Handbook of the Equity Risk Premium

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HANDBOOK OF THE EQUITY RISK PREMIUM

HANDBOOKS IN FINANCE Series Editor

WILLIAM T. ZIEMBA Advisory Editors

KENNETH J. ARROW GEORGE C. CONSTANTINIDES B. ESPEN ECKBO HARRY M. MARKOWITZ ROBERT C. MERTON STEWART C. MYERS PAUL A. SAMUELSON WILLIAM F. SHARPE

amsterdam • boston • heidelberg • london new york • oxford • paris • san diego san francisco • singapore • sydney • tokyo

HANDBOOK OF THE EQUITY RISK PREMIUM By

Rajnish Mehra

amsterdam • boston • heidelberg • london new york • oxford • paris • san diego san francisco • singapore • sydney • tokyo

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK First edition 2008 c 2008 Elsevier B.V. All rights reserved Copyright No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-444-50899-7 For information on all Elsevier publications visit our website at books.elsevier.com Printed and bound in the USA 08 09 10 11 10 9 8 7 6 5 4 3 2 1

Dedicated to my parents to Jyoti and Ravi to Neeru and to Chaitanya

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Contents

List of Contributors

xvii

Preface

xix

Introduction to the Series

xxiii

1

The Equity Premium: ABCs

1

Rajnish Mehra (UCSB) and Edward C. Prescott (Arizona State)

2

1. Introduction 1.1. An Important Preliminary Issue 1.2. Data Sources 1.3. Estimates of the Equity Premium 1.4. Variation in the Equity Premium Over Time 2. Is the Equity Premium Due to a Premium for Bearing Non-Diversifiable Risk? 2.1. Standard Preferences References Appendix A

2 2 3 6 9 11 14 25 29

Appendix B

29

Appendix C

35

Appendix D

35

Risk-Based Explanations of the Equity Premium

37

John B. Donaldson (Columbia) and Rajnish Mehra (UCSB)

Introduction 1. Alternative Preference Structures 1.1. Preliminaries 1.2. Coincidence of Risk and Time Preferences in CRRA utility 1.3. Separating Risk and Time Preferences: Epstein–Zin and others 1.4. Variation in the CRRA and EIS

39 41 41 44 46 52 vii

viii

Contents

2. 3. 4. 5. 6.

3

1.5. Habit Formation 1.6. Behavioral Models 1.7. Beyond One Good and a Representative Agent Production Economies Disaster Events and Survivorship Bias Market Incompleteness and Trading Frictions 4.1. Restricted Participation Model Uncertainty Concluding Comments References

Non-Risk-based Explanations of the Equity Premium

55 61 71 78 81 86 86 91 93 94

101

Rajnish Mehra (UCSB) and Edward C. Prescott (Arizona State)

4

Introduction 1. The Inappropriateness of Using T-Bills as a Proxy for the Intertemporal Marginal Rate of Substitution of Consumption 1.1. Liquidity 1.2. Transaction Balances 2. The Effect of Government Regulations and Rules 3. Taxes 4. Borrowing Constraints 5. The Impact of Agent Heterogeneity and Intermediation Costs 6. Concluding Comments References

102 102 104 104 106 107 110 113 114 114

Equity Premia with Benchmark Levels of Consumption: Closed-Form Results

117

Andrew B. Abel (Wharton)

1. Preferences 2. The Canonical Asset 2.1. The Price of the Canonical Asset 2.2. The Rate of Return on the Canonical Asset 3. Risk, Term, and Equity Premia 4. Log-Normality 5. Risk, Term, and Equity Premia Under Log-Normality with Consumption Externalities and Without Habit Formation 6. Linear Approximations to Risk, Term, and Equity Premia 7. Second Moments 7.1. Linear Approximations to Second Moments 8. Correlation of Dividend-Price Ratio and the Rate of Return on Stock 8.1. Correlation of Dividend-Price Ratio and the Excess Rate of Return on Stock

120 126 127 129 131 134 135 137 138 140 142 144

ix

Contents

9. Special Cases 9.1. Rational Expectations 9.2. Distorted Beliefs 10. Accuracy of Approximations 11. Summary References

5

146 146 151 153 156 156

Discussion: Francisco Gomes (LBS)

158

1. Introduction 2. Preferences with Benchmark Levels of Consumption 3. Changing the “Benchmark Level” of the Explanation 3.1. Aggregate Moments 3.2. Micro-Economic Implications 3.3. Micro-Economic Foundations and Aggregation 4. Leverage, Correlation between Dividends and Consumption, and distorted Beliefs 4.1. Levered Equity Claims and Correlation Between Dividends and Consumption 4.2. Non-Rational Expectations 5. Final Remarks References

158 159 161 161 162 163

Long-Run Risks and Risk Compensation in Equity Markets

163 163 164 165 165

167

Ravi Bansal (Duke)

1. Introduction 2. Long-Run Risks Model 2.1. Preferences and the Environment 2.2. Long-Run Growth Rate Risks 2.3. Long-Run Growth and Uncertainty Risks 2.4. Data and Model Implications 3. Cross-Sectional Implications 3.1. Value, Momentum, Size, and the Cross-Sectional Puzzle 4. Conclusion References

168 170 170 171 174 176 185 185 191 191

Discussion: John C. Heaton (Chicago)

194

1. 2. 3. 4. 5.

194 194 195 197 198 198

Summary A Low-Frequency Component in Consumption? Preferences Returns and Long-Run Cash Flows Conclusion References

x

Contents

6

The Loss Aversion/Narrow Framing Approach to the Equity Premium Puzzle 199 Nicholas Barberis (Yale) and Ming Huang (Cornell)

7

1. Introduction 2. Loss Aversion and Narrow Framing 3. The Equity Premium 3.1. Modeling Loss Aversion and Narrow Framing 3.2. Quantitative Implications 3.3. Attitudes to Large Monetary Gambles 3.4. Attitudes to Small Monetary Gambles 3.5. The Importance of Narrow Framing 4. Other Applications 5. Further Extensions 5.1. Dynamic Aspects of Loss Aversion 5.2. Other Forms of Narrow Framing 6. Conclusion and Future Directions References

201 203 207 207 212 216 218 220 224 225 225 226 227 228

Discussion: Xavier Gabaix (New York)

230

1. Work Out More Systematically the Preferences of PT vs. EU Investors—The “Equity Protection Puzzle” 2. Make Quantitative Predictions, Particularly About Equilibrium Market Phenomena, Rather than Just about Individual Trading Behavior 3. Do a Version of the Model in Continuous Time References

232 233 234

Discussion: Ravi Jagannathan (Northwestern)

235

Financial Markets and the Real Economy

237

230

John H. Cochrane (Chicago)

1. Introduction 1.1. Risk Premia 1.2. Who Cares? 1.3. The Mimicking Portfolio Theorem and the Division of Labor 2. Facts: Time Variation and Business Cycle Correlation of Expected Returns 2.1. Variation over Time 2.2. Variation Across Assets 2.3. Return Forecasts—Variation over Time 2.4. The Cross Section of Returns—Variation Across Assets 3. Equity Premium 3.1. Mehra and Prescott and the Puzzle 3.2. The Future of the Equity Premium

239 239 242 243 244 244 245 246 251 257 261 266

xi

Contents

4. Consumption Models 4.1. Hansen and Singleton; Power Utility 4.2. New Utility Functions 4.3. Empirics with New Utility Functions 4.4. Consumption and Factor Models 5. Production, Investment, and General Equilibrium 5.1. “Production-Based Asset Pricing” 5.2. General Equilibrium 6. Labor Income and Idiosyncratic Risk 6.1. Labor and Outside Income 6.2. Idiosyncratic Risk, Stockholding, and Micro Data 7. Challenges for the Future References Appendix

267 267 270 273 286 290 290 294 302 302 307 314 314 322

Discussion: Lars Peter Hansen (Chicago)

326

References

8

Understanding the Equity Risk Premium Puzzle

329

331

George M. Constantinides (Chicago)

1. 2. 3. 4. 5.

Introduction Habit Persistence Limited Stock Market Participation and Per Capita Consumption Incomplete Markets and Idiosyncratic Income Shocks Concluding Remarks References

332 337 345 349 355 356

Discussion: Hanno Lustig (UCLA)

360

1. Introduction 1.1. Environment 1.2. Preferences and Endowments 2. Complete Markets 2.1. Equilibrium 2.2. Equity Premium Puzzle 3. Missing Markets 3.1. Equilibrium 3.2. Mankiw’s Recipe for Generating Risk Premia 3.3. Constantinides and Duffie 3.4. Independence of Idiosyncratic Shocks from Aggregate Conditions 4. Missing Markets and State-Dependent Solvency Constraints 4.1. Incomplete Markets 4.2. Complete Markets

360 361 361 362 363 364 364 365 365 366 368 370 370 371

xii

Contents

9

5. Conclusion References A. Second-Order Taylor Expansion B. Constantinides and Duffie

372 372 373 374

Cash Flow Risk, Discounting Risk, and the Equity Premium Puzzle

377

Gurdip Bakshi (Maryland) and Zhiwu Chen (Yale)

1. Introduction 2. Economic Determinants of Equity Premium 2.1. Cash Flow Process 2.2. The Discounting Process 2.3. Dynamics of the Market Portfolio 2.4. Dynamics of the Equity Premium 3. Time-Series Data on S&P500 EPS, EPS Growth, and the Interest Rate 4. Implications of the Model for Equity Premium 4.1. How Large Is the Interest-Rate Risk Premium? 4.2. Maximum-Likelihood Estimation of the (Physical) Gt Process 4.3. Compensation for Cash Flow Risk and the Equity Premium 5. Concluding Remarks and Extensions Appendix References

379 381 381 382 383 385 387 389 389 391 392 396 398 400

Discussion: Vito D. Gala (LBS)

403

1. Discussion 1.1. Calibration and Estimation 1.2. Where Is the Equity Premium Puzzle? References

403 404 405 407

Discussion: Lior Menzly (Proxima)

409

1. Introduction 2. The Model 2.1. Pricing Kernel 2.2. Cash Flow Process 2.3. The Model—Solutions 3. Calibration 3.1. Calibrating the Model 3.2. Estimation Results 4. Two-Stage Procedure—An Empirical Concern 5. Conclusion References

410 410 410 411 411 412 412 412 412 414 414

xiii

Contents

10 Distribution Risk and Equity Returns

415

Jean-Pierre Danthine (Lausanne), John B. Donaldson (Columbia), and Paolo Siconolfi (Columbia)

1. Introduction 2. The Business Cycle and the Labor Market 2.1. The Stylized Facts of the Business Cycle 2.2. The Labor Market 3. The Model Economy 3.1. Workers 3.2. Shareholders 3.3. The Firm 3.4. Equilibrium 3.5. Numerical Procedures and Calibration 4. An Economy with Distribution Risk Only 5. Adding Aggregate Uncertainty 6. Comparative Dynamics and Welfare Assessment 6.1. Changes in the Correlation of Productivity and Distribution Shocks 6.2. Changes in Risk Aversion and the Conditional Mean Distribution Shock 6.3. Other Comparative Dynamic Tests 6.4. Welfare Considerations 6.5. Explaining the Market Value to National Income Ratio 7. Technology-Driven Variations in Factor Shares 8. Robustness 9. An Alternative Interpretation of the Sharing Mechanism 10. Related Literature 11. Concluding Comments References

Discussion: Urban J. Jermann (Wharton) References

11 The Worldwide Equity Premium: A Smaller Puzzle

417 418 418 421 423 423 424 425 427 429 430 432 436 437 438 440 441 442 443 446 448 452 459 460

463 466

467

Elroy Dimson (LBS), Paul Marsh (LBS), and Mike Stauhton (LBS)

1. Introduction 2. Prior Estimates of the Equity Premium 2.1. Expert Opinion 3. Long-Run International Data 3.1. The DMS Global Database: Composition and Start Date 3.2. The DMS Global Database: General Methodology and Guiding Principles 4. Long-Run Historical Rates of Return 4.1. Extremes of History 4.2. The Long-Run Perspective

469 471 472 474 475 477 479 480 483

xiv

Contents

5. New Global Evidence on the Equity Premium 5.1. The Equity Premium Around the World 5.2. A Smaller Risk Premium 5.3. Survivorship of Markets 5.4. Survivorship Bias Is Negligible 6. Decomposing the Historical Equity Premium 6.1. Unanticipated Success 6.2. Decomposition of the Equity Premium 6.3. From the Past to the Future 7. Conclusion References Appendix 1: Decomposition of the Equity Premium Appendix 2: Data Sources for the DMS Database

12 History and the Equity Risk Premium

486 487 489 490 492 493 493 495 497 500 501 505 507

515

William N. Goetzmann (Yale) and Roger G. Ibbotson (Yale)

1. 2. 3. 4. 5. 6.

Introduction Historical Conception and Measurement of the Equity Risk Premium Stocks, Bonds, Bills, and Inflation History as Written by the Winners? The Equity Premium Over the Very Long Term Conclusion References

Discussion: Stephen F. LeRoy (UCSB) References

13 Can Heterogeneity, Undiversified Risk, and Trading Frictions Solve the Equity Premium Puzzle

516 517 521 523 524 527 528

530 534

535

John C. Heaton (Chicago) and Deborah Lucas (Northwestern)

1. Introduction 2. Labor Income as Background Risk 2.1. Calibrating the Income Process 2.2. Adding Trading Frictions 3. Entrepreneurial Income as Background Risk 4. Limited Participation and Limited Diversification 5. Conclusions References

537 539 544 547 552 555 556 556

Discussion: Kjetil Storesletten (U Oslo)

558

1. Introduction 2. Labor Income Risk

558 559

xv

Contents

3. Transaction Costs 4. Concentrating Aggregate Risk on Fewer Hands 4.1. Entrepreneurial Risk 4.2. Limited Participation 5. Conclusion References

14 Asset Prices and Intergenerational Risk Sharing: The Role of Idiosyncratic Earnings Shocks

560 560 560 561 562 563

565

Kjetil Storesletten (U Oslo), Chris Telmer (CMU), and Amir Yaron (Wharton)

1. Introduction 2. An Analytical Example of the Constantinides–Duffie Model 2.1. Calibration of the Constantinides–Duffie Economy 2.2. Model Implications 3. Incorporating the Life Cycle 3.1. Calibration 4. Quantitative Results 4.1. Asset Pricing Implications 4.2. Sensitivity Analysis 5. Conclusions References A. Calibration Appendix B. Asset Pricing

567 569 570 571 573 576 577 580 581 581 584 587 590

Discussion: Darrell Duffie (Stanford)

591

References

Index

592

593



Andrew B. Abel, Department of Finance, 2315 Steinberg Hall-Dietrich Hall, The Wharton School, University of Pennsylvania, 3620 Locust Walk, Philadelphia, PA 19104-6367, USA Gurdip Bakshi, Department of Finance, Smith School of Business, University of Maryland, College Park, MD 20742, USA Ravi Bansal, Fuqua School of Business, Duke University, 1 Towerview Drive, Durham, NC 27708, USA Nicholas Barberis, Yale School of Management, 135 Prospect Street, New Haven CT 06511, USA Zhiwu Chen, Yale School of Management, 135 Prospect Street, New Haven, CT 06520, USA John H. Cochrane, Graduate School of Business, University of Chicago, 5807 S. Woodlawn, Chicago IL 60637, USA George M. Constantinides, Graduate School of Business, The University of Chicago, 5807 South Woodlawn Avenue, Chicago IL 60637, USA Jean-Pierre Danthine, University of Lausanne, Bldg Extranef, Dorigny, CH-1015 Lausanne, Switzerland Elroy Dimson, London Business School, Regents Park, London NW1 4SA, UK John B. Donaldson, Graduate School of Business, Columbia University, 3022 Broadway, New York, NY 10027-6989, USA Darrell Duffie, Graduate School of Business, Stanford University, 518 Memorial Way, Stanford, CA 94305-5015, USA Xavier Gabaix, Department of Finance, Stern School of Business, New York University, 44 West 4th Street, Suite 9-190, New York, NY 10012, USA Vito Gala, London Business School, Regent’s Park, London, NW1 4SA, UK

xvii

xviii


William N. Goetzmann, Yale School of Management, 135 Prospect Street, New Haven, CT 06511-3729, USA Francisco Gomes, Department of Finance, London Business School, Regent’s Park, London NW1 4SA, UK Lars Peter Hansen, Department of Economics, University of Chicago, 1126 East 59th St., Chicago, Illinois. 60637, USA John C. Heaton, The University of Chicago, Graduate School of Business, 5807 South Woodlawn Avenue, Chicago, IL 60637, USA Ming Huang, Johnson Graduate School of Management, Cornell University, 319 Sage Hall, Ithaca, NY 14853-6201, USA Roger G. Ibbotson, Yale School of Management, 135 Prospect Street, New Haven, CT 06511-3729, USA Ravi Jagannathan, Kellogg School of Management, Northwestern University, 2001 Sheridan Rd, Evanston, IL 60208, USA Urban J. Jermann, Department of Finance, The Wharton School of the University of Pennsylvania, 3620 Locust Walk, Philadelphia, PA 19104-6367, USA Stephen F. LeRoy, Department of Economics, University of California, Santa Barbara, CA 93106, USA Deborah Lucas, Kellogg School of Management, 2001 Sheridan Rd, Evanston, IL 60208, USA Hanno Lustig, Department of Economics, University of California at Los Angeles, Bunche Hall 8357, Box 951477, Los Angeles, CA 90095-1477, USA Paul Marsh, London Business School, Regents Park, London NW1 4SA, UK Rajnish Mehra, Department of Economics, 3014 North Hall, University of California, Santa Barbara, CA 93106-9210, USA Lior Menzly, Director of Quantitative Research and Risk Management, Proxima Alfa Investment (USA), 623 Fifth Ave, 14th Floor, New York, NY 10022, USA Edward C. Prescott, Department of Economics, W. P. Carey School of Business, Arizona State University, Tempe, AZ 85287-3806, USA Paolo Siconolfi, Columbia Business School, 3022 Broadway, Uris Hall 820, New York, NY 10027, USA Mike Staunton, London Business School, Regents Park, London NW1 4SA, UK Kjetil Storesletten, Department of Economics, University of Oslo, PO Box 1095 Blindern, N-0317 Oslo, Norway Chris Telmer, Tepper School of Business, Carnegie Mellon University, Posner Hall, Room 350, Tech and Frew Streets, Pittsburgh, PA 15213, USA Amir Yaron, The Wharton School, University of Pennsylvania, 2325 Steinberg Hall-Dietrich Hall, 3620 Locust Walk, Philadelphia, PA 19104-6367, USA

Preface

Historical data provide a wealth of evidence documenting that for more than a century, U.S. stock returns have been considerably higher than the returns for Treasury bills. The average annual real return (that is, the inflation-adjusted return) on the U.S. stock market for the past 110 years has been about 7.9 percent. In the same period, the real return on a relatively riskless security was a paltry 1.0 percent. The difference between these two returns, 6.9 percentage points, is the equity premium. The generally accepted tenet of the neoclassical paradigm has been that the observed differences in the rates of return to financial assets, in particular, the large difference between the average returns on corporate equity and T-bills, is a premium for bearing non-diversifiable aggregate risk. What came as a surprise to many economists and researchers in finance was the conclusion of a research paper that Edward Prescott and I wrote in 1979. We found that stocks and bonds pay off in approximately the same states of nature or economic scenarios and hence, they should command approximately the same rate of return. The historical U.S. equity premium was an order of magnitude greater than could be rationalized in the context of the standard neoclassical paradigm of financial economics. In fact, using standard theory to estimate risk-adjusted returns, we found that stocks on average should command, at most, a 1 percent return premium over bills. Since, for as long as we had reliable data, (about a hundred years), the mean premium on stocks over bills was considerably and consistently higher, we realized that we had a puzzle on our hands. It took us six more years to convince a skeptical profession and for “The Equity Premium: A Puzzle” to be published. I want to emphasize that the equity premium puzzle is a quantitative puzzle; standard theory is consistent with our notion of risk that, on average, stocks should return more than bonds. The puzzle cannot be dismissed lightly because much of our economic intuition is based on the very class of models that fall short so dramatically when confronted with financial data. It underscores the failure of paradigms central to financial and economic modelling to capture the characteristic that appears to make stocks comparatively riskier. Hence, the viability of using this class of models for any quantitative assessment—for instance, to gauge the welfare implications of alternative stabilization policies—is thrown open to question.

xix

xx

Preface

Creative research emerges at the fault lines where theory confronts observations. Over the past two decades, attempts to resolve the puzzle have become a major research impetus in finance and economics. Several generalizations of key features of the Mehra–Prescott (1985) model have been proposed to reconcile observation with theory. Consequently, we have a deeper understanding of the role and importance of the abstractions that contribute to the puzzle. While no single explanation has fully resolved the anomaly, considerable progress has been made and the equity premium is a lesser puzzle today than it was twenty years ago. This Handbook brings together fourteen papers by key researchers that span the spectrum of research efforts to resolve the Equity Premium Puzzle. I designed this volume to be a collection of essays by experts in the field, discussing their own work and contribution to the Equity Premium literature. My motivation is to give the profession a critical look at the subject through the eyes of the researchers that have made fundamental contributions to the field. Each essay is followed by an expert commentary. The expository style is intended to make the material accessible to doctoral students and beginning researchers in the field. In this spirit, individual chapters are self-contained and can be read independently. The three introductory chapters provide a summary and overview of research over the past 20 years. The first chapter explains why the equity premium is a puzzle. Chapter two is an extended survey of the risk-based explanations of the equity premium while chapter three surveys non-risk based explanations. The remaining chapters are arranged alphabetically by author. Each chapter has a descriptive title and an abstract. There is little value added in summarizing them here. This volume has taken the better part of four years to bring to press, as many essays went through two rounds of refereeing. I am grateful to the anonymous referees for their detailed comments, to the authors and discussants for their willingness to incorporate suggestions and for their diligence in adhering to our guidelines in preparing their contributions to this volume. Thanks are due to Francisco Azeredo, Alok Khare and Chaitanya Mehra, who contributed to this volume’s accessibility by reading and critiquing the chapters from a student’s perspective. Most of the papers were presented and discussed at “The Equity Premium Puzzle Conference,” held in October 2005 at the University of California, Santa Barbara, to commemorate the twentieth anniversary of the publication of “The Equity Premium: A Puzzle.” The conference was sponsored by the Laboratory for Aggregate Economics and Finance at UCSB and North Holland. I thank Finn Kydland, the director of LAEF, Scott Bentley, the executive editor and David Clark, the publishing director at North Holland for their generous support. Special thanks are due to Vijaisarath Parthasarathy, at Elsevier, Chennai, where this issue was typeset, for the patience and grace with which he handled the innumerable iterations involved in bringing this volume to press. The equity premium puzzle has its genesis in the research program initiated by Robert Lucas and Edward Prescott at Carnegie Mellon in the mid 70s, a program that transformed dynamic economics. I was fortunate to have learned economics from them, and to have witnessed that paradigmatic shift. I want to take this opportunity to express my gratitude to both of them.

Preface

xxi With Ed Prescott, our association has evolved from his role as mentor to that of co-author and colleague. The famous mathematician, Mark Kac, once made a distinction between two kinds of geniuses: the “ordinary” and the “magicians”. An ordinary genius is a fellow that you and I would be just as good as, if we were only many times better. It is different with magicians. The working of their minds is mysterious and profound and even after we understand what they have done, the creative process by which they have done it, is impossible to emulate. The last 30 odd years of collaborating with Ed Prescott have convinced me that he is, truly, a magician. It was at Carnegie too that I met John Donaldson and George Constantinides; over the years, they have been generous with their time and advice and have served as a sounding board for ideas. Both of them contributed immeasurably to this volume by their meticulous reading of and comments on many of the papers collected here. Finally, I thank my wife, Neeru, for editorial assistance. Her insightful comments and recommendations reflect her passion for words, versatility of expression and command of the language.


Introduction to the Series

Advisory Editors: Kenneth J. Arrow, Stanford University; George C. Constantinides, University of Chicago; B. Espen Eckbo, Dartmouth College; Harry M. Markowitz, University of California, San Diego; Robert C. Merton, Harvard University; Stewart C. Myers, Massachusetts Institute of Technology; Paul A. Samuelson, Massachusetts Institute of Technology; and William F. Sharpe, Stanford University. The Handbooks in Finance are intended to be a definitive source for comprehensive and accessible information. Each volume in the series presents an accurate, self-contained survey of a sub-field of finance, suitable for use by finance and economics professors and lecturers, professional researchers, graduate students and as a teaching supplement. The goal is to have a broad group of outstanding volumes in various areas of finance. William T. Ziemba University of British Columbia

xxiii


CHAPTER 1

The Equity Premium: ABCs Rajnish Mehra University of California, Santa Barbara, and NBER and

Edward C. Prescott Arizona State University and Federal Reserve Bank of Minneapolis

1. Introduction 1.1. An Important Preliminary Issue 1.2. Data Sources 1.3. Estimates of the Equity Premium 1.4. Variation in the Equity Premium over Time 2. Is the Equity Premium due to a Premium for Bearing Non-Diversifiable Risk? 2.1. Standard Preferences References Appendix A Appendix B Appendix C Appendix D

2 2 3 6 9 11 14 25 29 29 35 35

JEL Classification: G10, G12, D9 Keywords: asset pricing, equity risk premium, CAPM, consumption CAPM, risk free rate puzzle

We thank George Constantinides, John Donaldson and Viral Shah for helpful comments and Francisco Azeredo for excellent research assistance. HANDBOOK OF THE EQUITY RISK PREMIUM c 2008 by Rajnish Mehra and Edward C. Prescott. All rights of reproduction in any form reserved. Copyright

1

2

Chapter 1 • The Equity Premium: ABCs

1. INTRODUCTION The year 1978 saw the publication of Robert Lucas’ seminal paper “Asset Prices in an Exchange Economy” in Econometrica. Its publication transformed asset pricing and substantially raised the level of discussion, providing a theoretical construct to study issues that could not be addressed within the dominant paradigm at the time, the Capital Asset Pricing Model.1 A crucial input parameter for using the latter is the equity premium2 (the return earned by a broad market index in excess of that earned by a relatively risk-free security). Lucas’ asset pricing model allowed one to pose questions about the magnitude of the equity premium.3 In our paper, “The Equity Premium: A Puzzle,”4 we decided to address this issue. In this chapter we take a retrospective look at our original paper and show why we concluded that the equity premium is not a premium for bearing non-diversifiable risk.5 We critically evaluate the data sources used to document the puzzle and touch on other issues that may be of interest to the researcher who did not have a ringside seat 20 years ago. We stress that the perspective here captures the spirit of our original paper and not necessarily our current thinking on these issues.6 This and the subsequent two chapters are motivated by the intention to make this volume a self-contained reference for the beginning researcher in the field. The chapters that follow address the research efforts that have preoccupied the profession in an effort to explain the equity premium. This chapter is organized into two parts. Part 1 documents the historical equity premium in the United States and in selected countries with significant capital markets (in terms of market value) and comments on data sources. Part 2 examines the question, “Is the equity premium a premium for bearing non-diversifiable risk?”

1.1. An Important Preliminary Issue Any discussion of the equity premium raises the question of whether arithmetic or geometric returns should be used for summarizing historical return data. In Mehra and Prescott (1985), we used arithmetic averages. If returns are uncorrelated over time, the appropriate statistic is the arithmetic average because the expected future value of a $1 investment is obtained by compounding the mean returns. Thus, this is the appropriate statistic to report if one is interested in the mean terminal value of the investment.7 1 See

Mossin (1966) for a lucid articulation. was generally assumed to be in the 6–8 percent range. 3 To put this advance in perspective, an equivalent contribution in physics would be to come up with a model that enabled one to address the question of whether the value of Newton’s gravitational constant G (6.672 × 10−11 Nm2 /kg2 ) is reasonable, given other cosmological observations. Why is the value of G what we observe? 4 See Mehra and Prescott (1985). 5 This chapter draws on material in Mehra and Prescott (2003). All of the acknowledgements in that chapter continue to apply. 6 For an elaboration, see McGrattan and Prescott (2003, 2005) and Mehra and Prescott (2008) in this volume. 7 We present a simple proof in Appendix A. 2 This

Rajnish Mehra and Edward C. Prescott

3

The arithmetic average return exceeds the geometric average return. If returns are log-normally distributed, the difference between the two is one-half the variance of the returns. Since the annual standard deviation of the equity returns is about 20 percent, there is a difference of about 2 percent between the two measures. Using geometric averages significantly underestimates the expected future value of an investment. In this chapter, as in our 1985 paper, we report arithmetic averages. In instances where we cite the results of research when arithmetic averages are not available, we clearly indicate this.8

1.2. Data Sources A crucial consideration in a discussion of the historical equity premium has to do with the reliability of early data sources. The data we used in documenting the historical equity premium in the United States can be subdivided into three distinct subperiods, 1802–1871, 1871–1926, and 1926–present, with wide variation in the quality of the data over each subperiod. Data on stock prices for the 19th century is patchy, often necessarily introducing an element of arbitrariness to compensate for its incompleteness.

1.2.1. Subperiod 1802–1871 Equity Return Data The equity return data prior to 1871 is not particularly reliable. To the best of our knowledge, the stock return data used by all researchers for the period 1802–1871 is due to Schwert (1990), who gives an excellent account of the construction and composition of early stock market indexes. Schwert (1990) constructs a “spliced” index for the period 1802–1987; his index for the period 1802–1862 is based on the work of Smith and Cole (1935), who constructed a number of early stock indexes. For the period 1802–1820, their index was constructed from an equally weighted portfolio of seven bank stocks, and another index for 1815–1845 was composed of six bank stocks and one insurance stock. For the period 1834–1862, the index consisted of an equally weighted portfolio of (at most) 27 railroad stocks.9 They used one price quote, per stock, per month, from local newspapers. The prices used were the average of the bid and ask prices, rather than transaction prices, and the computation of returns ignores dividends. For the period 1863–1871, Schwert uses data from Macaulay (1938), who constructed a valueweighted index using a portfolio of 25 Northeast and Mid-Atlantic railroad stocks;10 this index also excludes dividends. Needless to say, it is difficult to assess how well this data proxies the “market,” since undoubtedly there were other industry sectors that were not reflected in the index. 8 In

this case an approximate estimate of the arithmetic average return can be obtained by adding one-half the variance of the returns to the geometric average. 9 “They chose stocks in hindsight . . . the sample selection bias caused by including only stocks that survived and were actively quoted for the whole period is obvious” (Schwert (1990)). 10 “It is unclear what sources Macaulay used to collect individual stock prices but he included all railroads with actively traded stocks.” Ibid.

4


Return on a Risk-Free Security Since there were no Treasury bills extant at the time, researchers have used the data set constructed by Siegel (2002) for this period, using highly rated securities with an adjustment for the default premium. Interestingly, based on this data set, the equity premium for the period 1802–1862 was zero. We conjecture that this may be due to the fact that since most financing in the first half of the 19th century was done through debt, the distinction between debt and equity securities was not very clear-cut.11

1.2.2. Subperiod 1871–1926 Equity Return Data Shiller (1989) is the definitive source for the equity return data for this period. His data is based on the work of Cowles (1939), which covers the period 1871–1938. Cowles used a value-weighted portfolio for his index, which consisted of 12 stocks12 in 1871 and ended with 351 in 1938. He included all stocks listed on the New York Stock Exchange, whose prices were reported in the Commercial and Financial Chronicle. From 1918 onward he used the Standard and Poor’s (S&P) industrial portfolios. Cowles reported dividends, so that, unlike the earlier indexes for the period 1802–1871, a total return calculation was possible. Return on a Risk-Free Security There is no definitive source for the short-term risk-free rate in the period before 1920, when Treasury certificates were first issued. In our 1985 study, we used short-term commercial paper as a proxy for a riskless short-term security prior to 1920 and Treasury certificates from 1920–1930.13 Our data prior to 1920 was taken from Homer (1963). Most researchers have used either our data set or Siegel’s.

1.2.3. Subperiod 1926–Present Equity Return Data This period is the “Golden Age” with regard to accurate financial data. The NYSE database at the Center for Research in Security Prices (CRSP) was initiated in 1926 and provides researchers with high-quality equity return data. The Ibbotson Associates Yearbooks14 are also a very useful compendium of post-1926 financial data.

11 The first actively traded stock was floated in the U.S. in 1791 and by 1801 there were over 300 corporations, although less than 10 were actively traded (Siegel (2002)). 12 It was only from February 16, 1885, that Dow Jones began reporting an index, initially composed of 12 stocks. The S&P index dates back to 1928, though for the period 1928–1957 it consisted of 90 stocks. The S&P 500 debuted in March 1957. 13 See Mehra and Prescott (2008) in this volume for a discussion on the choice of a proxy for the risk-free asset. 14 Ibbotson Associates, 2006. Stocks, bonds, bills and inflation. 2005 Yearbook. Ibbotson Associates, Chicago.

5


TABLE 1 U.S. Annual Real Growth Rate of Per Capita Consumption of Non-durables and Services 1889–1978

1889–2004

1889–1929

1930–1978

1930–2004

Mean

0.018

0.018

0.021

0.016

0.018

Std. Dev.

0.036

0.032

0.044

0.028

0.022

–0.140

–0.135

–0.463

0.520

0.450

Serial Correlation

Return on a Risk-Free Security Since the advent of Treasury bills in 1931, short-maturity bills have almost universally been used as proxy for a “real” risk-free security15 since the innovation in inflation is orthogonal to the path of real GNP growth.16 With the debut of Treasury Inflation Protected Securities (TIPS) on January 29, 1997, the return on these securities is the real risk-free rate.17

1.2.4. Consumption Data In our study, we used the Kuznets–Kendrik–NIA per capita real consumption of nondurables and services for the period 1889–1978. Our data source was Grossman and Shiller (1981). An updated version of this series is available in Shiller (1989).18 The initial series (the flow of perishable and semi-durable goods to consumers) for the period 1889–1919 was constructed by William Shaw.19 Simon Kuznets (1938, 1946) modified Shaw’s measure by incorporating transportation and distribution costs, and created a series (the flow of perishable and semi-durable goods to consumers) for the period 1919–1929. The final version of these series is available in an unpublished mimeograph underlying Tables R-27 and R-28 in Kuznets (1961). Kendrick (1961) made further adjustments to these series in order to make them comparable to the Department of Commerce’s personal consumption expenditure series. Kendrick’s adjustments are available in Tables A-IIa and A-IIb in Kendrick (1961). This is the data source that Grossman and Shiller (1981) used in constructing the1889–1929 subset of their series on per capita real consumption of non-durables and services. The post-1929 data is from the National Income and Product Accounts of the United States. Table 1 details the statistics on the growth rate of real per capita consumption of non-durables and services, while Figure 1 is a time-series plot. We used the statistics in the 1889–1978 column in our original study. 15 Mehra

and Prescott. Ibid. (1980) documents that in postwar data the innovation in inflation had a standard deviation of one-half of one percent. 17 Mehra and Prescott. Ibid. 18 Further updates are available on Robert Shiller’s website: www.econ.yale.edu/∼shiller. 19 See Shaw (1947). 16 Litterman

6

Chapter 1 • The Equity Premium: ABCs 0.15

0.10

0.05

0.00

20.05

FIGURE 1

2001

1997

1993

1989

1985

1981

1977

1973

1969

1965

1961

1957

1953

1949

1945

1941

1937

1933

1929

1925

1921

1917

1913

1909

1905

1901

1897

1889

20.15

1893

20.10

U.S. annual real growth rate of per capita consumption of non-durables and services

1889–2004. Source: Mehra and Prescott (1985), updated by the authors.

While the serial correlation of consumption growth for the entire sample is negative, Azeredo (2007) points out that for more than the last 70 years it has been positive. In addition, we note that the standard deviation has declined.20 We discuss the implications for the equity premium in Section 2.1.2 and further in Appendix B.

1.3. Estimates of the Equity Premium Historical data provides us with a wealth of evidence documenting that for over a century, stock returns have been considerably higher than those for Treasury bills. This is illustrated in Table 2, which reports the unconditional estimates21 for the U.S. equity premium based on the various data sets used in the literature, going back to 1802. The average annual real return (the inflation adjusted return) on the U.S. stock market over the last 116 years has been about 7.67 percent. Over the same period, the return on a relatively riskless security was a paltry 1.31 percent. The difference between these two returns, the “equity premium,” was 6.36 percent. 20 Christina Romer (1999) points out that the larger pre-1929 estimates may be an artifact of the early methodology rather than due to a change in the underlying stochastic process. 21 To obtain unconditional estimates, we use the entire data set to form our estimate. The Mehra–Prescott data set spans the longest time period for which both consumption and stock return data is available; the former is necessary to test the implications of consumption-based asset pricing models.

7


TABLE 2 U.S. Equity Premium Using Different Data Sets Real return on a market index (%)

Real return on a relatively riskless security (%)

Equity premium (%)

Data set

Mean

Mean

Mean

1802–2004 (Siegel)

8.38

3.02

5.36

1871–2005 (Shiller)

8.32

2.68

5.64

1889–2005 (Mehra–Prescott)

7.67

1.31

6.36

1926–2004 (Ibbotson)

9.27

0.64

8.63

TABLE 3 Equity Premium for Selected Countries Mean real return

Country

Period

Market index (%)

Relatively riskless security (%)

Equity premium (%)

United Kingdom

1900–2005

7.4

1.3

6.1

Japan

1900–2005

9.3

−0.5

9.8

Germany

1900–2005

8.2

−0.9

9.1

France

1900–2005

6.1

−3.2

9.3

Sweden

1900–2005

10.1

2.1

8.0

Australia

1900–2005

9.2

0.7

8.5

India

1991–2004

12.6

1.3

11.3

Source: Dimson et al. (2002) and Mehra (2007) for India.

Furthermore, this pattern of excess returns to equity holdings is not unique to the U.S. but is observed in every country with a significant capital market. The U.S. together with the U.K., Japan, Germany, and France accounts for more than 85 percent of the capitalized global equity value. The annual return on the British stock market was 7.4 percent over the last 106 years, an impressive 6.1 percent premium over the average bond return of 1.3 percent. Similar statistical differentials are documented for France, Germany, and Japan. Table 3 documents the equity premium for these countries.

8


The dramatic investment implications of this differential rate of return can be seen in Table 4, which maps the capital appreciation of $1 invested in different assets from 1802 to 2004 and from 1926 to 2004. One dollar invested in a diversified stock index yields an ending wealth of $655,348 versus a value of $293, in real terms, for $1 invested in a portfolio of T-bills for the period 1802–2004. The corresponding values for the 78-year period, 1926–2004, are $238.30 and $1.54. It is assumed that all payments to the underlying asset, such as dividend payments to stock and interest payments to bonds, are reinvested and that no taxes are paid. This long-term perspective underscores the remarkable wealth-building potential of the equity premium. It should come as no surprise, therefore, that the equity premium is of central importance in portfolio allocation decisions and estimates of the cost of capital and is front and center in the current debate about the advantages of investing Social Security Trust funds in the stock market. In Table 5 we document the premium for some interesting historical subperiods: 1889–1933, when the United States was on a gold standard; 1934–2005, when it was off the gold standard; and 1946–2005, the postwar period. Table 6 presents 30-year moving averages, similar to those reported by the U.S. meteorological service to document “normal” temperature.

TABLE 4 Terminal Value of $1 Invested in Stocks and Bonds Stocks

T-bills

Investment period

Real

Nominal

Real

Nominal

1802–2004

$655,348.00

$10,350,077.00

$293.00

$4,614.00

1926–2004

$238.30

$2,533.43

$1.54

$17.87

Source: Ibbotson (2006) and Siegel (2002).

TABLE 5 Equity Premium in Different Subperiods Real return on a market index (%)


Equity premium (%)

Time period

Mean

Mean

Mean

1889–1933

7.01

3.39

3.62

1934–2005

8.08

0.01

8.07

1946–2005

8.19

0.71

7.48

Source: Mehra and Prescott (1985). Updated by the authors.

9


TABLE 6 Equity Premium 30-Year Moving Averages Real return on a market index (%)


Equity premium (%)

Time period

Mean

Mean

Mean

1900–1950

7.45

2.95

4.50

1951–2005

8.53

1.11

7.42

Source: Mehra and Prescott (1985). Updated by the authors.

Although the premium has been increasing over time, this is primarily due to the diminishing return on the riskless asset, rather than a dramatic increase in the return on equity, which has been relatively constant. The low premium in the 19th century is largely due to the fact that the equity premium for the period 1802–1861 was zero.22 If we exclude this period, we find that difference in the premium in the second half of the 19th century relative to average values in the 20th century is less striking. We see a dramatic change in the equity premium in the post-1933 period—the premium rose from 3.62 percent to 8.07 percent, an increase of more than 125 percent. Since 1933 marked the end of the period when the U.S. was on the gold standard, this break can be seen as the change in the equity premium after the implementation of the new policy.

1.4. Variation in the Equity Premium Over Time The equity premium has varied considerably over time, as illustrated in Figures 2 and 3. Furthermore, the variation depends on the time horizon over which it is measured. There have even been periods when it has been negative. The low-frequency variation has been countercyclical. This is shown in Figure 4, where we have plotted the stock market value as a share of national income23 and the mean equity premium averaged over certain time periods. We have divided the time period from 1929 to 2005 into subperiods where the ratio market value of equity to national income (MV/NI) was greater than and when it was less than the mean value24 over the sample period. Historically, as the figure illustrates, subsequent to periods when this ratio was high, the realized equity premium was low. A similar result holds when 22 See

the earlier discussion on data. Mehra (1998) it is argued that the variation in this ratio is difficult to rationalize in the standard neoclassical framework since, over the same period, after-tax cash flows to equity as a share of national income are fairly constant. Here we do not address this issue and simply utilize the fact that this ratio has varied considerably over time. 24 Mean MV/NI for the period 1929–2005 was 0.91. 23 In

FIGURE 3 6

4

2

0 1989

1986

1983

1980

1977

1974

1971

1968

1965

1962

1959

1956

1953

1950

1947

1944

1941

1938

1935

1932

1929

Equity risk premium over the 20-year period 1926–2004. (Source: Ibbotson (2006))

2003

2004

2001

8 1998

10

2001

12 1995

14

1999

16 1992

18

1997

Realized equity risk premium per year: 1926–2004.

1995

1993

1991

1989

1987

1985

1983

1981

1979

1977

1975

1973

1971

1969

1967

1965

1963

1961

1959

1957

1955

1953

FIGURE 2

1951

1949

1926

260

1947

1945

Average Equity Risk Premium (Percent)

Equity Risk Premium (Percent)

10 Chapter 1 • The Equity Premium: ABCs

60

40

20

0

220

240

11

Rajnish Mehra and Edward C. Prescott 16.00

2.50

2.00 Average Equity Premium

12.00 10.00

1.50

8.00 1.00

6.00 4.00

0.50

Ratio of Market Value to National Income

14.00

2004

2001

1998

1995

1992

1989

1986

1983

1980

1977

1974

1971

1968

1965

1962

1959

1956

1953

1950

1947

1944

1941

1938

1935

1932

0.00

1929

2.00 0.00

FIGURE 4 Market value to national income ratio and average equity premium (average of subperiods when the MV/NI ratio is > or < avg. MV/NI ratio).

stock valuations are low relative to national income. In this case the subsequent equity premium is high. Since after-tax corporate profits as a share of national income are fairly constant over time, this translates into the observation that the realized equity premium was low subsequent to periods when the price/earnings ratio is high, and vice versa. This is the basis for the returns predictability literature in finance. In Figure 5 we have plotted stock market value as a share of national income and the subsequent three-year mean equity premium. This provides further conformation that, historically, periods of relatively high market valuation have been followed by periods when the equity premium was relatively low.

2. IS THE EQUITY PREMIUM DUE TO A PREMIUM FOR BEARING NON-DIVERSIFIABLE RISK? Why have stocks been such an attractive investment relative to bonds? Why has the rate of return on stocks been higher than that on relatively risk-free assets? One intuitive answer is that since stocks are “riskier” than bonds, investors require a larger premium for bearing this additional risk; and indeed, the standard deviation of the returns to stocks (about 20 percent per annum historically) is larger than that of the returns to T-bills (about 4 percent per annum), so, obviously they are considerably more risky than bills! But are they?

12

Chapter 1 • The Equity Premium: ABCs 16.00

2.50

2.00 Average Equity Premium

12.00 10.00

1.50

8.00 1.00

6.00 4.00

0.50

Ratio of Market Value to National Income

14.00

2.00

2004

2001

1998

1995

1992

1989

1986

1983

1980

1977

1974

1971

1968

1965

1962

1959

1956

1953

1950

1947

1944

1941

1938

1935

1932

0.00 1929

0.00

FIGURE 5

Market value to national income ratio and average 3-year ahead equity premium (average of subperiods when the MV/NI ratio is > or < avg. MV/NI ratio).

Figures 6 and 7 illustrate the variability of the annual real rate of return on the S&P 500 index and a relatively risk-free security over the period 1889–2005.25 To enhance and deepen our understanding of the risk-return trade-off in the pricing of financial assets, we take a detour into modern asset pricing theory and look at why different assets yield different rates of return. The deus ex machina of this theory is that assets are priced such that, ex-ante, the loss in marginal utility incurred by sacrificing current consumption and buying an asset at a certain price is equal to the expected gain in marginal utility, contingent on the anticipated increase in consumption when the asset pays off in the future. The operative emphasis here is the incremental loss or gain of utility of consumption and should be differentiated from incremental consumption. This is because the same amount of consumption may result in different degrees of well-being at different times. As a consequence, assets that pay off when times are good and consumption levels are high—when the marginal utility of consumption is low—are less desirable than those that pay off an equivalent amount when times are bad and additional consumption is more highly valued. Hence, consumption in period t has a different price if times are good than if times are bad. Let us illustrate this principle in the context of the standard, popular paradigm, the Capital Asset Pricing Model (CAPM). The model postulates a linear relationship between an asset’s “beta,” a measure of systematic risk, and its expected return. Thus, high-beta stocks yield a high expected rate of return. That is because in the CAPM, 25 The

index did not consist of 500 stocks for the entire period.

13

Rajnish Mehra and Edward C. Prescott 60

40

20

0

220

1965

1969

1973

1977

1981

1985

1989

1993

1997

2001

2005

1969

1973

1977

1981

1985

1989

1993

1997

2001

2005

1961

1957

1953

1949

1945

1941

1937

1933

1929

1925

1921

1917

1913

1909

1905

1965

FIGURE 6

1901

1897

1893

260

1889

240

Real annual return on S&P 500 Index (%) 1889–2005.

Source: Mehra and Prescott (1985). Data updated by the authors.

25 20 15 10 5 0 25 210 215

FIGURE 7

Real annual return on T-bills (%) 1889–2005.

Source: Mehra and Prescott (1985). Data updated by the authors.

1961

1957

1953

1949

1945

1941

1937

1933

1929

1925

1921

1917

1913

1909

1905

1901

1897

1893

225

1889

220

14


good times and bad times are captured by the return on the market. The performance of the market, as captured by a broad-based index, acts as a surrogate indicator for the relevant state of the economy. A high-beta security tends to pay off more when the market return is high—when times are good and consumption is plentiful; it provides less incremental utility than a security that pays off when consumption is low, is less valuable, and consequently sells for less. Thus, higher-beta assets that pay off in states of low marginal utility will sell for a lower price than similar assets that pay off in states of high marginal utility. Since rates of return are inversely proportional to asset prices, the lower beta assets will, on average, give a lower rate of return than the former. Another perspective on asset pricing emphasizes that economic agents prefer to smooth patterns of consumption over time. Assets that pay off a larger amount at times when consumption is already high “destabilize” these patterns of consumption, whereas assets that pay off when consumption levels are low “smooth” out consumption. Naturally, the latter are more valuable and thus require a lower rate of return to induce investors to hold these assets. (Insurance policies are a classic example of assets that smooth consumption. Individuals willingly purchase and hold them, despite their very low rates of return.) To return to the original question: are stocks that much riskier than T-bills so as to justify a 7-percentage differential in their rates of return? What came as a surprise to many economists and researchers in finance was the conclusion of our paper, written in 1979. Stocks and bonds pay off in approximately the same states of nature or economic scenarios and, hence, as argued earlier, they should command approximately the same rate of return. In fact, using standard theory to estimate risk-adjusted returns, we found that stocks on average should command, at most, a 1 percent return premium over bills. Since, for as long as we had reliable data (about 100 years), the mean premium on stocks over bills was considerably and consistently higher, we realized that we had a puzzle on our hands. It took us six more years to convince a skeptical profession and for our paper “The Equity Premium: A Puzzle” to be published (Mehra and Prescott (1985)).

2.1. Standard Preferences The neoclassical growth model and its stochastic variants are a central construct in contemporary finance, public finance, and business cycle theory. It has been used extensively by, among others, Abel et al. (1989), Auerbach and Kotlikoff (1987), Becker and Barro (1988), Brock (1979), Cox, Ingersoll, and Ross (1985), Donaldson and Mehra (1984), Kydland and Prescott (1982), Lucas (1978), and Merton (1971). In fact, much of our economic intuition is derived from this model class. A key idea of this framework is that consumption today and consumption in some future period are treated as different goods. Relative prices of these different goods are equal to people’s willingness to substitute between these goods and businesses’ ability to transform these goods into each other. The model has had some remarkable successes when confronted with empirical data, particularly in the stream of macroeconomic research referred to as Real Business Cycle Theory, where researchers have found that it easily replicates the essential

15


macroeconomic features of the business cycle. See, in particular, Kydland and Prescott (1982). Unfortunately, when confronted with financial market data on stock returns, tests of these models have led, without exception, to their rejection. Perhaps the most striking of these rejections is our 1985 paper.26 To illustrate this we employ a variation of Lucas’ (1978) endowment economy rather than the production economy studied in Prescott and Mehra (1980). This is an appropriate abstraction to use if it is the equilibrium relation between the consumption and asset returns that are being used to estimate the premium for bearing non-diversifiable risk, which is what we were doing. Introducing production would only complicate the selection of exogenous processes, which resulted in the observed process for consumption.27 To examine the role of other factors for mean asset returns, it would be necessary to introduce other features of reality such as taxes and intermediation costs as has recently been done.28 If the model had accounted for differences in average asset returns, the next step would have been to use the neoclassical growth model, which has intertemporal transformation opportunities through variations in the rate at which the capital stock is accumulated, to see if this abstraction accounted for the observed large differences in average asset returns. Since per capita consumption has grown over time, we assume that the growth rate of the endowment follows a Markov process. This is in contrast to the assumption in Lucas’ model that the endowment level follows a Markov process. Our assumption, which requires an extension of competitive equilibrium theory,29 enables us to capture the non-stationarity in the consumption series associated with the large increase in per capita consumption that occurred over the last century. We consider a frictionless economy that has a single representative “stand-in” household. This unit orders its preferences over random consumption paths by ∞ E0 β t U (c t) , 0 < β < 1, (1) t=0

where ct is the per capita consumption and the parameter β is the subjective time discount factor, which describes how impatient households are to consume. If β is small, people are highly impatient, with a strong preference for consumption now versus consumption in the future. As modeled, these households live forever, which implicitly means that the utility of parents depends on the utility of their children. In the real world, this is true for some people and not for others. However, economies with both types of people—those who care about their children’s utility and those who do not—have essentially the same implications for asset prices and returns.30 26 The

reader is referred to McGrattan and Prescott (2003) and Mehra and Prescott (2007 and 2008) for an alternative perspective. 27 In a production economy, consumption would be endogenously determined, restricting the class of consumption processes that could be considered. See Appendix C. 28 See McGrattan and Prescott (2003) and Mehra and Prescott (2007 and 2008). 29 This is accomplished in Mehra (1988). 30 See Constantinides, Donaldson, and Mehra (2002, 2005). Constantinides et al. (2007) explicitly model bequests.

16


We use this simple abstraction to build quantitative economic intuition about what the returns on equity and debt should be. E0 {·} is the expectations operator conditional upon information available at time zero (which denotes the present time), and U : R+ → R is the increasing, continuously differentiable concave utility function. We further restrict the utility function to be of the constant relative risk aversion (CRRA) class U (c, α) =

c1−α − 1 , 1−α

0 < α < ∞,

(2)

where the parameter α measures the curvature of the utility function. When α = 1, the utility function is defined to be logarithmic, which is the limit of the above representation as α approaches 1. The feature that makes this the “preference function of choice” in much of the literature in Growth and Real Business Cycle Theory is that it is scaleinvariant. This means that a household is more likely to accept a gamble if both its wealth and the gamble amount are scaled by a positive factor. Hence, although the level of aggregate variables such as capital stock have increased over time, the resulting equilibrium return process is stationary. A second attractive feature is that it is one of only two preference functions that allows for aggregation and a “stand-in” representative agent formulation that is independent of the initial distribution of endowments. One disadvantage of this representation is that it links risk preferences with time preferences. With CRRA preferences, agents who like to smooth consumption across various states of nature also prefer to smooth consumption over time, that is, they dislike growth. Specifically, the coefficient of relative risk aversion is the reciprocal of the elasticity of intertemporal substitution. There is no fundamental economic reason why this must be so. We will revisit this issue in the next chapter, where we examine preference structures that do not impose this restriction.31 We assume there is one productive unit, which produces output yt in period t, which is the period dividend. There is one equity share with price pt that is competitively traded; it is a claim to the stochastic process {yt}. Consider the intertemporal choice problem of a typical investor at time t. He equates the loss in utility associated with buying one additional unit of equity to the discounted expected utility of the resulting additional consumption in the next period. To carry over one additional unit of equity, pt units of the consumption good must be sacrificed, and the resulting loss in utility is pt U (ct). By selling this additional unit of equity in the next period, pt+1 + yt+1 additional units of the consumption good can be consumed and βEt {(pt+1 + yt+1)U (ct+1)} is the expected value of the incremental utility next period. At an optimum, these quantities must be equal. Hence, the fundamental relation that prices assets is pt U (ct) = βEt {(pt+1 + yt+1)U (ct+1)}. Versions of this expression can be found in Rubinstein (1976), Lucas (1978), Breeden (1979), and Prescott and Mehra (1980), among others. Excellent textbook treatments can be found in Cochrane (2005), Danthine and Donaldson (2005), Duffie (2001), and LeRoy and Werner (2001).

31 See

Epstein and Zin (1991) and Weil (1989).

17


We use it to price both stocks and risk-less one-period bonds. For equity we have U (ct+1 ) 1 = βEt Re,t+1 , U (ct )

(3)

where Re,t+1 =

pt+1 + yt+1 . pt

For the risk-less one-period bonds, the relevant expression is U (ct+1 ) Rf ,t+1 , 1 = βEt U (ct )

(4)

(5)

where the gross rate of return on the riskless asset is by definition Rf ,t+1 =

1 , qt

(6)

with qt being the price of the bond. Since U (c) is assumed to be increasing, we can rewrite (3) as 1 = βEt Mt+1 Re,t+1 , (7) where Mt+1 is a strictly positive stochastic discount factor. This guarantees that the economy will be arbitrage-free and the law of one price holds. A little algebra shows that −U (ct+1), Re,t+1 . (8) Et (Re,t+1) = Rf ,t+1 + Covt Et (U (ct+1)) The equity premium Et (Re,t+1) − Rf ,t+1 thus can be easily computed. Expected asset returns equal the risk-free rate plus a premium for bearing risk, which depends on the covariance of the asset returns with the marginal utility of consumption. Assets that co-vary positively with consumption—that is, they pay off in states when consumption is high and marginal utility is low—command a high premium since these assets “destabilize” consumption. The question we need to address is the following: is the magnitude of the covariance between the marginal utility of consumption large enough to justify the observed 6 percent equity premium in U.S. equity markets? To address this issue, we make some additional assumptions. While they are not necessary and were not, in fact, part of our original paper on the equity premium, we include them to facilitate exposition and because they result in closed-form solutions.32

32 The exposition below is based on Abel (1988), his unpublished notes, and Mehra (2003). See Appendix B for the analysis in our 1985 paper.

18


These assumptions are ct+1 1. the growth rate of consumption xt+1 ≡ is i.i.d. ct yt+1 is i.i.d. 2. the growth rate of dividends zt+1 ≡ yt 3. (xt , zt ) are jointly log-normally distributed. The consequences of these assumptions are that the gross return on equity Re,t (defined above) is i.i.d. and that (xt , Re,t) are jointly log-normal. Substituting U (ct) = ct−α in the fundamental pricing relation33 U (ct+1) , (9) pt = βEt (pt+1 + yt+1 ) U (ct ) we get

pt = βEt (pt+1 + yt+1)x−α t+1 .

(10)

As pt is homogeneous of degree 1 in y, we can represent it as pt = wyt , and hence Re,t+1 can be expressed as

w+1 w+1 yt+1 Re,t+1 = = · · zt+1 . w yt w

(11)

It is easily shown that w=

βEt {zt+1 x−α t+1 } 1 − βEt {zt+1 x−α t+1}

;

(12)

hence, Et {zt+1} . βEt {zt+1 x−α t+1} Analogously, the gross return on the riskless asset can be written as Et {Re,t+1} =

Rf ,t+1 =

1 1 . β Et {x−α t+1}

(13)

(14)

Since we have assumed the growth rate of consumption and dividends to be lognormally distributed, eμz + 2 σz 1

Et {Re,t+1} =

2

βeμz −αμx +1/2(σz +α 2

2 σ 2 −2ασ ) x,z x

(15)

33 In contrast to our approach, which is in the applied general equilibrium tradition, there is another tradition of testing Euler equations (such as Eq. (9)) and rejecting them. Hansen and Singleton (1982) and Grossman and Shiller (1981) exemplify this approach. See Appendix D for an elaboration.

19


and 1 ln Et {Re,t+1} = − ln β + αμx − α 2 σx2 + ασx,z , 2

(16)

where μx = E(ln x), σx2 = Var(ln x), σx,z = Cov(ln x, ln z), and ln x is the continuously compounded growth rate of consumption. The other terms involving z and Re are defined analogously. Similarly, Rf =

1 −αμx + 12 α 2 σx2

(17)

βe

and 1 ln Rf = − ln β + αμx − α 2 σx2 . 2 ∴

ln E{Re} − ln Rf = ασx,z .

(18)

(19)

From (11) it also follows that ln E{Re} − ln Rf = ασx,Re , where σx,Re = Cov(ln x, ln Re ).

(20)

The (log) equity premium in this model is the product of the coefficient of risk aversion and the covariance of the (continuously compounded) growth rate of consumption with the (continuously compounded) return on equity or the growth rate of dividends. If we impose the equilibrium condition that x = z, a consequence of which is the restriction that the return on equity is perfectly correlated to the growth rate of consumption, we get ln E{Re} − ln Rf = ασx2 ,

(21)

and the equity premium then is the product of the coefficient of relative risk aversion and the variance of the growth rate of consumption. As we see ahead, this variance is 0.001369, so unless the coefficient of risk aversion α is large, a high-equity premium is impossible. The growth rate of consumption just does not vary enough! In Mehra and Prescott (1985) we report the following sample statistics for the U.S. economy over the period 1889–1978: Risk-free rate Rf = 1.0080 Mean return on equity E{Re} = 1.0698

20


Mean growth rate of consumption E{x} = 1.0180 Standard deviation of the growth rate of consumption σ{x} = 0.0360 Mean equity premium E{Re} − Rf = 0.0618 In our calibration, we are guided by the tenet that model parameters should meet the criteria of cross-model verification: not only must they be consistent with the observations under consideration, but they should not be grossly inconsistent with other observations in growth theory, business cycle theory, labor market behavior, and so on. There is a wealth of evidence from various studies that the coefficient of risk aversion α is a small number, certainly less than 10.34 We can then pose a question: if we set the risk aversion coefficients α to be 10 and β to be 0.99, what are the expected rates of return and the risk premia using the parameterization above? Using the expressions derived earlier, we have 1 ln Rf = − ln β + αμx − α 2 σx2 = 0.124 2 or Rf = 1.132, that is, a risk-free rate of 13.2 percent! Since ln E{Re} = ln Rf + ασx2 = 0.136, we have E{Re} = 1.146, or a return on equity of 14.6 percent. This implies an equity risk premium of 1.4 percent, far lower than the 6.18 percent historically observed equity premium. In this calculation we have been liberal in choosing the values for α and β. Most studies indicate a value for α that is close to 3. If we pick a lower value for β, the risk-free rate will be even higher and the premium lower. So the 1.4 percent value represents the maximum equity risk premium that can be obtained in this class of models given the constraints on α and β. Since the observed equity premium is over 6 percent, we have a puzzle on our hands that risk considerations alone cannot account for.

2.1.1. The Risk-Free Rate Puzzle Philippe Weil (1989) has dubbed the high risk-free rate obtained above “the risk-free rate puzzle.” The short-term real rate in the U.S. averages less than 1 percent, while the high value of α required to generate the observed equity premium results in an unacceptably 34 A

number of these studies are documented in Mehra and Prescott (1985).

21


high risk-free rate. The risk-free rate as shown in Eq. (18) can be decomposed into three components: 1 ln Rf = − ln β + αμx − α 2 σx2 . 2 The first term, −ln β, is a time preference or impatience term. When β < 1, it reflects the fact that agents prefer early consumption to later consumption. Thus, in a world of perfect certainty and no growth in consumption, the unique interest rate in the economy will be Rf = 1/β. The second term, αμx , arises because of growth in consumption. If consumption is likely to be higher in the future, agents with concave utility would like to borrow against future consumption in order to smooth their lifetime consumption. The greater the curvature of the utility function and the larger the growth rate of consumption, the greater the desire to smooth consumption. In equilibrium, this will lead to a higher interest rate since agents in the aggregate cannot simultaneously increase their current consumption. The third term, 12 α 2 σx2 , arises due to a demand for precautionary saving. In a world of uncertainty, agents would like to hedge against future unfavorable consumption realizations by building “buffer stocks” of the consumption good. Hence, in equilibrium, the interest rate must fall to counter this enhanced demand for savings. Figure 8 plots ln Rf = − ln β + αμx − 21 α 2 σx2 calibrated to the U.S. historical values with μx = 0.0175 and σx2 = 0.00123 for various values of β. It shows that the precautionary savings effect is negligible for reasonable values of α (1 < α < 5). For α = 3 and β = 0.99, Rf = 1.65, which implies a risk-free rate of 6.5 percent— much higher than the historical mean rate of 0.8 percent. The economic intuition is 80 70 60 50 40 30 20 10

220 230 240 250 260 270 280 a

FIGURE 8

Mean risk-free rate vs. α.

49

47

45

43

41

39

37

35

33

31

29

27

25

23

21

19

17

15

13

9

11

7

5

210

3

1

0

b5 0.99 b 5 0.96 b 5 0.55

22


straightforward—with consumption growing at 1.8 percent a year with a standard deviation of 3.6 percent—agents with isoelastic preferences have a sufficiently strong desire to borrow in order to smooth consumption that it takes a high interest rate to induce them not to do so. The late Fischer Black35 proposed that α = 55 would solve the puzzle. Indeed, it can be shown that the 1889–1978 U.S. experience reported above can be reconciled with α = 48 and β = 0.55. To see this, observe that since var(x) = 0.00123 = ln 1 + [E(x)]2

σx2

and

1 μx = ln E(x) − σx2 = 0.0175, 2

this implies α=

ln E(R) − ln Rf σx2

= 48.4. Since 1 ln β = − ln Rf + αμx − α 2 σx2 2 = −0.60, this implies β = 0.55. Besides postulating an unacceptably high α, another problem is that this is a “knifeedge” solution. No other set of parameters will work, and a small change in α will lead to an unacceptable risk-free rate, as shown in Figure 8. An alternate approach is to experiment with negative time preferences; however, there seems to be no empirical evidence that agents do have such preferences.36 Figure 8 shows that for extremely high α, the precautionary savings term dominates and results in a “low” risk-free rate.37 However, then a small change in the growth rate of consumption will have a large impact on interest rates. This is inconsistent with a crosscountry comparison of real risk-free rates and their observed variability. For example, throughout the 1980s, South Korea had a much higher growth rate than the U.S., but real rates were not appreciably higher. Nor does the risk-free rate vary considerably over time, as would be expected if α was large. In Section 3 we show how alternative preference structures can help resolve the risk-free rate puzzle.

35 Private

communication, 1981. a model with growth, equilibrium can exist with β > 1. See Mehra (1988) for the restrictions on the parameters α and β for equilibrium to exist. 37 Kandel and Stambaugh (1991) have suggested this approach. 36 In

23


correl . 5 20.14 correl . 5 0.00 correl . 5 0.45

0.4

Equity Premium (Percent)

0.3

0.2

0.1

0

20.1 0

FIGURE 9

0.5

1

1.5 a

2

2.5

Equity Premium vs. α.

2.1.2. The Effect of Serial Correlation in the Growth Rate of Consumption The preceding analysis has assumed that the growth rate of consumption is i.i.d over time. However, for the sample period 1889–2004 it is slightly negative (−0.135), while for the sample period 1930–2004 the value is 0.45. The effect of this non-zero serial correlation on the equity premium can be analyzed using the framework in Appendix B. Figure 9 shows the effect of changes in the risk aversion parameter on the equity premium for different serial correlations.38 When the serial correlation of consumption is positive, the equity premium actually declines with increasing risk aversion, thus, further exacerbating the equity premium puzzle.39 An alternative perspective on the puzzle is provided by Hansen and Jagannathan (1991). The fundamental pricing equation can be written as Et (Re,t+1) = Rf ,t+1 − Covt

38 In

addition, see Figure 2B in Appendix B. Azeredo (2007) for a detailed discussion.

39 See

Mt+1 , Re,t+1 . Et (Mt+1)

(22)

24


This expression also holds unconditionally, so that E(Re,t+1) = Rf ,t+1 − σ(Mt+1)σ(Re,t+1)ρR,M /E(Mt+1)

(23)

E(Re,t+1) − Rf ,t+1 /σ(Re,t+1) = −σ(Mt+1)ρR,M /E(Mt+1),

(24)

or

and since −1 ≤ ρR,M ≤ 1,

E(Re,t+1) − Rf ,t+1 /σ(Re,t+1) ≤ σ(Mt+1)/E(Mt+1).

(25)

This inequality is referred to as the Hansen–Jagannathan lower bound on the pricing kernel. For the U.S. economy, the Sharpe ratio, E(Re,t+1) − Rf ,t+1 /σ(Re,t+1), can be calculated to be 0.37. Since E(Mt+1) is the expected price of a one-period risk-free bond, its value must be close to 1. In fact, for the parameterization discussed earlier, E(Mt+1) = 0.96 when α = 2. This implies that the lower bound on the standard deviation for the pricing kernel must be close to 0.3 if the Hansen–Jagannathan bound is to be satisfied. However, when this is calculated in the Mehra–Prescott framework, we obtain an estimate for σ(Mt+1) = 0.002, which is off by more than an order of magnitude. We would like to emphasize that the equity premium puzzle is a quantitative puzzle; standard theory is consistent with our notion of risk that, on average, stocks should return more than bonds. The puzzle arises from the fact that the quantitative predictions of theory are an order of magnitude different from what has been historically documented. The puzzle cannot be dismissed lightly, since much of our economic intuition is based on the very class of models that fall short so dramatically when confronted with financial data. It underscores the failure of paradigms central to financial and economic modeling to capture the characteristic that appears to make stocks comparatively so risky. Hence, the viability of using this class of models for any quantitative assessment, say, for instance, to gauge the welfare implications of alternative stabilization policies, is thrown open to question. For this reason, over the last 20 years or so, attempts to resolve the puzzle have become a major research impetus in finance and economics. Several generalizations of key features of the Mehra and Prescott (1985) model have been proposed to better reconcile observations with theory. These include alternative assumptions on preferences,40 modified probability distributions to admit rare but disastrous events,41 survival

40 For example, Abel (1990), Bansal and Yaron (2004), Benartzi and Thaler (1995), Boldrin, Christiano, and Fisher (2001), Campbell and Cochrane (1999), Constantinides (1990), Epstein and Zin (1991), and Ferson and Constantinides (1991). 41 See Rietz (1988) and Mehra and Prescott (1988).


25

bias,42 incomplete markets,43 and market imperfections.44 They also include attempts at modeling limited participation of consumers in the stock market45 and problems of temporal aggregation.46 We examine some of the research efforts to resolve the puzzle47 in the next two chapters.

References Abel, A. B. Stock prices under time varying dividend risk: An exact solution in an infinite horizon general equilibrium model. Journal of Monetary Economics 22 (1988): 375–394. Abel, A. B. Asset prices under habit formation and catching up with the Joneses. American Economic Review, Papers and Protocols 80 (1990): 38–42. Abel, A. B., N. G. Mankiw, L. H. Summers, and R. J. Zeckhauser. Assessing dynamic efficiency: Theory and evidence. Review of Economic Studies 56 (1989): 1–20. Aiyagari, S. R., and M. Gertler. Asset returns with transactions costs and uninsured individual risk. Journal of Monetary Economics 27 (1991): 311–331. Alvarez, F., and U. Jermann. Asset pricing when risk sharing is limited by default. Econometrica 48 (2000): 775–797. Attanasio, O. P., J. Banks, and S. Tanner. Asset holding and consumption volatility. Journal of Political Economy 110(4) (2002): 771–792. Auerbach, A. J., and L. J. Kotlikoff. Dynamic Fiscal Policy. Cambridge University Press, Cambridge (1987). Azeredo, F. Essays on aggregate economics and finance. Doctoral dissertation, University of California, Santa Barbara (2007). Bansal, R., and J. W. Coleman. A monetary explanation of the equity premium, term premium and risk-free rate puzzles. Journal of Political Economy 104 (1996): 1135–1171. Bansal, R., and A. Yaron. Risks for the long run: A potential resolution of asset pricing puzzles. Journal of Finance 59 (2004): 1481–1509. Basak, S., and D. Cuoco. An equilibrium model with restricted stock market participation. The Review of Financial Studies 11 (1998): 309–341. Becker, G. S., and R. J. Barro. A reformulation of the economic theory of fertility. Quarterly Journal of Economics 103(1) (1988): 1–25. Benartzi, S., and R. H. Thaler. Myopic loss aversion and the equity premium puzzle. Quarterly Journal of Economics 110 (1995): 73–92. Bewley, T. F. Thoughts on tests of the intertemporal asset pricing model. Working paper, Northwestern University (1982). Billingsley, P. Probability and Measure. John Wiley and Sons, New York (1995). Boldrin, M., L. J. Christiano, and J. D. M. Fisher. Habit persistence, asset returns, and the business cycle. American Economic Review 91 (2001): 149–166. 42 See

Brown, Goetzmann, and Ross (1995). example, Bewley (1982), Brav, Constantinides, and Geczy (2002), Constantinides and Duffie (1996), Heaton and Lucas (1997, 2000), Lucas (1994), Mankiw (1986), Mehra and Prescott (1985), Storesletten, Telmer, and Yaron (2004), and Telmer (1993). 44 For example, Aiyagari and Gertler (1991), Alvarez and Jerman (2000), Bansal and Coleman (1996), Basak and Cuoco (1998), Constantinides, Donaldson, and Mehra (2002), Danthine, Donaldson, and Mehra (1992), Daniel and Marshall (1997), He and Modest (1995), Heaton and Lucas (1996), Luttmer (1996), McGrattan and Prescott (2000), Sethi (1997), and Storesletten, Telmer, and Yaron (2006). 45 Attanasio, Banks, and Tanner (2002), Brav, Constantinides, and Geczy (2002), Mankiw and Zeldes (1991), and Vissing-Jorgensen (2002). 46 Gabaix and Laibson (2001), Heaton (1995), and Lynch (1996). 47 The reader is also referred to the excellent surveys by Kocherlakota (1996), Cochrane (1997), and Campbell (1999, 2001).

43 For

26

Chapter 1 • The Equity Premium: ABCs Brav, A., G. M. Constantinides, and C. C. Geczy. Asset pricing with heterogeneous consumers and limited participation: Empirical evidence. Journal of Political Economy 110 (2002): 793–824. Breeden, D. An intertemporal asset pricing model with stochastic consumption and investment opportunities. Journal of Financial Economics 7 (1979): 265–296. Brock, W. A. An integration of stochastic growth theory and the theory of finance, Part 1: The growth model, in J. Green, and J. Scheinkman, eds., General Equilibrium, Growth & Trade. Academic Press, New York (1979). Brown, S., W. Goetzmann, and S. Ross. Survival. Journal of Finance 50 (1995): 853–873. Campbell, J. Y. Asset prices, consumption, and the business cycle. Chapter 19 in J. B. Taylor, and M. Woodford, eds. Handbook of Macroeconomics 1. North–Holland, Amsterdam, (1999): 1231–1303. Campbell, J. Y. Asset pricing at the millennium. Journal of Finance 55 (2001): 1515–1567. Campbell, J. Y., and J. H. Cochrane. By force of habit: A consumption–based explanation of aggregate stock market behavior. Journal of Political Economy 107 (1999): 205–251. Cochrane, J. H. Where is the market going? Uncertain facts and novel theories. Economic Perspectives 21 (1997): 3–37. Cochrane, J. H. Asset Pricing. Princeton University Press, Princeton, NJ (2005). Constantinides, G. M. Habit formation: A resolution of the equity premium puzzle. Journal of Political Economy 98 (1990): 519–543. Constantinides, G. M., J. B. Donaldson, and R. Mehra. Junior can’t borrow: A new perspective on the equity premium puzzle. Quarterly Journal of Economics 118 (2002): 269–296. Constantinides, G. M., J. B. Donaldson, and R. Mehra. Junior must pay: Pricing the implicit put in privatizing Social Security. Annals of Finance 1 (2005): 1–34. Constantinides, G. M., J. B. Donaldson, and R. Mehra. Junior is rich: Bequests as consumption. Economic Theory 32 (2007): 125–155. Constantinides, G. M., and D. Duffie. Asset pricing with heterogeneous consumers. Journal of Political Economy 104 (1996): 219–240. Cowles, A., and Associates. Common Stock Indexes, 2d ed. Cowles Commission Monograph no. 3. Principia Press, Bloomington, IN (1939). Cox, J. C., J. E. Ingersoll, Jr., and S. A. Ross. A theory of the term structure of interest rates. Econometrica 53 (1985): 385–407. Daniel, K., and D. Marshall. The equity premium puzzle and the risk-free rate puzzle at long horizons. Macroeconomic Dynamics 1 (1997): 452–484. Danthine, J.-P., J. B. Donaldson, and R. Mehra. The equity premium and the allocation of income risk. Journal of Economic Dynamics and Control 16 (1992): 509–532. Danthine, J.-P., and J. B. Donaldson. Intermediate Financial Theory. Prentice Hall, Upper Saddle River, NJ (2005). Debreu, G. Valuation equilibrium and pareto optimum. Proceedings of the National Academy of Sciences. 40 (1954): 588–592. Dimson, E., P. Marsh, and M. Staunton. Triumph of the Optimists: 101 Years of Global Investment Returns. Princeton University Press, Princeton, NJ (2002). Donaldson, J. B., and R. Mehra. Comparative dynamics of an equilibrium intertemporal asset pricing model. Review of Economic Studies 51 (1984): 491–508. Duffie, D. Dynamic Asset Pricing Theory, 3rd ed. Princeton University Press, Princeton, NJ (2001). Epstein, L. G., and S. E. Zin. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: An empirical analysis. Journal of Political Economy 99 (1991): 263–286. Ferson, W. E., and G. M. Constantinides. Habit persistence and durability in aggregate consumption. Journal of Financial Economics 29 (1991): 199–240. Gabaix, X., and D. Laibson. The 6D bias and the equity premium puzzle, in B. Bernanke, and K. Rogoff, eds., NBER Macroeconomics Annual 2001. MIT Press, Cambridge, MA (2001). Grossman, S. J., and R. J. Shiller. The determinants of the variability of stock market prices. American Economic Review 71 (1981): 222–227. Hansen, L. P., and R. Jagannathan. Implications of security market data for models of dynamic economies. Journal of Political Economy 99 (1991): 225–262.


27

Hansen, L. P., and K. J. Singleton. Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica 50 (1982): 1269–1288. Hansen, L. P., and K. J. Singleton. Stochastic consumption, risk aversion and the intertemporal behavior of asset returns. Journal of Political Economy 91 (1983): 249–268. He, H., and D. M. Modest. Market frictions and consumption-based asset pricing. Journal of Political Economy 103 (1995): 94–117. Heaton, J. An empirical investigation of asset pricing with temporally dependent preference specifications. Econometrica 66 (1995): 681–717. Heaton, J., and D. J. Lucas. Evaluating the effects of incomplete markets on risk sharing and asset pricing. Journal of Political Economy 104 (1996): 443–487. Heaton, J., and D. J. Lucas. Market frictions, savings behavior and portfolio choice. Journal of Macroeconomic Dynamics 1 (1997): 76–101. Heaton, J. C., and D. J. Lucas. Portfolio choice and asset prices: The importance of entrepreneurial risk. Journal of Finance 55 (2000). Homer, S. A History of Interest Rates. Rutgers University Press, New Brunswick, NJ (1963). Ibbotson Associates. Stocks, bonds, bills and inflation. 2005 Yearbook. Ibbotson Associates, Chicago (2006). Kandel, S., and R. F. Stambaugh. Asset returns and intertemporal preferences. Journal of Monetary Economics 27 (1991): 39–71. Kendrick, J. W. Productivity trends in the United States, NBER 71 (1961). Kocherlakota, N. R. The equity premium: It’s still a puzzle. Journal of Economic Literature 34 (1996): 42–71. Kuznets, S. S. Commodity Flow and Capital Formation, NBER 34, New York (1938). Kuznets, S. S. National Product since 1869, National Bureau of Economic Research Number 46, New York (1946). Kuznets, S. S. Capital in the American economy: Its formation and financing studies in capital formation and Financing, NBER (1961). Kydland, F., and E. C. Prescott. Time to build and aggregate fluctuations. Econometrica 50 (1982): 1345– 1371. LeRoy, S. H., and J. Werner. Principles of Financial Economics. Cambridge University Press, New York (2001). Litterman, R. B. Bayesian procedure for forecasting with vector auto-regressions. Working paper, MIT (1980). Lucas, D. J. Asset pricing with undiversifiable risk and short sales constraints: Deepening the equity premium puzzle. Journal of Monetary Economics 34 (1994): 325–341. Lucas, R. E., Jr. Asset prices in an exchange economy. Econometrica 46 (1978): 1429–1445. Luttmer, E. G. J. Asset pricing in economies with frictions. Econometrica 64 (1996): 1439–1467. Lynch, A. W. Decision frequency and synchronization across agents: Implications for aggregate consumption and equity returns. Journal of Finance 51 (1996): 1479–1497. Macaulay, F. R. The movements of interest rates, bond yields and stock prices in the U.S. since 1856. National Bureau of Economic Research, New York (1938). Mankiw, N. G. The equity premium and the concentration of aggregate shocks. Journal of Financial Economics 17 (1986): 211–219. Mankiw, N. G., and S. P. Zeldes. The consumption of stockholders and nonstockholders. Journal of Financial Economics 29 (1991): 97–112. McGrattan, E. R., and E. C. Prescott. Is the stock market overvalued? Federal Reserve Bank of Minneapolis Quarterly Review (2000). McGrattan, E. R., and E. C. Prescott. Average debt and equity returns: Puzzling? American Economic Review 93 (2003): 392–397. McGrattan, E. R., and E. C. Prescott. Taxes, regulations, and the value of U.S. and U.K. corporations. Review of Economic Studies 92 (2005): 767–796. Mehra, R. On the existence and representation of equilibrium in an economy with growth and nonstationary consumption. International Economic Review 29 (1988): 131–135. Mehra, R. On the volatility of stock prices: An exercise in quantitative theory. International Journal of Systems Science 29 (1998): 1203–1211.

28

Chapter 1 • The Equity Premium: ABCs Mehra, R. The equity premium: Why is it a puzzle? Financial Analysts Journal (2003): 54–69. Mehra, R. The equity premium in India. Oxford Companion to Economics in India, B. Kaushik, ed. Oxford University Press, New York (2007). Mehra, R., and E. C. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15 (1985): 145–161. Mehra, R., and E. C. Prescott. The equity premium: A solution? Journal of Monetary Economics 22 (1988): 133–136. Mehra, R., and E. C. Prescott. The equity premium in retrospect. In Handbook of the Economics of Finance, G. M. Constantinides, M. Harris, and R. Stulz, eds. North-Holland, Amsterdam (2003). Mehra, R., and E. C. Prescott. Intermediated quantities and returns. Working paper, UCSB (2007). Mehra, R., and E. C. Prescott. Non-risk based explanations of the equity premium. Forthcoming in Handbook of the Equity Risk Premium, R. Mehra, ed. Amsterdam (2008). Merton, R. C. Optimum consumption and portfolio rules in a continuous time model. Journal of Theory 3 (1971): 373–413. Mossin, J. Equilibrium in a capital asset market. Econometrica 34 (1966): 768–783. Prescott, E. C., and R. Mehra. Recursive competitive equilibrium: The case of homogeneous households. Econometrica 48 (1980): 1365–1379. Rietz, T. A. The equity risk premium: A solution. Journal of Monetary Economics 22 (1988): 117–131. Romer, C. D. Changes in business cycles: Evidence and explanations. Journal of Economic Perspectives 13 (1999): 23–44. Rubinstein, M. The valuation of uncertain income streams and the pricing of options. Bell Journal of Economics 7 (1976): 407–425. Schwert, G. W. Indexes of U.S. stock prices from 1802 to 1987. Journal of Business 63 (1990): 399–426. Sethi, S. P. Optimal consumption and investment with bankruptcy. Kluwer Academic Publishers. Norwell, MA 1997. Shaw, W. H. The value of commodity output since 1869. NBER, 48 (1947). Shiller, R. J. Comovements in stock prices and comovements in dividends. Journal of Finance 44 (1989): 719–729. Siegel, J. Stocks for the Long Run, 3rd ed. Irwin, New York (2002). Smith, W. B., and A. H. Cole. Fluctuations in American business, 1790–1860. Harvard University Press, Cambridge, MA (1935). Storesletten, K., C. I. Telmer, and Y. Amir. Asset pricing with idiosyncratic risk and overlapping generations. Working paper. Carnegie Mellon University (2006). Review of Economic Dynamics, forthcoming. Storesletten, K., C. I. Telmer, and A. Yaron. Consumption and risk sharing over the life cycle. Journal of Monetary Economics 51(3) (2004): 609–633. Telmer, C. I. Asset-pricing puzzles and incomplete markets. Journal of Finance 49 (1993): 1803–1832. Vissing-Jorgensen, A. Limited asset market participation and the elasticity of intertemporal substitution. Journal of Political Economy, forthcoming. Weil, P. The equity premium puzzle and the risk-free rate puzzle. Journal of Monetary Economics 24 (1989): 401–421.

29


APPENDIX A Suppose the distribution of returns period by period is independently and identically distributed. Then, as the number of periods tends to infinity, the future value of the investment, computed at the arithmetic average of returns, tends to the expected value of the investment with probability 1. To see this, let T

VT = Π (1 + rt ) , t=1

where rt is the asset return in period t and VT is the terminal value of $1 at time T . Then

T

E(VT ) = E Π (1 + rt ) . i=1

Since the rt s are assumed to be uncorrelated, we have T

E(VT ) = Π E (1 + rt ) i=1

or T

E(VT ) = Π (1 + E (rt )) . i=1

Let the arithmetic average AA =

T 1 rt . T t=t

Then, by the strong law of large numbers (Theorem 22.1, Billingsley (1995)), T

E(VT ) → Π (1 + AA) i=1

as T → ∞

or E(VT ) → (1 + AA)T as the number of periods T becomes large.

APPENDIX B The Original Analysis of the Equity Premium Puzzle In this appendix we present our original analysis of the equity premium puzzle. Needless to say, it draws heavily from Mehra and Prescott (1985).

30

Chapter 1

•

The Equity Premium: ABCs

The Economy, Asset Prices and Returns We employ a variation of Lucas’ (1978) pure exchange model. Since per capita consumption has grown over time, we assume that the growth rate of the endowment follows a Markov process. This is in contrast to the assumption in Lucas’ model that the endowment level follows a Markov process. Our assumption, which requires an extension of competitive equilibrium theory, enables us to capture the non-stationarity in the consumption series associated with the large increase in per capita consumption that occurred in the 1889–1978 period. The economy we consider was judiciously selected so that the joint process governing the growth rates in aggregate per capita consumption and asset prices would be stationary and easily determined. The economy has a single representative “stand-in” household. This unit orders its preferences over random consumption paths by ∞ t E0 β U (ct ) , 0 < β < 1, (1B) t=0

where ct is per capita consumption, β is the subjective time discount factor, E{·} is the expectation operator conditional upon information available at time zero (which denotes the present time), and U : R+ → R is the increasing concave utility function. To ensure that the equilibrium return process is stationary, we further restrict the utility function to be of the constant relative risk aversion (CRRA) class U (c, α) =

c1−α − 1 , 1−α

0 < α < ∞.

(2B)

The parameter α measures the curvature of the utility function. When α is equal to one, the utility function is defined to be the logarithmic function, which is the limit of the above function as α approaches one. We assume there is one productive unit that produces output yt in period t, which is the period dividend. There is one equity share with price pt , that is competitively traded; it is a claim to the stochastic process {yt }. The growth rate in yt is subject to a Markov chain; that is, yt+1 = xt+1 yt ,

(3B)

where xt+1 ∈ {λ1 , . . . , λn } is the growth rate, and Pr{xt+1 = λj ; xt = λi} = φij .

(4B)

It is also assumed that the Markov chain is ergodic. The λi s are all positive and y0 > 0. The random variable yt is observed at the beginning of the period, at which time dividend payments are made. All securities are traded ex-dividend. We also assume that the matrix A with elements aij ≡ βφij λ1−α for i, j = 1, . . . , n is stable; that is, lim Am as j m → ∞ is zero. In Mehra (1988) it is shown that this is necessary and sufficient for the expected utility to exist if the stand-in household consumes yt every period. The paper

31


also defines and establishes the existence of a Debreu (1954) competitive equilibrium with a price system having a dot product representation under this condition. Next we formulate expressions for the equilibrium time t price of the equity share and the risk-free bill. We follow the convention of pricing securities ex-dividend or ex-interest payments at time t, in terms of the time t consumption good. For any security with process {ds } on payments, its price in period t is ∞ s−t U (ys )ds , (5B) β Pt = Et U (yt ) s=t+1

as the equilibrium consumption is the process {ys} and the equilibrium price system has a dot product representation. The dividend payment process for the equity share in this economy is {ys}. Consequently, using the fact that U (c) = c−α , Pte = P e (xt , yt) ∞ α s−t yt ys |xt , yt . =E β ysα s=t+1

(6B)

The variables xt and yt are sufficient relative to the entire history of shocks up to, and including, time t for predicting the subsequent evolution of the economy. They thus constitute legitimate state variables for the model. Since ys = yt xt+1 . . . xs , the price of the equity security is homogeneous of degree one in yt , which is the current endowment of the consumption good. As the equilibrium values of the economies being studied are time-invariant functions of the state (xt , yt ), the subscript t can be dropped. This is accomplished by redefining the state to be the pair (c, i), if yt = c and xt = λi . With this convention, the price of the equity share from (6B) satisfies pe (c, i) = β

n

φij (λj c)−α [pe (λj c, j) + λj c]cα .

(7B)

j=1

Using the result that pe (c, i) is homogeneous of degree one in c, we represent this function as pe (c, i) = wi c,

(8B)

where wi is a constant. Making this substitution in (7B) and dividing by c yields wi = β

n

(1−α)

φij λj

(wj + 1)

for i = 1, . . . , n.

(9B)

j=1

This is a system of n linear equations in n unknowns. The assumption that guaranteed the existence of equilibrium guarantees the existence of a unique positive solution to this system.

32


The period return if the current state is (c, i) and next period state (λj c, j) is pe (λj c, j) + λj c − pe (c, i) pe (c, i) λj (wj + 1) − 1. = wi

rije =

(10B)

The equity’s expected period return if the current state is i is Rie =

n

φij rije .

(11B)

j=1

Capital letters are used to denote the expected return. With the subscript i, it is the expected return conditional upon the current state being (c, i). Without this subscript, it is the expected return with respect to the stationary distribution. The superscript indicates the type of security. The other security considered is the one-period real bill or riskless asset, which pays one unit of the consumption good next period with certainty. From (6B), f

pi = pf (c, i) n U (λj c) =β φij U (c) j=1 =β φij λ−α j .

(12B)

The certain return on this riskless security is 1

f

Ri =

f pi

(13B)

−1

when the current state is (c, i). As mentioned earlier, the statistics that are probably most robust to the modeling specification are the means over time. Let π ∈ Rn be the vector of stationary probabilities on i. This exists because the chain on i has been assumed to be ergodic. The vector π is the solution to the system of equations π = φT π, with n

πi = 1

and

φT = {φji}.

i=1

The expected returns on the equity and the risk-free security are, respectively, Re =

n i=1

πi Rie

and

Rf =

n i=1

f

πi Ri .

(14B)

33


Time sample averages will converge in probability to these values given the ergodicity of the Markov chain. The risk premium for equity is, Re − Rf , a parameter that is used in the test. The parameters defining preferences are α and β, while the parameters defining technology are the elements of [φij ] and [λi ]. Our approach is to assume two states for the Markov chain and to restrict the process as follows: λ1 = 1 + μ + δ,

λ2 = 1 + μ − δ,

φ11 = φ22 = φ,

φ12 = φ21 = (1 − φ).

