTMENTS EXERCISES 1. Log on to finance.yahoo.com and enter the ticker symbol RRD in the Get Quotes box to find information about R.R. Donnelley & Sons. a. Click on Proﬁle. What is Donnelly’s main line of business? b. Now go to Key Statistics. How many shares of the company’s stock are outstanding? What is the total market value of the ﬁrm? What were its proﬁts in the most recent ﬁscal year? c. Look up Major Holders of the company’s stock. What fraction of total shares is held by insiders? d. Now go to Analyst Opinion. What is the average price target (i.e., the predicted stock price of the Donnelly shares) of the analysts covering this ﬁrm? How does that compare to the price at which the stock is currently trading? e. Look at the company’s Balance Sheet. What were its total assets at the end of the most recent ﬁscal year? 2. a. Go to the Securities and Exchange Commission Web site, www.sec.gov. What is the mission of the SEC? What information and advice does the SEC offer to beginning investors? b. Go to the NASD Web site, www.ﬁnra.org. What is its mission? What information and advice does it offer to beginners? c. Go to the IOSCO Web site, www.iosco.org. What is its mission? What information and advice does it offer to beginners?

SOLUTIONS TO CONCEPT CHECKS 1. a. Real b. Financial c. Real d. Real e. Financial

CHAPTER 1

The Investment Environment

27

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2. The central issue is the incentive to monitor the quality of loans when originated as well as over time. Freddie and Fannie clearly had incentive to monitor the quality of conforming loans that they had guaranteed, and their ongoing relationships with mortgage originators gave them opportunities to evaluate track records over extended periods of time. In the subprime mortgage market, the ultimate investors in the securities (or the CDOs backed by those securities), who were bearing the credit risk, should not have been willing to invest in loans with a disproportionate likelihood of default. If they properly understood their exposure to default risk, then the (correspondingly low) prices they would have been willing to pay for these securities would have imposed discipline on the mortgage originators and servicers. The fact that they were willing to hold such large positions in these risky securities suggests that they did not appreciate the extent of their exposure. Maybe they were led astray by overly optimistic projections for housing prices or by biased assessments from the credit-reporting agencies. In principle, either arrangement for default risk could have provided the appropriate discipline on the mortgage originators; in practice, however, the informational advantages of Freddie and Fannie probably made them the better “recipients” of default risk. The lesson is that information and transparency are some of the preconditions for well-functioning markets.

2

CHAPTER TWO

Asset Classes and Financial Instruments

PART I

YOU LEARNED IN Chapter 1 that the process of building an investment portfolio usually begins by deciding how much money to allocate to broad classes of assets, such as safe money market securities or bank accounts, longer term bonds, stocks, or even asset classes like real estate or precious metals. This process is called asset allocation. Within each class the investor then selects specific assets from a more detailed menu. This is called security selection. Each broad asset class contains many specific security types, and the many variations on a theme can be overwhelming. Our goal in this chapter is to introduce you to the important features of broad classes of securities. Toward this end, we organize our tour of financial instruments according to asset class. Financial markets are traditionally segmented into money markets and capital

markets. Money market instruments include short-term, marketable, liquid, low-risk debt securities. Money market instruments sometimes are called cash equivalents, or just cash for short. Capital markets, in contrast, include longer term and riskier securities. Securities in the capital market are much more diverse than those found within the money market. For this reason, we will subdivide the capital market into four segments: longer term bond markets, equity markets, and the derivative markets for options and futures. We first describe money market instruments. We then move on to debt and equity securities. We explain the structure of various stock market indexes in this chapter because market benchmark portfolios play an important role in portfolio construction and evaluation. Finally, we survey the derivative security markets for options and futures contracts.

CHAPTER 2

2.1

Asset Classes and Financial Instruments

The Money Market

The money market is a subsector of the fixed-income market. It consists of very shortterm debt securities that usually are highly marketable. Table2.1 lists outstanding volume in 2012 for some of the major instruments in this market. Many of these securities trade in large denominations, and so are out of the reach of individual investors. Money market funds, however, are easily accessible to small investors. These mutual funds pool the resources of many investors and purchase a wide variety of money market securities on their behalf.

Treasury Bills U.S. Treasury bills (T-bills, or just bills, for short) are the most marketable of all money market instruments. T-bills represent the simplest form of borrowing: The government raises money by selling bills to the public. Investors buy the bills at a discount from the stated maturity value. At the bill’s maturity, the holder receives from the government a payment equal to the face value of the bill. The difference between the purchase price and ultimate maturity value constitutes the investor’s earnings. T-bills are issued with initial maturities of 4, 13, 26, or 52 weeks. Individuals can purchase T-bills directly, at auction, or on the secondary market from a government securities dealer. T-bills are highly liquid; that is, they are easily converted to cash and sold at low transaction cost and with not much price risk. Unlike most other money market instruments, which sell in minimum denominations of $100,000, T-bills sell in minimum denominations of only $100, although $10,000 denominations are far more common. The income earned on T-bills is exempt from all state and local taxes, another characteristic distinguishing them from other money market instruments. Figure2.1 is a partial listing of T-bill rates. Rather than providing prices of each bill, the financial press reports yields based on those prices. You will see yields corresponding to both bid and ask prices. The ask price is the price you would have to pay to buy a T-bill from a securities dealer. The bid price is the slightly lower price you would receive if you wanted to sell a bill to a dealer. The bid–ask spread is the difference in these prices, which is the dealer’s source of profit. (Notice in Figure2.1 that the bid yield is higher than the ask yield. This is because prices and yields are inversely related.)

$ Billion Repurchase agreements Small-denomination time deposits and savings deposits* Large-denomination time deposits* Treasury bills Commercial paper Money market mutual funds

$1,141 7,202 1,603 1,478 1,445 2,645

*Small denominations are less than $100,000. Sources: Economic Report of the President, U.S. Government Printing Office, 2012; Flow of Funds Accounts of the United States, Board of Governors of the Federal Reserve System, September 2012.

Table 2.1 Major components of the money market

29

30

PART I

Introduction

The first two yields in Figure2.1 are reported using the bank-discount method. This means that the bill’s discount Treasury Bills from its maturity or face value is “annualized” based on DAYS TO ASK a 360-day year, and then reported as a percentage of face MATURITY MAT BID ASKED CHG YLD value. For example, for the highlighted bill maturing on Sep 06 12 51 0.080 0.075 0.000 0.076 December 20, 2012, days to maturity are 156 and the yield Oct 04 12 79 0.085 0.080 0.000 0.081 under the column labeled “ASKED” is given as .125%. This Nov 01 12 107 0.110 0.100 0.005 0.101 Nov 29 12 135 0.110 0.105 0.000 0.106 means that a dealer was willing to sell the bill at a discount Dec 20 12 156 0.130 0.125 0.005 0.127 from face value of .125%3(156/360)5.0542%. So a bill Apr 04 13 261 0.160 0.150 0.005 0.152 with $10,000 face value could be purchased for $10,000 3 (1 2 .000542) 5 $9,994.58. Similarly, on the basis of the bid yield of .130%, a dealer would be willing to purchase the Figure 2.1 Treasury bill yields bill for $10,0003(12.001303156/360)5$9,994.367. The bank discount method for computing yields has a Source: Compiled from data obtained from The Wall Street Journal Online, July 17, 2012. long tradition, but it is flawed for at least two reasons. First, it assumes that the year has only 360 days. Second, it computes the yield as a fraction of par value rather than of the price the investor paid to acquire the bill.1 An investor who buys the bill for the ask price and holds it until maturity will see her investment grow over 156 days by a multiple of $10,000/$9,994.5851.000542, for a gain of .0542%. Annualizing this gain using a 365-day year results in a yield of .0542%3365/1565.127%, which is the value reported in the last column under “ASK YLD.” This last value is called the Treasury-bill’s bond-equivalent yield.

Certificates of Deposit A certificate of deposit, or CD, is a time deposit with a bank. Time deposits may not be withdrawn on demand. The bank pays interest and principal to the depositor only at the end of the fixed term of the CD. CDs issued in denominations greater than $100,000 are usually negotiable, however; that is, they can be sold to another investor if the owner needs to cash in the certificate before its maturity date. Short-term CDs are highly marketable, although the market significantly thins out for maturities of 3 months or more. CDs are treated as bank deposits by the Federal Deposit Insurance Corporation, so they are currently insured for up to $250,000 in the event of a bank insolvency.

Commercial Paper Large, well-known companies often issue their own short-term unsecured debt notes rather than borrow directly from banks. These notes are called commercial paper. Very often, commercial paper is backed by a bank line of credit, which gives the borrower access to cash that can be used (if needed) to pay off the paper at maturity. Commercial paper maturities range up to 270 days; longer maturities would require registration with the Securities and Exchange Commission and so are almost never issued. Most often, commercial paper is issued with maturities of less than 1 or 2 months. Usually, it is issued in multiples of $100,000. Therefore, small investors can invest in commercial paper only indirectly, via money market mutual funds. Commercial paper is considered to be a fairly safe asset, because a firm’s condition presumably can be monitored and predicted over a term as short as 1 month. 1

Both of these “errors” were dictated by computational simplicity in precomputer days. It is easier to compute percentage discounts from a round number such as par value rather than purchase price. It is also easier to annualize using a 360-day year, because 360 is an even multiple of so many numbers.

CHAPTER 2

Asset Classes and Financial Instruments

While most commercial paper is issued by nonfinancial firms, in recent years there was a sharp increase in asset-backed commercial paper issued by financial firms such as banks. This was short-term commercial paper typically used to raise funds for the institution to invest in other assets, most notoriously, subprime mortgages. These assets were in turn used as collateral for the commercial paper—hence the label “asset backed.” This practice led to many difficulties starting in the summer of 2007 when the subprime mortgagors began defaulting. The banks found themselves unable to issue new commercial paper to refinance their positions as the old paper matured.

Bankers’ Acceptances A banker’s acceptance starts as an order to a bank by a bank’s customer to pay a sum of money at a future date, typically within 6 months. At this stage, it is similar to a postdated check. When the bank endorses the order for payment as “accepted,” it assumes responsibility for ultimate payment to the holder of the acceptance. At this point, the acceptance may be traded in secondary markets like any other claim on the bank. Bankers’ acceptances are considered very safe assets because traders can substitute the bank’s credit standing for their own. They are used widely in foreign trade where the creditworthiness of one trader is unknown to the trading partner. Acceptances sell at a discount from the face value of the payment order, just as T-bills sell at a discount from par value.

Eurodollars Eurodollars are dollar-denominated deposits at foreign banks or foreign branches of American banks. By locating outside the United States, these banks escape regulation by the Federal Reserve. Despite the tag “Euro,” these accounts need not be in European banks, although that is where the practice of accepting dollar-denominated deposits outside the United States began. Most Eurodollar deposits are for large sums, and most are time deposits of less than 6 months’ maturity. A variation on the Eurodollar time deposit is the Eurodollar certificate of deposit, which resembles a domestic bank CD except that it is the liability of a non-U.S. branch of a bank, typically a London branch. The advantage of Eurodollar CDs over Eurodollar time deposits is that the holder can sell the asset to realize its cash value before maturity. Eurodollar CDs are considered less liquid and riskier than domestic CDs, however, and thus offer higher yields. Firms also issue Eurodollar bonds, which are dollardenominated bonds outside the U.S., although bonds are not a money market investment because of their long maturities.

Repos and Reverses Dealers in government securities use repurchase agreements, also called “repos” or “RPs,” as a form of short-term, usually overnight, borrowing. The dealer sells government securities to an investor on an overnight basis, with an agreement to buy back those securities the next day at a slightly higher price. The increase in the price is the overnight interest. The dealer thus takes out a 1-day loan from the investor, and the securities serve as collateral. A term repo is essentially an identical transaction, except that the term of the implicit loan can be 30 days or more. Repos are considered very safe in terms of credit risk because the loans are backed by the government securities. A reverse repo is the mirror image of a repo. Here, the dealer finds an investor holding government securities and buys them, agreeing to sell them back at a specified higher price on a future date.

31

32

PART I

Introduction

Federal Funds Just as most of us maintain deposits at banks, banks maintain deposits of their own at a Federal Reserve bank. Each member bank of the Federal Reserve System, or “the Fed,” is required to maintain a minimum balance in a reserve account with the Fed. The required balance depends on the total deposits of the bank’s customers. Funds in the bank’s reserve account are called federal funds, or fed funds. At any time, some banks have more funds than required at the Fed. Other banks, primarily big banks in New York and other financial centers, tend to have a shortage of federal funds. In the federal funds market, banks with excess funds lend to those with a shortage. These loans, which are usually overnight transactions, are arranged at a rate of interest called the federal funds rate. Although the fed funds market arose primarily as a way for banks to transfer balances to meet reserve requirements, today the market has evolved to the point that many large banks use federal funds in a straightforward way as one component of their total sources of funding. Therefore, the fed funds rate is simply the rate of interest on very short-term loans among financial institutions. While most investors cannot participate in this market, the fed funds rate commands great interest as a key barometer of monetary policy.

Brokers’ Calls Individuals who buy stocks on margin borrow part of the funds to pay for the stocks from their broker. The broker in turn may borrow the funds from a bank, agreeing to repay the bank immediately (on call) if the bank requests it. The rate paid on such loans is usually about 1% higher than the rate on short-term T-bills.

The LIBOR Market The London Interbank Offered Rate (LIBOR) is the rate at which large banks in London are willing to lend money among themselves. This rate, which is quoted on dollardenominated loans, has become the premier short-term interest rate quoted in the European money market, and it serves as a reference rate for a wide range of transactions. For example, a corporation might borrow at a floating rate equal to LIBOR plus 2%. LIBOR interest rates may be tied to currencies other than the U.S. dollar. For example, LIBOR rates are widely quoted for transactions denominated in British pounds, yen, euros, and so on. There is also a similar rate called EURIBOR (European Interbank Offered Rate) at which banks in the euro zone are willing to lend euros among themselves. LIBOR is a key reference rate in the money market, and many trillions of dollars of loans and derivative assets are tied to it. Therefore, the 2012 scandal involving the fixing of LIBOR deeply shook these markets. The nearby box discusses those events.

Yields on Money Market Instruments Although most money market securities are of low risk, they are not risk-free. The securities of the money market promise yields greater than those on default-free T-bills, at least in part because of greater relative riskiness. In addition, many investors require more liquidity; thus they will accept lower yields on securities such as T-bills that can be quickly and cheaply sold for cash. Figure2.2 shows that bank CDs, for example, consistently have paid a premium over T-bills. Moreover, that premium increased with economic crises such as the energy price shocks associated with the two OPEC disturbances, the failure of Penn Square bank, the stock market crash in 1987, the collapse of Long Term Capital Management in 1998, and the credit crisis beginning with the breakdown of the market in subprime mortgages beginning in 2007. If you look back to Figure 1.1 in Chapter 1, you’ll see that

LIBOR was designed initially as a survey of interbank lending rates but soon became a key determinant of shortterm interest rates with far-reaching significance. Around $350 trillion of derivative contracts have payoffs tied to it, and perhaps another $400 trillion of loans and bonds with floating interest rates linked to LIBOR are currently outstanding. LIBOR is quoted for loans in several currencies, e.g., the dollar, yen, euro, and British pound, and for maturities ranging from a day to a year, although 3 months is the most common. However, LIBOR is not a rate at which actual transactions occur; instead, it is just a survey of “estimated” borrowing rates, and this has made it vulnerable to tampering. Several large banks are asked to report the rate at which they believe they can borrow in the interbank market. Outliers are trimmed from the sample of responses, and LIBOR is calculated as the average of the mid-range estimates. Over time, several problems surfaced. First, it appeared that banks understated the rates at which they claimed they could borrow in an effort to make themselves look financially stronger. Other surveys that asked for estimates of the rates at which other banks could borrow resulted in higher values. Moreover, LIBOR did not seem to reflect current market conditions. A majority of LIBOR submissions were unchanged from day to day even when other interest rates fluctuated, and LIBOR spreads showed surprisingly

low correlation with other measures of credit risk such as spreads on credit default swaps. Even worse, once the market came under scrutiny, it emerged that participating banks were colluding to manipulate their LIBOR submissions to enhance profits on their derivatives trades. Traders used e-mails and instant messages to tell each other whether they wanted to see higher or lower submissions. Members of this informal cartel essentially set up a “favor bank” to help each other move the survey average up or down depending on their trading positions. To date, around $2.5 billion of fines have been paid: Royal Bank of Scotland paid $612 million, Barclays $450 million, and UBS $1,500 million. Other banks remain under investigation. But government fines may be only the tip of the iceberg. Private lawsuits are sure to come, as anyone trading a LIBOR derivative against these banks or anyone who participated in a loan with an interest rate tied to LIBOR can claim to have been harmed. Several reforms have been suggested. The British Bankers Association, which until recently ran the LIBOR survey, yielded responsibility for LIBOR to British regulators. Other proposals are to increase the number of submissions to make collusion more difficult and to eliminate LIBOR in less active currencies and maturities where collusion is easier. More substantive proposals would replace the survey rate with one based on actual, verifiable, transactions—i.e., real loans among banks.

the TED spread, the difference between the LIBOR rate and Treasury bills, also peaked during periods of financial stress. Money market funds are mutual funds that invest in money market instruments and have become major sources of funding to that sector. The nearby box discusses the fallout of the credit crisis of 2008 on those funds.

5.0

OPEC I

Percentage Points

4.5 4.0

Credit Crisis

3.5

OPEC lI

3.0

Penn Square

2.5 Market Crash

2.0 1.5

LTCM

1.0 0.5 2014

2012

2010

2008

2006

2004

2002

2000

1998

1996

1994

1992

1990

1988

1986

1984

1982

1980

1978

1976

1974

1972

1970

0.0

Figure 2.2 The spread between 3-month CD and Treasury bill rates 33

WORDS FROM THE STREET

The LIBOR Scandals

WORDS FROM THE STREET

Money Market Funds and the Credit Crisis of 2008 Money market funds are mutual funds that invest in the short-term debt instruments that comprise the money market. In 2013, these funds had investments totaling about $2.6 trillion. They are required to hold only short-maturity debt of the highest quality: The average maturity of their holdings must be maintained at less than 3 months. Their biggest investments tend to be in commercial paper, but they also hold sizable fractions of their portfolios in certificates of deposit, repurchase agreements, and Treasury securities. Because of this very conservative investment profile, money market funds typically experience extremely low price risk. Investors for their part usually acquire checkwriting privileges with their funds and often use them as a close substitute for a bank account. This is feasible because the funds almost always maintain share value at $1.00 and pass along all investment earnings to their investors as interest. Until 2008, only one fund had “broken the buck,” that is, suffered losses large enough to force value per share below $1. But when Lehman Brothers filed for bankruptcy protection on September 15, 2008, several funds that had invested heavily in its commercial paper suffered large losses. The next day, the Reserve Primary Fund, the oldest money market fund, broke the buck when its value per share fell to only $.97. The realization that money market funds were at risk in the credit crisis led to a wave of investor redemptions similar to a run on a bank. Only three days after the Lehman bankruptcy, Putman’s Prime Money Market Fund announced that it was liquidating due to heavy redemptions. Fearing further outflows, the U.S. Treasury announced that it would make federal insurance available to money market funds willing to pay an insurance fee. This program would

2.2

thus be similar to FDIC bank insurance. With the federal insurance in place, the outflows were quelled. However, the turmoil in Wall Street’s money market funds had already spilled over into “Main Street.” Fearing further investor redemptions, money market funds had become afraid to commit funds even over short periods, and their demand for commercial paper had effectively dried up. Firms throughout the economy had come to depend on those markets as a major source of shortterm finance to fund expenditures ranging from salaries to inventories. Further breakdown in the money markets would have had an immediate crippling effect on the broad economy. To end the panic and stabilize the money markets, the federal government decided to guarantee investments in money market funds. The guarantee did in fact calm investors and end the run, but it put the government on the hook for a potential liability of up to $3 trillion—the assets held in money market funds at the time. To prevent another occurrence of this crisis, the SEC later proposed that money market funds no longer be allowed to “round off” value per share to $1, but instead be forced to recognize daily changes in value. Alternatively, funds wishing to maintain share value at $1 would be required to set aside reserves against potential investment losses. But the mutual fund industry lobbied vehemently against these reforms, arguing that their customers demanded stable share prices and that the proposed capital requirements would be so costly that the industry would no longer be viable. In the face of this opposition, the SEC commissioners voted in 2012 against the reforms, but they were given new life when the Financial Stability Oversight Council weighed in to support them. It is still too early to predict the final resolution of the debate.

The Bond Market The bond market is composed of longer term borrowing or debt instruments than those that trade in the money market. This market includes Treasury notes and bonds, corporate bonds, municipal bonds, mortgage securities, and federal agency debt. These instruments are sometimes said to comprise the fixed-income capital market, because most of them promise either a fixed stream of income or a stream of income that is determined according to a specific formula. In practice, these formulas can result in a flow of income that is far from fixed. Therefore, the term fixed income is probably not fully appropriate. It is simpler and more straightforward to call these securities either debt instruments or bonds.

Treasury Notes and Bonds The U.S. government borrows funds in large part by selling Treasury notes and Treasury bonds. T-notes are issued with maturities ranging up to 10 years, while bonds are issued with maturities ranging from 10 to 30 years. Both notes and bonds may be 34

CHAPTER 2

Asset Classes and Financial Instruments

35

issued in increments of $100 but far more commonly trade in denominations of $1,000. Both notes and bonds make Listing of Treasury Issues ASK semiannual interest payments called coupon payments, MATURITY COUPON BID ASKED CHG YLD a name derived from precomputer days, when investors 0.212 –0.0078 105.3438 105.3281 4.250 Nov 15 13 would literally clip coupons attached to the bond and pres0.398 –0.0859 113.5391 113.5078 4.500 Nov 15 15 ent a coupon to receive the interest payment. 0.729 –0.1406 115.1172 115.0703 3.500 Feb 15 18 1.107 –0.2734 154.4375 154.3906 8.500 Feb 15 20 Figure2.3 is a listing of Treasury issues. Notice the high1.809 –0.6641 158.7578 158.6797 6.875 Aug 15 25 lighted note that matures in November 2015. Its bid price 2.113 –0.8906 161.1875 161.1094 6.250 May 15 30 is 113.5078. (This is the decimal version of 11365/128. The 2.378 –0.9375 138.1250 138.0469 4.500 Feb 15 36 2.596 –0.9297 108.3594 108.2969 3.000 May 15 42 minimum tick size, or price increment in the Wall Street Journal listing, is generally 1/128 of a point.) Although bonds are typically traded in denominations of $1,000 par value, prices are quoted as a percentage of par. Thus, the bid price Figure 2.3 Listing of Treasury bonds and notes should be interpreted as 113.5078% of par, or $1,135.078 Source: Compiled from data obtained from the Wall Street for the $1,000 par value bond. Similarly, the ask price at Journal Online, July 17, 2012. which the bond could be sold to a dealer is 113.5391% of par, or $1,135.391. The 2.0859 change means that the closing price on this day fell by .0859% of par value (equivalently, by 11/128 of a point) from the previous day’s close. Finally, the yield to maturity based on the ask price is .398%. The yield to maturity reported in the financial pages is calculated by determining the semiannual yield and then doubling it, rather than compounding it for two half-year periods. This use of a simple interest technique to annualize means that the yield is quoted on an annual percentage rate (APR) basis rather than as an effective annual yield. The APR method in this context is also called the bond equivalent yield. We discuss the yield to maturity in more detail in Part Four.

CONCEPT CHECK

2.1

What were the bid price, ask price, and yield to maturity of the 4.5% February 2036 Treasury bond displayed in Figure2.3? What was its ask price the previous day?

Inflation-Protected Treasury Bonds The best place to start building an investment portfolio is at the least risky end of the spectrum. Around the world, governments of many countries, including the United States, have issued bonds that are linked to an index of the cost of living in order to provide their citizens with an effective way to hedge inflation risk. In the United States inflation-protected Treasury bonds are called TIPS (Treasury Inflation-Protected Securities). The principal amount on these bonds is adjusted in proportion to increases in the Consumer Price Index. Therefore, they provide a constant stream of income in real (inflation-adjusted) dollars. Yields on TIPS bonds should be interpreted as real or inflation-adjusted interest rates. We return to TIPS bonds in more detail in Chapter 14.

Federal Agency Debt Some government agencies issue their own securities to finance their activities. These agencies usually are formed to channel credit to a particular sector of the economy that Congress believes might not receive adequate credit through normal private sources.

36

PART I

Introduction

The major mortgage-related agencies are the Federal Home Loan Bank (FHLB), the Federal National Mortgage Association (FNMA, or Fannie Mae), the Government National Mortgage Association (GNMA, or Ginnie Mae), and the Federal Home Loan Mortgage Corporation (FHLMC, or Freddie Mac). The FHLB borrows money by issuing securities and lends this money to savings and loan institutions to be lent in turn to individuals borrowing for home mortgages. Although the debt of federal agencies was never explicitly insured by the federal government, it had long been assumed that the government would assist an agency nearing default. Those beliefs were validated when Fannie Mae and Freddie Mac encountered severe financial distress in September 2008. With both firms on the brink of insolvency, the government stepped in and put them both into conservatorship, assigned the Federal Housing Finance Agency to run the firms, but did in fact agree to make good on the firm’s bonds.

International Bonds Many firms borrow abroad and many investors buy bonds from foreign issuers. In addition to national capital markets, there is a thriving international capital market, largely centered in London. A Eurobond is a bond denominated in a currency other than that of the country in which it is issued. For example, a dollar-denominated bond sold in Britain would be called a Eurodollar bond. Similarly, investors might speak of Euroyen bonds, yen-denominated bonds sold outside Japan. Because the European currency is called the euro, the term Eurobond may be confusing. It is best to think of them simply as international bonds. In contrast to bonds that are issued in foreign currencies, many firms issue bonds in foreign countries but in the currency of the investor. For example, a Yankee bond is a dollardenominated bond sold in the United States by a non-U.S. issuer. Similarly, Samurai bonds are yen-denominated bonds sold in Japan by non-Japanese issuers.

Municipal Bonds Municipal bonds are issued by state and local governments. They are similar to Treasury and corporate bonds except that their interest income is exempt from federal income taxation. The interest income also is usually exempt from state and local taxation in the issuing state. Capital gains taxes, however, must be paid on “munis” when the bonds mature or if they are sold for more than the investor’s purchase price. General obligation bonds are backed by the “full faith and credit” (i.e., the taxing power) of the issuer, while revenue bonds are issued to finance particular projects and are backed either by the revenues from that project or by the particular municipal agency operating the project. Typical issuers of revenue bonds are airports, hospitals, and turnpike or port authorities. Obviously, revenue bonds are riskier in terms of default than general obligation bonds. Figure2.4 plots outstanding amounts of both types of municipal securities. An industrial development bond is a revenue bond that is issued to finance commercial enterprises, such as the construction of a factory that can be operated by a private firm. In effect, these private-purpose bonds give the firm access to the municipality’s ability to borrow at tax-exempt rates, and the federal government limits the amount of these bonds that may be issued.2 Like Treasury bonds, municipal bonds vary widely in maturity. A good deal of the debt issued is in the form of short-term tax anticipation notes, which raise funds to pay for 2

A warning, however. Although interest on industrial development bonds usually is exempt from federal tax, it can be subject to the alternative minimum tax if the bonds are used to finance projects of for-profit companies.

CHAPTER 2

Asset Classes and Financial Instruments

3,000

$ Billion

2,500 2,000

Industrial Revenue Bonds General Obligation

1,500 1,000 500

1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

Figure 2.4 Tax-exempt debt outstanding Source: Flow of Funds Accounts of the United States, Board of Governors of the Federal Reserve System, June 2012.

expenses before actual collection of taxes. Other municipal debt is long term and used to fund large capital investments. Maturities range up to 30 years. The key feature of municipal bonds is their tax-exempt status. Because investors pay neither federal nor state taxes on the interest proceeds, they are willing to accept lower yields on these securities. An investor choosing between taxable and tax-exempt bonds must compare after-tax returns on each bond. An exact comparison requires a computation of after-tax rates of return that explicitly accounts for taxes on income and realized capital gains. In practice, there is a simpler rule of thumb. If we let t denote the investor’s combined federal plus local marginal tax bracket and r denote the total before-tax rate of return available on taxable bonds, then r (1 2 t) is the after-tax rate available on those securities.3 If this value exceeds the rate on municipal bonds, rm, the investor does better holding the taxable bonds. Otherwise, the tax-exempt municipals provide higher after-tax returns. One way to compare bonds is to determine the interest rate on taxable bonds that would be necessary to provide an after-tax return equal to that of municipals. To derive this value, we set after-tax yields equal, and solve for the equivalent taxable yield of the tax-exempt bond. This is the rate a taxable bond must offer to match the after-tax yield on the tax-free municipal. r (1 2 t) 5 rm

(2.1)

r 5 rm /(12 t)

(2.2)

or

Thus the equivalent taxable yield is simply the tax-free rate divided by 1 2 t. Table2.2 presents equivalent taxable yields for several municipal yields and tax rates. 3

An approximation to the combined federal plus local tax rate is just the sum of the two rates. For example, if your federal tax rate is 28% and your state rate is 5%, your combined tax rate would be approximately 33%. A more precise approach would recognize that state taxes are deductible at the federal level. You owe federal taxes only on income net of state taxes. Therefore, for every dollar of income, your after-tax proceeds would be (1 2 tfederal) 3(1 2 tstate). In our example, your after-tax proceeds on each dollar earned would be (12.28)3(12.05)5.684, which implies a combined tax rate of 12.6845.316, or 31.6%.

37

38

PART I

Introduction

Table 2.2

Tax-Exempt Yield

Equivalent taxable yields corresponding to various tax-exempt yields

Marginal Tax Rate

1%

2%

3%

4%

5%

20% 30

1.25% 1.43

2.50% 2.86

3.75% 4.29

5.00% 5.71

6.25% 7.14

40

1.67

3.33

5.00

6.67

8.33

50

2.00

4.00

6.00

8.00

10.00

This table frequently appears in the marketing literature for tax-exempt mutual bond funds because it demonstrates to high-tax-bracket investors that municipal bonds offer highly attractive equivalent taxable yields. Each entry is calculated from Equation 2.2. If the equivalent taxable yield exceeds the actual yields offered on taxable bonds, the investor is better off after taxes holding municipal bonds. Notice that the equivalent taxable interest rate increases with the investor’s tax bracket; the higher the bracket, the more valuable the tax-exempt feature of municipals. Thus high-tax-bracket investors tend to hold municipals. We also can use Equation 2.1 or 2.2 to find the tax bracket at which investors are indifferent between taxable and tax-exempt bonds. The cutoff tax bracket is given by solving Equation 2.2 for the tax bracket at which after-tax yields are equal. Doing so, we find that t 5 12

rm r

(2.3)

Thus the yield ratio rm/r is a key determinant of the attractiveness of municipal bonds. The higher the yield ratio, the lower the cutoff tax bracket, and the more individuals will prefer to hold municipal debt. Figure2.5 plots the ratio of 20-year municipal debt yields to the

0.9

Ratio

0.8

0.7

0.6

Figure 2.5 Ratio of yields on municipal debt to corporate Baa-rated debt Source: Authors’ calculations, using data from www.federalreserve.gov/releases/h15/data.htm.

2013

2010

2007

2004

2001

1998

1995

1992

1989

1986

1983

1980

1977

1974

1971

1968

1965

1962

1959

1956

1953

0.5

CHAPTER 2

Asset Classes and Financial Instruments

yield on Baa-rated corporate debt. The default risk of these corporate and municipal bonds may be comparable, but certainly will fluctuate over time. For example, the sharp run-up in the ratio in 2011 probably reflects increased concern at the time about the precarious financial condition of several states and municipalities.

Example 2.1

Taxable versus Tax-Exempt Yields

Figure2.5 shows that in recent years, the ratio of tax-exempt to taxable yields has fluctuated around .70. What does this imply about the cutoff tax bracket above which tax-exempt bonds provide higher after-tax yields? Equation 2.3 shows that an investor whose tax bracket (federal plus local) exceeds 1 2.70 5 .30, or 30%, will derive a greater after-tax yield from municipals. Note, however, that it is difficult to control precisely for differences in the risks of these bonds, so the cutoff tax bracket must be taken as approximate.

CONCEPT CHECK

2.2

Suppose your tax bracket is 30%. Would you prefer to earn a 6% taxable return or a 4% tax-free return? What is the equivalent taxable yield of the 4% tax-free yield?

Corporate Bonds Corporate bonds are the means by which private firms borrow money directly from the public. These bonds are similar in structure to Treasury issues—they typically pay semiannual coupons over their lives and return the face value to the bondholder at maturity. They differ most importantly from Treasury bonds in degree of risk. Default risk is a real consideration in the purchase of corporate bonds, and Chapter 14 discusses this issue in considerable detail. For now, we distinguish only among secured bonds, which have specific collateral backing them in the event of firm bankruptcy; unsecured bonds, called debentures, which have no collateral; and subordinated debentures, which have a lower priority claim to the firm’s assets in the event of bankruptcy. Corporate bonds sometimes come with options attached. Callable bonds give the firm the option to repurchase the bond from the holder at a stipulated call price. Convertible bonds give the bondholder the option to convert each bond into a stipulated number of shares of stock. These options are treated in more detail in Chapter 14.

Mortgages and Mortgage-Backed Securities Because of the explosion in mortgage-backed securities, almost anyone can invest in a portfolio of mortgage loans, and these securities have become a major component of the fixed-income market. As described in Chapter 1, a mortgage-backed security is either an ownership claim in a pool of mortgages or an obligation that is secured by such a pool. Most pass-throughs have traditionally been comprised of conforming mortgages, which means that the loans must satisfy certain underwriting guidelines (standards for the credit-worthiness of the borrower) before they may be purchased by Fannie Mae

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8,000 7,000

Private Issuers Federal Agencies

$ Billions

6,000 5,000 4,000 3,000 2,000 1,000

2012

2009

2006

2003

2000

1997

1994

1991

1988

1985

1982

1979

Figure 2.6 Mortgage-backed securities outstanding Source: Flow of Funds Accounts of the United States, Board of Governors of the Federal Reserve System, June 2012.

or Freddie Mac. In the years leading up to the financial crisis, however, a large amount of subprime mortgages, that is, riskier loans made to financially weaker borrowers, were bundled and sold by “private-label” issuers. Figure 2.6 illustrates the explosive growth of both agency and private-label mortgage-backed securities, at least until the crisis. In an effort to make housing more affordable to low-income households, Fannie and Freddie had been encouraged to buy subprime mortgage securities. As we saw in Chapter1, these loans turned out to be disastrous, with trillion-dollar losses spread among banks, hedge funds and other investors, and Freddie and Fannie, which lost billions of dollars on the subprime mortgage pools they had purchased. You can see from Figure2.6 that starting in 2007, the market in private-label mortgage pass-throughs began to shrink rapidly. Agency pass-throughs shrank even more precipitously following an agreement for Freddie and Fannie to wind down purchases of mortgages for new pass-throughs. At the same time, existing pass-throughs shrank as healthy loans were paid off and delinquent loans were removed from outstanding pools. Despite these troubles, few believe that securitization itself will cease, although practices in this market are highly likely to become far more conservative than in previous years, particularly with respect to the credit standards that must be met by the ultimate borrower. Indeed, securitization has become an increasingly common staple of many credit markets. For example, car loans, student loans, home equity loans, credit card loans, and even debt of private firms now are commonly bundled into pass-through securities that can be traded in the capital market. Figure2.7 documents the rapid growth of nonmortgage asset-backed securities. The market expanded more than five-fold in the decade ending 2007. After the financial crisis, it contracted considerably as the perceived risks of credit card and home equity loans skyrocketed, but the asset-backed market is still substantial.

CHAPTER 2

Asset Classes and Financial Instruments

2,000 Automobile

1,800

Credit Card Equipment

1,600

Home Equity

$ Billion

1,400

Manufactured Housing

1,200

Student Loans

1,000 800 600 400 200

2012

2011

2010

2009

2008

2007

2006

2005

2004

2003

2002

2001

2000

1999

1998

1997

1996

Figure 2.7 Asset-backed securities outstanding Source: The Securities & Industry and Financial Markets Association, www.sifma.org.

2.3

Equity Securities

Common Stock as Ownership Shares Common stocks, also known as equity securities or equities, represent ownership shares in a corporation. Each share of common stock entitles its owner to one vote on any matters of corporate governance that are put to a vote at the corporation’s annual meeting and to a share in the financial benefits of ownership.4 The corporation is controlled by a board of directors elected by the shareholders. The board, which meets only a few times each year, selects managers who actually run the corporation on a day-to-day basis. Managers have the authority to make most business decisions without the board’s specific approval. The board’s mandate is to oversee the management to ensure that it acts in the best interests of shareholders. The members of the board are elected at the annual meeting. Shareholders who do not attend the annual meeting can vote by proxy, empowering another party to vote in their name. Management usually solicits the proxies of shareholders and normally gets a vast majority of these proxy votes. Thus, management usually has considerable discretion 4

A corporation sometimes issues two classes of common stock, one bearing the right to vote, the other not. Because of its restricted rights, the nonvoting stock might sell for a lower price.

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to run the firm as it sees fit—without daily oversight from the equityholders who actually own the firm. We noted in Chapter 1 that such separation of ownership and control can give rise to “agency problems,” in which managers pursue goals not in the best interests of shareholders. However, there are several mechanisms that alleviate these agency problems. Among these are compensation schemes that link the success of the manager to that of the firm; oversight by the board of directors as well as outsiders such as security analysts, creditors, or large institutional investors; the threat of a proxy contest in which unhappy shareholders attempt to replace the current management team; or the threat of a takeover by another firm. The common stock of most large corporations can be bought or sold freely on one or more stock exchanges. A corporation whose stock is not publicly traded is said to be closely held. In most closely held corporations, the owners of the firm also take an active role in its management. Therefore, takeovers are generally not an issue.

Characteristics of Common Stock The two most important characteristics of common stock as an investment are its residual claim and limited liability features. Residual claim means that stockholders are the last in line of all those who have a claim on the assets and income of the corporation. In a liquidation of the firm’s assets the shareholders have a claim to what is left after all other claimants such as the tax authorities, employees, suppliers, bondholders, and other creditors have been paid. For a firm not in liquidation, shareholders have claim to the part of operating income left over after interest and taxes have been paid. Management can either pay this residual as cash dividends to shareholders or reinvest CONCEPT CHECK 2.3 it in the business to increase the value of the shares. Limited liability means that the most shareholders a. If you buy 100 shares of IBM stock, to what can lose in the event of failure of the corporation is their are you entitled? original investment. Unlike owners of unincorporated b. What is the most money you can make on businesses, whose creditors can lay claim to the personal this investment over the next year? assets of the owner (house, car, furniture), corporate c. If you pay $180 per share, what is the most shareholders may at worst have worthless stock. They are money you could lose over the year? not personally liable for the firm’s obligations.

Stock Market Listings Figure2.8 presents key trading data for a small sample of stocks traded on the New York Stock Exchange. The NYSE is one of several markets in which investors may buy or sell shares of stock. We will examine these markets in detail in Chapter 3. To interpret Figure2.8, consider the highlighted listing for General Electric. The table provides the ticker symbol (GE), the closing price of the stock ($19.72), and its change (1$.13) from the previous trading day. About 45.3 million shares of GE traded on this day. The listing also provides the highest and lowest price at which GE has traded in the last 52 weeks. The .68 value in the Dividend column means that the last quarterly dividend payment was $.17 per share, which is consistent with annual dividend payments of $.17 34 5$.68. This corresponds to an annual dividend yield (i.e., annual dividend per dollar paid for the stock) of .68/19.725.0345, or 3.45%. The dividend yield is only part of the return on a stock investment. It ignores prospective capital gains (i.e., price increases) or losses. Low-dividend firms presumably offer greater prospects for capital gains, or investors would not be willing to hold these stocks in their portfolios. If you scan Figure2.8, you will see that dividend yields vary widely across companies.

CHAPTER 2

NAME

SYMBOL CLOSE NET CHG

Game Stop CI A Gannett Gap Gardner Denver Gartner GasLog GATX Gaylord Entertainment Gazit-Globe GenCorp Genco Shipping&Trading Generac Holdings General Cable General Dynamics General Electric

GME GCI GPS GDI IT GLOG GMT GET GZT GY GNK GNRC BGC GD GE

16.74 14.65 28.47 48.30 45.71 10.24 38.76 37.78 9.30 6.69 2.79 22.21 26.80 64.69 19.72

VOLUME 52 WK HIGH 52 WK LOW

0.17 1,584,470 20.04 4,277,615 0.50 5,952,210 0.08 797,258 20.14 580,056 0.08 126,666 0.27 226,825 0.44 273,954 20.11 2,152 20.01 321,785 0.04 355,441 20.14 169,519 0.47 322,159 0.65 1,359,120 0.13 45,307,178

26.66 16.26 29.23 92.93 46.69 13.34 45.50 40.37 11.07 7.27 10.14 30.61 45.20 74.54 21.00

16.36 8.28 15.08 45.54 31.98 8.76 28.90 17.39 8.41 3.74 2.72 15.41 20.21 53.95 14.02

Asset Classes and Financial Instruments

DIV YIELD

0.60 0.80 0.50 0.20 .... .... 1.20 .... 0.43 .... .... .... .... 2.04 0.68

P/E YTD% CHG

3.58 6.80 5.46 8.07 1.76 16.88 0.41 9.01 .... 30.98 .... .... 3.10 15.19 .... 105.03 .... 4.61 .... 93.96 .... ....dd .... 4.36 .... 19.43 3.15 9.31 3.45 16.01

230.63 9.57 53.48 237.32 31.46 217.49 211.22 56.50 21.59 25.75 258.73 220.76 7.16 22.59 10.11

Figure 2.8 Listing of stocks traded on the New York Stock Exchange Source: Compiled from data from The Wall Street Journal Online, July 18, 2012.

The P/E ratio, or price–earnings ratio, is the ratio of the current stock price to last year’s earnings per share. The P/E ratio tells us how much stock purchasers must pay per dollar of earnings that the firm generates. For GE, the ratio of price to earnings is 16.01. The P/E ratio also varies widely across firms. Where the dividend yield and P/E ratio are not reported in Figure2.8, the firms have zero dividends, or zero or negative earnings. We shall have much to say about P/E ratios in Chapter 18. Finally, we see that GE’s stock price has increased by 10.11% since the beginning of the year.

Preferred Stock Preferred stock has features similar to both equity and debt. Like a bond, it promises to pay to its holder a fixed amount of income each year. In this sense preferred stock is similar to an infinite-maturity bond, that is, a perpetuity. It also resembles a bond in that it does not convey voting power regarding the management of the firm. Preferred stock is an equity investment, however. The firm retains discretion to make the dividend payments to the preferred stockholders; it has no contractual obligation to pay those dividends. Instead, preferred dividends are usually cumulative; that is, unpaid dividends cumulate and must be paid in full before any dividends may be paid to holders of common stock. In contrast, the firm does have a contractual obligation to make the interest payments on the debt. Failure to make these payments sets off corporate bankruptcy proceedings. Preferred stock also differs from bonds in terms of its tax treatment for the firm. Because preferred stock payments are treated as dividends rather than interest, they are not tax-deductible expenses for the firm. This disadvantage is somewhat offset by the fact that corporations may exclude 70% of dividends received from domestic corporations in the computation of their taxable income. Preferred stocks therefore make desirable fixedincome investments for some corporations. Even though preferred stock ranks after bonds in terms of the priority of its claims to the assets of the firm in the event of corporate bankruptcy, preferred stock often sells at lower yields than do corporate bonds. Presumably, this reflects the value of the dividend exclusion, because the higher risk of preferred would tend to result in higher yields than those offered by bonds. Individual investors, who cannot use the 70% tax

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Introduction

exclusion, generally will find preferred stock yields unattractive relative to those on other available assets. Preferred stock is issued in variations similar to those of corporate bonds. It may be callable by the issuing firm, in which case it is said to be redeemable. It also may be convertible into common stock at some specified conversion ratio. Adjustable-rate preferred stock is another variation that, like adjustable-rate bonds, ties the dividend to current market interest rates.

Depository Receipts American Depository Receipts, or ADRs, are certificates traded in U.S. markets that represent ownership in shares of a foreign company. Each ADR may correspond to ownership of a fraction of a foreign share, one share, or several shares of the foreign corporation. ADRs were created to make it easier for foreign firms to satisfy U.S. security registration requirements. They are the most common way for U.S. investors to invest in and trade the shares of foreign corporations.

2.4

Stock and Bond Market Indexes Stock Market Indexes The daily performance of the Dow Jones Industrial Average is a staple portion of the evening news report. Although the Dow is the best-known measure of the performance of the stock market, it is only one of several indicators. Other more broadly based indexes are computed and published daily. In addition, several indexes of bond market performance are widely available. The ever-increasing role of international trade and investments has made indexes of foreign financial markets part of the general news as well. Thus foreign stock exchange indexes such as the Nikkei Average of Tokyo and the Financial Times index of London are fast becoming household names.

Dow Jones Averages The Dow Jones Industrial Average (DJIA) of 30 large, “blue-chip” corporations has been computed since 1896. Its long history probably accounts for its preeminence in the public mind. (The average covered only 20 stocks until 1928.) Originally, the DJIA was calculated as the average price of the stocks included in the index. Thus, one would add up the prices of the 30 stocks in the index and divide by 30. The percentage change in the DJIA would then be the percentage change in the average price of the 30 shares. This procedure means that the percentage change in the DJIA measures the return (excluding dividends) on a portfolio that invests one share in each of the 30 stocks in the index. The value of such a portfolio (holding one share of each stock in the index) is the sum of the 30 prices. Because the percentage change in the average of the 30 prices is the same as the percentage change in the sum of the 30 prices, the index and the portfolio have the same percentage change each day. Because the Dow corresponds to a portfolio that holds one share of each component stock, the investment in each company in that portfolio is proportional to the company’s share price. Therefore, the Dow is called a price-weighted average.

CHAPTER 2

Example 2.2

Asset Classes and Financial Instruments

Price-Weighted Average

Consider the data in Table2.3 for a hypothetical two-stock version of the Dow Jones Average. Let’s compare the changes in the value of the portfolio holding one share of each firm and the price-weighted index. Stock ABC starts at $25 a share and increases to $30. Stock XYZ starts at $100, but falls to $90. Portfolio:

Index:

Initial value5$251$1005$125 Final value5$301$905$120 Percentage change in portfolio value525/12552.04524% Initial index value5(251100)/2562.5 Final index value5(30190)/2560 Percentage change in index522.5/62.552.04524%

The portfolio and the index have identical 4% declines in value. Notice that price-weighted averages give higher-priced shares more weight in determining performance of the index. For example, although ABC increased by 20%, while XYZ fell by only 10%, the index dropped in value. This is because the 20% increase in ABC represented a smaller price gain ($5 per share) than the 10% decrease in XYZ ($10 per share). The “Dow portfolio” has four times as much invested in XYZ as in ABC because XYZ’s price is four times that of ABC. Therefore, XYZ dominates the average. We conclude that a high-price stock can dominate a price-weighted average.

Stock ABC XYZ

Initial Price

Final Price

Shares (million)

Initial Value of Outstanding Stock ($ million)

$25 100

$30 90

20 1

$500 100

$600 90

$600

$690

Total

Final Value of Outstanding Stock ($ million)

You might wonder why the DJIA is now (in early 2013) at a level of about 14,000 if it is supposed to be the average price of the 30 stocks in the index. The DJIA no longer equals the average price of the 30 stocks because the averaging procedure is adjusted whenever a stock splits or pays a stock dividend of more than 10%, or when one company in the group of 30 industrial firms is replaced by another. When these events occur, the divisor used to compute the “average price” is adjusted so as to leave the index unaffected by the event.

Example 2.3

Splits and Price-Weighted Averages

Suppose XYZ were to split two for one so that its share price fell to $50. We would not want the average to fall, as that would incorrectly indicate a fall in the general level of market prices. Following a split, the divisor must be reduced to a value that leaves the average unaffected. Table2.4 illustrates this point. The initial share price of XYZ, which

Table 2.3 Data to construct stock price indexes

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was $100 in Table 2.3, falls to $50 if the stock splits at the beginning of the period. Notice that the number of shares outstanding doubles, leaving the market value of the total shares unaffected. We find the new divisor as follows. The index value before the stock split5 125/2 5 62.5. We must find a new divisor, d, that leaves the index unchanged after XYZ splits and its price falls to $50. Therefore, we solve for d in the following equation: Price of ABC 1 Price of XYZ 25 1 50 5 5 62.5 d d which implies that the divisor must fall from its original value of 2.0 to a new value of 1.20. Because the split changes the price of stock XYZ, it also changes the relative weights of the two stocks in the price-weighted average. Therefore, the return of the index is affected by the split. At period-end, ABC will sell for $30, while XYZ will sell for $45, representing the same negative 10% return it was assumed to earn in Table2.3. The new value of the price-weighted average is (30 145)/1.20 562.5, the same as its value at the start of the year; therefore, the rate of return is zero, rather than the 24% return that we calculated in the absence of a split. The split reduces the relative weight of XYZ because its initial price is lower; because XYZ is the poorer performing stock, the performance of the average is higher. This example illustrates that the implicit weighting scheme of a priceweighted average is somewhat arbitrary, being determined by the prices rather than by the outstanding market values (price per share times number of shares) of the shares in the average.

Table 2.4 Data to construct stock price indexes after a stock split

Stock ABC XYZ Total

Initial Price

Final Price

Shares (million)

$25 50

$30 45

20 2

Initial Value of Outstanding Stock ($ million)

Final Value of Outstanding Stock ($ million)

$500 100 $600

$600 90 $690

In the same way that the divisor is updated for stock splits, if one firm is dropped from the average and another firm with a different price is added, the divisor has to be updated to leave the average unchanged by the substitution. By 2013, the divisor for the Dow Jones Industrial Average had fallen to a value of about .1302. Because the Dow Jones averages are based on small numbers of firms, care must be taken to ensure that they are representative of the broad market. As a result, the composition of the average is changed every so often to reflect changes in the economy. Table2.5 presents the composition of the Dow industrials in 1928 as well as its composition as of mid-2013. The table presents striking evidence of the changes in the U.S. economy in the last 85 years. Many of the “bluest of the blue chip” companies in 1928 no longer exist, and the industries that were the backbone of the economy in 1928 have given way to some that could not have been imagined at the time.

CHAPTER 2

CONCEPT CHECK

Asset Classes and Financial Instruments

2.4

Suppose the price of XYZ in Table 2.3 increases to $110, while ABC falls to $20. Find the percentage change in the price-weighted average of these two stocks. Compare that to the percentage return of a portfolio holding one share in each company.

Dow Industrials in 1928

Current Dow Companies

Ticker Symbol

Industry

Year Added to Index

Wright Aeronautical

3M

MMM

Diversified industrials

1976

Allied Chemical

Alcoa

AA

Aluminum

1959

North American

American Express

AXP

Consumer finance

1982

Victor Talking Machine

AT&T

T

Telecommunications

1999

International Nickel

Bank of America

BAC

Banking

2008

International Harvester

Boeing

BA

Aerospace & defense

1987 1991

Westinghouse

Caterpillar

CAT

Construction

Texas Gulf Sulphur

Chevron

CVX

Oil and gas

2008

General Electric

Cisco Systems

CSCO

Computer equipment

2009

American Tobacco

Coca-Cola

KO

Beverages

1987

Texas Corp

DuPont

DD

Chemicals

1935

Standard Oil (NJ)

ExxonMobil

XOM

Oil & gas

1928 1907

Sears Roebuck

General Electric

GE

Diversified industrials

General Motors

Hewlett-Packard

HPQ

Computers

1997

Chrysler

Home Depot

HD

Home improvement retailers

1999

Atlantic Refining

Intel

INTC

Semiconductors

1999

Paramount Publix

IBM

IBM

Computer services

1979

Bethlehem Steel

Johnson & Johnson

JNJ

Pharmaceuticals

1997

General Railway Signal

JPMorgan Chase

JPM

Banking

1991

Mack Trucks

McDonald’s

MCD

Restaurants

1985

Union Carbide

Merck

MRK

Pharmaceuticals

1979

American Smelting

Microsoft

MSFT

Software

1999

American Can

Pfizer

PFE

Pharmaceuticals

2004

Postum Inc

Procter & Gamble

PG

Household products

1932

Nash Motors

Travelers

TRV

Insurance

2009

American Sugar

UnitedHealth Group

UNH

Health insurance

2012

Goodrich

United Technologies

UTX

Aerospace

1939

Radio Corp

Verizon

VZ

Telecommunications

2004

Woolworth

Wal-Mart

WMT

Retailers

1997

U.S. Steel

Walt Disney

DIS

Broadcasting & entertainment

1991

Table 2.5 Companies included in the Dow Jones Industrial Average: 1928 and 2013

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Standard & Poor’s Indexes The Standard & Poor’s Composite 500 (S&P 500) stock index represents an improvement over the Dow Jones Averages in two ways. First, it is a more broadly based index of 500 firms. Second, it is a market-value-weighted index. In the case of the firms XYZ and ABC in Example 2.2, the S&P 500 would give ABC five times the weight given to XYZ because the market value of its outstanding equity is five times larger, $500 million versus $100 million. The S&P 500 is computed by calculating the total market value of the 500 firms in the index and the total market value of those firms on the previous day of trading. The percentage increase in the total market value from one day to the next represents the increase in the index. The rate of return of the index equals the rate of return that would be earned by an investor holding a portfolio of all 500 firms in the index in proportion to their market values, except that the index does not reflect cash dividends paid by those firms. Actually, most indexes today use a modified version of market-value weights. Rather than weighting by total market value, they weight by the market value of free float, that is, by the value of shares that are freely tradable among investors. For example, this procedure does not count shares held by founding families or governments. These shares are effectively not available for investors to purchase. The distinction is more important in Japan and Europe, where a higher fraction of shares are held in such nontraded portfolios.

Example 2.4

Value-Weighted Indexes

To illustrate how value-weighted indexes are computed, look again at Table 2.3. The final value of all outstanding stock in our two-stock universe is $690 million. The initial value was $600 million. Therefore, if the initial level of a market-value-weighted index of stocks ABC and XYZ were set equal to an arbitrarily chosen starting value such as 100, the index value at year-end would be 1003(690/600)5115. The increase in the index reflects the 15% return earned on a portfolio consisting of those two stocks held in proportion to outstanding market values. Unlike the price-weighted index, the value-weighted index gives more weight to ABC. Whereas the price-weighted index fell because it was dominated by higher-price XYZ, the value-weighted index rises because it gives more weight to ABC, the stock with the higher total market value. Note also from Tables2.3 and 2.4 that market-value-weighted indexes are unaffected by stock splits. The total market value of the outstanding XYZ stock decreases from $100 million to $90 million regardless of the stock split, thereby rendering the split irrelevant to the performance of the index.

CONCEPT CHECK

2.5

Reconsider companies XYZ and ABC from Concept Check 2.4. Calculate the percentage change in the market-value-weighted index. Compare that to the rate of return of a portfolio that holds $500 of ABC stock for every $100 of XYZ stock (i.e., an index portfolio).

A nice feature of both market-value-weighted and price-weighted indexes is that they reflect the returns to straightforward portfolio strategies. If one were to buy shares in each component firm in the index in proportion to its outstanding market value, the

CHAPTER 2

Asset Classes and Financial Instruments

value-weighted index would perfectly track capital gains on the underlying portfolio. Similarly, a price-weighted index tracks the returns on a portfolio comprised of an equal number of shares of each firm. Investors today can easily buy market indexes for their portfolios. One way is to purchase shares in mutual funds that hold shares in proportion to their representation in the S&P 500 or another index. These index funds yield a return equal to that of the index and so provide a low-cost passive investment strategy for equity investors. Another approach is to purchase an exchange-traded fund, or ETF, which is a portfolio of shares that can be bought or sold as a unit, just as one can buy or sell a single share of stock. Available ETFs range from portfolios that track extremely broad global market indexes all the way to narrow industry indexes. We discuss both mutual funds and ETFs in detail in Chapter 4. Standard & Poor’s also publishes a 400-stock Industrial Index, a 20-stock Transportation Index, a 40-stock Utility Index, and a 40-stock Financial Index.

Other U.S. Market-Value Indexes The New York Stock Exchange publishes a market-value-weighted composite index of all NYSE-listed stocks, in addition to subindexes for industrial, utility, transportation, and financial stocks. These indexes are even more broadly based than the S&P 500. The National Association of Securities Dealers publishes an index of more than 3,000 firms traded on the NASDAQ market. The ultimate U.S. equity index so far computed is the Wilshire 5000 index of the market value of essentially all actively traded stocks in the U.S. Despite its name, the index actually includes more than 5,000 stocks. The performance of many of these indexes appears daily in The Wall Street Journal.

Equally Weighted Indexes Market performance is sometimes measured by an equally weighted average of the returns of each stock in an index. Such an averaging technique, by placing equal weight on each return, corresponds to an implicit portfolio strategy that invests equal dollar values in each stock. This is in contrast to both price weighting (which requires equal numbers of shares of each stock) and market-value weighting (which requires investments in proportion to outstanding value). Unlike price- or market-value-weighted indexes, equally weighted indexes do not correspond to buy-and-hold portfolio strategies. Suppose that you start with equal dollar investments in the two stocks of Table 2.3, ABC and XYZ. Because ABC increases in value by 20% over the year while XYZ decreases by 10%, your portfolio no longer is equally weighted. It is now more heavily invested in ABC. To reset the portfolio to equal weights, you would need to rebalance: Sell off some ABC stock and/or purchase more XYZ stock. Such rebalancing would be necessary to align the return on your portfolio with that on the equally weighted index.

Foreign and International Stock Market Indexes Development in financial markets worldwide includes the construction of indexes for these markets. Among these are the Nikkei (Japan), FTSE (U.K.; pronounced “footsie”), DAX (Germany), Hang Seng (Hong Kong), and TSX (Canada). A leader in the construction of international indexes has been MSCI (Morgan Stanley Capital International), which computes over 50 country indexes and several regional indexes. Table2.6 presents many of the indexes computed by MSCI.

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Regional Indexes Developed Markets

Emerging Markets

Countries Developed Markets

Emerging Markets

EAFE (Europe, Australia, Far East) EASEA (EAFE excluding Japan)

Emerging Markets (EM) EM Asia

Australia Austria

Brazil

Europe

EM Far East

Belgium

China

European Monetary Union (EMU)

EM Latin America

Canada

Colombia

Far East

EM Eastern Europe

Denmark

Czech Republic

Kokusai (World excluding Japan)

EM Europe

Finland

Egypt

Nordic countries

EM Europe & Middle East

France

Hungary

North America

Germany

India

Pacific

Greece

Indonesia

World

Hong Kong

Korea

G7 countries

Ireland

Malaysia

World excluding U.S.

Israel

Mexico

Italy

Morocco

Japan

Peru

Netherlands

Poland

New Zealand

Russia

Norway

South Africa

Portugal

Taiwan

Singapore

Thailand

Spain

Turkey

Chile

Sweden Switzerland U.K. U.S.

Table 2.6 Sample of MSCI stock indexes Source: MSCI, www.msci.com. Used with permission.

Bond Market Indicators Just as stock market indexes provide guidance concerning the performance of the overall stock market, several bond market indicators measure the performance of various categories of bonds. The three most well-known groups of indexes are those of Merrill Lynch, Barclays (formerly, the Lehman Brothers index), and Salomon Smith Barney (now part of Citigroup). Figure2.9 shows the components of the U.S. fixed-income market in 2012. The major problem with bond market indexes is that true rates of return on many bonds are difficult to compute because the infrequency with which the bonds trade makes reliable up-to-date prices difficult to obtain. In practice, some prices must be estimated from bond-valuation models. These “matrix” prices may differ from true market values.

CHAPTER 2

Asset Classes and Financial Instruments

$1,049.3 $2,953.1 $10,827.5

Treasury Debt Federal Agency and Gov’t Sponsored Enterprise

$3,428.0

Corporate Bonds Tax-Exempt* Mortgage-Backed Securities Other Asset-Backed Securities $5,192.5

$6,202.0

Figure 2.9 The U.S. fixed-income market (values in $ billions) Source: Flow of Funds Accounts of the United States: Flows & Outstandings, Board of Governors of the Federal Reserve System, June 2012.

2.5

Derivative Markets

One of the most significant developments in financial markets in recent years has been the growth of futures, options, and related derivatives markets. These instruments provide payoffs that depend on the values of other assets such as commodity prices, bond and stock prices, or market index values. For this reason these instruments sometimes are called derivative assets. Their values derive from the values of other assets.

Options A call option gives its holder the right to purchase an asset for a specified price, called the exercise or strike price, on or before a specified expiration date. For example, a July call option on IBM stock with an exercise price of $180 entitles its owner to purchase IBM stock for a price of $180 at any time up to and including the expiration date in July. Each option contract is for the purchase of 100 shares. However, quotations are made on a pershare basis. The holder of the call need not exercise the option; it will be profitable to exercise only if the market value of the asset that may be purchased exceeds the exercise price. When the market price exceeds the exercise price, the option holder may “call away” the asset for the exercise price and reap a payoff equal to the difference between the stock price and the exercise price. Otherwise, the option will be left unexercised. If not exercised before the expiration date of the contract, the option simply expires and no longer has value. Calls therefore provide greater profits when stock prices increase and thus represent bullish investment vehicles. In contrast, a put option gives its holder the right to sell an asset for a specified exercise price on or before a specified expiration date. A July put on IBM with an exercise price of $180 thus entitles its owner to sell IBM stock to the put writer at a price of $180 at any time before expiration in July, even if the market price of IBM is lower than $180. Whereas profits on call options increase when the asset increases in value, profits on put options

51

52

PART I

Introduction

PRICES AT CLOSE, July 17, 2012

IBM (IBM) Call Open Interest Volume

Underlying stock price: 183.65 Put Volume Open Interest Last

Strike

Last

Jul

180.00

5.50

620

1998

2.11

3080

8123

Aug

180.00

6.85

406

2105

3.70

847

3621

Oct

180.00

9.70

424

6.85

245

4984

Jan

180.00

12.58

184 52

2372

10.25

76

3196

Jul

185.00

2.80

2231

3897

4.20

2725

7370

Aug

185.00

4.10

656

2656

6.26

634

3367

Oct

185.00

6.99

843

969

9.10

783

2692

Jan

185.00

9.75

135

3156

12.01

243

10731

Expiration

Figure 2.10 Stock options on IBM Source: Compiled from data downloaded from The Wall Street Journal Online, July 17, 2012.

increase when the asset value falls. The put is exercised only if its holder can deliver an asset worth less than the exercise price in return for the exercise price. Figure2.10 is an excerpt of the options quotations for IBM from the online edition of The Wall Street Journal. The price of IBM shares on this date was $183.65. The first two columns give the expiration month and exercise (or strike) price for each option. We have included listings for call and put options with exercise prices of $180 and $185 per share and with expiration dates in July, August, and October 2012 and January 2013. The next columns provide the closing prices, trading volume, and open interest (outstanding contracts) of each option. For example, 1,998 contracts traded on the July 2012 expiration call with an exercise price of $180. The last trade was at $5.50, meaning that an option to purchase one share of IBM at an exercise price of $180 sold for $5.50. Each option contract (on 100 shares) therefore costs $550. Notice that the prices of call options decrease as the exercise price increases. For example, the July expiration call with exercise price $185 costs only $2.80. This makes sense, because the right to purchase a share at a higher price is less valuable. Conversely, put prices increase with the exercise price. The right to sell IBM at a price of $180 in July costs $2.11, while the right to sell at $185 costs $4.20. Option prices also increase with time until expiration. Clearly, one would rather have the right to buy IBM for $180 at any time until October rather than at any time until July. Not surprisingly, this shows up in a higher price for the October expiration options. For example, the call with exercise price $180 expiring in October sells for $9.70 compared to only $5.50 for the July call. CONCEPT CHECK

2.6

What would be the profit or loss per share to an investor who bought the July 2012 expiration IBM call option with exercise price $180 if the stock price at the expiration date is $187? What about a purchaser of the put option with the same exercise price and expiration?

CHAPTER 2

Asset Classes and Financial Instruments

Futures Contracts A futures contract calls for delivery of an asset (or in some cases, its cash value) at a specified delivery or maturity date for an agreed-upon price, called the futures price, to be paid at contract maturity. The long position is held by the trader who commits to purchasing the asset on the delivery date. The trader who takes the short position commits to delivering the asset at contract maturity. Figure2.11 illustrates the listing of the corn futures contract on the Chicago Board of Trade for July 17, 2012. Each contract calls for delivery of 5,000 bushels of corn. Each row details prices for contracts expiring on various dates. The first row is for the nearest term or “front” contract, with maturity in September 2012. The most recent price was $7.95 per bushel. (The numbers after the apostrophe denote eighths of a cent.) That price is up $.155 from yesterday’s close. The next columns show the contract’s opening price as well as the high and low price during the trading day. Volume is the number of contracts trading that day; open interest is the number of outstanding contracts. The trader holding the long position profits from price increases. Suppose that at contract maturity, corn is selling for $7.97 per bushel. The long position trader who entered the contract at the futures price of $7.95 on July 17 would pay the previously agreed-upon $7.95 for each bushel of corn, which at contract maturity would be worth $7.97. Because each contract calls for delivery of 5,000 bushels, the profit to the long position would equal 5,0003($7.972$7.95)5$1,000. Conversely, the short position must deliver 5,000 bushels for the previously agreed-upon futures price. The short position’s loss equals the long position’s profit. The right to purchase the asset at an agreed-upon price, as opposed to the obligation, distinguishes call options from long positions in futures contracts. A futures contract obliges the long position to purchase the asset at the futures price; the call option, in contrast, conveys the right to purchase the asset at the exercise price. The purchase will be made only if it yields a profit. Clearly, a holder of a call has a better position than the holder of a long position on a futures contract with a futures price equal to the option’s exercise price. This advantage, of course, comes only at a price. Call options must be purchased; futures contracts are entered into without cost. The purchase price of an option is called the premium. It represents the compensation the purchaser of the call must pay for the ability to exercise the option only when it is profitable to do so. Similarly, the difference between a put option and a short futures position is the right, as opposed to the obligation, to sell an asset at an agreed-upon price.

Month

Last

Chg

Open

High

Low

Volume

Open Int

Sep 2012

795’0

15’4

780’0

797’0

763’4

83008

369243

Dec 2012

783’4

12’2

772’4

785’6

755’6

179014

499807

Mar 2013

783’2

11’4

784’4

757’2

24738

135778

May 2013

779’6

11’2

772’4 769’2

780’4

755’0

8119

21882

Jul 2013

773’4

10’6

763’2

774’0

749’0

12310

57618

Sep 2013

669’4

–1’6

670’0

673’0

660’0

1833

9120

Dec 2013

634’0

–1’0

633’0

637’0

625’0

4510

54205

Figure 2.11 Corn futures prices in the Chicago Board of Trade, July 17, 2012 Source: The Wall Street Journal Online, July 17, 2012.

53

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PART I

SUMMARY

Introduction

1. Money market securities are very short-term debt obligations. They are usually highly marketable and have relatively low credit risk. Their low maturities and low credit risk ensure minimal capital gains or losses. These securities trade in large denominations, but they may be purchased indirectly through money market funds. 2. Much of U.S. government borrowing is in the form of Treasury bonds and notes. These are coupon-paying bonds usually issued at or near par value. Treasury notes and bonds are similar in design to coupon-paying corporate bonds.

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3. Municipal bonds are distinguished largely by their tax-exempt status. Interest payments (but not capital gains) on these securities are exempt from federal income taxes. The equivalent taxable yield offered by a municipal bond equals rm/(12t), where rm is the municipal yield and t is the investor’s tax bracket. 4. Mortgage pass-through securities are pools of mortgages sold in one package. Owners of passthroughs receive the principal and interest payments made by the borrowers. The originator that issued the mortgage merely services it, simply “passing through” the payments to the purchasers of the mortgage. A federal agency may guarantee the payments of interest and principal on mortgages pooled into its pass-through securities, but these guarantees are absent in private-label pass-throughs. 5. Common stock is an ownership share in a corporation. Each share entitles its owner to one vote on matters of corporate governance and to a prorated share of the dividends paid to shareholders. Stock, or equity, owners are the residual claimants on the income earned by the firm. 6. Preferred stock usually pays fixed dividends for the life of the firm; it is a perpetuity. A firm’s failure to pay the dividend due on preferred stock, however, does not precipitate corporate bankruptcy. Instead, unpaid dividends simply cumulate. Newer varieties of preferred stock include convertible and adjustable-rate issues. 7. Many stock market indexes measure the performance of the overall market. The Dow Jones averages, the oldest and best-known indicators, are price-weighted indexes. Today, many broadbased, market-value-weighted indexes are computed daily. These include the Standard & Poor’s 500 stock index, the NYSE index, the NASDAQ index, the Wilshire 5000 index, and indexes of many non-U.S. stock markets.

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KEY TERMS

8. A call option is a right to purchase an asset at a stipulated exercise price on or before an expiration date. A put option is the right to sell an asset at some exercise price. Calls increase in value while puts decrease in value as the price of the underlying asset increases. 9. A futures contract is an obligation to buy or sell an asset at a stipulated futures price on a maturity date. The long position, which commits to purchasing, gains if the asset value increases while the short position, which commits to delivering, loses.

money market capital markets ask price bid price bid–ask spread certificate of deposit commercial paper banker’s acceptance Eurodollars repurchase agreements federal funds

London Interbank Offered Rate (LIBOR) Treasury notes Treasury bonds yield to maturity municipal bonds equivalent taxable yield equities residual claim limited liability capital gains

price–earnings ratio preferred stock price-weighted average market-value-weighted index index funds derivative assets call option exercise (strike) price put option futures contract

CHAPTER 2

Equivalent taxable yield:

Asset Classes and Financial Instruments

rmuni , where rmuni is the rate on tax-free municipal debt 1 2 tax rate

Cutoff tax rate (for indifference to taxable versus tax-free bonds): 1 2

55

KEY EQUATIONS

rmuni rtaxable

1. In what ways is preferred stock like long-term debt? In what ways is it like equity?

PROBLEM SETS

2. Why are money market securities sometimes referred to as “cash equivalents”?

Basic

3. Which of the following correctly describes a repurchase agreement?

4. What would you expect to happen to the spread between yields on commercial paper and Treasury bills if the economy were to enter a steep recession? 5. What are the key differences between common stock, preferred stock, and corporate bonds? 6. Why are high-tax-bracket investors more inclined to invest in municipal bonds than low-bracket investors?

Intermediate

7. Turn back to Figure2.3 and look at the Treasury bond maturing in May 2030. a. How much would you have to pay to purchase one of these bonds? b. What is its coupon rate? c. What is the yield to maturity of the bond? 8. Suppose investors can earn a return of 2% per 6 months on a Treasury note with 6 months remaining until maturity. What price would you expect a 6-month maturity Treasury bill to sell for? 9. Find the after-tax return to a corporation that buys a share of preferred stock at $40, sells it at year-end at $40, and receives a $4 year-end dividend. The firm is in the 30% tax bracket. 10. Turn to Figure2.8 and look at the listing for General Dynamics. a. b. c. d.

How many shares could you buy for $5,000? What would be your annual dividend income from those shares? What must be General Dynamics earnings per share? What was the firm’s closing price on the day before the listing?

11. Consider the three stocks in the following table. Pt represents price at time t, and Qt represents shares outstanding at time t. Stock C splits two for one in the last period.

A B C

P0

Q0

P1

Q1

P2

Q2

90 50 100

100 200 200

95 45 110

100 200 200

95 45 55

100 200 400

a. Calculate the rate of return on a price-weighted index of the three stocks for the first period (t50tot51). b. What must happen to the divisor for the price-weighted index in year 2? c. Calculate the rate of return for the second period (t51tot52).

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a. The sale of a security with a commitment to repurchase the same security at a specified future date and a designated price. b. The sale of a security with a commitment to repurchase the same security at a future date left unspecified, at a designated price. c. The purchase of a security with a commitment to purchase more of the same security at a specified future date.

56

PART I

Introduction 12. Using the data in the previous problem, calculate the first-period rates of return on the following indexes of the three stocks: a. A market-value-weighted index. b. An equally weighted index. 13. An investor is in a 30% tax bracket. If corporate bonds offer 9% yields, what must municipals offer for the investor to prefer them to corporate bonds? 14. Find the equivalent taxable yield of a short-term municipal bond currently offering yields of 4% for tax brackets of zero, 10%, 20%, and 30%. 15. What problems would confront a mutual fund trying to create an index fund tied to an equally weighted index of a broad stock market? 16. Which security should sell at a greater price? a. A 10-year Treasury bond with a 9% coupon rate versus a 10-year T-bond with a 10% coupon. b. A 3-month expiration call option with an exercise price of $40 versus a 3-month call on the same stock with an exercise price of $35. c. A put option on a stock selling at $50, or a put option on another stock selling at $60 (all other relevant features of the stocks and options may be assumed to be identical).

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17. Look at the futures listings for the corn contract in Figure2.11. a. Suppose you buy one contract for March delivery. If the contract closes in March at a level of 787.25, what will your profit be? b. How many March maturity contracts are outstanding? 18. Turn back to Figure2.10 and look at the IBM options. Suppose you buy a January 2013 expiration call option with exercise price $180. a. Suppose the stock price in January is $193. Will you exercise your call? What is the profit on your position? b. What if you had bought the January call with exercise price $185? c. What if you had bought a January put with exercise price $185? 19. Why do call options with exercise prices greater than the price of the underlying stock sell for positive prices? 20. Both a call and a put currently are traded on stock XYZ; both have strike prices of $50 and expirations of 6 months. What will be the profit to an investor who buys the call for $4 in the following scenarios for stock prices in 6 months? What will be the profit in each scenario to an investor who buys the put for $6? a. b. c. d. e.

Challenge

$40 $45 $50 $55 $60

21. Explain the difference between a put option and a short position in a futures contract. 22. Explain the difference between a call option and a long position in a futures contract.

1. A firm’s preferred stock often sells at yields below its bonds because a. b. c. d.

Preferred stock generally carries a higher agency rating. Owners of preferred stock have a prior claim on the firm’s earnings. Owners of preferred stock have a prior claim on a firm’s assets in the event of liquidation. Corporations owning stock may exclude from income taxes most of the dividend income they receive.

2. A municipal bond carries a coupon of 6¾% and is trading at par. What is the equivalent taxable yield to a taxpayer in a combined federal plus state 34% tax bracket?

CHAPTER 2

Asset Classes and Financial Instruments

57

3. Which is the most risky transaction to undertake in the stock index option markets if the stock market is expected to increase substantially after the transaction is completed? a. b. c. d.

Write a call option. Write a put option. Buy a call option. Buy a put option.

4. Short-term municipal bonds currently offer yields of 4%, while comparable taxable bonds pay 5%. Which gives you the higher after-tax yield if your tax bracket is: a. b. c. d.

Zero 10% 20% 30%

E-INVESTMENTS EXERCISES Barclays maintains a Web site at www.barcap.com/inflation/index.shtml with information about inflation around the world and tools to help issuers and investors understand the inflation-linked asset class. Inflation-linked bonds were issued by a number of countries after 1945, including Israel, Argentina, Brazil, and Iceland. However, the modern market is generally deemed to have been born in 1981, when the first index-linked gilts were issued in the U.K. The other large markets adopted somewhat different calculations to those used by the U.K., mostly copying the more straightforward model first employed by Canada in 1991. In chronological order, the markets are the U.K. (1981), Australia (1985), Canada (1991), Sweden (1994), the United States (1997), France (1998), Italy (2003), and Japan (2004).

SOLUTIONS TO CONCEPT CHECKS 1. The bid price of the bond is 138.0469% of par, or $1,380.469, and the ask price is 138.125% of par, or $1,381.25. This ask price corresponds to a yield of 2.378%. The ask price fell .9375 from its level yesterday, so the ask price then must have been 139.0625, or $1,390.625. 2. A 6% taxable return is equivalent to an after-tax return of 6(1 2.30) 54.2%. Therefore, you would be better off in the taxable bond. The equivalent taxable yield of the tax-free bond is 4/(1 2.30) 55.71%. So a taxable bond would have to pay a 5.71% yield to provide the same after-tax return as a tax-free bond offering a 4% yield. 3. a. You are entitled to a prorated share of IBM’s dividend payments and to vote in any of IBM’s stockholder meetings. b. Your potential gain is unlimited because IBM’s stock price has no upper bound. c. Your outlay was $18031005$18,000. Because of limited liability, this is the most you can lose. 4. The price-weighted index increases from 62.5 [i.e., (100 125)/2] to 65 [i.e., (110 120)/2], a gain of 4%. An investment of one share in each company requires an outlay of $125 that would increase in value to $130, for a return of 4% (i.e., 5/125), which equals the return to the priceweighted index. 5. The market-value-weighted index return is calculated by computing the increase in the value of the stock portfolio. The portfolio of the two stocks starts with an initial value

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5. The coupon rate on a tax-exempt bond is 5.6%, and the rate on a taxable bond is 8%. Both bonds sell at par. At what tax bracket (marginal tax rate) would an investor be indifferent between the two bonds?

58

PART I

Introduction of $100 million 1 $500 million 5 $600 million and falls in value to $110 million 1 $400 million 5 $510 million, a loss of 90/600 5 .15, or 15%. The index portfolio return is a weighted average of the returns on each stock with weights of 1⁄6 on XYZ and 5⁄6 on ABC (weights proportional to relative investments). Because the return on XYZ is 10%, while that on ABC is 220%, the index portfolio return is 1⁄6 310% 1 5⁄6 3(220%) 5 215%, equal to the return on the market-value-weighted index.

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6. The payoff to the call option is $7 per share at expiration. The option cost is $5.50 per share. The dollar profit is therefore $1.50. The put option expires worthless. Therefore, the investor’s loss is the cost of the put, or $2.11.

CHAPTER THREE

How Securities Are Traded

THIS CHAPTER WILL provide you with a broad introduction to the many venues and procedures available for trading securities in the United States and international markets. We will see that trading mechanisms range from direct negotiation among market participants to fully automated computer crossing of trade orders. The first time a security trades is when it is issued to the public. Therefore, we begin with a look at how securities are first marketed to the public by investment bankers, the midwives of securities. We turn next to a broad survey of how already-issued securities may be traded among investors, focusing on the differences

3.1

3

between dealer markets, electronic markets, and specialist markets. With this background, we consider specific trading arenas such as the New York Stock Exchange, NASDAQ, and several all-electronic markets. We compare the mechanics of trade execution and the impact of cross-market integration of trading. We then turn to the essentials of some specific types of transactions, such as buying on margin and short-selling stocks. We close the chapter with a look at some important aspects of the regulations governing security trading, including insider trading laws and the role of security markets as self-regulating organizations.

How Firms Issue Securities

PART I

Firms regularly need to raise new capital to help pay for their many investment projects. Broadly speaking, they can raise funds either by borrowing money or by selling shares in the firm. Investment bankers are generally hired to manage the sale of these securities in what is called a primary market for newly issued securities. Once these securities are issued, however, investors might well wish to trade them among themselves. For example, you may decide to raise cash by selling some of your shares in Apple to another investor. This transaction would have no impact on the total outstanding number of Apple shares. Trades in existing securities take place in the secondary market. Shares of publicly listed firms trade continually on well-known markets such as the New York Stock Exchange or the NASDAQ Stock Market. There, any investor can choose to buy shares for his or her portfolio. These companies are also called publicly traded, publicly owned, or just public companies. Other firms, however, are private corporations,

60

PART I

Introduction

whose shares are held by small numbers of managers and investors. While ownership stakes in the firm are still determined in proportion to share ownership, those shares do not trade in public exchanges. While many private firms are relatively young companies that have not yet chosen to make their shares generally available to the public, others may be more established firms that are still largely owned by the company’s founders or families, and others may simply have decided that private organization is preferable.

Privately Held Firms A privately held company is owned by a relatively small number of shareholders. Privately held firms have fewer obligations to release financial statements and other information to the public. This saves money and frees the firm from disclosing information that might be helpful to its competitors. Some firms also believe that eliminating requirements for quarterly earnings announcements gives them more flexibility to pursue long-term goals free of shareholder pressure. At the moment, however, privately held firms may have only up to 499 shareholders. This limits their ability to raise large amounts of capital from a wide base of investors. Thus, almost all of the largest companies in the U.S. are public corporations. When private firms wish to raise funds, they sell shares directly to a small number of institutional or wealthy investors in a private placement. Rule 144A of the SEC allows them to make these placements without preparing the extensive and costly registration statements required of a public company. While this is attractive, shares in privately held firms do not trade in secondary markets such as a stock exchange, and this greatly reduces their liquidity and presumably reduces the prices that investors will pay for them. Liquidity has many specific meanings, but generally speaking, it refers to the ability to buy or sell an asset at a fair price on short notice. Investors demand price concessions to buy illiquid securities. As firms increasingly chafe against the informational requirements of going public, federal regulators have come under pressure to loosen the constraints entailed by private ownership, and they are currently reconsidering some of the restrictions on private companies. They may raise beyond 499 the number of shareholders that private firms can have before they are required to disclose financial information, and they may make it easier to publicize share offerings. Trading in private corporations also has evolved in recent years. To get around the 499-investor restriction, middlemen have formed partnerships to buy shares in private companies; the partnership counts as only one investor, even though many individuals may participate in it. Very recently, some firms have set up computer networks to enable holders of privatecompany stock to trade among themselves. However, unlike the public stock markets regulated by the SEC, these networks require little disclosure of financial information and provide correspondingly little oversight of the operations of the market. For example, in the run-up to its 2012 IPO, Facebook enjoyed huge valuations in these markets, but skeptics worried that investors in these markets could not obtain a clear view of the firm, the interest among other investors in the firm, or the process by which trades in the firm’s shares were executed.

Publicly Traded Companies When a private firm decides that it wishes to raise capital from a wide range of investors, it may decide to go public. This means that it will sell its securities to the general public and allow those investors to freely trade those shares in established securities

CHAPTER 3

How Securities Are Traded

61

markets. The first issue of shares to the general public is called the firm’s initial public offering, or IPO. Later, the firm may go back to the public and issue additional shares. A seasoned equity offering is the sale of additional shares in firms that already are publicly traded. For example, a sale by Apple of new shares of stock would be considered a seasoned new issue. Public offerings of both stocks and bonds typically are marketed by investment bankers who in this role are called underwriters. More than one investment banker usually markets the securities. A lead firm forms an underwriting syndicate of other investment bankers to share the responsibility for the stock issue. Investment bankers advise the firm regarding the terms on which it should attempt to sell the securities. A preliminary registration statement must be filed with the Securities and Exchange Commission (SEC), describing the issue and the prospects of the company. When the statement is in final form and accepted by the SEC, it is called the prospectus. At this point, the price at which the securities will be offered to the public is announced. In a typical underwriting arrangement, the investment bankers purchase the securities from the issuing company and then resell them to the public. The issuing firm sells the securities to the underwriting syndicate for the public offering price less a spread that serves as compensation to the underwriters. This procedure is called a firm commitment. In addition to the spread, the investment banker also may receive shares of common stock or other securities of the firm. Figure3.1 depicts the relationships among the firm issuing the security, the lead underwriter, the underwriting syndicate, and the public.

Shelf Registration An important innovation in the issuing of securities CONCEPT CHECK 3.1 was introduced in 1982 when the SEC approved Rule 415, which allows firms to register securities and gradWhy does it make sense for shelf registration to ually sell them to the public for 2 years following the be limited in time? initial registration. Because the securities are already registered, they can be sold on short notice, with little additional paperwork. Moreover, they can be sold in small amounts without incurring substantial flotation costs. Issuing The securities are “on the shelf,” Firm ready to be issued, which has given rise to the term shelf registration. Lead Underwriter Underwriting Syndicate

Initial Public Offerings Investment bankers manage the issuance of new securities to the public. Once the SEC has commented on the registration statement and a preliminary prospectus has been distributed to interested investors, the investment bankers organize road shows in which they travel around the country to publicize

Investment Banker A

Investment Banker B

Investment Banker C

Investment Banker D

Private Investors

Figure 3.1 Relationships among a firm issuing securities, the underwriters, and the public

62

PART I

Introduction

the imminent offering. These road shows serve two purposes. First, they generate interest among potential investors and provide information about the offering. Second, they provide information to the issuing firm and its underwriters about the price at which they will be able to market the securities. Large investors communicate their interest in purchasing shares of the IPO to the underwriters; these indications of interest are called a book and the process of polling potential investors is called bookbuilding. The book provides valuable information to the issuing firm because institutional investors often will have useful insights about the market demand for the security as well as the prospects of the firm and its competitors. Investment bankers frequently revise both their initial estimates of the offering price of a security and the number of shares offered based on feedback from the investing community. Why do investors truthfully reveal their interest in an offering to the investment banker? Might they be better off expressing little interest, in the hope that this will drive down the offering price? Truth is the better policy in this case because truth telling is rewarded. Shares of IPOs are allocated across investors in part based on the strength of each investor’s expressed interest in the offering. If a firm wishes to get a large allocation when it is optimistic about the security, it needs to reveal its optimism. In turn, the underwriter needs to offer the security at a bargain price to these investors to induce them to participate in bookbuilding and share their information. Thus, IPOs commonly are underpriced compared to the price at which they could be marketed. Such underpricing is reflected in price jumps that occur on the date when the shares are first traded in public security markets. The November 2011 IPO of Groupon was a typical example of underpricing. The company issued about 35 million shares to the public at a price of $20. The stock price closed that day at $26.11, a bit more than 30% above the offering price. While the explicit costs of an IPO tend to be around 7% of the funds raised, such underpricing should be viewed as another cost of the issue. For example, if Groupon had sold its shares for the $26.11 that investors obviously were willing to pay for them, its IPO would have raised 30% more money than it actually did. The money “left on the table” in this case far exceeded the explicit cost of the stock issue. Nevertheless, underpricing seems to be a universal phenomenon. Figure3.2 presents average first-day returns on IPOs of stocks across the world. The results consistently indicate that IPOs are marketed to investors at attractive prices. Pricing of IPOs is not trivial and not all IPOs turn out to be underpriced. Some do poorly after issue. Facebook’s 2012 IPO was a notable disappointment. Within a week of its IPO, Facebook’s share price was 15% below the $38 offer price, and five months later, its shares were selling at about half the offer price. Interestingly, despite their typically attractive first-day returns, IPOs have been poor long-term investments. Ritter calculates the returns to a hypothetical investor who bought equal amounts of each U.S. IPO between 1980 and 2009 at the close of trading on the first day the stock was listed and held each position for three years. That portfolio would have underperformed the broad U.S. stock market on average by 19.8% for three-year holding periods and underperformed “style-matched” portfolios of firms with comparable size and ratio of book value to market value by 7.3%.1 Other IPOs cannot even be fully sold to the market. Underwriters left with unmarketable securities are forced to sell them at a loss on the secondary market. Therefore, the investment banker bears price risk for an underwritten issue. 1

Professor Jay Ritter’s Web site contains a wealth of information and data about IPOs: http://bear.warrington. ufl.edu/ritter/ipodata.htm.

CHAPTER 3

How Securities Are Traded

175%

Average First-Day Returns

150% 125% 100% 75% 50% 25%

Argentina Russia Canada Austria Chile Denmark Norway Netherlands Turkey France Turkey Spain Nigeria Portugal Israel Belgium Mexico Hong Kong Israel United Kingdom United States Finland Australia Indonesia New Zealand Philipines Iran Italy Poland Cyprus Ireland Greece Germany Singapore Sweden South Africa Thailand Taiwan Japan Brazil Korea Malaysia India China

0%

Figure 3.2 Average first-day returns on IPOs from around the world Source: Provided by Professor J. Ritter of the University of Florida, 2008. This is an updated version of the information contained in T. Loughran, J. Ritter, and K. Rydqvist, “Initial Public Offerings,” Pacific-Basin Finance Journal 2 (1994), pp. 165–199. Copyright 1994 with permission from Elsevier Science.

3.2

How Securities Are Traded

Financial markets develop to meet the needs of particular traders. Consider what would happen if organized markets did not exist. Any household wishing to invest in some type of financial asset would have to find others wishing to sell. Soon, venues where interested traders could meet would become popular. Eventually, financial markets would emerge from these meeting places. Thus, a pub in old London called Lloyd’s launched the maritime insurance industry. A Manhattan curb on Wall Street became synonymous with the financial world.

Types of Markets We can differentiate four types of markets: direct search markets, brokered markets, dealer markets, and auction markets. Direct Search Markets A direct search market is the least organized market. Buyers and sellers must seek each other out directly. An example of a transaction in such a market is the sale of a used refrigerator where the seller advertises for buyers in a local newspaper or on Craigslist. Such markets are characterized by sporadic participation and low-priced and nonstandard goods. Firms would find it difficult to profit by specializing in such an environment.

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Brokered Markets The next level of organization is a brokered market. In markets where trading in a good is active, brokers find it profitable to offer search services to buyers and sellers. A good example is the real estate market, where economies of scale in searches for available homes and for prospective buyers make it worthwhile for participants to pay brokers to help them conduct the searches. Brokers in particular markets develop specialized knowledge on valuing assets traded in that market. Notice that the primary market, where new issues of securities are offered to the public, is an example of a brokered market. In the primary market, investment bankers who market a firm’s securities to the public act as brokers; they seek investors to purchase securities directly from the issuing corporation. Dealer Markets When trading activity in a particular type of asset increases, dealer markets arise. Dealers specialize in various assets, purchase these assets for their own accounts, and later sell them for a profit from their inventory. The spreads between dealers’ buy (or “bid”) prices and sell (or “ask”) prices are a source of profit. Dealer markets save traders on search costs because market participants can easily look up the prices at which they can buy from or sell to dealers. A fair amount of market activity is required before dealing in a market is an attractive source of income. Most bonds trade in over-the-counter dealer markets. Auction Markets The most integrated market is an auction market, in which all traders converge at one place (either physically or “electronically”) to buy or sell an asset. The New York Stock Exchange (NYSE) is an example of an auction market. An advantage of auction markets over dealer markets is that one need not search across dealers to find the best price for a good. If all participants converge, they can arrive at mutually agreeable prices and save the bid–ask spread. Notice that both over-the-counter dealer markets and stock exchanges are secondary markets. They are organized for investors to trade existing securities among themselves. CONCEPT CHECK

3.2

Many assets trade in more than one type of market. What types of markets do the following trade in? a. Used cars b. Paintings c. Rare coins

Types of Orders Before comparing alternative trading practices and competing security markets, it is helpful to begin with an overview of the types of trades an investor might wish to have executed in these markets. Broadly speaking, there are two types of orders: market orders and orders contingent on price.

Market Orders Market orders are buy or sell orders that are to be executed immediately at current market prices. For example, our investor might call her broker and ask for the market price of FedEx. The broker might report back that the best bid price is $90 and the best ask price is $90.05, meaning that the investor would need to pay $90.05 to purchase a share, and could receive $90 a share if she wished to sell some of her own holdings of FedEx. The bid–ask spread in this case is $.05. So an order to buy 100 shares “at market” would result in purchase at $90.05, and an order to “sell at market” would be executed at $90. This simple scenario is subject to a few potential complications. First, the posted price quotes actually represent commitments to trade up to a specified number of shares. If the market order is for more than this number of shares, the order may be filled at multiple prices. For example, if the ask price is good for orders up to 1,000 shares, and the investor wishes to purchase 1,500 shares, it may be necessary to pay a slightly higher price for the last 500 shares. Figure3.3 shows the average depth of the markets for shares of stock (i.e., the total

CHAPTER 3

FDX

6,000 5,000 4,000 3,000 2,000 1,000

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Figure 3.3 Average market depth for large (S&P 500) and small (Russell 2000) firms Source: James J. Angel, Lawrence E. Harris, and Chester Spatt, ”Equity Trading in the 21st Century,” Quarterly Journal of Finance 1 (2011), pp. 1–53; Knight Capital Group.

FedEx Corporation

NYSE Arca. FDX

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Price-Contingent Orders Investors also may place orders specifying prices at which they are willing to buy or sell a security. A limit buy order may instruct the broker to buy some number of shares if and when FedEx may be obtained at or below a stipulated price. Conversely, a limit sell instructs the broker to sell if and when the stock price rises above a specified limit. A collection of limit orders waiting to be executed is called a limit order book. Figure3.4 is a portion of the limit order book for shares in FedEx taken from the

Average Depth (Sum of Shares Offered at Best Bid and Ask Prices)

number of shares offered for trading at the best bid and ask prices). Notice that depth is considerably higher for the large stocks in the S&P 500 than for the smaller stocks that constitute the Russell 2000 index. Depth is considered another component of liquidity. Second, another trader may beat our investor to the quote, meaning that her order would then be executed at a worse price. Finally, the best price quote may change before her order arrives, again causing execution at a price different from the one at the moment of the order.

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How Securities Are Traded

Go>>

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Figure 3.4 The limit order book for FedEx on the NYSE Arca market Source: New York Stock Exchange Euronext, www.nyse.com, June 22, 2012.

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NYSE Arca exchange (one of several electronic exchanges; more on these shortly). Notice that the best orders are at the top of the list: the offers to buy at the highest price and to sell at the lowest price. The buy and sell orders at the top of the list—$90.04 and $90.05— are called the inside quotes; they are the highest buy and lowest sell Limit-Buy Stop-Buy Buy orders. For FedEx, the inside spread at this time was only 1 cent. Order Order Note, however, that order sizes at the inside quotes can be fairly Stop-Loss Limit-Sell small. Therefore, investors interested in larger trades face an effective Sell Order Order spread greater than the nominal one because they cannot execute their entire trades at the inside price quotes. Stop orders are similar to limit orders in that the trade is not to be executed unless the stock hits a price limit. For stop-loss orders, Figure 3.5 Price-contingent orders the stock is to be sold if its price falls below a stipulated level. As the name suggests, the order lets the stock be sold to stop further losses from accumulating. Similarly, stop-buy orders specify that a stock should be bought when its price rises above a limit. These trades often accompany short sales (sales of securities you don’t own but have borrowed from your broker) and are used to limit potential losses from the short position. Short sales are discussed in greater detail later in this chapter. Figure 3.5 organizes these types of trades in a convenient matrix. Action

Condition Price below Price above the Limit the Limit

CONCEPT CHECK

3.3

What type of trading order might you give to your broker in each of the following circumstances? a. You want to buy shares of FedEx, to diversify your portfolio. You believe the share price is approximately at the “fair” value, and you want the trade done quickly and cheaply. b. You want to buy shares of FedEx, but believe that the current stock price is too high given the firm’s prospects. If the shares could be obtained at a price 5% lower than the current value, you would like to purchase shares for your portfolio. c. You plan to purchase a condominium sometime in the next month or so and will sell your shares of Intel to provide the funds for your down payment. While you believe that the Intel share price is going to rise over the next few weeks, if you are wrong and the share price drops suddenly, you will not be able to afford the purchase. Therefore, you want to hold on to the shares for as long as possible, but still protect yourself against the risk of a big loss.

Trading Mechanisms An investor who wishes to buy or sell shares will place an order with a brokerage firm. The broker charges a commission for arranging the trade on the client’s behalf. Brokers have several avenues by which they can execute that trade, that is, find a buyer or seller and arrange for the shares to be exchanged. Broadly speaking, there are three trading systems employed in the United States: overthe-counter dealer markets, electronic communication networks, and specialist markets. The best-known markets such as NASDAQ or the New York Stock Exchange actually use a variety of trading procedures, so before you delve into specific markets, it is useful to understand the basic operation of each type of trading system.

CHAPTER 3

How Securities Are Traded

Dealer Markets Roughly 35,000 securities trade on the over-the-counter or OTC market. Thousands of brokers register with the SEC as security dealers. Dealers quote prices at which they are willing to buy or sell securities. A broker then executes a trade by contacting a dealer listing an attractive quote. Before 1971, all OTC quotations were recorded manually and published daily on socalled pink sheets. In 1971, the National Association of Securities Dealers introduced its Automatic Quotations System, or NASDAQ, to link brokers and dealers in a computer network where price quotes could be displayed and revised. Dealers could use the network to display the bid price at which they were willing to purchase a security and the ask price at which they were willing to sell. The difference in these prices, the bid–ask spread, was the source of the dealer’s profit. Brokers representing clients could examine quotes over the computer network, contact the dealer with the best quote, and execute a trade. As originally organized, NASDAQ was more of a price-quotation system than a trading system. While brokers could survey bid and ask prices across the network of dealers in the search for the best trading opportunity, actual trades required direct negotiation (often over the phone) between the investor’s broker and the dealer in the security. However, as we will see, NASDAQ is no longer a mere price quotation system. While dealers still post bid and ask prices over the network, what is now called the NASDAQ Stock Market allows for electronic execution of trades, and the vast majority of trades are executed electronically. Electronic Communication Networks (ECNs) Electronic communication networks allow participants to post market and limit orders over computer networks. The limit-order book is available to all participants. An example of such an order book from NYSE Arca, one of the leading ECNs, appears in Figure3.4. Orders that can be “crossed,” that is, matched against another order, are done automatically without requiring the intervention of a broker. For example, an order to buy a share at a price of $50 or lower will be immediately executed if there is an outstanding ask price of $50. Therefore, ECNs are true trading systems, not merely price-quotation systems. ECNs offer several attractions. Direct crossing of trades without using a broker-dealer system eliminates the bid–ask spread that otherwise would be incurred. Instead, trades are automatically crossed at a modest cost, typically less than a penny per share. ECNs are attractive as well because of the speed with which a trade can be executed. Finally, these systems offer investors considerable anonymity in their trades. Specialist Markets Specialist systems have been largely replaced by electronic communication networks, but as recently as a decade ago, they were still a dominant form of market organization for trading in stocks. In these systems, exchanges such as the NYSE assign responsibility for managing the trading in each security to a specialist. Brokers wishing to buy or sell shares for their clients direct the trade to the specialist’s post on the floor of the exchange. While each security is assigned to only one specialist, each specialist firm makes a market in many securities. The specialist maintains the limit order book of all outstanding unexecuted limit orders. When orders can be executed at market prices, the specialist executes, or “crosses,” the trade. The highest outstanding bid price and the lowest outstanding ask price “win” the trade. Specialists are also mandated to maintain a “fair and orderly” market when the book of limit buy and sell orders is so thin that the spread between the highest bid price and lowest ask price becomes too wide. In this case, the specialist firm would be expected to offer to buy and sell shares from its own inventory at a narrower bid-ask spread. In this role, the specialist serves as a dealer in the stock and provides liquidity to other traders. In this context, liquidity providers are those who stand willing to buy securities from or sell securities to other traders.

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3.3

Introduction

The Rise of Electronic Trading When first established, NASDAQ was primarily an over-the-counter dealer market and the NYSE was a specialist market. But today both are primarily electronic markets. These changes were driven by an interaction of new technologies and new regulations. New regulations allowed brokers to compete for business, broke the hold that dealers once had on information about best-available bid and ask prices, forced integration of markets, and allowed securities to trade in ever-smaller price increments (called tick sizes). Technology made it possible for traders to rapidly compare prices across markets and direct their trades to the markets with the best prices. The resulting competition drove down the cost of trade execution to a tiny fraction of its value just a few decades ago. In 1975, fixed commissions on the NYSE were eliminated, which freed brokers to compete for business by lowering their fees. In that year also, Congress amended the Securities Exchange Act to create the National Market System to at least partially centralize trading across exchanges and enhance competition among different market makers. The idea was to implement centralized reporting of transactions as well as a centralized price quotation system to give traders a broader view of trading opportunities across markets. The aftermath of a 1994 scandal at NASDAQ turned out to be a major impetus in the further evolution and integration of markets. NASDAQ dealers were found to be colluding to maintain wide bid-ask spreads. For example, if a stock was listed at $30 bid—$30 1/2 ask, a retail client who wished to buy shares from a dealer would pay $30 1/2 while a client who wished to sell shares would receive only $30. The dealer would pocket the 1/2-point spread as profit. Other traders may have been willing to step in with better prices (e.g., they may have been willing to buy shares for $30 1/8 or sell them for $30 3/8), but those better quotes were not made available to the public, enabling dealers to profit from artificially wide spreads at the public’s expense. When these practices came to light, an antitrust lawsuit was brought against NASDAQ. In response to the scandal, the SEC instituted new order-handling rules. Published dealer quotes now had to reflect limit orders of customers, allowing them to effectively compete with dealers to capture trades. As part of the antitrust settlement, NASDAQ agreed to integrate quotes from ECNs into its public display, enabling the electronic exchanges to also compete for trades. Shortly after this settlement, the SEC adopted Regulation ATS (Alternative Trading Systems), giving ECNs the right to register as stock exchanges. Not surprisingly, they captured an ever-larger market share, and in the wake of this new competition, bid–ask spreads narrowed. Even more dramatic narrowing of trading costs came in 1997, when the SEC allowed the minimum tick size to fall from one-eighth of a dollar to one-sixteenth. Not long after, in 2001, “decimalization” allowed the tick size to fall to 1 cent. Bid–ask spreads again fell dramatically. Figure3.6 shows estimates of the “effective spread” (the cost of a transaction) during three distinct time periods defined by the minimum tick size. Notice how dramatically effective spread falls along with the minimum tick size. Technology was also changing trading practices. The first ECN, Instinet, was established in 1969. By the 1990s, exchanges around the world were rapidly adopting fully electronic trading systems. Europe led the way in this evolution, but eventually American exchanges followed suit. The National Association of Securities Dealers (NASD) spun off the NASDAQ Stock Market as a separate entity in 2000, which quickly evolved into a centralized limit-order matching system—effectively a large ECN. The NYSE acquired the electronic Archipelago Exchange in 2006 and renamed it NYSE Arca. In 2005, the SEC adopted Regulation NMS (for National Market System), which was fully implemented in 2007. The goal was to link exchanges electronically, thereby creating,

CHAPTER 3

How Securities Are Traded

0.200 0.175

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0.150 0.125 0.100 0.075 0.050 0.025 Sixteenths Regime

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Figure 3.6 The effective spread (measured in dollars per share) fell dramatically as the minimum tick size fell (value-weighted average of NYSE-listed shares) Source: Tarun Chordia, Richard Roll, and Avanidhar Subrahmanyam, “Liquidity and Market Efficiency,” Journal of Financial Economics 87 (2008), 249–268. Copyright © February 2008, with permission from Elsevier.

in effect, one integrated electronic market. The regulation required exchanges to honor quotes of other exchanges when they could be executed automatically. An exchange that could not handle a quote electronically would be labeled a “slow market” under Reg NMS and could be ignored by other market participants. The NYSE, which was still devoted to the specialist system, was particularly at risk of being passed over, and in response to this pressure, it moved aggressively toward automated execution of trades. Electronic trading networks and the integration of markets in the wake of Reg NMS made it much easier for exchanges around the world to compete; the NYSE lost its effective monopoly in the trading of its own listed stocks, and by the end of the decade, its share in the trading of NYSElisted stocks fell from about 75% to 25%. While specialists still exist, trading today is overwhelmingly electronic, at least for stocks. Bonds are still traded in more traditional dealer markets. In the U.S., the share of electronic trading in equities rose from about 16% in 2000 to over 80% by the end of the decade. In the rest of the world, the dominance of electronic trading is even greater.

3.4

U.S. Markets

The NYSE and the NASDAQ Stock Market remain the two largest U.S. stock markets. But electronic communication networks have steadily increased their market share. Figure3.7 shows the comparative trading volume of NYSE-listed shares on the NYSE and NASDAQ as well as on the major ECNs, namely, BATS, NYSE Arca, and Direct Edge. The “Other” category, which recently has risen above 30%, includes so-called dark pools, which we will discuss shortly.

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90% 80% 70%

NYSE

NYSE Arca

NASDAQ OMX

BATS

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60% 50% 40% 30% 20% 10% 0% Feb-02

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Figure 3.7 Market share of trading in NYSE-listed shares Source: James J. Angel, Lawrence E. Harris, and Chester Spatt, “Equity Trading in the 21st Century,” Quarterly Journal of Finance 1 (2011), 1–53.

NASDAQ The NASDAQ Stock Market lists around 3,000 firms. It has steadily introduced ever-more sophisticated trading platforms, which today handle the great majority of its trades. The current version, called the NASDAQ Market Center, consolidates NASDAQ’s previous electronic markets into one integrated system. NASDAQ merged in 2008 with OMX, a Swedish-Finnish company that controls seven Nordic and Baltic stock exchanges to form NASDAQ OMX Group. In addition to maintaining the NASDAQ Stock Market, it also maintains several stock markets in Europe as well as an options and futures exchange in the U.S. NASDAQ has three levels of subscribers. The highest, level 3 subscribers, are registered market makers. These are firms that make a market in securities, maintain inventories of securities, and post bid and ask prices at which they are willing to buy or sell shares. Level 3 subscribers can enter and change bid–ask quotes continually and have the fastest execution of trades. They profit from the spread between bid and ask prices. Level 2 subscribers receive all bid and ask quotes but cannot enter their own quotes. They can see which market makers are offering the best prices. These subscribers tend to be brokerage firms that execute trades for clients but do not actively deal in stocks for their own account. Level 1 subscribers receive only inside quotes (i.e., the best bid and ask prices), but do not see how many shares are being offered. These subscribers tend to be investors who are not actively buying or selling but want information on current prices.

CHAPTER 3

How Securities Are Traded

The New York Stock Exchange The NYSE is the largest U.S. stock exchange as measured by the value of the stocks listed on the exchange. Daily trading volume on the NYSE is about a billion shares. In 2006, the NYSE merged with the Archipelago Exchange to form a publicly held company called the NYSE Group, and then in 2007, it merged with the European exchange Euronext to form NYSE Euronext. The firm acquired the American Stock Exchange in 2008, which has since been renamed NYSE Amex and focuses on small firms. NYSE Arca is the firm’s electronic communications network, and this is where the bulk of exchange-traded funds trade. In 2012, NYSE Euronext agreed to be purchased by InternationalExchange (ICE), whose main business to date has been energy-futures trading. ICE plans to retain the NYSE Euronext name as well as the fabled trading floor on Wall Street. The NYSE was long committed to its specialist trading system, which relied heavily on human participation in trade execution. It began its transition to electronic trading for smaller trades in 1976 with the introduction of its DOT (Designated Order Turnaround), and later SuperDOT systems, which could route orders directly to the specialist. In 2000, the exchange launched Direct1, which could automatically cross smaller trades (up to 1,099 shares) without human intervention, and in 2004, it began eliminating the size restrictions on Direct1 trades. The change of emphasis dramatically accelerated in 2006 with the introduction of the NYSE Hybrid Market, which allowed brokers to send orders either for immediate electronic execution or to the specialist, who could seek price improvement from another trader. The Hybrid system allowed the NYSE to qualify as a fast market for the purposes of Regulation NMS, but still offer the advantages of human intervention for more complicated trades. In contrast, NYSE’s Arca marketplace is fully electronic.

ECNs Over time, more fully automated markets have gained market share at the expense of less automated ones, in particular, the NYSE. Some of the biggest ECNs today are Direct Edge, BATS, and NYSE Arca. Brokers that have an affiliation with an ECN have computer access and can enter orders in the limit order book. As orders are received, the system determines whether there is a matching order, and if so, the trade is immediately crossed. Originally, ECNs were open only to other traders using the same system. But following the implementation of Reg NMS, ECNs began listing limit orders on other networks. Traders could use their computer systems to sift through the limit order books of many ECNs and instantaneously route orders to the market with the best prices. Those cross-market links have become the impetus for one of the more popular strategies of so-called high-frequency traders, which seek to profit from even small, transitory discrepancies in prices across markets. Speed is obviously of the essence here, and ECNs compete in terms of the speed they can offer. Latency refers to the time it takes to accept, process, and deliver a trading order. BATS, for example, advertises latency times of around 200 microseconds, i.e., .0002 second.

3.5

New Trading Strategies

The marriage of electronic trading mechanisms with computer technology has had farranging impacts on trading strategies and tools. Algorithmic trading delegates trading decisions to computer programs. High frequency trading is a special class of algorithmic trading in which computer programs initiate orders in tiny fractions of a second, far faster than any human could process the information driving the trade. Much of the

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market liquidity that once was provided by brokers making a market in a security has been displaced by these high-frequency traders. But when high-frequency traders abandon the market, as in the so-called flash crash of 2010, liquidity can likewise evaporate in a flash. Dark pools are trading venues that preserve anonymity, but also affect market liquidity. We will address these emerging issues later in this section.

Algorithmic Trading Algorithmic trading is the use of computer programs to make trading decisions. Well more than half of all equity volume in the U.S. is believed to be initiated by computer algorithms. Many of these trades exploit very small discrepancies in security prices and entail numerous and rapid cross-market price comparisons that are well suited to computer analysis. These strategies would not have been feasible before decimalization of the minimum tick size. Some algorithmic trades attempt to exploit very short-term trends (as short as a few seconds) as new information about a firm becomes reflected in its stock price. Others use versions of pairs trading in which normal price relations between pairs (or larger groups) of stocks seem temporarily disrupted and offer small profit opportunities as they move back into alignment. Still others attempt to exploit discrepancies between stock prices and prices of stock-index futures contracts. Some algorithmic trading involves activities akin to traditional market making. The traders seek to profit from the bid–ask spread by buying a stock at the bid price and rapidly selling it at the ask price before the price can change. While this mimics the role of a market maker who provides liquidity to other traders in the stock, these algorithmic traders are not registered market makers and so do not have an affirmative obligation to maintain both bid and ask quotes. If they abandon a market during a period of turbulence, the shock to market liquidity can be disruptive. This seems to have been a problem during the flash crash of May 6, 2010, when the stock market encountered extreme volatility, with the Dow Jones average falling by 1,000 points before recovering around 600 points in intraday trading. The nearby box discusses this amazing and troubling episode.

High-Frequency Trading It is easy to see that many algorithmic trading strategies require extremely rapid trade initiation and execution. High-frequency trading is a subset of algorithmic trading that relies on computer programs to make extremely rapid decisions. High-frequency traders compete for trades that offer very small profits. But if those opportunities are numerous enough, they can accumulate to big money. We pointed out that one high-frequency strategy entails a sort of market making, attempting to profit from the bid–ask spread. Another relies on cross-market arbitrage, in which even tiny price discrepancies across markets allow the firm to buy a security at one price and simultaneously sell it at a slightly higher price. The competitive advantage in these strategies lies with the firms that are quickest to identify and execute these profit opportunities. There is a tremendous premium on being the first to “hit” a bid or ask price. Trade execution times for high-frequency traders are now measured in milliseconds, even microseconds. This has induced trading firms to “co-locate” their trading centers next to the computer systems of the electronic exchanges. When execution or latency periods are less than a millisecond, the extra time it takes for a trade order to travel from a remote location to a New York exchange would be enough to make it nearly impossible to win the trade.

CHAPTER 3

How Securities Are Traded

To understand why co-location has become a key issue, consider this calculation. Even light can travel only 186 miles in 1 millisecond, so an order originating in Chicago transmitted at the speed of light would take almost 5 milliseconds to reach New York. But ECNs today claim latency periods considerably less than 1 millisecond, so an order from Chicago could not possibly compete with one launched from a co-located facility. In some ways, co-location is a new version of an old phenomenon. Think about why, even before the advent of the telephone, so many brokerage firms originally located their headquarters in New York: They were “co-locating” with the NYSE so that their brokers could bring trades to the exchange quickly and efficiently. Today, trades are transmitted electronically, but competition among traders for fast execution means that the need to be near the market (now embodied in computer servers) remains.

Dark Pools Many large traders seek anonymity. They fear that if others see them executing a large buy or sell program, their intentions will become public and prices will move against them. Very large trades (called blocks, usually defined as a trade of more than 10,000 shares) traditionally were brought to “block houses,” brokerage firms specializing in matching block buyers and sellers. Part of the expertise of block brokers is in identifying traders who might be interested in a large purchase or sale if given an offer. These brokers discreetly arrange large trades out of the public eye, and so avoid moving prices against their clients. Block trading today has been displaced to a great extent by dark pools, trading systems in which participants can buy or sell large blocks of securities without showing their hand. Not only are buyers and sellers in dark pools hidden from the public, but even trades may not be reported, or if they are reported, they may be lumped with other trades to obscure information about particular participants. Dark pools are somewhat controversial because they contribute to the fragmentation of markets. When many orders are removed from the consolidated limit order book, there are fewer orders left to absorb fluctuations in demand for the security, and the public price may no longer be “fair” in the sense that it reflects all the potentially available information about security demand. Another approach to dealing with large trades is to split them into many small trades, each of which can be executed on electronic markets, attempting to hide the fact that the total number of shares ultimately to be bought or sold is large. This trend has led to rapid decline in average trade size, which today is less than 300 shares.

Bond Trading In 2006, the NYSE obtained regulatory approval to expand its bond-trading system to include the debt issues of any NYSE-listed firm. Until then, each bond needed to be registered before listing, and such a requirement was too onerous to justify listing most bonds. In conjunction with these new listings, the NYSE has expanded its electronic bond-trading platform, which is now called NYSE Bonds and is the largest centralized bond market of any U.S. exchange. Nevertheless, the vast majority of bond trading occurs in the OTC market among bond dealers, even for bonds that are actually listed on the NYSE. This market is a network of bond dealers such as Merrill Lynch (now part of Bank of America), Salomon Smith Barney (a division of Citigroup), and Goldman Sachs that is linked by a computer quotation

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WORDS FROM THE STREET

The Flash Crash of May 2010 At 2:42 New York time on May 6, 2010, the Dow Jones Industrial Average was already down about 300 points for the day. The market was demonstrating concerns about the European debt crisis, and nerves were already on edge. Then, in the next 5 minutes, the Dow dropped an additional 600 points. And only 20 minutes after that, it had recovered most of those 600 points. Besides the staggering intraday volatility of the broad market, trading in individual shares and ETFs was even more disrupted. The iShares Russell 1000 Value fund temporarily fell from $59 a share to 8 cents. Shares in the large consulting company Accenture, which had just sold for $38, traded at 1 cent only a minute or two later. At the other extreme, share prices of Apple and Hewlett-Packard momentarily increased to over $100,000. These markets were clearly broken. The causes of the flash crash are still debated. An SEC report issued after the trade points to a $4 billion sale of market index futures contracts by a mutual fund. As market prices began to tumble, many algorithmic trading programs withdrew from the markets, and those that remained became net sellers, further pushing down equity prices. As more and more of these algorithmic traders shut down, liquidity in these markets evaporated: Buyers for many stocks simply disappeared.

Finally, trading was halted for a short period. When it resumed, buyers decided to take advantage of many severely depressed stock prices, and the market rebounded almost as quickly as it had crashed. Given the intraday turbulence and the clearly distorted prices at which some trades had been executed, the NYSE and NASDAQ decided to cancel all trades that were executed more than 60% away from a “reference price” close to the opening price of the day. Almost 70% of those canceled trades involved ETFs. The SEC has since approved experimentation with new circuit breakers to halt trading for 5 minutes in large stocks that rise or fall by more than 10% in a 5-minute period. The idea is to prevent trading algorithms from moving share prices quickly before human traders have a chance to determine whether those prices are moving in response to fundamental information. The flash crash highlighted the fragility of markets in the face of huge variation in trading volume created by algorithmic traders. The potential for these highfrequency traders to withdraw from markets in periods of turbulence remains a concern, and many observers are not convinced that we are protected from future flash crashes.

system. However, because these dealers do not carry extensive inventories of the wide range of bonds that have been issued to the public, they cannot necessarily offer to sell bonds from their inventory to clients or even buy bonds for their own inventory. They may instead work to locate an investor who wishes to take the opposite side of a trade. In practice, however, the corporate bond market often is quite “thin,” in that there may be few investors interested in trading a specific bond at any particular time. As a result, the bond market is subject to a type of liquidity risk, for it can be difficult to sell one’s holdings quickly if the need arises.

3.6

Globalization of Stock Markets Figure3.8 shows that NYSE-Euronext is by far the largest equity market as measured by the total market value of listed firms. All major stock markets today are effectively electronic. Securities markets have come under increasing pressure in recent years to make international alliances or mergers. Much of this pressure is due to the impact of electronic trading. To a growing extent, traders view stock markets as computer networks that link them to other traders, and there are increasingly fewer limits on the securities around the world that they can trade. Against this background, it becomes more important for exchanges to provide the cheapest and most efficient mechanism by which trades can be executed and cleared. This argues for global alliances that can facilitate the nuts and bolts of cross-border trading and can benefit from economies of scale. Exchanges feel that they

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12,000 10,000

$ Billions

8,000 6,000 4,000 2,000

India

BME (Spanish)

Deutsche Börse

Australia

Brazil

Toronto

Hong Kong

Shanghai

Euronext (Europe)

London

Tokyo

NASDAQ-OMX

NYSE-Euronext (US)

Figure 3.8 The biggest stock markets in the world by domestic market capitalization in 2012 Source: World Federation of Exchanges, 2012.

eventually need to offer 24-hour global markets and platforms that allow trading of different security types, for example, both stocks and derivatives. Finally, companies want to be able to go beyond national borders when they wish to raise capital. These pressures have resulted in a broad trend toward market consolidation. In the last decade, most of the mergers were “local,” that is, involving exchanges operating on the same continent. In the U.S., the NYSE merged with the Archipelago ECN in 2006, and in 2008 acquired the American Stock Exchange. NASDAQ acquired Instinet (which operated another major ECN, INET) in 2005 and the Boston Stock Exchange in 2007. In the derivatives market, the Chicago Mercantile Exchange acquired the Chicago Board of Trade in 2007 and the New York Mercantile Exchange in 2008, thus moving almost all futures trading in the U.S. onto one exchange. In Europe, Euronext was formed by the merger of the Paris, Brussels, Lisbon, and Amsterdam exchanges and shortly thereafter purchased Liffe, the derivatives exchange based in London. The LSE merged in 2007 with Borsa Italiana, which operates the Milan exchange. There has also been a wave of intercontinental consolidation. The NYSE Group and Euronext merged in 2007. Germany’s Deutsche Börse and the NYSE Euronext agreed to merge in late 2011. The merged firm would be able to support trading in virtually every type of investment. However, in early 2012, the proposed merger ran aground when European Union antitrust regulators recommended that the combination be blocked. Still, the attempt at the merger indicates the thrust of market pressures, and other combinations

75

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Introduction

continue to develop. The NYSE and the Tokyo stock exchange have announced their intention to link their networks to give customers of each access to both markets. In 2007, the NASDAQ Stock Market merged with OMX, which operates seven Nordic and Baltic stock exchanges, to form NASDAQ OMX Group. In 2008, Eurex took over International Securities Exchange (ISE), to form a major options exchange.

3.7

Trading Costs Part of the cost of trading a security is obvious and explicit. Your broker must be paid a commission. Individuals may choose from two kinds of brokers: full-service or discount brokers. Full-service brokers who provide a variety of services often are referred to as account executives or financial consultants. Besides carrying out the basic services of executing orders, holding securities for safekeeping, extending margin loans, and facilitating short sales, brokers routinely provide information and advice relating to investment alternatives. Full-service brokers usually depend on a research staff that prepares analyses and forecasts of general economic as well as industry and company conditions and often makes specific buy or sell recommendations. Some customers take the ultimate leap of faith and allow a full-service broker to make buy and sell decisions for them by establishing a discretionary account. In this account, the broker can buy and sell prespecified securities whenever deemed fit. (The broker cannot withdraw any funds, though.) This action requires an unusual degree of trust on the part of the customer, for an unscrupulous broker can “churn” an account, that is, trade securities excessively with the sole purpose of generating commissions. Discount brokers, on the other hand, provide “no-frills” services. They buy and sell securities, hold them for safekeeping, offer margin loans, facilitate short sales, and that is all. The only information they provide about the securities they handle is price quotations. Discount brokerage services have become increasingly available in recent years. Many banks, thrift institutions, and mutual fund management companies now offer such services to the investing public as part of a general trend toward the creation of one-stop “financial supermarkets.” Stock trading fees have fallen steadily over the last decade, and discount brokerage firms such as Schwab, E*Trade, or TD Ameritrade now offer commissions below $10. In addition to the explicit part of trading costs—the broker’s commission—there is an implicit part—the dealer’s bid–ask spread. Sometimes the broker is also a dealer in the security being traded and charges no commission but instead collects the fee entirely in the form of the bid–ask spread. Another implicit cost of trading that some observers would distinguish is the price concession an investor may be forced to make for trading in quantities greater than those associated with the posted bid or ask price.

3.8

Buying on Margin When purchasing securities, investors have easy access to a source of debt financing called broker’s call loans. The act of taking advantage of broker’s call loans is called buying on margin. Purchasing stocks on margin means the investor borrows part of the purchase price of the stock from a broker. The margin in the account is the portion of the purchase price

CHAPTER 3

How Securities Are Traded

contributed by the investor; the remainder is borrowed from the broker. The brokers in turn borrow money from banks at the call money rate to finance these purchases; they then charge their clients that rate (defined in Chapter 2), plus a service charge for the loan. All securities purchased on margin must be maintained with the brokerage firm in street name, for the securities are collateral for the loan. The Board of Governors of the Federal Reserve System limits the extent to which stock purchases can be financed using margin loans. The current initial margin requirement is 50%, meaning that at least 50% of the purchase price must be paid for in cash, with the rest borrowed.

Example 3.1

Margin

The percentage margin is defined as the ratio of the net worth, or the “equity value,” of the account to the market value of the securities. To demonstrate, suppose an investor initially pays $6,000 toward the purchase of $10,000 worth of stock (100 shares at $100 per share), borrowing the remaining $4,000 from a broker. The initial balance sheet looks like this: Assets Value of stock

Liabilities and Owners’ Equity $10,000

Loan from broker Equity

$4,000 $6,000

The initial percentage margin is Margin 5

Equity in account $6,000 5 5 .60, or 60% Value of stock $10,000

If the price declines to $70 per share, the account balance becomes: Assets Value of stock

Liabilities and Owners’ Equity $7,000

Loan from broker Equity

$4,000 $3,000

The assets in the account fall by the full decrease in the stock value, as does the equity. The percentage margin is now Margin 5

Equity in account $3,000 5 5 .43, or 43% Value of stock $7,000

If the stock value in Example 3.1 were to fall below $4,000, owners’ equity would become negative, meaning the value of the stock is no longer sufficient collateral to cover the loan from the broker. To guard against this possibility, the broker sets a maintenance margin. If the percentage margin falls below the maintenance level, the broker will issue a margin call, which requires the investor to add new cash or securities to the margin account. If the investor does not act, the broker may sell securities from the account to pay off enough of the loan to restore the percentage margin to an acceptable level.

77

eXcel APPLICATIONS: Buying on Margin broker, i.e., the initial margin on your purchase is 25%. You pay an interest rate of 8% on margin loans.

he Online Learning Center (www.mhhe.com/bkm) contains the Excel spreadsheet model below, which makes it easy to analyze the impacts of different margin levels and the volatility of stock prices. It also allows you to compare return on investment for a margin trade with a trade using no borrowed funds.

T

a. How much of your own money do you invest? How much do you borrow from your broker? b. What will be your rate of return for the following stock prices at the end of a 1-year holding period? (1) $40, (2) $50, (3) $60.

Excel Questions

2. Repeat Question 1 assuming your initial margin was 50%. How does margin affect the risk and return of your position?

1. Suppose you buy 100 shares share of stock initially selling for $50, borrowing 25% of the necessary funds from your A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

B

C

D

E

Rate on Margin Loan Holding Period in Months Return on Investment Capital Gain on Stock Dividends Interest on Margin Loan Net Income Initial Investment Return on Investment

Example 3.2

$20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 80.00

G

H

Ending St Price

Ending St Price Initial Equity Investment Amount Borrowed Initial Stock Price Shares Purchased Ending Stock Price Cash Dividends During Hold Per. Initial Margin Percentage Maintenance Margin Percentage

F

−41.60% −121.60% −101.60% −81.60% −61.60% −41.60% −21.60% −1.60% 18.40% 38.40% 58.40% 78.40% 98.40% 118.40%

$20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00 65.00 70.00 75.00 80.00

−18.80% −58.80% −48.80% −38.80% −28.80% −18.80% −8.80% 1.20% 11.20% 21.20% 31.20% 41.20% 51.20% 61.20%

LEGEND: Enter data Value calculated

Maintenance Margin

Suppose the maintenance margin is 30%. How far could the stock price fall before the investor would get a margin call? Let P be the price of the stock. The value of the investor’s 100 shares is then 100P, and the equity in the account is 100P 2 $4,000. The percentage margin is (100P 2 $4,000)/100P. The price at which the percentage margin equals the maintenance margin of .3 is found by solving the equation 100P 2 4,000 5 .3 100 P which implies that P 5$57.14. If the price of the stock were to fall below $57.14 per share, the investor would get a margin call.

CONCEPT CHECK

3.4

Suppose the maintenance margin in Example 3.2 is 40%. How far can the stock price fall before the investor gets a margin call?

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CHAPTER 3

Change in Stock Price

End-of-Year Value of Shares

Repayment of Principal and Interest*

$26,000 20,000 14,000

$10,900 10,900 10,900

30% increase No change 30% decrease

How Securities Are Traded

Investor’s Rate of Return 51% 29 269

79

Table 3.1 Illustration of buying stock on margin

* Assuming the investor buys $20,000 worth of stock, borrowing $10,000 of the purchase price at an interest rate of 9% per year.

Why do investors buy securities on margin? They do so when they wish to invest an amount greater than their own money allows. Thus, they can achieve greater upside potential, but they also expose themselves to greater downside risk. To see how, let’s suppose an investor is bullish on FinCorp stock, which is selling for $100 per share. An investor with $10,000 to invest expects FinCorp to go up in price by 30% during the next year. Ignoring any dividends, the expected rate of return would be 30% if the investor invested $10,000 to buy 100 shares. But now assume the investor borrows another $10,000 from the broker and invests it in FinCorp, too. The total investment in FinCorp would be $20,000 (for 200 shares). Assuming an interest rate on the margin loan of 9% per year, what will the investor’s rate of return be now (again ignoring dividends) if FinCorp stock goes up 30% by year’s end? The 200 shares will be worth $26,000. Paying off $10,900 of principal and interest on the margin loan leaves $15,100 (i.e., $26,000 2$10,900). The rate of return in this case will be $15,000 2 $10,000 5 51% $10,000 The investor has parlayed a 30% rise in the stock’s price into a 51% rate of return on the $10,000 investment. Doing so, however, magnifies the downside risk. Suppose that, instead of going up by 30%, the price of FinCorp stock goes down by 30% to $70 per share. In that case, the 200 shares will be worth $14,000, and the investor is left with $3,100 after paying off the $10,900 of principal and interest on the loan. The result is a disastrous return of $3,100 2 $10,000 5 269% $10,000 Table3.1 summarizes the possible results of these hypothetical transactions. If there is no change in FinCorp’s stock price, the investor loses 9%, the cost of the loan.

CONCEPT CHECK

3.5

Suppose that in this margin example, the investor borrows only $5,000 at the same interest rate of 9% per year. What will the rate of return be if the price of FinCorp goes up by 30%? If it goes down by 30%? If it remains unchanged?

eXcel APPLICATIONS: Short Sale he Online Learning Center (www.mhhe.com/bkm) contains this Excel spreadsheet model, built using the text example for Dot Bomb. The model allows you to analyze the effects of returns, margin calls, and different levels of initial and maintenance margins. The model also includes a sensitivity analysis for ending stock price and return on investment.

T

Excel Questions 1. Suppose you sell short 100 shares of stock initially selling for $100 a share. Your initial margin requirement is 50% of the A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

3.9

B

value of the stock sold. You receive no interest on the funds placed in your margin account. a. How much do you need to contribute to your margin account? b. What will be your rate of return for the following stock prices at the end of a 1-year holding period? Assume the stock pays no dividends. (1) $90, (2) $100, (3) $110. 2. Repeat Question 1 (b) but now assume that the stock pays dividends of $2 per share at year-end. What is the relationship between the total rate of return on the stock and the return to your short position? C

D

E

Ending St Price Initial Investment Initial Stock Price Number of Shares Sold Short Ending Stock Price Cash Dividends Per Share Initial Margin Percentage Maintenance Margin Percentage

$50,000.00 $100.00 1,000 $70.00 $0.00 50.00% 30.00%

Return on Short Sale Capital Gain on Stock Dividends Paid Net Income Initial Investment Return on Investment

$30,000.00 $0.00 $30,000.00 $50,000.00 60.00%

Margin Positions Margin Based on Ending Price Price for Margin Call

114.29%

$170.00 160.00 150.00 140.00 130.00 120.00 110.00 100.00 90.00 80.00 70.00 60.00 50.00 40.00 30.00 20.00 10.00

60.00% −140.00% −120.00% −100.00% −80.00% −60.00% −40.00% −20.00% 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 120.00% 140.00% 160.00% 180.00%

$115.38 LEGEND: Enter data Value calculated

Short Sales Normally, an investor would first buy a stock and later sell it. With a short sale, the order is reversed. First, you sell and then you buy the shares. In both cases, you begin and end with no shares. A short sale allows investors to profit from a decline in a security’s price. An investor borrows a share of stock from a broker and sells it. Later, the short-seller must purchase a share of the same stock in order to replace the share that was borrowed. This is called covering the short position. Table3.2 compares stock purchases to short sales.2 The short-seller anticipates the stock price will fall, so that the share can be purchased later at a lower price than it initially sold for; if so, the short-seller will reap a profit. Short-sellers must not only replace the shares but also pay the lender of the security any dividends paid during the short sale. In practice, the shares loaned out for a short sale are typically provided by the shortseller’s brokerage firm, which holds a wide variety of securities of its other investors in 2

Naked short-selling is a variant on conventional short-selling. In a naked short, a trader sells shares that have not yet been borrowed, assuming that the shares can be acquired in time to meet any delivery deadline. While naked short-selling is prohibited, enforcement has been spotty, as many firms have engaged in it based on their “reasonable belief ” that they will be able to acquire the stock by the time delivery is required. Now the SEC is requiring that short-sellers have made firm arrangements for delivery before engaging in the sale.

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Table 3.2

Purchase of Stock Time 0 1

Action

Cash Flow*

Buy share Receive dividend, sell share

2Initial price Ending price1Dividend

Profit5(Ending price1Dividend)2Initial price Short Sale of Stock Time 0 1

Action

Cash Flow*

Borrow share; sell it Repay dividend and buy share to replace the share originally borrowed

1Initial price 2(Ending price1Dividend)

Profit5Initial price2(Ending price1Dividend) *A negative cash flow implies a cash outflow.

street name (i.e., the broker holds the shares registered in its own name on behalf of the client). The owner of the shares need not know that the shares have been lent to the shortseller. If the owner wishes to sell the shares, the brokerage firm will simply borrow shares from another investor. Therefore, the short sale may have an indefinite term. However, if the brokerage firm cannot locate new shares to replace the ones sold, the short-seller will need to repay the loan immediately by purchasing shares in the market and turning them over to the brokerage house to close out the loan. Finally, exchange rules require that proceeds from a short sale must be kept on account with the broker. The short-seller cannot invest these funds to generate income, although large or institutional investors typically will receive some income from the proceeds of a short sale being held with the broker. Short-sellers also are required to post margin (cash or collateral) with the broker to cover losses should the stock price rise during the short sale.

Example 3.3

Short Sales

To illustrate the mechanics of short-selling, suppose you are bearish (pessimistic) on Dot Bomb stock, and its market price is $100 per share. You tell your broker to sell short 1,000 shares. The broker borrows 1,000 shares either from another customer’s account or from another broker. The $100,000 cash proceeds from the short sale are credited to your account. Suppose the broker has a 50% margin requirement on short sales. This means you must have other cash or securities in your account worth at least $50,000 that can serve as margin on the short sale. Let’s say that you have $50,000 in Treasury bills. Your account with the broker after the short sale will then be: Assets

Liabilities and Owners’ Equity

Cash

$100,000

T-bills

50,000

Short position in Dot Bomb stock (1,000 shares owed) Equity

$100,000 50,000

Cash flows from purchasing versus short-selling shares of stock

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PART I

Introduction

Your initial percentage margin is the ratio of the equity in the account, $50,000, to the current value of the shares you have borrowed and eventually must return, $100,000: Percentage margin 5

Equity $50,000 5 5 .50 Value of stock owed $100,000

Suppose you are right and Dot Bomb falls to $70 per share. You can now close out your position at a profit. To cover the short sale, you buy 1,000 shares to replace the ones you borrowed. Because the shares now sell for $70, the purchase costs only $70,000.3 Because your account was credited for $100,000 when the shares were borrowed and sold, your profit is $30,000: The profit equals the decline in the share price times the number of shares sold short.

Like investors who purchase stock on margin, a short-seller must be concerned about margin calls. If the stock price rises, the margin in the account will fall; if margin falls to the maintenance level, the short-seller will receive a margin call.

Example 3.4

Margin Calls on Short Positions

Suppose the broker has a maintenance margin of 30% on short sales. This means the equity in your account must be at least 30% of the value of your short position at all times. How much can the price of Dot Bomb stock rise before you get a margin call? Let P be the price of Dot Bomb stock. Then the value of the shares you must pay back is 1,000P and the equity in your account is $150,00021,000P. Your short position margin ratio is equity/value of stock5(150,00021,000P)/1,000P. The critical value of P is thus Equity 15,000 2 1,000P 5 5 .3 Value of shares owed 1,000P which implies that P 5 $115.38 per share. If Dot Bomb stock should rise above $115.38 per share, you will get a margin call, and you will either have to put up additional cash or cover your short position by buying shares to replace the ones borrowed.

CONCEPT CHECK

3.6

a. Construct the balance sheet if Dot Bomb in Example 3.4 goes up to $110. b. If the short position maintenance margin in the Dot Bomb example is 40%, how far can the stock price rise before the investor gets a margin call?

You can see now why stop-buy orders often accompany short sales. Imagine that you short-sell Dot Bomb when it is selling at $100 per share. If the share price falls, you will profit from the short sale. On the other hand, if the share price rises, let’s say to $130, you will lose $30 per share. But suppose that when you initiate the short sale, you also enter a 3

Notice that when buying on margin, you borrow a given amount of dollars from your broker, so the amount of the loan is independent of the share price. In contrast, when short-selling you borrow a given number of shares, which must be returned. Therefore, when the price of the shares changes, the value of the loan also changes.

CHAPTER 3

How Securities Are Traded

stop-buy order at $120. The stop-buy will be executed if the share price surpasses $120, thereby limiting your losses to $20 per share. (If the stock price drops, the stop-buy will never be executed.) The stop-buy order thus provides protection to the short-seller if the share price moves up. Short-selling periodically comes under attack, particularly during times of financial stress when share prices fall. The last few years have been no exception to this rule. For example, following the 2008 financial crisis, the SEC voted to restrict short sales in stocks that decline by at least 10% on a given day. Those stocks may now be shorted on that day and the next only at a price greater than the highest bid price across national stock markets. The nearby box examines the controversy surrounding short sales in greater detail.

3.10 Regulation of Securities Markets Trading in securities markets in the United States is regulated by a myriad of laws. The major governing legislation includes the Securities Act of 1933 and the Securities Exchange Act of 1934. The 1933 act requires full disclosure of relevant information relating to the issue of new securities. This is the act that requires registration of new securities and issuance of a prospectus that details the financial prospects of the firm. SEC approval of a prospectus or financial report is not an endorsement of the security as a good investment. The SEC cares only that the relevant facts are disclosed; investors must make their own evaluation of the security’s value. The 1934 act established the Securities and Exchange Commission to administer the provisions of the 1933 act. It also extended the disclosure principle of the 1933 act by requiring periodic disclosure of relevant financial information by firms with already-issued securities on secondary exchanges. The 1934 act also empowers the SEC to register and regulate securities exchanges, OTC trading, brokers, and dealers. While the SEC is the administrative agency responsible for broad oversight of the securities markets, it shares responsibility with other regulatory agencies. The Commodity Futures Trading Commission (CFTC) regulates trading in futures markets, while the Federal Reserve has broad responsibility for the health of the U.S. financial system. In this role, the Fed sets margin requirements on stocks and stock options and regulates bank lending to security market participants. The Securities Investor Protection Act of 1970 established the Securities Investor Protection Corporation (SIPC) to protect investors from losses if their brokerage firms fail. Just as the Federal Deposit Insurance Corporation provides depositors with federal protection against bank failure, the SIPC ensures that investors will receive securities held for their account in street name by a failed brokerage firm up to a limit of $500,000 per customer. The SIPC is financed by levying an “insurance premium” on its participating, or member, brokerage firms. In addition to federal regulations, security trading is subject to state laws, known generally as blue sky laws because they are intended to give investors a clearer view of investment prospects. Varying state laws were somewhat unified when many states adopted portions of the Uniform Securities Act, which was enacted in 1956. The 2008 financial crisis also led to regulatory changes, some of which we detailed in Chapter 1. The Financial Stability Oversight Council (FSOC) was established by the Dodd-Frank Wall Street Reform and Consumer Protection Act to monitor the stability of the U.S. financial system. It is largely concerned with risks arising from potential failures of large, interconnected banks, but its voting members are the chairpersons of the main U.S. regulatory agencies, and therefore the FSOC serves a broader role to connect and coordinate key financial regulators.

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WORDS FROM THE STREET

Short-Selling Comes Under Fire—Again Short-selling has long been viewed with suspicion, if not outright hostility. England banned short sales for a good part of the 18th century. Napoleon called short-sellers enemies of the state. In the U.S., short-selling was widely viewed as contributing to the market crash of 1929, and in 2008, short-sellers were blamed for the collapse of the investment banks Bear Stearns and Lehman Brothers. With share prices of other financial firms tumbling in September 2008, the SEC instituted a temporary ban on short-selling of nearly 1,000 of those firms. Similarly, the Financial Services Authority, the financial regulator in the U.K., prohibited short sales on about 30 financial companies, and Australia banned shorting altogether. The rationale for these bans is that short sales put downward pressure on share prices that in some cases may be unwarranted: Rumors abound of investors who first put on a short sale and then spread negative rumors about the firm to drive down its price. More often, however, shorting is a legitimate bet that a share price is too high and is due to fall. Nevertheless, during the market stresses of late 2008, the widespread feeling was that even if short positions were legitimate, regulators should do what they could to prop up the affected institutions. Hostility to short-selling may well stem from confusion between bad news and the bearer of that news. Shortselling allows investors whose analysis indicates a firm is overpriced to take action on that belief—and to profit if they are correct. Rather than causing the stock price

to fall, shorts may be anticipating a decline in the stock price. Their sales simply force the market to reflect the deteriorating prospects of troubled firms sooner than it might have otherwise. In other words, short-selling is part of the process by which the full range of information and opinion—pessimistic as well as optimistic—is brought to bear on stock prices. For example, short-sellers took large (negative) positions in firms such as WorldCom, Enron, and Tyco even before these firms were exposed by regulators. In fact, one might argue that these emerging short positions helped regulators identify the previously undetected scandals. And in the end, Lehman and Bear Stearns were brought down by their very real losses on their mortgage-related investments—not by unfounded rumors. Academic research supports the conjecture that short sales contribute to efficient “price discovery.” For example, the greater the demand for shorting a stock, the lower its future returns tend to be; moreover, firms that attack short-sellers with threats of legal action or bad publicity tend to have especially poor future returns.1 Short-sale bans may in the end be nothing more than an understandable, but nevertheless misguided, impulse to “shoot the messenger.” 1

See, for example, C. Jones and O. A. Lamont, “Short Sale Constraints and Stock Returns,” Journal of Financial Economics, November 2002, pp. 207–39, or O. A. Lamont, “Go Down Fighting: Short Sellers vs. Firms,” Yale ICF Working Paper No. 04–20, July 2004.

Self-Regulation In addition to government regulation, the securities market exercises considerable selfregulation. The most important overseer in this regard is the Financial Industry Regulatory Authority (FINRA), which is the largest nongovernmental regulator of all securities firms in the United States. FINRA was formed in 2007 through the consolidation of the National Association of Securities Dealers (NASD) with the self-regulatory arm of the New York Stock Exchange. It describes its broad mission as the fostering of investor protection and market integrity. It examines securities firms, writes and enforces rules concerning trading practices, and administers a dispute-resolution forum for investors and registered firms. In addition to being governed by exchange regulation, there is also self-regulation among the community of investment professionals. For example, the CFA Institute has developed standards of professional conduct that govern the behavior of members with the Chartered Financial Analysts designation, commonly referred to as CFAs. The nearby box presents a brief outline of those principles.

The Sarbanes-Oxley Act The scandals of 2000–2002 centered largely on three broad practices: allocations of shares in initial public offerings, tainted securities research and recommendations put out to the

84

I. Professionalism •

Knowledge of law. Members must understand, have knowledge of, and comply with all applicable laws, rules, and regulations including the Code of Ethics and Standards of Professional Conduct.

•

Independence and objectivity. Members shall maintain independence and objectivity in their professional activities.

•

Misrepresentation. Members must not knowingly misrepresent investment analysis, recommendations, or other professional activities.

II. Integrity of Capital Markets •

Non-public information. Members must not exploit material non-public information.

•

Market manipulation. Members shall not attempt to distort prices or trading volume with the intent to mislead market participants.

IV. Duties to Employers • Loyalty. Members must act for the benefit of their employer. •

• Supervisors. Members must make reasonable efforts to detect and prevent violation of applicable laws and regulations by anyone subject to their supervision. V. Investment Analysis and Recommendations •

Diligence. Members must exercise diligence and have reasonable basis for investment analysis, recommendations, or actions.

•

Communication. Members must distinguish fact from opinion in their presentation of analysis and disclose general principles of investment processes used in analysis.

III. Duties to Clients •

Loyalty, prudence, and care. Members must place their clients’ interests before their own and act with reasonable care on their behalf.

•

Fair dealing. Members shall deal fairly and objectively with clients when making investment recommendations or taking actions.

•

Suitability. Members shall make a reasonable inquiry into a client’s financial situation, investment experience, and investment objectives prior to making appropriate investment recommendations.

•

Performance presentation. Members shall attempt to ensure that investment performance is presented fairly, accurately, and completely.

•

Confidentiality. Members must keep information about clients confidential unless the client permits disclosure.

Compensation. Members must not accept compensation from sources that would create a conflict of interest with their employer’s interests without written consent from all involved parties.

VI. Conflicts of Interest •

Disclosure of conflicts. Members must disclose all matters that reasonably could be expected to impair their objectivity or interfere with their other duties.

•

Priority of transactions. Transactions for clients and employers must have priority over transactions for the benefit of a member.

VII. Responsibilities as Member of CFA Institute •

Conduct. Members must not engage in conduct that compromises the reputation or integrity of the CFA Institute or CFA designation.

Source: “Code of Ethics and Standards of Professional Conduct,” reproduced with permission from the Standards of Practice Handbook, 10th Ed. 2010, CFA Institute, Copyright 2010. www.cfainstitute.org/ centre/codes/ethics

public, and, probably most important, misleading financial statements and accounting practices. The Sarbanes-Oxley Act was passed by Congress in 2002 in response to these problems. Among the key reforms are: • Creation of the Public Company Accounting Oversight Board to oversee the auditing of public companies. • Rules requiring independent financial experts to serve on audit committees of a firm’s board of directors. • CEOs and CFOs must now personally certify that their firms’ financial reports “fairly represent, in all material respects, the operations and financial condition of the company,” and are subject to personal penalties if those reports turn out to be misleading. Following the letter of the rules may still be necessary, but it is no longer sufficient accounting practice.

85

WORDS FROM THE STREET

Excerpts from CFA Institute Standards of Professional Conduct

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PART I

Introduction

• Auditors may no longer provide several other services to their clients. This is intended to prevent potential profits on consulting work from influencing the quality of their audit. • The Board of Directors must be composed of independent directors and hold regular meetings of directors in which company management is not present (and therefore cannot impede or influence the discussion). More recently, there has been a fair amount of pushback on Sarbanes-Oxley. Many observers believe that the compliance costs associated with the law are too onerous, especially for smaller firms, and that heavy-handed regulatory oversight is giving foreign locales an undue advantage over the United States when firms decide where to list their securities. Moreover, the efficacy of single-country regulation is being tested in the face of increasing globalization and the ease with which funds can move across national borders.

Insider Trading Regulations also prohibit insider trading. It is illegal for anyone to transact in securities to profit from inside information, that is, private information held by officers, directors, or major stockholders that has not yet been divulged to the public. But the definition of insiders can be ambiguous. While it is obvious that the chief financial officer of a firm is an insider, it is less clear whether the firm’s biggest supplier can be considered an insider. Yet a supplier may deduce the firm’s near-term prospects from significant changes in orders. This gives the supplier a unique form of private information, yet the supplier is not technically an insider. These ambiguities plague security analysts, whose job is to uncover as much information as possible concerning the firm’s expected prospects. The dividing line between legal private information and illegal inside information can be fuzzy. The SEC requires officers, directors, and major stockholders to report all transactions in their firm’s stock. A compendium of insider trades is published monthly in the SEC’s Official Summary of Securities Transactions and Holdings. The idea is to inform the public of any implicit vote of confidence or no confidence made by insiders. Insiders do exploit their knowledge. Three forms of evidence support this conclusion. First, there have been well-publicized convictions of principals in insider trading schemes. Second, there is considerable evidence of “leakage” of useful information to some traders before any public announcement of that information. For example, share prices of firms announcing dividend increases (which the market interprets as good news concerning the firm’s prospects) commonly increase in value a few days before the public announcement of the increase. Clearly, some investors are acting on the good news before it is released to the public. Share prices still rise substantially on the day of the public release of good news, however, indicating that insiders, or their associates, have not fully bid up the price of the stock to the level commensurate with the news. A third form of evidence on insider trading has to do with returns earned on trades by insiders. Researchers have examined the SEC’s summary of insider trading to measure the performance of insiders. In one of the best known of these studies, Jaffee4 examined the abnormal return of stocks over the months following purchases or sales by insiders. For months in which insider purchasers of a stock exceeded insider sellers of the stock by three or more, the stock had an abnormal return in the following 8 months of about 5%. Moreover, when insider sellers exceeded insider buyers, the stock tended to perform poorly. 4

Jeffrey E. Jaffee, “Special Information and Insider Trading,” Journal of Business 47 (July 1974).

CHAPTER 3

How Securities Are Traded

1. Firms issue securities to raise the capital necessary to finance their investments. Investment bankers market these securities to the public on the primary market. Investment bankers generally act as underwriters who purchase the securities from the firm and resell them to the public at a markup. Before the securities may be sold to the public, the firm must publish an SEC-accepted prospectus that provides information on the firm’s prospects.

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SUMMARY

3. Trading may take place in dealer markets, via electronic communication networks, or in specialist markets. In dealer markets, security dealers post bid and ask prices at which they are willing to trade. Brokers for individuals execute trades at the best available prices. In electronic markets, the existing book of limit orders provides the terms at which trades can be executed. Mutually agreeable offers to buy or sell securities are automatically crossed by the computer system operating the market. In specialist markets, the specialist acts to maintain an orderly market with price continuity. Specialists maintain a limit-order book, but also sell from or buy for their own inventories of stock. 4. NASDAQ was traditionally a dealer market in which a network of dealers negotiated directly over sales of securities. The NYSE was traditionally a specialist market. In recent years, however, both exchanges have dramatically increased their commitment to electronic and automated trading. Trading activity today is overwhelmingly electronic. 5. Buying on margin means borrowing money from a broker to buy more securities than can be purchased with one’s own money alone. By buying securities on a margin, an investor magnifies both the upside potential and the downside risk. If the equity in a margin account falls below the required maintenance level, the investor will get a margin call from the broker. 6. Short-selling is the practice of selling securities that the seller does not own. The short-seller borrows the securities sold through a broker and may be required to cover the short position at any time on demand. The cash proceeds of a short sale are kept in escrow by the broker, and the broker usually requires that the short-seller deposit additional cash or securities to serve as margin (collateral). 7. Securities trading is regulated by the Securities and Exchange Commission, by other government agencies, and through self-regulation of the exchanges. Many of the important regulations have to do with full disclosure of relevant information concerning the securities in question. Insider trading rules also prohibit traders from attempting to profit from inside information.

primary market secondary market private placement initial public offerings (IPOs) underwriters prospectus dealer markets auction market bid price ask price

bid–ask spread limit order stop orders over-the-counter (OTC) market NASDAQ Stock Market electronic communication networks (ECNs) specialist stock exchanges

latency algorithmic trading high-frequency trading blocks dark pools margin short sale inside information

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KEY TERMS

1. What are the differences between a stop-loss order, a limit sell order, and a market order?

PROBLEM SETS

2. Why have average trade sizes declined in recent years?

Basic

3. How do margin trades magnify both the upside potential and the downside risk of an investment position?

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2. Already-issued securities are traded on the secondary market, that is, on organized stock markets; on the over-the-counter market; and occasionally for very large trades, through direct negotiation. Only license holders of exchanges may trade on the exchange. Brokerage firms holding licenses to trade on the exchange sell their services to individuals, charging commissions for executing trades on their behalf.

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PART I

Introduction 4. A market order has: a. Price uncertainty but not execution uncertainty. b. Both price uncertainty and execution uncertainty. c. Execution uncertainty but not price uncertainty. 5. Where would an illiquid security in a developing country most likely trade? a. Broker markets. b. Electronic crossing networks. c. Electronic limit-order markets.

Intermediate

6. Dée Trader opens a brokerage account and purchases 300 shares of Internet Dreams at $40 per share. She borrows $4,000 from her broker to help pay for the purchase. The interest rate on the loan is 8%. a. What is the margin in Dée’s account when she first purchases the stock? b. If the share price falls to $30 per share by the end of the year, what is the remaining margin in her account? If the maintenance margin requirement is 30%, will she receive a margin call? c. What is the rate of return on her investment?

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7. Old Economy Traders opened an account to short sell 1,000 shares of Internet Dreams from the previous problem. The initial margin requirement was 50%. (The margin account pays no interest.) A year later, the price of Internet Dreams has risen from $40 to $50, and the stock has paid a dividend of $2 per share. a. What is the remaining margin in the account? b. If the maintenance margin requirement is 30%, will Old Economy receive a margin call? c. What is the rate of return on the investment? 8. Consider the following limit-order book for a share of stock. The last trade in the stock occurred at a price of $50. Limit Buy Orders

Limit Sell Orders

Price

Shares

Price

Shares

$49.75

500

$50.25

100

49.50

800

51.50

100

49.25

500

54.75

300

49.00

200

58.25

100

48.50

600

a. If a market buy order for 100 shares comes in, at what price will it be filled? b. At what price would the next market buy order be filled? c. If you were a security dealer, would you want to increase or decrease your inventory of thisstock?

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9. You are bullish on Telecom stock. The current market price is $50 per share, and you have $5,000 of your own to invest. You borrow an additional $5,000 from your broker at an interest rate of 8% per year and invest $10,000 in the stock. a. What will be your rate of return if the price of Telecom stock goes up by 10% during the next year? The stock currently pays no dividends. b. How far does the price of Telecom stock have to fall for you to get a margin call if the maintenance margin is 30%? Assume the price fall happens immediately.

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10. You are bearish on Telecom and decide to sell short 100 shares at the current market price of $50 per share. a. How much in cash or securities must you put into your brokerage account if the broker’s initial margin requirement is 50% of the value of the short position? b. How high can the price of the stock go before you get a margin call if the maintenance margin is 30% of the value of the short position?

CHAPTER 3

How Securities Are Traded

11. Suppose that Intel currently is selling at $20 per share. You buy 1,000 shares using $15,000 of your own money, borrowing the remainder of the purchase price from your broker. The rate on the margin loan is 8%.

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a. What is the percentage increase in the net worth of your brokerage account if the price of Intel immediately changes to: (i) $22; (ii) $20; (iii) $18? What is the relationship between your percentage return and the percentage change in the price of Intel? b. If the maintenance margin is 25%, how low can Intel’s price fall before you get a margin call? c. How would your answer to (b) change if you had financed the initial purchase with only $10,000 of your own money? d. What is the rate of return on your margined position (assuming again that you invest $15,000 of your own money) if Intel is selling after 1 year at: (i) $22; (ii) $20; (iii) $18? What is the relationship between your percentage return and the percentage change in the price of Intel? Assume that Intel pays no dividends. e. Continue to assume that a year has passed. How low can Intel’s price fall before you get a margin call?

a. If you earn no interest on the funds in your margin account, what will be your rate of return after 1 year if Intel stock is selling at: (i) $22; (ii) $20; (iii) $18? Assume that Intel pays no dividends. b. If the maintenance margin is 25%, how high can Intel’s price rise before you get a margin call? c. Redo parts (a) and (b), but now assume that Intel also has paid a year-end dividend of $1 per share. The prices in part (a) should be interpreted as ex-dividend, that is, prices after the dividend has been paid. 13. Here is some price information on Marriott:

Marriott

Bid

Ask

39.95

40.05

You have placed a stop-loss order to sell at $40. What are you telling your broker? Given market prices, will your order be executed? 14. Here is some price information on FinCorp stock. Suppose that FinCorp trades in a dealer market. Bid

Ask

55.25

55.50

a. Suppose you have submitted an order to your broker to buy at market. At what price will your trade be executed? b. Suppose you have submitted an order to sell at market. At what price will your trade be executed? c. Suppose you have submitted a limit order to sell at $55.62. What will happen? d. Suppose you have submitted a limit order to buy at $55.37. What will happen? 15. You’ve borrowed $20,000 on margin to buy shares in Disney, which is now selling at $40 per share. Your account starts at the initial margin requirement of 50%. The maintenance margin is 35%. Two days later, the stock price falls to $35 per share. a. Will you receive a margin call? b. How low can the price of Disney shares fall before you receive a margin call? 16. On January 1, you sold short one round lot (that is, 100 shares) of Four Sisters stock at $21 per share. On March 1, a dividend of $2 per share was paid. On April 1, you covered the short sale by buying the stock at a price of $15 per share. You paid 50 cents per share in commissions for each transaction. What is the value of your account on April 1?

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12. Suppose that you sell short 1,000 shares of Intel, currently selling for $20 per share, and give your broker $15,000 to establish your margin account.

90

PART I

Introduction

1. FBN Inc. has just sold 100,000 shares in an initial public offering. The underwriter’s explicit fees were $70,000. The offering price for the shares was $50, but immediately upon issue, the share price jumped to $53. a. What is your best guess as to the total cost to FBN of the equity issue? b. Is the entire cost of the underwriting a source of profit to the underwriters? 2. If you place a stop-loss order to sell 100 shares of stock at $55 when the current price is $62, how much will you receive for each share if the price drops to $50? a. b. c. d.

$50. $55. $54.87. Cannot tell from the information given.

3. Specialists on the New York Stock Exchange do all of the following except:

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a. b. c. d.

Act as dealers for their own accounts. Execute limit orders. Help provide liquidity to the marketplace. Act as odd-lot dealers.

E-INVESTMENTS EXERCISES When you are choosing which brokerage firm(s) to use to execute your trades, you should consider several factors. Also, a wide range of services claim to objectively recommend brokerage firms. However, many are actually sponsored by the brokerage firms themselves. Go to the Web site www.consumersearch.com/online-brokers/reviews and read the information provided under “Our Sources.” Then follow the link for the Barron’s ratings. Here you can read the Barron’s annual broker survey and download the “How the Brokers Stack Up” report, which contains a list of fees. Suppose that you have $3,000 to invest and want to put it in a non-IRA account. 1. Are all of the brokerage firms suitable if you want to open a cash account? Are they all suitable if you want a margin account? 2. Choose two of the firms listed. Assume that you want to buy 200 shares of LLY stock using a market order. If the order is filled at $42 per share, how much will the commission be for the two firms if you place an online order? 3. Are there any maintenance fees associated with the account at either brokerage firm? 4. Now assume that you have a margin account and the balance is $3,000. Calculate the interest rate you would pay if you borrowed money to buy stock.

SOLUTIONS TO CONCEPT CHECKS 1. Limited-time shelf registration was introduced because its cost savings outweighed the disadvantage of slightly less up-to-date disclosures. Allowing unlimited shelf registration would circumvent “blue sky” laws that ensure proper disclosure as the financial circumstances of the firm change over time. 2. a. Used cars trade in dealer markets (used-car lots or auto dealerships) and in direct search markets when individuals advertise in local newspapers or on the Web. b. Paintings trade in broker markets when clients commission brokers to buy or sell art for them, in dealer markets at art galleries, and in auction markets. c. Rare coins trade mostly in dealer markets in coin shops, but they also trade in auctions and in direct search markets when individuals advertise they want to buy or sell coins.

CHAPTER 3

How Securities Are Traded

91

3. a. You should give your broker a market order. It will be executed immediately and is the cheapest type of order in terms of brokerage fees. b. You should give your broker a limit-buy order, which will be executed only if the shares can be obtained at a price about 5% below the current price. c. You should give your broker a stop-loss order, which will be executed if the share price starts falling. The limit or stop price should be close to the current price to avoid the possibility of large losses. 4. Solving 100P 2 $4,000 5 .4 100P yields P5$66.67 per share. 5. The investor will purchase 150 shares, with a rate of return as follows: Year-End Change in Price

Year-End Value of Shares

Repayment of Principal and Interest

Investor’s Rate of Return

30% No change

$19,500 15,000

$5,450 5,450

24.5

230%

10,500

5,450

249.5

6. a. Once Dot Bomb stock goes up to $110, your balance sheet will be: Assets

Liabilities and Owner’s Equity

Cash

$100,000

T-bills

50,000

Short position in Dot Bomb Equity

b. Solving $150,000 2 1,000P 5 .4 1,000P yields P5$107.14 per share.

$110,000 40,000

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40.5%

4

CHAPTER FOUR

Mutual Funds and Other Investment Companies

THE PREVIOUS CHAPTERintroduced you to the mechanics of trading securities and the structure of the markets in which securities trade. Commonly, however, individual investors do not trade securities directly for their own accounts. Instead, they direct their funds to investment companies that purchase securities on their behalf. The most important of these financial intermediaries are openend investment companies, more commonly known as mutual funds, to which we devote most of this chapter. We also touch briefly on other types of investment companies such as unit investment trusts, hedge funds, and closed-end funds. We begin the chapter by describing and comparing the various

PART I

4.1

types of investment companies available to investors. We then examine the functions of mutual funds, their investment styles and policies, and the costs of investing in these funds. Next we take a first look at the investment performance of these funds. We consider the impact of expenses and turnover on net performance and examine the extent to which performance is consistent from one period to the next. In other words, will the mutual funds that were the best past performers be the best future performers? Finally, we discuss sources of information on mutual funds, and we consider in detail the information provided in the most comprehensive guide, Morningstar’s Mutual Fund Sourcebook.

Investment Companies Investment companies are financial intermediaries that collect funds from individual investors and invest those funds in a potentially wide range of securities or other assets. Pooling of assets is the key idea behind investment companies. Each investor has a claim to the portfolio established by the investment company in proportion to the amount invested. These companies thus provide a mechanism for small investors to “team up” to obtain the benefits of large-scale investing. Investment companies perform several important functions for their investors: 1. Record keeping and administration. Investment companies issue periodic status reports, keeping track of capital gains distributions, dividends, investments, and redemptions, and they may reinvest dividend and interest income for shareholders.

CHAPTER 4

Mutual Funds and Other Investment Companies

93

2. Diversification and divisibility. By pooling their money, investment companies enable investors to hold fractional shares of many different securities. They can act as large investors even if any individual shareholder cannot. 3. Professional management. Investment companies can support full-time staffs of security analysts and portfolio managers who attempt to achieve superior investment results for their investors. 4. Lower transaction costs. Because they trade large blocks of securities, investment companies can achieve substantial savings on brokerage fees and commissions. While all investment companies pool assets of individual investors, they also need to divide claims to those assets among those investors. Investors buy shares in investment companies, and ownership is proportional to the number of shares purchased. The value of each share is called the net asset value, or NAV. Net asset value equals assets minus liabilities expressed on a per-share basis: Net asset value 5

Example 4.1

Market value of assets minus liabilities Shares outstanding

Net Asset Value

Consider a mutual fund that manages a portfolio of securities worth $120 million. Suppose the fund owes $4 million to its investment advisers and owes another $1 million for rent, wages due, and miscellaneous expenses. The fund has 5 million shares outstanding. Net asset value 5

CONCEPT CHECK

$120 million 2 $5 million 5 $23 per share 5 million shares

4.1

Consider these data from the March 2012 balance sheet of Vanguard’s Growth and Income Fund. What was the net asset value of the fund? Assets: $2,877.06 million Liabilities: $ 14.73 million Shares: 95.50 million

4.2

Types of Investment Companies

In the United States, investment companies are classified by the Investment Company Act of 1940 as either unit investment trusts or managed investment companies. The portfolios of unit investment trusts are essentially fixed and thus are called “unmanaged.” In contrast, managed companies are so named because securities in their investment portfolios continually are bought and sold: The portfolios are managed. Managed companies are further classified as either closed-end or open-end. Open-end companies are what we commonly call mutual funds.

Unit Investment Trusts Unit investment trusts are pools of money invested in a portfolio that is fixed for the life of the fund. To form a unit investment trust, a sponsor, typically a brokerage firm, buys a

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PART I

Introduction

portfolio of securities that are deposited into a trust. It then sells shares, or “units,” in the trust, called redeemable trust certificates. All income and payments of principal from the portfolio are paid out by the fund’s trustees (a bank or trust company) to the shareholders. There is little active management of a unit investment trust because once established, the portfolio composition is fixed; hence these trusts are referred to as unmanaged. Trusts tend to invest in relatively uniform types of assets; for example, one trust may invest in municipal bonds, another in corporate bonds. The uniformity of the portfolio is consistent with the lack of active management. The trusts provide investors a vehicle to purchase a pool of one particular type of asset that can be included in an overall portfolio as desired. Sponsors of unit investment trusts earn their profit by selling shares in the trust at a premium to the cost of acquiring the underlying assets. For example, a trust that has purchased $5 million of assets may sell 5,000 shares to the public at a price of $1,030 per share, which (assuming the trust has no liabilities) represents a 3% premium over the net asset value of the securities held by the trust. The 3% premium is the trustee’s fee for establishing the trust. Investors who wish to liquidate their holdings of a unit investment trust may sell the shares back to the trustee for net asset value. The trustees can either sell enough securities from the asset portfolio to obtain the cash necessary to pay the investor, or they may instead sell the shares to a new investor (again at a slight premium to net asset value). Unit investment trusts have steadily lost market share to mutual funds in recent years. Assets in such trusts declined from $105 billion in 1990 to only $60 billion in 2012.

Managed Investment Companies There are two types of managed companies: closed-end and open-end. In both cases, the fund’s board of directors, which is elected by shareholders, hires a management company to manage the portfolio for an annual fee that typically ranges from .2% to 1.5% of assets. In many cases the management company is the firm that organized the fund. For example, Fidelity Management and Research Corporation sponsors many Fidelity mutual funds and is responsible for managing the portfolios. It assesses a management fee on each Fidelity fund. In other cases, a mutual fund will hire an outside portfolio manager. For example, Vanguard has hired Wellington Management as the investment adviser for its Wellington Fund. Most management companies have contracts to manage several funds. Open-end funds stand ready to redeem or issue shares at their net asset value (although both purchases and redemptions may involve sales charges). When investors in open-end funds wish to “cash out” their shares, they sell them back to the fund at NAV. In contrast, closed-end funds do not redeem or issue shares. Investors in closed-end funds PREM/ 52-WEEK FUND NAV MKT PRICE DISC % MARKET RETURN % who wish to cash out must sell their shares to other inves28.31 17.44 15.99 1.59 Gabelli Div & Inc Tr (GDV) tors. Shares of closed-end funds are traded on organized 1.72 5.22 5.31 21.89 Gabelli Equity Trust (GAB) 214.37 31.73 27.17 22.49 General Amer Investors (GAM) exchanges and can be purchased through brokers just like 28.68 19.13 17.47 5.34 Guggenheim Enh Eq Strat (GGE) 27.25 20.15 18.69 19.89 J Hancock Tx-Adv Div Inc (HTD) other common stock; their prices, therefore, can differ 29.92 5.04 4.54 26.59 Liberty All-Star Equity (USA) from NAV. In early 2013, about $265 billion of assets were 27.14 4.20 3.90 28.96 Liberty All-Star Growth (ASG) 27.27 11.01 10.21 0.73 Nuveen Tx-Adv TR Strat (JTA) held in closed-end funds. Figure4.1 is a listing of closed-end funds. The first column gives the name and ticker symbol of the fund. The next two columns give the fund’s most recent net asset Figure 4.1 Closed-end mutual funds value and closing share price. The premium or discount Source: Data compiled from The Wall Street Journal in the next column is the percentage difference between Online, July 24, 2012. price and NAV: (Price – NAV)/NAV. Notice that there are

CHAPTER 4

Mutual Funds and Other Investment Companies

more funds selling at discounts to NAV (indicated by negative differences) than premiums. Finally, the 52-week return based on the percentage change in share price plus dividend income is presented in the last column. The common divergence of price from net asset value, often by wide margins, is a puzzle that has yet to be fully explained. To see why this is a puzzle, consider a closedend fund that is selling at a discount from net asset value. If the fund were to sell all the assets in the portfolio, it would realize proceeds equal to net asset value. The difference between the market price of the fund and the fund’s NAV would represent the per-share increase in the wealth of the fund’s investors. Moreover, fund premiums or discounts tend to dissipate over time, so funds selling at a discount receive a boost to their rate of return as the discount shrinks. Pontiff estimates that a fund selling at a 20% discount would have an expected 12-month return more than 6% greater than funds selling at net asset value.1 Interestingly, while many closed-end funds sell at a discount from net asset value, the prices of these funds when originally issued are often above NAV. This is a further puzzle, as it is hard to explain why investors would purchase these newly issued funds at a premium to NAV when the shares tend to fall to a discount shortly after issue. In contrast to closed-end funds, the price of open-end funds cannot fall below NAV, because these funds stand ready to redeem shares at NAV. The offering price will exceed NAV, however, if the fund carries a load. A load is, in effect, a sales charge. Load funds are sold by securities brokers and directly by mutual fund groups. Unlike closed-end funds, open-end mutual funds do not trade on organized exchanges. Instead, investors simply buy shares from and liquidate through the investment company at net asset value. Thus the number of outstanding shares of these funds changes daily.