The parameters μ, φ, and δ now define the technology. We require δ > 0 and 0 < φ < 1. This particular parameterization was selected because it permitted us to independently vary the average growth rate of output by changing μ, the variability of consumption by altering δ, and the serial correlation of growth rates by adjusting φ. The parameters were selected so that the average growth rate of per capita consumption, the standard deviation of the growth rate of per capita consumption, and the first-order serial correlation of this growth rate, all with respect to the model’s stationary distribution, matched the sample values for the U.S. economy between 1889–1978. The sample values for the U.S. economy were 0.018, 0.036, and −0.14, respectively. The resulting parameters’ values were μ = 0.018, δ = 0.036, and φ = 0.43. Given these values, the nature of the test is to search for parameters α and β for which the model’s averaged risk-free rate and equity risk premium match those observed for the U.S. economy over this 90-year period. The parameter α, which measures people’s willingness to substitute consumption between successive yearly time periods, is an important one in many fields of economics. As mentioned in the text, there is a wealth of evidence from various studies that the coefficient of risk aversion α is a small number, certainly less than 10. A number of these studies are documented in Mehra and Prescott (1985). This is an important restriction, for with a large α virtually any pair of average equity and risk-free returns can be obtained by making small changes in the process on consumption. Given the estimated process on consumption, Figure 1B depicts the set of values of the average risk-free rate and equity risk premium, which are both consistent with the model and result in average real risk-free rates between zero and four percent. These are values that can be obtained by varying preference parameters α between 0 and 10 and β between 0 and 1. The observed real return of 0.80 percent and equity premium of 6 percent are clearly inconsistent with the predictions of the model. The largest premium obtainable with the model is 0.35 percent, which is not close to the observed value. An advantage of our approach is that we can easily test the sensitivity of our results to such distributional assumptions. With α less than 10, we found that our results were essentially unchanged for very different consumption processes, provided that the mean and variances of growth rates equaled the historically observed values. We use this fact in motivating the discussion in the text.

34


Average Risk Premia (Percent)

R e2R f

2

1 Admissible Region

0

1

2

3

Rf

4

Average Risk-Free Rate (Percent)

FIGURE 1B 0.3

Equity Premium (Percent)

0.25

Set of admissible average equity risk premia and real returns.

correl. 5 20.14 correl. 5 0.45

0.2 0.15 0.1 0.05 0

1.5

2

2.5

3

3.5

4

4.5

Risk-Free Rate (Percent)

FIGURE 2B

Set of admissible average equity risk premia and real returns.

As mentioned earlier in the text, the serial correlation of the growth rate of consumption for the period 1930–2004 is 0.45. Figure 2B shows the resulting feasible region in this case (we have also included the region from Figure 1B for comparison). The conclusion that the premium for bearing non-diversifiable aggregate risk is small remains unchanged.

35


APPENDIX C Expanding the set of technologies in a pure exchange, Arrow–Debreu economy to admit capital accumulation and production as in Brock (1979), Prescott and Mehra (1980), or Donaldson and Mehra (1984) does not increase the set of joint equilibrium processes on consumption and asset prices. Since the set of equilibria in a production company is a subset of those in an exchange economy, it follows immediately that if the equity premium cannot be accounted for in an exchange economy, modifying the technology to incorporate production will not alter this conclusion.48 To see this, let θ denote preferences, τ technologies, E the set of the exogenous processes on the aggregate consumption good, P the set of technologies with production opportunities, and m(θ, τ) the set of equilibria for economy (θ, τ). Theorem ∪ m(θ, τ) ⊃ ∪ m(θ, τ)

τεE

τεP

Proof. For θ0 εθ and τ0 εP , let (a0 , c0 ) be a joint equilibrium process on asset prices and consumption. A necessary condition for equilibrium is that the asset prices a0 be consistent with c0 , the optimal consumption for the household with preferences θ0 . Thus, if (a0 , c0 ) is an equilibrium, then a0 = g(c0 , θ), where g is defined by the first-order necessary conditions for household maximization. This functional relation must hold for all equilibria, regardless of whether they are for a pure exchange or a production economy. Let (a0 , c0 ) be an equilibrium for some economy (θ0 , τ0 ) with τ0 εP . Consider the pure exchange economy with θ1 = θ0 and τ1 = c0 . Our contention is that (a0 , c0 ) is a joint equilibrium process for asset prices and consumption for the pure exchange economy (θ1 , τ1 ). For all pure exchange economies, the equilibrium consumption process is τ, so c1 = τ1 = c0 , given that more is preferred to less. If c0 is the equilibrium process, the corresponding asset price must be g(c0 , θ1 ). But θ1 = θ0 so g(c0 , θ1 ) = g(c0 , θ0 ) = a0 . Hence, a0 is the equilibrium for the pure exchange economy (θ1 , τ1 ), proving the theorem.

APPENDIX D Estimating the Equity Risk Premium Versus Estimating the Risk Aversion Parameter Estimating or measuring the relative risk aversion parameter using statistical tools is very different than estimating the equity risk premium. Mehra and Prescott (1985), as 48 The

discussion below is based on Mehra (1998).

36


discussed earlier, use an extension of Lucas’ (1978) asset pricing model to estimate how much of the historical difference in yields on Treasury bills and corporate equity is a premium for bearing aggregate risk. Crucial to their analysis is their use of micro observations to restrict the value of the risk aversion parameter. They did not estimate either the risk aversion parameter or the discount rate parameters. Mehra and Prescott (1985) reject extreme risk aversion based upon observations on individual behavior. These observations include the small size of premia for jobs with uncertain income and the limited amount of insurance against idiosyncratic income risk. Another observation is that people with limited access to capital markets make investments in human capital that result in very uneven consumption over time. A sharp estimate for the magnitude of the risk aversion parameter comes from macroeconomics. The evidence is that the basic growth model, when restricted to be consistent with the growth facts, generates business cycle fluctuations if and only if this risk aversion parameter is near zero. (This corresponds to the log case in standard usage.) The point is that the risk aversion parameter comes up in a wide variety of observations at both the household and the aggregate level and is not found to be large. For all values of the risk aversion coefficient less than 10, which is an upper bound number for this parameter, Mehra and Prescott find that a premium for bearing aggregate risk accounts for little of the historic equity premium. This finding has stood the test of time. Another tradition is to use consumption and stock market data to estimate the degree of relative risk aversion parameter and the discount factor parameter. This is what Grossman and Shiller report they did in their American Economic Review Papers and Proceedings article (1981, p. 226). In a paper in which they develop “a method for estimating nonlinear rational expectations models directly from stochastic Euler equations,” Hansen and Singleton illustrate their methods by estimating the risk aversion parameter and the discount factor using stock dividend consumption prices (1981, p. 1269). What the work of Grossman and Shiller (Ibid.) and Hansen and Singleton (1982, 1983) establish is that using consumption and stock market data and assuming frictionless capital markets is a bad way to estimate the risk aversion and discount factor parameters. It is analogous to estimating the force of gravity near the earth’s surface by dropping a feather from the top of the Leaning Tower of Pisa, under the assumption that friction is zero. A tradition related to statistical estimation is to statistically test whether the stochastic Euler equation arising from the stand-in household’s intertemporal optimization holds. Both Grossman and Shiller (1983) and Hansen and Singleton (1982) reject this relation. The fact that this relation is inconsistent with the U.S. time-series data is no reason to conclude that the model economy used by Mehra and Prescott to estimate how much of the historical equity premium is a premium for bearing aggregate risk is not a good one for that purpose. Returning to the analogy from physics, it would be silly to reject Newtonian mechanics as a useful tool for drawing scientific inference because the distance traveled by the feather did not satisfy 1/2gt2 .

CHAPTER 2

Risk-Based Explanations of the Equity Premium John Donaldson Columbia University and

Rajnish Mehra University of California, Santa Barbara, and NBER

Introduction 1. Alternative Preference Structures 1.1. Preliminaries 1.2. Coincidence of Risk and Time Preferences in CRRA utility 1.3. Separating Risk and Time Preferences: Epstein–Zin and others 1.4. Variation in the CRRA and EIS 1.5. Habit Formation 1.6. Behavioral Models 1.7. Beyond One Good and a Representative Agent 2. Production Economies 3. Disaster Events and Survivorship Bias 4. Market Incompleteness and Trading Frictions 4.1. Restricted Participation 5. Model Uncertainty 6. Concluding Comments References

39 41 41 44 46 52 55 61 71 78 81 86 86 91 93 94

We are grateful to Francisco Azeredo, Ravi Bansal, George Constantinides, Vito Gala, Alok Khare, and Edward Prescott for their comments on earlier drafts of this chapter.

HANDBOOK OF THE EQUITY RISK PREMIUM c 2008 by John B. Donaldson and Rajnish Mehra. All rights of reproduction in any form reserved. Copyright

37

38

Chapter 2 • Risk-Based Explanations of the Equity Premium

Abstract This essay reviews the family of models that seek to provide aggregate risk-based explanations for the empirically observed equity premium. Theories based on nonexpected utility preference structures, limited financial market participation, model uncertainty, and the small probability of enormous losses are detailed. We impose the additional requirements that candidate models yield consistent intertemporal portfolio choice and that a representative agent can be constructed that is independent of the underlying heterogeneous economy’s initial wealth distribution. While many models are able to replicate a wide variety of financial statistics including the premium, few satisfy these latter criteria as well. JEL Classification: D50, D52, E30, E32, G00, G11, G12, D90, D91, D10, D11 Keywords: equity premium puzzle, habit persistence, borrowing constraints, limited participation, incomplete markets, idiosyncratic income shocks, life-cycle savings, non-expected utility preferences, prospect theory, rank-dependent, expected utility, disaster events, survivorship bias

John Donaldson and Rajnish Mehra

39

INTRODUCTION Mehra and Prescott (1985) demonstrated that the empirical equity premium (the return earned on a well-diversified “market portfolio” in excess of that earned by a risk-free T-bill) is an order of magnitude greater than can be rationalized as a premium for bearing non-diversifiable risk in the context of the standard neoclassical representative agent paradigms of financial economics.1 Inevitably, the subsequent literature has principally focused on modifying the standard paradigm to generate much larger non-diversifiable premia.2 As a companion exercise, these efforts have adopted an ancillary goal of replicating the risk-free rate, rate of return volatilities, return autocorrelations, return cross correlations, and return correlations with consumption growth. In this chapter, we review this literature, which falls into four broad categories: 1. Preference-based theories: These theories seek to modify an agent’s preferences so that he is acutely sensitive in some way to poor consumption outcomes. Outcomes may be “poor” in either an absolute sense or in a relative sense vis-à-vis some external benchmark, as in the habit formation literature. Illustrations sometimes require that agent preferences be defined over objects directly reflective of the agent’s portfolio experience, such as his recent gains or losses. While this latter feature is outside the standard expected utility construct, axiomatic foundations are nevertheless sometimes available. Section 1 of this essay focuses on these theories.3 2. Disaster scenarios: Researchers have also explored models where utility is of the standard CRRA variety, yet in which the worst outcomes are much more disastrous than the worst cases possible in the standard Mehra and Prescott (1985) context. Rietz (1988) and Barro (2006), in particular, fall into this category.4 Literatures 1 and 2 are dual to one another: either agents in the model must effectively be very sensitive to bad outcomes, or it is the outcomes themselves that must be very bad. 3. Trading frictions: Under either of the categories above, the premium will still be counterfactually small if agents can somehow insure against their worst consumption outcomes. Since the standard Mehra and Prescott (1985) representative agent construct is implicitly one of complete markets, perfect insurance opportunities are available in that context. Furthermore, for models parameterized to the per capita aggregate consumption process of the U.S. economy, agents do not find themselves in situations of extreme consumption risk. There must then 1 The following chapter (Mehra and Prescott (2008b)) surveys the literature on non-risk-based explanations of the equity premium. 2 Mehra and Prescott (1985) can match the premium with a very high CRRA. Alternatively, we can view the literature as seeking to match the premium (and other financial quantities) with a low CRRA. 3 For example, Abel (1990), Bansal and Yaron (2004), Benartzi and Thaler (1995), Boldrin, Christiano, and Fisher (2001), Campbell and Cochrane (1999), Constantinides (1990), Epstein and Zin (1991), and Ferson and Constantinides (1991). 4 See also Mehra and Prescott (1988) and Brown, Goetzman, and Ross (1995).

40


be added features to the model, which deny at least some of the agents with risky consumption profiles access to a full set of state claims. Incomplete markets (whether in the form of restricted financial market participation for some agents or in the form of allowing trade in only a restricted set of assets) and collateral constraints are possibilities that have been explored.5,6,7 4. Model uncertainty: Nearly all the asset pricing literature presumes rational expectations; the relevant agents in the economy know the true probability distributions governing the relevant state variables: dividends, wages, etc. The model uncertainty literature presumes that this is not the case; it thereby introduces another source of uncertainty, a feature that gives rise to an additional risk premium for equity securities. Barrillas et al. (2006) and Weitzman (2007) are cases in point. We do not address this entire literature in full detail, as many of the topics are dealt with in specific chapters of the present volume. Category 3, in particular, is thoroughly considered in the essays by Constantinides and by Heaton and Lucas, particularly as regards classical incompleteness (too few securities are traded).8 We thus consider only the impact of borrowing constraints. Loss aversion and narrow framing are carefully considered in the Barberis and Huang chapter.9 As it constitutes a preference-related modification, we consider it in the present chapter but in an abbreviated form. As a field of study, model uncertainty as applied to the premium is in its infancy. While potentially very fruitful, the literature is small and our discussion will be correspondingly brief. Most research has followed the preference route, and this is the literature that we will emphasize.10 Whenever possible, we cast the discussion within the original Mehra and Prescott (1985) discrete-time representative agent setting.

5 For incomplete market formulations, see Bewley (1982), Brav, Constantinides and Geczy (2002), Constantinides and Duffie (1996), Heaton and Lucas (1997, 2000), Krebs (2000), Lucas (1994), Lustig and van Nieuwerburgh (2005), Mankiw (1986), Mehra and Prescott (1985), Storesletten, Telmer, and Yaron (2007), and Telmer (1993). 6 For restricted participation and collateral constraints, see Aiyagari and Gertler (1991), Alvarez and Jerman (2000), Bansal and Coleman (1996), Basak and Cuoco (1998), Constantinides, Donaldson, and Mehra (2002), Daniel and Marshall (1997), Danthine, Donaldson, and Mehra (1992), He and Modest (1995), Heaton and Lucas (1996), and Luttmer (1996), McGrattan and Prescott (2001), and Storesletten, Telmer, and Yaron (2004). 7 Attanasio, Banks, and Tanner (2002), Brav, Constantinides, and Geczy (2002), Brav and Geczy (1995), Mankiw and Zeldes (1991), and Vissing-Jorgensen (2002) are models that also incorporate restricted participation. 8 Constantinides (2008) and Heaton and Lucas (2008). 9 Barberis and Huang (2008). 10 Gabaix and Laibson (2001) and Heaton (1995). The reader is also referred to the excellent surveys by Kocherlakota (1996), Cochrane (1997), Campbell (2001, 2003), and DeLong and Magin (2007), preliminary draft. These papers review the proposed explanations for the magnitude of the premium, and the associated models. For an up-to-date international perspective on the empirical magnitude of the premium, see Dimson, Marsh, and Staunton (2008). Goetzmann and Ibbotson (2008) review the conceptual issues in its measurement over the long term.


41

1. ALTERNATIVE PREFERENCE STRUCTURES 1.1. Preliminaries The focus of this section is to assess the extent to which plausible preference structures alone can rationalize the basic stylized facts as they relate to asset returns and the equity premium. The analysis in the previous chapter shows that the isoelastic CRRA preferences used in Mehra and Prescott (1985) can be made consistent with the observed equity premium only if the coefficient of relative risk aversion is implausibly large. Before proceeding with our discussion of alternative preference structures, we examine in detail some especially attractive features of the CRRA class, which makes it the “preference function of choice” in modern finance and macroeconomics. These features, while desirable in their own right for logical or empirical reasons, also constitute common properties that any reasonable preference specification should display. (i) Equilibrium Return Stationarity CRRA preferences result in a stationary equilibrium return process. This property is readily demonstrated in the context of the Mehra and Prescott (1985) model, where both the equity return and the return on the risk-free asset follow stationary processes, despite the fact that the level of output in the economy is growing over time and hence is itself non-stationary.11 This is consistent with the statistical evidence on the time series of asset returns over the past 100 years. Asset returns appear to be stationary, although the level of stock prices and the magnitude of aggregate dividends have grown enormously. Any serious preference structure should yield equilibrium return series with this feature. (ii) Aggregation Actual asset prices are formed via the trading behavior of large numbers of heterogeneous investors as each attempts to maintain his individual optimal portfolio composition, given his information on the future distribution of returns. Equilibria in such economies are difficult to characterize. If financial markets are competitive and complete, and agent preferences are expected utility, there will, in general, exist, by construction, a representative (single-agent) economy with the same aggregate consumption series as the heterogeneous agent economy and the same asset price functions. These economies are comparatively easy to analyze. In addition, if the representative agent can be constructed in a manner that is independent of the underlying heterogeneous agent economy’s initial wealth distribution, we say the economy displays aggregation. Aggregation (vis-à-vis the existence of a representative agent) is the stronger, more restrictive, and more desirable property. It implies that assets may be priced in the representative agent economy without knowledge of the wealth distribution in the 11 See

Eqs. (8)–(11) in Mehra and Prescott (1985) or Appendix B of Chapter 1 of this volume.

42


underlying heterogeneous agent counterpart. Under aggregation, results derived in the representative agent economy are general and robust. Aggregation also permits the use of the representative agent for welfare comparisons.12 In what follows, we provide a brief description of how the representative agent is constructed and then point out the additional structure imposed by aggregation. Consider an exchange economy of k = 1, 2, . . . K agents. We make the following notational identifications: ctk represents agent k’s period t consumption, while qte and qtb , respectively, denote the period t prices of the equity and one period risk-free security. The period t price of a state claim to one unit of consumption in period t + j, if state θ θ occurs, is denoted by qt,j . By analogy, the corresponding period t desired holdings of e,k b,k θ,k these securities by agent k are, respectively, zt , zt , and zt,j . For notational simplicity, we have suppressed the dependence of these quantities and prices on the period t state. With this notation in mind, the period t decision problem faced by an arbitrary agent k is ∞ max E β t u ctk t=0

subject to e,k

b,k

ctk + zt+1 qte + zt+1 qtb +

∞ j=1

e,k e qt

θ,k

θ zt+1,j qt,j ≤ zt

∞ b,k θ,k θ,k θ + Yt + zt + zt + zt,j qt,j .

θ

j=1

θ

In equilibrium, both goods and asset markets must clear: K

ctk = Yt ,

k=1 K

e,k

zt = 1,

k=1 K

b,k

zt

= 0,13

k=1 K

θ,k

zt,j = 0 for all j, θ.

k=1

Under competitive market conditions, equilibrium is Pareto optimal. This means the equilibrium consumption allocations above are also the solution to

12 Suppose, alternatively, that aggregation fails. Then each possible initial wealth distribution, via its associated representative agent, will display its own asset pricing characteristics. No general statements may be made. 13 The assumption that bonds are in zero net supply is not innocuous, although it is customary in the literature. In particular, it allows representative agent models to match the risk-free rate more easily. For a recent perspective see Gomes and Michaelides (2006).

43


max E

K ∞

λk u ctk

t=0 k=1

subject to K

1

K

ctk ≤ Yt ,

k=1

in general directly depend on the initial endowment where the weights λ , . . . , λ

θ,k b,1 b,K e,1 e,K , k = 1, . . . , K . z0,j and distributions z0 , . . . , z0 , z0 , . . . , z0 θ,j For the given λ1 , . . . , λK we define the utility function of the representative K K

agent as u(Ct ) = λk u ctk , where Ct = Yt = ctk .14 These weights correspond to k=1

k=1

the inverses of the Lagrange multipliers on the consumers’ decentralized problems. Roughly speaking, consumers with greater endowments are accorded higher λs. The equilibrium stock and bond prices in the heterogeneous agent economy are then identical to those arising in a homogeneous agent economy where a “representative agent,” constructed as above, maximizes15 ∞ t E0 β u (Ct ) , 0 < β < 1, t=0

subject to

Ct + zet+1 qte + zbt+1 qtb ≤ qte + Yt zet + zbt ,

where, in equilibrium, zet = 1 and zbt = 0 ∀t. Under this equivalence, the representative agent is a stand-in for the weighted average of the economy’s members. As we have noted, the weights and thus the asset price series itself will, in general, depend on the initial wealth distribution. This is an enormously complicating feature as it suggests that asset prices in any period are in significant ways dependent on wealth distributions many years previously. If all agents in the decentralized economy have preferences of the CRRA type, however, the weights 1 multi-agent λ , . . . , λK will be independent of the initial wealth distribution, a fact that makes the use of representative agent models for asset pricing exercises more plausible. More formally, aggregation requires that the period utility function must be of the 1−1/γ k , where the α k are parameters form u ctk = e−α ct or u ctk = (γ/γ − 1) α k + γctk unique to agent k. To emphasize this point again, even if the individual preferences assume neither of these forms (but are still expected utility), there will continue to 14 We use capital letters to denote economy wide aggregates. For representative agent economies, individual and aggregate quantities coincide, so we use the aggregate notation in those contexts. In situations of clear agent heterogeneity, for example in OLG models, we denote individual agent quantities by lower case variables. 15 See Prescott and Mehra (1980) for an early articulation of this and Constantinides (1982) for a formal analysis of this decentralization perspective.

44


exist a representative agent whose preferences are the weighted average of the individual agent preferences. However, they cannot be used to make any statements about “off-equilibrium” paths.16 (iii) Time-Consistent Planning Time consistency implies that the optimal future-contingent portfolio decisions made at t = 0 remain the optimal decisions even as uncertainty resolves and intermediate consumption is experienced. When considering multi-period decision problems, time consistency is a natural property to propose. In its absence, one would observe portfolio rebalancing not motivated by any event or information flow but rather simply motivated by the (unobservable) changes in the investor’s preference ordering as time passed. Asset trades would be motivated by endogenous and unobservable preference characteristics and would thus be mysterious and unexplainable. To understand the utility restrictions that time consistency imposes, let us consider a two-period context, t = 0, 1, where any one of s ∈ S possible states may occur the next period. Denote by c1 (s) the investor’s consumption at date t = 1 contingent on state “s” occurring. Johnsen and Donaldson (1985) demonstrate that if the overall (both periods’ consumption) utility function is to exhibit time-consistent planning, there must exist continuous and monotone increasing functions f ( ) and {us ( ) : s ∈ S} such that u c0 , c1 s : s ∈ S = f c0 , us c0 , c1 s : s ∈ S , where us (c0 , c1 (s)) is the state s contingent utility function. The recursive preference structure of Kreps and Porteus (1978), which makes explicit the preference for the timing of uncertainty resolution, is closely related. Note that under this formulation, preferences in any future state sˆ may be conditioned on consumption today c0 , but not on consumption planned for states s = sˆ that, de facto, have not occurred. In effect, the utility function must be of a form such that utility representations in future states can be recursively nested as individual arguments of the overall utility function, a condition fulfilled by the expected utility representation. We are thus, assuredly, in a time-consistent planning context under the expected utility assumption.17

1.2. Coincidence of Risk and Time Preferences in CRRA utility One restriction imposed by the CRRA class of preferences is that the coefficient of risk aversion is rigidly linked to the elasticity of intertemporal substitution. One is the reciprocal of the other. If an individual is averse to variation of consumption across different states at a particular point of time, then this feature implies he will 16 Within the context of seeking to replicate the equity premium, the statement is intended to remind the reader that, without aggregation, the asset pricing properties may become unique to the particular underlying initial wealth distribution. As a result, the same asset pricing results may not apply across various initial wealth distributions and associated representative agents. No statements of any generality can be made. 17 Time-consistent planning is satisfied in the Mehra and Prescott (1985) setup, because they assume an expected utility preference representation, in addition to the CRRA period utility specification.

45


be averse to consumption variation over time as well. Since, on average, consumption is growing over time, the agents in the Mehra and Prescott (1985) setup have little incentive to save. The demand for bonds is low, and as a consequence the risk-free rate is counterfactually high. We illustrate this in the context of a deterministic two-period model where agents have preferences of the form u(c, γ) =

c1−γ 1−γ

and we solve the following problem: max u(c0 , γ) + β u(c1 , γ) subject to c0 + s0 ≤ Y0 , c1 ≤ s0 (1 + r). The solution is

1 β(1 + r)

1γ

=

Y0 − s0 (1 + r)s0

=

c0 . c1

As γ → ∞ (greater CRRA), c0 c1 → 1, the agent wishes for an extremely smooth consumption profile across time, a preference that is maintained under uncertainty. In a setting without uncertainty, this effectively means that the agent dislikes growth in his intertemporal consumption profile. In an asset pricing environment in which the representative agent’s consumption is growing exogenously through time (in order that it be properly calibrated), the agent will thus demand a very high rate of return in order to hold securities, the very possession of which increases his intertemporal consumption discrepancy. From the perspective of intertemporal consumption smoothing, the parameter 1/γ is referred to as the elasticity of intertemporal substitution (hereafter EIS). While indeed increasing the premium, increasing the CRRA has the counterfactual consequence of increasing mean returns to a level much in excess of what is observed, especially in the case of the risk-free security. In order to confirm these effects, let us explore the mean returns and volatilities for a number of CRRAs in the context of the basic Mehra and Prescott (1985) model and calibration.18 From Table 1 there is no doubt that increasing risk aversion increases the equilibrium equity premium: as the agent increasingly dislikes consumption variation of any sort, he must be paid an increasing premium in order to be willing to hold risky securities in preference to risk-free ones. It also improves the volatilities of the respective 18 This

model is described in greater detail in Appendix B to Chapter 1 (Mehra and Prescott (2008a)).

46


TABLE 1i The Mehra and Prescott (1985) Model: Various CRRAs (Rates of return annualized in percent) U.S. Dataii

γ=2

γ=3

γ=5

γ=9 18.25

Ere

6.98

7.84

9.58

12.83

SDr e

16.54

4.34

4.99

6.36

9.24

Erf

0.80

7.56

9.10

11.85

15.93

SDr f

5.67

1.06

1.61

2.73

4.99

Erp

6.18

0.28

0.48

0.97

2.31

SDr p

16.67

4.20

4.70

5.69

7.61

i Same ii Data

calibration as Mehra and Prescott (1985), but for various γ. for the period 1889–1978.

return series. But the cost in other counterfactuals is very great. In particular, even as the CRRA is not yet absurdly large (γ = 9), the mean equity and risk-free returns become much too high, at, respectively, 18.25 and 15.93 percent annualized. This is a direct consequence of the single-preference parameter construct: in order to induce the representative agent to hold either type of security—an action that reduces the extent of his intertemporal consumption smoothing—security prices must fall to very low levels. Equivalently, the returns on all types of securities must be very high.19 There is no a priori reason that the desire to smooth consumption across states within a period and the desire to smooth consumption intertemporally should be tightly bound together. After all, one may speak of intertemporal consumption smoothing in a no-risk environment (in a context, say, of motivating savings behavior), and one may as well speak of risk aversion in an uncertain, atemporal environment. The situation considered previously requires that the same parameter describe the extent of sensitivity to both of these variations. Empirical studies seem to suggest, in fact, that investors are more desirous of a smooth intertemporal consumption stream than they are of an atemporal risk-free environment.

1.3. Separating Risk and Time Preferences: Epstein–Zin and others Epstein and Zin (1989, 1991, 2001) and Kreps and Porteus (1978) have emphasized a class of preferences that they term “generalized expected utility” (GEU), which 19 Mehra and Prescott’s (1985) calibration admits a small degree of negative autocorrelation in the growth rate of consumption. Azeredo (2007) argues that the Mehra and Prescott (1985) consumption autocorrelation estimates may be due to consumption data mismeasurement prior to 1929 and that mild positive autocorrelation is more accurate. If the stochastic process governing the growth rate of consumption is positively serial correlated, then as the risk aversion increases progressively, the equity premium eventually declines and turns negative. The earlier assertion that a higher CRRA leads to a larger premium is therefore consumption process-specific. The consumption process underlying most tables and model evaluations in this survey, however, is that of Mehra and Prescott (1985) in order to benchmark a basis of comparison.

47


allows independent specification of the coefficient of risk aversion and the elasticity of intertemporal substitution.20 These preferences separate time and risk preferences in a way that preserves the time-consistency property of the previous section.21 The basic notions involved in separating time and risk preferences are roughly summarized as follows: first characterize preferences over riskless consumption sequences (C0 , C1 , C2 , . . .) with a Koopmans’ (1960) time aggregation V ( ) where U (t C) = V (u(Ct ), U (t+1 C)).22,23

(1)

In the above expression, U ( ) is the overall utility representation, Ct is the period t consumption, u( ) is the period utility function, and (t C) denotes a continuous consumption sequence (Ct , Ct+1 . . .). This framework is then generalized to evaluate uncertain consumption sequences essentially by replacing the second argument in V (, ) by the period t certainty equivalent of the probability distribution over all possible consumption continuations. The resultant class of recursive preferences may be notationally characterized as Ut = V (u (Ct ), μt (Ut+1 )) ,

(2)

where μt ( ) describes the certainty equivalent function based on the conditional probability distribution over consumption sequences beginning in period t + 1. Such preferences are dynamically consistent. If the certainty equivalent is obtained via expected utility, the preferences fall into the Kreps and Porteus (1978) family. If it is obtained via a more general risk aggregator of the Chew–Deckel class (e.g., disappointment aversion risk preferences to be considered shortly), the preferences are said to be of the more general Epstein and Zin (1989) variety. These preferences have axiomatic underpinnings as well.

20 This preference construct has as its intellectual antecedents the time-consistent recursive preference structures of Kreps and Porteus (1978) and Johnsen and Donaldson (1985), and the Chew (1983, 1989) and Dekel (1986) preference generalization, which allows the independence assumption of classical expected utility to be relaxed. The latter is important for at least two reasons. First, there are widely reported violations of independence in the experimental literature; the Allais (1979) paradox is perhaps the most celebrated example. Second, this generalization has permitted the creation of preference structures that allow greater utility weight on bad outcomes than would arise in an expected utility context; see, in particular, Gul (1991). Not only is there empirical support for the hypothesis of high sensitivity to bad outcomes, but it is also a plausible feature that allows consumption-based asset pricing models to better replicate the premium. The reader is directed to Backus et al. (2004) for an elegant and detailed discussion of this preference class. 21 More recently, Kihlstrom (2007) proposes an alternative dynamic choice model. 22 We use the same notation as Backus et al. (2004) and Johnsen and Donaldson (1985). See the former study for a careful discussion of all the issues involved. 23 Preferences over deterministic consumption sequences that have the form (1) satisfy three reasonable properties (by construction): (i) history independence (preferences over consumption sequences (C0 , C1 , C2 , . . .) do not depend on prior consumption); (ii) future independence (preferences for period t consumption, Ct ) are independent of future consumption (t+1 C); and (iii) stationarity (preferences are the same at all dates).

48


In their most basic analysis, Epstein and Zin explore the CES-like specialized preference ordering that is detailed below: 1− 1ρ 1− 1ρ 1− ρ1 ut ≡ u(Ct , CEt+1 ) = (1 − β)Ct + β(CEt+1 )

(3)

with 0 < β < 1, 1 = γ > 0, ρ > 0, or, in the case of ρ = 1, u(Ct , CEt+1 ) = (1 − β) logCt + β log(CEt+1 )

(4)

where CEt+1 = CE(u˜ t+1 ), the certainty equivalent of next period’s utility, is calculated according to (5) [CE (u˜ t+1 )]1−γ = Et (u˜ t+1 )1−γ , 1 = γ > 0, or log(CE (u˜ t+1 )) = Et (log u˜ t+1 ) , γ = 1.

(6)

In the above specification, the elasticity of intertemporal substitution ρ may be specified independently of the CRRA γ. Note that if γ = 1/ρ, recursive substitution to eliminate ut+1 yields ut = (1 − β) Et

∞

1 1−γ

β

j

1−γ Ct+j

,

(7)

j=0

which represents the same preference ordering over uncertain consumptions streams as the ordering given by ∞ 1−γ Et β j Ct+j , j=0

a fact that suggests we may unambiguously view the parameter γ of (3) and (4) as the CRRA (Epstein and Zin (1989) provide a more formal argument). Similarly, as the parameter ρ becomes smaller, the agent becomes less willing to substitute utility, and thus consumption, intertemporally. The parameter β can be viewed as the agent’s customary time preference parameter. Weil (1989) uses these preferences in a setting otherwise identical to that of Mehra and Prescott (1985) to compute the resultant premium. Even as specialized to the Epstein–Zin (1989) preference structure, Weil (1989) demonstrates that the Mehra and Prescott (1985) solution for the equilibrium price functions, q e Yt , xi = νi Yt and q b Yt , xi = ωi , still applies with the proviso that the constants νi , ωi now must satisfy a system of non-linear equations. Using an endowment process {Yt } otherwise identical to that of Mehra and Prescott (1985), Weil (1989) obtains the results detailed in Table 2.

49


TABLE 2i Equilibrium Mean Rates of Return Under Epstein–Zin Preferences: Various Parameters/Selections (Rates of return annualized in percent. The first entry is the premium; the second is the risk-free rate.) EIS

CRRA(γ)

(ρ)

0

0.5

1

5

10

20

45

∞

0.00

0.05

0.10

0.48

0.94

1.77

3.01

5.25

5.24

5.21

5.01

4.78

4.40

4.09

0.01

0.06

0.11

0.51

1.01

1.89

3.14

6.20

6.16

6.12

5.79

5.40

4.73

3.93

0.01

0.07

0.12

0.55

1.08

2.00

3.27

7.14

7.08

7.03

6.56

6.02

5.06

3.76

2

1

0.2

0.1

0.05

1/45

i This

0.10

0.18

0.26

0.88

1.64

2.91

4.34

15.02

14.81

14.61

13.02

11.11

7.75

2.45

0.24

0.35

0.45

1.31

2.33

4.04

5.72

25.73

25.32

24.96

21.68

17.87

11.23

0.85

0.56

0.72

0.87

2.12

3.66

6.25

8.66

50.51

49.55

48.61

41.26

32.80

18.65

−2.23

1.13

1.36

1.60

3.58

6.22

11.22

17.11

138.91

135.56

132.25

107.44

80.69

40.39

−9.22

is Table 1 of Weil (1989).

It is readily apparent that the Epstein–Zin preference construct can, per se, provide some progress toward the equity premium puzzle resolution, but not a complete resolution. In particular, if the CRRA is set at 45 and ρ at 0.1, then a reasonable match to the data (mean values) is obtained. But if a much more reasonable CRRA of 1 is hypothesized, the premium is a scant 0.45 percent, while the mean risk-free rate is 25 percent.24 Both of these features are counterfactual. As Weil (1989) argues, generalizing preferences in the direction of Epstein and Zin (1989) seems only to deepen the puzzle: (1) in abstract economies such as Mehra and Prescott’s (1985), calibrated to observed per capita income dynamics (Ct = Yt ), there is insufficient consumption 24 Consistent with these findings yet using a very different methodological perspective, Kocherlakota (1990) demonstrates that when the growth rate of consumption is i.i.d. (an assumption with substantial empirical support), asset pricing models that separate the CRRA and the EIS as per (1)–(4) have no more explanatory power in empirical tests than those preferences for which these parameters coincide as in the standard Mehra and Prescott (1985) paradigm.

50


growth risk to justify a substantial premium unless agents are implausibly risk-averse. (2) Furthermore, per capita consumption growth is too high (μ = 1.8 percent) for the equilibrium risk-free rate to be low, when agents are averse to intertemporal consumption substitution to the extent observed. In hindsight, these results are not entirely surprising. Even under Epstein–Zin’s (1989, 1991) preferences, there is one parameter alone, γ, which is given the entire role of characterizing the degree of risk aversion. Furthermore, by separating time and risk preferences, their structure tends to strengthen the influence of the intertemporal substitution parameter 1/ρ. As a result, agents must once again be offered a counterfactually high risk-free rate in order to induce them to save more in an environment when mean consumption growth is exogenously fixed at 1.8 percent. Note that Weil’s findings do not disqualify the entire Epstein–Zin family since other approaches to obtaining certainty equivalents may apply. They do suggest, however, that the difficulty may be in separating time and risk preferences where the agent does not need to be too risk-averse to match the data. Epstein–Zin preference constructs have been employed more successfully by Bansal and Yaron (2004) in a richer economic environment. These authors postulate processes on consumption growth and the dividend which are distinct but share a small, but highly persistent time varying expected growth component. The resultant ‘long run consumption risk’ is specially onerous for Epstein–Zin style representative agents who desire an early resolution of uncertainty.25 This latter feature demands a preference specification where the CRRA > (1/EIS), a precondition that the Bansal and Yaron (2004) calibration easily satisfies. For the γ = 10 and ρ = 1.5 case, they are able to replicate all the stylized facts quite well: Ere = 6.84%, Erf = 0.93%, σre = 18.65% and σrr = 0.57%. (Table 4 in Bansal and Yaron (2004)).26 That the persistent growth component is small lends credibility to their basic consumption growth hypothesis, as it is very difficult, empirically, to distinguish this possibility from the more customary i.i.d. consumption growth assumption. While an EIS as high as 1.5 is disputable, there is no prevailing consensus estimate of this quantity, even as regards to it being greater, equal to, or less than one. (See Attanasio and Vissing-Jorgensen (2007), Hall (1988), Guvenen (2006) and Yogo (2004).

25 See

also the chapter by Ravi Bansal in this volume (Bansal (2008)). the presence of long run consumption risk should be an especially important factor in the Epstein and Zin preference context is more apparent if we examine the corresponding process on the marginal rate of substitution, 26 Why

Mt+1 =

U1(Ct+1 ) Ct+1 θ e : ln Mt+1 = θ ln β + ln − (1 − θ) rt+1 U1(Ct ) ρ Ct

Wt+1 + Ct+1 1−γ e = ln , and rt+1 is the period t + 1 return on the agent’s wealth Wt 1 − 1/ρ 1 portfolio, Wt . If γ = ρ , the standard power utility case, the expression collapses whereby only consumption growth matters for the MRS variation. If γ = 1/ρ, the wealth component adds another source of risk. Long run persistence in dividend growth affects this latter term. where θ =

51


We view Bansal and Yaron (2004), its predecessors and successors, as important for a number of reasons.27 First and foremost, it emphasizes the importance of long run growth variation for the asset pricing literature. Their paper also suggests that a full resolution of the equity premium puzzle may demand an improved understanding not only of investor preference structures, but also of the long run statistical characteristics of the principal aggregate macro series.28 More recently, Kallenbrunner and Lochstoer (2006) have demonstrated that the consumption growth process postulated by Bansal and Yaron (2004) will arise endogenously in a fairly standard production style real business cycle model, purely as a result of the consumption smoothing efforts of the representative agent armed with the same Epstein-Zin preference specification. While the authors match a fairly limited set of business cycle facts, their study is important not only for lending further credibility to the Bansal and Yaron (2004) perspective but also for the critical issue of cross model verification. Although standard Epstein and Zin preferences do not necessarily go very far in resolving the equity premium puzzle, it is nevertheless informative to study the full implication of these preferences for the properties of security returns beyond the equity premium. Within the standard Mehra-Prescott (1985) context, Epstein and Zin (1989, 1991) develop an elaborate dynamic programming argument to derive the following asset pricing equation. ⎫ ⎧ ⎨ C˜ − 1ρ θ 1 1−θ ⎬ j t+1 1 + r˜t+1 ≡ 1, Et β e ⎭ ⎩ Ct 1 + r˜t+1

(8)

j

e denotes the period t return on the agent’s wealth portfolio and rt+1 the period where r˜t+1 t return on some asset j included within it.29 In the above expression, the pricing kernel is of the form

1−θ − 1 θ C˜ t+1 ρ 1 β e Ct 1 + r˜t+1 where, θ = (1 − γ)/(1 − 1/ρ).

27 We include in the list of predecessors Backus and Zin (1994) and Cochrane and Hansen (1992). Other implementations of the Epstein–Zin preference construct are Epstein and Zin (1989), Yogo (2006) and Malloy et al. (2005). 28 Bakshi and Chen, (2008) modify the standard model by identifying dividend processes as a fixed fraction of earnings plus an i.i.d error. Potential errors in consumption measurement are there by avoided. They also assume risk free interest rates follow a mean reverting process with constant volatility. As with Bansal & Yaron (2004) they derive closed form solutions and match the premium. 29 Cochrane derives this relationship in the Appendix to his essay in this volume (2008).

52


This is a geometric average (with weights θ and 1 − θ, respectively) of the kernel of the standard CCAPM, − 1 C˜ t+1 ρ , β Ct and the reciprocal of the gross return on the wealth portfolio.30 Epstein and Zin (1991) next consider a linear approximation to the prior geometric average, − 1 1 C˜ t+1 ρ + (1 − θ) , (9) θ β e Ct 1 + r˜t+1 which, when substituted into (8), yields ⎧ ⎫ ⎪ ⎪ − 1ρ ⎨ ⎬ C˜ t+1 1 j j Et θ β 1 + r˜t+1 + (1 − θ) 1 + r ˜ =1 e t+1 ⎪ ⎪ Ct 1 + r˜t+1 ⎩ ⎭

(10)

As is well known, the standard CAPM relates the undiversifiable risk of an asset to the covariance of its returns with the returns on the market portfolio. When time and risk preferences are distinct, Eq. (10) suggests that both covariances matter: the covariance of an individual asset’s return with the market portfolio captures its essential undiversifiable risk, while the covariance of its returns with the consumption growth rate fundamentally captures its risk across time periods. With separate time and risk preferences, it is natural that both should be separately and individually present. But, as we are aware, this alone does not, in general, solve the puzzle.31

1.4. Variation in the CRRA and EIS But what about the basic Epstein–Zin construct generalized to admit time variation in the CRRA and EIS parameters? Several authors, in particular, have suggested that countercyclical risk aversion may allow for resolving the puzzle (cf. Campbell and Cochrane (1999), Gordon and St. Amour (2000, 2003), and Danthine et al. (1992)). Note, however, that such modifications should still respect time-consistent planning and aggregation while also generating stationary equilibrium security returns. 30 The

difficulty in having a pricing kernel that is dependent on the return to the agent’s wealth portfolio is the ambiguity that is introduced in the corresponding empirical tests of the model: what is a reasonable proxy for the wealth portfolio? Epstein and Zin (1991) use the traditional “market portfolio” as that proxy, but it can reasonably be argued that this overstates the covariance of investor wealth and individual asset returns. In particular, there is no labor income in their model, and thus their abstraction implies that human capital wealth is a negligible component of total wealth. 31 Azeredo (2007) confirms that mild positive autocorrelation in the growth rate of consumption also leads to negative equity premia for reasonably parameterized Epstein–Zin utility (EIS near zero) in a standard Mehra and Prescott (1985) setting. For mild positive serial correlation and EIS near 1, the equity premium is positive but still very small.

53


Simple generalizations to admit countercyclical risk aversion are, first and foremost, problematic along the return stationarity dimension. Consider a simple expected utility, representative agent preference specification of the form E

∞

t

β u (Ct , γt ) ,

t=0

where 1−γ

u (Ct , γt ) =

Ct t 1 − γt

with the stochastic process on γt , the period CRRA, chosen so that corr (γt , gt ) < 0 (gt denoting the output growth rate, gt = Yt /Yt−1 − 1). An exploration of this non-trivial generalization of the standard Mehra and Prescott (1985) paradigm is revealing (see Danthine et al. (2004) for details). In particular, returns are non-stationary. This latter assertion can be seen from a straightforward calculation of the representative agent’s equilibrium (Ct = Yt ) intertemporal marginal rate of substitution (IMRS): IMRSt, t+1 = β

(Yt ) γt − γt+1 u1 (Yt+1 , γt+1 ) =β . u1 (Yt , γt ) (gt+1 )γt+1

(11)

Clearly, the IMRSt,t+1 depends not only on a transformation of the growth rate of output alone, as in Mehra and Prescott (1985), but also directly on the level of output Yt . The presence of Yt in expression (11) guarantees that risk-free asset prices and the equilibrium risk-free return will not be a stationary series (the same is true of the risky asset). We notice also that the presence of γt and γt+1 introduces another source of uncertainty into the agent’s economic environment, albeit one that he correctly anticipates (rational expectations). It turns out that this single-parameter curvature variation has big consequences for the equity premium and risk-free rate generated by the model. This is presented in Table 3, where the output growth process is otherwise identical to that found in Mehra and Prescott (1985). Note that under these parameterizations, the problem is not that returns are insufficiently volatile or that the premium is too small, but that both these quantities are too large! Returns on both securities, but especially the equity security, are also much too large relative to the data. The point is not that the model fails, for the idea that one parameter—even if it now assumes two distinct values—should allow enough flexibility to explain a myriad of financial statistics remains a bit preposterous. Rather, what is of interest is the qualitative fact that parameter uncertainty appears to have an enormous influence on the representative agent’s demand for securities. But can this effect play itself out more constructively in an Epstein–Zin context? Melino and Yang (2003) explore a generalized version of Epstein and Zin (1989) that admits parameter uncertainty in the CRRA for “timeless gambles,” the EIS, and the

54


TABLE 3 Variation in the CRRA: Representative Cases (Summary return statistics in percent, annualized; unless otherwise indicated, β = 0.96, π = 0.47, μ = 0.018, σ = 0.036, N = 120, Y 0 = 1)i,ii (1) Mehra and Prescott (1985) γ1 = γ2 = 3

(2) γ 1 = 1.0 γ 2 = 1.5

(3) γ 1 = 0.5 γ 2 = 1.0

(4) γ 1 = 4.0 γ 2 = 4.5

(5) γ 1 = 1.5 γ 2 = 1.0

(6) γ 1 = 1.0 γ 2 = 1.5

Ere

9.58

16.95

16.14

21.5

19.56

18.77

SDr e

4.99

53.04

53.17

51.94

59.81

58.36

Erf

9.10

8.47

7.19

15.74

6.84

7.79

SDr f

1.61

34.21

33.90

35.78

34.61

35.75

Erp

0.48

8.48

5.76

12.72

10.98

SDr p

4.7

34.92

32.75

43.21

40.32

8.945 35.26

i Panel 1 reports results from the original Mehra and Prescott (1985) model when the CRRA is fixed at γ = 3. ii The number N indicates the length of the time series of prices and dividends that was used to compute return statistics. N = 120 corresponds to 30 years of data. Reported mean returns and standard deviations represent averages of estimates obtained from 1000 independently generated time series of this length.

agent’s subjective discount factor:32 1 ρ γt ! γtt ρt ρ ut = ct t + βt Et ut+1 .

(12)

They also allow γt ∈ {γ H, γ L }, ρt ∈ {ρH, ρL }, and βt ∈ {β H, β L }, where all parameters superscripted by H occur simultaneously, and so also for those superscripted by L (there are two parameter sets); H and L are also coincident with the high and low growth states, respectively. Roughly speaking, they derive expressions for the equilibrium Ere, SDr e, Erf, and SDr f in the context of a consumption growth process identical to that of Mehra and Prescott (1985) and then ask what sets of parameters allow the resolution of these equations to satisfy the observed values of Ere , SDr e , etc. Melino and Yang (2003) find that certain reasonable sets of parameters do allow the exact replication of the four required historical moments. Generally speaking, this is accomplished with “modest procyclical variation in the EIS, but strong countercyclical 32 More precisely, under this specification, 1 − γ is the CRRA , EIS is defined by 1/(1 − ρ ), and β is the t t t t t customary (though now stochastic) subjective discount factor. Formulation (11) leads to an equilib!1−γ(st )/ρ(st ) −1 γ(s )−1 Qt /β(st )) rium IMRSt of the form IMRSt+1 = β (st ) gt+1t [1 + Qt+1 ][γ(st )/ρ(st+1 )] , where st e denotes the underlying state influencing β, γ, and ρ; gt is the period t growth rate, and Qt = qt /ct , the “priceto-earnings ratio.” From our discussion of Epstein and Zin (1989, 1991), we would expect such a term to be present in the IMRSt .

55


TABLE 4i Parameter Values that Replicate Ere, SDr e, Er f, SDr f (Countercyclical CRRA 1 − γ and procyclical EIS 1 / (1 − ρt ); constant β) β

γL

γH

ρL

ρH

0.950

−25.00

0.89

1.25

1.31

0.960

−51.89

1.85

0.16

0.17

0.965

−18.91

0.67

−0.38

−0.40

0.970

−21.21

0.76

−0.92

−0.97

0.980

−22.25

0.79

−1.98

−2.10

0.990

−22.57

0.81

−3.04

−3.22

i This

is Table 4 of Melino and Yang (2003).

risk aversion on the part of the representative agent.”33 Variation in the subjective discount factor is found to have only negligible effects. A sample of the relevant parameters is found in Table 4. The reader may judge for herself whether any of the reported combinations constitute a reasonable resolution of the equity premium and associated puzzles. We venture only to comment that, for all cases, the degree of risk aversion implied by the low growth state seems high, especially in a context where the probability that the low growth state will continue for more than one period is less than 50 percent. There is also the added complication that risk tolerance cannot be uniquely identified with the γ parameter; rather, it appears that β, γ, and ρ all enter into the preference for random consumption sequences. Melino and Yang (2003) is important, for it presents with clarity the importance of countercyclical risk aversion at least as a (joint with procyclical EIS) sufficient quantity for resolving the first and second moments of the equity and risk-free securities.

1.5. Habit Formation A second approach to modifying preferences, that of incorporating habit formation, was initiated by Constantinides (1990) and Sundaresan (1989).34

1.5.1. The Concept Under this specification, the agent’s period-by-period utility is a function not only of his current consumption but also of his past consumption history. In its simplest form, the representative agent is postulated as maximizing E

∞

β t u(Ct − α Ct−1 ),

(13)

t=0 33 Melino

and Yang (2003), Section 7. et al. (2004) develop a notion of habits being formed at the level of individual goods. It is a notion that has yet to be applied, to our knowledge, in the equity premium literature. 34 Ravn

56


where Ct−1 denotes the agent’s prior period consumption and α measures the intensity with which the prior period’s consumption experience influences utility today. Note that preferences are no longer time separable under this specification: by assumption, an increase in the agent’s current consumption, while lowering the marginal utility of his consumption today, increases the marginal utility of his consumption tomorrow. Informally, if the “agent eats more today, he ends up being hungrier when he wakes up tomorrow.” A number of variations on this basic model are considered in the literature. Under the most obvious of these generalizations, the agent’s “stock” of habit is represented as some known function of his entire consumption history: u( ) = u(Ct , St ), where St = S(Ct−1 , Ct−2 , . . .) denotes an aggregator function that summarizes the agent’s past consumption experience. Alternatively, the effect of the habit may be captured as the ratio of current consumption to its habit or benchmark level as per St , u1 ( ) > 0, u2 ( ) > 0. u(Ct , St ) = u Ct , Ct Both of these representations are referred to as internal habit models, as the magnitude of the habit is determined by the actions of the individual agent alone.35 Pollock (1970) and subsequently Abel (1990) postulate a habit that is external to the agent. Since Abel (1990), this perspective has been referred to as “keeping up with the Joneses.”36 It suggests that the agent is primarily concerned with his own consumption relative to the consumption of others in society; for example, u( ) = u(Ct − αC ¯ t−1 ), where C t−1 denotes the societal average consumption level in the previous period and α¯ the measure of his sensitivity to it. From the agent’s perspective, C t−1 is exogenous. Under this interpretation the agent wakes up feeling hungrier if those around him had all enjoyed a large meal. It is an indirect way of modeling an agent’s preference for relative “status.” In a representative agent equilibrium context Ct = C t , but this does not imply that internal and external habit formations lead to equivalent asset pricing relationships.37 35 Habits that enter as a ratio to current consumption are compatible with balanced growth (see King and Rebelo (1988)), whereas difference representations such as (11) are not. In order to restore balanced growth in the latter case, the difference must be normalized by some series that grows on average at the same rate as consumption. Such a normalization will appear in most preference specifications going forward. 36 See Andrew Abel’s chapter in this volume (Abel (2008)). 37 To demonstrate the difference, we need only write out the equilibrium equations that the equity price must satisfy: in the case of an internal habit, Ct−1 = C t−1 , qte satisfies " ! e + dt+1 + u1 (Ct+1 − αCt ) αqte dF ( ), a. u1 (Ct − αCt−1 ) qte = β u1 (Ct+1 − αCt ) qt+1 where the final term reflects the fact that higher asset purchases in period t reduce the habit in t + 1. In the case of an otherwise " identical externalehabit, ! b. u1 (Ct − αCt−1 ) qte = β u1 (Ct+1 − α Ct ) qt+1 + dt+1 dF ( ).

With a finite number of possible dividends (and thus a finite number of habit levels), in either case the state-contingent asset prices can be obtained as the solution to a system of linear equations (leading to a level stationary equilibrium). This is directly analogous to the Mehra and Prescott (1985) situation.

57


We summarize these various alternatives by the tree structure below: habit formation external measured as a difference

internal measured as a ratio

difference

ratio

Strictly speaking, preferences structures of this form lie outside the family of recursive preference structures that admit consistent planning for a subtle reason: utility today depends on past consumption, which may depend on consumption plans that are no longer feasible. The way around this is to view past consumption as a habit state variable and view utility today as conditional upon it. In this way, consistency of choice is preserved. Aggregation is also preserved under this simple habit formation, provided u( ) is of the CRRA form.

1.5.2. Relationship to the Equity Premium As noted elsewhere in this volume, one source of the equity premium puzzle is the very low volatility in per capita consumption growth. If the risky asset is to command a high rate of return in an environment of very little (per capita) consumption risk, agents must be extremely risk-averse—much more so than micro studies to date have been able to identify. Incorporating a habit feature sidesteps this issue by postulating that agents are averse to variations in habit-adjusted levels of consumption rather than to consumption variation itself. Especially in circumstances where the current consumption level and the habit do not differ significantly, small changes in consumption growth can generate very large variations in habit-adjusted consumption growth. Even in the presence of modest overall habit-adjusted risk aversion, such variation may contribute to a resolution of the puzzle. Another way to present the mechanism underlying habit models is to compute the agent’s effective (or local) CRRA in the special case where the habit is well proxied by a fixed subsistence level of consumption:38 u( ) = u Ct − C =

38 See

also the discussion in Weil (1989).

Ct − C

1−γ

1−γ

.

(14)

58


Under these circumstances, −Ct u11 (Ct ) γ = . u1 (Ct ) 1 − C/Ct

(15)

Note that if the subsistence level of consumption is high relative to actual consumption (a high threshold for status), then the local CRRA can be very large. For example, if γ = 3 and C/Ct ≈ 0.9, then the effective CRRA ≈ 30. Under this habit specification, effective risk aversion is time-varying: higher consumption in period t − 1 provokes the agent to be more risk-averse, ceteris paribus, in period t. It is also countercyclical: when Ct (and output) is low relative to C in a recession, C/Ct will be high and thus the effective CRRA high. With effective risk aversion not directly observable, these theories will be difficult to test empirically. Nevertheless, these simple relationships give us some idea of what is going on. They also provide an early warning sign that naive habit formation may not necessarily represent a “cure-all” that allows otherwise standard stochastic equilibrium paradigms to explain the full range of financial phenomena. The basic ingredient of the model, per capita consumption, in particular, does not display much variation in the data. This suggests that under a proper calibration Ct and Ct−1 will typically be similar in value. With an α close to 1 and standard CRRA utility, the argument Ct − αCt−1 will frequently be very small and be on the most concave portion of the agent’s utility surface. Marginal utility will be highly volatile and, as a consequence, the risk-free rate is likely to be counterfactually highly volatile as well.39 This is borne out in many studies for a wide range of settings (see, e.g., Constantinides (1990), Jermann (1998), etc.). Taming this excessive volatility is a particular focus of the habit specification described in the next section.

1.5.3. The Campbell and Cochrane (1999) Mechanism Campbell and Cochrane (1999) postulate a process on an external habit, Xt , and a period utility function of the simple CRRA form: u(Ct , Xt ) =

(Ct − St )1−γ , 1−γ

(16)

where Ct in this case denotes the individual agent’s consumption and Ct − Xt St = , Ct his surplus consumption habit. Note that from the agent’s perspective, St is exogenous although in equilibrium this will not be so. They next postulate that (1) consumption growth follows an i.i.d. log-normal process, 39 The

reader is also referred to the section on habit formation in Mehra and Prescott (2008b).

59


Δct+1 = log Ct+1 − log Ct ≡ g + ν˜t where ν˜t ∼ i.i.d. N 0, σν2 ,

(17)

and (2) the log of the surplus consumption ratio log St = st also follows a log-normal process to be consistent with (1): st+1 = (1 − φ) s¯ + φ st + λ (st ) (Δct+1 − g) ,

(18)

where φ, s, ¯ and g are parameters, s¯ = log S (S denotes the steady-state surplus consumption ratio), and λ (st ) is a prespecified sensitivity function. As a result, the period t, t + 1 pricing kernel is of the form −γ u1 (Ct+1 , St+1 ) St+1 Ct+1 · MRSt, t+1 = β =β U1 (Ct , St ) St Ct ¯ t )+(1+λ(st ))Δct+1 −λ(st )g] = βe−γ[(1−φ)(s−s ¯ t )]ν¯t+1 ] = βe−γ[(φ−1)(st −s)+[1+λ(s

(19)

by identification (1) above. Under this specification it is apparent that the standard deviation of the MRS, as well as its correlation with consumption growth (both important determinants of the premium), will be determined by the form of the sensitivity function. Campbell and Cochrane (1999) go on to make the following assumption on the form of λ (st ): ⎧ ⎨1# 1 − 2(st − s) ¯ − 1 st ≤ smax (20) λ(st ) = S ⎩ 0 s ≥s t

max

# where S = σ β/(1 − φ) and smax = s¯ + (1/2) 1 − S 2 . These requirements allow the habit formulation not only to reflect certain key features of the data but also to generate a plausible habit process.40 In doing so, the authors attempt to set an upper bound on what can be achieved with a habit model in an exchange setting; that is, they seek to provide a sort of “performance standard” that other competing habit formulations must exceed. No axiomatic foundations for the postulated surplus consumption process are proposed, however. Lastly, an i.i.d. process on dividend growth is postulated, one that is imperfectly correlated with growth in consumption: Δ dt+1 = g + ω˜ t+1 , ω˜ t+1 ∼ i.i.d. N 0, σω2 with corr(ω˜ t , ν˜t ) = ρ. (21) As before, dt denotes the log of the actual period t dividend, Dt . With an independently specified dividend process, the Ct = Yt identity characteristic of Mehra and a first-order approximation, (14) and (15) imply Xt+1 = φXt + (1 − φ) (1 − S)Ct , which appears fairly conventional.

40 To

60


Prescott (1985) is broken, allowing an implicit wage process, levered capital structure, etc. The parameters ρ and σω remain to be specified. The model—and, in particular, the surplus consumption process—is designed to replicate the observed low risk-free rate volatility. Taking advantage of the well-known properties of the log-normal return process in conjunction with model specifications (16)–(19), it becomes a matter of straightforward manipulation to derive the implied f expression for log rt : f

ln rt = −lnβ + γg − γ(1 − φ) (st − s) ¯ −

γ 2σ2 (1 + λ (st ))2 . 2

(22)

Under substitution into (18), expression (22) becomes f

ln rt = −lnβ + γg −

γ (1 − φ) , 2

(23)

a constant. This clearly is an extreme case of low volatility, and Campbell and Cochrane (1999) offer generalizations by which the precise real risk-free rate volatility can be matched as well. Plausibility dictates that the habit should not be predetermined, except possibly at the steady state, and formulations (18) and (20) also satisfy this requirement. Indeed, the fact that Δct+1 appears in the expression for the period t + 1 surplus consumption ratio st+1 dictates that current consumption growth has some influence on the habit.41 Since the habit is external, it represents the influence on an individual’s preference ordering of the average consumption plan of others in the society. Even in an environment of instant communication, it is perhaps unreasonable to presume an immediate and fully external habit adjustment process. The sensitivity parameter λ (st ) is also increasing in st by construction. This guarantees countercyclical risk aversion and, as the authors demonstrate, a countercyclical market price of risk. By making the agent very risk-averse when consumption is already low, this feature accentuates the perceived riskiness of a variable consumption stream. Table 5 presents some sample results of this elaborate exercise. From these results it is clear that habit formation as articulated by Campbell and Cochrane (1999) is a powerful mechanism for influencing security returns in a dynamic stochastic general equilibrium model. Its flexibility is even greater than what Table 5 suggests since Campbell and Cochrane (1999) subject the model to other tests (e.g., predictability), which it easily transcends. Yet we remain uneasy, as there is no empirical evidence to support the assertion that surplus consumption habits evolve in the way proposed. Furthermore, for the reported simulation, the agent’s effective CRRA varies between 1 and 100, which is arguably extreme. The surplus consumption habit process in Campbell and Cochrane (1999) is highly specialized in other ways. Ljungqvist and Uhlig (1999), in particular, point out that under their specification, the representative agent would experience substantial welfare 41 Including current consumption as a part of the habit formation process may be used to ensure that period t’s consumption never falls below period t’s habit. Otherwise, period utility may not be well defined.

61


TABLE 5 Equilibrium Security Returns and the Premium in a Campbell–Cochrane Model (Rates of return, growth rates annualized in percent; g = 1.89, σ = 1.50, lnr f = 0.94, Φ = 0.87, ρ = 0.20, σω = 11.2, β = 0.89, S = 0.057, Smax = 0.094) Postwar data

Simulated moments

Erf

0.94

0.094

SDr f

(not reported)

0

Ere

7.58

7.630

SDr e

(not reported)

15.200

Erp

6.69

6.640

SDr p

15.7

15.200

EΔct

1.89

1.890

SDΔct

1.22

1.220

Source: Campbell and Cochrane (1999), various tables.

gains if 10 percent of her endowment were periodically destroyed. The basic intuition is straightforward: while utility is diminished in the period in which consumption endowment is destroyed, future utility gains result since the habit is correspondingly lower. If the former loss is more than compensated for by the latter gains, the overall result is welfare-enhancing. While this is never the case under standard linear habit evolution, this is possible under the Campbell and Cochrane (1999) construct.42 These anomalies suggest the need for an axiomatic treatment of habit formation.

1.6. Behavioral Models Behavioral models are ones for which the postulated agents are not “fully rational.” While this can mean that they do not have rational expectations43 or that they do not 42 These observations are not as general as would initially appear. Consider the representative agent utility

function β t {g (xt ) + v (ct , xt )}, where xt is the aggregate consumption history, ct is the agent’s time-t consumption and v(ct , xt ) is increasing and concave in ct ; g(xt ) and v(ct , xt ) together constitute the agent’s period utility function. With an external habit, marginal utility is given by ∂v (ct , xt ) /∂ct , independent of g(xt ). The class of utility functions with common v(ct , xt ) but different g(xt ) support the same equilibrium but may, in general, have different welfare implications. Campbell and Cochrane indeed focus on equilibrium implications driven exclusively by marginal utility ∂v (ct , xt ) /∂ct . It is not the case that all utility functions in the class will necessarily exhibit identical welfare implications and therefore the criticism leveled by Ljungquist and Uhlig (1999) against Campbell–Cochrane cannot be viewed as a general statement. We thank George Constantinides for pointing this out to us. 43 By this we mean, in the context of this essay, that agents know the exact probability distribution governing all random outcomes—returns, consumption, etc.—driving the model.

62


update their beliefs using Bayes’ rule, in the context of this essay it will be taken to mean that the agents make decisions in accordance with preferences that are not expected utility defined over consumption or wealth (indirect utility). In this sense, “habit formation” preferences are behavioral, although their successful use in explaining a wide class of phenomena has given them a degree of acceptance not typical of the broader behavioral finance literature. While the proposed family of behavioral preference structures is otherwise quite large, most have not been applied to the study of the “equity premium” and associated puzzles.44 For this reason, we restrict our attention to four: (1) the happiness maintenance preferences of Falato (2003), (2) the disappointment aversion and generalized disappointment aversion of Gul (1991) and Routledge and Zin (2004), (3) the prospect theory of Kahneman and Tversky (1979) as applied, most especially, by Barberis et al. (2001), and (4) the rank-dependent expected utility preference structures of Quiggin (1982, 1993).

1.6.1. Happiness Maintenance Consistent with a large body of experimental evidence (see Isen (1989) for a review), Falato (2003) builds on the assumption that risk aversion may be procyclical. The phenomenon he seeks to capture is the following: investors who are experiencing high utility—due either to a favorable income or to wealth shock—become more risk-averse, as they do not wish these favorable circumstances to be lost; that is, they seek “happiness maintenance.” Falato’s model postulates that investors derive direct utility from both consumption and the consumption-to-wealth ratio. (As with Melino and Yang (2003), the associated MRS in his model also necessarily contains as an argument the consumption-to-wealth ratio.) In particular, Falato (2003) assumes that the agent’s period utility is defined over a composite good u( ) = u(gt ) = 1−θt

where g(Ct , Wt ; θt ) = Ct

(g (Ct , Wt ; θt ))1−γ , 1−γ

θ

Wt t and Wt , wealth, satisfies Wt = qte + Yt zet

(the setting is otherwise a pure Mehra and Prescott (1985) exchange economy). The agent’s full preference ordering over random consumption sequences is thus given by E0

∞ t=0

44 For

1−γ

βt

Ct 1−γ

Wt Ct

(1−γ)θt ,

the wider classes that have been used for macroeconomic studies, see Backus et al. (2004).

(24)

63


where θt = θ x¯ t (n) and n 1 Yt−τ . x¯ t (n) = n+1 Yt−τ−1

(25)

t=0

If n = 0, Eq. (25) reduces to

Yt θt = θ Yt−1

.

It is this latter feature that captures the “happiness maintenance” phenomenon underlying Falato’s (2003) work: high average dividend (consumption) growth in the recent past, x¯ t (n), leads the agent to be more risk-averse over wealth in the current period via the exponent (1 − γ)θt of the wealth-to-consumption ratio. Falato (2003) proves that equilibrium prices exist for a Mehra and Prescott (1985) economy when the representative agent has “happiness maintenance” preferences.45 Furthermore, the postulated ordering admits consistent planning and generates stationary return series. Preliminary analysis suggests that a representative agent may be constructed. The issue of full aggregation remains open, however. As with Melino and Yang (2003), the methodology required to calculate statecontingent asset prices and returns involves the solution to a non-linear system of equations. Table 6 contains the results of this exercise for a sample set of parameters. We are drawn particularly to the γ = 3, θ = 0.24 case, which matches nearly all the moments exceptionally well, with the sole qualification that the SD of the equity return is about 30 percent too large. This is furthermore accomplished in a model with fewer degrees of freedom than, e.g., Melino and Yang (2003). The presence of the consumption-to-wealth ratio as an argument of the period utility function is not entirely surprising given earlier work by Lettau and Ludvigson (2001). What is perhaps more striking (but not reported in the table) is the ability of Falato’s model also to explain the predictable feature of stock returns (see, e.g., Campbell and Shiller (1988)) at least as well as the habit construct of Campbell and Cochrane (1999).46 Countercyclical risk aversion is clearly not a necessary condition for this class of models to replicate, acceptably, the equity premium and allied statistics. Furthermore, the addition of the (Ct /Wt ) ratio does not require the presence of a scaling factor since the consumption-to-wealth ratio series is endogenously stationary. The remaining behavioral approaches rely directly on countercyclical risk aversion. In both of them, the investing agents are concerned especially with bad outcomes relative to some benchmark, and it is how this latter quantity is defined that sets them apart.47 45 Such an exercise is non-trivial. The main challenge to establishing existence derives from the endogenous nature of the pricing kernel, which is, in turn, the result of the utility function’s dependence on wealth. 46 In Falato’s (2003) model, the cost of the business cycle is also about ten times the original estimate of Lucas (1985). See also Alvarez and Jermann (2004). 47 As such, they are direct descendants of Reitz (1988). Reitz (1988) retains the expected utility representation and postulates a low-probability-very-low-consumption state. In Routledge and Zin (2004) and Barberis et al. (2001), the “disaster” is measured relative to a benchmark. This fact facilitates a more reasonable model calibration.

64


TABLE 6 Financial Statistics under Happiness Maintenancei (All cases: β = 0.99, n = 2, π = 0.43, μ = 0.018, δ = 0.036; various γ, θ; returns annualized in percentage terms)ii E(rf )

θ = 0.24

θ = 0.18

θ = 0.12

θ = 0.06

θ = 0.00

E(re )

E(rp )

SD(rf )

SD(r e )

SD(rp )

γ = 1.1

2.36

2.56

0.19

0.71

4.04

3.98

γ=2

2.85

3.75

0.90

1.87

8.12

7.90

γ=3

0.84

6.95

6.11

5.55

23.17

22.57

γ = 1.1

2.48

2.66

0.17

0.68

3.94

3.88

γ=2

3.27

3.89

0.63

1.64

6.66

6.45

γ=3

3.46

5.60

2.15

3.69

12.69

12.14

γ = 1.1

2.60

2.76

0.16

0.63

3.85

3.80

γ=2

3.62

4.09

0.46

1.39

5.59

5.41

γ=3

4.49

5.59

1.09

2.62

8.38

7.95

γ = 1.1

2.72

2.87

0.15

0.60

3.77

3.72

γ=2

3.94

4.30

0.36

1.20

4.83

4.67

γ=3

5.16

5.85

0.69

2.00

6.18

5.85

γ = 1.1

2.85

2.99

0.14

0.56

3.69

3.65

γ=2

4.27

4.55

0.29

1.04

4.23

4.10

γ=3

5.74

6.22

0.48

1.57

4.87

4.60

i Table ii The

4.1 in Falato (2003). μ, δ parameters refer to the output growth process in Mehra and Prescott (1985).

We emphasize the Routledge and Zin (2004) generalization of Gul (1991) since the latter model does not satisfactorily explain the premium in the context of a straightforward Lucas (1978)-Mehra and Prescott (1985) tree economy while the former does.48

1.6.2. Generalized Disappointment Aversion Disappointment-averse preferences place a greater utility weight on outcomes that disappoint. For Gul (1991), this threshold is to be identified with consumption realizations that lie below the (conditional) certainty equivalent consumption level, while for Routledge and Zin (2004) outcomes more distant from the certainty equivalent are given special weighting.49

48 Bonomo and Garcia (1993), however, combine a joint Markov switching endowments model with a disappointment version to match the first and second moments of the equity premium and the risk-free rate. 49 Gul (1991) disappointment aversion (DA) preferences in a standard Mehra–Prescott (1985) economy do not per se resolve the premium. Bonomo and Garcia (1993) enhance the construct by incorporating a joint process on consumption and dividends that follows a Markov switching model—in addition to (DA) preferences. They are able to match the risk-free rate and the first two moments of the premium.

65


As in the Epstein and Zin (1989) utility construct, time and risk preferences are separated. Their recursive intertemporal utility function U(Ct , CEt+1 ) has the form U (Ct , CEt+1 ) = (1 −

1 1+ρ

γ Ct

1 γ + CE 1 + ρ t+1

1/γ ,

(26)

where CEt+1 is the certainty equivalent pertaining to random consumption in the following period. In functional form (26), ρ is the marginal rate of time preference and 1/1 − γ the elasticity of intertemporal substitution. Risk preferences and the certainty equivalent computation are specified as follows. Let us presume a pure Mehra and Prescott (1985) setting with a finite number of states and suppose, in some period t, that the output level is Yti = Y i. Temporal risk preferences are represented as a period utility function of the form u(C) = C α /α (α = 0) or u(C) = log C(α = 0). The period t + 1 certainty equivalent consumption level CEt+1 = CEt+1 Y˜ i is then endogenously determined as the solution to the following definitional equation: α CEt+1 Y˜ i i ˜ u CEt+1 Y = α j α α j α δCEt+1 Y i Y Y πi j = . πi j −θ − α α α j Y j ≤ δCEt+1 (Y i ) (27) If θ = 0, structure (27) reduces to the definition of a certainty equivalent in a pure expected utility context.50 Otherwise, there is a penalty, proportional to θ, for consumption realizations that lie sufficiently below the endogenously determined certainty equivalent. If δ = 1, the certainty equivalent computation follows according to Gul (1991); if δ < 1, the context is that of Routledge and Zin (2004). Note that there is no mention of any external reference point as in the (external) habit formation literature. Nevertheless, the model generates countercyclical risk aversion, a feature that bodes well for its asset pricing characteristics. Furthermore, GDA preferences are homothetic, are linear in probabilities, and have the same aggregation properties as CRRA expected utility structures. They admit consistent planning, and Routledge and Zin (2004) provide the axiomatic underpinnings. Table 7 gives a quantitative idea of how well GDA preferences perform in the standard Mehra and Prescott (1985) environment. A number of these results are worthy of explicit mention. When dividend volatility is low, GDA preferences give only modest premia; they become quite large only when SD(Yt ) = 0.10. For the data period underlying the empirical background to the 50 When θ = 0, the Gul-Routledge-Zin structures relax the independence assumption that underlies the expected utility representation. Independence fails because mixing in some arbitrary payoff lottery with two lotteries under comparison may fundamentally change the tail properties of each.

66


TABLE 7i Equilibrium Security Returns and the Premium: GDA Preferences (Various parameter combinations; all returns in percent) Parameter choices A

B

C

D

E

F

−0.5000

α

1

1

0.5000

−2

1

γ

1

1

1

1

0.5000

θ

9

24

9

9

9

9

δ

0.9692

0.9692

0.9692

0.9692

0.9692

0.9692

ρ

0.0300

0.0300

0.0300

0.0300

0.0300

0.0300

1

SD(Yt ) = 0.05 A

B

C

D

E

F

Erf

1.74

1.56

1.72

1.62

2.45

3.89

SDr f

1.26

1.44

1.25

1.23

1.21

1.11

Ere

4.30

4.68

4.33

4.46

5.07

6.63

SDr e

5.72

6.14

5.68

5.52

5.86

6.14

Erp

2.56

3.12

2.61

3.84

3.86

2.74

F

SDr p

not reported SD(Yt ) = 0.10 Aii

B

C

D

E

Erf

1.74

1.56

1.72

1.62

2.45

SDr f

1.26

1.44

1.25

1.23

1.21

1.11

Ere

8.07

9.66

8.17

8.59

8.86

10.46

SDr e

14.37

16.26

14.27

13.80

14.55

14.91

Erp

6.33

8.10

6.45

6.97

6.65

9.35

SDr p

3.89

not reported

i Table

7 is an amalgamation of selected results reported in Tables 1 and 2 in Routledge and Zin (2004). return statistics for the risk-free rate are identical for the SD(Yt ) = 0.05 and 0.10 cases because risk-free securities do not result in disappointments. ii The

article, de-trended dividend volatility is on the order of 12 percent, which advocates for the model with SD(Yt ) = 0.10; Campbell (1999) argues that the SD of dividend should be as high as 28 percent. Qualitative results generally match intuition; comparing columns A and B, the result of a greater penalty is to raise the risk-free asset prices and lower equity ones: risky securities are less desirable and the premium rises. Comparing columns B and D, the representative agent becomes atemporally more risk-averse and the premium naturally increases. In general, equity volatilities (SD(Yt ) = 0.10 case) match the data quite well; risk-free rate volatilities are generally too small. Although

67


not part of Table 7, it can also be shown that the risk-free rate is procyclical. Overall, the results are very good, and the model is advantaged by retaining a utility specification defined only over the agent’s consumption. Note also that countercyclical risk aversion is generated endogenously within the (GDA) construct. To see this, consider the cases of columns A and B: disappointment aversion is the only source of risk aversion (α = 1). As a result, the pricing kernel is risk-neutral in the high growth states (no disappointing outcomes) but substantially risk-averse in the low growth states (see Routledge and Zin (2004) for a more detailed analysis). There is one other major structure that provides countercyclical risk aversion: prospect theory as applied to asset pricing.

1.6.3. Prospect Theories The major reference is Barberis et al. (2001). See also the essay by Barberis and Huang in this volume. The idea is to postulate investors who derive utility not only from their period-by-period real consumption, but also over equity portfolio gains and losses relative to a plausible benchmark.51 Roughly speaking, these investors max U(C) + V(G/L), where G/L denotes gains or losses. That utility is defined over only equity gains/losses rather than over aggregate gains and losses to total wealth is an illustration of the notion of “narrow framing.” Furthermore, investors are more sensitive to losses than to gains (this is the essence of the notion of “loss aversion”), with the extent of the loss sensitivity depending on the agent’s prior portfolio experience. Equity gains and losses are measured with respect to a benchmark, and if the investor’s past experience has been to sustain losses relative to his benchmark, then he is modeled as being more acutely sensitive to further ones. If his recent experience has been one of equity portfolio gains, then the agent is modeled as being less sensitive to losses (provided they are not so severe as to negate past gains). These requirements lead to a kink in the agent’s utility curve at gain = loss = 0. Roughly speaking, loss aversion as captured in Barberis et al. (2001) implies linear utility over gains and losses to the equity portion of an investor’s portfolio of the form Xt+1 , Xt+1 ≥ 0, V (Xt+1 , St , zt ) = λ (zt ) Xt+1 , Xt+1 < 0, where f

e Xt+1 = St Rt+1 − Rt St . f

e and Rt , designate, respectively, the gross rates Following our customary notation, Rt+1 of return on stock and risk-free bonds from period t to t + 1 and St the value of the stock portion of the investor’s portfolio. The latter is subscripted by t alone, as the risk-free

51 See also Benartzi and Thaler (1995). These authors postulated investors with loss-averse preferences over variations in their financial wealth only. Without any direct connection to consumption, it is impossible to ascertain how their model might describe the joint processes on equilibrium returns and consumption growth, for example.

68

V(Xt 11, St, zt )


zt . 1

zt 5 1 Xt 11,0

FIGURE 1

458

zt ,1

Xt1150

Xt 11 . 0

Xt 11

V (Xt+1 , St , zt ): Utility of equity portfolio gains or losses. Source: From Barberis et al. (2001).

rate from period t to t + 1 is solely determined by the risk-free bond’s price in period t. Gains or losses are thus measured relative to what would have been earned had the stock portion St of his portfolio been invested in risk-free assets. The slope parameter λ(zt ) measures the degree of loss sensitivity which is dependent on a variable, zt , that recalls past portfolio experience, where zt = 1 captures a case of no prior gains or losses, zt < 1 means prior gains, and zt > 1 means prior losses. A representative graphical portrayal is found in Figure 1.52 More formally, in a discrete-time asset pricing context, the representative agent undertakes to assemble portfolios of the stock market index with value St in conjunction with an aggregate valuation of risk-free bonds, Bt , so as to solve ∞ Ct1−γ t+1 t (28) + bt β V (Xt+1 , St , zt ) , max E β 1−γ t=0

where the first term is the standard present discounted utility of consumption under CRRA utility, and the second, V( ), captures the utility/disutility of gains and losses, Xt+1 , in the stock market portion of the optimal portfolio, St ; the zt -term is as indicated above. The standard period t budget constraint and the constraint governing the evolution of wealth are, respectively, Ct + St + Bt ≤ Wt ,

(29) f

e Wt+1 = St Rt+1 + Bt Rt ,

(30)

where Wt represents period t wealth and St , Bt the amounts of it allocated, respectively, to stock and risk-free bonds post-consumption, Ct , in period t. It remains to detail the precise form of V( ). Barberis et al. (2001) postulate 52 An amplified version of this same representation may be found in the companion chapter by Barberis and Huang (2008).

69


V (Xt+1 , St , z)t =

⎧ ⎪ Xt+1 ⎪ ⎪ ⎪ ⎪ ⎪ f f f e ⎪ ⎪ ⎨St (zt Rt − Rt ) + λSt (Rt+1 − zt Rt ), ⎪ Xt+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩λ(zt )Xt+1

f

e Rt+1 ≥ zt Rt , and zt ≤ 1, f

e Rt+1 < zt Rt , and zt ≤ 1, f

e Rt+1 ≥ Rt , and zt > 1, f

e Rt+1 < Rt , and zt > 1,

(31) where λ(zt ) and zt+1 evolve according to λ(zt ) = λ + k(zt − 1) , and

zt+1

Re = η zt e Rt+1

(32)

+ (1 − η)1,

(33)

where λ > 1, η > 0, and k > 0 are constants and zt is an index that captures whether the investor has recently experienced gains (zt ≤ 1) or losses (zt > 1). Some careful interpretation is in order. Equation (33) describes the evolution of the historical experience variable, zt , where Re amounts to the average stock return and η is a parameter that captures the memory in the adjustment process.53 If equity returns e > 1), this experience tends to increase zt+1 > zt are much less than average (Re /Rt+1 since η ≥ 0. If η = 0, Eq. (31) suggests that zt ≈ 1 or that prior gains and losses have an immediate effect on the investor’s risk attitude. If η = 1, then the benchmark adjusts slowly. Note that λ(zt ), the loss sensitivity, is increasing in zt . Otherwise, bt = b0 (C)−γ, where (C)−γ is purely a scaling factor that is included to maintain stationarity in the equilibrium asset return distributions and b0 determines the relative utility of gains and losses vs. the utility of consumption. There is, however, no formal proof that investor preferences of this type survive aggregation and no axiomatic foundation for their existence. It remains to close the economy by specifying the exogenous dividend and consumption processes (as in any exchange model, there is one equity security outstanding, and risk-free bonds are priced in zero net supply). Barberis et al. (2001) consider a number of scenarios. We first report the one that is in the spirit of a pure Mehra and Prescott (1985) setting: Dt+1 Ct+1 = log = gC + σC ε˜t+1 , log Ct Dt where ε˜t+1 ∼ i.i.d. N (1, 0), where gC = 1.84 and σC = 3.79 (both expressed in percent) as estimated from data for the period 1926–1995. The results are presented in Table 8. 53 Re is actually determined within the model in order that, at equilibrium, the median value of Z ≡ 1. This t turns out, not surprisingly, to be approximately the average equity return.

70


TABLE 8i The Equity Premium in a Model with Loss Aversion (γ = 1, β = 0.98, λ = 2.25, η = 0.9, gC = 1.84%, σC = 3.79%; various b0 , k; annualized returns in percent) Data

log rf

0.58

b0 = 0.7

b0 = 2

b0 = 100

b0 = 7

b0 = 2

b0 = 100

k=3

k=3

k=3

k = 150

k = 100

k = 50

3.79

3.79

3.79

3.79

3.79

3.79ii

E log rP

6.03

0.63

0.88

1.26

3.50

3.66

3.28

SD log rP

20.02

4.77

5.17

5.62

10.43

10.22

9.35

0.30

0.13

0.17

0.22

0.34

0.36

0.35

2.25

2.25

2.25

10.70

7.50

4.40

Sharpe ratio Average loss aversion i This

is Table II of Barberis et al. (2001). their specification it can be shown that log rf is constant.

ii Under

TABLE 9i The Equity Premium in a Model with Loss Aversion and Distinct Consumption and Dividend Processes (gD = gC = 1.84, σC = 3.79, σD = 0.12, w = 0.15, η = 0.9, γ = 1, β = 0.98, λ = 2.25; all return measures in percent) Data

b0 = 0.7 k=3

b0 = 2 k=3

b0 = 100 k=3

b0 = 0.7 k = 20

b0 = 2 k = 10

b0 = 100 k=8

log rf

0.58

3.79

3.79

3.79

3.79

3.79

3.79

E log rP

6.03

1.30

2.62

3.68

5.17

5.02

5.88

20.02

17.39

20.87

20.47

25.85

23.84

24.04

0.30

0.07

0.13

0.18

0.20

0.21

0.24

0.10

0.15

0.15

0.15

0.15

0.15

0.15

2.25

2.25

2.25

5.80

3.50

3.20

SD log rP Sharpe ratio Ct+1 P ρ ,r Ct t+1 Average loss aversion i This

table is drawn from Table III in Barberis et al. (2001).

Note that for very large k, which leads via Eq. (30) to very high loss aversion in some states of the world, the premium can rise to a level of 3.50 percent in conjunction with about half (10.4 percent) of the observed historical equity return volatility. Increases in the value of the parameter b0 , which place greater overall relative weight on the utility of gains and losses, also lend to increases in the premium, but only to 1.26 percent when b0 = 100. Clearly, it is the loss aversion parameter that has the power. Table 9 presents results when the consumption and dividend processes are specified independently: Ct+1 = gC + σC ξ˜t+1 , log Ct

71


log

Dt+1 Dt

= gD + σD ε˜t+1 ,

0 ξt ∼ i.i.d N , 0 εt

1 w

w 1

.

Comparing Tables 8 and 9, it is apparent that separating consumption from dividends has a substantial effect on raising the premium as well as on increasing return volatilities. This is not qualitatively entirely surprising, since there is now much greater dividend volatility–which makes equities more volatile–and this in turn affects an agent’s V ( ) utility much more significantly. Although stock returns are now less highly correlated with consumption (Table 9), this does not matter since the investor is concerned about stock market volatility per se. This latter fact more than compensates for the correlation diminution. By way of contrast, this feature is relatively unproductive in habit-style models as the increased volatility of returns is offset by its reduced correlation with consumption; the net consequence is very little change in the premium. An increase in the parameter k, by increasing loss aversion, has a similarly salutary effect on the premium as it does in Table 8. Loss aversion/narrow framing is an appealing idea, and Barberis et al. (2001) analyze its equilibrium asset pricing implications in a careful and through way. There is, however, a sense in which their study is premature. In particular, we as yet lack choice theoretic underpinnings, and the aggregation properties are as yet unconfirmed (strict aggregation will not hold).

1.7. Beyond One Good and a Representative Agent All of the models considered in the prior sections of the review have been essentially one good, representative agent models. In this section we first highlight elements of the multi-goods literature, which otherwise retains the representative agent construct. Subsequently, we explore a one good heterogeneous agent model where preferences are postulated to have a non-standard expected utility form (rank-dependent expected utility).

1.7.1. Multiple Goods A basic reference here is Piazzesi et al. (2007), which considers preferences defined over two goods, all non-housing consumption (this category comprises non-durable goods consumption and the flow of services from non-housing durables) and housing services. Adopting an otherwise standard Lucas-(1978) style exchange setting, the authors postulate the representative agent’s preferences to be of the form max E

∞ t=0

t

β u(Ct ) ,

(34)

72


where 1− 1

Ct σ , u(Ct ) = 1 − 1/σ (ε−1)/ε

Ct = (Ct

(35)

(ε−1)/ε ε/(ε−1)

+ ωst

)

.

(36)

In (34)–(36), ct denotes non-housing consumption and st the flow of housing services; accordingly, σ is the intertemporal elasticity of substitution and ε is the corresponding intratemporal elasticity. For high ε values, the agent is very willing to substitute goods within a period; as ε → ∞, they become perfect substitutes. In this way, the level of housing services directly affects the marginal utility of non-housing consumption (the numeraire), and therein lies an additional source of consumption risk. In severe recessions where both housing services and non-housing consumption are simultaneously low, for example, the marginal utility of non-housing consumption is especially high as the agent tries to substitute future non-housing consumption for the anticipated shortfall in future housing services. This increases MRS volatility. An important methodological insight allows the equilibrium pricing kernel to be expressed in terms of non-housing consumption and the share of income going to housing services, an insight that avoids the challenge of constructing a time series for the somewhat imprecise notion of “housing services.” For the parameterization 1/σ = 5, β = 0.99, and ε = 1.05, they obtain a premium of 3.5 percent in conjunction with a risk-free return of 1.8 percent and a premium standard deviation of 11 percent. None of these parameter values is a priori objectionable.54 Another study in the multi-good tradition is Ait-Sahalia et al. (2004), which characterizes investor consumption behavior using time series of luxury-goods purchases. Roughly speaking, the idea is that since the “rich” own a vastly disproportionate share of non-pension equity and debt, their consumption behavior should matter for equity pricing. For this group, furthermore, the marginal consumption items are likely to be luxury goods (proxied by sales of expensive automobiles, the sales of high-end Manhattan co-operative apartments, etc.). The distinction between luxury and basic consumption goods is captured in their utility specification: u (ct , Lt ) =

(ct − a)1−φ (Lt + b)1−ϕ + 1−φ 1−ϕ

(37)

with constants a, b > 0 and φ > ϕ. The former assignment implies that subsistence consumption is positive for basic goods and negative (−b) for luxury goods. In equilibrium, luxury goods are not consumed by those with low consumption levels (the poor). In fact, there exists 54 Piazzesi et al. (2007) rely on two sources of uncertainty: consumption growth and the log expenditure ratio for housing. For that study it is important that they be statistically independent, a property that is assumed. Others have questioned its validity.

73

John Donaldson and Rajnish Mehra 1/φ

a c = a + bϕ/φ Pt > a such that Lt = 0 for ct ≤ c (Pt is the relative price for luxury goods vis-à-vis basic ones). Finally, the rich consume an increasing fraction of luxury goods as their incomes rise: ct = 0, (38) lim xt → ∞ Xt where Xt = ct + Pt Lt denotes total consumption expenditure. Of relevance to this essay is the fact that this specification leads to two unconditional asset pricing equations of the form −φ ct+1 − a f e = 0, (39) rt+1 − rt+1 E β ct − a −ϕ Lt+1 + b Pt f e E β = 0. rt+1 − rt+1 Lt + b Pt+1

(40)

Ait-Sahalia et al. (2004) then focus their model on a number of issues related to and inclusive of the equity premium puzzle. Using various self-constructed time series, each corresponding to the aggregate purchase of some type of luxury goods (e.g., luxury cars, wine, apartments, etc.), as proxy consumption sequences, the authors estimate, respectively, the implied CRRAs and equity premia, and explore the ability of the model to replicate the cross-sectional returns of the 25 Fama and French (1993) portfolios. Broadly speaking, they find that certain of these series co-vary strongly with the observed return on equity and lead to high premia (7.8 percent, using an index of Manhattan co-op apartment prices). In most cases, the CRRA estimates are on the order of one-tenth that obtained using standard NIPA aggregate consumption measures: since this latter series hardly varies at all and is little correlated with equity returns. The idea is certainly an appealing one, and the results are impressive. Yet the by-now-familiar questions remain: what are the axiomatic foundations, in what sense is aggregation allowed, etc.? A less radical foray into multiple-goods territory is found in Yogo (2006). As such, Yogo (2006) provides another illustration of both the potential and the limitations of the Epstein–Zin utility specification. He considers a model in which the representative agent’s intratemporal utility level is a CES function of non-durable goods consumption, Ct , and the stock of durable goods, Dt :55 ut = u (Ct , Dt ) = (1 −

1− 1ρ α) Ct

+

1− ρ1 αDt

1 1− 1ρ

.