Other Investment Organizations Some intermediaries are not formally organized or regulated as investment companies, but nevertheless serve similar functions. Three of the more important are commingled funds, real estate investment trusts, and hedge funds. Commingled Funds Commingled funds are partnerships of investors that pool funds. The management firm that organizes the partnership, for example, a bank or insurance company, manages the funds for a fee. Typical partners in a commingled fund might be trust or retirement accounts with portfolios much larger than those of most individual investors, but still too small to warrant managing on a separate basis. Commingled funds are similar in form to open-end mutual funds. Instead of shares, though, the fund offers units, which are bought and sold at net asset value. A bank or insurance company may offer an array of different commingled funds, for example, a money market fund, a bond fund, and a common stock fund. Real Estate Investment Trusts (REITs) A REIT is similar to a closed-end fund. REITs invest in real estate or loans secured by real estate. Besides issuing shares, they raise capital by borrowing from banks and issuing bonds or mortgages. Most of them are highly leveraged, with a typical debt ratio of 70%. There are two principal kinds of REITs. Equity trusts invest in real estate directly, whereas mortgage trusts invest primarily in mortgage and construction loans. REITs generally are established by banks, insurance companies, or mortgage companies, which then serve as investment managers to earn a fee. 1

Jeffrey Pontiff, “Costly Arbitrage: Evidence from Closed-End Funds,” Quarterly Journal of Economics 111 (November 1996), pp. 1135–51.

95

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PART I

Introduction

Hedge Funds Like mutual funds, hedge funds are vehicles that allow private investors to pool assets to be invested by a fund manager. Unlike mutual funds, however, hedge funds are commonly structured as private partnerships and thus subject to only minimal SEC regulation. They typically are open only to wealthy or institutional investors. Many require investors to agree to initial “lock-ups,” that is, periods as long as several years in which investments cannot be withdrawn. Lock-ups allow hedge funds to invest in illiquid assets without worrying about meeting demands for redemption of funds. Moreover, because hedge funds are only lightly regulated, their managers can pursue investment strategies involving, for example, heavy use of derivatives, short sales, and leverage; such strategies typically are not open to mutual fund managers. Hedge funds by design are empowered to invest in a wide range of investments, with various funds focusing on derivatives, distressed firms, currency speculation, convertible bonds, emerging markets, merger arbitrage, and so on. Other funds may jump from one asset class to another as perceived investment opportunities shift. Hedge funds enjoyed great growth in the last several years, with assets under management ballooning from about $50 billion in 1990 to just about $2 trillion in 2012. We devote all of Chapter 26 to these funds.

4.3

Mutual Funds Mutual funds are the common name for open-end investment companies. This is the dominant investment company today, accounting for more than 90% of investment company assets. Assets under management in the U.S. mutual fund industry were approximately $13.5 trillion in early 2013, and approximately another $13 trillion was held in non-U.S. funds.

Investment Policies Each mutual fund has a specified investment policy, which is described in the fund’s prospectus. For example, money market mutual funds hold the short-term, low-risk instruments of the money market (see Chapter 2 for a review of these securities), while bond funds hold fixed-income securities. Some funds have even more narrowly defined mandates. For example, some bond funds will hold primarily Treasury bonds, others primarily mortgage-backed securities. Management companies manage a family, or “complex,” of mutual funds. They organize an entire collection of funds and then collect a management fee for operating them. By managing a collection of funds under one umbrella, these companies make it easy for investors to allocate assets across market sectors and to switch assets across funds while still benefiting from centralized record keeping. Some of the most well-known management companies are Fidelity, Vanguard, Barclays, and T. Rowe Price. Each offers an array of open-end mutual funds with different investment policies. In 2013, there were nearly 8,000 mutual funds in the U.S., which were offered by a bit more than 700 fund complexes. Funds are commonly classified by investment policy into one of the following groups. Money Market Funds These funds invest in money market securities such as commercial paper, repurchase agreements, or certificates of deposit. The average maturity of

CHAPTER 4

Mutual Funds and Other Investment Companies

these assets tends to be a bit more than 1 month. Money market funds usually offer checkwriting features, and net asset value is fixed at $1 per share,2 so that there are no tax implications such as capital gains or losses associated with redemption of shares. Equity Funds Equity funds invest primarily in stock, although they may, at the portfolio manager’s discretion, also hold fixed-income or other types of securities. Equity funds commonly will hold between 4% and 5% of total assets in money market securities to provide the liquidity necessary to meet potential redemption of shares. Stock funds are traditionally classified by their emphasis on capital appreciation versus current income. Thus, income funds tend to hold shares of firms with consistently high dividend yields. Growth funds are willing to forgo current income, focusing instead on prospects for capital gains. While the classification of these funds is couched in terms of income versus capital gains, in practice, the more relevant distinction concerns the level of risk these funds assume. Growth stocks, and therefore growth funds, are typically riskier and respond more dramatically to changes in economic conditions than do income funds. Sector Funds Some equity funds, called sector funds, concentrate on a particular industry. For example, Fidelity markets dozens of “select funds,” each of which invests in a specific industry such as biotechnology, utilities, energy, or telecommunications. Other funds specialize in securities of particular countries. Bond Funds As the name suggests, these funds specialize in the fixed-income sector. Within that sector, however, there is considerable room for further specialization. For example, various funds will concentrate on corporate bonds, Treasury bonds, mortgagebacked securities, or municipal (tax-free) bonds. Indeed, some municipal bond funds invest only in bonds of a particular state (or even city!) to satisfy the investment desires of residents of that state who wish to avoid local as well as federal taxes on interest income. Many funds also specialize by maturity, ranging from short-term to intermediate to long-term, or by the credit risk of the issuer, ranging from very safe to high-yield, or “junk,” bonds. International Funds Many funds have international focus. Global funds invest in securities worldwide, including the United States. In contrast, international funds invest in securities of firms located outside the United States. Regional funds concentrate on a particular part of the world, and emerging market funds invest in companies of developing nations. Balanced Funds Some funds are designed to be candidates for an individual’s entire investment portfolio. These balanced funds hold both equities and fixed-income securities in relatively stable proportions. Life-cycle funds are balanced funds in which the asset mix can range from aggressive (primarily marketed to younger investors) to conservative (directed at older investors). Static allocation life-cycle funds maintain a stable mix across stocks and bonds, while targeted-maturity funds gradually become more conservative as the investor ages. Many balanced funds are in fact funds of funds. These are mutual funds that primarily invest in shares of other mutual funds. Balanced funds of funds invest in equity and bond funds in proportions suited to their investment goals. 2

The box in Chapter 2 noted that money market funds are able to maintain NAV at $1.00 because they invest in short-maturity debt of the highest quality with minimal price risk. In only the rarest circumstances have any funds incurred losses large enough to drive NAV below $1.00. In September 2008, however, Reserve Primary Fund, the nation’s oldest money market fund, “broke the buck” when it suffered losses on its holding of Lehman Brothers commercial paper, and its NAV fell to $.97.

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Asset Allocation and Flexible Funds These funds are similar to balanced funds in that they hold both stocks and bonds. However, asset allocation funds may dramatically vary the proportions allocated to each market in accord with the portfolio manager’s forecast of the relative performance of each sector. Hence these funds are engaged in market timing and are not designed to be low-risk investment vehicles. Index Funds An index fund tries to match the performance of a broad market index. The fund buys shares in securities included in a particular index in proportion to each security’s representation in that index. For example, the Vanguard 500 Index Fund is a mutual fund that replicates the composition of the Standard & Poor’s 500 stock price index. Because the S&P 500 is a value-weighted index, the fund buys shares in each S&P 500 company in proportion to the market value of that company’s outstanding equity. Investment in an index fund is a low-cost way for small investors to pursue a passive investment strategy—that is, to invest without engaging in security analysis. About 15% of equity funds in 2012 were indexed. Of course, index funds can be tied to nonequity indexes as well. For example, Vanguard offers a bond index fund and a real estate index fund. Table 4.1 breaks down the number of mutual funds by investment orientation. Sometimes a fund name describes its investment policy. For example, Vanguard’s GNMA fund invests in mortgage-backed securities, the Municipal Intermediate fund invests in intermediate-term municipal bonds, and the High-Yield Corporate bond fund invests

Table 4.1 U.S. mutual funds by investment classification

Equity funds Capital appreciation focus World/international Total return Total equity funds Bond funds Corporate High yield World Government Strategic income Single-state municipal National municipal Total bond funds Hybrid (bond/stock) funds Money market funds Taxable Tax-exempt Total money market funds Total

Assets ($ billion)

% of Total Assets

Number of Funds

$ 2,356 1,359 1,490 $ 5,205

20.3% 11.7 12.8 44.8%

2,686 1,285 610 4,581

$

3.9% 1.8 2.2 2.2 10.4 1.4 2.9 24.8%

252 179 205 246 484 347 216 1,929

839

7.2%

495

$ 2,400 292 $ 2,692 $11,621

20.7% 2.5 23.2% 100.0%

431 201 632 7,637

452 212 259 261 1,204 159 338 $ 2,885 $

Note: Column sums subject to rounding error. Source: Investment Company Institute, 2012 Mutual Fund Fact Book.

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in large part in speculative grade, or “junk,” bonds with high yields. However, names of common stock funds often reflect little or nothing about their investment policies. Examples are Vanguard’s Windsor and Wellington funds.

How Funds Are Sold Mutual funds are generally marketed to the public either directly by the fund underwriter or indirectly through brokers acting on behalf of the underwriter. Direct-marketed funds are sold through the mail, various offices of the fund, over the phone, or, more so, over the Internet. Investors contact the fund directly to purchase shares. About half of fund sales today are distributed through a sales force. Brokers or financial advisers receive a commission for selling shares to investors. (Ultimately, the commission is paid by the investor. More on this shortly.) Investors who rely on their broker’s advice to select their mutual funds should be aware that brokers may have a conflict of interest with regard to fund selection. This can arise from a practice called revenue sharing, in which fund companies pay the brokerage firm for preferential treatment when making investment recommendations. Many funds also are sold through “financial supermarkets” that sell shares in funds of many complexes. Instead of charging customers a sales commission, the broker splits management fees with the mutual fund company. Another advantage is unified record keeping for all funds purchased from the supermarket, even if the funds are offered by different complexes. On the other hand, many contend that these supermarkets result in higher expense ratios because mutual funds pass along the costs of participating in these programs in the form of higher management fees.

4.4

Costs of Investing in Mutual Funds

Fee Structure An individual investor choosing a mutual fund should consider not only the fund’s stated investment policy and past performance but also its management fees and other expenses. Comparative data on virtually all important aspects of mutual funds are available in Morningstar’s Mutual Fund Sourcebook, which can be found in many academic and public libraries. You should be aware of four general classes of fees. Operating Expenses Operating expenses are the costs incurred by the mutual fund in operating the portfolio, including administrative expenses and advisory fees paid to the investment manager. These expenses, usually expressed as a percentage of total assets under management, may range from 0.2% to 2%. Shareholders do not receive an explicit bill for these operating expenses; however, the expenses periodically are deducted from the assets of the fund. Shareholders pay for these expenses through the reduced value of the portfolio. The simple average of the expense ratio of equity funds in the U.S. was 1.43% in 2011. But larger funds tend to have lower expense ratios, so the average expense ratio weighted by assets under management is considerably smaller, 0.79%. Not surprisingly, the average expense ratio of actively managed funds is considerably higher than that of indexed funds, .93% versus .14% (weighted by assets under management). In addition to operating expenses, many funds assess fees to pay for marketing and distribution costs. These charges are used primarily to pay the brokers or financial advisers who sell the funds to the public. Investors can avoid these expenses by buying shares directly from the fund sponsor, but many investors are willing to incur these distribution fees in return for the advice they may receive from their broker.

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Front-End Load A front-end load is a commission or sales charge paid when you purchase the shares. These charges, which are used primarily to pay the brokers who sell the funds, may not exceed 8.5%, but in practice they are rarely higher than 6%. Low-load funds have loads that range up to 3% of invested funds. No-load funds have no front-end sales charges. Loads effectively reduce the amount of money invested. For example, each $1,000 paid for a fund with a 6% load results in a sales charge of $60 and fund investment of only $940. You need cumulative returns of 6.4% of your net investment (60/9405.064) just to break even. Back-End Load A back-end load is a redemption, or “exit,” fee incurred when you sell your shares. Typically, funds that impose back-end loads start them at 5% or 6% and reduce them by 1 percentage point for every year the funds are left invested. Thus an exit fee that starts at 6% would fall to 4% by the start of your third year. These charges are known more formally as “contingent deferred sales charges.” 12b-1 Charges The Securities and Exchange Commission allows the managers of so-called 12b-1 funds to use fund assets to pay for distribution costs such as advertising, promotional literature including annual reports and prospectuses, and, most important, commissions paid to brokers who sell the fund to investors. These 12b-1 fees are named after the SEC rule that permits use of these plans. Funds may use 12b-1 charges instead of, or in addition to, front-end loads to generate the fees with which to pay brokers. As with operating expenses, investors are not explicitly billed for 12b-1 charges. Instead, the fees are deducted from the assets of the fund. Therefore, 12b-1 fees (if any) must be added to operating expenses to obtain the true annual expense ratio of the fund. The SEC requires that all funds include in the prospectus a consolidated expense table that summarizes all relevant fees. The 12b-1 fees are limited to 1% of a fund’s average net assets per year.3 Many funds offer “classes” that represent ownership in the same portfolio of securities, but with different combinations of fees. For example, Class A shares might have front-end loads while Class B shares rely on 12b-1 fees.

Example 4.2

Fees for Various Classes

Here are fees for different classes of the Dreyfus High Yield Fund in 2012. Notice the trade-off between the front-end loads versus 12b-1 charges in the choice between Class A and Class C shares. Class I shares are sold only to institutional investors and carry lower fees.

Front-end load Back-end load 12b-1 feesc Expense ratio

Class A

Class C

Class I

0–4.5%a 0 .25% .70%

0 0–1%b 1.0% .70%

0 0%b 0% .70%

a

Depending on size of investment. Depending on years until holdings are sold. Including service fee.

b c

3

The maximum 12b-1 charge for the sale of the fund is .75%. However, an additional service fee of .25% of the fund’s assets also is allowed for personal service and/or maintenance of shareholder accounts.

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Each investor must choose the best combination of fees. Obviously, pure no-load no-fee funds distributed directly by the mutual fund group are the cheapest alternative, and these will often make most sense for knowledgeable investors. However, as we have noted, many investors are willing to pay for financial advice, and the commissions paid to advisers who sell these funds are the most common form of payment. Alternatively, investors may choose to hire a fee-only financial manager who charges directly for services instead of collecting commissions. These advisers can help investors select portfolios of low- or noload funds (as well as provide other financial advice). Independent financial planners have become increasingly important distribution channels for funds in recent years. If you do buy a fund through a broker, the choice between paying a load and paying 12b-1 fees will depend primarily on your expected time horizon. Loads are paid only once for each purchase, whereas 12b-1 fees are paid annually. Thus, if you plan to hold your fund for a long time, a one-time load may be preferable to recurring 12b-1 charges.

Fees and Mutual Fund Returns The rate of return on an investment in a mutual fund is measured as the increase or decrease in net asset value plus income distributions such as dividends or distributions of capital gains expressed as a fraction of net asset value at the beginning of the investment period. If we denote the net asset value at the start and end of the period as NAV0 and NAV1, respectively, then Rate of return 5

NAV1 2 NAV0 1 Income and capital gain distributions NAV0

For example, if a fund has an initial NAV of $20 at the start of the month, makes income distributions of $.15 and capital gain distributions of $.05, and ends the month with NAV of $20.10, the monthly rate of return is computed as Rate of return 5

$20.10 2 $20.00 1 $.15 1 $.05 5 .015, or 1.5% $20.00

Notice that this measure of the rate of return ignores any commissions such as front-end loads paid to purchase the fund. On the other hand, the rate of return is affected by the fund’s expenses and 12b-1 fees. This is because such charges are periodically deducted from the portfolio, which reduces net asset value. Thus the investor’s rate of return equals the gross return on the underlying portfolio minus the total expense ratio.

Example 4.3

Fees and Net Returns

To see how expenses can affect rate of return, consider a fund with $100 million in assets at the start of the year and with 10 million shares outstanding. The fund invests in a portfolio of stocks that provides no income but increases in value by 10%. The expense ratio, including 12b-1 fees, is 1%. What is the rate of return for an investor in the fund? The initial NAV equals $100 million/10 million shares5$10 per share. In the absence of expenses, fund assets would grow to $110 million and NAV would grow to $11 per share, for a 10% rate of return. However, the expense ratio of the fund is 1%. Therefore, $1 million will be deducted from the fund to pay these fees, leaving the portfolio worth only $109 million, and NAV equal to $10.90. The rate of return on the fund is only 9%, which equals the gross return on the underlying portfolio minus the total expense ratio.

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Table 4.2

Cumulative Proceeds (All Dividends Reinvested)

Impact of costs on investment performance

Fund A

Fund B

Fund C

$10,000 17,234

$10,000 16,474

$ 9,200 15,502

10 years

29,699

27,141

26,123

15 years

51,183

44,713

44,018

20 years

88,206

73,662

74,173

Initial investment* 5 years

*After front-end load, if any. Notes: 1. Fund A is no-load with .5% expense ratio. 2. Fund B is no-load with 1.5% expense ratio. 3. Fund C has an 8% load on purchases and a 1% expense ratio. 4. Gross return on all funds is 12% per year before expenses.

Fees can have a big effect on performance. Table4.2 considers an investor who starts with $10,000 and can choose among three funds that all earn an annual 12% return on investment before fees but have different fee structures. The table shows the cumulative amount in each fund after several investment horizons. Fund A has total operating expenses of .5%, no load, and no 12b-1 charges. This might represent a low-cost producer like Vanguard. Fund B has no load but has 1% in management expenses and .5% in 12b-1 fees. This level of charges is fairly typical of actively managed equity funds. Finally, Fund C has 1% in management expenses, has no 12b-1 charges, but assesses an 8% front-end load on purchases. Note the substantial return advantage of low-cost Fund A. Moreover, that differential is greater for longer investment horizons. CONCEPT CHECK

4.2

The Equity Fund sells Class A shares with a front-end load of 4% and Class B shares with 12b-1 fees of .5% annually as well as back-end load fees that start at 5% and fall by 1% for each full year the investor holds the portfolio (until the fifth year). Assume the rate of return on the fund portfolio net of operating expenses is 10% annually. What will be the value of a $10,000 investment in Class A and Class B shares if the shares are sold after (a) 1 year, (b) 4 years, (c) 10 years? Which fee structure provides higher net proceeds at the end of each investment horizon?

Although expenses can have a big impact on net investment performance, it is sometimes difficult for the investor in a mutual fund to measure true expenses accurately. This is because of the practice of paying for some expenses in soft dollars. A portfolio manager earns soft-dollar credits with a brokerage firm by directing the fund’s trades to that broker. On the basis of those credits, the broker will pay for some of the mutual fund’s expenses, such as databases, computer hardware, or stock-quotation systems. The soft-dollar arrangement means that the stockbroker effectively returns part of the trading commission to the fund. Purchases made with soft dollars are not included in the fund’s expenses, so funds with extensive soft-dollar arrangements may report artificially low expense ratios to the public. However, the fund may have paid its broker needlessly high commissions to obtain its soft-dollar “rebate.” The impact of the higher trading commission shows up in net investment performance rather than the reported expense ratio.

CHAPTER 4

4.5

Mutual Funds and Other Investment Companies

103

Taxation of Mutual Fund Income

Investment returns of mutual funds are granted “pass-through status” under the U.S. tax code, meaning that taxes are paid only by the investor in the mutual fund, not by the fund itself. The income is treated as passed through to the investor as long as the fund meets several requirements, most notably that virtually all income is distributed to shareholders. A fund’s short-term capital gains, long-term capital gains, and dividends are passed through to investors as though the investor earned the income directly. The pass-through of investment income has one important disadvantage for individual investors. If you manage your own portfolio, you decide when to realize capital gains and losses on any security; therefore, you can time those realizations to efficiently manage your tax liabilities. When you invest through a mutual fund, however, the timing of the sale of securities from the portfolio is out of your control, which reduces your ability to engage in tax management.4 A fund with a high portfolio turnover rate can be particularly “tax inefficient.” Turnover is the ratio of the trading activity of a portfolio to the assets of the portfolio. It measures the fraction of the portfolio that is “replaced” each year. For example, a $100 million portfolio with $50 million in sales of some securities and purchases of other securities would have a turnover rate of 50%. High turnover means that capital gains or losses are being realized constantly, and therefore that the investor cannot time the realizations to manage his or her overall tax obligation. Turnover rates in equity funds in the last decade have typically been around 60% when weighted by assets under management. By contrast, a low-turnover fund such as an index fund may have turnover as low as 2%, which is both tax-efficient and economical with respect to trading costs. CONCEPT CHECK

4.3

An investor’s portfolio currently is worth $1 million. During the year, the investor sells 1,000 shares of FedEx at a price of $80 per share and 4,000 shares of Cisco at a price of $20 per share. The proceeds are used to buy 800 shares of IBM at $200 per share. a. What was the portfolio turnover rate? b. If the shares in FedEx originally were purchased for $70 each and those in Cisco were purchased for $17.50, and the investor’s tax rate on capital gains income is 20%, how much extra will the investor owe on this year’s taxes as a result of these transactions?

4.6

Exchange-Traded Funds

Exchange-traded funds (ETFs), first introduced in 1993, are offshoots of mutual funds that allow investors to trade index portfolios just as they do shares of stock. The first ETF was the “spider,” a nickname for SPDR, or Standard & Poor’s Depository Receipt, which is a unit investment trust holding a portfolio matching the S&P 500 Index. Unlike mutual 4

An interesting problem that an investor needs to be aware of derives from the fact that capital gains and dividends on mutual funds are typically paid out to shareholders once or twice a year. This means that an investor who has just purchased shares in a mutual fund can receive a capital gain distribution (and be taxed on that distribution) on transactions that occurred long before he or she purchased shares in the fund. This is particularly a concern late in the year when such distributions typically are made.

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A. ETF Sponsors Sponsor

Product Name

BlackRock Global Investors Merrill Lynch StateStreet/Merrill Lynch Vanguard

iShares HOLDRS (Holding Company Depository Receipts: “Holders”) Select Sector SPDRs (S&P Depository Receipts: “Spiders”) Vanguard ETF

B. Sample of ETF Products Name

Ticker

Index Tracked

Broad U.S. indexes Spiders Diamonds Cubes iShares Russell 2000 Total Stock Market (Vanguard)

SPY DIA QQQ IWM VTI

S&P 500 Dow Jones Industrials NASDAQ 100 Russell 2000 Wilshire 5000

Industry indexes Energy Select Spider iShares Energy Sector Financial Sector Spider iShares Financial Sector

XLE IYE XLF IYF

S&P 500 energy companies Dow Jones energy companies S&P 500 financial companies Dow Jones financial companies

International indexes WEBS United Kingdom WEBS France WEBS Japan

EWU EWQ EWJ

MSCI U.K. Index MSCI France Index MSCI Japan Index

Table 4.3 ETF sponsors and products

funds, which can be bought or sold only at the end of the day when NAV is calculated, investors can trade spiders throughout the day, just like any other share of stock. Spiders gave rise to many similar products such as “diamonds” (based on the Dow Jones Industrial Average, ticker DIA), “cubes” (based on the NASDAQ 100 index, ticker QQQ), and “WEBS” (World Equity Benchmark Shares, which are shares in portfolios of foreign stock market indexes). By 2012, about $1 trillion was invested in more than 1,100 U.S. ETFs. Table4.3, panel A presents some of the major sponsors of ETFs, and panel B gives a small flavor of the types of funds offered. Figure 4.2 shows the rapid growth in the ETF market since 1998. Until 2008, most ETFs were required to track specified indexes, and ETFs tracking broad indexes still dominate the industry. However, there are dozens of industry-sector ETFs, and as Figure4.2 makes clear, commodity, bond, and international ETFs have grown especially dramatically in recent years. While only $1 billion was invested in commodity ETFs in 2004, by 2011 this value had grown to $109 billion. Gold and silver ETFs dominate this sector, accounting for about three-quarters of commodity-based funds. Indeed, ETFs have become the main way for investors to speculate in precious metals. Figure 4.3 shows that by 2011 ETFs had captured a significant portion of the assets under management in the investment company universe.

CHAPTER 4

Mutual Funds and Other Investment Companies

1000 900

ETF Assets ($ million)

800 700

Bond Commodities Global/Int’l Equity U.S. Equity: Sector U.S. Equity: Broad Index

600 500 400 300 200 100 0 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

Figure 4.2 Growth of U.S. ETFs over time Source: Investment Company Institute, 2012 Investment Company Fact Book.

$239

$60

$1,047

Mutual Funds Exchange-Traded Funds Closed-End Funds Unit Investment Trusts $11,600

Figure 4.3 Investment company assets under management, 2011 ($ billion) Source: Investment Company Institute, 2012 Investment Company Fact Book.

Barclays Global Investors was long the market leader in the ETF market, using the product name iShares. Since Barclays’s 2009 merger with Blackrock, iShares has operated under the Blackrock name. The firm sponsors ETFs for several dozen equity index funds, including many broad U.S. equity indexes, broad international and single-country funds, and U.S. and global industry sector funds. Blackrock also offers several bond ETFs and

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a few commodity funds such as ones for gold and silver. For more information on these funds, go to www.iShares.com. More recently, a variety of new ETF products have been devised. Among these are leveraged ETFs, with daily returns that are a targeted multiple of the returns on an index, and inverse ETFs, which move in the opposite direction to an index. In addition, there is now a small number of actively managed ETF funds that, like actively managed mutual funds, attempt to outperform market indexes. But these account for only about 3% of assets under management in the ETF industry. Other even more exotic variations are so-called synthetic ETFs such as exchangetraded notes (ETNs) or exchange-traded vehicles (ETVs). These are nominally debt securities, but with payoffs linked to the performance of an index. Often that index measures the performance of an illiquid and thinly traded asset class, so the ETF gives the investor the opportunity to add that asset class to his or her portfolio. However, rather than invest in those assets directly, the ETF achieves this exposure by entering a “total return swap” with an investment bank in which the bank agrees to pay the ETF the return on the index in exchange for a relatively fixed fee. These have become controversial, as the ETF is then exposed to risk that in a period of financial stress the investment bank will be unable to fulfill its obligation, leaving investors without the returns they were promised. ETFs offer several advantages over conventional mutual funds. First, as we just noted, a mutual fund’s net asset value is quoted—and therefore, investors can buy or sell their shares in the fund—only once a day. In contrast, ETFs trade continuously. Moreover, like other shares, but unlike mutual funds, ETFs can be sold short or purchased on margin. ETFs also offer a potential tax advantage over mutual funds. When large numbers of mutual fund investors redeem their shares, the fund must sell securities to meet the redemptions. This can trigger capital gains taxes, which are passed through to and must be paid by the remaining shareholders. In contrast, when small investors wish to redeem their position in an ETF, they simply sell their shares to other traders, with no need for the fund to sell any of the underlying portfolio. Large investors can exchange their ETF shares for shares in the underlying portfolio; this form of redemption also avoids a tax event. ETFs are often cheaper than mutual funds. Investors who buy ETFs do so through brokers rather than buying directly from the fund. Therefore, the fund saves the cost of marketing itself directly to small investors. This reduction in expenses may translate into lower management fees. There are some disadvantages to ETFs, however. First, while mutual funds can be bought at no expense from no-load funds, ETFs must be purchased from brokers for a fee. In addition, because ETFs trade as securities, their prices can depart from NAV, at least for short periods, and these price discrepancies can easily swamp the cost advantage that ETFs otherwise offer. While those discrepancies typically are quite small, they can spike unpredictably when markets are stressed. Chapter 2 briefly discussed the socalled flash crash of May 6, 2010, when the Dow Jones Industrial Average fell by 583 points in seven minutes, leaving it down nearly 1,000 points for the day. Remarkably, the index recovered more than 600 points in the next 10 minutes. In the wake of this incredible volatility, the stock exchanges canceled many trades that had gone off at what were viewed as distorted prices. Around one-fifth of all ETFs changed hands on that day at prices less than one-half of their closing price, and ETFs accounted for about two-thirds of all canceled trades.

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Mutual Funds and Other Investment Companies

At least two problems were exposed in this episode. First, when markets are not working properly, it can be hard to measure the net asset value of the ETF portfolio, especially for ETFs that track less liquid assets. And, reinforcing this problem, some ETFs may be supported by only a very small number of dealers. If they drop out of the market during a period of turmoil, prices may swing wildly.

4.7

Mutual Fund Investment Performance: A First Look

We noted earlier that one of the benefits of mutual funds for the individual investor is the ability to delegate management of the portfolio to investment professionals. The investor retains control over the broad features of the overall portfolio through the asset allocation decision: Each individual chooses the percentages of the portfolio to invest in bond funds versus equity funds versus money market funds, and so forth, but can leave the specific security selection decisions within each investment class to the managers of each fund. Shareholders hope that these portfolio managers can achieve better investment performance than they could obtain on their own. What is the investment record of the mutual fund industry? This seemingly straightforward question is deceptively difficult to answer because we need a standard against which to evaluate performance. For example, we clearly would not want to compare the investment performance of an equity fund to the rate of return available in the money market. The vast differences in the risk of these two markets dictate that year-by-year as well as average performance will differ considerably. We would expect to find that equity funds outperform money market funds (on average) as compensation to investors for the extra risk incurred in equity markets. How then can we determine whether mutual fund portfolio managers are performing up to par given the level of risk they incur? In other words, what is the proper benchmark against which investment performance ought to be evaluated? Measuring portfolio risk properly and using such measures to choose an appropriate benchmark is far from straightforward. We devote all of Parts Two and Three of the text to issues surrounding the proper measurement of portfolio risk and the trade-off between risk and return. In this chapter, therefore, we will satisfy ourselves with a first look at the question of fund performance by using only very simple performance benchmarks and ignoring the more subtle issues of risk differences across funds. However, we will return to this topic in Chapter 11, where we take a closer look at mutual fund performance after adjusting for differences in the exposure of portfolios to various sources of risk. Here we use as a benchmark for the performance of equity fund managers the rate of return on the Wilshire 5000 index. Recall from Chapter 2 that this is a value-weighted index of essentially all actively traded U.S. stocks. The performance of the Wilshire 5000 is a useful benchmark with which to evaluate professional managers because it corresponds to a simple passive investment strategy: Buy all the shares in the index in proportion to their outstanding market value. Moreover, this is a feasible strategy for even small investors, because the Vanguard Group offers an index fund (its Total Stock Market Portfolio) designed to replicate the performance of the Wilshire 5000 index. Using the Wilshire 5000 index as a benchmark, we may pose the problem of evaluating the performance of mutual fund portfolio managers this way: How does the typical performance of actively managed equity mutual funds compare to the performance of a passively managed portfolio that simply replicates the composition of a broad index of the stock market?

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40

Rate of Return (%)

30 20 10 0 210 220

2011

2007

Wilshire 5000 2003

1995

1991

1987

1983

1979

1975

Active Funds 1971

240

1999

230

Figure 4.4 Rates of return on actively managed equity funds versus Wilshire 5000 index, 1971–2011 Source: For Wilshire returns, see www.wilshire.com. Used with permission of Wilshire Associates. For active fund returns, see http:// www.fiscalisadvisory.com/assets/pdfs/spiva_report_year_end_2011.pdf.

Casual comparisons of the performance of the Wilshire 5000 index versus that of professionally managed mutual funds reveal disappointing results for active managers. Figure4.4 shows that the average return on diversified equity funds was below the return on the Wilshire index in 25 of the 41 years from 1971 to 2011. The average annual return on the index was 11.75%, which was 1% greater than that of the average mutual fund.5 This result may seem surprising. After all, it would not seem unreasonable to expect that professional money managers should be able to outperform a very simple rule such as “hold an indexed portfolio.” As it turns out, however, there may be good reasons to expect such a result. We explore them in detail in Chapter 11, where we discuss the efficient market hypothesis. Of course, one might argue that there are good managers and bad managers, and that good managers can, in fact, consistently outperform the index. To test this notion, we examine whether managers with good performance in one year are likely to repeat that performance in a following year. Is superior performance in any particular year due to luck, and therefore random, or due to skill, and therefore consistent from year to year? To answer this question, we can examine the performance of a large sample of equity mutual fund portfolios, divide the funds into two groups based on total investment return, and ask: “Do funds with investment returns in the top half of the sample in one period continue to perform well in a subsequent period?” 5

Of course, actual funds incur trading costs while indexes do not, so a fair comparison between the returns on actively managed funds versus those on a passive index should first reduce the return on the Wilshire 5000 by an estimate of such costs. Vanguard’s Total Stock Market Index portfolio, which tracks the Wilshire 5000, charges an expense ratio of less than .10%, and, because it engages in little trading, incurs low trading costs. Therefore, it would be reasonable to reduce the returns on the index by about .15%. This reduction would not erase the difference in average performance.

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Mutual Funds and Other Investment Companies

Successive Period Performance Initial Period Performance

Top Half

Bottom Half

A. Malkiel study, 1970s Top half

65.1%

34.9%

Bottom half

35.5

64.5

Top half

51.7

48.3

Bottom half

47.5

52.5

B. Malkiel study, 1980s

Source: Burton G. Malkiel, “Returns from Investing in Equity Mutual Funds 1971–1991,” Journal of Finance 50 (June 1995), pp. 549–72. Used with permission of John Wiley and Sons, via Copyright Clearance Center.

Table4.4 presents such an analysis from a study by Malkiel.6 The table shows the fraction of “winners” (i.e., top-half performers) in each year that turn out to be winners or losers in the following year. If performance were purely random from one period to the next, there would be entries of 50% in each cell of the table, as top- or bottom-half performers would be equally likely to perform in either the top or bottom half of the sample in the following period. On the other hand, if performance were due entirely to skill, with no randomness, we would expect to see entries of 100% on the diagonals and entries of 0% on the off-diagonals: Top-half performers would all remain in the top half while bottom-half performers similarly would all remain in the bottom half. In fact, the table shows that 65.1% of initial top-half performers fall in the top half of the sample in the following period, while 64.5% of initial bottom-half performers fall in the bottom half in the following period. This evidence is consistent with the notion that at least part of a fund’s performance is a function of skill as opposed to luck, so that relative performance tends to persist from one period to the next.7 On the other hand, this relationship does not seem stable across different sample periods. While initial-year performance predicts subsequent-year performance in the 1970s (panel A), the pattern of persistence in performance virtually disappears in the 1980s (panel B). To summarize, the evidence that performance is consistent from one period to the next is suggestive, but it is inconclusive. Other studies suggest that there is little performance persistence among professional managers, and if anything, bad performance is more likely to persist than good performance.8 This makes some sense: It is easy to identify fund characteristics that will result in consistently poor investment performance, notably high expense ratios, and high turnover ratios with associated trading costs. It is far harder to identify the secrets of successful stock picking. (If it were easy, we would all be rich!) Thus the consistency we do observe in fund performance may be due in large part to the poor performers. This suggests that the real value of past performance data is to avoid truly poor funds, even if identifying the future top performers is still a daunting task. 6

Burton G. Malkiel, “Returns from Investing in Equity Mutual Funds 1971–1991,” Journal of Finance 50 (June 1995), pp. 549–72. 7 Another possibility is that performance consistency is due to variation in fee structure across funds. We return to this possibility in Chapter 11. 8 See for example, Mark M. Carhart, “On Persistence in Mutual Fund Performance,” Journal of Finance 52 (1997), 57–82. Carhart’s study also addresses survivorship bias, the tendency for better-performing funds to stay in business and thus remain in the sample. We return to his study in Chapter 11.

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Table 4.4 Consistency of investment results

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PART I

CONCEPT CHECK

Introduction

4.4

Suppose you observe the investment performance of 400 portfolio managers and rank them by investment returns during the year. Twenty percent of all managers are truly skilled, and therefore always fall in the top half, but the others fall in the top half purely because of good luck. What fraction of this year’s top-half managers would you expect to be top-half performers next year?

4.8

Information on Mutual Funds The first place to find information on a mutual fund is in its prospectus. The Securities and Exchange Commission requires that the prospectus describe the fund’s investment objectives and policies in a concise “Statement of Investment Objectives” as well as in lengthy discussions of investment policies and risks. The fund’s investment adviser and its portfolio manager are also described. The prospectus also presents the costs associated with purchasing shares in the fund in a fee table. Sales charges such as front-end and back-end loads as well as annual operating expenses such as management fees and 12b-1 fees are detailed in the fee table. Funds provide information about themselves in two other sources. The Statement of Additional Information (SAI), also known as Part B of the prospectus, includes a list of the securities in the portfolio at the end of the fiscal year, audited financial statements, a list of the directors and officers of the fund—as well as their personal investments in the fund, and data on brokerage commissions paid by the fund. However, unlike the fund prospectus, investors do not receive the SAI unless they specifically request it; one industry joke is that SAI stands for “something always ignored.” The fund’s annual report also includes portfolio composition and financial statements, as well as a discussion of the factors that influenced fund performance over the last reporting period. With thousands of funds to choose from, it can be difficult to find and select the fund that is best suited for a particular need. Several publications now offer “encyclopedias” of mutual fund information to help in the search process. One prominent source is Morningstar’s Mutual Fund Sourcebook. Morningstar’s Web site, www.morningstar.com, is another excellent source of information, as is Yahoo!’s site, finance.yahoo.com/funds. The Investment Company Institute (www.ici.org), the national association of mutual funds, closed-end funds, and unit investment trusts, publishes an annual Directory of Mutual Funds that includes information on fees as well as phone numbers to contact funds. To illustrate the range of information available about funds, we consider Morningstar’s report on Wells Fargo’s Advantage Growth Fund, reproduced in Figure4.5. The table on the left labeled “Performance” first shows the fund’s quarterly returns in the last few years, and just below that, returns over longer periods. You can compare returns to both its standard index (the S&P 500) and its category index (the Russell 1000) in the rows labeled 1/2 Index, as well as its percentile rank within its comparison group, or category. Continuing down the left column we see data on fees and expenses, as well as several measures of the fund’s risk and return characteristics. (We will discuss all of these measures in Part 2 of the text.) The fund has provided good returns compared to risk, earning it the coveted Morningstar 5-star rating. Of course, we are all accustomed to the disclaimer that “past performance is not a reliable measure of future results,” and this is true as well of Morningstar’s star ratings. Consistent with this disclaimer, past results have little predictive power for future performance, as we saw in Table4.4.

CHAPTER 4

Mutual Funds and Other Investment Companies

Figure 4.5 Morningstar report Source: Morningstar Mutual Funds, © 2012 Morningstar, Inc. All rights reserved. Used with permission.

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Introduction

More data on the performance of the fund are provided in the graph near the top of the figure. The line graph compares the growth of $10,000 invested in the fund versus its two comparison indexes over the last 10 years. Below the graph are boxes for each year that depict the relative performance of the fund for that year. The shaded area on the box shows the quartile in which the fund’s performance falls relative to other funds with the same objective. If the shaded band is at the top of the box, the firm was a top quartile performer in that period, and so on. The table below the bar charts presents historical data on the yearby-year performance of the fund. Below the table, the “Portfolio Analysis” table shows the asset allocation of the fund, and then Morningstar’s well-known style box. In this box, Morningstar evaluates style along two dimensions: One dimension is the size of the firms held in the portfolio as measured by the market value of outstanding equity; the other dimension is a value/growth measure. Morningstar defines value stocks as those with low ratios of market price per share to various measures of value. It puts stocks on a growth-value continuum based on the ratios of stock price to the firm’s earnings, book value, sales, cash flow, and dividends. Value stocks are those with a low price relative to these measures of value. In contrast, growth stocks have high ratios, suggesting that investors in these firms must believe that the firm will experience rapid growth to justify the prices at which the stocks sell. The shaded box shows that the fund tends to hold larger firms (top row) and growth stocks (right column). Finally, the tables in the right column provide information on the current composition of the portfolio. You can find the fund’s 15 “Top Holdings” there as well as the weighting of the portfolio across various sectors of the economy.

SUMMARY

1. Unit investment trusts, closed-end management companies, and open-end management companies are all classified and regulated as investment companies. Unit investment trusts are essentially unmanaged in the sense that the portfolio, once established, is fixed. Managed investment companies, in contrast, may change the composition of the portfolio as deemed fit by the portfolio manager. Closed-end funds are traded like other securities; they do not redeem shares for their investors. Open-end funds will redeem shares for net asset value at the request of the investor. 2. Net asset value equals the market value of assets held by a fund minus the liabilities of the fund divided by the shares outstanding. 3. Mutual funds free the individual from many of the administrative burdens of owning individual securities and offer professional management of the portfolio. They also offer advantages that are available only to large-scale investors, such as discounted trading costs. On the other hand, funds are assessed management fees and incur other expenses, which reduce the investor’s rate of return. Funds also eliminate some of the individual’s control over the timing of capital gains realizations. 4. Mutual funds are often categorized by investment policy. Major policy groups include money market funds; equity funds, which are further grouped according to emphasis on income versus growth; fixed-income funds; balanced and income funds; asset allocation funds; index funds; and specialized sector funds. 5. Costs of investing in mutual funds include front-end loads, which are sales charges; back-end loads, which are redemption fees or, more formally, contingent-deferred sales charges; fund operating expenses; and 12b-1 charges, which are recurring fees used to pay for the expenses of marketing the fund to the public. 6. Income earned on mutual fund portfolios is not taxed at the level of the fund. Instead, as long as the fund meets certain requirements for pass-through status, the income is treated as being earned by the investors in the fund.

Mutual Funds and Other Investment Companies

7. The average rate of return of the average equity mutual fund in the last four decades has been below that of a passive index fund holding a portfolio to replicate a broad-based index like the S&P 500 or Wilshire 5000. Some of the reasons for this disappointing record are the costs incurred by actively managed funds, such as the expense of conducting the research to guide stock-picking activities, and trading costs due to higher portfolio turnover. The record on the consistency of fund performance is mixed. In some sample periods, the better-performing funds continue to perform well in the following periods; in other sample periods they do not.

12b-1 fees soft dollars turnover exchange-traded funds

closed-end fund load hedge fund funds of funds

investment company net asset value (NAV) unit investment trust open-end fund

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KEY TERMS

1. Would you expect a typical open-end fixed-income mutual fund to have higher or lower operating expenses than a fixed-income unit investment trust? Why?

PROBLEM SETS

2. What are some comparative advantages of investing in the following:

Basic

a. Unit investment trusts. b. Open-end mutual funds. c. Individual stocks and bonds that you choose for yourself. 3. Open-end equity mutual funds find it necessary to keep a significant percentage of total investments, typically around 5% of the portfolio, in very liquid money market assets. Closed-end funds do not have to maintain such a position in “cash equivalent” securities. What difference between open-end and closed-end funds might account for their differing policies? 4. Balanced funds, life-cycle funds, and asset allocation funds all invest in both the stock and bond markets. What are the differences among these types of funds? 5. Why can closed-end funds sell at prices that differ from net asset value while open-end funds do not? 6. What are the advantages and disadvantages of exchange-traded funds versus mutual funds? 7. An open-end fund has a net asset value of $10.70 per share. It is sold with a front-end load of 6%. What is the offering price? 8. If the offering price of an open-end fund is $12.30 per share and the fund is sold with a frontend load of 5%, what is its net asset value? 9. The composition of the Fingroup Fund portfolio is as follows: Stock

Shares

Price

A B C D

200,000 300,000 400,000 600,000

$35 40 20 25

The fund has not borrowed any funds, but its accrued management fee with the portfolio manager currently totals $30,000. There are 4 million shares outstanding. What is the net asset value of the fund? 10. Reconsider the Fingroup Fund in the previous problem. If during the year the portfolio manager sells all of the holdings of stock D and replaces it with 200,000 shares of stock E at $50 per share and 200,000 shares of stock F at $25 per share, what is the portfolio turnover rate?

Intermediate

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CHAPTER 4

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PART I

Introduction 11. The Closed Fund is a closed-end investment company with a portfolio currently worth $200 million. It has liabilities of $3 million and 5 million shares outstanding. a. What is the NAV of the fund? b. If the fund sells for $36 per share, what is its premium or discount as a percent of net asset value? 12. Corporate Fund started the year with a net asset value of $12.50. By year-end, its NAV equaled $12.10. The fund paid year-end distributions of income and capital gains of $1.50. What was the (pretax) rate of return to an investor in the fund? 13. A closed-end fund starts the year with a net asset value of $12.00. By year-end, NAV equals $12.10. At the beginning of the year, the fund was selling at a 2% premium to NAV. By the end of the year, the fund is selling at a 7% discount to NAV. The fund paid year-end distributions of income and capital gains of $1.50.

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a. What is the rate of return to an investor in the fund during the year? b. What would have been the rate of return to an investor who held the same securities as the fund manager during the year? 14. a. Impressive Fund had excellent investment performance last year, with portfolio returns that placed it in the top 10% of all funds with the same investment policy. Do you expect it to be a top performer next year? Why or why not? b. Suppose instead that the fund was among the poorest performers in its comparison group. Would you be more or less likely to believe its relative performance will persist into the following year? Why? 15. Consider a mutual fund with $200 million in assets at the start of the year and with 10 million shares outstanding. The fund invests in a portfolio of stocks that provides dividend income at the end of the year of $2 million. The stocks included in the fund’s portfolio increase in price by 8%, but no securities are sold, and there are no capital gains distributions. The fund charges 12b-1 fees of 1%, which are deducted from portfolio assets at year-end. What is net asset value at the start and end of the year? What is the rate of return for an investor in the fund? 16. The New Fund had average daily assets of $2.2 billion last year. The fund sold $400 million worth of stock and purchased $500 million during the year. What was its turnover ratio? 17. If New Fund’s expense ratio (see the previous problem) was 1.1% and the management fee was .7%, what were the total fees paid to the fund’s investment managers during the year? What were other administrative expenses? 18. You purchased 1,000 shares of the New Fund at a price of $20 per share at the beginning of the year. You paid a front-end load of 4%. The securities in which the fund invests increase in value by 12% during the year. The fund’s expense ratio is 1.2%. What is your rate of return on the fund if you sell your shares at the end of the year? 19. Loaded-Up Fund charges a 12b-1 fee of 1.0% and maintains an expense ratio of .75%. Economy Fund charges a front-end load of 2% but has no 12b-1 fee and an expense ratio of .25%. Assume the rate of return on both funds’ portfolios (before any fees) is 6% per year. How much will an investment in each fund grow to after: a. 1 year. b. 3 years. c. 10 years. 20. City Street Fund has a portfolio of $450 million and liabilities of $10 million. a. If 44 million shares are outstanding, what is net asset value? b. If a large investor redeems 1 million shares, what happens to the portfolio value, to shares outstanding, and to NAV? 21. The Investments Fund sells Class A shares with a front-end load of 6% and Class B shares with 12b-1 fees of .5% annually as well as back-end load fees that start at 5% and fall by 1% for each full year the investor holds the portfolio (until the fifth year). Assume the portfolio rate of return

CHAPTER 4

Mutual Funds and Other Investment Companies

115

net of operating expenses is 10% annually. If you plan to sell the fund after 4 years, are Class A or Class B shares the better choice for you? What if you plan to sell after 15 years? 22. You are considering an investment in a mutual fund with a 4% load and expense ratio of .5%. You can invest instead in a bank CD paying 6% interest. a. If you plan to invest for 2 years, what annual rate of return must the fund portfolio earn for you to be better off in the fund than in the CD? Assume annual compounding of returns. b. How does your answer change if you plan to invest for 6 years? Why does your answer change? c. Now suppose that instead of a front-end load the fund assesses a 12b-1 fee of .75% per year. What annual rate of return must the fund portfolio earn for you to be better off in the fund than in the CD? Does your answer in this case depend on your time horizon?

24. You expect a tax-free municipal bond portfolio to provide a rate of return of 4%. Management fees of the fund are .6%. What fraction of portfolio income is given up to fees? If the management fees for an equity fund also are .6%, but you expect a portfolio return of 12%, what fraction of portfolio income is given up to fees? Why might management fees be a bigger factor in your investment decision for bond funds than for stock funds? Can your conclusion help explain why unmanaged unit investment trusts tend to focus on the fixed-income market? 25. Suppose you observe the investment performance of 350 portfolio managers for 5 years and rank them by investment returns during each year. After 5 years, you find that 11 of the funds have investment returns that place the fund in the top half of the sample in each and every year of your sample. Such consistency of performance indicates to you that these must be the funds whose managers are in fact skilled, and you invest your money in these funds. Is your conclusion warranted?

E-INVESTMENTS EXERCISES Go to www.morningstar.com. In the Morningstar Tools section, click on the link for the Mutual Fund Screener. Set the criteria you desire, then click on the Show Results tab. If you get no funds that meet all of your criteria, choose the criterion that is least important to you and relax that constraint. Continue the process until you have several funds to compare. 1. Examine all of the views available in the drop-down box menu (Snapshot, Performance, Portfolio, and Nuts and Bolts) to answer the following questions: a. Which fund has the best expense ratio? b. Which funds have the lowest Morningstar Risk rating? c. Which fund has the best 3-year return? Which has the best 10-year return? d. Which fund has the lowest turnover ratio? Which has the highest? e. Which fund has the longest manager tenure? Which has the shortest? f. Do you need to eliminate any of the funds from consideration due to a minimum initial investment that is higher than you are capable of making? 2. Based on what you know about the funds, which one do you think would be the best one for your investment? 3. Select up to five funds that are of the most interest to you. Click on the button that says Score These Results. Customize the criteria listed by indicating their importance to you. Examine the score results. Does the fund with the highest score match the choice you made in part 2?

Challenge

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23. Suppose that every time a fund manager trades stock, transaction costs such as commissions and bid–ask spreads amount to .4% of the value of the trade. If the portfolio turnover rate is 50%, by how much is the total return of the portfolio reduced by trading costs?

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Introduction

SOLUTIONS TO CONCEPT CHECKS $2,877.06 2 $14.73 5 $29.97 95.50 2. The net investment in the Class A shares after the 4% commission is $9,600. If the fund earns a 10% return, the investment will grow after n years to $9,6003(1.10)n. The Class B shares have no front-end load. However, the net return to the investor after 12b-1 fees will be only 9.5%. In addition, there is a back-end load that reduces the sales proceeds by a percentage equal to (5 – years until sale) until the fifth year, when the back-end load expires.

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1. NAV 5

Class A Shares

Class B Shares

Horizon

$9,6003(1.10)n

$10,0003(1.095)n3(12percentageexitfee)

1 year 4 years 10 years

$10,560 $14,055 $24,900

$10,0003(1.095)3(12.04) 5 $10,512 $10,0003(1.095)43(12.01) 5 $14,233 $10,0003(1.095)10 5 $24,782

For a very short horizon such as 1 year, the Class A shares are the better choice. The front-end and back-end loads are equal, but the Class A shares don’t have to pay the 12b-1 fees. For moderate horizons such as 4 years, the Class B shares dominate because the front-end load of the Class A shares is more costly than the 12b-1 fees and the now-smaller exit fee. For long horizons of 10 years or more, Class A again dominates. In this case, the one-time front-end load is less expensive than the continuing 12b-1 fees. 3. a. Turnover5$160,000 in trades per $1 million of portfolio value516%. b. Realized capital gains are $1031,0005$10,000 on FedEx and $2.5034,0005$10,000 on Cisco. The tax owed on the capital gains is therefore .203$20,0005$4,000. 4. Twenty percent of the managers are skilled, which accounts for .23400580 of those managers who appear in the top half. There are 120 slots left in the top half, and 320 other managers, so the probability of an unskilled manager “lucking into” the top half in any year is 120/320, or .375. Therefore, of the 120 lucky managers in the first year, we would expect .375 3120 545 to repeat as top-half performers next year. Thus, we should expect a total of 80 145 5125, or 62.5%, of the better initial performers to repeat their top-half performance.

CHAPTER FIVE

Risk, Return, and the Historical Record

1

is even more difficult than forecasting the past.) Moreover, in learning from a historical record we face what has become known as the “black swan” problem.1 No matter how long a historical record, there is never a guarantee that it exhibits the worst (and best) that nature can throw at us in the future. This problem is particularly daunting when considering the risk of long-run investments. In this chapter, we present the essential tools for estimating expected returns and risk from the historical record and consider implications for future investments. We begin with interest rates and investments in safe assets and examine the history of risk-free investments in the U.S over the last 86 years. Moving to risky assets, we begin with scenario analysis of risky investments and the data inputs necessary to conduct it. With this in mind, we develop statistical tools needed to make inferences from historical time series of portfolio returns. We present a global view of the history of stock and bond returns worldwide. We end with implications of the historical record for future investments and risk measures commonly used in the industry.

Black swans are a metaphor for highly improbable—but highly impactful—events. Until the discovery of Australia, Europeans, having observed only white swans, believed that a black swan was outside the realm of reasonable possibility or, in statistical jargon, an extreme “outlier” relative to their “sample” of observations.

PART II

CASUAL OBSERVATION ANDformal research both suggest that investment risk is as important to investors as expected return. While we have theories about the relationship between risk and expected return that would prevail in rational capital markets, there is no theory about the levels of risk we should find in the marketplace. We can at best estimate the level of risk likely to confront investors from historical experience. This situation is to be expected because prices of investment assets fluctuate in response to news about the fortunes of corporations, as well as to macroeconomic developments. There is no theory about the frequency and importance of such events; hence we cannot determine a “natural” level of risk. Compounding this difficulty is the fact that neither expected returns nor risk are directly observable. We observe only realized rates of return. Hence, to make forecasts about future expected returns and risk, we must learn how to “forecast” their past values, that is, the expected returns and risk that investors actually anticipated, from historical data. (There is an old saying that forecasting the future

5

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5.1

PART II

Portfolio Theory and Practice

Determinants of the Level of Interest Rates Interest rates and forecasts of their future values are among the most important inputs into an investment decision. For example, suppose you have $10,000 in a savings account. The bank pays you a variable interest rate tied to some short-term reference rate such as the 30-day Treasury bill rate. You have the option of moving some or all of your money into a longer-term certificate of deposit that offers a fixed rate over the term of the deposit. Your decision depends critically on your outlook for interest rates. If you think rates will fall, you will want to lock in the current higher rates by investing in a relatively longterm CD. If you expect rates to rise, you will want to postpone committing any funds to long-term CDs. Forecasting interest rates is one of the most notoriously difficult parts of applied macroeconomics. Nonetheless, we do have a good understanding of the fundamental factors that determine the level of interest rates: 1. The supply of funds from savers, primarily households. 2. The demand for funds from businesses to be used to finance investments in plant, equipment, and inventories (real assets or capital formation). 3. The government’s net demand for funds as modified by actions of the Federal Reserve Bank. Before we elaborate on these forces and resultant interest rates, we need to distinguish real from nominal interest rates.

Real and Nominal Rates of Interest An interest rate is a promised rate of return denominated in some unit of account (dollars, yen, euros, or even purchasing power units) over some time period (a month, a year, 20 years, or longer). Thus, when we say the interest rate is 5%, we must specify both the unit of account and the time period. Assuming there is no default risk, we can refer to the promised rate of interest as a risk-free rate for that particular unit of account and time period. But if an interest rate is risk-free for one unit of account and time period, it will not be risk-free for other units or periods. For example, interest rates that are absolutely safe in dollar terms will be risky when evaluated in terms of purchasing power because of inflation uncertainty. To illustrate, consider a 1-year dollar (nominal) risk-free interest rate. Suppose exactly 1 year ago you deposited $1,000 in a 1-year time deposit guaranteeing a rate of interest of 10%. You are about to collect $1,100 in cash. What is the real return on your investment? That depends on what money can buy these days, relative to what you could buy a year ago. The consumer price index (CPI) measures purchasing power by averaging the prices of goods and services in the consumption basket of an average urban family of four. Suppose the rate of inflation (the percent change in the CPI, denoted by i) for the last year amounted to i56%. This tells you that the purchasing power of money is reduced by 6% a year. The value of each dollar depreciates by 6% a year in terms of the goods it can buy. Therefore, part of your interest earnings are offset by the reduction in the purchasing power of the dollars you will receive at the end of the year. With a 10% interest rate, after you net out the 6% reduction in the purchasing power of money, you are left with a net increase in purchasing power of about 4%. Thus we need to distinguish between a nominal interest rate—the growth rate of your money—and a real interest rate—the growth rate

CHAPTER 5

Risk, Return, and the Historical Record

of your purchasing power. If we call rn the nominal rate, rr the real rate, and i the inflation rate, then we conclude rr < rn 2 i

(5.1)

In words, the real rate of interest is the nominal rate reduced by the loss of purchasing power resulting from inflation. In fact, the exact relationship between the real and nominal interest rate is given by 1 1 rr 5

1 1 rn 11i

(5.2)

This is because the growth factor of your purchasing power, 1 1 rr, equals the growth factor of your money, 11rn, divided by the growth factor of prices, 11i. The exact relationship can be rearranged to rr 5

rn 2 i 11i

(5.3)

which shows that the approximation rule (Equation 5.1) overstates the real rate by the factor 11i.

Example 5.1

Approximating the Real Rate

If the nominal interest rate on a 1-year CD is 8%, and you expect inflation to be 5% over the coming year, then using the approximation formula, you expect the real rate of interest to be rr 5 8% 2 5% 5 3%. Using the exact formula, the real rate .08 2 .05 is rr 5 5 .0286, or 2.86%. Therefore, the approximation rule overstates the 1 1 .05 expected real rate by .14% (14 basis points). The approximation rule is more exact for small inflation rates and is perfectly exact for continuously compounded rates. We discuss further details in the next section.

Notice that conventional certificates of deposit offer a guaranteed nominal rate of interest. Thus you can only infer the expected real rate on these investments by adjusting for your expectation of the rate of inflation. It is always possible to calculate the real rate after the fact. The inflation rate is published by the Bureau of Labor Statistics (BLS). The future real rate, however, is unknown, and one has to rely on expectations. In other words, because future inflation is risky, the real rate of return is risky even when the nominal rate is risk-free.2

The Equilibrium Real Rate of Interest Three basic factors—supply, demand, and government actions—determine the real interest rate. The nominal interest rate is the real rate plus the expected rate of inflation. And thus a fourth factor affecting the nominal interest rate is expected inflation. Although there are many different interest rates economywide (as many as there are types of debt securities), these rates tend to move together, so economists frequently talk as if 2

You can find the real rate for a desired maturity from inflation-indexed bonds issued by the U.S. Treasury, called TIPS. (See Chapter 14 for a fuller discussion.) The difference between TIPS yields and yields on comparable nominal Treasury bonds provides an estimate of the market’s expectation of future inflation.

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there were a single representative rate. We can use this Real Interest Rate abstraction to gain some Supply insights into the real rate of interest if we consider the E' supply and demand curves for funds. Equilibrium Figure 5.1 shows a E Real Rate of Interest downward-sloping demand curve and an upwardDemand sloping supply curve. On the horizontal axis, we measure the quantity of funds, Funds and on the vertical axis, Equilibrium Funds Lent we measure the real rate of interest. The supply curve slopes Figure 5.1 Determination of the equilibrium real rate of interest up from left to right because the higher the real interest rate, the greater the supply of household savings. The assumption is that at higher real interest rates households will choose to postpone some current consumption and set aside or invest more of their disposable income for future use.3 The demand curve slopes down from left to right because the lower the real interest rate, the more businesses will want to invest in physical capital. Assuming that businesses rank projects by the expected real return on invested capital, firms will undertake more projects the lower the real interest rate on the funds needed to finance those projects. Equilibrium is at the point of intersection of the supply and demand curves, point E in Figure5.1. The government and the central bank (the Federal Reserve) can shift these supply and demand curves either to the right or to the left through fiscal and monetary policies. For example, consider an increase in the government’s budget deficit. This increases the government’s borrowing demand and shifts the demand curve to the right, which causes the equilibrium real interest rate to rise to point E9. That is, a forecast that indicates higher than previously expected government borrowing increases expected future interest rates. The Fed can offset such a rise through an expansionary monetary policy, which will shift the supply curve to the right. Thus, although the fundamental determinants of the real interest rate are the propensity of households to save and the expected profitability of investment in physical capital, the real rate can be affected as well by government fiscal and monetary policies.

The Equilibrium Nominal Rate of Interest We’ve seen that the nominal rate of return on an asset is approximately equal to the real rate plus inflation. Because investors should be concerned with real returns—the increase in purchasing power—we would expect higher nominal interest rates when inflation is 3

Experts disagree significantly on the extent to which household saving increases in response to an increase in the real interest rate.

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higher. This higher nominal rate is necessary to maintain the expected real return offered by an investment. Irving Fisher (1930) argued that the nominal rate ought to increase one-for-one with expected inflation, E(i ). The so-called Fisher equation is rn 5 rr 1 E(i)

(5.4)

The equation implies that when real rates are reasonably stable, changes in nominal rates ought to predict changes in inflation rates. This claim has been debated and empirically investigated with mixed results. Although the data do not strongly support the Fisher equation, nominal interest rates seem to predict inflation as well as alternative methods, in part because we are unable to forecast inflation well with any method. It is difficult to determine the empirical validity of the Fisher hypothesis because real rates also change unpredictably over time. Nominal interest rates can be viewed as the sum of the required real rate on nominally risk-free assets, plus a “noisy” forecast of inflation. In Chapter 15 we discuss the relationship between short- and long-term interest rates. Longer rates incorporate forecasts for long-term inflation. For this reason alone, interest rates on bonds of different maturity may diverge. In addition, we will see that prices of longer-term bonds are more volatile than those of short-term bonds. This implies that expected returns on longer-term bonds may include a risk premium, so that the expected real rate offered by bonds of varying maturity also may vary. CONCEPT CHECK

5.1

a. Suppose the real interest rate is 3% per year and the expected inflation rate is 8%. According to the Fisher hypothesis, what is the nominal interest rate? b. Suppose the expected inflation rate rises to 10%, but the real rate is unchanged. What happens to the nominal interest rate?

Taxes and the Real Rate of Interest Tax liabilities are based on nominal income and the tax rate determined by the investor’s tax bracket. Congress recognized the resultant “bracket creep” (when nominal income grows due to inflation and pushes taxpayers into higher brackets) and mandated indexlinked tax brackets in the Tax Reform Act of 1986. Index-linked tax brackets do not provide relief from the effect of inflation on the taxation of savings, however. Given a tax rate (t) and a nominal interest rate, rn, the after-tax interest rate is rn(12t). The real after-tax rate is approximately the after-tax nominal rate minus the inflation rate: rn(1 2 t) 2 i 5 (rr 1 i)(1 2 t) 2 i 5 rr (1 2 t) 2 it

(5.5)

Thus the after-tax real rate falls as inflation rises. Investors suffer an inflation penalty equal to the tax rate times the inflation rate. If, for example, you are in a 30% tax bracket and your investments yield 12%, while inflation runs at the rate of 8%, then your beforetax real rate is approximately 4%, and you should, in an inflation-protected tax system, net after taxes a real return of 4%(1 2.3) 52.8%. But the tax code does not recognize that the first 8% of your return is just compensation for inflation—not real income—and hence your after-tax return is reduced by 8%3.352.4%, so that your after-tax real interest rate, at .4%, is almost wiped out.

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Comparing Rates of Return for Different Holding Periods Consider an investor who seeks a safe investment, say, in U.S. Treasury securities.4 We observe zero-coupon Treasury securities with several different maturities. Zero-coupon bonds, discussed more fully in Chapter 14, are sold at a discount from par value and provide their entire return from the difference between the purchase price and the ultimate repayment of par value.5 Given the price, P(T), of a Treasury bond with $100 par value and maturity of T years, we calculate the total risk-free return available for a horizon of T years as the percentage increase in the value of the investment. rf ( T ) 5

100 21 P(T )

(5.6)

For T51, Equation 5.6 provides the risk-free rate for an investment horizon of 1 year.

Example 5.2

Annualized Rates of Return

Suppose prices of zero-coupon Treasuries with $100 face value and various maturities are as follows. We find the total return of each security by using Equation 5.6:

Horizon, T Half-year 1 year 25 years

Price, P(T )

[100/P(T )] 21

$97.36 $95.52 $23.30

100/97.36215 .0271 100/95.52215 .0469 100/23.302153.2918

Risk-Free Return for Given Horizon rf (.5) 52.71% rf (1) 54.69% rf (25)5329.18%

Not surprisingly, longer horizons in Example 5.2 provide greater total returns. How should we compare returns on investments with differing horizons? This requires that we express each total return as a rate of return for a common period. We typically express all investment returns as an effective annual rate (EAR), defined as the percentage increase in funds invested over a 1-year horizon. For a 1-year investment, the EAR equals the total return, rf (1), and the gross return, (1 1 EAR), is the terminal value of a $1 investment. For investments that last less than 1 year, we compound the per-period return for a full year. For the 6-month bill in Example5.2, we compound 2.71% half-year returns over two semiannual periods to obtain a terminal value of 11EAR5(1.0271)251.0549, implying that EAR55.49%. For investments longer than a year, the convention is to express the EAR as the annual rate that would compound to the same value as the actual investment. For example, the

4

Yields on Treasury bills and bonds of various maturities are widely available on the Web, for example at Yahoo! Finance, MSN Money, or directly from the Federal Reserve. 5 The U.S. Treasury issues T-bills, which are pure discount (or zero-coupon) securities with maturities of up to 1 year. However, financial institutions create zero-coupon Treasury bonds called Treasury strips with maturities up to 30 years by buying coupon-paying T-bonds, “stripping” off the coupon payments, and selling claims to the coupon payments and final payment of face value separately. See Chapter 14 for further details.

CHAPTER 5

Risk, Return, and the Historical Record

investment in the 25-year bond in Example 5.2 grows by its maturity by a factor of 4.2918 (i.e., 113.2918), so its EAR is found by solving (1 1 EAR)25 5 4.2918 1 1 EAR 5 4.29181/25 5 1.0600 In general, we can relate EAR to the total return, rf (T), over a holding period of length T by using the following equation: 1 1 EAR 5 3 1 1 rf (T ) 4 1/T

(5.7)

We illustrate with an example.

Example 5.3

Effective Annual Rate versus Total Return

For the 6-month Treasury in Example 5.2, T5½, and 1/T52. Therefore, 1 1 EAR 5 (1.0271)2 5 1.0549 and EAR 5 5.49% For the 25-year Treasury in Example 5.2, T525. Therefore, 1 1 EAR 5 4.29181/25 5 1.060 and EAR 5 6.0%

Annual Percentage Rates Annualized rates on short-term investments (by convention, T,1year) often are reported using simple rather than compound interest. These are called annual percentage rates, or APRs. For example, the APR corresponding to a monthly rate such as that charged on a credit card is reported as 12 times the monthly rate. More generally, if there are n compounding periods in a year, and the per-period rate is rf (T ), then the APR5n3rf (T ). Conversely, you can find the per-period rate from the APR as rf (T )5T3APR. Using this procedure, the APR of the 6-month bond in Example 5.2 with a 6-month rate of 2.71% is 2 32.71 55.42%. To generalize, note that for short-term investments of length T, there are n 51/T compounding periods in a year. Therefore, the relationship among the compounding period, the EAR, and the APR is 1 1 EAR 5 3 1 1 rf (T )4 n 5 3 1 1 rf (T )4 1/T 5 3 1 1 T 3 APR 4 1/T Equivalently, APR 5

Example 5.4

(5.8)

(1 1 EAR)T 2 1 T

EAR versus APR

In Table5.1 we use Equation 5.8 to find the APR corresponding to an EAR of 5.8% with various compounding periods. Conversely, we find values of EAR implied by an APR of 5.8%.

Continuous Compounding It is evident from Table5.1 (and Equation 5.8) that the difference between APR and EAR grows with the frequency of compounding. This raises the question: How far will these

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EAR5[11rf (T )]1/T215.058

Compounding Period 1 year 6 months 1 quarter 1 month 1 week 1 day Continuous

APR5rf (T)*(1/T)5.058

T

rf (T)

APR5[(11EAR)^T21]/T

rf(T)

EAR5(11APR*T)^(1/T)21

1.0000 0.5000 0.2500 0.0833 0.0192 0.0027

.0580 .0286 .0142 .0047 .0011 .0002

.05800 .05718 .05678 .05651 .05641 .05638 rcc5ln(11EAR)5.05638

.0580 .0290 .0145 .0048 .0011 .0002

.05800 .05884 .05927 .05957 .05968 .05971 EAR5exp(rcc )215.05971

Table 5.1 Annual percentage rates (APR) and effective annual rates (EAR). In the first set of columns, we hold the equivalent annual rate (EAR) fixed at 5.8% and find APR for each holding period. In the second set of columns, we hold APR fixed at 5.8% and solve for EAR.

eXcel Please visit us at www.mhhe.com/bkm

two rates diverge as the compounding frequency continues to grow? Put differently, what is the limit of [11T3APR]1/ T, as T gets ever smaller? As T approaches zero, we effectively approach continuous compounding (CC), and the relation of EAR to the annual percentage rate, denoted by rcc for the continuously compounded case, is given by the exponential function 1 1 EAR 5 exp(rcc) 5 e rcc

(5.9)

where e is approximately 2.71828. To find rcc from the effective annual rate, we solve Equation 5.9 for rcc as follows: ln(1 1 EAR) 5 rcc where ln(•) is the natural logarithm function, the inverse of exp(•). Both the exponential and logarithmic functions are available in Excel, and are called EXP(•) and LN(•), respectively.

Example 5.5

Continuously Compounded Rates

The continuously compounded annual percentage rate, rcc, that provides an EAR of 5.8% is 5.638% (see Table5.1). This is virtually the same as the APR for daily compounding. But for less frequent compounding, for example, semiannually, the APR necessary to provide the same EAR is noticeably higher, 5.718%. With less frequent compounding, a higher APR is necessary to provide an equivalent effective return.

While continuous compounding may at first seem to be a mathematical nuisance, working with such rates can actually simplify calculations of expected return and risk. For example, given a continuously compounded rate, the total return for any period T, rcc(T), is simply exp(T 3 rcc).6 In other words, the total return scales up in direct proportion to the time period, T. This is far simpler than working with the exponents that arise using discrete This follows from Equation 5.9. If 1 1 EAR 5 ercc, then (1 1 EAR)T 5 erccT.

6

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Risk, Return, and the Historical Record

125

period compounding. As another example, look again at Equation 5.1. There, the relationship between the real rate rr, the nominal rate rn, and the inflation rate i, rr 1) = .27 the borrowing rate is r Bf 5 9%. Then E(rP) = 15% in the borrowing range, the reward-tovolatility ratio, the slope of the CAL, will rƒB = 9% S(y ≤ 1) = .36 B rƒ = 7% be 3 E(rP) 2 rf 4 /sP 5 6/22 5 .27. The CAL will therefore be “kinked” at point P, as shown in Figure 6.5. To the left σ of P the investor is lending at 7%, and the σP = 22% slope of the CAL is .36. To the right of P, where y.1, the investor is borrowing at 9% to finance extra investments in the Figure 6.5 The opportunity set with differential borrowing risky asset, and the slope is .27. and lending rates In practice, borrowing to invest in the risky portfolio is easy and straightforward if you have a margin account with a broker. All you have to do is tell your broker that you want to buy “on margin.” Margin purchases may not exceed 50% of the purchase value. Therefore, if your net worth in the account is $300,000, the broker is allowed to lend you up to $300,000 to purchase additional stock.4 You would then have $600,000 on the asset side of your account and $300,000 on the liability side, resulting in y52.0.

CONCEPT CHECK

6.6

Suppose that there is an upward shift in the expected rate of return on the risky asset, from 15% to 17%. If all other parameters remain unchanged, what will be the slope of the CAL for y≤1 and y.1?

4

Margin purchases require the investor to maintain the securities in a margin account with the broker. If the value of the securities falls below a “maintenance margin,” a “margin call” is sent out, requiring a deposit to bring the net worth of the account up to the appropriate level. If the margin call is not met, regulations mandate that some or all of the securities be sold by the broker and the proceeds used to reestablish the required margin. See Chapter3, Section 3.6, for further discussion.

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Risk Tolerance and Asset Allocation We have shown how to develop the CAL, the graph of all feasible risk–return combinations available for capital allocation. The investor confronting the CAL now must choose one optimal portfolio, C, from the set of feasible choices. This choice entails a trade-off between risk and return. Individual differences in risk aversion lead to different capital allocation choices even when facing an identical opportunity set (that is, a risk-free rate and a reward-to-volatility ratio). In particular, more risk-averse investors will choose to hold less of the risky asset and more of the risk-free asset. The expected return on the complete portfolio is given by Equation 6.3: E(rC) 5 rf 1 y[E(rP) 2 rf]. Its variance is, from Equation 6.4, s2C 5 y2s2P. Investors choose the allocation to the risky asset, y, that maximizes their utility function as given by Equation 6.1: U5 E(r) 2½ As2. As the allocation to the risky asset increases (higher y), expected return increases, but so does volatility, so utility can increase or decrease. Table 6.4 shows utility levels corresponding to different values of y. Initially, utility increases as y increases, but eventually it declines. Figure6.6 is a plot of the utility function from Table6.4. The graph shows that utility is highest at y5.41. When y is less than .41, investors are willing to assume more risk to increase expected return. But at higher levels of y, risk is higher, and additional allocations to the risky asset are undesirable—beyond this point, further increases in risk dominate the increase in expected return and reduce utility. To solve the utility maximization problem more generally, we write the problem as follows: U 5 E(rC) 2 ½ As2C 5 rf 1 y 3 E(rP) 2 rf 4 2 ½ Ay2s2P Max y Students of calculus will recognize that the maximization problem is solved by setting the derivative of this expression to zero. Doing so and solving for y yields the optimal position for risk-averse investors in the risky asset, y*, as follows:5 E(rP) 2 rf y* 5 (6.7) As2P Table 6.4 Utility levels for various positions in risky assets (y) for an investor with risk aversion A54

5

(1) y

(2) E(rC)

(3) sC

(4) U 5 E(r) 2 ½As2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

.070 .078 .086 .094 .102 .110 .118 .126 .134 .142 .150

0 .022 .044 .066 .088 .110 .132 .154 .176 .198 .220

.0700 .0770 .0821 .0853 .0865 .0858 .0832 .0786 .0720 .0636 .0532

The derivative with respect to y equals E(rP) 2 rf 2 yAs2P. Setting this expression equal to zero and solving for y yields Equation 6.7.

CHAPTER 6

Capital Allocation to Risky Assets

.10 .09 .08

Utility

.07 .06 .05 .04 .03 .02 .01 0 0

0.2

0.4 0.6 0.8 Allocation to Risky Asset, y

1

1.2

Figure 6.6 Utility as a function of allocation to the risky asset, y

This solution shows that the optimal position in the risky asset is inversely proportional to the level of risk aversion and the level of risk (as measured by the variance) and directly proportional to the risk premium offered by the risky asset.

Example 6.4

Capital Allocation

Using our numerical example [rf57%,E(rP)515%, and sP522%], and expressing all returns as decimals, the optimal solution for an investor with a coefficient of risk aversion A54 is y* 5

.15 2 .07 5 .41 4 3 .222

In other words, this particular investor will invest 41% of the investment budget in the risky asset and 59% in the risk-free asset. As we saw in Figure6.6, this is the value of y at which utility is maximized. With 41% invested in the risky portfolio, the expected return and standard deviation of the complete portfolio are E(rC) 5 7 1 3 .41 3 (15 2 7) 4 5 10.28% sC 5 .41 3 22 5 9.02% The risk premium of the complete portfolio is E(rC)2rf53.28%, which is obtained by taking on a portfolio with a standard deviation of 9.02%. Notice that 3.28/9.02=.36, which is the reward-to-volatility (Sharpe) ratio of any complete portfolio given the parameters of this example.

A graphical way of presenting this decision problem is to use indifference curve analysis. To illustrate how to build an indifference curve, consider an investor with risk aversion A 54 who currently holds all her wealth in a risk-free portfolio yielding

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rf55%. Because the variance of such a portfolio is zero, Equation 6.1 tells us that its utility value is U5.05. Now we find the expected return the investor would require to maintain the same level of utility when holding a risky portfolio, say, with s51%. We use Equation 6.1 to find how much E(r) must increase to compensate for the higher value of s: U 5 E(r) 2 ½ 3 A 3 s2 .05 5 E(r) 2 ½ 3 4 3 .012 This implies that the necessary expected return increases to Required E(r) 5 .05 1 ½ 3 A 3 s2 5 .05 1 ½ 3 4 3 .012 5 .0502

(6.8)

We can repeat this calculation for other levels of s, each time finding the value of E(r) necessary to maintain U5.05. This process will yield all combinations of expected return and volatility with utility level of .05; plotting these combinations gives us the indifference curve. We can readily generate an investor’s indifference curves using a spreadsheet. Table6.5 contains risk–return combinations with utility values of .05 and .09 for two investors, one with A 5 2 and the other with A 5 4. The plot of these indifference curves appears in Figure6.7. Notice that the intercepts of the indifference curves are at .05 and .09, exactly the level of utility corresponding to the two curves. Any investor would prefer a portfolio on the higher indifference curve with a higher certainty equivalent (utility). Portfolios on higher indifference curves offer a higher expected return for any given level of risk. For example, both indifference curves for A52 have the same shape, but for any level of volatility, a portfolio on the curve with utility of .09 offers an expected return 4% greater than the corresponding portfolio on the lower curve, for which U5.05. Figure 6.7 demonstrates that more risk-averse investors have steeper indifference curves than less risk-averse investors. Steeper curves mean that investors require a greater increase in expected return to compensate for an increase in portfolio risk.

Table 6.5 Spreadsheet calculations of indifference curves (Entries in columns 2–4 are expected returns necessary to provide specified utility value.)

A52 s 0 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50

A54

U5.05

U5.09

U5.05

U5.09

.0500 .0525 .0600 .0725 .0900 .1125 .1400 .1725 .2100 .2525 .3000

.0900 .0925 .1000 .1125 .1300 .1525 .1800 .2125 .2500 .2925 .3400

.050 .055 .070 .095 .130 .175 .230 .295 .370 .455 .550

.090 .095 .110 .135 .170 .215 .270 .335 .410 .495 .590

CHAPTER 6 Higher indifference curves correspond to higher levels of utility. The investor thus attempts to find the complete portfolio on the highest possible indifference curve. When we superimpose plots of indifference curves on the investment opportunity set represented by the capital allocation line as in Figure 6.8, we can identify the highest possible indifference curve that still touches the CAL. That indifference curve is tangent to the CAL, and the tangency point corresponds to the standard deviation and expected return of the optimal complete portfolio. To illustrate, Table 6.6 provides calculations for four indifference curves (with utility levels of .07, .078, .08653, and .094) for an investor with A 5 4. Columns (2)–(5) use Equation 6.8 to calculate the expected return that must be paired

E(r)

E(r) .60

A=4 A=4

.40 A=2 A=2 .20

U = .09 U = .05 .10

.20

.30

and A54

CAL P C E(rc) = .1028 rf = .07

σ 0

σc = .0902

.40

.50

σ

Figure 6.7 Indifference curves for U5.05 and U5.09 with A52

U = .094 U = .08653 U = .078 U = .07

E(rP) = .15

185

Capital Allocation to Risky Assets

σP = .22

Figure 6.8 Finding the optimal complete portfolio by using indifference curves

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Table 6.6 Expected returns on four indifference curves and the CAL. Investor’s risk aversion is A54.

Portfolio Theory and Practice

s 0 .02 .04 .06 .08 .0902 .10 .12 .14 .18 .22 .26 .30

U5.07

U5.078

U5.08653

U5.094

CAL

.0700 .0708 .0732 .0772 .0828 .0863 .0900 .0988 .1092 .1348 .1668 .2052 .2500

.0780 .0788 .0812 .0852 .0908 .0943 .0980 .1068 .1172 .1428 .1748 .2132 .2580

.0865 .0873 .0897 .0937 .0993 .1028 .1065 .1153 .1257 .1513 .1833 .2217 .2665

.0940 .0948 .0972 .1012 .1068 .1103 .1140 .1228 .1332 .1588 .1908 .2292 .2740

.0700 .0773 .0845 .0918 .0991 .1028 .1064 .1136 .1209 .1355 .1500 .1645 .1791

with the standard deviation in column (1) to provide the utility value corresponding to each curve. Column (6) uses Equation 6.5 to calculate E(rC) on the CAL for the standard deviation sC in column (1): sC sC E(rC) 5 rf 1 3 E(rP) 2 rf 4 5 7 1 3 15 2 7 4 sP 22 Figure6.8 graphs the four indifference curves and the CAL. The graph reveals that the indifference curve with U5.08653 is tangent to the CAL; the tangency point corresponds to the complete portfolio that maximizes utility. The tangency point occurs at sC59.02% and E(rC)510.28%, the risk–return parameters of the optimal complete portfolio with y*50.41. These values match our algebraic solution using Equation 6.7. We conclude that the choice for y*, the fraction of overall investment funds to place in the risky portfolio, is determined by risk aversion (the slope of indifference curves) and the Sharpe ratio (the slope of the opportunity set). In sum, capital allocation determines the complete portfolio, which constitutes the investor’s entire wealth. Portfolio P represents all-wealth-at-risk. Hence, when returns are normally distributed, standard deviation is the appropriate measure of risk. In future chapters we will consider augmenting P with “good” additions, meaning assets that improve the feasible risk-return trade-off. The risk of these potential additions will have to be measured by their incremental effect on the standard deviation of P.

Nonnormal Returns In the foregoing analysis we assumed normality of returns by taking the standard deviation as the appropriate measure of risk. But as we discussed in Chapter 5, departures from normality could result in extreme losses with far greater likelihood than would be plausible under a normal distribution. These exposures, which are typically measured by value at risk (VaR) or expected shortfall (ES), also would be important to investors. Therefore, an appropriate extension of our analysis would be to present investors with forecasts of VaR and ES. Taking the capital allocation from the normal-based analysis as a benchmark, investors facing fat-tailed distributions might consider reducing their allocation to the risky portfolio in favor of an increase in the allocation to the risk-free vehicle.

CHAPTER 6

Capital Allocation to Risky Assets

187

There are signs of advances in dealing with extreme values (in addition to new techniques to handle transaction data mentioned in Chapter 5). Back in the early 20th century, Frank Knight, one of the great economists of the time, distinguished risk from uncertainty, the difference being that risk is a known problem in which probabilities can be ascertained while uncertainty is characterized by ignorance even about probabilities (reminiscent of the black swan problem). Hence, Knight argued, we must use different methods to handle uncertainty and risk. Probabilities of moderate outcomes in finance can be readily assessed from experience because of the high relative frequency of such observations. Extreme negative values are blissfully rare, but for that very reason, accurately assessing their probabilities is virtually impossible. However, the Bayesian statistics that took center stage in decision making in later periods rejected Knight’s approach on the argument that even if probabilities are hard to estimate objectively, investors nevertheless have a notion, albeit subjective, of what they may be and must use those beliefs to make economic decisions. In the Bayesian framework, these so-called priors must be used even if they apply to unprecedented events that characterize uncertainty. Accordingly, in this school of thought, the distinction between risk and uncertainty is deemed irrelevant. Economists today are coming around to Knight’s position. Advanced utility functions attempt to distinguish risk from uncertainty and give these uncertain outcomes a larger role in the choice of portfolios. These approaches have yet to enter everyday practice, but as they are developed, practical measures are certain to follow.

CONCEPT CHECK

6.7

a. If an investor’s coefficient of risk aversion is A53, how does the optimal asset mix change? What are the new values of E(rC) and sC? b. Suppose that the borrowing rate, r fB 5 9% is greater than the lending rate, rf57%. Show graphically how the optimal portfolio choice of some investors will be affected by the higher borrowing rate. Which investors will not be affected by the borrowing rate?

6.6

Passive Strategies: The Capital Market Line

The CAL is derived with the risk-free and “the” risky portfolio, P. Determination of the assets to include in P may result from a passive or an active strategy. A passive strategy describes a portfolio decision that avoids any direct or indirect security analysis.6 At first blush, a passive strategy would appear to be naive. As will become apparent, however, forces of supply and demand in large capital markets may make such a strategy the reasonable choice for many investors. In Chapter 5, we presented a compilation of the history of rates of return on different portfolios. The data are available at Professor Kenneth French’s Web site, mba.tuck .dartmouth.edu/pages/faculty/ken.french/data_library.html. We can use these data to examine various passive strategies. A natural candidate for a passively held risky asset would be a well-diversified portfolio of common stocks such as “All U.S.” described in Chapter 5. Because a passive strategy 6

By “indirect security analysis” we mean the delegation of that responsibility to an intermediary such as a professional money manager.

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requires that we devote no resources to acquiring information on any individual stock or group of stocks, we must follow a “neutral” diversification strategy. One way is to select a diversified portfolio of stocks that mirrors the value of the corporate sector of the U.S. economy. This results in a portfolio in which, for example, the proportion invested in Microsoft stock will be the ratio of Microsoft’s total market value to the market value of all listed stocks. The most popular value-weighted index of U.S. stocks is the Standard & Poor’s Composite Index of 500 large capitalization U.S. corporations (the S&P 500). Table 6.7 summarizes the performance of the S&P 500 portfolio over the 87-year period 1926–2012, as well as for four subperiods. Table 6.7 shows the average return for the portfolio, the return on rolling over 1-month T-bills for the same period, as well as the resultant average excess return and its standard deviation. The Sharpe ratio was .40 for the overall period, 1926–2012. In other words, stock market investors enjoyed a .40% average excess return over the T-bill rate for every 1% of standard deviation. The large standard deviation of the excess return (20.48%) is one reason we observe a wide range of average excess returns and Sharpe ratios across subperiods (varying from .21 to .74). Using the statistical distribution of the difference between the Sharpe ratios of two portfolios, we can estimate the probability of observing a deviation of the Sharpe ratio for a particular subperiod from that of the overall period, assuming the latter is the true value. The last column of Table 6.7 shows that the probabilities of finding such widely different Sharpe ratios over the subperiods are actually quite substantial. We call the capital allocation line provided by 1-month T-bills and a broad index of common stocks the capital market line (CML). A passive strategy generates an investment opportunity set that is represented by the CML. How reasonable is it for an investor to pursue a passive strategy? We cannot answer such a question without comparing the strategy to the costs and benefits accruing to an active portfolio strategy. Some thoughts are relevant at this point, however. First, the alternative active strategy is not free. Whether you choose to invest the time and cost to acquire the information needed to generate an optimal active portfolio of risky assets, or whether you delegate the task to a professional who will charge a fee, constitution of an active portfolio is more expensive than a passive one. The passive portfolio requires negligible cost to purchase T-bills and management fees to either an exchange-traded fund

Average Annual Returns

S&P 500 Portfolio

Period

S&P 500 Portfolio

1-Month T-Bills

Risk Premium

Standard Deviation

Sharpe Ratio (Reward-toVolatility)

Probability*

1926–2012 1989–2012 1968–1988 1947–1967 1926–1946

11.67 11.10 10.91 15.35 9.40

3.58 3.52 7.48 2.28 1.04

8.10 7.59 3.44 13.08 8.36

20.48 18.22 16.71 17.66 27.95

0.40 0.42 0.21 0.74 0.30

— 0.94 0.50 0.24 0.71

Table 6.7 Average annual return on large stocks and 1-month T-bills; standard deviation and Sharpe ratio of large stocks over time *The probability that the estimate of the Sharpe ratio over 1926–2012 equals the true value and that we observe the reported, or an even more different Sharpe ratio for the subperiod.

Investors are jumping out of mutual funds managed by professional stock pickers and shifting massive amounts of money into lower-cost funds that echo the broader market. Through November 2012, investors pulled $119.3 billion from so-called actively managed U.S. stock funds according to the latest data from research firm Morningstar Inc. At the same time, they poured $30.4 billion into U.S. stock exchange-traded funds. The move reflects the fact that many money managers of stock funds, which charge fees but also dangle the prospect of higher returns, have underperformed the benchmark stock indexes. As a result, more investors are choosing simply to invest in funds tracking the indexes, which carry lower fees and are perceived as having less risk. The mission of stock pickers in a managed mutual fund is to outperform the overall market by actively trading individual stocks or bonds, with fund managers receiving higher fees for their effort. In an ETF (or indexed mutual fund), managers balance the share makeup of the fund so it accurately reflects the performance of its underlying index, charging lower fees.

Morningstar says that when investors have put money in stock funds, they have chosen low-cost index funds and ETFs. Some index ETFs cost less than 0.1% of assets a year, while many actively managed stock funds charge 1% a year or more. While the trend has put increasing pressure lately on stock pickers, it is shifting the fortunes of some of the biggest players in the $14 trillion mutual-fund industry. Fidelity Investments and American Funds, among the largest in the category, saw redemptions or weak investor interest compared with competitors, according to an analysis of mutual-fund flows done for The Wall Street Journal by research firm Strategic Insight, a unit of New York-based Asset International. At the other end of the spectrum, Vanguard, the world’s largest provider of index mutual funds, pulled in a net $141 billion last year through December, according to the company. Many investors say they are looking for a way to invest cheaply, with less risk. Source: Adapted from Kirsten Grind, The Wall Street Journal, January 3, 2013. Reprinted with permission. © 2013 Dow Jones & Company, Inc. All Rights Reserved Worldwide.

or a mutual fund company that operates a market index fund. Vanguard, for example, operates the Index 500 Portfolio that mimics the S&P 500 index fund. It purchases shares of the firms constituting the S&P 500 in proportion to the market values of the outstanding equity of each firm, and therefore essentially replicates the S&P 500 index. The fund thus duplicates the performance of this market index. It has one of the lowest operating expenses (as a percentage of assets) of all mutual stock funds precisely because it requires minimal managerial effort. A second reason to pursue a passive strategy is the free-rider benefit. If there are many active, knowledgeable investors who quickly bid up prices of undervalued assets and force down prices of overvalued assets (by selling), we have to conclude that at any time most assets will be fairly priced. Therefore, a well-diversified portfolio of common stock will be a reasonably fair buy, and the passive strategy may not be inferior to that of the average active investor. (We will elaborate on this argument and provide a more comprehensive analysis of the relative success of passive strategies in later chapters.) The nearby box points out that passive index funds have actually outperformed most actively managed funds in the past decades and that investors are responding to the lower costs and better performance of index funds by directing their investments into these products. To summarize, a passive strategy involves investment in two passive portfolios: virtually risk-free short-term T-bills (or, alternatively, a money market fund) and a fund of common stocks that mimics a broad market index. The capital allocation line representing such a strategy is called the capital market line. Historically, based on 1926 to 2012 data, the passive risky portfolio offered an average risk premium of 8.1% and a standard deviation of 20.48%, resulting in a reward-to-volatility ratio of .40. Passive investors allocate their investment budgets among instruments according to their degree of risk aversion. We can use our analysis to deduce a typical investor’s riskaversion parameter. From Table 1.1 in Chapter 1, we estimate that approximately 65.6% 189

WORDS FROM THE STREET

Investors Sour on Pro Stock Pickers

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of net worth is invested in a broad array of risky assets.7 We assume this portfolio has the same reward–risk characteristics that the S&P 500 has exhibited since 1926, as documented in Table 6.7. Substituting these values in Equation 6.7, we obtain E(rM) 2 rf .081 y* 5 5 5 .656 2 AsM A 3 .20482 which implies a coefficient of risk aversion of

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A5

.081 5 2.94 .656 3 .20482

Of course, this calculation is highly speculative. We have assumed that the average investor holds the naive view that historical average rates of return and standard deviations are the best estimates of expected rates of return and risk, looking to the future. To the extent that the average investor takes advantage of contemporary information in addition to simple historical data, our estimate of A52.94 would be an unjustified inference. Nevertheless, a broad range of studies, taking into account the full range of available assets, places the degree of risk aversion for the representative investor in the range of 2.0 to 4.0.8

CONCEPT CHECK

6.8

Suppose that expectations about the S&P 500 index and the T-bill rate are the same as they were in 2012, but you find that a greater proportion is invested in T-bills today than in 2012. What can you conclude about the change in risk tolerance over the years since 2012?

7

We include in the risky portfolio real assets, half of pension reserves, corporate and noncorporate equity, and half of mutual fund shares. This portfolio sums to $50.05 trillion, which is 65.6% of household net worth. (See Table 1.1.) 8 See, for example, I. Friend and M. Blume, “The Demand for Risky Assets,” American Economic Review 64 (1974); or S. J. Grossman and R. J. Shiller, “The Determinants of the Variability of Stock Market Prices,” American Economic Review 71 (1981).

SUMMARY

1. Speculation is the undertaking of a risky investment for its risk premium. The risk premium has to be large enough to compensate a risk-averse investor for the risk of the investment. 2. A fair game is a risky prospect that has a zero risk premium. It will not be undertaken by a riskaverse investor. 3. Investors’ preferences toward the expected return and volatility of a portfolio may be expressed by a utility function that is higher for higher expected returns and lower for higher portfolio variances. More risk-averse investors will apply greater penalties for risk. We can describe these preferences graphically using indifference curves. 4. The desirability of a risky portfolio to a risk-averse investor may be summarized by the certainty equivalent value of the portfolio. The certainty equivalent rate of return is a value that, if it is received with certainty, would yield the same utility as the risky portfolio.

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5. Shifting funds from the risky portfolio to the risk-free asset is the simplest way to reduce risk. Other methods involve diversification of the risky portfolio and hedging. We take up these methods in later chapters. 6. T-bills provide a perfectly risk-free asset in nominal terms only. Nevertheless, the standard deviation of real rates on short-term T-bills is small compared to that of other assets such as long-term bonds and common stocks, so for the purpose of our analysis we consider T-bills as the risk-free asset. Money market funds hold, in addition to T-bills, short-term relatively safe obligations such as CP and CDs. These entail some default risk, but again, the additional risk is small relative to most other risky assets. For convenience, we often refer to money market funds as risk-free assets.

8. The investor’s degree of risk aversion is characterized by the slope of his or her indifference curve. Indifference curves show, at any level of expected return and risk, the required risk premium for taking on one additional percentage point of standard deviation. More risk-averse investors have steeper indifference curves; that is, they require a greater risk premium for taking on more risk. 9. The optimal position, y*, in the risky asset, is proportional to the risk premium and inversely proportional to the variance and degree of risk aversion: y* 5

E(rP) 2 rf As2P

Graphically, this portfolio represents the point at which the indifference curve is tangent to the CAL. 10. A passive investment strategy disregards security analysis, targeting instead the risk-free asset and a broad portfolio of risky assets such as the S&P 500 stock portfolio. If in 2012 investors took the mean historical return and standard deviation of the S&P 500 as proxies for its expected return and standard deviation, then the values of outstanding assets would imply a degree of risk aversion of about A52.94 for the average investor. This is in line with other studies, which estimate typical risk aversion in the range of 2.0 through 4.0.

risk premium fair game risk averse utility certainty equivalent rate

risk neutral risk lover mean-variance (M-V) criterion indifference curve complete portfolio

Utility score: U = E(r) – ½ As2 Optimal allocation to risky portfolio: y* 5

E(rP) 2 rf As2P

risk-free asset capital allocation line reward-to-volatility ratio passive strategy capital market line

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KEY TERMS

KEY EQUATIONS

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7. An investor’s risky portfolio (the risky asset) can be characterized by its reward-to-volatility ratio, S 5 [E(rP) 2 rf]/sP. This ratio is also the slope of the CAL, the line that, when graphed, goes from the risk-free asset through the risky asset. All combinations of the risky asset and the risk-free asset lie on this line. Other things equal, an investor would prefer a steeper-sloping CAL, because that means higher expected return for any level of risk. If the borrowing rate is greater than the lending rate, the CAL will be “kinked” at the point of the risky asset.

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PROBLEM SETS

Basic

Portfolio Theory and Practice

1. Which of the following choices best completes the following statement? Explain. An investor with a higher degree of risk aversion, compared to one with a lower degree, will prefer investment portfolios a. b. c. d. e.

with higher risk premiums. that are riskier (with higher standard deviations). with lower Sharpe ratios. with higher Sharpe ratios. None of the above is true.

2. Which of the following statements are true? Explain. a. A lower allocation to the risky portfolio reduces the Sharpe (reward-to-volatility) ratio. b. The higher the borrowing rate, the lower the Sharpe ratios of levered portfolios. c. With a fixed risk-free rate, doubling the expected return and standard deviation of the risky portfolio will double the Sharpe ratio. d. Holding constant the risk premium of the risky portfolio, a higher risk-free rate will increase the Sharpe ratio of investments with a positive allocation to the risky asset.

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3. What do you think would happen to the expected return on stocks if investors perceived higher volatility in the equity market? Relate your answer to Equation 6.7.

Intermediate

4. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $70,000 or $200,000 with equal probabilities of .5. The alternative risk-free investment in T-bills pays 6% per year. a. If you require a risk premium of 8%, how much will you be willing to pay for the portfolio? b. Suppose that the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the portfolio? c. Now suppose that you require a risk premium of 12%. What is the price that you will be willing to pay? d. Comparing your answers to (a) and (c), what do you conclude about the relationship between the required risk premium on a portfolio and the price at which the portfolio will sell? 5. Consider a portfolio that offers an expected rate of return of 12% and a standard deviation of 18%. T-bills offer a risk-free 7% rate of return. What is the maximum level of risk aversion for which the risky portfolio is still preferred to bills? 6. Draw the indifference curve in the expected return–standard deviation plane corresponding to a utility level of .05 for an investor with a risk aversion coefficient of 3. (Hint: Choose several possible standard deviations, ranging from 0 to .25, and find the expected rates of return providing a utility level of .05. Then plot the expected return–standard deviation points so derived.) 7. Now draw the indifference curve corresponding to a utility level of .05 for an investor with risk aversion coefficient A 5 4. Comparing your answer to Problem 6, what do you conclude? 8. Draw an indifference curve for a risk-neutral investor providing utility level .05. 9. What must be true about the sign of the risk aversion coefficient, A, for a risk lover? Draw the indifference curve for a utility level of .05 for a risk lover. For Problems 10 through 12: Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 85 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has been about 20% per year. Assume these values are representative of investors’ expectations for future performance and that the current T-bill rate is 5%.

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10. Calculate the expected return and variance of portfolios invested in T-bills and the S&P 500 index with weights as follows: Wbills

Windex

0 0.2 0.4 0.6 0.8 1.0

1.0 0.8 0.6 0.4 0.2 0

11. Calculate the utility levels of each portfolio of Problem 10 for an investor with A52. What do you conclude? 12. Repeat Problem 11 for an investor with A53. What do you conclude? Use these inputs for Problems 13 through 19: You manage a risky portfolio with expected rate of return of 18% and standard deviation of 28%. The T-bill rate is 8%.

14. Suppose that your risky portfolio includes the following investments in the given proportions: Stock A Stock B Stock C

25% 32% 43%

What are the investment proportions of your client’s overall portfolio, including the position in T-bills? 15. What is the reward-to-volatility ratio (S) of your risky portfolio? Your client’s? 16. Draw the CAL of your portfolio on an expected return–standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund’s CAL. 17. Suppose that your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall portfolio will have an expected rate of return of 16%. a. What is the proportion y? b. What are your client’s investment proportions in your three stocks and the T-bill fund? c. What is the standard deviation of the rate of return on your client’s portfolio? 18. Suppose that your client prefers to invest in your fund a proportion y that maximizes the expected return on the complete portfolio subject to the constraint that the complete portfolio’s standard deviation will not exceed 18%. a. What is the investment proportion, y? b. What is the expected rate of return on the complete portfolio? 19. Your client’s degree of risk aversion is A53.5. a. What proportion, y, of the total investment should be invested in your fund? b. What is the expected value and standard deviation of the rate of return on your client’s optimized portfolio? 20. Look at the data in Table 6.7 on the average risk premium of the S&P 500 over T-bills, and the standard deviation of that risk premium. Suppose that the S&P 500 is your risky portfolio. a. If your risk-aversion coefficient is A54 and you believe that the entire 1926–2012 period is representative of future expected performance, what fraction of your portfolio should be allocated to T-bills and what fraction to equity? b. What if you believe that the 1968–1988 period is representative? c. What do you conclude upon comparing your answers to (a) and (b)?

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13. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. What is the expected value and standard deviation of the rate of return on his portfolio?

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Portfolio Theory and Practice 21. Consider the following information about a risky portfolio that you manage, and a risk-free asset: E(rP)511%,sP515%,rf55%. a. Your client wants to invest a proportion of her total investment budget in your risky fund to provide an expected rate of return on her overall or complete portfolio equal to 8%. What proportion should she invest in the risky portfolio, P, and what proportion in the risk-free asset? b. What will be the standard deviation of the rate of return on her portfolio? c. Another client wants the highest return possible subject to the constraint that you limit his standard deviation to be no more than 12%. Which client is more risk averse? 22. Investment Management Inc. (IMI) uses the capital market line to make asset allocation recommendations. IMI derives the following forecasts: • Expected return on the market portfolio: 12%. • Standard deviation on the market portfolio: 20%. • Risk-free rate: 5%.

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Samuel Johnson seeks IMI’s advice for a portfolio asset allocation. Johnson informs IMI that he wants the standard deviation of the portfolio to equal half of the standard deviation for the market portfolio. Using the capital market line, what expected return can IMI provide subject to Johnson’s risk constraint? For Problems 23 through 26: Suppose that the borrowing rate that your client faces is 9%. Assume that the S&P 500 index has an expected return of 13% and standard deviation of 25%, that rf55%, and that your fund has the parameters given in Problem 21. 23. Draw a diagram of your client’s CML, accounting for the higher borrowing rate. Superimpose on it two sets of indifference curves, one for a client who will choose to borrow, and one who will invest in both the index fund and a money market fund. 24. What is the range of risk aversion for which a client will neither borrow nor lend, that is, for which y51? 25. Solve Problems 23 and 24 for a client who uses your fund rather than an index fund. 26. What is the largest percentage fee that a client who currently is lending (y,1) will be willing to pay to invest in your fund? What about a client who is borrowing (y.1)?

Challenge

For Challenge Problems 27, 28, and 29: You estimate that a passive portfolio, that is, one invested in a risky portfolio that mimics the S&P 500 stock index, yields an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The risk-free rate is 8%. 27. Draw the CML and your funds’ CAL on an expected return–standard deviation diagram. a. What is the slope of the CML? b. Characterize in one short paragraph the advantage of your fund over the passive fund. 28. Your client ponders whether to switch the 70% that is invested in your fund to the passive portfolio. a. Explain to your client the disadvantage of the switch. b. Show him the maximum fee you could charge (as a percentage of the investment in your fund, deducted at the end of the year) that would leave him at least as well off investing in your fund as in the passive one. (Hint: The fee will lower the slope of his CAL by reducing the expected return net of the fee.) 29. Consider again the client in Problem 19 with A53.5. a. If he chose to invest in the passive portfolio, what proportion, y, would he select? b. Is the fee (percentage of the investment in your fund, deducted at the end of the year) that you can charge to make the client indifferent between your fund and the passive strategy affected by his capital allocation decision (i.e., his choice of y)?

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Use the following data in answering CFA Problems 1–3: Utility Formula Data Investment

Expected Return, E(r)

Standard Deviation, s

.12 .15 .21 .24

.30 .50 .16 .21

1 2 3 4

U5E(r)2½As2,whereA54

1. On the basis of the utility formula above, which investment would you select if you were risk averse with A54? 2. On the basis of the utility formula above, which investment would you select if you were risk neutral? 3. The variable (A) in the utility formula represents the: investor’s return requirement. investor’s aversion to risk. certainty equivalent rate of the portfolio. preference for one unit of return per four units of risk.

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a. b. c. d.

Use the following graph to answer CFA Problems 4 and 5. Expected Return, E(r) G

4 3 2

H

E

4 3

1

F

2

Capital Allocation Line (CAL)

1 Risk,

4. Which indifference curve represents the greatest level of utility that can be achieved by the investor? 5. Which point designates the optimal portfolio of risky assets? 6. Given $100,000 to invest, what is the expected risk premium in dollars of investing in equities versus risk-free T-bills on the basis of the following table? Action Invest in equities Invest in risk-free T-bills

Probability

Expected Return

.6 .4 1.0

$50,000 2$30,000 $ 5,000

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Portfolio Theory and Practice 7. The change from a straight to a kinked capital allocation line is a result of the: a. b. c. d.

Reward-to-volatility ratio increasing. Borrowing rate exceeding the lending rate. Investor’s risk tolerance decreasing. Increase in the portfolio proportion of the risk-free asset.

8. You manage an equity fund with an expected risk premium of 10% and an expected standard deviation of 14%. The rate on Treasury bills is 6%. Your client chooses to invest $60,000 of her portfolio in your equity fund and $40,000 in a T-bill money market fund. What is the expected return and standard deviation of return on your client’s portfolio? 9. What is the reward-to-volatility ratio for the equity fund in CFA Problem 8?

E-INVESTMENTS EXERCISES

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There is a difference between an investor’s willingness to take risk and his or her ability to take risk. Take the quizzes offered at the Web sites below and compare the results. If they are significantly different, which one would you use to determine an investment strategy? http://mutualfunds.about.com/library/personalitytests/blrisktolerance.htm http://mutualfunds.about.com/library/personalitytests/blriskcapacity.htm

SOLUTIONS TO CONCEPT CHECKS 1. The investor is taking on exchange rate risk by investing in a pound-denominated asset. If the exchange rate moves in the investor’s favor, the investor will benefit and will earn more from the U.K. bill than the U.S. bill. For example, if both the U.S. and U.K. interest rates are 5%, and the current exchange rate is $2 per pound, a $2 investment today can buy 1 pound, which can be invested in England at a certain rate of 5%, for a year-end value of 1.05 pounds. If the year-end exchange rate is $2.10 per pound, the 1.05 pounds can be exchanged for 1.053$2.105$2.205 for a rate of return in dollars of 1 1 r 5 $2.205/$2 5 1.1025, or r 5 10.25%, more than is available from U.S. bills. Therefore, if the investor expects favorable exchange rate movements, the U.K. bill is a speculative investment. Otherwise, it is a gamble. 2. For the A54 investor the utility of the risky portfolio is U 5 .02 2 (½ 3 4 3 .32) 5 .02 while the utility of bills is U 5 .07 2 (½ 3 4 3 0) 5 .07 The investor will prefer bills to the risky portfolio. (Of course, a mixture of bills and the portfolio might be even better, but that is not a choice here.) Even for the A52 investor, the utility of the risky portfolio is U 5 .20 2 (½ 3 2 3 .32) 5 .11 while the utility of bills is again .07. The less risk-averse investor prefers the risky portfolio. 3. The less risk-averse investor has a shallower indifference curve. An increase in risk requires less increase in expected return to restore utility to the original level.

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E(r)

More Risk Averse Less Risk Averse E(rP)

P

4. Holding 50% of your invested capital in Ready Assets means that your investment proportion in the risky portfolio is reduced from 70% to 50%. Your risky portfolio is constructed to invest 54% in E and 46% in B. Thus the proportion of E in your overall portfolio is .5 3 54% 5 27%, and the dollar value of your position in E is $300,0003.275$81,000. 5. In the expected return–standard deviation plane all portfolios that are constructed from the same risky and risk-free funds (with various proportions) lie on a line from the risk-free rate through the risky fund. The slope of the CAL (capital allocation line) is the same everywhere; hence the reward-to-volatility ratio is the same for all of these portfolios. Formally, if you invest a proportion, y, in a risky fund with expected return E(rP) and standard deviation sP, and the remainder, 12y, in a risk-free asset with a sure rate rf , then the portfolio’s expected return and standard deviation are E(rC) 5 rf 1 y 3 E(rP) 2 rf 4 sC 5 ysP and therefore the reward-to-volatility ratio of this portfolio is SC 5

E(r C ) 2 r f

5

sC

y 3 E(rP) 2 rf 4 ysP

5

E(rP) 2 rf sP

which is independent of the proportion y. 6. The lending and borrowing rates are unchanged at rf 5 7%, rfB 5 9%. The standard deviation of the risky portfolio is still 22%, but its expected rate of return shifts from 15% to 17%. The slope of the two-part CAL is E(rP) 2 rf sP

for the lending range

E(rP) 2 r Bf sP

for the borrowing range

Thus in both cases the slope increases: from 8/22 to 10/22 for the lending range, and from 6/22 to 8/22 for the borrowing range.

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σ

σP

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Portfolio Theory and Practice 7. a. The parameters are rf5.07,E(rP)5.15,sP5.22. An investor with a degree of risk aversion A will choose a proportion y in the risky portfolio of y5

E(rP) 2 rf As2P

With the assumed parameters and with A53 we find that y5

.15 2 .07 5 .55 3 3 .0484

When the degree of risk aversion decreases from the original value of 4 to the new value of 3, investment in the risky portfolio increases from 41% to 55%. Accordingly, both the expected return and standard deviation of the optimal portfolio increase:

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E(rC) 5 .07 1 (.55 3 .08) 5 .114 (before: .1028) sC 5 .55 3 .22 5 .121 (before: .0902) b. All investors whose degree of risk aversion is such that they would hold the risky portfolio in a proportion equal to 100% or less (y≤1.00) are lending rather than borrowing, and so are unaffected by the borrowing rate. The least risk-averse of these investors hold 100% in the risky portfolio (y51). We can solve for the degree of risk aversion of these “cut off” investors from the parameters of the investment opportunities: y515

E(rP) 2 rf As2P

5

.08 .0484 A

which implies A5

.08 5 1.65 .0484

Any investor who is more risk tolerant (that is, A,1.65) would borrow if the borrowing rate were 7%. For borrowers, E(rP) 2 rfB y5 As2P Suppose, for example, an investor has an A of 1.1. When rf 5 r Bf 5 7%, this investor chooses to invest in the risky portfolio: y5

.08 5 1.50 1.1 3 .0484

which means that the investor will borrow an amount equal to 50% of her own investment capital. Raise the borrowing rate, in this case to rBf 5 9%, and the investor will invest less in the risky asset. In that case: y5

.06 5 1.13 1.1 3 .0484

and “only” 13% of her investment capital will be borrowed. Graphically, the line from rf to the risky portfolio shows the CAL for lenders. The dashed part would be relevant if the borrowing rate equaled the lending rate. When the borrowing rate exceeds the lending rate, the CAL is kinked at the point corresponding to the risky portfolio. The following figure shows indifference curves of two investors. The steeper indifference curve portrays the more risk-averse investor, who chooses portfolio C0, which involves lending. This investor’s choice is unaffected by the borrowing rate. The more risk-tolerant investor is portrayed by the shallower-sloped indifference curves. If the lending rate equaled the borrowing rate, this investor would choose portfolio C1 on the dashed part of the CAL. When the borrowing rate goes up, this investor chooses portfolio C2 (in the borrowing range of the kinked CAL), which involves less borrowing than before. This investor is hurt by the increase in the borrowing rate.

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E(r)

C1 C2

E(rP)

rfB C0 rf s

8. If all the investment parameters remain unchanged, the only reason for an investor to decrease the investment proportion in the risky asset is an increase in the degree of risk aversion. If you think that this is unlikely, then you have to reconsider your faith in your assumptions. Perhaps the S&P 500 is not a good proxy for the optimal risky portfolio. Perhaps investors expect a higher real rate on T-bills.

APPENDIX A: Risk Aversion, Expected Utility, and the St. Petersburg Paradox We digress in this appendix to examine the rationale behind our contention that investors are risk averse. Recognition of risk aversion as central in investment decisions goes back at least to 1738. Daniel Bernoulli, one of a famous Swiss family of distinguished mathematicians, spent the years 1725 through 1733 in St. Petersburg, where he analyzed the following coin-toss game. To enter the game one pays an entry fee. Thereafter, a coin is tossed until the first head appears. The number of tails, denoted by n, that appears until the first head is tossed is used to compute the payoff, $R, to the participant, as R(n) 5 2n The probability of no tails before the first head (n50) is 1/2 and the corresponding payoff is 205$1. The probability of one tail and then heads (n51) is 1/231/2 with payoff 215$2, the probability of two tails and then heads (n52) is 1/231/231/2, and so forth. The following table illustrates the probabilities and payoffs for various outcomes: Tails

Probability

Payoff5$R(n)

Probability3Payoff

0 1 2 3

1/2 1/4 1/8 1/16

$1 $2 $4 $8

$1/2 $1/2 $1/2 $1/2

…

…

…

…

n

(1/2)n11

$2n

$1/2

The expected payoff is therefore `

E(R) 5 a Pr(n)R(n) 5 1@2 1 1@2 1 c5 ` n50

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sP

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The evaluation of this game is called the “St. Petersburg Paradox.” Although the expected payoff is infinite, participants obviously will be willing to purchase tickets to play the game only at a finite, and possibly quite modest, entry fee. Bernoulli resolved the paradox by noting that investors do not assign the same value per dollar to all payoffs. Specifically, the greater their wealth, the less their “appreciation” for each extra dollar. We can make this insight mathematically precise by assigning a welfare or utility value to any level of investor wealth. Our utility function should increase as wealth is higher, but each extra dollar of wealth should increase utility by progressively smaller amounts.9 (Modern economists would say that investors exhibit “decreasing marginal utility” from an additional payoff dollar.) One particular function that assigns a subjective value to the investor from a payoff of $R, which has a smaller value per dollar the greater the payoff, is the function ln(R) where ln is the natural logarithm function. If this function measures utility values of wealth, the subjective utility value of the game is indeed finite, equal to .693.10 The certain wealth level necessary to yield this utility value is $2.00, because ln(2.00) 5 .693. Hence the certainty equivalent value of the risky payoff is $2.00, which is the maximum amount that this investor will pay to play the game. Von Neumann and Morgenstern adapted this approach to investment theory in a complete axiomatic system in 1946. Avoiding unnecessary technical detail, we restrict ourselves here to an intuitive exposition of the rationale for risk aversion. Imagine two individuals who are identical twins, except that one of them is less fortunate than the other. Peter has only $1,000 to his name while Paul has a net worth of $200,000. How many hours of work would each twin be willing to offer to earn one extra dollar? It is likely that Peter (the poor twin) has more essential uses for the extra money than does Paul. Therefore, Peter will offer more hours. In other words, Peter derives a greater personal welfare or assigns a greater “utility” value to the 1,001st dollar than Paul does to the 200,001st. Figure6A.1 depicts graphically the relationship between the wealth and the utility value of wealth that is consistent with this notion of decreasing marginal utility. Individuals have different rates of decrease in their marginal utility of wealth. What is constant is the principle that the per-dollar increment to utility decreases with wealth. Functions that exhibit the property of decreasing per-unit value as the number of units grows are called concave. A simple example is the log function, familiar from high school mathematics. Of course, a log function will not fit all investors, but it is consistent with the risk aversion that we assume for all investors. Now consider the following simple prospect: p5½

$150,000

12p5½

$50,000

$100,000

This is a fair game in that the expected profit is zero. Suppose, however, that the curve in Figure6A.1 represents the investor’s utility value of wealth, assuming a log utility function. Figure6A.2 shows this curve with numerical values marked. Figure6A.2 shows that the loss in utility from losing $50,000 exceeds the gain from winning $50,000. Consider the gain first. With probability p5.5, wealth goes from $100,000 9

This utility is similar in spirit to the one that assigns a satisfaction level to portfolios with given risk and return attributes. However, the utility function here refers not to investors’ satisfaction with alternative portfolio choices but only to the subjective welfare they derive from different levels of wealth. 10 If we substitute the “utility” value, ln(R), for the dollar payoff, R, to obtain an expected utility value of the game (rather than expected dollar value), we have, calling V(R) the expected utility, `

`

n50

n50

V(R) 5 a Pr(n) ln 3 R(n) 4 5 a (1/2)n11 ln(2n) 5 .693

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201

W

Figure 6A.1 Utility of wealth with a log utility function

to $150,000. Using the log utility function, utility goes from ln(100,000) 5 11.51 to ln(150,000)511.92, the distance G on the graph. This gain is G511.92211.515.41. In expected utility terms, then, the gain is pG5.53.415.21. Now consider the possibility of coming up on the short end of the prospect. In that case, wealth goes from $100,000 to $50,000. The loss in utility, the distance L on the graph, is L5ln(100,000) 2ln(50,000)511.51 210.825.69. Thus the loss in expected utility

U(W) = ln(W) U(150,000) = 11.92

G

U(100,000) = 11.51 E[U(W)] = 11.37 Y

L

U(50,000) = 10.82

W W1 = 50,000

WCE E(W) = 100,000

Figure 6A.2 Fair games and expected utility

W2 = 150,000

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U (W)

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terms is (1 2 p)L5.5 3.695.35, which exceeds the gain in expected utility from the possibility of winning the game. We compute the expected utility from the risky prospect: E 3 U(W)4 5 pU(W1) 1 (1 2 p)U(W2) 5 ½ ln(50,000) 1 ½ ln(150,000) 5 11.37 If the prospect is rejected, the utility value of the (sure) $100,000 is ln(100,000)511.51, greater than that of the fair game (11.37). Hence the risk-averse investor will reject the fair game. Using a specific investor utility function (such as the log utility function) allows us to compute the certainty equivalent value of the risky prospect to a given investor. This is the amount that, if received with certainty, she would consider equally attractive as the risky prospect. If log utility describes the investor’s preferences toward wealth outcomes, then Figure6A.2 can also tell us what is, for her, the dollar value of the prospect. We ask, What sure level of wealth has a utility value of 11.37 (which equals the expected utility from the prospect)? A horizontal line drawn at the level 11.37 intersects the utility curve at the level of wealth WCE. This means that ln(WCE) 5 11.37 which implies that WCE 5 e11.37 5 $86,681.87 WCE is therefore the certainty equivalent of the prospect. The distance Y in Figure6A.2 is the penalty, or the downward adjustment, to the expected profit that is attributable to the risk of the prospect. Y 5 E(W) 2 WCE 5 $100,000 2 $86,681.87 5 $13,318.13 This investor views $86,681.87 for certain as being equal in utility value as $100,000 at risk. Therefore, she would be indifferent between the two.

CONCEPT CHECK

6A.1

Suppose the utility function is U(W ) 5 "W a. What is the utility level at wealth levels $50,000 and $150,000? b. What is expected utility if p still equals .5? c. What is the certainty equivalent of the risky prospect? d. Does this utility function also display risk aversion? e. Does this utility function display more or less risk aversion than the log utility function?

PROBLEMS: APPENDIX A

1. Suppose that your wealth is $250,000. You buy a $200,000 house and invest the remainder in a risk-free asset paying an annual interest rate of 6%. There is a probability of .001 that your house will burn to the ground and its value will be reduced to zero. With a log utility of end-of-year wealth, how much would you be willing to pay for insurance (at the beginning of the year)? (Assume that if the house does not burn down, its end-of-year value still will be $200,000.) 2. If the cost of insuring your house is $1 per $1,000 of value, what will be the certainty equivalent of your end-of-year wealth if you insure your house at: a. ½ its value. b. Its full value. c. 1½ times its value.

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The utility function of an individual investor allows us to measure the subjective value the individual would place on a dollar at various levels of wealth. Essentially, a dollar in bad times (when wealth is low) is more valuable than a dollar in good times (when wealth is high). Suppose that all investors hold the risky S&P 500 portfolio. Then, if the portfolio value falls in a worse-than-expected economy, all investors will, albeit to different degrees, experience a “low-wealth” scenario. Therefore, the equilibrium value of a dollar in the lowwealth economy would be higher than the value of a dollar when the portfolio performs better than expected. This observation helps explain the apparently high cost of portfolio insurance that we encountered when considering long-term investments in the previous chapter. It also helps explain why an investment in a stock portfolio (and hence in individual stocks) has a risk premium that appears to be so high and results in probability of shortfall that is so low. Despite the low probability of shortfall risk, stocks still do not dominate the lower-return risk-free bond, because if an investment shortfall should transpire, it will coincide with states in which the value of dollar returns is high. Does revealed behavior of investors demonstrate risk aversion? Looking at prices and past rates of return in financial markets, we can answer with a resounding yes. With remarkable consistency, riskier bonds are sold at lower prices than are safer ones with otherwise similar characteristics. Riskier stocks also have provided higher average rates of return over long periods of time than less risky assets such as T-bills. For example, over the 1926 to 2012 period, the average rate of return on the S&P 500 portfolio exceeded the T-bill return by around 8% per year. It is abundantly clear from financial data that the average, or representative, investor exhibits substantial risk aversion. For readers who recognize that financial assets are priced to compensate for risk by providing a risk premium and at the same time feel the urge for some gambling, we have a constructive recommendation: Direct your gambling impulse to investment in financial markets. As Von Neumann once said, “The stock market is a casino with the odds in your favor.” A small risk-seeking investment may provide all the excitement you want with a positive expected return to boot!

APPENDIX C:The Kelly Criterion To take a step upwards from the gamble of the St. Petersburg Paradox, consider a sequence of identical one-period investment prospects, each with two possible payoffs (with rates of return expressed as decimals): a positive excess return, b, with probability p, and a negative excess return, 2a (a . 0), with probability q 5 1 2 p. J.L. Kelly11 considered this a basic form of a capital allocation problem and determined the optimal investment in such a sequence of bets for an investor with a log utility function (described in Appendix A). Investing a fraction y in the prospect and the remainder in the risk-free asset provides a total rate of return of 1 1 r 1 by with probability p, or 1 1 r – ay with probability q. Because Kelly employs a log utility function, the expected utility of the prospect, per dollar of initial wealth, is: E 3 U(y) 4 5 p ln(1 1 r 1 yb) 1 q ln(1 1 r 2 ay) 11

(6.C.1)

J.L. Kelly Jr., “A New Interpretation of Information Rate,” Bell System Technical Journal 35 (1956), 917–56.

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APPENDIX B: Utility Functions and Equilibrium Prices of Insurance Contracts

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The investment that maximizes the expected utility has become known as the Kelly criterion (or Kelly formula). The criterion states that the fraction of total wealth invested in the risky prospect is independent of wealth and is given by:

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q p y 5 (1 1 r ) a 2 b a b

(6.C.2)

This will be the investor’s asset allocation in each period. The Kelly formula calls for investing more in the prospect when p and b are large and less when q and a are large. Risk aversion stands out since, when the gains and losses are equal, i.e., when a 5 b, y 5 (1 1 r)(p 2 q)/a, the larger the win/loss spread (corresponding to larger values of a and b), the smaller the fraction invested. A higher interest rate also increases risk taking (an income effect). Kelly’s rule is based on the log utility function. One can show that investors who have such a utility function will, in each period, attempt to maximize the geometric mean of the portfolio return. So the Kelly formula also is a rule to maximize geometric mean, and it has several interesting properties: (1) It never risks ruin, since the fraction of wealth in the risky asset in Equation 6C.2 never exceeds 1/a. (2) The probability that it will outperform any other strategy goes to 1 as the investment horizon goes to infinity. (3) It is myopic, meaning the optimal strategy is the same regardless of the investment horizon. (4) If you have a specified wealth goal (e.g., $1 million), the strategy has the shortest expected time to that goal. Considerable literature has been devoted to the Kelly criterion.12

SOLUTION TO CONCEPT CHECK A.1. a. U(W) 5 "W U(50,000) 5 "50,000 5 223.61 U(150,000) 5 387.30 b. E(U)5(.53223.61)+(.53387.30)5305.45 c. We must find WCE that has utility level 305.45. Therefore "WCE 5 305.45 WCE 5 305.452 5 $93,301 d. Yes. The certainty equivalent of the risky venture is less than the expected outcome of $100,000. e. The certainty equivalent of the risky venture to this investor is greater than it was for the log utility investor considered in the text. Hence this utility function displays less risk aversion.

12

See, for example, L.C. MacLean, E.O. Thorp, W.T. Ziemba, Eds., The Kelly Capital Growth Criterion: Theory and Practice (World Scientific Handbook in Financial Economic Series), Singapore: World Scientific Publishing Co., 2010.

CHAPTER SEVEN

Optimal Risky Portfolios

examine the process of efficient diversification from the ground up, starting with an investment menu of only two risky assets, then adding the risk-free asset, and finally, incorporating the entire universe of available risky securities. We learn how diversification can reduce risk without affecting expected returns. This accomplished, we re-examine the hierarchy of capital allocation, asset allocation, and security selection. Finally, we offer insight into the power of diversification by drawing an analogy between it and the workings of the insurance industry. The portfolios we discuss in this and the following chapters are of a short-term-horizon— even if the overall investment horizon is long, portfolio composition can be rebalanced or updated almost continuously. For these short horizons, the assumption of normality is sufficiently accurate to describe holding-period returns, and we will be concerned only with portfolio means and variances. In Appendix A, we demonstrate how construction of the optimal risky portfolio can easily be accomplished with Excel. AppendixB provides a review of portfolio statistics with emphasis on the intuition behind covariance and correlation measures. Even if you have had a good quantitative methods course, it may well be worth skimming.

PART II

THE INVESTMENT DECISIONcan be viewed as a top-down process: (i) Capital allocation between the risky portfolio and risk-free assets, (ii) asset allocation in the risky portfolio across broad asset classes (e.g., U.S. stocks, international stocks, and long-term bonds), and (iii) security selection of individual assets within each asset class. Capital allocation, as we saw in Chapter6, determines the investor’s exposure to risk. The optimal capital allocation is determined by risk aversion as well as expectations for the risk–return trade-off of the optimal risky portfolio. In principle, asset allocation and security selection are technically identical; both aim at identifying that optimal risky portfolio, namely, the combination of risky assets that provides the best risk–return trade-off. In practice, however, asset allocation and security selection are typically separated into two steps, in which the broad outlines of the portfolio are established first (asset allocation), while details concerning specific securities are filled in later (security selection). After we show how the optimal risky portfolio may be constructed, we will consider the costs and benefits of pursuing this two-step approach. We first motivate the discussion by illustrating the potential gains from simple diversification into many assets. We then proceed to

7

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Diversification and Portfolio Risk Suppose your portfolio is composed of only one stock, say, Dell Inc. What would be the sources of risk to this “portfolio”? You might think of two broad sources of uncertainty. First, there is the risk that comes from conditions in the general economy, such as the business cycle, inflation, interest rates, and exchange rates. None of these macroeconomic factors can be predicted with certainty, and all affect the rate of return on Dell stock. In addition to these macroeconomic factors there are firm-specific influences, such as Dell’s success in research and development, and personnel changes. These factors affect Dell without noticeably affecting other firms in the economy. Now consider a naive diversification strategy, in which you include additional securities in your portfolio. For example, place half your funds in ExxonMobil and half in Dell. What should happen to portfolio risk? To the extent that the firm-specific influences on the two stocks differ, diversification should reduce portfolio risk. For example, when oil prices fall, hurting ExxonMobil, computer prices might rise, helping Dell. The two effects are offsetting and stabilize portfolio return. But why end diversification at only two stocks? If we diversify into many more securities, we continue to spread out our exposure to firm-specific factors, and portfolio volatility should continue to fall. Ultimately, however, even with a large number of stocks we cannot avoid risk altogether, because virtually all securities are affected by the common macroeconomic factors. For example, if all stocks are affected by the business cycle, we cannot avoid exposure to business cycle risk no matter how many stocks we hold. When all risk is firm-specific, as in Figure7.1, panel A, diversification can reduce risk to arbitrarily low levels. The reason is that with all risk sources independent, the exposure to any particular source of risk is reduced to a negligible level. The reduction of risk to very low levels in the case of independent risk sources is sometimes called the insurance principle, because of the notion that an insurance company depends on the risk reduction achieved through diversification when it writes many policies insuring against many independent sources of risk, each policy being a small part of the company’s overall portfolio. (See Section 7.5 for a discussion of the insurance principle.) When common sources of risk affect all firms, however, even extensive diversification cannot eliminate risk. In Figure7.1, panel B, portfolio standard deviation falls as the number of securities increases, but it cannot be reduced to zero. The risk that remains even after extensive diversification is called market risk, risk that is attributable to marketwide risk sources. Such risk is also called systematic risk, or nondiversifiable risk. In contrast, the risk that can be eliminated by diversification is called unique risk, firm-specific risk, nonsystematic risk, or diversifiable risk. This analysis is borne out by empirical studies. Figure7.2 shows the effect of portfolio diversification, using data on NYSE stocks.1 The figure shows the average standard deviation of equally weighted portfolios constructed by selecting stocks at random as a function of the number of stocks in the portfolio. On average, portfolio risk does fall with diversification, but the power of diversification to reduce risk is limited by systematic or common sources of risk.

1

Meir Statman, “How Many Stocks Make a Diversified Portfolio?” Journal of Financial and Quantitative Analysis 22 (September 1987).

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A

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B

σ

σ

Unique Risk

Market Risk n

n

100

50 45 40 35 30 25 20 15 10 5 0

75 50 40

2

4 6 8 10 12 14 16 Number of Stocks in Portfolio

18

20

0 100 200 300 400 500 600 700 800 900 1,000

Risk Compared to a One-Stock Portfolio (%)

Average Portfolio Standard Deviation (%)

Figure 7.1 Portfolio risk as a function of the number of stocks in the portfolio Panel A: All risk is firm specific. Panel B: Some risk is systematic, or marketwide.

Figure 7.2 Portfolio diversification. The average standard deviation of returns of portfolios composed of only one stock was 49.2%. The average portfolio risk fell rapidly as the number of stocks included in the portfolio increased. In the limit, portfolio risk could be reduced to only 19.2%. Source: From Meir Statman, “How Many Stocks Make a Diversified Portfolio?” Journal of Financial and Quantitative Analysis 22 (September 1987). Reprinted by permission.

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Portfolios of Two Risky Assets In the last section we considered naive diversification using equally weighted portfolios of several securities. It is time now to study efficient diversification, whereby we construct risky portfolios to provide the lowest possible risk for any given level of expected return. Portfolios of two risky assets are relatively easy to analyze, and they illustrate the principles and considerations that apply to portfolios of many assets. It makes sense to think about a two-asset portfolio as an asset allocation decision, and so we consider two mutual funds, a bond portfolio specializing in long-term debt securities, denoted D, and a stock fund that specializes in equity securities, E. Table7.1 lists the parameters describing the rate-of-return distribution of these funds. A proportion denoted by wD is invested in the bond fund, and the remainder, 1 2 wD, denoted wE, is invested in the stock fund. The rate of return on this portfolio, rp, will be2 rp 5 wD rD 1 wE rE

(7.1)

where rD is the rate of return on the debt fund and rE is the rate of return on the equity fund. The expected return on the portfolio is a weighted average of expected returns on the component securities with portfolio proportions as weights: E (rp ) 5 wD E (rD ) 1 wE E (rE )

(7.2)

The variance of the two-asset portfolio is s 2p 5 w 2D s 2D 1 w 2E s 2E 1 2wDwE Cov(rD, rE )

(7.3)

Our first observation is that the variance of the portfolio, unlike the expected return, is not a weighted average of the individual asset variances. To understand the formula for the portfolio variance more clearly, recall that the covariance of a variable with itself is the variance of that variable; that is Cov(rD, rD) 5

a Pr(scenario)3 rD 2 E(rD)4 3 rD 2 E(rD)4

scenarios

5

2 a Pr (scenario)3 rD 2 E (rD )4

(7.4)

scenarios

5 s2D Therefore, another way to write the variance of the portfolio is s 2p 5 wD wD Cov (rD , rD) 1 wE wE Cov (rE , rE ) 1 2wDwE Cov(rD , rE )

Table 7.1 Descriptive statistics for two mutual funds

2

Expected return, E(r) Standard deviation, s Covariance, Cov(rD,rE) Correlation coefficient, rDE

See Appendix B of this chapter for a review of portfolio statistics.

(7.5)

Debt

Equity

8% 12%

13% 20% 72 .30

CHAPTER 7

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Table 7.2

A. Bordered Covariance Matrix Portfolio Weights

wD

wD Cov(rD,rD) wE Cov(rE,rD) B. Border-Multiplied Covariance Matrix

wE Cov(rD,rE) Cov(rE,rE)

Portfolio Weights

wD

wE

wD wE

wDwDCov(rD,rD) wEwDCov(rE,rD)

wDwECov(rD,rE) wEwECov(rE,rE)

wD1wE51

wDwDCov(rD,rD)1wEwDCov(rE,rD)

Portfolio variance

209

Computation of portfolio variance from the covariance matrix

wDwE Cov(rD,rE)1wEwECov(rE,rE)

wDwDCov(rD,rD)1wEwDCov(rE,rD)1wDwE Cov(rD,rE)1wEwE Cov(rE,rE)

In words, the variance of the portfolio is a weighted sum of covariances, and each weight is the product of the portfolio proportions of the pair of assets in the covariance term. Table7.2 shows how portfolio variance can be calculated from a spreadsheet. Panel A of the table shows the bordered covariance matrix of the returns of the two mutual funds. The bordered matrix is the covariance matrix with the portfolio weights for each fund placed on the borders, that is, along the first row and column. To find portfolio variance, multiply each element in the covariance matrix by the pair of portfolio weights in its row and column borders. Add up the resultant terms, and you have the formula for portfolio variance given in Equation 7.5. We perform these calculations in panel B, which is the border-multiplied covariance matrix: Each covariance has been multiplied by the weights from the row and the column in the borders. The bottom line of panel B confirms that the sum of all the terms in this matrix (which we obtain by adding up the column sums) is indeed the portfolio variance in Equation 7.5. This procedure works because the covariance matrix is symmetric around the diagonal, that is, Cov(rD,rE)5Cov(rE,rD). Thus each covariance term appears twice. This technique for computing the variance from the border-multiplied covariance matrix is general; it applies to any number of assets and is easily implemented on a spreadsheet. Concept Check 1 asks you to try the rule for a three-asset portfolio. Use this problem to verify that you are comfortable with this concept.

CONCEPT CHECK

7.1

a. First confirm for yourself that our simple rule for computing the variance of a two-asset portfolio from the bordered covariance matrix is consistent with Equation 7.3. b. Now consider a portfolio of three funds, X, Y, Z, with weights wX,wY, and wZ. Show that the portfolio variance is wX2 sX2 1 wY2 sY2 1 wZ2 sZ2 1 2wX wY Cov(rX , rY) 1 2wX w Z Cov(rX , rZ ) 1 2wY wZ Cov (rY , rZ )

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Equation 7.3 reveals that variance is reduced if the covariance term is negative. It is important to recognize that even if the covariance term is positive, the portfolio standard deviation still is less than the weighted average of the individual security standard deviations, unless the two securities are perfectly positively correlated. To see this, notice that the covariance can be computed from the correlation coefficient, rDE, as Cov(rD, rE) 5 rDE sDsE

(7.6)

sp2 5 wD2 sD2 1 w2E sE2 1 2wDwE sD sE rDE

(7.7)

Therefore,

Other things equal, portfolio variance is higher when rDE is higher. In the case of perfect positive correlation, rDE 51, the right-hand side of Equation 7.7 is a perfect square and simplifies to sp2 5 (wD sD 1 wE sE )2

(7.8)

sp 5 wD sD 1 wE sE

(7.9)

or

Therefore, the standard deviation of the portfolio with perfect positive correlation is just the weighted average of the component standard deviations. In all other cases, the correlation coefficient is less than 1, making the portfolio standard deviation less than the weighted average of the component standard deviations. A hedge asset has negative correlation with the other assets in the portfolio. Equation 7.7 shows that such assets will be particularly effective in reducing total risk. Moreover, Equation 7.2 shows that expected return is unaffected by correlation between returns. Therefore, other things equal, we will always prefer to add to our portfolios assets with low or, even better, negative correlation with our existing position. Because the portfolio’s expected return is the weighted average of its component expected returns, whereas its standard deviation is less than the weighted average of the component standard deviations, portfolios of less than perfectly correlated assets always offer some degree of diversification benefit. The lower the correlation between the assets, the greater the gain in efficiency. How low can portfolio standard deviation be? The lowest possible value of the correlation coefficient is 21, representing perfect negative correlation. In this case, Equation 7.7 simplifies to sp2 5 (wD sD 2 wE sE )2

(7.10)

and the portfolio standard deviation is sp 5 Absolute value (wDsD 2 wE sE )

(7.11)

When r521, a perfectly hedged position can be obtained by choosing the portfolio proportions to solve wD sD 2 wE sE 5 0 The solution to this equation is sE sD 1 sE sD wE 5 5 1 2 wD sD 1 sE

wD 5

(7.12)

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211

These weights drive the standard deviation of the portfolio to zero.

Example 7.1

Portfolio Risk and Return

Let us apply this analysis to the data of the bond and stock funds as presented in Table7.1. Using these data, the formulas for the expected return, variance, and standard deviation of the portfolio as a function of the portfolio weights are E(rp) 5 8wD 1 13wE sp2 5 122wD2 1 202wE2 1 2 3 12 3 20 3 .3 3 wDwE 5 144wD2 1 400wE2 1 144wD wE sp 5 "s2p

We can experiment with different portfolio proportions to observe the effect on portfolio expected return and variance. Suppose we change the proportion invested in bonds. The effect on expected return is tabulated in Table7.3 and plotted in Figure7.3. When the proportion invested in debt varies from zero to 1 (so that the proportion in equity varies from 1 to zero), the portfolio expected return goes from 13% (the stock fund’s expected return) to 8% (the expected return on bonds). What happens when wD.1 and wE,0? In this case portfolio strategy would be to sell the equity fund short and invest the proceeds of the short sale in the debt fund. This will decrease the expected return of the portfolio. For example, when wD 52 and wE 5 21, expected portfolio return falls to 2381(21)31353%. At this point the value of the bond fund in the portfolio is twice the net worth of the account. This extreme position is financed in part by short-selling stocks equal in value to the portfolio’s net worth.

Portfolio Standard Deviation for Given Correlation wD

wE

E(rP)

r 5 21

r50

r 5 .30

r51

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

13.00 12.50 12.00 11.50 11.00 10.50 10.00 9.50 9.00 8.50 8.00

20.00 16.80 13.60 10.40 7.20 4.00 0.80 2.40 5.60 8.80 12.00

20.00 18.04 16.18 14.46 12.92 11.66 10.76 10.32 10.40 10.98 12.00

20.00 18.40 16.88 15.47 14.20 13.11 12.26 11.70 11.45 11.56 12.00

20.00 19.20 18.40 17.60 16.80 16.00 15.20 14.40 13.60 12.80 12.00

Minimum Variance Portfolio wD wE E(rP) sP

0.6250 0.3750 9.8750 0.0000

0.7353 0.2647 9.3235 10.2899

0.8200 0.1800 8.9000 11.4473

— — — —

Table 7.3 Expected return and standard deviation with various correlation coefficients

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Expected Return

13%

Equity Fund

8%

Debt Fund

w (stocks) − 0.5

1.0

2.0

1.5

1.0

−1.0

w (bonds) = 1 − w (stocks)

Figure 7.3 Portfolio expected return as a function of investment proportions

Portfolio Standard Deviation (%) ρ = −1

35

ρ=0 ρ = .30

30 25

ρ=1

20 15 10 5 0 −.50

.50

1.0 1.50 Weight in Stock Fund

Figure 7.4 Portfolio standard deviation as a function of investment proportions

3

The reverse happens when wD , 0 and wE.1. This strategy calls for selling the bond fund short and using the proceeds to finance additional purchases of the equity fund. Of course, varying investment proportions also has an effect on portfolio standard deviation. Table 7.3 presents portfolio standard deviations for different portfolio weights calculated from Equation 7.7 using the assumed value of the correlation coefficient, .30, as well as other values of r. Figure7.4 shows the relationship between standard deviation and portfolio weights. Look first at the solid curve for rDE 5.30. The graph shows that as the portfolio weight in the equity fund increases from zero to 1, portfolio standard deviation first falls with the initial diversification from bonds into stocks, but then rises again as the portfolio becomes heavily concentrated in stocks, and again is undiversified. This pattern will generally hold as long as the correlation coefficient between the funds is not too high.3 For a pair of assets with a large positive correlation of

As long as r,sD/sE, volatility will initially fall when we start with all bonds and begin to move into stocks.

CHAPTER 7

Optimal Risky Portfolios

returns, the portfolio standard deviation will increase monotonically from the low-risk asset to the high-risk asset. Even in this case, however, there is a positive (if small) benefit from diversification. What is the minimum level to which portfolio standard deviation can be held? For the parameter values stipulated in Table7.1, the portfolio weights that solve this minimization problem turn out to be4 wMin(D) 5 .82 wMin(E) 5 1 2 .82 5 .18 This minimum-variance portfolio has a standard deviation of sMin 5 3(.822 3 122) 1 (.182 3 202) 1 (2 3 .82 3 .18 3 72)4 1/2 5 11.45% as indicated in the last line of Table7.3 for the column r5.30. The solid colored line in Figure7.4 plots the portfolio standard deviation when r5.30 as a function of the investment proportions. It passes through the two undiversified portfolios of wD 51 and wE 51. Note that the minimum-variance portfolio has a standard deviation smaller than that of either of the individual component assets. This illustrates the effect of diversification. The other three lines in Figure7.4 show how portfolio risk varies for other values of the correlation coefficient, holding the variances of each asset constant. These lines plot the values in the other three columns of Table7.3. The solid dark straight line connecting the undiversified portfolios of all bonds or all stocks, wD 51 or wE 51, shows portfolio standard deviation with perfect positive correlation, r 51. In this case there is no advantage from diversification, and the portfolio standard deviation is the simple weighted average of the component asset standard deviations. The dashed colored curve depicts portfolio risk for the case of uncorrelated assets, r50. With lower correlation between the two assets, diversification is more effective and portfolio risk is lower (at least when both assets are held in positive amounts). The minimum portfolio standard deviation when r50 is 10.29% (see Table7.3), again lower than the standard deviation of either asset. Finally, the triangular broken line illustrates the perfect hedge potential when the two assets are perfectly negatively correlated (r 5 21). In this case the solution for the minimum-variance portfolio is, by Equation 7.12, w Min( D; r 5 21) 5

sE 20 5 5 .625 sD 1 sE 12 1 20

wMin (E; r 5 21) 5 1 2 .625 5 .375 and the portfolio variance (and standard deviation) is zero. We can combine Figures7.3 and 7.4 to demonstrate the relationship between portfolio risk (standard deviation) and expected return—given the parameters of the available assets. 4

This solution uses the minimization techniques of calculus. Write out the expression for portfolio variance from Equation 7.3, substitute 12wD for wE, differentiate the result with respect to wD, set the derivative equal to zero, and solve for wD to obtain wMin(D) 5

sE2 2 Cov(rD, rE) sD2 1 sE2 2 2 Cov(rD, rE )

Alternatively, with a spreadsheet program such as Excel, you can obtain an accurate solution by using the Solver to minimize the variance. See Appendix A for an example of a portfolio optimization spreadsheet.

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This is done in Figure7.5. For any pair of investment proportions, wD, wE, we read the expected Expected Return (%) return from Figure7.3 and the standard deviation from Figure 7.4. The resulting pairs of expected 14 return and standard deviation are tabulated in E 13 Table7.3 and plotted in Figure7.5. 12 The solid colored curve in Figure7.5 shows the ρ = −1 portfolio opportunity set for r 5 .30. We call it 11 ρ=0 the portfolio opportunity set because it shows all 10 ρ = .30 ρ = 1 combinations of portfolio expected return and stan9 dard deviation that can be constructed from the two available assets. The other lines show the portfolio 8 D opportunity set for other values of the correlation 7 coefficient. The solid black line connecting the two 6 funds shows that there is no benefit from diversifi5 cation when the correlation between the two is perfectly positive (r 5 1). The opportunity set is not 0 2 4 6 8 10 12 14 16 18 20 “pushed” to the northwest. The dashed colored line Standard Deviation (%) demonstrates the greater benefit from diversification when the correlation coefficient is lower than .30. Finally, for r 5 21, the portfolio opportunity Figure 7.5 Portfolio expected return as a function of set is linear, but now it offers a perfect hedging standard deviation opportunity and the maximum advantage from diversification. To summarize, although the expected return of any portfolio is simply the weighted average of the asset expected returns, this is not true of the standard deviation. Potential benefits from diversification arise when correlation is less than perfectly positive. The lower the correlation, the greater the potential benefit from diversification. In the extreme case of perfect negative correlation, we have a perfect hedging opportunity and can construct a zero-variance portfolio. Suppose now an investor wishes to select the optimal portfolio from the opportunity set. The best portfolio will depend on risk aversion. Portfolios to the northeast in Figure7.5 provide higher rates of return but impose greater risk. CONCEPT CHECK 7.2 The best trade-off among these choices is a matter of personal preference. Investors with greater risk aversion will Compute and draw the portfolio opportunity set prefer portfolios to the southwest, with lower expected for the debt and equity funds when the correlareturn but lower risk.5 tion coefficient between them is r5.25.

5 Given a level of risk aversion, one can determine the portfolio that provides the highest level of utility. Recall from Chapter 6 that we were able to describe the utility provided by a portfolio as a function of its expected return, E(rp), and its variance, s2p, according to the relationship U 5 E(rp) 2 0.5As2p. The portfolio mean and variance are determined by the portfolio weights in the two funds, wE and wD, according to Equations 7.2 and 7.3. Using those equations and some calculus, we find the optimal investment proportions in the two funds. A warning: To use the following equation (or any equation involving the risk aversion parameter, A), you must express returns in decimal form.

wD 5

E (rD) 2 E (rE ) 1 A(sE2 2 sDsE rDE )

A(sD2 1 sE2 2 2sDsE rDE ) wE 5 1 2 wD

Here, too, Excel’s Solver or similar software can be used to maximize utility subject to the constraints of Equations 7.2 and 7.3, plus the portfolio constraint that wD1wE51 (i.e., that portfolio weights sum to 1).

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7.3

Optimal Risky Portfolios

215

Asset Allocation with Stocks, Bonds, and Bills

When optimizing capital allocation, we want to work with the capital allocation line (CAL) offering the highest slope or Sharpe ratio. The steeper the CAL, the greater is the expected return corresponding to any level of volatility. Now we proceed to asset allocation: constructing the risky portfolio of major asset classes, here a bond and a stock fund, with the highest possible Sharpe ratio. The asset allocation decision requires that we consider T-bills or another safe asset along with the risky asset classes. The reason is that the Sharpe ratio we seek to maximize is defined as the risk premium in excess of the risk-free rate, divided by the standard deviation. We use T-bill rates as the risk-free rate in evaluating the Sharpe ratios of all possible portfolios. The portfolio that maximizes the Sharpe ratio is the solution to the asset allocation problem. Using only stocks, bonds, and bills is actually not so restrictive, as it includes all three major asset classes. As the nearby box emphasizes, most investment professionals recognize that “the really critical decision is how to divvy up your money among stocks, bonds, and super-safe investments such as Treasury bills.”

Asset Allocation with Two Risky Asset Classes What if our risky assets are still confined to the bond and stock funds, but we can also invest in risk-free T-bills yielding 5%? We start with a graphical solution. Figure7.6 shows the opportunity set based on the properties of the bond and stock funds, using the data from Table7.1 and assuming that r 5 .3. Two possible capital allocation lines (CALs) are drawn from the risk-free rate (rf55%) to two feasible portfolios. The first possible CAL is drawn through the minimum-variance portfolio A, which is invested 82% in bonds and 18% in stocks (Table7.3, bottom panel, last column). Portfolio A’s expected return is 8.90%, and its standard deviation is 11.45%. With a T-bill rate of 5%, its Sharpe ratio, which is the slope of the CAL, is SA 5

E (r A ) 2 r f sA

Expected Return (%)

5

8.9 2 5 5 .34 11.45

Now consider the CAL that uses portfolio B instead of A. Portfolio B invests 70% in bonds and 30% in stocks. Its expected return is 9.5% (a risk premium of 4.5%), and its standard deviation is 11.70%. Thus the Sharpe ratio on the CAL supported by portfolio B is SB 5

9.5 2 5 5 .38 11.7

which is higher than the Sharpe ratio of the CAL using the minimum-variance portfolio and T-bills. Hence, portfolio B dominates A. But why stop at portfolio B? We can continue to ratchet the CAL upward until it ultimately reaches the point of tangency with the investment opportunity set. This must yield the CAL with the highest feasible Sharpe ratio. Therefore, the tangency portfolio, labeled P in

E

13 12

CAL(A)

11 10

B CAL(B)

9 8

A D

7 6 5 0

5

10

15 20 25 Standard Deviation (%)

Figure 7.6 The opportunity set of the debt and equity funds and two feasible CALs

WORDS FROM THE STREET

Recipe for Successful Investing: First, Mix Assets Well First things first. If you want dazzling investment results, don’t start your day foraging for hot stocks and stellar mutual funds. Instead, say investment advisers, the really critical decision is how to divvy up your money among stocks, bonds, and supersafe investments such as Treasury bills. In Wall Street lingo, this mix of investments is called your asset allocation. “The asset-allocation choice is the first and most important decision,” says William Droms, a finance professor at Georgetown University. “How much you have in [the stock market] really drives your results.” “You cannot get [stock market] returns from a bond portfolio, no matter how good your security selection is or how good the bond managers you use,” says William John Mikus, a managing director of Financial Design, a Los Angeles investment adviser. For proof, Mr. Mikus cites studies such as the 1991 analysis done by Gary Brinson, Brian Singer and Gilbert Beebower. That study, which looked at the 10-year results for 82 large pension plans, found that a plan’s asset-allocation policy explained 91.5% of the return earned.

DESIGNING A PORTFOLIO Because your asset mix is so important, some mutual fund companies now offer free services to help investors design their portfolios. Gerald Perritt, editor of the Mutual Fund Letter, a Chicago newsletter, says you should vary your mix of assets depending on how long you plan to invest. The further

away your investment horizon, the more you should have in stocks. The closer you get, the more you should lean toward bonds and money-market instruments, such as Treasury bills. Bonds and money-market instruments may generate lower returns than stocks. But for those who need money in the near future, conservative investments make more sense, because there’s less chance of suffering a devastating short-term loss.

SUMMARIZING YOUR ASSETS “One of the most important things people can do is summarize all their assets on one piece of paper and figure out their asset allocation,” says Mr. Pond. Once you’ve settled on a mix of stocks and bonds, you should seek to maintain the target percentages, says Mr. Pond. To do that, he advises figuring out your asset allocation once every six months. Because of a stock-market plunge, you could find that stocks are now a far smaller part of your portfolio than you envisaged. At such a time, you should put more into stocks and lighten up on bonds. When devising portfolios, some investment advisers consider gold and real estate in addition to the usual trio of stocks, bonds and money-market instruments. Gold and real estate give “you a hedge against hyperinflation,” says Mr. Droms. Source: Jonathan Clements, “Recipe for Successful Investing: First, Mix Assets Well,” The Wall Street Journal, October 6, 1993. Reprinted by permission of The Wall Street Journal, © 1993 Dow Jones & Company, Inc. All rights reserved worldwide.

Figure 7.7, is the optimal risky portfolio to mix with T-bills. We can read the expected return and standard deviation of portfolio P from the graph in Figure7.7: E(rP)511% and sP514.2%. In practice, when we try to construct optimal risky portfolios from more than two risky assets, we need to rely on a spreadsheet (which we present in Appendix A) or another computer program. To start, however, we will demonstrate the solution of the portfolio construction problem with only two risky assets and a risk-free asset. In this simpler case, we can find an explicit formula for the weights of each asset in the optimal portfolio, making it easier to illustrate general issues. The objective is to find the weights wD and wE that result in the highest slope of the CAL. Thus our objective function is the Sharpe ratio: Sp 5

E(rp) 2 rf sp

For the portfolio with two risky assets, the expected return and standard deviation of portfolio p are E(rp) 5 wD E (rD) 1 wE E (rE) 5 8wD 1 13wE sp 5 3 wD2 sD2 1 wE2 sE2 1 2wDwE Cov (rD, rE)4 1/2 216

5 3 144wD2 1 400wE2 1 (2 3 72wD wE )4 1/2

CHAPTER 7

Optimal Risky Portfolios

Expected Return (%) 18 CAL(P)

16 14

E

12 P

10 8

Opportunity Set of Risky Assets

D

6 rf = 5% 4 2 0 0

5

10

15

20 25 30 Standard Deviation (%)

Figure 7.7 The opportunity set of the debt and equity funds with the optimal CAL and the optimal risky portfolio

When we maximize the objective function, Sp, we have to satisfy the constraint that the portfolio weights sum to 1.0, that is, wD 1 wE 51. Therefore, we solve an optimization problem formally written as E (rp ) 2 rf Max S 5 p wi sp subject to Swi51. This is a maximization problem that can be solved using standard tools of calculus. In the case of two risky assets, the solution for the weights of the optimal risky portfolio, P, is given by Equation 7.13. Notice that the solution employs excess returns (denoted R) rather than total returns (denoted r).6 E ( RD )sE2 2 E ( RE ) Cov ( RD, RE) (7.13) wD 5 E ( RD )sE2 1 E ( RE )sD2 2 3 E ( RD ) 1 E ( RE )4 Cov ( RD, RE ) wE 5 1 2 wD

Example 7.2

Optimal Risky Portfolio

Using our data, the solution for the optimal risky portfolio is wD 5

(8 2 5)400 2 (13 2 5)72 5 .40 (8 2 5)400 1 (13 2 5)144 2 (8 2 5 1 13 2 5)72

wE 5 1 2 .40 5 .60 6

The solution procedure for two risky assets is as follows. Substitute for E(rP) from Equation 7.2 and for sP from Equation 7.7. Substitute 1 2 wD for wE. Differentiate the resulting expression for Sp with respect to wD, set the derivative equal to zero, and solve for wD.

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The expected return and standard deviation of this optimal risky portfolio are E (rP ) 5 (.4 3 8) 1 (.6 3 13) 5 11% sP 5 [(.42 3 144) 1 (.62 3 400) 1 (2 3 .4 3 .6 3 72)]1/2 5 14.2% This asset allocation produces an optimal risky portfolio whose CAL has a slope of SP 5

11 2 5 5 .42 14.2

which is the Sharpe ratio of portfolio P. Notice that this slope exceeds the slope of any of the other feasible portfolios that we have considered, as it must if it is to be the slope of the best feasible CAL.

In Chapter 6 we found the optimal complete portfolio given an optimal risky portfolio and the CAL generated by a combination of this portfolio and T-bills. Now that we have constructed the optimal risky portfolio, P, we can use the individual investor’s degree of risk aversion, A, to calculate the optimal proportion of the complete portfolio to invest in the risky component.

Example 7.3

The Optimal Complete Portfolio

Now that asset allocation is decided, we can find each investor’s optimal capital allocation. An investor with a coefficient of risk aversion A 54 would take a position in portfolio P of7 y5

E (rP ) 2 rf AsP2

5

.11 2 .05 5 .7439 4 3 .1422

(7.14)

Thus the investor will invest 74.39% of his or her wealth in portfolio P and 25.61% in T-bills. Portfolio P consists of 40% in bonds, so the fraction of wealth in bonds will be ywD 5.4 3.7439 5.2976, or 29.76%. Similarly, the investment in stocks will be ywE 5.6 3.7439 5.4463, or 44.63%. The graphical solution of this asset allocation problem is presented in Figures7.8 and 7.9.

Once we have reached this point, generalizing to the case of many risky assets is straightforward. Before we move on, let us briefly summarize the steps we followed to arrive at the complete portfolio. 1. Specify the return characteristics of all securities (expected returns, variances, covariances). 2. Establish the risky portfolio (asset allocation): a. Calculate the optimal risky portfolio, P (Equation 7.13). b. Calculate the properties of portfolio P using the weights determined in step (a) and Equations 7.2 and 7.3. 3. Allocate funds between the risky portfolio and the risk-free asset (capital allocation): a. Calculate the fraction of the complete portfolio allocated to portfolio P (the risky portfolio) and to T-bills (the risk-free asset) (Equation 7.14). b. Calculate the share of the complete portfolio invested in each asset and in T-bills. 7

Notice that we express returns as decimals in Equation 7.14. This is necessary when using the risk aversion parameter, A, to solve for capital allocation.

CHAPTER 7

Optimal Risky Portfolios

219

Expected Return (%) 18 CAL(P)

16

Portfolio P 74.39%

Indifference Curve 14 E

12 P

C

10 8

Optimal Risky Portfolio

D

6 rf = 5% 4

Opportunity Set of Risky Assets Stocks 44.63%

Optimal Complete Portfolio

2

Bonds 29.76%

T-bills 25.61%

0 0

5

10

15

20

25

30

Standard Deviation (%)

Figure 7.9 The proportions of the optimal complete portfolio

Figure 7.8 Determination of the optimal complete portfolio

Recall that our two risky assets, the bond and stock mutual funds, are already diversified portfolios. The diversification within each of these portfolios must be credited for a good deal of the risk reduction compared to undiversified single securities. For example, the standard deviation of the rate of return on an average stock is about 50% (see Figure7.2). In contrast, the standard deviation of our stock-index fund is only 20%, about equal to the historical standard deviation of the S&P 500 portfolio. This is evidence of the importance of diversification within the asset class. Optimizing the asset allocation between bonds and stocks contributed incrementally to the improvement in the Sharpe ratio of the complete portfolio. The CAL using the optimal combination of stocks and bonds (see Figure7.8) shows that one can achieve an expected return of 13% (matching that of the stock portfolio) with a standard deviation of 18%, which is less than the 20% standard deviation of the stock portfolio. CONCEPT CHECK

7.3

The universe of available securities includes two risky stock funds, A and B, and T-bills. The data for the universe are as follows: Expected Return A B T-bills

Standard Deviation

10% 30 5

20% 60 0

The correlation coefficient between funds A and B is 2.2. a. Draw the opportunity set of funds A and B. b. Find the optimal risky portfolio, P, and its expected return and standard deviation. c. Find the slope of the CAL supported by T-bills and portfolio P. d. How much will an investor with A55 invest in funds A and B and in T-bills?

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Portfolio Theory and Practice

The Markowitz Portfolio Optimization Model Security Selection

E(r)

We can generalize the portfolio construction problem to the case of many risky securities and a risk-free asset. Efﬁcient Frontier As in the two risky assets example, the problem has three parts. First, we identify the risk–return combinaIndividual Global tions available from the set of risky Assets Minimumassets. Next, we identify the optimal Variance Minimum-Variance Frontier portfolio of risky assets by finding Portfolio the portfolio weights that result in the steepest CAL. Finally, we choose an appropriate complete portfolio by mixing the risk-free asset with the optimal risky portfolio. Before σ describing the process in detail, let us first present an overview. The first step is to determine the risk–return opportunities available to Figure 7.10 The minimum-variance frontier of risky assets the investor. These are summarized by the minimum-variance frontier of risky assets. This frontier is a graph of the lowest possible variance that can be attained for a given portfolio expected return. Given the input data for expected returns, variances, and covariances, we can calculate the minimum-variance portfolio for any targeted expected return. The plot of these expected return–standard deviation pairs is presented in Figure7.10. Notice that all the individual assets lie to the right inside the frontier, at least when we allow short sales in the construction of risky portfolios.8 This tells us that risky portfolios comprising only a single asset are inefficient. Diversifying investments leads to portfolios with higher expected returns and lower standard deviations. All the portfolios that lie on the minimum-variance frontier from the global minimumvariance portfolio and upward provide the best risk–return combinations and thus are candidates for the optimal portfolio. The part of the frontier that lies above the global minimum-variance portfolio, therefore, is called the efficient frontier of risky assets. For any portfolio on the lower portion of the minimum-variance frontier, there is a portfolio with the same standard deviation and a greater expected return positioned directly above it. Hence the bottom part of the minimum-variance frontier is inefficient. The second part of the optimization plan involves the risk-free asset. As before, we search for the capital allocation line with the highest Sharpe ratio (that is, the steepest slope) as shown in Figure7.11. The CAL that is supported by the optimal portfolio, P, is tangent to the efficient frontier. This CAL dominates all alternative feasible lines (the broken lines that are drawn through the frontier). Portfolio P, therefore, is the optimal risky portfolio. 8

When short sales are prohibited, single securities may lie on the frontier. For example, the security with the highest expected return must lie on the frontier, as that security represents the only way that one can obtain a return that high, and so it must also be the minimum-variance way to obtain that return. When short sales are feasible, however, portfolios can be constructed that offer the same expected return and lower variance. These portfolios typically will have short positions in low-expected-return securities.

eXcel APPLICATIONS: Two–Security Model he accompanying spreadsheet can be used to measure the return and risk of a portfolio of two risky assets. The model calculates the return and risk for varying weights of each security along with the optimal risky and minimum-variance portfolio. Graphs are automatically generated for various model inputs. The model allows you to specify a target rate of return and solves for optimal combinations using the risk-free asset and the optimal risky portfolio. The spreadsheet is constructed with the

T

A

B

C

D

E

1

Asset Allocation Analysis: Risk and Return

2

Expected

Standard

Correlation

3

Return

Deviation

Coefficient

4

Security 1

0.08

0.12

5

Security 2

0.13

0.2

6

T-Bill

0.05

0.3

two-security return data from Table7.1. This spreadsheet is available at www.mhhe.com/bkm.

Excel Question 1. Suppose your target expected rate of return is 11%. a. What is the lowest-volatility portfolio that provides that expected return? b. What is the standard deviation of that portfolio? c. What is the composition of that portfolio?

F

Expected Return (%)

Covariance 0.0072 11

7 8

Weight

Weight

Expected

Standard

Reward to

9

Security 1

Security 2

Return

Deviation

Volatility

10

1

0.08000

0.12000

0.25000

11

0.9

0.1

0.08500

0.11559

0.30281

12

0.8

0.2

0.09000

0.11454

0.34922

13

0.7

0.3

0.09500

0.11696

0.38474

14

0.6

0.4

0.10000

0.12264

0.40771

5

0 0

5

10

15

20

25

30

35

Standard Deviation (%)

Finally, in the last part of the problem the individual investor chooses the appropriate mix between the optimal risky portfolio P and T-bills, exactly as in Figure7.8. Now let us consider each part of the portfolio construction problem in more detail. In the first part of the problem, risk–return analysis, the portfolio manager needs as inputs a set of estimates for the expected returns of each security and a set of estimates for the covariance matrix. (In Part Five on security analysis we will examine the security valuation techniques and methods of financial analysis that analysts use. For now, E(r) we will assume that analysts already CAL(P) have spent the time and resources to Efﬁcient Frontier prepare the inputs.) The portfolio manager is now armed with the n estimates of E(ri) P and the n3n estimates of the covariance matrix in which the n diagonal elements are estimates of the variances, s2i , and the n22n5n(n21) off-diagonal elements are the estirf mates of the covariances between each pair of asset returns. (You can verify this from Table7.2 for the case n52.) We know that each covariance σ appears twice in this table, so actually we have n(n 21)/2 different covariance estimates. If our portfolio manFigure 7.11 The efficient frontier of risky assets with the optimal CAL agement unit covers 50 securities, 221

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our security analysts need to deliver 50 estimates of expected returns, 50 estimates of variances, and 50349/251,225 different estimates of covariances. This is a daunting task! (We show later how the number of required estimates can be reduced substantially.) Once these estimates are compiled, the expected return and variance of any risky portfolio with weights in each security, wi, can be calculated from the bordered covariance matrix or, equivalently, from the following extensions of Equations 7.2 and 7.3: n

E (rp) 5 a wi E (ri )

(7.15)

i51

n

n

sp2 5 a a wi wj Cov (ri, rj )

(7.16)

i51 j51

An extended worked example showing how to do this using a spreadsheet is presented in Appendix A of this chapter. We mentioned earlier that the idea of diversification is age-old. The phrase “don’t put all your eggs in one basket” existed long before modern finance theory. It was not until 1952, however, that Harry Markowitz published a formal model of portfolio selection embodying diversification principles, thereby paving the way for his 1990 Nobel Prize in Economics.9 His model is precisely step one of portfolio management: the identification of the efficient set of portfolios, or the efficient frontier of risky assets. The principal idea behind the frontier set of risky portfolios is that, for any risk level, we are interested only in that portfolio with the highest expected return. Alternatively, the frontier is the set of portfolios that minimizes the variance for any target expected return. Indeed, the two methods of computing the efficient set of risky portfolios are equivalent. To see this, consider the graphical representation of these procedures. Figure 7.12 shows the minimum-variance frontier. E(r)

Efficient Frontier of Risky Assets E(r3)

E(r2)

Global MinimumE(r1) Variance Portfolio σ σA

σB

σC

Figure 7.12 The efficient portfolio set 9

Harry Markowitz, “Portfolio Selection,” Journal of Finance, March 1952.

CHAPTER 7

Optimal Risky Portfolios

The points marked by squares are the result of a variance-minimization program. We first draw the constraints, that is, horizontal lines at the level of required expected returns. We then look for the portfolio with the lowest standard deviation that plots on each horizontal line—we look for the portfolio that will plot farthest to the left (smallest standard deviation) on that line. When we repeat this for many levels of required expected returns, the shape of the minimum-variance frontier emerges. We then discard the bottom (dashed) half of the frontier, because it is inefficient. In the alternative approach, we draw a vertical line that represents the standard deviation constraint. We then consider all portfolios that plot on this line (have the same standard deviation) and choose the one with the highest expected return, that is, the portfolio that plots highest on this vertical line. Repeating this procedure for many vertical lines (levels of standard deviation) gives us the points marked by circles that trace the upper portion of the minimum-variance frontier, the efficient frontier. When this step is completed, we have a list of efficient portfolios, because the solution to the optimization program includes the portfolio proportions, wi, the expected return, E(rp), and the standard deviation, sp. Let us restate what our portfolio manager has done so far. The estimates generated by the security analysts were transformed into a set of expected rates of return and a covariance matrix. This group of estimates we shall call the input list. This input list is then fed into the optimization program. Before we proceed to the second step of choosing the optimal risky portfolio from the frontier set, let us consider a practical point. Some clients may be subject to additional constraints. For example, many institutions are prohibited from taking short positions in any asset. For these clients the portfolio manager will add to the optimization program constraints that rule out negative (short) positions in the search for efficient portfolios. In this special case it is possible that single assets may be, in and of themselves, efficient risky portfolios. For example, the asset with the highest expected return will be a frontier portfolio because, without the opportunity of short sales, the only way to obtain that rate of return is to hold the asset as one’s entire risky portfolio. Short-sale restrictions are by no means the only such constraints. For example, some clients may want to ensure a minimal level of expected dividend yield from the optimal portfolio. In this case the input list will be expanded to include a set of expected dividend yields d1,..., dn and the optimization program will include an additional constraint that ensures that the expected dividend yield of the portfolio will equal or exceed the desired level, d. Portfolio managers can tailor the efficient set to conform to any desire of the client. Of course, any constraint carries a price tag in the sense that an efficient frontier constructed subject to extra constraints will offer a Sharpe ratio inferior to that of a less constrained one. The client should be made aware of this cost and should carefully consider constraints that are not mandated by law. Another type of constraint is aimed at ruling out investments in industries or countries considered ethically or politically undesirable. This is referred to as socially responsible investing, which entails a cost in the form of a lower Sharpe ratio on the resultant constrained, optimal portfolio. This cost can be justifiably viewed as a contribution to the underlying cause.