(41)

55 In models of this type, the non-durable good is always the numeraire. Technically, period utility is a function of non-durable goods consumption and the service flow from the stock of durable goods. Yugo (2006) assumes the service flow is directly proportional to the stock.

74


This is coupled with an Epstein–Zin intertemporal utility specification of the form Ut =

(1 − β) [u (Ct , Dt )]

1− σ1

% κ1 1−1σ1 $ 1−γ + β Et Ut+1 .

The corresponding MRSt,t+1 is thus κ 1 1 − σ1 v(Dt+1 /Ct+1 ) ρ − σ Ct+1 1− κ1 MRSt,t+1 = ρ · Rw,t+1 , Ct v(Dt /Ct ) where

(42)

(43)

κ = (1 − γ)/(1 − 1/σ) % 11 $ 1 v (Dt /Ct ) = 1 − α + α (Dt /Ct )1− ρ 1− ρ

and Rw,t+1 is the return on wealth from the optimal portfolio. Following Hansen and Singleton (1982), Yogo (2006) then estimates the parameter values (α, β, ρ, σ, γ) that are consistent with (1) the above model and (2) the historical returns on the 25 Fama–French portfolios.56 He obtains the estimates σ = 0.024, α = 0.827, ρ = 0.52, β = 0.9 and γ = 191, all of which seem plausible except for the risk aversion estimate and, possibly, the relative intratemporal emphasis on durable goods (α = 0.827). That these values can, via (41)–(43), match the data quite well is attributable to the additional terms in the IMRSt,t+1 expression, which are dependent on the (Dt /Ct ) ratio. In the data, this ratio is highly procyclical and thus magnifies the countercyclicality of the marginal utility of consumption. Marginal utility becomes very high at business cycle troughs where both consumption itself and the (Dt /Ct ) ratio are low. Nevertheless, because durable consumption is so smooth, the required CRRA estimate is still excessive relative to that obtained in standard microeconomic applications. While the Epstein and Zin construct is flexible in accommodating many phenomena, it does not necessarily do so for reasonable parameter choices. Note that for this story to work, ρ < σ. In comparing financial statistics, it is the low value of the EIS that allows the model to replicate the historically observed risk-free rate. Multiple goods are not, however, any guarantee of puzzle resolution. Although Yogo (2006) obtains generally attractive results in a model with durable and nondurable goods within the Epstein–Zin preference specification, Giannikos (2007) derives equilibrium premia similar to that obtained in Mehra and Prescott (1985) using standard CES preferences defined over non-durable goods consumption and various representations of the durable goods service flow. His results are driven by positive autocorrelation in the estimated non-durable consumption dividend, a fact that leads to a negative premium on the non-durable goods firm (recall Azeredo (2007)). This negative premium overwhelms the positive premium to the durable goods—producing firm, giving an overall market premium that is counterfactually small and is negative for some parameterizations. 56 As

such, Yogo’s (2006) model also displays a form of composition risk (via the Dt /Ct ratio).

75


1.7.2. Agent Heterogeneity There already exists a substantial literature that explores the magnitude of the premium in incomplete market settings where heterogeneity is captured via the experience of idiosyncratic income shocks. In this section, we focus alternatively on preference heterogeneity, its implications for the equity premium in a trading environment, and, ultimately, the usefulness of the representative agent construct. The literature is extremely small: we found only two directly relevant papers, Chan and Kogan (2002) and Chapman and Polkovnichenko (2006).57 Each studies an exchange equilibrium where financial markets are complete. Chan and Kogan (2002) postulate agents with ratio habit preferences of the form u (ct , Xt ; γ) =

1 (ct /Xt )1−γ 1−γ

where ct is the consumption of an individual agent with CRRA parameter value γ. The Xt term, which is their measure of the economy wide living standard, is defined as a weighted average of past aggregate endowment realizations: xt = ln(Xt ) satisfies &t xt = x0 e−λt + λt e−λ(t−s) ys ds 0

where yt = ln(Yt ), and Yt is the period t economy wide endowment. The latter follows a standard Brownian motion process. Note that the choice of the parameter λ determines how strongly past endowment experience influences the living standard measure. Chan and Kogan (2002) construct a representative agent and solve for equilibrium by solving the associated planning problem. Changes in the cross sectional distribution of wealth lead to changes in the economy wide level of risk aversion in the following way. Since the risk averse agents hold most of their wealth in stocks, when endowment is low and the stock’s price low, the fraction of the economy’s wealth controlled by these agents diminishes, leading to increased economy wide risk aversion. The resultant countercyclical risk aversion allows their model to match nearly all of the Campbell and Cochrane (1999) results except for the volatility of the risk free rate. In Chan and Kogan (2002) it is too high relative to what is observed in the data; in Campbell and Cochrane (1999) it matches the data by construction. Chan and Kogan (2002) is important particularly for generating countercyclical risk aversion endogenously in a way that works to resolve the puzzle. In Chapman and Polkovnichenko (2006), preference heterogeneity works against these same stated goals. They postulate a setting where investors have different rank dependent expected utility preferences (RDEU). With its surprising implications, the paper demands a somewhat detailed review. Rather than weighing various utility outcomes by their objective probabilities, RDEU preferences weight outcomes by some 57 We

thank Nick Barberis for bringing these papers to our attention.

76


non-linear weighting function defined over the cumulative probabilities of outcomes that have been utility ranked from worst to best. Typical parameterizations end up over weighting bad outcomes. In general, a RDEU welfare functional will have kinks at allocations where consumption outcomes are equal across any two states. Consider an environment of i = 1, . . . , n states of nature and K agents where all agents have the same CRRA utility function over consumption outcomes in particular states, k 1−γ ci k k , U ci = 1−γ where cik represents agent k’s consumption in state i. Let C k, P k denote a consumption lottery where, respectively, consumption across states is ranked from worst to best, and P denotes the vector of cumulative probabilities. More specifically, C k, P k = k

k k c1 . . . , cN , P1k , . . . , PNk where cik < ci+1 and Pik = πj where πj is the objective j≤i

probability of the state corresponding to the consumption allocation cjk . Note that different outcomes will have different associated vectors of cumulative probabilities. In the Chapman and Polkovnichenko (2006) formulation, the RDEU value function of agent k is given by U k ({C, P }) =

N

qk (P ) uk cik ,

(44)

i=1

φk k φk where qk (P ) = Pik − Pi−1 is the weighting function. Note that if φk ≡ 1, the expression collapses to one of expected utility, and if φk < 1, the agent is pessimistic. In general equilibrium, the qk (P ) weights will be endogenously determined, just as the consumption outcomes across states are themselves determined endogenously in equilibrium. The authors then go on to consider a two-agent, two-date equilibrium where the agents differ in their pessimism parameters, φk . With φ1 = 1 (expected utility) and φ2 = 0.4 (substantial pessimism), they find that the equity risk premium is only 17 percent of the risk premium in an economy where the consumers are identical (φ1 = φ2 = 0.4). It is, however, twice the level obtained for the corresponding homogeneous preference economy, where everyone has expected utility (φ1 = φ2 = 1). In solving for the equilibrium, they construct a representative agent and solve accordingly. In equilibrium, REDU economies display limited risk sharing since the preference structure implies that for small risk premia, the more pessimistic agent may not choose to invest in the risky asset at all. Aggregate risk is then priced only by the less pessimistic agent. What is present is endogenous participation in the equity markets. In a dynamic equilibrium multi-agent RDEU model, we could imagine a situation of some agents entering and exiting the equity markets voluntarily as the economy passes from expansions to recessions. Would such an economy endogenously exhibit countercyclical risk aversion? Would the attractive business cycle/equity premium characteristics of, e.g., the Guvenen (2005) and Danthine and Donaldson (2002) models be preserved, despite

77


the fact that participation in equity markets is not fixed a priori? These are reasonable issues to explore. From the perspective of this paper, the results are especially striking. In particular, Chapman and Polkovnichenko (2006) remind us that using the representative agent construct in a non-standard heterogeneous preference setting may be uninformative as it ignores important endogenous risk-sharing effects. Before leaving this body of material, some comparative remarks might be useful. Table 10 summarizes the properties of the various preference specifications considered thus far. We first note that a number of open questions remain. Many involve aggregation. Without this property, the results of Chapman and Polkovnichenko (2006) may recur in other contexts. The habit formation construct of Campbell and Cochrane (1999), the prospect theory of Barberis et al. (2001), and the GDA preferences of Routledge and Zin (2004) all yield effective countercyclical risk aversion, and in many respects, their results with regard to the equity premium are similar. A number of features of the GDA construct would seem to give it an edge, however. Relative to simple expected utility, it

TABLE 10 Characteristics of Principal Asset Pricing Paradigms Attribute

Preference type A B C D E F G H I Conventional Epstein– Melino– Constantinides– Campbell– Falato Zinn– Barberis– RDEU CRRA Zin, Yang Sundaresan Cochrane Routledge Huang Chew– (linear habit) Deckel

1. Does a representative agent exist?

Yes

Yes

?i

Yes

No

Yes

Yes

No

Yes

2. Is strict aggregation a possibility?

Yes

No

No

?

No

No

Yes

No

Yes

3. Does its use in portfolio problems lead to time consistent planning?

Yes

Yes

?

Yes

?

?

Yes

?

?

4. Do choicetheoretic axiomatic foundations exist?

Yes

Yes

No

No

No

No

Yes

No

?

5. Is countercyclical risk aversion a principal feature?

No

Yes

Yes

Yes

Yes

No

Yes

Yes

?

i The

question mark indicates that the answer to the question is unknown or not fully understood.

78


generates the fewest additional parameters and is the only structure of the three (to date) with axiomatic foundations. Relative to the two alternatives cited above, they are the only structure that allows the separation of risk and time preferences. (More recent work by Barberis and Huang (2001) allows this as well.) Furthermore, there is no need for external scaling or an external reference point to maintain return stationarity. In short, GDA preferences have the advantage of economy and simplicity. It is unclear, however, if any of these constructs will be compatible with the stylized financial and business cycle facts—or, indeed, with generally accepted consumption processes—when placed in a production setting. Numerous other behavioral representations are modeled in the literature. Constantinides et al. (2007), for example, argue for a period utility representation of the form U (ct ) + LV (Bt ) , 1−γ

where ct denotes the period consumption and V (Bt ) = Bt B /(1 − γB ) the utility of bequests, Bt . The L parameter assumes a value of 0 except in the final period of the agent’s life when L = 1 (the context is obviously one of finitely lived overlapping generations). Since Bt measures the value of bequests, it is a quantity that will be highly correlated with the return on the stock market. As a result, agents who are substantially bequest risk-averse will demand a high premium to hold equity securities in their final period of life. While this modeling device does lead to a high premium, other counterfactuals arise.

2. PRODUCTION ECONOMIES The production-based asset pricing literature is oriented to constructing models that successfully replicate both the stylized facts of the business cycle and the financial markets. To review how the business cycle is characterized in the data and in dynamic, stochastic, general equilibrium models, see the section by Danthine, Donaldson, and Siconolfi in this volume (Danthine et al. (2008)). Relevant papers with a primary focus on the equity premium puzzle include Jermann (1998), Boldrin et al. (2001), Danthine et al. (1992), Danthine and Donaldson (2002), and Guvenen (2005). Lettau and Uhlig (2000) focus only on the relationship of habit formation to consumption volatility, although the implications of their results for the premium are straightforward. Production models display the feature that the dividend series and the asset pricing agent’s consumption series are endogenously determined within the model.58 All the mechanisms reviewed in this section will have consequences for the statistical properties of these series. Habit formation in a production context such as Hansen (1985), in particular, will induce the model’s representative agent to tailor his investment 58 Accordingly,

assets are then priced using the standard Lucas (1978) asset pricing perspective.

79


and labor service decisions in such a way as to reduce, dramatically, his endogenous consumption volatility vis-à-vis an otherwise identically parameterized construct without the habit feature: consumption as a pure macroeconomic quantity becomes counterfactually smooth, and the associated premium falls essentially to zero. A successful model must thus display not only a strong desire on the part of agents for consumption smoothing but also severe (but plausible) technological restrictions that inhibit their ability to do so. Within the production literature, most models require some variation on habit formation to generate a high and countercyclical level of effective risk aversion. The complementary features that interfere with the consequent consumption smoothing are myriad. We consider Jermann (1998) as an important illustrative case. Jermann’s equilibrium allocations of consumption, investment, etc. solve ∞ t max E β u (Ct − αCt−1 ) , {It }

t=0

subject to Kt+1 = (1 − Ω) Kt + ϕ (It /Kt ) Kt Ct + It ≤ f (Kt , Pt Nt ) λt λt = ρλt−1 + ε˜t , where f (Kt , Pt Nt ) λt = K θ (Pt Nt )1−θ λt , with θ = 0.64, β = 0.99, ρ = 0.99, u (Ct − αCt ) =

(Ct − αCt−1 )1−γ , 1−γ

with γ = 5, Ω = 0.025, and Pt = (1.005) Pt−1 ; i.e., a constant growth rate of labor productivity of 0.5 percent per quarter (the calibrated time period).59 Clearly, it is the habit formation feature most prominently that causes the agent to display a strong desire for consumption smoothing. But what features stand in his way from doing so? First and foremost is the cost of capital adjustment function, φ

It Kt

Kt =

a1 1−ξ

It Kt

1−1/ξ + a2 ,

where ξ = 0.23, and a1 , a2 are constants chosen so that the balanced growth path is invariant to ξ. Second, and less transparently, there is no ability of the agent to vary his labor supply, despite the fact that his period preference ordering manifests no disutility from doing so. The results of Jermann’s (1998) exercise are presented in Table 11. All the claims we have made in this section are manifest in Table 11. Consider column A’s statistics. This corresponds to an unfettered production model: no habit and 59 Under

a growth scenario β = 0.99 = βˆ (1.005)1−γ, where βˆ is the subjective discount factor.

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TABLE 11i The Equity Premium in the Production Economy of Jermann (All parameter values as in text, except as noted; various scenarios; all return statistics in percent, annualized) Parameter choices E

Fiii

3.91

0.82

1.580

0.61

11.46

0.460

4.23

4.58

7.00

1.590

2.90

1.21

6.09

19.86

0.480

0.02

0.26

0.03

0.67

6.18

0.001

0.02

0.26

0.03

0.67

6.18

0.001

A

B

C

D

Erf

4.26

3.36

4.20

SDr f

0.62

0.76

0.59

Ere

4.28

3.66

SDr e

1.02

Erp Erp SDr p SD ΔC/SD ΔY (ii)

not reported 0.77

0.78

0.33

1.14

0.49

A. Standard model: α = 0, a1 = a2 = 0 – no habit, no COA. B. Same as case A but with γ = 10. C. Habit formation, no COA. D. Costs of adjustment, no habit formation. E. Habit and adjustment costs. F. Habit and adjustment costs but variable labor supply. i Columns A–E are from Table 1 in Jermann (1998), while column F is from Boldrin et al. (2001). ii This is the SD of the consumption growth rate relative to the SD of the output growth rate. iii Preferences as noted in text; α = 0.9, ξ = 0.23; otherwise as in Jermann (1998).

no adjustment costs, so the agent is totally free to adjust his investment to smooth out his consumption. As a result, the premium is a miserly 0.02 percent, which is even lower than its exchange economy counterpart. Column B corresponds to an otherwise identical case but for γ = 10. The premium does rise, but by an insignificant 0.24 percent, and return standard deviations remain much too low (compare this result with those in Table 1). In column C, the agent displays habit formation with a habit parameter α = 0.82. This greatly amplified effective risk aversion induces the agent to tailor his investment plan to elicit greater consumption smoothing. As a result, the SD of the growth rate of consumption declines by more than one-half. Despite the added local risk aversion, the smoother consumption series draws the premium—and the return volatilities—more or less back to their column A values. Costs of adjustment (column D) make it substantially more expensive for the agent to effect consumption smoothing via investment volatility. With this obstruction, the premium rises but still falls short even of 1 percent. When habit formation is incorporated in conjunction with COA (column E), the premium matches its historical benchmark, confirming the earlier intuition; equity return volatility is quite reasonable as well, as is the mean risk-free rate. Only the risk-free rate volatility is mismatched, with the model producing an SD more than twice the historical level. This is not entirely surprising given the knife-edged nature of the habit formation mechanism and the enormous swings in risk aversion it tends to generate.

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Column F, from an early version of Boldrin et al. (2001), contains the results for a slightly modified Jermann (1998) model enriched by incorporating an agent whose utility is linear in leisure (risk-neutral), essentially as in Hansen (1985). Such an agent is unconcerned with variations in his labor supply; under this specification, the agent’s period utility function is of the form u(Ct , Ct−1 , 1 − nt ) = log(Ct − αCt−1 ) + Bnt , Unfortunately, this added flexibility causes the premium to collapse to a mere 0.001 percent, with the equity and risk-free standard deviations not far behind at, respectively, 0.46 and 0.48 percent. This example fully illustrates the difficulty in incorporating standard preference mechanisms into production settings. As of this writing, we are unaware of any attempts to incorporate “loss aversion” or “happiness maintenance” into a production context, but the consequences are likely to be similar. Note that nearly all recent production models rely in some way on habit formation. Eschewing this feature, Campanale et al. (2006) explore the business cycle and agent pricing properties of a production economy very similar to Jermann (1998) (for instance, there is a cost to adjusting capital) where the representative agent possesses Epstein–Zin preferences and the certainty equivalent is obtained using GDA risk preferences and special cases thereof (expected utility and (DA), disappointment aversion preferences). Solving the model using traditional discrete state-space methodologies, they are able to replicate a very wide class of stylized financial and business cycle facts, thereby effectively challenging the habit formation device. Perhaps most strikingly, they do so for implied risk aversion that is similar to a logarithmic expected utility agent. As with the earlier habit-based literature, risk-free rate volatility remains excessive.60 It would also be of interest to know the consequences for their results of admitting a variable labor supply. The paper is important not only for introducing these richer preference structures into a production context but also for more broadly re-energizing the asset pricing production research program. Production-based asset pricing models are important because they afford an opportunity for cross-model verification. Since the actions of the same agents give rise to both macroeconomic and financial phenomena, a comprehensive, reliable model should be able to explain both sets of stylized facts equally well. At the current level of understanding, this is not yet the case.

3. DISASTER EVENTS AND SURVIVORSHIP BIAS Reitz (1988) proposed a solution to the equity premium puzzle that incorporates a very small probability of a very large drop in consumption (a “disaster state”). Beyond the incorporation of this third growth state, the structure of his model is identical to that of 60 Kallenbruner and Lochstoer (2004) demonstrate that a reasonably calibrated production model with Epstein–Zin preferences can jointly explain important macroeconomic and financial stylized facts without the excessive risk-free rate volatility characteristic of habit formulations.

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Mehra and Prescott (1985). Since disaster states are ones of extremely high marginal utility of consumption, we might expect their incorporation to push up risk-free asset prices and diminish risky ones. As a result, the premium should rise. Table 12 presents a sample of scenarios drawn from Reitz (1988). The incorporation of the disaster scenario, even with small conditional or unconditional probability, does allow for a resolution of the premium. This is not in dispute although the plausibility of such a huge drop in consumption is open to question. The model requires a 1-in-100 chance of a 25 percent decline in consumption to reconcile the equity premium with a risk aversion parameter of 10. Following on these results, Barro (2006) details all major cross-country consumption declines post-1913. While certain European countries experienced consumption declines of as much as 64 percent during World War II (e.g., Germany and Greece), even in the worst years of the Great Depression, U.S. consumption only declined by about 30 percent. In light of these figures, the percentage declines postulated by Reitz (1988) do not seem so extreme. One must assess other potential implications of the model, however. One is that the real interest rate and the probability of the occurrence of the extreme event move inversely. For example, the perceived probability of a recurrence of a depression was probably very high just after World War II and subsequently declined over time. If real interest rates rose significantly as the war years receded, that evidence would support the Rietz hypothesis. Similarly, if the low probability event precipitating the large decline in consumption were a nuclear war, the perceived probability of such an event has surely

TABLE 12 The Equity Premium and Risk-Free Rate: Possible Disaster State (All returns in percent, annualized) Panel Ai Disaster: Output falls to half its previous value Probability of entering

γ

β

Erf

Erp

0.0008

7.00

0.997

0.83

6.18

0.0030

5.30

0.980

0.89

6.15

Disaster

Panel

Bii

Disaster: Output falls to three-fourths its previous value Probability of entering

γ

β

Erf

Erp

0.010

9.80

0.999

0.74

6.95

0.014

8.85

0.999

0.84

6.49

Disaster

i Taken ii Taken

from Table 3 of Reitz (1988). from Table 6 of Reitz (1988).

83


varied over the last 100 years. It must have been low before 1945, the first and only year the atom bomb was used. And it must have been higher before the Cuban Missile Crisis than after it. If real interest rates had moved as predicted, that would have supported Rietz’s disaster scenario. But they did not. The reader is referred to Mehra and Prescott (1988) for a detailed response to Reitz (1988) along these lines. Barro (2006) presents a model in the same spirit as Reitz (1988) but in a context where more attention may be given to calibration and the historical record. The historical justification for his calibration is, in fact, fascinating to read. The point of departure of the paper is a postulated stochastic process for aggregate output growth that allows for rare events in a generalized Rietz setting. Specifically, he models aggregate output growth as a random walk with drift, whose innovations are of three possible types: 1. “diffusive” shocks (i.i.d. normal shocks), 2. jump shocks of “type v”—shocks that represent situations where output contracts sharply but there is no occurrence of a default on debt, and 3. jump shocks of “type w”—shocks that represent situations where output contracts sharply and a default on debt ensues.61 In particular, Barro assumes that the log of output Yt ≡ Ct grows according to the random walk with drift process: log Yt+1 = log Yt + g + u˜ t+1 + v˜t+1 ,

(45)

where u˜ t+1 is an i.i.d random variable distributed N (0, σu2 ) and v˜t+1 ( jump shock “type v”) captures low probability downward jumps in GDP. If a disaster occurs, with probability p, output declines by the fraction b. Very roughly speaking, a reasonable calibration requires b to be large and p small. More formally, −p 0, with probability e , −p (46) v˜t+1 = ˜ log 1 − b , with probability 1 − e . In his calibration exercise, b˜ is a random variable whose probability density function coincides with the frequency distribution reflecting the size of contractions in 35 countries in the 20th century (b˜ ranges from 0.17 to 0.62). Barro also admits the possibility of default on government debt of proportion d, with probability q whenever a “type v” jump occurs. In other words, whenever a jump shock of “type v” occurs, there is a probability q ofa “type w” jump. Table 13 presents a sample of results for the calibration exercises b˜ = d˜ . In all cases, preferences are as in Mehra and Prescott (1985) with CRRA = 4 and time preferences parameter β = e−ρ = e−0.03 = 0.9704. The results presented here are intriguing and lend support to a resurrection of the disaster scenario as a viable justification for the premium. The distinction between “type v” events and “type w” events is theoretically appealing in terms of their different implications for quantities and prices. However, we feel 61 Barro

(2006) does not explicitly define a type w shock. We introduce it here as an aid to exposition.

84


TABLE 13i Financial Statistics: Barro’s (2006) Model (All rates of return annualized in percent; all cases θ = 4, σu = 0.02, q = 0.4; time preference e−ρ, ρ = 0.03) g = 0.025

g = 0.025

g = 0.020

g = 0.025

p = 0.017

p = 0.025

p = 0.017

p = 0.017

Ere

7.1

4.4

5.1

6.1

Erf

3.5

−0.7

1.5

2.5

Erp

3.6

5.2

3.6

3.6

this distinction is largely a consequence of the specific functional forms used to model the two types of rare events and the assumption that they are independent—something we intuitively think is not very plausible. To support the observed equity premium (in the baseline model), a decline in GDP growth of 50 percent is needed. The plausibility of such a substantial decline in per capita GDP growth is justified by the figures in Barro (2006, Table 1), where this decline is shown to be 64 percent in Germany and Greece (during World War II) or 31 percent in the U.S. (during the Great Depression). These observations, however, are not at an annual frequency.62 Looking at data at an annual frequency, we see little evidence of such a substantial decline either in GDP growth or in the growth rate of consumption. If the equity premium is estimated at an annual frequency, then the relevant growth rates should also be at annual frequencies. We argue that calibrations should be done at the same frequency as observations. As is standard in the equity premium literature, Barro calibrates his model using output growth rates rather than consumption growth rates since in a pure exchange setting they are identical. While this distinction may not matter in a stylized economy without disaster states, the two quantities might empirically be quite different in the event of an extreme disaster. Consumption per capita will probably be a much smoother series than GDP per capita. In the event that such states occur, consumption is unlikely to remain a constant proportion of the output. Capital investment and capital utilization rates will change and act as channels to absorb the negative productivity shocks (see also Section 2), enabling consumption smoothing over time and across states. For this reason, since we do live in a production economy, it would be interesting to explore the robustness of Barro’s (2006) results using cross-country data on the growth rates of per capita consumption instead of output within that more general context.

62 In a sense the results are obtained using the 4-year drop in consumption as if it is a 1-year effect. The claimed equity premium should be compared to the observed 4-year effect (4 × 6 = 24 percent) and not the 1-year 6 percent.


85

A closely related literature concerns “peso problems.” This means that returns are higher than justified by current experience because investors assign a positive probability to some as-yet-unrealized disaster state. In the context of this essay, it suggests the possibility that the observed equity premium in the U.S. may result from expectations of disaster events that happen not to have materialized in the sample period of observations. Recent work by Goetzmann and Jorion (1999) supports this perspective: after concluding that the high U.S. historical premium is in some sense unique, they suggest that it may be attributable to the fact that disastrous events (most especially WWII) affecting many financial markets have largely spared the U.S. economy. In a production setting similar to Jermann’s (1988) but with a variable labor supply, no costs of adjustment and no habit (a model that is in some sense set up with a bias against resolving the puzzle), Danthine and Donaldson (2002) replicate all first moments of returns, nearly exactly, in a peso-style model where the disaster is anticipated but never realized. The required ingredients mandate a debt-to-equity ratio of 0.3, an anticipated disaster state where output falls by 50 percent, a CRRA of 3, and a (conditional) probability of entering the disaster state of 0.008. Return volatilities are much too low, however (this is to a lesser extent true for Barro (2006) as well). A similar but more detailed exercise is undertaken by Cecchetti et al. (1993, 2000) in an exchange setting. All this is to suggest that unrealized beliefs may be critical to the puzzle’s resolution and that the results in Reitz (1988) and others can be strengthened if reported returns are generated from time series in which the actual disaster state does not appear. Veronesi (2004) goes one step further by postulating a peso-style model with representative agent learning. Dividends follow a simple linear stochastic process where the drift coefficient assumes one of two values, a normal level of θ¯ and a low, ¯ there is a small probability of shifting from θ¯ to θ¯ and a high probrecession level of θ; ability of returning. The drift coefficient is not observed directly, however, and Veronesi (2004) derives the period-by-period posterior probability of being in the normal growth state. This estimate itself evolves stochastically. Veronesi (2004) then explores the asset pricing characteristics of this economy for time paths where the low growth state is never actually realized. He derives high price sensitivity to dividend changes, increased return volatility in perceived recessions, etc. as well as significant equity premia. Unfortunately, theories built on beliefs other than pure rational expectations are difficult to test empirically. Another attempt at resolving the puzzle proposed by Brown et al. (1995) focuses on the related notion of survival bias. The central thesis here is that the ex-post measured returns reflect the premium, in the U.S., on a stock market that has successfully weathered the vicissitudes of fluctuating financial fortunes. Many other exchanges were unsuccessful, and hence the ex-ante equity premium was low. Since it was not known a priori which exchanges would survive, for this explanation to work, stock and bond markets must be differentially influenced by a financial crisis. Governments have expropriated much of the real value of nominal debt by the mechanism of unanticipated inflation. Five historical instances come readily to mind: during the German hyperinflation, holders of bonds denominated in Reichmarks lost virtually all value invested in those assets. During the Poincaré administration in France in the 1920s, bond holders

86


lost nearly 90 percent of the value invested in nominal debt. And in the 1980s, Mexican holders of dollar-denominated debt lost a sizable fraction of its value when the Mexican government, in a period of rapid inflation, converted the debt to pesos and limited the rate at which these funds could be withdrawn. Czarist bonds in Russia and Chinese debt holdings (subsequent to the fall of the Nationalists) suffered a similar fate under communist regimes. The above examples demonstrate that in times of financial crisis, bonds are as likely to lose value as stocks. Although a survival bias may impact on the levels of both the return on equity and debt, there is no evidence to support the assertion that these crises impact differentially on the returns to stocks and bonds; hence, the equity premium should not necessarily be materially affected. In every instance where trading equity has been suspended, due to political upheavals, etc., governments have either reneged on their debt obligations or expropriated much of the real value of nominal debt through the mechanism of unanticipated inflation. The difficulty that several model classes have collectively had in explaining the equity premium as a compensation for bearing risk leads us to conclude that perhaps it is not a “risk premium” but rather due to other factors. We consider these in the next section.

4. MARKET INCOMPLETENESS AND TRADING FRICTIONS Market incompleteness assumes one of two different forms: (1) certain securities are not traded, and/or (2) certain individuals are for some reason excluded from financial market participation. A complete market structure where all investors can trade any security is necessary for the construction of the representative agent. In this case, the equilibrium in a heterogeneous full information economy is isomorphic in its asset pricing implications to the equilibrium of a representative agent, full information economy, if households have von Neumann–Morgenstern preferences. It is this paradigm that has provided the foundation to the essay so far. In this section we briefly review context (2), above, since the first is more than adequately considered in the essays by Constantinides and Heaton and Lucas. In our discussion of (2), we focus on restricted participation resulting from borrowing constraints imposed on certain agents.

4.1. Restricted Participation In infinite horizon models, the effect of borrowing constraints and transaction costs is to force investors to hold an inventory of bonds (precautionary demand) to smooth their consumption. As a result, agents come close to equalizing their marginal rates of substitution with little effect on the equity premium: the economy is essentially one of complete markets.63 Recent attempts to resolve the puzzle by incorporating both 63 This

is true unless the supply of bonds is unrealistically low. See Aiyagari and Gertler (1991).


87

borrowing constraints and consumer heterogeneity have thus focused on finite horizon OLG economies. One such example is proposed in Constantinides, Donaldson, and Mehra (2002).64 The authors construct an overlapping-generations (OLG) exchange economy in which consumers live for three periods. In the first period, a period of human capital acquisition, the consumer receives a relatively low endowment income. In the second period, the consumer is employed and receives wage income subject to large uncertainty. In the third period, the consumer retires and consumes the assets accumulated in the second period. The implications of a borrowing constraint are explored by deriving and contrasting the stationary equilibria in two versions of the economy. In the borrowing-constrained version, the young are prohibited from borrowing or from selling equity short. This is not an artificial constraint since loans and short sales require collateral (accumulated wealth), which the young typically lack. The borrowing-unconstrained economy differs from the borrowing-constrained one only in that the borrowing constraint and the shortsale constraint for the young are absent. There are two types of securities in the model, bonds, and equity with ex-coupon and ex-dividend prices qtb and qte , respectively. Bonds are a claim to a coupon payment b every period, while the equity is a claim to the dividend stream {dt }. The consumer born in period t receives deterministic wage income w 0 > 0 in period t, when young; 1 stochastic wage income wt+1 > 0 in period t + 1, when middle-aged; and zero wage income in period t + 2, when old. The consumer purchases zet,0 shares of stock and zbt,0 bonds when young. The consumer adjusts these holdings to zet,1 and zbt,1 , respectively, when middle-aged. The consumer liquidates his/her entire portfolio when old. Thus, zet,2 = 0 and zbt,2 = 0. An unconstrained representative agent’s maximization problem is formulated as follows. The agent born in period t solves 2 i β U (ct,i ) emaxb E zt,i , zt,i i=0 subject to ct,0 + qte zet,1 + qtb zbt,1 ≤ w 0 , e b e b 1 ct,1 + qt+1 zet,2 + qt+1 zbt,2 ≤ qt+1 + dt+1 zet,1 + qt+1 + b zbt,1 + wt+1 , e b b e ct,2 ≤ qt+2 + dt+2 zt,2 + qt+2 + b zt,2 , where ct,j is the consumption in period t + j (j = 0, 1, 2) of a consumer born in period t. When considering the borrowing constrained equilibrium, the following additional constraints are imposed: zet,0 > 0 and zbt,0 > 0. The model introduces two forms of restricted participation. First, consumers of one generation are prohibited from trading claims against their future wage income 64 See

also the companion papers by Constantinides et al. (2005, 2007) and the paper by Storesletten et al. (2008) in this volume.

88


with consumers of another generation.65 Second, consumers of one generation are prohibited from trading bonds and equity with consumers of an unborn generation. As discussed earlier in Section 2, absent a complete set of contingent claims, consumer heterogeneity in the form of uninsurable, persistent, and heteroscedastic idiosyncratic income shocks, with countercyclical conditional variance, can potentially contribute to the resolution of empirical difficulties encountered by representative-consumer models.66 The novelty lies in incorporating a life-cycle feature to study asset pricing. As discussed earlier, the attractiveness of equity as an asset depends on the correlation between consumption and equity income. If equity pays off in states of high marginal utility of consumption, it will command a higher price (and consequently a lower rate of return) than if its payoff is in states where marginal utility is low. Since the marginal utility of consumption varies inversely with consumption itself, equity will command a high rate of return if it pays off in states when consumption is high, and vice versa.67 In and of itself, this is the standard intuition. A key insight of the paper is the observation that as the correlation of equity income with consumption changes over the life cycle of an individual, so does the attractiveness of equity as an asset. Consumption can be decomposed into the sum of wages and equity income. A young person looking forward has uncertain future wage and equity income; furthermore, the correlation of his equity income with his consumption will not be particularly high, as long as stock and wage income are not highly correlated. This is empirically the case, as documented by Davis and Willen (2000). Equity will thus be a hedge against fluctuations in wages and a “desirable” asset to hold as far as the young are concerned. The same asset (equity) has a very different characteristic for the middle-aged. Their wage uncertainty has been entirely resolved: their future retirement wage income is either zero or deterministic, and the innovations (fluctuations) in their consumption occur from fluctuations in equity income. At this stage of the life cycle, equity income is highly correlated with consumption. Consumption is high when equity income is high, and equity is no longer a hedge against fluctuations in consumption; hence, for this group, it requires a higher rate of return. The characteristics of equity as an asset therefore change, depending on who the predominant holder of the equity happens to be. Life-cycle considerations thus become crucial for asset pricing. If equity is a “desirable” asset for the marginal investor in the economy, then the observed equity premium will be low, relative to an economy where the marginal investor finds it unattractive to hold equity. The deus ex machina is the stage in the life cycle of the marginal investor. 65 Being homogeneous within their generation, consumers have no incentive to trade claims with consumers of their own generation. 66 See Mankiw (1986) and Constantinides and Duffie (1996). 67 This is precisely the reason, why high-beta stocks in the simple CAPM framework have a high rate of return. In that model, the return on the market is a proxy for consumption. High-beta stocks pay off when the market return is high, i.e., when marginal utility is low; hence, their price is (relatively) low and their rate of return high.

89


The authors argue that the young, who should be holding equity in an economy without frictions and with complete contracting, are effectively shut out of this market because of borrowing constraints. The young are characterized by low wages; ideally, they would like to smooth lifetime consumption by borrowing against future wage income (consuming a part of the loan and investing the rest in higher-return equity). However, they are prevented from doing so because human capital alone does not collateralize major loans in modern economies for reasons of moral hazard and adverse selection. In the presence of borrowing constraints, equity is thus exclusively priced by the middle-aged investors, since the young are effectively excluded from the equity markets, and we observe a high equity premium. If the borrowing constraint is relaxed, the young will borrow to purchase equity, thereby raising the bond yield. The increase in the bond yield induces the middle-aged to shift their portfolio holdings from equity to bonds. The increase in demand for equity by the young and the decrease in the demand for equity by the middle-aged work in opposite directions. On balance, the effect is to increase both the equity and the bond return while simultaneously shrinking the equity premium. Furthermore, the relaxation of the borrowing constraint reduces the net demand for bonds, and the risk-free rate puzzle re-emerges.68 In a subsequent paper, Da Silva (2006) generalizes the Constantinides et al. (2002) restricted participation model to admit habit formation where the habit coefficients of the old and middle-aged differ.69 Using the notation of this section, the objective function of the period t-born representative agent is 2 i max E β u ct,i − ηi ct−1, i , (47) {zet,i , zbt,i } i=1 with η1 , η2 the habit parameters. It follows that the MRS and the effective level of risk aversion of the middle-aged agent are, respectively, βu1 ct,2 − η2 ct,1 (48) , u1 ct,1 − η1 ct,0 − η2 βE u1 ct,2 − η2 ct,1 γct,1 . ct,1 − η1 ct,0

(49)

the latter expression corresponding to the standard CRRA period utility function, u(c) = (c1−γ /1 − γ), interacting with the habit. It is immediately apparent from (48) that both η1 and η2 appear in the MRS and that, by (49), the effective CRRA is thus time-varying. 68 Bear

in mind, however, that the returns reported for this construct are annualized 20-year returns. Silva and Giannikos (2006) explore the same construct where the CRRA of the old agent is greater than the corresponding value for the middle-aged agent. 69 Da

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Generally speaking, Da Silva and Giannikos (2006) plausibly hypothesize that η1 > η2 , suggesting that the elderly more readily accept consumption levels altered relative to that of their middle-aged working years than do the middle-aged relative to their youth. The model yields surprisingly good results. In particular, for γ = 2 and (η1 , η2 ) = (0.7, 0.2), the premium exceeds 4 percent. This figure is associated with standard deviations that match their empirical counterparts quite well (20 and 7 percent for the equity and risk-free securities, respectively); average returns, however, are somewhat too high across the board. By (49), the effective CRRA of the middle-aged agent is 3.37 (2.48 for the old agent), figures that, per se, will not raise eyebrows. Compared to the effective risk aversion coefficient in Campbell and Cochrane (1999), these figures are astonishingly low. Generally speaking, a high equilibrium premium can result even if η1 and η2 are both small, a fact that is not necessarily true of infinite horizon constructs. A premium of 2.96 is obtained, for example, from a parameter combination of γ = 2, η1 = 0.30 and η2 = 0.05.70 Da Silva’s work is an example of the case where a small amount of agent heterogeneity goes a long way toward matching the data. As in representative agent habit models, risk aversion is effectively stochastic. Nevertheless, returns are stationary since the MRS depends only on consumption growth rates, which themselves are stationary series.71 The level dependence effects noted in Danthine et al. (2004) are thus absent. Guvenen (2005) invokes limited participation but within a traditional infinite horizon production context. In his model, shareholders can trade stocks and bonds to smooth out their own consumption, while workers can only trade risk-free bonds. Thus limited in their trading opportunities, workers can only self-insure by acquiring bonds, thereby bidding up their prices, and leading to a low risk-free rate. Although shareholders have multiple consumption smoothing devices (capital shares, bonds), in equilibrium they end up issuing bonds to workers and thus indirectly provide insurance to them. Not only does Guvenen’s (2005) abstraction lead to a replication of all the basic financial stylized facts, but he is also able to demonstrate a near equivalence with the habit formation construct of Campbell and Cochrane (1999). The restricted participation construct of Guvenen (2005) is related to the hypothesis that firms’ shareholders directly provide partial income insurance to their workers in the context of a labor contract (see Danthine et al. (2008) in this volume). In either case, the equilibrium outcome is one in which the wealthier agents with greater access to consumption smoothing devices end up insuring the less wealthy, restricted-access agents. With regard to replicating the stylized facts, it does not seem important whether the insurance is provided within the firm or in the financial market. In the former case, 70 All

figures are obtained in a context where the endowment process is identical to that hypothesized in Constantinides et al. (2002). 71 For CRRA preferences, u(c) = c1−γ /1 − γ, the MRS is ⎛ ⎞−γ β

u (ct+1 − ct ) (ct+1 − ct )−γ ⎜ gt,t+1 − 1 ⎟ =β =⎝ ⎠ u (ct − ct−1 ) (ct − ct−1 ) 1 1− gt−1 ,t


91

shareholder consumption is more variable as they are residual claimants: wages essentially leverage their dividend income streams. In the latter case (Guvenen (2005)), shareholders have, on net, negative bond positions: they hold leveraged portfolios and receive as a result equally high risk consumption streams. Surely, these alternative mechanisms can be assessed on empirical grounds—Mehra and Prescott’s (2007) balance sheet analysis of the U.S. economy emphasizes the latter alternative since wealthy shareholders are found to be net borrowers in the economy. The actual fact is likely to be a bit of both. Both alternatives are more appealing—based on micro-foundations—than the somewhat higher-level abstraction, e.g., of Campbell and Cochrane (1999). In an important, very recent paper, Gomes and Michaelides (2006) go beyond Constantinides et al. (2005) by endogenizing stock market participation (there is a fixed cost to participation and only as agents become sufficiently wealthy are they willing to pay it) in a production based OLG model where agents experience idiosyncratic labor income shocks and borrowing constraints (they cannot borrow against their stochastic life cycle earnings profile). They find that the equity risk premium is relatively unaffected by the limited participation, and is almost entirely determined by incomplete risk sharing within the shareholder class. Non stock market participants are more realistically less wealthy than their counterparts in Guvenen (2005). In their baseline parameterization they derive an equlibrium premium of around 4% with a 2.5% risk free rate. The model is itself a new benchmark and reminds us how the progressive endogenization of previously fixed decisions can change our perspective on the fundamental drivers of financial phenomena. It lacks only a standard macro-style labor leisure choice, something that experience has shown will not be helpful for the task at hand. Nevertheless, Gomes and Michaelides (2006), appears to be the most comprehensive production model to date.

5. MODEL UNCERTAINTY The original Mehra and Prescott (1985) model, as well as all the modifications considered heretofore in this essay, are rational expectations formulations. As such, the agents in the model know exactly the environment in which they operate, including precise knowledge of the stochastic process governing the evolution of output (also the dividend and consumption) and any other state variables that arise in more elaborate versions. At best, this is a very strong assumption. In reality, the exact form of the relevant stochastic processes is not known to investors, at least with regard to distribution parameters, and it is reasonable to expect that this “model uncertainty” would itself command a premium. Three recent papers in this tradition are Mehra and Sah (2002), Barillas et al. (2006), and Weitzman (2007). Mehra and Sah (2002) analyze the consequences of human mood fluctuations on financial markets. The consequences turn out to be surprisingly large. They address a

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very specific question: can small fluctuations in the subjective discount factor (β) and the risk aversion coefficient (γ) of agents result in large price movements in equity markets? They consider the type of fluctuations that occur over a short period of time rather than as secular changes in individual tastes. On a given day, for example, an individual may be in one of several possible preference states, but he is unaware of these taste fluctuations and thus does not implicitly incorporate their consequences into his optimization decisions. In a variant of the Mehra-Prescott (1985) paradigm, they show that even changes of small magnitude in γ and β can have a significant impact on the evolution of asset prices. Their analysis assumes that taste fluctuations occur around temporally stable preferences, and it employs two simplifying assumptions. First, the growth rate of dividends is assumed to evolve according to a geometric Brownian motion. This allows them to derive a closed-form representation for the equity price in terms of the subjective parameters of interest and the drift and volatility parameters of the Brownian motion. Second, they assume that although the parameters change stochastically, the agent’s demands are based on the assumption that the current realization will prevail forever. Thus, the agent, in his demand for securities, behaves myopically with respect to changes in these parameters; their analysis has been criticized on this account. Weitzman (2007) addresses the equity premium, the risk-free rate, and the excess volatility puzzles in a pure exchange economy where the representative agent has standard CRRA preferences. The continuously compounded growth rate of consumption is assumed to be i.i.d with a normal distribution. Its mean and variance, however, are uncertain. While the first three specifications are standard, the last differs from the setting in Mehra and Prescott (1985), where the mean and variance are treated as known parameters that are calibrated to past sample averages. The structural uncertainty in the mean and variance of the growth rate of aggregate consumption introduces a form of Bayesian posterior background risk that is inherited from the prior and does not converge to zero as the number of subsequent observations increases to infinity. This background risk fattens the tails of the posterior distribution of future consumption growth rates and increases significantly the value of both the equity premium and excess volatility, while simultaneously decreasing the risk-free rate (as the motivation for precautionary savings becomes relatively more important than intertemporal substitution). In his framework, expected utility fails to exist except when preferences are logarithmic (γ = 1).72 Weitzman circumvents this problem by assuming that the precision w = 1/σ 2 is a truncated-gamma p.d.f. with truncation parameter δ > 0, which represents a lower bound for the support of the prior distribution of w. The assumption that 1/δ > σ 2 guarantees both the existence of finite expected utility and that the posterior distribution is in the same form as the prior and is subject to the same bounding constraint. 72 See

Geweke (2001).


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In general, with Bayesian learning it should not matter what initial values are assigned to the parameters characterizing the priors since, as the sample size grows, they converge to the true values. In Weitzman’s (2007) context, this holds true for all model parameters except δ, which controls the convergence properties of expected utility. The structural parameter δ never loses its critical impact on the subsequent behavior of agents, regardless of the amount of data accumulated over time. One natural concern (and a potential criticism) about the model is that there is no learning about fundamentals in this setting. If δ does not decline as the sample size grows, then in what sense is the representative agent learning through time? In this model the representative agent can never precisely assess the volatility of consumption growth even if the sample size is infinity. A second concern is that there is, as yet, no refutable hypothesis. Is the calibrated optimal value of δ a reasonable one? More importantly, how can one assess whether a value of δ is reasonable? For instance, to account for the equity premium, it might well be the case that Weitzman is using a low (reasonable) value of the risk aversion coefficient, but an unrealistic value of δ. In Barillas et al. (2006), the agent is in the possession of an approximating model that he does not entirely trust. This latter fact manifests itself in the agent postulating a set of unspecified models—which are statistically similar, as measured by entropy— to his benchmark and which he believes will govern the data. The consequence of this model uncertainty is to allow an otherwise standard construct to satisfy the Hansen and Jagannathan (1991) bounds.

6. CONCLUDING COMMENTS Is the resolution of the equity premium and other financial puzzles dependent on a wise choice of preference structures alone? There is ample indication, in many of the models considered in this essay, that certain behavioral style constructs can come close to such resolution: they work. In many cases, however, we do not know the axiomatic foundations underlying them or their behavior in more general settings. In that sense, we have very incomplete knowledge. Yet, the significance of the equity premium and related puzzles cannot be overstated. The consistency of neoclassical intertemporal economics would seem to rest, in large measure, on its eventual resolution. Our sense is that the context of infinitely lived agents may not ultimately be the most fruitful setting. If one lives forever in a stationary return environment of complete markets, there is an intuitive sense that small risks should not matter. Properly measured and accounted for, they should average out over long horizons. Investors should therefore be largely indifferent to them, implying a very low premium. For finitely lived agents nearing retirement, however, the same risks are critical: a downturn in the stock market is potentially crippling for life, for example. It is our view that a fully satisfactory resolution of the puzzle is likely to await more tractable multi-agent models with a richer demographic structure.

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References Abel, A. B. Asset prices under habit formation and catching up with the Joneses. American Economic Review, Papers and Proceedings 80 (1990): 38–42. Abel, A. B. Equity premia with benchmark levels of consumption: Closed-form results, in R. Mehra, ed., Handbook of the Equity Risk Premium. Elsevier, Amsterdam (2008). Ait-Sahalia, Y., J. Parker, and M. Yogo. Luxury goods and the equity premium. Journal of Finance 59 (2004): 2959–3004. Aiyagari, S. R., and M. Gertler. Asset returns with transactions costs and uninsured individual risk. Journal of Monetary Economics 27 (1991): 311–331. Allais, M. The foundations of a positive theory of choice involving risk and a criticism of the postulates and axioms of the American School, in M. Allais, and O. Hagen, eds., Expected Utility Hypothesis and the Allais Paradox. D. Reidel Publishing Co., Dordrecht, Holland (1979). Alvarez, F., and U. Jermann. Asset pricing when risk sharing is limited by default. Econometrica 48 (2000): 775–797. Alvarez, F., and U. Jermann. Using asset prices to measure the cost of business cycles. Journal of Political Economy 112 (2004): 1223–1256. Attanasio, O. P., J. Banks, and S. Tanner. Asset holding and consumption volatility. Journal of Political Economy 110 (2002): 771–92. Attanasio, O. P., and A. Vissing-Jorgensen. Stock market participation, intertemporal substitution and risk aversion. American Economic Review, Papers and Proceedings (2007). Azeredo, F. Essays on aggregate economics and finance. Doctoral dissertation, University of California, Santa Barbara (2007). Backus, D. K., B. R. Routledge, and S. E. Zin. Exotic preferences for macroeconomists, in M. Gertler, and K. Rogoff, eds., NBER Macroeconomics Annual 2004. MIT Press, Cambridge, MA (2004). Backus, D. K., and S. E. Zin. Reverse engineering the yield curve. NBER Working paper no. 4676 (1994). Bakshi, G., and Z. Chen. Can compensation for cash flow risk and discounting risk reconcile the equity premium puzzle: A quantitative analysis, in R. Mehra, ed., Handbook of the Equity Risk Premium. Elsevier, Amsterdam (2008). Bansal, R. Risk compensation in equity markets, in R. Mehra, ed., Handbook of the Equity Risk Premium. Elsevier, Amsterdam (2008). Bansal, R., and J. W. Coleman. A monetary explanation of the equity premium, term premium and risk free rate puzzles. Journal of Political Economy 104 (1996): 1135–1171. Bansal, R., and A. Yaron. Risks for the long run: A potential resolution of asset pricing puzzles. Journal of Finance 59 (2004): 1481–1509. Barberis, N., and M. Huang. Mental accounting, loss aversion, and individual stock returns. The Journal of Finance, LVI(4) (2001): 1247. Barberis, N., and M. Huang. The loss aversion/narrow framing approach to the stock market pricing and participation puzzles, in R. Mehra, ed., Handbook of the Equity Risk Premium. Elsevier, Amsterdam (2008). Barberis, N., M. Huang, and T. Santos. Prospect theory and asset prices. The Quarterly Journal of Economics, CXVI (1) (2001): 1–53. Barrillas, F., L. Hansen, and T. Sargent. Risk-sensitive valuations of aggregate consumption fluctuations: Risk or model uncertainty? Mimeo, New York University (2006). Barro, R. Rare disasters and asset markets in the twentieth century. Quarterly Journal of Economics 121 (2006): 823–866. Basak, S., and D. Cuoco. An equilibrium model with restricted stock market participation. The Review of Financial Studies 11 (1998): 309–341. Benartzi, S., and R. H. Thaler. Myopic loss aversion and the equity premium puzzle. Quarterly Journal of Economics 110 (1995): 73–92. Bewley, T. F. Thoughts on tests of the intertemporal asset pricing model. Working paper, Northwestern University (1982).


95

Boldrin, M., L. Christiano, and J. Fisher. Habit persistence, asset returns, and the business cycle. American Economic Review 91 (2001): 149–166. Bonomo, M., and R. Garcia. Disappointment aversion as a solution to the equity premium and the risk-free rate puzzles. Working paper, Université de Montréal (1993). Brav, A., G. M. Constantinides, and C. C. Geczy. Asset pricing with heterogeneous consumers and limited participation: Empirical evidence. Journal of Political Economy 110 (2002): 793–824. Brav, A., and C. C. Geczy. An empirical resurrection of the simple consumption CAPM with power utility. Working paper, University of Chicago (1995). Brown, S., W. Goetzmann, and S. Ross. Survival. Journal of Finance 50 (1995): 853–873. Campanale, C., R. Castro, and G. L. Clementi. Asset pricing in a production economy with Chew-Dekel preferences. Working Paper, New York University Stern School of Business (2006). Campbell, J. Y. Asset pricing at the millennium. Journal of Finance 55 (2001): 1515–1567. Campbell, J. Y. Consumption-based asset pricing, in G. M., Constantinides, M. Harris, and R. Stulz, eds., Handbook of the Economics of Finance. North-Holland Elsevier, Amsterdam, (2003) pages 803–887. Campbell, J. Y., and J. H. Cochrane. By force of habit: A consumption-based explanation of aggregate stock market behaviour. Journal of Political Economy 107 (1999): 205–251. Campbell, J. Y., and R. Shiller. The dividend-price ratio and expectations of future dividends and discount factors. Review of Financial Studies 1 (1988): 195–228. Cecchetti, S. J., P. Lam, and N. C. Mark. The equity premium and the risk free rate: Matching the moments. Journal of Monetary Economics 31 (1993): 21–46. Cecchetti, S. J., P. Lam, and N. C. Mark. Asset pricing with distorted beliefs: Are equity returns too good to be true? American Economic Review 90 (2000): 787–805. Chan, Y. L., and L. Kogan. Catching up with the Joneses: Heterogeneous preferences and the dynamics of asset pricing. Journal of Political Economy 110 (2002): 1255–1285. Chapman, D., and V. Polkovnichenko. Heterogeneity in preferences and asset market outcomes. Working paper, Boston College (2006). Cochrane, J. H. Where is the market going? Uncertain facts and novel theories. Economic Perspectives 21 (1997): 3–37. Cochrane, J. H. Financial markets and the real economy, in R. Mehra, ed., Handbook of the Equity Risk Premium. Elsevier, Amsterdam (2008). Cochrane, J., and L. Hansen. Asset pricing explorations for macroeconomics, in NBER Macroeconomics Annual (1992) pages 115–165. Constantinides, G. M. Intertemporal asset pricing with heterogeneous consumers and without demand aggregation. Journal of Business 55 (1982): 253–267. Constantinides, G. M. Habit formation: A resolution of the equity premium puzzle. Journal of Political Economy 98 (1990): 519–543. Constantinides, G. M. Understanding the equity risk premium puzzle, in R. Mehra, ed., Handbook of the Equity Risk Premium. Elsevier, Amsterdam (2008). Constantinides, G. M., J. B. Donaldson, and R. Mehra. Junior can’t borrow: A new perspective on the equity premium puzzle. Quarterly Journal of Economics 118 (2002): 269–296. Constantinides, G. M., J. B. Donaldson, and R. Mehra. Junior must pay: Pricing the implicit put in privatizing social security. Annals of Finance 1 (2005): 1–34. Constantinides, G. M., J. B. Donaldson, and R. Mehra. Junior is rich: Bequests as consumption. Economic Theory, 32 (2007) 125–155. Constantinides, G. M., and D. Duffie. Asset pricing with heterogeneous consumers. Journal of Political Economy 104 (1996): 219–240. Chew, S. H. A generalization of the quasilinear mean with applications to the measurement of income inequality and decision theory resolving the Allais paradox. Econometrica 51(4) (1983): 1065–1092. Chew, S. H. Axiomatic utility theories with the betweenness property. Annals of Operations Research 19 (1989): 273–298.

96

Chapter 2 • Risk-Based Explanations of the Equity Premium Daniel, K., and D. Marshall. The equity premium puzzle and the risk-free rate puzzle at long horizons. Macroeconomic Dynamics 1 (1997): 452–484. Danthine, J.-P., and J. B. Donaldson. Labour relations and asset pricing. Review of Economic Studies 69 (2002): 41–64. Danthine, J.-P., J. B. Donaldson, and R. Mehra. The equity premium and the allocation of income risk. Journal of Economic Dynamics and Control 16 (1992): 509–532. Danthine, J.-P., J. B. Donaldson, C. Giannikos, and H. Guirguis. On the consequences of state dependent preferences for the pricing of financial assets. Financial Research Letters 1 (2004): 143–153. Danthine, J.-P., J. B. Donaldson, and P. Siconolfi. Distribution risk and equity returns, in R. Mehra, ed., Handbook of the Equity Risk Premium. Elsevier, Amsterdam (2008). Da Silva, A. Habit formation in an overlapping generations model with borrowing constraints. Working paper, Columbia University Department of Economics (2006). Da Silva, A., and C. Giannikos. Higher risk aversion in older agents: Its asset pricing implications. Working paper, Columbia University Department of Economics (2006). DeLong, B., and K. Magin. The U.S. equity premium: Past, present and future. Working paper, U.C. Berkeley, Department of Economics (2007). Davis, S. J., and P. Willen. Occupation-level income shocks and asset returns: Their covariance and implications for portfolio choice. CRSP working papers 523, Center for Research in Security Prices, Graduate School of Business, University of Chicago (2000). Dekel, E. An axiomatic characterization of preferences under uncertainty. Journal of Economic Theory 40 (1986): 304–318. Dimson, E., P. Marsh, and R. Staunton. Global evidence on the equity risk premium, in R. Mehra, ed., Handbook of the Equity Risk Premium. Elsevier, Amsterdam (2008). Epstein, L. G., and S. E. Zin. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica 57(4) (1989): 937–969. Epstein, L. G., and S. E. Zin. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: An empirical analysis. Journal of Political Economy 99 (1991): 263–286. Epstein, L. G., and S. E. Zin. The independence axiom and asset returns. Journal of Empirical Finance 8 (2001): 537–572. Ferson, W. E., and G. M. Constantinides. Habit persistence and durability in aggregate consumption. Journal of Financial Economics 29 (1991): 199–240. Falato, A. Happiness maintenance and asset prices. Mimeo, Columbia University (2003). Fama, E. F., and K. R. French. Common risk factors in the returns of stocks and bonds. Journal of Financial Economics 33 (1993): 3–56. Gabaix, X., and D. Laibson. The 6D bias and the equity premium puzzle, in B. Bernanke, and K. Rogoff, eds., NBER Macroeconomics Annual 2001. MIT Press, Cambridge, MA (2001). Geweke, J. A note on some limitations of CRRA utility. Economic Letters 71 (2001): 341–345. Giannikos, C. Asset pricing in the presence of durable goods. Working paper, Columbia University, Department of Economics (2007). Goetzmann, W., and P. Jorion. Global stock markets in the twentieth century. Journal of Finance 54 (1999): 953–980. Goetzmann, W., and R. Ibbotson. History and the equity risk premium, in R. Mehra, ed., Handbook of the Equity Risk Premium. Elsevier, Amsterdam (2008). Gomes, F., and A. Michaelides. Asset pricing with limited risk sharing and heterogeneous agents. Working paper, London Business School, (2006) forthcoming Review of Financial Studies. Gordon, S., and P. St-Amour. A preference regime model of bull and bear markets. American Economic Review 90(4) (2000): 1019–1033. Gordon, S., and P. St-Amour. Asset returns and state-dependent preferences. CIRPEE Working Paper No. 03–16 (2003). Gul, F. A theory of disappointment aversion. Econometrica 59(3) (1991): 667–686. Guvenen, F. A parsimonious macroeconomic model for asset pricing: Habit formation of cross-sectional heterogeneity. Mimeo, University of Texas, Dept. of Economics (2005).


97

Guvenen, F. Reconciling conflicting evidence on the elasticity of intertemporal substitution: A macroeconomic perspective. Journal of Monetary Economics 53(7) (2006): 1451–1472. Hall, R. E. Intertemporal substitution in consumption. Journal of Political Economy 96(2) (1988): 339–357. Hansen, G. Indivisible labor and the business cycle. Journal of Monetary Economics 16 (1985): 309–327. Hansen, L. P., and R. Jagannathan. Implications of security market data for models of dynamic economies. Journal of Political Economy 99 (1991): 225–262. Hansen, L. P., and K. J. Singleton. Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica 50 (1982): 1269–1288. He, H., and D. M. Modest. Market frictions and consumption-based asset pricing. Journal of Political Economy 103 (1995): 94–117. Heaton, J. An empirical investigation of asset pricing with temporally dependent preference specifications. Econometrica 66 (1995): 681–717. Heaton, J. C., and D. J. Lucas. Evaluating the effects of incomplete markets on risk sharing and asset pricing. Journal of Political Economy 104 (1996): 443–487. Heaton, J. C., and D. J. Lucas. Market frictions, savings behavior and portfolio choice. Journal of Macroeconomic Dynamics 1 (1997): 76–101. Heaton, J. C., and D. J. Lucas. Portfolio choice and asset prices: The importance of entrepreneurial risk. Journal of Finance 55 (2000). Heaton, J. C., and D. J. Lucas. Can heterogeneity, undiversifiable risk, and trading frictions explain the equity premium? in R. Mehra, ed., Handbook of the Equity Risk Premium. Elsevier, Amsterdam (2008). Isen, A. M. Positive affect and decision making, in M. Lewis, and J. M. Haviland, eds., Handbook of Emotion. The Guilford Press, New York (1989). Jermann, U. Asset pricing in production economies. Journal of Monetary Economics 41 (1998): 257–275. Johnsen, T., and J. B. Donaldson. The structure of intertemporal preferences under uncertainty and time consistent plans. Econometrica 53 (1985): 1451–1458. Kahneman, D., and A. Tervsky. Prospect theory: An analysis of decision under risk. Econometrica 47 (1979): 263–292. Kallenbrunner, G., and L. Lochstoer. Long run risk through consumption smoothing. Working paper, London Business School (2006). Kihlstrom, R. Risk aversion and the elasticity of substitution in general dynamic portfolio theory: Consistent planning by forward looking, expected utility maximising investors. Working paper, Finance Department, The Wharton School, University of Pennsylvania (2007). King, R. G., and S. Rebelo. Public policy and economic growth: Developing neoclassical implications. RCER Working Papers 225, University of Rochester-Center for Economic Research (RCER) (1988). Kocherlakota, N. Disentangling the coefficient of relative risk aversion from the elasticity of intertemporal substitution: An irrelevance result. Journal of Finance 45 (1990): 175–190. Kocherlakota, N. The equity premium: It’s still a puzzle. Journal of Economic Literature 34 (1996): 42–76. Koopmans, T. C. Stationary ordinal utility and impatience. Econometrica 28 (1960): 287–309. Krebs, T. Consumption-based asset pricing with incomplete markets. Working paper, Brown University (2000). Kreps, D. M., and E. L. Porteus. Temporal resolution of uncertainty and dynamic choice theory. Econometrica 46 (1978): 185–200. Lettau, M., and S. C. Ludvigson. Consumption, aggregate wealth, and expected stock returns. Journal of Finance 53 (2001): 815–850. Lettau, M., and H. Uhlig. Can habit formation be reconciled with business cycle facts? Review of Economic Dynamics 3 (2000): 79–99. Ljungqvist, L., and H. Uhlig. On consumption bunching under Campbell-Cochrane habit formation. SFB 649 Discussion Paper, Berlin (1999). http://sfb649.wiwi.hu-berlin.de/discussionPapers de.php. Lucas, D. J. Asset pricing with undiversifiable risk and short sales constraints: Deepening the equity premium puzzle. Journal of Monetary Economics 34 (1994): 325–341. Lucas, R. E., Jr. Asset prices in an exchange economy. Econometrica 46 (1978): 1429–1445.

98

Chapter 2 • Risk-Based Explanations of the Equity Premium Lucas, R. E., Jr. Models of Business Cycles. Yrjo Jahnsson Lectures, Basil Blackwell, Oxford (1985). Lustig, H., and S. G. van Nieuwerburgh. Housing collateral, consumption insurance and the risk premia: An empirical perspective, Journal of Finance 60 (2005): 1167–1219. Luttmer, E. G. J. Asset pricing in economies with frictions. Econometrica 64 (1996): 1439–1467. Malloy, C., T. Moskowitz, and A. Vissing-Jorgensen. Long run stockholders consumption risk and asset returns. Working paper (2005). Mankiw, N. G. The equity premium and the concentration of aggregate shocks. Journal of Financial Economics 17 (1986): 211–219. Mankiw, N. G., and P. Z. Stephen. The consumption of stockholders and nonstockholders. Journal of Financial Economics 29 (1991): 97–112. McGrattan, E. R., and E. C. Prescott. Taxes, regulations, and asset prices. Working paper 610, Federal Reserve Bank of Minneapolis (2001). Mehra, R., and E. C. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15 (1985): 145–161. Mehra, R., and E. C. Prescott. The equity premium: A solution? Journal of Monetary Economics 22 (1988): 133–136. Mehra, R., and E. C. Prescott. The equity premium: ABCs, in R. Mehra, ed., Handbook of the Equity Risk Premium. Elsevier, Amsterdam (2008a). Mehra, R., and E. C. Prescott. Non risk based explanations of the equity premium puzzle, in R. Mehra, ed., Handbook of the Equity Risk Premium. Elsevier, Amsterdam (2008b). Mehra, R, and E. C. Prescott. Intermediated quantities and returns. UCSB working paper. (2007). Mehra, R., and R. Sah. Mood fluctuations, projection bias, and volatility of equity prices. Journal of Economic Dynamics and Control 26 (May) (2002): 869–887. Melino, A., and A. X. Yang. State dependent preferences can explain the equity premium puzzle. Review of Economic Dynamics 6(2) (2003): 806–830. Piazzesi, M., M. Schneider, and S. Tuzel. Housing, consumption and asset prices. Journal of Financial Economics 83 (2007): 531–569. Pollack, R. Habit formation and dynamic demand functions. Journal of Political Economy 78 (1970): 745–763. Prescott, E. C., and R. Mehra. Recursive competitive equilibrium: The case of homogeneous households. Econometrica 48 (1980): 1365–1379. Quiggin, J. A theory of anticipated utility. Journal of Economic Behavior and Organisation 3 (1982): 323–343. Quiggin, J. Generalized Expected Utility Theory/The Rank-Dependent Expected Utility Model, KluwerNijhoff, Amsterdam (1993). Ravn, M., S. Schmitt-Groh, and M. Uribe. Deep habits. CEPR Working paper no. 4269 (2004). Reitz, T. The equity premium: A solution. Journal of Monetary Economics 22 (1988): 117–131. Routledge, B., and S. Zin. Generalized disappointment aversion and asset prices. Mimeo, Carnegie-Mellon University (2004). Storesletten, K., C. I. Telmer, and A. Yaron. Consumption and risk sharing over the lifecycle. Journal of Monetary Economics 51 (2004): 609–633. Storesletten, K., C. I. Telmer, and A. Yaron. Asset pricing with idiosyncratic risk and overlapping generations. Review of Economic Dynamics (2007). Storesletten, K., C. I. Telmer, and A. Yaron. Asset prices and intergenerational risk sharing: The role of idiosyncratic earnings shocks, in R. Mehra, ed., Handbook of the Equity Risk Premium. Elsevier, Amsterdam (2008). Sundaresan, S. M. Intertemporally dependent preferences and the volatility of consumption and wealth. Review of Financial Studies 2 (1989): 73–89. Telmer, C. I. Asset-pricing puzzles and incomplete markets. Journal of Finance 49 (1993): 1803–1832.


99

Veronesi, P. The peso problem hypothesis and stock market returns. Journal of Economic Dynamics and Control 28 (2004). Vissing-Jorgensen, A. Limited asset market participation and the elasticity of intertemporal substitution. Journal of Political Economy 110 (2002): 825–853. Weil, P. The equity premium puzzle and the risk-free rate puzzle. Journal of Monetary Economics 24 (1989): 401–421. Weitzman, M. Subjective expectations and asset return puzzles (2007). Forthcoming in American Economic Review. Yogo, M. Estimating the elasticity of intertemporal substitution when instruments are weak. The Review of Economics and Statistics 86(3) (2004): 797–810. Yogo, M. A consumption based explanation of expected stock returns. Journal of Finance 61 (2006): 539–580.


CHAPTER 3

Non-Risk-based Explanations of the Equity Premium Rajnish Mehra University of California, Santa Barbara, and NBER and

Edward C. Prescott Arizona State University, and Federal Reserve Bank of Minneapolis

Introduction 1. The Inappropriateness of Using T-Bills as a Proxy for the Intertemporal Marginal Rate of Substitution of Consumption 1.1. Liquidity 1.2. Transaction Balances 2. The Effect of Government Regulations and Rules 3. Taxes 4. Borrowing Constraints 5. The Impact of Agent Heterogeneity and Intermediation Costs 6. Concluding Comments References

102 102 104 104 106 107 110 113 114 114

JEL Classification: G10, G12, G28, N42, D9 Keywords: asset pricing, equity risk premium, liquidity, risk free rate, transaction balances, government regulations, taxes, agent heterogeneity, intermediation costs

We thank John Donaldson and Ellen McGrattan for invaluable comments. HANDBOOK OF THE EQUITY RISK PREMIUM c 2008 by Rajnish Mehra and Edward C. Prescott. All rights of reproduction in any form reserved. Copyright

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Chapter 3 • Non-Risk-based Explanations of the Equity Premium

INTRODUCTION A widely accepted tenet of the neoclassical paradigm has been that the observed differences in the rates of return on financial assets, in particular, the large difference between the average returns on corporate equity and T-bills, are a premium for bearing nondiversifiable aggregate risk. The findings in our paper, “The Equity Premium: A Puzzle” that only a small part of this difference is a premium for bearing aggregate risk came as a surprise to the profession. Over the last 20 years or so, attempts to successfully account for the equity premium have become a major research impetus in finance and economics. A majority of these explanations have focused on mechanisms to increase the premium for bearing non-diversifiable risk.1 In this chapter, in contrast, we review the nascent literature that takes as given the findings in Mehra and Prescott (1985) and tries to account for the observed magnitude of the equity premium by using factors other than aggregate risk. Much of this literature re-examines the appropriateness of the abstractions and assumptions made in our original paper. In particular, the appropriateness of using T-bills as a proxy for the intertemporal marginal rate of substitution of consumption, the impact of government regulations, the abstractions from taxes, diversification and intermediation costs, borrowing constraints and agent heterogeneity have been explored. We consider each in turn and examine their impact on the equity premium.

1. THE INAPPROPRIATENESS OF USING T-BILLS AS A PROXY FOR THE INTERTEMPORAL MARGINAL RATE OF SUBSTITUTION OF CONSUMPTION In the two decades since “The Equity Premium: A Puzzle” was published, the way we think about economic phenomena has undergone a fundamental change, influenced in large measure by the development of Real Business Cycle theory by Kydland and Prescott (1982). One finding of the real business cycle literature is that the real after-tax return on capital, that is the after tax capital income divided by the cost of reproducing the tangible capital stock, averages 4.5 percent, with modest variation, over time. A key question that arises is, what is a good empirical counterpart to the household’s real interest rate? Mehra and Prescott (1985) used the highly liquid T-bill rate, corrected for expected inflation, as a proxy for the return on the riskless asset that is used by agents to smooth consumption. The appropriateness of this assumption was questioned by McGrattan and Prescott (2003) and more recently by Mehra and Prescott (2007). The remainder of this section closely follows the discussion in these papers. An assumption implicit in Mehra and Prescott (1985) is that agents use both equity and the riskless asset to intertemporally smooth consumption. This is as a direct consequence of the first-order condition (see ahead) for the representative household in their 1 Many

of the chapters in this volume elaborate on these efforts. The previous chapter (Donaldson and Mehra (2008) surveys this literature.

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TABLE 1 Household Assets and Liabilities (Average of 2000 and 2005) Assets (GDP)

Liabilities (GDP)

Tangible household

1.65

Liabilities

0.7

Corporate equity

0.85

Net worth

4.15

Non corporate equity

0.5

Pension and life insurance reserves

1.0

Debt assets

0.85

Total

4.85

Total

4.85

model, which saves by optimally allocating resources between equity and risk less debt. Equation (1) is the standard asset pricing equation in macroeconomics and finance. Uc (ct+s ) e d 0 = Et ) , (1) (rt,t+s − rt,t+s Uc (ct ) e d where Uc (ct+s ) is the marginal utility of consumption at time t + s, rt,t+s and rt,t+s are, respectively, the return on equity and the risk less asset over the period t, t + s and Et is the expectation conditional on the agent’s information set at time t. Is it reasonable to assume that the return on T-bills is equal to the real interest rate that households use to save for retirement and smooth consumption? Do households actually hold T-bills to finance their retirement? If this were empirically true it would be reasonable to equate their expected marginal rate of substitution of consumption to the rate of return on T-bills. This question cannot be answered in the abstract, without reference to the asset holdings of households. A natural next step then is to examine the assets held by households. Table 1 details these holdings for American households. The four big asset-holding categories of households are tangible assets, pension and life insurance holdings, equity (both corporate and non corporate), and debt assets. In the year 2000, privately held government debt was only 0.30 GDP, a third of which was held by foreigners. The amount of interest bearing government debt with maturity less than a year was only 0.085 GDP, which is a small fraction of the total household net worth. Virtually no T-bills are directly owned by households.2 Approximately one-third of the outstanding T-bills are held by foreigners, in particular foreign central banks, and two-thirds by American financial institutions. Although there are large amounts of debt assets held, most of these are in the form of pension fund and life insurance reserves. Some are in the form of demand deposits for which free services are provided. Most of the government debt is held indirectly; however, some is held in the form of savings bonds that people gift to their grandchildren.

2 Table

B-89, Economic Report of the President 2005.

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Thus, much of intertemporal saving is in debt assets such as annuities and mortgage debt, held in retirement accounts and as pension fund reserves. Assets other than T-bills, are held to finance consumption when old. Two natural questions then are who holds T-bills and do the holders use them to intertemporally smooth consumption? A large fraction of the stock of short-term U.S. T-bills are held by foreign central banks and are used to smooth exchange rates. These central banks are not using these assets to equate the marginal utility of consumption today to the expected discounted marginal utility of consumption tomorrow. U.S. households do hold a sizable quantity of close substitutes for T-bills. These including money market accounts, and bank CDs with maturities of one year, or less. They are of the order of magnitude of 0.50 times GDP. However, these short-term liquid assets with low expected returns are held, in most part, for reasons other than saving for retirement. These include liquidity and the maintence of transaction balances.

1.1. Liquidity Part of these assets are held for liquidity purposes and as precautionary balances which serve as a substitute for insurance against idiosyncratic risk. The latter is very costly to insure against because of the associated moral hazard and administrative costs. When this is the case, in the household’s maximization problem there are Lagrange multipliers, which are not zero, and as a result the expected returns are less than that predicted by a theory that abstracts from the costs of insuring against idiosyncratic risk. One of the first attempts to quantify this liquidity premium was the work of Bansal and Coleman (1996). In their model, some assets other than money play a key feature by facilitating transactions. This affects the rate of return they offer in equilibrium. Considering the role of a variety of assets in facilitating transactions, they argue that, on the margin, the transaction service return of money relative to interest-bearing checking accounts should be the interest rate paid on these accounts. They estimate this to be 6 percent, based on the rate offered on NOW accounts for the period they analyze. Since this is a substantial number, they suggest that other money-like assets may also implicitly include a transaction service component to their return. Insofar as T-bills and equity have a different service component built into their returns, this may offer an explanation for the observed equity premium.

1.2. Transaction Balances Large transaction costs are associated with moving into and out of high-yielding assets, with a major component being record keeping. With assets whose prices vary, there are capital gains or losses associated with every sale of these assets which must be reported on the annual tax form. This is why money market accounts hold debt securities until they mature. Thus, there are no capital gains or losses associated with the transaction when an individual buys or sells shares in his money market account. All that the money market mutual fund need report to its shareholders is an annual statement of the interest they received on the 1099 form. This is a major reason that sizable quantities of currency and non-interest-bearing demand deposits are held. We note that Ml is about 15 percent of GDP.

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Brokerage charges were (and perhaps continue to be for some) another significant cost associated with getting into and out of higher-yielding securities such as stocks and bonds. It is true that discount brokers now exist, but gaining the prerequisite knowledge is costly in terms of time. If the holding period is short, even for a risk-neutral investor it makes no sense for the investor to buy and later sell stocks unless the expected return on stocks is huge. Based on the above insights, both McGrattan and Prescott (2003) and Mehra and Prescott (2007) conclude that returns on T-bills and short-term debt are not a reasonable empirical counterpart of the household interest rate in Eq. (1). Hence, it would be inappropriate to equate the return on these assets to the marginal rate of substitution for an important group of agents. McGrattan and Prescott (2003), who are concerned with the difference in the returns on debt and equity held to finance retirement, use long term high grade municipal bonds as their measure of the long term household interest rate during the gold standard period of 1880–1934. They note that as these bonds were tax exempt and were held directly by individuals, no further adjustment for taxes and intermediation costs is required. They do not adjust for inflationary expectations, arguing that prices were expected to be stable during this period. For the post World War II and Korean War period (1960–2002), they use “the highest grade corporate bonds which were held primarily in tax-deferred pension funds.” To account for inflationary expectations they subtract the inflation rate for the previous 10 years. Since government regulations during the interwar years affected bond returns (see ahead), they used NIPA capital returns for these years. Their results are plotted in Figure 1. The returns for the entire period are seen to be remarkably close to 4 percent, considerably higher than the 0.8 percent return for T-bills reported by Mehra and Prescott (1985). 10% 8

Gold standard period

Postwar period

6

4 2 0 22

High-grade bonds NIPA capital

24 1880

1900

FIGURE 1 Source: McGrattan and Prescott (2003).

1920

1940

1960

1980

2000

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Mehra and Prescott (2007) argue that an inflation indexed default-free bond portfolio would essentially be a risk free asset that could be used for savings to finance consumption when old. Since TIPS have only recently (1997) been introduced in the U.S. capital markets, it is difficult to get accurate estimates of the mean return on this asset class. Due to the small quantity supplied, these average returns are not necessarily the same as they would be, had the quantity been larger. The average real return for the 1997–2005 period was 3.7 percent. An alternative would be to use assets such as indexed mortgages guaranteed by Ginnie Mae or issued by Fannie Mae. Recent work by Krishnamurthy and Vissing-Jorgensen (2007) provides evidence for a convenience yield on all treasury securities (not limited to T-bills). They document that the debt/GDP ratio in the U.S. is negatively correlated with the spread between corporate bond yields and Treasury bond yields and claim that the result holds even when controlling for the default risk of corporate bonds. They argue that the corporate bond spread reflects a convenience yield that investors attribute to Treasury debt. This yield, which varies with the debt/GDP ratio, is in the 0.2–1.6 percent range. This finding, if it holds up, provides additional evidence that returns on even longdated Treasury securities probably underestimate the marginal rate of intertemporal substitution of consumption and further strengthen the arguments in McGrattan and Prescott (2003).

2. THE EFFECT OF GOVERNMENT REGULATIONS AND RULES McGrattan and Prescott (2003) argue that the estimated expected return on debt assets in the U.S. (including T-bills) over the period 1926–2000 is biased downwards because of various regulations (in particular, W and X) that helped the Treasury keep nominal rates below 2.5 percent during the 1941–54 period. Table 2 shows that the return on debt securities during that period was considerably lower than their long-term average value. This serves as a reminder that governments can pursue regulatory policies that result in negative interest rates over an extended period of time. Clearly, these rates have little to do with the agents’ marginal rate of substitution that would be inferred were there no regulations. Such regulatory periods should be excluded in estimating the long-term average rates on debt securities. TABLE 2 U.S. Inflation Adjusted Average Return on Debt Period

1926–2004

1926–40 and 1955–2004

1941–1954

U.S. Treasury bills

0.74%

1.68%

−3.59%

Intermediate-term government bonds

2.48%

3.60%

−2.71%

Long-term government bonds

2.84%

3.86%

−1.88%

Long-term corporate bonds

3.25%

4.36%

−1.89%

Source: Ibbotson 2005.

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The third column in Table 2 shows how the conventionally used numbers (in column 2) change when the 1941–54 period is excluded. The estimated average rates increase by about 1 percent for all asset classes. In the case of T-bills, a further adjustment needs to be made to the returns in the 1930s. During that period, in some states, T-bills were exempt from personal property taxes while cash in the form of bank notes was not. This created an additional demand for the T-bills, suggesting they were sold at a premium. Again, these rates on return have little to do with the marginal rate of substitution of consumption over time. The effect of these adjustments is to further reduce the magnitude of the equity premium relative to T-bills. To summarize: using the return on a relative risk-free asset that is used for life cycle saving as a proxy for the intertemporal marginal rate of substitution of consumption (instead of a T-bill return) can significantly reduce the equity premium. Adjusting debt returns for government regulations further reduces the premium by 1 percent irrespective of the debt asset used as a benchmark.

3. TAXES McGrattan and Prescott (2003) point out that although taxes on equity income were low until the mid-1930s, they rose dramatically during the war years. They argue that equity returns should be adjusted to reflect this, especially because of regulations that discouraged insurance companies and pension funds from holding equity in tax deferred accounts. They use IRS data to compute the marginal tax rates on equity income distributions. Figure 2 is a plot of the marginal tax rates on equity income distributions. These rates peaked at about 50 percent. 60% 50 40 30 20 10 0

1920

1940

1960


1980

2000

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Consider the case where the tax on corporate distributions is increased from 0 percent to 50 percent and the additional revenue collected is returned to households in a lump sum. If all, or almost all investments are financed by retained earnings, the value of the stock would be twice as high when the distribution tax rate is zero as when this tax rate is 50 percent. A consequence of this is that the return on equity computed in the standard way will be twice as high when the tax on distribution is 50 percent. This is important because there have been large changes in the tax rate on corporate distributions to owners. Figure 2 plots the distribution tax rate for the United States. McGrattan and Prescott (2003) use this to compute the after tax equity returns to households after making a number of other adjustments to account for inflation, diversification costs and the fact that some equity was not taxed or was tax deferred. Figure 3 is a plot of their estimate of the after tax return on equity. An interesting question that arises from their analysis is, why was the return on corporate equity so high in the 1960–2000 period? McGrattan and Prescott (2000, 2005) answer this in the process of estimating the fundamental value of the stock market in 1962 and 2000. They chose these two points in time because after-tax corporate earnings, net corporate debt, and corporate tangible capital stock were approximately the same relative to GDP and the tax system had been stable for a number of years. Further, at neither point in time was there any fear of full or partial expropriation of capital. What differed was that the value of the stock market relative to GDP in 2000 was nearly twice as large as in 1962. What changed between 1962 and 2000 were the tax and legal-regulatory systems. The marginal tax rate on corporate distributions was 43 percent in the 1955–1962 period

10% 8 6 4 2 0 22

Large company stocks NIPA capital

24 1880

1900


1920

1940

1960

1980

2000

109


and only 17 percent in the 1987–2000. This marginal tax rate on dividends does not have consequences for steady-state after-tax earnings or steady-state corporate capital, if tax revenues are returned lump sum to households. This tax rate does, however, have consequences for the value of corporate equity. The important changes in the legal-regulatory system, most of which occurred in the late 1970s and early 1980s, were that corporate equity was permitted to be held as pension fund reserves and that individuals could invest on a before-tax basis in individual retirement accounts that include equity. The threat of a lawsuit is why debt assets and not equity with higher returns were held as pension fund reserves prior to the late 1970s. At that time, little equity was held in defined contribution retirement accounts because the total assets in these accounts were then a small number. Thus, debt and not equity could, and was, held tax free in 1962. In 2000, both could be held tax-free in defined benefit and defined contribution pension funds and in individual retirement accounts. Not surprisingly, the assets held in tax deferred retirement accounts were large in 2000, being approximately 1.3 GDP (McGrattan and Prescott (2000)). In determining whether the stock market was over or undervalued vis-à-vis standard growth theory, McGrattan and Prescott (2000, 2005) exploit the fact that the value of a set of real assets is the sum of the values of the individual assets in the set. They develop a method for estimating the value of intangible corporate capital, something that is not reported on balance sheets and, like tangible capital, adds to the value of corporations. Their method uses only national account data and the equilibrium condition that aftertax returns are equated across assets. They also incorporate the most important features of the U.S. and U.K. tax systems into the model, in particular, the fact that capital gains are only taxed upon realization. The formula they develop for the fundamental value of corporate equities V is V = (1 − τd )KT + (1 − τ d ) (1 − τc )KI

(2)

where τd is the tax rate on distributions, τc is the tax rate on corporate income, KT is the end-of-period tangible corporate capital stock at reproduction cost, and KI is the end-of-period intangible corporate capital stock at reproduction cost. The reasons for the tax factors are as follows. Corporate earnings significantly exceed corporate investment, and, as a result, aggregate corporate distributions are large and positive. Historically, these distributions have been in the form of dividends though buybacks have come to be an increasingly important form of distribution in recent years. Therefore, the cost of a unit of tangible capital on margin is only 1 − τd units of forgone consumption. In the case of intangible capital, the consumption cost of a unit of capital is even smaller because investments in intangible capital reduce corporate tax liabilities.3 The tricky part of the calculation is in constructing a measure of intangible capital. These investments reduce current accounting profits and the returns on the existing stock 3 In fact, formula (1) must be adjusted if economic depreciation and accounting depreciation are not equal and if there is an investment tax credit. See McGrattan and Prescott (2005).

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of intangible capital but increase future accounting profits. The formula for steady-state before tax accounting profits is π=

i KT + iKI − gKI , 1 − τc

(3)

where g is the steady-state growth rate of the economy and i the steady-state after-tax real interest rate. Note that gKI is the steady-state net investment in intangible capital, which reduces accounting profits because it is expensed. Note also that all the variables in formula (3) are reported in the system of national accounts with the exception of i and KI . McGrattan and Prescott (2005) estimate i using national income data. Their estimate of i is the after-tax real return on capital in the non-corporate sector, which has as much capital as does the corporate sector. They find that the stock market was neither overvalued nor undervalued in 1962 and 2000. The primary reason for the low valuation in 1962 relative to GDP and high valuation in 2000 relative to GDP is that τd was much higher in 1962 than it was in 2000. A second reason is that the value of foreign subsidiaries of U.S. corporations grew in the period. An increase in the size of the corporate intangible capital stock was also a contributing factor. McGrattan and Prescott (2005) find that in the economically and politically stable 1960–2000 period, the after-tax real return on holding corporate equity was as predicted by theory if the changes in the tax and regulatory system were not anticipated. These unanticipated changes led to a large unanticipated capital gain on holding corporate equity. Evidence of the importance of these changes is that the share of corporate equity held in retirement accounts and as pension fund reserves increased, from essentially zero in 1962 to slightly over 50 percent in 2000. This share is significantly higher for publicly traded securities as Schedule S corporations and small private Schedule C corporations are not held in retirement accounts. This is important because it means that half of corporate dividends are now subject to zero taxation. In periods of economic uncertainty, such as those that prevailed in the 1930–1955 period with the Great Depression, World War II, and the fear of another great depression, the survival of the capitalistic system was in doubt. In such times, low equity prices and high real returns on holding equity are not surprising. This is the Brown, Goetzmann, and Ross (1995) explanation of the equity premium. By 1960, the fears of another great depression and of an abandonment of the capitalistic system in the United States had vanished, and clearly other factors gave rise to the high return on equity in the 1960–2000 period.

4. BORROWING CONSTRAINTS4 In models with borrowing constraints and transaction costs, the effect is to force investors to hold an inventory of bonds (precautionary demand) to smooth consumption. 4 The reader is also referred to the section on Market Incompleteness and Trading Frictions in Chapter 2 in this volume.

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Hence, in infinite horizon models without borrowing constraints, agents come close to equalizing their marginal rates of substitution with little effect on the equity premium.5 Some recent attempts to resolve the puzzle incorporating both borrowing constraints and consumer heterogeneity appear promising. One approach, which departs from the representative agent model, has been proposed in Constantinides, Donaldson and Mehra (2002). In order to systematically illustrate these ideas, the authors construct an overlappinggenerations (OLG) exchange economy in which consumers live for three periods. In the first period, a period of human capital acquisition, the consumer receives a relatively low endowment income. In the second period, the consumer is employed and receives wage income subject to large uncertainty. In the third period, the consumer retires and consumes the assets accumulated in the second period. The authors explore the implications of a borrowing constraint by deriving and contrasting the stationary equilibria in two versions of the economy. In the borrowingconstrained version, the young are prohibited from borrowing and from selling equity short. The borrowing-unconstrained economy differs from the borrowing-constrained one only in that the borrowing constraint and the short-sale constraint are absent. An unconstrained representative agent’s maximization problem is formulated as follows. An agent born in period t solves: 2 i β U (Ct,i ) (4) max E {zet,i ,zbt,i } i=0 subject to ct,0 + qte zet,0 + qtb zbt,0 ≤ w 0

(5)

e b e b 1 ct,1 + qt+1 zet,1 + qt+1 zbt,1 ≤ qt+1 + dt+1 zet,0 + qt+1 + b zbt,0 + wt+1 e b ct,2 ≤ qt+2 + dt+2 zet,1 + qt+2 + b zbt,1

(6)

ct,j is the consumption in period t + j(j = 0, 1, 2) of a consumer born in period t. There are two types of securities in the model, bonds, and equity with ex-coupon and ex-dividend prices qtb and qte respectively. Bonds are a claim to a coupon payment b every period, while the equity is a claim to the dividend stream {dt }. The consumer born in period t receives deterministic wage income w 0 > 0 in period t, when young; stochas1 tic wage income wt+1 > 0 in period t + 1, when middle-aged; and zero wage income in period t + 2, when old. The consumer purchases zet,0 shares of stock and zbt,0 bonds when young. The consumer adjusts these holdings to zet,1 and zbt,1 , respectively, when middle-aged. The consumer liquidates his/her entire portfolio when old. Thus zet,2 = 0 and zbt,2 = 0. 5 This

is true unless the supply of bonds is unrealistically low. See Aiyagari and Gertler (1991).