Capital Allocation and the Separation Property Now that we have the efficient frontier, we proceed to step two and introduce the riskfree asset. Figure7.13 shows the efficient frontier plus three CALs representing variousportfolios from the efficient set. As before, we ratchet up the CAL by selecting different

223

eXcel APPLICATIONS: Optimal Portfolios spreadsheet model featuring optimal risky portfolios is available on the Online Learning Center at www .mhhe.com/bkm. It contains a template that is similar to the template developed in this section. The model can be used to find optimal mixes of securities for targeted levels of returns for both restricted and unrestricted portfolios. Graphs of the efficient frontier are generated for each set of inputs. The example available at our Web site applies the model to portfolios constructed from equity indexes (called WEBS securities) of several countries.

A

A 1

B

C

Excel Questions 1. Find the optimal risky portfolio formed from the eight country index portfolios using the data provided in this box. What is the mean and variance of that portfolio’s rate of return? 2. Does the optimal risky portfolio entail a short position in any index? If it does, redo Question 1 but now impose a constraint barring short positions. Explain why this constrained portfolio offers a less attractive risk-return trade-off than the unconstrained portfolio in Question 1.

D

E

F

Efficient Frontier for World Equity Benchmark Securities (WEBS)

2 Mean

Standard

4

WEBS

Return

Deviation

5

EWD

15.5393

26.4868

Sweden

6

EWH

6.3852

41.1475

Hong Kong

7

EWI

26.5999

26.0514

Italy

8

EWJ

1.4133

26.0709

Japan Switzerland

3

Country

9

EWL

18.0745

21.6916

10

EWP

18.6347

25.0779

Spain

11

EWW

16.2243

38.7686

Mexico

12

S&P 500

17.2306

17.1944

portfolios until we reach portfolio P, which is the tangency point of a line from F to the efficient frontier. Portfolio P maximizes the Sharpe ratio, the slope of the CAL from F to portfolios on the efficient frontier. At this point our portfolio manager is done. Portfolio P is the optimal risky portfolio for the manager’s clients. There is yet another way to find the best risky portfolio, achievable by introducing the risk-free E(r) (T-bill) rate from the outset. In this approach, we CAL(P) ask the spreadsheet program to maximize the Sharpe Efficient Frontier ratio of portfolio P. The reason this is worth mentionCAL(A) of Risky Assets ing is that we can skip the charting of the efficient frontier altogether and proceed directly to find the P CAL(G) portfolio that produces the steepest CAL. The program maximizes the Sharpe ratio with no constraint A on expected return or variance at all (using just the G (Global Minimum-Variance Portfolio) F feasibility constraint that portfolio weights sum to 1.0). Examination of Figure7.13 shows that the solution strategy is to find the portfolio producing the highest slope of the CAL (Sharpe ratio) regardless of σ expected return or SD. Expected return and standard deviation are easily computed from the optimal portfolio weights applied to the input list in Equations 7.15 and 7.16. Figure 7.13 Capital allocation lines with various While this last approach does not immediately portfolios from the efficient set produce the entire minimum-variance frontier, this 224

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shortcoming can be rectified by directly identifying two portfolios on the frontier. The first is the already familiar Global Minimum Variance portfolio, identified in Figure 7.12 as G. Portfolio G is achieved by minimizing variance without any constraint on the expected return; check this in Figure7.13. The expected return on portfolio G is higher than the riskfree rate (its risk premium will be positive). Another portfolio that will be of great interest to us later is the portfolio on the inefficient portion of the minimum-variance frontier with zero covariance (or correlation) with the optimal risky portfolio. We will call this portfolio Z. Once we identify portfolio P, we can find portfolio Z by solving in Excel for the portfolio that minimizes standard deviation subject to having zero covariance with P. In later chapters we will return to this portfolio. An important property of frontier portfolios is that any portfolio formed by combining two portfolios from the minimum-variance frontier will also be on that frontier, with location along the frontier depending on the weights of that mix. Therefore, portfolio P plus either G or Z can be used to easily trace out the entire efficient frontier. This is a good time to ponder our results and their implementation. The most striking conclusion of all this analysis is that a portfolio manager will offer the same risky portfolio, P, to all clients regardless of their degree of risk aversion.10 The degree of risk aversion of the client comes into play only in capital allocation, the selection of the desired point along the CAL. Thus the only difference between clients’ choices is that the more riskaverse client will invest more in the risk-free asset and less in the optimal risky portfolio than will a less risk-averse client. However, both will use portfolio P as their optimal risky investment vehicle. This result is called a separation property; it tells us that the portfolio choice problem may be separated into two independent tasks.11 The first task, determination of the optimal risky portfolio, is purely technical. Given the manager’s input list, the best risky portfolio is the same for all clients, regardless of risk aversion. However, the second task, capital allocation, depends on personal preference. Here the client is the decision maker. The crucial point is that the optimal portfolio P that the manager offers is the same for all clients. Put another way, investors with varying degrees of risk aversion would be satisfied with a universe of only two mutual funds: a money market fund for risk-free investments and a mutual fund that holds the optimal risky portfolio, P, on the tangency point of the CAL and the efficient frontier. This result makes professional management more efficient and hence less costly. One management firm can serve any number of clients with relatively small incremental administrative costs. In practice, however, different managers will estimate different input lists, thus deriving different efficient frontiers, and offer different “optimal” portfolios to their clients. The source of the disparity lies in the security analysis. It is worth mentioning here that the universal rule of GIGO (garbage in–garbage out) also applies to security analysis. If the quality of the security analysis is poor, a passive portfolio such as a market index fund will result in a higher Sharpe ratio than an active portfolio that uses low-quality security analysis to tilt portfolio weights toward seemingly favorable (mispriced) securities. One particular input list that would lead to a worthless estimate of the efficient frontier is based on recent security average returns. If sample average returns over recent years are 10

Clients who impose special restrictions (constraints) on the manager, such as dividend yield, will obtain another optimal portfolio. Any constraint that is added to an optimization problem leads, in general, to a different and inferior optimum compared to an unconstrained program. 11 The separation property was first noted by Nobel laureate James Tobin, “Liquidity Preference as Behavior toward Risk,” Review of Economic Statistics 25 (February 1958), pp. 65–86.

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used as proxies for the future return on the security, the noise in those estimates will make the resultant efficient frontier virtually useless for portfolio construction. Consider a stock with an annual standard deviation of 50%. Even if one were to use a 10-year average to estimate its expected return (and 10 years is almost ancient history in the life of a corporation), the standard deviation of that estimate would still be 50/"10 5 15.8%. The chances that this average represents expected returns for the coming year are negligible.12 In Chapter 25, we demonstrate that efficient frontiers constructed from past data may be wildly optimistic in terms of the apparent opportunities they offer to improve Sharpe ratios. As we have seen, optimal risky portfolios for different clients also may vary because of portfolio constraints such as dividend-yield requirements, tax considerations, or other client preferences. Nevertheless, this analysis suggests that a limited number of portfolios may be sufficient to serve the demands of a wide range of investors. This is the theoretical basis of the mutual fund industry. The (computerized) optimization technique is the easiest part of the portfolio construction problem. The real arena of competition among portfolio managers is in sophisticated security analysis. This analysis, as well as its proper interpretation, is part of the art of portfolio construction.13

CONCEPT CHECK

7.4

Suppose that two portfolio managers who work for competing investment management houses each employ a group of security analysts to prepare the input list for the Markowitz algorithm. When all is completed, it turns out that the efficient frontier obtained by portfolio manager A dominates that of manager B. By dominate, we mean that A’s optimal risky portfolio lies northwest of B’s. Hence, given a choice, investors will all prefer the risky portfolio that lies on the CAL of A. a. What should be made of this outcome? b. Should it be attributed to better security analysis by A’s analysts? c. Could it be that A’s computer program is superior? d. If you were advising clients (and had an advance glimpse at the efficient frontiers of various managers), would you tell them to periodically switch their money to the manager with the most northwesterly portfolio?

The Power of Diversification Section 7.1 introduced the concept of diversification and the limits to the benefits of diversification resulting from systematic risk. Given the tools we have developed, we can reconsider this intuition more rigorously and at the same time sharpen our insight regarding the power of diversification. 12

Moreover, you cannot avoid this problem by observing the rate of return on the stock more frequently. In Chapter 5 we pointed out that the accuracy of the sample average as an estimate of expected return depends on the length of the sample period, and is not improved by sampling more frequently within a given sample period. 13 You can find a nice discussion of some practical issues in implementing efficient diversification in a white paper prepared by Wealthcare Capital Management at this address: www.financeware.com/ruminations/ WP_EfficiencyDeficiency.pdf. A copy of the report is also available at the Online Learning Center for this text, www.mhhe.com/bkm.

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Recall from Equation 7.16, restated here, that the general formula for the variance of a portfolio is n

n

sp2 5 a a wi wj Cov (ri , rj )

(7.16)

i51 j51

Consider now the naive diversification strategy in which an equally weighted portfolio is constructed, meaning that wi 51/n for each security. In this case Equation 7.16 may be rewritten as follows, where we break out the terms for which i 5 j into a separate sum, noting that Cov(ri, ri) 5 si2 : n n 1 n 1 1 s2p 5 n a s2i 1 a a 2 Cov (ri , rj) n i51 j51 i51 n

(7.17)

j2i

Note that there are n variance terms and n(n21) covariance terms in Equation 7.17. If we define the average variance and average covariance of the securities as s2 5 Cov 5

1 n 2 a si n i51

n n 1 a a Cov (ri, rj ) n (n 2 1) j51 i51

(7.18)

(7.19)

j2i

we can express portfolio variance as sp2 5

1 2 n21 s 1 Cov n n

(7.20)

Now examine the effect of diversification. When the average covariance among security returns is zero, as it is when all risk is firm-specific, portfolio variance can be driven to zero. We see this from Equation 7.20. The second term on the right-hand side will be zero in this scenario, while the first term approaches zero as n becomes larger. Hence when security returns are uncorrelated, the power of diversification to reduce portfolio risk is unlimited. However, the more important case is the one in which economywide risk factors impart positive correlation among stock returns. In this case, as the portfolio becomes more highly diversified (n increases), portfolio variance remains positive. Although firm-specific risk, represented by the first term in Equation 7.20, is still diversified away, the second term simply approaches Cov as n becomes greater. [Note that (n 2 1)/n 5 1 2 1/n, which approaches 1 for large n.] Thus the irreducible risk of a diversified portfolio depends on the covariance of the returns of the component securities, which in turn is a function of the importance of systematic factors in the economy. To see further the fundamental relationship between systematic risk and security correlations, suppose for simplicity that all securities have a common standard deviation, s, and all security pairs have a common correlation coefficient, r. Then the covariance between all pairs of securities is rs2, and Equation 7.20 becomes sp2 5

1 2 n21 2 s 1 rs n n

(7.21)

The effect of correlation is now explicit. When r 50, we again obtain the insurance principle, where portfolio variance approaches zero as n becomes greater. For r.0, however, portfolio variance remains positive. In fact, for r 51, portfolio variance equals s2 regardless of n, demonstrating that diversification is of no benefit: In the case of perfect

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Table 7.4

r50

Risk reduction of equally weighted portfolios in correlated and uncorrelated universes

Universe Size n

Portfolio Weights w51/n(%)

Standard Deviation (%)

1 2 5 6 10 11 20 21 100 101

100 50 20 16.67 10 9.09 5 4.76 1 0.99

50.00 35.36 22.36 20.41 15.81 15.08 11.18 10.91 5.00 4.98

r 5 .40

Reduction in s

Standard Deviation (%)

14.64

50.00 41.83 36.06 35.36 33.91 33.71 32.79 32.73 31.86 31.86

1.95 0.73 0.27 0.02

Reduction in s 8.17 0.70 0.20 0.06 0.00

correlation, all risk is systematic. More generally, as n becomes greater, Equation 7.21 shows that systematic risk becomes rs2. Table7.4 presents portfolio standard deviation as we include ever-greater numbers of securities in the portfolio for two cases, r50 and r5.40. The table takes s to be 50%. As one would expect, portfolio risk is greater when r5.40. More surprising, perhaps, is that portfolio risk diminishes far less rapidly as n increases in the positive correlation case. The correlation among security returns limits the power of diversification. Note that for a 100-security portfolio, the standard deviation is 5% in the uncorrelated case—still significant compared to the potential of zero standard deviation. For r 5.40, the standard deviation is high, 31.86%, yet it is very close to undiversifiable systematic risk in the infinite-sized security universe, "rs 2 5 ".4 3 50 2 5 31.62%. At this point, further diversification is of little value. Perhaps the most important insight from the exercise is this: When we hold diversified portfolios, the contribution to portfolio risk of a particular security will depend on the covariance of that security’s return with those of other securities, and not on the security’s variance. As we shall see in Chapter 9, this implies that fair risk premiums also should depend on covariances rather than total variability of returns. CONCEPT CHECK

7.5

Suppose that the universe of available risky securities consists of a large number of stocks, identically distributed with E(r)515%,s560%, and a common correlation coefficient of r5.5. a. What are the expected return and standard deviation of an equally weighted risky portfolio of 25 stocks? b. What is the smallest number of stocks necessary to generate an efficient portfolio with a standard deviation equal to or smaller than 43%? c. What is the systematic risk in this security universe? d. If T-bills are available and yield 10%, what is the slope of the CML? (Because of the symmetry assumed for all securities in the investment universe, the market index in this economy will be an equally weighted portfolio of all stocks.)

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Asset Allocation and Security Selection As we have seen, the theories of security selection and asset allocation are identical. Both activities call for the construction of an efficient frontier, and the choice of a particular portfolio from along that frontier. The determination of the optimal combination of securities proceeds in the same manner as the analysis of the optimal combination of asset classes. Why, then, do we (and the investment community) distinguish between asset allocation and security selection? Three factors are at work. First, as a result of greater need and ability to save (for college educations, recreation, longer life in retirement, health care needs, etc.), the demand for sophisticated investment management has increased enormously. Second, the widening spectrum of financial markets and financial instruments has put sophisticated investment beyond the capacity of many amateur investors. Finally, there are strong economies of scale in investment analysis. The end result is that the size of a competitive investment company has grown with the industry, and efficiency in organization has become an important issue. A large investment company is likely to invest both in domestic and international markets and in a broad set of asset classes, each of which requires specialized expertise. Hence the management of each asset-class portfolio needs to be decentralized, and it becomes impossible to simultaneously optimize the entire organization’s risky portfolio in one stage, although this would be prescribed as optimal on theoretical grounds. In future chapters we will see how optimization of decentralized portfolios can be mindful as well of the entire portfolio of which they are a part. The practice is therefore to optimize the security selection of each asset-class portfolio independently. At the same time, top management continually updates the asset allocation of the organization, adjusting the investment budget allotted to each asset-class portfolio.

Optimal Portfolios and Nonnormal Returns The portfolio optimization techniques we have used so far assume normal distributions of returns in that standard deviation is taken to be a fully adequate measure of risk. However, potential nonnormality of returns requires us to pay attention as well to risk measures that focus on worst-case losses such as value at risk (VaR) or expected shortfall (ES). In Chapter 6 we suggested that capital allocation to the risky portfolio should be reconsidered in the face of fat-tailed distributions that can result in extreme values of VaR and ES. Specifically, forecasts of greater than normal VaR and ES should encourage more moderate capital allocations to the risky portfolio. Accounting for the effect of diversification on VaR and ES would be useful as well. Unfortunately, the impact of diversification on tail risk cannot be easily anticipated. A practical way to estimate values of VaR and ES in the presence of fat tails is bootstrapping (described in Section 5.9). We start with a historical sample of returns of each asset in our prospective portfolio. We compute the portfolio return corresponding to a draw of one return from each asset’s history. We thus calculate as many of these random portfolio returns as we wish. Fifty thousand returns produced in this way can provide a good estimate of VaR and ES values. The forecasted values for VaR and ES of the mean-variance optimal portfolio can then be compared to other candidate portfolios. If these other portfolios yield sufficiently better VaR and ES values, we may prefer one of those to the meanvariance efficient portfolio.

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Risk Pooling, Risk Sharing, and the Risk of Long-Term Investments* Diversification means that we spread our investment budget across a variety of assets and thus limit overall risk. Sometimes it is argued that spreading investments across time, so that average performance reflects returns in several investment periods, offers an analogous benefit dubbed “time diversification.” A common belief is that time diversification can make long-term investing safer. Is this extension of diversification to investments over time valid? The question of how risk increases when the horizon of a risky investment lengthens is analogous to risk pooling, the process by which an insurance company aggregates a large portfolio (or pool) of uncorrelated risks. However, the application of risk pooling to investment risk is widely misunderstood, as is the application of “the insurance principle” to long-term investments. In this section, we try to clarify these issues and explore the appropriate extension of the insurance principle to investment risk.

Risk Pooling and the Insurance Principle Risk pooling means merging uncorrelated, risky projects as a means to reduce risk. Applied to the insurance business, risk pooling entails selling many uncorrelated insurance policies. This application of risk pooling has come to be known as the insurance principle. Conventional wisdom holds that risk pooling reduces risk, and that such pooling is the driving force behind risk management for the insurance industry. But even brief reflection should convince you that risk pooling cannot be the entire story. How can adding bets that are independent of your other bets reduce your total exposure to risk? This would be little different from a gambler in Las Vegas arguing that a few more trips to the roulette table will reduce his total risk by diversifying his overall “portfolio” of wagers. You would immediately realize that the gambler now has more money at stake, and his overall potential swing in wealth is clearly wider: While his average gain or loss per bet may become more predictable as he repeatedly returns to the table, his total proceeds become less so. As we will see, the insurance principle is sometimes similarly misapplied to long-term investments by incorrectly extending what it implies about average returns to predictions about total returns. Imagine a rich investor, Warren, who holds a $1 billion portfolio, P. The fraction of the portfolio invested in a risky asset, A, is y, leaving the fraction 1 2 y invested in the riskfree rate. Asset A’s risk premium is R, and its standard deviation is s. From Equations 6.3 and 6.4, the risk premium of the complete portfolio P is RP5yR, its standard deviation is sP 5 ys, and the Sharpe ratio is SP 5 R/s. Now Warren identifies another risky asset, B, with the same risk premium and standard deviation as A. Warren estimates that the correlation (and therefore covariance) between the two investments is zero, and he is intrigued at the potential this offers for risk reduction through diversification. Given the benefits that Warren anticipates from diversification, he decides to take a position in asset B equal in size to his existing position in asset A. He therefore transfers another fraction, y, of wealth from the risk-free asset to asset B. This leaves his total portfolio allocated as follows: The fraction y is still invested in asset A, an additional investment of y is invested in B, and 1 22y is in the risk-free asset. Notice that this strategy is *The material in this section is more challenging. It may be skipped without impairing the ability to understand later chapters.

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analogous to pure risk pooling; Warren has taken on additional risky (albeit uncorrelated) bets, and his risky portfolio is larger than it was previously. We will denote Warren’s new portfolio as Z. We can compute the risk premium of portfolio Z from Equation 7.2, its variance from Equation 7.3, and thus its Sharpe ratio. Remember that capital R denotes the risk premium of each asset and the risk premium of the risk-free asset is zero. When calculating portfolio variance, we use the fact that covariance is zero. Thus, for Portfolio Z: RZ 5 yR 1 yR 1 (1 2 2y)0 5 2yR sZ2 5 y 2s 2 1 y 2s 2 1 0 5 2y 2s 2 sZ 5 "s2Z 5 ys"2 SZ 5 RZ /sZ 5 2yR/( ys"2) 5 "2R/s

(double RP) (double the variance of P) ("2 5 1.41 times the standard deviation of P) ("2 5 1.41 times Sharpe ratio of P)

The good news from these results is that the Sharpe ratio of Z is higher than that of P by the factor "2. Its excess rate of return is double that of P, yet its standard deviation is only "2 times larger. The bad news is that by increasing the scale of the risky investment, the standard deviation of the portfolio also increases by "2. We might now imagine that instead of two uncorrelated assets, Warren has access to many. Repeating our analysis, we would find that with n assets the Sharpe ratio under strategy Z increases (relative to its original value) by a factor of "n to "n 3 R/s. But the total risk of the pooling strategy Z will increase by the same multiple, to s"n. This analysis illustrates both the opportunities and limitations of pure risk pooling: Pooling increases the scale of the risky investment (from y to 2y) by adding an additional position in another, uncorrelated asset. This addition of another risky bet also increases the size of the risky budget. So risk pooling by itself does not reduce risk, despite the fact that it benefits from the lack of correlation across policies. The insurance principle tells us only that risk increases less than proportionally to the number of policies insured when the policies are uncorrelated; hence profitability—in this application, the Sharpe ratio—increases. But this effect does not actually reduce risk. This might limit the potential economies of scale of an ever-growing portfolio such as that of a large insurer. You can interpret each “asset” in our analysis as one insurance policy. Each policy written requires the insurance company to set aside additional capital to cover potential losses. The insurance company invests its capital until it needs to pay out on claims. Selling more policies entails increasing the total position in risky investments and therefore the capital that must be allocated to those policies. As the company invests in more uncorrelated assets (insurance policies), the Sharpe ratio continuously increases (which is good), but since more funds are invested in risky policies, the overall risk of the portfolio rises (which is bad). As the number of policies grows, the risk of the pool will certainly grow—despite “diversification” across policies. Eventually, that growing risk will overwhelm the company’s available capital. Insurance analysts often think in terms of probability of loss. Their mathematically correct interpretation of the insurance principle is that the probability of loss declines with risk pooling. This interpretation relates to the fact that the Sharpe ratio (profitability) increases with risk pooling. But to equate the declining probability of loss to reduction in total risk is erroneous; the latter is measured by overall standard deviation, which increases with risk pooling. (Again, think about the gambler in Las Vegas. As he returns over and over again to the roulette table, the probability that he will lose becomes ever more certain, but the magnitude of potential dollar gains or losses becomes ever greater.) Thus risk pooling allows neither investors nor insurance companies to shed risk. However, the increase in risk can be overcome when risk pooling is augmented by risk sharing, as discussed in the next subsection.

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Risk Sharing Now think about a variation on the risk pooling portfolio Z. Imagine that Warren has identified several attractive insurance policies and wishes to invest in all of them. For simplicity, we will look at the case of two policies, so the pool will have the same properties as portfolio Z. We saw that if Warren invested in this two-policy pool, his total risk would be sZ 5 ys"2. But if this is more risk than he is willing to bear, what might he do? His solution is risk sharing, the act of selling shares in an attractive risky portfolio to limit risk and yet maintain the Sharpe ratio (profitability) of the resultant position. Suppose that every time a new risky asset is added to the portfolio, Warren sells off a portion of his investment in the pool to maintain the total funds invested in risky assets unchanged. For example, when a second asset is added, he sells half of his position to other investors. While the total investment budget directed into risky assets is therefore unchanged, it is equally divided between assets A and B, with weights in each of y/2. In following this strategy, the risk-free component of his complete portfolio remains fixed with weight 12y. We will call this strategy V. If you compare the risk-pooling strategy Z with the risk-pooling-plus-risk-sharing strategy V, you will notice that they both entail an investment in the pool of two assets; the only difference between them is that the risk-sharing strategy sells off half the combined pool to maintain a risky portfolio of fixed size. While the weight of the total risky pool in strategy Z is 2y, in the risk-sharing strategy, the risky weight is only one-half that level. Therefore, we can find the properties of the risk-sharing portfolio by substituting y for 2y in each formula or, equivalently, substituting y/2 for y in the following table. Risk Pooling: Portfolio Z

Risk Sharing: Portfolio V

RZ 5 2yR 2 2 2 Z 5 2y 2 2 Z 5 Z 5 y SZ 5 R Z / Z 5 2yR/ ( y

RV 5 2(y/2)R 5 yR 2 2 2 5 y2 V 5 2 y /2 2 2 V 5 V 5 y / SV 5 R V / V 5 2R/

2)5

2R/

2

/2

We observe that portfolio V matches the attractive Sharpe ratio of portfolio Z, but with lower volatility. Thus risk sharing combined with risk pooling is the key to the insurance industry. True diversification means spreading a portfolio of fixed size across many assets, not merely adding more risky bets to an ever-growing risky portfolio. To control his total risk, Warren had to sell off a fraction of the pool of assets. This implies that a portion of those assets must now be held by someone else. For example, if the assets are insurance policies, other investors must be sharing the risk, perhaps by buying shares in the insurance company. Alternatively, insurance companies commonly “reinsure” their risk by selling off portions of the policies to other investors or insurance companies, thus explicitly sharing the risk. We can easily generalize Warren’s example to the case of more than two assets. Suppose the risky pool has n assets. Then the volatility of the risk-sharing portfolio will be sV 5 ys/"n, and its Sharpe ratio will be "nR/s. Clearly, both of these improve as n increases. Think back to our gambler at the roulette wheel one last time. He was wrong to argue that diversification means that 100 bets are less risky than 1 bet. His intuition would be correct, however, if he shared those 100 bets with 100 of his friends. A 1/100 share of 100 bets is in fact less risky than one bet. Fixing the amount of his total money at risk as that money is spread across more independent bets is the way for him to reduce risk.14 14

For the Las Vegas gambler, risk sharing makes the gambles ever more certain to produce a negative rate of return, highlighting the illness that characterizes compulsive gambling.

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With risk sharing, one can set up an insurance company of any size, amassing a large portfolio of policies and limiting total risk by selling shares among many investors. As the Sharpe ratio steadily increases with the number of policies written, while the risk to each diversified shareholder falls, the size of ever-more-profitable insurance companies appears unlimited. In reality, however, two problems put a damper on this process. First, burdens related to problems of managing very large firms will sooner or later eat into the increased gross margins. More important, the issue of “too big to fail” may emerge. The possibility of error in assessing the risk of each policy or misestimating the correlations across losses on the pooled policies (or worse yet, intentional underestimation of risk) can cause an insurance company to fail. As we saw in Chapter 1, too big to fail means that such failure can lead to related failures among the firm’s trading partners. This is similar to what happened in the financial crisis of 2008. The jury is still out on the role of lack of scruples in this affair. It is hoped that future regulation will put real limits on exaggerated optimism concerning the power of diversification to limit risk, despite the appealing mitigation of risk sharing.

Investment for the Long Run Now we turn to the implications of risk pooling and risk sharing for long-term investing. Think of extending an investment horizon for another period (which adds the uncertainty of that period’s risky return) as analogous to adding another risky asset or insurance policy to a pool of assets. Examining the impact of an extension of the investment horizon requires us to clarify what the alternative is. Suppose you consider an investment in a risky portfolio over the next 2 years, which we’ll call the “long-term investment.” How should you compare this decision to a “short-run investment”? We must compare these two strategies over the same period, that is, 2 years. The short-term investment therefore must be interpreted as investing in the risky portfolio over 1 year and in the risk-free asset over the other. Once we agree on this comparison, and assuming the risky return on the first year is uncorrelated with that of the second, it becomes clear that the “long-term” strategy is analogous to portfolio Z. This is because holding on to the risky investment in the second year (rather than withdrawing to the risk-free rate) piles up more risk, just as selling another insurance policy does. Put differently, the long-term investment may be considered analogous to risk pooling. While extending a risky investment to the long run improves the Sharpe ratio (as does risk pooling), it also increases risk. Thus “time diversification” is not really diversification. The more accurate analogy to risk sharing for a long-term horizon is to spread the risky investment budget across each of the investment periods. Compare the following three strategies applied to the whole investment budget over a 2-year horizon: 1. Invest the whole budget at risk for one period, and then withdraw the entire proceeds, placing them in a risk-free asset in the other period. Because you are invested in the risky asset for only 1 year, the risk premium over the whole investment period is R, the 2-year SD is s, and the 2-year Sharpe ratio is S5R/s. 2. Invest the whole budget in the risky asset for both periods. The 2-year risk premium is 2R (assuming continuously compounded rates), the 2-year variance is 2s2, the 2-year SD is s"2, and the 2-year Sharpe ratio is S 5 R"2/s. This is analogous to risk pooling, taking two “bets” on the risky portfolio instead of one (as in Strategy 1). 3. Invest half the investment budget in the risky position in each of two periods, placing the remainder of funds in the risk-free asset. The 2-year risk premium is R,

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the 2-year variance is 23(½ s)25s2/2, the SD is s/"2, and the Sharpe ratio is S 5 R"2/s. This is analogous to risk sharing, taking a fractional position in each year’s investment return.

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Strategy 3 is less risky than either alternative. Its expected total return equals Strategy 1’s, yet its risk is lower and therefore its Sharpe ratio is higher. It achieves the same Sharpe ratio as Strategy 2 but with standard deviation reduced by a factor of 2. In summary, its Sharpe ratio is at least as good as either alternative and, more to the point, its total risk is less than either. We conclude that risk does not fade in the long run. An investor who can invest in an attractive portfolio for only one period, and chooses to invest a given budget in that period, would find it preferable to put money at risk in that portfolio in as many periods as allowed but will decrease the risky budget in each period. Simple risk pooling, or in this case, time diversification, does not reduce risk.

SUMMARY

1. The expected return of a portfolio is the weighted average of the component security expected returns with the investment proportions as weights. 2. The variance of a portfolio is the weighted sum of the elements of the covariance matrix with the product of the investment proportions as weights. Thus the variance of each asset is weighted by the square of its investment proportion. The covariance of each pair of assets appears twice in the covariance matrix; thus the portfolio variance includes twice each covariance weighted by the product of the investment proportions in each of the two assets. 3. Even if the covariances are positive, the portfolio standard deviation is less than the weighted average of the component standard deviations, as long as the assets are not perfectly positively correlated. Thus portfolio diversification is of value as long as assets are less than perfectly correlated. 4. The greater an asset’s covariance with the other assets in the portfolio, the more it contributes to portfolio variance. An asset that is perfectly negatively correlated with a portfolio can serve as a perfect hedge. The perfect hedge asset can reduce the portfolio variance to zero. 5. The efficient frontier is the graphical representation of a set of portfolios that maximize expected return for each level of portfolio risk. Rational investors will choose a portfolio on the efficient frontier. 6. A portfolio manager identifies the efficient frontier by first establishing estimates for asset expected returns and the covariance matrix. This input list is then fed into an optimization program that reports as outputs the investment proportions, expected returns, and standard deviations of the portfolios on the efficient frontier. 7. In general, portfolio managers will arrive at different efficient portfolios because of differences in methods and quality of security analysis. Managers compete on the quality of their security analysis relative to their management fees. 8. If a risk-free asset is available and input lists are identical, all investors will choose the same portfolio on the efficient frontier of risky assets: the portfolio tangent to the CAL. All investors with identical input lists will hold an identical risky portfolio, differing only in how much each allocates to this optimal portfolio and to the risk-free asset. This result is characterized as the separation principle of portfolio construction. 9. Diversification is based on the allocation of a fixed portfolio across several assets, limiting the exposure to any one source of risk. Adding additional risky assets to a portfolio, thereby increasing the total amounts invested, does not reduce dollar risk, even if it makes the rate of return more predictable. This is because that uncertainty is applied to a larger investment base.

Optimal Risky Portfolios

Nor does investing over longer horizons reduce risk. Increasing the investment horizon is analogous to investing in more assets. It increases total risk. Analogously, the key to the insurance industry is risk sharing—the spreading of risk across many investors, each of whom takes on only a small exposure to any given source of risk. Risk pooling—the assumption of ever-more sources of risk—may increase rate of return predictability, but not the predictability of total dollar returns.

diversification insurance principle market risk systematic risk nondiversifiable risk unique risk firm-specific risk

efficient frontier of risky assets input list separation property risk pooling risk sharing

nonsystematic risk diversifiable risk minimum-variance portfolio portfolio opportunity set Sharpe ratio optimal risky portfolio minimum-variance frontier

The expected rate of return on a portfolio: E(rp)5wDE(rD)1wEE(rE) The variance of the return on a portfolio: The Sharpe ratio of a portfolio: Sp 5

sp2

5 ( w D sD

)2

1 ( w E sE

)2

235

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KEY TERMS

KEY EQUATIONS

1 2 (wD sD )(wE sE ) rDE

E (r p ) 2 r f sp

Sharpe ratio maximizing portfolio weights with two risky assets (D and E) and a risk-free asset: wD 5

3 E (rD ) 2 rf 4 sE2 2 3 E (rE ) 2 rf 4 sD sE rDE 3 E (rD ) 2 rf 4 s2E 1 3 E (rE ) 2 rf 4 sD2 2 3 E (rD) 2 rf 1 E (rE ) 2 rf 4 sDs E rDE

wE 5 1 2 wD Optimal capital allocation to the risky asset, y:

E(rp) 2 rf As2p

1. Which of the following factors reflect pure market risk for a given corporation? a. b. c. d. e.

Increased short-term interest rates. Fire in the corporate warehouse. Increased insurance costs. Death of the CEO. Increased labor costs.

2. When adding real estate to an asset allocation program that currently includes only stocks, bonds, and cash, which of the properties of real estate returns affect portfolio risk? Explain. a. Standard deviation. b. Expected return. c. Correlation with returns of the other asset classes. 3. Which of the following statements about the minimum variance portfolio of all risky securities are valid? (Assume short sales are allowed.) Explain. a. b. c. d.

Its variance must be lower than those of all other securities or portfolios. Its expected return can be lower than the risk-free rate. It may be the optimal risky portfolio. It must include all individual securities.

PROBLEM SETS

Basic

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236

PART II

Intermediate

Portfolio Theory and Practice The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 8%. The probability distribution of the risky funds is as follows:

Stock fund (S) Bond fund (B)

Expected Return

Standard Deviation

20% 12

30% 15

The correlation between the fund returns is .10. 4. What are the investment proportions in the minimum-variance portfolio of the two risky funds, and what is the expected value and standard deviation of its rate of return? 5. Tabulate and draw the investment opportunity set of the two risky funds. Use investment proportions for the stock fund of zero to 100% in increments of 20%. 6. Draw a tangent from the risk-free rate to the opportunity set. What does your graph show for the expected return and standard deviation of the optimal portfolio?

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7. Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio. 8. What is the Sharpe ratio of the best feasible CAL? 9. You require that your portfolio yield an expected return of 14%, and that it be efficient, on the best feasible CAL. a. What is the standard deviation of your portfolio? b. What is the proportion invested in the T-bill fund and each of the two risky funds? 10. If you were to use only the two risky funds, and still require an expected return of 14%, what would be the investment proportions of your portfolio? Compare its standard deviation to that of the optimized portfolio in Problem 9. What do you conclude? 11. Stocks offer an expected rate of return of 18%, with a standard deviation of 22%. Gold offers an expected return of 10% with a standard deviation of 30%. a. In light of the apparent inferiority of gold with respect to both mean return and volatility, would anyone hold gold? If so, demonstrate graphically why one would do so. b. Given the data above, reanswer (a) with the additional assumption that the correlation coefficient between gold and stocks equals 1. Draw a graph illustrating why one would or would not hold gold in one’s portfolio. Could this set of assumptions for expected returns, standard deviations, and correlation represent an equilibrium for the security market? 12. Suppose that there are many stocks in the security market and that the characteristics of stocks A and B are given as follows: Stock

Expected Return

Standard Deviation

A B

10% 15 Correlation 5 21

5% 10

Suppose that it is possible to borrow at the risk-free rate, rf. What must be the value of the riskfree rate? (Hint: Think about constructing a risk-free portfolio from stocks A and B.) 13. Assume that expected returns and standard deviations for all securities (including the risk-free rate for borrowing and lending) are known. In this case all investors will have the same optimal risky portfolio. (True or false?) 14. The standard deviation of the portfolio is always equal to the weighted average of the standard deviations of the assets in the portfolio. (True or false?) 15. Suppose you have a project that has a .7 chance of doubling your investment in a year and a .3 chance of halving your investment in a year. What is the standard deviation of the rate of return on this investment?

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16. Suppose that you have $1 million and the following two opportunities from which to construct a portfolio: a. Risk-free asset earning 12% per year. b. Risky asset with expected return of 30% per year and standard deviation of 40%. If you construct a portfolio with a standard deviation of 30%, what is its expected rate of return? The following data are for Problems 17 through 19: The correlation coefficients between pairs of stocks are as follows: Corr(A, B)5.85;Corr(A, C )5.60;Corr(A, D)5.45. Each stock has an expected return of 8% and a standard deviation of 20%. 17. If your entire portfolio is now composed of stock A and you can add some of only one stock to your portfolio, would you choose (explain your choice): a. B. b. C.

c. D. d. Need more data.

18. Would the answer to Problem 17 change for more risk-averse or risk-tolerant investors? Explain. 19. Suppose that in addition to investing in one more stock you can invest in T-bills as well. Would you change your answers to Problems 17 and 18 if the T-bill rate is 8%?

Small-company stocks Large-company stocks Long-term government Intermediate-term government Treasury bills Inflation

Challenge

1920s*

1930s

1940s

1950s

1960s

1970s

1980s

1990s

23.72% 18.36 3.98 3.77

7.28% 21.25 4.60 3.91

20.63% 9.11 3.59 1.70

19.01% 19.41 0.25 1.11

13.72% 7.84 1.14 3.41

8.75% 5.90 6.63 6.11

12.46% 17.60 11.50 12.01

13.84% 18.20 8.60 7.74

3.56 21.00

0.30 22.04

0.37 5.36

1.87 2.22

3.89 2.52

9.00 5.10

5.02 2.93

6.29 7.36

*Based on the period 1926–1929.

20. Input the data from the table into a spreadsheet. Compute the serial correlation in decade returns for each asset class and for inflation. Also find the correlation between the returns of various asset classes. What do the data indicate? 21. Convert the asset returns by decade presented in the table into real rates. Repeat the analysis of Challenge Problem 20 for the real rates of return. The following information applies to Problems 22 through 26: Greta, an elderly investor, has a degree of risk aversion of A53 when applied to return on wealth over a 3-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of 3-year strategies. (All rates are annual, continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a SD of 20%. The hedge fund risk premium is estimated at 10% with a SD of 35%. The return on each of these portfolios in any year is uncorrelated with its return or the return of any other portfolio in any other year. The hedge fund management claims the correlation coefficient between the annual returns on the S&P 500 and the hedge fund in the same year is zero, but Greta believes this is far from certain. 22. Compute the estimated 3-year risk premiums, SDs, and Sharpe ratios for the two portfolios. 23. Assuming the correlation between the annual returns on the two portfolios is indeed zero, what would be the optimal asset allocation? What should be Greta’s capital allocation? 24. If the correlation coefficient between annual portfolio returns is 0.3, what is the annual covariance? 25. With correlation of 0.3, what is the covariance between the 3-year returns? 26. Repeat Problem 15 using an annual correlation of 0.3. (If you cannot calculate the 3-year covariance in Problem 17, assume it is 0.05.)

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The following table of compound annual returns by decade applies to Challenge Problems 20 and 21.

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Portfolio Theory and Practice

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The following data apply to CFA Problems 1 through 3: Hennessy & Associates manages a $30 million equity portfolio for the multimanager Wilstead Pension Fund. Jason Jones, financial vice president of Wilstead, noted that Hennessy had rather consistently achieved the best record among the Wilstead’s six equity managers. Performance of the Hennessy portfolio had been clearly superior to that of the S&P 500 in 4 of the past 5 years. In the one less-favorable year, the shortfall was trivial. Hennessy is a “bottom-up” manager. The firm largely avoids any attempt to “time the market.” It also focuses on selection of individual stocks, rather than the weighting of favored industries. There is no apparent conformity of style among Wilstead’s six equity managers. The five managers, other than Hennessy, manage portfolios aggregating $250 million made up of more than 150 individual issues. Jones is convinced that Hennessy is able to apply superior skill to stock selection, but the favorable returns are limited by the high degree of diversification in the portfolio. Over the years, the portfolio generally held 40–50 stocks, with about 2%–3% of total funds committed to each issue. The reason Hennessy seemed to do well most years was that the firm was able to identify each year 10 or 12 issues that registered particularly large gains. On the basis of this overview, Jones outlined the following plan to the Wilstead pension committee: Let’s tell Hennessy to limit the portfolio to no more than 20 stocks. Hennessy will double the commitments to the stocks that it really favors, and eliminate the remainder. Except for this one new restriction, Hennessy should be free to manage the portfolio exactly as before. All the members of the pension committee generally supported Jones’s proposal because all agreed that Hennessy had seemed to demonstrate superior skill in selecting stocks. Yet the proposal was a considerable departure from previous practice, and several committee members raised questions. Respond to each of the following questions. 1. a. Will the limitation to 20 stocks likely increase or decrease the risk of the portfolio? Explain. b. Is there any way Hennessy could reduce the number of issues from 40 to 20 without significantly affecting risk? Explain. 2. One committee member was particularly enthusiastic concerning Jones’s proposal. He suggested that Hennessy’s performance might benefit further from reduction in the number of issues to 10. If the reduction to 20 could be expected to be advantageous, explain why reduction to 10 might be less likely to be advantageous. (Assume that Wilstead will evaluate the Hennessy portfolio independently of the other portfolios in the fund.) 3. Another committee member suggested that, rather than evaluate each managed portfolio independently of other portfolios, it might be better to consider the effects of a change in the Hennessy portfolio on the total fund. Explain how this broader point of view could affect the committee decision to limit the holdings in the Hennessy portfolio to either 10 or 20 issues. 4. Which one of the following portfolios cannot lie on the efficient frontier as described by Markowitz? a. b. c. d.

Portfolio

Expected Return (%)

Standard Deviation (%)

W X Z Y

15 12 5 9

36 15 7 21

5. Which statement about portfolio diversification is correct? a. Proper diversification can reduce or eliminate systematic risk. b. Diversification reduces the portfolio’s expected return because it reduces a portfolio’s total risk. c. As more securities are added to a portfolio, total risk typically would be expected to fall at a decreasing rate. d. The risk-reducing benefits of diversification do not occur meaningfully until at least 30 individual securities are included in the portfolio.

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6. The measure of risk for a security held in a diversified portfolio is: a. b. c. d.

Specific risk. Standard deviation of returns. Reinvestment risk. Covariance.

7. Portfolio theory as described by Markowitz is most concerned with: a. b. c. d.

The elimination of systematic risk. The effect of diversification on portfolio risk. The identification of unsystematic risk. Active portfolio management to enhance return.

a. b. c. d.

Decline more when the investor buys Mac. Decline more when the investor buys Green. Increase when either Mac or Green is bought. Decline or increase, depending on other factors.

9. Stocks A, B, and C have the same expected return and standard deviation. The following table shows the correlations between the returns on these stocks. Stock A

Stock B

Stock B

11.0 10.9 10.1

11.0 20.4

11.0

Stock A Stock B Stock C

Given these correlations, the portfolio constructed from these stocks having the lowest risk is a portfolio: a. b. c. d.

Equally invested in stocks A and B. Equally invested in stocks A and C. Equally invested in stocks B and C. Totally invested in stock C.

10. Statistics for three stocks, A, B, and C, are shown in the following tables. Standard Deviations of Returns Stock:

A

B

C

Standard deviation (%):

40

20

40

Correlations of Returns Stock A B C

A

B

C

1.00

0.90 1.00

0.50 0.10 1.00

Only on the basis of the information provided in the tables, and given a choice between a portfolio made up of equal amounts of stocks A and B or a portfolio made up of equal amounts of stocks B and C, which portfolio would you recommend? Justify your choice.

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8. Assume that a risk-averse investor owning stock in Miller Corporation decides to add the stock of either Mac or Green Corporation to her portfolio. All three stocks offer the same expected return and total variability. The covariance of return between Miller and Mac is 2.05 and between Miller and Green is 1.05. Portfolio risk is expected to:

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PART II

Portfolio Theory and Practice 11. George Stephenson’s current portfolio of $2 million is invested as follows: Summary of Stephenson’s Current Portfolio

Short-term bonds Domestic large-cap equities Domestic small-cap equities Total portfolio

Value

Percent of Total

Expected Annual Return

Annual Standard Deviation

$ 200,000 600,000 1,200,000 $2,000,000

10% 30 60 100%

4.6% 12.4 16.0 13.8

1.6% 19.5 29.9 23.1

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Stephenson soon expects to receive an additional $2 million and plans to invest the entire amount in an index fund that best complements the current portfolio. Stephanie Coppa, CFA, is evaluating the four index funds shown in the following table for their ability to produce a portfolio that will meet two criteria relative to the current portfolio: (1) maintain or enhance expected return and (2) maintain or reduce volatility. Each fund is invested in an asset class that is not substantially represented in the current portfolio. Index Fund Characteristics Index Fund Fund A Fund B Fund C Fund D

Expected Annual Return

Expected Annual Standard Deviation

15% 11 16 14

Correlation of Returns with Current Portfolio

25% 22 25 22

10.80 10.60 10.90 10.65

State which fund Coppa should recommend to Stephenson. Justify your choice by describing how your chosen fund best meets both of Stephenson’s criteria. No calculations are required. 12. Abigail Grace has a $900,000 fully diversified portfolio. She subsequently inherits ABC Company common stock worth $100,000. Her financial adviser provided her with the following forecast information: Risk and Return Characteristics Expected Monthly Returns Original Portfolio ABC Company

0.67% 1.25

Standard Deviation of Monthly Returns 2.37% 2.95

The correlation coefficient of ABC stock returns with the original portfolio returns is .40. a. The inheritance changes Grace’s overall portfolio and she is deciding whether to keep the ABC stock. Assuming Grace keeps the ABC stock, calculate the: i. Expected return of her new portfolio which includes the ABC stock. ii. Covariance of ABC stock returns with the original portfolio returns. iii. Standard deviation of her new portfolio, which includes the ABC stock. b. If Grace sells the ABC stock, she will invest the proceeds in risk-free government securities yielding .42% monthly. Assuming Grace sells the ABC stock and replaces it with the government securities, calculate the i. Expected return of her new portfolio, which includes the government securities. ii. Covariance of the government security returns with the original portfolio returns. iii. Standard deviation of her new portfolio, which includes the government securities. c. Determine whether the systematic risk of her new portfolio, which includes the government securities, will be higher or lower than that of her original portfolio. d. On the basis of conversations with her husband, Grace is considering selling the $100,000 of ABC stock and acquiring $100,000 of XYZ Company common stock instead. XYZ stock

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has the same expected return and standard deviation as ABC stock. Her husband comments, “It doesn’t matter whether you keep all of the ABC stock or replace it with $100,000 of XYZ stock.” State whether her husband’s comment is correct or incorrect. Justify your response. e. In a recent discussion with her financial adviser, Grace commented, “If I just don’t lose money in my portfolio, I will be satisfied.” She went on to say, “I am more afraid of losing money than I am concerned about achieving high returns.” i. Describe one weakness of using standard deviation of returns as a risk measure for Grace. ii. Identify an alternate risk measure that is more appropriate under the circumstances. 13. Dudley Trudy, CFA, recently met with one of his clients. Trudy typically invests in a master list of 30 equities drawn from several industries. As the meeting concluded, the client made the following statement: “I trust your stock-picking ability and believe that you should invest my funds in your five best ideas. Why invest in 30 companies when you obviously have stronger opinions on a few of them?” Trudy plans to respond to his client within the context of modern portfolio theory.

E-INVESTMENTS EXERCISES Go to the www.investopedia.com/articles/basics/03/050203.asp Web site to learn more about diversification, the factors that influence investors’ risk preferences, and the types of investments that fit into each of the risk categories. Then check out www. investopedia.com/articles/pf/05/061505.asp for asset allocation guidelines for various types of portfolios from conservative to very aggressive. What do you conclude about your own risk preferences and the best portfolio type for you? What would you expect to happen to your attitude toward risk as you get older? How might your portfolio composition change?

SOLUTIONS TO CONCEPT CHECKS 1. a. The first term will be wD 3 wD 3 sD2 , because this is the element in the top corner of the matrix (sD2 ) times the term on the column border (wD) times the term on the row border (wD). Applying this rule to each term of the covariance matrix results in the sum wD2 sD2 1 wD wE Cov ( rE , rD ) 1 wE wD Cov ( rD , rE ) 1 wE2 sE2 , which is the same as Equation 7.3, because Cov(rE,rD)5Cov(rD,rE ). b. The bordered covariance matrix is wX

wY

wZ

wX

sx2

Cov(rX,rY)

Cov(rX,rZ)

wY

Cov(rY,rX)

sY2

Cov(rY,rZ)

wZ

Cov(rZ,rX)

Cov(rZ,rY)

sZ2

There are nine terms in the covariance matrix. Portfolio variance is calculated from these nine terms: sP2 5 wX2 sX2 1 wY2 sY2 1 wZ2 sZ2 1 wX wY Cov(rX , rY) 1 wY wX Cov(rY , rX) 1 wX wZ Cov(rX , rZ) 1 wZ wX Cov(rZ , rX) 1 wY wZ Cov (rY , rZ) 1 wZ wY Cov(rZ , rY) 5 wX2 sX2 1 wY2 sY2 1 wZ2 sZ2 1 2wX wY Cov(rX , rY) 1 2wX wZ Cov(rX , rZ) 1 2wY wZ Cov(rY , rZ)

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a. Contrast the concepts of systematic risk and firm-specific risk, and give an example of each type of risk. b. Critique the client’s suggestion. Discuss how both systematic and firm-specific risk change as the number of securities in a portfolio is increased.

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PART II

Portfolio Theory and Practice 2. The parameters of the opportunity set are E(rD) 58%, E(rE) 513%, sD 512%, sE 520%, and r(D,E) 5.25. From the standard deviations and the correlation coefficient we generate the covariance matrix: Fund

D

E

D E

144 60

60 400

The global minimum-variance portfolio is constructed so that wD 5 5

sD2

sE2 2 Cov(rD , rE ) 1 sE2 2 2 Cov (rD , rE )

400 2 60 5 .8019 (144 1 400) 2 (2 3 60)

wE 5 1 2 wD 5 .1981

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Its expected return and standard deviation are E(rP) 5 (.8019 3 8) 1 (.1981 3 13) 5 8.99% sP 5 3 wD2 sD2 1 wE2 sE2 1 2 wD wE Cov (rD , rE ) 4 1/2 5 3(.80192 3 144) 1 (.19812 3 400) 1 (2 3 .8019 3 .1981 3 60) 4 1/2 5 11.29% For the other points we simply increase wD from .10 to .90 in increments of .10; accordingly, wE ranges from .90 to .10 in the same increments. We substitute these portfolio proportions in the formulas for expected return and standard deviation. Note that when wE 51.0, the portfolio parameters equal those of the stock fund; when wD 51, the portfolio parameters equal those of the debt fund. We then generate the following table: wE

wD

E(r)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1981

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.8019

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 8.99

s 12.00 11.46 11.29 11.48 12.03 12.88 13.99 15.30 16.76 18.34 20.00 11.29 minimum variance portfolio

You can now draw your graph. 3. a. The computations of the opportunity set of the stock and risky bond funds are like those of Question 2 and will not be shown here. You should perform these computations, however, in order to give a graphical solution to part a. Note that the covariance between the funds is Cov(rA , rB) 5 r(A, B) 3 sA 3 sB 5 2.2320360 5 2240 b. The proportions in the optimal risky portfolio are given by wA 5

(10 2 5)602 2 (30 2 5)(2240) (10 2 5)602 1 (30 2 5)202 2 30(2240)

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5 .6818 wB 5 1 2 wA 5 .3182 The expected return and standard deviation of the optimal risky portfolio are E(rP) 5 (.6818 3 10) 1 (.3182 3 30) 5 16.36% sP 5 {(.68182 3 202) 1 (.31822 3 602) 1 [2 3 .6818 3 .3182(2240)]}1/2 5 21.13% Note that portfolio P is not the global minimum-variance portfolio. The proportions of the latter are given by wA 5

602 2 (2240) 5 .8571 60 1 202 2 2(2240) 2

wB 5 1 2 wA 5 .1429 With these proportions, the standard deviation of the minimum-variance portfolio is s ( min) 5 (.85712 3 202) 1 (.14292 3 602 ) 1 3 2 3 .8571 3 .1429 3 (2240) 4 1/2 which is less than that of the optimal risky portfolio. c. The CAL is the line from the risk-free rate through the optimal risky portfolio. This line represents all efficient portfolios that combine T-bills with the optimal risky portfolio. The slope of the CAL is S5

E (r P ) 2 r f sP

5

16.36 2 5 5 .5376 21.13

d. Given a degree of risk aversion, A, an investor will choose a proportion, y, in the optimal risky portfolio of (remember to express returns as decimals when using A): y5

E (r P ) 2 r f AsP2

5

.1636 2 .05 5 .5089 5 3 .21132

This means that the optimal risky portfolio, with the given data, is attractive enough for an investor with A 55 to invest 50.89% of his or her wealth in it. Because stock A makes up 68.18% of the risky portfolio and stock B makes up 31.82%, the investment proportions for this investor are Stock A: .5089 3 68.18 5 34.70% Stock B: .5089 3 31.82 5 16.19% Total 50.89% 4. Efficient frontiers derived by portfolio managers depend on forecasts of the rates of return on various securities and estimates of risk, that is, the covariance matrix. The forecasts themselves do not control outcomes. Thus preferring managers with rosier forecasts (northwesterly frontiers) is tantamount to rewarding the bearers of good news and punishing the bearers of bad news. What we should do is reward bearers of accurate news. Thus if you get a glimpse of the frontiers (forecasts) of portfolio managers on a regular basis, what you want to do is develop the track record of their forecasting accuracy and steer your advisees toward the more accurate forecaster. Their portfolio choices will, in the long run, outperform the field. 5. The parameters are E(r)515,s560, and the correlation between any pair of stocks is r5.5. a. The portfolio expected return is invariant to the size of the portfolio because all stocks have identical expected returns. The standard deviation of a portfolio with n525 stocks is sP 5 3 s2/n 1 r 3 s2(n 2 1) /n 4 1/2 5 3 602/25 1 .5 3 602 3 24/25 4 1/2 5 43.27%

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5 17.57%

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PART II

Portfolio Theory and Practice b. Because the stocks are identical, efficient portfolios are equally weighted. To obtain a standard deviation of 43%, we need to solve for n: 602 602(n 2 1) 1 .5 3 n n 1,849n 5 3,600 1 1,800n 2 1,800 1,800 5 36.73 n5 49 432 5

Thus we need 37 stocks and will come in with volatility slightly under the target. c. As n gets very large, the variance of an efficient (equally weighted) portfolio diminishes, leaving only the variance that comes from the covariances among stocks, that is

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sP 5 "r 3 s2 5 ".5 3 602 5 42.43% Note that with 25 stocks we came within .84% of the systematic risk, that is, the nonsystematic risk of a portfolio of 25 stocks is only .84%. With 37 stocks the standard deviation is 43%, of which nonsystematic risk is .57%. d. If the risk-free is 10%, then the risk premium on any size portfolio is 15 210 55%. The standard deviation of a well-diversified portfolio is (practically) 42.43%; hence the slope of the CAL is S 5 5/42.43 5 .1178

APPENDIX A:A Spreadsheet Model for Efficient Diversification Several software packages can be used to generate the efficient frontier. We will demonstrate the method using Microsoft Excel. Excel is far from the best program for this purpose and is limited in the number of assets it can handle, but working through a simple portfolio optimizer in Excel can illustrate concretely the nature of the calculations used in more sophisticated “black-box” programs. You will find that even in Excel, the computation of the efficient frontier is fairly easy. We apply the Markowitz portfolio optimization program to a practical problem of international diversification. We take the perspective of a portfolio manager serving U.S. clients, who wishes to construct for the next year an optimal risky portfolio of large stocks in the U.S and six developed capital markets (Japan, Germany, U.K., France, Canada, and Australia). First we describe the input list: forecasts of risk premiums and the covariance matrix. Next, we describe Excel’s Solver, and finally we show the solution to the manager’s problem.

The Covariance Matrix To capture recent risk parameters the manager compiles an array of 60 recent monthly (annualized) rates of return, as well as the monthly T-bill rates for the same period. The standard deviations of excess returns are shown in Spreadsheet 7A.1 (column C). They range from 14.93% (U.K. large stocks) to 22.7% (Germany). For perspective on how these parameters can change over time, standard deviations for the period 1991–2000 are also shown (column B). In addition, we present the correlation coefficient between large stocks in the six foreign markets with U.S. large stocks for the same two periods. Here we see that correlations are higher in the more recent period, consistent with the process of globalization. The covariance matrix shown in Spreadsheet 7A.2 was estimated from the array of 60 returns of the seven countries using the COVARIANCE function from the dialog box of Data Analysis in Excel’s Tools menu. Due to a quirk in the Excel software, the covariance

CHAPTER 7 A 1 2 3 4 5 6 7 8 9

B

C

D

E

F

Optimal Risky Portfolios G

245

eXcel

H

Please visit us at www.mhhe.com/bkm 7A.1 Country Index Statistics and Forecasts of Excess Returns Correlation with the Standard Deviation U.S. Country 1991–2000 2001–2005 1991–2000 2001–2005 0.1295 0.1495 1 1 US 0.1466 0.1493 0.64 0.83 UK 0.1741 0.2008 0.54 0.83 France 0.1538 0.2270 0.53 0.85 Germany

10 Australia 11 Japan 12 Canada

A

Average Excess Return 1991–2000 2001–2005 0.1108 −0.0148 0.0536 0.0094 0.0837 0.0247 0.0473 0.0209

Forecast 2006 0.0600 0.0530 0.0700 0.0800

0.1808

0.1617

0.52

0.81

0.0468

0.1225

0.0580

0.2432

0.1878

0.41

0.43

−0.0177

0.0398

0.0450

0.1687

0.1727

0.72

0.79

0.0727

0.1009

0.0590

F

G

B

C

D

E

H

I

13 14

7A.2 The Bordered Covariance Matrix

Portfolio 16 Weights 17 18 19 20 21 22 23 24 25 26 27 28 29

1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0600 0.1495 0.4013

US UK France Germany Australia Japan Canada

1.0000 US 0.0224 0.0184 0.0250 0.0288 0.0195 0.0121 0.0205 0.0224

0.0000 France 0.0250 0.0275 0.0403 0.0438 0.0259 0.0177 0.0273 0.0000

0.0000 UK 0.0184 0.0223 0.0275 0.0299 0.0204 0.0124 0.0206 0.0000

0.0000 Germany 0.0288 0.0299 0.0438 0.0515 0.0301 0.0183 0.0305 0.0000

0.0000 Australia 0.0195 0.0204 0.0259 0.0301 0.0261 0.0147 0.0234 0.0000

0.0000 Japan 0.0121 0.0124 0.0177 0.0183 0.0147 0.0353 0.0158 0.0000

0.0000 Canada 0.0205 0.0206 0.0273 0.0305 0.0234 0.0158 0.0298 0.0000

Mean SD Slope

30 Cell A18 - A24

A18 is set arbitrarily to 1 while A19 to A24 are set to 0

31 Formula in cell C16

=A18

32 Formula in cell A25

=SUM(A18:A24)

33 Formula in cell C25

=C16*SUMPRODUCT($A$18:$A$24,C18:C24)

34 Formula in cell D25-I25

Copied from C25 (note the absolute addresses)

35 Formula in cell A26

=SUMPRODUCT($A$18:$A$24,H6:H12)

36 Formula in cell A27

=SUM(C25:I25)^0.5

37 Formula in cell A28

=A26/A27

...

=A24

Formula in cell I16

38

39

A B C 7A.3 The Efficient Frontier

D

E

F

G

H

I

J

K

L

40 41 Cell to store constraint on risk premium 42 43 44 45 46 47 48 49 50 51 52 53 54 55

0.0400

Min Var 0.0500 0.0450 0.0400 0.0383 0.1238 0.1168 0.1135 0.1132 0.1 0.4037 0.3853 0.3525 0.3386 0.6696 0.6446 0.6195 0.6112 US UK 0.3900 0.5992 0.8083 0.8778 France −0.1357 −0.1693 −0.2029 −0.2140 Germany −0.1679 −0.3144 −0.4610 −0.5097 Australia 0.1067 0.0907 0.0748 0.0695 Japan 0.1575 0.1781 0.1987 0.2055 Canada −0.0203 −0.0288 −0.0374 −0.0402 CAL* 0.0411 0.0509 0.0480 0.0466 0.0465 *Risk premium on CAL = SD ⫻ slope of optimal risky portfolio Mean SD Slope

Spreadsheets 7A.1, 7A.2, 7A.3 Spreadsheet model for international diversiﬁcation

0.0550 0.1340 0.4104 0.6947 0.1809 −0.1021 −0.0213 0.1226 0.1369 −0.0118 0.0550

Optimum 0.0564 0.1374 0.4107 0.7018 0.1214 −0.0926 0.0205 0.1271 0.1311 −0.0093 0.0564

0.0575 0.1401 0.4106 0.7073 0.0758 −0.0852 0.0524 0.1306 0.1266 −0.0075 0.0575

0.0600 0.1466 0.4092 0.7198 −0.0283 −0.0685 0.1253 0.1385 0.1164 −0.0032 0.0602

0.0700 0.1771 0.3953 0.7699 −0.4465 −0.0014 0.4185 0.1704 0.0752 0.0139 0.0727

0.0800 0.2119 0.3774 0.8201 −0.8648 0.0658 0.7117 0.2023 0.0341 0.0309 0.0871

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15

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matrix is not corrected for degrees-of-freedom bias; hence, each of the elements in the matrix was multiplied by 60/59 to eliminate downward bias.

Expected Returns While estimation of the risk parameters (the covariance matrix) from excess returns is a simple technical matter, estimating the risk premium (the expected excess return) is a daunting task. As we discussed in Chapter 5, estimating expected returns using historical data is unreliable. Consider, for example, the negative average excess returns on U.S. large stocks over the period 2001–2005 (cell G6) and, more generally, the big differences in average returns between the 1991–2000 and 2001–2005 periods, as demonstrated in columns F and G. In this example, we simply present the manager’s forecasts of future returns as shown in column H. In Chapter 8 we will establish a framework that makes the forecasting process more explicit.

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The Bordered Covariance Matrix and Portfolio Variance The covariance matrix in Spreadsheet 7A.2 is bordered by the portfolio weights, as explained in Section 7.2 and Table7.2. The values in cells A18–A24, to the left of the covariance matrix, will be selected by the optimization program. For now, we arbitrarily input 1.0 for the U.S. and zero for the others. Cells A16–I16, above the covariance matrix, must be set equal to the column of weights on the left, so that they will change in tandem as the column weights are changed by Excel’s Solver. Cell A25 sums the column weights and is used to force the optimization program to set the sum of portfolio weights to 1.0. Cells C25–I25, below the covariance matrix, are used to compute the portfolio variance for any set of weights that appears in the borders. Each cell accumulates the contribution to portfolio variance from the column above it. It uses the function SUMPRODUCT to accomplish this task. For example, row 33 shows the formula used to derive the value that appears in cell C25. Finally, the short column A26–A28 below the bordered covariance matrix presents portfolio statistics computed from the bordered covariance matrix. First is the portfolio risk premium in cell A26, with formula shown in row 35, which multiplies the column of portfolio weights by the column of forecasts (H6–H12) from Spreadsheet 7A.1. Next is the portfolio standard deviation in cell A27. The variance is given by the sum of cells C25–I25 below the bordered covariance matrix. Cell A27 takes the square root of this sum to produce the standard deviation. The last statistic is the portfolio Sharpe ratio, cell A28, which is the slope of the CAL (capital allocation line) that runs through the portfolio constructed using the column weights (the value in cell A28 equals cell A26 divided by cell A27). The optimal risky portfolio is the one that maximizes the Sharpe ratio.

Using the Excel Solver Excel’s Solver is a user-friendly, but quite powerful, optimizer. It has three parts: (1) an objective function, (2) decision variables, and (3) constraints. Figure 7A.1 shows three pictures of the Solver. For the current discussion we refer to picture A. The top panel of the Solver lets you choose a target cell for the “objective function,” that is, the variable you are trying to optimize. In picture A, the target cell is A27, the portfolio standard deviation. Below the target cell, you can choose whether your objective is to maximize, minimize, or set your objective function equal to a value that you specify. Here we choose to minimize the portfolio standard deviation.

CHAPTER 7

A

Optimal Risky Portfolios

247

B

Figure 7A.1 Solver dialog box

The next panel contains the decision variables. These are cells that the Solver can change in order to optimize the objective function in the target cell. Here, we input cells A18–A24, the portfolio weights that we select to minimize portfolio volatility. The bottom panel of the Solver can include any number of constraints. One constraint that must always appear in portfolio optimization is the “feasibility constraint,” namely, that portfolio weights sum to 1.0. When we bring up the constraint dialogue box, we specify that cell A25 (the sum of weights) be set equal to 1.0.

Finding the Minimum Variance Portfolio It is helpful to begin by identifying the global minimum variance portfolio (G). This provides the starting point of the efficient part of the frontier. Once we input the target cell, the decision variable cells, and the feasibility constraint, as in picture A, we can select “solve” and the Solver returns portfolio G. We copy the portfolio statistics and weights to our output Spreadsheet 7A.3. Column C in Spreadsheet 7A.3 shows that the lowest standard deviation (SD) that can be achieved with our input list is 11.32%. Notice that the SD of portfolio G is considerably lower than even the lowest SD of the individual indexes. From the risk premium of portfolio G (3.83%) we begin building the efficient frontier with ever-larger risk premiums.

Charting the Efficient Frontier of Risky Portfolios We determine the desired risk premiums (points on the efficient frontier) that we wish to use to construct the graph of the efficient frontier. It is good practice to choose more points in the neighborhood of portfolio G because the frontier has the greatest curvature in that region. It is sufficient to choose for the highest point the highest risk premium from the

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C

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input list (here, 8% for Germany). You can produce the entire efficient frontier in minutes following this procedure. 1. Input to the Solver a constraint that says: Cell A26 (the portfolio risk premium) must equal the value in cell E41. The Solver at this point is shown in picture B of Figure 7A.1. Cell E41 will be used to change the required risk premium and thus generate different points along the frontier. 2. For each additional point on the frontier, you input a different desired risk premium into cell E41, and ask the Solver to solve again. 3. Every time the Solver gives you a solution to the request in (2), copy the results into Spreadsheet 7A.3, which tabulates the collection of points along the efficient frontier. For the next step, change cell E41 and repeat from step 2.

Finding the Optimal Risky Portfolio on the Efficient Frontier

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Now that we have an efficient frontier, we look for the portfolio with the highest Sharpe ratio. This is the efficient frontier portfolio that is tangent to the CAL. To find it, we just need to make two changes to the Solver. First, change the target cell from cell A27 to cell A28, the Sharpe ratio of the portfolio, and request that the value in this cell be maximized. Next, eliminate the constraint on the risk premium that may be left over from the last time you used the Solver. At this point the Solver looks like picture C in Figure 7A.1. The Solver now yields the optimal risky portfolio. Copy the statistics for the optimal portfolio and its weights to your Spreadsheet 7A.3. In order to get a clean graph, place the column of the optimal portfolio in Spreadsheet 7A.3 so that the risk premiums of all portfolios in the spreadsheet are steadily increasing from the risk premium of portfolio G (3.83%) all the way up to 8%. The efficient frontier is graphed using the data in cells C45–I45 (the horizontal or x-axis is portfolio standard deviation) and C44–I44 (the vertical or y-axis is portfolio risk premium). The resulting graph appears in Figure 7A.2.

The Optimal CAL It is instructive to superimpose on the graph of the efficient frontier in Figure 7A.2 the CAL that identifies the optimal risky portfolio. This CAL has a slope equal to the Sharpe ratio of

.0900

Risk Premium

.0800

Efficient Frontier Capital Allocation Line Eff Frontier — No Short sales

.0700

Risk Premium SD Efficient Pf No short Min. Var pf No short

.0600 .0500

0.0564 0.0575 0.0383 0.0540

0.1374 0.1401 0.1132 0.1350

Slope 0.4107 0.4104 0.3386 0.3960

.0400 .0300 .1000

.1200

.1400

.1600

.1800

.2000

Standard Deviation

Figure 7A.2 Efficient frontier and CAL for country stock indexes

.2200

.2400

CHAPTER 7

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249

the optimal risky portfolio. Therefore, we add at the bottom of Spreadsheet 7A.3 a row with entries obtained by multiplying the SD of each column’s portfolio by the Sharpe ratio of the optimal risky portfolio from cell H46. This results in the risk premium for each portfolio along the CAL efficient frontier. We now add a series to the graph with the standard deviations in B45–I45 as the x-axis and cells B54–I54 as the y-axis. You can see this CAL in Figure 7A.2.

The Optimal Risky Portfolio and the Short-Sales Constraint

APPENDIX B:Review of Portfolio Statistics We base this review of scenario analysis on a two-asset portfolio. We denote the assets D and E (which you may think of as debt and equity), but the risk and return parameters we use in this appendix are not necessarily consistent with those used in Section 7.2.

Expected Returns We use “expected value” and “mean” interchangeably. For an analysis with n scenarios, where the rate of return in scenario i is r(i) with probability p(i), the expected return is n

E (r ) 5 a p (i )r (i )

(7B.1)

i51

If you were to increase the rate of return assumed for each scenario by some amount D, then the mean return will increase by D. If you multiply the rate in each scenario by a factor w, the new mean will be multiplied by that factor: n

n

n

i51 n

i51

i51

n

i51

i51

a p(i) 3 3 r (i) 1 D 4 5 a p(i) 3 r (i) 1 D a p(i) 5 E(r) 1 D

(7B.2)

a p(i) 3 3 wr(i)4 5 w a p(i) 3 r(i) 5 wE(r)

Example 7B.1

Expected Rates of Return

Column C of Spreadsheet 7B.1 shows scenario rates of return for debt, D. In column D we add 3% to each scenario return and in column E we multiply each rate by .4. The spreadsheet shows how we compute the expected return for columns C, D, and E. It is evident that the mean increases by 3% (from .08 to .11) in column D and is multiplied by .4 (from .08 to 0.032) in column E.

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With the input list used by the portfolio manager, the optimal risky portfolio calls for significant short positions in the stocks of France and Canada (see column H of Spreadsheet7A.3). In many cases the portfolio manager is prohibited from taking short positions. If so, we need to amend the program to preclude short sales. To accomplish this task, we repeat the exercise, but with one change. We add to the Solver the following constraint: Each element in the column of portfolio weights, A18– A24, must be greater than or equal to zero. You should try to produce the short-sale constrained efficient frontier in your own spreadsheet. The graph of the constrained frontier is also shown in Figure 7A.2.

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D

E

A

B

C

3

Scenario

Probability

rD(i)

rD(i) + 0.03

0.4*rD(i)

4

1

0.14

-0.10

-0.07

-0.040

5

2

0.36

0.00

0.03

0.000

6

3

0.30

0.10

0.13

0.040

7

4

0.20

0.32

0.35

0.128

0.110

0.032

F

G

1 Scenario rates of return

2

8

Mean

0.080

9

Cell C8

=SUMPRODUCT($B$4:$B$7,C4:C7)

10 11 12

Spreadsheet 7B.1 Scenario analysis for bonds

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Now let’s construct a portfolio that invests a fraction of the investment budget, w(D), in bonds and the fraction w(E) in stocks. The portfolio’s rate of return in each scenario and its expected return are given by rP(i) 5 wDrD(i) 1 wErE(i)

(7B.3)

E(rP) 5 a p(i)3 wDrD(i) 1 wErE(i) 4 5 a p(i)wDrD(i) 1 a p(i)wErE(i) 5 wDE(rD) 1 wEE(rE) The rate of return on the portfolio in each scenario is the weighted average of the component rates. The weights are the fractions invested in these assets, that is, the portfolio weights. The expected return on the portfolio is the weighted average of the asset means.

Example 7B.2

Portfolio Rate of Return

Spreadsheet 7B.2 lays out rates of return for both stocks and bonds. Using assumed weights of .4 for debt and .6 for equity, the portfolio return in each scenario appears in column L. Cell L8 shows the portfolio expected return as .1040, obtained using the SUMPRODUCT function, which multiplies each scenario return (column L) by the scenario probability (column I) and sums the results.

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H

I

J

K

L

3

Scenario

Probability

rD(i)

rE(i)

Portfolio return 0.4*rD(i)+0.6*rE(i)

4

1

0.14

-0.10

-0.35

-0.2500

5

2

0.36

0.00

0.20

0.1200

6

3

0.30

0.10

0.45

0.3100

7

4

0.20

0.32

-0.19

0.0140

0.12

0.1040

1 Scenario rates of return

2

8

Mean

9

Cell L4

0.08 =0.4*J4+0.6*K4

10

Cell L8

=SUMPRODUCT($I$4:$I$7,L4:L7)

11 12

Spreadsheet 7B.2 Scenario analysis for bonds and stocks

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251

Variance and Standard Deviation The variance and standard deviation of the rate of return on an asset from a scenario analysis are given by15 n

s2(r) 5 a p(i)3 r(i) 2 E(r) 4 2

(7B.4)

i51

s(r) 5 "s2(r)

n

n

Var (wr ) 5 a p (i ) 3 3 wr (i ) 2 E (wr )4 2 5 w 2 a p (i )3 r (i ) 2 E (r )4 2 5 w 2s 2 i51

i51

SD(wr) 5 "w s 5 ws(r) 2 2

(7B.5)

Excel does not have a direct function to compute variance and standard deviation for a scenario analysis. Its STDEV and VAR functions are designed for time series. We need to calculate the probability-weighted squared deviations directly. To avoid having to first compute columns of squared deviations from the mean, however, we can simplify our problem by expressing the variance as a difference between two easily computable terms: s2(r) 5 E[r 2 E(r)]2 5 E{r2 1 [E(r)]2 2 2rE(r)} 5 E (r 2 ) 1 3 E (r)4 2 2 2E (r)E (r)

(7B.6)

n

n

i51

i51

5 E(r2) 2 3 E(r)4 2 5 a p(i)r(i)2 2 B a p(i)r(i) R

Example 7B.3

2

Calculating the Variance of a Risky Asset in Excel

You can compute the first expression, E(r2), in Equation 7B.6 using Excel’s SUMPRODUCT function. For example, in Spreadsheet 7B.3, E(r2) is first calculated in cell C21 by using SUMPRODUCT to multiply the scenario probability times the asset return times the asset return again. Then [E(r)]2 is subtracted (notice the subtraction of C202 in cell C21), to arrive at variance.

15

Variance (here, of an asset rate of return) is not the only possible choice to quantify variability. An alternative would be to use the absolute deviation from the mean instead of the squared deviation. Thus, the mean absolute deviation (MAD) is sometimes used as a measure of variability. The variance is the preferred measure for several reasons. First, working with absolute deviations is mathematically more difficult. Second, squaring deviations gives more weight to larger deviations. In investments, giving more weight to large deviations (hence, losses) is compatible with risk aversion. Third, when returns are normally distributed, the variance is one of the two parameters that fully characterize the distribution.

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Notice that the unit of variance is percent squared. In contrast, standard deviation, the square root of variance, has the same dimension as the original returns, and therefore is easier to interpret as a measure of return variability. When you add a fixed incremental return, D, to each scenario return, you increase the mean return by that same increment. Therefore, the deviation of the realized return in each scenario from the mean return is unaffected, and both variance and SD are unchanged. In contrast, when you multiply the return in each scenario by a factor w, the variance is multiplied by the square of that factor (and the SD is multiplied by w):

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D

E

A

B

C

15

Scenario

Probability

rD(i)

rD(i) + 0.03

0.4*rD(i)

16

1

0.14

-0.10

-0.07

-0.040

17

2

0.36

0.00

0.03

0.000

18

3

0.30

0.10

0.13

0.040

19

4

0.20

0.32

0.35

0.128

F

G

13 Scenario rates of return

14

20

Mean

0.0800

0.1100

0.0240

21

Variance

0.0185

0.0185

0.0034

22

SD

0.1359

0.1359

0.0584

23 Cell C21

=SUMPRODUCT($B$16:$B$19,C16:C19,C16:C19)-C20^2

24 Cell C22

=C21^0.5

Spreadsheet 7B.3

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Scenario analysis for bonds

The variance of a portfolio return is not as simple to compute as the mean. The portfolio variance is not the weighted average of the asset variances. The deviation of the portfolio rate of return in any scenario from its mean return is rP 2 E (rP) 5 wD rD(i ) 1 wE rE (i ) 2 3 wDE (rD) 1 wE E (rE )4 5 wD 3 rD (i ) 2 E (rD )4 1 wE 3 rE (i ) 2 E (rE )4 5 wD d (i ) 1 wE e (i )

(7B.7)

where the lowercase variables denote deviations from the mean: d(i ) = rD(i) 2 E(rD) e(i ) 5 rE (i ) 2 E(rE ) We express the variance of the portfolio return in terms of these deviations from the mean in Equation 7B.7: n

n

sP2 5 a p (i )3 rP 2 E (rP)4 2 5 a p (i )3 wD d (i ) 1 wE e (i )4 2 i51 n

i51

5 a p (i )3 wD2 d (i )2 1 wE2 e (i ) 2 1 2wDwE d (i )e (i )4 i51

5

n wD2 a p (i ) d (i ) 2 i51

1

n wE2 a p (i )e (i ) 2 i51 n

n

(7B.8)

1 2wD wE a p (i ) d (i )e (i ) i51

5 wD2 sD2 1 wE2 sE2 1 2wD wE a p (i ) d (i )e (i ) i51

The last line in Equation 7B.8 tells us that the variance of a portfolio is the weighted sum of portfolio variances (notice that the weights are the squares of the portfolio weights), plus an additional term that, as we will soon see, makes all the difference. Notice also that d(i)3e(i) is the product of the deviations of the scenario returns of the two assets from their respective means. The probability-weighted average of this product is its expected value, which is called covariance and is denoted Cov(rD,rE). The covariance between the two assets can have a big impact on the variance of a portfolio.

Covariance The covariance between two variables equals Cov(rD, rE) 5 E(d 3 e) 5 E 5 3 rD 2 E(rD)4 3 rE 2 E(rE)46 5 E (rD rE ) 2 E (rD)E (rE )

(7B.9)

CHAPTER 7 A

B

1

C

3

Probability 0.25

4

0.50

5

0.25

6

2

6

D

Rates of Return

Mean:

Bonds -2

E

F

G

Deviation from Mean Stocks 20

Optimal Risky Portfolios

253

eXcel

H Product of Deviations -160

Stocks 30

Bonds -8

6

10

14

-10

8

-20

-160

10

-80

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Spreadsheet 7B.4

The covariance is an elegant way to quantify the covariation of two variables. This is easiest seen through a numerical example. Imagine a three-scenario analysis of stocks and bonds as given in Spreadsheet 7B.4. In scenario 1, bonds go down (negative deviation) while stocks go up (positive deviation). In scenario 3, bonds are up, but stocks are down. When the rates move in opposite directions, as in this case, the product of the deviations is negative; conversely, if the rates moved in the same direction, the sign of the product would be positive. The magnitude of the product shows the extent of the opposite or common movement in that scenario. The probability-weighted average of these products therefore summarizes the average tendency for the variables to co-vary across scenarios. In the last line of the spreadsheet, we see that the covariance is 280 (cell H6). Suppose our scenario analysis had envisioned stocks generally moving in the same direction as bonds. To be concrete, let’s switch the forecast rates on stocks in the first and third scenarios, that is, let the stock return be 210% in the first scenario and 30% in the third. In this case, the absolute value of both products of these scenarios remains the same, but the signs are positive, and thus the covariance is positive, at 180, reflecting the tendency for both asset returns to vary in tandem. If the levels of the scenario returns change, the intensity of the covariation also may change, as reflected by the magnitude of the product of deviations. The change in the magnitude of the covariance quantifies the change in both direction and intensity of the covariation. If there is no comovement at all, because positive and negative products are equally likely, the covariance is zero. Also, if one of the assets is risk-free, its covariance with any risky asset is zero, because its deviations from its mean are identically zero. The computation of covariance using Excel can be made easy by using the last line in Equation 7B.9. The first term, E(rD 3 rE), can be computed in one stroke using Excel’s SUMPRODUCT function. Specifically, in Spreadsheet 7B.4, SUMPRODUCT(A3:A5, B3:B5, C3:C5) multiplies the probability times the return on debt times the return on equity in each scenario and then sums those three products. Notice that adding D to each rate would not change the covariance because deviations from the mean would remain unchanged. But if you multiply either of the variables by a fixed factor, the covariance will increase by that factor. Multiplying both variables results in a covariance multiplied by the products of the factors because Cov (wD rD , wE rE ) 5 E 5 3 wD rD 2 wD E (rD)4 3 wE rE 2 wE E (rE )46 5 wDwE Cov(rD, rE)

(7B.10)

The covariance in Equation 7B.10 is actually the term that we add (twice) in the last line of the equation for portfolio variance, Equation 7B.8. So we find that portfolio variance is the weighted sum (not average) of the individual asset variances, plus twice their covariance weighted by the two portfolio weights (wD3wE). Like variance, the dimension (unit) of covariance is percent squared. But here we cannot get to a more easily interpreted dimension by taking the square root, because the average product of deviations can be negative, as it was in Spreadsheet 7B.4. The solution in this case is to scale the covariance by the standard deviations of the two variables, producing the correlation coefficient.

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Three-scenario analysis for stocks and bonds

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Correlation Coefficient Dividing the covariance by the product of the standard deviations of the variables will generate a pure number called correlation. We define correlation as follows: Corr (rD , rE ) 5

Cov (rD , rE ) sD sE

(7B.11)

The correlation coefficient must fall within the range [21,1]. This can be explained as follows. What two variables should have the highest degree comovement? Logic says a variable with itself, so let’s check it out.

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Cov (rD , rD) 5 E 5 3 rD 2 E (rD)4 3 3 rD 2 E (rD )46 5 E 3 rD 2 E (rD )4 2 5 sD2 sD2 Cov (rD , rD ) Corr (rD , rD) 5 5 2 51 sD sD sD

(7B.12)

Similarly, the lowest (most negative) value of the correlation coefficient is 21. (Check this for yourself by finding the correlation of a variable with its own negative.) An important property of the correlation coefficient is that it is unaffected by both addition and multiplication. Suppose we start with a return on debt, rD, multiply it by a constant, wD, and then add a fixed amount D. The correlation with equity is unaffected: Corr ( D 1 wD rD , rE ) 5

Cov (D 1 wDrD, rE )

"Var ( D 1 wD rD ) 3 sE wD Cov (rD , rE ) wD Cov (rD , rE ) 5 5 wD sD 3 sE "wD2 sD2 3 sE 5 Corr(rD, rE)

(7B.13)

Because the correlation coefficient gives more intuition about the relationship between rates of return, we sometimes express the covariance in terms of the correlation coefficient. Rearranging Equation 7B.11, we can write covariance as Cov (rD , rE ) 5 sD sE Corr (rD , rE )

Example 7B.4

(7B.14)

Calculating Covariance and Correlation

Spreadsheet 7B.5 shows the covariance and correlation between stocks and bonds using the same scenario analysis as in the other examples in this appendix. Covariance is calculated using Equation 7B.9. The SUMPRODUCT function used in cell J22 gives us E(rD 3 rE), from which we subtract E(rD) 3 E(rE) (i.e., we subtract J203K20). Then we calculate correlation in cell J23 by dividing covariance by the product of the asset standard deviations.