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When considering the borrowing constrained equilibrium the following additional constraints are imposed: zet,0 = 0 and zbt,0 = 0. The model introduces two constraints on consumer trades. First, consumers of one generation remove cannot trade claims against their future wage income with consumers of another generation.6 Second, consumers of one generation cannot trade bonds and equity with consumers of an unborn generation. They build on the observation that absent a complete set of contingent claims, consumer heterogeneity in the form of uninsurable, persistent and heteroscedastic idiosyncratic income shocks, with countercyclical conditional variance, can potentially resolve empirical difficulties encountered by representative-consumer models.7 The novelty of their paper lies in incorporating a life-cycle feature to study asset pricing. The idea is appealingly simple. As discussed earlier, the attractiveness of equity as an asset depends on the correlation between consumption and equity income. If equity pays off in states of high marginal utility of consumption, it will command a higher price, (and consequently a lower rate of return), than if its payoff is in states where marginal utility is low. Since the marginal utility of consumption varies inversely with consumption, equity will command a high rate of return if it pays off in states when consumption is high, and vice versa.8 A key insight of their paper is that as the correlation of equity income with consumption changes over the life cycle of an individual, so does the attractiveness of equity as an asset. Consumption can be decomposed into the sum of wages and equity income. A young person looking forward at his life has uncertain future wage and equity income; furthermore, the correlation of equity income with consumption will not be particularly high, as long as stock and wage income are not highly correlated. This is empirically the case, as documented by Davis and Willen (2000). Equity will thus be a hedge against fluctuations in wages and a “desirable” asset to hold as far as the young are concerned. The same asset (equity) has a very different characteristic for the middle-aged. Their wage uncertainty has largely been resolved. Their future retirement wage income is either zero or deterministic and the innovations (fluctuations) in their consumption occur from fluctuations in equity income. At this stage of the life cycle, equity income is highly correlated with consumption. Consumption is high when equity income is high, and equity is no longer a hedge against fluctuations in consumption; hence, for this group, it requires a higher rate of return. The characteristics of equity as an asset therefore change, depending on who the predominant holder of the equity is. Life cycle considerations thus become crucial for asset pricing. If equity is a “desirable” asset for the marginal investor in the economy, then the observed equity premium will be low, relative to an economy where the marginal 6 Being homogeneous within their generation, consumers have no incentive to trade claims with consumers of their own generation. 7 See Mankiw (1986) and Constantinides and Duffie (1996). 8 This is precisely the reason why high-beta stocks in the simple CAPM framework have a high rate of return. In that model, the return on the market is a proxy for consumption. High-beta stocks pay off when the market return is high, i.e. when marginal utility is low, hence their price is (relatively) low and their rate of return high.


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investor finds it unattractive to hold equity. The deus ex machina is the stage in the life cycle of the marginal investor. The authors argue that the young, who should be holding equity in an economy with markets for all event-contingent commodities, are effectively shut out of this market because of borrowing constraints. The young are characterized by low wages; ideally they would like to smooth lifetime consumption by borrowing against future wage income (consuming a part of the loan and investing the rest in higher return equity). However, they are prevented from doing so because human capital alone does not collateralize major loans in modern economies for reasons of moral hazard and adverse selection. In the presence of borrowing constraints, equity is thus exclusively priced by the middle-aged investors, since the young are effectively excluded from the equity markets and we observe a high equity premium. If the borrowing constraint is relaxed, the young will borrow to purchase equity, thereby raising the bond yield. The increase in the bond yield induces the middle-aged to shift their portfolio holdings from equity to bonds. The increase in demand for equity by the young and the decrease in the demand for equity by the middle-aged work in opposite directions. On balance, the effect is to increase both the equity and the bond return while simultaneously shrinking the equity premium. Furthermore, the relaxation of the borrowing constraint reduces the net demand for bonds and the risk-free rate puzzle re-emerges.

5. THE IMPACT OF AGENT HETEROGENEITY AND INTERMEDIATION COSTS A limitation of the homogenous household construct is that it precludes the modelling of borrowing and lending among agents. In equilibrium, the shadow price of consumption at date t + 1 in terms of consumption at date t is such that the amount of borrowing and lending is zero. Homogenous household models are thus incapable of matching the quantities of assets held and intermediated. To address this issue, Mehra and Prescott (2007) construct a model economy that incorporates agent heterogeneity in the form of differences in the strength of the bequest motive. In light of their earlier finding (1985) that the premium for bearing nondiversifiable aggregate risk is small, their analysis abstracts from aggregate risk. The only uncertainty that agents face is idiosyncratic risk about the duration of their lifetime after retirement. Agents have identical preferences for consumption; however, they differ with respect to their intensity for bequests. In equilibrium, those with a strong bequest motive accumulate equity assets and, when retired, live off the income of these assets. The households with no bequest motive buy annuities during their working years and consume the annuity benefits over their retirement years. We emphasize that annuities are mostly defined benefit pension plans. The incorporation of agent heterogeneity allows them to capture a key empirical fact—that there is a large amount of borrowing and lending between households, in

114


particular, between older households. This borrowing is done either directly, by issuing mortgages to finance owner occupied housing or indirectly, by owning partially debt financed rental properties through direct or limited partnerships or REITS. Mehra and Prescott (2007) abstract from that small amount of borrowing and lending directly between households and assume that all of it is intermediated through financial institutions such as banks and pension funds. For the United States, in 2005 the amount intermediated was approximately 1.6 times the GDP. Their intermediation technology is constant returns to scale with intermediation costs being proportional to the amount intermediated. To calibrate the constant of proportionality, they use Flow of Funds statistics and data from National Income and Product Accounts. The calibrated value of this parameter equals the net interest income of financial intermediaries, divided by the quantity of intermediated debt and is a little over 2 percent. In the absence of aggregate uncertainty, the return on equity and the borrowing rate are identical, since the agents who borrow are also marginal in equity markets. In their framework, government debt is not intermediated and thus its return is equal to the lending rate. The equity premium relative to government debt is the intermediation spread. The divergence between borrowing and lending rates gives rise to a 2 percent equity premium even in a world without aggregate uncertainty.

6. CONCLUDING COMMENTS In this essay, we present plausible arguments that using an appropriate benchmark for the households’ real interest rate, incorporating the effect of government regulations, taxes and intermediation costs can account for a large part of the observed equity premium. As a result, we have a deeper understanding of the role and importance of the abstractions that contribute to the puzzle. While no single explanation fully resolved the anomaly, considerable progress has been made and the equity premium is a lesser puzzle today than it was twenty years ago.

References Aiyagari, S. R., and M. Gertler. Asset returns with transactions costs and uninsured individual risk. Journal of Monetary Economics 27 (1991): 311–331. Bansal, R., and J. W. Coleman. A monetary explanation of the equity premium, term premium and risk free rate puzzles. Journal of Political Economy 104 (1996): 1135–1171. Brown, S., W. Goetzmann, and S. Ross. Survival. Journal of Finance 50 (1995): 853–873. Constantinides, G. M., J. B. Donaldson, and R. Mehra. Junior can’t borrow: A new perspective on the equity premium puzzle. Quarterly Journal of Economics 118 (2002): 269–296. Constantinides, G. M., and D. Duffie. Asset pricing with heterogeneous consumers. Journal of Political Economy 104 (1996): 219–240. Davis, S. J., and P. Willen. Using financial assets to hedge labor income risk: Estimating the benefits. Working paper, University of Chicago (2000). Donaldson, J. B., and R. Mehra. Risk based explanations of the equity premium forthcoming in Handbook of the Equity Risk Premium, R. Mehra, ed. Amsterdam (2008).


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Economic Report of the President (2005). Council of Economic Advisors. United States Government Printing Office. Washington. Ibbotson Associates. Stocks, bonds, bills and inflation. 2005 yearbook. Ibbotson Associates, Chicago (2006). Krishnamurthy, A., and A. Vissing-Jorgensen. The demand for treasury debt. Working paper, Northwestern University (2007). Kydland, F., and E. C. Prescott. Time to build and aggregate fluctuations. Econometrica 50 (1982): 1345– 1371. Mankiw, N. G. The equity premium and the concentration of aggregate shocks. Journal of Financial Economics 17 (1986): 211–219. McGrattan, E. R., and E. C. Prescott. Is the stock market overvalued? Federal Reserve Bank of Minneapolis Quarterly Review (2000). McGrattan, E. R., and E. C. Prescott. Average debt and equity returns: Puzzling? American Economic Review 93 (2003): 392–397. McGrattan, E. R., and E. C. Prescott. Taxes, regulations, and the value of U.S. and U.K. corporations. Review of Economic Studies 92 (2005): 767–796. Mehra, R., and E. C. Prescott. The equity premium: A Puzzle. Journal of Monetary Economics 15 (1985): 145–161. Mehra, R., and E. C. Prescott. Intermediated quantities and returns. Working paper, UCSB (2007).


CHAPTER 4

Equity Premia with Benchmark Levels of Consumption: Closed-Form Results∗ Andrew B. Abel The Wharton School of the University of Pennsylvania and NBER

1. Preferences 2. The Canonical Asset 2.1. The Price of the Canonical Asset 2.2. The Rate of Return on the Canonical Asset 3. Risk, Term, and Equity Premia 4. Log-Normality 5. Risk, Term, and Equity Premia Under Log-Normality with Consumption Externalities and Without Habit Formation 6. Linear Approximations to Risk, Term, and Equity Premia 7. Second Moments 7.1. Linear Approximations to Second Moments 8. Correlation of Dividend-Price Ratio and the Rate of Return on Stock 8.1. Correlation of Dividend-Price Ratio and the Excess Rate of Return on Stock 9. Special Cases 9.1. Rational Expectations 9.2. Distorted Beliefs 10. Accuracy of Approximations 11. Summary References

120 126 127 129 131 134 135 137 138 140 142 144 146 146 151 153 156 156

∗ I thank Martin Lettau, Sydney Ludvigson, Rajnish Mehra, Jessica Wachter, Amir Yaron, Jianfeng Yu, participants in the Finance Seminar at the Stern School, the European Summer Symposium in Financial Markets, Gerzensee, Switzerland, and the Penn Macro Lunch Group for helpful comments.

HANDBOOK OF THE EQUITY RISK PREMIUM c 2008 by Elsevier B.V. All rights of reproduction in any form reserved. Copyright

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118

Chapter 4 • Equity Premia with Benchmark Levels of Consumption: Closed-Form Results

Abstract I calculate exact expressions for risk premia, term premia, and the premium on levered equity in a framework that includes habit formation, keeping/catching up with the Joneses, and possible departures from rational expectations. Closed-form expressions for the first and second moments of returns and for the R2 of a regression of stock returns on the dividend-price ratio are derived under log-normality for the case that includes keeping/catching up with the Joneses. Linear approximations illustrate how these moments of returns are affected by parameter values and illustrate quantitatively how well the model can account for values of the equity premium, the term premium, and the standard deviations of the riskless return and the rate of return on levered equity. For empirically relevant parameter values, the linear approximations yield values of the various moments that are close to those obtained from the exact solutions. JEL Classification: G12 Keywords: equity premium, term premium, risk premium, habit formation, catching up with the Joneses

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Andrew B. Abel

Mehra and Prescott (1985) showed that over a period of almost a century, the equity premium—the excess of the rate of return on stocks over the rate of return on riskless bills—averaged 6.18 percent per year. They then calibrated a general equilibrium asset pricing model of the sort introduced by Lucas (1978) and showed that such a model, with conventional values of the coefficient of relative risk aversion, cannot come close to accounting for the historically observed equity premium. In the two decades since Mehra and Prescott proclaimed an equity premium puzzle, a large body of research has been devoted to closing the gap between theoretical asset pricing models and empirically observed asset returns. The seminal study by Mehra and Prescott used a simple general equilibrium model with a representative consumer with constant relative risk aversion transacting in frictionless asset markets to determine the prices of unlevered equity and riskless bills. Subsequent research has extended this simple model to allow for heterogeneous consumers, trading frictions, more general preferences, leverage, a richer time series of the endowment of consumption, and departures from rational expectations. Since the equity premium puzzle is a quantitative puzzle, many of these studies provide numerical solutions rather than closed-form solutions for equilibrium prices and rates of return. In this chapter, I will provide closed-form solutions for the equilibrium price and rate of return on a canonical asset. The canonical asset is general enough to include riskless bills and risky stocks and thus allows analysis of the equity premium. I will extend the basic framework used by Mehra and Prescott to allow for preferences that display habit formation and a specific form of externalities. I will also allow for leverage so that I can examine the return on levered equity. Finally, I will allow for departures from rational expectations, though I will set up the framework so that rational expectations is a special—indeed focal—case. Instead of enriching the time-series structure used by Mehra and Prescott, I will simplify it by restricting attention to growth rates of consumption and dividends that are i.i.d. over time, though unlike Mehra and Prescott, I will allow consumption and dividends to differ from each other. As in Mehra and Prescott, I will assume that there is a representative consumer who transacts in frictionless markets. An advantage of closed-form solutions is that they provide a precise description of how equilibrium rates of return depend on various parameters. Though this description is precise, it is not always transparent. To help understand the effects of various parameters on the equilibrium rates of return, I will provide linear approximations to the closed-form solutions. I will show that, for empirically relevant parameter values, these linear approximations tend to be very close to the values obtained by evaluating the exact closed-form expressions. In this chapter, the major departure from the standard model1 is the specification of preferences to include a benchmark level of consumption that captures habit formation

1 In

this context, the standard model is the Lucas (1978) fruit-tree model, generalized by Mehra and Prescott (1985) to be stationary in growth rates of consumption. The standard model has a representative agent with time-separable utility who can trade assets frictionlessly without restrictions, such as short sale constraints.

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and consumption externalities that are sometimes called “keeping up with the Joneses” or “catching up with the Joneses.” I will introduce this more general specification of preferences in Section 1. In Section 2, I introduce the canonical asset and compute its equilibrium price and rate of return. The expression for the rate of return on the canonical asset can be used to determine rates of return on riskless bills, levered equity, and a claim on a single dividend to be received one period in the future. In Section 3, I use these rates of return to define and compute the risk premium, term premium, and equity premium. Beginning in Section 4, I restrict attention to log-normally distributed growth rates. In Section 4, I provide identities that will be helpful in computing expected returns under log-normality. Beginning in Section 5, I generally restrict attention to the case without habit formation, though I allow consumption externalities of the keeping/catching up with the Joneses variety. I compute closed-form expressions for risk, term, and equity premia in Section 5, and in Section 6, I compute linear approximations to these expressions. In Section 7, I compute exact closed-form expressions for the variances of the rates of return, and then I compute linear approximations to these expressions. Although the vector of exogenous growth rates of consumption and dividends is independently and identically distributed over time, the rate of return on stocks is predictable if preferences display the catching up with the Joneses feature. In fact, the dividend-price ratio can be used to predict the rate of return on stocks, and in Section 8, I compute the R2 of a regression of stock returns on the dividend-price ratio. In Section 9, I examine various special cases of the model and focus on the quantitative predictions for the equity premium, the term premium, the standard deviations of the riskless rate and the rate of return on stock, and the R2 of the regression of stock returns on the dividend-price ratio. The quantitative implications in Section 9 are based on the approximations of these features of assets, so I show in Section 10 that the approximations are, in fact, close to the values obtained by evaluating the exact closed-form solutions.

1. PREFERENCES Consider a discrete-time economy with a continuum of identical infinitely-lived consumers. Each period the economy receives an endowment of a homogeneous non-storable good. Because the good is non-storable, in equilibrium all of it is consumed in the period in which it arrives. The quantity of this good is exogenous and stochastic. The representative consumer maximizes a utility function that differs in two important ways from the standard time-separable isoelastic utility function commonly used in asset pricing models. First, the utility flow in each period depends on a benchmark level of consumption that evolves endogenously over time. Second, the representative consumer does not necessarily have rational expectations and thus may not use the objective distributions of random variables when evaluating lifetime utility. For the sake of obtaining closed-form solutions for asset prices, I specify a simple parametric utility

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function that embodies these two departures from standard preferences. At time t, an individual consumer maximizes ⎧ 1−α ⎫ j ∞ ⎬ ⎨ c 1 1 t+j , (1) Ut = Et∗ γ ⎭ ⎩ 1+δ 1 − α νt+j j=0 where ct+j is the individual’s own consumption in period t + j, νt+j is the benchmark level of consumption that I will specify in Eq. (2), 0 ≤ γ ≤ 1, the operator Et∗{} is the conditional expectation based on the consumer’s subjective distribution of exogenous stochastic variables as of time t, δ ≥ 0 is the rate of time preference, and α > 0 is a curvature parameter. Beginning in Section 5, I will confine attention to cases in which the benchmark defined in Eq. (2) does not depend on a consumer’s own past consumption; in these cases, α is the coefficient of relative risk aversion. In the special case in which γ = 0 and in which the subjective distributions equal the true objective distributions, the utility function in Eq. (1) is the standard time-separable isoelastic utility function with rational expectations. The parameter γ measures the extent to which utility flow in period t depends on ct /νt , the ratio of the consumer’s own consumption to the benchmark level of consumpγ tion νt . Specifically, utility flow in period t is an isoelastic function of ct /νt , which 1−γ can be written as an isoelastic function of ct (ct /νt )γ . Thus, utility flow in period t is an isoelastic function of a weighted geometric average of ct , the consumer’s own consumption in period t, and ct /νt , the ratio of the consumer’s own consumption to the benchmark level of consumption. In the extreme case in which γ = 1, utility flow in period t depends only on the ratio of the consumer’s own consumption to the benchmark level of consumption.2,3 Suppose that the benchmark level of consumption in period t is

1−D θ θ1 D Ct 0 Ct−1 Gt , νt ≡ ct−1

(2) γ

2 The benchmark ν affects utility flow in period t in the “ratio form” c /ν , rather than in the “dift t t γ ference form” ct − νt as in Constantinides (1990) and Campbell and Cochrane (1999). The ratio form facilitates the finding of closed-form expressions for equilibrium asset prices and returns. As Campbell and Cochrane point out, the difference form of the specification allows the coefficient of relative risk aversion to vary over time, whereas the ratio form does not allow this variation. However, the difference form requires that the current level of consumption never falls below the current benchmark level, which requires additional restrictions on the processes generating consumption and the benchmark. 3 In an economy with production and capital accumulation (Abel (2005)), I rule out the case in which γ = 1. If ∞ a given sequence of consumption and benchmarks between ∞γ = 1, consumers would be indifferent ct∗ , νt∗ t=−∞ and, for instance, a sequence 2ct∗ , 2νt∗ t=−∞ in which all consumptions and benchmarks are twice as high. That is, if γ = 1, consumers are indifferent to the scale of the economy. In a production economy with capital accumulation, the scale of the economy is endogenous, and the indifference to scale would create a problem for a benevolent social planner. However, such a problem does not arise in the endowment economy I study here because the scale of the economy is exogenous.

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where ct−1 is the individual’s own consumption in the previous period, Ct is the average consumption per person in the current period, Ct−1 is the average consumption per person in the previous period, 0 ≤ D ≤ 1, θ0 ≥ 0, θ1 ≥ 0, θ0 + θ1 ≤ 1, and G ≥ 1. The assumption that G ≥ 1 allows for the possibility that the benchmark level of consumption can grow exogenously. In equilibrium in this representative consumer economy, the individual’s own consumption in period t, ct , equals the average consumption per person in period t, Ct . While an individual consumer treats his own consumption, ct , as a choice variable, the average consumption per person, Ct , is exogenous to the individual and thus is treated as a parameter in the individual’s decision problem. If D = 1, then the benchmark is simply νt ≡ ct−1 , which is an example of habit formation. In this case, an individual’s own consumption in period t directly affects utility flow in period t and also affects the benchmark level of consumption in period t + 1. (I will use the term “habit formation” to indicate that the benchmark depends on that person’s own past consumption. Campbell and Cochrane (1999) have extended this terminology to include cases in which the benchmark in a person’s utility function depends on past levels of aggregate consumption per capita, but not directly on past levels of the person’s own consumption. They recognize that this terminology is unusual and so refer to such preferences as displaying “external habit formation.” I will reserve the term “habit formation” for cases in which the benchmark depends on the consumer’s own past consumption. I will use the term “consumption externalities” to refer to what Campbell and Cochrane call external habit formation.)4 The case without habit formation, but with consumption externalities, is represented θ θ1 by D = 0, so that the benchmark is νt ≡ Ct 0 Ct−1 G t , which is exogenous to an indiθ0 θ1 vidual consumer. The term Ct Ct−1 captures the notion that the benchmark level of consumption depends on the current and/or recent level of consumption per person in the economy. For instance, if θ0 + θ1 = 1 and G = 1, the benchmark level of consumpθ 1−θ tion is νt = Ct 0 Ct−1 0 , which is simply a weighted geometric average of current and lagged consumption per person. Within this restricted class of benchmark functions, if θ0 = 1, so that θ1 = 0, the benchmark level of consumption is simply νt = Ct , as in Gali (1994). Alternatively, if θ1 = 1, so that θ0 = 0, then the benchmark level of consumption is νt = Ct−1 , as in the specification of the “catching up with the Joneses” preferences in Abel (1990).5 As a step toward calculating the pricing kernel, which is the equilibrium value of the representative consumer’s marginal rate of substitution between current consumption, 4 See also the usage in the chapter by George Constantinides in this volume, where he distinguishes between “internal” and “external” habit. 5 In Abel (1990) I used the term “catching up with the Joneses” to describe the case in which ν = C t t−1 , because the benchmark level of consumption depends on lagged rather than current consumption. With this terminology, one might describe the case in which νt = Ct as a case of “keeping up with the Joneses,” because the benchmark level of consumption depends on the contemporaneous consumption of others. Dupor and Liu (2003) refine the terminology for various features of consumption externalities. If utility flow is u(c, b), where c is the individual’s own consumption and b is a benchmark that equals the average level of consumption by others, they define jealousy as ∂u/∂b < 0 and keeping up with the Joneses as ∂2 u/∂c∂b > 0. The utility function in Eq. (1) displays jealousy. It also displays keeping up with the Joneses for the case in which the curvature parameter α, introduced in Eq. (1), is greater than one.

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ct , and next period’s consumption, ct+1 , differentiate Eq. (1) with respect to ct to obtain ⎧ 1−α ⎫ 1−α ⎬ 1 ∗ ⎨ ct ct+1 ∂Ut 1 = Et − γD . (3) γ ⎩ νtγ ⎭ ∂ct ct 1 + δ νt+1 Because all consumers are identical, ct = Ct in equilibrium. Let xt+1 ≡ Ct+1 /Ct = ct+1 /ct be the (gross) growth rate of consumption from period t to period t + 1, and use this definition of xt+1 and the definition of the benchmark level of consumption in Eq. (2) to rewrite Eq. (3) as ∂Ut γ(α−1) Ht , = ct−α νt ∂ct

(4)

where 1−η Ht ≡ 1 − DγβEt∗ xt+1 xυt ,

(5)

and6 1 G (1−D)γ(α−1), 1+δ η ≡ α + θ0 (1 − D) γ (1 − α) > 0, υ ≡ γ (α − 1) [D + θ1 (1 − D)] > −1.

β≡

(6) (7) (8)

Note that the factor β can be larger than one. However, β cannot be so large that certain series fail to converge. I will provide a more specific upper bound for β in Section 2.1. The pricing kernel is the intertemporal marginal rate of substitution, Mt+1 ≡

1 ∂Ut+1 /∂ct+1 . 1 + δ ∂Ut /∂ct

(9)

Use Eq. (4) and the definition of the benchmark in Eq. (2) to rewrite Eq. (9) as −η

Mt+1 = βxυt xt+1

Ht+1 . Ht

(10)

inequalities in Eqs. (7) and (8) are derived as follows. Rewrite η as η = [1 − θ0 (1 − D) γ] α + [θ0 (1 − D) γ]. The assumptions that θ0 ≥ 0, θ1 ≥ 0, and θ0 + θ1 ≤ 1 imply that 0 ≤ θ0 ≤ 1 and 0 ≤ θ1 ≤ 1. Also, since 0 ≤ D ≤ 1 and 0 ≤ γ ≤ 1, both of the terms in square brackets in the expression for η are nonnegative, and since the terms in square brackets sum to one, at least one of them must be positive. Since α > 0, it follows that η > 0. Rewrite υ as υ = αγ [D + θ1 (1 − D)] − γ [D + θ1 (1 − D)], so that υ + 1 = {αγ [D + θ1 (1 − D)]} + {1 − γ [D + θ1 (1 − D)]}. Since 0 ≤ D ≤ 1, 0 ≤ γ ≤ 1, and 0 ≤ θ1 ≤ 1, it follows that 0 ≤ γ [D + θ1 (1 − D)] ≤ 1. Therefore, since α > 0, both of the terms in curly brackets in the expression for υ + 1 are non-negative and at least one of them is positive. Therefore, υ + 1 > 0, so υ > −1.

6 The

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In the standard case of time-separable isoelastic utility, which is represented by γ = 0, we have β = 1/(1 + δ), η = α, υ = 0, and Ht ≡ 1. Therefore, the marginal rate of substitution in Eq. (10) is simply Mt+1 = βx−α t+1 ,

if γ = 0.

(11)

If we introduce a benchmark level of consumption into the utility function (γ > 0), but confine attention to the case with D = 0, so that there may be consumption externalities but no habit formation, the marginal rate of substitution in Eq. (10) is −η

Mt+1 = βxυt xt+1 ,

if D = 0,

(12)

since Ht ≡ 1 when D = 0. If the growth rate of consumption, xt , is independently distributed over time, then in the standard isoelastic case, Mt+1 = βx−α t+1 is also independently distributed over time. However, with a benchmark level of consumption, the pricing kernel Mt in Eq. (12) is, in general, not independently distributed over time. In fact, since η > 0, the serial correlation of Mt has the opposite sign of υ, which can be positive, negative, or zero. Under rational expectations, Backus, Gregory, and Zin (1989) have shown that the term premium has the opposite sign of the serial correlation of the pricing kernel. Therefore, the term premium has the same sign as υ, as I will show under log-normality of xt (and without habit formation) in Section 5. In the standard case of time-separable utility, υ = 0, so the term premium is zero if the growth rate of consumption, xt , is independently distributed over time. In the case considered by Gali (1994) (θ1 = 0 = D, so that υ = 0; and G = 1), the −η marginal rate of substitution in Eq. (12) becomes Mt+1 = βxt+1 , which is the pricing kernel for standard time-separable isoelastic utility in which the coefficient of relative risk aversion is constant and equal to η. Gali points out an interesting special case in which γ = 1 = θ0 , so that the benchmark is simply νt = Ct and, as in Abel (1990), consumers care only about the ratio of their own consumption to the benchmark. In this special case, η = 1, so that regardless of the value of the curvature parameter α, the pricing kernel is Mt+1 = βx−1 t+1 , which is the marginal rate of substitution for a standard logarithmic utility function without benchmark levels of consumption (α = 1 and γ = 0). Thus, asset prices in this special case of Gali’s specification would be identical to asset prices that would arise in the simple case of logarithmic utility without benchmark levels of consumption. The specification of the utility function has seven fundamental parameters: the curvature paramater α; the rate of time preference δ; the importance of the consumptionbenchmark ratio as measured by γ; the relative weight D of the consumer’s own past consumption in the benchmark; the weights θ0 and θ1 on contemporaneous and lagged aggregate consumption in the benchmark; and the exogenous growth rate G in the benchmark. Beginning in Section 5, I generally restrict attention to the case without habit formation, though I allow consumption externalities of the keeping/catching up with the Joneses variety. In this case, D = 0, which implies Ht ≡ 1. With Ht ≡ 1, Eq. (10) shows that the pricing kernel depends on only the three

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“derived” preference parameters β, η, and υ in Eqs. (6), (7), and (8), respectively. Thus, one could not recover the seven fundamental parameters by observing data on asset returns. The reason for starting with seven fundamental parameters is to illustrate that many different representations of preferences can be captured by the three derived parameters β, η, and υ. These parameters are defined precisely in Eqs. (6), (7), and (8), respectively. Loosely speaking, β is an adjusted time-preference discount factor, η is a curvature parameter related to risk aversion, and υ is a parameter reflecting the role of past (aggregate or individual) consumption. Table 1 summarizes the relationships among the fundamental preference parameters and the derived preference parameters in interesting special cases. To see how the marginal rate of substitution determines asset prices and rates of return, let Rt+1 be the one-period gross rate of return on an asset from period t to period t + 1. Consider a consumer who reduces consumption in period t by one unit and purchases one unit of the asset in period t. In period t + 1, the consumer collects any payoff (for instance, dividends or coupons) and then sells the asset and uses the proceeds to

TABLE 1 Preference Parameters and Pricing Kernel in Various Cases Time separable

Habit formation

Keeping up with the Joneses

Catching up with the Joneses

Fundamental preference parameters γ

0

00

>0 ≥1

G

—

—

≥1

D

—

1

0

0

θ0

—

—

1

0

θ1

—

—

0

1

β

1 1+δ

1 1+δ

1 γ(α−1) 1+δ G

1 γ(α−1) 1+δ G

η

α

α

α + γ (1 − α)

α

υ

0

0

γ (α − 1)

Ht

1

γ (α − 1) γ(α−1) 1 − βγEt∗ x1−α t+1 xt

Mt+1

βx−α t+1

Derived preference parameters and pricing kernel

Ht Mt+1

— —

γ(α−1) −α Ht+1 βxt xt+1 Ht

1

1

−α−γ(1−α) βxt+1

γ(α−1) −α βxt xt+1

Special case in which γ = 1 α−1 1 − βEt∗ x1−α 1 t+1 xt Ht+1 βxtα−1 x−α t+1 Ht

βx−1 t+1

1 βxtα−1 x−α t+1

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increase consumption in period t + 1. Since the payoff plus the proceeds from the sale of the asset equal Rt+1 units of consumption, the consumer can increase ct+1 by Rt+1 . Optimality from the consumer’s point of view in period t requires that the subjective expected reduction in utility from reducing ct by one unit equals the subjective expected increase in utility from increasing ct+1 by Rt+1 units, so that 1 ∂Ut+1 ∂Ut ∗ = Et Rt+1 . ∂ct 1 + δ ∂ct+1

(13)

The first-order condition in Eq. (13) can be rewritten using the definition of the marginal rate of substitution in Eq. (9) as Et∗{Mt+1 Rt+1 } = 1.

(14)

Equation (14) is the familiar fundamental condition of asset pricing: the product of the marginal rate of substitution and the gross rate of return on any frictionlessly traded asset has a conditional expectation equal to one. The only unfamiliar aspect of Eq. (14) is that the conditional expectation is based on the consumer’s subjective distribution, which may differ from the objective distribution. Using the marginal rate of substitution in Eq. (10), the fundamental condition of asset pricing can be written as βxυt ∗ −η Et xt+1 Ht+1 Rt+1 = 1. Ht

(15)

In the next section, I will define a canonical asset and will use Eq. (15) to determine the equilibrium price of this asset.

2. THE CANONICAL ASSET The fundamental condition for asset pricing in Eq. (15) can be used to determine the price of any frictionlessly traded asset in the absence of borrowing constraints and other restrictions such as short sale constraints. To avoid the need for separate derivations for the equilibrium prices of, for instance, riskless bills and stocks, I will calculate the equilibrium price of a canonical asset, introduced in Abel (1999), that is general enough to include these assets as special cases. The canonical asset is an n-period asset, where n may be infinite. In period t, the λ λ λ canonical asset is a claim to a stream of n payoffs an−1 yt+1 , an−2 yt+2 , . . . , a0 yt+n , where a0 > 0, aj ≥ 0, j = 1, . . . , n − 1 are constants, yt+j > 0 is a random variable, and λ ≥ 0 is a constant. The (gross) growth rate of the random variable yt is zt+1 ≡ yt+1 /yt . Recall that xt+1 is the (gross) growth rate of consumption per capita. I will assume that the vector (xt+1 , zt+1 ) is i.i.d. over time and that even when the subjective distributions differ from the corresponding objective distributions, the subjective distributions

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incorporate the fact that (xt+1 , zt+1 ) is i.i.d. over time. The definition of Ht in Eq. (5) and the assumption that (xt+1 , zt+1 ) is i.i.d. over time imply that (xt+1 , zt+1 , Ht+1 ) is i.i.d. over time. When λ = 0, the canonical asset is a fixed-income security, with a deterministic stream of payoffs an−j in period t + j, for j = 1, . . . , n. For instance, a coupon bond with face value F , maturity n, and coupon d is represented by a0 = d + F and aj = d, for j = 1, . . . , n − 1. A pure discount bond is represented by d = 0. A consol that pays a constant coupon d forever is represented by aj = d, for j = 1, 2, 3, . . . . Securities with risky payoffs are represented by λ = 0. For instance, in Lucas (1978), unlevered equity pays a dividend per capita in each period that equals consumption per capita. Thus, the payoff in period t + j is Ct+j , so unlevered equity can be represented by the canonical asset with λ = 1, n = ∞, aj = 1, and yt+j ≡ Ct+j , for j = 1, 2, 3, . . . . As explained in Abel (1999), levered equity is represented by λ > 1.7 This formulation of levered equity has been adopted by, for example, Bansal and Yaron (2004) and Lettau, Ludvigson, and Wachter (2004).

2.1. The Price of the Canonical Asset Let pt (n, λ) be the ex-payment (e.g., ex-dividend for equity, ex-coupon for bonds) price of the canonical asset in period t. The price also depends on the sequence of the constants aj , j = 0, . . . , n − 1, and on the properties of the stochastic process for yt , but, to reduce clutter, I have not included this dependence in the notation. A consumer can buy a unit of the canonical asset at price pt (n, λ) in period t and then in period t + 1 λ can collect the payoff an−1 yt+1 and sell the asset at price pt+1 (n − 1, λ) when the asset becomes an (n − 1)-period asset. Therefore, the gross rate of return on the asset from period t to period t + 1 is Rt+1 (n, λ) =

λ + pt+1 (n − 1, λ) an−1 yt+1

pt (n, λ)

.

(16)

Substituting the rate of return on the canonical asset from Eq. (16) into the fundamental condition of asset pricing in Eq. (15), recognizing that pt (n, λ) is known at time t 7 To

illustrate levered equity most simply, consider a heuristic example with one-period levered equity. Define q ≡ y − B, where y is the payoff to owners of equity before making any payment to bondholders, and 0 ≤ B < E{y} is the payment promised to bondholders. The payoff to one-period levered equity is max [q, 0]. If the probability of default is small enough to allow us to ignore default, the payoff to levered equity is simply q, with mean E{q} = E{y} − B, standard deviation sd{q} = sd{y}, and coefficient of variation equal to sd{y}/(E{y} − B) = λ(sd{y}/E{y}), where λ ≡ 1/ (1 − (B/E{y})) ≥ 1. That is, the coefficient of variation of the levered payoff (ignoring bankruptcy) is λ times as large as the coefficient of variation of the unlevered payoff y. If y is log-normally distributed, then the coefficient of variation of y λ is approximately λ times as large as the coefficient of variation of y, so y λ with λ > 1 reflects the fact that leverage increases the coefficient of variation of the payoff to equity. If we interpret B/E{y} as the leverage ratio, then λ is an increasing function of the leverage ratio. Note that without leverage, i.e., with B = 0, λ = 1. This description is heuristic and, importantly, has ignored the role of default. In Abel (1999), I take explicit account of bankruptcy and show that for a broad range of values of the leverage ratio B/E{y}, the rate of return on a one-period asset paying y λ is very close to the rate of return on a one-period asset paying max [y − B, 0].

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and multiplying both sides by pt (n, λ), yields a recursive expression for the price of the canonical asset pt (n, λ) =

βxυt ∗ −η λ Et xt+1 Ht+1 an−1 yt+1 + pt+1 (n − 1, λ) . Ht

(17)

I will hypothesize that the price of the canonical asset, which must satisfy Eq. (17), is pt (n, λ) = ω(n, λ)

xυt ytλ , Ht

(18)

where ω(n, λ) is a function to be determined. As a step toward determining the function ω(n, λ), substitute Eq. (18) into Eq. (17) and use the facts that an−1 and ω(n − 1, λ) are υ−η nonstochastic and that (xt , zt ) is i.i.d. (so that the unconditional means of xt+1 zλt+1 and −η xt+1 Ht+1 zλt+1 equal the means of the corresponding conditional distributions at time t) to simplify and obtain ω(n, λ) = A(λ) ω(n − 1, λ) + B(λ) an−1 , where

and

for n ≥ 1,

(19)

υ−η A(λ) ≡ βE ∗ xt+1 zλt+1 > 0

(20)

−η B(λ) ≡ βE ∗ xt+1 Ht+1 zλt+1 > 0.

(21)

Equation (19) is a first-order linear difference equation. In order to guarantee that this difference equation converges as n grows without bound, I will assume that β and the subjective distribution of (xt+1 , zt+1 ) are such that A(λ) < 1. This assumption is υ−η −1 equivalent to β < E ∗ xt+1 zλt+1 , which allows β to exceed one to the extent that ∗ υ−η λ E xt+1 zt+1 is less than one. Equation (19) has a simple boundary condition associated with it. After an asset has paid its last payment, it is formally a zero-period asset, and its price is zero. That is, pt (0, λ) = 0 so ω(0, λ) = 0. Using this boundary condition, evaluate the difference equation in Eq. (19) for n = 1 to obtain ω(1, λ) = B(λ) a0 .

(22)

For arbitrary n, the solution to the difference equation in Eq. (19) that satisfies the boundary condition in Eq. (22) is ω(n, λ) = B(λ)

n

ai−1 [A(λ)]n−i .

(23)

i=1

The function ω(n, λ) can be increasing or decreasing in the maturity n. For a pure discount bond with face value F , a0 = F and aj = 0 for j = 1, . . . , n − 1. In this case,

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ω(n, λ) = B(λ) [A(λ)]n−1 F , which is strictly decreasing in n because A(λ) < 1. For a security with a level stream of payoff coefficients aj = d for j = 0, . . . , n − 1, we have n ω(n, λ) = B(λ) 1 1−−[A(λ)] A(λ) d, which is increasing in maturity n. A coupon bond is a composite of a discount bond with a face value of F and a security that promises a level stream of payments d, so a0 = d + F and aj = d, for j = 1, . . . , n − 1. Therefore, for an

n-period coupon bond, ω(n, λ) = B(λ) [A(λ)]n−1 F + 1 − A(λ) A(λ) , decreasing in n if d/F

1 − [A(λ)]n 1 − A(λ) d

, which is increasing

1 − A(λ) A(λ) , and invariant to n if d/F

A(λ) < = 1 −A(λ) . in n if d/F > That is, if the coupon is small relative to the face value, the coupon bond is more like a discount bond in which ω(n, λ) is decreasing in n. If the coupon is large relative to the face value, the coupon bond is more like a security with a flat stream of payoffs, and ω(n, λ) is increasing in maturity n. The expression for ω(n, λ) in Eq. (23) completes the solution of the equilibrium price of the canonical asset for arbitrary values of n and λ. The functions A(λ) and B(λ) are unconditional moments of the subjective distributions of exogenous random variables. Given the subjective distributions of the exogenous random variables, Eqs. (20) and (21) give the values of A(λ) and B(λ), which can be used along with the sequence of payoff coefficients aj , j = 0, . . . , n − 1, in Eq. (23) to obtain the value of ω(n, λ). Substituting this value of ω(n, λ) into Eq. (18) and using the definition of Ht in Eq. (5) gives the price of the canonical asset for any n and λ.

2.2. The Rate of Return on the Canonical Asset To calculate the rate of return on the canonical asset, substitute the expression for the price of the canonical asset from Eq. (18) into the expression for the rate of return in Eq. (16) to obtain an−1 + ω(n − 1, λ) Rt+1 (n, λ) =

xυt+1 Ht+1

ω(n, λ)

λ Ht x−υ t zt+1 .

(24)

It will be convenient to rewrite the rate of return on the canonical asset in Eq. (24) as Rt+1 (n, λ) = J (xt+1 , zt+1 , Ht+1 ; n, λ) × K(xt , Ht ) ,

(25)

where J (xt+1 , zt+1 , Ht+1 ; n, λ) ≡

λ υ an−1 λ ω(n − 1, λ) xt+1 zt+1 zt+1 + ω(n, λ) ω(n, λ) Ht+1

(26)

and K(xt , Ht ) ≡ x−υ t Ht .

(27)

Equation (25) expresses the rate of return on the canonical asset as the product of J (xt+1 , zt+1 , Ht+1 ; n, λ) and K(xt , Ht ), which are independent of each other because

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the vector of exogenous random variables (xt , zt , Ht ) is independently distributed over time. I will confine attention to assets for which aj = 1, for j = 0, 1, 2, . . . , n − 1. In this case, Eq. (23) implies that 1 1 − A(λ) 1 , = ω(n, λ) B(λ) 1 − [A(λ)]n

(28)

ω(n − 1, λ) 1 − [A(λ)]n−1 . = ω(n, λ) 1 − [A(λ)]n

(29)

which implies that

Substitute Eqs. (28) and (29) into the definition of J (xt+1 , zt+1 , Ht+1 ; n, λ) in Eq. (26), set an−1 = 1, and rearrange to obtain J (xt+1 , zt+1 , Ht+1 ; n, λ) υ λ 1 1 − A(λ) λ −1 n−1 z H x = + 1 − z [A(λ)] t+1 t+1 t+1 . B(λ) t+1 1 − [A(λ)]n

(30)

Because (xt , zt , Ht ) is i.i.d. over time, J (xt+1 , zt+1 , Ht+1 ; n, λ) is i.i.d. over time. Therefore, the objective conditional expectation of J (xt+1 , zt+1 , Ht+1 ; n, λ) equals the objective unconditional expectation of J (xt+1 , zt+1 , Ht+1 ; n, λ). Now define E H −1 xυ zλ B(λ) Ψ (λ) ≡ , A(λ) E zλ

(31)

where the operator E{} (without an asterisk) denotes the expectation using the objective distribution of the relevant random variables. I have suppressed the time subscripts from the variables inside the expectation operator, with the understanding that any variables inside the operator (such as H and x) are contemporaneous with each other (i.e., Ht and xt ). I will show later that Ψ(λ) helps determine the term premium. Use Eq. (31) to rewrite Eq. (30) as E zλ 1 J (xt+1 , zt+1 , Ht+1 ; n, λ) = 1 − [A(λ)]n B(λ) −1 λ υ z H x zλt+1 t+1 t+1 t+1 × [1 − A(λ)] + 1 − [A(λ)]n−1 Ψ (λ) A(λ) . E zλ E xυ zλ H −1

(32)

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Andrew B. Abel

Now compute the objective unconditional expectations of both sides of Eq. (32) to obtain λ 1 − [A(λ)]n−1 E z . (33) E{J (xt+1 , zt+1 , Ht+1 ; n, λ)} = 1 + A(λ) [Ψ (λ) − 1] B(λ) 1 − [A(λ)]n

The expectation of K(xt , Ht ) is calculated directly from the definition in Eq. (27) to obtain E{K(xt , Ht )} ≡ E x−υ H .

(34)

Finally, calculate the expected value of Rt+1 (n, λ) in Eq. (25) using the independence of J (xt+1 , zt+1 , Ht+1 ; n, λ) and K(xt , Ht ) and Eqs. (33) and (34) to obtain λ 1 − [A(λ)]n−1 E z E x−υ H . E{Rt+1 (n, λ)} = 1 + A(λ) [Ψ (λ) − 1] B(λ) 1 − [A(λ)]n

(35)

Equation (35) gives the unconditional expected rate of return on a canonical asset with arbitary λ and arbitrary n, provided that a0 = . . . = an−1 = 1. This expression simplifies considerably for one-period canonical assets. With n = 1, E zλ E x−υ H . E{Rt+1 (1, λ)} = B(λ)

(36)

The expected rates of return in Eqs. (35) and (36) are expectations based on the objective distributions of equilibrium rates of return. In the case in which subjective distributions differ from objective distributions, the subjective distributions guide behavior, and equilibrium rates of return depend on subjective distributions. However, in a long time series of observations on equilibrium rates of return, the sample average will correspond to the objective expectations of these rates of return (which depend on subjective distributions through A(λ), B(λ), and Ψ (λ) in Eqs. (20), (21), and (31), respectively) as in Eqs. (35) and (36). I will use these equations to calculate risk, term, and equity premia, which I will define in Section 3.

3. RISK, TERM, AND EQUITY PREMIA I will define the risk premium for the set of canonical assets with a given value of λ as the excess of the expected rate of return on the one-period version of the canonical

132


asset with that value of λ relative to the rate of return on a one-period riskless bill. Specifically, I will define the unconditional risk premium RP (λ) as RP (λ) ≡

E{Rt+1 (1, λ)} − 1. E{Rt+1 (1, 0)}

(37)

Use the expression for the expected rate of return on one-period canonical assets from Eq. (36) to rewrite the risk premium in Eq. (37) as RP (λ) =

B(0) λ E z − 1. B(λ)

(38)

Equation (38) can be rewritten by using the definition of B(λ) in Eq. (21) to obtain E ∗ x−η H E zλ RP (λ) = − 1. E ∗ x−η Hzλ

(39)

To interpret the sources of the risk premium in Eq. (39), rewrite this equation as E ∗ x−η H E ∗ zλ E zλ 1 + RP (λ) = × . E ∗ x−η Hzλ E ∗ zλ

(40)

This risk premium—more precisely, 1 + RP (λ)—is the product of two ratios on the right-hand side of Eq. (40). The first ratio differs from one to the extent that there is a nonzero subjective covariance between the risky payoff in period t + 1, zλt+1 , and the part −η of the pricing kernel in Eq. (10) that is unpredictable one period in advance, xt+1 Ht+1 . If −η λ zt+1 is (subjectively) uncorrelated with xt+1 Ht+1 , this component of 1 + RP (λ) equals one, and thus it does not contribute to the risk premium. If zλt+1 is negatively correlated −η with xt+1 Ht+1 under the subjective distribution, so that the risky payoff is high when the marginal utility of consumption is low, then the risk premium is positive. Under rational expectations, the second ratio on the right-hand side of Eq. (40) equals one, and this term does not contribute to the risk premium. However, if the representative consumer is pessimistic, in the sense that the subjective expectation E ∗ zλ is less λ than the objective expectation E z , this ratio exceeds one, which contributes to a positive risk premium. Pessimistic consumers will underprice the risky asset, leading to an increase in the realized rate of return on this asset. Equivalently, with E zλ > E ∗ zλ , the realized rate of return on the risky asset is higher than the representative consumer expects. For any given value of λ, I will define the term premium as the excess of the expected rate of return on a security with maturity n > 1 relative to the expected rate of return

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on a security with the same value of λ and with a maturity of one period. Specifically, I will define the unconditional term premium T P (n, λ) as T P (n, λ) ≡

E{Rt+1 (n, λ)} − 1. E{Rt+1 (1, λ)}

(41)

Substituting the expected rate of return on the canonical asset from Eq. (35) into the definition of the term premium in Eq. (41) and using Eq. (36) yields T P (n, λ) = A(λ) [Ψ (λ) − 1]

1 − [A(λ)]n−1 . 1 − [A(λ)]n

(42)

The sign of the term premium is the same as the sign of Ψ (λ) − 1. The magnitude of the term premium is increasing in maturity n. For an infinite-maturity asset, such as a stock or a consol, the term premium is computed by taking the limit as n → ∞ to obtain T P (∞, λ) = A(λ) [Ψ (λ) − 1] .

(43)

I will define the equity premium as the excess of the expected rate of return on the canonical asset with n = ∞ and λ ≥ 1 relative to the rate of return on a one-period riskless bill (n = 1, λ = 0). Specifically, I will define the unconditional equity premium EP (λ) as EP (λ) ≡

E{Rt+1 (∞, λ)} − 1. E{Rt+1 (1, 0)}

(44)

The definitions of the risk premium in Eq. (37) and the term premium in Eq. (41) imply that EP (λ) = (1 + RP (λ)) (1 + T P (∞, λ)) − 1.

(45)

Substitute the risk premium from Eq. (39) and the term premium from Eq. (43) into Eq. (45) to obtain E ∗ x−η H E zλ EP (λ) = [1 − A(λ) + Ψ (λ) A(λ)] − 1. E ∗ x−η Hzλ

(46)

I have defined the risk, term, and equity premia as ratios of unconditional expectations of rates of return on various assets. The values of these premia would be unchanged if they were defined as ratios of conditional expectations of the same rates of return.

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Specifically, Eq. (25) and the fact that (xt , zt , Ht ) is i.i.d. over time imply that for any two canonical assets characterized by (n1 , λ1 ) and (n2 , λ2 ), respectively, Et {Rt+1 (n1 , λ1 )} Et {J (xt+1 , zt+1 , Ht+1 ; n1 , λ1 )} E{Rt+1 (n1 , λ1 )} = = . Et {Rt+1 (n2 , λ2 )} Et {J (xt+1 , zt+1 , Ht+1 ; n2 , λ2 )} E{Rt+1 (n2 , λ2 )}

(47)

Thus, for instance, the conditional equity premium, Et {Rt+1 (∞, λ)}/Et {Rt+1 (1, 0)} − 1, equals the unconditional equity premium defined in Eq. (44). Similarly, the conditional risk and term premia equal the respective unconditional risk and term premia defined in Eqs. (37) and (41).

4. LOG-NORMALITY Suppose that the objective of the vector (ln x, ln z) is N (μ, Σ) where μ = 2 distribution σx σxz (μx , μz ) and Σ = . It is straightforward to show that for constants a, b, σxz σz2 and λ, E{xa } E zλ = exp (a − b) μx + 0.5 a2 − b2 σx2 − bλσxz b λ E xz

(48)

E xa zλ E xb zλ = exp −abσx2 . a+b λ λ E x z E z

(49)

and

∗ ∗ Suppose that the subjective distribution of the vector (ln x, ln z) is N (μ , Σ ), ∗2 ∗ σx σxz

. Let θx ≡ μ∗x − μx and θz ≡ μ∗z − μz , so where μ∗ = μ∗x , μ∗z and Σ∗ = ∗ σxz σz∗2

θ ≡ θx θz is the amount by which the mean of the subjective distribution, μ∗ , Ωxx Ωxz ≡ Ω ≡ Σ∗ − Σ exceeds the mean of the objective distribution, μ. Let Ωxz Ωzz be the amount by which the variance-covariance matrix of the subjective distribution exceeds that of the objective distribution. It is straightforward to show that

E ∗ xa zλ = exp a a λ E x z

λ θ + 0.5 a

a λ Ω . λ

(50)

I will use Eqs. (48), (49), and (50) to derive simple expressions for the risk, term, and equity premia under log-normality.

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Andrew B. Abel

5. RISK, TERM, AND EQUITY PREMIA UNDER LOG-NORMALITY WITH CONSUMPTION EXTERNALITIES AND WITHOUT HABIT FORMATION Equation (35) is an analytic expression for the unconditional objective expectation of the rate of return on the canonical asset for arbitrary n and λ. Given the values of various moments of the objective and subjective distributions of the exogenous stochastic variables xt , zt , and Ht , it is a simple matter to compute A(λ), B(λ), and Ψ(λ) (from Eqs. (20), (21), and (31), respectively) and substitute the values into Eq. (35) to obtain the unconditional objective expectation of the rate of return on the canonical asset. Similarly, it is a simple matter to compute the risk premium RP (λ) in Eq. (39) and the term premium T P (∞, λ) in Eq. (43). In order to derive easily interpreted expressions for the expected rate of return, the risk premium, the term premium, and the equity premium expressed as functions of the parameters of the distributions of the exogenous stochastic variables, henceforth I will restrict attention to cases in which (1) the exogenous growth rates xt and zt are jointly log-normal (and, as before, i.i.d. over time) and (2) D = 0, so that preferences do not depend on the individual’s own lagged consumption, though, of course, they may depend on current or lagged aggregate consumption. With D = 0, we have Ht ≡ 1. To calculate the risk premium under log-normality with D = 0, rearrange Eq. (39) and set H ≡ 1 to obtain E x−η E zλ E ∗ x−η E x−η zλ (51) RP (λ) = − 1. E{x−η } E ∗ x−η zλ E x−η zλ Now use Eqs. (48) and (50) to calculate the ratios of moments in Eq. (51) to obtain RP (λ) = exp ηλσxz − λθz + ηλΩxz − 0.5λ2 Ωzz − 1.

(52)

As a step toward calculating the term premium, I will calculate Ψ(λ) under lognormality and the assumption that D = 0. Substitute A(λ) from Eq. (20) and B(λ) from Eq. (21) into the expression for Ψ(λ) in Eq. (31), and use the fact that Ht ≡ 1 when D = 0 to obtain E xυ−η zλ E ∗ x−η zλ E xυ zλ E x−η zλ (53) Ψ(λ) = × × . E zλ E xυ−η zλ E ∗ xυ−η zλ E x−η zλ The expression for Ψ(λ) in Eq. (53) is the product of three factors. The first factor is a ratio in which the numerator is the product of two expectations and the denominator is the product of two expectations. If the variables x and z were deterministic, the numerator and the denominator of this ratio would be equal and this factor would equal one. The second factor is the ratio of the objective expectation of a random variable to the subjective expectation of the same variable. The third factor is the ratio of the subjective expectation of a different random variable to the objective expectation of that variable.

136


Under rational expectations, in which the subjective expectations of variables equal the objective expectations of those variables, the second and third factors are both equal to one. Under the assumption of log-normality, the first factor on the right-hand side of Eq. (53) can be calculated using Eq. (49) and the second and third factors on the right-hand side of Eq. (53) can be calculated using Eq. (50) to obtain Ψ(λ) = exp υησx2 − υθx + υ[η − 0.5υ] Ωxx − υλΩxz .

(54)

Ψ(λ) is one component of the term premium in Eq. (43). The other component of the term premium is A(λ), which is defined in Eq. (20). Under log-normality, A(λ) is given by A(λ) = β exp (υ −

η) μ∗x

+

λμ∗z

+ 0.5 υ − η

λ Σ

∗

υ−η λ

.

(55)

Note that under log-normality with D = 0, so that Ht ≡ 1, the definition of B(λ) in Eq. (21) implies that ∗ B(λ) = β exp −ημ∗x + λμ∗z + 0.5 η2 σx∗2 + λ2 σz∗2 − 2ηλσxz .

(56)

Substituting Ψ(λ) from Eq. (54) into the expression for the term premium in Eq. (43) gives a closed-form expression for the term premium of an infinite-maturity canonical asset in terms of the parameters of the subjective and objective log-normal distributions, T P (∞, λ) = A(λ) exp υησx2 − υθx + υ [η − 0.5υ] Ωxx − υλΩxz − 1 ,

(57)

where A(λ) is given by Eq. (55). Under rational expectations, θ = 0 and Ω = 0, so the term premium is simply T P (∞, λ) = A(λ) exp υησx2 − 1 , which has the same sign as υησx2 . Since ησx2 > 0, the term premium has the same sign as υ. Under standard time-separable preferences, γ = 0 so that υ = 0 and the term premium is zero. With a benchmark level of consumption, γ > 0, so that υ and the term premium have the same sign as α − 1, which can be positive, zero, or negative. To compute the equity premium under log-normality without habit formation, i.e., with D = 0, set Ht ≡ 1 and apply Eqs. (48) and (50) to Eq. (46) to obtain EP (λ) = [1 − A(λ) + Ψ(λ) A(λ)] exp ηλσxz − λθz + ηλΩxz − 0.5λ2 Ωzz − 1, (58) where A(λ) is given by Eq. (55) and Ψ(λ) is given by Eq. (54).

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6. LINEAR APPROXIMATIONS TO RISK, TERM, AND EQUITY PREMIA I have derived exact expressions for the risk premium, term premium, and equity premium under log-normality and D = 0. These expressions are very easy to implement, but they are cumbersome to interpret. To understand the effects of various parameters on these premia, I will linearize the exact expressions as functions of the parameters. Specifically, I will treat each of these premia as functions of β, μ, Σ, θ, Ω, and I will linearize each of these premia around β = 1, μ = θ = 0, and Σ = Ω = 0. Define φ ≡ β, μx , μz , σx2 , σxz , σz2 , θx , θz , Ωxx , Ωxz , Ωzz as a vector containing the effective discount factor β, the parameters of the objective log-normal distribution, and the difference between the parameter values of the subjective and objective lognormal distributions. Let φ0 = (1, 0, . . . , 0) be the value of φ around which I linearize the risk premium, term premium, and equity premium. Equations (54), (55), and (56) imply that Ψ(λ; φ0 ) = A(λ; φ0 ) = B(λ; φ0 ) = 1.

(59)

Linearizing the risk premium in Eq. (52) yields RP (λ) ∼ = ηλσxz − λθz + ηλΩxz − 0.5λ2 Ωzz .

(60)

Under rational expectations, θ = 0 and Ω = 0, and the linearized risk premium is simply ηλσxz , which is (up to a logarithmic approximation) the negative of the conditional −η covariance of the pricing kernel (in which xt+1 is unpredictable one period in advance) λ and the growth rate, zt+1 , of the risky payoff. I will indicate that the term premium on an infinite-maturity canonical asset depends on the parameter vector φ by rewriting Eq. (43) as T P (∞, λ; φ) = (Ψ(λ; φ) − 1) A(λ; φ) .

(61)

Recall from Eq. (59) that Ψ(λ; φ0 ) = A(λ; φ0 ) = 1. Therefore, T P (∞, λ; φ0 ) = 0, and the first-order linear approximation of the term premium is ∂Ψ(λ; φ0 ) (φ − φ0 ) . T P (∞, λ; φ) ∼ = ∂φ

(62)

Linearizing the expression for Ψ(λ) in Eq. (54) yields the following linear approximation to the term premium on an infinite-maturity asset: T P (∞, λ; φ) ∼ = υησx2 − υθx + υ [η − 0.5υ] Ωxx − υλΩxz .

(63)

Under rational expectations, θ = 0 and Ω = 0, so the linearized term premium is simply υησx2 .

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Equation (45) implies that the linear approximation to the equity premium is the sum of the linear approximations to the risk premium and to the term premium on an infinite-maturity asset. Therefore, Eqs. (60) and (63) imply EP (λ) ∼ = ηλσxz + υησx2 − υθx − λθz + υ(η − 0.5υ) Ωxx + (η − υ) λΩxz − 0.5λ2 Ωzz .

(64)

Under rational expectations, θ = 0 and Ω = 0, so the linearized equity premium is simply ηλσxz + υησx2 .

7. SECOND MOMENTS In this section I calculate the variances of the conditional and unconditional objective distributions of the rate of return on the canonical asset under log-normality and without habit formation, so that D = 0. Since Ht ≡ 1 when D = 0, Eq. (25) implies Rt+1 (n, λ) = J (xt+1 , zt+1 , 1; n, λ) K(xt , 1).

(65)

The objective variance of Rt+1 (n, λ) conditional on information available at date t is Vart {Rt+1 (n, λ)} = [K(xt , 1)]2 Var{J (xt+1 , zt+1 , 1; n, λ)}.

(66)

The variance of J (xt+1 , zt+1 , 1; n, λ) on the right-hand side of Eq. (66) is an unconditional variance because J (xt+1 , zt+1 , 1; n, λ) is i.i.d., so its conditional and unconditional variances are identical. I will confine attention to two special cases of Eq. (66), corresponding to riskless one-period bills and to equity. In the case of riskless one-period bills, n = 1 and λ = 0, so Eq. (32) reveals that J (xt+1 , zt+1 , 1; 1, 0) is constant. Thus, the conditional variance of the riskless one-period return is zero. Indeed, this conditional variance must be zero in order for the bill to be (conditionally) riskless. In the case of equity, n = ∞ and the conditional variance of the rate of return depends on the (unconditional) variance of J (xt+1 , zt+1 , 1; ∞, λ), since Ht ≡ 1 when there is no habit formation (i.e., when D = 0). Setting n = ∞ and Ht = 1 in Eq. (32) yields E zλ zλt+1 xυt+1 zλt+1 (1 − A(λ)) + Ψ(λ) A(λ) J (xt+1 , zt+1 , 1; ∞, λ) = . (67) B(λ) E zλ E xυ zλ Therefore, E {J (xt+1 , zt+1 , 1; ∞, λ)} =

E zλ B(λ)

[1 − A(λ) + Ψ(λ) A(λ)] ,

(68)

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which is consistent with Eq. (33) with n = ∞. Use the facts that ⎧ 2 ⎫ ⎬ ⎨ λ z = exp λ2 σz2 , E ⎭ ⎩ E zλ ⎧ ⎨ E

υ λ

xz ⎩ E xυ zλ

and

E

zλ E zλ

2 ⎫ ⎬

⎭

= exp υ 2 σx2 + λ2 σz2 + 2υλσxz ,

xυ zλ E xυ zλ

= exp λ2 σz2 + υλσxz

along with Eq. (67) to obtain E [J (xt+1 , zt+1 , 1; ∞, λ)]2 ⎤ ⎡ 2 (1 − A(λ))2 exp λ2 σz2 λ E z ⎥ ⎢ 2 = ⎣ + [Ψ (λ) A(λ)] exp λ2 σz2 + υ 2 σx2 + 2υλσxz ⎦. B(λ) + 2 (1 − A(λ)) Ψ (λ) A(λ) exp λ2 σz2 + υλσxz

(69)

(70)

(71)

(72)

Use Eqs. (68) and (72) along with the fact that Var{u} = E u2 − [E{u}]2 to obtain Var{J (xt+1 , zt+1 , 1; ∞, λ)} =

2 E zλ B(λ)

V (λ) ,

(73)

where V (λ) ≡ (1 − A(λ))2 exp λ2 σz2 − 1 + [Ψ (λ) A(λ)]2 exp λ2 σz2 + υ 2 σx2 + 2υλσxz − 1 + 2(1 − A(λ)) Ψ(λ) A(λ) exp λ2 σz2 + υλσxz − 1 .

(74)

imply that the Therefore, Eqs. (66) and (73) and the fact that K(xt , 1) = x−υ t conditional variance of the rate of return on an infinite-maturity asset, such as equity, is 2 E zλ −υ V (λ). (75) Vart {Rt+1 (∞, λ)} = x B(λ) t In order to calculate the unconditional variance of the rate of return on the canonical asset in Eq. (25), I will use the following identity for independent random variables K

140


(which is abbreviated notation for K(xt , Ht ) in Eq. (27)) and J (which is abbreviated notation for J (xt+1 , zt+1 , Ht+1 ; n, λ) in Eq. (26)): Var{K × J } = E K 2 E J 2 − [E{K}]2 [E{J }]2.

(76)

Rearrange Eq. (76) and use the fact that Rt+1 (∞, λ) = K × J to obtain Var{Rt+1 (∞, λ)} = Var{K} Var{J } + [E{K}]2 Var{J } + [E{J }]2 Var{K}.

(77)

Equation (27) implies that when D = 0, K(xt , Ht ) = x−υ t , which is i.i.d. Under lognormality with D = 0, E{K(xt , 1)} = exp −υμx + 0.5υ 2 σx2

(78)

Var{K(xt , 1)} = exp υ 2 σx2 − 1 exp −2υμx + υ 2 σx2 .

(79)

and

To summarize, the objective unconditional variance of the rate of return on equity under log-normality with D = 0 is given by Eq. (77), where E{K} is given by Eq. (78), Var{K} is given by Eq. (79), E{J } is given by Eq. (68), and Var{J } is given by Eq. (73). For riskless one-period bills (n = 1, λ = 0), the unconditional variance is particularly simple. Equation (22) together with a0 = 1 implies ω(1, 0) = B(0), so that Eq. (24) implies Rt+1 (1, 0) =

1 −υ x B(0) t

(80)

when D = 0, so that Ht ≡ 1. Therefore, the unconditional variance of the rate of return on the one-period riskless asset is Var{Rt+1 (1, 0)} = [B(0)]−2 exp υ 2 σx2 − 1 exp −2υμx + υ 2 σx2 .

(81)

7.1. Linear Approximations to Second Moments I will linearize the conditional and unconditional variances of the rate of return on the canonical asset around the point φ = φ0 , where, as in Section 6, φ ≡ (β, μx , μz , σx2 , σxz , σz2 , θx , θz , Ωxx , Ωxz , Ωzz and φ0 = (1, 0, . . . , 0). For infinite-maturity

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Andrew B. Abel

assets, such as stocks or consols, the linear approximation to the conditional variance is obtained by first linearizing Var{J (xt+1 , zt+1 , 1; ∞, λ)} in Eq. (73), using Eq. (59), to obtain ' } ≡ υ 2 σx2 + λ2 σz2 + 2υλσxz . Var{J (xt+1 , zt+1 , 1; ∞, λ)} ∼ = Var{J

(82)

Substituting Eq. (82) into Eq. (66) and using the fact that K(xt , 1) = x−υ t yields the linearized conditional variance 2 2 υ σx + λ2 σz2 + 2υλσxz . Vart {Rt+1 (∞, λ)} ∼ = x−2υ t

(83)

As a step toward linearizing the unconditional variance of the return on an infinitematurity canonical asset, observe from Eq. (78) that E{K(xt , 1; φ0 )} = 1.

(84)

Using Eq. (59), Eq. (68) implies that E{J (xt+1 , zt+1 , 1; ∞, λ; φ0 )} = 1.

(85)

Equations (73) and (79) imply that Var{J (xt+1 , zt+1 , 1; ∞, λ; φ0 )} = Var{K(xt , 1; φ0 )} = 0.

(86)

Linearizing Eq. (77) around φ = φ0 , and using Eqs. (84), (85), and (86) yields ' } + Var{K}, ' Var{Rt+1 (∞, λ)} ∼ = Var{J

(87)

' } is the linearized variance of J (xt+1 , zt+1 , 1; ∞, λ) in Eq. (82), and where Var{J ' Var{K} ≡ υ 2 σx2 is the linearized variance of K(xt , 1), obtained by linearizing Eq. (79). Therefore, Var{Rt+1 (∞, λ)} ∼ = 2υ 2 σx2 + λ2 σz2 + 2υλσxz .

(88)

The linearized unconditional variance of the one-period riskless rate is calculated by linearizing Eq. (81) to obtain Var{Rt+1 (1, 0)} ∼ = υ 2 σx2 .

(89)

Notice that the linearized variances (conditional and unconditional) of the riskless rate and the rate of return on equity depend on the moments of the objective distributions,

142


but do not depend on any deviation of the moments of the subjective distributions from the corresponding moments of the objective distributions. Of course, the linearizations are expansions around φ = φ0 where the means and second moments of the objective and subjective distributions are both equal to zero.

8. CORRELATION OF DIVIDEND-PRICE RATIO AND THE RATE OF RETURN ON STOCK Let ϕt (λ) ≡ ytλ /pt (∞, λ) be the dividend-price ratio of an infinite-maturity asset, such as equity, for which aj = 1 for all j. Equation (18), the limit as n → ∞ of Eq. (28), and the definition of K(xt , Ht ) in Eq. (27) imply ϕt (λ) =

1 − A(λ) K(xt , Ht ). B(λ)

(90)

Recall from Eq. (25) that, with n = ∞, the rate of return on stock is Rt+1 (∞, λ) = J (xt+1 , zt+1 , Ht+1 ; ∞, λ) K(xt , Ht ), where J (xt+1 , zt+1 , Ht+1 ; ∞, λ) and K(xt , Ht ) are independent of each other. To the extent that K(xt , Ht ) exhibits variation, the dividend-price ratio will vary, and this variation will help predict the rate of return on stock. To measure the correlation between the dividend-price ratio and the rate of return on stock, I will use the following lemma. Lemma 1 Let X and Y be independent random variables and let a = 0 be an arbitrary constant. Let νX ≡ Var {X}/[E{X}]2 be the square of the coefficient of variation of X and let νY ≡ Var {Y }/[E{Y }]2 be the square of the coefficient of variation of Y . Then the square of the correlation of aX and XY is R2 = νX /(νX νY + νX + νY ). 2 Proof. R2 = E aX 2 Y − E{aX} E{XY } / (Var {aX} Var {XY }). Since X and Y are independent, R2 = [E{Y }]2 [Var {X}]2 / (Var {X} Var {XY }) = [E{Y }]2 Var {X}/ Var {XY }. Use the identity for independent variables in Eq. (77) to obtain an expres2 2 sion for Var {XY }, and rewrite R2 as R = [E{Y }] Var {X}/ (Var {X} Var {Y } + 2 2 [E{Y }] Var {X} + [E{X}] Var {Y } . Divide the numerator and denominator by [E{X}]2 [E{Y }]2 and use the definition of the coefficient of variation to obtain R2 = νX / (νX νY + νX + νY ). Let νK be the square of the coefficient of variation of K(xt , Ht ) and νJ be the square of the coefficient of variation of J (xt+1 , zt+1 , Ht+1 ; ∞, λ). Applying Lemma 1 A(λ) with X = K(xt , Ht ), Y = J (xt+1 , zt+1 , Ht+1 ; ∞, λ), and a = 1 −B(λ) , so that ϕt (λ) = aX and Rt+1 (∞, λ) = XY , the R2 of a regression of the rate of return on equity on the dividend-price ratio is R2 =

νK . νK νJ + νK + νJ

(91)

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The expression for R2 in Eq. (91) is an exact expression and holds for the case with habit formation (so that Ht is endogenous) as well as for the case in which Ht ≡ 1. Now, as in Sections 5, 6, and 7, I will confine attention to the case with log-normality and without habit formation so that Ht ≡ 1. When Ht ≡ 1, K(xt , Ht ) = x−υ t , so that under log-normality, the squared coefficient of variation of K(xt , Ht ) is νK = exp υ 2 σx2 − 1.

(92)

Substituting Eq. (92) into Eq. (91) yields 2 2 exp υ σx − 1 . R2 = (νJ + 1) exp υ 2 σx2 − 1

(93)

In the case of standard preferences without a benchmark level of consumption, γ = 0, which implies that υ = 0, so that R2 in Eq. (93) is zero. That is, the dividend-price ratio has no predictive power for the risky rate of return in the case with standard preferences. However, the introduction of a benchmark level of consumption allows υ to differ from zero and thus enables the dividend-price ratio to predict the rate of return on stock. Now use the expression for the mean of J (xt+1 , zt+1 , 1; ∞, λ) in Eq. (68) and the variance of J (xt+1 , zt+1 , 1; ∞, λ) in Eqs. (73) and (74) to obtain (νJ + 1) exp υ 2 σx2 =

Υ(λ) [1 − A(λ) + Ψ(λ) A(λ)]2

,

(94)

where Υ(λ) ≡ (1 − A(λ))2 exp υ 2 σx2 + λ2 σz2 + [Ψ(λ) A(λ)]2 exp λ2 σz2 + 2υ 2 σx2 + 2υλσxz + 2 (1 − A(λ)) Ψ (λ) A(λ) exp υ 2 σx2 + λ2 σz2 + υλσxz .

(95)

Finally, substitute Eq. (95) into Eq. (93) and multiply the numerator and denominator of the resulting expression by [1 − A(λ) + Ψ(λ) A(λ)]2 to obtain R = 2

[1 − A(λ) + Ψ(λ) A(λ)]2 exp υ 2 σx2 − 1 Υ (λ) − [1 − A (λ) + Ψ (λ) A (λ)]2

.

(96)

144


I will approximate R2 around φ = φ0 by taking the ratio of the linearized numerator of Eq. (96) to the linearized denominator of Eq. (96) to obtain8 R2 ∼ =

υ 2 σx2 λ2 σz2 + 2υ 2 σx2 + 2υλσxz

(97)

.

As with the approximate variances of the riskless rate and the rate of return on equity, the approximate value of R2 in Eq. (97) depends on the objective moments of the distribution of (xt , zt ) but does not depend on the subjective moments of the distribution of (xt , zt ).

8.1. Correlation of Dividend-Price Ratio and the Excess Rate of Return on Stock I have shown that the dividend-price ratio at time t can be used to predict the rate of return on stock from period t to period t + 1. The ability to forecast the rate of return on stock in this model does not reflect any ability to forecast dividend growth because dividend growth, zλt+1 , is i.i.d. Instead, the predictability of the rate of return on stock is the direct result of the predictability of the intertemporal marginal rate of substitution in the case in which υ = 0. The predictable component of the intertemporal marginal rate of substitution is captured by K(xt , Ht ) ≡ x−υ t Ht . Since, as shown in Eq. (90), the dividend-price ratio is proportional to K(xt , Ht ), and since, as shown in Eq. (25), the rate of return on the canonical asset is proportional to K(xt , Ht ), variation in K(xt , Ht ) allows the dividend-price ratio to predict the rate of return on any asset that is a special case of the canonical asset. For instance, the dividend-price ratio can predict the rate of return on stock and the rate of return on one-period riskless bills. However, the dividendprice ratio cannot predict the excess rate of return on stock relative to one-period riskless bills, when the excess rate of return is defined as the ratio of the gross rates of return on

8 Rewrite

Eq. (96) as R2 =

where

2 RN 2 RD

,

2 RN = [1 − A(λ) + Ψ (λ) A(λ)]2 exp υ 2 σx2 − 1

is the numerator of R2 and 2 RD = Υ (λ) − [1 − A(λ) + Ψ (λ) A(λ)]2 2 2 2 is the denominator of R2 . When φ = φ0 , RN = 0 and RD = 0. Linearizing RN around φ = φ0 yields 2 ∼ 2 2 2 2 ∼ ( RN = υ σx and linearizing RD around φ = φ0 yields RD 2Ψ + 2A( + λ2 σz2 + 2υ 2 σx2 + 2υλσxz − 2A( − =

( + A( , where Ψ ( is the linearized value of Ψ (λ) and A( is the linearized value of A(λ). There2 −A( + Ψ 2 ∼ 2 2 2 2 fore, R = λ σ + 2υ σ + 2υλσxz . Taking the ratio of the approximate value of R2 to the approximate D

z

x

2 value of RD yields Eq. (97) in the text.

N

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Andrew B. Abel

the two assets. Specifically, using Eq. (25), the excess rate of return on stocks, expressed in ratio form, is J (xt+1 , zt+1 , Ht+1 ; ∞, λ) Rt+1 (∞, λ) = , Rt+1 (1, 0) J (xt+1 , zt+1 , Ht+1 ; 1, 0)

(98)

which is i.i.d., and hence is unpredictable. Indeed, the fact that the ratio of returns in Eq. (98) is i.i.d. accounts for the fact that conditional and unconditional premia are equal in this model. An alternative formulation of the excess rate of return is the arithmetic difference S ≡ between the rates of return on stocks and one-period riskless bills. Define Jt+1 b b S E J (xt+1 , zt+1 , Ht+1 ; ∞, λ), Jt+1 ≡ J (xt+1 , zt+1 , Ht+1 ; 1, 0), and Jt+1 ≡ Jt+1 − Jt+1 , so S × K (xt , Ht ), the rate of return on one-period that the rate of return on stock is Jt+1 b riskless bills is Jt+1 × K(xt , Ht ), and the arithmetic excess rate of return on stocks is E Rt+1 (∞, λ) − Rt+1 (1, 0) = Jt+1 × K (xt , Ht ).

(99)

The excess rate of return in Eq. (99) is proportional to K(xt , Ht ) and thus is predictable by the dividend-price ratio, which is also proportional to K(xt , Ht ). However, for empirically plausible values, the R2 of the regression of the arithmetic excess return in Eq. (99) on the dividend-price ratio is extremely small. Applying Eq. (91), the R2 of the regression of the arithmetic excess stock return on the dividend-price ratio is R2 =

νK νJ E

νK , + νK + νJ E

(100)

b E . Since Jt+1 ≡ where is νJ E is the square of the coefficient of variation of Jt+1 E S J (xt+1 , zt+1 , Ht+1 ; 1, 0) = a0 /ω(1, 0) is constant, Var Jt+1 = Var Jt+1 , so

νJ E

S Var Jt+1 = b 2 . S E Jt+1 − E Jt+1

(101)

Rearrange S the right-hand side of Eq. (101) using b the facts that E{Rt+1 (∞, λ)} = E Jt+1 E{K(xt , Ht )} and E{Rt+1 (1, 0)} = E Jt+1 E{K(xt , Ht )} to obtain νJ E = νJ S Q, where Q ≡

E{Rt+1(∞, λ)} E{Rt+1(∞, λ)} − E{Rt+1(1, 0)}

(102)

2 and νJ S is the squared coefficient of variation of

S . Equations (100) and (102) and the fact that νK νJ E ≥ 0 imply that Jt+1

R2 ≤

νK 1 = . ν S νK + νJ E 1 + νJK Q

(103)

To get a sense of the size of the upper bound on R2 on the right-hand side of Eq. (103), consider an average (gross) rate of return on stocks, E{Rt+1 (∞, λ)}, of 1.0698 per year

146


and an average equity premium, E{Rt+1 (∞, λ)} − E{Rt+1 (1, 0)}, of 0.0618 per year, as in Mehra and Prescott (1985). In this case, Q = 300. Empirically, the ratio νJ S /νK is greater than one. If νJ S /νK were as low as one, the upper bound on R2 would be 1/(1 + 300) = 0.003. In calibrations reported later in this chapter, νJ S /νK is greater than 5, so the upper bound on R2 is smaller than 1/(1 + 1500) = 0.00067.

9. SPECIAL CASES In this section, I will examine the characteristics of rates of return in various special cases. I will adopt the normalization that the variance of ln z equals the variance of ln x, i.e., that σz ≡ σx . With this normalization, λ captures any difference in the standard deviations of the “un-normalized” values of ln z and ln x. Also, with this normalization, the covariance σxz = ρσx2 , where −1 ≤ ρ ≤ 1 is the correlation of ln x and ln z.

9.1. Rational Expectations Under rational expectations (RE), subjective distributions are identical to the corresponding objective distributions, so that θ = 0 and Ω = 0. In this case, the approximate equity premium in Eq. (64) simplifies to EP (λ) ∼ = (λρ + υ) ησx2 ,

under RE,

(104)

which is the sum of the approximate risk premium from Eq. (60) RP (λ) ∼ = λρησx2 ,

under RE,

(105)

and the approximate term premium from Eq. (63) T P (∞, λ) ∼ = υησx2 ,

under RE.

(106)

The approximate unconditional variance of the rate of return on equity in Eq. (88) simplifies to Var{Rt+1 (∞, λ)} ∼ under RE. (107) = 2υ 2 + λ2 + 2υλρ σx2 , The approximate unconditional variance of the one-period riskless rate under rational expectations is the same as in Eq. (89). Finally, the approximate value of R2 in Eq. (97) simplifies to R2 ∼ =

υ2 , λ2 + 2υ 2 + 2υλρ

under RE.

(108)

The value of R2 in Eq. (108) applies to a regression of the equity rate of return (rather than the excess equity rate of return) on the dividend-price ratio. As emphasized in subsection 8.1, excess returns are not predictable in the model presented here, so the

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Andrew B. Abel

model cannot account for the empirically observed predictability of excess returns. Nevertheless, the model can account for predictability of equity returns, and in the calculations throughout this section I will focus on the R2 of the regressions of the equity rate of return on the dividend-price ratio.

9.1.1. Perfectly Correlated ln x and ln z In the original Lucas (1978) model of asset pricing in an exchange economy, equilibrium consumption per capita, Ct , is identically equal to the dividend per capita on risky stock, ytλ . In the current framework, the identity of consumption per capita and dividend per capita is represented by Ct ≡ yt and λ = 1. Because Ct ≡ yt , the growth rates of consumption and dividends are identically equal. That is, xt ≡ zt , so ρ = 1. In this subsection, I will maintain the assumption that ρ = 1, but I will allow λ to differ from one to take account of levered equity. In the remainder of this subsection on asset returns under rational expectations, I will examine values of the risk premium, term premium, equity premium, standard deviations of rates of return on one-period riskless bills and levered equity, and the R2 of regressions of the equity rate of return on the dividend-price ratio. Table 2 presents the approximate values of these six features of asset returns for five particular cases. The final column of Table 2, labeled “Data,” presents the empirical values of these features of the data. TABLE 2 Approximate Features of Rates of Return Standard

Consumption externalities

Data

(1)

(2)

(3)

(4)

(5)

υ

0

3.85

1.25

1.25

1.7

η

48.5

10

10

10

20

Leverage

Yes

Utility function

No

No

Yes

Yes

λ

1

1

3.6

3.6

3.6

ρ

1

1

1

0.2

0.2

6.18%

1.27%

4.59%

0.92%

1.84%

(6)

RP (λ)

λρησx2

T P (∞, λ)

υησx2

0

4.91%

1.59%

1.59%

4.33%

1.70%

EP (∞, λ)

(λρ + υ) ησ 2x

6.18%

6.18%

6.18%

2.51%

6.17%

6.18%

|υ| σx ) σx 2υ 2 +λ2 + 2υλρ

0

13.74%

4.46%

4.46%

6.07%

5.67%

3.57%

s.d.{Rriskless } s.d. Rrisky R2

υ2 λ2 + 2υ 2 + 2υλρ

0

22.11% 17.88% 15.10% 16.43% 16.54% 38.7%

6.2%

8.7%

13.6%

9%

Sources for data: Equity premium, standard deviations of riskless and risky rates from Mehra and Prescott (1985), Table 1. Term premium reported in Abel (1999) based on Ibbotson Associates (1994, Exhibit 9, p. 31). R2 from Fama and French (1988), Table 3.

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With ρ = 1, the approximate equity premium in Eq. (104) simplifies to EP (λ) ∼ = (λ + υ) ησx2 ,

under RE, if ρ = 1,

(109)

which is the sum of the approximate risk premium from Eq. (105) RP (λ) ∼ = λησx2 ,

under RE, if ρ = 1,

(110)

and the approximate term premium from Eq. (106), which is invariant to ρ. The expression for the approximate unconditional variance of the rate of return on equity in Eq. (107) simplifies to (111) Var{Rt+1 (∞, λ)} ∼ = (υ + λ)2 + υ 2 σx2 , under RE, if ρ = 1, and Eq. (108) implies that the R2 of a regression of the rate of return on equity, Rt+1 (∞, λ), on the dividend-price ratio, ϕt (λ), is approximately R2 ∼ =

υ2 (λ + υ)2 + υ 2

,

under RE, if ρ = 1.

(112)

Standard Utility Function and No Leverage With a standard utility function that does not include benchmark levels of consumption, γ = 0, which implies υ = 0 and η = α, where α is the coefficient of relative risk aversion. The seminal papers of Lucas (1978) and Mehra and Prescott (1985) analyze unlevered equity, which is represented here by the canonical asset with n = ∞ and λ = 1. In this case, the approximate equity premium in Eq. (109) simplifies to EP (λ) ∼ = ασx2 ,

under RE, if ρ = 1, γ = 0, and λ = 1.

(113)

In this case, since υ = 0, the term premium is zero, and the equity premium is composed entirely of the risk premium. Eq. (113) is the essence of the Mehra–Prescott equity premium puzzle. Over the time period examined by Mehra and Prescott, 1889–1978, the average value of the equity premium was 0.0618 per year and the annual standard deviation of the growth rate of consumption per capita, measured here by σx , was 0.0357. Thus, if the coefficient of relative risk aversion, α, equals 10, which is a very high value in the judgment of Mehra and Prescott, the equity premium would be only (10)(0.0357)2 = 0.0127, i.e., 1.27 percent per year. In order for the model to match the historical equity premium of 0.0618, the coefficient of relative risk aversion, α, would need to equal 0.0618/(0.0357)2 = 48.5, which is far higher than the upper bound of 10 imposed by Mehra and Prescott on the value of α.9 Note from Eq. (112) that with standard preferences, which imply υ = 0, the approximate value of R2 is zero. Indeed, when υ = 0, the exact expression for R2 in Eq. (96) is also equal to zero. Therefore, the dividend-price ratio cannot help predict the rate of 9 Kandel and Stambaugh (1991) provide a compelling challenge to arguments that the coefficient of relative risk aversion is no larger than 10.

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return on equity in a linear regression. In addition, with υ = 0, the rate of return on one-period riskless bills is constant, which is indicated by a value of zero for s.d.{Rriskless } in column 1 of Table 2. Consumption Externalities and No Leverage Consumption externalities, by allowing υ to be positive, can increase the equity premium associated with any given value of η, introduce variability in the one-period riskless rate of return and a positive term premium, and help account for the ability of the dividend-price ratio to predict equity returns, though as discussed in subsection 8.1, they cannot account for the predictability of excess returns. In the absence of leverage, λ = 1. Therefore, in this case, the approximate equity premium in Eq. (109) becomes EP (λ) ∼ = (1 + υ) ησx2 ,

under RE, if ρ = 1

and

λ = 1.

(114)

Thus, for given values of η and σx2 , the equity premium implied by the model increases by a factor 1 + υ. Therefore, the introduction of a positive υ allows the model to match the historical value of the equity premium with a smaller value of η. For instance, if η = 10 (which would arise if, for instance, α = 10 and θ0 = 0), the model will deliver (λ) an approximate equity premium equal to its historical average value if (1 + υ) = EP 10σx2 = 0.0618/ 10 (0.0357)2 = 4.85, or equivalently, if υ = 3.85, as in column 2 of Table 2. In Abel (1990), I suggested that this formulation of utility might help account for the historically observed equity premium, but I acknowledged that the implied variability of the riskless rate was far larger than the historical variability of the riskless rate. Equation (89) implies that the approximate standard deviation of the one-period riskless rate is |υ| σx , which equals (3.85)(0.0357) = 13.74 percent. However, the historical standard deviation of the riskless rate reported by Mehra and Prescott (1985) is only 5.67 percent per year. The fact that υ can differ from zero when utility depends on a benchmark level of consumption implies that benchmark utility introduces a term premium and also allows the dividend-price ratio to help predict the rate of return on equity. Substituting υ = 3.85 into Eq. (106) yields a term premium of (3.85)(10)(0.0357)2 = 0.0491, or 4.91 percent, which is much higher than the empirical value. For instance, in Abel (1999) I report, based on data from Ibbotson Associates (1994, Exhibit 9, p. 31), that over the period 1926–1993 the term premium on long-term government bonds relative to short-term riskless bills was about 170 basis points per year, or 0.0170. The term premium in the model is for infinite-maturity assets with risky payoffs, whereas the data applies to finite-maturity riskless claims. However, I have argued in Abel (1999) that these differences in maturity and risk may increase the term premium in the model, but by only a small fraction of one percent per year. Thus, the 491 basis point term premium produced by the model in this case is too high compared to the empirical term premium. Finally, substituting υ = 3.85 into Eq. (112), with λ = 1, shows that a linear regression of the rate of return on equity on the dividend-price ratio has an approximate R2 of 0.387, which is much higher than typical values of R2 in such regressions. For

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Chapter 4

•

Equity Premia with Benchmark Levels of Consumption: Closed-Form Results

instance, using annual real returns on the value-weighted NYSE portfolio, Fama and French (1988, Table 3) report R2 = 0.09 for the period 1941–1986. Thus, consumption externalities without leverage can account for the observed equity premium with a value of η = 10. However, the implied value of υ, 3.85, leads to too much variability in the one-period riskless rate of return, to too high a term premium, and to too high a value of R2 in a regression of the equity rate of return on the dividend-price ratio. The next challenge is to account for the observed value of the equity premium with a lower value of υ. Consumption Externalities with Leverage Adding leverage to the model with consumption externalities allows the model to match the historical equity premium with a lower value of υ. The lower value of υ reduces the variability of the riskless rate, the term premium, and the value of R2 in the model toward more realistic values. The standard deviation of the growth rate of dividends, zλt+1 , is approximately λ times the standard deviation of zt+1 . Since zt+1 ≡ xt+1 , the standard deviation of the growth rate of dividends is approximately λ times σx , the standard deviation of the growth rate of consumption. Using data from Table 2, p. 402, of Cecchetti, Lam, and Mark (1990), the standard deviation of dividend growth is 3.6 times the standard deviation of consumption growth, so I will set λ = 3.6. With λ = 3.6, the approximate equity premium in Eq. (109) will equal the historical value of 0.0618 with η = 10 and σx = 0.0357 if υ = 0.0618/ 10 (0.0357)2 − 3.6 = 1.25. Because the introduction of leverage permits a reduction in the value of υ, it also reduces the variability of the one-period riskless rate, the term premium, and the value of R2 . As shown in column 3 of Table 2, with υ = 1.25, the approximate standard deviation of the one-period riskless rate in Eq. (89) is (1.25) (0.0357) = 0.0446, which is much closer to its historical value of 5.67 percent, as reported by Mehra and Prescott (1985). Also, with υ = 1.25, the approximate term premium is (1.25)(10)(0.0357)2 = 0.0159, which is much closer to the empirical value of the term premium of 0.0170 mentioned earlier. With λ = 3.6 and υ = 1.25, Eq. (112) implies that the R2 of a regression of the rate of return on equity on the dividend-price ratio is approximately 0.062, which is more in line with the values reported by Fama and French (1988). Finally, from Eq. (111), the approximate unconditional standard deviation of the rate of return on equity when λ = 3.6 and υ = 1.25 is 0.1788, which is not far from the value of 16.54 percent reported by Mehra and Prescott (1985).

9.1.2. Imperfectly Correlated ln x and ln z I have shown that with η = 10, υ = 1.25, and λ = 3.6, the rational expectations solution of the model with ρ = 1 generates realistic values for five features of returns: the equity premium; the term premium; the unconditional standard deviation of the risky rate of return; the unconditional standard deviation of the riskless rate of return; and the R2 of the regression of the equity rate of return on the dividend-price ratio. However, with ρ = 1, consumption growth and dividend growth are perfectly correlated. I will show

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Andrew B. Abel

that reducing the value of ρ to a more realistic value reduces the ability of the model to match the equity premium, unless the value of η is allowed to be greater than 10. Campbell and Cochrane (1999) argue that the correlation between consumption growth and dividend growth is difficult to estimate accurately and ultimately settle on a value of 0.2 to use in their calculations. I will follow their lead and set ρ = 0.2. Since the equity premium under rational expectations approximately equals (λρ + υ) ησx2 , reducing the correlation ρ to 0.2 from 1.0 substantially reduces the equity premium when λ = 3.6 and υ = 1.25. In fact, when these values of ρ, λ, and υ are combined with η = 10 and σx = 0.0357, the approximate equity premium is only 0.0251, which is less than half of the historical average equity premium. As shown in column 4 of Table 2, the other four features of the returns implied by these parameter values are not as far from empirical values. Specifically, the approximate standard deviation of the rate of return on stocks is 0.1510, and the approximate R2 of the regression of the equity rate of return on the dividend-price ratio is 0.087. The approximate term premium and the approximate standard deviation of the riskless rate are invariant to the correlation ρ, and hence these features are the same in column 4 as in column 3. Doubling the value of the curvature parameter η to 20 and increasing the value of υ slightly to 1.7, while keeping λ = 3.6, ρ = 0.2, and σx = 0.0357, allows the model to match the empirical value of the equity premium and produces standard deviations of the riskless rate and the rate of return on equity that are close to their empirical values. Specifically, as shown in column 5 of Table 2, these parameter values lead to an equity premium of 0.0617, a standard deviation of the riskless rate of 0.0607, and a standard deviation of the rate of return on equity of 0.1643. However, these parameter values lead to a value of the term premium of 0.0433, which is too high compared to the data, and an R2 of 0.136, which is a bit high.

9.2. Distorted Beliefs In order to focus on the role of distorted beliefs on rates of return, I will restrict attention to the case in which xt ≡ zt , so that σx2 = σxz = σz2 . I assume that the subjective distributions take account of the fact that xt ≡ zt , so that θx = θz and Ωxx = Ωxz = Ωzz . In this case, the approximate equity premium in Eq. (64) simplifies to EP (λ) ∼ = (υ + λ) ησx2 − θx + [η − 0.5 (υ + λ)] Ωxx ,

if xt ≡ zt .

(115)

Note that υ + λ > 0 because υ > −1 (Eq. (8)) and λ ≥ 1. Although the distortion of beliefs, represented here by nonzero values of θx and Ωxx , affects the approximate equity premium, Eqs. (89), (88), and (97) show, as mentioned earlier, that the distortion of beliefs does not affect the approximate unconditional variances of the riskless and risky rates of return, nor does it affect the approximate R2 of a regression of the risky rate of return on the dividend-price ratio.

152


9.2.1. Standard Utility Function and No Leverage In the case with standard utility (γ = 0, so υ = 0) and no leverage (λ = 1), the approximate equity premium in Eq. (115) simplifies to EP (1) ∼ = η σx2 + Ωxx − (θx + 0.5Ωxx ),

if xt ≡ zt , γ = 0, and λ = 1.

(116)

In Abel (2002), I used the term pessimism to describe a situation in which the subjective distribution of the growth rate is first-order stochastically dominated by the objective distribution of the growth rate. When the growth rate is log-normal, I measure the amount of pessimism about the growth rate by Δ1 , where Δ1 satisfies E ∗ {x} = e−Δ1 E{x}, which implies Δ1 = − (θx + 0.5Ωxx ) .

(117)

Substituting Eq. (117) into Eq. (116) yields EP (1) ∼ = η σx2 + Ωxx + Δ1 ,

if xt ≡ zt , γ = 0, and λ = 1.

(118)

It is evident from Eq. (118) that pessimism, measured by Δ1 , increases the equity premium. The economic reason for this effect is that pessimism reduces the price of the risky asset, which increases the rate of return to anyone who buys this asset. On average, the realized rate of return is higher than the amount required by consumers to compensate them for the risk of holding equity because the realized rate of return on equity is higher, on average, than consumers expect. Indeed, in the special case with standard utility (γ = 0) and no leverage (λ = 0), an increase in pessimism about the expected growth rate causes an equal-sized increase in the objective average equity premium. In Abel (2002), I used the term doubt to describe a situation in which the subjective distribution of the growth rate is a mean-preserving spread of the objective distribution of the growth rate. When the growth rate is log-normal, an increase in doubt is represented by an increase in Ωxx ≡ σx∗2 − σx2 , holding constant θx + 0.5Ωxx = −Δ1 . Equation (118) shows that an increase in doubt increases the equity premium. The economic reason is that an increase in doubt increases the perceived riskiness of equity and thus increases the equity premium that consumers require in order to hold equity.

9.2.2. Consumption Externalities with Leverage In the presence of consumption externalities and/or leverage, we can interpret the effects on the equity premium of distorted beliefs by generalizing the measure of pessimism. Instead of comparing subjective and objective distributions of the growth rate xt+1 , I will compare the subjective and objective distributions of xυ+λ t+1 (which, in the special case with standard utility (υ = 0) and no leverage (λ = 1), is simply the growth rate xt+1 ).

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Now when the growth rate is log-normal, I will the amount υ+λ υ+λ measure of pessimism about ∗ −Δυ+λ xυ+λ by Δ , where Δ satisfies E E x x = e , which implies υ+λ υ+λ t+1 Δυ+λ = − (υ + λ) θx − 0.5 (υ + λ)2 Ωxx .

(119)

Substituting Eq. (119) into Eq. (115) yields EP (λ) ∼ = (υ + λ) η σx2 + Ωxx + Δυ+λ ,

if xt ≡ zt ,

(120)

which has the same form and interpretation as Eq. (118). Again, an increase in pessimism Δυ+λ increases the equity premium, and an increase in doubt, represented as an increase in Ωxx holding Δυ+λ constant, increases the equity premium.

10. ACCURACY OF APPROXIMATIONS I have conducted the analysis in two distinct parts. First, I set up the model and derived exact solutions for the equilibrium price and expected rate of return on the canonical asset. Because the canonical asset is sufficiently general, I used it to calculate rates of return on various assets, including one-period riskless bills and infinite-horizon levered equity. To obtain closed-form expressions for these exact solutions, I restricted attention to the case without habit formation (but with consumption externalities including keeping/catching up with the Joneses) and with log-normally distributed growth rates of consumption and dividends. The second part of the analysis started with the exact solutions derived in the first part, and derived linear approximations to various moments of asset returns. The rationale for approximating these expressions was to help interpret closed-form solutions, in particular, to understand the effects of various parameters on the means and variances of returns. In Section 9.1, I examined some quantitative features of the model under rational expectations. For clarity, I used the approximate expressions for the moments of asset returns. In this section, I will briefly explore the accuracy of these approximations by comparing the values obtained by evaluating the approximate expressions with the values obtained by evaluating the corresponding exact expressions. Before comparing the approximate and exact solutions, I need to address an issue regarding the parameter values. As discussed in Section 1, the primitive specification of preferences includes seven parameters: the curvature paramater α, the rate of time preference δ, the importance of the consumption-benchmark ratio as measured by γ, the relative weight D of the consumer’s own past consumption in the benchmark, the weights θ0 and θ1 on contemporaneous and lagged aggregate consumption in the benchmark, and the exogenous growth rate G in the benchmark. For the purposes of asset pricing in the absence of habit formation (i.e., with D = 0), these seven preference parameters can be summarized by the three parameters β, η, and υ defined in Eqs. (6), (7), and (8), respectively. The approximate expressions for the equity premium, the term premium, the variances of the riskless rate of return and the rate of return on stocks, and

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the value of R2 are invariant to β. Thus, in Section 9.1 I did not specify a value for β. However, the exact expressions for these features of asset returns depend on β. Thus, to compare the exact and approximate expressions, I need to specify a value for β. I will choose a value for β by calibrating the model to the exact expression for the riskless rate of return under rational expectations. In the case without habit formation, D = 0, which implies Ht ≡ 1, so setting λ = 0 in Eq. (36) implies that the average one-period riskless rate is E x−υ . (121) E{Rt+1 (1, 0)} = B(0) Setting λ = 0 in Eq. (21) to obtain B(0), imposing rational expectations, and substituting the result into Eq. (121) yields E x−υ E{Rt+1 (1, 0)} = . (122) βE{x−η } Let Rf be the empirical average value of the one-period riskless rate. The model will yield an expected one-period riskless rate equal to Rf if β is chosen to set the right-hand side of Eq. (122) equal to Rf . Under log-normality, the value of the β that will allow the model to match this value of the expected riskless rate is exp (η − υ) μx + 0.5 υ 2 − η2 σx2 . (123) β= Rf In the case with consumption externalities and leverage under rational expectations with perfectly correlated consumption growth and dividend growth in Section 9.1.1, I have set μx = 0.018, σx = 0.0357, η = 10, and υ = 1.25. Mehra and Prescott (1985) report Rf = 1.0080. Substituting these values into Eq. (123) yields β = 1.09. At first blush, a value of β greater than one might lead to two concerns. One concern is that β > 1 might imply that the rate of time preference δ is negative. However, the definition of β in Eq. (6) shows that β can exceed one with a positive rate of time preference δ if G > 1 and (1 − D) γ (α − 1) > 0. The second concern is that some important infinite sums will fail to converge. However, as shown by Mehra (1988, Appendix) and Kocherlakota (1990), β can be slightly greater than one in a growing economy without causing this problem. However, for the particular parameter values in this example, including λ = 3.6, the value of A(λ) exceeds one when β = 1.09. In order to make A(λ) less than one, so that the difference equation in Eq. (19) will converge, I will set β = 1.075.10 10 The

values of derived preference parameters β, η, and υ are consistent with various combinations of the seven fundamental preference parameters. If, for example, δ = 0.01, G = 1.01279, D = 0, α = 14, γ = 0.4979, θ0 = 0.618, and θ1 = 0.193, then the values of the three derived parameters are β = 1.075, η = 10, and υ = 1.25, as in column 1 in Table 3. If, for example, δ = 0.01, G = 1.0076, D = 0, α = 25, γ = 0.3759, θ0 = 0.554, and θ1 = 0.188, then the values of the three derived parameters are β = 1.060, η = 20, and υ = 1.70, as in column 2 of Table 3.

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Of course, reducing the value of β is equivalent to increasing the rate of time preference, and as a result of the increased rate of time preference, the (net) expected riskless rate increases to 0.0227, or 2.27 percent per year. Having chosen a value for β, I can compute the exact as well as the approximate values of the equity premium, the standard deviations of the riskless rate of return and the rate of return on stock, and the value of R2 in a regression of stock returns on the dividend-price ratio. As illustrated in column 1 of Table 3, the approximations are quite close to the exact values for the equity premium, the standard deviation of the riskless rate, and the value of R2 . The approximation for the standard deviation of the risky rate of return is not as good, but is within 10 percent of the value obtained from the exact expression. Table 3 also reports approximate and exact values for the case with consumption externalities and leverage under rational expectations with less than perfectly correlated consumption growth and dividend growth. As in Section 9.1.2, I have set the correlation ρ = 0.2. As I discussed earlier, the model is better able to deliver empirically realistic values if η is increased to 20 and if υ is increased to 1.7. In order for A(λ) to be smaller than one, as required for the difference equation in Eq. (19) to converge, I have reduced β to 1.060. Despite this reduction in β, relative to its value in column 1, the expected riskless rate is only 1.82 percent per year, which is lower than its value in column 1. Again, the approximate values of the equity premium, the standard deviation of riskless

TABLE 3 Exact and Approximate Moments of Returns (1)

(2)

β

1.075

1.060

υ

1.25

Preference parameters

1.70

10

η Leverage parameter, λ

20

3.6

3.6

Growth rate parameters μx = μz

0.018

0.018

σx = σz

0.0357

0.0357

ρ

1.0

0.2

Moments of returns from model

Exact

Approx.

Exact

Approx.

Equity premium

6.37%

6.18%

6.36%

6.17%

s.d.{Rriskless } s.d. Rrisky

4.57%

4.46%

6.19%

6.07%

19.59%

17.88%

17.91%

16.43%

R2

6.1%

6.2%

13.5%

13.6%

Expected (net) riskless rate

2.27%

1.82%

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rate, and the value of R2 are very close to the values obtained from the exact expressions. And the approximate value of the standard deviation of the risky rate is within 10 percent of the value obtained from the exact expression.

11. SUMMARY I have derived exact expressions for the price and rate of return on a canonical asset in a framework that extends the standard utility function to include habit formation, keeping/catching up with the Joneses, and possible departures from rational expectations. The canonical asset is general enough to include one-period assets and infinite-maturity levered equity, and thus can be used to calculate risk premia, term premia, and equity premia. To calculate exact closed-form expressions for the first and second moments of returns, I have restricted attention to the case that excludes habit formation, but includes keeping/catching up with the Joneses, and I have assumed that the growth rates of consumption and dividends are jointly i.i.d. log-normal. After deriving the exact expressions for the first and second moments of returns, I have derived linear approximations to these expressions to show clearly how these moments of returns are affected by various parameters. I have used these linear approximations to show, quantitatively, how well the model can account for the empirical values of the equity premium, the term premium, the standard deviations of the riskless return and the rate of return on levered equity, and the value of R2 of a regression of stock returns on the dividend-price ratio. Finally, I showed that the approximations yield values of these moments that are close to those obtained from the exact solutions, for empirically relevant parameter values.

References Abel, A. B. Asset prices under habit formation and catching up with the Joneses. American Economic Review 80(2) (1990): 38–42. Abel, A. B. Risk premia and term premia in general equilibrium. Journal of Monetary Economics 43, (February) (1999): 3–33. Abel, A. B. An exploration of the effects of pessimism and doubt on asset returns. Journal of Economic Dynamics and Control 26(7–8) (2002): 1075–1092. Abel, A. B. Optimal taxation when consumers have endogenous benchmark levels of consumption. The Review of Economic Studies 72(1) (2005): 21–42. Backus, D. K., A. W. Gregory, and S. E. Zin. Risk premiums in the term structure: Evidence from artificial economies. Journal of Monetary Economics 24(3) (1989): 371–399. Bansal, R., and A. Yaron. Risks for the long run: A potential resolution of asset pricing puzzles. Journal of Finance 59(4) (2004): 1481–1509. Campbell, J. Y., and J. H. Cochrane. By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107(2) (1999): 205–251. Cecchetti, S., P. Lam, and N. Mark. Mean reversion in equilibrium asset prices. American Economic Review 80(3) (1990): 398–418. Constantinides, G. M. Habit formation: A resolution of the equity premium puzzle. Journal of Political Economy 98(3) (1990): 519–543. Dupor, B., and W.-F. Liu. Jealousy and equilibrium overconsumption. American Economic Review 93(1) (2003): 423–428.

Andrew B. Abel

157 Fama, E. F., and K. R. French. Dividend yields and expected stock returns. Journal of Financial Economics 22(1) (1988): 3–25. Gali, J. Keeping up with the Joneses: consumption externalities, portfolio choice, and asset prices. Journal of Money, Credit and Banking 26(1) (1994): 1–8. Ibbotson Associates. Stocks, Bonds, Bills, Inflation, 1994 Yearbook. Ibbotson Associates, Chicago (1994). Kandel, S., and R. F. Stambaugh. Asset returns and intertemporal preferences. Journal of Monetary Economics 27(1) (1991): 39–71. Kocherlakota, N. R. On the “Discount” factor in growth economies. Journal of Monetary Economics 25(1) (1990): 43–47. Lettau, M., S. C. Ludvigson, and J. A. Wachter. The declining equity premium: What role does macroeconomic risk play? April (2004). Lucas, R. E., Jr. Asset prices in an exchange economy. Econometrica 46(6) (1978): 1429–1445. Mehra, R. On the existence and representation of equilibrium in an economy with growth and nonstationary consumption. International Economic Review 29(1) (1988): 131–135. Mehra, R., and E. C. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15(2) (1985): 145–161.

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Discussion: Equity Premia with Benchmark Levels of Consumption: Closed-Form Results Francisco Gomes London Business School

1. INTRODUCTION Andy Abel’s paper discusses the role of preferences with consumption externalities, leveraged equity claims, and non-rational expectations in explaining the equity premium puzzle. The “equity premium puzzle” is a term that is now used to define a 20-year-old literature, started by Mehra and Prescott (1985). In many ways, the search for an answer to this puzzle did for the asset pricing field what the famous search for the proof of Fermat’s last theorem did for the field of mathematics. Searching for the proof of Fermat’s last theorem lasted more than 300 years, drawing the attention of many of the greatest minds in the field, as well as scientists from related fields. In a similar way, many finance academics and several economists with very limited (if any) previous interest in the field contributed to the equity premium puzzle debate. In both cases, the search for answers led to the development of multiple new ideas and techniques, which in turn contributed to their respective fields in many (often unexpected) ways, sometimes even spawning whole new areas of research. This was particularly the case for asset pricing. The field underwent a significant revolution in the last two decades, and a sizable number of those changes can be traced back to contributions toward the equity premium puzzle debate. Preferences with “benchmark levels of consumption,” which Andy discusses in detail in his paper, is a very good example of this. The basic idea was borrowed from economics, where it has existed since at least Duesenberry (1949), but in an attempt to explain the equity premium puzzle, the concept was significantly expanded and transformed. As a byproduct, these preferences have since been used in several other asset pricing areas, and in other fields of economics. Before discussing Andy’s paper in more detail, I will conclude this comparison by pointing out two very important differences, to which I shall return later on. First,

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while Fermat’s last theorem is by itself of very limited consequence for the field of mathematics, the equity premium debate is extremely important, not only for asset pricing and finance, but for many other fields of economics as well. The equity premium puzzle revealed a fundamental weakness in the ability of extant economic models to price non-hedgeable risk. As such, it impacts every field of economics dealing with behavior under uncertainty and potentially questions the results in those areas, in particular those of a quantitative nature. Second, while the mathematicians’ quest had a clear ending (i.e., finding the proof), ours did/does not. It is not just about matching the (historical) equity premium; it is also about matching the riskless rate, the volatility of consumption growth, the ex-post volatility of the riskless rate, the correlation between consumption and stock returns, the term premium, the volatility of investment, etc. As soon as we explain one piece of the puzzle, we immediately add another one. That’s what makes this search an ongoing saga, and one that contributes to the continuing progress of the field. After all, if we just want a model that will match the observed equity premium, that is not very hard: Mehra and Prescott themselves showed us that it can be done with the Lucas (1978) model and power utility with a risk aversion coefficient of 50. If that were a satisfactory answer, the equity premium puzzle literature would never have been born! In my discussion I will start by summarizing the results presented in the paper, and then point out some of the (yet) unanswered questions.

2. PREFERENCES WITH BENCHMARK LEVELS OF CONSUMPTION The main theme in Andy’s paper is the concept of benchmark levels of consumption. More precisely, and following the notation in the paper, the utility function of the representative investor is given by U = Et

j ∞ 1 u(ct , vt ) , 1+δ j=0

where vt denotes a benchmark level of consumption, such that 1. a higher consumption benchmark decreases utility: ∂u < 0; ∂v 2. a higher consumption benchmark increases the marginal utility of current consumption: ∂2 u > 0. ∂c∂v


160

The benchmark level of consumption, υt , can be an arbitrary function of past consumption levels (habit formation preferences, e.g., Constantinides (1990)), or it can depend on the consumption of a reference group (consumption externality, also referred to as “catching up with the Joneses,” or “external habit formation,” e.g., Abel (1990)). In this paper Andy uses a very nice general framework that allows him to nest these two formulations. In addition, the literature has typically focused on the following parametric representations: ⎧ ⎨u[ct /g(υt )] “ratio” or “multiplicative” form (e.g., Abel (1990)), u(ct , υt ) = u[ct − g(υt )] “difference” or “additive” form (e.g., Constantinides ⎩ (1990)). Andy’s paper focus exclusively on “ratio” habit models, which allows him to derive closed-form solutions for asset prices. One important feature of this formulation is that it doesn’t affect the local curvature of the utility function. For example, take the common power utility specification u[ct /g(υt )] =

[ct /g(υt )]1−α ; 1−α

(1)

then the relative risk aversion is −c

∂2 u/∂c2 = α, ∂u/∂c

just as in the “standard” model with constant υt .1 This result is visible in column 2 of Table 2 in the paper. The consumption externality model generates a modest risk premium: 1.27 percent, even with a local curvature of 10. Preferences with benchmark levels of consumption increase the agent’s willingness to smooth consumption over time, and thus (typically) decrease the riskless rate and increase the term premia. Column 2 in Table 2 shows that, through this channel, the model’s implied equity premium actually matches the data. However, this is the result of a counterfactually high term premia: 4.91 percent. In addition, the volatility of marginal utility increases (for most standard specifications of g(υt ), ct /g(υt ) is more volatile than ct ) and therefore the volatility of the riskless rate is also higher than in the data.

1 More

precisely, the model with ∂u = 0. ∂υt

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The “difference” models provide more flexibility but are usually less tractable. Let me take the power utility specification from Eq. (1) but now write it as u[ct − g(υt )] =

[ct − g(υt )]1−α . 1−α

(2)

The local risk aversion coefficient in this model is −c

∂2 u/∂c2 αct = . ct − g(υt ) ∂u/∂c

So, the agent’s risk aversion is always higher than α and, moreover, as ct falls toward g(υt ), the agent’s risk aversion coefficient increases (toward infinity).2 Naturally, since the representative agent in this model is more risk-averse, the implied risk premium is mechanically higher. This is hardly surprising and could have already been obtained by setting a high risk aversion coefficient in the “standard” model (i.e., without a benchmark level of consumption). However, since risk aversion is time-varying and countercyclical, this leads to a time-varying risk premium (and Sharpe ratio). Therefore, the degree of (average) risk aversion required to match the equity premium is not as high as in the model without a benchmark level of consumption.3 Unfortunately, Campbell and Cochrane (1999) show that this effect is quantitatively very small: the model with benchmark level of consumption still requires a very high (average) coefficient of relative risk aversion to match the observed equity premium.

3. CHANGING THE “BENCHMARK LEVEL” OF THE EXPLANATION 3.1. Aggregate Moments As successive models managed to explain the equity premium and the riskless rate, a new benchmark level of explanation was set: match the equity premium, the riskless rate, the term premium, the volatility of the riskless rate, the correlation between consumption growth and dividend growth, etc. The successive papers in this literature have tried to address these different issues. For example, Campbell and Cochrane’s (1999) model tackles a standard problem with these preferences, the high volatility of the riskless rate, by reverse-engineering a process for g(υt ) that prevents large fluctuations in marginal utility. Andy’s paper does a very good job at describing the successes and failures of these models with respect to aggregate moments. Moreover, as discussed in the introduction, these preferences have now been used to explain other asset pricing features (see Wachter (2006), Bekaert et al. (2005), or Abel (1999), this case, the standard model can be obtained by setting g(υt ) = 0. mechanism can also be obtained with the “ratio” formulation, in a model with preference heterogeneity (see Chan and Kogan (2002)). 2 In

3 This


162

among others), and in applications outside finance as well (see Carroll et al. (2000), or Furher (2000), among others).4 The latest generations of these models perform relatively well at matching several aggregate moments, although they do require a very high value of risk aversion . . . which was the basic puzzle in the first place!5 I will now argue that the latest challenges for these preferences are at the micro level.

3.2. Micro-Economic Implications The initial idea of preferences with a benchmark level of consumption was essentially a “micro-foundations”-driven concept. By simple introspection we would probably agree that once we get used to a certain consumption level we find it difficult to “cut back.” Likewise, we often see households buy certain items just because some of their neighbors or co-workers have bought them. Moreover, there is a large volume of supporting experimental and psychological evidence for these effects, and many marketing or even political campaigns are frequently based on those exact premises. It is therefore surprising that while these models have been relatively successful at matching aggregate quantities, they have performed much worse when confronted with micro-level evidence. These preferences make agents more averse to fluctuations in consumption across time and (in some cases) across states. In a model of micro-economic behavior, this implies that these agents will have a much higher demand for both precautionary and retirement savings, since they will “suffer more” from a drop in consumption. The wealth accumulation implied by these preferences is therefore counterfactually high, particularly for younger households (see Gomes and Michaelides (2003) or Polkovnichenko (2005)). In the finance literature this makes it extremely hard to explain the participation puzzle.6 Gomes and Michaelides (2003) show that agents with these preferences would have very high savings ratios, and it would take extremely unrealistic transaction/participation costs to exclude them from investing even a small fraction of that into equities. Tests for the presence of benchmark levels of consumption by estimating reduced-form Euler equations on micro-consumption data have produced mixed results. Dynan (2000) finds evidence against these preferences, while Ravina (2005) finds supporting evidence for both habits and catching up with the Joneses. In addition, cross-sectional data for individual households and investors present a special challenge for the “difference” or “additive” form of consumption externalities in Eq. (2). Under the “additive” form, the utility function must be specified to guarantee that ct − g(υt ) is positive for everyone in the cross section. Unless we allow the 4 Chapman

(1998) discusses some counterfactual macro-economic implications of these models.

5 Note that this is still an important advance relative to the “basic model,” with power utility and no benchmark

level of consumption. Such a model would also require a high value of risk aversion to match the equity premium, and even then it would fail on many other dimensions (e.g., it would deliver a counterfactually high riskless rate; Weil (1989)). 6 The participation puzzle refers to the difficulty in explaining why 50 percent of the U.S. population does not own any equity (with similar or larger numbers reported in most OECD countries).

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external benchmark υt to differ across households, this requires that g(υt ) is always smaller than the lowest level of consumption in the cross section. As a result, g(υt ) would be negligible relative to the consumption of the majority of households, and thus it would have a negligible impact on their behavior. The “ratio” model, or the “habit models,” do not suffer from this problem.

3.3. Micro-Economic Foundations and Aggregation The (majority of the) asset pricing models that use these preferences only assume that they apply to the representative investor in the economy. This is an important point because the aggregation exercise is potentially non-trivial. Even if all individual investors have these preferences, the representative investor’s utility function might take on a different form, particularly in the case of the “difference model” outlined above (Eq. (2)). On the one hand, this is actually a good defense against the previously discussed “negative micro-evidence.” These representative agent asset pricing models do not imply counterfactual micro behavior, because they are not equivalent to a model with heterogenous agents with those same preferences. However, on the other hand, such a defense immediately raises the question: “But what then is the motivation for assuming such preferences for the representative investor?” More generally, what are the micro-foundations of preferences with benchmark levels of consumption? Recent developments in the literature have proposed some potential answers, but this still remains an important open question.

4. LEVERAGE, CORRELATION BETWEEN DIVIDENDS AND CONSUMPTION, AND DISTORTED BELIEFS The remainder of Andy’s paper studies the role of levered equity claims, imperfect correlation between dividends and consumption, and distorted beliefs. I will now discuss those in sequence.

4.1. Levered Equity Claims and Correlation Between Dividends and Consumption In the Lucas (1978) model, there is only one source of wealth in the economy: the “tree” that produces the aggregate dividend flow. There is no storage technology and/or no productive sector; thus the representative agent’s consumption (ct ) must equal aggregate dividends (dt ) in every period: ct = dt

∀t .

(3)

So, in equilibrium, equity is essentially a consumption claim: every period it pays the aggregate consumption as its dividend. Abel (1999) introduces levered equity claims in

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the model by specifying instead dt = (ct )λ

∀t ,

(4)

where λ > 1. While Eq. (3) equates the volatilities of consumption and dividend growth, Eq. (4) allows them to differ. Using those two moments to calibrate λ, we find that the risk premium increases from 1.27 percent to 4.59 percent (column 3 in Table 2 of the paper). The logic is quite straightforward. The only source of priced risk in this economy is aggregate consumption risk. Both claims (the one in Eq. (4) and the one in Eq. (3)) are perfectly correlated with aggregate consumption, but since λ > 1, the second claim has a higher loading on this unique factor. However, if we use Eq. (4), then dividends are no longer equal to consumption, which now becomes an exogenous quantity in the model. This equation must therefore be viewed as a reduced form for a model that explicitly incorporates additional sources of income/wealth. A similar issue arises when calibrating the correlation between consumption growth and dividend growth, which is equal to 1, under both Eqs. (3) and (4), while in the data that number is much smaller. Relaxing this assumption also requires the introduction of additional sources of income in the model, or alternatively we can write a different reduced-form representation that will produce a lower correlation such as dt = (ct )λ + εt

∀t .

(5)

For a given volatility of dt , a lower correlation between dividends and consumption significantly decreases the risk premium, since equity is no longer perfectly correlated with aggregate risk. This is visible in column 4 of Table 2 in the paper, where the risk premium now falls from 4.59 percent to 0.92 percent when the correlation decreases from 1 to 0.2. Production economy models provide a natural framework for endogeneizing this separation between consumption and dividends, by modeling labor income that enters explicitly into the investor’s budget constraint. Jermann (1998) incorporates preferences with benchmark levels of consumption in a production economy model where, endogenously, consumption growth and dividend growth have different volatility and are no longer perfectly correlated. The results are not as strong as in the pure exchange economy model with exogenous consumption.7

4.2. Non-Rational Expectations The final part of the paper explores departures from the rational expectations paradigm, by introducing distorted beliefs. More precisely, Andy explores two sources of distortion, “pessimism” and “doubt.” Under “pessimism,” the agent’s subjective probability distribution of future payoffs (F ) is first-order stochastically dominated by the 7 For examples of exchange economy models where investors explicitly receive an exogenous labor income flow in addition to the equity dividend, see Abel (1994) or Santos and Veronesi (2006).

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objective probability distribution (F ). Under “doubt,” F is a mean-preserving spread of F . The risk premium increases with both “pessimism” and “doubt” since, for a given price, investors assign either a lower expected return to the asset or a higher volatility, respectively. Of course, important questions are how much “pessimism” and/or “doubt” is required, and are those values credible? This is the greatest hurdle for these explanations: they require measures of expectations, for which we have only proxies. Those proxies are either based on limited survey evidence or inferred indirectly from market prices and thus are conditional on a pricing model and/or estimation technique.

5. FINAL REMARKS Preferences with benchmark levels of consumption have significantly improved our ability to match asset pricing data, but they still fail to solve the essence of the puzzle: match the historical risk premium with a low coefficient of relative risk aversion. In addition, the micro-economic foundations for these preferences are not as straightforward as initially thought. Either they are the result of aggregating households with these same preferences, in which case several counterfactual micro-economic results must be explained, or the micro-foundation question is left unanswered. Introducing leverage improves the performance of the model, but less so if the separation between consumption and dividends is modeled explicitly. Non-rational expectations may be a route to explore, but more empirical work measuring real expectations is still required. Only then will a realistic calibration of those models be possible.

References Abel, A. Asset prices under habit formation and catching up with the Joneses. American Economic Review Papers and Proceedings 80 (1990): 38–42. Abel, A. Exact solutions for expected rates of return under Markov regime switching: Implications for the equity premium. Journal of Money, Credit and Banking 26(3) (1994): 345–361. Abel, A. Risk premia and term premia in general equilibrium. Journal of Monetary Economics 43(1) (1999): 3–33. Bekaert, G., E. Engstrom, and S. Grenadier. Stock and bond returns with moody investors. Working paper, Columbia University (2005). Campbell, J. Y., and J. Cochrane. By force of habit: A consumption based explanation of aggregate stock market behavior. Journal of Political Economy 107(2) (1999): 205–251. Carroll, C., J. Overland, and D. Weil. Saving and growth with habit formation. American Economic Review 90(3) (2000): 341–355. Chan, Y. L., and L. Kogan. Heterogeneous preferences, catching up with the Joneses and the dynamics of asset prices. Journal of Public Economics 110(6) (2002): 1255–1285. Chapman, D. Habit formation and aggregate consumption. Econometrica 66 (1998): 1223–1230. Constantinides, G. Habit formation: A resolution of the equity premium puzzle. Journal of Political Economy 98 (1990): 519–543. Duesenberry, J. Income, Saving, and the Theory of Consumer Behavior, Harvard University Press, Cambridge (1949).

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Chapter 4 • Equity Premia with Benchmark Levels of Consumption: Closed-Form Results Dynan, K. Habit formation in consumer preferences: Evidence from panel data. American Economic Review 90 (2000): 391–406. Fuhrer, J. C. An optimizing model for monetary policy: Can habit formation help? American Economic Review 90(3) (2000): 367–390. Gomes, F., and A. Michaelides. Portfolio choice with internal habit formation: A life cycle model with uninsurable labor income risk. Review of Economic Dynamics 6 (2003): 729–766. Jermann, U. Asset pricing in production economies. Journal of Monetary Economics 41 (1998): 257–275. Lucas, R., Jr. Asset prices in an exchange economy. Econometrica 46(6) (1978): 1429–1445. Mehra, R., and E. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15(2) (1985): 145–161. Polkovnichenko, V. Life cycle consumption and portfolio choice with additive habit formation preferences and uninsurable labor income risk. Review of Financial Studies (2005). Ravina, E. Habit formation and keeping up with the Joneses: Evidence from micro data. Working paper, Stern School of Business (2005). Santos, T., and P. Veronesi. Labor income and predictable stock returns. Review of Financial Studies 19 (2006): 1–44. Wachter, J. A consumption-based model of the term structure of interest rates. Journal of Financial Economics 79 (2006): 365–399. Weil, P. The equity premium and the risk-free rate puzzle. Journal of Monetary Economics 24 (1989): 401–421.

CHAPTER 5

Long-Run Risks and Risk Compensation in Equity Markets Ravi Bansal∗ Fuqua School of Business, Duke University, and NBER

1. Introduction 2. Long-Run Risks Model 2.1. Preferences and the Environment 2.2. Long-Run Growth Rate Risks 2.3. Long-Run Growth and Uncertainty Risks 2.4. Data and Model Implications 3. Cross-Sectional Implications 3.1. Value, Momentum, Size, and the Cross-Sectional Puzzle 4. Conclusion References

168 170 170 171 174 176 185 185 191 191

Abstract What drives the compensation in equity markets? This article shows that long-run growth and economic uncertainty in the economy play an important role in determining the risk in equity markets. The size of the market risk premium, the level of the risk-free rate, the volatility of asset prices, and differences in the risk compensation across assets are shown to be related to risks pertaining to the long-run growth and uncertainty in the economy. JEL Classification: G12, E44 Keywords: long-run risks, volatility risks, intertemporal elasticity of substitution ∗I

thank Amir Yaron, Dana Kiku, John Heaton, and Lars Hansen, and the Editor, Rajnish Mehra, for their comments. The usual disclaimer applies. HANDBOOK OF THE EQUITY RISK PREMIUM c 2008 by Elsevier B.V. All rights of reproduction in any form reserved. Copyright

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Chapter 5 • Long-Run Risks and Risk Compensation in Equity Markets

1. INTRODUCTION Several aspects of asset markets are puzzling. The work of Mehra and Prescott (1985) shows that the magnitude of risk compensation in equity markets is a puzzle—Treasury bills offer a return of about 1 percent per annum while the equity market portfolio offers 7.5 percent. What risks justify such a sizable compensation for holding equity? Another equally puzzling, and related, dimension is the large difference in the average returns across equity portfolios. For example, the return to value firms exceeds that of growth firms by about 7 percent per annum. In addition to these return puzzles, Shiller (1981) and Leroy and Porter (1981) document the volatility puzzle—it is hard to explain the high volatility of equity prices. This article highlights the ideas developed in Bansal and Lundblad (2002), Bansal and Yaron (2004), Bansal, Khatchatrian, and Yaron (2005), Bansal, Dittmar, and Lundblad (2002, 2005), and Bansal, Gallant, and Tauchen (2007) pertaining to various financial market anomalies. This research argues that the magnitudes of asset returns and volatility are a natural outcome of risks associated with the long-run growth prospects and changing economic uncertainty in the economy. Bansal and Yaron (2004) argue that investors care about the long-run growth prospects and the level of economic uncertainty. Changes in these fundamentals drive the risks and volatility in asset prices. They document that consumption and dividend growth rates contain a small long-run component. That is, current shocks to expected growth alter expectations about future economic growth not only for short horizons but also for the very long run. Agents care a lot about these long-run components as small revisions in them lead to large changes in asset prices. Any adverse movements in the long-run growth components lower asset prices and concomitantly the wealth and consumption of investors. This makes holding equity very risky for investors, making them demand a high equity risk compensation. Bansal and Yaron further argue that time variation in expected excess returns is due to variation in economic uncertainty. They model this uncertainty by incorporating time-varying consumption volatility in the consumption process. Empirical motivation for this channel is provided by Bansal, Khatchatrian, and Yaron (2005)—their robust empirical finding is that current consumption volatility predicts future asset valuations and that current asset valuations predict future consumption volatility. Both projection coefficients are significantly negative. A rise in economic uncertainty lowers asset prices—that is, asset markets dislike economic uncertainty. Bansal and Yaron derive the result that volatility shocks, in equilibrium, carry a separate risk premium; this is a novel feature of their model. Epstein and Zin (1989) preferences play an important role in the Bansal and Yaron model. These preferences allow for separation between risk aversion and the intertemporal elasticity of substitution (IES) of investors. An IES larger than one is required for the wealth to consumption ratio to rise with expected consumption growth; when the IES is smaller than one, high expected growth lowers the wealth to consumption ratio. The equity price to dividend ratio mirrors the behavior of the wealth to consumption ratio. This ensures that equity payoffs are high when consumption and corporate profits rise, leading to a positive equity risk premium. An IES larger than one is also required to capture the data feature that asset markets dislike economic uncertainty.

Ravi Bansal

169 The magnitude of the IES is a key empirical issue. Hansen and Singleton (1982), Attanasio and Weber (1989), and Attanasio and Vissing-Jorgensen (2003) estimate the IES to be well in excess of one. Hall (1988) and Campbell (1999), on the other hand, estimate its value to be well below one. Bansal and Yaron argue that the estimates for the IES in Hall and Campbell are not a robust guide for the magnitude of the IES parameter. They show that even if the population value of the IES is larger than one, the estimation methods used by Hall would measure the IES to be close to zero. That is, there is a severe downward bias in the point estimates of the IES. Bansal and Yaron as well as Bansal, Khatchatrian, and Yaron (2005) further argue that the economic implications when the IES is less than one—a rise in consumption volatility and/or a drop is expected growth raises asset valuations, are counterfactual, making the low magnitude of the IES suspect. The arguments presented in Bansal and Yaron also have immediate implications for the cross-sectional differences in mean returns across assets. Firms whose expected cash-flow (profits) growth rates move with the economy are more exposed to longrun risks and hence should carry a higher risk compensation. In Breeden (1979), Lucas (1978), and Hansen and Singleton (1982), the riskiness of the asset is determined by the consumption beta of the asset. However, the consumption beta of an asset is not exogenous—in equilibrium, it is determined by the systematic risks in cash flows and the preference parameters of the representative agent. That is, cross-sectional differences in betas of assets reflect differences in the systematic risks in cash flows. Risks in cash flows should consequently contain information about differences in mean returns across assets. We review the work of Bansal, Dittmar, and Lundblad (2002, 2005), who show that systematic risks in cash flows can account for the cross-sectional differences in risk premia of assets. Specifically, their cash-flow betas can account for the puzzling value, size, and the momentum spread in the cross section of assets. Bansal and Lundblad (2002) rely on long-run components and varying risk premia to address issues in international equity markets. Developed markets’ asset prices and returns show a high degree of correlation; however, dividends and earnings growth across these economies are virtually uncorrelated. Bansal and Lundblad show that high asset price volatility and correlation across national equity markets are due to the long-run component in dividend growth rates and time-varying systematic risk. The argument that long-term economic growth and uncertainty are the key drivers of risks in equity markets is distinct from the arguments presented in Campbell and Cochrane (1999). They argue that equity market risks are driven largely (even exclusively) by fluctuations in the ex-ante rate of discount (cost of capital) through external habit formation. Sorting out which of the channels is critical for explaining the risk compensation in equity markets, consequently, is largely an empirical issue. Using the Efficient Method of Moments (EMM) estimation technique, Bansal, Gallant, and Tauchen (2007) document the differences between the Bansal and Yaron and Campbell and Cochrane models. A unique dimension of the Bansal, Gallant, and Tauchen paper is that consumption and dividends in their model are cointegrated—this feature is typically missing in earlier work on asset market models. The remainder of the article has three sections. Section 2 discusses the long-run risks model of Bansal and Yaron. Section 3 discusses the issue of cross-sectional differences in returns across asset portfolios. Section 4 presents concluding comments.

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2. LONG-RUN RISKS MODEL 2.1. Preferences and the Environment We consider a representative agent with the generalized preferences developed in Epstein and Zin (1989). The logarithm of the Intertemporal Marginal Rate of Substitution (IMRS), mt+1 , for these preferences, as shown in the Epstein and Zin paper, is mt+1 = θ log δ −

θ gt+1 + (θ − 1)ra,t+1 . ψ

(1)

If Rt+1 is the gross return on the asset, then log (Rt+1) ≡ rt+1 is the continuous return on the asset. Using the standard asset pricing restriction for any continuous return ri,t+1 , it follows that1 θ Et exp θ log δ − gt+1 + (θ − 1)ra,t+1 + ri,t+1 = 1, ψ

(2)

where gt+1 equals log(Ct+1 /Ct )—the log growth rate of aggregate consumption. The return, ra,t+1 , is the log of the return (i.e., continuous return) on an asset that delivers aggregate consumption as its dividends each time period. The time discount factor is δ and the parameter θ ≡ 1 − γ/(1 − 1/ψ), where γ ≥ 0 is the risk aversion (sensitivity) parameter, and ψ ≥ 0 is the intertemporal elasticity of substitution. The sign of θ is determined by the magnitudes of the risk aversion and the elasticity of substitution.2 Note that when θ equals one, the above IMRS and the asset pricing implications collapse to the usual case of power utility considered in Mehra and Prescott (1985). The return to the aggregate consumption claim, ra,t+1 , is not observed in the data while the return on the dividend claim corresponds to the observed return on the market portfolio rm,t+1 . The levels of market dividends and consumption are not equal; aggregate consumption is much larger than aggregate dividends. The difference is financed by labor income. In the model, aggregate consumption and aggregate dividends are treated as two separate processes and the difference between them implicitly defines the agent’s labor income process.

1 Note

that the standard asset pricing condition in frictionless markets is Et [exp(mt+1 + rt+1)] = 1,

where the intertemporal marginal rate of substitution is Mt+1 and log(Mt+1 ) ≡ mt+1 . The logarithm of the gross return for an asset equals rt+1 . 2 In particular, if ψ > 1 and γ > 1, then θ will be negative. Note that when θ = 1, that is, γ = (1/ψ), the above recursive preferences collapse to the standard case of expected utility. Further, when θ = 1 and in addition γ = 1, we get the standard case of log utility.

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The key ideas of the model are developed and the intuition is provided via approximate analytical solutions. However, all the quantitative results reported in the paper are based on numerical solutions of the model. To derive the approximate analytical solutions for the model, we use the standard first-order Taylor series approximation developed in Campbell and Shiller (1988),3 ra,t+1 = κ0 + κ1 zt+1 − zt + gt+1 ,

(3)

where lowercase letters refer to variables in logs, in particular, ra,t+1 = log(Ra,t+1 ) is the continuous return on the consumption claim, and zt ≡ log (Pt /Ct ) is the log price to consumption ratio. Analogously, rm,t+1 and zm,t correspond to the continuous return on the dividend claim and its log price-dividend ratio. As Pt + Ct /Ct is the agent’s wealth to consumption ratio, fluctuations in the price to consumption ratio, consequently, also correspond to movements in the wealth to consumption ratio. Parameters κ0 and κ1 are approximating constants that both depend only on the average level of z.4 From Eq. (1) it follows that the innovation in the IMRS, mt+1 , is driven by the innovations in gt+1 and ra,t+1 . Covariation with the innovation in mt+1 determines the risk premium for any asset. The simpler model specification, with only long-run growth rate risks, is discussed first. The full model that incorporates long-run growth rate and economic uncertainty risks is presented after that.

2.2. Long-Run Growth Rate Risks The agents’ IMRS depends on the endogenous consumption return, ra,t+1 . The risk compensation on all assets depends on this return, which itself is determined by the process for consumption growth. The dividend process is needed for determining the return on the market portfolio. To capture long-run risks, consumption and dividend growth rates, gt+1 and gd,t+1 , are modeled to contain a small persistent predictable component xt, xt+1 = ρxt + ϕe σet+1 , gt+1 = μ + xt + σηt+1 , gd,t+1 = μd + φxt + ϕd σut+1 , et+1 , ut+1 , ηt+1 ∼ N.i.i.d.(0, 1), 3 Any

(4)

equity return can be written as

Pt+1 Dt+1 Dt+1 Rt+1 = . Pt Dt Dt The approximate return expression follows from taking the log of the gross return and then taking a first-order Pt+1 Taylor series approximation of log 1 + D around the average value of the log of the price-dividend ratio, t+1 which is referenced as z in Eq. (3). The approximating constants κ0 and κ1 are solely determined by the average value of z. 4 The Campbell–Shiller approximate return follows from using a first-order Taylor series expansion of the ¯ + exp(z)). ¯ The value of κ1 is set at 0.997, which is consistent continuous return. Note that κ1 = exp(z)/(1 with the magnitude of z¯ in our sample and with the magnitudes used in Campbell and Shiller (1988). 1+

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with the three shocks, et+1 , ut+1 , and ηt+1 , assumed to be mutually independent. The volatility of the consumption growth rate innovation is σ. Similarly, the volatility of the innovation in xt+1 and gd,t+1 is ϕe σ and ϕd σ, respectively. The parameter ρ determines the persistence of the expected growth rate process. First, note that when ϕe = 0, the processes gt and gd,t+1 are i.i.d. Second, if et+1 = ηt+1 , the process for consumption is the ARMA(1,1) used in Campbell (1999), Cecchetti, Lam, and Mark (1993), and Bansal and Lundblad (2002). If, in addition, ϕe = ρ, then consumption growth corresponds to an AR(1) process used in Mehra and Prescott (1985). Two additional parameters, φ > 1 and ϕd > 1, calibrate the overall volatility of dividends and its correlation with consumption. The parameter φ can be interpreted, as in Abel (1999), as the leverage ratio on expected consumption growth. Alternately, this says that corporate profits, relative to consumption, are more sensitive to changing expected economic growth conditions. That is, any fluctuation in xt leads to larger changes in expected dividend growth relative to expected consumption growth. The maintained assumption is that the three innovations are uncorrelated. It is straightforward to allow the three shocks to be correlated; however, to maintain parsimony in the number of parameters, they are assumed to be independent. Note that consumption and dividends are not cointegrated in the above specification—Bansal, Gallant, and Tauchen (2007) develop a specification that does allow for cointegration between consumption and dividends. Asset prices reflect expectations of future growth rates. To develop some intuition about long-run risks, consider the quantity

Et

∞

j κ1 gt+j ,

j=1

with κ1 less than one, this expectation equals κ1 xt /(1 − κ1 ρ). Even if the variance of x is tiny, but ρ fairly high, then shocks to x can alter growth rate expectations for the long run, leading to volatile asset prices. Bansal and Lundblad (2002) and Bansal and Yaron (2004) provide empirical support for the existence of this long-run component in observed growth rates.

2.2.1. Equilibrium and Asset Prices The consumption and dividend growth rates processes are exogenous in this endowment economy. Further, the IMRS depends on an endogenous return ra,t+1 . To characterize the IMRS and the behavior of asset returns, a solution for the log price to consumption ratio zt and the log price-dividend ratio zm,t is needed. The relevant state variable for deriving the solution for zt and zm,t is the expected growth rate of consumption xt . Exploiting the Euler equation (2), the approximate solution for the log priceconsumption zt has the form zt = A0 + A1 xt . An analogous expression holds for

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the log price-dividend ratio zm,t .5 Bansal and Yaron (2004) show that the solution coefficients are A1 =

1−

1 ψ

1 − κ1 ρ

,

A1,m =

φ−

1 ψ

1 − κ1,m ρ

.

(5)

It follows that A1 is positive if the IES, ψ, is greater than one. In this case the intertemporal substitution effect dominates the wealth effect. In response to higher expected growth, agents buy more assets, and consequently the wealth to consumption ratio rises. The level of consumption rises due to a rise in expected growth; however, wealth rises more than consumption. In the standard power utility model with risk aversion larger than one, the IES is less than one, and hence A1 is negative—a rise in expected growth potentially lowers asset valuations. That is, the wealth effect dominates the substitution effect.6 Corporate payouts (i.e., dividends), with φ > 1, are more sensitive to long-run risks (i.e., A1,m > A1 ), and changes in expected growth rate lead to a larger reaction in the price of the dividend claim than in the price of the consumption claim. Equation (3), the solution in (5), and the dynamics for the consumption and dividend growth rates provide a complete characterization for the endogenous returns on the consumption and the dividend asset.

2.2.2. Pricing of Long-Run Growth Risks Substituting the equilibrium return for ra,t+1 into the IMRS, Bansal and Yaron show that the innovation in mt+1 is

θ 1 ϕe σet+1 − θ + 1 σηt+1 − (1 − θ) κ1 1 − mt+1 − Et (mt+1 ) = − ψ ψ 1 − κ1 ρ = −λm,η σηt+1 − λm,e σet+1 .

(6)

The parameters λm,e and λm,η determine the risk compensation for expected growth rate shock and the independent consumption shock ηt+1 . The risk compensation for the ηt+1 shocks is very standard as λm,η equals the risk aversion parameter γ.7 In addition, with power utility, that is, when θ equals one, λm,e = 0. Long-run risks are priced only when θ differs from one, that is, when risk aversion is not the reciprocal of the IES—this highlights the importance of the generalized preferences of Epstein and Zin (1989). The market price of long-run risks is sensitive to the magnitude of the permanence parameter ρ. The risk compensation for 5 The expression for the intercept terms A for the valuation ratio for the consumption claim, and A 0 0,m for the dividend claim are not important for our qualitative results. 6 An alternative interpretation with the power utility model is that higher expected growth rates increase the risk-free rate to an extent that discounting dominates the effects of higher expected growth rates. This leads to a fall in asset prices. 7 This follows by substituting the expression for θ and simplifying the expression [θ/ψ − θ + 1].

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long-run risks, λm,e , rises as the permanence parameter ρ rises. The conditional volatility of the pricing kernel is constant, as all risk sources have constant conditional variances. As asset returns and the pricing kernel in this model economy are conditionally log-normal, the risk premium on any asset i is Et [ri,t+1 − rf ,t ] = −covt (mt+1 , ri,t+1 ) − 0.5σr2i ,t . Given the solutions for A1 and A1,m , it is straightforward to derive the equity premium on the market portfolio, E(rm,t+1 − rf ,t ) = βm,η λm,η σ 2 + βm,e λm,e σ 2 − 0.5Var(rm,t ). The market portfolio’s beta with respect to the long-run risk component is

ϕe 1 . βm,e ≡ κ1,m φ − ψ 1 − κ1,m ρ

(7)

(8)

The exposure of the market return to long-run risk is βm,e , and the market price of the long-run risk is λm,e . The expressions for these variables reveal that a rise in ρ increases both βm,e and λm,e . Consequently, the risk premium on the asset also increases with ρ. The market portfolio’s beta with respect to the short-run risk component ηt+1 is determined by the exposure of the dividend’s innovation to ηt+1 . The assumption that the dividend innovation of the market portfolio is independent of the short-run shock in consumption ηt+1 implies that βm,η will be zero in our calibration exercise. As all shocks have a constant conditional variance, the conditional risk premium on the market portfolio and its conditional volatility are constant. The ratio of the two, namely the Sharpe ratio, is also constant. In order to address issues that pertain to timevarying risk premia and predictability of risk premia, Bansal and Yaron augment the above model by incorporating time-varying economic uncertainty.

2.3. Long-Run Growth and Uncertainty Risks Bansal and Yaron model fluctuating economic uncertainty as time-varying volatility of consumption growth. The consumption and dividends dynamics that incorporate stochastic volatility are xt+1 = ρxt + ϕe σt et+1 , gt+1 = μ + xt + σt ηt+1 , gd,t+1 = μd + φxt + ϕd σt ut+1 , 2 σt+1

(9)

− σ ) + σw wt+1 , =σ + et+1 , ut+1 , ηt+1 , wt+1 ∼ N.i.i.d.(0, 1), 2

ν1 (σt2

2

where σt+1 , the conditional volatility of consumption growth, represents the timevarying economic uncertainty incorporated in consumption growth rate. The unconditional mean of the time-varying variance of consumption growth is σ 2 , and σw determines the volatility of shocks to consumption uncertainty. The parameter ν1 determines the persistence of shocks to consumption variance. To maintain parsimony, it is assumed that the shocks are uncorrelated and only one source of time-varying economic uncertainty affects consumption and dividends.

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The relevant state variables in solving for the equilibrium price-consumption (and price-dividend) ratio are now xt and σt2 . Thus, the approximate solution for the price-consumption ratio is zt = A0 + A1 xt + A2 σt2 . The solution for A1 is unchanged (Eq. (5)). The solution coefficient A2 for measuring the sensitivity of the priceconsumption ratio to volatility fluctuations is 2 2 θ 0.5 θ − ψ + (θA1 κ1 ϕe ) A2 = . (10) θ(1 − κ1 ν1 ) An analogous coefficient for the market price-dividend ratio, A2,m , is provided in Bansal and Yaron (2004). The expression for A2 provides two valuable insights. First, if the IES and risk aversion are larger than one, then θ and consequently A2 are negative. In this case, a rise in consumption volatility lowers asset valuations and increases the risk premia on all assets. To capture the intuition that a rise in economic uncertainty lowers asset valuations requires that the IES be larger than one. Bansal, Khatchatrian, and Yaron (2005) present robust empirical evidence that asset markets dislike economic uncertainty—that is, A2 is negative. This empirical evidence, given the expression for A2 , has a direct bearing on the plausible magnitude for the IES. Second, an increase in the permanence of volatility shocks, that is ν1 , magnifies the effects of volatility shocks on valuation ratios as changes in economic uncertainty are perceived by investors as being long-lasting.

2.3.1. Pricing of Uncertainty Risks As the wealth to consumption ratio is affected by consumption volatility shocks, so are the return ra,t+1 and the IMRS. Specifically, the innovation in mt+1 is mt+1 − Et (mt+1 ) = −λm,η σt ηt+1 − λm,e σt et+1 − λm,w σw wt+1 ,

(11)

where λm,η , λm,e , and λm,w are the market prices of risks for the short-run, long-run, and volatility risks. The market prices of systematic risks, including the compensation for stochastic volatility risk in consumption, can be expressed in terms of underlying preferences and parameters that govern the evolution of consumption growth as λm,η = γ,

1 κ1 ϕ e , λm,e = γ − ψ 1 − κ1 ρ

κ1 (1 + ( κ1 ϕe )2 ) 1 1−κ1 ρ . (1 − γ) λm,w = γ − ψ 2 (1 − κ1 ν1 )

(12)

Expression (11) is similar to the earlier model (see Eq. (6)) save for the inclusion of wt+1 , the shocks to consumption volatility. In the special case of power utility, when

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θ = 1 or more specifically, where γ = 1/ψ, the risk compensation parameters λm,e and λm,w are zero. The long-run risks and volatility risks are not reflected in the innovation of the pricing kernel. With power utility there is no separate risk compensation for longrun growth rate risks and volatility risks—with Epstein and Zin preferences, both risks are priced. The pricing of long-run and volatility risks is an important and novel feature of the Bansal and Yaron model. The equity premium in the presence of time-varying economic uncertainty is Et (rm,t+1 − rf ,t ) = βm,η λm,η σt2 + βm,e λm,e σt2 + βm,w λm,w σw2 − 0.5Vart (rm,t+1 ),

(13)

where βm,w ≡ κ1,m A2,m . The first β corresponds to the exposure to short-run risks, and the second to long-run risks. The last beta, βm,w , is the exposure of the asset to volatility risks. The risk premium on the market portfolio is time-varying as σt fluctuates. The ratio of the conditional risk premium to the conditional volatility of the market portfolio fluctuates with σt , and hence the Sharpe ratio is time-varying. The maximal Sharpe ratio in this model economy, which approximately equals the conditional volatility of the log IMRS, also varies with σt .8 This implies that during periods of high economic uncertainty all risk premia will rise. The first-order effects on the level of the risk-free rate, as discussed in Bansal and Yaron (2006), are the rate of time preference and the average consumption growth rate, divided by the IES. Increasing the IES keeps the level low. The variance of the risk-free rate is determined by the volatility of the expected consumption growth rate and the IES. Increasing the IES lowers the volatility of the risk-free rate. In addition, incorporating economic uncertainty leads to an interesting channel for interpreting fluctuations in the real risk-free rate. In particular, Bansal and Yaron show that this has serious implications for the measurement of the IES in the data. In the presence of varying volatility, the estimates of the IES based on the projections considered in Hall (1988) and Campbell (1999) are seriously biased downwards.

2.4. Data and Model Implications 2.4.1. Data and the Growth Rate Dynamics Bansal and Yaron (2004) calibrate the model described in (4) and (9) at the monthly frequency. From this monthly model they derive time-aggregated annual growth rates of consumption and dividends to match key aspects of annual aggregate consumption and dividends data. Further, as in Campbell and Cochrane (1999) and in Kandel and Stambaugh (1991), they assume that the decision interval of the agent is monthly, but the targeted data to match is annual.9 8 Given the conditional normality of the logarithm of the IMRS, the maximal Sharpe ratio is simply the conditional standard deviation of the logarithm of the IMRS. 9 The evidence regarding the model is based on numerical solutions using standard polynomial-based projection methods discussed in Judd (1998). The numerical results are quite close to those based on the approximate analytical solutions.

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For consumption, BEA data on real per capita annual consumption growth of non-durables and services for the period 1929–1998 is utilized. This is the longest single data source of consumption data. Dividends and the value-weighted market return data are taken from the CRSP. All nominal quantities are deflated using the CPI. To facilitate comparisons between the model, which is calibrated to a monthly decision interval, and the annual data, the monthly model is time-aggregated to the annual frequency to derive annual statistics. The annual real per capita consumption growth mean is 1.8 percent, and its standard deviation is about 2.9 percent. This volatility is somewhat lower for our sample than for the period considered in Mehra and Prescott (1985), Kandel and Stambaugh (1991), and Abel (1999). Table 1, adapted from Bansal and Yaron (2004), shows that in the data, consumption growth has a large first-order autocorrelation coefficient and a small second-order one. The standard errors in the data for these autocorrelations are sizeable. An alternative way to view the long horizon properties of the consumption and dividend growth rates is to use variance ratios, which themselves are determined by the autocorrelations (see Cochrane (1988)). In the data the variance ratios first rise significantly and at about 7 years start to decline. The standard errors on these variance ratios, not surprisingly, are quite substantial. There is considerable evidence of small sample biases in estimating autoregression coefficients and variance ratios (see Hurwicz (1950) and Ansley and Newbold (1980)). To account for any small sample biases, Bansal and Yaron also report statistics based on 1000 Monte Carlo experiments TABLE 1 Time-Series Properties of Data Variable

Estimate

S.E.

σ(g)

2.93

(0.69)

AC(1)

0.49

(0.14)

AC(2)

0.15

(0.22)

AC(5)

−0.08

(0.10)

AC(10)

0.05

(0.09)

V R(2)

1.61

(0.34)

V R(5)

2.01

(1.23)

V R(10)

1.57

(2.07)

11.49

(1.98)

σ(gd ) AC(1)

0.21

(0.13)

corr(g, gd )

0.55

(0.34)

Table 1 displays the time-series properties of aggregate consumption and dividend growth rates: g and gd , respectively. The statistics are based on annual observations from 1929 to 1998. Consumption is real per capita consumption of non-durables and services; dividends are the sum of real dividends across all CRSP firms. AC(j) is the jth autocorrelation, VR(j) is the jth variance ratio, σ is the volatility, and corr denotes the correlation. Standard errors are Newey and West (1987) corrected using 10 lags.

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each with 840 monthly observations corresponding to the 70 annual observations available in the annual data set. In terms of the specific parameters, Bansal and Yaron (2004) calibrate ρ at 0.979, which determines the persistence in the long-run component in growth rates. Their choice of ϕe and σ ensures that the model matches the unconditional variance and the autocorrelation function of annual consumption growth. The standard deviation of the one-step-ahead innovation in consumption, that is σ, equals 0.0078. This parameter configuration implies that the predictable variation in monthly consumption growth is very small, as the implied R2 is only 4.4 percent. The exposure of the corporate sector to long-run risks is governed by φ, and its magnitude is similar to that in Abel (1999). The standard deviation of the monthly innovation in dividends, ϕd σ, is 0.0351. Bansal and Yaron also consider the consumption and dividend dynamics that incorporate time-varying volatility (see Eq. (9)). The parameters of the volatility process are chosen to capture the persistence in consumption volatility. Based on the evidence of slow decay in volatility shocks, they calibrate ν1 , the parameter governing the persistence of conditional volatility, at 0.987. The shocks to the volatility process have very small volatility, and σw is calibrated at 0.23 × 10−5 . Bansal and Yaron show that with this configuration, the assumed consumption and dividend growth rates very closely match the key consumption and dividends data features reported in Table 1. Table 2 presents the targeted asset market data for 1929 to 1998. The equity risk premium is 6.33 percent per annum, and the real risk-free rate is 0.9 percent. The annual market return volatility is 19.42 percent, and that of the real risk-free is quite small, TABLE 2 Asset Market Data Variable

Estimate

S.E.

Returns E(rm − rf )

6.33

(2.15)

E(rf )

0.86

(0.42)

σ(rm )

19.42

(3.07)

σ(rf )

0.97

(0.28)

Price-dividend ratio E(exp(p − d))

26.56

(2.53)

σ(p − d)

0.29

(0.04)

AC1(p − d)

0.81

(0.09)

AC2(p − d)

0.64

(0.15)

Table 2, adapted from Bansal and Yaron (2004), presents descriptive statistics of asset market data. E(rm − rf ) and E(rf ) are, respectively, the annualized equity premium and mean risk free-rate. σ(rm ), σ(rf ), and σ(p − d) are the annualized volatilities of the market return, the risk-free rate, and the log price-dividend, respectively. AC1 and AC2 denote the first and second autocorrelations. Standard errors are Newey and West (1987) corrected using 10 lags.

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about 1 percent per annum. The volatility of the price-dividend ratio is quite high, and it is a very persistent series. In addition to these data dimensions, Bansal and Yaron also evaluate the ability of the model to capture the predictability of returns and the new evidence (see Bansal, Khatchatrian, and Yaron (2005)) that price-dividend ratios are negatively correlated with consumption volatility at long leads and lags. It is often argued that consumption and dividend growth, in the data, is close to being i.i.d. Bansal and Yaron show that their model of consumption and dividends is also consistent with the observed data on consumption and dividends growth rates. However, while the financial market data is hard to interpret from the perspective of the i.i.d. growth rate dynamics, Bansal and Yaron show that it is interpretable from the perspective of the growth rate dynamics that incorporate long-run risks. This issue is further considered in Shephard and Harvey (1990), Barsky and Delong (1993), and Bansal and Lundblad (2002), who show that discrimination across the i.i.d. growth rate specification and the one that incorporates long-run components is extremely difficult in finite samples. Given these difficulties in discrimination across models, Anderson, Hansen, and Sargent (2003) utilize features of the long-run growth rate dynamics developed in Bansal and Yaron for motivating economic models that incorporate robust control.

2.4.2. Preference Parameters The preference parameters take account of economic considerations. The time preference parameter δ < 1 and the risk aversion parameter γ in Bansal and Yaron is either 7.5 or 10. Mehra and Prescott (1985) do not entertain risk aversion values larger than 10. Bansal and Yaron focus on an IES of 1.5—an IES value larger than one is important for their quantitative results. There is considerable debate about the magnitude of the IES. Hansen and Singleton (1982) and Attanasio and Weber (1989) estimate the IES to be well in excess of 1. More recently, Guvenen (2006) and Attanasio and Vissing-Jorgensen (2003) also estimate the IES over one—they show that their estimates are close to that used in Bansal and Yaron. However, Hall (1988) and Campbell (1999) estimate the IES to be well below one. Bansal and Yaron (2004) argue that the low IES estimates of Hall and Campbell are based on a model without time-varying volatility. They show that ignoring the effects of time-varying consumption volatility leads to a serious downwards bias in the estimates of the IES. If the population value of the IES in the Bansal and Yaron model is 1.5, then the estimated value of the IES using Hall estimation methods will be less than 0.3. Bansal and Yaron show that with fluctuating consumption volatility, the projection of consumption growth on the level of the risk-free rate does not equal the IES, leading to the downwards bias. This suggests that Hall and Campbell’s estimates are not a robust guide for calibrating the IES. In addition to the above arguments, the empirical evidence in Bansal, Khatchatrian, and Yaron (2005) shows that a rise in consumption volatility sharply lowers asset prices at long leads and lags and that higher asset valuations today predict higher corporate earnings growth. Figures 1 to 4 use data from the U.S., the U.K., Germany, and Japan

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to highlight the volatility channel. The asset valuation measure is the price to earnings ratio, and the consumption volatility measure is constructed by averaging eight lags of the absolute value of consumption residuals; see Bansal, Khatchatrian, and Yaron (2005) for additional details. It is evident from the graphs that a rise in consumption volatility lowers asset valuations for all counties under consideration—this highlights the volatility channel and motivates the specification of the IES larger than one. In terms of growth rate predictability, Ang and Bekaert (2007) and Bansal, Khatchatrian, and Yaron (2005) report a positive relation between asset valuations and expected earnings growth. These data features, as discussed in the theory sections above, again require an IES larger than one.

2.4.3. Asset Pricing Implications To underscore the importance of two key aspects of the model, preferences and longrun risks, first consider the genesis of the risk premium on ra,t+1 —the return on the asset that delivers aggregate consumption as its dividends. The market risk premium magnifies these risk compensations due to leverage and, consequently, larger exposure to the various risk components.

3 2 1 0 21 22 23 55

60

65

70

75

Cons. Volatility (12 qrt.) FIGURE 1

80

85

90

95

log price-earnings ratio

P/E ratio and consumption volatility for the U.S. Both series are standardized.

(Source: Bansal, Khatchatrian, and Yaron (2005)).

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Ravi Bansal

3 2 1 0 21 22 23 76

78

80

82

84

86

88

92

94

96

98



90

P/E ratio and consumption volatility for the U.K. Both series are standardized.

3 2 1 0 21 22 23 76

78

80

82

84

86


88

90

92

94

96

98


P/E ratio and consumption volatility for Germany. Both series are standardized.

182


3

2

1

0

21

22 76

78

80

82

84

86


88

90

92

94

96

98


P/E ratio and consumption volatility for Japan. Both series are standardized.

Table 3 shows the market price of risk and the breakdown of the risk premium from various risk sources. Column 1 considers the case of power utility as the IES equals the reciprocal of the risk aversion parameter. The prices of long-run risks and economic uncertainty are zero—with power utility, long-run risks and volatility risks are not priced separately. In this case, the risk premium on the consumption asset equals γσ 2 and is 0.7 percent per annum—that is, only short-run risks are priced. Column 2 of Table 3 considers the case of Epstein and Zin preferences with an IES less than one (set at 0.5). The price of long-run growth rate risks is positive and negative for volatility risks. However, the consumption asset’s beta for the long-run risks is negative. This, as discussed earlier, is because A1 is negative (see Eq. (5)), implying that a rise in expected growth lowers the wealth to consumption ratio. Consequently, long-run risks in this case contribute a negative risk premium of −1.96 percent per annum. The market price of volatility risk is negative and small; however, the asset’s beta for this risk source is large and positive, reflecting the fact that asset prices rise when economic uncertainty rises (see Eq. (10)). In all, when the IES is less than one, the risk premium on the consumption asset is negative, which is highly counterintuitive. Column 3 of Table 3 shows that when the IES is larger than one (set at 1.5), the price of long-run growth risk rises. More importantly, the asset’s beta with respect to the long-run growth risk is positive and that for volatility risk is negative—hence, both risk sources contribute toward a positive risk premium. The risk premium from long-run growth is 0.76 percent and that for the short-run consumption shock is 0.73 percent. The overall risk premium for this consumption asset is 1.52 percent. This evidence shows

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TABLE 3 Risk Components and Risk Compensation ψ = 0.1

ψ = 0.5

ψ = 1.5

mprη

93.60

93.60

93.60

mpre

0.00

137.23

160.05

mprw

0.00

−27.05

−31.56

βη

1.00

1.00

1.00

βe

−16.49

−1.83

0.61

βw

11026.45

1225.16

−408.39

prmη

0.73

0.73

0.73

prme

0.00

−1.96

0.76

prmw

0.00

−0.08

0.03

Table 3 presents model-implied components of the risk premium on the consumption asset for different values of the intertemporal elasticity of the substitution parameter, ψ. All entries are based on γ = 10. The parameters that govern the dynamics of the consumption process in Eq. (9) are identical to Bansal and Yaron (2004): ρ = 0.979, σ = 0.0078, ϕe = 0.044, ν1 = 0.987, σw = 0.23 × 10−5 , and κ1 = 0.997. The first three rows report the annualized percentage prices of risk for innovations in consumption, the expected growth risk, and the consumption volatility risk—mprη , mpre , and mprw , respectively. These prices of risks correspond to annualized percentage values for λm,η σ, λm,e σ, λm,w σw in Eq. (11). The exposures of the consumption asset to the three systematic risks, βη , βe , and βw , are presented in the middle part of the table. Total risk compensation in annual percentage terms for each risk is reported as prm∗ and equals the product of the price of risk, the standard deviation of the shock, and the beta for the specific risk.

that an IES larger than one is required for the long-run and volatility risks to carry to a positive risk premium. It is clear from Table 3 that the price of risk is highest for the long-run risks (see columns 2 and 3) and smallest for the volatility risks. A comparison of columns 2 and 3 also shows that raising the IES increases the prices of long-run and volatility risks in absolute value. The magnitudes reported in Table 3 are with ρ = 0.979—lowering this persistence parameter also lowers the prices of long-run and volatility risks (in absolute value). Increasing the risk aversion parameter increases the prices of all consumption risks, as shown in Eq. (12). Hansen and Jagannathan (1991) document the importance of the maximal Sharpe ratio, determined by the volatility of the IMRS, in assessing asset pricing models. Bansal and Yaron show that incorporating long-run risks increases the maximal Sharpe ratio for their model, and it satisfies the non-parametric bounds of Hansen and Jagannathan (1991). The risk premium on the market portfolio (i.e., the dividend asset) is also affected by the presence of long-run risks. To underscore their importance, assume that consumption and dividend growth rates are i.i.d. This shuts off the long-run risk channel. The market risk premium in this case is Et (rm,t+1 − rf ,t ) = γCov(gt+1 , gd,t+1 ) − 0.5Var(gd,t+1 ),

(14)

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and market return volatility equals the dividend growth rate volatility. If shocks to consumption and dividends are uncorrelated, then the geometric risk premium is negative and equals −0.5Var(gd,t+1 ). If the correlation between monthly consumption and dividend growth is 0.25, then the equity premium is 0.08 percent per annum—similar to the evidence documented in Mehra and Prescott (1985) and Weil (1989). Bansal and Yaron show that incorporating long-run growth rate risks (see Eq. (4)) produces an annual equity risk premium of 4.2 percent and a risk-free rate of 1.34 percent along with market return and price-dividend volatility of 16.21 percent and 0.16, respectively. These are fairly comparable to what we see in the data (see Table 2) and highlight the importance of long-run growth rate risks. Bansal and Yaron show that the full model that incorporates long-run growth rate risks and fluctuating economic uncertainty provides a very close match to the asset market data reported in Table 2. That is, this model can account for the low risk-free rate, high equity premium, high asset price volatility, and low risk-free rate volatility. This model also quantitatively matches additional data features, such as (i) predictability of returns at short and long horizons using the dividend yield as a predictive variable, (ii) time-varying and persistent market return volatility, (iii) negative correlation between market return and volatility shocks, i.e., the volatility feedback effect, (iv) negative relation between consumption volatility and asset prices at long leads and lags, documented in Bansal, Khatchatrian, and Yaron (2005). In all, this evidence shows that incorporating long-run risks in growth rates and fluctuating economic uncertainty can help interpret a wide array of the asset market puzzles.

2.4.4. Value of Contingent Claims and Macro Markets Using the Efficient Method of Moments (EMM), Bansal, Gallant, and Tauchen (2007) consider the implications of alternative asset pricing models presented in Bansal and Yaron and in Campbell and Cochrane (1999). A unique dimension of this paper is that they model the consumption and dividends as being cointegrated, a feature that is missing in earlier work of Campbell and Cochrane (1999) and Bansal and Yaron (2004). Bansal, Gallant, and Tauchen evaluate the value of contingent claims on the aggregate wealth of the economy. This is a first step in assessing the plausibility of introducing contingent claims on macro variables for better risk sharing, as espoused by Shiller (1998). This valuation exercise also helps understand the different channels operating in these models. Bansal, Gallant, and Tauchen document large differences in the prices of put options on the consumption claim across the two asset pricing models. The volatility of the consumption claim in the Bansal and Yaron model is about one-fourth that of the market return, while in the Campbell and Cochrane model it is about as volatile as the market portfolio.

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3. CROSS-SECTIONAL IMPLICATIONS 3.1. Value, Momentum, Size, and the Cross-Sectional Puzzle Table 4, taken from Bansal, Dittmar, and Lundblad (2005), shows that there are sizeable differences in mean real returns across assets. Bansal, Dittmar, and Lundblad (2005) use quarterly data from 1967 to 2001. They rely on standard book-to-market, size, and momentum sorted portfolios. For the first two sorts, firms are sorted into different deciles once a year, and the subsequent return on these portfolios is used for empirical work. For momentum assets, CRSP-covered NYSE and AMEX firms are sorted on the basis of their cumulative return over months t − 12 through t − 1. The loser portfolio (M1) includes firms with the worst performance over the last year, and the winner portfolio (M10) includes firms with the best performance. The data shows that subsequent returns on these portfolios have a large spread (i.e., M10 return − M1 return), of about 4.62 percent per quarter—this is the momentum spread puzzle. Similarly, the highest book-to-market firms (B10) earn average real quarterly returns of 3.27 percent, whereas the lowest book-to-market (B1) firms average 1.54 percent per quarter. The value spread (return on B10 − return on B1) is about 2 percent per quarter— this is the value spread puzzle. What explains these big differences in mean returns across portfolios?

TABLE 4 Portfolio Returns Mean

Std. Dev.

Mean

Std. Dev.

Mean

Std. Dev.

S1

0.0230

0.1370

B1

0.0154

0.1058

M1

−0.0104

0.1541

S2

0.0231

S3

0.0233

0.1265

B2

0.0199

0.1200

B3

0.0211

0.0956

M2

0.0070

0.1192

0.0921

M3

0.0122

S4

0.0233

0.1174

B4

0.1089

0.0218

0.0915

M4

0.0197

S5

0.0242

0.1112

0.0943

B5

0.0200

0.0798

M5

0.0135

S6

0.0207

0.0869

0.1050

B6

0.0234

0.0813

M6

0.0160

0.0876

S7 S8

0.0224

0.1041

B7

0.0237

0.0839

M7

0.0200

0.0886

0.0219

0.1001

B8

0.0259

0.0837

M8

0.0237

0.0825

S9

0.0207

0.0913

B9

0.0273

0.0892

M9

0.0283

0.0931

S10

0.0181

0.0827

B10

0.0327

0.1034

M10

0.0358

0.1139

Table 4, reported in Bansal, Dittmar, and Lundblad (2005), presents descriptive statistics for the returns on the 30 characteristic-sorted decile portfolios. Value-weighted returns are presented for portfolios formed on momentum (M), market capitalization (S), and book-to-market ratio (B). M1 represents the lowest momentum (loser) decile, S1 the lowest size (small firms) decile, and B1 the lowest book-to-market decile. Data are converted to real using the PCE deflator. The data are sampled at the quarterly frequency, and cover the first quarter of 1967 through the fourth quarter of 2001.

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The central idea, in theory, is that differences in exposure to systematic risk should justify this puzzling difference in mean returns. The static CAPM (see Sharpe (1964)) implies that these differences mirror differences in market betas of assets, while Lucas (1978) and Breeden (1979) argue that they reflect differences in exposure to aggregate consumption movements. Evidence presented in Hansen and Singleton (1982) for the consumption-based models, and in Fama and French (1992) for the CAPM, shows that these models have considerable difficulty in accounting for the differences in observable rates of return across assets. Consequently, identifying economic sources of risks that justify differences in the measured risk premia continues to be an important economic issue. Bansal, Dittmar, and Lundblad (2002, 2005) connect systematic risks to cash-flow risks. They build on the intuition developed in the Bansal and Yaron (2004) model. This intuition is provided in Eq. (8). Assume that the dividend growth rate dynamics for equity i follow Eq. (4). In equilibrium, the differences in the risk premium across assets mirror the differences in their long-run risks beta, βi , βi =

φi ϕe 1 − κ1 ρ

−

1 ϕe ψ 1 − κ1 ρ

κ1 .

(15)

An important insight is that βi risk measure differs across assets only because of the differences in φi —the exposure of dividends to long-run risks. In the cross section of assets, the systematic risks in cash flows determine the differences in the systematic risks in the asset. Assets with higher φi are more exposed to systematic risks, and agents, consequently, should demand a higher risk premium of such assets. Empirical work for cross-sectional differences in mean returns, typically, explores the link between cross-sectional mean return and the systematic risk measure, βi . That is, it considers the projection E(ri,t+1 ) = λ0 + λc βi .

(16)

As the βi and the cash-flow risks measures, φi , are correlated in the cross section, it follows that cross-sectional differences in mean returns must mirror differences in φi . Given this connection, Bansal, Dittmar, and Lundblad ask if cash-flow betas can explain differences in mean return across assets.

3.1.1. Measuring Risks in Cash Flows To measure consumption risks in cash flows (i.e., cash-flow betas), Bansal, Dittmar, and Lundblad model the joint dynamics of observed cash flow and aggregate consumption growth rates as an autoregression (VAR). I briefly discuss this below.

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Ravi Bansal

For any asset i, using the return approximation in Eq. (3), the log price minus log cash flow, pi,t − di,t , satisfies pi,t − di,t =

j κi,0 j + Et κi,1 gi,t+1+j − κi,1 ri,t+1+j . 1 − κi,1 ∞

∞

j=0

j=0

(17)

The log price − cash flow ratio is determined by the discounted expected cash-flow growth rates and discounted expected returns. The discount rate, κi,1 , is less than one by construction. Exploiting (3) and (17), it follows that return innovations are related to innovations in expectations of future cash flows and returns: ∞

ri,t − Et−1 [ri,t ] = {Et − Et−1 }

∞ j j κi,1 gi,t+j − {Et − Et−1 } κi,1 ri,t+j

j=0

j=1

= ηgi ,t − ηei ,t .

(18)

j The piece ηgi ,t = {Et − Et−1 }[ ∞ j=0 κi,1 gi,t+j ] is cash-flow news and represents the revision in expectations of the sum of future dividend growth rates. Analogously, ηei ,t represents discount rate news. Given the return decomposition, the consumption beta can be described as βi =

Cov(ηgi ,t − ηei ,t , ηt ) Cov(ri,t − Et−1 (ri,t ), ηt ) = = βi,g − βi,e , Var(ηt ) Var(ηt )

(19)

where ηt is the time t innovation in consumption growth. The consumption beta is governed by two components—the cash-flow beta and the beta of discount rate news. Bansal, Dittmar, and Lundblad ask if the cash-flow beta, βi,g , can explain differences in risk premia across assets. To estimate the cash-flow betas, they model the de-meaned log consumption growth, gc,t , as a simple AR(1) process: gc,t = ρc gc,t−1 + ηt ,

(20)

with ρc being the AR(1) coefficient and ηt the consumption news at date t. Further, they assume that the relationship between de-meaned dividend and consumption growth rates is gi,t = φi ui,t =

K 1

L j=1

K

gc,t−k + ui,t ,

(21)

k=1

ρj,i ui,t−j + ζi,t .

(22)

188


The expression 1/K K k=1 gc,t−k represents a trailing K-period moving average of past consumption growth—this a measure of xt discussed above in the Bansal and Yaron (2004) model. The parameter φi measures the leverage of the dividends, as discussed in Eq. (4). This specification allows for cash-flow growth rates to depend on the current consumption innovation through the process for ζi,t . This contemporaneous covariance is reflected in the measured cash-flow betas. Equations (20), (21), and (22) characterize a simple VAR. The q-vector, Yt , is Yt = [gi,t ui,t · · · ui,t−(L−1) gc,t · · · gc,t−(K−1) ] ,

(23)

and the dynamics of the state variables and portfolio cash flow growth can then be expressed as Yt = AYt−1 + vt ,

(24)

where A is the q × q matrix of coefficients. Let the first element of Yt be gi,t such that e1 zt = gi,t , where e1 is a q × 1 vector with first element 1 and remaining elements 0. From Eq. (18), it follows that ηgi ,t is equal to ∞

ηgi ,t = {Et − Et−1 }

j

κi,1 gi,t+j

j=0

= e1

∞

j

κi,1 Aj vt

j=0

−1 = e1 I − κi,1 A vt .

(25)

This residual represents the innovation to current and expected future cash-flow growth rates. The exposure of this innovation to consumption growth is measured by projecting it on the innovation in consumption growth, specifically, ηgi ,t = βi,g ηt + ξgi ,t .

(26)

The resulting projection coefficient, βi,g , is the asset’s cash-flow beta developed in Bansal, Dittmar, and Lundblad (2002, 2005). Note that if ui,t is uncorrelated with consumption innovation, then the cross-sectional differences in the cash-flow beta based on Eq. (26) solely reflect differences in φi . Hence, if one imposes the restriction that ui,t is uncorrelated with consumption innovations, then it is sufficient to focus on φi . Given the cash flow’s consumption beta, Bansal, Dittmar, and Lundblad run the cross-sectional regression, E[Ri,t ] = λ0 + βi,g λc to evaluate the empirical plausibility of the cash-flow beta model.

(27)

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Ravi Bansal

3.1.2. Dividends and Cash Flows Bansal, Dittmar, and Lundblad measure the cash flows as dividends for each portfolio in a standard manner, specifically, Rt+1 = ht+1 + yt+1 ,

(28)

where ht+1 is the price appreciation and yt+1 the dividend yield (i.e., dividends at date t + 1 per dollar invested at date t). We observe Rt+1 (RET in CRSP terminology) and the price gain series ht+1 (RET X) for each portfolio; hence, yt+1 = Rt+1 − ht+1 . The level of the dividends we use in the paper is computed as Dt+1 = yt+1 Vt ,

(29)

Vt+1 = ht+1 Vt ,

(30)

where with V0 = 1. Hence, the dividend series that we use, Dt , corresponds to the total cash dividends given out by a mutual fund at t that extracts the dividends and reinvests the capital gains. The ex-dividend value of the mutual fund is Vt , and the per dollar return for the investors in the mutual fund is Rt+1 =

Vt+1 + Dt+1 = ht+1 + yt+1 . Vt

(31)

From this equation, it is evident that Vt is the discounted value of the dividends that we use. Bansal, Dittmar, and Lundblad (2005) also use repurchase adjusted dividends and earnings as a measure of cash flows. The empirical evidence is similar to that found using cash dividends. More recently, Bansal and Yaron (2006) use market clearing restrictions as a way to identify the appropriate trading strategy to use for measuring aggregate dividends. This measure, relative to the per share-based traditionally used measure (as in Eq. (29)), incorporates the relative shift in scale of the different sectors and yields different insights into the sources of the asset price variation.

3.1.3. Performance of Cash-Flow Beta Model As predicted by the Bansal and Yaron (2004) model, the consumption leverage of divided growth rates has great explanatory power. Bansal, Dittmar, and Lundblad show that using φi or βi,g as the risk measure yields a highly positive and significant price of risk estimates of λc , and the cross-sectional R2 are well in excess of 60 percent. Using an alternative asset menu of 25 portfolios based on the Fama–French 5 × 5 two-way sort on market capitalization and book-to-market value also yields comparable results. Bansal, Dittmar, and Lundblad (2002, 2005) also ask if the cash-flow risks are related to the long-run risks of Bansal and Yaron (2004). They show that when, in Eq. (21), K = 1, the consumption risks are largely short-run risks and the cash-flow betas are not able to capture the cross-sectional differences in risk premia. Increasing K, and hence

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exaggerating the long-run risks, is important for the success of the cash-flow betas in capturing differences in mean returns across assets. In contrast, alternative models find it quite hard to explain the differences in mean returns for the 30 asset menus used in Bansal, Dittmar, and Lundblad. The standard consumption betas (i.e., C-CAPM) and the market-based CAPM asset betas have close to zero explanatory power. The R2 for the C-CAPM is 2.7 percent, and that for the market CAPM is 6.5 percent, with an implausible negative slope coefficient. The Fama and French three-factor empirical specification also generates point estimates with negative, and difficult to interpret, price of risk for the market and size factors—the cross-sectional R2 is about 36 percent. Compared to all these models, the cash-flow risks model of Bansal, Dittmar, and Lundblad is able to capture a significant portion of the differences in risk premia across assets. In addition, Bansal, Dittmar, and Lundblad (2001) and Bansal, Dittmar, and Kiku (2007) also consider a risk measure based on stochastic cointegration between the log level of dividends and consumption. This risk measure is estimated via the projection of the deterministically de-trended level of log dividends on the de-trended level of log consumption. Bansal, Dittmar, and Kiku (2007) derive new results that link this cointegration parameter to consumption betas by investment horizon and evaluate the ability of their model to explain differences in mean returns for different horizons. Hansen, Heaton, and Li (2006) inquire about the robustness of the stochastic cointegration-based risk measures considered in Bansal, Dittmar, and Lundblad (2001). Bansal, Dittmar, and Kiku provide new evidence regarding the robustness of the stochastic cointegration-based measures of permanent risks in equity markets. Parker and Julliard (2005) evaluate if long-run risks in aggregate consumption can account for the cross section of expected returns. Malloy, Moskowitz, and Vissing-Jørgensen (2006) evaluate if long-run risks in the consumption of stockholders has greater ability to explain the cross section of equity returns, relative to aggregate consumption measures. Bansal, Dittmar, and Lundblad (2005) also discuss why cash-flow risks may capture the differences in risk premia when standard consumption betas fail to do so. The Bansal and Yaron model helps explain why this may be the case—in this model of multiple risks, the consumption beta is not sufficient to capture differences in risk premia across assets. Imagine that risk premia are determined by Eq. (13), that is, there are multiple sources of risks. Let the shocks to the various risk sources be correlated, and then the standard consumption beta will measure a weighted linear combination of the three different betas. While each individual beta may be important in capturing the risk premia across assets—a weighted linear combination may fail to do so. Bansal, Dittmar, and Lundblad (2005) provide simulation evidence wherein the C-CAPM betas fail to explain the difference in mean returns across assets, while the cash-flow betas capture a sizable portion of the cross-sectional differences in mean returns. The approach pursued in Bansal, Dittmar, and Lundblad (2005) derives constant risk measures. Jagannathan and Wang (1996) and Lettau and Ludvigson (2001) focus on time-varying return betas. Deriving and measuring time-varying cash-flow betas would be a valuable extension for future research.

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4. CONCLUSION The work of Bansal and Lundblad (2002), Bansal and Yaron (2004), and Bansal, Gallant, and Tauchen (2007) show that long-run growth rate risks and varying economic uncertainty are important for quantitatively interpreting financial markets. These papers argue that investors care about the long-run growth prospects and the uncertainty surrounding the growth rate. Risks associated with changing long-run growth prospects and varying economic uncertainty drive the level of returns and asset price volatility in financial markets. A key issue in terms of the preferences of investors is their attitude toward risk as measured by risk aversion and the magnitude of the parameter that determines intertemporal substitution. Based on recent empirical evidence on the magnitude of the IES, and on its economic implications for asset markets, Bansal and Yaron argue that the IES should be larger than one. They show that only when the IES is larger than one does increased economic uncertainty translate into a drop in asset prices. Bansal, Khatchatrian, and Yaron (2005) find very robust empirical evidence that a rise in economic uncertainty lowers asset prices across different samples and countries. This evidence suggests that the IES may indeed be larger than one. Bansal, Dittmar, and Lundblad (2002, 2005) show that long-run risks in cash flows of portfolios should explain differences in mean returns across assets. They measure cash-flow risks via estimating cash-flow betas and find that the consumption risks in cash flows of portfolios explain very well the differences in mean returns across equity portfolios. Using cash flows and consumption, they devise ways to measure cash-flow betas that capture systematic risks in cash flows. These risk measures, they find, provide very sharp information about differences in risk premia across assets. They show that the cash-flow betas can explain the differences in mean returns for value, size, and momentum sorted portfolios. All of this evidence and the economics underlying it support the view that the longrun risks and uncertainty channel contain very valuable insights about the workings of financial markets.

References Abel, A. B. Risk premia and term premia in general equilibrium. Journal of Monetary Economics 43 (1999): 3–33. Anderson, E., L. P. Hansen, and T. Sargent. A quartet of semi-groups for model specification, robustness, prices of risk and model detection. Journal of the European Economic Association 1 (2003): 68–123. Ang, A., and G. Bekaert. Stock return predictability: Is it there? Review of Financial Studies 20 (2007): 651–707. Ansley, C. F., and P. Newbold. Finite sample properties of estimators for autoregressive moving average models. Journal of Econometrics 13 (1980): 159–184. Attanasio, P. O., and A. Vissing-Jorgensen. Stock market participation, intertemporal substitution and risk aversion. American Economic Review 93 (2003): 383–391. Attanasio, P. O., and G. Weber. Intertemporal substitution, risk aversion and the Euler equation for consumption. Economic Journal 99 (1989): 59–73.

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Chapter 5 • Long-Run Risks and Risk Compensation in Equity Markets Bansal, R., R. F. Dittmar, and D. Kiku. Cointegration and consumption risks in asset returns. Review of Financial Studies forthcoming (2007). Bansal, R., R. F. Dittmar, and C. Lundblad. Consumption, dividends and the cross-section of equity returns. Working paper. Duke University (2001). Bansal, R., R. F. Dittmar, and C. Lundblad. Interpreting risk premia across size, value, and industry portfolios. Working paper. Duke University (2002). Bansal, R., R. F. Dittmar, and C. Lundblad. Consumption, dividends and the cross-section of equity returns. Journal of Finance 60 (2005): 1639–1672. Bansal, R., R. A. Gallant, and G. Tauchen. Rational pessimism, rational exuberance, and asset pricing models. Review of Economic Studies forthcoming (2007). Bansal, R., V. Khatchatrian, and A. Yaron. Interpretable asset markets? European Economic Review 49 (2005): 531–560. Bansal, R., and C. Lundblad. Market efficiency, asset returns, and the size of the risk premium in global equity markets. Journal of Econometrics 109 (2002): 195–237. Bansal, R., and A. Yaron. Risks for the long run: A potential resolution of asset pricing puzzles. Journal of Finance 59 (2004): 1481–1509. Bansal, R., and A. Yaron. The asset pricing-macro nexus and return-cash flow predictability. Working paper. Duke University (2006). Barsky, R., and B. J. DeLong. Why does the stock market fluctuate? Quarterly Journal of Economics 108 (1993): 291–312. Breeden, D. T. An intertemporal asset pricing model with stochastic consumption and investment opportunities. Journal of Financial Economics 7 (1979): 265–296. Campbell, J. Y. Asset prices, consumption and the business cycle, in J. B. Taylor, and M. Woodford, eds., Handbook of Macroeconomics, Volume 1. Elsevier Science, North-Holland, Amsterdam (1999). Campbell, J. Y., and J. H. Cochrane. By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107 (1999): 205–251. Campbell, J. Y., and R. J. Shiller. The dividend-price ratio and expectations of future dividends and discount factors. Review of Financial Studies 1 (1988): 195–227. Cecchetti, S. G., P.-S., Lam, and N. C. Mark. The equity premium and the risk free rate: Matching the moments. Journal of Monetary Economics 31 (1993): 21–46. Cochrane, J. H. How big is the random walk in GNP? Journal of Political Economy 96 (1988): 893–920. Epstein, L., and S. Zin. Substitution, risk aversion and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica 57 (1989): 937–969. Fama, E., and K. French. The cross-section of expected stock returns. Journal of Finance 59 (1992): 427–465. Guvenen, F. Reconciling conflicting evidence on the elasticity of intertemporal substitution: A macroeconomic perspective. Journal of Monetary Economics 53 (2006): 1451–1472. Hall, R. E. Intertemporal substitution in consumption. Journal of Political Economy 96 (1988): 339–357. Hansen, L. P., J. Heaton, and N. Li. Consumption strikes back? Measuring long-run risk. Working paper. University of Chicago (2006). Hansen, L. P., and R. Jagannathan. Implications of security market data for models of dynamic economies, Journal of Political Economy 99 (1991): 225–262. Hansen, L. P., and K. Singleton. Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica 50 (1982): 1269–1286. Hurwicz, L. Least square bias in time series, in T. Koopmans, ed., Statistical Inference in Dynamic Economic Models. Wiley Press, New York (1950). Jagannathan, R., and Z. Wang. The conditional CAPM and the cross-section of expected returns. Journal of Finance 51 (1996): 3–54. Judd, K. Numerical Methods in Economics. MIT Press, Cambridge, MA (1998). Kandel, S., and R. F. Stambaugh. Asset returns and intertemporal preferences. Journal of Monetary Economics 27 (1991): 39–71. LeRoy, S., and R. Porter. The present value relation: Tests based on implied variance bounds. Econometrica 49 (1981): 555–574.