Portfolio Variance We have seen in Equation 7B.8, with the help of Equation 7B.10, that the variance of a two-asset portfolio is the sum of the individual variances multiplied by the square of the

CHAPTER 7 H

I

J

K

L

Optimal Risky Portfolios

M

Scenario rates of return

15

Scenario

Probability

rD(i)

rE(i)

16

1

0.14

-0.10

-0.35

17

2

0.36

0.00

0.20

18

3

0.30

0.10

0.45

19

4

0.20

0.32

-0.19

20

Mean

21

SD

22

Covariance

-0.0034

23

Correlation

-0.0847

24 Cell J22 25 Cell J23

=SUMPRODUCT(I16:I19,J16:J19,K16:K19)-J20*K20

0.08

0.12

0.1359

0.2918

eXcel Please visit us at www.mhhe.com/bkm

13 14

255

=J22/(J21*K21)

Spreadsheet 7B.5 Scenario analysis for bonds and stocks

Example 7B.5

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portfolio weights, plus twice the covariance between the rates, multiplied by the product of the portfolio weights: sP2 5 wD2 sD2 1 wE2 sE2 1 2wD wE Cov (rD , rE ) (7B.15) 2 2 2 2 ( ) 5 wDsD 1 wE sE 1 2wD wE sD sE Corr rD , rE

Calculating Portfolio Variance

We calculate portfolio variance in Spreadsheet 7B.6. Notice there that we calculate the portfolio standard deviation in two ways: once from the scenario portfolio returns (cell E35) and again (in cell E36) using the first line of Equation 7B.15. The two approaches yield the same result. You should try to repeat the second calculation using the correlation coefficient from the second line in Equation 7B.15 instead of covariance in the formula for portfolio variance. Suppose that one of the assets, say, E, is replaced with a money market instrument, that is, a risk-free asset. The variance of E is then zero, as is the covariance with D. In that case, as seen from Equation 7B.15, the portfolio standard deviation is just wDsD. In other words, when we mix a risky portfolio with the risk-free asset, portfolio standard deviation equals the risky asset’s standard deviation times the weight invested in that asset. This result was used extensively in Chapter 6. D E A B C 25 26 27 28 Scenario rates of return Portfolio return rE(i) 0.4*rD(i)+0.6rE(i) rD(i) 29 Scenario Probability 0.14 1 -0.10 -0.35 -0.25 30 0.36 2 0.00 0.20 0.12 31 0.30 3 0.10 0.45 0.31 32 0.20 4 0.32 -0.19 0.014 33 Mean 0.08 0.12 0.1040 34 SD 0.1359 0.2918 0.1788 35 Covariance SD: 0.1788 -0.0034 36 Correlation -0.0847 37 38 Cell E35 =SUMPRODUCT(B30:B33,E30:E33,E30:E33)-E34^2)^0.5 39 Cell E36 =(0.4*C35)^2+(0.6*D35)^2+2*0.4*0.6*C36)^0.5

Spreadsheet 7B.6 Scenario analysis for bonds and stocks

F

G

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PART II

8

THE MARKOWITZ PROCEDURE introduced in the preceding chapter suffers from two drawbacks. First, the model requires a huge number of estimates to fill the covariance matrix. Second, the model does not provide any guideline to the forecasting of the security risk premiums that are essential to construct the efficient frontier of risky assets. Because past returns are unreliable guides to expected future returns, this drawback can be telling. In this chapter we introduce index models that simplify estimation of the covariance matrix and greatly enhance the analysis of security risk premiums. By allowing us to explicitly decompose risk into systematic and firm-specific components, these models also shed considerable light on both the power and the limits of diversification. Further, they allow us to measure these components of risk for particular securities and portfolios. We begin the chapter by describing a single-factor security market and show how it can justify a single-index model of security returns. Once its properties are analyzed, we proceed to an extensive example of estimation of the single-index model. We review the

CHAPTER EIGHT

Index Models

statistical properties of these estimates and show how they relate to the practical issues facing portfolio managers. Despite the simplification they offer, index models remain true to the concepts of the efficient frontier and portfolio optimization. Empirically, index models are as valid as the assumption of normality of the rates of return on available securities. To the extent that short-term returns are well approximated by normal distributions, index models can be used to select optimal portfolios nearly as accurately as the Markowitz algorithm. Finally, we examine optimal risky portfolios constructed using the index model. While the principles are the same as those employed in the previous chapter, the properties of the portfolio are easier to derive and interpret in this context. We illustrate how to use the index model by constructing an optimal risky portfolio using a small sample of firms. This portfolio is compared to the corresponding portfolio constructed from the Markowitz model. We conclude with a discussion of several practical issues that arise when implementing the index model.

CHAPTER 8

8.1

Index Models

A Single-Factor Security Market

The Input List of the Markowitz Model The success of a portfolio selection rule depends on the quality of the input list, that is, the estimates of expected security returns and the covariance matrix. In the long run, efficient portfolios will beat portfolios with less reliable input lists and consequently inferior reward-to-risk trade-offs. Suppose your security analysts can thoroughly analyze 50 stocks. This means that your input list will include the following: n5 n5

50 estimates of expected returns

50 estimates of variances (n 2 n)/2 5 1,225 estimates of covariances 1,325 total estimates 2

This is a formidable task, particularly in light of the fact that a 50-security portfolio is relatively small. Doubling n to 100 will nearly quadruple the number of estimates to 5,150. If n 53,000, roughly the number of NYSE stocks, we need more than 4.5 million estimates. Another difficulty in applying the Markowitz model to portfolio optimization is that errors in the assessment or estimation of correlation coefficients can lead to nonsensical results. This can happen because some sets of correlation coefficients are mutually inconsistent, as the following example demonstrates:1 Correlation Matrix

Asset

Standard Deviation (%)

A

B

C

A B C

20 20 20

1.00 0.90 0.90

0.90 1.00 0.00

0.90 0.00 1.00

Suppose that you construct a portfolio with weights 21.00; 1.00; 1.00, for assets A; B; C, respectively, and calculate the portfolio variance. You will find that the portfolio variance appears to be negative (2200). This of course is not possible because portfolio variances cannot be negative: We conclude that the inputs in the estimated correlation matrix must be mutually inconsistent. Of course, true correlation coefficients are always consistent.2 But we do not know these true correlations and can only estimate them with some imprecision. Unfortunately, it is difficult to determine at a quick glance whether a correlation matrix is inconsistent, providing another motivation to seek a model that is easier to implement. Introducing a model that simplifies the way we describe the sources of security risk allows us to use a smaller, consistent set of estimates of risk parameters and risk premiums. The simplification emerges because positive covariances among security returns arise from common economic forces that affect the fortunes of most firms. Some examples of common economic factors are business cycles, interest rates, and the cost of natural resources. The unexpected changes in these variables cause, simultaneously, unexpected 1

We are grateful to Andrew Kaplin and Ravi Jagannathan, Kellogg Graduate School of Management, Northwestern University, for this example. 2 The mathematical term for a correlation matrix that cannot generate negative portfolio variance is “positive definite.”

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changes in the rates of return on the entire stock market. By decomposing uncertainty into these systemwide versus firm-specific sources, we vastly simplify the problem of estimating covariance and correlation.

Normality of Returns and Systematic Risk We can always decompose the rate of return on any security, i, into the sum of its expected plus unanticipated components: ri 5 E (ri ) 1 ei

(8.1)

where the unexpected return, ei , has a mean of zero and a standard deviation of si that measures the uncertainty about the security return. When security returns can be well approximated by normal distributions that are correlated across securities, we say that they are joint normally distributed. This assumption alone implies that, at any time, security returns are driven by one or more common variables. When more than one variable drives normally distributed security returns, these returns are said to have a multivariate normal distribution. We begin with the simpler case where only one variable drives the joint normally distributed returns, resulting in a single-factor security market. Extension to the multivariate case is straightforward and is discussed in later chapters. Suppose the common factor, m, that drives innovations in security returns is some macroeconomic variable that affects all firms. Then we can decompose the sources of uncertainty into uncertainty about the economy as a whole, which is captured by m, and uncertainty about the firm in particular, which is captured by ei. In this case, we amend Equation 8.1 to accommodate two sources of variation in return: ri 5 E(ri ) 1 m 1 ei

(8.2)

The macroeconomic factor, m, measures unanticipated macro surprises. As such, it has a mean of zero (over time, surprises will average out to zero), with standard deviation of sm. In contrast, ei measures only the firm-specific surprise. Notice that m has no subscript because the same common factor affects all securities. Most important is the fact that m and ei are uncorrelated, that is, because ei is firm-specific, it is independent of shocks to the common factor that affect the entire economy. The variance of ri thus arises from two uncorrelated sources, systematic and firm specific. Therefore, s 2i 5 s 2m 1 s 2 (ei )

(8.3)

The common factor, m, generates correlation across securities, because all securities will respond to the same macroeconomic news, while the firm-specific surprises, captured by ei , are assumed to be uncorrelated across firms. Because m is also uncorrelated with any of the firm-specific surprises, the covariance between any two securities i and j is Cov (ri, rj) 5 Cov (m 1 ei , m 1 ej ) 5 s 2m

(8.4)

Finally, we recognize that some securities will be more sensitive than others to macroeconomic shocks. For example, auto firms might respond more dramatically to changes in general economic conditions than pharmaceutical firms. We can capture this refinement by assigning each firm a sensitivity coefficient to macro conditions. Therefore, if we denote the sensitivity coefficient for firm i by the Greek letter beta, bi, we modify Equation 8.2 to obtain the single-factor model: ri 5 E (ri ) 1 bi m 1 e i

(8.5)

CHAPTER 8

Index Models

Equation 8.5 tells us the systematic risk of security i is determined by its beta coefficient. “Cyclical” firms have greater sensitivity to the market and therefore higher systematic risk. The systematic risk of security i is b 2i s 2m , and its total risk is s 2i 5 b 2i s 2m 1 s 2 (ei )

(8.6)

The covariance between any pair of securities also is determined by their betas: Cov (ri , rj ) 5 Cov (b i m 1 e i , b j m 1 ej ) 5 b i b j s 2m

(8.7)

In terms of systematic risk and market exposure, this equation tells us that firms are close substitutes. Equivalent beta securities give equivalent market exposures. Up to this point we have used only statistical implications from the joint normality of security returns. Normality of security returns alone guarantees that portfolio returns are also normal (from the “stability” of the normal distribution discussed in Chapter 5) and that there is a linear relationship between security returns and the common factor. This greatly simplifies portfolio analysis. Statistical analysis, however, does not identify the common factor, nor does it specify how that factor might operate over a longer investment period. However, it seems plausible (and can be empirically verified) that the variance of the common factor usually changes relatively slowly through time, as do the variances of individual securities and the covariances among them. We seek a variable that can proxy for this common factor. To be useful, this variable must be observable, so we can estimate its volatility as well as the sensitivity of individual securities returns to variation in its value.

8.2

The Single-Index Model

A reasonable approach to making the single-factor model operational is to assert that the rate of return on a broad index of securities such as the S&P 500 is a valid proxy for the common macroeconomic factor. This approach leads to an equation similar to the singlefactor model, which is called a single-index model because it uses the market index to proxy for the common factor.

The Regression Equation of the Single-Index Model Because rates of return on market indexes such as the S&P 500 can be observed, we have a considerable amount of past data with which to estimate systematic risk. We denote the market index by M, with excess return of RM 5 rM 2 rf , and standard deviation of sM. Because the index model is linear, we can estimate the sensitivity (or beta) coefficient of a security on the index using a single-variable linear regression. We regress the excess return of a security, Ri5ri2rf , on the excess return of the index, RM. To estimate the regression, we collect a historical sample of paired observations, Ri (t) and RM (t), where t denotes the date of each pair of observations (e.g., the excess returns on the stock and the index in a particular month).3 The regression equation is Ri (t ) 5 a i 1 b i R M ( t ) 1 ei ( t )

(8.8)

The intercept of this equation (denoted by the Greek letter alpha, or a) is the security’s expected excess return when the market excess return is zero. The slope coefficient, bi, is 3

Practitioners often use a “modified” index model that is similar to Equation 8.8 but that uses total rather than excess returns. This practice is most common when daily data are used. In this case the rate of return on bills is on the order of only about .01% per day, so total and excess returns are almost indistinguishable.

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the security beta. Beta is the security’s sensitivity to the index: It is the amount by which the security return tends to increase or decrease for every 1% increase or decrease in the return on the index. ei is the zero-mean, firm-specific surprise in the security return in time t, also called the residual.

The Expected Return–Beta Relationship Because E (ei ) 50, if we take the expected value of E (Ri ) in Equation 8.8, we obtain the expected return–beta relationship of the single-index model: E (Ri ) 5 a i 1 b i E (RM )

(8.9)

The second term in Equation 8.9 tells us that part of a security’s risk premium is due to the risk premium of the index. The market risk premium is multiplied by the relative sensitivity, or beta, of the individual security. We call this the systematic risk premium because it derives from the risk premium that characterizes the entire market, which proxies for the condition of the full economy or economic system. The remainder of the risk premium is given by the first term in the equation, a. Alpha is a nonmarket premium. For example, a may be large if you think a security is underpriced and therefore offers an attractive expected return. Later on, we will see that when security prices are in equilibrium, such attractive opportunities ought to be competed away, in which case a will be driven to zero. But for now, let’s assume that each security analyst comes up with his or her own estimates of alpha. If managers believe that they can do a superior job of security analysis, then they will be confident in their ability to find stocks with nonzero values of alpha. We will see shortly that the index model decomposition of an individual security’s risk premium to market and nonmarket components greatly clarifies and simplifies the operation of macroeconomic and security analysis within an investment company.

Risk and Covariance in the Single-Index Model Remember that one of the problems with the Markowitz model is the overwhelming number of parameter estimates required to implement it. Now we will see that the index model simplification vastly reduces the number of parameters that must be estimated. Equation 8.8 yields the systematic and firm-specific components of the overall risk of each security, and the covariance between any pair of securities. Both variances and covariances are determined by the security betas and the properties of the market index: Total risk 5 Systematic risk 1 Firm-specific risk s 2i 5 b 2i s 2M 1 s 2 ( ei ) Covariance 5 Product of betas 3 Market-index risk Cov (ri , rj ) 5 b i b j s 2M

(8.10)

Correlation 5 Product of correlations with the market index Corr (ri , rj ) 5

b i b j s 2M sisj

5

b i s 2M b j s 2M sisMsjsM

5 Corr (ri , rM ) 3 Corr (rj , rM )

Equations 8.9 and 8.10 imply that the set of parameter estimates needed for the singleindex model consists of only a,b, and s(e) for the individual securities, plus the risk premium and variance of the market index.

CHAPTER 8

CONCEPT CHECK

Index Models

8.1

The data below describe a three-stock financial market that satisfies the single-index model.

Stock

Capitalization

Beta

Mean Excess Return

Standard Deviation

A B C

$3,000 $1,940 $1,360

1.0 0.2 1.7

10% 2 17

40% 30 50

The standard deviation of the market-index portfolio is 25%. a. What is the mean excess return of the index portfolio? b. What is the covariance between stock A and stock B? c. What is the covariance between stock B and the index? d. Break down the variance of stock B into its systematic and firm-specific components.

The Set of Estimates Needed for the Single-Index Model We summarize the results for the single-index model in the table below. Symbol 1. The stock’s expected return if the market is neutral, that is, if the market’s excess return, rM2rf, is zero 2. The component of return due to movements in the overall market; bi is the security’s responsiveness to market movements 3. The unexpected component of return due to unexpected events that are relevant only to this security (firm specific) 4. The variance attributable to the uncertainty of the common macroeconomic factor 5. The variance attributable to firm-specific uncertainty

ai bi (rM2rf) ei b2i s2M s2(ei)

These calculations show that if we have: • • • • •

n estimates of the extra-market expected excess returns, ai n estimates of the sensitivity coefficients, bi n estimates of the firm-specific variances, s2(ei ) 1 estimate for the market risk premium, E (RM) 1 estimate for the variance of the (common) macroeconomic factor, s 2M

then these (3n12) estimates will enable us to prepare the entire input list for this singleindex-security universe. Thus for a 50-security portfolio we will need 152 estimates rather than 1,325; for the entire New York Stock Exchange, about 3,000 securities, we will need 9,002 estimates rather than approximately 4.5 million! It is easy to see why the index model is such a useful abstraction. For large universes of securities, the number of estimates required for the Markowitz procedure using the index model is only a small fraction of what otherwise would be needed. Another advantage is less obvious but equally important. The index model abstraction is crucial for specialization of effort in security analysis. If a covariance term had to be

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calculated directly for each security pair, then security analysts could not specialize by industry. For example, if one group were to specialize in the computer industry and another in the auto industry, who would have the common background to estimate the covariance between IBM and GM? Neither group would have the deep understanding of other industries necessary to make an informed judgment of co-movements among industries. In contrast, the index model suggests a simple way to compute covariances. Covariances among securities are due to the influence of the single common factor, represented by the market index return, and can be easily estimated using the regression Equation 8.8. The simplification derived from the index model assumption is, however, not without cost. The “cost” of the model lies in the restrictions it places on the structure of asset return uncertainty. The classification of uncertainty into a simple dichotomy—macro versus micro risk—oversimplifies sources of real-world uncertainty and misses some important sources of dependence in stock returns. For example, this dichotomy rules out industry events, events that may affect many firms within an industry without substantially affecting the broad macroeconomy. This last point is potentially important. Imagine that the single-index model is perfectly accurate, except that the residuals of two stocks, say, British Petroleum (BP) and Royal Dutch Shell, are correlated. The index model will ignore this correlation (it will assume it is zero), while the Markowitz algorithm (which accounts for the full covariance between every pair of stocks) will automatically take the residual correlation into account when minimizing portfolio variance. If the universe of securities from which we must construct the optimal portfolio is small, the two models will yield substantively different optimal portfolios. The portfolio of the Markowitz algorithm will place a smaller weight on both BP and Shell (because their mutual covariance reduces their diversification value), resulting in a portfolio with lower variance. Conversely, when correlation among residuals is negative, the index model will ignore the potential diversification value of these securities. The resulting CONCEPT CHECK 8.2 “optimal” portfolio will place too little weight on these securities, resulting in an unnecessarily high variance. Suppose that the index model for the excess The optimal portfolio derived from the single-index returns of stocks A and B is estimated with the model therefore can be significantly inferior to that of following results: the full-covariance (Markowitz) model when stocks with correlated residuals have large alpha values and account RA 5 1.0% 1 .9RM 1 eA for a large fraction of the portfolio. If many pairs of the RB 5 22.0% 1 1.1RM 1 eB covered stocks exhibit residual correlation, it is possible sM 5 20% that a multi-index model, which includes additional facs(eA) 5 30% tors to capture those extra sources of cross-security cors(eB) 5 10% relation, would be better suited for portfolio analysis and construction. We will demonstrate the effect of correlated Find the standard deviation of each stock and residuals in the spreadsheet example in this chapter, and the covariance between them. discuss multi-index models in later chapters.

The Index Model and Diversification The index model, first suggested by Sharpe,4 also offers insight into portfolio diversification. Suppose that we choose an equally weighted portfolio of n securities. The excess rate of return on each security is given by Ri 5 a i 1 b i R M 1 e i 4

William F. Sharpe, “A Simplified Model of Portfolio Analysis,” Management Science, January 1963.

CHAPTER 8

Index Models

Similarly, we can write the excess return on the portfolio of stocks as RP 5 a P 1 bP R M 1 eP

(8.11)

We now show that, as the number of stocks included in this portfolio increases, the part of the portfolio risk attributable to nonmarket factors becomes ever smaller. This part of the risk is diversified away. In contrast, market risk remains, regardless of the number of firms combined into the portfolio. To understand these results, note that the excess rate of return on this equally weighted portfolio, for which each portfolio weight wi51/n, is n 1 n 1 n RP 5 a wi Ri 5 n a Ri 5 n a (ai 1 bi R M 1 e i ) i51 i51 i51

1 n 1 n 1 n 5 n a ai 1 a n a bi bRM 1 n a ei i51 i51 i51

(8.12)

Comparing Equations 8.11 and 8.12, we see that the portfolio has a sensitivity to the market given by 1 n bP 5 n a bi i51

(8.13)

which is the average of the individual bis. It has a nonmarket return component of 1 n aP 5 n a ai i51

(8.14)

which is the average of the individual alphas, plus the zero mean variable 1 n eP 5 n a ei i51

(8.15)

which is the average of the firm-specific components. Hence the portfolio’s variance is s 2P 5 b 2P s 2M 1 s 2 (eP)

(8.16)

The systematic risk component of the portfolio variance, which we defined as the component that depends on marketwide movements, is b 2P s 2M and depends on the sensitivity coefficients of the individual securities. This part of the risk depends on portfolio beta and s 2M and will persist regardless of the extent of portfolio diversification. No matter how many stocks are held, their common exposure to the market will be reflected in portfolio systematic risk.5 In contrast, the nonsystematic component of the portfolio variance is s2(eP) and is attributable to firm-specific components, ei. Because these eis are independent, and all have zero expected value, the law of averages can be applied to conclude that as more and more stocks are added to the portfolio, the firm-specific components tend to cancel out, resulting in ever-smaller nonmarket risk. Such risk is thus termed diversifiable. To see this more rigorously, examine the formula for the variance of the equally weighted “portfolio” of firm-specific components. Because the eis are uncorrelated, n 1 2 1 –2 (e) s 2( e P ) 5 a a n b s 2 (e i ) 5 n s i51

(8.17)

– 2(e) is the average of the firm-specific variances. Because this average is indepenwhere s dent of n, when n gets large, s2(eP) becomes negligible. 5 One can construct a portfolio with zero systematic risk by mixing negative b and positive b assets. The point of our discussion is that the vast majority of securities have a positive b, implying that well-diversified portfolios with small holdings in large numbers of assets will indeed have positive systematic risk.

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σ 2P

Diversiﬁable Risk

σ2 (eP) 5 σ2(e)/ n β2P σ2M Systematic Risk n

Figure 8.1 The variance of an equally weighted portfolio with risk coefficient bP in the single-factor economy

To summarize, as diversification increases, the total variance of a portfolio approaches the systematic variance, defined as the variance of the market factor multiplied by the square of the portfolio sensitivity coefficient, b 2P. This is shown in Figure8.1. Figure8.1 shows that as more and more securities are combined into a portfolio, the portfolio variance decreases because of the diversification of firm-specific risk. However, the power of diversification is limited. Even for very large n, part of the risk remains because of the exposure of virtually all assets to the common, or market, factor. Therefore, this systematic risk is CONCEPT CHECK 8.3 said to be nondiversifiable. This analysis is borne out by empirical evidence. We Reconsider the two stocks in Concept Check 2. saw the effect of portfolio diversification on portfolio Suppose we form an equally weighted portfolio standard deviations in Figure 7.2. These empirical results of A and B. What will be the nonsystematic stanare similar to the theoretical graph presented here in dard deviation of that portfolio? Figure8.1.

8.3

Estimating the Single-Index Model Armed with the theoretical underpinnings of the single-index model, we now provide an extended example that begins with estimation of the regression equation (8.8) and continues through to the estimation of the full covariance matrix of security returns. To keep the presentation manageable, we focus on only six large U.S. corporations: Hewlett-Packard and Dell from the information technology (IT) sector of the S&P 500, Target and Walmart from the retailing sector, and British Petroleum and Royal Dutch Shell from the energy sector.

CHAPTER 8

265

Index Models

We work with monthly observations of rates of return for the six stocks, the S&P 500 portfolio, and T-bills over a 5-year period (60 observations). As a first step, the excess returns on the seven risky assets are computed. We start with a detailed look at the preparation of the input list for Hewlett-Packard (HP), and then proceed to display the entire input list. Later in the chapter, we will show how these estimates can be used to construct the optimal risky portfolio.

The Security Characteristic Line for Hewlett-Packard The index model regression Equation 8.8 restated for Hewlett-Packard (HP) is RHP(t)5aHP1bHPRS&P500 (t)1eHP (t) The equation describes the (linear) dependence of HP’s excess return on changes in the state of the economy as represented by the excess returns of the S&P 500 index portfolio. The regression estimates describe a straight line with intercept aHP and slope bHP, which we call the security characteristic line (SCL) for HP. Figure8.2 shows a graph of the excess returns on HP and the S&P 500 portfolio over the 60-month period. The graph shows that HP returns generally follow those of the index, but with much larger swings. Indeed, the annualized standard deviation of the excess return on the S&P 500 portfolio over the period was 13.58%, while that of HP was 38.17%. The swings in HP’s excess returns suggest a greater-than-average sensitivity to the index, that is, a beta greater than 1.0. The relationship between the returns of HP and the S&P 500 is made clearer by the scatter diagram in Figure8.3, where the regression line is drawn through the scatter. The vertical distance of each point from the regression line is the value of HP’s residual, eHP(t), corresponding to that particular month. The rates in Figure 8.2 are not annualized, and the scatter diagram shows monthly swings of over 6 30% for HP, but returns in the range of 211% to 8.5% for the S&P 500. The regression analysis output obtained by using Excel is shown in Table 8.1. .4

S&P 500 HP

The Explanatory Power of the SCL for HP Considering the top panel of Table 8.1 first, we see that the correlation of HP with the S&P 500 is quite high (.7238), telling us that HP tracks changes in the returns of the S&P 500 fairly closely. The R-square (.5239) tells us that variation in the S&P 500 excess returns explains about 52% of the variation in the HP series. The adjusted R-square (which is slightly smaller) corrects for an upward bias in R-square that arises because we use the fitted values of two parameters,6 the slope

Excess Returns (%)

.3 .2 .1 .0 −.1 −.2 −.3 −.4 0

10

20

30

40

Observation Month

Figure 8.2 Excess returns on HP and S&P 500

n21 , where k n2k21 is the number of independent variables (here, k51). An additional degree of freedom is lost to the estimate of the intercept. In general, the adjusted R-square (R 2A) is derived from the unadjusted by R2A 5 1 2 (1 2 R2)

6

50

60

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.4 .3

Excess Return, HP

.2 .1 0 −.15

−.10

−.05

.05

.10

−.1 −.2 −.3 −.4 Excess Returns, S&P 500

Figure 8.3 Scatter diagram of HP, the S&P 500, and the security characteristic line (SCL) for HP

(beta) and intercept (alpha), rather than their true, but unobservable, values. With 60 observations, this bias is small. The standard error of the regression is the standard deviation of the residual, which we discuss in more detail shortly. This is a measure of the slippage in the average relationship between the stock and the index due to the impact of firm-specific factors, and is based on in-sample data. A more severe test is to look at returns from periods after the one covered by the regression sample and test the power of the independent variable (the S&P500) to predict the dependent variable (the return on HP). Correlation between regression forecasts and realizations of out-of-sample data is almost always considerably lower than in-sample correlation.

Analysis of Variance A

The next panel of Table 8.1 shows the analysis of variance (ANOVA) for the SCL. The sum of squares (SS) of the regression (.3752) is the portion of the variance of the dependent variable (HP’s return) that is explained by the independent variable (the S&P 500 return); it is equal to b2HPs2S&P 500. The MS column for the residual (.0059) shows the variance of the unexplained portion of HP’s return, that is, the portion of return that is independent of the market index. The square root of this value is the standard error (SE) of the regression (.0767) reported in the first panel. If you divide the total SS of the regression (.7162) by 59,

Table 8.1 Excel output: Regression statistics for the SCL of Hewlett-Packard

Regression Statistics Multiple R R-square Adjusted R-square Standard error Observations

.7238 .5239 .5157 .0767 60

ANOVA

Regression Residual Total

Intercept S&P 500

df

SS

MS

1 58 59

.3752 .3410 .7162

.3752 .0059

Coefficients

Standard Error

t-Stat

p-Value

0.0086 2.0348

.0099 .2547

0.8719 7.9888

.3868 .0000

CHAPTER 8

Index Models

you will obtain the estimate of the variance of the dependent variable (HP), .012 per month, equivalent to a monthly standard deviation of 11%. When it is annualized,7 we obtain an annualized standard deviation of 38.17%, as reported earlier. Notice that the R-square (the ratio of explained to total variance) equals the explained (regression) SS divided by the total SS.8

The Estimate of Alpha We move to the bottom panel. The intercept (.0086 5.86% per month) is the estimate of HP’s alpha for the sample period. Although this is an economically large value (10.32% on an annual basis), it is statistically insignificant. This can be seen from the three statistics next to the estimated coefficient. The first is the standard error of the estimate (0.0099).9 This is a measure of the imprecision of the estimate. If the standard error is large, the range of likely estimation error is correspondingly large. The t-statistic reported in the bottom panel is the ratio of the regression parameter to its standard error. This statistic equals the number of standard errors by which our estimate exceeds zero, and therefore can be used to assess the likelihood that the true but unobserved value might actually equal zero rather than the estimate derived from the data.10 The intuition is that if the true value were zero, we would be unlikely to observe estimated values far away (i.e., many standard errors) from zero. So large t-statistics imply low probabilities that the true value is zero. In the case of alpha, we are interested in the average value of HP’s return net of the impact of market movements. Suppose we define the nonmarket component of HP’s return as its actual return minus the return attributable to market movements during any period. Call this HP’s firm-specific return, which we abbreviate as Rfs. Rfirm-specific 5 Rfs 5 RHP 2 bHPRS&P500 If Rfs were normally distributed with a mean of zero, the ratio of its estimate to its standard error would have a t-distribution. From a table of the t-distribution (or using Excel’s TINV function) we can find the probability that the true alpha is actually zero or even lower given the positive estimate of its value and the standard error of the estimate. This is called the level of significance or, as in Table 8.1, the probability or p-value. The conventional cutoff for statistical significance is a probability of less than 5%, which requires a t-statistic of about 2.0. The regression output shows the t-statistic for HP’s alpha

7

When monthly data are annualized, average return and variance are multiplied by 12. However, because variance is multiplied by 12, standard deviation is multiplied by "12. R-Square 5

8

b2HPs2S&P500 2 2 bHPsS&P500 1 s2(eHP)

5

.3752 5 .5239 .7162

Equivalently, R-square equals 1 minus the fraction of variance that is not explained by market returns, i.e., 1 minus the ratio of firm-specific risk to total risk. For HP, this is 12

s2(eHP) b2HPs2S&P500

1 s2(eHP)

512

.3410 5 .5239 .7162

9

We can relate the standard error of the alpha estimate to the standard error of the residuals as follows: SE(aHP) 5 s(eHP)

10

(AvgS&P500)2 1 1 Var(S&P500) 3 (n 2 1) Ån

The t-statistic is based on the assumption that returns are normally distributed. In general, if we standardize the estimate of a normally distributed variable by computing its difference from a hypothesized value and dividing by the standard error of the estimate (to express the difference as a number of standard errors), the resulting variable will have a t-distribution. With a large number of observations, the bell-shaped t-distribution approaches the normal distribution.

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to be .8719, indicating that the estimate is not significantly different from zero. That is, we cannot reject the hypothesis that the true value of alpha equals zero with an acceptable level of confidence. The p-value for the alpha estimate (.3868) indicates that if the true alpha were zero, the probability of obtaining an estimate as high as .0086 (given the large standard error of .0099) would be .3868, which is not so unlikely. We conclude that the sample average of Rfs is too low to reject the hypothesis that the true value of alpha is zero. But even if the alpha value were both economically and statistically significant within the sample, we still would not use that alpha as a forecast for a future period. Overwhelming empirical evidence shows that 5-year alpha values do not persist over time, that is, there seems to be virtually no correlation between estimates from one sample period to the next. In other words, while the alpha estimated from the regression tells us the average return on the security when the market was flat during that estimation period, it does not forecast what the firm’s performance will be in future periods. This is why security analysis is so hard. The past does not readily foretell the future. We elaborate on this issue in Chapter 11 on market efficiency.

The Estimate of Beta The regression output in Table 8.1 shows the beta estimate for HP to be 2.0348, more than twice that of the S&P 500. Such high market sensitivity is not unusual for technology stocks. The standard error (SE) of the estimate is .2547.11 The value of beta and its SE produce a large t-statistic (7.9888), and a p-value of practically zero. We can confidently reject the hypothesis that HP’s true beta is zero. A more interesting t-statistic might test a null hypothesis that HP’s beta is greater than the marketwide average beta of 1. This t-statistic would measure how many standard errors separate the estimated beta from a hypothesized value of 1. Here too, the difference is easily large enough to achieve statistical significance: Estimated value 2 Hypothesized value 2.03 2 1 5 5 4.00 Standard error .2547 However, we should bear in mind that even here, precision is not what we might like it to be. For example, if we wanted to construct a confidence interval that includes the true but unobserved value of beta with 95% probability, we would take the estimated value as the center of the interval and then add and subtract about two standard errors. This produces a range between 1.43 and 2.53, which is quite wide.

Firm-Specific Risk The monthly standard deviation of HP’s residual is 7.67%, or 26.6% annually. This is quite large, on top of HP’s already high systematic risk. The standard deviation of systematic risk is b3s (S&P500)52.03313.58527.57%. Notice that HP’s firm-specific risk is as large as its systematic risk, a common result for individual stocks.

Correlation and Covariance Matrix Figure8.4 graphs the excess returns of the pairs of securities from each of the three sectors with the S&P 500 index on the same scale. We see that the IT sector is the most variable, followed by the retail sector, and then the energy sector, which has the lowest volatility. Panel 1 in Spreadsheet8.1 shows the estimates of the risk parameters of the S&P 500 portfolio and the six analyzed securities. You can see from the high residual standard SE(b) 5

11

s (eHP) sHP"n 2 1

CHAPTER 8

Index Models

A

Monthly Rates (%)

.3 .2 .1

S&P 500 HP DELL

.0 −.1 −.2 −.3 0

10

20

30 Month

40

50

60

B

Monthly Rates (%)

.3 .2 S&P 500 WMT TARGET

.1 .0 −.1 −.2 −.3 0

10

20

30 Month

40

50

60

C

Monthly Rates (%)

.3 .2 S&P 500 BP SHELL

.1 .0 −.1 −.2 −.3 0

10

20

30 Month

40

50

60

Figure 8.4 Excess returns on portfolio assets

deviations (column E) how important diversification is. These securities have tremendous firm-specific risk. Portfolios concentrated in these (or other) securities would have unnecessarily high volatility and inferior Sharpe ratios. Panel 2 shows the correlation matrix of the residuals from the regressions of excess returns on the S&P 500. The shaded cells show correlations of same-sector stocks, which are as high as .7 for the two oil stocks (BP and Shell). This is in contrast to the assumption of the index model that all residuals are uncorrelated. Of course, these correlations are, to

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A 1 2

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

B

C

D

E

F

G

H

I

J

Panel 1: Risk Parameters of the Investable Universe (annualized) SD of Excess Return S&P 500 HP DELL WMT TARGET BP SHELL

0.1358 0.3817 0.2901 0.1935 0.2611 0.1822 0.1988

Beta 1.00 2.03 1.23 0.62 1.27 0.47 0.67

Correlation SD of with the Systematic SD of S&P 500 Component Residual 0.1358 1 0 0.2762 0.2656 0.72 0.1672 0.2392 0.58 0.0841 0.1757 0.43 0.1720 0.1981 0.66 0.0634 0.1722 0.35 0.0914 0.1780 0.46

Panel 2: Correlation of Residuals HP HP DELL WMT TARGET BP SHELL

1 0.08 20.34 20.10 20.20 20.06

DELL

WMT

1 0.17 0.12 20.28 20.19

TARGET

1 0.50 20.19 20.24

BP

1 20.13 20.22

1 0.70

DELL 1.23 0.0227 0.0462 0.0842 0.0141 0.0288 0.0106 0.0153

WMT 0.62 0.0114 0.0232 0.0141 0.0374 0.0145 0.0053 0.0077

Panel 3: The Index Model Covariance Matrix

Beta 1.00 2.03 1.23 0.62 1.27 0.47 0.67

S&P 500 HP DELL WMT TARGET BP SHELL

HP 2.03 0.0375 0.1457 0.0462 0.0232 0.0475 0.0175 0.0253

S&P 500 1.00 0.0184 0.0375 0.0227 0.0114 0.0234 0.0086 0.0124

TARGET 1.27 0.0234 0.0475 0.0288 0.0145 0.0682 0.0109 0.0157

BP 0.47 0.0086 0.0175 0.0106 0.0053 0.0109 0.0332 0.0058

SHELL 0.67 0.0124 0.0253 0.0153 0.0077 0.0157 0.0058 0.0395

Cells on the diagonal (shadowed) equal to variance formula in cell C26 5 B4^2 Off-diagonal cells equal to covariance ^ formula in cell C27 5 C$25*$B27*$B$4 2 multiplies beta from row and column by index variance

Panel 4: Macro Forecast and Forecasts of Alpha Values

Alpha Risk premium

S&P 500 0 0.0600

HP 0.0150 0.1371

DELL 20.0100 0.0639

WMT 20.0050 0.0322

TARGET 0.0075 0.0835

BP 0.012 0.0400

SHELL 0.0025 0.0429

Panel 5: Computation of the Optimal Risky Portfolio S&P 500 Active Pf A

s2(e) a/s2(e) w0(i) [w0(i)]2 aA s2(eA) w 0A w*(Risky portf) Beta Risk premium SD Sharpe ratio

HP 0.0705

DELL

WMT

TARGET

BP

SHELL

0.0572

0.0309

0.0392

0.0297

0.0317

0.5505

0.2126

20.1748

20.1619

0.1911

0.4045

0.0789

1.0000

0.3863

20.3176

20.2941

0.3472

0.7349

0.1433

0.1492

0.1009

0.0865

0.1205

0.5400

0.0205

2.0348 0.1371

1.2315 0.0639

0.6199 0.0322

1.2672 0.0835

0.4670 0.0400

0.6736 0.0429

Overall Pf

0.0222 0.0404 0.1691 0.8282 1 0.06 0.1358 0.44

0.1718 1.0922 0.0878 0.2497 0.35

Spreadsheet 8.1

eXcel

Implementing the index model

Please visit us at www.mhhe.com/bkm

1.0158 0.0648 0.1422 0.46

CHAPTER 8

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a great extent, high by design, because we selected pairs of firms from the same industry. Cross-industry correlations are typically far smaller, and the empirical estimates of correlations of residuals for industry indexes (rather than individual stocks in the same industry) would be far more in accord with the model. In fact, a few of the stocks in this sample actually seem to have negatively correlated residuals. Of course, correlation also is subject to statistical sampling error, and this may be a fluke. Panel 3 produces covariances derived from Equation 8.10 of the single-index model. Variances of the S&P 500 index and the individual covered stocks appear on the diagonal. The variance estimates for the individual stocks equal b2i s2M 1 s 2(ei). The off-diagonal 2 . terms are covariance values and equal bi bj sM

8.4

Portfolio Construction and the Single-Index Model

In this section, we look at the implications of the index model for portfolio construction.12 We will see that the model offers several advantages, not only in terms of parameter estimation, but also for the analytic simplification and organizational decentralization that it makes possible.

Alpha and Security Analysis Perhaps the most important advantage of the single-index model is the framework it provides for macroeconomic and security analysis in the preparation of the input list that is so critical to the efficiency of the optimal portfolio. The Markowitz model requires estimates of risk premiums for each security. The estimate of expected return depends on both macroeconomic and individual-firm forecasts. But if many different analysts perform security analysis for a large organization such as a mutual fund company, a likely result is inconsistency in the macroeconomic forecasts that partly underlie expectations of returns across securities. Moreover, the underlying assumptions for market-index risk and return often are not explicit in the analysis of individual securities. The single-index model creates a framework that separates these two quite different sources of return variation and makes it easier to ensure consistency across analysts. We can lay down a hierarchy of the preparation of the input list using the framework of the single-index model. 1. Macroeconomic analysis is used to estimate the risk premium and risk of the market index. 2. Statistical analysis is used to estimate the beta coefficients of all securities and their residual variances, s2(ei). 3. The portfolio manager uses the estimates for the market-index risk premium and the beta coefficient of a security to establish the expected return of that security absent any contribution from security analysis. The market-driven expected return is conditional on information common to all securities, not on information gleaned from security analysis of particular firms. This market-driven expected return can be used as a benchmark. 4. Security-specific expected return forecasts (specifically, security alphas) are derived from various security-valuation models (such as those discussed in Part Five). Thus, the alpha value distills the incremental risk premium attributable to private information developed from security analysis. 12

The use of the index model to construct optimal risky portfolios was originally developed in Jack Treynor and Fischer Black, “How to Use Security Analysis to Improve Portfolio Selection,” Journal of Business, January 1973.

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In the context of Equation 8.9, the risk premium on a security not subject to security analysis would be biE(RM). In other words, the risk premium would derive solely from the security’s tendency to follow the market index. Any expected return beyond this benchmark risk premium (the security alpha) would be due to some nonmarket factor that would be uncovered through security analysis. The end result of security analysis is the list of alpha values. Statistical methods of estimating beta coefficients are widely known and standardized; hence, we would not expect this portion of the input list to differ greatly across portfolio managers. In contrast, macro and security analysis are far less of an exact science and therefore provide an arena for distinguished performance. Using the index model to disentangle the premiums due to market and nonmarket factors, a portfolio manager can be confident that macro analysts compiling estimates of the market-index risk premium and security analysts compiling alpha values are using consistent estimates for the overall market. In the context of portfolio construction, alpha is more than just one of the components of expected return. It is the key variable that tells us whether a security is a good or a bad buy. Consider an individual stock for which we have a beta estimate from statistical considerations and an alpha value from security analysis. We easily can find many other securities with identical betas and therefore identical systematic components of their risk premiums. Therefore, what really makes a security attractive or unattractive to a portfolio manager is its alpha value. In fact, we’ve suggested that a security with a positive alpha is providing a premium over and above the premium it derives from its tendency to track the market index. This security is a bargain and therefore should be overweighted in the overall portfolio compared to the passive alternative of using the market-index portfolio as the risky vehicle. Conversely, a negative-alpha security is overpriced and, other things equal, its portfolio weight should be reduced. In more extreme cases, the desired portfolio weight might even be negative, that is, a short position (if permitted) would be desirable.

The Index Portfolio as an Investment Asset The process of charting the efficient frontier using the single-index model can be pursued much like the procedure we used in Chapter 7, where we used the Markowitz model to find the optimal risky portfolio. Here, however, we can benefit from the simplification the index model offers for deriving the input list. Moreover, portfolio optimization highlights another advantage of the single-index model, namely, a simple and intuitively revealing representation of the optimal risky portfolio. Before we get into the mechanics of optimization in this setting, however, we start by considering the role of the index portfolio in the optimal portfolio. Suppose the prospectus of an investment company limits the universe of investable assets to only stocks included in the S&P 500 portfolio. In this case, the S&P 500 index captures the impact of the economy on the large stocks the firm may include in its portfolio. Suppose that the resources of the company allow coverage of only a relatively small subset of this so-called investable universe. If these analyzed firms are the only ones allowed in the portfolio, the portfolio manager may well be worried about limited diversification. A simple way to avoid inadequate diversification is to include the S&P 500 portfolio as one of the assets of the portfolio. Examination of Equations 8.8 and 8.9 reveals that if we treat the S&P 500 portfolio as the market index, it will have a beta of 1.0 (its sensitivity to itself), no firm-specific risk, and an alpha of zero—there is no nonmarket component in its expected return. Equation 8.10 shows that the covariance of any security, i, with the index is bi s 2M. To distinguish the S&P 500 from the n securities covered by the firm, we will designate it the (n11)th asset. We can think of the S&P 500 as a passive portfolio that the manager would select in the absence of security analysis. It gives broad market exposure without the need for expensive security analysis. However, if the manager is willing to

CHAPTER 8

Index Models

engage in such research, she may devise an active portfolio that can be mixed with the index to provide an even better risk–return trade-off.

The Single-Index-Model Input List If the portfolio manager plans to compile a portfolio from a list of n actively researched firms plus a passive market-index portfolio, the input list will include the following estimates: 1. Risk premium on the S&P 500 portfolio. 2. Standard deviation of the S&P 500 portfolio. 3. n sets of estimates of (a) beta coefficients, (b) stock residual variances, and (c)alpha values. (The alpha values, together with the risk premium of the S&P 500 and the beta of each security, determine the expected return on each security.)

The Optimal Risky Portfolio in the Single-Index Model The single-index model allows us to solve for the optimal risky portfolio directly and to gain insight into the nature of the solution. First we confirm that we easily can set up the optimization process to chart the efficient frontier in this framework along the lines of the Markowitz model. With the estimates of the beta and alpha coefficients, plus the risk premium of the index portfolio, we can generate the n 1 1 expected returns using Equation 8.9. With the estimates of the beta coefficients and residual variances, together with the variance of the index portfolio, we can construct the covariance matrix using Equation 8.10. Given a column of risk premiums and the covariance matrix, we can conduct the optimization program described in Chapter 7. We can take the description of how diversification works in the single-index framework of Section 8.2 a step further. We showed earlier that the alpha, beta, and residual variance of an equally weighted portfolio are the simple averages of those parameters across component securities. This result is not limited to equally weighted portfolios. It applies to any portfolio, where we need only replace “simple average” with “weighted average,” using the portfolio weights. Specifically, n11

aP 5 a wiai

and for the index, an11 5 aM 5 0

i51

n11

bP 5 a wibi

and for the index, bn11 5 bM 5 1

(8.18)

i51

n11

s 2 (eP) 5 a w2i s 2(ei)and for the index, s 2(en11) 5 s 2 (eM) 5 0 i51

The objective is to maximize the Sharpe ratio of the portfolio by using portfolio weights, w1,...,wn11. With this set of weights, the expected return, standard deviation, and Sharpe ratio of the portfolio are n11

n11

E (RP) 5 aP 1 E (RM)bP 5 a wiai 1 E (RM) a wi bi i51

n11

i51 2

n11

2 2 sP 5 3 b2P sM 1 s 2(eP)4 1/2 5 BsM ¢ a wibi ≤ 1 a w2i s2(ei) R i51

SP 5

E (RP) sP

i51

1/2

(8.19)

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At this point, as in the Markowitz procedure, we could use Excel’s optimization program to maximize the Sharpe ratio subject to the adding-up constraint that the portfolio weights sum to 1. However, this is not necessary because when returns follow the index model, the optimal portfolio can be derived explicitly, and the solution for the optimal portfolio provides insight into the efficient use of security analysis in portfolio construction. It is instructive to outline the logical thread of the solution. We will not show every algebraic step, but will instead present the major results and interpretation of the procedure. Before delving into the results, let us first explain the basic trade-off the model reveals. If we were interested only in diversification, we would just hold the market index. Security analysis gives us the chance to uncover securities with a nonzero alpha and to take a differential position in those securities. The cost of that differential position is a departure from efficient diversification, in other words, the assumption of unnecessary firm-specific risk. The model shows us that the optimal risky portfolio trades off the search for alpha against the departure from efficient diversification. The optimal risky portfolio turns out to be a combination of two component portfolios: (1) an active portfolio, denoted by A, comprised of the n analyzed securities (we call this the active portfolio because it follows from active security analysis), and (2) the marketindex portfolio, the (n11)th asset we include to aid in diversification, which we call the passive portfolio and denote by M. Assume first that the active portfolio has a beta of 1. In that case, the optimal weight in the active portfolio would be proportional to the ratio aA/s2(eA). This ratio balances the contribution of the active portfolio (its alpha) against its contribution to the portfolio vari2 ance (via residual variance). The analogous ratio for the index portfolio is E (RM) /sM , and hence the initial position in the active portfolio (i.e., if its beta were 1) is aA w0A 5

s2A E (RM)

(8.20)

s2M Next, we amend this position to account for the actual beta of the active portfolio. For any level of sA2, the correlation between the active and passive portfolios is greater when the beta of the active portfolio is higher. This implies less diversification benefit from the passive portfolio and a lower position in it. Correspondingly, the position in the active portfolio increases. The precise modification for the position in the active portfolio is:13 w*A 5

w0A 1 1 (1 2 bA)w0A

(8.21)

Notice that when bA 5 1, w*A 5 w0A.

The Information Ratio Equations 8.20 and 8.21 yield the optimal position in the active portfolio once we know its alpha, beta, and residual variance. With w*A in the active portfolio and 1 2 w*A invested in the index portfolio, we can compute the expected return, standard deviation, and Sharpe ratio of the optimal risky portfolio. The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy). The exact relationship is sM Cov (RA, RM) 5 bA . Therefore, given the ratio of SD, sAsM sA a higher beta implies higher correlation and smaller benefit from diversification than when b51 in Equation 8.20. This requires the modification of Equation 8.21. The definition of correlation implies that r (RA, RM) 5

13

CHAPTER 8 S 2P 5 S 2M 1 B

aA 2 R s(eA)

Index Models

(8.22)

Equation 8.22 shows us that the contribution of the active portfolio (when held in its optimal weight, w*A) to the Sharpe ratio of the overall risky portfolio is determined by the ratio of its alpha to its residual standard deviation. This important ratio is called the information ratio. It measures the extra return we can obtain from security analysis compared to the firm-specific risk we incur when we over- or underweight securities relative to the passive market index. Equation 8.22 therefore implies that to maximize the overall Sharpe ratio, we must maximize the information ratio of the active portfolio. It turns out that the information ratio of the active portfolio will be maximized if we invest in each security in proportion to its ratio of ai/s2(ei). Scaling this ratio so that the total position in the active portfolio adds up to w*A , the weight in each security is ai w*i 5 w*A

2(

s ei) ai a s2(e ) i51 i n

(8.23)

With this set of weights, the contribution of each security to the information ratio of the active portfolio is the square of its own information ratio, that is, B

n ai 2 aA 2 R 5 aB R s(eA) i51 s(ei)

(8.24)

The model thus reveals the central role of the information ratio in efficiently taking advantage of security analysis. The positive contribution of a security to the portfolio is made by its addition to the nonmarket risk premium (its alpha). Its negative impact is to increase the portfolio variance through its firm-specific risk (residual variance). In contrast to alpha, the market (systematic) component of the risk premium, biE(RM), is offset by the security’s nondiversifiable (market) risk, b2i sM2 , and both are driven by the same beta. This trade-off is not unique to any security, as any security with the same beta makes the same balanced contribution to both risk and return. Put differently, the beta of a security is neither vice nor virtue. It is a property that simultaneously affects the risk and risk premium of a security. Hence we are concerned only with the aggregate beta of the active portfolio, rather than the beta of each individual security. We see from Equation 8.23 that if a security’s alpha is negative, the security will assume a short position in the optimal risky portfolio. If short positions are prohibited, a negativealpha security would simply be taken out of the optimization program and assigned a portfolio weight of zero. As the number of securities with nonzero alpha values (or the number with positive alphas if short positions are prohibited) increases, the active portfolio will itself be better diversified and its weight in the overall risky portfolio will increase at the expense of the passive index portfolio. Finally, we note that the index portfolio is an efficient portfolio only if all alpha values are zero. This makes intuitive sense. Unless security analysis reveals that a security has a nonzero alpha, including it in the active portfolio would make the portfolio less attractive. In addition to the security’s systematic risk, which is compensated for by the market risk premium (through beta), the security would add its firm-specific risk to portfolio variance. With a zero alpha, however, the latter is not compensated by an addition to the nonmarket risk premium. Hence, if all securities have zero alphas, the optimal weight in the active portfolio will be zero, and the weight in the index portfolio will be 1. However, when

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security analysis uncovers securities with nonmarket risk premiums (nonzero alphas), the index portfolio is no longer efficient.

Summary of Optimization Procedure Once security analysis is complete, the optimal risky portfolio is formed from the indexmodel estimates of security and market index parameters using these steps: 1. Compute the initial position of each security in the active portfolio as w0i 5 ai/s 2 (ei ). 2. Scale those initial positions to force portfolio weights to sum to 1 by dividing by their sum, that is, w i 5

w 0i n

.

0 a wi

i51

n

3. Compute the alpha of the active portfolio: aA 5 a wi ai. i51 n

4. Compute the residual variance of the active portfolio: s 2 (eA) 5 a w2i s 2 (ei). i51 A

5. Compute the initial position in the active portfolio:

2

(eA ) . E ( RM )

w0A

2 M

n

6. Compute the beta of the active portfolio: bA 5 a wi bi. i51 7. Adjust the initial position in the active portfolio: w*A 5

w0A 1 1 (1 2 bA) w0A

.

8. Note: the optimal risky portfolio now has weights: w*M 5 1 2 w*A ; w*i 5 w*Awi . 9. Calculate the risk premium of the optimal risky portfolio from the risk premium of the index portfolio and the alpha of the active portfolio: E (RP) 5 (w*M 1 w*AbA) E (RM) 1 w*AaA. Notice that the beta of the risky portfolio is w*M 1 w*AbA because the beta of the index portfolio is 1. 10. Compute the variance of the optimal risky portfolio from the variance of the index portfolio and the residual variance of the active portfolio: 2 sP2 5 (w*M 1 w*AbA)2sM 1 3 w*As (eA)4 2.

An Example We can illustrate the implementation of the index model by constructing an optimal portfolio from the S&P 500 index and the six stocks for which we analyzed risk parameters in Section 8.3. This example entails only six analyzed stocks, but by virtue of selecting three pairs of firms from the same industry with relatively high residual correlations, we put the index model to a severe test. This is because the model ignores the correlation between residuals when producing estimates for the covariance matrix. Therefore, comparison of results from the index model with the full-blown covariance (Markowitz) model should be instructive. Risk Premium Forecasts Panel 4 of Spreadsheet 8.1 contains estimates of alpha and the risk premium for each stock. These alphas would be the most important production of the investment company in a real-life procedure. Statistics plays a small role here; in this arena, macro/security analysis is king. In this example, we simply use illustrative values to demonstrate the portfolio construction process and possible results. You may wonder why we

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277

have chosen such small, forecast alpha values. The reason is that even when security analysis uncovers a large apparent mispricing, that is, large alpha values, these forecasts must be substantially trimmed to account for the fact that such forecasts are subject to large estimation error. We discuss the important procedure of adjusting actual forecasts in Chapter 27.

Risk Premium

The Optimal Risky Portfolio Panel 5 of Spreadsheet 8.1 displays calculations for the optimal risky portfolio. They follow the summary procedure of Section 8.4 (you should try to replicate these calculations in your own spreadsheet). In this example we allow short sales. Notice that the weight of each security in the active portfolio (see row 52) has the same sign as the alpha value. Allowing short sales, the positions in the active portfolio are quite large (e.g., the position in BP is .7349); this is an aggressive portfolio. As a result, the alpha of the active portfolio (2.22%) is larger than that of any of the individual alpha forecasts. However, this aggressive stance also results in a large residual variance (.0404, which corresponds to a residual standard deviation of 20%). Therefore, the position in the active portfolio is scaled down (see Equation 8.20) and ends up quite modest (.1718; cell C57), reinforcing the notion that diversification considerations are paramount in the optimal risky portfolio. The optimal risky portfolio has a risk premium of 6.48%, standard deviation of 14.22%, and a Sharpe ratio of .46 (cells J58–J61). By comparison, the Sharpe ratio of the index portfolio is .06/.1358 5.44 (cell B61), which is quite close to that of the optimal risky portfolio. The small improvement is a result of the modest alpha forecasts that we used. In Chapter 11 on market efficiency and Chapter 24 on performance evaluation we demonstrate that such results are common in the mutual fund industry. Of course, a few portfolio managers can and do produce portfolios with better performance. The interesting question here is the extent to which the index model produces results that are inferior to that of the full-covariance (Markowitz) model. Figure 8.5 shows the efficient frontiers from the two models with the example data. We find that the difference is in fact small. Table 8.2 compares the compositions and expected performance of the global minimum variance (G) and the optimal risky portfolios derived from the two models. The standard deviations of efficient portfolios produced from the Markowitz model and the index model are calculated from the covariance matrixes used in each model. As discussed earlier, we cannot be sure that the covariance estimates from the full covariance model are more accurate than those from the more restrictive single-index model. .08 However, by assuming the full covariance model to be more accurate, we get an idea .07 of how far off the two models can be. Figure 8.5 shows that for conservative .06 portfolios (closer to the minimum-variance portfolio G), the index model underesti.05 mates the volatility and hence overestimates Efﬁcient Frontier (Full Covariance) performance. The reverse happens with Efﬁcient Frontier (Index Model) .04 portfolios that are riskier than the index, S&P500 which also include the region near the opti.03 mal portfolio. Despite these differences, .10 .11 .12 .13 .14 .15 .16 .17 what stands out from this comparison is Standard Deviation that the outputs of the two models are in fact extremely similar, with the index model perhaps calling for a more conservative Figure 8.5 Efficient frontiers with the index model and position. This is where we would like to be full-covariance matrix with a model relying on approximations.

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Table 8.2 Comparison of portfolios from the single-index and fullcovariance models

Global Minimum Variance Portfolio Full-Covariance Model Mean SD Sharpe ratio

.0371 .1089 .3409

Index Model .0354 .1052 .3370

Optimal Portfolio Full-Covariance Model

Index Model

.0677 .1471 .4605

.0649 .1423 .4558

Portfolio Weights S&P 500 HP DELL WMT TARGET BP SHELL

8.5

.88 2.11 2.01 .23 2.18 .22 2.02

.83 2.17 2.05 .14 2.08 .20 .12

.75 .10 2.04 2.03 .10 .25 2.12

.83 .07 2.06 2.05 .06 .13 .03

Practical Aspects of Portfolio Management with the Index Model The tone of our discussions in this chapter indicates that the index model may be preferred for the practice of portfolio management. Switching from the Markowitz to an index model is an important decision and hence the first question is whether the index model really is inferior to the Markowitz full-covariance model.

Is the Index Model Inferior to the Full-Covariance Model? This question is partly related to a more general question of the value of parsimonious models. As an analogy, consider the question of adding additional explanatory variables in a regression equation. We know that adding explanatory variables will in most cases increase R-square, and in no case will R-square fall. But this does not necessarily imply a better regression equation.14 A better criterion is contribution to the predictive power of the regression. The appropriate question is whether inclusion of a variable that contributes to insample explanatory power is likely to contribute to out-of-sample forecast precision. Adding variables, even ones that may appear significant, sometimes can be hazardous to forecast precision. Put differently, a parsimonious model that is stingy about inclusion of independent variables is often superior. Predicting the value of the dependent variable depends on two factors, the precision of the coefficient estimates and the precision of the forecasts of the independent variables. When we add variables, we introduce errors on both counts. This problem applies as well to replacing the single-index with the full-blown Markowitz model, or even a multi-index model of security returns. To add another index, we need both a forecast of the risk premium of the additional index portfolio and estimates of security betas with respect to that additional factor. The Markowitz model allows far 14

In fact, the adjusted R-square may fall if the additional variable does not contribute enough explanatory power to compensate for the extra degree of freedom it uses.

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more flexibility in our modeling of asset covariance structure compared to the single-index model. But that advantage may be illusory if we can’t estimate those covariances with a sufficient degree of accuracy. Using the full-covariance matrix invokes estimation risk of thousands of terms. Even if the full Markowitz model would be better in principle, it is very possible that the cumulative effect of so many estimation errors will result in a portfolio that is actually inferior to that derived from the single-index model. Against the potential superiority of the full-covariance model, we have the clear practical advantage of the single-index framework. Its aid in decentralizing macro and security analysis is another decisive advantage.

The Industry Version of the Index Model Not surprisingly, the index model has attracted the attention of practitioners. To the extent that it is approximately valid, it provides a convenient benchmark for security analysis. A portfolio manager who has neither special information about a security nor insight that is unavailable to the general public will take the security’s alpha value as zero, and, according to Equation 8.9, will forecast a risk premium for the security equal to biRM. If we restate this forecast in terms of total returns, one would expect E(ri ) 5 rf 1 bi 3 E(rM) 2 rf 4

(8.25)

A portfolio manager who has a forecast for the market index, E(rM), and observes the risk-free T-bill rate, rf , can use the model to determine the benchmark expected return for any stock. The beta coefficient, the market risk, s 2M , and the firm-specific risk, s2(e), can be estimated from historical SCLs, that is, from regressions of security excess returns on market index excess returns. There are several proprietary sources for such regression results, sometimes called “beta books.” The Web sites for this chapter at the Online Learning Center (www.mhhe.com/bkm) also provide security betas. Table8.3 is a sample of a typical page from a beta book. Beta books typically use the S&P 500 as the proxy for the market portfolio. They commonly employ the 60 most recent monthly observations to calculate regression parameters, and use total returns, rather than excess returns (deviations from T-bill rates) in the regressions. In this way they estimate a variant of our index model, which is r 5 a 1 brM 1 e*

(8.26)

r 2 rf 5 a 1 b (rM 2 rf) 1 e

(8.27)

instead of

To see the effect of this departure, we can rewrite Equation 8.27 as r 5 rf 1 a 1 brM 2 brf 1 e 5 a 1 rf (1 2 b) 1 brM 1 e

(8.28)

Comparing Equations 8.26 and 8.28, you can see that if rf is constant over the sample period, both equations have the same independent variable, rM, and residual, e. Therefore, the slope coefficient will be the same in the two regressions.15 However, the intercept that beta books call ALPHA, as in Table8.3, is really an estimate of a1rf(12b). The apparent justification for this procedure is that, on a monthly basis, rf(12b) is small and is likely to be swamped by the volatility of actual stock returns. But it is worth noting that for b≠1, the regression intercept in Equation 8.26 will not equal the index model a as it does when excess returns are used as in Equation 8.27. 15

Actually, rf does vary over time and so should not be grouped casually with the constant term in the regression. However, variations in rf are tiny compared with the swings in the market return. The actual volatility in the T-bill rate has only a small impact on the estimated value of b.

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Ticker Symbol AMZN F NEM INTC MSFT DELL BA MCD PFE DD DIS XOM IBM WMT HNZ LTD ED GE

Portfolio Theory and Practice

Security Name

BETA

ALPHA

RSQ

Residual Std Dev

Std Error Beta

Standard Error Alpha

Adjusted Beta

Amazon.com Ford Newmont Mining Corp. Intel Corporation Microsoft Corporation Dell Inc. Boeing Co. McDonald’s Corp. Pfizer Inc. DuPont Walt Disney Co. ExxonMobil Corp. IBM Corp. Walmart HJ Heinz Co. Limited Brands Inc. Consolidated Edison Inc. General Electric Co. MEAN STD DEVIATION

2.25 1.64 0.44 1.60 0.87 1.36 1.42 0.92 0.65 0.97 0.91 0.87 0.88 0.06 0.43 1.30 0.15 0.65 0.97 0.56

0.006 20.012 0.002 20.010 0.001 20.014 0.004 0.016 20.006 20.002 0.005 0.011 0.004 0.002 0.009 0.001 0.004 20.002 0.001 0.008

0.238 0.183 0.023 0.369 0.172 0.241 0.402 0.312 0.131 0.311 0.278 0.216 0.248 0.002 0.110 0.216 0.101 0.173 0.207 0.109

0.1208 0.1041 0.0853 0.0627 0.0569 0.0723 0.0517 0.0409 0.0504 0.0434 0.0440 0.0497 0.0459 0.0446 0.0368 0.0741 0.0347 0.0425 0.0589 0.0239

0.5254 0.4525 0.3709 0.2728 0.2477 0.3143 0.2250 0.1777 0.2191 0.1887 0.1913 0.2159 0.1997 0.1941 0.1599 0.3223 0.1509 0.1850 0.2563 0.1039

0.0156 0.0135 0.0110 0.0081 0.0074 0.0094 0.0067 0.0053 0.0065 0.0056 0.0057 0.0064 0.0059 0.0058 0.0048 0.0096 0.0045 0.0055 0.0076 0.0031

1.84 1.43 0.62 1.40 0.91 1.24 1.28 0.95 0.77 0.98 0.94 0.91 0.92 0.38 0.62 1.20 0.43 0.77 0.98 0.37

Table 8.3 Market sensitivity statistics: Regressions of total stock returns on total S&P 500 returns over 60 months, 2004–2008 Source: Compiled from CRSP (University of Chicago) database.

CONCEPT CHECK

8.4

What was Intel’s index-model a per month during the period covered by the Table8.3 regression if during this period the average monthly rate of return on T-bills was .2%?

Always remember as well that these alpha estimates are ex post (after the fact) measures. They do not mean that anyone could have forecast these alpha values ex ante (before the fact). In fact, the name of the game in security analysis is to forecast alpha values ahead of time. A well-constructed portfolio that includes long positions in future positivealpha stocks and short positions in future negative-alpha stocks will outperform the market index. The key term here is “well constructed,” meaning that the portfolio has to balance concentration on high-alpha stocks with the need for risk-reducing diversification as discussed earlier in the chapter. Much of the other output in Table8.3 is similar to the Excel output (Table 8.1) that we discussed when estimating the index model for Hewlett-Packard. The R-square statistic is the ratio of systematic variance to total variance, the fraction of total volatility attributable to market movements. For most firms, R-square is substantially below .5, indicating that stocks have far more firm-specific than systematic risk. This highlights the practical importance of diversification.

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The Resid Std Dev column is the standard deviation of the monthly regression residuals, also sometimes called the standard error of the regression. The standard errors of the alpha and beta estimates allow us to evaluate the precision of the estimates. Notice that the standard errors of alpha tend to be far greater multiples of the estimated value of alpha than is the case for beta estimates. Intel’s Resid Std Dev is 6.27% per month and its R2 is .369. This tells us that 2 ( ) sIntel e 5 6.272 5 39.31 and, because R2 51 2 s2(e)/s2, we can solve for Intel’s total standard deviation by rearranging as follows: sIntel 5 B

s2Intel(e)

R 1 2 R2

1/2

5a

39.31 1/2 b 5 7.89% per month .631

This is Intel’s monthly standard deviation for the sample period. Therefore, the annualized standard deviation for that period was 7.89 "12 5 27.33%. The last column is called Adjusted Beta. The motivation for adjusting beta estimates is that, on average, the beta coefficients of stocks seem to move toward 1 over time. One explanation for this phenomenon is intuitive. A business enterprise usually is established to produce a specific product or service, and a new firm may be more unconventional than an older one in many ways, from technology to management style. As it grows, however, a firm often diversifies, first expanding to similar products and later to more diverse operations. As the firm becomes more conventional, it starts to resemble the rest of the economy even more. Thus its beta coefficient will tend to change in the direction of 1. Another explanation for this phenomenon is statistical. We know that the average beta over all securities is 1. Thus, before estimating the beta of a security, our best forecast would be that it is 1. When we estimate this beta coefficient over a particular sample period, we sustain some unknown sampling error of the estimated beta. The greater the difference between our beta estimate and 1, the greater is the chance that we incurred a large estimation error and that beta in a subsequent sample period will be closer to 1. The sample estimate of the beta coefficient is the best guess for that sample period. Given that beta has a tendency to evolve toward 1, however, a forecast of the future beta coefficient should adjust the sample estimate in that direction. Table8.3 adjusts beta estimates in a simple way.16 It takes the sample estimate of beta and averages it with 1, using weights of two-thirds and one-third: Adjusted beta 5 2⁄ 3 sample beta 1 1⁄ 3 (1)

Example 8.1

(8.29)

Adjusted Beta

For the 60 months used in Table8.3, Intel’s beta was estimated at 1.60. Therefore, its adjusted beta is 2⁄ 331.6011⁄ 3 5 1.40, taking it a third of the way toward 1. In the absence of special information concerning Intel, if our forecast for the market index is 10% and T-bills pay 4%, we learn from the beta book that the forecast for the rate of return on Intel stock is E(rIntel) 5 rf 1 Adjusted beta 3 3 E(rM) 2 rf 4 5 4 1 1.40(10 2 4) 5 12.40% The sample period regression alpha is 21.0%. Because Intel’s beta is greater than 1, we know that this means that the estimate of the index model a is somewhat larger.

16

A more sophisticated method is described in Oldrich A. Vasicek, “A Note on Using Cross-Sectional Information in Bayesian Estimation of Security Betas,” Journal of Finance 28 (1973), pp. 1233–39.

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As in Equation 8.28, we have to subtract (1 2 b) rf from the regression alpha to obtain the index model a. In any event, the standard error of the alpha estimate is .81%. The estimate of alpha is far less than twice its standard error. Consequently, we cannot reject the hypothesis that the true alpha is zero.

Predicting Betas Adjusted betas are a simple way to recognize that betas estimated from past data may not be the best estimates of future betas: Betas seem to drift toward 1 over time. This suggests that we might want a forecasting model for beta. One simple approach would be to collect data on beta in different periods and then estimate a regression equation: Current beta 5 a 1 b (Past beta)

(8.30)

Given estimates of a and b, we would then forecast future betas using the rule Forecast beta 5 a 1 b (Current beta)

(8.31)

There is no reason, however, to limit ourselves to such simple forecasting rules. Why not also investigate the predictive power of other financial variables in forecasting beta? For example, if we believe that firm size and debt ratios are two determinants of beta, we might specify an expanded version of Equation 8.30 and estimate Current beta 5 a 1 b1(Past beta) 1 b2(Firm size) 1 b3(Debt ratio) Now we would use estimates of a and b1 through b3 to forecast future betas. Such an approach was followed by Rosenberg and Guy,17 who found the following variables to help predict betas: 1. 2. 3. 4. 5. 6.

Variance of earnings. Variance of cash flow. Growth in earnings per share. Market capitalization (firm size). Dividend yield. Debt-to-asset ratio.

Rosenberg and Guy also found that even after controlling for a firm’s financial characteristics, industry group helps to predict beta. For example, they found that the beta values of gold mining companies are on average .827 lower than would be predicted based on financial characteristics alone. This should not be surprising; the 2.827 “adjustment factor” for the gold industry reflects the fact that gold values are inversely related to market returns. Table 8.4 presents beta estimates and adjustment factors for a subset of firms in the Rosenberg and Guy study. CONCEPT CHECK

8.5

Compare the first five and last four industries in Table8.4. What characteristic seems to determine whether the adjustment factor is positive or negative? 17

Barr Rosenberg and J. Guy, “Prediction of Beta from Investment Fundamentals, Parts 1 and 2,” Financial Analysts Journal, May–June and July–August 1976.

For those who believe in efficient markets, the recent explosion in the number of exchange-traded funds represents a triumph. ETFs are quoted securities that track a particular index, for a fee that is normally just a fraction of a percentage point. They enable investors to assemble a low-cost portfolio covering a wide range of assets from international equities, through government and corporate bonds, to commodities. But as fast as the assets of ETFs and index-tracking mutual funds are growing, another section of the industry seems to be flourishing even faster. Watson Wyatt, a firm of actuaries, estimates that “alternative asset investment” (ranging from hedge funds through private equity to property) grew by around 20% in 2005, to $1.26 trillion. Investors who take this route pay much higher fees in the hope of better performance. One of the fastest-growing assets, funds of hedge funds, charge some of the highest fees of all. Why are people paying up? In part, because investors have learned to distinguish between the market return, dubbed beta, and managers’ outperformance, known as alpha. “Why wouldn’t you buy beta and alpha separately?” asks Arno Kitts of Henderson Global Investors, a fund-management firm. “Beta is a commodity and alpha is about skill.” Clients have become convinced that no one firm can produce good performance in every asset class. That has

led to a “core and satellite” model, in which part of the portfolio is invested in index trackers with the rest in the hands of specialists. But this creates its own problems. Relations with a single balanced manager are simple. It is much harder to research and monitor the performance of specialists. That has encouraged the middlemen—managers of managers (in the traditional institutional business) and funds-of-funds (in the hedge-fund world), which are usually even more expensive. That their fees endure might suggest investors can identify outperforming fund managers in advance. However, studies suggest this is extremely hard. And even where you can spot talent, much of the extra performance may be siphoned off into higher fees. “A disproportionate amount of the benefits of alpha go to the manager, not the client,” says Alan Brown at Schroders, an asset manager. In any event, investors will probably keep pursuing alpha, even though the cheaper alternatives of ETFs and tracking funds are available. Craig Baker of Watson Wyatt, says that, although above-market returns may not be available to all, clients who can identify them have a “first mover” advantage. As long as that belief exists, managers can charge high fees. Source: The Economist, September 14, 2006. Copyright © The Economist Newspaper Limited, London.

Index Models and Tracking Portfolios Suppose a portfolio manager believes she has identified an underpriced portfolio. Her security analysis team estimates the index model equation for this portfolio (using the S&P 500 index) in excess return form and obtains the following estimates: RP 5 .04 1 1.4RS&P500 1 eP

Industry

Beta

Adjustment Factor

Agriculture Drugs and medicine Telephone Energy utilities Gold Construction Air transport Trucking Consumer durables

0.99 1.14 0.75 0.60 0.36 1.27 1.80 1.31 1.44

2.140 2.099 2.288 2.237 2.827 .062 .348 .098 .132

(8.32)

Table 8.4 Industry betas and adjustment factors

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Alpha Betting

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Therefore, P has an alpha value of 4% and a beta of 1.4. The manager is confident in the quality of her security analysis but is wary about the performance of the broad market in the near term. If she buys the portfolio, and the market as a whole turns down, she still could lose money on her investment (which has a large positive beta) even if her team is correct that the portfolio is underpriced on a relative basis. She would like a position that takes advantage of her team’s analysis but is independent of the performance of the overall market. To this end, a tracking portfolio (T) can be constructed. A tracking portfolio for portfolio P is a portfolio designed to match the systematic component of P’s return. The idea is for the portfolio to “track” the market-sensitive component of P’s return. This means the tracking portfolio must have the same beta on the index portfolio as P and as little nonsystematic risk as possible. This procedure is also called beta capture. A tracking portfolio for P will have a levered position in the S&P 500 to achieve a beta of 1.4. Therefore, T includes positions of 1.4 in the S&P 500 and 2.4 in T-bills. Because T is constructed from the index and bills, it has an alpha value of zero. Now consider buying portfolio P but at the same time offsetting systematic risk by assuming a short position in the tracking portfolio. The short position in T cancels out the systematic exposure of the long position in P: the overall combined position is thus market neutral. Therefore, even if the market does poorly, the combined position should not be affected. But the alpha on portfolio P will remain intact. The combined portfolio, C, provides an excess return per dollar of RC 5 RP 2 RT 5 (.04 1 1.4RS&P500 1 eP) 2 1.4RS&P500 5 .04 1 eP

(8.33)

While this portfolio is still risky (due to the residual risk, eP), the systematic risk has been eliminated, and if P is reasonably well-diversified, the remaining nonsystematic risk will be small. Thus the objective is achieved: The manager can take advantage of the 4% alpha without inadvertently taking on market exposure. The process of separating the search for alpha from the choice of market exposure is called alpha transport. This “long-short strategy” is characteristic of the activity of many hedge funds. Hedge fund managers identify an underpriced security and then try to attain a “pure play” on the perceived underpricing. They hedge out all extraneous risk, focusing the bet only on the perceived “alpha” (see the box on p. 283). Tracking funds are the vehicle used to hedge the exposures to which they do not want exposure. Hedge fund managers use index regressions such as those discussed here, as well as more-sophisticated variations, to create the tracking portfolios at the heart of their hedging strategies.

SUMMARY

1. A single-factor model of the economy classifies sources of uncertainty as systematic (macroeconomic) factors or firm-specific (microeconomic) factors. The index model assumes that the macro factor can be represented by a broad index of stock returns. 2. The single-index model drastically reduces the necessary inputs in the Markowitz portfolio selection procedure. It also aids in specialization of labor in security analysis. 2 3. According to the index model specification, the systematic risk of a portfolio or asset equals b2sM 2 and the covariance between two assets equals b i bj sM .

4. The index model is estimated by applying regression analysis to excess rates of return. The slope of the regression curve is the beta of an asset, whereas the intercept is the asset’s alpha during the sample period. The regression line is also called the security characteristic line.

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285

5. Optimal active portfolios constructed from the index model include analyzed securities in proportion to their information ratios. The full risky portfolio is a mixture of the active portfolio and the passive market-index portfolio. The index portfolio is used to enhance the diversification of the overall risky position.

7. Betas show a tendency to evolve toward 1 over time. Beta forecasting rules attempt to predict this drift. Moreover, other financial variables can be used to help forecast betas.

single-factor model single-index model regression equation

residuals security characteristic line scatter diagram

information ratio tracking portfolio

Single-index model (in excess returns): Ri (t) 5 ai 1 biRM (t) 1 ei (t)

Related Web sites for this chapter are available at www. mhhe.com/bkm

KEY TERMS

KEY EQUATIONS

Security risk in index model: Total risk 5 Systematic risk 1 firm-specific risk s2 5 b2s2M 1 s2(e) 2 Covariance 5 Cov (ri , rj ) 5 Product of betas 3 Market-index risk 5 bi bj sM

Active portfolio management in the index model Sharpe ratio of optimal risky portfolio: S2P 5 S2M 1 B

aA 2 R s(eA)

ai Asset weight in active portfolio: w*i 5 w*A

Information ratio of active portfolio: B

2(

s ei) ai a s2(e ) i51 i n

n ai 2 aA 2 R 5 aB R s(eA) i51 s(ei)

1. What are the advantages of the index model compared to the Markowitz procedure for obtaining an efficiently diversified portfolio? What are its disadvantages?

PROBLEM SETS

2. What is the basic trade-off when departing from pure indexing in favor of an actively managed portfolio?

Basic

3. How does the magnitude of firm-specific risk affect the extent to which an active investor will be willing to depart from an indexed portfolio? 4. Why do we call alpha a “nonmarket” return premium? Why are high-alpha stocks desirable investments for active portfolio managers? With all other parameters held fixed, what would happen to a portfolio’s Sharpe ratio as the alpha of its component securities increased?

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6. Practitioners routinely estimate the index model using total rather than excess rates of return. This makes their estimate of alpha equal to a1rf(12b).

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Portfolio Theory and Practice 5. A portfolio management organization analyzes 60 stocks and constructs a mean-variance efficient portfolio using only these 60 securities. a. How many estimates of expected returns, variances, and covariances are needed to optimize this portfolio? b. If one could safely assume that stock market returns closely resemble a single-index structure, how many estimates would be needed? 6. The following are estimates for two stocks. Stock

Expected Return

Beta

Firm-Specific Standard Deviation

A B

13% 18

0.8 1.2

30% 40

The market index has a standard deviation of 22% and the risk-free rate is 8%.

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a. What are the standard deviations of stocks A and B? b. Suppose that we were to construct a portfolio with proportions: Stock A: Stock B: T-bills:

.30 .45 .25

Compute the expected return, standard deviation, beta, and nonsystematic standard deviation of the portfolio. 7. Consider the following two regression lines for stocks A and B in the following figure. rA − rf

rB − rf

rM − rf

a. b. c. d. e.

Which stock has higher firm-specific risk? Which stock has greater systematic (market) risk? Which stock has higher R2? Which stock has higher alpha? Which stock has higher correlation with the market?

8. Consider the two (excess return) index model regression results for A and B: RA51%11.2RM R-square5.576 Residual standard deviation510.3% RB522%1.8RM R-square5.436 Residual standard deviation59.1%

rM − rf

CHAPTER 8 a. b. c. d.

Index Models

287

Which stock has more firm-specific risk? Which has greater market risk? For which stock does market movement explain a greater fraction of return variability? If rf were constant at 6% and the regression had been run using total rather than excess returns, what would have been the regression intercept for stock A?

Use the following data for Problems 9 through 14. Suppose that the index model for stocks A and B is estimated from excess returns with the following results: RA 5 3% 1 .7RM 1 eA RB 5 22% 1 1.2RM 1 eB sM 5 20%; R-squareA 5 .20; R-squareB 5 .12 9. What is the standard deviation of each stock? 10. Break down the variance of each stock to the systematic and firm-specific components. 11. What are the covariance and correlation coefficient between the two stocks? 12. What is the covariance between each stock and the market index?

15. A stock recently has been estimated to have a beta of 1.24: a. What will a beta book compute as the “adjusted beta” of this stock? b. Suppose that you estimate the following regression describing the evolution of beta over time: bt 5 .3 1 .7bt21 What would be your predicted beta for next year? 16. Based on current dividend yields and expected growth rates, the expected rates of return on stocks A and B are 11% and 14%, respectively. The beta of stock A is .8, while that of stock B is 1.5. The T-bill rate is currently 6%, while the expected rate of return on the S&P 500 index is 12%. The standard deviation of stock A is 10% annually, while that of stock B is 11%. If you currently hold a passive index portfolio, would you choose to add either of these stocks to your holdings? 17. A portfolio manager summarizes the input from the macro and micro forecasters in the following table: Micro Forecasts Asset Stock A Stock B Stock C Stock D

Expected Return (%)

Beta

Residual Standard Deviation (%)

20 18 17 12

1.3 1.8 0.7 1.0

58 71 60 55

Macro Forecasts Asset T-bills Passive equity portfolio

Expected Return (%)

Standard Deviation (%)

8 16

0 23

a. Calculate expected excess returns, alpha values, and residual variances for these stocks. b. Construct the optimal risky portfolio. c. What is Sharpe’s measure for the optimal portfolio and how much of it is contributed by the active portfolio? d. What should be the exact makeup of the complete portfolio for an investor with a coefficient of risk aversion of 2.8?

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13. For portfolio P with investment proportions of .60 in A and .40 in B, rework Problems 9, 10, and 12. 14. Rework Problem 13 for portfolio Q with investment proportions of .50 in P, .30 in the market index, and .20 in T-bills.

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Portfolio Theory and Practice 18. Recalculate Problem 17 for a portfolio manager who is not allowed to short sell securities. a. What is the cost of the restriction in terms of Sharpe’s measure? b. What is the utility loss to the investor (A52.8) given his new complete portfolio? 19. Suppose that on the basis of the analyst’s past record, you estimate that the relationship between forecast and actual alpha is: Actual abnormal return 5 .3 3 Forecast of alpha Use the alphas from Problem 17. How much is expected performance affected by recognizing the imprecision of alpha forecasts?

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Challenge

20. Suppose that the alpha forecasts in row 44 of Spreadsheet 8.1 are doubled. All the other data remain the same. Recalculate the optimal risky portfolio. Before you do any calculations, however, use the Summary of Optimization Procedure to estimate a back-of-the-envelope calculation of the information ratio and Sharpe ratio of the newly optimized portfolio. Then recalculate the entire spreadsheet example and verify your back-of-the-envelope calculation.

1. When the annualized monthly percentage rates of return for a stock market index were regressed against the returns for ABC and XYZ stocks over a 5-year period ending in 2013, using an ordinary least squares regression, the following results were obtained: Statistic

ABC

XYZ

Alpha

23.20%

7.3%

Beta R2 Residual standard deviation

0.60 0.35 13.02%

0.97 0.17 21.45%

Explain what these regression results tell the analyst about risk–return relationships for each stock over the sample period. Comment on their implications for future risk–return relationships, assuming both stocks were included in a diversified common stock portfolio, especially in view of the following additional data obtained from two brokerage houses, which are based on 2 years of weekly data ending in December 2013. Brokerage House

Beta of ABC

Beta of XYZ

A B

.62 .71

1.45 1.25

2. Assume the correlation coefficient between Baker Fund and the S&P 500 Stock Index is .70. What percentage of Baker Fund’s total risk is specific (i.e., nonsystematic)? 3. The correlation between the Charlottesville International Fund and the EAFE Market Index is 1.0. The expected return on the EAFE Index is 11%, the expected return on Charlottesville International Fund is 9%, and the risk-free return in EAFE countries is 3%. Based on this analysis, what is the implied beta of Charlottesville International? 4. The concept of beta is most closely associated with: a. b. c. d.

Correlation coefficients. Mean-variance analysis. Nonsystematic risk. Systematic risk.

5. Beta and standard deviation differ as risk measures in that beta measures: a. b. c. d.

Only unsystematic risk, while standard deviation measures total risk. Only systematic risk, while standard deviation measures total risk. Both systematic and unsystematic risk, while standard deviation measures only unsystematic risk. Both systematic and unsystematic risk, while standard deviation measures only systematic risk.

CHAPTER 8

Index Models

289

E-INVESTMENTS EXERCISES

SOLUTIONS TO CONCEPT CHECKS 1. a. Total market capitalization is 3,000 11,940 11,360 56,300. Therefore, the mean excess return of the index portfolio is 3,000 1,940 1,360 3 10 1 321 3 17 5 9.05% 5 .0905 6,300 6,300 6,300 b. The covariance between stocks A and B equals Cov (RA, RB) 5 bAbB s2M 5 1 3 .2 3 .252 5 .0125 c. The covariance between stock B and the index portfolio equals Cov (RB, RM) 5 bB s2M 5 .2 3 .252 5 .0125 d. The total variance of B equals s 2B 5 Var (bB RM 1 eB) 5 b 2B s 2M 1 s2 (eB) Systematic risk equals b2B s2M 5 .22 3 .252 5 .0025. Thus the firm-specific variance of B equals s 2 (eB) 5 s2B 2 b2B s2M 5 .302 2 .22 3 .252 5 .0875 2. The variance of each stock is b2 s2M 1 s2(e). For stock A, we obtain s2A 5 .92(20)2 1 302 5 1,224 sA 5 35% For stock B, s2B 5 1.12(20)2 1 102 5 584 sB 5 24% The covariance is bAbB s2M 5 .9 3 1.1 3 202 5 396

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Go to http://finance.yahoo.com and click on Stocks link under the Investing tab. Look for the Stock Screener link under Research Tools. The Java Yahoo! Finance Screener lets you create your own screens. In the Click to Add Criteria box, find Trading and Volume on the menu and choose Beta. In the Conditions box, choose ,5and in the Values box, enter 1. Hit the Enter key and then request the top 200 matches in the Return Top_Matches box. Click on the Run Screen button. Select the View Table tab and sort the results to show the lowest betas at the top of the list by clicking on the Beta column header. Which firms have the lowest betas? In which industries do they operate? Select the View Histogram tab and when the histogram appears, look at the bottom of the screen to see the Show Histogram for box. Use the menu that comes up when you click on the down arrow to select beta. What pattern(s), if any, do you see in the distributions of betas for firms that have betas less than 1?

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PART II

Portfolio Theory and Practice 3. s2(eP)5(1⁄2)2[s2(eA)1s2(eB)] 51⁄4 (.3021.102) 5.0250 Therefore s(eP)5.158515.8% 4. The regression ALPHA is related to the index-model a by ALPHA 5 aindex model 1 (1 2 b)rf For Intel, ALPHA521.0%,b51.60, and we are told that rf was .2%. Thus aindex model 5 21.0% 2 (1 2 1.60).2% 5 2.88% Intel’s return was somewhat disappointing. It underperformed its “benchmark” return by an average of .88% per month.

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5. The industries with positive adjustment factors are most sensitive to the economy. Their betas would be expected to be higher because the business risk of the firms is higher. In contrast, the industries with negative adjustment factors are in business fields with a lower sensitivity to the economy. Therefore, for any given financial profile, their betas are lower.

CHAPTER NINE

The Capital Asset Pricing Model

THE CAPITAL ASSET pricing model, almost always referred to as the CAPM, is a centerpiece of modern financial economics. The model gives us a precise prediction of the relationship that we should observe between the risk of an asset and its expected return. This relationship serves two vital functions. First, it provides a benchmark rate of return for evaluating possible investments. For example, if we are analyzing securities, we might be interested in whether the expected return we forecast for a stock is more or less than its

9.1

“fair” return given its risk. Second, the model helps us to make an educated guess as to the expected return on assets that have not yet been traded in the marketplace. For example, how do we price an initial public offering of stock? How will a major new investment project affect the return investors require on a company’s stock? Although the CAPM does not fully withstand empirical tests, it is widely used because of the insight it offers and because its accuracy is deemed acceptable for important applications.

The Capital Asset Pricing Model

1

William Sharpe, “Capital Asset Prices: A Theory of Market Equilibrium,” Journal of Finance, September 1964. John Lintner, “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets,” Review of Economics and Statistics, February 1965. 2

Jan Mossin, “Equilibrium in a Capital Asset Market,” Econometrica, October 1966.

PART III

The capital asset pricing model is a set of predictions concerning equilibrium expected returns on risky assets. Harry Markowitz laid down the foundation of modern portfolio management in 1952. The CAPM was published 12 years later in articles by William Sharpe,1 John Lintner,2 and Jan Mossin.3 The time for this gestation indicates that the leap from Markowitz’s portfolio selection model to the CAPM is not trivial. Shooting straight to the heart of the CAPM, suppose all investors optimized their portfolios á la Markowitz. That is, each investor uses an input list (expected returns and covariance matrix) to draw an efficient frontier employing all available risky assets and identifies an efficient risky portfolio, P, by drawing the tangent CAL (capital allocation line) to the frontier as in Figure9.1, panel A (which is just a reproduction of Figure 7.11). As a result, each investor holds securities in the investable universe with weights arrived at by the Markowitz optimization process.

3

9

292

PART III

Equilibrium in Capital Markets

A: The Efﬁcient Frontier of Risky Assets with the Optimal CAL E(r) CAL(P) Efﬁcient Frontier

E(rP)

P

rf

σP B: The Efﬁcient Frontier and the Capital Market Line

σ

E(r) CML Efﬁcient Frontier

E(rM)

M

rf

σM

σ

The CAPM asks what would happen if all investors shared an identical investable universe and used the same input list to draw their efficient frontiers. Obviously, their efficient frontiers would be identical. Facing the same risk-free rate, they would then draw an identical tangent CAL and naturally all would arrive at the same risky portfolio, P. All investors therefore would choose the same set of weights for each risky asset. What must be these weights? A key insight of the CAPM is this: Because the market portfolio is the aggregation of all of these identical risky portfolios, it too will have the same weights. Therefore, if all investors choose the same risky portfolio, it must be the market portfolio, that is, the valueweighted portfolio of all assets in the investable universe. Therefore, the capital allocation line based on each investor’s optimal risky portfolio will in fact also be the capital market line, as depicted in Figure 9.1, panel B. This implication will allow us to say much about the risk–return trade-off.

Why Do All Investors Hold the Market Portfolio?

What is the market portfolio? When we sum over, or aggregate, Figure 9.1 Capital allocation line and the capital market line the portfolios of all individual investors, lending and borrowing will cancel out (because each lender has a corresponding borrower), and the value of the aggregate risky portfolio will equal the entire wealth of the economy. This is the market portfolio, M. The proportion of each stock in this portfolio equals the market value of the stock (price per share times number of shares outstanding) divided by the sum of the market value of all stocks.4 This implies that if the weight of GE stock, for example, in each common risky portfolio is 1%, then GE also will constitute 1% of the market portfolio. The same principle applies to the proportion of any stock in each investor’s risky portfolio. As a result, the optimal risky portfolio of all investors is simply a share of the market portfolio in Figure9.1. 4

We use the term “stock” for convenience; the market portfolio properly includes all assets in the economy.

CHAPTER 9

The Capital Asset Pricing Model

293

Now suppose that the optimal portfolio of our investors does not include the stock of some company, such as Delta Airlines. When all investors avoid Delta stock, the demand is zero, and Delta’s price takes a free fall. As Delta stock gets progressively cheaper, it becomes ever more attractive and other stocks look relatively less attractive. Ultimately, Delta reaches a price where it is attractive enough to include in the optimal stock portfolio. Such a price adjustment process guarantees that all stocks will be included in the optimal portfolio. It shows that all assets have to be included in the market portfolio. The only issue is the price at which investors will be willing to include a stock in their optimal risky portfolio.

The Passive Strategy Is Efficient In Chapter 6 we defined the CML as the CAL that is constructed from a money market account (or T-bills) and the market portfolio. Perhaps now you can fully appreciate why the CML is an interesting CAL. In the simple world of the CAPM, M is the optimal tangency portfolio on the efficient frontier. In this scenario, the market portfolio held by all investors is based on the common input list, thereby incorporating all relevant information about the universe of securities. This means that investors can skip the trouble of doing security analysis and obtain an efficient portfolio simply by holding the market portfolio. (Of course, if everyone were to follow this strategy, no one would perform security analysis and this result would no longer hold. We discuss this issue in greater depth in Chapter 11 on market efficiency.) Thus the passive strategy of investing in a market-index portfolio is efficient. For this reason, we sometimes call this result a mutual fund theorem. The mutual fund theorem is another incarnation of the separation property discussed in Chapter 7. If all investors would freely choose to hold a common risky portfolio identical to the market portfolio, they would not object if all stocks in the market were replaced with shares of a single mutual fund holding that market portfolio. In reality, different investment managers do create risky portfolios that differ from the market index. We attribute this in part to the use of different input lists in the formation of their optimal risky portfolios. Nevertheless, the practical significance of the mutual fund theorem is that a passive CONCEPT CHECK 9.1 investor may view the market index as a reasonable first approximation to an efficient risky portfolio. If there are only a few investors who perform The nearby box contains a parable illustrating the argu- security analysis, and all others hold the market ment for indexing. If the passive strategy is efficient, then portfolio, M, would the CML still be the efficient attempts to beat it simply generate trading and research CAL for investors who do not engage in security costs with no offsetting benefit, and ultimately inferior analysis? Why or why not? results.

The Risk Premium of the Market Portfolio In Chapter 6 we discussed how individual investors go about deciding capital allocation. If all investors choose to invest in portfolio M and the risk-free asset, what can we deduce about the equilibrium risk premium of portfolio M? Recall that each individual investor chooses a proportion y, allocated to the optimal portfolio M, such that E ( rM ) 2 rf y5 (9.1) As2M where E(rM) 2 rf 5 E(RM) is the risk premium (expected excess return) on the market portfolio.

WORDS FROM THE STREET

The Parable of the Money Managers Some years ago, in a land called Indicia, revolution led to the overthrow of a socialist regime and the restoration of a system of private property. Former government enterprises were reformed as corporations, which then issued stocks and bonds. These securities were given to a central agency, which offered them for sale to individuals, pension funds, and the like (all armed with newly printed money). Almost immediately a group of money managers came forth to assist these investors. Recalling the words of a venerated elder, uttered before the previous revolution (“Invest in Corporate Indicia”), they invited clients to give them money, with which they would buy a cross-section of all the newly issued securities. Investors considered this a reasonable idea, and soon everyone held a piece of Corporate Indicia. Before long the money managers became bored because there was little for them to do. Soon they fell into the habit of gathering at a beachfront casino where they passed the time playing roulette, craps, and similar games, for low stakes, with their own money. After a while, the owner of the casino suggested a new idea. He would furnish an impressive set of rooms which would be designated the Money Managers’ Club. There the members could place bets with one another about the fortunes of various corporations, industries, the level of the Gross Domestic Product, foreign trade, etc. To make the betting more exciting, the casino owner suggested that the managers use their clients’ money for this purpose. The offer was immediately accepted, and soon the money managers were betting eagerly with one another. At the end of each week, some found that they had won money for their clients, while others found that they had lost. But the losses always exceeded the gains, for a certain

amount was deducted from each bet to cover the costs of the elegant surroundings in which the gambling took place. Before long a group of professors from Indicia U. suggested that investors were not well served by the activities being conducted at the Money Managers’ Club. “Why pay people to gamble with your money? Why not just hold your own piece of Corporate Indicia?” they said. This argument seemed sensible to some of the investors, and they raised the issue with their money managers. A few capitulated, announcing that they would henceforth stay away from the casino and use their clients’ money only to buy proportionate shares of all the stocks and bonds issued by corporations. The converts, who became known as managers of Indicia funds, were initially shunned by those who continued to frequent the Money Managers’ Club, but in time, grudging acceptance replaced outright hostility. The wave of puritan reform some had predicted failed to materialize, and gambling remained legal. Many managers continued to make their daily pilgrimage to the casino. But they exercised more restraint than before, placed smaller bets, and generally behaved in a manner consonant with their responsibilities. Even the members of the Lawyers’ Club found it difficult to object to the small amount of gambling that still went on. And everyone but the casino owner lived happily ever after. Source: William F. Sharpe, “The Parable of the Money Managers,” The Financial Analysts’ Journal 32 (July/August 1976), p. 4. Copyright 1976, CFA Institute. Reproduced from The Financial Analysts’ Journal with permission from the CFA Institute. All rights reserved.

In the simplified CAPM economy, risk-free investments involve borrowing and lending among investors. Any borrowing position must be offset by the lending position of the creditor. This means that net borrowing and lending across all investors must be zero, and therefore, substituting the representative investor’s risk aversion, A, for A, the average position in the risky portfolio is 100%, or y 5 1. Setting y 51 in Equation9.1 and rearranging, we find that the risk premium on the market portfolio is related to its variance by the average degree of risk aversion: E ( RM ) 5 As2M (9.2) CONCEPT CHECK

9.2

Data from the last eight decades for the S&P 500 index yield the following statistics: average excess return, 7.9%; standard deviation, 23.2%. a. To the extent that these averages approximated investor expectations for the period, what must have been the average coefficient of risk aversion? b. If the coefficient of risk aversion were actually 3.5, what risk premium would have been consistent with the market’s historical standard deviation?

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Expected Returns on Individual Securities The CAPM is built on the insight that the appropriate risk premium on an asset will be determined by its contribution to the risk of investors’ overall portfolios. Portfolio risk is what matters to investors and is what governs the risk premiums they demand. Remember that in the CAPM, all investors use the same input list, that is, the same estimates of expected returns, variances, and covariances. To calculate the variance of the market portfolio, we use the bordered covariance matrix with the market portfolio weights, as discussed in Chapter 7. We highlight GE in this depiction of the n stocks in the market portfolio so that we can measure the contribution of GE to the risk of the market portfolio. Recall that we calculate the variance of the portfolio by summing over all the elements of the covariance matrix, first multiplying each element by the portfolio weights from the row and the column. The contribution of one stock to portfolio variance therefore can be expressed as the sum of all the covariance terms in the column corresponding to the stock, where each covariance is first multiplied by both the stock’s weight from its row and the weight from its column.5

Portfolio Weights w1 w2 . . . wGE . . . wn

w1

w2

...

wGE

...

wn

Cov(R1,R1) Cov(R2,R1) . . . Cov(RGE,R1) . . . Cov(Rn,R1)

Cov(R1,R2) Cov(R2,R2) . . . Cov(RGE,R2) . . . Cov(Rn,R2)

... ...

Cov(R1,RGE) Cov(R2,RGE) . . . Cov(RGE,RGE) . . . Cov(Rn,RGE)

... ...

Cov(R1,Rn) Cov(R2,Rn) . . . Cov(RGE,Rn) . . . Cov(Rn,Rn)

...

...

...

...

Thus, the contribution of GE’s stock to the variance of the market portfolio is wGE 3 w1Cov ( R1, RGE ) 1 w2Cov ( R2, RGE ) 1 . . . 1 wGECov ( RGE, RGE ) 1 . . . 1 wnCov ( Rn, RGE )4

(9.3)

Notice that every term in the square brackets can be slightly rearranged as follows: wi Cov(Ri,RGE )5Cov(wiRi,RGE ). Moreover, because covariance is additive, the sum of the terms in the square brackets is n

n

n

a wi Cov ( Ri, RGE ) 5 a Cov (wi Ri, RGE ) 5 Cov a a wi Ri, RGEb

i51

5

i51

(9.4)

i51

An alternative approach would be to measure GE’s contribution to market variance as the sum of the elements in the row and the column corresponding to GE. In this case, GE’s contribution would be twice the sum in Equation9.3. The approach that we take in the text allocates contributions to portfolio risk among securities in a convenient manner in that the sum of the contributions of each stock equals the total portfolio variance, whereas the alternative measure of contribution would sum to twice the portfolio variance. This results from a type of double-counting, because adding both the rows and the columns for each stock would result in each entry in the matrix being added twice.

295

296

PART III

Equilibrium in Capital Markets n

But because a wiRi 5 RM, Equation 9.4 implies that i51

n

a wi Cov (Ri, RGE ) 5 Cov (RM, RGE )

i51

and therefore, GE’s contribution to the variance of the market portfolio (Equation 9.3) may be more simply stated as wGECov(RM,RGE ). This should not surprise us. For example, if the covariance between GE and the rest of the market is negative, then GE makes a “negative contribution” to portfolio risk: By providing excess returns that move inversely with the rest of the market, GE stabilizes the return on the overall portfolio. If the covariance is positive, GE makes a positive contribution to overall portfolio risk because its returns reinforce swings in the rest of the portfolio.6 We also observe that the contribution of GE to the risk premium of the market portfolio is wGEE(RGE). Therefore, the reward-to-risk ratio for investments in GE can be expressedas GE’s contribution to risk premium wGE E ( RGE ) E ( RGE ) 5 5 GE’s contribution to variance wGECov( RGE, RM) Cov ( RGE, RM) The market portfolio is the tangency (efficient mean-variance) portfolio. The reward-torisk ratio for investment in the market portfolio is Market risk premium E ( RM ) 5 Market variance s2M

(9.5)

The ratio in Equation 9.5 is often called the market price of risk because it quantifies the extra return that investors demand to bear portfolio risk. Notice that for components of the efficient portfolio, such as shares of GE, we measure risk as the contribution to portfolio variance (which depends on its covariance with the market). In contrast, for the efficient portfolio itself, variance is the appropriate measure of risk.7 A basic principle of equilibrium is that all investments should offer the same rewardto-risk ratio. If the ratio were better for one investment than another, investors would rearrange their portfolios, tilting toward the alternative with the better trade-off and shying away from the other. Such activity would impart pressure on security prices until the ratios were equalized. Therefore we conclude that the reward-to-risk ratios of GE and the market portfolio should be equal: E ( RGE ) E ( RM ) 5 Cov ( RGE, RM ) s2M

(9.6)

To determine the fair risk premium of GE stock, we rearrange Equation 9.6 slightly to obtain E ( RGE ) 5

Cov ( RGE, RM ) s2M

E ( RM )

(9.7)

6

A positive contribution to variance doesn’t imply that diversification isn’t beneficial. Excluding GE from the portfolio would require that its weight be assigned to the remaining stocks, and that reallocation would increase variance even more. Variance is reduced by including more stocks and reducing the weight of all (i.e., diversifying), despite the fact that each positive-covariance security makes some contribution to variance. 7 Unfortunately the market portfolio’s Sharpe ratio E(rM) 2 rf sM sometimes is referred to as the market price of risk, but it is not. The unit of risk is variance, and the price of risk relates risk premium to variance (or to covariance for incremental risk).

CHAPTER 9

The Capital Asset Pricing Model

The ratio Cov(RGE, RM)/s2M measures the contribution of GE stock to the variance of the market portfolio as a fraction of the total variance of the market portfolio. The ratio is called beta and is denoted by b. Using this measure, we can restate Equation 9.7 as E (rGE ) 5 rf 1 bGE 3 E ( rM ) 2 rf 4

(9.8)

This expected return–beta (or mean-beta) relationship is the most familiar expression of the CAPM to practitioners. If the expected return–beta relationship holds for any individual asset, it must hold for any combination of assets. Suppose that some portfolio P has weight wk for stock k, where k takes on values 1, ... , n. Writing out the CAPM Equation 9.8 for each stock, and multiplying each equation by the weight of the stock in the portfolio, we obtain these equations, one for each stock: w1E ( r1 ) 5 w1rf 1 w1b1 3 E ( rM ) 2 rf 4 1w2E ( r2 ) 5 w2rf 1 w2b2 3 E ( rM ) 2 rf 4 c5 c 1 1wnE ( rn ) 5 wnrf 1 wnbn 3 E ( rM ) 2 rf 4 E ( rP ) 5 rf 1 bP 3 E ( rM ) 2 rf 4 Summing each column shows that the CAPM holds for the overall portfolio because E(rP) 5 g wkE(rk ) is the expected return on the portfolio, and bP 5 g wk bk is the portfolio k

k

beta. Incidentally, this result has to be true for the market portfolio itself, E( rM ) 5 rf 1 bM 3 E ( rM ) 2 rf 4 Indeed, this is a tautology because bM51, as we can verify by noting that bM 5

Cov ( RM, RM ) s2M

5

s2M s2M

This also establishes 1 as the weighted-average value of beta across all assets. If the market beta is 1, and the market is a portfolio of all assets in the economy, the weighted-average beta of all assets must be 1. Hence betas greater than 1 are considered aggressive in that investment in high-beta stocks entails above-average sensitivity to market swings. Betas below 1 can be described as defensive. A word of caution: We often hear that well-managed firms will provide high rates of return. We agree this is true if one measures the firm’s return on its investments in plant and equipment. The CAPM, however, predicts returns on investments in the securities of the firm. Let’s say that everyone knows a firm is well run. Its stock price will therefore be bid up, and consequently returns to stockholders who buy at those high prices will not be excessive. Security prices, in other words, already reflect public information about a firm’s prospects; therefore only the risk of the company (as measured by beta in the context of the CAPM) should affect expected returns. In a well-functioning market, investors receive high expected returns only if they are willing to bear risk. Investors do not directly observe or determine expected returns on securities. Rather, they observe security prices and bid those prices up or down. Expected rates of return are determined by the prices investors must pay compared to the cash flows those investments might garner.

297

298

PART III

CONCEPT CHECK

Equilibrium in Capital Markets

9.3

Suppose that the risk premium on the market portfolio is estimated at 8% with a standard deviation of 22%. What is the risk premium on a portfolio invested 25% in Toyota and 75% in Ford, if they have betas of 1.10 and 1.25, respectively?

The Security Market Line We can view the expected return–beta relationship as a reward–risk equation. The beta of a security is the appropriate measure of its risk because beta is proportional to the risk the security contributes to the optimal risky portfolio. Risk-averse investors measure the risk of the optimal risky portfolio by its variance. Hence, we would expect the risk premium on individual assets to depend on the contribution of the asset to the risk of the portfolio. The beta of a stock measures its contribution to the variance of the market portfolio and therefore the required risk premium is a function of beta. The CAPM confirms this intuition, stating further that the security’s risk premium is directly proportional to both the beta and the risk premium of the market portfolio; that is, the risk premium equals b[E(rM)2 rf]. The expected return–beta relationship can be portrayed graphically as the security market line (SML) in Figure9.2. Because the market’s beta is 1, the slope is the risk premium of the market portfolio. At the point on the horizontal axis where b 51, we can read off the vertical axis the expected return on the market portfolio. It is useful to compare the security market line to the capital market line. The CML graphs the risk premiums of efficient portfolios (i.e., portfolios composed of the market and the risk-free asset) as a function of portfolio standard deviation. This is appropriate because standard deviation is a valid measure of risk for efficiently diversified portfolios that are candidates for an investor’s overall portfolio. The SML, in contrast, graphs individual asset risk premiums as a function of asset risk. The relevant E(r) measure of risk for individual assets held as parts of well-diversified portfolios is not the asset’s standard SML deviation or variance; it is, instead, the contribution of the asset to the portfolio variance, which we measure by the asset’s beta. The SML is valid for both efficient portfolios and individual assets. E(rM) The security market line provides a benchmark for the evaluation of investment performance. Given the risk of an investment, as measured by its beta, the E(rM) 2 rf 5 Slope of SML SML provides the required rate of return necessary to compensate investors for risk as well as the time value rf of money. 1 Because the security market line is the graphic representation of the expected return–beta relationship, β “fairly priced” assets plot exactly on the SML; that is, their expected returns are commensurate with their βM 51.0 risk. All securities must lie on the SML in market equilibrium. We see here how the CAPM may be of use in the money-management industry. Suppose that Figure 9.2 The security market line the SML relation is used as a benchmark to assess the

CHAPTER 9

The Capital Asset Pricing Model

fair expected return on a risky asset. Then security E(r) (%) analysis is performed to calculate the return actually expected. (Notice that we depart here from the simple SML CAPM world in that some investors now apply their own unique analysis to derive an “input list” that may differ from their competitors’.) If a stock is perceived Stock 17 to be a good buy, or underpriced, it will provide an α 15.6 expected return in excess of the fair return stipulated M 14 by the SML. Underpriced stocks therefore plot above the SML: Given their betas, their expected returns are greater than dictated by the CAPM. Overpriced stocks plot below the SML. The difference between the fair and actually expected rates of return on a stock is called the stock’s 6 alpha, denoted by a. For example, if the market return is expected to be 14%, a stock has a beta of 1.2, and the T-bill rate is 6%, the SML would predict an expected β return on the stock of 6 11.2(14 26) 515.6%. If 1.0 1.2 one believed the stock would provide an expected return of 17%, the implied alpha would be 1.4% (see Figure9.3). Figure 9.3 The SML and a positive-alpha stock One might say that security analysis (which we treat in Part Five) is about uncovering securities with nonzero alphas. This analysis suggests that the starting point of portfolio management can be a passive market-index portfolio. The portfolio manager will then increase the weights of securities with positive alphas and decrease the weights of securities with negative alphas. We showed one strategy for adjusting the portfolio weights in such a manner in Chapter 8. The CAPM is also useful in capital budgeting decisions. For a firm considering a new project, the CAPM can provide the required rate of return that the project needs to yield, based on its beta, to be acceptable to investors. Managers can use the CAPM to obtain this cutoff internal rate of return (IRR), or “hurdle rate” for the project. The nearby box describes how the CAPM can be used in capital budgeting. It also discusses some empirical anomalies concerning the model, which we address in detail in Chapters 11–13.

Example 9.1

Using the CAPM

Yet another use of the CAPM is in utility rate-making cases.8 In this case the issue is the rate of return that a regulated utility should be allowed to earn on its investment in plant and equipment. Suppose that the equityholders have invested $100 million in the firm and that the beta of the equity is .6. If the T-bill rate is 6% and the market risk premium is 8%, then the fair profits to the firm would be assessed as 61.638510.8% of the $100 million investment, or $10.8 million. The firm would be allowed to set prices at a level expected to generate these profits.

8

This application is becoming less common, as many states are in the process of deregulating their public utilities and allowing a far greater degree of free market pricing. Nevertheless, a considerable amount of rate setting still takes place.

299

WORDS FROM THE STREET

Tales From the Far Side Financial markets’ evaluation of risk determines the way firms invest. What if the markets are wrong? Investors are rarely praised for their good sense. But for the past two decades a growing number of firms have based their decisions on a model which assumes that people are perfectly rational. If they are irrational, are businesses making the wrong choices? The model, known as the “capital-asset pricing model,” or CAPM, has come to dominate modern finance. Almost any manager who wants to defend a project—be it a brand, a factory or a corporate merger—must justify his decision partly based on the CAPM. The reason is that the model tells a firm how to calculate the return that its investors demand. If shareholders are to benefit, the returns from any project must clear this “hurdle rate.” Although the CAPM is complicated, it can be reduced to five simple ideas: 1. Investors can eliminate some risks—such as the risk that workers will strike, or that a firm’s boss will quit—by diversifying across many regions and sectors.

Beta Power Return

Market Return

rA

A

Risk-Free Return

2. Some risks, such as that of a global recession, cannot be eliminated through diversification. So even a basket of all of the stocks in a stock market will still be risky. 3. People must be rewarded for investing in such a risky basket by earning returns above those that they can get on safer assets, such as Treasury bills. 4. The rewards on a specific investment depend only on the extent to which it affects the market basket’s risk. 5. Conveniently, that contribution to the market basket’s risk can be captured by a single measure—dubbed “beta”—which expresses the relationship between the investment’s risk and the market’s. Beta is what makes the CAPM so powerful. Although an investment may face many risks, diversified investors should care only about those that are related to the market

CONCEPT CHECK

B

rB

b 12

1

2

basket. Beta not only tells managers how to measure those risks, but it also allows them to translate them directly into a hurdle rate. If the future profits from a project will not exceed that rate, it is not worth shareholders’ money. The diagram shows how the CAPM works. Safe investments, such as Treasury bills, have a beta of zero. Riskier investments should earn a premium over the risk-free rate which increases with beta. Those whose risks roughly match the market’s have a beta of one, by definition, and should earn the market return. So suppose that a firm is considering two projects, A and B. Project A has a beta of ½: when the market rises or falls by 10%, its returns tend to rise or fall by 5%. So its risk premium is only half that of the market. Project B’s

9.4 and 9.5

Stock XYZ has an expected return of 12% and risk of b 51. Stock ABC has expected return of 13% and b 51.5. The market’s expected return is 11%, and rf 55%. a. According to the CAPM, which stock is a better buy? b. What is the alpha of each stock? Plot the SML and each stock’s risk–return point on one graph. Show the alphas graphically. The risk-free rate is 8% and the expected return on the market portfolio is 16%. A firm considers a project that is expected to have a beta of 1.3. a. What is the required rate of return on the project? b. If the expected IRR of the project is 19%, should it be accepted?

300

risk premium is twice that of the market, so it must earn a higher return to justify the expenditure.

NEVER KNOWINGLY UNDERPRICED But there is one small problem with the CAPM: Financial economists have found that beta is not much use for explaining rates of return on firms’ shares. Worse, there appears to be another measure which explains these returns quite well. That measure is the ratio of a firm’s book value (the value of its assets at the time they entered the balance sheet) to its market value. Several studies have found that, on average, companies that have high book-to-market ratios tend to earn excess returns over long periods, even after adjusting for the risks that are associated with beta. The discovery of this book-to-market effect has sparked a fierce debate among financial economists. All of them agree that some risks ought to carry greater rewards. But they are now deeply divided over how risk should be measured. Some argue that since investors are rational, the book-to-market effect must be capturing an extra risk factor. They conclude, therefore, that managers should incorporate the book-to-market effect into their hurdle rates. They have labeled this alternative hurdle rate the “new estimator of expected return,” or NEER. Other financial economists, however, dispute this approach. Since there is no obvious extra risk associated with a high book-to-market ratio, they say, investors must be mistaken. Put simply, they are underpricing high bookto-market stocks, causing them to earn abnormally high returns. If managers of such firms try to exceed those inflated hurdle rates, they will forgo many profitable investments. With economists now at odds, what is a conscientious manager to do? Jeremy Stein, an economist at the Massachusetts Institute of Technology’s business school, offers a paradoxical

answer.* If investors are rational, then beta cannot be the only measure of risk, so managers should stop using it. Conversely, if investors are irrational, then beta is still the right measure in many cases. Mr. Stein argues that if beta captures an asset’s fundamental risk—that is, its contribution to the market basket’s risk—then it will often make sense for managers to pay attention to it, even if investors are somehow failing to. Often, but not always. At the heart of Mr. Stein’s argument lies a crucial distinction—that between (a) boosting a firm’s long-term value and (b) trying to raise its share price. If investors are rational, these are the same thing: any decision that raises long-term value will instantly increase the share price as well. But if investors are making predictable mistakes, a manager must choose. For instance, if he wants to increase today’s share price—perhaps because he wants to sell his shares, or to fend off a takeover attempt—he must usually stick with the NEER approach, accommodating investors’ misperceptions. But if he is interested in long-term value, he should usually continue to use beta. Showing a flair for marketing, Mr. Stein labels this far-sighted alternative to NEER the “fundamental asset risk”—or FAR—approach. Mr. Stein’s conclusions will no doubt irritate many company bosses, who are fond of denouncing their investors’ myopia. They have resented the way in which CAPM—with its assumption of investor infallibility—has come to play an important role in boardroom decision making. But it now appears that if they are right, and their investors are wrong, then those same far-sighted managers ought to be the CAPM’s biggest fans. *Jeremy Stein, “Rational Capital Budgeting in an Irrational World,” The Journal of Business, October 1996. Source: “Tales from the FAR Side,” The Economist Group, Inc. November 16, 1996, p. 8. © The Economist Newspaper Limited, London.

The CAPM and the Single-Index Market The key implications of the CAPM can be summarized by these two statements: 1. The market portfolio is efficient. 2. The risk premium on a risky asset is proportional to its beta. While these two statements often are thought of as complementary, they are actually substitutes because one can be derived from the other (one is true if and only if the other is as well). We have focused on one direction, proceeding from the efficiency of the market portfolio to the mean-beta equation. We now proceed from the mean return–beta relationship to the efficiency of the market portfolio using the index-model market structure we described in Chapter 8. Deriving the CAPM is even more intuitive when starting from a single-index market. Rather than beginning with investors who all apply the Markowitz algorithm to identical input lists, suppose instead that they all face a market where excess stock returns, Ri, are 301

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normally distributed and driven by one systematic factor. The effect of the macro factor is assumed captured by the return on a broad, value-weighted stock-index portfolio, M. The excess return on any stock is described by Equation 8.11 and restated here. Ri 5 ai 1 biRM 1 ei

(9.9)

Each firm-specific, zero-mean residual, ei, is uncorrelated across stocks and uncorrelated with the market factor, RM. Residuals represent diversifiable, nonsystematic, or unique risk. The total risk of a stock is then just the sum of the variance of the systematic component, biRM, and the variance of ei. In sum, the risk premium (mean excess return) and variance are: E ( Ri ) 5 ai 1 biE ( RM ) s2i 5 b2i s2M 1 s2( ei )

(9.10)

The return on a portfolio, Q, constructed from N stocks (ordered by k51,...,N) with a set of weights, wk, must satisfy Equation 9.11, which states that the portfolio alpha, beta, and residual will be the weighted average of the respective parameters of the component securities. N

N

N

RQ 5 a wkak 1 a wkbkRM 1 a wkek 5 aQ 1 bQRM 1 eQ k51 k51 k51

(9.11)

Investors have two considerations when forming their portfolios: First, they can diversify N nonsystematic risk. Since the residuals are uncorrelated, residual risk, s2(eQ) 5 g k 51wk2s 2 ( ek ), becomes ever smaller as diversification reduces portfolio weights. Second, by choosing stocks with positive alpha, or taking short positions in negative-alpha stocks, the risk premium on Q can be increased.9 As a result of these considerations, investors will relentlessly pursue positive alpha stocks, and shun (or short) negative-alpha stocks. Consequently, prices of positive alpha stocks will rise and prices of negative alpha stocks will fall. This will continue until all alpha values are driven to zero. At this point, investors will be content to minimize risk by completely eliminating unique risk, that is, by holding the broadest possible, market portfolio. When all stocks have zero alphas, the market portfolio is the optimal risky portfolio.10

9.2

Assumptions and Extensions of the CAPM Now that we understand the basic insights of the CAPM, we can more explicitly identify the set of simplifying assumptions on which it relies. A model consists of (i) a set of assumptions, (ii) logical/mathematical development of the model through manipulation of those assumptions, and (iii) a set of predictions. Assuming the logical/mathematical manipulations are free of errors, we can test a model in two ways, normative and positive. Normative tests examine the assumptions of the model, while positive tests examine the predictions. 9

The systematic part of the portfolio is of no relevance in this endeavor, since, if desired, the beta of Q can be increased by leverage (borrow and invest in M), or decreased by including in Q a short position in M. The proceeds from the short position in M can be invested in the risk-free asset, thus leaving the alpha and nonsystematic risk unchanged. 10 Recall from Chapter 8 that the weight of a stock in an active portfolio will be zero if its alpha is zero (see Equation 8.20); hence if all alphas are zero, the passive market portfolio will be the optimal risky portfolio.

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If a model’s assumptions are valid, and the development is error-free, then the predictions of the model must be true. In this case, testing the assumptions is synonymous with testing the model. But few, if any, models can pass the normative test. In most cases, as with the CAPM, the assumptions are admittedly invalid—we recognize that we have simplified reality, and therefore to this extent are relying on “untrue” assumptions. The motivation for invoking unrealistic assumptions is clear; we simply cannot solve a model that is perfectly consistent with the full complexity of real-life markets. As we’ve noted, the need to use simplifying assumptions is not peculiar to economics—it characterizes all of science. Assumptions are chosen first and foremost to render the model solvable. But we prefer assumptions to which the model is “robust.” A model is robust with respect to an assumption if its predictions are not highly sensitive to violation of the assumption. If we use only assumptions to which the model is robust, the model’s predictions will be reasonably accurate despite its shortcomings. The upshot of all this is that tests of models are almost always positive—we judge a model on the success of its empirical predictions. This standard brings statistics into any science and requires us to take a stand on what are acceptable levels of significance and power.11 Because the nonrealism of the assumptions precludes a normative test, the positive test is really a test of the robustness of the model to its assumptions.

Assumptions of the CAPM Table9.1 enumerates the list of assumptions underlying the CAPM. In our discussion so far, we have cited explicitly only these three assumptions: 1.a. Investors are rational, mean-variance optimizers. 1.c. Investors use identical input lists, referred to as homogeneous expectations. 2.a. All assets are publicly traded (short positions are allowed) and investors can borrow or lend at a common risk-free rate. The first assumption is far-reaching. Its “visible” part is that investors are not concerned with higher moments (skew and kurtosis) that may “fatten” the left tail of the return distribution. We can ascertain the validity of this assumption from statistical tests of the normality of return distributions as we did in Chapter 5. Less visible is that, by assuming that only the mean and variance of wealth matter to investors, Assumption 1(a) rules out concern with the correlation of asset returns with either inflation or prices of important consumption items such as housing or energy. The extra demand for assets that can be used to hedge these “extra market” risks would increase their prices and reduce their risk premiums relative to the prediction of the CAPM. 11

To illustrate the meanings of significance and power, consider a test of the efficacy of a new drug. The agency testing the drug may make two possible errors. The drug may be useless (or even harmful), but the agency may conclude that it is useful. This is called a “Type I” error. The significance level of a test is the probability of a Type I error. Typical practice is to fix the level of significance at some low level, for example, 5%. In the case of drug testing, for example, the first goal is to avoid introducing ineffective or harmful treatments. The other possible error is that the drug is actually useful, but the testing procedure concludes it is not. This mistake, called “Type II” error, would lead us to discard a useful treatment. The power of the test is the probability of avoiding Type II error (i.e., one minus the probability of making such an error), that is, the probability of accepting the drug if it is indeed useful. We want tests that, at a given level of significance, have the most power, so we will admit effective drugs with high probability. In social sciences in particular, available tests often have low power, in which case they are susceptible to Type II error and will reject a correct model (a “useful drug”) with high frequency. “The drug is useful” is analogous in the CAPM to alphas being zero. When the test data reject the hypothesis that observed alphas are zero at the desired level of significance, the CAPM fails. However, if the test has low power, the probability that we accept the model when not true is too high.

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Table 9.1

1. Individual behavior a. Investors are rational, mean-variance optimizers. b. Their planning horizon is a single period. c. Investors have homogeneous expectations (identical input lists).

The assumptions of the CAPM

2. Market structure a. All assets are publicly held and trade on public exchanges, short positions are allowed, and investors can borrow or lend at a common risk-free rate. b. All information is publicly available. c. No taxes. d. No transaction costs.

Similar extra-market risk factors would arise in a multiperiod model, which requires the addition of Assumption 1(b), limiting investors to a common single-period horizon. Consider a possible decline in future interest rates. Investors would be unhappy about this event to the extent that it would reduce the expected income their investments could throw off in the future. Assets whose returns are negatively correlated with interest rates (e.g., long-term bonds) would hedge this risk and thus command higher prices and lower risk premiums. Because of such hedging demands, correlation with any parameter describing future investment opportunities can result in violations of the CAPM mean-beta equation (and therefore with the efficiency of the market portfolio). A single-period investor horizon eliminates these possibilities. Interestingly, Assumption 1(c) (investors optimize with the same input list), appears ominously restrictive, but it actually is not all that problematic. With the addition of Assumption 2(b) (all information is public), investors generally will be close to agreement. Moreover, trades of investors who derive different input lists will offset and prices will reflect consensus expectations. We will later allow for the likelihood that some investors expend resources to obtain private information and exploit prices that don’t reflect the insights derived from this information. But regardless of their success, it is reasonable to assert that, absent private information, investors should assume alpha values are zero. The assumption that all assets are tradable (2a) is essential for identical input lists. It allows us to ignore federal and state assets and liabilities. More importantly, privately held but nontraded assets such as human capital and private business can create large differences in investor portfolios. Consider owners of a family business. Prudence dictates that they avoid assets that are highly correlated with their businesses. Similarly, investors should avoid stock returns that are positively correlated with their personal income; for example, Boeing employees should avoid investing in the airline and related businesses. Differential demands arising from this consideration can lead to violation of the mean-beta equation and derail the mean-variance efficiency of the index portfolio. Restrictions on borrowing (or significantly higher rates on borrowed funds), which violates Assumption 2(a), also can create problems for the CAPM, because borrowers and lenders will arrive at different tangency portfolios and thus different optimal risky portfolios. Taxes create conditions in which two investors can realize different after-tax returns from the same stock. Such distortions could, in principle, lead to different after-tax optimal risky portfolios to different investors; hence Assumption 2(c) (no taxes). Despite an extension to the CAPM that incorporates personal taxes on dividends and capital gains,12 12

Michael J. Brennan, “Taxes, Market Valuation, and Corporate Finance Policy,” National Tax Journal, December 1973.

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there is no decisive evidence that taxes are a major factor in stock returns. A plausible explanation for this negative finding relies on “clientele” and supply effects. If high taxbracket investors shy away from high-yield (dividend-paying) stocks and thus force down their prices, tax-exempt investors will view the stocks as a bargain and take up the slack in demand. On the other end, if corporations see that high dividend yields reduce stock prices, they simply will substitute stock repurchases for dividends, reinforcing the clientele effect in neutralizing tax effects. Finally, transaction costs inhibit trades and thus the reaction to changes in information; hence Assumption 2(d) (no transaction costs). While in reality trading costs have fallen, remaining differentials in trading costs may still play an important role in stock returns.

Challenges and Extensions to the CAPM Which assumptions are most worrisome? We start with the fact that short positions are not as easy to take as long ones for three reasons: 1. The liability of investors who hold a short position in an asset is potentially unlimited, since the price may rise without limit. Hence a large short position requires large collateral, and proceeds cannot be used to invest in other risky assets. 2. There is a limited supply of shares of any stock to be borrowed by would-be short sellers. It often happens that investors simply cannot find shares to borrow in order to short. 3. Many investment companies are prohibited from short sales. The U.S. and other countries further restrict short sales by regulation. Why are short sales important? Notice that Assumption 1(a) begins with “investors are rational...” When investors exhibit “irrational exuberance” (excessive optimism) about an asset and, as a result, prices rise above intrinsic values, rational investors will take short positions, thus holding down the price. But with effective restrictions, short sales can fail to prevent prices rising to unsustainable levels that are precursors to a correction or even a crash. This really defines a “bubble.” Three unrealistic assumptions, 2(a) (all assets trade) and 2(d) (there are no transaction costs), combined with 1(b) (single-period horizon), generate the major challenges to the model. These challenges have motivated a set of extensions that are, even today, still “under construction” in one way or another. For this reason, none of the extensions has decisively superseded the simple CAPM in the industry. It is an impressive phenomenon, that despite failing many empirical tests, the compelling logic of the CAPM keeps it at the center of the investments industry. However, for better insight to the CAPM, it is useful to understand the extensions of the model.

The Zero-Beta Model Efficient frontier portfolios have a number of interesting characteristics, independently derived by Merton and Roll.13 Two of these are 1. Any portfolio that is a combination of two frontier portfolios is itself on the efficient frontier. 2. Every portfolio on the efficient frontier, except for the global minimum-variance portfolio, has a “companion” portfolio on the bottom (inefficient) half of the frontier 13

Robert C. Merton, “An Analytic Derivation of the Efficient Portfolio Frontier,” Journal of Financial and Quantitative Analysis, 1972. Richard Roll, “A Critique of the Asset Pricing Theory’s Tests: Part I: On Past and Potential Testability of the Theory,” Journal of Financial Economics 4 (1977).

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with which it is uncorrelated. Because it is uncorrelated, the companion portfolio is referred to as the zero-beta portfolio of the efficient portfolio. If we choose the market portfolio M and its zero-beta companion portfolio Z, then we obtain a CAPM-like equation E ( ri ) 2 E ( rZ ) 5 3 E ( RM ) 2 E ( RZ )4

Cov ( ri, rM ) s2M

5 bi 3 E ( rM ) 2 E ( rZ )4

(9.12)

Equation 9.12 resembles the SML of the CAPM, except that the risk-free rate is replaced with the expected return on the zero-beta companion of the market-index portfolio. Fischer Black used these properties to show that Equation 9.12 is the CAPM equation that results when investors face restrictions on borrowing.14 In this case, at least some investors will choose portfolios on the high risk-premium portion of the efficient frontier. Put differently, investors who would otherwise wish to borrow and leverage their portfolios but who find it impossible or costly will instead tilt their portfolios toward high-beta stocks and away from low-beta ones. As a result, prices of high beta stocks will rise, and their risk premiums will fall. The SML will be flatter than in the simple CAPM. You see from Equation 9.12 that the risk premium on the market portfolio is smaller (because the expected return on the zero-beta portfolio is greater than the risk-free rate) and therefore the reward to bearing beta risk is smaller.

Labor Income and Nontraded Assets Two important asset classes that are not traded are human capital and privately held businesses. The discounted value of future labor income exceeds the total market value of traded assets. The market value of privately held corporations and businesses is of the same order of magnitude. Human capital and private enterprises are different types of assets with possibly different implications for equilibrium returns on traded securities. Privately held businesses may be the lesser of the two sources of departures from the CAPM. Suppose that privately held businesses have risk characteristics similar to those of traded assets. In this case, individuals can partially offset the diversification problems posed by their nontraded entrepreneurial assets by reducing their portfolio demand for securities of similar, traded assets. Thus, the CAPM expected return–beta equation may not be greatly disrupted by the presence of entrepreneurial income. To the extent that risk characteristics of private enterprises differ from those of traded securities, a portfolio of traded assets that best hedges the risk of typical private business would enjoy excess demand from the population of private business owners. The price of assets in this portfolio will be bid up relative to the CAPM considerations, and the expected returns on these securities will be lower in relation to their systematic risk. Conversely, securities highly correlated with such risk will have high equilibrium risk premiums and may appear to exhibit positive alphas relative to the conventional SML. In fact, Heaton and Lucas show that adding proprietary income to a standard asset-pricing model improves its predictive performance.15 The size of labor income and its special nature is of greater concern for the validity of the CAPM. The possible effect of labor income on equilibrium returns can be appreciated 14

Fischer Black, “Capital Market Equilibrium with Restricted Borrowing,” Journal of Business, July 1972. John Heaton and Deborah Lucas, “Portfolio Choice and Asset Prices: The Importance of Entrepreneurial Risk,” Journal of Finance 55 (June 2000). This paper offers evidence of the effect of entrepreneurial risk on both portfolio choice and the risk–return relationship. 15

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from its important effect on personal portfolio choice. Despite the fact that an individual can borrow against labor income (via a home mortgage) and reduce some of the uncertainty about future labor income via life insurance, human capital is less “portable” across time and may be more difficult to hedge using traded securities than nontraded business. This may induce pressure on security prices and result in departures from the CAPM expected return–beta equation. Thus, the demand for stocks of labor-intensive firms with high wage expenses may be good hedges for uncertain labor income, and these stocks may require a lower expected return than predicted by the CAPM. Mayers16 derives the equilibrium expected return–beta equation for an economy in which individuals are endowed with labor income of varying size relative to their nonlabor capital. The resultant SML equation is Cov ( Ri, RM ) 1 E ( Ri ) 5 E ( RM ) s2M 1

PH Cov ( Ri, RH ) PM

PH Cov ( RM, RH ) PM

(9.13)

where PH5value of aggregate human capital PM5market value of traded assets (market portfolio) RH5excess rate of return on aggregate human capital The CAPM measure of systematic risk, beta, is replaced in the extended model by an adjusted beta that also accounts for covariance with the portfolio of aggregate human capital. Notice that the ratio of human capital to market value of all traded assets, PH/PM, may well be greater than 1, and hence the effect of the covariance of a security with labor income, Cov(Ri, RH), relative to the average, Cov(RM, RH), is likely to be economically significant. When Cov(Ri,RH) is positive, the adjusted beta is greater when the CAPM beta is smaller than 1, and vice versa. Because we expect Cov(Ri,RH) to be positive for the average security, the risk premium in this model will be greater, on average, than predicted by the CAPM for securities with beta less than 1, and smaller for securities with beta greater than 1. The model thus predicts a security market line that is less steep than that of the standard CAPM. This may help explain the average negative alpha of high-beta securities and positive alpha of low-beta securities that lead to the statistical failure of the CAPM equation. In Chapter 13 on empirical evidence we present additional results along these lines.

A Multiperiod Model and Hedge Portfolios Robert C. Merton revolutionized financial economics by using continuous-time models to extend models of asset pricing.17 While his (Nobel Prize–winning) contributions to optionpricing theory and financial engineering (along with those of Fischer Black and Myron Scholes) may have had greater impact on the investment industry, his solo contribution to portfolio theory was equally important for our understanding of the risk–return relationship. In his basic model, Merton relaxes the “single-period” myopic assumptions about investors. He envisions individuals who optimize a lifetime consumption/investment plan, and who continually adapt consumption/investment decisions to current wealth and planned retirement age. When uncertainty about portfolio returns is the only source of risk and 16

David Mayers, “Nonmarketable Assets and Capital Market Equilibrium under Uncertainty,” in Studies in the Theory of Capital Markets, ed. M. C. Jensen (New York: Praeger, 1972).

17

Merton’s classic works are collected in Continuous-Time Finance (Oxford, U.K.: Basil Blackwell, 1992).

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investment opportunities remain unchanged through time, that is, there is no change in the risk-free rate or the probability distribution of the return on the market portfolio or individual securities, Merton’s so-called intertemporal capital asset pricing model (ICAPM) predicts the same expected return–beta relationship as the single-period equation.18 But the situation changes when we include additional sources of risk. These extra risks are of two general kinds. One concerns changes in the parameters describing investment opportunities, such as future risk-free rates, expected returns, or the risk of the market portfolio. Suppose that the real interest rate may change over time. If it falls in some future period, one’s level of wealth will now support a lower stream of real consumption. Future spending plans, for example, for retirement spending, may be put in jeopardy. To the extent that returns on some securities are correlated with changes in the risk-free rate, a portfolio can be formed to hedge such risk, and investors will bid up the price (and bid down the expected return) of those hedge assets. Investors will sacrifice some expected return if they can find assets whose returns will be higher when other parameters (in this case, the real risk-free rate) change adversely. The other additional source of risk concerns the prices of the consumption goods that can be purchased with any amount of wealth. Consider inflation risk. In addition to the expected level and volatility of nominal wealth, investors must be concerned about the cost of living—what those dollars can buy. Therefore, inflation risk is an important extramarket source of risk, and investors may be willing to sacrifice some expected return to purchase securities whose returns will be higher when the cost of living changes adversely. If so, hedging demands for securities that help to protect against inflation risk would affect portfolio choice and thus expected return. One can push this conclusion even further, arguing that empirically significant hedging demands may arise for important subsectors of consumer expenditures; for example, investors may bid up share prices of energy companies that will hedge energy price uncertainty. These sorts of effects may characterize any assets that hedge important extramarket sources of risk. More generally, suppose we can identify K sources of extramarket risk and find K associated hedge portfolios. Then, Merton’s ICAPM expected return–beta equation would generalize the SML to a multi-index version: K

E ( Ri ) 5 biME ( RM ) 1 a bikE ( Rk )

(9.14)

k51

where biM is the familiar security beta on the market-index portfolio, and bik is the beta on the kth hedge portfolio. Other multifactor models using additional factors that do not arise from extramarket sources of risk have been developed and lead to SMLs of a form identical to that of the ICAPM. These models also may be considered extensions of the CAPM in the broad sense. We examine these models in the next chapter.

A Consumption-Based CAPM The logic of the CAPM together with the hedging demands noted in the previous subsection suggest that it might be useful to center the model directly on consumption. Such models were first proposed by Mark Rubinstein, Robert Lucas, and Douglas Breeden.19 18

Eugene F. Fama also made this point in “Multiperiod Consumption-Investment Decisions,” American Economic Review 60 (1970).

19

Mark Rubinstein, “The Valuation of Uncertain Income Streams and the Pricing of Options,” Bell Journal of Economics and Management Science 7 (1976), pp. 407–25; Robert Lucas, “Asset Prices in an Exchange Economy,” Econometrica 46 (1978), pp. 1429–45; Douglas Breeden, “An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities,” Journal of Financial Economics 7 (1979), pp. 265–96.

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In a lifetime consumption plan, the investor must in each period balance the allocation of current wealth between today’s consumption and the savings and investment that will support future consumption. When optimized, the utility value from an additional dollar of consumption today must be equal to the utility value of the expected future consumption that can be financed by that additional dollar of wealth.20 Future wealth will grow from labor income, as well as returns on that dollar when invested in the optimal complete portfolio. Suppose risky assets are available and you wish to increase expected consumption growth by allocating some of your savings to a risky portfolio. How would we measure the risk of these assets? As a general rule, investors will value additional income more highly during difficult economic times (when resources are scarce) than in affluent times (when consumption is already abundant). An asset will therefore be viewed as riskier in terms of consumption if it has positive covariance with consumption growth—in other words, if its payoff is higher when consumption is already high and lower when consumption is relatively restricted. Therefore, equilibrium risk premiums will be greater for assets that exhibit higher covariance with consumption growth. Developing this insight, we can write the risk premium on an asset as a function of its “consumption risk” as follows: E ( Ri ) 5 biCRPC

(9.15)

where portfolio C may be interpreted as a consumption-tracking portfolio (also called a consumption-mimicking portfolio), that is, the portfolio with the highest correlation with consumption growth; biC is the slope coefficient in the regression of asset i’s excess returns, Ri, on those of the consumption-tracking portfolio; and, finally, RPC is the risk premium associated with consumption uncertainty, which is measured by the expected excess return on the consumption-tracking portfolio: RPC 5 E ( RC ) 5 E ( rC ) 2 rf

(9.16)

Notice how similar this conclusion is to the conventional CAPM. The consumptiontracking portfolio in the CCAPM plays the role of the market portfolio in the conventional CAPM. This is in accord with its focus on the risk of consumption opportunities rather than the risk and return of the dollar value of the portfolio. The excess return on the consumption-tracking portfolio plays the role of the excess return on the market portfolio, M. Both approaches result in linear, single-factor models that differ mainly in the identity of the factor they use. In contrast to the CAPM, the beta of the market portfolio on the market factor of the CCAPM is not necessarily 1. It is perfectly plausible and empirically evident that this beta is substantially greater than 1. This means that in the linear relationship between the market-index risk premium and that of the consumption portfolio, E ( RM ) 5 aM 1 bMCE ( RC ) 1 eM

(9.17)

where aM and áM allow for empirical deviation from the exact model in Equation 9.15, and bMC is not necessarily equal to 1. Because the CCAPM is so similar to the CAPM, one might wonder about its usefulness. Indeed, just as the CAPM is empirically flawed because not all assets are traded, so is the 20

Wealth at each point in time equals the market value of assets in the balance sheet plus the present value of future labor income. These models of consumption and investment decisions are often made tractable by assuming investors exhibit constant relative risk aversion, or CRRA. CRRA implies that an individual invests a constant proportion of wealth in the optimal risky portfolio regardless of the level of wealth. You might recall that our prescription for optimal capital allocation in Chapter 6 also called for an optimal investment proportion in the risky portfolio regardless of the level of wealth. The utility function we employed there also exhibited CRRA.

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CCAPM. The attractiveness of this model is in that it compactly incorporates consumption hedging and possible changes in investment opportunities, that is, in the parameters of the return distributions in a single-factor framework. There is a price to pay for this compactness, however. Consumption growth figures are published infrequently (monthly at the most) compared with financial assets, and are measured with significant error. Nevertheless, recent empirical research21 indicates that this model is more successful in explaining realized returns than the CAPM, which is a reason why students of investments should be familiar with it. We return to this issue, as well as empirical evidence concerning the CCAPM, in Chapter 13.

Liquidity and the CAPM Despite Assumption 2(d) saying that securities can be traded costlessly, the CAPM has little to say about trading activity. In the equilibrium of the CAPM, all investors share all available information and demand identical portfolios of risky assets. The awkward implication of this result is that there is no reason for trade. If all investors hold identical portfolios of risky assets, then when new (unexpected) information arrives, prices will change commensurately, but each investor will continue to hold a piece of the market portfolio, which requires no exchange of assets. How do we square this implication with the observation that on a typical day, trading volume amounts to several billion shares? One obvious answer is heterogeneous expectations, that is, beliefs not shared by the entire market. Diverse beliefs will give rise to trading as investors attempt to profit by rearranging portfolios in accordance with their now-heterogeneous demands. In reality, trading (and trading costs) will be of great importance to investors. The liquidity of an asset is the ease and speed with which it can be sold at fair market value. Part of liquidity is the cost of engaging in a transaction, particularly the bid–ask spread. Another part is price impact—the adverse movement in price one would encounter when attempting to execute a larger trade. Yet another component is immediacy—the ability to sell the asset quickly without reverting to fire-sale prices. Conversely, illiquidity can be measured in part by the discount from fair market value a seller must accept if the asset is to be sold quickly. A perfectly liquid asset is one that would entail no illiquidity discount. Liquidity (or the lack of it) has long been recognized as an important characteristic that affects asset values. In legal cases, courts have routinely applied very steep discounts to the values of businesses that cannot be publicly traded. But liquidity has not always been appreciated as an important factor in security markets, presumably due to the relatively small trading cost per transaction compared with the large costs of trading assets such as real estate. The breakthrough came in the work of Amihud and Mendelson22 and today, liquidity is increasingly viewed as an important determinant of prices and expected returns. We supply only a brief synopsis of this important topic here and provide empirical evidence in Chapter 13. One important component of trading cost is the bid–ask spread. For example, in electronic markets, the limit-order book contains the “inside spread,” that is, the difference between the highest price at which some investor will purchase any shares and the lowest 21

Ravi Jagannathan and Yong Wang, “Lazy Investors, Discretionary Consumption, and the Cross-Section of Stock Returns,” Journal of Finance 62 (August 2007), pp. 1633–61. 22

Yakov Amihud and Haim Mendelson, “Asset Pricing and the Bid–Ask Spread,” Journal of Financial Economics 17 (1986). A summary of the ensuing large body of literature on liquidity can be found in Yakov Amihud, Haim Mendelson, and Lasse Heje Pedersen, Market Liquidity: Asset Pricing Risk and Crises, Cambridge University Press, New York: 2013.

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price at which another investor is willing to sell. The effective bid–ask spread will also depend on the size of the desired transaction. Larger purchases will require a trader to move deeper into the limit-order book and accept less-attractive prices. While inside spreads on electronic markets often appear extremely low, effective spreads can be much larger, because most limit orders are good for only small numbers of shares. There is greater emphasis today on the component of the spread due to asymmetric information. Asymmetric information is the potential for one trader to have private information about the value of the security that is not known to the trading partner. To see why such an asymmetry can affect the market, think about the problems facing someone buying a used car. The seller knows more about the car than the buyer, so the buyer naturally wonders if the seller is trying to get rid of the car because it is a “lemon.” At the least, buyers worried about overpaying will shave the prices they are willing to pay for a car of uncertain quality. In extreme cases of asymmetric information, trading may cease altogether.23 Similarly, traders who post offers to buy or sell at limit prices need to be worried about being picked off by better-informed traders who hit their limit prices only when they are out of line with the intrinsic value of the firm. Broadly speaking, we may envision investors trading securities for two reasons. Some trades are driven by “noninformational” motives, for example, selling assets to raise cash for a big purchase, or even just for portfolio rebalancing. These sorts of trades, which are not motivated by private information that bears on the value of the traded security, are called noise trades. Security dealers will earn a profit from the bid–ask spread when transacting with noise traders (also called liquidity traders because their trades may derive from needs for liquidity, i.e., cash). Other transactions are initiated by traders who believe they have come across information that a security is mispriced. But if that information gives them an advantage, it must be disadvantageous to the other party in the transaction. In this manner, information traders impose a cost on both dealers and other investors who post limit orders. Although on average dealers make money from the bid–ask spread when transacting with liquidity traders, they will absorb losses from information traders. Similarly, any trader posting a limit order is at risk from information traders. The response is to increase limit-ask prices and decrease limit-bid orders—in other words, the spread must widen. The greater the relative importance of information traders, the greater the required spread to compensate for the potential losses from trading with them. In the end, therefore, liquidity traders absorb most of the cost of the information trades because the bid–ask spread that they must pay on their “innocent” trades widens when informational asymmetry is more severe. The discount in a security price that results from illiquidity can be surprisingly large, far larger than the bid–ask spread. Consider a security with a bid–ask spread of 1%. Suppose it will change hands once a year for the next 3 years and then will be held forever by the third buyer. For the last trade, the investor will pay for the security 99.5% or .995 of its fair price; the price is reduced by half the spread that will be incurred when the stock is sold. The second buyer, knowing the security will be sold a year later for .995 of fair value, and having to absorb half the spread upon purchase, will be willing to pay .9952.005/1.055.9902 (i.e., 99.02% of fair value), if the spread from fair value is discounted at a rate of 5%. Finally, the current buyer, knowing the loss next year, when the stock will be sold for .9902 of fair value (a discount of .0098), will pay for the security only .9952.0098/1.055.9857. Thus the discount has ballooned from .5% to 1.43%. In other 23

The problem of informational asymmetry in markets was introduced by the 2001 Nobel laureate George A. Akerlof and has since become known as the lemons problem. A good introduction to Akerlof’s contributions can be found in George A. Akerlof, An Economic Theorist’s Book of Tales (Cambridge, U.K.: Cambridge University Press, 1984).

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words, the present values of all three future trading costs (spreads) are discounted into the current price.24 To extend this logic, if the security will be traded once a year forever, its current illiquidity cost will equal immediate cost plus the present value of a perpetuity of .5%. At an annual discount rate of 5%, this sum equals .0051.005/.055.105, or 10.5%! Obviously, liquidity is of potentially large value and should not be ignored in deriving the equilibrium value of securities. As trading costs are higher, the illiquidity discount will be greater. Of course, if someone can buy a share at a lower price, the expected rate of return will be higher. Therefore, we should expect to see less-liquid securities offer higher average rates of return. But this illiquidity premium need not rise in direct proportion to trading cost. If an asset is less liquid, it will be shunned by frequent traders and held instead by longer term traders who are less affected by high trading costs. Hence in equilibrium, investors with long holding periods will, on average, hold more of the illiquid securities, while short-horizon investors will prefer liquid securities. This “clientele effect” mitigates the effect of the bid–ask spread for illiquid securities. The end result is that the liquidity premium should increase with trading costs (measured by the bid–ask spread) at a decreasing rate. Figure9.4 confirms this prediction. So far, we have shown that the expected level of liquidity can affect prices, and therefore expected rates of return. What about unanticipated changes in liquidity? In some circumstances, liquidity can unexpectedly dry up. For example, in the financial crisis of 2008, as many investors attempted to reduce leverage and cash out their positions, finding buyers for some assets became difficult. Many mortgage-backed securities stopped trading Average Monthly Return (% per month) 1.2 1 0.8 0.6 0.4 0.2 Bid–Ask Spread (%)

0 0

0.5

1

1.5

2

2.5

3

3.5

Figure 9.4 The relationship between illiquidity and average returns Source: Derived from Yakov Amihud and Haim Mendelson, “Asset Pricing and the Bid–Ask Spread,” Journal of Financial Economics 17 (1986), pp. 223–49. Copyright 1986, with permission from Elsevier.

24

We will see another instance of such capitalization of trading costs in Chapter 13, where one explanation for large discounts on closed-end funds is the substantial present value of a stream of apparently small per-period expenses.

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altogether. Liquidity had evaporated. Nor was this an unheard-of phenomenon. The market crash of 1987, as well as the failure of Long-Term Capital Management in 1998, also saw large declines in liquidity across broad segments of the market. In fact, several studies have investigated variation in a number of measures of liquidity for large samples of stocks and found that when liquidity in one stock decreases, it tends to decrease in other stocks at the same time; thus liquidity across stocks shows significant correlation.25 In other words, variation in liquidity has an important systematic component. Not surprisingly, investors demand compensation for exposure to liquidity risk. The extra expected return for bearing liquidity risk modifies the CAPM expected return–beta relationship. Following up on this insight, Amihud demonstrates that firms with greater liquidity uncertainty have higher average returns.26 Later studies focus on exposure to marketwide liquidity risk, as measured by a “liquidity beta.” Analogously to a traditional market beta, the liquidity beta measures the sensitivity of a firm’s returns to changes in market liquidity (whereas the traditional beta measures return sensitivity to the market return). Firms that provide better returns when market liquidity falls offer some protection against liquidity risk, and thus should be priced higher and offer lower expected returns. In fact, we will see in Chapter 13 that firms with high liquidity betas have offered higher average returns, just as theory predicts.27 Moreover, the liquidity premium that emerges from these studies appears to be of roughly the same order of magnitude as the market risk premium, suggesting that liquidity should be a first-order consideration when thinking about security pricing.

9.3

The CAPM and the Academic World

The thorn in the side of academic researchers is Assumption 1(a) (all assets trade) that leads to the result that the efficient portfolio must include all risky assets in the economy. In reality, we cannot even observe all the assets that do trade, let alone properly account for those that do not. The theoretical market portfolio, which is central to the CAPM, is impossible to pin down in practice. Since the theoretical CAPM market portfolio cannot be observed, tests of the CAPM must be directed at the mean-beta relationship as applied to all observed assets with respect to an observed, but perhaps inefficient, stock index portfolio. These tests face surprisingly difficult hurdles. The objective is to test the SML equation, E(Ri)5biRM. We do so with a regression of excess returns of a sample of stocks (i51,...,N) over a given period, t, against the betas of each stock: Ri,t 5 l0 1 l1bi 1 l2s2ei 1 hi,t

(9.18)

The CAPM predicts that (1) l050, that is, the average alpha in the sample will be zero; (2) l1 5 RM, that is, the slope of the SML equals the market-index risk premium; and (3) l2 50, that is, unique risk, s2ei, doesn’t earn a risk premium. hi is the zero-mean residual of this regression. 25

See, for example, Tarun Chordia, Richard Roll, and Avanidhar Subrahmanyam, “Commonality in Liquidity,” Journal of Financial Economics 56 (2000), pp. 3–28, or J. Hasbrouck and D. H. Seppi, “Common Factors in Prices, Order Flows and Liquidity,” Journal of Financial Economics 59 (2001), pp. 383–411. 26

Yakov Amihud, “Illiquidity and Stock Returns: Cross-Section and Time-Series Effects,” Journal of Financial Markets 9 (2002), pp. 31–56. 27 See L. Pástor and R. F. Stambaugh, “Liquidity Risk and Expected Stock Returns,” Journal of Political Economy 111 (2003), pp. 642–685, or V. V. Acharya and L. H. Pedersen, “Asset Pricing with Liquidity Risk,” Journal of Financial Economics 77 (2005), pp. 375–410.

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Where, you may ask, do we obtain the beta coefficients and residual variances for the N stocks in the regression? We have to estimate this pair for each stock from a time series of stock returns. And therein lies the snag: We estimate these parameters with large errors. Moreover, these errors may be correlated: First, beta may be correlated with the residual variance of each stock (as well as errors in these estimates), and second, the error terms in the regression may be correlated across stocks. These measurement errors can result in a downward bias in the slope of the SML (l1), and an upward bias in the average alpha (l0). We can’t even predict the sign of the bias in (l2). An example of this hazard was pointed out in an early paper by Miller and Scholes,28 who demonstrated how econometric problems could lead one to reject the CAPM even if it were perfectly valid. They considered a checklist of difficulties encountered in testing the model and showed how these problems potentially could bias conclusions. To prove the point, they simulated rates of return that were constructed to satisfy the predictions of the CAPM and used these rates to test the model with standard statistical techniques of the day. The result of these tests was a rejection of the model that looks surprisingly similar to what we find in tests of returns from actual data—this despite the fact that the data were constructed to satisfy the CAPM. Miller and Scholes thus demonstrated that econometric technique alone could be responsible for the rejection of the model in actual tests. Moreover, both coefficients, alpha and beta, as well as residual variance, are likely time varying. There is nothing in the CAPM that precludes such time variation, but standard regression techniques rule it out and thus may lead to false rejection of the model. There are now well-known techniques to account for time-varying parameters. In fact, Robert Engle won the Nobel Prize for his pioneering work on econometric techniques to deal with time-varying volatility, and a good portion of the applications of these new techniques have been in finance.29 Moreover, betas may vary not purely randomly over time, but in response to changing economic conditions. A “conditional” CAPM allows risk and return to change with a set of “conditioning variables.”30 As importantly, Campbell and Vuolteenaho31 find that the beta of a security can be decomposed into two components, one that measures sensitivity to changes in corporate profitability and another that measures sensitivity to changes in the market’s discount rates. These are found to be quite different in many cases. Improved econometric techniques such as those proposed in this short survey may help resolve part of the empirical failure of the simple CAPM. A strand of research that has not yet yielded fruit is the search for portfolios that hedge the price risk of specific consumption items, as in Merton’s Equation 9.14. But the jury is still out on the empirical content of this equation with respect to future investment opportunities. As mentioned in Chapter 5, Fama and French documented the explanatory power of size and book-to-market ratios (B/M). They interpret portfolios formed to align with these characteristics as hedging portfolios in the context of Equation 9.14. Following their lead, other papers have now suggested a number of other extra-market risk factors (discussed in the next chapter). But we don’t really know what uncertainties in future investment opportunities are hedged by these factors, leading many to be skeptical of empirically driven identification of extra-market hedging portfolios. 28

Merton H. Miller and Myron Scholes, “Rates of Return in Relations to Risk: A Re-examination of Some Recent Findings,” in Studies in the Theory of Capital Markets, Michael C. Jensen, ed. (New York: Praeger, 1972). 29 Engle’s work gave rise to the widespread use of so-called ARCH models. ARCH stands for autoregressive conditional heteroskedasticity, which is a fancy way of saying that volatility changes over time, and that recent levels of volatility can be used to form optimal estimates of future volatility. 30 There is now a large literature on conditional models of security market equilibrium. Much of it derives from Ravi Jagannathan and Zhenyu Wang, “The Conditional CAPM and the Cross-Section of Expected Returns,” Journal of Finance 51 (March 1996), pp. 3–53. 31

John Campbell and Tuomo Vuolteenaho, “Bad Beta, Good Beta,” American Economic Review 94 (December 2004), pp. 1249–75.

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The bottom line is that in the academic world the single-index CAPM is considered passé. We don’t yet know, however, what shape the successful extension to replace it will take. Stay tuned for future editions of this text.

9.4

The CAPM and the Investment Industry

While academics have been riding multiple-index models in search of a CAPM that best explains returns, the industry has steadfastly stayed with the single-index CAPM. This interesting phenomenon can be explained by a “test of the non-testable.” Presumably, the CAPM tenet that the market portfolio is efficient cannot be tested because the true market portfolio cannot be observed in the first place. But as time has passed, it has become ever more evident that consistently beating a (not very broad) index portfolio such as the S&P 500 appears to be beyond the power of most investors. Indirect evidence on the efficiency of the market portfolio can be found in a study by Burton Malkiel,32 who estimates alpha values for a large sample of equity mutual funds. The results, which appear in Figure9.5, show that the distribution of alphas is roughly bell shaped, with a mean that is slightly negative but statistically indistinguishable from zero. On average, it does not appear that mutual funds outperform the market index (the S&P 500) on a risk-adjusted basis.33

36 32 28 Frequency

24 20 16 12 8 4 0

−3

−2

−1

1

2

Alpha (%)

Figure 9.5 Estimates of individual mutual fund alphas, 1972–1991. This is a plot of the frequency distribution of estimated alphas for all-equity mutual funds with 10-year continuous records. Source: Burton G. Malkiel, “Returns from Investing in Equity Mutual Funds 1971–1991,” Journal of Finance 50 (June 1995), pp. 549–72. Used with permission of John Wiley and Sons, via Copyright Clearance Center.

32

Burton G. Malkiel, “Returns from Investing in Equity Mutual Funds 1971–1991,” Journal of Finance 50 (June 1995), pp. 549–72. 33 Notice that the study included all mutual funds with at least 10 years of continuous data. This suggests the average alpha from this sample would be upward biased because funds that failed after less than 10 years were ignored and omitted from the left tail of the distribution. This survivorship bias makes the finding that the average fund underperformed the index even more telling. We discuss survivorship bias further in Chapter 11.

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This result is quite meaningful. While we might expect realized alpha values of individual securities to center around zero, professionally managed mutual funds might be expected to demonstrate average positive alphas. Funds with superior performance (and we do expect this set to be nonempty) should tilt the sample average to a positive value. The small impact of superior funds on this distribution suggests the difficulty in beating the passive strategy that the CAPM deems to be optimal. From the standpoint of the industry, an index portfolio that can be beaten by only a small fraction of professional managers over a 10-year period may well be taken as ex-ante efficient for all practical purposes, that is, to be used as: (1) a diversification vehicle to mix with an active portfolio from security analysis (discussed in Chapter 8); (2) a benchmark for performance evaluation and compensation (discussed in Chapter 24); (3) a means to adjudicate law suits about fair compensation to various risky enterprises; and (4) a means to determine proper prices in regulated industries, allowing shareholders to earn a fair rate of return on their investments, but no more.

SUMMARY

1. The CAPM assumes that investors are single-period planners who agree on a common input list from security analysis and seek mean-variance optimal portfolios. 2. The CAPM assumes that security markets are ideal in the sense that: a. b. c. d.

They are large, and investors are price-takers. There are no taxes or transaction costs. All risky assets are publicly traded. Investors can borrow and lend any amount at a fixed risk-free rate.

3. With these assumptions, all investors hold identical risky portfolios. The CAPM holds that in equilibrium the market portfolio is the unique mean-variance efficient tangency portfolio. Thus a passive strategy is efficient. 4. The CAPM market portfolio is a value-weighted portfolio. Each security is held in a proportion equal to its market value divided by the total market value of all securities. 5. If the market portfolio is efficient and the average investor neither borrows nor lends, then the risk premium on the market portfolio is proportional to its variance, s2M, and to the average coefficient of risk aversion across investors, A: E ( rM ) 2 rf 5 As2M 6. The CAPM implies that the risk premium on any individual asset or portfolio is the product of the risk premium on the market portfolio and the beta coefficient: E ( r i ) 2 r f 5 bi 3 E ( r M ) 2 r f 4 where the beta coefficient is the covariance of the asset with the market portfolio as a fraction of the variance of the market portfolio: bi 5

Cov ( ri, rM ) s2M

7. When risk-free investments are restricted but all other CAPM assumptions hold, then the simple version of the CAPM is replaced by its zero-beta version. Accordingly, the risk-free rate in the expected return–beta relationship is replaced by the zero-beta portfolio’s expected rate of return: E( ri ) 5 E(rZ) 1 bi[E(rM) 2 E(rZ)]

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317

8. The simple version of the CAPM assumes that investors have a single-period time horizon. When investors are assumed to be concerned with lifetime consumption and bequest plans, but investors’ tastes and security return distributions are stable over time, the market portfolio remains efficient and the simple version of the expected return–beta relationship holds. But if those distributions change unpredictably, or if investors seek to hedge nonmarket sources of risk to their consumption, the simple CAPM will give way to a multifactor version in which the security’s exposure to these nonmarket sources of risk command risk premiums. 9. The consumption-based capital asset pricing model (CCAPM) is a single-factor model in which the market portfolio excess return is replaced by that of a consumption-tracking portfolio. By appealing directly to consumption, the model naturally incorporates consumptionhedging considerations and changing investment opportunities within a single-factor framework.

11. Liquidity costs and liquidity risk can be incorporated into the CAPM relationship. Investors demand compensation for expected costs of illiquidity as well as the risk surrounding those costs.

market portfolio mutual fund theorem market price of risk beta

expected return–beta (or mean-beta) relationship security market line (SML) alpha

homogeneous expectations zero-beta portfolio liquidity illiquidity

2 Market risk premium: E ( RM ) 5 AsM

Beta: b i 5

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KEY TERMS

KEY EQUATIONS

Cov ( Ri , RM ) sM2

Security market line: E ( ri ) 5 rf 1 bi 3 E ( rM ) 2 rf 4 Zero-beta SML: E ( ri ) 5 E ( rZ ) 1 bi 3 E ( rM ) 2 E ( rZ )4 K

Multifactor SML (in excess returns): E ( Ri ) 5 biM E ( RM ) 1 a E ( Rk ) k51

1. What must be the beta of a portfolio with E(rP)518%, if rf56% and E(rM)514%? 2. The market price of a security is $50. Its expected rate of return is 14%. The risk-free rate is 6% and the market risk premium is 8.5%. What will be the market price of the security if its correlation coefficient with the market portfolio doubles (and all other variables remain unchanged)? Assume that the stock is expected to pay a constant dividend in perpetuity. 3. Are the following true or false? Explain. a. Stocks with a beta of zero offer an expected rate of return of zero. b. The CAPM implies that investors require a higher return to hold highly volatile securities. c. You can construct a portfolio with beta of .75 by investing .75 of the investment budget in T-bills and the remainder in the market portfolio.

PROBLEM SETS

Basic

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10. The security market line of the CAPM must be modified to account for labor income and other significant nontraded assets.

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Equilibrium in Capital Markets 4. Here are data on two companies. The T-bill rate is 4% and the market risk premium is 6%. Company

$1 Discount Store

Everything $5

12% 8% 1.5

11% 10% 1.0

Forecasted return Standard deviation of returns Beta

What would be the fair return for each company, according to the capital asset pricing model (CAPM)? 5. Characterize each company in the previous problem as underpriced, overpriced, or properly priced. 6. What is the expected rate of return for a stock that has a beta of 1.0 if the expected return on the market is 15%? a. 15%. b. More than 15%. c. Cannot be determined without the risk-free rate.

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7. Kaskin, Inc., stock has a beta of 1.2 and Quinn, Inc., stock has a beta of .6. Which of the following statements is most accurate? a. The expected rate of return will be higher for the stock of Kaskin, Inc., than that of Quinn, Inc. b. The stock of Kaskin, Inc., has more total risk than Quinn, Inc. c. The stock of Quinn, Inc., has more systematic risk than that of Kaskin, Inc.

Intermediate

8. You are a consultant to a large manufacturing corporation that is considering a project with the following net after-tax cash flows (in millions of dollars): Years from Now

After-Tax Cash Flow

0 1–10

240 15

The project’s beta is 1.8. Assuming that rf 5 8% and E(rM) 5 16%, what is the net present value of the project? What is the highest possible beta estimate for the project before its NPV becomes negative? 9. Consider the following table, which gives a security analyst’s expected return on two stocks for two particular market returns: Market Return

Aggressive Stock

Defensive Stock

5% 25

22% 38

6% 12

a. What are the betas of the two stocks? b. What is the expected rate of return on each stock if the market return is equally likely to be 5% or 25%? c. If the T-bill rate is 6% and the market return is equally likely to be 5% or 25%, draw the SML for this economy. d. Plot the two securities on the SML graph. What are the alphas of each? e. What hurdle rate should be used by the management of the aggressive firm for a project with the risk characteristics of the defensive firm’s stock? For Problems 10 to 16: If the simple CAPM is valid, which of the following situations are possible? Explain. Consider each situation independently. 10. Portfolio A B

Expected Return

Beta

20 25

1.4 1.2

11. Portfolio

Expected Return

Standard Deviation

A B

30 40

35 25

Portfolio

Expected Return

Standard Deviation

Risk-free Market A

10 18 16

0 24 12

Portfolio

Expected Return

Standard Deviation

Risk-free Market A

10 18 20

0 24 22

12.

13.

14. Portfolio Risk-free Market A

15. Portfolio Risk-free Market A

Expected Return

Beta

10 18 16

0 1.0 1.5

Expected Return

Beta

10 18 16

0 1.0 0.9

16. Portfolio

Expected Return

Standard Deviation

Risk-free Market A

10 18 16

0 24 22

The Capital Asset Pricing Model

For Problems 17 to 19 assume that the risk-free rate of interest is 6% and the expected rate of return on the market is 16%. 17. A share of stock sells for $50 today. It will pay a dividend of $6 per share at the end of the year. Its beta is 1.2. What do investors expect the stock to sell for at the end of the year? 18. I am buying a firm with an expected perpetual cash flow of $1,000 but am unsure of its risk. If I think the beta of the firm is .5, when in fact the beta is really 1, how much more will I offer for the firm than it is truly worth? 19. A stock has an expected rate of return of 4%. What is its beta? 20. Two investment advisers are comparing performance. One averaged a 19% rate of return and the other a 16% rate of return. However, the beta of the first investor was 1.5, whereas that of the second was 1. a. Can you tell which investor was a better selector of individual stocks (aside from the issue of general movements in the market)? b. If the T-bill rate were 6% and the market return during the period were 14%, which investor would be the superior stock selector? c. What if the T-bill rate were 3% and the market return were 15%?

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Equilibrium in Capital Markets 21. Suppose the rate of return on short-term government securities (perceived to be risk-free) is about 5%. Suppose also that the expected rate of return required by the market for a portfolio with a beta of 1 is 12%. According to the capital asset pricing model: a. What is the expected rate of return on the market portfolio? b. What would be the expected rate of return on a stock with b50? c. Suppose you consider buying a share of stock at $40. The stock is expected to pay $3 dividends next year and you expect it to sell then for $41. The stock risk has been evaluated at b52.5. Is the stock overpriced or underpriced? 22. Suppose that borrowing is restricted so that the zero-beta version of the CAPM holds. The expected return on the market portfolio is 17%, and on the zero-beta portfolio it is 8%. What is the expected return on a portfolio with a beta of .6? 23. a. A mutual fund with beta of .8 has an expected rate of return of 14%. If rf 55%, and you expect the rate of return on the market portfolio to be 15%, should you invest in this fund? What is the fund’s alpha? b. What passive portfolio comprised of a market-index portfolio and a money market account would have the same beta as the fund? Show that the difference between the expected rate of return on this passive portfolio and that of the fund equals the alpha from part (a).

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Challenge

24. Outline how you would incorporate the following into the CCAPM: a. Liquidity. b. Nontraded assets. (Do you have to worry about labor income?)