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193 Lettau, M., and S. Ludvigson. Resurrecting the (C)CAPM: A cross-sectional test when risk premia are time-varying. Journal of Political Economy 109 (2001): 1238–1287. Lucas, R. Asset prices in an exchange economy. Econometrica 46 (1978): 1429–1446. Malloy, C. J., T. J. Moskowitz, and A. Vissing-Jorgensen. Long-run stockholder consumption risk and asset returns. Working paper. University of Chicago (2006). Mehra, R., and E. C. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15 (1985): 145–161. Newey, W. K., and K. D. West. A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55 (1987): 703–708. Parker, J., and C. Julliard. Consumption risk and the cross-section of expected returns. Journal of Political Economy 113 (2005): 185–222. Sharpe, W. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance 19 (1964): 425–442. Shephard, N. G., and A. C. Harvey. On the probability of estimating a deterministic component in the local level model, Journal of Time Series Analysis 11 (1990): 339–347. Shiller, R. J. Do stock prices move too much to be justified by subsequent changes in dividends? American Economic Review 71 (1981): 421–436. Shiller, R. J. Macro markets: Creating institutions for managing society’s largest economic risks. Oxford University Press, New York (1998). Weil, P. The equity premium puzzle and the risk-free rate puzzle. Journal of Monetary Economics 24 (1989): 401–421.


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Discussion: Long-Run Risks and Risk Compensation in Equity Markets John Heaton Graduate School of Business, University of Chicago, and NBER

1. SUMMARY Bansal and his co-authors have produced a series of important and provocative papers that demonstrate how low-frequency risk can provide a justification for observed risk premia. Bansal summarizes this work and helps us understand the linkages between long-run risk in consumption and long-run risk in financial securities. Since he does such a nice job of explaining the economics of the approach, I focus on a few issues raised by Bansal’s work and by some of my own.

2. A LOW-FREQUENCY COMPONENT IN CONSUMPTION? An important aspect of the model of Bansal and Yaron (2004) is the presence of a lowfrequency component in aggregate consumption. Shocks to the level of consumption are persistent, as are shocks to volatility. A substantial and time-varying equity premium results when dividends display exposure to the shocks. It is natural to ask whether there is any empirical support for the assumed model of consumption. As Bansal argues, one piece of evidence may come from financial markets. Agents could use the model’s structure along with signals from asset prices to detect the low-frequency component. This may take the rational expectations assumption to an untenable position, however. Unless agents directly observe the low-frequency shocks driving consumption, it is not clear how the shocks could be reflected in security prices. Appealing to the standard idea that the econometrician observes a smaller information set than agents is also a little delicate in this context. In the model of preferences considered by Bansal and Yaron (2004), the exact conditioning information of the agents is needed in order to derive the implications of the model.

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An interesting way of introducing a concern for low-frequency components, even when agents have a hard time detecting those components, is provided by the recent work of Hansen and Sargent (2007). In their work the representative agent is uncertain about the probability model generating consumption. There are two alternative models: one where consumption is not predictable and one where consumption is predictable. Because the agent is worried about model uncertainty, she acts as if there is a very high probability attached to the model with predictable consumption. This model generates both the high-risk premia consistent with a low-frequency component in consumption, and consumption dynamics that are difficult to distinguish from an i.i.d. model. There is evidence in the literature for an important predictable low-frequency component to consumption. This evidence is typically obtained using additional variables to predict consumption along with plausible economic linkages across variables. Examples include the work of Fisher (2006) and Mulligan (2002). In Hansen et al. (2006a) we consider a bivariate model for aggregate consumption and aggregate corporate earnings. We present evidence that corporate earnings are co-integrated with aggregate consumption as predicted by most models of business cycles and economic growth. Under the co-integration assumption, corporate earnings reveal important long-run shocks to consumption. To illustrate this, Figure 1 reports the impulse response of aggregate consumption to the two shocks in the model.1 We call the first shock a “consumption shock”; it impacts both consumption and earnings contemporaneously. In contrast, the second shock impacts earnings contemporaneously but has no immediate impact on consumption. We call this shock an “earnings shock.” Shocks to corporate earnings predict consumption over many quarters and reveal an important low-frequency component of consumption similar to the setup considered by Bansal and Yaron (2004).

3. PREFERENCES In his work, Bansal employs the recursive specification of preferences developed by Kreps and Porteus (1978), Epstein and Zin (1989b), Weil (1990), and others. In the context of models with long-run consumption risk, this model of preferences is useful because it induces a concern for the resolution of uncertainty. In a standard model with time-additive CRRA utility, risk prices are determined by risk aversion and the oneperiod or instantaneous impact of shocks on consumption. With recursive preferences, the long-run impact of shocks on consumption also influences risk prices. Shocks that strongly predict future consumption have larger risk premia. Calibration of the models considered by Bansal does run into the problem that consumption is not that volatile even in the long run. Large levels of risk aversion are typically needed to fit aggregate and cross-sectional risk premia. An advantage of the assumption of recursive preference is that the effects of high risk aversion on the level of 1 This

figure is taken from Hansen et al. (2006b).


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Impulse Response

0.006 0.005 0.004 0.003 0.002 0.001 0 0

5

10 Horizon (years)

depicts response to a consumption shock

FIGURE 1

15

20

depicts response to an earnings shock

Impulse responses implied by the VAR of the Hansen–Heaton–Li model.

interest rates can be controlled by assuming the intertemporal elasticity of substitution (IES) is close to 1. In the calibration exercise considered by Bansal and Yaron (2004), the IES parameter is assumed to be greater than 1 so that shocks that forecast increased future consumption also increase the price-to-dividend ratio or wealth-to-consumption ratio. This produces the desired effect that shocks that forecast future consumption have a magnified risk premium. In the model, this is captured by variation in the market return, which becomes part of the stochastic discount factor with recursive utility. There is some controversy over the value of the IES parameter used by Bansal and Yaron (2004) because of other evidence in the literature. In particular, the typical assumption is that the IES parameter is less than 1. As Bansal points out, however, many of the existing estimates in the literature are sensitive to the assumption that consumption is homoskedastic. This identification problem is potentially magnified if the true underlying preferences are of the recursive form as shown by Hansen et al. (2006b). An issue related to the assumption about the IES parameter is the solution method used by Bansal and many others in the literature. The general stochastic discount factor induced by recursive preferences has a term in consumption growth and a term in the

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“continuation value” due to future consumption. Following earlier work by Epstein and Zin (1989a), shocks to the “continuation value” are replaced by shocks to the return to holding a claim on aggregate consumption (the “market” return). This generates a proper solution to the model except for the case of logarithmic intertemporal risk preferences (IES = 1). At this point the wealth-to-consumption ratio is constant so that the market return covaries exactly with consumption. Shocks to future consumption still matter to the consumer, but the continuation value from future consumption plans must be calculated directly. Shocks to the derived continuation value then enter the stochastic discount factor. In Hansen et al. (2006a), we show how the continuation value can be calculated in a log-linear stochastic environment with logarithmic intertemporal utility. Further, we develop an approximation to the case of a more general value of the IES parameter that works well for values of the IES near 1. This approach is extended to an environment with stochastic volatility in Hansen et al. (2006b). An advantage of this method of solution is that it allows for the consideration of values for the IES parameter close to 1, where the more typical use of the return to the market portfolio breaks down.

4. RETURNS AND LONG-RUN CASH FLOWS In his work with Lundlad and Dittmar, Bansal develops the idea of a “cash flow” beta where the long-run covariance between consumption and dividends is assumed to drive risk premia. Measured differences in this long-run covariance across portfolio cash flows do coincide with observed differences in average returns to the portfolios. This is an important finding because there is very little heterogeneity in the contemporaneous covariance between shocks to consumption and the returns to the portfolios considered by Bansal. This is reflected in the well-established fact that the consumption CAPM cannot explain the value premium, the size effect, and other observed risk premia. The contrast between the cash flow beta measure of risk and the contemporaneous covariance between returns and shocks to consumption creates a tension. One potential resolution of this tension could be that the short-run dynamics of risk exposure are difficult to model or measure. For example, there could be important market frictions, behavioral biases, or general model misspecification. In these situations, the long-run covariances between returns and consumption (as in Daniel and Marshall (1997)), or between long-run shocks to consumption and portfolio cash flows may be more appropriate measures of risk. If short-run implications are to be ignored, what do we mean by the “short run?” One way to answer this question is to specify a complete model of pricing and then examine the model’s predictions for the pricing of a portfolio’s cash flows at different frequencies. For example, the holding period returns for each future cash flow can be calculated. The one-period portfolio return is just a weighted average of these individual returns. By ignoring the contribution from short-term cash flows, a long-run return is calculated. Alternatively, the rate at which the price of future cash flows declines relative to the predicted growth in cash flows can be used to infer a long-run return. These


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alternative specifications of long-run returns and their long-run limits are developed in Hansen et al. (2006a).2 We apply our analysis to portfolios of growth and value stocks. Consistent with the results in Bansal et al. (2005), we find that portfolios of value stocks are predicted to have high long-run returns.

5. CONCLUSION Bansal’s paper does a nice job of summarizing his work and showing why the consideration of long-run risk can potentially help us to understand observed security prices. The paper also points the way toward the work to be done both in modeling the dynamics of consumption and in understanding how long-run risk is priced.

References Bansal, R., and A. Yaron. Risks for the long run: A potential resolution of asset pricing puzzles. Journal of Finance 59 (2004): 1481–1509. Bansal, R., R. F. Dittmar, and C. T. Lundblad. Consumption, dividends, and the cross-section of equity returns. Journal of Finance (2005). Daniel, K., and D. Marshall. Equity-premium and risk-free-rate puzzles at long horizons. Macroeconomic Dynamics 1 (1997): 452–484. Epstein, L., and S. Zin. Substitution, risk aversion and the temporal behavior of stock returns: An empirical investigation. Journal of Political Economy 99 (1989a): 263–286. Epstein, L. G., and S. E. Zin. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica 57 (1989b): 937–969. Fisher, J. The dynamic effects of neutral and investment-specific technology shocks. Journal of Political Economy (2006). Hansen, L. P., and T. J. Sargent. Fragile beliefs and the price of model uncertainty. Working paper. University of Chicago (2007). Hansen, L. P., and J. Scheinkman. Long term risk: An operator approach. Working paper. University of Chicago (2006). Hansen, L. P., J. C. Heaton, and N. Li. Consumption strikes back?: Measuring long run risk. Working paper. University of Chicago (2006a). Hansen, L. P., J. Heaton, N. Roussanov, and J. Lee. Intertemporal substitution and risk aversion. Handbook of Econometrics, volume 6 forthcoming (2006b). Kreps, D. M., and E. L. Porteus, Temporal resolution of uncertainty and dynamic choice. Econometrica 46 (1978): 185–200. Mulligan, C. Capital, interest, and aggregate intertemporal substitution. Working paper. NBER (2002). Weil, P. Nonexpected utility in macroeconomics. Quarterly Journal of Economics 105 (1990): 29–42.

2 The

analysis in Hansen et al. (2006a) is conducted in a log-linear environment. Hansen and Scheinkman (2006) extend this to more general environments.

CHAPTER 6

The Loss Aversion/Narrow Framing Approach to the Equity Premium Puzzle Nicholas Barberis and Ming Huang Yale University and Cornell University/SUFE

1. Introduction 2. Loss Aversion and Narrow Framing 3. The Equity Premium 3.1. Modeling Loss Aversion and Narrow Framing 3.2. Quantitative Implications 3.3. Attitudes to Large Monetary Gambles 3.4. Attitudes to Small Monetary Gambles 3.5. The Importance of Narrow Framing 4. Other Applications 5. Further Extensions 5.1. Dynamic Aspects of Loss Aversion 5.2. Other Forms of Narrow Framing 6. Conclusion and Future Directions References

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We thank Rajnish Mehra for inviting us to contribute a chapter, and are grateful to him and to participants in the “20th Anniversary of the Equity Premium” conference in Santa Barbara, California, for valuable feedback. Huang acknowledges financial support from the National Natural Science Foundation of China under grant 70432002.


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Chapter 6 • The Loss Aversion/Narrow Framing Approach to the Equity Premium Puzzle

Abstract We review a recent approach to understanding the equity premium puzzle. The key elements of this approach are loss aversion and narrow framing, two well-known features of decision-making under risk in experimental settings. In equilibrium, models that incorporate these ideas can generate a large equity premium and a low and stable risk-free rate, even when consumption growth is smooth and only weakly correlated with the stock market. Moreover, they can do so for parameter values that correspond to sensible attitudes to independent monetary gambles. We conclude by suggesting some possible directions for future research. JEL Classification: G10, G11, D9 Keywords: asset pricing, equity risk premium, CAPM, consumption CAPM, risk-free rate puzzle

Nicholas Barberis and Ming Huang

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1. INTRODUCTION One of the best-known stock market puzzles is the equity premium puzzle, which asks why stocks historically earned a higher average return, relative to T-bills, than seems justified by standard measures of risk (Mehra and Prescott (1985)). In this essay, we discuss a recent approach to addressing this puzzle. The broad theme of this approach is that we may be able to improve our understanding of how people evaluate stock market risk, and hence our understanding of the equity premium, by looking at how people evaluate risk in experimental settings. Specifically, this approach argues that loss aversion and narrow framing, two of the most important ideas to emerge from the experimental literature on decision-making under risk, may play an important role in the way some people think about the stock market. Loss aversion is a central feature of Kahneman and Tversky’s (1979) prospect theory—a descriptive theory, based on extensive experimental evidence, of how people evaluate risk. In this theory, the carriers of value are not absolute levels of wealth, but rather, gains and losses measured relative to a reference point. Loss aversion is the finding that people are much more sensitive to losses—even small losses—than to gains of the same magnitude. To understand narrow framing, recall that under traditional utility functions defined over consumption or total wealth, the agent evaluates a new gamble by first mixing it with the other risks he is already facing and then checking whether the combination is attractive. Narrow framing, by contrast, is the phenomenon documented in experimental settings whereby, when people are offered a new gamble, they sometimes evaluate it in isolation, separately from their other risks. In other words, they act as if they get utility directly from the outcome of the gamble, even if the gamble is just one of many that determine their overall wealth risk. This behavior is at odds with traditional utility functions, under which the agent only gets utility from the outcome of a new gamble indirectly, via its contribution to his total wealth. Motivated by these ideas, some recent papers propose that people are loss-averse over changes in the value of their stock market holdings. In other words, even if stock market risk is just one of many risks that determine their overall wealth risk—others being labor income risk and housing risk, say—people still get utility directly from stock market fluctuations (narrow framing) and are more sensitive to losses than to gains (loss aversion). For reasons we discuss later, most implementations also assume that people focus on annual gains and losses. Informally, then, people evaluate stock market risk by saying, “Well, stocks could go up over the next year, and that would feel good; but they could also go down, and since I’m more sensitive to losses than to gains, that would be really painful. Overall, the stock market doesn’t look like an attractive risk to me.” According to the approach we describe in this essay, it is this sort of thinking that leads the investing population to demand a high equity premium. Why should economists be interested in this particular approach to the equity premium puzzle? What are its selling points? In this survey, we emphasize two. First, a model that incorporates loss aversion and narrow framing can generate a high equity premium while also matching other aspects of the data, such as the low and stable

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risk-free rate, the low volatility of consumption growth, and the low correlation of stock returns and consumption growth. With some additional structure, it can also match the high volatility and time-series predictability of stock returns. A second benefit of the framework described here is that it can address the equity premium puzzle for preference parameters that are “reasonable,” by which we mean parameters that correspond to sensible attitudes to independent monetary gambles. This is important because it was, in part, the difficulty researchers encountered in reconciling the high average return on stocks with reasonable attitudes to large-scale monetary gambles that launched the equity premium literature in the first place. The approach we survey here was first proposed by Benartzi and Thaler (1995). In their framework, the investor is loss-averse over fluctuations in the value of his financial wealth, which, since financial wealth is just one component of total wealth, constitutes narrow framing. One drawback of this framework is that, since the investor gets no direct utility at all from consumption or total wealth, consumption plays no role, making it hard to check how well the model describes the joint properties of stock returns and consumption growth. Benartzi and Thaler’s work therefore opens up a new challenge: to build and evaluate more realistic models in which, even if the investor gets utility from fluctuations in the value of one component of his wealth, he also gets some utility from consumption. In large part, this essay surveys the progress that has been made on this front, drawing primarily on the analysis of Barberis, Huang, and Santos (2001) and Barberis and Huang (2004). The story we tell in this essay is a simple one: investors require a high equity premium because any drop in the stock market over the next year will bring them direct disutility. To some readers, this story may be too simple, in that the distance between assumption and conclusion may appear too close for comfort. We are aware of this concern and agree that if the loss aversion/narrow framing framework is to gain currency, its predictions must be tested and confirmed. Fortunately, tests of the framework are starting to appear, and we discuss some of them at the end of the essay. Even before the outcome of these tests is known, however, there is a methodological contribution in the research surveyed here that even a skeptical reader can appreciate: the papers we discuss show how loss aversion and narrow framing can be incorporated into more traditional models of asset pricing, thereby helping us understand the predictions of these features of decision-making. In Section 2, we discuss loss aversion and narrow framing in more detail, examining both the evidence they are inferred from and the interpretations they are given. In Section 3, we show that, once embedded into more traditional utility functions, these features can generate a high equity premium and a low and stable risk-free rate, even when consumption growth is smooth and only weakly correlated with stock returns; and moreover, that they can do so for parameter values that correspond to sensible attitudes to both large-scale and small-scale monetary gambles. We highlight the importance of the narrow framing assumption by showing that, without this feature, the results are very different. In Section 4, we note that the analysis also has implications for a portfolio puzzle, the stock market participation puzzle. Section 5 considers various


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extensions of the basic framework, while Section 6 concludes and discusses possible directions for future research. Since loss aversion and narrow framing are the defining features of the approach we describe here, the framework should, strictly speaking, be called the “loss aversion and narrow framing” approach to the equity premium puzzle. Given that narrow framing is the more distinctive of the two ingredients, we sometimes abbreviate this to the “narrow framing” approach.1

2. LOSS AVERSION AND NARROW FRAMING Loss aversion is a central feature of Kahneman and Tversky’s (1979) prospect theory, a prominent descriptive theory of decision-making under risk. In this theory, the carriers of value are not absolute wealth levels but, rather, gains and losses measured relative to a reference point. Loss aversion is a greater sensitivity to losses—even small losses—than to gains of the same magnitude and is represented by a kink in the utility function. The most basic evidence for loss aversion is the fact that people tend to reject gambles of the form

1 1 , (1) 110, ; −100, 2 2 to be read as “win $110 with probability 1/2, lose $100 with probability 1/2, independent of other risks” (Kahneman and Tversky (1979), Tversky and Kahneman (1992)). It is hard to explain this evidence with differentiable utility functions, whether expected utility or non-expected utility, because the very high local risk aversion required to do so typically predicts an implausibly high level of aversion to large-scale gambles (Epstein and Zin (1990), Rabin (2000), Barberis, Huang, and Thaler (2006)).2 For reasons of tractability, the asset pricing models we describe later do not incorporate all features of prospect theory. However, even if it were possible to solve a dynamic asset pricing model that did incorporate all of prospect theory’s features, we would not expect the implications for the equity premium to be very different. Under prospect theory, attitudes to a gamble like the stock market, which entails a moderate probability of a gain or of a loss, are largely determined by loss aversion alone.3 1 Benartzi and Thaler (1995) use the label “myopic loss aversion.” By using this phrase, they emphasize the investor’s sensitivity to losses (loss aversion) and his focus on annual gains and losses (myopia), but not the narrow framing. As we will see, narrow framing is more crucial to the results than the annual evaluation of gains and losses. We therefore prefer to emphasize the narrow framing while playing down the myopia. 2 There is also strong evidence of what Thaler (1980) calls an “endowment effect,” which can be thought of as loss aversion in the absence of uncertainty. Kahneman, Knetsch, and Thaler (1990) conduct a series of experiments in which subjects are either given some object such as a coffee mug and then asked if they would be willing to sell it, or not given a mug and then offered the chance to buy one. The authors find that mug owners demand more than twice as much to sell their mugs as non-owners are willing to pay to acquire one. 3 The asset pricing implications of other features of prospect theory are studied, in simple settings, by Barberis and Huang (2005), who focus on the probability weighting function; and by Barberis and Xiong (2005) and Gomes (2005), who focus on the concavity (convexity) of the value function over gains (losses).

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The classic demonstration of narrow framing is due to Tversky and Kahneman (1981), who ask 150 subjects the following question: Imagine that you face the following pair of concurrent decisions. First examine both decisions, and then indicate the options you prefer. Choice I. Choose between A. a sure gain of $240, B. a 25 percent chance to gain $1,000 and a 75 percent chance to gain nothing. Choice II. Choose between C. a sure loss of $750, D. a 75 percent chance to lose $1,000 and a 25 percent chance to lose nothing. Tversky and Kahneman (1981) report that 84 percent of subjects chose A, with only 16 percent choosing B, and that 87 percent chose D, with only 13 percent choosing C. In particular, 73 percent of subjects chose the combination A&D, namely a 25% chance to win $240, a 75% chance to lose $760,

(2)

which is surprising, given that this choice is dominated by the combination B&C, namely a 25% chance to win $250, a 75% chance to lose $750.

(3)

It appears that instead of focusing on the combined outcome of decisions I and II—in other words, on the outcome that determines their final wealth—subjects are focusing on the outcome of each decision separately. Indeed, subjects who are asked only about decision I do overwhelmingly choose A; and subjects asked only about decision II do overwhelmingly choose D. In more formal terms, it appears that we cannot model the typical subject as maximizing a utility function defined only over total wealth. Rather, his utility function appears to depend directly on the outcome of each of decisions I and II, rather than just indirectly, via the contribution of each decision to overall wealth. As such, this is an example of narrow framing. More recently, Barberis, Huang, and Thaler (2006) have argued that the commonly observed rejection of the gamble in Eq. (1) is not only evidence of loss aversion, but of narrow framing as well. To see why, note that most of the subjects who are offered this gamble are typically already facing other kinds of risk, such as labor income risk, housing risk, or financial market risk. In the absence of narrow framing, they must therefore evaluate the 110/100 gamble by mixing it with these other risks and then checking if the combination is attractive. It turns out that the combination is almost always attractive: since the 110/100 gamble is independent of other risks, it offers useful diversification benefits, which, even if loss averse, people can enjoy. The rejection of the


205

110/100 gamble therefore suggests that people are not fully merging the gamble with their other risks, but that, to some extent, they are evaluating it in isolation; in other words, that they are framing it narrowly. By the same token, any evidence of aversion to a small, independent, actuarially favorable risk points to a possible role for narrow framing. Examples of such evidence in the field are the high premia consumers pay for telephone wiring insurance and the low deductibles chosen in automobile insurance contracts (Cicchetti and Dubin (1994), Rabin and Thaler (2001), Cohen and Einav (2005)).4 Motivated by these ideas, some recent papers propose that people are loss-averse over changes in the value of their stock market holdings. In other words, even if stock market risk is just one of many risks that determine their overall wealth risk—others being labor income risk and housing risk, say—people still get utility directly from stock market fluctuations (narrow framing) and are more sensitive to losses than to gains (loss aversion). Is it plausible that people might frame stock market risk narrowly? To answer this, it is helpful to first think about the underlying sources of narrow framing. One view is that narrow framing stems from non-consumption utility, such as regret. Regret is the pain we feel when we realize that we would be better off today if we had taken a different action in the past. Even if a gamble that an agent accepts is just one of many risks that he faces, it is still linked to a specific decision, namely the decision to accept the gamble. As a result, it exposes the agent to possible future regret: if the gamble turns out badly, he may regret the decision to accept it. Consideration of non-consumption utility therefore leads quite naturally to preferences that depend directly on the outcomes of specific gambles the agent faces. A second interpretation of narrow framing is proposed by Kahneman (2003). He argues that it occurs when decisions are made intuitively, rather than through effortful reasoning. Since intuitive thoughts are by nature spontaneous, they are heavily shaped by the features of the situation at hand that come to mind most easily; to use the technical term, by the features that are most “accessible.” When an agent is offered a new gamble, the distribution of the gamble, considered separately, is much more accessible than the distribution of his overall wealth once the new gamble has been merged with his other risks. As a result, if the agent thinks about the gamble intuitively, the distribution of the gamble, taken alone, may play a more important role in decision-making than would be predicted by traditional utility functions defined only over wealth or consumption. In Tversky and Kahneman’s (1981) example, the outcome of each one of choices A, B, C, or D is highly accessible. Much less accessible, though, is the overall outcome once two choices—A&D, say, or B&C—are combined: the distributions in (2) and (3) are less “obvious” than the distributions of A, B, C, and D given in the original question. As a result, the outcome of each of decisions I and II may play a bigger role in decisionmaking than predicted by traditional utility functions. Similar reasoning applies in the case of the 110/100 gamble. 4 For more discussion and evidence of narrow framing, see Kahneman and Tversky (1983), Tversky and Kahneman (1986), Redelmeier and Tversky (1992), Kahneman and Lovallo (1993), and Read, Loewenstein, and Rabin (1999).

206


It seems to us that both the “regret” and “accessibility” interpretations of narrow framing could be relevant when investors—even sophisticated investors—think about stock market risk. Allocating some fraction of his wealth to the stock market constitutes a specific action on the part of the agent—one that he may later regret if his stock market gamble turns out poorly.5 Alternatively, given our daily exposure, through newspapers, books, and other media, to large amounts of information about the distribution of stock market risk, such information is very accessible. Much less accessible is information about the distribution of future outcomes once stock risk is merged with the other kinds of risk that people face. Judgments about how much to invest in stocks might therefore be made, at least in part, using a narrow frame. The accessibility interpretation of narrow framing also provides a rationale for why investors might focus on annual gains and losses in the stock market. Much of the public discussion about the historical performance of different asset classes is couched in terms of annual returns, making the annual return distribution particularly accessible.6 While Tversky and Kahneman’s (1981) experiment provides conclusive evidence of narrow framing, it is also somewhat extreme in that, in this example, narrow framing leads subjects to choose a dominated alternative. In general, narrow framing does not necessarily lead to violations of dominance. All the same, Tversky and Kahneman’s (1981) example does raise the concern that, when applied to asset pricing, narrow framing might give rise to arbitrage opportunities. To ensure that this does not happen, the analysis in Section 3 focuses on applications to absolute pricing—in other words, to the pricing of assets, like the aggregate stock market, which lack perfect substitutes. Since the substitutes are imperfect, there are no riskless arbitrage opportunities in the economies we construct. We would not expect narrow framing to have much useful application to relative pricing: in this case, any impact that narrow framing had on prices would create an arbitrage opportunity that could be quickly exploited. While the regret and accessibility interpretations both suggest that narrow framing may play a role when people evaluate stock market risk, they make different predictions as to how long-lasting this role will be. Under the regret interpretation, the agent simply gets utility from things other than consumption and takes this into account when making decisions. Since he is acting optimally, there is no reason to expect his behavior 5 Of course, investing in T-bills may also lead to regret if the stock market goes up in the meantime. Regret is thought to be stronger, however, when it stems from having taken an action—for example, actively moving one’s savings from the default option of a riskless bank account to the stock market—than from having not taken an action—for example, leaving one’s savings in place at the bank. In short, errors of commission are more painful than errors of omission (Kahneman and Tversky (1982)). 6 Clever tests of this logic can be found in Gneezy and Potters (1997) and Thaler et al. (1997). The latter paper, for example, asks subjects how they would allocate between a risk-free asset and a risky asset over a long time horizon such as 30 years. The key manipulation is that some subjects are shown draws from the distribution of asset returns over short horizons—the distribution of monthly returns, say—while others are shown draws from a long-term return distribution—the distribution of 30-year returns, say. Since they have the same decision problem, the two groups of subjects should make similar allocation decisions: subjects who see short-term returns should simply use them to infer the more directly relevant long-term returns. In fact, these subjects allocate substantially less to the risky asset, suggesting that they are simply falling back on the returns that are most accessible to them, namely the short-term returns they were shown. Since losses occur more often in high-frequency data, they perceive the risky asset to be especially risky and allocate less to it.


207

to change over time. Narrow framing is therefore likely to be a permanent feature of preferences, and if it leads the agent to demand a high equity premium today, then it will lead him to demand a high equity premium in the future as well. Suppose, however, that narrow framing instead stems from intuitive thinking and from basing decisions only on “accessible” information. In this case, the agent would be happier with a different decision rule, but has failed to go through the effortful reasoning required to uncover that rule. We would therefore expect the agent’s behavior to change over time, as he learns that his intuitive thinking is leading him astray, and either through his own efforts, or by observing the actions of others, discovers a better decision rule. If accessibility-based narrow framing is driving the equity premium, we would expect the premium to fall over time as investors gradually switch away from narrow framing. Our discussion has treated loss aversion and narrow framing as two distinct phenomena. Recent work, however, suggests that they may form a natural pair, because in those situations where people exhibit loss aversion, they often also exhibit narrow framing. For example, as noted above, the rejection of the 110/100 gamble in (1) points not only to loss aversion, but to narrow framing as well. Kahneman (2003) suggests an explanation for why loss aversion and narrow framing might appear in combination like this. He argues that prospect theory captures the way people act when making decisions intuitively, rather than through effortful reasoning. Since narrow framing is also thought to derive, at least in part, from intuitive decisionmaking, it is natural that prospect theory, and therefore also loss aversion, would be used in parallel with narrow framing.

3. THE EQUITY PREMIUM In this section, we discuss various ways of modeling loss aversion and narrow framing and then demonstrate the advantages, from the perspective of addressing the equity premium puzzle, of a model that incorporates these features of decision-making. Specifically, in Section 3.2, we show that such a model can generate a high equity premium at the same time as a low and stable risk-free rate, even when consumption growth is smooth and only weakly correlated with stock returns; and then, in Section 3.3, that it can do so for preference parameters that correspond to reasonable attitudes to large-scale monetary gambles.

3.1. Modeling Loss Aversion and Narrow Framing Benartzi and Thaler (1995) are the first to apply loss aversion and narrow framing in the context of the aggregate stock market. They consider an investor who is loss-averse over changes in the value of his financial wealth, defined here as holdings of T-bills and stocks. Since financial wealth is just one component of overall wealth—others being human capital and housing wealth—defining utility directly over fluctuations in financial wealth constitutes narrow framing.

208


Benartzi and Thaler (1995) argue that, in equilibrium, their investor charges a high equity premium. The reason is that the high volatility of stock returns leads to substantial volatility in returns on financial wealth. Given that he is more sensitive to losses than to gains, these fluctuations in his financial wealth cause the investor substantial discomfort. As a result, he only holds the market supply of stocks if compensated by a high average return. A weakness of Benartzi and Thaler’s (1995) framework is that, since the investor gets direct utility only from changes in the value of his financial wealth, and none at all from consumption or total wealth, consumption plays no role, making it hard to check how well the model describes the joint properties of stock returns and consumption growth. An important challenge therefore remains: to build and evaluate a more realistic model in which, even if the investor gets utility from fluctuations in the value of one component of his wealth, he also gets some utility from consumption. Barberis, Huang, and Santos (2001) take up this challenge. Before presenting their specification, we introduce the basic economic structure that will apply throughout our essay. At time t, the investor, whose wealth is denoted Wt , chooses a consumption level Ct and allocates his post-consumption wealth, Wt − Ct , across three assets. The first asset is risk-free and earns a gross return of Rf ,t between t and t + 1. The second asset is the stock market, which earns a gross return of RS,t+1 over the same interval, and the third is a non-financial asset, such as human capital or housing wealth, which earns a gross return of RN,t+1 . The investor’s wealth therefore evolves according to Wt+1 = (Wt − Ct )((1 − θS,t − θN,t )Rf ,t + θS,t RS,t+1 + θN,t RN,t+1 ) ≡ (Wt − Ct )RW ,t+1 ,

(4)

where θS,t (θN,t ) is the fraction of post-consumption wealth allocated to the stock market (the non-financial asset) and RW ,t+1 is the gross return on wealth between t and t + 1. A stripped-down version of Barberis, Huang, and Santos’ (2001) framework can be written as follows. The investor maximizes

E0

∞ t=0

1−γ C −γ β t t + b0 β t+1 C t υ(GS,t+1 ) , 1−γ

(5)

subject to the standard budget constraint, where GS,t+1 = θS,t (Wt − Ct )(RS,t+1 − 1), x for x ≥ 0, v(x) = λx for x < 0, where λ > 1, and where C t is aggregate per capita consumption.

(6) (7)

209


The first term inside the brackets in Eq. (5) ensures that, as in traditional models, the investor gets utility directly from consumption. Consumption utility takes the standard, time-additive, power form analyzed by Mehra and Prescott (1985). The parameter β is the time discount factor, while γ > 0 controls the curvature of the utility function. The second term introduces narrow framing and loss aversion. The variable GS,t+1 is the change in the value of the investor’s stock market holdings, computed as stock market wealth at time t, θS,t (Wt − Ct ), multiplied by the net stock market return, RS,t+1 − 1; υ(GS,t+1 ) represents utility from this change in value. Narrow framing is therefore introduced by letting the agent get utility directly from changes in the value of just one component of his total wealth, with b0 controlling the degree of narrow framing. Loss aversion is introduced via the piecewise linear form of υ(·), which makes the investor −γ more sensitive to declines in stock market value than to increases. Finally, C t is a neutral scaling term that ensures stationarity in equilibrium. Equation (6) is the simplest way of specifying the stock market’s “gains” and “losses” that the investor is loss-averse over. Here, so long as θS,t > 0, a positive net return is considered a gain and, from (7), is assigned positive utility; a negative net return is considered a loss and is assigned negative utility. Barberis, Huang, and Santos (2001) work primarily with another, possibly more realistic formulation, GS,t+1 = θS,t (Wt − Ct )(RS,t+1 − Rf ,t ),

(8)

in which a stock market return is only considered a gain, and hence is only assigned positive utility, if it exceeds the risk-free rate. In Section 2, we noted that even though narrow framing has mainly been documented in experimental settings, both the “regret” and “accessibility” interpretations suggest that people may frame the stock market narrowly as well. One could argue that they also suggest that people will frame their non-financial assets narrowly: for example, on the grounds that the distribution of those assets’ returns is also very accessible. The specification in (5) can certainly accommodate such behavior, but we have found that doing so has little effect on our results. For simplicity, then, we assume that only stock market risk is framed narrowly. The preferences in (5) are a simplified version of Barberis, Huang, and Santos’ (2001) specification. In an effort to understand not only the equity premium, but also the volatility and time-series predictability of stock returns, their original model captures not only loss aversion, but also some dynamic evidence on loss aversion, sometimes known as the “house money” effect, whereby prior gains and losses affect current sensitivity to losses. The specification in (5) strips out this dynamic effect, leaving only the core features of loss aversion and narrow framing. We discuss the full model in Section 5.7 7 Barberis, Huang, and Santos (2001) also consider the case in which the investor gets utility from changes in total wealth, rather than in stock market wealth, so that there is no narrow framing.

210


The first-order conditions of optimality for the preferences in (5), (7), and (8) can be derived using straightforward perturbation arguments. They are 1 = βRf ,t Et 1 = βEt RS,t+1

C t+1

−γ ,

(9)

Ct C t+1 Ct

−γ

+ b0 βEt v RS,t+1 − Rf ,t .

(10)

When there is no narrow framing, so that b0 = 0, these equations reduce to those derived from a standard asset pricing model with time-additive power utility over consumption, such as that of Mehra and Prescott (1985). Introducing narrow framing, so that b0 > 0, has no effect on the first-order condition for the risk-free rate, condition (9): consuming a little less today and investing the savings in the risk-free rate does not change the investor’s exposure to losses in the stock market. Narrow framing does, however, introduce a second term in the first-order condition for the stock market, condition (10): consuming less today and investing the proceeds in the stock market exposes the investor to potentially greater disutility from a drop in the stock market. Barberis, Huang, and Santos (2001) assign the preferences in (5), (7), and (8) to the representative agent in a simple endowment economy, and, using conditions (9)–(10), show that, when the model is calibrated to annual data, the narrow framing term can generate a substantial equity premium and a low and stable risk-free rate, even when consumption growth is smooth and only weakly correlated with stock returns. Much as in Benartzi and Thaler (1995), the intuition is that, since the investor gets direct utility from changes in the value of his stock market holdings and is more sensitive to losses than to gains, he perceives the stock market to be very risky and only holds the market supply if compensated by a high average return. Of course, in assigning the utility function in (5) to a representative agent, Barberis, Huang, and Santos (2001) are assuming that the key features of these preferences survive aggregation. Intuitively, if all investors are loss-averse over annual fluctuations in stock market wealth, it is hard to see why this would “wash out” in the aggregate. However, this point has not yet been formalized. While the preference specification in (5) yields a number of insights, it also has some limitations. First, it does not admit an explicit value function. This makes it hard to compute attitudes to independent monetary gambles and therefore to check whether the preference parameters (γ, λ, b0 ) used to generate a high equity premium are reasonable or not. Second, the preferences in (5) are intractable in partial equilibrium settings and so cannot be used to investigate the implications of narrow framing for portfolio choice. Finally, to ensure stationarity, the narrow framing component has to be scaled by an ad-hoc factor based on aggregate consumption. Recently, Barberis and Huang (2004) propose a new preference specification that overcomes these limitations. Their starting point is a non-expected utility formulation

211


known as recursive utility, in which the agent’s time t utility, Vt , is given by Vt = W (Ct , μ(Vt+1 |It )),

(11)

where μ(Vt+1 |It ) is the certainty equivalent of the distribution of future utility, Vt+1 , conditional on time t information It , and W (·, ·) is an aggregator function that aggregates current consumption Ct with the certainty equivalent of future utility to give current utility (see Epstein and Zin (1989), for a detailed discussion). Most implementations of recursive utility assign W (·, ·) the form 1

W (C, y) = ((1 − β)C ρ + βy ρ ) ρ ,

0 < β < 1,

0 = ρ < 1,

(12)

where β is a time discount factor and ρ controls the elasticity of intertemporal substitution. Most implementations also assume homogeneity of μ(·). If a certainty equivalent functional is homogeneous, it is necessarily homogeneous of degree one, so that μ(kz) = kμ(z),

k > 0.

(13)

In its current form, the specification in Eq. (11) does not allow for narrow framing: an investor with these preferences only cares about the outcome of a gamble he is offered to the extent that that outcome affects his overall wealth risk. Barberis and Huang (2004) show, however, that these preferences can be extended to accommodate narrow framing. They specify their utility function in a general context, but for the specific three-asset setting introduced earlier, their formulation reduces to Vt = W (Ct , μ(Vt+1 |It ) + b0 Et (v(GS,t+1 ))),

(14)

where 1

W (C, y) = ((1 − β)C ρ + βy ρ ) ρ , μ(kz) = kμ(z), k > 0,

0 < β < 1,

GS,t+1 = θS,t (Wt − Ct )(RS,t+1 − Rf ,t ), x for x ≥ 0, v(x) = λx for x < 0, where λ > 1.

0 = ρ < 1,

(15) (16) (17) (18)

Relative to the usual recursive specification in Eq. (11), this new formulation maintains the standard assumptions for W (·, ·) and μ(·). The difference is that a new term, which captures loss aversion and narrow framing, has been added to the second argument of W (·, ·). As before, GS,t+1 represents changes in the value of the investor’s stock market holdings, measured relative to the risk-free rate. By letting the investor get direct utility v(GS,t+1 ) from changes in the value of this one component of his wealth, we are

212


introducing narrow framing, with the degree of narrow framing again controlled by b0 . Loss aversion is introduced through the piecewise linearity of v(·), just as in the earlier specification in (5).8 Since our focus is on the effects of narrow framing, we give the certainty equivalent functional μ(·) the simplest possible form, namely 1

μ(z) = (E(zζ )) ζ ,

(19)

where the exponent ζ is set to the same value as the exponent in the aggregator function, ρ. We denote this common value 1 − γ, so that ρ = ζ = 1 − γ.

(20)

3.2. Quantitative Implications We now use the specification in Eq. (14) to illustrate two benefits of the narrow framing approach in more detail: first, that it can generate a high equity premium at the same time as a low and stable risk-free rate, even when consumption growth is smooth and only weakly correlated with stock returns; and in Section 3.3, that it can do so while also making sensible predictions about attitudes to large-scale monetary gambles. To see the first result, consider a simple economy with a representative agent who has the preferences in Eq. (14). As before, there are three assets: a risk-free asset in zero net supply, and two risky assets, a stock market and a non-financial asset, both in positive net supply. Barberis and Huang (2004) show that, in this setting, the first-order conditions of optimality are

1 = βRf ,t Et Et 0=

Ct+1 Ct

−γ βEt

Ct+1 Ct

γ

1−γ

−γ

,

(21)

Et (v(RS,t+1 − Rf ,t )),

(22)

RW ,t+1

−γ Ct+1 (RS,t+1 − Rf ,t ) Ct

−γ

Ct+1 Et Ct

+ b0 Rf ,t

β 1−β

1 1−γ

1 − αt αt

−γ

1−γ

8 It is straightforward to also allow for the narrow framing of the non-financial asset. Doing so does not have a significant effect on our results.

213


Et 0=

−γ Ct+1 (RW ,t+1 − Rf ,t ) Ct

−γ

Ct+1 Et Ct

+ b0 Rf ,t

β 1−β

1 1−γ

1 − αt αt

−γ

1−γ

θS,t Et (v(RS,t+1 − Rf ,t )),

(23)

where αt ≡ Ct /Wt is the consumption to wealth ratio, and where RW ,t+1 is defined in Eq. (4). We consider an equilibrium in which (i) the risk-free rate is a constant Rf ; (ii) consumption growth and stock returns are distributed as Ct+1 = gC + σC εC,t+1 , Ct log RS,t+1 = gS + σS εS,t+1 , log

where

εC,t εS,t

∼N

0 1 , ρCS 0

ρCS 1

, i.i.d. over time;

(24) (25)

(26)

(iii) the consumption to wealth ratio αt is a constant α, which, using RW ,t+1 =

Wt+1 1 Ct+1 = , Wt − Ct 1 − α Ct

(27)

implies that log RW ,t+1 = gW + σW εW ,t+1 ,

(28)

where gW = gC + log σ W = σC , εW ,t+1 = εC,t+1 ;

1 , 1−α

(29) (30) (31)

and (iv) the fraction of total wealth made up by the stock market, θS,t , is a constant over time, θS , so that θS,t =

St = θS , ∀t, S t + Nt

(32)

where St and Nt are the total market value of the stock and of the non-financial asset, respectively. Barberis and Huang (2004) demonstrate that this structure, while

214


restrictive, can be embedded in a general equilibrium framework with endogeneous production. Barberis and Huang (2004) also show that, under this structure, Eqs. (21)–(23) simplify to α=

1−γ 1 2 γ γ 1 − β Rf e(1−γ)σC /2

1 1−γ β 1 b0 Rf

(33)

−γ − α 1−γ gS +σS2 /2 0= e − Rf 1−β α 2 2 + (λ − 1) egS +σS /2 N εS − σS − Rf N εS + egS +σS /2−γσS σC ρCS − Rf , (34)

1

−γ 1−γ β 1 − α 1−γ gS +σS2 /2 − Rf θS e 0 = b0 Rf 1−β α 2 + (λ − 1) egS +σS /2 N εS − σS − Rf N εS +

1 gC +σC2 /2−γσC2 − Rf , e 1−α (35)

where εS =

log Rf − gS . σS

(36)

We use Eqs. (33)–(35) to compute the equilibrium equity premium. First, we set the return and consumption process parameters to the values in Table 1. These values are estimated from annual data spanning the 20th century and are standard in the literature. Then, for given preference parameters β, γ, b0 , and λ, and for a given stock market fraction of total wealth θS , Eqs. (33)–(35) can be solved for α, Rf , and gS , thereby giving us the equity premium. Table 2 presents the results. We take β = 0.98 and θS = 0.2, and consider various values of γ, λ, and b0 . The parameter β has little effect on attitudes to risk; setting it to TABLE 1 Parameter Values for a Representative Agent Equilibrium Model gC and σC are the mean and standard deviation of log consumption growth, σS is the standard deviation of log stock returns, and ρCS is the correlation of log consumption growth and log stock returns. Parameter gC

1.84%

σC

3.79%

σS

20.00%

ρCS

0.10

215


TABLE 2 Equity Premia and Attitudes to Large-scale Gambles The table shows, for given aversion to consumption risk γ, sensitivity to narrowly framed losses λ, and degree of narrow framing b0 , the risk-free rate Rf and equity premium EP generated by narrow framing in a simple representative agent economy. πL is the premium the representative agent would pay, given his equilibrium holdings of risky assets and wealth of $75,000, to avoid a 50:50 bet to win or lose $25,000. Rf (%)

EP(%)

πL ($)

0

4.7

0.12

6,371

0.05

3.7

3.72

6,285

0.10

3.4

4.63

6,269

3

0

4.7

0.12

6,371

3

0.05

2.7

7.00

8,027

1.5

3

0.10

2.3

8.12

8,188

3

2

0

7.1

0.24

11,754

3

2

0.05

5.3

3.29

8,318

3

2

0.10

4.7

4.37

7,383

3

3

0

7.1

0.24

11,754

3

3

0.05

3.3

6.65

8,981

3

3

0.10

2.4

8.08

8,601

γ

λ

1.5

2

1.5

2

1.5

2

1.5 1.5

b0

0.98 ensures that the risk-free rate is not too high. Our results are quantitatively similar for a range of values of θS . The table confirms that narrow framing of stocks can generate a substantial equity premium at the same time as a low risk-free rate, even when, as shown in Table 1, consumption growth is smooth and only weakly correlated with stock returns. For example, the parameter values (γ, λ, b0 ) = (1.5, 2, 0.1) produce an equity premium of 4.63 percent and a risk-free rate of 3.4 percent, while (γ, λ, b0 ) = (1.5, 3, 0.1) produce a premium as high as 8.12 percent with a risk-free rate of only 2.3 percent. The intuition is the same as in Benartzi and Thaler (1995) and Barberis, Huang, and Santos (2001): if the agent gets utility directly from changes in the value of the stock market and, via the parameter λ, is more sensitive to losses than to gains, he perceives the stock market to be very risky and only holds the available supply if compensated by a high average return. The assumption that the agent evaluates stock market gains and losses on an annual basis is important for our results, but not critical. Table 3 reports equity premia for an investor with the preferences in Eq. (14), but who evaluates stock market gains and losses at intervals other than a year. The table shows that, even though the equity premium declines as the interval grows, long evaluation periods can still generate substantial equity premia at the same time as a low risk-free rate.

216


TABLE 3 Equity Premia for Different Evaluation Periods The table shows, for given aversion to consumption risk γ, sensitivity to narrowly framed losses λ, and degree of narrow framing b0 , the risk-free rate Rf and equity premium EP generated by narrow framing in a simple representative agent economy. πL is the premium the representative agent would pay, given his equilibrium holdings of risky assets and wealth of $75,000, to avoid a 50:50 bet to win or lose $25,000. T is the interval, in years, over which stock market gains and losses are measured. T

γ

λ

b0

Rf (%)

EP(%)

πL ($)

0.5

1.5

2

0.1

2.5

7.59

6,257

1

1.5

2

0.1

3.4

4.63

6,269

2

1.5

2

0.1

4.0

2.53

6,288

3

1.5

2

0.1

4.3

1.70

6,301

The intuition for why the equity premium is lower for longer evaluation periods, first pointed out by Benartzi and Thaler (1995), is straightforward. Since the distribution of stock returns has a positive mean, the probability of seeing a drop in the stock market falls as returns are aggregated at longer intervals. While annual stock returns might be negative 40 percent of the time, five-year returns are negative less often. A loss-averse agent is therefore less scared of stocks when he evaluates their returns at longer intervals and, as a result, he demands a lower equity premium.

3.3. Attitudes to Large Monetary Gambles We now demonstrate another attractive feature of the preference specification in Eq. (14), namely that it can deliver a high equity premium for parameterizations that are “reasonable,” in the sense that they correspond to sensible attitudes to independent monetary gambles. This is important because it was, in part, the difficulty researchers encountered in reconciling the equity premium with attitudes to monetary gambles that launched the equity premium literature in the first place. Economists are primarily interested in attitudes to large-scale monetary gambles, so we begin with those. In Section 3.4, we also consider attitudes to small-scale gambles. The literature has suggested a number of thought experiments involving large-scale gambles. Epstein and Zin (1990) and Kandel and Stambaugh (1991) consider an individual with wealth of $75,000 and ask what premium he would pay to avoid a 50:50 chance of losing $25,000 or gaining the same amount; in Kandel and Stambaugh’s (1991) view, a premium of $24,000 is too high, but a premium of $8,333 is more reasonable. Mankiw and Zeldes (1991) think about the value of x for which an agent would be indifferent between certain consumption of $x and a 50:50 chance of $50,000 consumption or $100,000 consumption. Rabin (2000) suggests a mild condition, namely that an agent should accept a clearly attractive large gamble such as a 50:50 bet to win $20 million against a $10,000 loss.

217


It does not matter, for our results, which of these thought experiments we use. In what follows, we focus on the one suggested by Epstein and Zin (1990) and Kandel and Stambaugh (1991). In our view, a reasonable condition to impose is9 Condition L: An individual with wealth of $75,000 should not pay a premium higher than $15,000 to avoid a 50:50 chance of losing $25,000 or gaining the same amount. Barberis and Huang (2004) show that, to avoid a gamble g offering an equal chance to win or lose x, an investor with the preferences in Eq. (14) would pay a premium equal to 1 A Wt − (E(Wt + g)1−γ ) 1−γ + b0 x2 (λ − 1) , (37) π= A + b0 λ where γ

A = (1 − β) 1−γ α − 1−γ , 1

(38)

with α already computed in Eqs. (33)–(35). In this calculation, they make the simplest possible assumption, namely that, whatever degree of narrow framing b0 and level of loss aversion λ the investor uses when thinking about stock market risk, he also uses when thinking about the independent monetary gamble g. When b0 = 0, Eq. (37) gives the premium that would be charged by an agent with standard power utility preferences. When b0 > 0, the premium in Eq. (37) reflects the fact that, to some extent, the investor is framing gamble g narrowly. For large b0 , Eq. (37) reduces to π=

x (λ − 1), 2λ

(39)

the premium that would be charged by an agent who evaluates gamble g completely in isolation and who is λ times as sensitive to losses as to gains. Using Eq. (37), the rightmost columns in Tables 2 and 3 show, for each parameterization, the amount that the representative agent would pay, given his equilibrium holdings of risky assets, to avoid the symmetric bet in condition L. The rows in which b0 = 0 reproduce a well-known result; that for power utility preferences, those values of γ low enough to make sensible predictions about attitudes to large-scale monetary gambles inevitably generate too low an equity premium. Table 2 shows, however, that as soon as narrow framing is allowed—in other words, as soon as b0 > 0—it is easy to find parameterizations that give a high equity premium while also satisfying condition L. When (γ, λ, b0 ) = (1.5, 2, 0.1), for example, the investor charges a substantial equity premium of 4.63 percent, and a reasonable $6,269 to avoid the ±$25,000 gamble. How is it that the preference specification in Eq. (14) can reconcile attitudes to stock market risk and to the large-scale monetary gamble in condition L when other specifications have trouble doing so? To see how, note first that, in the simple representative 9 We

use the label “condition L” to emphasize that we are thinking about large-scale gambles.

218


agent economy described by conditions (i)–(iv) in Section 3.2, the equity premium is determined by the agent’s attitude, in equilibrium, to adding a small amount of stock market risk to a portfolio that is only weakly correlated with the stock market. Why can we say “weakly” correlated? Since representative agent economies are calibrated to aggregate data, the correlation of stock returns and consumption growth, ρCS , must be set to a low value; given that the consumption to wealth ratio is constant, this immediately implies a low correlation between stock returns and returns on total wealth.10 To generate a substantial equity premium, then, we need the agent to be strongly averse or, at the very least, moderately averse, to a small, weakly correlated gamble. To satisfy condition L, we need the agent to be mildly averse or, at most, moderately averse, to a large, independent gamble. Now consider the two functions in the second argument of W (·, ·) in Eq. (14), namely μ(·) and v(·). For a γ of 1.5, the μ(·)-term, by virtue of its local risk-neutrality, produces only mild aversion to a small, weakly correlated gamble, but moderate aversion to a large, independent gamble. For a λ of 2, the v(·)-term, by virtue of being piecewise linear, produces moderate aversion both to a small, weakly correlated gamble and to a large, independent gamble. For a degree of narrow framing b0 that is high enough, the two terms therefore generate moderate aversion to a small, weakly correlated gamble—thereby giving a substantial equity premium—and moderate aversion to a large, independent gamble, thereby satisfying condition L.

3.4. Attitudes to Small Monetary Gambles In Section 3.3, we saw that the preferences in Eq. (14), capturing both loss aversion and narrow framing, can generate a large equity premium for preference parameters that also correspond to sensible attitudes to large-scale monetary gambles, in that they satisfy condition L. In fact, condition L does not put very sharp restrictions on the range of equity premia that we can generate: as Table 2 shows, it can be consistent with premia as low as 0.12 percent or as high as 8.12 percent. In this section, we show that by requiring the preference specification in Eq. (14) to also make sensible predictions about attitudes to small-scale gambles, we can put tighter bounds on the range of equity premia that narrow framing can plausibly generate. The intuition is straightforward. As argued earlier, in the simple representative agent economy of Section 3.2, the equity premium is determined by the agent’s attitude, in equilibrium, to adding a small amount of weakly correlated stock market risk to the rest of his portfolio. If we impose constraints on the investor’s attitude to a small, independent risk, it is likely that we will also constrain his attitude to a small, weakly correlated risk and thereby, also, the equity premium he will charge. What condition should we impose on attitudes to small-scale gambles? As with large-scale gambles, the earlier literature has suggested a number of relevant thought experiments. For consistency with our earlier discussion, we return to Epstein and Zin 10 Of course, in more general representative agent economies, the consumption to wealth ratio need not be constant. So long as the ratio is sufficiently stable, however, it should still follow that stock returns and returns on total wealth are only weakly correlated.

219


PREFERENCES WITH NARROW FRAMING

0.1 0.09 0.08 0.07

b0

0.06 0.05 0.04 0.03 0.02 0.01 0

1.5

2

2.5 ␭

3

3.5

4

FIGURE 1

The “×” signs mark the parameter values for which an agent who is loss-averse over stock market risk would charge an equity premium higher than 5 percent in a simple representative agent economy. The “+” signs mark the parameter values for which the agent would pay a premium below $40 to avoid a 50:50 bet to win or lose $250 at a wealth level of $75,000.

(1990), who ask how much an individual with wealth of $75,000 would pay to avoid a 50:50 bet to lose $250 or to win the same amount. In our view, a reasonable condition to impose here is11 Condition S: An individual with wealth of $75,000 should not pay a premium higher than $40 to avoid a 50:50 chance of losing $250 or gaining the same amount. Figure 1 shows how condition S sharply restricts the range of equity premia that can be generated by the preferences in Eq. (14). The “×” signs show, for γ = 1.5, the values of λ and b0 that produce equity premia higher than 5 percent. Clearly, either a high sensitivity to losses λ or a high degree of narrow framing b0 , or both, is required to generate equity premia as large as 5 percent. Note that our earlier condition on attitudes to large-scale gambles, condition L, is satisfied by all values of λ and b0 spanned by the graph—in other words, by all pairs (λ, b0 ) ∈ [0, 4] × [0, 0.1]. If condition L were 11 We

use the label “condition S” to emphasize that we are thinking about small-scale gambles.

220


TABLE 4 Equity Premia and Attitudes to Small-scale Gambles The table shows, for given aversion to consumption risk γ, sensitivity to narrowly framed losses λ, and degree of narrow framing b0 , the risk-free rate Rf and equity premium EP generated by narrow framing in a simple representative agent economy. πL (πS ) is the premium the representative agent would pay, given his equilibrium holdings of risky assets and wealth of $75,000, to avoid a 50:50 bet to win or lose $25,000 ($250). γ

λ

b0

Rf (%)

EP(%)

πL ($)

πS ($)

1.5

2

0.035

3.8

3.14

6,297

38.7

1.5

3

0.012

3.8

3.06

7,252

37.7

3

2

0.050

5.3

3.29

8,318

39.5

3

3

0.016

5.4

3.10

10,259

37.1

the only condition constraining our choice of preference parameters, we could therefore easily obtain premia higher than 5 percent. The “+” signs in the figure show the values of λ and b0 that satisfy condition S. Imposing this condition severely restricts the range of feasible values of λ and b0 . In fact, we cannot obtain an equity premium as high as 5 percent without violating it. Even though condition S does restrict the feasible parameter set, it still allows for sizable equity premia. Table 4 lists some parameter values that satisfy both condition L and condition S and yet still produce equity premia above 3 percent.

3.5. The Importance of Narrow Framing While narrow framing is admittedly an unusual feature of preferences, it is crucial to our results. To demonstrate this, we now show that, in the absence of narrow framing, it becomes much harder to replicate some of the attractive features of the preferences in Eq. (14)—much harder, for example, to reconcile a high equity premium with reasonable attitudes to large-scale monetary gambles and, in particular, with the attitudes imposed by condition L. Consider a model in which the agent is loss-averse over annual changes in total wealth, rather than in stock market wealth. Such a model maintains the assumptions of loss aversion and of annual evaluation of gains and losses, but by changing the focus from gains and losses in stock market wealth to gains and losses in total wealth, it removes the narrow framing. One such preference specification is the following: Vt = W (Ct , μ(Vt+1 |It )),

(40)

where 1

W (C, y) = ((1 − β)C ρ + βy ρ ) ρ ,

0 < β < 1,

0 = ρ < 1,

(41)

and where the certainty equivalent functional μ(·) takes a form proposed by Gul (1991), often referred to as “disappointment aversion”:

221


μ(V )1−γ = E(V 1−γ ) + (λ − 1)E((V 1−γ − μ(V )1−γ )1(V < μ(V ))), 0 < γ = 1, λ > 1.

(42)

While the specification in (42) looks somewhat messy, it is simply a function with a kink, which makes the investor more sensitive to losses than to gains. The parameter λ controls the relative sensitivity to losses.12 We consider a simple economy with a representative agent who has the preferences in Eqs. (40)–(42). The market structure is the same as before. There are three risky assets: a risk-free asset, in zero net supply, and two risky assets, a stock market and a non-financial asset, each in positive net supply. Epstein and Zin (1989) show that the first-order conditions of optimality are

ρ−1 1 ρ 1 C t+1 ρ RW ,t+1 , (43) 0 = Et φ β ρ Ct

ρ−1

ρ−1 1 ρ ρ 1 C C t+1 t+1 ρ RW ,t+1 (RS,t+1 − Rf ,t ) , (44) 0 = Et φ β ρ Ct RW ,t+1

ρ−1

ρ−1 1 ρ ρ 1 C C t+1 t+1 ρ 0 = Et φ β ρ RW ,t+1 (RW ,t+1 − Rf ,t ) , (45) Ct RW ,t+1 where

⎧ 1−γ −1 x ⎪ ⎨ for x ≥ 1, 1 − γ φ(x) = (46) 1−γ −1 ⎪ ⎩λx for x < 1. 1−γ We look for a simple equilibrium in which conditions (i)–(iii) of Section 3.2 hold.13 Under these conditions, Eqs. (43)–(45) become

1−γ

1−γ ρ ρ β β 0= C1 − 1 + (λ − 1) C1 N ( εC − (1 − γ)σC ) 1−α 1−α − (λ − 1)N ( εC ),

(47)

0 = C2 − Rf C3 + (λ − 1)(C2 N ( εC − σS ρCS + γσC ) − Rf C3 N ( εC + γσC )), 0=

1 C1 − Rf C3 1−α

C1 + (λ − 1) N ( εC − (1 − γ)σC ) − Rf C3 N ( εC + γσC ) , 1−α

(48)

(49)

12 Epstein and Zin (2001) and Ang, Bekaert, and Liu (2005) discuss the implementation of disappointment aversion in dynamic environments. 13 It is straightforward to show that such a structure can be embedded in a general equilibrium model.

222


where

1 C1 = exp (1 − γ)gC + (1 − γ)2 σC2 , 2

1 2 2 2 C2 = exp gS − γgC + σ − 2γσS σC ρCS + γ σC , 2 S

1 C3 = exp −γgC + γ 2 σC2 , 2

1 1 β εC = − gC + log , σC ρ 1−α

(50) (51) (52) (53)

and where N (·) is the cumulative normal distribution function. We use Eqs. (47)–(49) to compute the equilibrium equity premium. As before, we set the return and consumption process parameters to the values in Table 1. Then, for given preference parameters β, ρ, and γ, we use Eq. (47) to compute the consumption to wealth ratio α, Eq. (49) to compute the risk-free rate Rf , and Eq. (48) to compute the mean log stock return gS . To check whether the parameters corresponding to any particular equity premium are reasonable—in other words, whether they satisfy condition L—we need to know the premium an agent with the preferences in Eqs. (40)–(42) would pay to avoid a gamble to win or lose x with equal chance. Following the analysis in Epstein and Zin (1989), it can be shown that the premium π is given by 1 ⎛

1−γ

1−γ ⎞1−γ x x +λ 1− ⎜ 1+ W ⎟ Wt π t ⎜ ⎟ =1−⎜ (54) ⎟ . Wt 1+λ ⎝ ⎠ The parameters β and ρ have little effect on attitudes to risk. We set them to 0.95 and −1, respectively, to ensure that the risk-free rate is not too high. The area shaded with “+” signs in Figure 2 shows the values of γ and λ for which the representative agent satisfies condition L; in other words, the values for which, given his equilibrium holdings of risky assets and wealth of $75,000, he pays a premium below $15,000 to avoid a 50:50 chance of losing $25,000 or winning the same amount. The area shaded with “×” signs shows the values of γ and λ for which the representative agent charges an equity premium higher than 2 percent. There is no overlap between the two regions: in fact, the largest equity premium that we can generate with this preference specification under condition L is 0.93 percent, far smaller than the equity premia derived from narrow framing in Table 2.14 14 Epstein and Zin (1990) and Epstein and Zin (2001) obtain comparable results. For example, in Table 5 of their paper, Epstein and Zin (2001) report that, for parameterizations of the preferences in (40)–(42) that match the historical equity premium, the representative agent may charge as much as $23,000 to avoid the ±$25, 000 bet. See also Bekaert, Hodrick, and Marshall (1997).

223


DISAPPOINTMENT AVERSION PREFERENCES 10 9 8 7 equity premium . 2%

␭

6 5 4

pay ,$15,000 to avoid a 6$25,000 gamble at W 5 $75,000

3 2 1 1

2

3

4

5

6

7

8

9

10

␥ FIGURE 2 The “×” signs mark the parameter values for which an agent with a recursive utility function with Gul (1991)-type certainty equivalent would charge an equity premium higher than 2 percent in a simple representative agent economy. The “+” signs mark the parameter values for which the agent would pay a premium below $15,000 to avoid a 50:50 bet to win or lose $25,000 at a wealth level of $75,000.

To see the intuition for this result, recall from Section 3.3 that, in the simple representative agent economy considered here, the equity premium is determined by the agent’s attitude, in equilibrium, to adding an extra dollar of stock market risk to a portfolio that is only weakly correlated with the stock market. In the absence of narrow framing, the agent evaluates this extra risk by merging it with his other risks and checking if the combination is attractive. Since the stock market is only weakly correlated with his other risks, it diversifies those other risks and so the combination is attractive: even a loss-averse agent enjoys diversification. As a result, he charges a low equity premium. To generate a large premium, we would need to push up aversion to overall wealth risk, but this would immediately lead to a violation of condition L. As we saw in Section 3.3, a simple way out of this difficulty is to argue that, when the agent evaluates stock market risk, he does not fully merge it with his other risks but, rather, evaluates it in isolation; in other words, that he frames stock market risk narrowly.

224


4. OTHER APPLICATIONS Barberis, Huang, and Thaler (2006) argue that the preferences in Eq. (14) can also address a portfolio puzzle that is closely related to the equity premium puzzle, namely the stock market participation puzzle: the fact that, even though stocks have a high mean return, many households have historically been unwilling to allocate any money to them. Mankiw and Zeldes (1991) report, for example, that in 1984, only 28 percent of households held any stock at all, and only 12 percent held more than $10,000 in stock. Non-participation was not simply the result of not having any liquid assets. Even among households with more than $100,000 in liquid assets, only 48 percent held stocks (see also Haliassos and Bertaut (1995)). One approach to this puzzle is to argue that there are transaction costs of investing in the stock market; another is to examine whether non-stockholders have background risk that is somewhat correlated with the stock market (Heaton and Lucas (1997, 2000), Vissing-Jorgensen (2002)). A third approach is preference-based, and this is the one Barberis, Huang, and Thaler (2006) focus on. Specifically, they show that the preferences in Eq. (14) can generate stock market non-participation and, mirroring the results for the equity premium, that they can do so for preference parameterizations that are reasonable, in other words, that make sensible predictions about attitudes to large-scale monetary gambles by, for example, satisfying condition L. It is easy to see how the preferences in (14) generate non-participation: if the agent gets direct utility from fluctuations in the value of any stocks that he owns, and if he is loss-averse over these fluctuations, he is naturally going to be averse to stock market risk and may refuse to participate. How is it that Barberis, Huang, and Thaler (2006) generate non-participation for reasonable parameter values? An agent who refuses to participate in the stock market is effectively refusing to take on a small amount of a risk that is, according to Heaton and Lucas’ (2000) estimates, relatively uncorrelated with his other risks. To generate such attitudes at the same time as reasonable attitudes to large-scale gambles, we need preferences that generate moderate aversion to a small, weakly correlated risk—thereby leading to stock market non-participation—at the same time as moderate aversion to a large, independent risk, thereby satisfying condition L. As discussed in Section 3.3, the preferences in Eq. (14) can achieve exactly this. Without narrow framing, it becomes much harder to find preference specifications that can generate non-participation for reasonable parameter values. In the absence of narrow framing, the agent decides whether to participate by mixing a small amount of stock market risk with his other risks and checking whether the combination is attractive. Since stock market risk is only weakly correlated with his other risks, it is diversifying, and so the combination is, quite generally, attractive. To prevent the agent from participating, we need to impose very high aversion to overall wealth risk, but this typically leads to implausible aversion to large-scale gambles and, in particular, to violations of condition L. This logic has been confirmed by Heaton and Lucas (2000), Haliassos and Hassapis (2001), and Barberis, Huang, and Thaler (2006), who consider a number of different specifications without narrow framing—including


225

specifications that incorporate loss aversion—and find that all of them have trouble generating non-participation for reasonable parameter values.15

5. FURTHER EXTENSIONS 5.1. Dynamic Aspects of Loss Aversion In equilibrium, the preferences in (5) and (14) can easily deliver a high equity premium and a low and stable risk-free rate, but they have a harder time matching the empirical volatility of returns. Under these preferences, the volatility of returns is typically very similar to the volatility of dividend growth and is therefore too low. Barberis, Huang, and Santos (2001) show that incorporating dynamic aspects of loss aversion into the specification in (5) can help match the empirical volatility of returns.16 Drawing on a number of different experimental tests, Thaler and Johnson (1990) argue that the degree of loss aversion is not constant over time, but depends on prior gains and losses. In particular, they present evidence that losses are less painful than usual after prior gains, perhaps because those gains cushion any subsequent loss, and that losses after prior losses are more painful than usual, perhaps because people have only limited capacity for dealing with bad news. Barberis, Huang, and Santos (2001) capture this evidence by making v(·) in (5) a function not only of the current stock market return RS,t+1 but also of prior gains and losses in the stock market. They then show that this raises the volatility of returns relative to the volatility of dividend growth: on good dividend news, the stock market goes up, giving the investor a cushion of prior gains and making him less sensitive to future losses; as a result, he perceives stocks to be less risky and discounts their future cash flows at a lower rate, thereby pushing prices still higher and raising the volatility of returns. The same mechanism also generates predictability in the time series: after prior gains, the investor perceives the stock market to be less risky and so pushes the price of stocks up relative to dividends; but from this point on, average returns will be lower, as the investor needs less compensation for the lower perceived risk. Price-dividend ratios therefore predict returns. One attractive feature of this mechanism is that it preserves the low correlation of stock returns and consumption growth seen in the models of Section 3: since movements in the price-dividend ratio are driven by innovations to dividends, the correlation of stock returns and consumption growth is similar to the correlation of dividend growth and consumption growth and is therefore low. This contrasts with other models of stock market volatility, such as that of Campbell and Cochrane (1999), in which movements in the price-dividend ratio are driven by innovations to consumption. 15 An alternative preference-based approach to the stock market participation puzzle is based on ambiguity aversion (Epstein and Schneider (2002)). This approach has some similarities to the narrow framing approach, in that it works by inducing something akin to loss aversion over the stock market gamble itself. 16 A similar analysis can be conducted with the specification in Eq. (14).

226


These models inevitably predict a high correlation of stock returns and consumption growth.

5.2. Other Forms of Narrow Framing In the economy described in Section 3, there were only two risky assets: the stock market and a non-financial asset. There were therefore only a limited number of ways in which narrow framing could manifest itself. The investor could get direct utility from stock market fluctuations, direct utility from fluctuations in the value of the nonfinancial asset, or both. A more realistic model would allow the investor to trade not only a broad stock market index, but individual stocks as well. Narrow framing could then, in principle, mean that the investor gets direct utility from fluctuations in the value of individual stocks that he owns. What effect would this have? Barberis and Huang (2001) investigate this issue by extending the preferences in (5) to allow the agent to frame several assets narrowly.17 Among other implications, they find that, if investors engage in the more extreme form of narrow framing whereby they frame even individual stocks narrowly, the equity premium can be even higher than in the case studied in Section 3, where they frame only their overall portfolio of stocks narrowly: if investors worry about fluctuations in highly volatile individual stocks rather than about fluctuations in the less volatile aggregate stock market, they perceive stocks to be very risky and charge a very high equity premium in equilibrium. Is it plausible that people might frame individual stocks narrowly? From a theoretical perspective, it is hard to tell. Consider Kahneman’s (2003) “accessibility” theory of framing. It is true that for most investors, information about the return distribution of individual stocks is highly accessible. But so too is information about the return distribution of diversified stock portfolios, and it seems that given a choice between the broader frame and the narrower one, people will choose the normatively more acceptable frame, namely the broader one, for their decision-making. Under the alternative theory that narrow framing is related to non-consumption utility such as regret, framing at the level of individual stocks becomes more plausible. If one of the investor’s stocks performs poorly, he may regret the specific decision to buy that stock. Gains and losses on individual stocks can therefore be carriers of utility in their own right, and the investor may take this into account when making decisions. The framing of individual stocks is also supported by the well-known disposition effect—the fact that, when individual investors sell stocks in their portfolios, they tend to sell stocks that have gone up in value since purchase, rather than stocks that have gone down in value (Shefrin and Statman (1985), Odean (1998)). A common explanation for this finding is that people get direct utility from realizing a loss on an individual stock that they own and that this leads them to postpone selling a losing stock for as long as possible.18 17 A

similar analysis can be conducted with the specification in Eq. (14). considerations point to the selling of prior losers, so they cannot explain the disposition effect. Nor can the effect be explained by a rational belief in mean reversion: the stocks that individual investors sell actually outperform the ones they buy (Odean (1998)).

18 Tax


227

6. CONCLUSION AND FUTURE DIRECTIONS In this essay, we review a recent approach to addressing the equity premium puzzle. The broad theme of this approach is that we may be able to improve our understanding of how people evaluate stock market risk by looking at how they evaluate risk in experimental settings. Specifically, this approach argues that loss aversion and narrow framing, two of the most important ideas to emerge from the experimental literature on decision-making under risk, may also play an important role in the way some people think about the stock market. We discuss various ways of incorporating loss aversion and narrow framing into more traditional utility functions and show that models with these features may indeed offer an attractive way of thinking about the historical equity premium. For example, they can generate a high equity premium and a low and stable risk-free rate, even when consumption growth is smooth and only weakly correlated with the stock market; moreover, they can do this for parameter values that are reasonable, in other words, parameter values that correspond to sensible attitudes to independent monetary gambles. A parallel result holds in the case of the stock market participation puzzle: loss aversion and narrow framing can generate non-participation for reasonable parameter values. While these initial results are promising, much work remains. The most obvious direction for future research is to think about testable implications of the loss aversion/narrow framing view. For example, while narrow framing makes a blanket prediction of non-participation in the stock market, does it also make more detailed predictions about what kinds of people are more likely to participate than others? Does it predict changes in participation over time, perhaps due to changes in framing? Are there any real-world situations in which people are asked to make a certain financial decision after seeing some data, and that have the feature that, while everyone sees the same data, some people see it presented somewhat differently than others? The differences in the way the data is presented could lead people to frame future outcomes differently, and therefore to make different choices. Researchers have already begun testing the view of the equity premium and participation puzzles presented here. Dimmock (2005) describes a survey, conducted in the Netherlands, in which subjects were given a decision problem involving riskless choice. Responses to this decision problem can be used to extract estimates of individual loss aversion. After extracting these estimates from the data, Dimmock (2005) finds that individuals with greater loss aversion are indeed less likely to participate in the stock market. Narrow framing is harder to measure than loss aversion, but successful tests of narrow framing in other settings suggest that progress can also be made in the context of the equity premium and stock market participation puzzles. Kumar and Lim (2004), for example, test the idea that narrow framing is behind the disposition effect by checking whether individual investors who engage less in narrow framing also exhibit the disposition effect less. They identify these investors as those who tend to execute more than one trade on any given day, and who might therefore pay less attention to the outcome of

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any one transaction. They find that these investors do indeed exhibit less of a disposition effect. Our attempt to bring psychology into economics has also served to highlight some areas of psychology where more research would be valuable. While there is evidence that, in some situations, people frame narrowly, we still do not fully understand when people frame narrowly and when they do not, nor what the underlying causes of narrow framing are. Similarly, while loss aversion itself is a robust and well-documented phenomenon, much less is known about its dynamic aspects: for example, about how past gains and losses affect subsequent loss aversion. Thaler and Johnson (1990) provide some valuable evidence on this point, but it is hard to believe that theirs is the last word. A better understanding of these issues, perhaps through more experimental research, may eventually help us craft better models of how people evaluate stock market risk.

References Ang, A., G. Bekaert, and J. Liu. Why stocks may disappoint. Journal of Financial Economics 76 (2005): 471–508. Barberis, N., and M. Huang. Mental accounting, loss aversion, and individual stock returns. Journal of Finance 56 (2001): 1247–1292. Barberis, N., and M. Huang. Preferences with frames: A new utility specification that allows for the framing of risks. Working paper, Yale University (2004). Barberis, N., and M. Huang. Stocks as lotteries: The implications of probability weighting for security prices. Working paper, Yale University (2005). Barberis, N., M. Huang, and T. Santos. Prospect theory and asset prices. Quarterly Journal of Economics 116 (2001): 1–53. Barberis, N., M. Huang, and R. Thaler. Individual preferences, monetary gambles, and stock market participation: A case for narrow framing. American Economic Review 96 (2006): 1069–1090. Barberis, N., and W. Xiong. What drives the disposition effect? An analysis of a long-standing preferencebased explanation, Working paper, Yale University (2005). Bekaert, G., R. Hodrick, and D. Marshall. The implications of first-order risk aversion for asset market risk premiums. Journal of Monetary Economics 40 (1997): 3–39. Benartzi, S., and R. Thaler. Myopic loss aversion and the equity premium puzzle. Quarterly Journal of Economics 110 (1995): 75–92. Campbell, J. Y., and J. Cochrane. By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107 (1999): 205–251. Cicchetti, C., and J. Dubin. A microeconometric analysis of risk aversion and the decision to self insure. Journal of Political Economy 102 (1994): 169–186. Cohen, A., and L. Einav. Estimating risk preferences from deductible choice. Working paper, Stanford University (2005). Dimmock, S. Loss aversion and household portfolio choice. Working paper, Michigan State University (2005). Epstein, L., and M. Schneider. Learning under ambiguity. Working paper, University of Rochester (2002). Epstein, L., and S. Zin. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical perspective. Econometrica 57 (1989): 937–969. Epstein, L., and S. Zin. First-order risk aversion and the equity premium puzzle. Journal of Monetary Economics 26 (1990): 387–407. Epstein, L., and S. Zin. The independence axiom and asset returns. Journal of Empirical Finance 8 (2001): 537–572. Gneezy, U., and J. Potters. An experiment on risk taking and evaluation periods. Quarterly Journal of Economics 112 (1997): 631–645.


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Gomes, F. Portfolio choice and trading volume with loss averse investors. Journal of Business 78 (2005): 675–706. Gul, F. A theory of disappointment aversion. Econometrica 59 (1991): 667–686. Haliassos, M., and C. Bertaut. Why do so few hold stocks? Economic Journal 105 (1995): 1110–1129. Haliassos, M., and C. Hassapis. Non-expected utility, saving, and portfolios. Economic Journal 111 (2001): 69–102. Heaton, J., and D. Lucas. Market frictions, savings behavior, and portfolio choice. Macroeconomic Dynamics 1 (1997): 76–101. Heaton, J., and D. Lucas. Portfolio choice in the presence of background risk. Economic Journal 110 (2000): 1–26. Kahneman, D. Maps of bounded rationality: Psychology for behavioral economics. American Economic Review 93 (2003): 1449–1475. Kahneman, D., A. Knetsch, and R. Thaler. Experimental tests of the endowment effect and the Coase theorem. Journal of Political Economy 98 (1990): 1325–1348. Kahneman, D., and D. Lovallo. Timid choices and bold forecasts: A cognitive perspective on risk taking. Management Science 39 (1993): 17–31. Kahneman, D., and A. Tversky. Prospect theory: An analysis of decision under risk. Econometrica 47 ( 1979): 263–291. Kahneman, D., and A. Tversky. The psychology of preferences. Scientific American 246 (1982): 160–173. Kahneman, D., and A. Tversky. Choices, values, and frames. American Psychologist 39 (1983): 341–350. Kandel, S., and R. Stambaugh. Asset returns and intertemporal preferences. Journal of Monetary Economics 27 (1991): 39–71. Kumar, A., and S. Lim. One trade at a time: Narrow framing and stock investment decisions of individual investors. Working paper, University of Notre Dame (2004). Mankiw, N. G., and S. Zeldes. The consumption of stockholders and non-stockholders. Journal of Financial Economics 29 (1991): 97–112. Mehra, R., and E. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15 (1985): 145–161. Odean, T. Are investors reluctant to realize their losses? Journal of Finance 53 (1998): 1775–1798. Rabin, M. Risk aversion and expected utility theory: A calibration theorem. Econometrica 68 (2000): 1281–1292. Rabin, M., and R. Thaler. Anomalies: Risk aversion. Journal of Economic Perspectives 15 (2001): 219–232. Read, D., G. Loewenstein, and M. Rabin. Choice bracketing. Journal of Risk and Uncertainty 19 (1999): 171–197. Redelmeier, D., and A. Tversky. On the framing of multiple prospects. Psychological Science 3 (1992): 191–193. Shefrin, H., and M. Statman. The disposition to sell winners too early and ride losers too long. Journal of Finance 40 (1985): 777–790. Thaler, R. Towards a positive theory of consumer choice. Journal of Economic Behavior and Organization 1 (1980): 39–60. Thaler, R., and E. Johnson. Gambling with the house money and trying to break even: The effects of prior outcomes on risky choice. Management Science 36 (1990): 643–660. Thaler, R., A. Tversky, D. Kahneman, and A. Schwartz. The effect of myopia and loss aversion on risk taking: An experimental test. Quarterly Journal of Economics 112 (1997): 647–661. Tversky, A., and D. Kahneman. The framing of decisions and the psychology of choice. Science 211 (1981): 453–458. Tversky, A., and D. Kahneman. Rational choice and the framing of decisions. Journal of Business 59 (1986): 251–278. Tversky, A., and D. Kahneman. Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty 5 (1992): 297–323. Vissing-Jorgensen, A. Towards an explanation of household portfolio choice heterogeneity: Nonfinancial income and participation cost structures. Working paper, Northwestern University (2002).

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Discussion: The Loss Aversion/Narrow Framing Approach to the Equity Premium Puzzle Xavier Gabaix∗ New York University and NBER

This is a great research program. It is motivated by clear evidence on decision making, for first-order risk aversion (a particular case of which is loss aversion), and narrow framing. The Barberis, et al. (2001) paper is on many syllabi as an example of a workedout, general equilibrium behavioral finance theory. Barberis and Huang’s chapter shows further progress. One could wish more people were working on general equilibrium, behavioral finance—or macro, for that matter. One reason why this is the case is that technically tooled people seem to typically prefer the rational framework, perhaps because it offers more clearly formulated technical challenges to solve. I will try to spell out some research questions that seem directly doable with the available tools. I will not talk about the dimensions that require drastic new conceptual innovations, for instance those linked to bounded rationality, even though, of course, they are the most crucial ones in many respects.

1. WORK OUT MORE SYSTEMATICALLY THE PREFERENCES OF PT VS. EU INVESTORS—THE “EQUITY PROTECTION PUZZLE” In the Barberis and Huang framework, there is just one type of agent. In the real world, there are likely to be at least two types of agents: prospect theoretic (perhaps, ∗ I thank Rajnish Mehra for inviting me to write this piece, and Nick Barberis and Ming Huang for their feedback. email: [email protected].

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individuals) and expected utility (perhaps, institutions). This allows us to predict some patterns of trading. For instance, the prospect theory (PT) individual should buy investment protection from the expected utility (EU) agents, so as to diminish their losses in the bad states of the world. To see the point more clearly, suppose now that there are two agents, one EU, with utility function u(c), and one PT, with v(c). At time 0, agents can trade a complete set of financial assets. At time 1, the state of the world is realized (for instance, stock market returns), and they consume cEU(z) and cP T(z), respectively, the consumption of the EU and PT agents. As they trade at time 0, the ratio of their marginal utilities at time 1 will be equalized, i.e., there is a ξ such that, for all states z of the world, u cEU(z) = ξv cP T(z) . For concreteness, we use the functional form u (c) = c−γ . The utility of the PT agent is equal to v(c) = u(c), c > c∗ = u(c∗ ) + λ(u(c) − u(c∗ )),

c ≤ c∗ ,

where c∗ is the reference point, and λ > 1 the coefficient of loss aversion. Goldstein, et al. (2006) use this utility function, and also the idea of complete markets, albeit without interaction with PT agents. The function v is different from the one studied by Barberis and Huang, and it is chosen for tractability. v exhibits loss aversion, but it is concave; hence, it does not generate risk seeking in the domain of losses. That is probably a good feature when dealing with equilibrium. If the PT function v is convex in the domain of losses, and if the probabilities are linearly weighted, then rather than accepting a loss, a PT agent prefers to randomize, with gambles that can make him lose all his wealth with positive probability. Often the equilibrium does not even exist. A solution may involve appealing to a non-linear probability weighting function, but then tractability breaks down, and new difficulties arise. Hence, it is hard, conceptually and technically, to reconcile a convex part of the PT value function with a market equilibrium. However, with the concave function v above, everything is well defined. Defining Λ = λ1/γ and κ = ξ 1/γ , we get c > C ∗ , cP T/cEU = κ for cP T > C ∗ , and P T EU c /c = Λκ for cP T < C ∗ . All in all, we get cP T = κcEU = C∗ = ΛκC EU

for cEU ∈ [C ∗/κ, ∞] for cEU ∈ [C ∗/(Λ), C ∗ /κ] for cEU ∈ [0, C ∗/(2κ)],

which Figure 1 plots. Market prices, and budget constraints, affect the consumption ratios via just one number, κ. Compared to their normal consumption, the PT agent consumes Λ (say, twice) as much, in the states of the world where loss aversion matters, as the EU agent. On top of that, there is a whole range of events where the PT agent consumes exactly the reference


232 c PT

Slope 5 k C*

Slope 5 Lk C */(Lk )

C */k

c EU

FIGURE 1

Relation between the consumptions of prospect theory (PT) and expected utility (EU) agents, when they can trade a complete set of financial assets.

amount of consumption, C ∗ . In other terms, the EU agents sells an “equity protection” to the PT agent, of the specific type above. This seems like a robust prediction. Does it hold empirically? We do observe some of that, in the form of “equity linked” or “investment protected” assets. But the market looks particularly small. Prima facie, this is surprising. One might call a “equity protection puzzle” the fact that, contrary to PT, people don’t seem to trade a lot of equity protection. Of course, a similar remark can be applied to other behavioral biases: often, it seems that there are fewer market responses to the biases that one might predict. For instance, there are few debiasing services. Relatively few institutions offer self-control devices— housing does, but perhaps as a side effect of an institution that exists for other reasons. Perhaps this just reflects a lack of awareness for the bias on the consumers’ part, or people’s lack of confidence in their own ability to maximize their PT function. Still, this is puzzling.

2. MAKE QUANTITATIVE PREDICTIONS, PARTICULARLY ABOUT EQUILIBRIUM MARKET PHENOMENA, RATHER THAN JUST ABOUT INDIVIDUAL TRADING BEHAVIOR Barberis and Huang offer predictions, but they are mostly about loss aversion and the amount of stocks held at the individual level. As loss aversion is correlated with risk aversion, one does not immediately get a very distinctive prediction from loss aversion. One would like to see more predictions that are distinctive of PT. The model does predict a countercyclical premium, but so do other models, such as Campbell–Cochrane’s (1999). Perhaps there is a way to make them sharper.

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Let me venture a way in which this might be done. In the macro-based model, a` la Cambell–Cochrane, the effective risk aversion is predicted by consumption risk. In a narrow framing approach, the returns of the stock market, or even an individual stock, are the predictors of effective risk aversion of accumulated “pain from losses in the stock market.” For instance, the loss aversion could be Λt = exp

s>0

δs g(−rt−s ) ,

for some function g that could just be g(x) = x, or something more subtle or microfounded. Λt could be some other type of moving average such as the one used by Barberis and Huang. Of course, it would be useful to think more deeply about functional forms for the evaluation of “current loss aversion.” In any case, once one has a measure Λt of current loss aversion, one could see if this variable has some explanatory power for equity premia. For instance, if one regresses future returns (of a stock, or of the market) on book/market, and accumulated loss aversion Λt , does accumulated loss aversion have some good predictive power? Making those predictions seems particularly important. Other views do make predictions. For instance, the view that equity premium is about long-term risk, rather than short-term risk, makes a concrete, and empirically corroborated, prediction: the covariance between return and consumption increases with the horizon, a prediction that seems to hold empirically (Gabaix and Laibson (2002), Parker (2001)). Theories based on a small probability of a rare disaster, a` la Barro (2006) and Rietz (1988), predict higher equity premia when deep out of the money puts are more valuable—a prediction not yet tested to my knowledge. So far, much of the current work by Barberis and Huang is based on psychological and conceptual appeal, and calibrations, rather than predictions. It would be nice to have more concrete predictions, and see other authors test them.

3. DO A VERSION OF THE MODEL IN CONTINUOUS TIME The Barberis, Huang, and Santos model is a bit untractable right now, though the new version with recursive utility is more tractable. As continuous time is simpler, it seems very desirable to study a version of the Barberis and Huang approach in continuous time. It seems that there is a payoff here, first, to make the theory more complete; second, to be able to think more clearly about dynamics, e.g., about trading dynamics for loss averse agents. For instance, is there a way to say something sensible and useful about the evaluation horizon, rather than just postulate it, or, set it to help the model fit? I will stop here. I want to congratulate the authors for their most interesting enterprise, and hope that in a next handbook we will have even more tractable, worked-out models and more direct empirical evidence for the quantitative importance of narrow framing and loss aversion in market data.

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References Barberis, N., H. Huang, and T. Santos. Prospect theory and asset prices. Quarterly Journal of Economics 116(1) (2001): 1–53. Barro, R. Rare disasters and asset markets in the twentieth century. Quarterly Journal of Economics 121(3) (2006). Campbell, J. Y., and J. Cochrane. By force of habit: A consumption based explanation of aggregate stock market behaviour. Journal of Political Economy 107 (1999): 205–251 Gabaix, X., and D. Laibson. The 6D bias and the equity premium puzzle. In B. Bernanke and, K. Rogoff, eds. NBER Macroeconomics Annual 2001, Vol. 16, MIT Press, Cambridge, MA, (2002): pages 257–312. Goldstein, D. G., E. J. Johnson, and W. F. Sharpe. Measuring consumer risk-return tradeoffs. London Business School working paper (2006). Parker, J. The consumption risk of the stock market. Brookings Papers on Economic Activity, (2001): 279–333. Rietz, T. A. The equity risk premium: A solution. Journal of Monetary Economics 22 (1988): 117–131.

Ravi Jagannathan

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Discussion: The Loss Aversion/Narrow Framing Approach to the Equity Premium Puzzle1 Ravi Jagannathan Northwestern University and NBER.

The authors make a nice case for modifying the utility function of the representative investor in the standard model to incorporate two research findings that characterize individual decision making in experimental settings: loss aversion and narrow framing (LANF). The authors show how this can be done in a parsimonious way so that the investor’s optimization problem can be solved in a rational expectations general equilibrium framework. With this modification alone, they are able to reconcile the low risk-free rate with the high equity risk premium at moderate levels of risk aversion. In addition, they are able to explain limited participation in the stock market. This is an achievement, and the authors must be complemented. I have just one observation to make. LANF, possibly an inherited trait, might have been a desirable characteristic in primitive societies. Those who were willing to fight for avoiding a potential loss, however small when taken in a broader context, in equilibrium probably most often ended up avoiding the loss without actually fighting. However, LANF, probably because it makes individuals react instinctively to situations, can lead to potentially inferior decisions in the modern world. In those situations where LANF leads to high social costs, I expect institutions to develop endogenously to train and educate individuals to minimize the undesirable impact of LANF on their decisions. Dickhaut, et al. compared risk versus ambiguity in gain and loss situations on subjects. They found that choice under risk/loss generates more use of the calculational part of the brain (neocortical dorsomedial system), i.e., the more developed part of the brain

1 The

views expressed are my own and not necessarily those of the institutions I represent.

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takes over. The ventromedial system (that arose phylogenetically earlier, supporting decision making in animals) plays more of a role in risk/gain, ambiguity/gain, and ambiguity/loss situations. Their findings suggest that individuals are likely to react instinctively when they encounter unfamiliar situations that they do not understand well, with the less developed part of the brain playing a more important role. With education, individuals are likely to process alternatives in a more calculated manner instead of relying on intuition, with the more developed part of the brain taking over. Therefore, the degree of LANF exhibited by an individual may not remain constant over time and is likely to decline with the level of education and experience. In order to fully understand the implications of narrow framing and risk aversion on security prices, it is necessary to allow for heterogeneity in investors’ preference parameters, with some investors having standard preferences not subject to the influence of LANF.

Reference Dickhaut, J., K. McCabe, J. Pardo, and K. Smith. Neuronal substrates for choice under ambiguity, risk, certainty, gains, and losses. Management Science 48 (2002): 711–718.

CHAPTER 7

Financial Markets and the Real Economy John H. Cochrane∗ University of Chicago

1. Introduction 1.1. Risk Premia 1.2. Who Cares? 1.3. The Mimicking Portfolio Theorem and the Division of Labor 2. Facts: Time Variation and Business Cycle Correlation of Expected Returns 2.1. Variation over Time 2.2. Variation Across Assets 2.3. Return Forecasts—Variation over Time 2.4. The Cross Section of Returns—Variation Across Assets 3. Equity Premium 3.1. Mehra and Prescott and the Puzzle 3.2. The Future of the Equity Premium 4. Consumption Models 4.1. Hansen and Singleton; Power Utility 4.2. New Utility Functions 4.3. Empirics with New Utility Functions 4.4. Consumption and Factor Models 5. Production, Investment, and General Equilibrium 5.1. “Production-Based Asset Pricing” 5.2. General Equilibrium

239 239 242 243 244 244 245 246 251 257 261 266 267 267 270 273 286 290 290 294

∗ This

is a substantially reworked version of two papers that appeard under the same title, Cochrane (2005b, 2006a). I gratefully acknowledge research support from the NSF in a grant administered by the NBER and from the CRSP. I thank Ron Balvers, Frederico Belo, John Campbell, George Constantinides, Hugo Garduno, François Gourio, Robert Ditmar, Lars Hansen, John Heaton, Hanno Lustig, Rajnish Mehra, Marcus Opp, Dino Palazzo, Monika Piazzesi, Nick Roussanov, Alsdair Scott, Luis Viceira, Mike Wickens, and Motohiro Yogo for comments.


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Chapter 7 • Financial Markets and the Real Economy

6. Labor Income and Idiosyncratic Risk 6.1. Labor and Outside Income 6.2. Idiosyncratic Risk, Stockholding, and Micro Data 7. Challenges for the Future References Appendix

302 302 307 314 314 322

Abstract I survey work on the intersection between macroeconomics and finance. The challenge is to find the right measure of “bad times,” rises in the marginal value of wealth, so that we can understand high average returns or low prices as compensation for assets’ tendency to pay off poorly in “bad times.” I cover the time-series and cross-sectional facts, the equity premium, consumption-based models, general equilibrium models, and labor income/idiosyncratic risk approaches. JEL Classification: G12, E44 Keywords: asset pricing, cross-sectional tests, empirical work, intangible capital, long horizons, macroeconomics, microdata, proprietary income, risk premia, time series tests, risk aversion, state variables, bad times

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1. INTRODUCTION 1.1. Risk Premia Some assets offer higher average returns than other assets, or, equivalently, they attract lower prices. These “risk premia” should reflect aggregate, macroeconomic risks; they should reflect the tendency of assets to do badly in bad economic times. I survey research on the central question: what is the nature of macroeconomic risk that drives risk premia in asset markets? The central idea of modern finance is that prices are generated by expected discounted payoffs, pit = Et mt+1 xit+1 , (1) where xit+1 is a random payoff of a specific asset i, and mt+1 is a stochastic discount factor. Using the definition of covariance and the real risk-free rate Rf = 1/E(m), we can write the price as pit

=

Et xit+1 f Rt

+ Covt mt+1 , xit+1 .

(2)

The first term is the risk-neutral present value. The second term is the crucial discount for risk—a large negative covariance generates a low or “discounted” price. Applied to excess returns Rei (short or borrow one asset, invest in another), this statement becomes1 ei ei Et Rt+1 , mt+1 . = −Covt Rt+1 (3) The expected excess return or “risk premium” is higher for assets that have a large negative covariance with the discount factor. 1 From

(1), we have for gross returns R, 1 = E(mR),

and for a zero-cost excess return Re = Ri − Rj , 0 = E mRe . Using the definition of covariance, and 1 = E(m)Rf for a real risk-free rate, 0 = E(m)E Re + Cov m, Re , e E R = −Rf Cov m, Re . For small time intervals Rf ≈ 1, so we have E Re = −Cov m, Re . This equation holds exactly in continuous time.

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The discount factor mt+1 is equal to growth in the marginal value of wealth, mt+1 =

VW (t + 1) . VW (t)

This is a simple statement of an investor’s first-order conditions. The marginal value of wealth2 VW answers the question, “How much happier would you be if you found a dollar on the street?” It measures “hunger”—marginal utility, not total utility. The discount factor is high at t + 1 if you desperately want more wealth at t + 1—and would be willing to give up a lot of wealth in other dates or states to get it. Equation (3) thus says that the risk premium E(Rei ) is driven by the covariance of returns with the marginal value of wealth.3 Given that an asset must do well sometimes and do badly at other times, investors would rather it did well when they are otherwise desperate for a little bit of extra wealth, and that it did badly when they do not particularly value extra wealth. Thus, investors want assets whose payoffs have a positive covariance with hunger, and they will avoid assets with a negative covariance. Investors will drive up the prices and drive down the average returns of assets that covary positively with hunger, and vice versa, generating the observed risk premia. These predictions are surprising to newcomers for what they do not say. More volatile assets do not necessarily generate a higher risk premium. The variance of the return Rei or payoff xi is irrelevant per se and does not measure risk or generate a risk premium. Only the covariance of the return with “hunger” matters. Also, many people do not recognize that Eqs. (2) and (3) characterize an equilibrium. They describe a market after everyone has settled on their optimal portfolios. They do not generate portfolio advice. Deviations from (2) and (3), if you can find them, can give portfolio advice. It’s natural to think that high expected return assets are “good” and one should buy more of them. But the logic goes the other way: “Good” assets pay off well in bad times when investors are hungry. Since investors all want them, those assets get lower average returns and command higher prices in equilibrium. High average return assets are forced to pay those returns, or equivalently to suffer low prices, because they are so “bad”—because they pay off badly precisely when investors are most hungry. In the end, there is no “good” or “bad.” Equations (2) and (3) describe an equilibrium in which the quality of the asset and its price are exactly balanced. To make these ideas operational, we need some procedure to measure the growth in the marginal value of wealth or “hunger” mt+1 . The traditional theories of finance, 2 Formally,

the value of wealth is the achieved level of utility given the investor has wealth W , ∞ β j u ct+j , V (Wt ) = max Et j=0

subject to an appropriate budget constraint that is limited by initial wealth Wt . It can be a function V(Wt , zt ) of other “state variables” zt , for example, the expected returns of assets or the amount of outside income the investor expects to recieve, since higher values of these variables allow the investor to generate more utility. 3m t+1 really measures the growth in marginal utility or “hunger.” However, from the perspective of time t, VW (t) is fixed, so what counts is how the realization of the return covaries with the realization of time t + 1 marginal value of wealth VW (t + 1).

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CAPM, ICAPM, and APT, measure hunger by the behavior of large portfolios of assets. For example, in the CAPM a high average return is balanced by a large tendency of an asset to fall just when the market as a whole falls—a high “beta.” In equations, ei ei m Et Rt+1 , Rt+1 = Covt Rt+1 × γ, where Re denote excess returns, γ is a constant of proportionality equal to the average investor’s risk aversion, and Rm is the market portfolio.4 Multifactor models such as the popular Fama–French (1996) three-factor model use returns on multiple portfolios to measure the marginal value of wealth. Research connecting financial markets to the real economy—the subject of this survey—goes one step deeper. It asks what are the fundamental, economic determinants of the marginal value of wealth? I start with the consumption-based model, ei ct+1 ei × γ, , c Et Rt+1 = Covt Rt+1 t

which states that assets must offer high returns if they pay off badly in “bad times” as measured by consumption growth.5 As we will see, this simple and attractive model does not (yet) work very well. The research in this survey is aimed at improving that performance. It aims to find better measures of the marginal value of wealth, rooted 4 To derive this expression of the CAPM, assume the investor lives one period and has quadratic utility u(ct+1 ) = − 12 (c∗ − ct+1 )2 . The investor’s problem is ⎛ ⎞ N 2 1 ∗ ej p f ⎝ max E − c − ct+1 subject to ct+1 = Rt+1 Wt = R + wj Rt+1 ⎠Wt , 2 j=1

where

Re

Rf

denotes and is the risk-free rate. Taking the derivative with respect to wj , we $ excess returns ej % p c∗ − Rt+1 Wt Rt+1 . Using the definition of covariance,

obtain 0 = E

E

ej Rt+1

$ ej % p Cov c∗ − Rt+1 Wt , Rt+1 Wt ej p = Cov Rt+1 , Rt+1 =− p . p ∗ E c − Rt+1 Wt c∗ − E Rt+1 Wt

The risk aversion coefficient is γ = −cu (c)/u (c) = c/(c∗ − c) . Thus, we can express the term multiplying the covariance as the local risk aversion coefficient γ, at a value of consumption cˆ given by 1/c = p (1/Wt ) − (E(Rt+1 ) − 1/b). If consumers are enough alike, then the indvidual portfolio is the market portfolio, p m R =R . 5 One may derive this expression quickly by a Taylor expansion of the investor’s first-order conditions, and using Rf = 1/E(m) ≈ 1 for short horizons, u (ct+1 ) ei 0 = E mRei = E β Rt+1 , u (ct ) (c ei u t+1 ) ei , E Rt+1 = −Rf Cov Rt+1 u (ct ) −ct u (ct ) ct+1 − ct ct+1 ei ei , , ≈ Cov Rt+1 = Cov R × γ. t+1 c u (ct ) ct t

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in measures of economic conditions such as aggregate consumption, that explain the ei pattern by which mean returns Et (Rt+1 ) vary across assets i and over time t.

1.2. Who Cares? Why is this important? What do we learn by connecting asset returns to macroeconomic events in this way? Why bother, given that “reduced form” or portfolio-based models like the CAPM are guaranteed to perform better?

1.2.1. Macroeconomics Understanding the marginal value of wealth that drives asset markets is most obviously important for macroeconomics. The centerpieces of dynamic macroeconomics are the equation of savings to investment, the equation of marginal rates of substitution to marginal rates of transformation, and the allocation of consumption and investment across time and states of nature. Asset markets are the mechanism that does all this equating. If we can learn the marginal value of wealth from asset markets, we have a powerful measurement of the key ingredient of all modern, dynamic, intertemporal macroeconomics. In fact, the first stab at this piece of economics is a disaster, in a way first made precise by the “equity premium” puzzle. The marginal value of wealth needed to make sense of the most basic stock market facts is orders of magnitude more volatile than that specified in almost all macroeconomic models. Clearly, finance has a lot to say about macroeconomics, and it says that something is desperately wrong with most macroeconomic models. In response to this challenge, many macroeconomists simply dismiss asset market data. “Something’s wacky with stocks,” they say, or perhaps “stocks are driven by fads and fashions disconnected from the real economy.” That might be true, but if so, by what magic are marginal rates of substitution and transformation equated? It makes no sense to say “markets are crazy” and then go right back to market-clearing models with wildly counterfactual asset pricing implications. If asset markets are screwed up, so is the equation of marginal rates of substitution and transformation in every macroeconomic model, so are those models’ predictions for quantities, and so are their policy and welfare implications.

1.2.2. Finance Many financial economists return the compliment, and dismiss macroeconomic approaches to asset pricing because portfolio-based models “work better”—they provide smaller pricing errors. This dismissal of macroeconomics by financial economists is just as misguided as the dismissal of finance by macroeconomists. First, a good part of the better performance of portfolio-based models simply reflects Roll’s (1977) theorem: we can always construct a reference portfolio that perfectly fits all asset returns: the sample mean-variance efficient portfolio. The only content to empirical

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John H. Cochrane

work in asset pricing is what constraints the author put on his fishing expedition to avoid rediscovering Roll’s theorem. The instability of many “anomalies” and the ever-changing nature of factor models that “explain” them (Schwert (2003)) lends some credence to this worry. The main fishing constraint one can imagine is that the factor portfolios are in fact mimicking portfolios for some well-understood macroeconomic risk. Fama (1991) famously labeled the ICAPM and similar theories “fishing licenses,” but his comment cuts in both directions. Yes, current empirical implementations do not impose much structure from theory, but no, you still can’t fish without a license. For example, momentum has yet to acquire the status of a factor despite abundant empirical success, because it has been hard to come up with stories that it corresponds to some plausible measure of the marginal utility of wealth. Second, much work in finance is framed as answering the question of whether markets are “rational” and “efficient” or not. No amount of research using portfolios on the right-hand side can ever address this question. The only possible content to the “rationality” question is whether the “hunger” apparent in asset prices—the discount factor, marginal value of wealth, etc.—mirrors macroeconomic conditions correctly. If Mars has perfectly smooth consumption growth, then prices that are perfectly “rational” on volatile Earth would be “irrational” on Mars. Price data alone cannot answer the question, because you can’t tell from the prices which planet you’re on. In sum, the program of understanding the real, macroeconomic risks that drive asset prices (or the proof that they do not do so at all) is not some weird branch of finance; it is the trunk of the tree. As frustratingly slow as progress is, this is the only way to answer the central questions of financial economics, and a crucial and unavoidable set of uncomfortable measurements and predictions for macroeconomics.

1.3. The Mimicking Portfolio Theorem and the Division of Labor Portfolio-based models will always be with us. The “mimicking portfolio” theorem states that if we have the perfect model of the marginal utility of wealth, then a portfolio formed by its regression on to asset returns will work just as well.6 And this “mimicking portfolio” will have better-measured and more frequent data, so it will work better in sample and in practice. It will be the right model to recommend for many applications.

6 Start

with the true model, 1 = E(mR),

where R denotes a vector of returns. Consider a regression of the discount factor on the returns, with no constant, m = bR + ε. By construction, E(Rε) = 0, so

1 = E[(b R)R].

Therefore, the payoff bR is a discount factor as well.

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This theorem is important for doing and evaluating empirical work. First, together with the Roll theorem, it warns us that it is pointless to engage in an alpha contest between real and portfolio-based models. Ad-hoc portfolio models must always win this contest—even the true model would be beat by its own mimicking portfolio because of measurement issues, and it would be beaten badly by an ad-hoc portfolio model that could slide a bit toward the sample mean-variance frontier. Thus, the game “see if macro factors do better than the Fama–French three-factor model” in pricing the Fama–French 25 portfolios is rather pointless. Even if you do succeed, a “small-growth/large-value” fourth factor or the increasingly popular momentum factor can always come back to trump any alpha successes. Portfolio-based models are good for relative pricing; for describing one set of asset returns given another set. The CAPM describes average returns of stock portfolios given the market premium. The Fama–French model describes average returns of 25 size and book/market sorted portfolios given the average returns of the three-factor portfolios. But why is the average market return what it is? Why are the average returns of the Fama–French value and size portfolios what they are? Why does the expected market return vary over time? By their nature, portfolio models cannot answer these questions. Macroeconomic models are the only way to answer these questions. With this insight, we can achieve a satisfying division of labor, rather than a fruitless alpha-fishing contest. Portfolio models document whether expected returns of a large number of assets or dynamic strategies can be described in terms of a few sources of common movement. Macro models try to understand why the common factors (market, hml, smb) are priced. Such an understanding will of course ultimately pay off for pure portfolio questions, by helping us to understand which apparent risk premia are stable rewards for risk, and which were chimeric features of the luck in one particular sample.

2. FACTS: TIME VARIATION AND BUSINESS CYCLE CORRELATION OF EXPECTED RETURNS We start with the facts. What is the pattern by which expected returns vary over time and across assets? What is the variation on the left-hand side of (3) that we want to explain by understanding the marginal value of wealth on the right-hand side of (3)?

2.1. Variation over Time First, a number of variables forecast aggregate stock, bond, and foreign exchange returns. Thus, expected returns vary over time. The central technique is simple forecasting regression: if we find |b| > 0 in Rt+1 = a + bxt + εt+1 , then we know that Et (Rt+1 ) varies over time. The forecasting variables xt typically have a suggestive business cycle correlation. Expected returns are high in “bad times,” when we might well suppose people are less willing to hold risks. For example, Table 1 reports regressions of excess returns on dividend-price ratios. A one percentage point higher dividend yield leads to a four percentage point higher

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John H. Cochrane

TABLE 1 OLS Regressions of Excess Returns (value-weighted NYSE—Treasury bill) and Real Dividend Growth on the Value-Weighted NYSE Dividend-Price Ratio Horizon k (years)

Dt+k Dt

e t Rt→t+k = a + bD Pt + εt+k

t = a + bD Pt + εt+k

b

t(b)

R2

1

4.0

2.7

0.08

0.07

0.06

0.0001

2

7.9

3.0

0.12

−0.42

−0.22

0.0010

3

12.6

3.0

0.20

0.16

0.13

0.0001

5

20.6

2.6

0.22

2.42

1.11

0.0200

b

t(b)

R2

e Sample 1927–2005, annual data. Rt→t+k denotes the total excess return from time t to time t + k. Standard errors use GMM (Hansen–Hodrick) to correct for heteroskedasticity and serial correlation.

return. This is a surprisingly large number. If there were no price adjustment, a one percentage point higher dividend yield would only lead to a one percentage point higher return. The conventional “random walk” view implies a price adjustment that takes this return away. Apparently, prices adjust in the “wrong” direction, reinforcing the higher dividend yield. Since the right-hand variable (dividend-price ratio) is very persistent, long-horizon forecasts are even more dramatic, with larger coefficients and R2 -values. The second set of regressions in Table 1 is just as surprising. A high dividend yield means a “low” price, and it should signal a decline in future dividends. We see tiny and completely insignificant coefficients, and tiny R2 -values. Apparently, variation in price-dividend ratios does not come from news about future dividends. This pattern is not unique to stocks. Bond and foreign exchange returns are also predictable, meaning that expected returns vary through time. The same pattern holds in each case: a “yield” or “yield spread” (dividend yield, bond yields, international interest rate differential) forecasts excess returns; it does so because something that should be forecastable to offset the variation in expected returns (dividend growth, short-term interest rates, exchange rates) does not move, or does not move quickly enough; and the high-expected return signal (high dividend yield, upward sloping yield curve, low interest rates relative to foreign) typically comes in bad macroeconomic times. A large number of additional variables also forecast returns.

2.2. Variation Across Assets Second, expected returns vary across assets. Stocks earn more than bonds of course. In addition, a large number of stock characteristics are now associated with average returns. The book/market ratio is the most famous example: stocks with low prices (market value) relative to book value seem to provide higher subsequent average returns. A long list of other variables including size (market value), sales growth, past returns, past volume, accounting ratios, short sale restrictions, and corporate actions such as

246


investment, equity issuance, and repurchases are also associated with average returns going forward. We can think of all these phenomena as similar regression forecasts applied to individual assets or characteristic-sorted portfolios: the basic finding is that there exist many variables xi,t that give significant coefficients in f

i − Rt = a + bxi,t + εi,t+1 . Rt+1

This variation in expected returns across assets would not cause any trouble for traditional finance theory if the characteristics associated with high average returns were also associated with large market betas. Alas, they often are not. Instead, the empirical finance literature has associated these patterns in expected returns with betas on new “factors.” (Cochrane (1999a) is an easily accessible review paper that synthesizes current research on both the time-series and the cross-sectional issues. Chapter 20 of Asset Pricing by Cochrane (2004) is a somewhat expanded version, with more emphasis on the relationship between various time-series representations. Campbell (2003) also has a nice summary of the facts.)

2.3. Return Forecasts—Variation over Time Return forecasts have a long history. The classic view that “stocks follow a random walk,” meaning that the expected return is constant over time, was first challenged in the late 1970s. Fama and Schwert (1977) found that expected stock returns did not increase one-for-one with inflation. They interpreted this result to say that expected returns are higher in bad economic times, since people are less willing to hold risky assets, and are lower in good times. Inflation is lower in bad times and higher in good times, so lower expected returns in times of high inflation are not a result of inflation, but a coincidence. To us, the association with inflation that motivated Fama and Schwert is less interesting, but the core finding that expected returns vary over time, and are correlated with business cycles (high in bad times, low in good times), remains the central fact. Fama and Gibbons (1982) added investment to the economic modeling, presaging the investment and equilibrium models we study later. In the early 1980s, we learned that bond and foreign exchange expected excess returns vary over time—that the classic “expectations hypothesis” is false. Hansen and Hodrick (1980) and Fama (1984a) documented the predictability of foreign exchange returns by running regressions of those returns on forward-spot spread or interest rate differentials across countries. If the foreign interest rate is unusually higher than the domestic interest rate, it turns out that the foreign currency does not tend to depreciate, and thus an adverse currency movement does not, on average, wipe out the apparently attractive return to investing abroad. (“Unusually” is an important qualifier. If you just invest in high interest rate countries, you end up investing in high inflation countries, and depreciation does wipe out any gains. The phenomenon requires you to invest in countries with a higher-than-usual interest rate spread, i.e., following a regression of

John H. Cochrane

247 returns on interest rate spreads over time, with a constant. What “usual” means, i.e., the fact of an estimated constant in these regressions, is still a bit of an open question. See Engel (1996) for a review.) Fama (1984b) documented the predictability of short-term bond returns, and Fama and Bliss (1987) the predictability of long-term bond returns, by running regressions of bond returns on forward-spot spreads or yield differentials. Shiller, Campbell, and Schoenholtz (1983) and Campbell and Shiller (1991) analogously rejected the expectations hypothesis by regressions of future yields on current yields; their regressions imply time-varying expected returns. Campbell (1995) is an excellent summary of this line of research. While the expectations hypothesis had been rejected before,7 these papers focused a lot of attention on the problem. In part, they did so by applying a simple and easily interpretable regression methodology rather than more indirect tests: just forecast tomorrow’s excess returns from today’s yields or other forecasting variables. They also regressed changes in prices (returns) or yields on today’s yield or forward-rate spreads. The expectations hypothesis looks pretty good if you just regress (say) the ex-post spot rate on the ex-ante forward rate to test the prediction that the forward rate is equal to the expected spot rate. But this is not a very powerful test. For example, if you forecast tomorrow’s temperature by just quoting today’s temperature, you will also get a nice 1.0 coefficient and a high R2 , as overall temperature varies over the year. To see a good weather forecaster, you have to check whether he can predict the difference of tomorrow’s temperature over today’s temperature. Similarly, we see the failure of the expectations hypothesis by seeing that the difference between the forward rate and this year’s spot rate does not forecast a change in the spot rate from this year to next year. Finally, when looked at this way, these papers showed the striking magnitude and character of expectations-hypothesis failures. If the forward rate is one percentage point higher than the spot rate, Fama and Bliss showed that expected returns rise by a full percentage point, and the one-year short rate forecast does not change at all. Foreign exchange forecasts are even larger: a one percentage point interest differential seems to signal an increase in expected returns larger than one percentage point. The latter findings have been extended and stand up well over time. Stambaugh (1988) extended the results for short-term bonds and Cochrane and Piazzesi (2005) did so for long-term bonds. Both papers ran bond returns from t to t + 1 on all forward rates available at time t, and substantially raised the forecast R2 . The Cochrane and Piazzesi bond return forecasting variable also improves on the yield spread’s ability to forecast stock returns, and we emphasize that a single “factor” seems to forecast bond returns for all maturities. During this period, we also accumulated direct regression evidence that expected excess returns vary over time for the stock market as a whole. Rozeff (1984), Shiller (1984), Keim and Stambaugh (1986), Campbell and Shiller (1988), and Fama and 7 Evidence against the expectations hypothesis of bond yields goes back at least to Macaulay (1938). Shiller, Campbell, and Schoenholtz (1983) Hansen and Sargent (1981), Roll (1970), Sargent (1978, 1972), and Shiller (1979). Fama says, “The existing literature generally finds that forward rates . . . are poor forecasts of future spot rates” and cites Hamburger and Platt (1975), Fama (1976), and Shiller, Campbell, and Shoenholtz (1983).

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French (1988b) showed that dividend/price ratios forecast stock market returns. Fama and French really dramatized the importance of the D/P effect by emphasizing long horizons, at which the R2 rise to 60 percent. (The lower R2 -values in Table 1 reflect my use of both the pre-1947 and post-1988 data.) This observation emphasized that stock return forecastability is an economically interesting phenomenon that cannot be dismissed as another little anomaly that might be buried in transactions costs. Long horizon forecastability is not really a distinct phenomenon; it arises mechanically as the result of a small short horizon variability and a slow-moving right-hand variable (D/P ). Fama and French (1989) is an excellent summary and example of the large body of work that documents variation of expected returns over time. This paper shows how dividend-price ratios, term spreads (long bond yield less short bond yield), and default spreads forecast stock and bond returns. The paper emphasizes the comforting link between stock and bond markets: the term “spread” forecasts stock returns much as it forecasts bond returns. If returns are predictable from variables such as dividend yields, it stands to reason that returns should also be predictable from past returns. The way the dividend yield changes after all is by having a good sequence of returns so dividends are divided by a larger price. Such “mean reversion” in returns has the powerful implication that the variance of returns grows less than linearly with horizon, so stocks really are “safer in the long run.” Initially, this did seem to be the case. Poterba and Summers (1988) and Fama and French (1988a) documented that past stock market returns forecast subsequent returns at long horizons. However, this effect seems to have vanished, and the current consensus is that although variables such as dividend yields forecast returns, univariate forecastability or mean reversion are small (see, for example, Cochrane (2004), pp. 413–415). This is not a logical contradiction. For example, the weather can be i.i.d. and thus not forecastable from its own past, yet still may be forecastable the day ahead by meteorologists who look at more data than past weather. Similarly, stock returns can be forecastable by other variables such as dividend yields, yet unforecastable by their own past. A related literature including Campbell and Shiller (1988) and Cochrane (1991a) (summarized in Cochrane (1999)) connects the time-series predictability of stock −1 returns to stock price volatility. Linearizing and iterating the identity 1 = Rt+1 Rt+1 , we can obtain an identity that looks a lot like a present value model, pt − dt = k + Et

∞

! ρj−1 Et Δdt+j − Et rt+j + lim ρj pt+j − dt+j , j→∞

(4)

j=1

where small letters are logs of capital letters, and k and ρ = (P /D)/(1 + (P /D)) ≈ 0.96 are constants related to the point P /D about which we linearize. If price-dividend ratios vary at all, then, then either (1) price-dividend ratios forecast dividend growth, (2) price-dividend ratios forecast returns, or (3) prices must follow a “bubble” in which the price-dividend ratio is expected to rise without bound.

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It would be lovely if variation in price-dividend ratios corresponded to dividendgrowth forecasts. Investors, knowing future dividends will be higher than they are today, bid up stock prices relative to current dividends; then today’s high price-dividend ratio forecasts the subsequent rise in dividends. It turns out that price-dividend ratios do not forecast aggregate dividends at all, as shown in the right-hand panel of Table 1. This is the “excess volatility” found by Shiller (1981) and LeRoy and Porter (1981). However, prices can also be high if this is a time of temporarily low expected returns; then the same dividends are discounted at a lower rate, and a high price-dividend ratio forecasts low returns. It turns out that the return forecastability we see in regressions such as the left-hand side of Table 1 is just enough to completely account for the volatility of pricedividend ratios through (4). (This is a main point of Cochrane (1991a).) Thus, return forecastability and “excess volatility” are exactly the same phenomenon. Since pricedividend ratios are stationary (Craine (1993)) and since the return forecastability does neatly account for price-dividend volatility, we do not need to invoke the last “rational bubble” term. Alas, the fact that almost all stock price movements are due to changing expected excess returns rather than to changing expectations of future dividend growth means that we have to tie stock market movements to the macroeconomy entirely through harder-to-measure time-varying risk premia rather than easier-to-understand cash flows.

2.3.1. Macro Variables and Forecastability The forecasting variables in return regressions are so far all based on market prices, which seems to take us away from our macroeconomic quest. However, as emphasized by Fama and French (1989) with a nice series of plots, the prices that forecast returns are correlated with business cycles, with higher expected returns in bad times. A number of authors, including Estrella and Hardouvelis (1991) and more recently Ang, Piazzesi, and Wei (2004), document that the price variables that forecast returns also forecast economic activity. One can, of course, run regressions of returns on macroeconomic variables. A number of macroeconomic variables forecast stock returns, including the investment/capital ratio (Cochrane (1991b)), the dividend-earnings ratio (Lamont (1998)), investment plans (Lamont (2000)), the ratio of labor income to total income (Menzly, Santos, and Veronesi (2004)), the ratio of housing to total consumption (Piazzesi, Schneider, and Tuzel (2007)), an “output gap” formed from the Federal Reserve capacity index (Cooper and Priestley (2005)), and the ratio of consumption to wealth (Lettau and Ludvigson (2001a)). The investment to capital ratio and consumption to wealth ratios are particularly attractive variables. The Q theory of investment says that firms will invest more when expected returns are low; the investment to capital regressions verify this fact. Similarly, optimal consumption out of wealth is smaller when expected returns are larger. In this way, both variables exploit agents’ quantity decisions to learn their expectations, and exploit natural cointegrating vectors to measure longterm forecasts. For example, Cochrane (1994) showed that consumption provides a natural “trend” for income, and so we see long-run mean reversion in income most

250


easily by watching the consumption to income ratio. I also showed that dividends provide a natural “trend” for stock prices, so we see long-run mean reversion in stock prices most easily by watching the dividend-price ratio. Lettau and Ludvigson nicely put the two pieces together, showing how consumption relative to income and wealth has a cross-over prediction for long-run stock returns. Lettau and Ludvigson (2004) show that the consumption to wealth ratio also forecasts dividend growth. This is initially surprising. So far, very little has forecast dividend growth. And if anything does forecast dividend growth, why is a high dividend forecast not reflected in and hence forecast by higher prices? Lettau and Ludvigson answer this puzzle by noting that the consumption to wealth ratio forecasts returns, even in the presence of D/P. In the context of (4), the consumption to wealth ratio sends dividend growth and returns in the same direction, so its effects on the price to dividend ratio are offset. Thus, on second thought, the observation is natural. If anything forecasts dividend growth, it must also forecast returns to account for the fact that price-dividend ratios do not forecast dividend growth. Conversely, if anything has additional explanatory power for returns, it must also forecast dividend growth. And it makes sense. In the bottom of a recession, both returns and dividend growth will be strong as we come out of the recession, with offsetting effects on prices. So we end up with a new variable, and an opening for additional variables, that forecast both returns and cash flows, giving stronger links from macroeconomics to finance.

2.3.2. Statistics Return forecastability has come with a long statistical controversy. The first round of statistical investigation asked whether the impressive long horizon regressions (the extra rows of Table 1) capture any information not present in one-period regressions (the first row). Given the large persistence of the dividend yield and related forecasting variables, the first answer was that, by and large, they do not. Hodrick (1992) put the point nicely: the multiyear regression amounts to a test of the moment E[(rt+1 + rt+2 )xt ] = 0, where x is the forecasting variable and r are log returns. But this is the same moment as a one-year regression using a moving average right-hand variable, E[rt+1 (xt + xt−1 )]. Given the extreme persistence of the right-hand variables such as dividend yield, one can naturally see that this moment is no more powerful than E(rt+1 xt ) = 0—noone would think that lags of the dividend yield have much marginal forecast power. Campbell and Shiller (1988) also make this point by emphasizing that multiyear regressions are implied by one-year regressions. If xt+1 = ρxt + υt+1 , rt+1 = bxt + εt+1 , then rt+1 + rt+2 = b(1 + ρ)xt + (εt+1 + bυt+1 + εt+2 ).

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All of the information in multiyear regressions can be recovered from one-year regressions, which is what maximum likelihood would have you look at anyway. More seriously, even the one-period regressions are suspect. The t-statistics in Table 1 are already not that large given the long time span. In addition, the dividend yield is very persistent, and innovations in returns are highly correlated with innovations in dividend yields, since a change in prices moves both variables. As a result, the return-forecasting coefficient inherits near-unit-root properties of the dividend yield. It is biased upward, and its t-statistic is biased toward rejection. Other forecasting variables have similar characteristics. Perhaps even the forecastability as seen in the first row is really not there in the first place. Following this idea, Goetzmann and Jorion (1993) and Nelson and Kim (1993) find the distribution of the return-forecasting coefficient by simulation, and find greatly reduced evidence for return forecastability. Stambaugh (1999) derives the finite-sample properties of the return-forecasting regression, showing the bias in the return-forecasting coefficient and the standard errors, and shows that the apparent forecastability disappears once one takes account of the biases. More recently, Goyal and Welch (2003, 2005) show that return forecasts based on dividend yields and a menagerie of other variables do not work out of sample. They compare forecasts in which one estimates the regression using data up to time t to forecast returns at t + 1 with forecasts using the sample mean in the same period. They find that the sample mean produces a better out-of-sample prediction than do the return-forecasting regressions. Does this mean we should abandon forecastability and go back to the random walk, i.i.d. return view of the world? I think not, since there is still not a shred of evidence that price ratios forecast dividend (or earning or cash flow) growth. If prices vary, they must forecast something—we cannot hold the view that both returns and dividend growth are i.i.d., since in that case price-dividend ratios are constant. Thus, the lack of dividend forecastability is important evidence for return forecastability, and this is ignored in the statistical studies. In Cochrane (2006b), I formalize this argument. I show that return forecastability is still highly significant, including small-sample biases, when one takes into account both pieces of evidence. (The paper also contains a more complete bibliography on this statistical issue.) I also show that long horizon return forecasts can add important statistical evidence for return forecastability and that long horizon return forecasts are closely related to dividend growth forecasts.

2.4. The Cross Section of Returns—Variation Across Assets Fama and French (1996) is an excellent crystallization of how average returns vary across stocks. Fama and French start by summarizing for us the “size” and “value” effects; the fact that small stocks and stocks with low market values relative to book values tend to have higher average returns than other stocks.8 See the average returns in their Table 1 panel A, reproduced in Table 2. Again, this pattern is not by itself a puzzle. High expected returns should be revealed by low market values (see Eq. (4)). The puzzle is that the value and small firms do not 8 These expected-return findings go back a long way, including Ball (1978), Basu (1983), Banz (1981), DeBondt and Thaler (1985), and Fama and French (1992, 1993).

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TABLE 2 Summary Statistics and Three-Factor Regressions for Simple Monthly Percent Excess Returns on 25 Portfolios Formed on Size and BE/ME: 7/63–12/93, 366 Months Book-to-market equity (BE/ME) quintiles Size

Low

2

3

4

High

Low

2

3

4

High

Panel A: Summary statistics Means

Standard deviations

Small

0.31

0.70

0.82

0.95

1.08

7.67

6.74

6.14

5.85

6.14

2

0.48

0.71

0.91

0.93

1.09

7.13

6.25

5.71

5.23

5.94

3

0.44

0.68

0.75

0.86

1.05

6.52

5.53

5.11

4.79

5.48

4

0.51

0.39

0.64

0.80

1.04

5.86

5.28

4.97

4.81

5.67

Big

0.37

0.39

0.36

0.58

0.71

4.84

4.61

4.28

4.18

4.89

Fama and French (1996), Table 1.

have higher market betas. As panel B of Fama and French’s Table 1 shows, all of the market betas are about one. Market betas vary across portfolios a little more in single regressions without hml and smb as additional right-hand variables, but here the result is worse: the high average return “value” portfolios have lower market betas. Fama and French then explain the variation in mean returns across the 25 portfolios by variation in regression slope coefficients on two new “factors,” the hml portfolio of value minus growth firms and the smb portfolio of small minus large firms. Looking across the rest of their Table 1, you see regression coefficients b, s, h rising in panel B (see Table 2), where expected returns rise in panel A. Replacing the CAPM with this “three-factor model” is the central point of Fama and French’s paper. (Keep in mind that the point of the factor model is to explain the variation in average returns across the 25 portfolios. The fact that the factors “explain” a large part of the return variance—the high R2 in the time-series regressions of Table 1—is not the central success of an asset pricing model.) This argument is not as circular as it sounds. Fama and French say that value stocks earn more than growth stocks not because they are value stocks (a characteristic) but because they all move with a common risk factor. This comovement is not automatic. For example, if we split stocks into 26 portfolios based on the first letter of the ticker symbol and subtract the market return, we would not expect to see a 95 percent R2 in a regression of the A portfolio on an A–L minus M–Z “factor,” because we would expect no common movement among the A, B, C, etc. portfolios. Stocks with high average returns should move together. Otherwise, one could build a diversified portfolio of high expected return (value) stocks, short a portfolio of low expected return (growth) stocks, and make huge profits with no risk. This strategy remains risky and does not attract massive capital, which would wipe out the anomaly,

253

John H. Cochrane Table 2–Continued Book-to-market equity (BE/ME) quintiles Size

Low

2

3

4

High

Low

2

3

4

High

Panel B: Regressions: Ri − Rf = ai + bi RM − Rf + st smb + hi hml + et a

t(a)

Small

−0.45

−0.16

−0.05

0.04

0.02

−4.19

−2.04

−0.82

0.69

0.29

2

−0.07

−0.04

0.09

0.07

0.03

−0.80

−0.59

1.33

1.13

0.51

3

−0.08

0.04

−0.00

0.06

0.07

−1.07

0.47

−0.06

0.88

0.89

4

0.14

−0.19

−0.06

0.02

0.06

1.74

−2.43

−0.73

0.27

0.59

Big

0.20

−0.04

−0.10

−0.08

−0.14

3.14

−0.52

−1.23

−1.07

−1.17

b

t(b)

Small

1.03

1.01

0.94

0.89

0.94

39.10

50.89

59.93

58.47

57.71

2

1.10

1.04

0.99

0.97

1.08

52.94

61.14

58.17

62.97

65.58

3

1.10

1.02

0.98

0.97

1.07

57.08

55.49

53.11

55.96

52.37

4

1.07

1.07

1.05

1.03

1.18

54.77

54.48

51.79

45.76

46.27

Big

0.96

1.02

0.98

0.99

1.07

60.25

57.77

47.03

53.25

37.18

s

t(s)

Small

1.47

1.27

1.18

1.17

1.23

39.01

44.48

52.26

53.82

52.65

2

1.01

0.97

0.88

0.73

0.90

34.10

39.94

36.19

32.92

38.17

3

0.75

0.63

0.59

0.47

0.64

27.09

24.13

22.37

18.97

22.01

4

0.36

0.30

0.29

0.22

0.41

12.87

10.64

10.17

6.82

11.26

−0.16

−0.13

−0.25

−0.16

−0.03

−6.97

−5.12

−8.45

−6.21

−0.77

Big

h

t(h)

Small

−0.27

0.10

0.25

0.37

0.63

−6.28

3.03

9.74

15.16

23.62

2

−0.49

0.00

0.26

0.46

0.69

−14.66

0.34

9.21

18.14

25.59

3

−0.39

0.03

0.32

0.49

0.68

−12.56

0.89

10.73

17.45

20.43

4

−0.44

0.03

0.31

0.54

0.72

−13.98

0.97

9.45

14.70

17.34

Big

−0.47

0.00

0.20

0.56

0.82

−18.23

0.18

6.04

18.71

17.57

R2

s(e)

Small

0.93

0.95

0.96

0.96

0.96

1.97

1.49

1.18

1.13

1.22

2

0.95

0.96

0.95

0.95

0.96

1.55

1.27

1.28

1.16

1.23

3

0.95

0.94

0.93

0.93

0.92

1.44

1.37

1.38

1.30

1.52

4

0.94

0.92

0.91

0.88

0.89

1.46

1.47

1.51

1.69

1.91

Big

0.94

0.92

0.87

0.89

0.81

1.19

1.32

1.55

1.39

2.15

Fama and French (1996), Table 1.

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precisely because there is a common component to value stocks, captured by the Fama–French hml factor. Fama and French go further, showing that the size and book-to-market factors explain average returns formed by other characteristics. Sales growth is an impressive example, since it is a completely non-financial variable. Stocks with high past sales growth have lower subsequent returns (“too high prices”) than stocks with low sales growth, a fact that turns conventional investment advice on its head. They do not have higher market betas, but they do have higher betas on the Fama–French factors. In this sense, the Fama–French three-factor model “explains” this additional pattern in expected returns. In this kind of application, the Fama–French three-factor model has become the standard model replacing the CAPM for risk adjusting returns. The Fama–French paper has also, for better or worse, defined the methodology for evaluating asset pricing models for the last 10 years. A generation of papers studies the Fama–French 25 size and book-to-market portfolios to see whether alternative factor models can explain their average returns. Empirical papers now routinely form portfolios by sorting on other characteristics, and then run time-series regressions like Fama and French’s to see which factors explain the spread in average returns, as revealed by small regression intercepts. Most importantly, where in the 1980s papers would focus entirely on the probability value of some overall statistic, Fama and French rightly got people to focus on the spread in average returns, the spread in betas, and the economic size of the pricing errors. Remarkably, this, the most successful model since the CAPM, is decisively rejected by formal tests. Fama and French taught us to pay attention to more important things than test statistics. Macro-modelers have gotten into the habit of evaluating models on the Fama–French 25 portfolios, just as Fama and French did. I think that, in retrospect, this is a misreading of the point of Fama and French’s paper. The central point of the paper is that all of the important cross-sectional information in the 25 portfolios is captured by the three-factor portfolios. This is true of both returns (high R2 ) and expected returns. One could state the result that three dominant eigenvalues in the covariance matrix of the 25 portfolios explain the vast majority of the correlation structure of the portfolios, and expected returns are almost completely described by betas on these three portfolios. To the extent that the Fama–French three-factor model is successful in describing average returns, macro-modelers need only worry about why the value (hml) and smalllarge (smb) portfolio have nonzero expected returns. Given these factors, the expected returns of the 25 portfolios (and any other, different, portfolios that are explained by the three-factor model) follow automatically. The point of the 25 portfolios is to show “nonparametrically” that the three-factor portfolios account for all information in stocks sorted by size and book to market. The point of the 25 portfolios is not to generate a good set of portfolios that captures 25 degrees of freedom in the cross section of all stocks. There are really not 25 degrees of freedom in the Fama–French portfolios: there are 3 degrees of freedom. This is very bad news for models that explain the Fama– French portfolios with 4, 5, and sometimes 10 factors! This is the central point of Daniel and Titman (2005) and Lewellen, Nagel, and Shanken (2006).

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The Fama–French model is rejected in the 25 portfolios, however. The rejection of the three-factor model in the 25 portfolios is caused primarily by small-growth portfolios, and Fama and French’s Table 1 shows the pattern. Small-growth stocks earn about the same average returns as large-growth portfolios—see Table 1 “means” left column—but they have much larger slopes s. A larger slope that does not correspond to a larger average return generates a pricing error a. In addition, the R2 are so large in these regressions, and the residuals correspondingly so small, that economically small pricing errors are statistically significant. α Σ−1 α is large if α is small, but Σ is even smaller. A fourth “small growth–large value” factor eliminates this pricing error as well, but I don’t think Fama and French take the anomaly that seriously. For the division of labor and the use of 25 portfolios, however, this fact means that models that improve on the Fama–French factors in the 25 Fama–French portfolios do so by better pricing the small-growth puzzle and other very small discrepancies of the model. One must ask whether those discrepancies are at all meaningful. The Fama–French model seems to take us away from economic explanation of risk premia. After all, hml and smb are just other portfolios of stocks. Fama and French speculate suggestively on the macroeconomic foundations of the value premium (p. 77): One possible explanation is linked to human capital, an important asset for most investors. Consider an investor with specialized human capital tied to a growth firm (or industry or technology). A negative shock to the firm’s prospects probably does not reduce the value of the investor’s human capital; it may just mean that employment in the firm will expand less rapidly. In contrast, a negative shock to a distressed firm more likely implies a negative shock to the value of specialized human capital since employment in the firm is more likely to contract. Thus, workers with specialized human capital in distressed firms have an incentive to avoid holding their firms’ stocks. If variation in distress is correlated across firms, workers in distressed firms have an incentive to avoid the stocks of all distressed firms. The result can be a state-variable risk premium in the expected returns of distressed stocks. Much of the work described ahead tries to formalize this kind of intuition and measure the required correlations in the data. A large body of empirical research asks whether the size and book-to-market factors do in fact represent macroeconomic phenomena via rather astructural methods. It is natural to suppose that value stocks—stocks with low prices relative to book value, thus stocks that have suffered a sequence of terrible shocks—should be more sensitive to recessions and “distress” than other stocks, and that the value premium should naturally emerge as a result. Initially, however, efforts to link value stocks and value premia to economic or financial trouble did not bring much success. Fama and French (1997a, 1997b) were able to link value effects to individual cash flows and “distress,” but getting a premium requires a link to aggregate bad times, a link that Lakonishok, Shleifer, and Vishny (1994) did not find. However, in the 1990s and early 2000s, value stocks moved much more closely with the aggregate economy, so more recent estimates do show a

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Chapter 7

•

Financial Markets and the Real Economy

significant and heartening link between value returns and macroeconomic conditions. In this context, Liew and Vassalou (2000) show that Fama and French’s size and bookto-market factors forecast output growth, and thus are “business cycle” variables. The Fama–French paper closes with a puzzle. Though the three-factor model captures the expected returns from many portfolio sorts, it fails miserably on momentum. If you form portfolios of stocks that have gone up in the last year, this portfolio continues to do well in the next year, and vice versa (Jegadeesh and Titman (1993); see Fama and French’s Table VI). Again, this result by itself would not be a puzzle if the “winner” portfolio had higher market, smb, or hml betas than the loser portfolios. Alas (Fama and French, Table VII), the winner portfolio actually has lower hml slopes than the loser portfolio; winners act, sensibly enough, like high-price growth stocks that should have low mean returns in the three-factor model. The three-factor model is worse than useless at capturing the expected returns of this “momentum” strategy, just as the CAPM is worse than useless at explaining the average returns of book-to-market portfolios. Now, the returns of these 10 momentum-sorted portfolios can be explained by an additional “momentum factor” umd of winner stocks less loser stocks. You cannot form a diversified portfolio of momentum stocks and earn high returns with no risk; a common component to returns shows up once again. Yet Fama and French did not take the step of adding this fourth factor, and thus claiming a model that would explain all the known anomalies of its day. This reluctance is understandable. First, Fama and French worry (p. 81) whether the momentum effect is real. They note that the effect is much weaker before 1963, and they call for more out-of-sample verification. They may also have worried that the effect would not survive transactions costs. Exploiting the momentum anomaly requires high-frequency trading, and shorting small losing stocks can be difficult. Equivalently, momentum is, like long horizon regression, a way to enhance the economic size of a well-known statistical anomaly, as a tiny positive autocorrelation of returns can generate the observed momentum profits. Last year’s 1/10 best winners typically have gone up a tremendous amount, often 100 percent or more. It only takes a small, 0.1 or less, autocorrelation or 0.01 forecasting R2 to turn such past returns to 10 percent expected future returns. (See Cochrane (1999a) for a more detailed calculation.) Can one really realize profits that result from 0.01 forecast R2 ? Second, having just swallowed hml and smb, one might naturally be reluctant to add a new factor for every new anomaly, and to encourage others to do so. Third, and perhaps most importantly, Fama and French had at least a good story for the macroeconomic underpinnings of size and value effects, as expressed in the above quotation. They had no idea of a macroeconomic underpinning for a momentum premium, and in fact in their view (p. 81) there isn’t even a coherent behavioral story for such a premium. They know that having some story is the only “fishing license” that keeps one from rediscovering the Roll theorem. Still, they acknowledge (p. 82) that if the effect survives scrutiny, another “factor” may soon be with us. In the time since Fama and French wrote, many papers have examined the momentum effect in great detail. I do not survey that literature here, since it takes us

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away from our focus of macroeconomic understanding of premia rather than exploration of the premia themselves. However, momentum remains an anomaly. One can begin to imagine macroeconomic stories for momentum. Good cash-flow news could bring growth options into the money, and this event could increase the systematic risk (betas) of the winner stocks. Of course, then a good measure of “systematic risk” and good measurements of conditional betas should explain the momentum effect. Momentum is correlated with value, so it’s tempting to extend a macroeconomic interpretation of the value effect to the momentum effect. Alas, the sign is wrong. Last year’s winners act like growth stocks, but they get high, not low, average returns. Hence, the component of a momentum factor orthogonal to value must have a very high risk premium, and its variation is orthogonal to whatever macroeconomic effects underlie value. In any case, the current crop of papers that try to measure macroeconomic risks follow Fama and French by trying to explain the value and size premium, or the Fama– French 25 portfolios, and so far largely exclude the momentum effect. The momentum factor is much more commonly used in performance evaluation applications, following Carhart (1997). In order to evaluate whether, say, fund managers have stock-picking skill, it does not matter whether the factor portfolios correspond to real risks or not, and whether or not the average returns of the factor portfolios continue out of sample. One only wants to know whether a manager did better in a sample period than a mechanical strategy. I suspect that if the momentum effect survives its continued scrutiny, macro-finance will add momentum to the list of facts to be explained. A large number of additional expected-return anomalies have also popped up, which will also make it to the macrofinance list of facts if they survive long enough. We are thus likely to face many new “factors.” After all, each new expected-return sort must fall into one of the following categories: (1) a new expected-return sort might be explained by betas on existing factors, so once you understand the existing factors you understand the new anomaly, and it adds nothing. This is how, for example, sales growth behaves for the Fama– French model. (2) The new expected-return sort might correspond to a new dimension of comovement in stock returns, and thus be “explained” (maybe “summarized” is a better word) by a new factor. (3) If a new expected-return sort does not fall into 1 and 2, it corresponds to an arbitrage opportunity, which is most unlikely to be real—and, if real, to survive longer than a chicken in a crocodile pond. Thus, any expected return variation that is both real and novel must correspond to a new “factor.”

3. EQUITY PREMIUM With the basic facts in mind, we are ready to see what theories can match the facts; what specifications of the marginal utility of wealth VW can link asset prices to macroeconomics.

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The most natural starting point is the classic consumption-based asset pricing model. It states that expected excess returns should be proportional to the covariance of returns with consumption growth, with risk aversion as the constant of proportionality. If the utility function is of the simple time-separable form Et

∞

β ju(ct+j ),

j=0

then the marginal value of wealth is equal to the marginal utility of consumption—a marginal dollar spent gives the same utility as a marginal dollar saved—and our basic asset pricing equation (3) becomes9 ei u (ct+1 ) e , Et Rt+1 = −Covt Rt+1 , u (ct )

(5)

or, with the popular power utility function u (c) = c−γ , (or using that form as a local approximation), ei ct+1 e Et Rt+1 = γ × Covt Rt+1 , . (6) ct This model is a natural first place to link asset returns to macroeconomics. It has a great economic and intuitive appeal. Assets should give a high premium if they pay off badly in “bad times.” What better measure of “bad times” than consumption? People may complain, or seem to be in bad straits, but if they’re going out to fancy dinners you can tell that times aren’t so bad after all. More formally, consumption subsumes or reveals all we need to know about wealth, income prospects, etc. in a wide class of models starting with the Permanent Income Hypothesis. In every formal derivation of the CAPM, ICAPM, and every other factor model (at least all the ones I know of ), the marginal utility of consumption growth is a single factor that should subsume all the others. They are all special cases of the consumption-based model, not alternatives to it. The equity premium puzzle points out that this consumption-based model cannot explain the most basic premium, that of the market portfolio over the risk-free rate. (Again, notice in this exercise the proper role of macro models—the CAPM takes the 9 In

discrete time, the actual equation is ei 1 u (ct+1 ) e = − f Covt Rt+1 Et Rt+1 ,β , u (ct ) R

with 1 f

Rt

u (ct+1 ) ≡ Et β . u (ct )

The simpler form of Eq. (5) results in the continuous-time limit.

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mean market return as exogenously given. We are asking what are the economics behind the mean market return.) From (6) write E Rei = γσ Rei σ(Δc)ρ Δc, Rei , so, since ρ < 1,

- ei -E R σ(Rei )

< γσ(Δc).

(7)

(8)

The left-hand side of (8) is the “Sharpe ratio,” a common measure of the ratio of reward to risk in asset markets. In postwar U.S. data, the mean return of stocks over bonds is about 8 percent, with a standard deviation of about 16 percent, so the Sharpe ratio is about 0.5. Longer time series and other countries give somewhat lower values, but numbers above 0.2–0.3 are characteristic of most times and markets. Other investments (such as value stocks or some dynamic strategies in bond markets) can sometimes give much larger numbers, up to Sharpe ratios of 1.0. Aggregate non-durable and services consumption volatility is much smaller, about 1.5 percent per year in the postwar U.S. To get from σ(Δc) = 0.015 to a Sharpe ratio of 0.5, we need a risk aversion of at least 0.5/0.015 = 33, which seems much larger than most economists find plausible. One initial reaction is that the problem is not so much high stock average returns but low interest rates. Perhaps something is wrong with bonds, perhaps traceable to monetary policy, liquidity, etc. Alas, this solution does not work. The key to the calculation in (8) is the Sharpe ratio on the left-hand side. There are large Sharpe ratios between stocks (as in the value-growth premium studied by Fama and French) ignoring bonds all together. High sample Sharpe ratios are pervasive in finance and not limited to the difference between stocks and bonds. One might simply accept high risk aversion, but the corresponding equation for the risk-free rate, from the continuous-time limit of 1 + rf = 1/E(e−δ (u (ct+1 )/u (ct ))), is 1 rf = δ + γE(Δc) − γ(γ + 1)σ 2 (Δc). 2

(9)

If we accept γ = 33, with about 1 percent expected consumption growth E(Δc) = 0.01 and σ 2 (Δc) = 0.0152 , we predict a risk-free rate of rf = δ + 33 × 0.01 −

1 × 33 × 34 × 0.0152 2

= δ + 0.33 − 0.13. Thus, with δ = 0, the model predicts a 20 percent interest rate. To generate a (say) 5 percent interest rate, we need a negative 15 percent discount rate δ. Worse, (9) with γ = 33 predicts that the interest rate will be extraordinarily sensitive to changes in

260


expected consumption growth or consumption volatility. Therefore, the puzzle is often known as the “equity premium–risk-free rate” puzzle. The puzzle is a lower bound, and more information makes it worse. Among other observations, we do know something about the correlation of consumption and asset returns, and we know it is less than one. Using the sample correlation of ρ = 0.2 in postwar quarterly data, i.e., using (7) or using the sample covariance in (6), raises the required risk aversion by a factor of 5, to 165! Even using ρ = 0.41, the largest correlation among many consumption definitions (you get this with 4th quarter to 4th quarter real chain-weighted non-durable consumption), the required risk aversion rises to 33/0.41 = 80. The equity premium puzzle, and the larger failure of the consumption-based model that it crystallizes, is quantitative, not qualitative. The signs are right. The stock market does covary positively with consumption growth, so the market should give a positive risk premium. The problem is that the risk premium is quantitatively too large to be explained given sensible risk aversion and the observed volatility of consumption growth. Also, the puzzle necessarily unites macroeconomic and financial analysis. Finance models always had consumption hidden in them, and that consumption process had huge volatility. Consumption is proportional to wealth in the derivation of the CAPM, so the CAPM predicts that consumption should inherit the large 16 percent or so volatility of the stock market. You don’t notice this prediction though unless you ask for the implicit consumption volatility and you check it against consumption data. Equivalently, the standard optimal portfolio calculation says that the weight in risky assets should be w=

1 E(Re ) . γ σ 2 (Re )

Using an 8 percent mean and a 16 percent standard deviation, this calculation predicts 100 percent equities (w = 1) at γ = 0.08/0.162 = 3.125, which seems like a nice, sensible risk aversion. (In fact, this calculation was often cited—miscited, in my view— as evidence for low risk aversion.) The problem with the calculation is that the standard portfolio model also says consumption should be proportional to wealth, and thus consumption should also have a 16 percent standard deviation. That consumption is so much smoother than wealth remains a deep insight for understanding economic dynamics, one whose implications have not been fully explored. For example, it implies that one of consumption or wealth must have substantial dynamics. If wealth increases 16 percent in a typical 1σ year and consumption moves 2 percent in the same 1σ year, either consumption must eventually rise 14 percent or wealth must eventually decline 14 percent, as the consumption to wealth ratio is stable in the long run. This is a powerful motivation for Lettau and Ludvigson’s (2001a, 2001b) use of consumption and wealth as a forecasting variable. It means that timevarying expected returns, “excess” stock volatility, and the equity premium puzzle are all linked in ways that are still not fully exploited.

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3.1. Mehra and Prescott and the Puzzle The ink spilled on the equity premium would sink the Titanic, so there is no way here to do justice to all who contributed to or extended the puzzle, or even to summarize the huge literature. My quick overview takes the approach of Cochrane and Hansen’s (1992) review paper, “Asset Pricing Explorations for Macroeconomics.” The fundamental idea there, Eq. (8), is similar to a relation derived by Shiller (1982) (see p. 221) and much elaborated on by Hansen and Jagannathan (1991), who also provide many deep insights into the representation of asset prices. Cochrane and Hansen (1992) discuss the bounds including correlation as above and a large number of additional extensions. Weil (1989) points out the risk-free rate part of the puzzle. Chapters 1 and 21 of Asset Pricing (Cochrane (2004)) gives a review of the equity premium and related puzzles. Campbell (2000, 2003) and Kocherlakota (1996) are also excellent recent reviews. Mehra and Prescott (1985) named and announced the “puzzle” and launched the literature devoted to “explaining” it. Mehra and Prescott take a different approach from my simple synthesis: they specify an explicit two-state Markov process for consumption growth; they calculate the price of the consumption claim and risk-free rate; and they point out that the mean stock excess return calculated in this “calibrated economy” is much too low unless risk aversion is raised to apparently implausible values (55, in their model). The history of the equity premium puzzle is an interesting case study for how ideas form, catch on, and evolve in economics and finance. The pattern does not fit well into the familiar stylized models of intellectual evolution such as Kuhn (1962) or McCloskey (1983). Like many famous papers, this one has precursors. Shiller (1982) derived the first bound on discount-factor volatility. On p. 221, Shiller writes, It is also possible to arrive at a lower bound on the standard deviation of the marginal rate of substitution . . . by using data on asset returns alone. . . . One finds that σ(S) ≥

E(Rj ) − E(Ri ) σ(Ri )E(Rj ) − σ(Rj )E(Ri )

[Shiller uses S for what I have denoted m]. This inequality puts a lower bound on the standard deviation of S in terms only of the means and standard deviations [of returns]. . . . This inequality asserts that if two assets have very different average returns and their standard deviations are not sufficiently large, then σ(S) must be large if the covariance [of returns] with S is to explain the difference in average returns. If one uses the Standard & Poor’s portfolio as the jth asset, prime 4–6 month commercial paper as the ith asset and the sample means and sample standard deviations of after-tax real one-year returns for 1891 to 1980 in the right-hand side of the above inequality, then the lower bound on σ(S) is 0.20. . . . The large standard deviation for S arises because of the large difference between the after-tax average real return on stocks (. . . 5.7 percent per year for 1891 to 1980), and

262


[the] average after-tax real return on commercial paper (. . . 1.4 percent per year for 1891 to 1980), while the standard deviations of the real after-tax returns are not sufficiently high (0.154 for stocks and 0.059 for commercial paper) to account for the average return spread unless σ(S) is very high. A high σ(S) suggests a high coefficient of relative risk aversion A (γ) since σ(S) ≈ Aσ(ΔC/C). For 1891 to 1980 σ(ΔC/C) was 0.035 so a lower bound for σ(S) of 0.20 suggests A be over five. . . . . . . the conventional notion that stocks have a much higher return than does short term debt, coupled with the notion that pretax stock real returns have a standard deviation in the vicinity of 20 percent per year (commercial paper much less) implies that the standard deviation of S is very high. There it is, in a nutshell. Interestingly, we have come full circle, as my summary boils the calculation down to much the same sort of inequality Shiller started with. This work appeared in the context of a number of studies in the early 1980s that found very high risk aversion popping up in estimates of consumption-based first-order conditions, and Grossman and Shiller (1981) and Hansen and Singelton (1983) in particular, but the latter do not have as clear a statement of the puzzle. It’s interesting that Mehra and Prescott’s more complex approach was so much more influential. (A quick count in the Social Sciences Citation index gives 679 citations to Mehra and Prescott (1985), and only 35 to Shiller (1982).) Mostly, it seems to me that Mehra and Prescott were the first to argue and to persuade others that this puzzle, among so many in fitting the consumption-based model to data, is particularly important and that solving it would lead to some fundamental revision of the economics in the consumption-based model. This really is their distinctive, and central, contribution. Columbus “discovered” America, though Leif Ericson and a thousand Basque fishermen had been there before. Shiller’s (1982) result is presented in Section IV of a long survey paper, most of which covers volatility tests. The equity premium is, to Shiller, one of many interesting aspects of fitting the consumption-based model to data, and not the most important. The introduction makes no mention of the calculation. Instead, it advises that “the bulk of this paper will be an exploratory data analysis,” and will present “the broadest possible array of evidence relevant to judging the plausibility of the model.” It advertises that the paper will focus on . . . “three substantive questions,” the business-cycle behavior of interest rates, the accuracy of consumption data and the fact that few consumers hold stocks, and whether prices are too volatile—and does not include risk aversion and the equity premium in this list. Section IV first reviews other risk aversion estimates, gives a reminder of a different, volatility-test-based discount factor volatility calculation in Shiller (1981), and only then presents the result quoted above. The conclusion (p. 231) briefly mentions the calculation, among many others, but phrases it as “encouraging for the model” since large σ(S) can rationalize volatile prices, not noting that large σ(S) and smooth σ(ΔC) imply huge risk aversion. It is not a surprise that readers did not seize on the puzzle and run with it as they did after reading Mehra and Prescott. (Hansen’s (1982) comment on Shiller did notice the bounds on the volatility of marginal rates of

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substitution, and sharpened and extended Shiller’s calculations; one can see the roots of the Hansen–Jagannathan (1991) bounds here very clearly.) Grossman and Shiller (1981) devote almost their entire paper to tests of price volatility. Only in the very last paragraph, in a section titled “Further research,” do they write We have some preliminary results on the estimation of A (γ) and β. Estimates of both parameters can be derived using expression (3) (1 = E(mRi )) for two different assets which we took as stocks and short-term bonds. Unfortunately, the estimates of A for the more recent sub-periods seem implausibly high. They attribute the result to the divergence between P ∗ and P since the early 1950’s as well as the extremely low real returns on short-term bonds in this period. There was an enormous rise in stock prices in that period . . . They do not present the actual estimates or document them in any more detail than these sentences, though one may surmise that working paper versions of this paper presented more details. It would have been truly extraordinary if a verbal report of “preliminary” and “implausible” results, attributed to peculiarities of one data sample, at the end of a Papers and Proceedings elaboration of volatility tests, were to launch the equity-premium ship. (Volatility tests are also an important contribution, and with 211 citations this is a highly influential paper. The point here is not to diminish volatility tests but to track down why this paper did not also launch the equity premium.) Grossman, Melino, and Shiller (1987) is the other published work to result from Grossman and Shiller’s early 1980s’, risk aversion estimates. This paper starts with a simple table (Table 1, p. 318) of risk aversion estimates based on E(Re ) ≈ γ Cov(Re , Δc), and reports estimates between 13.8 and 398, depending on data set. “Table 1 shows that the mean excess return on stocks is associated with a relatively small covariance with consumption changes. If we ignore sampling and measurement error, this can be justified only by an implausibly high estimate of the risk-aversion parameter (see also Mehra and Prescott 1985).” This calculation shows that the low correlation between consumption growth and returns is another part of the problem, already extending the puzzle. At this point, though, the paper has become an explain-the-equitypremium paper, devoted to the question of whether a sophisticated treatment of time aggregation in consumption will overturn the result, and coming to the conclusion that it doesn’t do so. Hansen and Singleton (1983) also report a high risk aversion estimate. Hansen and Singleton describe the result in Table 5 thus: Consistent with their [Grossman and Shiller’s] results, we found α ˆ [risk aversion, γ in the above notation] to be very large with a correspondingly large standard error when NLAG = 0. Consistent with our other findings α ˆ is approximately one when the serial correlation in the time-series data

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is taken into account in estimation. This shows the extent to which the precision and magnitude of our estimates rely on the restrictions across the serial correlation parameters of the respective time series. Clearly, the point of this paper is to introduce instruments and to study varying conditioning information and how that conditioning information can be used to sharpen estimates. The bulk of this paper studies intertemporal substitution, how consumptiongrowth forecasts line up with interest rate forecasts, which involves one asset at a time and many instruments. The introduction (p. 250) summarizes the crucial idea of the paper as “The predictable components of the logarithms of asset returns are proportional to the predictable components of the change in the logarithm of consumption, with the proportionality factor being minus the coefficient of relative risk aversion.” Table 5 is the only table in this paper or in Hansen and Singleton (1982, 1984) that does not have instruments, or that does given a high risk aversion estimate. These are groundbreaking contributions, as I discuss in detail ahead, but again it’s clear how readers can easily miss the equity premium, introduced only as “for the sake of comparison” with Grossman and Shiller, buried in Table 5, summarized as an illustration of the sensitivity of the method to serial correlation, and the finding of high risk aversion needed to explain the unconditional equity premium ignored in the introduction or conclusion. By contrast, Mehra and Prescott (1985) claim that high risk aversion is a robust and unavoidable feature of any method for matching the model to data. They also argue that the puzzle is important because it will require fundamental changes in macroeconomic modeling. Compare the previous quotes to these, from the first page of Mehra and Prescott: The question addressed in this paper is whether this large differential in average yields can be accounted for by models that abstract from transactions costs, liquidity constraints and other frictions absent in the Arrow-Debreu setup. Our finding is that it cannot be, at least not for the class of economies considered. Our conclusion is that most likely some equilibrium model with a friction will be the one that successfully accounts for the large average equity premium. In sum, while it’s clear the central result can be found in Shiller (1982), Grossman and Shiller (1981), and Hansen and Singleton (1983), it is also pretty clear why readers missed it there. Part of Mehra and Prescott’s influence might also be traced to things they left out. Mehra and Prescott completely avoided √ inference or standard errors. Alas, the equity premium is not that well measured. σ/ T with σ √ ≈ 20 percent means that in 50 years of data the sample mean is estimated with a 20/ 50 = 2.8 percent standard error, so a 6 percent equity premium is barely two standard errors above zero. By ignoring standard errors, they focused attention on an economically interesting moment. But standard

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errors are not that hard. Shiller (1982, p. 221) already had them, directly below the above paragraph: Of course expected returns and standard deviations of returns are not precisely measured, even in a hundred years of data. An asymptotic standard error for the estimate of the right hand side of the inequality . . . was 0.078. Thus, the estimated lower bound for σ(S) is only two and a half standard deviations from zero. Hansen and Singleton (1983) also calculate standard errors. In fact, it is exactly the greater precision of estimates based on predictable movements in consumption growth and returns that drives them to pay more attention to moments with one return and many instruments and their indications of low risk aversion (which we now label “intertemporal substitution”) rather than the apparently less well measured moment consisting of stock and bond returns and no instruments, which is central to the equity premium. In fact, even reading Mehra and Prescott as saying “one needs high risk aversion to explain the equity premium” involves some hindsight. The introduction does not mention high risk aversion, it simply says that the equity premium “cannot” be accounted for by frictionless Arrow–Debreu models. The text on p. 155 documents this fact, in their two-state model, for risk aversion “calibrated” to be less than 10. The possibility that the model might work with high risk aversion is only acknowledged in a footnote describing a private communication with Fischer Black, and stated in the context of a different model. Mehra and Prescott also gave a structure that many people found useful for thinking about variations on the puzzle. A very large number of alternative explicitly-calculated two-state endowment economies followed Mehra and Prescott, though we now understand that the equity premium point really only needs first-order conditions as Shiller derived them and as I summarized earlier. Even the latter approach needed Hansen and Jagannathan’s (1991) paper to be revived. It took another army of papers calculating Hansen–Jagannathan bounds to come back in the end to the simple sorts of calculations in Shiller’s (1982) original article. Leaving a complex structure for others to play with seems to be a crucial piece of generating followers. Answering a question too quickly is dangerous to your influence. Mehra and Prescott’s general equilibrium modeling imposes extra discipline on this kind of research and has a separate and fully justified place of honor as the progenitor of the general equilibrium models described ahead. In a general equilibrium model, the covariance of consumption with returns is generated endogenously. You can’t just take Cov(R, Δc) as given and crank up γ (see Eq. (6)) to get any premium you want. Thus, seemingly normal specifications of the model can generate unexpected results. For example, positive consumption growth autocorrelation and risk aversion greater than one generate a negative equity premium because it generates a negative covariance of consumption growth with returns. Working out a general equilibrium model, one also notices that many other predictions go awry. For example, Mehra and Prescott’s model does not generate nearly enough return variance and measures to increase the

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equity premium or return variance dramatically and counterfactually increase the variation in the risk-free rate over time. These basic moments remain quite difficult for general equilibrium models to capture, but you cannot notice that they are a problem if you only look at first-order conditions.

3.2. The Future of the Equity Premium My view of the literature is that work “explaining the equity premium puzzle” is dying out. We have several preferences consistent with equity premium and risk-free rates, including habits and Epstein–Zin preferences. These preferences, described in more detail later, break the link between risk aversion and intertemporal substitution, so there is no connection to a “risk-free rate” puzzle any more, and we can coherently describe the data with high risk aversion. No model has yet been able to account for the equity premium with low risk aversion, and Campbell and Cochrane (1999) offer some reasons why this is unlikely ever to be achieved. So we may have to accept high risk aversion, at least for reconciling aggregate consumption with market returns in this style of model. (Frictions, as advocated by Mehra and Prescott (1985), have not emerged as the consensus answer to the puzzle. In part, this is because high Sharpe ratios occur between pairs of stocks as well as between stocks and bonds.) At the same time, many economists’ beliefs about the size of the equity premium are declining from the 8 percent postwar average, past the 6 percent average in longer samples, down to 2 or 3 percent or less. The U.S. economy and others with high sample equity premia may simply have been lucky. Did people in 1947 really think that the stock market would gain 8 percent per year more than bonds, and shy away from buying more stocks in the full knowledge of this mean, because the 16 percent annual standard deviation of stock returns seemed like too much risk? Or was the 8 percent mean return largely a surprise? Putting the argument a little more formally, we can separate the achieved average stock return into (1) the initial dividend yield (dividend payment/initial price), (2) increases in the price-dividend ratio, and (3) growth in dividends, giving growth in prices at the same price-dividend ratio. Dividend yields were about 4 percent and have declined to about 2 percent. Dividend yields are known ahead of time and so cannot contribute to a “surprise” return. The price-dividend ratio has about doubled in the postwar era, and this increase could well be a surprise. But this doubling happened over 50 years, contributing only 1.4 percent (compounded; 21/50 = 1.014) to the equity return. If there is a surprise, then, the surprise is that economic growth was so strong in the postwar era, resulting in surprisingly strong dividend growth. (In the long run, all of the return must be dividend growth since price-dividend ratios are stationary.) And, of course, economic growth was surprisingly good in the postwar era. Most people in 1947 expected a return to depression. For these reasons, as well as perhaps simple boredom in the face of intractable questions, research attention is moving to understanding stock return dynamics and the cross section, either ignoring the equity premium or simply allowing high risk aversion to

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account for it. One never can tell when a striking new insight will emerge, but I can tell that new twists in the standard framework are attracting less attention.

4. CONSUMPTION MODELS Really, the most natural thing to do with the consumption-based model is to estimate it and test it, as one would do for any economic model. Logically, this investigation comes before “puzzles,” which throw away information (correlation, multiple assets, time-variation of moments). The puzzles are not tests; they are useful diagnostics for why tests fail. We start here with Hansen and Singleton’s (1982, 1984) classic investigation of the consumption-based model. Alas, they decisively reject the model; among other things, they find the “equity premium puzzle” result that the model cannot explain the spread between stock and bond returns with low interest rates. The following 20 years have seen an enormous effort aimed at the consumptionbased model. There are, of course, all sorts of issues to address. What utility function should one use? How should one treat time aggregation and consumption data? How about multiple goods? What asset returns and instruments are informative? Asset pricing empirical work has moved from industry or beta portfolios and lagged returns and consumption growth as instruments to the use of size, book-to-market and momentum portfolios, and to the dividend-price ratio, term spreads, and other more powerful instruments. How does the consumption-based model fare against this higher bar? As I see it, there were 10 years of depressing rejection after rejection, followed by 10 years of increasing success. This is heartening. At some level, the consumption-based model must be right if economics is to have any hope of describing stock markets. The data may be poor enough that practitioners will still choose “reduced-form” financial models, but economic understanding of the stock market must be based on the idea that people fear stocks, and hence do not buy more despite attractive returns, because people fear that stocks will fall in “bad times.” At some point “bad times” must be mirrored in a decision to cut back on consumption.

4.1. Hansen and Singleton; Power Utility The classic consumption-based model test is due to Hansen and Singleton (1982, 1984). The influence of this paper is hard to overstate. It gives a clear exposition of the GMM methodology, which has pretty much taken over estimation and testing. (At least it has for me. Asset Pricing, by Cochrane (2004) maps all standard asset pricing estimates into GMM and shows how they can and should be easily generalized using GMM to account for heteroskedasticity and autocorrelation.) Also, with this work (generalizing Hall’s 1978 test for a random walk in consumption), macroeconomists and financial economists realized they did not need to write complete models before going to the data; they could examine the first-order conditions of investors without specifying technology, model solution, and a complete set of shocks.

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Hansen and Singleton examine the discrete-time nonlinear consumption-based model with power utility, −γ ct+1 i Rt+1 = 1. Et β ct

(10)

The method is astonishingly simple. Multiply both sides both sides of (10) by instruments—any variable zt observed at time t—and take unconditional expectations, yielding −γ ct+1 i (11) E β Rt+1 − 1 zt = 0. ct Then, take sample averages, and search numerically for values of β, γ that make these “moment conditions” (equivalently, pricing errors) as small as possible. GMM gives a distribution theory for the parameter estimates and a test statistic based on the idea that these pricing errors should not be too big. Hansen and Singleton’s (1984) results provide a useful baseline. If we take a single asset and multiply it by instruments (Hansen and Singleton’s Table 1), we are asking whether movements in returns predictable by some instrument zt —as in regressions of Rt+1 on zt —are matched by movements in consumption growth or by the product of consumption growth and returns as predicted by the same instrument. The results give sensible parameter estimates; small coefficients of risk aversion γ and discount factors less than one. However, the standard errors on the risk aversion coefficients are pretty large, and the estimates are not that stable across specifications. The problem, or rather the underlying fact, is that Hansen and Singleton’s instruments—lags of consumption and returns—don’t forecast either consumption growth or returns very well. Consumption and stock prices are, in fact, pretty close to random walks, especially when forecast by their own lags. To the extent that these instruments do forecast consumption and returns, they forecast them by about the same amount, leading to risk aversion coefficients near one. Simplifying somewhat, consider the linearized risk-free rate equation, 1 f rt = δ + γEt (Δct+1 ) − γ(γ + 1)σt2 (Δct+1 ). 2

(12)

If risk premia are not well forecast by these instruments (and they aren’t) and consumption is homoskedastic (pretty close), then the main thing underlying estimates of (11) with a single asset and many instruments is whether predictable movements in consumption growth line up with predictable movements in interest rates. The answer for Hansen and Singleton is that they do, with a constant of proportionality (γ) near one. (Hansen and Singleton’s (1983) study this linearized version of the consumption-based model, and their Table 4 studies this interest rate equation explicitly.)

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TABLE 3 Consumption data

Lags

1.001 (0.0462)

Nondurable

0

0.205

0.999

Nondurable

4

3

58.25 (66.57)

1.088 (0.0687)

ND & Services

0

4

0.209

1.000

ND& Services

4

γ∗

β∗

1

30.58 (34.06)

2

Model

χ 2†

Degrees of freedom Just identified

170.25 (0.9999)

24 Just identified

366.22 (0.9999)

24

Estimates of the consumption-based model using the value-weighted NYSE return and the Treasury bill return. Lags is the number of lags of consumption growth and returns used as instruments. Source: Hansen and Singleton (1983), Table 5. ∗ Standard errors in parentheses. † Probability values in parentheses.

If we take multiple assets, the picture changes, however. The middle panel of Hansen and Singleton’s (1984) Table III uses one stock and one bond return, and a number of instruments. It finds small, well-measured risk aversion coefficients—but the tests all decisively reject the model. Hansen and Singleton’s (1983) Table 5, reproduced here, makes the story clear. If we just use the unconditional moments—no instruments, the “lags = 0” rows—we find a very large value of the risk aversion coefficient. The covariance of consumption growth with stock returns is small, so it takes a very large risk aversion coefficient to explain the large mean stock excess return. This finding is the equity premium in a nutshell. (Using more recent data and the full nonlinear model, the smallest pricing error occurs around γ = 50, but there is no choice of γ that sets the moment to zero, even though the model is just identified.) The β slightly greater than one is the risk-free rate puzzle. The data are monthly, so even a β slightly greater than one is puzzling. If we use instruments as well, in the lags = 4 rows, then the estimate is torn between a small value of γ to match the roughly one-for-one movement of predicted consumption growth and returns (using past consumption growth and returns as predictors) and the very large value of γ necessary to explain the equity premium. Efficient methods weigh the evidence provided by different moments according to the statistical significance of those moments. Here, the moments corresponding to predictable movements are better measured, so the estimate of γ is close to those values. But the test statistic gives a huge rejection, as in Hansen and Singleton (1984). That huge test statistic tells us that there is a tension over the value of γ. The value of γ that makes sense of the equity premium (unconditional returns) is much larger than the value that makes sense of the conditional moments (forecasted returns vs. consumption growth), so one set of moments or pricing errors is left very large in the end.

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4.1.1. Risk Aversion and Intertemporal Substitution—More Recent Estimates The fact that quite high risk aversion is required to digest the equity premium is robust in consumption-based model estimation, as the equity premium discussion above makes clear. The parameter needed to understand the behavior of a single asset over time, and in particular to line up variation in expected consumption growth with variation in interest rates, is less certain. This number (or more precisely its inverse, how much consumption growth changes when interest rates go up 1 percent) is usually called the intertemporal substitution elasticity since it captures how much people are willing to defer consumption when presented with a large return opportunity. While Hansen and Singleton found numbers near one, Hall (1988) argued the estimate should be closer to zero, i.e., a very high risk aversion coefficient here as well. Hall emphasizes the difficulties of measuring both real interest rates and especially consumption growth. A good deal of the more recent macro literature has tended to side with Hall. Campbell (2000, 2003) gives an excellent summary with estimates. Real interest rates have moved quite a bit, and slowly, over time, especially in the period since the early 1980s when Hansen and Singleton wrote. Thus, there is a good deal of predictable variation in real interest rates. After accounting for time aggregation and other problems, consumption growth is only very poorly predictable. Lining up the small movements in expected consumption growth against large movements in real interest rates, we see a small intertemporal substitution elasticity, or a large risk aversion coefficient. At least now both moments consistently demand the same puzzlingly high number!

4.2. New Utility Functions Given problems with the consumption-based model, the most natural place to start is by questioning the utility function. Functional form is not really an issue, since linearized and nonlinear models already behave similarly. Different arguments of the utility function are a more likely source of progress. Perhaps the marginal utility of consumption today depends on variables other than today’s consumption. To get this effect, the utility function must be non-separable. If a utility function is separable, u(c, x) = υ(c) + w(x), then ∂u(c, x)/∂c = υ (c) and x does not matter for asset pricing. This is the implicit assumption that allowed us to use only non-durable consumption rather than total consumption in the first place. To have marginal utility of consumption depend on something else, we must have a functional form that does not add up in this way, so that ∂u(c, x)/∂c is a function of x, too. The first place to look for non-separability is across goods. Perhaps the marginal utility of non-durable consumption is affected by durables, or by leisure. Also, business cycles are much clearer in durables purchases and employment data, so business cycle risk in stock returns may correlate better with these variables than with non-durable and services consumption. One problem with this generalization is that we don’t have much intuition for which way the effect should go. If you work harder, does that make a TV more valuable as a

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break from all that work, or less valuable since you have less time to enjoy it? will you believe an estimate that relies strongly on one or the other effect? We can also consider non-separability over time. This was always clear for durable goods. If you bought a car last year, it still provides utility today. One way to model this non-separability is to posit a separable utility over the services and a durable goods stock that depreciates over time; U=

β t u(kt ); kt+1 = (1 − δ)kt + ct+1 .

t

This expression is equivalent to writing down a utility function in which last year’s purchases give utility directly today: U=

βtu

t

∞

(1 − δ)j ct−j .

j=0

If u (·) is concave, this function is non-separable, so marginal utility at t is affected by consumption (purchases) at t − j. At some horizon, all goods are durable. Yesterday’s pizza lowers the marginal utility for another pizza today. Following this line also leads us to thinking about the opposite direction: habits. If good times lead people to acquire a “taste for the good life,” higher consumption in the past might raise rather than lower the marginal utility of consumption today. A simple formulation is to introduce the “habit level” or “subsistence level” of consumption xt , and then let β t u(ct − θxt ); xt = ρxt−1 + ct U= t

or, directly, U=

t

β u ct − θ t

∞

j

ρ ct−j .

j=0

Again, you see how this natural idea leads to a non-separable utility function in which past consumption can affect marginal utility today. A difficulty in adding multiple goods is that if the non-separability is strong enough to affect asset prices, it tends to affect other prices as well. People start to care a lot about the composition of their consumption stream. Therefore, if we hold quantities fixed (as in the endowment-economy GMM tradition), such models tend to predict lots of relative price and interest rate variation; if we hold prices fixed, such models tend to predict lots of quantity variation, including serial correlation in consumption growth. An investigation with multiple goods needs to include the first-order condition for allocation across goods, and this often causes trouble. Finally, utility could be non-separable across states of nature. Epstein and Zin (1991) pioneered this idea in the asset pricing literature, following the thoretical development

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by Kreps and Porteus (1978). The expected utility function adds over states, just as separable utility adds over goods, π(s)u [c(s)]. Eu(c) = s

Epstein and Zin propose a recursive formulation of utility: Ut =

(1 −

1−ρ β)ct

1 1−ρ 1−ρ 1−γ 1−γ , + β Et Ut+1

(13)

which, 1 other things, abandons separability across states of nature. The term among 1−γ 1−γ is sometimes called a “risk adjustment” or the “certain equivalent” Et Ut+1 of future utility. The Epstein–Zin formulation separates the coefficient of risk aversion γ from the inverse of the elasticity of intertemporal substitution ρ. Equation (13) reduces to power utility for ρ = γ. Models with non-time separable utilities (habits, durables) also distinguish risk aversion and intertemporal substitution, but not in such a simple way. The stochastic discount factor/marginal rate of substitution is ⎛ ⎞ρ−γ −ρ ⎜ ⎟ Ut+1 ct+1 ⎜ ⎟ . (14) mt+1 = β 1 ⎠ ⎝ ct 1−γ 1−γ Et Ut+1 (The Appendix contains a short derivation.) If ρ = γ, we see a second term; expected returns depend on covariances of returns with the utility index, capturing news about the investor’s future prospects, as well as on covariances of returns with consumption growth. As we will see, a large number of modifications to the standard setup lead to a marginal rate of substitution that is the old power formula times a multiplicative new term. The utility index itself is not directly measurable, so to make this formula operational we need some procedure for its measurement. It turns out that the utility index is proportional to the value of the wealth portfolio (the claim to the consumption stream), so one can write the discount factor 1−θ −ρ θ 1 ct+1 , (15) mt+1 = β ct RW t+1

where θ=

1−γ . 1−ρ

(This formula is also derived in the Appendix.) This effect provides a route to including stock returns in the asset pricing model alongside consumption growth, which of course

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can give a much improved fit. This was the central theoretical and empirical point of Epstein and Zin (1991). However, this modification stands a bit on shaky ground: the substitution only works for the entire wealth portfolio (claim to future consumption), including non-traded assets such as real estate and the present value of labor income, not the stock market return alone. Furthermore, wealth and consumption do not move independently; news about consumption growth moves the wealth return. To emphasize the latter point, we can think of the discount factor in terms only of current and future consumption. In the discount factor (14), the utility index is a function of the distribution of future consumption, so the essence of the discount factor is that news about future consumption matters as well as current consumption in the discount factor. To see this effect more concretely, we can derive the discount factor for the case ρ = 1, and log-normal heteroskedastic consumption. I present the algebra in the Appendix. The result is (Et+1 − Et )ln mt+1 = − γ(Et+1 − Et )(Δct+1 ) ∞ j + (1 − γ) β (Et+1 − Et )(Δct+1j ) ,

(16)

j=1

where Δc is log consumption growth, Δct = ln ct − ln ct−1 . News about future long horizon consumption growth enters the current period marginal rate of substitution. Shocks to variables that predict future consumption growth will appear as additional risk factors even with (perfectly measured) current consumption growth. (Campbell (1996, p. 306) pursues the mirror-image expression, in which assets are priced by covariance with current and future wealth-portfolio returns, substituting out consumption. Restoy and Weil (1998, p. 10) derive an approximation similar to (16) and make this point. Hansen, Heaton, and Li (2006) and Hansen, Heaton, Lee, and Roussanov (2006) derive (16) and show how to make similar approximations for ρ = 1.)

4.3. Empirics with New Utility Functions 4.3.1. Non-separabilities Across Goods Eichenbaum, Hansen, and Singleton (1988) is an early paper that combined nonseparability over time and across goods. They used a utility function (my notation) U=

β

t

∗θ ∗1−θ 1−γ −1 ct lt

ct∗ = ct + αct−1 , lt∗ = lt + blt−1

1−γ

or

;

lt∗ = lt + b

∞ j=0

ηj lt−j ,

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where l denotes leisure. However, they only test the model on the Treasury bill return, not the equity premium or certainly not the Fama–French portfolios. They also focus on parameter estimates and test statistics rather than pricing errors. Clearly, it is still an open and interesting question of whether this extension of the consumption-based model can address what we now understand are the interesting questions.10 Eichenbaum and Hansen (1990) investigate a similar model with non-separability between durables and non-durables. This is harder because one needs also to model the relation between observed durable purchases and the service flow that enters the utility function. Also, any model with multiple goods gives rise to an intratemporal first-order condition, marginal utility of non-durables/marginal utility of durables = relative price. Eichenbaum and Hansen solve both problems. However, they again only look at consumption and interest rates, leaving open how well this model does at explaining our current understanding of cross-sectional risk premia. See also Ogaki and Reinhart (1998). In the consumption-based revival, Yogo (2004) reconsiders non-separability across goods by looking again at durable goods. He examines the utility function 11 1 1 u(C, D) = (1 − α)C 1− ρ + αD1− ρ 1− ρ . He embeds this specification in an Epstein–Zin aggregator (13) over time. This framework allows Yogo to use quite high risk aversion without the implication of wildly varying interest rates. Following tradition in the Epstein–Zin literature, he uses the market portfolio return to proxy for the wealth portfolio or utility index, which appears in the marginal rate of substitution. Estimating the model on the Fama–French 25 size and book-to-market portfolios, along with the 3-month T-bill rate, and including the intratemporal first-order condition for durables vs. non-durables, he estimates high (γ = 191; 1/γ = 0.005) risk aversion, as is nearly universal in models that account for the equity premium. He estimates a larger elasticity of intertemporal substitution σ = 0.024 to explain a low and relatively constant interest rate, and a modest 0.54 − 0.79 (depending on method) elasticity of substitution between durables and non-durables. As in the discussion of Piazzesi, Schneider, and Tuzel ahead, the difference between this modest elasticity and the much smaller σ and 1/γ means that the non-separabilities matter, and durables do affect the marginal utility of consumption. Yogo linearizes this model giving a discount factor linear in consumption growth, durable consumption growth, and the market return: mt+1 ≈ a − b1 Δct+1 − b2 Δdt+1 − b3 rW t+1. 10 Lettau’s (2003) Footnote 2 points out that consumption and leisure are negatively correlated (people work and consume more in expansions). The product c × l and the resulting marginal rate of substitution are then typically less volatile than with c alone, making the equity premium puzzle worse. However, the greater correlation of labor with asset returns may still make asset pricing work better, especially if one admits a large risk aversion coefficient.