1. a. John Wilson is a portfolio manager at Austin & Associates. For all of his clients, Wilson manages portfolios that lie on the Markowitz efficient frontier. Wilson asks Mary Regan, CFA, a managing director at Austin, to review the portfolios of two of his clients, the Eagle Manufacturing Company and the Rainbow Life Insurance Co. The expected returns of the two portfolios are substantially different. Regan determines that the Rainbow portfolio is virtually identical to the market portfolio and concludes that the Rainbow portfolio must be superior to the Eagle portfolio. Do you agree or disagree with Regan’s conclusion that the Rainbow portfolio is superior to the Eagle portfolio? Justify your response with reference to the capital market line. b. Wilson remarks that the Rainbow portfolio has a higher expected return because it has greater nonsystematic risk than Eagle’s portfolio. Define nonsystematic risk and explain why you agree or disagree with Wilson’s remark. 2. Wilson is now evaluating the expected performance of two common stocks, Furhman Labs Inc. and Garten Testing Inc. He has gathered the following information: • • • •

The risk-free rate is 5%. The expected return on the market portfolio is 11.5%. The beta of Furhman stock is 1.5. The beta of Garten stock is .8.

Based on his own analysis, Wilson’s forecasts of the returns on the two stocks are 13.25% for Furhman stock and 11.25% for Garten stock. Calculate the required rate of return for Furhman Labs stock and for Garten Testing stock. Indicate whether each stock is undervalued, fairly valued, or overvalued. 3. The security market line depicts: a. b. c. d.

A security’s expected return as a function of its systematic risk. The market portfolio as the optimal portfolio of risky securities. The relationship between a security’s return and the return on an index. The complete portfolio as a combination of the market portfolio and the risk-free asset.

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4. Within the context of the capital asset pricing model (CAPM), assume: • • • •

Expected return on the market515%. Risk-free rate58%. Expected rate of return on XYZ security517%. Beta of XYZ security51.25.

Which one of the following is correct? a. b. c. d.

XYZ is overpriced. XYZ is fairly priced. XYZ’s alpha is 2.25%. XYZ’s alpha is .25%.

5. What is the expected return of a zero-beta security? a. b. c. d.

Market rate of return. Zero rate of return. Negative rate of return. Risk-free rate of return.

a. b. c. d.

Economic factors. Specific risk. Systematic risk. Diversification.

7. According to CAPM, the expected rate of return of a portfolio with a beta of 1.0 and an alpha of 0 is: a. b. c. d.

Between rM and rf . The risk-free rate, rf . b(rM2rf ). The expected return on the market, rM.

The following table shows risk and return measures for two portfolios. Portfolio

Average Annual Rate of Return

Standard Deviation

Beta

R S&P 500

11% 14%

10% 12%

0.5 1.0

8. When plotting portfolio R on the preceding table relative to the SML, portfolio R lies: a. b. c. d.

On the SML. Below the SML. Above the SML. Insufficient data given.

9. When plotting portfolio R relative to the capital market line, portfolio R lies: a. b. c. d.

On the CML. Below the CML. Above the CML. Insufficient data given.

10. Briefly explain whether investors should expect a higher return from holding portfolio A versus portfolio B under capital asset pricing theory (CAPM). Assume that both portfolios are well diversified.

Systematic risk (beta) Specific risk for each individual security

Portfolio A

Portfolio B

1.0 High

1.0 Low

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6. Capital asset pricing theory asserts that portfolio returns are best explained by:

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Equilibrium in Capital Markets 11. Joan McKay is a portfolio manager for a bank trust department. McKay meets with two clients, Kevin Murray and Lisa York, to review their investment objectives. Each client expresses an interest in changing his or her individual investment objectives. Both clients currently hold well-diversified portfolios of risky assets. a. Murray wants to increase the expected return of his portfolio. State what action McKay should take to achieve Murray’s objective. Justify your response in the context of the CML. b. York wants to reduce the risk exposure of her portfolio but does not want to engage in borrowing or lending activities to do so. State what action McKay should take to achieve York’s objective. Justify your response in the context of the SML. 12. Karen Kay, a portfolio manager at Collins Asset Management, is using the capital asset pricing model for making recommendations to her clients. Her research department has developed the information shown in the following exhibit.

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Forecast Returns, Standard Deviations, and Betas

Stock X Stock Y Market index Risk-free rate

Forecast Return

Standard Deviation

Beta

14.0% 17.0 14.0 5.0

36% 25 15

0.8 1.5 1.0

a. Calculate expected return and alpha for each stock. b. Identify and justify which stock would be more appropriate for an investor who wants to i. add this stock to a well-diversified equity portfolio. ii. hold this stock as a single-stock portfolio.

E-INVESTMENTS EXERCISES Fidelity provides data on the risk and return of its funds at www.fidelity.com. Click on the Research link, then choose Mutual Funds from the submenu. In the Fund Evaluator section, search over all open no-load funds. On the next screen, click on Risk/Volatility Measures and indicate that you want to screen for funds with betas less than or equal to .50. Click Search Funds to see the results. Select five funds from the resulting list and click Compare. Rank the five funds according to their betas and then according to their standard deviations. Do both lists rank the funds in the same order? How would you explain any difference in the rankings? Repeat the exercise to compare five funds that have betas greater than or equal to 1.50. Why might the degree of agreement when ranking funds by beta versus standard deviation differ when using high versus low beta funds?

SOLUTIONS TO CONCEPT CHECKS 1. We can characterize the entire population by two representative investors. One is the “uninformed” investor, who does not engage in security analysis and holds the market portfolio, whereas the other optimizes using the Markowitz algorithm with input from security analysis. The uninformed investor does not know what input the informed investor uses to make portfolio purchases. The uninformed investor knows, however, that if the other investor is informed, the market portfolio proportions will be optimal. Therefore, to depart from these proportions would constitute an uninformed bet, which will, on average, reduce the efficiency of diversification with no compensating improvement in expected returns.

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2. a. Substituting the historical mean and standard deviation in Equation 9.2 yields a coefficient of risk aversion of E ( r M) 2 r f

.079 5 1.47 .2322 b. This relationship also tells us that for the historical standard deviation and a coefficient of risk aversion of 3.5 the risk premium would be A5

s2M

5

E(rM) 2 rf 5 As2M 5 3.5 3 .2322 5 .188 5 18.8% 3. For these investment proportions, w Ford ,w Toyota , the portfolio b is bP 5 w Fordb Ford 1 w Toyotab Toyota 5 (.75 3 1.25) 1 (.25 3 1.10) 5 1.2125 As the market risk premium, E(rM)2rf , is 8%, the portfolio risk premium will be E (rP) 2 rf 5 bP 3 E (rM ) 2 rf 4 5 1.2125 3 8 5 9.7%

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4. The alpha of a stock is its expected return in excess of that required by the CAPM. a 5 E (r ) 2 5 rf 1 b 3 E (rM ) 2 rf 46 a XYZ 5 12 2 3 5 1 1.0 (11 2 5 ) 4 5 1% a ABC 5 13 2 3 5 1 1.5 (11 2 5 ) 4 5 21% ABC plots below the SML, while XYZ plots above. E(r), Percent

SML 14 XYZ

12 E(rM) ⫽ 11

αXYZ ⬎ 0

ABC

αABC ⬍ 0

Market

rf ⫽ 5

b .5

1

1.5

5. The project-specific required return is determined by the project beta coupled with the market risk premium and the risk-free rate. The CAPM tells us that an acceptable expected rate of return for the project is rf 1 b 3 E ( rM ) 2 rf 4 5 8 1 1.3(16 2 8) 5 18.4% which becomes the project’s hurdle rate. If the IRR of the project is 19%, then it is desirable. Any project with an IRR equal to or less than 18.4% should be rejected.

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PART III

Arbitrage Pricing Theory and Multifactor Models of Risk and Return

THE EXPLOITATION OFsecurity mispricing in such a way that risk-free profits can be earned is called arbitrage. It involves the simultaneous purchase and sale of equivalent securities in order to profit from discrepancies in their prices. Perhaps the most basic principle of capital market theory is that equilibrium market prices are rational in that they rule out arbitrage opportunities. If actual security prices allow for arbitrage, the result will be strong pressure to restore equilibrium. Therefore, security markets ought to satisfy a “no-arbitrage condition.” In this chapter, we show how such no-arbitrage conditions together with the factor model introduced in Chapter 8 allow us to generalize the security market line of the CAPM to gain richer insight into the risk–return relationship. We begin by showing how the decomposition of risk into market versus firm-specific influences that we introduced in earlier chapters can be extended to deal with the multifaceted nature of systematic risk. Multifactor models of security returns can be used to measure and manage exposure to each of many economywide factors such as business-cycle risk, interest or inflation rate

risk, energy price risk, and so on. These models also lead us to a multifactor version of the security market line in which risk premiums derive from exposure to multiple risk sources, each with their own risk premium. We show how factor models combined with a no-arbitrage condition lead to a simple relationship between expected return and risk. This approach to the risk–return tradeoff is called arbitrage pricing theory, or APT. In a single-factor market where there are no extra-market risk factors, the APT leads to a mean return–beta equation identical to that of the CAPM. In a multifactor market with one or more extra-market risk factors, the APT delivers a mean-beta equation similar to Merton’s intertemporal extension of the CAPM (his ICAPM). We ask next what factors are likely to be the most important sources of risk. These will be the factors generating substantial hedging demands that brought us to the multifactor CAPM introduced in Chapter 9. Both the APT and the CAPM therefore can lead to multiple-risk versions of the security market line, thereby enriching the insights we can derive about the risk–return relationship.

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Arbitrage Pricing Theory and Multifactor Models of Risk and Return

10.1 Multifactor Models: An Overview The index model introduced in Chapter 8 gave us a way of decomposing stock variability into market or systematic risk, due largely to macroeconomic events, versus firm-specific or idiosyncratic effects that can be diversified in large portfolios. In the single-index model, the return on a broad market-index portfolio summarized the impact of the macro factor. In Chapter 9 we introduced the possibility that asset-risk premiums may also depend on correlations with extra-market risk factors, such as inflation, or changes in the parameters describing future investment opportunities: interest rates, volatility, market-risk premiums, and betas. For example, returns on an asset whose return increases when inflation increases can be used to hedge uncertainty in the future inflation rate. Its risk premium may fall as a result of investors’ extra demand for this asset. Risk premiums of individual securities should reflect their sensitivities to changes in extra-market risk factors just as their betas on the market index determine their risk premiums in the simple CAPM. When securities can be used to hedge these factors, the resulting hedging demands will make the SML multifactor, with each risk source that can be hedged adding an additional factor to the SML. Risk factors can be represented either by returns on these hedge portfolios (just as the index portfolio represents the market factor), or more directly by changes in the risk factors themselves, for example, changes in interest rates orinflation.

Factor Models of Security Returns We begin with a familiar single-factor model like the one introduced in Chapter 8. Uncertainty in asset returns has two sources: a common or macroeconomic factor and firmspecific events. The common factor is constructed to have zero expected value, because we use it to measure new information concerning the macroeconomy, which, by definition, has zero expected value. If we call F the deviation of the common factor from its expected value, bi the sensitivity of firm i to that factor, and ei the firm-specific disturbance, the factor model states that the actual excess return on firm i will equal its initially expected value plus a (zero expected value) random amount attributable to unanticipated economywide events, plus another (zero expected value) random amount attributable to firm-specific events. Formally, the single-factor model of excess returns is described by Equation 10.1: Ri 5 E (Ri ) 1 bi F 1 ei

(10.1)

where E(Ri ) is the expected excess return on stock i. Notice that if the macro factor has a value of 0 in any particular period (i.e., no macro surprises), the excess return on the security will equal its previously expected value, E(Ri ), plus the effect of firm-specific events only. The nonsystematic components of returns, the eis, are assumed to be uncorrelated across stocks and with the factor F.

Example 10.1

Factor Models

To make the factor model more concrete, consider an example. Suppose that the macro factor, F, is taken to be news about the state of the business cycle, measured by the unexpected percentage change in gross domestic product (GDP), and that the consensus is that GDP will increase by 4% this year. Suppose also that a stock’s b value is 1.2.

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If GDP increases by only 3%, then the value of F would be 21%, representing a 1% disappointment in actual growth versus expected growth. Given the stock’s beta value, this disappointment would translate into a return on the stock that is 1.2% lower than previously expected. This macro surprise, together with the firm-specific disturbance, ei, determines the total departure of the stock’s return from its originally expected value.

CONCEPT CHECK

10.1

Suppose you currently expect the stock in Example 10.1 to earn a 10% rate of return. Then some macroeconomic news suggests that GDP growth will come in at 5% instead of 4%. How will you revise your estimate of the stock’s expected rate of return?

The factor model’s decomposition of returns into systematic and firm-specific components is compelling, but confining systematic risk to a single factor is not. Indeed, when we motivated systematic risk as the source of risk premiums in Chapter 9, we noted that extra market sources of risk may arise from a number of sources such as uncertainty about interest rates, inflation, and so on. The market return reflects macro factors as well as the average sensitivity of firms to those factors. It stands to reason that a more explicit representation of systematic risk, allowing for different stocks to exhibit different sensitivities to its various components, would constitute a useful refinement of the single-factor model. It is easy to see that models that allow for several factors—multifactor models—can provide better descriptions of security returns. Apart from their use in building models of equilibrium security pricing, multifactor models are useful in risk management applications. These models give us a simple way to measure investor exposure to various macroeconomic risks and construct portfolios to hedge those risks. Let’s start with a two-factor model. Suppose the two most important macroeconomic sources of risk are uncertainties surrounding the state of the business cycle, news of which we will again measure by unanticipated growth in GDP and changes in interest rates. We will denote by IR any unexpected change in interest rates. The return on any stock will respond both to sources of macro risk and to its own firm-specific influences. We can write a two-factor model describing the excess return on stock i in some time period as follows: Ri 5 E (Ri ) 1 biGDPGDP 1 biIRIR 1 ei

(10.2)

The two macro factors on the right-hand side of the equation comprise the systematic factors in the economy. As in the single-factor model, both of these macro factors have zero expectation: They represent changes in these variables that have not already been anticipated. The coefficients of each factor in Equation 10.2 measure the sensitivity of share returns to that factor. For this reason the coefficients are sometimes called factor loadings or, equivalently, factor betas. An increase in interest rates is bad news for most firms, so we would expect interest rate betas generally to be negative. As before, ei reflects firmspecific influences. To illustrate the advantages of multifactor models, consider two firms, one a regulated electric-power utility in a mostly residential area, the other an airline. Because residential demand for electricity is not very sensitive to the business cycle, the utility has a low beta

CHAPTER 10

Arbitrage Pricing Theory and Multifactor Models of Risk and Return

on GDP. But the utility’s stock price may have a relatively high sensitivity to interest rates. Because the cash flow generated by the utility is relatively stable, its present value behaves much like that of a bond, varying inversely with interest rates. Conversely, the performance of the airline is very sensitive to economic activity but is less sensitive to interest rates. It will have a high GDP beta and a lower interest rate beta. Suppose that on a particular day, a news item suggests that the economy will expand. GDP is expected to increase, but so are interest rates. Is the “macro news” on this day good or bad? For the utility, this is bad news: Its dominant sensitivity is to rates. But for the airline, which responds more to GDP, this is good news. Clearly a one-factor or single-index model cannot capture such differential responses to varying sources of macroeconomic uncertainty.

Example 10.2

Risk Assessment Using Multifactor Models

Suppose we estimate the two-factor model in Equation 10.2 for Northeast Airlines and find the following result: R5.133 1 1.2(GDP)2.3(IR)1e This tells us that, based on currently available information, the expected excess rate of return for Northeast is 13.3%, but that for every percentage point increase in GDP beyond current expectations, the return on Northeast shares increases on average by 1.2%, while for every unanticipated percentage point that interest rates increases, Northeast’s shares fall on average by .3%.

Factor betas can provide a framework for a hedging strategy. The idea for an investor who wishes to hedge a source of risk is to establish an opposite factor exposure to offset that particular source of risk. Often, futures contracts can be used to hedge particular factor exposures. We explore this application in Chapter 22. As it stands, however, the multifactor model is no more than a description of the factors that affect security returns. There is no “theory” in the equation. The obvious question left unanswered by a factor model like Equation 10.2 is where E(R ) comes from, in other words, what determines a security’s expected excess rate of return. This is where we need a theoretical model of equilibrium security returns. We therefore now turn to arbitrage pricing theory to help determine the expected value, E(R), in Equations 10.1 and 10.2.

10.2 Arbitrage Pricing Theory Stephen Ross developed the arbitrage pricing theory (APT) in 1976.1 Like the CAPM, the APT predicts a security market line linking expected returns to risk, but the path it takes to the SML is quite different. Ross’s APT relies on three key propositions: (1) security returns can be described by a factor model; (2) there are sufficient securities to diversify away idiosyncratic risk; and (3) well-functioning security markets do not allow for the persistence of arbitrage opportunities. We begin with a simple version of Ross’s model, which assumes that only one systematic factor affects security returns. 1

Stephen A. Ross, “Return, Risk and Arbitrage,” in I. Friend and J. Bicksler, eds., Risk and Return in Finance (Cambridge, MA: Ballinger, 1976).

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Arbitrage, Risk Arbitrage, and Equilibrium An arbitrage opportunity arises when an investor can earn riskless profits without making a net investment. A trivial example of an arbitrage opportunity would arise if shares of a stock sold for different prices on two different exchanges. For example, suppose IBM sold for $195 on the NYSE but only $193 on NASDAQ. Then you could buy the shares on NASDAQ and simultaneously sell them on the NYSE, clearing a riskless profit of $2 per share without tying up any of your own capital. The Law of One Price states that if two assets are equivalent in all economically relevant respects, then they should have the same market price. The Law of One Price is enforced by arbitrageurs: If they observe a violation of the law, they will engage in arbitrage activity—simultaneously buying the asset where it is cheap and selling where it is expensive. In the process, they will bid up the price where it is low and force it down where it is high until the arbitrage opportunity is eliminated. The idea that market prices will move to rule out arbitrage opportunities is perhaps the most fundamental concept in capital market theory. Violation of this restriction would indicate the grossest form of market irrationality. The critical property of a risk-free arbitrage portfolio is that any investor, regardless of risk aversion or wealth, will want to take an infinite position in it. Because those large positions will quickly force prices up or down until the opportunity vanishes, security prices should satisfy a “no-arbitrage condition,” that is, a condition that rules out the existence of arbitrage opportunities. There is an important difference between arbitrage and risk–return dominance arguments in support of equilibrium price relationships. A dominance argument holds that when an equilibrium price relationship is violated, many investors will make limited portfolio changes, depending on their degree of risk aversion. Aggregation of these limited portfolio changes is required to create a large volume of buying and selling, which in turn restores equilibrium prices. By contrast, when arbitrage opportunities exist, each investor wants to take as large a position as possible; hence it will not take many investors to bring about the price pressures necessary to restore equilibrium. Therefore, implications for prices derived from no-arbitrage arguments are stronger than implications derived from a risk–return dominance argument. The CAPM is an example of a dominance argument, implying that all investors hold mean-variance efficient portfolios. If a security is mispriced, then investors will tilt their portfolios toward the underpriced and away from the overpriced securities. Pressure on equilibrium prices results from many investors shifting their portfolios, each by a relatively small dollar amount. The assumption that a large number of investors are mean-variance optimizers is critical. In contrast, the implication of a no-arbitrage condition is that a few investors who identify an arbitrage opportunity will mobilize large dollar amounts and quickly restore equilibrium. Practitioners often use the terms arbitrage and arbitrageurs more loosely than our strict definition. Arbitrageur often refers to a professional searching for mispriced securities in specific areas such as merger-target stocks, rather than to one who seeks strict (risk-free) arbitrage opportunities. Such activity is sometimes called risk arbitrage to distinguish it from pure arbitrage. To leap ahead, in Part Four we will discuss “derivative” securities such as futures and options, whose market values are determined by prices of other securities. For example, the value of a call option on a stock is determined by the price of the stock. For such securities, strict arbitrage is a practical possibility, and the condition of no-arbitrage leads to exact pricing. In the case of stocks and other “primitive” securities whose values are not determined strictly by another bundle of assets, no-arbitrage conditions must be obtained by appealing to diversification arguments.

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329

Well-Diversified Portfolios Consider the risk of a portfolio of stocks in a single-factor market. We first show that if a portfolio is well diversified, its firm-specific or nonfactor risk becomes negligible, so that only factor (or systematic) risk remains. The excess return on an n-stock portfolio with weights wi,Swi51, is RP 5 E (RP) 1 bP F 1 eP

(10.3)

where bP 5 g w i b i ; E ( RP) 5 g wi E (Ri ) are the weighted averages of the bi and risk premiums of the n securities. The portfolio nonsystematic component (which is uncorrelated with F ) is eP5Swi ei , which similarly is a weighted average of the ei of the n securities. We can divide the variance of this portfolio into systematic and nonsystematic sources: s 2P 5 b 2P s 2F 1 s2 (eP ) where s2F is the variance of the factor F and s2(eP) is the nonsystematic risk of the portfolio, which is given by s2 (eP ) 5 Variance ( g wi ei ) 5 g w2i s2 (ei ) Note that in deriving the nonsystematic variance of the portfolio, we depend on the fact that the firm-specific eis are uncorrelated and hence that the variance of the “portfolio” of nonsystematic eis is the weighted sum of the individual nonsystematic variances with the square of the investment proportions as weights. If the portfolio were equally weighted, wi51/n, then the nonsystematic variance would be 2 2 1 1 1 s (ei ) s2(eP) 5 Variance ( g wiei ) 5 a a n b s2(ei ) 5 n a n 5 n s2 (ei )

where the last term is the average value of nonsystematic variance across securities. In words, the nonsystematic variance of the portfolio equals the average nonsystematic variance divided by n. Therefore, when n is large, nonsystematic variance approaches zero. This is the effect of diversification. This property is true of portfolios other than the equally weighted one. Any portfolio for which each wi becomes consistently smaller as n gets large (more precisely, for which each w2i approaches zero as n increases) will satisfy the condition that the portfolio nonsystematic risk will approach zero. This property motivates us to define a well-diversified portfolio as one with each weight, wi , small enough that for practical purposes the nonsystematic variance, s2(eP), is negligible.

CONCEPT CHECK

10.2

a. A portfolio is invested in a very large number of shares (n is large). However, one-half of the portfolio is invested in stock 1, and the rest of the portfolio is equally divided among the other n21 shares. Is this portfolio well diversified? b. Another portfolio also is invested in the same n shares, where n is very large. Instead of equally weighting with portfolio weights of 1/n in each stock, the weights in half the securities are 1.5/n while the weights in the other shares are .5/n. Is this portfolio well diversified?

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Because the expected value of eP for any well-diversified portfolio is zero, and its variance also is effectively zero, we can conclude that any realized value of eP will be virtually zero. Rewriting Equation 10.1, we conclude that, for a well-diversified portfolio, for all practical purposes RP 5 E (RP ) 1 bP F The solid line in Figure 10.1, panel A plots the excess return of a well-diversified portfolio A with E(RA)510% and bA51 for various realizations of the systematic factor. The expected return of portfolio A is 10%; this is where the solid line crosses the vertical axis. At this point the systematic factor is zero, implying no macro surprises. If the macro factor is positive, the portfolio’s return exceeds its expected value; if it is negative, the portfolio’s return falls short of its mean. The excess return on the portfolio is therefore E (RA ) 1 bA F 5 10% 1 1.0 3 F Compare panel A in Figure 10.1 with panel B, which is a similar graph for a single stock(S) with bs 5 1. The undiversified stock is subject to nonsystematic risk, which is seen in a scatter of points around the line. The well-diversified portfolio’s return, in contrast, is determined completely by the systematic factor. In a single-factor world, all pairs of well-diversified portfolios are perfectly correlated: Their risk is fully determined by the same systematic factor. Consider a second welldiversified portfolio, Portfolio Q, with RQ 5 E(RQ) 1 bQF. We can compute the standard deviations of P and Q, as well as the covariance and correlation between them: sP 5 bP sF; sQ 5 bQ sF Cov ( RP , RQ ) 5 Cov (bP F, bQ F ) 5 bP bQ s2F r PQ 5

Cov (RP, RQ ) 51 sP sQ

Excess Return (%)

Excess Return (%) A

S

10

10

F

A

F

B

Figure 10.1 Excess returns as a function of the systematic factor. Panel A, Well-diversified portfolio A. Panel B, Single stock (S).

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Arbitrage Pricing Theory and Multifactor Models of Risk and Return

Perfect correlation means that in a plot of expected return versus standard deviation (such as Figure 7.5), any two well-diversified portfolios lie on a straight line. We will see later that this common line is the CML.

Diversification and Residual Risk in Practice What is the effect of diversification on portfolio residual SD in practice, where portfolio size is not unlimited? In reality, we may find (annualized) residual SDs as high as 50% for large stocks and even 100% for small stocks. To illustrate the impact of diversification, we examine portfolios of two configurations. One portfolio is equally weighted; this achieves the highest benefits of diversification with equal-SD stocks. For comparison, we form the other portfolio using far-from-equal weights. We select stocks in groups of four, with relative weights in each group of 70%, 15%, 10%, and 5%. The highest weight is 14 times greater than the lowest, which will severely reduce potential benefits of diversification. However, extended diversification in which we add to the portfolio more and more groups of four stocks with the same relative weights will overcome this problem because the highest portfolio weight still falls with additional diversification. In an equally weighted 1,000-stock portfolio, each weight is 0.1%; in the unequally weighted portfolio, with 1,000/4 5250 groups of four stocks, the highest and lowest weights are 70%/25050.28% and 5%/25050.02%, respectively. What is a large portfolio? Many widely held ETFs each include hundreds of stocks, and some funds such as the Wilshire 5000 hold thousands. These portfolios are accessible to the public since the annual expense ratios of investment companies that offer such funds are of the order of only 10 basis points. Thus a portfolio of 1,000 stocks is not unheard of, but a portfolio of 10,000 stocks is. Table10.1 shows portfolio residual SD as a function of the number of stocks. Equally weighted, 1,000-stock portfolios achieve small but not negligible standard deviations of 1.58% when residual risk is 50% and 3.16% when residual risk is 100%. The SDs for the unbalanced portfolios are about double these values. For 10,000-stock portfolios, the SDs are negligible, verifying that diversification can eliminate risk even in very unbalanced portfolios, at least in principle, if the investment universe is large enough.

Residual SD of Each Stock550%

Residual SD of Each Stock5100%

N N SD(eP) Equal weights: wi51/N 4 25.00 4 60 6.45 60 200 3.54 200 1,000 1.58 1,000 10,000 0.50 10,000 Sets of four relative weights:w150.65,w250.2,w350.1,w450.05 4 36.23 4 60 9.35 60 200 5.12 200 1,000 2.29 1,000 10,000 0.72 10,000

Table 10.1 Residual variance with even and uneven portfolio weights

SD(eP) 50.00 12.91 7.07 3.16 1.00 72.46 18.71 10.25 4.58 1.45

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Executing Arbitrage Imagine a single-factor market where the well-diversified portfolio, M, represents the market factor, F, of Equation 10.1. The excess return on any security is given by Ri5ai1bi RM1ei , and that of a well-diversified (therefore zero residual) portfolio, P, is RP 5 aP 1 bP RM

(10.4)

E (RP) 5 aP 1 bP E (RM)

(10.5)

Now suppose that security analysis reveals that portfolio P has a positive alpha.2 We also estimate the risk premium of the index portfolio, M, from macro analysis. Since neither M nor portfolio P have residual risk, the only risk to the returns of the two portfolios is systematic, derived from their betas on the common factor (the beta of the index is 1.0). Therefore, you can eliminate the risk of P altogether: Construct a zero-beta portfolio, called Z, from P and M by appropriately selecting weights wP and wM512wP on each portfolio: bZ 5 wPbP 1 (1 2 wP)bM 5 0 bM 5 1 wP 5

(10.6)

2bP 1 ;wM 5 1 2 wP 5 1 2 bP 1 2 bP

Therefore, portfolio Z is riskless, and its alpha is aZ 5 wP aP 1 (1 2 wP) aM 5 wP aP

(10.7)

The risk premium on Z must be zero because the risk of Z is zero. If its risk premium were not zero, you could earn arbitrage profits. Here is how: Since the beta of Z is zero, Equation 10.5 implies that its risk premium is just its alpha. Using Equation 10.7, its alpha is wP aP, so E (RZ ) 5 wP aP 5

1 a 1 2 bP P

(10.8)

You now form a zero-net-investment arbitrage portfolio: If bP,1 and the risk premium of Z is positive (implying that Z returns more than the risk-free rate), borrow and invest the proceeds in Z. For every borrowed dollar invested in Z, you get a net return (i.e., net of 1 paying the interest on your loan) of a . This is a money machine, which you would 1 2 bP P 3 work as hard as you can. Similarly if bP . 1, Equation 10.8 tells us that the risk premium is negative; therefore, sell Z short and invest the proceeds at the risk-free rate. Once again, a money machine has been created. Neither situation can persist, as the large volume of trades from arbitrageurs pursuing these strategies will push prices until the arbitrage opportunity disappears (i.e., until the risk premium of portfolio Z equals zero). 2

If the portfolio alpha is negative, we can still pursue the following strategy. We would simply switch to a short position in P, which would have a positive alpha of the same absolute value as P’s, and a beta that is the negative of P’s. 3

The function in Equation 10.8 becomes unstable at bP51. For values of bP near 1, it becomes infinitely large with the sign of aP. This isn’t an economic absurdity, since in that case, the sizes of your long position in P and short position in M will be almost identical, and the arbitrage profit you earn per dollar invested will be nearly infinite.

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Arbitrage Pricing Theory and Multifactor Models of Risk and Return

The No-Arbitrage Equation of the APT We’ve seen that arbitrage activity will quickly pin the risk premium of any zero-beta welldiversified portfolio to zero.4 Setting the expression in Equation 10.8 to zero implies that the alpha of any well-diversified portfolio must also be zero. From Equation 10.5, this means that for any well-diversified P, E (RP) 5 bP E (RM)

(10.9)

In other words, the risk premium (expected excess return) on portfolio P is the product of its beta and the market-index risk premium. Equation 10.9 thus establishes that the SML of the CAPM applies to well-diversified portfolios simply by virtue of the “no-arbitrage” requirement of the APT. Another demonstration that the APT results in the same SML as the CAPM is more graphical in nature. First we show why all well-diversified portfolios with the same beta must have the same expected return. Figure10.2 plots the returns on two such portfolios, A and B, both with betas of 1, but with differing expected returns: E (rA)510% and E (rB)58%. Could portfolios A and B coexist with the return pattern depicted? Clearly not: No matter what the systematic factor turns out to be, portfolio A outperforms portfolio B, leading to an arbitrage opportunity. If you sell short $1 million of B and buy $1 million of A, a zero-net-investment strategy, you would have a riskless payoff of $20,000, as follows: (.10 1 1.0 3 F ) 3 $1 million from long position in A 2(.08 1 1.0 3 F ) 3 $1 million from short position in B .02 3 $1 million 5 $20,000 net proceeds Your profit is risk-free because the factor risk cancels out across the long and short positions. Moreover, the strategy requires zero-net-investment. You should pursue it on an

Return (%) A B

10 8

F (Realization of Macro Factor)

Figure 10.2 Returns as a function of the systematic factor: an arbitrage opportunity

4

As an exercise, show that when aP ,0 you reverse the position of P in Z, and the arbitrage portfolio will still earn a riskless excess return.

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Expected Return (%)

A

10

rf = 4

Risk Premium

D

7 6

C F

.5

1

β (With Respect to Macro Factor)

Figure 10.3 An arbitrage opportunity

infinitely large scale until the return discrepancy between the two portfolios disappears. Well-diversified portfolios with equal betas must have equal expected returns in market equilibrium, or arbitrage opportunities exist. What about portfolios with different betas? Their risk premiums must be proportional to beta. To see why, consider Figure 10.3. Suppose that the risk-free rate is 4% and that a well-diversified portfolio, C, with a beta of .5, has an expected return of 6%. Portfolio C plots below the line from the risk-free asset to portfolio A. Consider, therefore, a new portfolio, D, composed of half of portfolio A and half of the risk-free asset. Portfolio D’s beta will be (.5 3 0 1 .5 3 1.0) 5 .5, and its expected return will be (.5341.5310)57%. Now portfolio D has an equal beta but a greater expected return than portfolio C. From our analysis in the previous paragraph we know that this constitutes an arbitrage opportunity. We conclude that, to preclude arbitrage opportunities, the expected return on all well-diversified portfolios must lie on the straight line from the risk-free asset in Figure10.3. Notice in Figure10.3 that risk premiums are indeed proportional to portfolio betas. The risk premium is depicted by the vertical arrow, which measures the distance between the risk-free rate and the expected return on the portfolio. As in the simple CAPM, the risk premium is zero for b50 and rises in direct proportion to b.

10.3 The APT, the CAPM, and the Index Model Equation 10.9 raises three questions: 1. Does the APT also apply to less-than-well-diversified portfolios? 2. Is the APT as a model of risk and return superior or inferior to the CAPM? Do we need both models?

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3. Suppose a security analyst identifies a positive-alpha portfolio with some remaining residual risk. Don’t we already have a prescription for this case from the TreynorBlack (T-B) procedure applied to the index model (Chapter 8)? Is this framework preferred to the APT?

The APT and the CAPM The APT is built on the foundation of well-diversified portfolios. However, we’ve seen, for example in Table10.1, that even large portfolios may have non-negligible residual risk. Some indexed portfolios may have hundreds or thousands of stocks, but active portfolios generally cannot, as there is a limit to how many stocks can be actively analyzed in search of alpha. How does the APT stand up to these limitations? Suppose we order all portfolios in the universe by residual risk. Think of Level 0 portfolios as having zero residual risk; in other words, they are the theoretically well-diversified portfolios of the APT. Level 1 portfolios have very small residual risk, say up to 0.5%. Level 2 portfolios have yet greater residual SD, say up to 1%, and so on. If the SML described by Equation 10.9 applies to all well-diversified Level 0 portfolios, it must at least approximate the risk premiums of Level 1 portfolios. Even more important, while Level 1 risk premiums may deviate slightly from Equation 10.9, such deviations should be unbiased, with alphas equally likely to be positive or negative. Deviations should be uncorrelated with beta or residual SD and should average to zero. We can apply the same logic to portfolios of slightly higher Level 2 residual risk. Since all Level 1 portfolios are still well approximated by Equation 10.9, so must be risk premiums of Level 2 portfolios, albeit with slightly less accuracy. Here too, we may take comfort in the lack of bias and zero average deviations from the risk premiums predicted by Equation 10.9. But still, the precision of predictions of risk premiums from Equation 10.9 consistently deteriorates with increasing residual risk. (One might ask why we don’t transform Level 2 portfolios into Level 1 or even Level 0 portfolios by further diversifying, but as we’ve pointed out, this may not be feasible in practice for assets with considerable residual risk when active portfolio size or the size of the investment universe is limited.) If residual risk is sufficiently high and the impediments to complete diversification are too onerous, we cannot have full confidence in the APT and the arbitrage activities that underpin it. Despite this shortcoming, the APT is valuable. First, recall that the CAPM requires that almost all investors be mean-variance optimizers. We may well suspect that they are not. The APT frees us of this assumption. It is sufficient that a small number of sophisticated arbitrageurs scour the market for arbitrage opportunities. This alone produces an SML, Equation 10.9, that is a good and unbiased approximation for all assets but those with significant residual risk. Perhaps even more important is the fact that the APT is anchored by observable portfolios such as the market index. The CAPM is not even testable because it relies on an unobserved, all-inclusive portfolio. The reason that the APT is not fully superior to the CAPM is that at the level of individual assets and high residual risk, pure arbitrage may be insufficient to enforce Equation 10.9. Therefore, we need to turn to the CAPM as a complementary theoretical construct behind equilibrium risk premiums. It should be noted, however, that when we replace the unobserved market portfolio of the CAPM with an observed, broad index portfolio that may not be efficient, we no longer can be sure that the CAPM predicts risk premiums of all assets with no bias. Neither model therefore is free of limitations. Comparing the APT arbitrage strategy to maximization of the Sharpe ratio in the context of an index model may well be the more useful framework for analysis.

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The APT and Portfolio Optimization in a Single-Index Market The APT is couched in a single-factor market5 and applies with perfect accuracy to welldiversified portfolios. It shows arbitrageurs how to generate infinite profits if the risk premium of a well-diversified portfolio deviates from Equation 10.9. The trades executed by these arbitrageurs are the enforcers of the accuracy of this equation. In effect, the APT shows how to take advantage of security mispricing when diversification opportunities are abundant. When you lock in and scale up an arbitrage opportunity you’re sure to be rich as Croesus regardless of the composition of the rest of your portfolio, but only if the arbitrage portfolio is truly risk-free! However, if the arbitrage position is not perfectly well diversified, an increase in its scale (borrowing cash, or borrowing shares to go short) will increase the risk of the arbitrage position, potentially without bound. The APT ignores this complication. Now consider an investor who confronts the same single factor market, and whose security analysis reveals an underpriced asset (or portfolio), that is, one whose risk premium implies a positive alpha. This investor can follow the advice weaved throughout Chapters 6–8 to construct an optimal risky portfolio. The optimization process will consider both the potential profit from a position in the mispriced asset, as well as the risk of the overall portfolio and efficient diversification. As we saw in Chapter 8, the TreynorBlack (T-B) procedure can be summarized as follows.6 1. Estimate the risk premium and standard deviation of the benchmark (index) portfolio, RPM and sM. 2. Place all the assets that are mispriced into an active portfolio. Call the alpha of the active portfolio aA, its systematic-risk coefficient bA, and its residual risk s(eA). Your optimal risky portfolio will allocate to the active portfolio a weight, w*A: A

w0A

2

(eA ) * ; wA E ( RM ) 2 M

w0A 1 1 w0A (1 2 bA)

The allocation to the passive portfolio is then, w*M 5 1 2 w*A. With this allocation, the increase in the Sharpe ratio of the optimal portfolio, SP, over that of the passive portfolio, SM, depends on the size of the information ratio of the active portfolio, IR A5aA/s(eA). The optimized portfolio can attain a Sharpe ratio of SP 5 "S 2M 1 IR 2A. 3. To maximize the Sharpe ratio of the risky portfolio, you maximize the IR of the active portfolio. This is achieved by allocating to each asset in the active portfolio a portfolio weight proportional to: wAi5ai/s2(ei). When this is done, the square of the information ratio of the active portfolio will be the sum of the squared individual information ratios: IR 2A 5 a IR 2i . Now see what happens in the T-B model when the residual risk of the active portfolio is zero. This is essentially the assumption of the APT, that a well-diversified portfolio (with zero residual risk) can be formed. When the residual risk of the active portfolio goes to zero, the position in it goes to infinity. This is precisely the same implication as the APT: When portfolios are well-diversified, you will scale up an arbitrage position without 5

The APT is easily extended to a multifactor market as we show later. The tediousness of some of the expressions involved in the T-B method should not deter anyone. The calculations are pretty straightforward, especially in a spreadsheet. The estimation of the risk parameters also is a relatively straightforward statistical task. The real difficulty is to uncover security alphas and the macro-factor risk premium, RPM. 6

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bound. Similarly, when the residual risk of an asset in the active T-B portfolio is zero, it will displace all other assets from that portfolio, and thus the residual risk of the active portfolio will be zero and elicit the same extreme portfolio response. When residual risks are nonzero, the T-B procedure produces the optimal risky portfolio, which is a compromise between seeking alpha and shunning potentially diversifiable risk. The APT ignores residual risk altogether, assuming it has been diversified away. Obviously, we have no use for the APT in this context. When residual risk can be made small through diversification, the T-B model prescribes very aggressive (large) positions in mispriced securities that exert great pressure on equilibrium risk premiums to eliminate nonzero alpha values. The T-B model does what the APT is meant to do but with more flexibility in terms of accommodating the practical limits to diversification. In this sense, Treynor and Black anticipated the development of the APT.

Example 10.3

Exploiting Alpha

Table10.2 summarizes a rudimentary experiment that compares the prescriptions and predictions of the APT and T-B model in the presence of realistic values of residual risk. We use relatively small alpha values (1 and 3%), three levels of residual risk consistent with values in Table10.1 (2, 3, and 4%), and two levels of beta (0.5 and 2) to span the likely range of reasonable parameters. The first set of columns in Table 10.2, titled Active Portfolio, show the parameter values in each example. The second set of columns, titled Zero-Net-Investment, Arbitrage (Zero-Beta), shows the weight in the active portfolio and resultant information ratio of the active portfolio. This would be the Sharpe ratio if the arbitrage position (the positivealpha, zero-beta portfolio) made up the entire risky portfolio (as would be prescribed

Index Risk Premium57

Index SD520

Index Sharpe Ratio50.35

Active Portfolio

Zero-Net-Investment, Arbitrage (Zero-Beta) Portfolio

Treynor-Black Procedure

Alpha (%)

Residual SD

Beta

w in Active

Info Ratio

w(beta50)

w(beta)

1 1 1 1 1 1 3 3 3 3 3 3

4 4 3 3 2 2 4 4 3 3 2 2

0.5 2 0.5 2 0.5 2 0.5 2 0.5 2 0.5 2

2 1 2 1 2 1 2 1 2 1 2 1

0.25 0.25 0.33 0.33 0.50 0.50 0.75 0.75 1.00 1.00 1.50 1.50

3.57 3.57 6.35 6.35 14.29 14.29 10.71 10.71 19.05 19.05 42.86 42.86

1.28 1.00 1.52 1.00 1.75 1.00 1.69 1.00 1.81 1.00 1.91 1.00

Table 10.2 Performance of APT vs. Index Model when diversification of residual SD is incomplete

Sharpe Incremental Ratio Sharpe Ratio 0.43 0.43 0.48 0.48 0.61 0.61 0.83 0.83 1.06 1.06 1.54 1.54

0.18 0.18 0.15 0.15 0.11 0.11 0.08 0.08 0.06 0.06 0.04 0.04

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by the APT). The last set of columns shows the T-B position in the active portfolio that maximizes the Sharpe ratio of the overall risky portfolio. The final column shows the increment to the Sharpe ratio of the T-B portfolio relative to the APT portfolio. Keep in mind that even when the two models call for a similar weight in the active portfolio (compare w in Active for the APT model to w(beta) for the T-B model), they nevertheless prescribe a different overall risky portfolio. The APT assumes zero investment beyond what is necessary to hedge out the market risk of the active portfolio. In contrast, the T-B procedure chooses a mix of active and index portfolios to maximize the Sharpe ratio. With identical investment in the active portfolio, the T-B portfolio can still include additional investment in the index portfolio. To obtain the Sharpe ratio of the risky portfolio, we need the Sharpe ratio of the index portfolio. As an estimate, we use the average return and standard deviation of the broad market index (NYSE 1AMEX 1NASDAQ) over the period 1926–2012. The top row (over the column titles) of Table10.2 shows an annual Sharpe ratio of 0.35. The rows of the table are ordered by the information ratio of the active portfolio. Table10.2 shows that the T-B procedure noticeably improves the Sharpe ratio beyond the information ratio of the APT (for which the IR is also the Sharpe ratio). However, as the information ratio of the active portfolio increases, the difference in the T-B and APT active portfolio positions declines, as does the difference between their Sharpe ratios. Put differently, the higher the information ratio, the closer we are to a risk-free arbitrage opportunity, and the closer are the prescriptions of the APT and T-B models.

10.4 A Multifactor APT We have assumed so far that only one systematic factor affects stock returns. This simplifying assumption is in fact too simplistic. We’ve noted that it is easy to think of several factors driven by the business cycle that might affect stock returns: interest rate fluctuations, inflation rates, and so on. Presumably, exposure to any of these factors will affect a stock’s risk and hence its expected return. We can derive a multifactor version of the APT to accommodate these multiple sources of risk. Suppose that we generalize the single-factor model expressed in Equation 10.1 to a two-factor model: Ri 5 E (Ri ) 1 bi1 F1 1 bi2 F2 1 ei

(10.10)

In Example 10.2, factor 1 was the departure of GDP growth from expectations, and factor 2 was the unanticipated change in interest rates. Each factor has zero expected value because each measures the surprise in the systematic variable rather than the level of the variable. Similarly, the firm-specific component of unexpected return, ei , also has zero expected value. Extending such a two-factor model to any number of factors is straightforward. We can now generalize the simple APT to a more general multifactor version. But first we must introduce the concept of a factor portfolio, which is a well-diversified portfolio constructed to have a beta of 1 on one of the factors and a beta of zero on any other factor. We can think of a factor portfolio as a tracking portfolio. That is, the returns on such a portfolio track the evolution of particular sources of macroeconomic risk but are uncorrelated with other sources of risk. It is possible to form such factor portfolios because we have a large number of securities to choose from, and a relatively small number of factors. Factor portfolios will serve as the benchmark portfolios for a multifactor security market line. The multidimensional SML predicts that exposure to each risk factor contributes to the security’s total risk premium by an amount equal to the factor beta times the risk premium of the factor portfolio tracking that source of risk. We illustrate with an example.

CHAPTER 10

Example 10.4

Arbitrage Pricing Theory and Multifactor Models of Risk and Return

Multifactor SML

Suppose that the two factor portfolios, portfolios 1 and 2, have expected returns E (r1) 510% and E (r2) 512%. Suppose further that the risk-free rate is 4%. The risk premium on the first factor portfolio is 10%24%56%, whereas that on the second factor portfolio is 12%24%58%. Now consider a well-diversified portfolio, portfolio A, with beta on the first factor, bA15.5, and beta on the second factor, bA25.75. The multifactor APT states that the overall risk premium on this portfolio must equal the sum of the risk premiums required as compensation for each source of systematic risk. The risk premium attributable to risk factor 1 should be the portfolio’s exposure to factor 1, bA1, multiplied by the risk premium earned on the first factor portfolio, E (r1) 2 rf. Therefore, the portion of portfolio A’s risk premium that is compensation for its exposure to the first factor is bA1 [E (r1) 2 rf] 5 .5(10% 2 4%) 5 3%, whereas the risk premium attributable to risk factor 2 is bA2[E (r2)2rf]5.75(12%24%)56%. The total risk premium on the portfolio should be 3%16%59% and the total return on the portfolio should be 4%19%513%.

To generalize the argument in Example 10.4, note that the factor exposures of any portfolio, P, are given by its betas, bP1 and bP2. A competing portfolio, Q, can be formed by investing in factor portfolios with the following weights: bP1 in the first factor portfolio, bP2 in the second factor portfolio, and 12bP12bP2 in T-bills. By construction, portfolio Q will have betas equal to those of portfolio P and expected return of E (rQ ) 5 bP1E (r1) 1 bP2 E (r2) 1 (1 2 bP1 2 bP2) rf 5 rf 1 bP1 3 E (r1) 2 rf 4 1 bP2 3 E (r2) 2 rf 4

(10.11)

Using the numbers in Example 10.4: E (rQ ) 5 4 1 .5 3 (10 2 4) 1 .75 3 (12 2 4) 5 13%

Example 10.5

Mispricing and Arbitrage

Suppose that the expected return on portfolio A from Example 10.4 were 12% rather than 13%. This return would give rise to an arbitrage opportunity. Form a portfolio from the factor portfolios with the same betas as portfolio A. This requires weights of .5 on the first factor portfolio, .75 on the second factor portfolio, and 2.25 on the risk-free asset. This portfolio has exactly the same factor betas as portfolio A: It has a beta of .5 on the first factor because of its .5 weight on the first factor portfolio, and a beta of .75 on the second factor. (The weight of 2.25 on risk-free T-bills does not affect the sensitivity to either factor.) Now invest $1 in portfolio Q and sell (short) $1 in portfolio A. Your net investment is zero, but your expected dollar profit is positive and equal to $13 E(rQ)2$13E(rA)5$13.132$13.125$.01 Moreover, your net position is riskless. Your exposure to each risk factor cancels out because you are long $1 in portfolio Q and short $1 in portfolio A, and both of these well-diversified portfolios have exactly the same factor betas. Thus, if portfolio A’s expected return differs from that of portfolio Q’s, you can earn positive risk-free profits on a zero-net-investment position. This is an arbitrage opportunity.

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Because portfolio Q in Example 10.5 has precisely the same exposures as portfolio A to the two sources of risk, their expected returns also ought to be equal. So portfolio A also ought to have an expected return of 13%. If it does not, then there will be an arbitrage opportunity.7 We conclude that any well-diversified portfolio with betas bP1 and bP2 must have the return given in Equation 10.11 if arbitrage opportunities are to be precluded. Equation 10.11 simply generalizes the one-factor SML. Finally, the extension of the multifactor SML of Equation 10.11 to individual assets is precisely the same as for the one-factor APT. Equation 10.11 cannot be satisfied by every well-diversified portfolio unless it is satisfied approximately by individual securities. Equation 10.11 thus represents the multifactor SML for an economy with multiple sources of risk. We pointed out earlier that one application of the CAPM is to provide “fair” rates of return for regulated utilities. The multifactor APT can be used to the same ends. The nearby box summarizes a study in which the APT was applied to find the cost of capital for regulated electric companies. Notice that CONCEPT CHECK 10.3 empirical estimates for interest rate and inflation risk premiums in the box are negative, as Using the factor portfolios of Example 10.4, find the equilibwe argued was reasonable in our discussion rium rate of return on a portfolio with b15.2 and b251.4. of Example 10.2.

10.5 The Fama-French (FF) Three-Factor Model The currently dominant approach to specifying factors as candidates for relevant sources of systematic risk uses firm characteristics that seem on empirical grounds to proxy for exposure to systematic risk. The factors chosen are variables that on past evidence seem to predict average returns well and therefore may be capturing risk premiums. One example of this approach is the Fama and French three-factor model and its variants, which have come to dominate empirical research and industry applications:8 Rit 5 a i 1 biM RMt 1 biSMB SMBt 1 biHMLHMLt 1 eit (10.12) where SMB 5 Small Minus Big, i.e., the return of a portfolio of small stocks in excess of the return on a portfolio of large stocks. HML 5 High Minus Low, i.e., the return of a portfolio of stocks with a high book-to-market ratio in excess of the return on a portfolio of stocks with a low book-to-market ratio. Note that in this model the market index does play a role and is expected to capture systematic risk originating from macroeconomic factors. These two firm-characteristic variables are chosen because of long-standing observations that corporate capitalization (firm size) and book-to-market ratio predict deviations 7

The risk premium on portfolio A is 9% (more than the historical risk premium of the S&P 500) despite the fact that its betas, which are both below 1, might seem defensive. This highlights another distinction between multifactor and single-factor models. Whereas a beta greater than 1 in a single-factor market is aggressive, we cannot say in advance what would be aggressive or defensive in a multifactor economy where risk premiums depend on the sum of the contributions of several factors. 8 Eugene F. Fama and Kenneth R. French, “Multifactor Explanations of Asset Pricing Anomalies,” Journal of Finance 51 (1996), pp. 55–84.

Elton, Gruber, and Mei* use the APT to derive the cost of capital for electric utilities. They assume that the relevant risk factors are unanticipated developments in the term structure of interest rates, the level of interest rates, inflation rates, the business cycle (measured by GDP), foreign exchange rates, and a summary measure they devise to measure other macro factors. Their first step is to estimate the risk premium associated with exposure to each risk source. They accomplish this in a two-step strategy (which we will describe in considerable detail in Chapter 13): 1. Estimate “factor loadings” (i.e., betas) of a large sample of firms. Regress returns of 100 randomly selected stocks against the systematic risk factors. They use a time-series regression for each stock (e.g., 60 months of data), therefore estimating 100 regressions, one for each stock. 2. Estimate the reward earned per unit of exposure to each risk factor. For each month, regress the return of each stock against the five betas estimated. The coefficient on each beta is the extra average return earned as beta increases, i.e., it is an estimate of the risk premium for that risk factor from that month’s data. These estimates are of course subject to sampling error. Therefore, average the risk premium estimates across the 12 months in each year. The average response of return to risk is less subject to sampling error. The risk premiums are in the middle column of the table at the top of the next column. Notice that some risk premiums are negative. The interpretation of this result is that risk premium should be positive for risk factors you don’t want exposure to, but negative for factors you do want exposure to. For example, you should desire securities that have higher returns when inflation increases and be willing to accept lower expected returns on such securities; this shows up as a negative risk premium.

Factor Term structure Interest rates Exchange rates Business cycle Inﬂation Other macro factors

Factor Risk Premium

Factor Betas for Niagara Mohawk

.425 2.051 2.049 .041 2.069 .530

1.0615 22.4167 1.3235 .1292 2.5220 .3046

Therefore, the expected return on any security should be related to its factor betas as follows: rf 1 .425 bterm struc 2 .051 bint rate 2.049 bex rate 1 .041 bbus cycle 2 .069 binflation 1 .530 bother Finally, to obtain the cost of capital for a particular firm, the authors estimate the firm’s betas against each source of risk, multiply each factor beta by the “cost of factor risk” from the table above, sum over all risk sources to obtain the total risk premium, and add the risk-free rate. For example, the beta estimates for Niagara Mohawk appear in the last column of the table above. Therefore, its cost of capital is Cost of capital 5 rf 1 .425 3 1.0615 2 .051(22.4167) 2.049(1.3235) 1 .041(.1292) 2.069(2.5220) 1 .530(.3046) 5 rf 1 .72 In other words, the monthly cost of capital for Niagara Mohawk is .72% above the monthly risk-free rate. Its annualized risk premium is therefore .72%31258.64%. *Edwin J. Elton, Martin J. Gruber, and Jianping Mei, “Cost of Capital Using Arbitrage Pricing Theory: A Case Study of Nine New York Utilities,” Financial Markets, Institutions, and Instruments 3 (August 1994), pp. 46–68.

of average stock returns from levels consistent with the CAPM. Fama and French justify this model on empirical grounds: While SMB and HML are not themselves obvious candidates for relevant risk factors, the argument is that these variables may proxy for yet-unknown more-fundamental variables. For example, Fama and French point out that firms with high ratios of book-to-market value are more likely to be in financial distress and that small stocks may be more sensitive to changes in business conditions. Thus, these variables may capture sensitivity to risk factors in the macroeconomy. More evidence on the Fama-French model appears in Chapter 13. The problem with empirical approaches such as the Fama-French model, which use proxies for extramarket sources of risk, is that none of the factors in the proposed models can be clearly identified as hedging a significant source of uncertainty. Black9 points out that 9

Fischer Black, “Beta and Return,” Journal of Portfolio Management 20 (1993), pp. 8–18.

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when researchers scan and rescan the database of security returns in search of explanatory factors (an activity often called data-snooping), they may eventually uncover past “patterns” that are due purely to chance. Black observes that return premiums to factors such as firm size have proven to be inconsistent since first discovered. However, Fama and French have shown that size and book-to-market ratios have predicted average returns in various time periods and in markets all over the world, thus mitigating potential effects of data-snooping. The firm-characteristic basis of the Fama-French factors raises the question of whether they reflect a multi-index ICAPM based on extra-market hedging demands or just represent yet-unexplained anomalies, where firm characteristics are correlated with alpha values. This is an important distinction for the debate over the proper interpretation of the model, because the validity of FF-style models may signify either a deviation from rational equilibrium (as there is no rational reason to prefer one or another of these firm characteristics per se), or that firm characteristics identified as empirically associated with average returns are correlated with other (yet unknown) risk factors. The issue is still unresolved and is discussed in Chapter 13.

SUMMARY

1. Multifactor models seek to improve the explanatory power of single-factor models by explicitly accounting for the various systematic components of security risk. These models use indicators intended to capture a wide range of macroeconomic risk factors. 2. Once we allow for multiple risk factors, we conclude that the security market line also ought to be multidimensional, with exposure to each risk factor contributing to the total risk premium of the security. 3. A (risk-free) arbitrage opportunity arises when two or more security prices enable investors to construct a zero-net-investment portfolio that will yield a sure profit. The presence of arbitrage opportunities will generate a large volume of trades that puts pressure on security prices. This pressure will continue until prices reach levels that preclude such arbitrage. 4. When securities are priced so that there are no risk-free arbitrage opportunities, we say that they satisfy the no-arbitrage condition. Price relationships that satisfy the no-arbitrage condition are important because we expect them to hold in real-world markets. 5. Portfolios are called “well-diversified” if they include a large number of securities and the investment proportion in each is sufficiently small. The proportion of a security in a well-diversified portfolio is small enough so that for all practical purposes a reasonable change in that security’s rate of return will have a negligible effect on the portfolio’s rate of return. 6. In a single-factor security market, all well-diversified portfolios have to satisfy the expected return–beta relationship of the CAPM to satisfy the no-arbitrage condition. If all well-diversified portfolios satisfy the expected return–beta relationship, then individual securities also must satisfy this relationship, at least approximately. 7. The APT does not require the restrictive assumptions of the CAPM and its (unobservable) market portfolio. The price of this generality is that the APT does not guarantee this relationship for all securities at all times. 8. A multifactor APT generalizes the single-factor model to accommodate several sources of systematic risk. The multidimensional security market line predicts that exposure to each risk factor contributes to the security’s total risk premium by an amount equal to the factor beta times the risk premium of the factor portfolio that tracks that source of risk.

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9. A multifactor extension of the single-factor CAPM, the ICAPM, is a model of the risk–return trade-off that predicts the same multidimensional security market line as the APT. The ICAPM suggests that priced risk factors will be those sources of risk that lead to significant hedging demand by a substantial fraction of investors.

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arbitrage pricing theory arbitrage Law of One Price risk arbitrage

single-factor model multifactor model factor loading factor beta

well-diversified portfolio factor portfolio

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KEY EQUATIONS

Single factor model: Ri5E(Ri)1b1F1ei Multifactor model (here, 2 factors, F1 and F2): Ri5E(Ri)1b1F11b2F21ei Single-index model: Ri5ai1bi RM1ei Multifactor SML (here, 2 factors, labeled 1 and 2) E(ri) 5 rf 1 b1 3 E(r1) 2 rf 4 1 b2 3 E(r2 ) 2 rf 4 5 rf 1 b1E(R1) 1 b2 E(R2)

1. Suppose that two factors have been identified for the U.S. economy: the growth rate of industrial production, IP, and the inflation rate, IR. IP is expected to be 3%, and IR 5%. A stock with a beta of 1 on IP and .5 on IR currently is expected to provide a rate of return of 12%. If industrial production actually grows by 5%, while the inflation rate turns out to be 8%, what is your revised estimate of the expected rate of return on the stock?

PROBLEM SETS

2. The APT itself does not provide guidance concerning the factors that one might expect to determine risk premiums. How should researchers decide which factors to investigate? Why, for example, is industrial production a reasonable factor to test for a risk premium?

Basic

3. If the APT is to be a useful theory, the number of systematic factors in the economy must be small. Why? 4. Suppose that there are two independent economic factors, F1 and F2. The risk-free rate is 6%, and all stocks have independent firm-specific components with a standard deviation of 45%. The following are well-diversified portfolios: Portfolio

Beta on F1

Beta on F2

Expected Return

1.5 2.2

2.0 20.2

31% 27%

A B

What is the expected return–beta relationship in this economy? 5. Consider the following data for a one-factor economy. All portfolios are well diversified. Portfolio

E(r)

Beta

A F

12% 6%

1.2 0.0

Suppose that another portfolio, portfolio E, is well diversified with a beta of .6 and expected return of 8%. Would an arbitrage opportunity exist? If so, what would be the arbitrage strategy? 6. Assume that both portfolios A and B are well diversified, that E(rA) 512%, and E(rB) 59%. If the economy has only one factor, and bA 51.2, whereas bB 5.8, what must be the riskfree rate?

Intermediate

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where the risk premiums on the two factor portfolios are E(R1) and E(R2 )

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Equilibrium in Capital Markets 7. Assume that stock market returns have the market index as a common factor, and that all stocks in the economy have a beta of 1 on the market index. Firm-specific returns all have a standard deviation of 30%. Suppose that an analyst studies 20 stocks, and finds that one-half have an alpha of 12%, and the other half an alpha of 22%. Suppose the analyst buys $1 million of an equally weighted portfolio of the positive alpha stocks, and shorts $1 million of an equally weighted portfolio of the negative alpha stocks. a. What is the expected profit (in dollars) and standard deviation of the analyst’s profit? b. How does your answer change if the analyst examines 50 stocks instead of 20 stocks? 100 stocks? 8. Assume that security returns are generated by the single-index model, Ri 5 a i 1 bi RM 1 ei

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where Ri is the excess return for security i and RM is the market’s excess return. The risk-free rate is 2%. Suppose also that there are three securities A, B, and C, characterized by the following data: Security

bi

E (Ri )

s(ei )

A B C

0.8 1.0 1.2

10% 12 14

25% 10 20

a. If sM520%, calculate the variance of returns of securities A, B, and C. b. Now assume that there are an infinite number of assets with return characteristics identical to those of A, B, and C, respectively. If one forms a well-diversified portfolio of type A securities, what will be the mean and variance of the portfolio’s excess returns? What about portfolios composed only of type B or C stocks? c. Is there an arbitrage opportunity in this market? What is it? Analyze the opportunity graphically. 9. The SML relationship states that the expected risk premium on a security in a one-factor model must be directly proportional to the security’s beta. Suppose that this were not the case. For example, suppose that expected return rises more than proportionately with beta as in the figure below. E(r)

B

C

A

β

a. How could you construct an arbitrage portfolio? (Hint: Consider combinations of portfolios A and B, and compare the resultant portfolio to C.) b. Some researchers have examined the relationship between average returns on diversified portfolios and the b and b2 of those portfolios. What should they have discovered about the effect of b2 on portfolio return?

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10. Consider the following multifactor (APT) model of security returns for a particular stock. Factor

Factor Beta

Inflation Industrial production Oil prices

Factor Risk Premium

1.2 0.5 0.3

6% 8 3

a. If T-bills currently offer a 6% yield, find the expected rate of return on this stock if the market views the stock as fairly priced. b. Suppose that the market expected the values for the three macro factors given in column 1 below, but that the actual values turn out as given in column 2. Calculate the revised expectations for the rate of return on the stock once the “surprises” become known. Expected Rate of Change

Inflation Industrial production Oil prices

Actual Rate of Change

5% 3 2

4% 6 0

11. Suppose that the market can be described by the following three sources of systematic risk with associated risk premiums. Factor

Risk Premium

Industrial production (I) Interest rates (R) Consumer confidence (C)

6% 2 4

The return on a particular stock is generated according to the following equation: r 5 15% 1 1.0 I 1 .5R 1 .75C 1 e Find the equilibrium rate of return on this stock using the APT. The T-bill rate is 6%. Is the stock over- or underpriced? Explain. 12. As a finance intern at Pork Products, Jennifer Wainwright’s assignment is to come up with fresh insights concerning the firm’s cost of capital. She decides that this would be a good opportunity to try out the new material on the APT that she learned last semester. She decides that three promising factors would be (i) the return on a broad-based index such as the S&P 500; (ii) the level of interest rates, as represented by the yield to maturity on 10-year Treasury bonds; and (iii) the price of hogs, which are particularly important to her firm. Her plan is to find the beta of Pork Products against each of these factors by using a multiple regression and to estimate the risk premium associated with each exposure factor. Comment on Jennifer’s choice of factors. Which are most promising with respect to the likely impact on her firm’s cost of capital? Can you suggest improvements to her specification? Use the following information to Answer Problems 13–16: Orb Trust (Orb) has historically leaned toward a passive management style of its portfolios. The only model that Orb’s senior management has promoted in the past is the capital asset pricing model (CAPM). Now Orb’s management has asked one of its analysts, Kevin McCracken, CFA, to investigate the use of the arbitrage pricing theory (APT) model. McCracken believes that a two-factor APT model is adequate, where the factors are the sensitivity to changes in real GDP and changes in inflation. McCracken has concluded that the factor risk premium for real GDP is 8% while the factor risk premium for inflation is 2%. He estimates for Orb’s High Growth Fund that the sensitivities to these two factors are 1.25 and 1.5, respectively.

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Factor

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Equilibrium in Capital Markets Using his APT results, he computes the expected return of the fund. For comparison purposes, he then uses fundamental analysis to also compute the expected return of Orb’s High Growth Fund. McCracken finds that the two estimates of the Orb High Growth Fund’s expected return are equal. McCracken asks a fellow analyst, Sue Kwon, to provide an estimate of the expected return of Orb’s Large Cap Fund based on fundamental analysis. Kwon, who manages the fund, says that the expected return is 8.5% above the risk-free rate. McCracken then applies the APT model to the Large Cap Fund. He finds that the sensitivities to real GDP and inflation are .75 and 1.25, respectively. McCracken’s manager at Orb, Jay Stiles, asks McCracken to compose a portfolio that has a unit sensitivity to real GDP growth but is not affected by inflation. McCracken is confident in his APT estimates for the High Growth Fund and the Large Cap Fund. He then computes the sensitivities for a third fund, Orb’s Utility Fund, which has sensitivities equal to 1.0 and 2.0, respectively. McCracken will use his APT results for these three funds to accomplish the task of creating a portfolio with a unit exposure to real GDP and no exposure to inflation. He calls the fund the “GDP Fund.” Stiles says such a GDP Fund would be good for clients who are retirees who live off the steady income of their investments. McCracken says that the fund would be a good choice if upcoming supply side macroeconomic policies of the government are successful.

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13. According to the APT, if the risk-free rate is 4%, what should be McCracken’s estimate of the expected return of Orb’s High Growth Fund? 14. With respect to McCracken’s APT model estimate of Orb’s Large Cap Fund and the information Kwon provides, is an arbitrage opportunity available? 15. The GDP Fund composed from the other three funds would have a weight in Utility Fund equal to (a) 22.2; (b) 23.2; or (c) .3. 16. With respect to the comments of Stiles and McCracken concerning for whom the GDP Fund would be appropriate: a. McCracken was correct and Stiles was wrong. b. Both were correct. c. Stiles was correct and McCracken was wrong.

Challenge

17. Assume a universe of n (large) securities for which the largest residual variance is not larger than ns2M. Construct as many different weighting schemes as you can that generate welldiversified portfolios. 18. Derive a more general (than the numerical example in the chapter) demonstration of the APT security market line: a. For a single-factor market. b. For a multifactor market. 19. Small firms will have relatively high loadings (high betas) on the SMB (small minus big) factor. a. Explain why. b. Now suppose two unrelated small firms merge. Each will be operated as an independent unit of the merged company. Would you expect the stock market behavior of the merged firm to differ from that of a portfolio of the two previously independent firms? How does the merger affect market capitalization? What is the prediction of the Fama-French model for the risk premium on the combined firm? Do we see here a flaw in the FF model?

1. Jeffrey Bruner, CFA, uses the capital asset pricing model (CAPM) to help identify mispriced securities. A consultant suggests Bruner use arbitrage pricing theory (APT) instead. In comparing CAPM and APT, the consultant made the following arguments: a. Both the CAPM and APT require a mean-variance efficient market portfolio. b. Neither the CAPM nor APT assumes normally distributed security returns. c. The CAPM assumes that one specific factor explains security returns but APT does not.

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State whether each of the consultant’s arguments is correct or incorrect. Indicate, for each incorrect argument, why the argument is incorrect. 2. Assume that both X and Y are well-diversified portfolios and the risk-free rate is 8%. Portfolio X Y

Expected Return 16% 12

Beta 1.00 0.25

In this situation you would conclude that portfolios X and Y: a. b. c. d.

Are in equilibrium. Offer an arbitrage opportunity. Are both underpriced. Are both fairly priced.

a. b. c. d.

The expected return of the portfolio equals zero. The capital market line is tangent to the opportunity set. The Law of One Price remains unviolated. A risk-free arbitrage opportunity exists.

4. According to the theory of arbitrage: a. b. c. d.

High-beta stocks are consistently overpriced. Low-beta stocks are consistently overpriced. Positive alpha investment opportunities will quickly disappear. Rational investors will pursue arbitrage consistent with their risk tolerance.

5. The general arbitrage pricing theory (APT) differs from the single-factor capital asset pricing model (CAPM) because the APT: a. b. c. d.

Places more emphasis on market risk. Minimizes the importance of diversification. Recognizes multiple unsystematic risk factors. Recognizes multiple systematic risk factors.

6. An investor takes as large a position as possible when an equilibrium price relationship is violated. This is an example of: a. b. c. d.

A dominance argument. The mean-variance efficient frontier. Arbitrage activity. The capital asset pricing model.

7. The feature of the general version of the arbitrage pricing theory (APT) that offers the greatest potential advantage over the simple CAPM is the: a. Identification of anticipated changes in production, inflation, and term structure of interest rates as key factors explaining the risk–return relationship. b. Superior measurement of the risk-free rate of return over historical time periods. c. Variability of coefficients of sensitivity to the APT factors for a given asset over time. d. Use of several factors instead of a single market index to explain the risk–return relationship. 8. In contrast to the capital asset pricing model, arbitrage pricing theory: a. b. c. d.

Requires that markets be in equilibrium. Uses risk premiums based on micro variables. Specifies the number and identifies specific factors that determine expected returns. Does not require the restrictive assumptions concerning the market portfolio.

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3. A zero-investment portfolio with a positive alpha could arise if:

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E-INVESTMENTS EXERCISES One of the factors in the APT model specified by Chen, Roll, and Ross is the percent change in unanticipated inflation. Who gains and who loses when inflation changes? Go to http://hussmanfunds.com/rsi/infsurprises.htm to see a graph of the Inflation Surprise Index and Economists’ Inflation Forecasts.

SOLUTIONS TO CONCEPT CHECKS 1. The GDP beta is 1.2 and GDP growth is 1% better than previously expected. So you will increase your forecast for the stock return by 1.2 31% 51.2%. The revised forecast is for an 11.2% return.

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2. a. This portfolio is not well diversified. The weight on the first security does not decline as n increases. Regardless of how much diversification there is in the rest of the portfolio, you will not shed the firm-specific risk of this security. b. This portfolio is well diversified. Even though some stocks have three times the weight of other stocks (1.5/n versus .5/n), the weight on all stocks approaches zero as n increases. The impact of any individual stock’s firm-specific risk will approach zero as n becomes ever larger. 3. The equilibrium return is E (r) 5 rf 1 bP1[E (r1) 2 rf ] 1 bP2 [E (r2) 2 rf ]. Using the data in Example 10.4: E(r) 5 4 1 .2 3 (10 2 4) 1 1.4 3 (12 2 4) 5 16.4%

CHAPTER ELEVEN

11 1

The Efficient Market Hypothesis

ONE OF THE early applications of computers in economics in the 1950s was to analyze economic time series. Business cycle theorists felt that tracing the evolution of several economic variables over time would clarify and predict the progress of the economy through boom and bust periods. A natural candidate for analysis was the behavior of stock market prices over time. Assuming that stock prices reflect the prospects of the firm, recurrent patterns of peaks and troughs in economic performance ought to show up in those prices. Maurice Kendall examined this proposition in 1953.1 He found to his great surprise that he could identify no predictable patterns in stock prices. Prices seemed to evolve randomly. They were as likely to go up as they were to go down on any particular day, regardless of past performance. The data provided no way to predict price movements.

Maurice Kendall, “The Analysis of Economic Time Series, Part I: Prices,” Journal of the Royal Statistical Society 96 (1953).

PART III

1

At first blush, Kendall’s results were disturbing to some financial economists. They seemed to imply that the stock market is dominated by erratic market psychology, or “animal spirits”—that it follows no logical rules. In short, the results appeared to confirm the irrationality of the market. On further reflection, however, economists came to reverse their interpretation of Kendall’s study. It soon became apparent that random price movements indicated a well-functioning or efficient market, not an irrational one. In this chapter we explore the reasoning behind what may seem a surprising conclusion. We show how competition among analysts leads naturally to market efficiency, and we examine the implications of the efficient market hypothesis for investment policy. We also consider empirical evidence that supports and contradicts the notion of market efficiency.

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11.1 Random Walks and the Efficient Market Hypothesis Suppose Kendall had discovered that stock price changes are predictable. What a gold mine this would have been. If they could use Kendall’s equations to predict stock prices, investors would reap unending profits simply by purchasing stocks that the computer model implied were about to increase in price and by selling those stocks about to fall in price. A moment’s reflection should be enough to convince yourself that this situation could not persist for long. For example, suppose that the model predicts with great confidence that XYZ stock price, currently at $100 per share, will rise dramatically in 3 days to $110. What would all investors with access to the model’s prediction do today? Obviously, they would place a great wave of immediate buy orders to cash in on the prospective increase in stock price. No one holding XYZ, however, would be willing to sell. The net effect would be an immediate jump in the stock price to $110. The forecast of a future price increase will lead instead to an immediate price increase. In other words, the stock price will immediately reflect the “good news” implicit in the model’s forecast. This simple example illustrates why Kendall’s attempt to find recurrent patterns in stock price movements was likely to fail. A forecast about favorable future performance leads instead to favorable current performance, as market participants all try to get in on the action before the price jump. More generally, one might say that any information that could be used to predict stock performance should already be reflected in stock prices. As soon as there is any information indicating that a stock is underpriced and therefore offers a profit opportunity, investors flock to buy the stock and immediately bid up its price to a fair level, where only ordinary rates of return can be expected. These “ordinary rates” are simply rates of return commensurate with the risk of the stock. However, if prices are bid immediately to fair levels, given all available information, it must be that they increase or decrease only in response to new information. New information, by definition, must be unpredictable; if it could be predicted, then the prediction would be part of today’s information. Thus stock prices that change in response to new (that is, previously unpredicted) information also must move unpredictably. This is the essence of the argument that stock prices should follow a random walk, that is, that price changes should be random and unpredictable.2 Far from a proof of market irrationality, randomly evolving stock prices would be the necessary consequence of intelligent investors competing to discover relevant information on which to buy or sell stocks before the rest of the market becomes aware of that information. Don’t confuse randomness in price changes with irrationality in the level of prices. If prices are determined rationally, then only new information will cause them to change. Therefore, a random walk would be the natural result of prices that always reflect all current knowledge. Indeed, if stock price movements were predictable, that would be damning evidence of stock market inefficiency, because the ability to predict prices would indicate that

2

Actually, we are being a little loose with terminology here. Strictly speaking, we should characterize stock prices as following a submartingale, meaning that the expected change in the price can be positive, presumably as compensation for the time value of money and systematic risk. Moreover, the expected return may change over time as risk factors change. A random walk is more restrictive in that it constrains successive stock returns to be independent and identically distributed. Nevertheless, the term “random walk” is commonly used in the looser sense that price changes are essentially unpredictable. We will follow this convention.

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The Efficient Market Hypothesis

351

Cumulative Abnormal Return (%)

all available information was not already reflected in stock prices. Therefore, the 36 notion that stocks already reflect all avail32 able information is referred to as the 28 efficient market hypothesis (EMH).3 24 Figure 11.1 illustrates the response 20 of stock prices to new information in an 16 efficient market. The graph plots the price 12 response of a sample of firms that were 8 targets of takeover attempts. In most take4 overs, the acquiring firm pays a substan0 tial premium over current market prices. −4 Therefore, announcement of a takeover −8 attempt should cause the stock price to −12 jump. The figure shows that stock prices −16 jump dramatically on the day the news −135 −120 −105 −90 −75 −60 −45 −30 −15 0 15 30 becomes public. However, there is no Days Relative to Announcement Date further drift in prices after the announcement date, suggesting that prices reflect the new information, including the likely Figure 11.1 Cumulative abnormal returns before takeover magnitude of the takeover premium, by attempts: target companies the end of the trading day. Source: Arthur Keown and John Pinkerton, “Merger Announcements and Even more dramatic evidence of Insider Trading Activity,” Journal of Finance 36 (September 1981). Used with permission of John Wiley and Sons, via Copyright Clearance Center. Updates rapid response to new information may courtesy of Jinghua Yan. be found in intraday prices. For example, Patell and Wolfson show that most of the stock price response to corporate dividend or earnings announcements occurs within 10 minutes of the announcement.4 A nice illustration of such rapid adjustment is provided in a study by Busse and Green, who track minute-by-minute stock prices of firms that are featured on CNBC’s “Morning” or “Midday Call” segments.5 Minute 0 in Figure 11.2 is the time at which the stock is mentioned on the midday show. The top line is the average price movement of stocks that receive positive reports, while the bottom line reports returns on stocks with negative reports. Notice that the top line levels off, indicating that the market has fully digested the news within 5 minutes of the report. The bottom line levels off within about 12 minutes.

Competition as the Source of Efficiency Why should we expect stock prices to reflect “all available information”? After all, if you are willing to spend time and money on gathering information, it might seem reasonable that you could turn up something that has been overlooked by the rest of the investment community. When information is costly to uncover and analyze, one would expect investment analysis calling for such expenditures to result in an increased expected return. 3

Market efficiency should not be confused with the idea of efficient portfolios introduced in Chapter 7. An informationally efficient market is one in which information is rapidly disseminated and reflected in prices. An efficient portfolio is one with the highest expected return for a given level of risk. 4 J. M. Patell and M. A. Wolfson, “The Intraday Speed of Adjustment of Stock Prices to Earnings and Dividend Announcements,” Journal of Financial Economics 13 (June 1984), pp. 223–52. 5 J. A. Busse and T. C. Green, “Market Efficiency in Real Time,” Journal of Financial Economics 65 (2002), pp. 415–37. You can find an intraday movie version of this figure at www.bus.emory.edu/cgreen/docs/cnbc/ cnbc.html.

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0.75

Cumulative Return (%)

0.50

Midday-Positive Midday-Negative

0.25 0.00 −0.25 −0.50 −0.75 −1.00 −1.25 −1.50 −15

−10

−5 0 5 Minutes Relative to Mention

10

15

Figure 11.2 Stock price reaction to CNBC reports. The figure shows the reaction of stock prices to on-air stock reports during the “Midday Call” segment on CNBC. The chart plots cumulative returns beginning 15 minutes before the stock report. Source: Reprinted from J. A. Busse and T. C. Green, “Market Efficiency in Real Time,” Journal of Financial Economics 65 (2002), p. 422. Copyright 2002, with permission from Elsevier.

Example 11.1

This point has been stressed by Grossman and Stiglitz.6 They argued that investors will have an incentive to spend time and resources to analyze and uncover new information only if such activity is likely to generate higher investment returns. Thus, in market equilibrium, efficient information-gathering activity should be fruitful. Moreover, it would not be surprising to find that the degree of efficiency differs across various markets. For example, emerging markets that are less intensively analyzed than U.S. markets or in which accounting disclosure requirements are less rigorous may be less efficient than U.S. markets. Small stocks that receive relatively little coverage by Wall Street analysts may be less efficiently priced than large ones. Still, while we would not go so far as to say that you absolutely cannot come up with new information, it makes sense to consider and respect your competition.

Rewards for Incremental Performance

Consider an investment management fund currently managing a $5 billion portfolio. Suppose that the fund manager can devise a research program that could increase the portfolio rate of return by one-tenth of 1% per year, a seemingly modest amount. This program would increase the dollar return to the portfolio by $5billion 3.001, or $5 million. Therefore, the fund would be willing to spend up to $5 million per year on research to increase stock returns by a mere tenth of 1% per year. With such large rewards for such small increases in investment performance, it should not be surprising that professional portfolio managers are willing to spend large sums on industry analysts, computer support, and research effort, and therefore that price changes are, generally speaking, difficult to predict. With so many well-backed analysts willing to spend considerable resources on research, easy pickings in the market are rare. Moreover, the incremental rates of return on research activity may be so small that only managers of the largest portfolios will find them worth pursuing.

Although it may not literally be true that “all” relevant information will be uncovered, it is virtually certain that there are many investigators hot on the trail of most leads that seem likely to improve investment performance. Competition among these many well-backed, 6

Sanford J. Grossman and Joseph E. Stiglitz, “On the Impossibility of Informationally Efficient Markets,” American Economic Review 70 (June 1980).

The most precious commodity on Wall Street is information, and informed players can charge handsomely for providing it. An industry of so-called expert network providers has emerged for selling access to experts with unique insights about a wide variety of firms and industries to investors who need that information to make decisions. These firms have been dubbed matchmakers for the information age. Experts can range from doctors who help predict the release of blockbuster drugs to meteorologists who forecast weather that can affect commodity prices to business executives who can provide specialized insight about companies and industries. But some of those experts have peddled prohibited inside information. In 2011, Winifred Jiau, a consultant for Primary Global Research, was convicted of selling information about Nvidia and Marvell Technologies to the hedge fund SAC Capital Advisors. Several other employees of Primary Global also were charged with insider trading. Expert firms are supposed to provide only public information, along with the expert’s insights and perspective. But the temptation to hire experts with inside information and charge handsomely for access to them is obvious.

The SEC has raised concerns about the boundary between legitimate and illegal services, and several hedge funds in 2011 shut down after raids searched for evidence of such illicit activity. In the wake of increased scrutiny, compliance efforts of both buyers and sellers of expert information have mushroomed. The largest network firm is Gerson Lehrman Group with a stable of 300,000 experts. It now maintains records down to the minute of which of its experts talks to whom and the topics they have discussed.7 These records could be turned over to authorities in the event of an insider trading investigation. For their part, some hedge funds have simply ceased working with expert-network firms or have promulgated clearer rules for when their employees may talk with consultants. Even with these safeguards, however, there remains room for trouble. For example, an investor may meet an expert through a legitimate network and then the two may establish a consulting relationship on their own. This legal matchmaking becomes the precursor to the illegal selling of insider tips. Where there is a will to cheat, there usually will be a way.

highly paid, aggressive analysts ensures that, as a general rule, stock prices ought to reflect available information regarding their proper levels. Information is often said to be the most precious commodity on Wall Street, and the competition for it is intense. Sometimes the quest for a competitive advantage can tip over into a search for illegal inside information. In 2011, Raj Rajaratnam, the head of the Galleon Group hedge fund which once managed $6.5 billion, was convicted on insider trading charges for soliciting tips from a network of corporate insiders and traders. Rajaratnam’s was only one of several major insider trading cases working their way through the courts in 2011. While Galleon’s practices were egregious, drawing a clear line separating legitimate and prohibited sources of information often can be difficult. For example, a large industry of expert network firms has emerged in the last decade to connect (for a fee) investors to industry experts who can provide unique perspective on a company. As the nearby box discusses, this sort of arrangement can easily cross the line into insider trading.

Versions of the Efficient Market Hypothesis It is common to distinguish among three versions of the EMH: the weak, semistrong, and strong forms of the hypothesis. These versions differ by their notions of what is meant by the term “all available information.” The weak-form hypothesis asserts that stock prices already reflect all information that can be derived by examining market trading data such as the history of past prices, trading volume, or short interest. This version of the hypothesis implies that trend analysis is fruitless. Past stock price data are publicly available and virtually costless to obtain. The weak-form hypothesis holds that if such data ever conveyed reliable signals about future performance, all investors already would have learned to exploit the signals. Ultimately, 7

“Expert Networks Are the Matchmakers for the Information Age,” The Economist, June 16, 2011.

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the signals lose their value as they become widely known because a buy signal, for instance, would result in an immediate price increase. The semistrong-form hypothesis states that all publicly available information regarding the prospects of a firm must be reflected already in the stock price. Such information includes, in addition to past prices, fundamental data on the firm’s product line, quality of management, balance sheet composition, patents held, earning forecasts, and accounting practices