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This linearized model prices the Fama–French 25 portfolios (except the small growth portfolio, left out of many studies) with a large cross-sectional R2 . By linearizing, Yogo is able to display that there is a substantial spread in betas, addressing the concern that a model prices well by an insignificant spread in betas and a huge risk premium. Yogo also shows some evidence that variation in conditional mean returns lines up with varying conditional covariances on these three factors. Pakos (2004) also considers durables vs. non-durables, using the non-linear specification, dealing with the intratemporal first-order condition (durable vs. non-durable and their relative price), and considering the level of the interest rate as well as the equity premium and the Fama–French 25 portfolios. Pakos needs an extreme unwillingness to substitute durable for non-durable consumption in order to make quantitatively important differences to asset pricing. To keep the durable vs. non-durable first-order condition happy, given the downward trend in the ratio of durables to non-durables, he adds an income elasticity (non-homothetic preferences).

4.3.2. Habits Ferson and Constantinides (1991) took the lead in estimating a model with temporal non-separabilities. One has to face parameter profusion in such models; they do it by limiting the non-separability to one lag, so the utility function is u(ct − bct−1 ).

(17)

This is one of the first papers to include an interesting cross section of assets, including the market (equity premium) and some size portfolios, along with a modern set of instruments, including dividend-price ratio and T-bill rate, that actually forecast returns. However, much of the model’s apparently good performance comes down to larger standard errors rather than smaller pricing errors. Heaton (1993, 1995) considers the joint effects of time aggregation, habit persistence, and durability on the time-series process for consumption and on consumptionbased asset pricing models. The 1993 paper focuses on consumption, showing how the random walk in consumption that occurs with quadratic utility and constant real rates is replaced by interesting autocorrelation patterns with time aggregation, habit persistence, and durability. Heaton (1995) then integrates these ideas into the specification of consumption-based asset pricing models, not an easy task. In particular, Heaton gives us a set of tools with which to address time aggregation, and Campbell and Cochrane (2000) argue in a simulation model that time aggregation helps a lot to explain consumption-based model failures. Sensibly, Heaton finds signs of both durability and habit persistence, with durability dominating at short horizons (even a pizza is durable at a one-minute horizon) and habit persistence at longer horizons. However, he only considers the value-weighted stock market and T-bill rate as assets. Campbell and Cochrane (1999) adapt a habit persistence model to generate a number of asset pricing facts. We replace the utility function u(C) with u(C − X), where X

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denotes the level of habits: E

∞ t=0

δt

(Ct − Xt )1−γ − 1 . 1−γ

Habits move slowly in response to consumption. The easiest specification to capture this observation would be an AR(1), Xt = ρXt−1 + λCt .

(18)

(Small letters denote the logs of large letters throughout this section, ct = ln Ct , etc.) This specification means that habit can act as a “trend” line for consumption; as consumption declines relative to the “trend” in a recession, people will become more risk-averse, stock prices will fall, expected returns will rise, and so on. The idea is not implausible (well, not to us at least). Anyone who has had a large pizza dinner or smoked a cigarette knows that what you consumed yesterday can have an impact on how you feel about more consumption today. Might a similar mechanism apply for consumption in general and at a longer time horizon? Perhaps we get used to an accustomed standard of living, so a fall in consumption hurts after a few years of good times, even though the same level of consumption might have seemed very pleasant if it arrived after years of bad times. This thought can at least explain the perception that recessions are awful events, even though a recession year may be just the second- or third-best year in human history rather than the absolute best. Law, custom, and social insurance also insure against falls in consumption as much or more than low levels of consumption. But it seems more sensible that habits move slowly in response to consumption experience rather than with the one-period lag of many specifications. In addition, slow-moving habits will generate the slow-moving state variables we seem to see in return forecastability. We specify a non-linear version of (18). This non-linear version allows us to avoid an Achilles heel of many habit models, a huge variation in interest rates. When consumers have habits, they are anxious in bad times (consumption close to habit) to borrow against coming good times (consumption grows away from habit). This anxiety results in a high interest rate, and vice versa in good times. The nonlinear version of (18) allows us to offset this “intertemporal substitution” effect with a “precautionary savings” effect. In bad times, consumers are also more risk-averse, so rather than borrow to push consumption above habit today, they save to make more sure that consumption does not fall even more tomorrow. The nonlinear version of (18) allows us to control these two effects. In Campbell and Cochrane (1999), we make the interest rate constant. The working paper version (Campbell and Cochrane (1995)) showed how to make interest rates vary with the state and thus create an interesting term structure model with time-varying risk premia. This sort of reverse engineering is important in a wide variety of models. Devices that increase the volatility of the discount factor or marginal rate of substitution across states of nature σt (mt+1 ), to generate a large equity premium, also tend to increase the

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volatility of the marginal rate of substitution over time σ(Et (mt+1 )), thus generating counterfactually large interest rate variation. To be empirically plausible, it takes some care to set up a model so that it has a lot of the former variation with little of the latter. We examine the model’s behavior by a combination of simulation and simple moment-matching rather than a full-blown estimation on an interesting cross section of portfolios, as do Constantinides (1990), Abel (1990), and Sundaresan’s (1989) habit persistence investigations. We let aggregate consumption follow a random walk, we calibrate the model to match sample means including the equity premium, and we then compare the behavior of common time-series tests in our artificial data to their outcome in real data. The model matches the time-series facts mentioned above quite well. In particular, the dividend-price ratio forecasts stock returns, and variance decompositions find all variation in stock prices is due to changing expected returns. In this model, the marginal rate of substitution—growth in the marginal value of wealth or discount factor—between dates t and t + k depends on change in the ratio of consumption to habit as well as on consumption growth, mt+1 = β

Ct+1 Ct

−γ

St+1 St

−γ ,

(19)

where St = (Ct − Xt )/Ct and Xt is habit. A large number of models amount to something like Eq. (19), in which the discount factor generalizes the power utility case by adding another state variable. The basic question is, why do people fear stocks so much? This model’s answer is not so much that they fear that stocks will decline when consumption is low in absolute terms (C); the answer is that they fear stocks will decline in future recessions, times when consumption falls low relative to habits (S). There is a danger in models of the form (19) that they often work well for short-run returns, but not in the long run. The trouble is that S is stationary, while consumption of course is a random walk. Now, to generate a large Sharpe ratio, we need a large volatility of the discount factor σ(m), and to generate a large Sharpe ratio in long-run returns we need the variance of the discount factor to increase linearly with horizon. If the second term S −γ is stationary, it may contribute a lot to the volatility of one-period discount factors, but in the long run we will be right back to the power utility model and all its problems, since the variance of a stationary variable approaches a limit while the variance of the random walk consumption component increases without bounds. The Campbell–Cochrane model turns out not to suffer from this problem: while −γ St is stationary, the conditional variance of St grows without bound. Thus, at any horizon the equity premium is generated by covariance with S −γ , not so much by covariance with consumption growth. This result stems from our non-linear habit accumulation process. It may not be there in many superficially attractive simplifications or linearizations of the habit model. However, though the maximum Sharpe ratio, driven by σ(mt,t+k ) remains high at long horizons, this fact does not necessarily mean that the average returns of all assets remain high at long horizons. For example, a consumption claim gets a high premium at a one-year horizon, since Ct+1 and St+1 are correlated, so the consumption claim payoff

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covaries a great deal with the discount factor. However, at long horizons, consumption −γ and St+k become uncorrelated, so a long-term consumption claim will not attain the still-high Sharpe ratio bound. Simulation is a prequel to empirical work, not a substitute for it, so this sort of model needs to be evaluated in a modern cross-sectional setting, for example in the Fama– French 25 size and book-to-market portfolios. Surprisingly, no one has tried this (including Campbell and myself ). The closest effort is Chen and Ludvigson (2004). They evaluate a related habit model using the Fama–French 25 size and book-to-market portfolios. They use a “nonparametric” (really, highly parametric) three-lag version of the MA habit specification (17) rather than the slow-moving counterpart (18). Comparing models based on Hansen–Jagannathan (1997) distance, which is a sum of squared pricing errors weighted by the inverse of the second-moment matrix of returns, they find that the resulting consumption-based model performs quite well, even better than the Fama–French three-factor model. Within this structure, they find that the “internal habit” version of the model performs better than the “external habit” version in which each person’s habit is set by the consumption of his neighbors. (I add the qualifier “within this structure” because in other structures internal and external habits are observationally indistinguishable.) The “internal habit” specification may be able to exploit the correlation of returns with subsequent consumption growth, which is also the key to Parker and Julliard (2005), discussed later. Wachter (2004) extends the habit model to think seriously about the term structure of interest rates, in particular adding a second shock and making a quantitative comparison to the empirical findings of the term structure literature such as Fama and Bliss’ (1987) finding that forward-spot spreads forecast excess bond returns. Verdelhan (2004) extends the habit model to foreign exchange premia. Here the puzzle is that high foreign interest rates relative to domestic interest rates signal higher returns in foreign bonds, even after including currency risk. His explanation is straightforward. The first part of the puzzle is, why should (say) the Euro/dollar exchange rate covary at all with U.S. consumption growth, generating a risk premium? His answer is to point out that in complete markets the exchange rate is simply determined by the ratio of foreign to domestic marginal utility growth, so the correlation pops out naturally. The second part of the puzzle is, why should this risk premium vary over time? In the habit model, recessions, times when consumption is close to habit, are times of low interest rates and also times of high risk premium (people are more risk-averse when consumption is near habit). Voilá, the interest rate spread forecasts a time-varying exchange rate risk premium. More generally, these papers pave the way to go beyond equity, value, size, and momentum premiums to start thinking about bond risk premia and foreign exchange risk premia.

4.3.3. Related Models The essence of these models really does not hinge on habits per se, as a large number of microeconomic mechanisms can give rise to a discount factor of the form (19), where C is aggregate consumption and S is a slow-moving business cycle-related state variable.

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Constantinides and Duffie (1996), discussed aheads, generate a discount factor of the form (19), in a model with power utility but idiosyncratic shocks. The “S” component is generated by the cross-sectional variance of the idiosyncratic shocks. In Piazzesi, Schneider, and Tuzel (2004), the share of housing consumption in total consumption plays the role of habits. They specify that utility is non-separable between non-housing consumption and consumption of housing services; you need a roof to enjoy the new TV. Thus, the marginal rate of substitution or stochastic discount factor is

mt+1

Ct+1 =β Ct

− σ1

αt+1 αt

ε−σ σ(ε−1)

.

(20)

Here, α is the expenditure share of non-housing services, which varies slowly over the business cycle just like S in (19). Housing services are part of the usual non-durable and services aggregate, of course, and the fact that utility is non-separable across two components of the index does not invalidate the theory behind the use of aggregate consumption. Therefore, the paper essentially questions the accuracy of price indices used to aggregate housing services into overall services. Does more housing raise or lower the marginal utility of other consumption, and do we trust this effect? Piazzesi, Schneider, and Tuzel calibrate the elasticity of substitution ε from the behavior of the share and relative prices, exploiting the static first-order condition. If ε = 1, the share of housing is the same for all prices. They find that ε = 1.27: when housing prices rise, the quantity falls enough that the share of housing expenditure actually falls slightly. This does not seem like an extreme value. As (20) shows though, whether the housing share enters positively or negatively in marginal utility depends on the substitutability of consumption over time and states σ as well as the substitutability of housing for other consumption ε. Like others, they calibrate to a relatively large risk premium, hence small σ. This calibration means that the housing share enters negatively in the marginal rate of substitution; a lower housing share makes you “hungrier” for other consumption. Most of Piazzesi, Schneider, and Tuzel’s empirical work consists of a simulation model. They use an i.i.d. consumption growth process, and they fit an AR(1) to the housing share. They then simulate artificial data on the stock price as a levered claim to consumption. The model works very much like the Campbell–Cochrane model. Expected returns are high, matching the equity premium, because investors are afraid that stocks will fall when the housing share α is low in recessions. (They also document the correlation between α and stock returns in real data.) Interest rates are low, from a precautionary savings effect due to the volatility of α and due to the mean α growth. Interest rates vary over time, since α moves slowly over time and there are periods of predictable α growth. Variation in the conditional moments of α generates a time-varying risk premium. Thus, the model generates returns predictable from price-dividend ratios and from housing share ratios. They verify the latter prediction, adding to the list of macro variables that forecast returns. (See Tables 4 and 5.) Finally, the model generates slow-moving variation in pricedividend ratios and stock return volatility, all coming from risk premia rather than

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dividend growth. However, the second term is stationary in their model, so it is likely that this model does not produce a long-run equity premium or any high long-run Sharpe ratios. Lustig and Van Niewerburgh (2004a, 2004b) explore a similar model. Here, variations in housing collateral play the role of the “habit.” Consumer-investors (-homeowners) whose housing collateral declines become effectively more riskaverse. Lustig and Van Niewerburgh show that variations in housing collateral predict stock returns in the data, as the surplus consumption ratio predicts stock returns in the Cambpell–Cochrane model. They also show that a conditional consumption CAPM using housing collateral as a conditioning variable explains the value-size cross-sectional effects, as implied by their model, in the same manner as with the Lettau–Ludvigson (2001a, 2001b) cay state variable. Chetty and Szeidl (2004) show how consumption commitments mimic habits. If in good times you buy a house, it is difficult to unwind that decision in bad times. Nonhousing consumption must therefore decline disproportionately. They also show that people who have recently moved for exogenous reasons hold a smaller proportion of stocks, acting in more risk-averse manner.

4.3.4. Long Horizons Nobody expects the consumption-based model (and data) to work at arbitrarily high frequencies. We do not calibrate purchasing an extra cup of coffee against the last hour’s stock returns. Even if consumers act “perfectly” (i.e., ignoring all transaction, information, etc. costs), high-frequency data is unreliable. If Δct and rt are perfectly correlated but independent over time, a one-period timing error, in which you mistakenly line up Δct−1 with rt , will show no correlation at all. The methods for collecting quantity data are not attuned to getting high-frequency timing just right, and the fact that returns are much better correlated with macro variables one or two quarters later than they are with contemporaneous macro variables is suggestive. The data definitions break down at a high frequency, too. Clothing is “non-durable.” In sum, at some high frequency, we expect consumption and return data to be de-linked. Conversely, at some low enough frequency, we know consumption and stock market values must move one for one; both must eventually track the overall level of the economy, and the consumption to wealth ratio will neither grow without bound nor decline to zero. Thus, some form of the consumption model may well hold at a longenough horizon. Following this intuition, a number of authors have found germs of truth in long-run relations between consumption and returns. Daniel and Marshall (1997) showed that consumption growth and aggregate returns become more correlated at longer frequencies. They don’t do a formal estimation, but they do conclude that the equity premium is less of a puzzle at longer frequencies. Brainard, Nelson, and Shapiro (1991) show that the consumption CAPM performance gets better in some dimensions at longer horizons. However, these greater correlations do not mean the model is a total success, as other moments still do not line up. For

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example, Cochrane and Hansen (1992) find that long horizon consumption performs worse in Hansen–Jagannathan bounds. There are fewer consumption declines in long horizon data, and the observation that (Ct+k /Ct )−γ can enter a Hansen–Jagannathan bound at high risk aversion depends on consumption declines raised to a large power to bring up the mean discount factor and solve the risk-free rate puzzle. Most recently and most spectacularly, Jagannathan and Wang (2005) find that by using fourth quarter to fourth quarter non-durable and services consumption, the simple consumption-based model can account for the Fama–French 25 size and book-to-market portfolios. Figure 1 captures this result dramatically. On reflection, this is a natural result. A lot of purchases happen at Christmas, and with an annual planning horizon. Time aggregation and seasonal adjustment alone would make it unlikely that monthly average consumption would line up with end-of-month returns. And it is a stunning result: the simple power utility consumption-based model does work quite well after all, at least for one horizon (annual). Of course, not everything works. The model is linearized (Jagannathan and Wang examine average returns vs. betas on consumption growth), the slope coefficient of average returns on betas does imply an admittedly rather high risk aversion coefficient, and there are still many moments for which the model does not work. But it is a delightful sign that at least one sensible moment does work, and delightful to see an economic connection to the puzzling value premium. Parker and Julliard (2005) similarly examine whether size and book-to-market portfolios can be priced by their exposure to “long-run” consumption risk. Specifically, they examine whether a multiperiod return formed by investing in stocks for one period and then transforming to bonds for k − 1 periods is priced by k period consumption growth. They study the multiperiod moment condition −γ f f f k Ct+k Rt+1 Rt+1 Rt+2 ...Rt+k−1 . 1 = Et β Ct

(21)

They argue that this moment condition is robust to measurement errors in consumption and simple “errors” by consumers. For example, they argue that if consumers adjust consumption slowly to news, this moment will work while the standard one will not. Parker and Julliard find that this model accounts for the value premium. Returns at date t + 1 forecast subsequent consumption growth very slightly, and this forecastability accounts for the results. In addition to selecting one of many possible long-run moment conditions, Parker and Julliard leave the moment condition for the level of the interest rate out, thus avoiding equity premium puzzles. Lustig and Verdelhan (2004) do a standard consumption-beta test on foreign exchange returns at an annual horizon and find, surprisingly, that the standard consumption-based model works quite well. One of their clever innovations is to use portfolios, formed by going in to high interest rate countries and out of low interest rate countries. As in the rest of asset pricing, portfolios can isolate the effect one is after and can offer a stable set of returns.

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Consumption CAPM

Realized Excess Return %

20

R2 5 0.73

15 14

15

24 34 23 13 44

25 35 12 45

43

10

22 53 42 41 52 31 51 21 11

33 32 55

54

5

0

0

Realized Excess Return %

20

10 15 5 Fitted Excess Return % Fama–French 3 Factor Model R2 5 0.8

15 14 25

15

10 53 51

5

0 0

20

41

54 42

24 35 34 23 12 44 13 45 43 32 33 22 55

31 52 21 11

10 15 5 Fitted Excess Return %

20

FIGURE 1 Top panel: Average returns of Fama–French 25 portfolios vs. predictions of the linearized consumption-based model (essentially, consumption betas) and vs. predictions of the Fama–French 3 factor model. Fourth-quarter to fourth-quarter data, 1954–2003. (Source: Jagannathan and Wang (2005), Figure 2.)

4.3.5. Epstein and Zin and the Long Run Epstein and Zin (1991) is the classic empirical investigation of preferences that are non-separable across states. Ambitiously, for the time, they have some cross section of returns, five industry portfolios. The instruments are lags of consumption and market returns. But industry portfolios don’t show much variation in expected returns to begin with, and we now know that variables such as D/P and consumption/wealth have much

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more power to forecast returns. In essence, their empirical work, using the discount factor 1−γ −ρ 1−ρ ρ−γ ct+1 W 1−ρ 1+γ−ρ , Rt+1 mt+1 = β ct amounted to showing that by using the stock market portfolio as a proxy for the utility index the consumption-based model could perform as well as the CAPM, W mt+1 = a − bRt+1 .

Alas, now we know the CAPM doesn’t perform that well on a more modern set of portfolios and instruments. How these preferences work in a consumption-based estimation with a more modern setup has yet to be investigated. The Epstein–Zin framework has made a dramatic comeback along with the renewed interest in long-run phenomena. As discussed above, the model ties the discount factor to news about future consumption as well as to current consumption; in the ρ = 1 log-normal homoskedastic case, (Et+1 − Et )ln mt+1 = − γ(Et+1 − Et )(Δct+1 ) ∞ + (1 − γ)(Et+1 − Et ) β j (Δct+1+j ) .

(22)

j=1

Hansen, Heaton, and Li (2007) point out that this expression gives another interpretation to Parker and Julliard (2005). The resulting moment condition is almost exactly the f same as that in (21); the only difference is the string of Rt+j in (21), and they are typically small and relatively constant. If the return at t + 1 predicts a string of small changes in consumption growth Δct+j , the finding underlying Parker and Julliard’s result, then the second term in this expression of the Epstein–Zin discount factor will pick it up. Bansal and Yaron (2004) exploit (22) in a simulation economy context. Concentrating on the behavior of the market return, they hypothesize that consumption, rather than being a random walk, continues to grow after a shock. Together with an assumption of conditional heteroskedasticity, the second term in (22) can then act as an “extra factor” to generate a high equity premium, a return volatility, and the fact that returns are forecastable over time. Bansal, Dittmar, and Lundblad (2005) also argue that average returns of value vs. growth stocks can be understood by different covariances with long-run consumption growth in this framework. They examine long-run covariances of earnings with consumption, rather than those of returns. This is an interesting innovation; eventually finance must relate asset prices to the properties of cash flows rather than “explain” today’s price by the covariance of tomorrow’s price with a factor (β). Also, longrun returns must eventually converge to long-run dividend and earnings growth, since valuation ratios are stationary. However, Hansen, Heaton, and Li (2006) show that Bansal, Dittmar, and Lundblad’s evidence that value stocks have much different long-run consumption betas than do

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growth stocks depends crucially on the inclusion of a time trend in the regression of earnings on consumption. In the data, earnings and consumption move about one for one, as one might expect. With a time trend, a strong time trend and a strong opposing regression coefficient offset each other, leading to Bansal, Dittmar, and Lundblad’s finding of a strong beta to explain value premia. Without the time trend, all the betas are about one. Piazzesi and Schneider (2006) have started to apply the framework to bonds. They generate risk premia in the term structure by the ability of state variables to forecast future consumption growth.

4.3.6. Questions The central questions for the empirical importance of the Epstein–Zin framework are (1) is the elasticity of intertemporal substitution really that different from the coefficient of risk aversion? and (2) Are there really important dynamics in consumption growth? As discussed earlier, the evidence on the intertemporal substitution elasticity is not yet decisive, since there just isn’t that much time variation in real interest rates and expected consumption growth to correlate. On intuitive grounds, it’s not obvious why people would strongly resist substitution of consumption across states of nature, but happily accept substitution of consumption over time. Why would you willingly put off going out to dinner for a year in exchange for a free drink (high intertemporal elasticity), but refuse a bet of that dinner for one at the fanciest restaurant in town (high risk aversion)? Consumption dynamics are vital. If consumption growth is unpredictable, then Epstein–Zin utility is observationally equivalent to power utility, a point made by Kocherlakota (1990). This is clear in (22), but it is true more generally. If there is no information about future consumption growth at t + 1, then Ut+1 depends only on ct+1 ; there are no other state variables. Now, consumption growth is the least forecastable of all macroeconomic time series, for good reasons that go back to Hall’s (1978) random walk finding, especially if one takes out the effects of time aggregation, slightly durable goods, seasonal adjustment, and measurement error. Parker and Julliard (2005) provide evidence on the central question: how

much do current returns Rt+1 forecast long horizon future consumption growth kj=1 Δct+j ? Alas, they include Δct+1 , so we do not know from the table how important is the Epstein–Zin innovation, forecasts of kj=2 Δct+k , and they give unweighted truncated

j forecasts rather than an estimate of the weighted infinite horizon forecast ∞ j=2 β Δct+j . Still, one can infer from their table the general result: the forecastability of future consumption growth by current returns is economically tiny, statistically questionable, and certainly poorly measured. The returns hmlt+1 and smbt+1 together generate a maximum forecast R2 of 3.39 percent at a one-year horizon. That R2 is a good deal lower at longer horizons we are interested in, 1.23 percent at 3 years and 0.15 percent at nearly 4 years, and some of that predictability comes from the 1.78 percent R2 from explaining Δct+1 from returns at time t + 1.

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Long-run properties of anything are hard to measure, as made clear in this context by the Hansen, Heaton, and Li (2007) sensitivity analysis. Now, one may imagine interesting long-run properties of consumption growth, and one may find that specifications within one standard error of the very boring point estimates have important asset pricing effects, which is essentially what Bansal and Yaron (2004) do. But without strong direct evidence for the required long-run properties of consumption growth, the conclusions will always be a bit shaky. Without independent measurements, movements in long-run consumption growth forecasts (the second term in (22)) act like unobservable shifts in marginal utility, or shifts in “sentiment,” which are always suspicious explanations for anything. At a minimum, explanation-based, difficult-to-observe shifts in long-run consumption growth should parsimoniously tie together many asset pricing phenomena. Epstein–Zin utility has another unfortunate implication, that we really have to consider all components of consumption. We usually focus on non-durable and services consumption, ignoring durables. This is justified if the utility function is separable across goods, u(cnds ) + υ(cd ), where cnds is consumption of nondurables and services, and cd is the flow of services from durables. Alas, even if the period utility function is separable in this way, the resulting Epstein–Zin utility index responds to news about future durables consumption. In this way, the non-separability across states induces a non-separability across goods, which really cannot be avoided (see Uhlig (2006)).

4.3.7. A Final Doubt An alternative strand of thought says we don’t need new utility functions at all in order to match the aggregate facts. If the conditional moments of consumption growth vary enough over time, then we can match the aggregate facts with a power utility model. Campbell and Cochrane (1999) start with the premise that aggregate consumption is a pure random walk, so any dynamics must come from preferences. Kandel and Stambaugh (1990, 1991) construct models in which time-varying consumption e e moments do all the work. For example, from Et (Rt+1 )/σt (Rt+1 ) ≈ γσt (Δct+1 ), conditional heteroskedasticity in consumption growth can generate a time-varying Sharpe ratio. The empirical question is again whether consumption growth really is far enough from i.i.d. to generate the large variations in expected returns that we see. There isn’t much evidence for conditional heteroskedasticity in consumption growth, but with high risk aversion you might not need a lot, so one might be able to assume a consumption process less than one standard error from point estimates that generates all sorts of interesting asset pricing behavior. The Epstein–Zin literature is to some extent going back to this framework. Bansal and Yaron (2004), for example, add conditional heteroskedasiticty in consumption growth to generate time-varying risk premiums just as Kandel and Stambaugh do. The Epstein–Zin framework gives another tool—properties of long-run consumption Et β j Δct+j —to work with, but the philosophy is in many respects the same.

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4.4. Consumption and Factor Models A second tradition also has re-emerged with some empirical success. Breeden, Gibbons, and Litzenberger (1989) examine a linearized version of the consumptionbased model, a form more familiar to financial economists. Breeden, Gibbons, and Litzenberger simply ask whether average returns line up with betas computed relative to consumption growth, after correcting for a number of problems with consumption data and using a set of industry portfolios. They find the consumption-based model does about as well as the CAPM. This work, along with Breeden (1979) and other theoretical presentations, was important in bringing the consumption-based model to the finance community. Breeden emphasized that consumption should stand in for all of the other factors including wealth, state variables for investment opportunities, non-traded income, and so forth that pervade finance models. More recent empirical research has raised the bar somewhat: industry portfolios show much less variation in mean returns than size and book-to-market portfolios that dominate cross-sectional empirical work. In addition, we typically use instruments variables such as the dividend price ratio that forecast returns much better than lagged returns. Lettau and Ludvigson (2001b) is the first modern re-examination of a consumptionbased factor model, the first recent paper that finds some success in pricing the value premium from a macro-based model, and nicely illustrates current trends in how we evaluate models. Lettau and Ludvigson examine a conditional version of the linearized consumption-based model in this modern testing ground. In our notation, they specify that the stochastic discount factor or growth in marginal utility of wealth is mt+1 = a + (b0 + b1 zt ) × Δct+1 . They also examine a conditional CAPM, w mt+1 = a + (b0 + b1 zt ) × Rt+1 .

The innovation is to allow the slope coefficient b, which acts as the risk aversion coefficient in the model, to vary over time. They use the consumption to wealth ratio to measure zt . In traditional finance language, this specification is equivalent to a factor model in which both betas and factor risk premia vary over time: ei Et Rt+1 = βi,Δc,t λt . Though consumption is the only factor, the unconditional mean returns from such a model can be related to an unconditional multiple-factor model, in which the most important additional factor is the product of consumption growth and the forecasting variable, ei E Rt+1 = βi,zt λ1 + βi,Δct+1 λ2 + βi,(zt ×Δct+1 ) λ3 . (See Cochrane (2004) for a derivation.) Thus, a conditional one-factor model may be behind empirical findings for an unconditional multifactor model.

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Lettau and Ludvigson’s Figure 1, reproduced here as Figure 2, makes a strong case for the performance of the model. Including the scaled consumption factor, they are able to explain the cross section of 25 size and book-to-market portfolios about as well as does the Fama–French three-factor model. A model that uses labor income rather than consumption as a factor does almost as well. This is a tremendous success. This was the first paper to even try to price the value effect with macroeconomic factors. This paper also set a style for many that

FIGURE 2

Lettau and Ludvigson’s Figure 1.

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followed: evaluate a macro model by pricing the Fama–French 25 size and book-tomarket portfolios, and present the results in graphical form of actual mean returns vs. model predictions. We now are focusing on the pricing errors themselves, and less on whether a test statistic formed by a quadratic form of pricing errors is large or small by statistical standards. A “rejected” model with 0.1 percent pricing errors is a lot more interesting than a “non-rejected” model with 10 percent pricing errors, and the pattern of pricing errors across portfolios is revealing. (Cochrane (1996) also has graphs, but only uses size portfolios. Fama and French (1996) also encourage this shift in attention by presenting average returns and pricing errors across portfolios, but in tabular rather than graphical format.) Following Lettau and Ludvigson, so many papers have found high cross-sectional R2 in the Fama–French 25 portfolios using ad-hoc macro models (m = linear functions of macro variables with free coefficients), that it is worth remembering the limitations of the technique. Cross-sectional R2 (average returns on predicted average returns) can be a dangerous statistic. First, the cross-sectional R2 rises automatically as we add factors. With (say) 10 factors in 25 portfolios, a high sample R2 is not that surprising. In addition, to the extent that the Fama–French three-factor model works, the information in the 25 portfolios is really all contained in the three-factor portfolios, so there are really that much fewer degrees of freedom. Second, the cross-sectional R2 and the corresponding visual look of plots like Lettau and Ludvigson’s Figure 1 are not invariant to portfolio formation (Roll and Ross (1994), Kandel and Stambaugh (1995)). We can take linear combinations of the original portfolios to make the plots look as good or as bad as we want. Third, cross-sectional R2 depends a lot on the estimation method. R2 is only well defined for an OLS cross-sectional regression of average returns on betas with a free intercept. For any other estimation technique, and in particular for the popular timeseries regression as used by Fama and French, various ways of computing R2 can give wildly different results.11 These criticisms are, of course, solved by statistical measures; test statistics based on α Cov(α, α )−1 α, where α is a vector of pricing errors, are invariant to portfolio formation and take account of degrees of freedom. However, one can respond that the original portfolios are the interesting ones; the portfolios that modify R2 a lot have unnatural and large long-short positions, and we certainly don’t want to go back to the old days of simply displaying p-values and ignoring these much more revealing measures of model fit. Surely the answer is to present both formal test statistics and carefully chosen diagnostics such as the R2 . Once the game goes past “do as well as the Fama–French three-factor model in the Fama–French 25 portfolios” and moves on to “do better than Fama–French in 11 In

a regression y = a + xb + ε, identities such as R2 =

Var(xb) Var(ε) Var(xb) =1− = Var(y) Var(y) Var(xb) + Var(ε)

only hold when b is the OLS estimate. Some of these calculations can give R2 greater than one or less than zero when applied to other estimation techniques.

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pricing these portfolios,” that means pricing Fama and French’s failures. The Fama– French model does not do well on small growth and large value stocks. Any model that improves on the Fama–French cross-sectional R2 does so by better pricing the smallgrowth/large-value stocks. But is this phenomenon real? Is it interesting? As above, I think it would be better for macro models to focus on pricing the three Fama–French factors rather than the highly cross-correlated 25 portfolios, which really add no more credible information. Macro models also suffer from the fact that real factors are much less correlated with asset returns than are portfolio-based factors. The time series R2 are necessarily lower, so test results can depend on a few data points (Menzly (2001)). This isn’t a defect; it’s exactly what we should expect from a macro model. But it does make inference less reliable. Lewellen and Nagel (2004) have also criticized macro models for having too small a spread in betas; this means that the factor risk premia are unreliably large and the spread in betas may be spurious. Correctly-done standard errors will reveal this problem. Finally, these linearized macro models almost always leave as free parameters the betas, factor risk premia, and (equivalently) the coefficients linking the discount factor to data, hiding the economic interpretation of these parameters. This observation also applies to current models on the investment side such as Cochrane (1996) and Li, Vassalou, and Ying (2003) and to most ICAPM style work such as Vassalou (2003), who shows that variables that forecast GDP growth can price the Fama–French 25 portfolios. Let’s not repeat the mistake of the CAPM that hid the implied 16 percent volatility of consumption growth or extroardinary risk aversion for so many years.

4.4.1. What Next, Then? Many people have the impression that consumption-based models were tried and failed. I hope this review leaves exactly the opposite impression. Despite 30 years of effort, the consumption-based model and its variants have barely been tried. The playing field for empirical work has changed since the classic investigations of the consumption-based model and its extension to non-separable utility functions. We now routinely check any model in the size and book-to-market (and, increasingly, momentum) cross section rather than industry or beta portfolios, since the former show much more variation in average returns. When we use instruments, we use a few lags of powerful instruments known to forecast returns rather than many lags of returns or consumption growth, which are very weak instruments. We worry about time aggregation (or at least we should!). Above all, we focus on pricing errors rather than p-values, as exemplified by Fama–French-style tables of mean returns, betas, and alphas across portfolios, or by equivalent plots of actual mean returns vs. predicted mean returns. We are interested when models capture some moments quite well, even admitting that they fail on others. We recognize that simulation models, in which artificial data display many patterns of real data, are interesting, even though those models may miss other patterns in the data (such as the prediction of perfect correlations) that are easily rejected by formal statistical tests.

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This change is part of a larger, dramatic, and unheralded change in the style of empirical work in finance. The contrast between, say, Hansen and Singleton (1983) and Fama and French (1996), each possibly the most important asset pricing paper of its decade, could not be starker. Both models are formally rejected. But the Fama and French paper persuasively shows the dimensions in which the model does work; it shows there is a substantial and credible spread in average returns to start with (not clear in many asset pricing papers), and it shows how betas line up with average returns and how the betas make the pricing errors an order of magnitude smaller than the average return spread. In the broader scheme of things, much of macroeconomics has gone from “testing” to “calibration” in which we examine economically interesting predictions of models that are easily statistically rejected (though the “calibration” literature’s resistance to so much as displaying a standard error is a bit puzzling). Of course, we cannot expect authors of 20 years ago to do things as we would today. But it remains true that we are only beginning to know how the standard consumptionbased model and its extensions to simple non-separability across time, goods, and states behave in this modern testing ground. There is still very much to do to understand where the consumption-based model works, where it doesn’t work, and how it might be improved. In all these cases, I have pointed out the limitations, including specializations and linearizations of the models, and selection of which moments to look at and which to ignore. This is progress, not criticism. We’ve already rejected the model taken literally, i.e., using arbitrary assets, instruments, and monthly data; there is no need to do that again. But we learn something quite valuable from knowing which assets, horizons, specifications, and instruments do work, and it is gratifying to know that there are some.

5. PRODUCTION, INVESTMENT, AND GENERAL EQUILIBRIUM If we want to link asset prices to macroeconomics, consumption seems like a weak link. Aggregate nondurable and services consumption is about the smoothest and least cyclical of all economic time series. Macroeconomic shocks are seen in output, investment, employment and unemployment, and so forth. Consumers themselves are a weak link; we have to think about which predictions of the model are robust to small costs of information, transaction or attention. For example, a one-month delay in adjusting consumption would destroy a test in monthly data, yet it would have trivial utility costs, or equivalently it could result from perfect optimization with trivially small transaction and information costs (Cochrane 1989).

5.1. “Production-based Asset Pricing” These thoughts led me to want to link asset prices to production through firm first-order conditions in Cochrane (1991b). This approach should allow us to link stock returns

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to genuine business cycle variables, and firms may do a better job of optimization, i.e., small information and transactions cost frictions from which our models abstract may be less important for firms.

5.1.1. Time-Series Tests A production technology defines an “investment return,” the (stochastic) rate of return that results from investing a little more today and then investing a little less tomorrow. With a constant returns to scale production function, the investment return should equal the stock return, data point for data point. The major empirical result in Cochrane (1991b) is that investment returns—functions only of investment data—are highly correlated with stock returns. The prediction is essentially a first-differenced version of the Q theory of investment. The stock return is pretty much the change in stock price or Q, and the investment return is pretty much the change in investment to capital ratio. Thus, the finding is essentially a first-differenced version of the Q theory prediction that investment should be high when stock prices are high. This view bore up well even through the gyrations of the late 1990s. When internet stock prices were high, investment in internet technology boomed. Pastor and Veronesi (2004) show how the same sort of idea can account for the boom in Internet IPOs as internet stock prices rose. The formation of new firms responds to market prices much as does investment by old firms. The Q theory also says that investment should be high when expected returns (the cost of capital) are low, because stock prices are high in such times. Cochrane (1991b) confirms this prediction: investment to capital ratios predict stock returns. There has been a good deal of additional work on the relation between investment and stock returns. Lamont (2000) cleverly uses a survey data set on investment plans. Investment plans data are great forecasters of actual investment. Investment plans also can avoid some of the timing issues that make investment expenditures data hard to use. If the stock price goes up today, it takes time to conceive a new factory, draw the plans, design the machinery, issue stock, etc., so investment expenditures can only react with a lag. Investment plans can react almost instantly. Lamont finds that investment plans also forecast stock returns, even better than the investment to capital ratios in Cochrane (1991). Kogan (2004), inspired by a model with irreversible investment (an asymmetric adjustment cost, really), finds that investment forecasts the variance of stock returns as well. Zhang (2004) uses the Q theory to “explain” many cross-sectional asset pricing anomalies. Firms with high prices (low expected returns or cost of capital) will invest more, issue more stock, and go public; firms with low prices (high expected returns) will repurchase stock. We see the events, followed by low or high returns, which constitutes the “anomaly.” Merz and Yashiv (2005) extend the Q theory to include adjustment costs to labor as well as to capital. Hiring lots of employees takes time and effort and gets in the way of production and investment. This fact means that gross labor flows and their interaction with investment should also enter into the Q-theory prediction for stock prices and stock

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returns. Merz and Yashiv find that the extended model substantially improves the fit; the labor flow and in particular the interaction of labor and investment correlate well with aggregate stock market variations. The model matches slow movements in the level of stock prices, such as the events of the late 1990s, not just the returns or first differences on which my 1991 paper focused (precisely because it could not match the slow movements of the level). Merz and Yashiv’s Figure 2 summarizes this central finding well.

5.1.2. Cross-Sectional Tests Cochrane (1996) is an attempt to extend the “production-based” ideas to describe a cross section of returns rather than a single (market) return. I use multiple production technologies, and I investigate the question of whether the investment returns from these technologies span stock returns, i.e., whether a discount factor of the form (1)

(2)

mt+1 = a + b1Rt+1 + b2 Rt+1 satisfies 1 = E(mt+1 Rt+1 ) (i)

for a cross section of asset returns Rt+1 . Here Rt+1 denote the investment returns, (i) i i functions of investment, and capital only, i.e., Rt+1 = f (It+1 /Kt+1 , Iti /Kti ). The paper also explores scaled factors and returns to incorporate conditioning information, (though Cochrane (2004) does a better job of summarizing this technique) and plots predicted vs. actual mean returns to evaluate the model. I only considered size portfolios, not the now-standard size and book-to-market or other portfolio sorts. Li, Vassalou, and Xing (2003) find that an extended version of the model with four technological factors does account for the Fama–French 25 size and book-to-market portfolios, extending the list of macro models that can account for the value effect.

5.1.3. Really “Production-based” Asset Pricing These papers do not achieve the goal of a “production-based asset pricing model,” which links macro variables to asset returns independently of preferences. The trouble is that the technologies we are used to writing down allow firms to transform goods across time, but not across states of nature. We write functions like yt+1 (s) = θt+1 (s)f (kt ), where s indexes states at time t + 1. More kt results in more yt+1 in all states, but there is no action the firm can take to increase output yt+1 in one state

and reduce it in another state. By contrast, the usual utility function E[u(c)] = s π(s)u[c(s)] defines marginal rates of substitution across all dates and states; mrss1 ,s2 = {π(s1 )u [c(s1 )]}/{π(s2 )u [c(s2 )]}. Production functions are kinked (Leontief) across states of nature, so we cannot read contingent claim prices from outputs as we can read contingent claim prices from state-contingent consumption.

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Cochrane (1993) explains the issue and suggests three ways to put marginal rates of transformation into economic models. The dynamic spanning literature in asset pricing naturally suggests the first two approaches: allow continuous trading or a large number of underlying technologies. For example, with one field that does well in rainy weather and one that does well in sunshine, a farmer can span all [rain, shine] contingent claims. Jermann (2005) pursues the idea of spanning across two states of nature with two technologies, and constructs a simulation model that reproduces the equity premium based on output data. Third, we can directly write technologies that allow marginal rates of transformation across states. Equivalently, we can allow the firm to choose the distribution of its technology shock process as it chooses capital and labor. If the firm’s objective is max E[mt+1 εt+1 f (kt )] − kt = πs ms εs f (kt ) − kt , {kt ,εt+1 ∈Θ}

s

where m denotes contingent claim prices, then the first-order conditions with respect to εs identify ms in strict analogy to the consumption-based model. For example, we can use the standard CES aggregator, α α1 α α1 εs εt+1 = πs = 1, Θ: E θt+1 θ s s

(23)

where θt+1 is an exogenously given shock. As an interpretation, nature hands the firm a production shock θt+1 , but the firm can take actions to increase production in one state relative to another from this baseline. Then, the firm’s first-order conditions with respect to εs give ms f (kt ) = λ

εsα−1 θsα

or mt+1 = λ

α−1 yt+1 α θt+1 f (kt )α

.

(24)

Naturally, the first-order conditions say that the firm should arrange its technology shocks to produce more in high-contingent-claim-price states of nature, and produce less in states of nature for which its output is less valuable. This extension of standard theory is not that strange. The technologies we write down, of the form yt+1 (s) = ε(s)f (kt ), are a historical accident. We started writing technologies for non-stochastic models and then tacked on shocks. They did not come from a detailed microeconomic investigation that persuasively argued that firms in fact have absolutely no way to transform output across states of nature, or no choice at all about the distribution of the shocks they face. Putting the choice of the shock distribution back into production theory, restoring its symmetry with utility theory, will give us marginal rates of transformation that we can compare to asset prices.

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Belo (2005) takes a crucial step to making this approach work, by proposing a solution to the problem of identifying θt+1 in (24). He imposes a restriction that the sets Θ from which firms can choose their technology shocks are related. Belo shows that the resulting form of the production-based model for pricing excess returns is the same as a standard linear macro-factor model, mt+1 = 1 + bi Δyi,t+1 , i

where y denotes output. The derivation produces the typical result in the data that the bi have large magnitudes and opposing sign. Thus, the standard relative success of macrofactor models in explaining the Fama–French 25 can be claimed as a success for a truly “production-based” model as well.

5.2. General Equilibrium Most efforts to connect stock returns to a fuller range of macroeconomic phenomena instead construct general equilibrium models. These models include the consumptionbased first-order condition but also include a full production side. In a general equilibrium model, we can go through consumers and connect returns to the determinants of consumption, basically substituting decision rules c(I, Y , . . .) in mt+1 = βu (ct+1 )/u (ct ) to link m to I, Y , etc. The consumption model predictions are still there, but if we throw them out, perhaps citing measurement issues, we are left with interesting links between asset returns and business cycle variables. While vast numbers of general equilibrium asset pricing models have been written down, I focus here on a few models that make quantitative connections between asset pricing phenomena and macroeconomics.

5.2.1. Market Returns and Macroeconomics Urban Jermann’s (1998) “Asset Pricing in Production Economies” really got this literature going. This paper starts with a standard real business cycle (one sector stochastic growth) model and verifies that its asset pricing implications are a disaster. Capital can be instantaneously transferred to and from consumption—the technology is of the form yt = θt f (kt ); kt+1 = (1 − δ)kt + (yt − ct ). This feature means that the relative price of stocks—Q, or the market-to-book ratio—is always exactly one. Stock returns still vary a bit, since productivity θt is random giving random dividends, but all the stock price fluctuation that drives the vast majority of real-world return variation is absent. Jermann therefore adds adjustment costs, as in the Q theory. Now there is a wedge between the price of “installed” (stock market) capital and “uninstalled” (consumption) capital. That wedge is larger when investment in larger. This specification leads to a good deal of equilibrium price variation. Jermann also includes habit persistence in preferences. He finds that both ingredients are necessary to give any sort of match to the data. Without habit persistence, marginal rates of substitution do not vary much at all—there is no equity premium—and

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295 expected returns do not vary over time. Without adjustment costs, the habit-persistence consumers can use the production technology to provide themselves very smooth consumption paths. In Jermann’s words, “They [consumers] have to care, and they have to be prevented from doing anything [much] about it.” The challenge is to see if this kind of model can match asset pricing facts, while at the same time maintaining if not improving on the real business cycle model’s ability to match quantity fluctuations. This is not a small challenge: given a production technology, consumers will try to smooth out large fluctuations in consumption used by endowment economies to generate stock price fluctuation, and the impediments to transformation across states or time necessary to give adequate stock price variation could well destroy those mechanisms’ ability to generate business cycle facts such as the relative smoothness of consumption relative to investment and output. Jermann’s model makes progress on both tasks, but leaves much for the rest of us to do. He matches the equity premium and relative volatilities of consumption and output and investment. However, he does not evaluate predictability in asset returns, make a detailed comparison of correlation properties (impulse responses) of macro time series, or begin work on the cross section of asset returns. Jermann also points out the volatility of the risk-free rate. This is a central and important problem in this sort of model. Devices such as adjustment costs and habits that raise the variation of marginal rates of substitution across states, and hence generate the equity premium, tend also to raise the variation of marginal rates of substitution over time, and thus give rise to excessive risk-free rate variation. On the preference side, the non-linear habit in Campbell and Cochrane (1999) is one device for quelling interest rate volatility with a high equity premium; a move to Epstein–Zin preferences is another common ingredient for solving this puzzle. Adding a second linear technology might work, but might give back the excessive smoothness of consumption growth. Production technologies such as (23) may allow us to separately control the variability of marginal rates of transformation across states and marginal rates of transformation over time. In the meantime, we learn that checking interest rate volatility is an important question to ask of any general equilibrium model in finance. Boldrin, Christiano, and Fisher (2001) is a good example of more recent work in this area. Obviously, one task is to fit more facts with the model. Boldrin, Christiano, and Fisher focus on quantity dynamics. Habit persistence and adjustment costs or other frictions to investment constitute a dramatic change relative to standard real business cycle models, and one would suspect that they would radically change the dynamics of output, consumption, investment, and so forth. Boldrin, Christiano, and Fisher’s major result is the opposite: the frictions they introduce actually improve on the standard model’s description of quantity dynamics, in particular the model’s ability to replicate hump-shaped dynamics rather than simple exponential decay. Rather than adjustment costs, Boldrin, Christiano, and Fisher have a separate capitalgoods production sector with declining returns to scale. This specification has a similar effect: one cannot transform consumption costlessly to capital, so the relative prices of capital (stocks) and consumption goods can vary. They include additional frictions, in particular that labor must be fixed one period in advance. Like Jermann, they include

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only the one-period habit ct − bct−1 rather than the autoregressive habit (18). They replicate the equity premium, though again with a bit too much interest rate volatility. The big improvement in this paper comes on the quantity side. The next obvious step in this program is to unite the relative success of the Campbell– Cochrane (1999) habit specification with a fleshed-out production technology, in the style of Jermann (1998) or Boldrin, Christiano, and Fisher (2001). Such a paper would present a full set of quantity dynamics as it matches the equity premium, a relatively stable risk-free rate, and time-varying expected returns and return predictability. As far as I know, nobody has put these elements together yet.

5.2.2. Does the Divorce Make Sense? Tallarini (2000) goes after a deep puzzle in this attempt to unite general equilibrium macroeconomics and asset pricing. If asset pricing phenomena require such a complete overhaul of equilibrium business cycle models, why didn’t anybody notice all the missing pieces before? Why did a generation of macroeconomists trying to match quantity dynamics not find themselves forced to adopt long-lasting habit persistence in preferences and adjustment costs or other frictions in technology? Of course, one answer, implicit in Boldrin, Christiano, and Fisher (2001), is that they should have; that these ingredients help the standard model to match the hump-shaped dynamics of impulse-response functions that real business cycle models have so far failed to match well. Tallarini explores a different possibility, one that I think we should keep in mind; that maybe the divorce between real business cycle macroeconomics and finance isn’t that short-sighted after all (at least leaving out welfare questions, in which case models with identical dynamics can make wildly different predictions). Tallarini adapts Epstein–Zin preferences to a standard RBC model; utility is ! β Ut = log Ct + θ log Lt + σ log Et(eσUt+1 ) , where L denotes leisure. Output is a standard production function with no adjustment costs, 1−α α Yt = Xtα Kt−1 Nt , Kt+1 = (1 − δ)Kt + It,

where X is stochastic productivity and N is labor. The Epstein–Zin preferences allow him to raise risk aversion while keeping intertemporal substitution constant. As he does so, he is better able to account for the market price of risk or Sharpe ratio of the stock market (mean stock-bond return/standard deviation), but the quantity dynamics remain almost unchanged. In Tallarini’s world, macroeconomists might well not have noticed the need for large risk aversion. There is a strong intuition for Tallarini’s result. In the real business cycle model without adjustment costs, risk comes entirely from the technology shock, and there is

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nothing anyone can do about it, since as above, production sets are Leontief across states of nature. The production function allows relatively easy transformation over time, however, with a little bit of interest rate variation as ∂f (K, N)/∂K varies a small amount. Thus, if you raise the intertemporal substitution elasticity, you can get quite different business cycle dynamics as agents choose more or less smooth consumption paths. But if you raise the risk aversion coefficient without changing intertemporal substitution, saving, dissaving, or working can do nothing to mitigate the now frightful technology shocks, so quantity dynamics are largely unaffected. The real business cycle model is essentially an endowment economy across states of nature. With this intuition we can see that Tallarini does not quite establish that “macroeconomists safely go on ignoring finance.” First of all, the welfare costs of fluctuations rise with risk aversion. Lucas’ famous calculation that welfare costs of fluctuations are small depends on small risk aversion, and Lucas’s model with power utility and low risk aversion is a disaster on asset pricing facts including the equity premium and return volatility. Tallarini’s observational equivalence cuts both ways: business cycle facts tell you nothing about risk aversion. You have to look to prices for risk aversion, and they say risk aversion, and hence the cost of fluctuations, is large. (See Alvarez and Jermann (2004) for an explicit calculation along these lines.) Second, the equity premium is Tallarini’s only asset pricing fact. In particular, with no adjustment costs, he still has Q = 1 at all times, so there is no stock price variation. Even when there is a high Sharpe ratio, both the mean stock return and its standard deviation are low. Papers that want to match more facts, including the mean and standard deviation of returns separately, price-dividend ratio variation, return predictability and cross-sectional value/growth effects, are driven to add habits and adjustment costs or the more complex ingredients. In these models, higher risk premia may well affect investment/consumption decisions and business cycle dynamics, as suggested by Boldrin, Christiano, and Fisher. For these reasons, I think that we will not end up with a pure “separation theorem” of quantity and price dynamics. I certainly hope not! But the simple form of the observation given by Tallarini is worth keeping in mind. The spillovers may not be as strong as we think, and we may well be able to excuse macroeconomists for not noticing the quantity implications of ingredients we need to add to understand asset prices and the joint evolution of asset prices and quantities. Or perhaps we should chide them further for continuing to ignore the asset-market predictions of their models.

5.2.3. Intangible Capital If prices and quantities in standard models and using standard measurement conventions resist lining up, perhaps those models or measurements are at fault. Hall (2001) is a provocative paper suggesting this view. In thinking about the extraordinary rise of stock values in the late 1990s, we so far have thought of a fairly stable quantity of capital multiplied by a large change in the relative price of (installed) capital. Yes, there was a surge of measured investment, but the resulting increase in the quantity of capital did not come close to accounting for the large increase in stock market valuations.

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The stock market values profit streams, however, not just physical capital. A firm is bricks and mortar to be sure, but it is also ideas, organizations, corporate culture, and so on. All of these elements of “intangible capital” are crucial to profits, yet they do not show up on the books, and nor does the output of “intangible goods” that are accumulated to “intangible capital.” Could the explosion of stock values in the late 1990s reflect a much more normal valuation of a huge, unmeasured stock of “intangible capital,” accumulated from unmeasured “intangible output?” Hall pursues the asset pricing implications of this view. (This is the tip of an iceberg of work in macroeconomics and accounting on the effects of potential intangible capital. Among others, see Hansen, Heaton, and Li (2005).) Hall allows for adjustment costs and some variation in the price of installed vs. uninstalled capital, and backs out the size of those costs from investment data and reasonable assumptions for the size of adjustment costs. These are not sufficient, so he finds that the bulk of stock market values in the late 1990s came from a large quantity of intangible capital. This is a provocative paper, throwing into question much of the measurement underlying all of the macroeconomic models so far. It has its difficulties—it’s hard to account for the large stock market declines as loss of “organizational capital”—but it bears thinking about.

5.2.4. The Cross Section of Returns Obviously, the range of asset pricing phenomena addressed by this sort of generalequilibrium model needs to be expanded, in particular to address cross-sectional results such as the value and growth effects. Menzly, Santos, and Veronesi (2004) approach the question through a “multipleendowment” economy. They model the cash flows of the multiple technologies, but not the investment and labor decisions that go behind these cash flows. They specify a clever model for the shares of each cash flow in consumption so that the shares add up to one and the model is easy to solve for equilibrium prices. They specify a longlived autoregressive habit, which can generate long horizon return predictability and slow movement of the price-dividend ratio as in Campbell and Cochrane (1999). They generate value and growth effects in cross-sectional average returns from the interaction between the changes in aggregate risk premium and the variation in shares. When a cash flow is temporarily low, the duration of that cash flow is longer since more of the expected cash flows are pushed out to the future. This makes the cash flow more exposed to the aggregate risk premium, giving it a higher expected return and a lower price. The obvious next step is to amplify the model’s underpinnings to multiple production functions, allowing us understand the joint determination of asset prices with output, investment, labor, etc., moving from a “multiple-endowment” economy to “multiple production” economies just as the single representative firm literature did in moving from Mehra and Prescott’s endowment model to the production models discussed above. Berk, Green, and Naik (1999) and Gomes, Kogan, and Zhang (2003) derive size and book-to-market effects in general equilibrium models with a bit more explicit, but also fairly stylized, technologies. For example, Gomes, Kogan, and Zhang envision

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“projects” that arrive continuously; firms can decide to undertake a project by paying a cost, but then the scale of the project is fixed forever. Zhang (2005) uses a multiplesector technology of the usual y = θf(k) form with adjustment costs and both aggregate and idiosyncratic shocks, but specifies the discount factor process exogenously, rather than via a utility function and consumption that is driven by the output of the firms in his model. Gourio (2004) generates book-to-market effects in an economy with relatively standard adjustment cost technology and finds some interesting confirmation in the data. Gala (2006) is the latest addition to this line of research. This is a full general equilibrium model—the discount factor comes from consumption via a utility function—with a relatively standard production function. He includes adjustment costs and irreversibilities. The model produces value and growth effects. Fast-growing firms are investing, and so are on the positive, adjustment cost side of the investment function. Value firms are shrinking and up against irreversibility constraints. Thus, when a shock comes, the growth firms can adjust production plans more than value firms can, so value firms are more affected by the shocks. Gala has one non-standard element; there is an “externality” in that investment is easier (lower adjustment costs) for small firms. This solves a technical aggregation problem, and also produces size effects that would be absent in a completely homogenous model.

5.2.5. Challenges for General Equilibrium Models of the Cross Section Bringing multiple firms in at all is the first challenge for a general equilibrium model that wants to address the cross section of returns. Since the extra technologies represent non-zero net supply assets, each “firm” adds another state variable to the equilibrium. Some papers circumvent this problem by modeling the discount factor directly as a function of shocks rather than specify preferences and derive the discount factor from the equilibrium consumption process. Then each firm can be valued in isolation. This is a fine shortcut in order to learn about useful specifications of technology, but in the end of course we don’t really understand risk premia until they come from the equilibrium consumption process fed through a utility function. Other papers are able cleverly to prune the state space or find sufficient statistics for the entire distribution of firms in order to make the models tractable. The second challenge is to produce “value” and “growth” firms that have low and high valuations. Furthermore, the low valuations of “value” firms must correspond to high expected returns, not entirely low cash-flow prospects, and vice versa for growth. This challenge has largely been met, too. The third challenge is to reproduce the failures of the CAPM, as in the data. Again, the puzzle is not so much the existence of value and growth firms but the fact that these characteristics do not correspond to betas. A model in which some firms have high-beta cash flows and some firms have low-beta cash flows will generate a spread in expected returns, and prices will be lower for the high expected-return firms so we will see value and growth effects. But these effects will be explained by the betas. Few of the current

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models really achieve this step. Most models price assets by a conditional CAPM or a conditional consumption-based model; the “value” firms do have higher conditional betas. Any failures of the CAPM in the models are due to omitting conditioning information or the fact that the stock market is imperfectly correlated with consumption. In most models, these features do not account quantitatively for the failures of the CAPM or consumption-based model in the data. Fourth, a model must produce the comovement of value and growth firm returns that lies behind the Fama–French factors. Most models still have a single aggregate shock. And we haven’t started talking about momentum or other anomalies. Finally, let us not forget the full range of aggregate asset pricing facts including equity premium, low and smooth risk-free rate, return predictability, price-dividend ratio volatility and so forth, along with quantity dynamics that are at least as good as the standard real business cycle model. I remain a bit worried about the accuracy of approximations in general equilibrium model solutions. Most papers solve their models by making a linear-quadratic approximation about a non-stochastic steady state. But the central fact of life that makes financial economics interesting is that risk premia are not at all second order. The equity premium of 8 percent is much larger than the interest rate of 1 percent. Thinking of risk as a “second-order” effect, expanding around a 1 percent interest rate in a perfect foresight model, seems very dangerous. There is an alternative but less popular approach, exemplified by Hansen (1987). Rather than specify a non-linear and unsolvable model, and then find a solution by linear-quadratic approximation, Hansen writes down a linear-quadratic (approximate) model, and then quickly finds an exact solution. This technique, emphasized in a large number of papers by Hansen and Sargent, might avoid many approximation and computation issues, especially as the state space expands with multiple firms. Hansen (1987) is also a very nice exposition of how general equilibrium asset pricing economies work and is well worth reading on those grounds alone. Clearly, there is much to do in the integration of asset pricing and macroeconomics. It’s tempting to throw up one’s hands and go back to factor fishing, or partial equilibrium economic models. They are, however, only steps on the way. We will not be able to say we understand the economics of asset prices until we have a complete model that generates artificial time series that look like those in the data. What does it mean to say that we “explain” a high expected return Et (Rt+1 ) “because” the return covaries strongly with consumption growth or the market return m Covt (Rt+1 , Δct+1 ) or Covt (Rt+1 , Rt+1 )? Isn’t the covariance of the return, formed from the covariance of tomorrow’s price with a state variable, every bit as much an endogenous variable as the expected return, formed from the level of today’s price? I think we got into this habit by historical accident. In a one-period model, the covariance is driven by the exogenous liquidating dividend, so it makes a bit more sense to treat the covariance as exogenous and today’s price or expected return as endogenous. If the world had constant expected returns, so that innovations in tomorrow’s price were simple reflections of tomorrow’s dividend news, it’s almost as excusable. But given that so much price variation is driven by expected return variation, reading the standard one-period

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first-order condition as a causal relation from covariance or betas to expected returns makes no sense at all. General equilibrium models force us to avoid this sophistry. They force us to generate the covariance of returns with state variables endogenously along with all asset prices; they force us to tie asset prices, returns, expected returns, and covariances all back to the behavior of fundamental cash flows and consumption, and they even force us to trace those “fundamentals” back to truly exogenous shocks that propagate through technology and utility by optimal decisions. General equilibrium models force us (finally) to stop treating tomorrow’s price as an exogenous variable; to focus on pricing rather than one-period returns. This feature provides great discipline to the general equilibrium modeler, and it makes reverse engineering a desired result much harder, perhaps accounting for slow progress and technically demanding papers. As a simple example, think about raising the equity premium in the Mehra–Prescott economy. This seems simple enough; the e e first-order condition is Et (Rt+1 ) ≈ γ Covt (Rt+1 , Δct+1 ), so just raise the risk aversion coefficient γ. If you try this, in a sensible calibration that mimics the slight positive autocorrelation of consumption growth in postwar data, you get a large negative equity premium. The problem is that the covariance is endogenous in this model; it does not sit still as you change assumptions. With positive serial correlation of consumption growth, good news about today’s consumption growth implies good news about future consumption growth. With a large risk aversion coefficient, good news about future consumption growth lowers the stock price, since the “discount rate” effect is larger than the “wealth” effect.12 In this way, the model endogenously generates a negative covariance term. To boost the equity premium, you have also to change assumptions on the consumption process (or the nature of preferences) to raise the risk aversion coefficient without destroying the covariance. As this survey makes clear, we have only begun to scratch the surface of explicit general equilibrium models—models that start with preferences, technology, shocks, market structure—that can address basic asset pricing and macroeconomic facts including the equity premium, predictable returns, and value, size, and similar effects in the cross section of returns. 12 The

price of a consumption claim is Pt = Et

∞

βj

j=1

Ct+j Ct

−γ Ct+j

or, dividing by current consumption, ∞ Ct+j 1−γ Pt = Et βj . Ct Ct j=1

With γ > 1, a rise in Ct+j /Ct lowers Pt /Ct .

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6. LABOR INCOME AND IDIOSYNCRATIC RISK The basic economics we are chasing is the idea that assets must pay a higher average return if they do badly in “bad times,” and we are searching for the right macroeconomic measure of “bad times.” A natural idea in this context is to include labor income risks in our measure of “bad times.” Surely people will avoid stocks that do badly when they have just lost their jobs, or are at great risk for doing so. Here, I survey models that emphasize overall employment as a state variable (“labor income”) and then models that emphasize increases in individual risk from non-market sources (“idiosyncratic risk”).

6.1. Labor and Outside Income The economics of labor income as a state variable are a little tricky. If utility is separable between consumption and leisure, then consumption should summarize labor income information as it summarizes all other economically relevant risks. If someone loses their job and this is bad news, they should consume less as well, and consumption should therefore reveal all we need to know about the risk. Labor hours can also enter, as above, if utility is non-separable between consumption and leisure. However, current work on labor income work does not stress this possibility, perhaps again because we don’t have much information about the cross-elasticity. Does more leisure make you hungrier, or does it substitute for other goods? A better motivation for labor income risk, as for most traditional factor models in finance, is the suspicion that consumption data are poorly measured or otherwise correspond poorly to the constructs of the model. The theory of finance from the CAPM on consists of various tricks for using determinants of consumption such as wealth (CAPM) or news about future investment opportunities (ICAPM) in place of consumption itself; not because anything is wrong with the consumption-based model in the theory, but on the supposition that it is poorly measured in practice. With that motivation, labor income is one big determinant of consumption or one big source of wealth that is not included in stock market indices. Many investors also have privately held businesses, and the income from those businesses affects their asset demands exactly as does labor income, so we can think of the two issues simultaneously. Measurement is still tricky. The present value of labor income, or the value of “human capital,” belongs most properly in asset pricing theory. Consumption does not decline (marginal utility of wealth does not rise) if you lose your job and you know you can quickly get a better one. Now, one can certainly cook up a theory in which labor income itself tells us a lot about the present value of labor income. An AR(1) time-series model and constant discount rates are the standard assumptions, but they are obviously implausible. For example, the same procedure applied to stocks says that today’s dividend tells us all we need to know about stock prices; that a beta on dividend growth would give the same answer as a beta on returns, that price-dividend ratios are exact functions of each period’s dividend growth. We would laugh at any paper that did this for stocks, yet it is standard practice for labor income.

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Still, the intuition for the importance of labor income risk is strong. The paragraph from Fama and French (1996, p. 77) quoted earlier combines some of the “labor income” risk here and the “idiosyncratic risk” that follows. What remains is to find evidence in the data for these mechanisms.

6.1.1. Labor Income Growth in Linear Discount Factor Models Jagannathan and Wang (1996) is so far the most celebrated recent model that includes a labor income variable. (See also the successful extension in Jagannathan, Kubota, and Takehara (1998).) The main model is a three-factor model, prem

E(Ri ) = c0 + cvw βiV W + cprem βi

+ clabor βilabor

where the betas are defined as usual from time-series regressions, prem

Rti = a + βiV W V Wt + βi

premt + βilabor labort + εit ;

where V W is the value-weighted market return, prem is the previous month’s BAA-AAA yield spread, and labor is the previous month’s growth in a two-month moving average of labor income. prem is included as a conditioning variable; this is a restricted specification of a conditional CAPM. (“Restricted” because in general one would include prem × V W and prem × labor as factors, as in Lettau and Ludvigson’s (2001b) conditional CAPM.) With V W and prem alone, Jagannathan and Wang report only 30 percent crosssectional R2 (average return on betas), presumably because the yield spread does not forecast returns as well as the cay variable used in a similar fashion by Lettau and Ludvigson (2001b). Adding labor income, they obtain up to 55 percent crosssectional13 R2 . Alas, the testing ground is not portfolios sorted by book-to-market ratio, but 100 portfolios sorted by beta and size. Jagannathan and Wang do check (Table VI) that the Fama–French three-factor model does no better (55 percent cross-sectional R2 ) on their portfolios, but we don’t know from the paper if labor income prices the book-to-market sorted portfolios. Furthermore, the paper makes the usual assumption that labor income is a random walk and is valued with a constant discount rate so that the current change in labor income measures the change in its present value (p. 14, “We assume that the return on human capital is an exact linear function of the growth rate in per capita labor income”). Finally, the labor income factor labort = (Lt−1 + Lt−2 ) /(Lt−2 − Lt−3 ) means that the factor is really news about aggregate labor income, since Lt−1 data is released at time t, rather than actual labor income as experienced by workers. Much of Jagannathan and Wang’s empirical point can be seen in Table 1 of Lettau and Ludvigson (2001b), reproduced here as Table 4. Δy is labor income growth, 13 Again, I pass on these numbers with some hesitation. Unless the model is fit by an cross-sectional regression, the R2 depends on technique and even on how you calculate it. Only under OLS is Var(xβ)/Var(y) = 1 − Var(ε)/Var(y). Yet cross-sectional R2 is a popular statistic to report, even for models not fit by OLS crosssectional regression.

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this time measured contemporaneously. Lettau and Ludvigson use the consumption to wealth ratio cay rather than the bond premium as the conditioning variable, which may account for the better results. Most importantly, they also examine the Fama–French 25 size and book-to-market portfolios, which allows us better to compare across models in this standard playground. They actually find reasonable performance (58 percent R2 ) in an unconditional model that includes only the market return and labor income growth as factors. Adding the scaled factors of the conditional model, i.e., VW VW mt+1 = a + b1 Rt+1 + b2 Δyt+1 + b3 cayt + b4 cayt × Rt+1 + b5 (cayt × Δyt+1 ), they achieve essentially the same R2 as the Fama–French three-factor model. The take-away point, then, is that a large number of macroeconomic variables can be added to ad-hoc linear factor models (mt+1 = a − bft+1 ) to price the Fama–French 25 portfolios, including consumption, investment, and now labor income. Of course, the usual caveat applies that there are really only three independent assets in the Fama– French 25 portfolios (market, hml, smb), so one should be cautious about models with many factors.

6.1.2. Explicit Modeling of Labor Income in a VAR Framework Campbell (1996) uses labor income in a three-factor model. His factors are (1) the market return, (2) innovations in variables that help to forecast future market returns, and (3) innovations in variables that help to forecast future labor income. The analysis starts from a vector autoregression including the market return, real labor income growth, and as forecasting variables the dividend/price ratio, a de-trended interest rate and a credit spread. This paper has many novel and distinguishing features. First, despite the nearly 40 years that have passed since Merton’s (1973) theoretical presentation of the ICAPM, only a very small number of empirical papers have ever checked that their proposed factors do, in fact, forecast market returns. This is one of the rare exceptions. (Ferson and Harvey (1999) and Brennan, Xia, and Wang (2005) are the only other ones I know of.) Campbell’s factors also forecast current and future labor income, again taking one big step closer to innovations in human capital rather than just the flow of labor income. Finally, parameters are tied to estimates of fundamental parameters such as risk aversion, rather than being left unexamined as is the usual practice. Alas, this paper came out before that much attention was lavished on the book-tomarket effect, so the test portfolios are an intersection of size and industry portfolios. Size really does little more than sort on market beta, and industry portfolios give little variation in expected returns, as seen in Campbell’s Table 5. As one might suspect, most variation in the present value of labor income and return comes not from current labor income or changing forecasts of future labor income, but from a changing discount rate applied to labor income. However, the discount rate here is the same as the stock market discount rate. On one hand, we expect discount rate variation to dominate the present value of labor income, as it does in stock prices. This model serves as a good

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John H. Cochrane

TABLE 4 Fama–MacBeth (1972) Regressions Using 25 Fama–French Portfolios:λj Coefficient Estimates on Betas in Cross–Sectional Regression c. ay t · Factorst+1

Factorst+1 Row 1

2

3

4

5

6

7

Constant

c. ay t

Ruw

4.18

−.32

(4.47)

(−.27)

(4.45)

(−.27)

Δy

smb

hml

Ruw

Δy

2 R 2 R

.01 −.03

3.21

−1.41

1.26

.58

(3.37)

(−1.20)

(3.42)

.54

(1.87)

(−.67)

(1.90)

1.87

1.33

.47

1.46

.80

(1.31)

(.83)

(.94)

(3.24)

.77

(1.21)

(.76)

(.86)

(2.98)

3.70

−.52

−.06

1.14

.31

(3.88)

(−.22)

(−.05)

(3.59)

.21

(2.61)

(−.15)

(−.03)

(2.41)

3.70

−.08

1.16

.31

(3.86)

(−.07)

(3.58)

.25

(2.60)

(−.44)

(2.41)

5.18

−.44

−1.99

.56

.34

−.17

.77

(5.59)

(−1.60)

(−1.73)

(2.12)

(1.67)

(−2.40)

.71

(3.32)

(−.95)

(−1.02)

(1.26)

(.99)

(−1.42)

3.81

−2.22

.59

.63

−.08

.75

(4.02)

(−1.88)

(2.20)

(2.79)

(−2.52)

.70

(2.80)

(−1.31)

(1.53)

(1.94)

(−1.75)

N OTE.–The table presents ! ! λ estimates from cross-sectional Fama–MacBeth regressions using returns of 25 Fama–French portfolios: E Ri,t+1 = E R0,t + β λ. The individual λj estimates (from the second-pass cross-sectional regression) for the beta of the factor listed in the column heading are reported. In the first stage, the time-series betas β are computed in one multiple regression of the portfolio returns on the factors. The term Ruw is the return of the value-weighted CRSP index, Δyt+1 is labor income growth, and SMB and HML are the Fama–French mimicking portfolios related to size and book-market equity ratios. The scaling variable is c. ay. The table reports the Fama–MacBeth cross-sectional regression coefficient; in parentheses are two t-statistics for each coefficient estimate. The top statistic uses uncorrected Fama–MacBeth standard errors; the bottom statistic uses the Shanken (1992) correction. The term R2 denotes the unadjusted cross-sectional R2 statistic, and R2 adjusts for the degrees of freedom. Lettau and Ludvigson’s (2001b) Table 1.

warning to the vast majority of researchers who blithely use current labor income to proxy for the present value of future labor income. On the other hand, though, it’s not obvious that the stock discount rate should apply to labor income, and at a data level it means that labor income is really not a new factor. The bottom line is on p. 336: the CAPM is pretty good on size portfolios, and other factors do not seem that important.

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Campbell and Vuolteenaho (2004) follow on the ICAPM component of Campbell (1996). They break the standard CAPM beta into two components, a “bad” cash-flow beta that measures how much an asset return declines if expected future market cash flows decline, and “good” return beta that measures how much an asset return declines if a rise in future expected returns lowers prices today. The latter beta is “good” because in an ICAPM world (long-lived investors) it should have a lower risk premium. Ignoring the troubling small-growth portfolio, the improvement of the two-beta model over the CAPM on the Fama–French 25 portfolios can be seen quickly in their Figure 1. Petkova (2006) also estimates an ICAPM-like model on the Fama–French 25 portfolios, finding that innovations to the dividend yield, term spread, default spread, and level of the interest rate, all variables known to forecast the market return, can account for the average returns of the Fama–French 25. Ultimately, ICAPM models should be part of macro finance as well, since the “state variables” must forecast consumption as well as the market return in order to influence prices.

6.1.3. Proprietary Income Heaton and Lucas (2000) note that proprietary income—the income from non-marketed businesses—should be as, if not more, important to asset pricing than labor income as measured by Jagannathan and Wang (1996). For rich people who own stocks, fluctuations in proprietary income are undoubtedly a larger concern than are fluctuations in wages. They find that individuals with more and more volatile proprietary income in fact hold less stocks. They also replicate Jagannathan and Wang’s investigation (using the same 100 industry/beta portfolios) using proprietary income. Using Jagannathan and Wang’s timing, they find that proprietary income is important, but more importantly the proprietary income series still works using “normal” timing rather than the one-period lag in Jagannathan and Wang.

6.1.4. Micro Data Malloy, Moskowitz, and Vissing-Jorgenson (2005) take another big step in the labor income direction. Among other refinements, they check whether their model explains portfolios sorted on book-to-market, size, and momentum as well as individual stocks; they use measures of hiring and firing rather than the quite smooth average earnings data; and they measure the permanent component of labor income, which at least gets one step closer to the present value of human capital that should matter in theory. They find good performance of the model in book-to-market sorted portfolios, suggesting that labor income risk (or associated macroeconomic risk) really is behind the “value effect.”

6.1.5. A Model Santos and Veronesi (2005) study a two-sector version of the model in Menzly, Santos, and Veronesi (2004). They think of the two sectors as labor income (human capital) vs. market or dividend income, corresponding to physical capital. A conditional CAPM

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John H. Cochrane

holds in the model in which the ratio of labor income to total income is a conditioning variable—expected returns etc. vary as this ratio varies. In addition, the relevant market return is the total wealth portfolio including human capital, and so shocks to the value of labor income are priced as well. This completely solved model nicely shows the potential effects of labor income on asset pricing. One part of Santos and Veronesi’s empirical work checks that the ratio of labor to total income forecasts aggregate returns; it does, and better than the dividend price ratio, adding to evidence that macro variables forecast stock returns. The second part of the empirical work checks whether the factors can account for the average returns of the 25 Fama–French size and book-to-market portfolios (Santos and Veronesi’s Table 6). Here, adding the ratio of labor to total income as a conditioning variable helps a lot, raising the cross-sectional R2 from nearly zero for the CAPM to 50 percent for this conditional CAPM, in line with Lettau and Ludvigson’s (2001b) conditional labor income model that uses cay as a conditioning variable. Alas, adding shocks to the present value of labor income (measured here by changes in wages, with all the usual warnings) as a factor does not help much, either alone or in combination with the conditioning variables. The major success with this specification comes then as a conditioning variable rather than as a risk factor.

6.2. Idiosyncratic Risk, Stockholding, and Micro Data In most of our thinking about macroeconomics and finance, we use a “representative consumer.” We analyze economy-wide aggregates, making a first approximation that the distribution across consumers, while important and interesting, does not affect the evolution of aggregate prices or quantities. We say that a “tax cut” or “interest rate reduction” may increase “consumption” or “savings,” thereby affecting “employment” and “output,” but we ignore the possibility that the effect is different if it hits people differently. Of course, the theory needed to justify perfectly this simplification is extreme, but seems a quite sensible first approximation. Macroeconomics and finance are thus full of investigations of whether crosssectional distributions matter. Two particular strains of this investigation are important for us. First, perhaps idiosyncratic risk matters. Perhaps people fear stocks not because they might fall at a time when total employment or labor income falls, but because they might fall at a time when the cross-sectional risk of unemployment or labor income increases. Second, most people don’t hold any stocks at all. Therefore, their consumption may be de-linked from the stock market, and models that connect the stock market only to those who actually hold stocks might be more successful. Both considerations suggest examining our central asset pricing conditions using individual household data rather than aggregate consumption data.

6.2.1. Constantinides and Duffie and Idiosyncratic Risk Basically, Constantinides and Duffie (1996) prove a constructive existence theorem: there is a specification of idiosyncratic income risk that can explain any premium, using

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only power (constant relative risk aversion, time-separable) utility, and they show you how to construct that process. This is a brilliant contribution as a decade of research into idiosyncratic risk had stumbled against one after another difficulty and had great trouble to demonstrate even the possibility of substantial effects. Constantinides and Duffie’s Equation (11) gives the central result, which I reproduce with a slight change of notation: −γ γ(γ + 1) 2 Ct+1 exp (25) yt+1 Rt+1 = 1. Et β Ct 2 2 is the cross-sectional variance of individual log consumption growth taken Here, yt+1 after aggregates at time t + 1 are known. Equation (25) adds the exponential term to the standard consumption-based asset pricing equation. Since you can construct a discount factor (term before Rt+1 ) to represent any asset pricing anomaly, you can con2 struct a idiosyncratic risk process yt+1 to rationalize any asset pricing anomaly. For example, DeSantis (2005) constructs a model in which the conditional variance of 2 yt+1 varies slowly over time, acting in many ways like the Campbell–Cochrane surplus consumption ratio (19) and generating the same facts in a simulation economy. The nonlinearity of marginal utility is the key to the Constantinides–Duffie result. You might have thought that idiosyncratic risk cannot matter. Anything idiosyncratic must be orthogonal to aggregates, including the market return, so E(mt+1 + εit+1 , Rt+1 ) = E(mt+1 , Rt+1 ). But the shocks should be to consumption or income, not to marginal utility, and marginal utility is a!non-linear function of consumption. Examining E(mit+1 Rt+1 ) = E E(mit+1 |Rt+1 )Rt+1 , we see that a non-linear m will lead to a Jensen’s inequality 1/2σ 2 -term, which is exactly the exponential term in (25). Thus, if the cross-sectional variance of idiosyncratic shocks is higher when the returns Rt+1 are higher, we will see a premium that does not make sense from aggregate consumption. The derivation of (25) follows exactly this logic and doesn’t take much extra algebra.14 Idiosyncratic consumption growth risk yt+1 plays the part of consumption growth in the standard models. In order to generate risk premia, then, we need the distribution of idiosyncratic risk to vary over time; it must widen when high-average-return securities (stocks vs. bonds, value stocks vs. growth stocks) decline. It needs to widen unexpectedly, to generate a covariance with returns, and so as not to generate a lot of variation in interest rates. And, if we are to avoid high risk aversion, it needs to widen a lot.

14 Individual

consumption is generated from N (0, 1) idiosyncratic shocks ηi,t+1 by i Ct+1 Ct+1 1 2 . ln + ηi,t+1 yt+1 − yt+1 = ln i Ct 2 Ct

(26)

You can see by inspection that yt+1 is the cross-sectional variance of individual log consumption growth. 2 term is there exactly for Aggregate consumption really is the sum of individual consumption—the −1/2yt+1 this reason: i / Ct+1 / Ct+1 ηi,t+1 yt+1 − 1 y2 Ct+1 / Ct+1 2 t+1 E E e . = = / i / C Ct Ct t Ct

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As with the equity premium, the challenge for the idiosyncratic risk view is about quantities, not about signs. The usual Hansen–Jagannathan calculation, σ(m) E(Re ) ≥ , E(m) σ(Re ) means that the discount factor m must vary by 50 percent or so. (E(Re ) ≈ 8 percent, σ(Re ) ≈ 16 percent, Rf = 1/E(m) ≈ 1.01.) We can make some back-of-the-envelope calculations with the approximation γ(γ + 1) 2 γ(γ + 1) 2 ≈ yt+1 σ yt+1 . σ exp 2 2

(27)

2 ) = 0.5. Now, if the level of the cross-sectional variWith γ = 1, then, we need σ(yt+1 √ ance were 0.5, that would mean a cross-sectional standard deviation of 0.5 = 0.71. This number seems much too large. Can it be true that if aggregate consumption growth is 2 percent, the typical person you meet either has +73 percent or −69 percent consumption growth? But the problem is worse than this, because 0.71 does not describe the level of idiosyncratic consumption growth; it must represent the unexpected increase or decrease in idiosyncratic risk in a typical year. Slow, business cycle-related variation 2 in idiosyncratic risk yt+1 will give rise to changes in interest rates, not a risk premium. Based on this sort of simple calculation, the reviews in Cochrane (1997) and Cochrane (2004) suggest that an idiosyncratic risk model will have to rely on high risk aversion, just like the standard consumption model, to fit the standard asset pricing facts. Again, I am not criticizing the basic mechanism or the plausibility of the signs. My only point is that in order to get anything like plausible magnitudes, idiosyncratic risk models seem destined to need high risk aversion just like standard models.

Now, start with the individual’s first-order conditions: 1 = Et β

i Ct+1

Cti

−γ

Rt+1

−γ / /C / t+1 = Et βE Rt+1 / / Ct Cti −γ / −γ i /C Ct+1 /Ct+1 Ct+1 / t+1 E Rt+1 = Et β / / Ct Ct Cti /Ct

i Ct+1

−γ 2 Ct+1 −γ ηi,t+1 yt+1 − 12 yt+1 e Rt+1 =E β Ct −γ 1 2 1 2 2 Ct+1 =E β e 2 γyt+1 + 2 γ yt+1 Rt+1 Ct −γ γ(γ+1) 2 Ct+1 e 2 yt+1 Rt+1 . =E β Ct

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The situation gets worse as we think about different time horizons. The required volatility of individual consumption growth, and the size of unexpected changes in that 2 volatility σt (yt+h ) must explode as the horizon shrinks. The Sharpe ratio Et (Re )/σt (Re ) declines with the square root of horizon, so σt (mt,t+h ) must decline with the square 2 root of horizon h. But yt+h governs the variance of individual consumption growth, not 2 its standard deviation, and variances usually decline linearly with horizon. If σt (yt+h ) 2 declines only with the square root of horizon, then typical values of the level of yt+h 2 must also decline only with the square root of horizon, since yt+h must remain positive. That fact means that the annualized variance of individual consumption growth must rise unboundedly as the observation interval shrinks. In sum, neither consumption nor the conditional variance of consumption growth yt2 can follow diffusion (random walklike) processes. Both must instead follow a jump process in order to allow enormous variance at short horizons. (Of course, they may do so. We are used to using diffusions, but the sharp breaks in individual income and consumption on rare big events like being fired may well be better modeled by a jump process.) In a sense, we knew that individual consumption would have to have extreme variance at short horizons to get this mechanism to work. Grossman and Shiller (1982) showed that marginal utility is linear in continuous-time models when consumption and asset prices follow diffusions; it’s as if utility were quadratic. The basic pricing equation is, in continuous time, dCti f (28) Et (dRt ) − rt dt = γEt dRt i , Ct where dRt = dPt /Pt + Dt /Pt dt is the instantaneous total return. The average of dC i /C i across people must equal the aggregate, dC/C, so we have dCt f Et (dRt ) − rt dt = γEt dRt . Ct Aggregation holds even with incomplete markets and non-linear utility, and the Constantinides–Duffie effect has disappeared. It has disappeared into terms of order dzdt and higher, of course. To keep the Constantinides–Duffie effect, one must suppose that dC i /C i has variance larger than order dz, i.e., that it does not follow a diffusion.15 Conversely, we may anticipate the same generic problem that many models have at long horizons. Like many models (see the Cambpbell–Cochrane discussion earlier), the Constantinides–Duffie model (25) adds a multiplicative term to the standard power utility discount factor. To generate an equity premium at long horizons, the extra term must also have a variance that grows linearly with time, as does the variance of consumption growth, and functions of stationary variables, such as the cross-sectional variance of f

f

is another logical possibility. Et (dRt ) = rt dt does not imply Et (Rt+1 ) = Rt if interest rates vary strongly over time, so one could construct a Constantinides–Duffie discrete-time model with consumption that follows a diffusion, and hence no infinitesimal risk premium, but instead strong instantaneous interest rate variation. I don’t think anyone would want to do so. 15 There

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idiosyncratic shocks, usually do not grow with horizon, leaving us back to the power utility model at long horizons.

6.2.2. Empirical Work Of course, empirical arguments should be made with data, not on the backs of envelopes. Empirical work on whether variation in the cross-sectional distribution of income and consumption is important for asset pricing is just beginning. Most investigations find some support for the basic effect—consumption and income do become more volatile across people in recessions and at times when the stock market declines. However, they confirm that the magnitudes are not large enough to explain the equity or value premia without high risk aversion. Heaton and Lucas (1996) calibrate an income process from the PSID and find it does not have the required volatility or correlation with stock market declines. Cogley (2002) examines the cross-sectional properties of consumption from the consumer expenditure survey. He finds that “crosssectional factors”—higher moments of the cross-sectional distribution of consumption growth—“are indeed weakly correlated with stock returns, and they generate equity premia of 2 percent or less when the coefficient of relative risk aversion is below 5.” Even ignoring the distinction between consumption and income, Lettau (2002) finds that the cross-sectional distribution of idiosyncratic income does not vary enough to explain the equity premium puzzle without quite high risk aversion. Storesletten, Telmer, and Yaron (2005) document greater dispersion in labor income across households in PSID in recessions, but they do not connect that greater dispersion to asset pricing. Constantinides and Duffie’s model also requires a substantial permanent component to idiosyncratic labor income, in order to keep consumers from smoothing it by saving and dissaving. Yet standard calibrations such as in Heaton and Lucas (1996) don’t find enough persistence in the data. Of course, abundant measurement error in micro data will give the false appearance of mean reversion, but if labor income were really very volatile and persistent, then the distribution of income would fan out quickly and counterfactually over time. In contrast, Brav, Constantinides, and Geczy (2002) report some asset pricing success. They use household consumption data from the consumer expenditure survey and consider measurement error extensively. They examine one central implication, whether by aggregating marginal utility rather than aggregating consumption, they can explain the equity premium and (separately) the value premium, 0 = E(mRe ). Specifically, remember that the individual first-order conditions still hold: i ) u (Ct+1 (29) Rt+1 . 1=E β u (Cti ) We therefore can always “aggregate” by averaging marginal utilities: i 1 u (Ct+1 ) β 1=E Rt+1 . N i u (Cti )

(30)

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We cannot in general aggregate by averaging consumption: i u N1 i Ct+1 1 = E β 1 i Rt+1 . u N i Ct

(31)

Brav, Constantinides, and Geczy contrast calculations of (30) with those of (31). This analysis also shows again how important non-linearities in marginal utility are to generating an effect: if marginal utility were linear, as it is under quadratic utility or in continuous time, then of course averaging consumption would work and would give the same answer as aggregating marginal utility. This estimation is exactly identified; one moment E(mR) and one parameter γ. Brav, Constantinides, and Geczy find that by aggregating marginal utilities, E(mR) = 1, they are able to find a γ between 2 and 5 that matches the equity premium, i.e., satisfies the single moment restriction. By contrast, using aggregate consumption data, the best fit requires very high risk aversion, and there is no risk aversion parameter γ that satisfies this single moment for the equity premium. (One equation and one unknown do not guarantee a solution.) I hope that future work will analyze this result more fully. What are the time-varying cross-sectional moments that drive the result, and why did Brav, Constantinides, and Geczy find them where Cogley and Lettau did not, and my back-of-the-envelope calculations suggest that the required properties are extreme? How will this approach work as we extend the number of assets to be priced, and to be priced simultaneously? Jacobs and Wang (2004) take a good step in this direction. They use the Fama– French 25 size and book-to-market portfolios as well as some bond returns, and they look at the performance of a two-factor model that includes aggregate consumption plus the cross-sectional variance of consumption, constructed from consumer expenditure survey data. They find that the cross-sectional variance factor is important (i.e., should be included in the discount factor), and the two consumption factors improve on the (disastrous, in this data) CAPM. Not surprisingly, of course, the Fama–French ad-hoc factors are not driven out, and the overall pricing errors remain large.

6.2.3. Micro Data Of course, individuals still price assets exactly as before. Equation (29) still holds for each individual’s consumption in all these models. So, once we have opened the CES or PSID database, we could simply test whether asset returns are correctly related to household level consumption with (29) and forget about aggregation either of consumption (31) or of marginal utility (30). With micro data, we can also isolate stockholders or households more likely to own stocks (older, wealthier) and see if the model works better among these. Alas, this approach is not so easy either: individual consumption data is full of measurement error as well as idiosyncratic risk, and raising measurement error to a large −γ power can swamp the signal (see Brav, Constantinides, and Geczy for an extended discussion). In addition, individual behavior may not be stationary over time, where

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313 aggregates are. For just this reason (betas vary over time), we use characteristic-sorted portfolios rather than individual stock data to test asset pricing models. It may make sense to aggregate the m in 1 = E(mR) just as we aggregate the R into portfolios. Also, typical data sets are short and do not include a long panel dimension; we do not track individual households over long periods of time. Finally, equity premium problems are just as difficult for (correctly measured) individual consumption as for aggregate consumption. For example, the Hansen–Jagannathan bound says that the volatility of marginal utility growth must exceed 50 percent per year (and more, to explain the value premium). For log utility, that means consumption growth must vary by 50 percentage points per year. This is non-durable consumption and the flow of durables services, not durables purchases. Buying a house once in 10 years or a car once in 3 does not count toward this volatility. Furthermore, only the portion of consumption (really marginal utility) volatility correlated with the stock market counts. Purely idiosyncratic volatility (due to individual job loss, illness, divorce, etc.) does not count. Despite these problems, there are some empirical successes in microdata. Mankiw and Zeldes (1991) find that stockholder’s consumption is more volatile and more correlated with the stock market than that of nonstockholders, a conclusion reinforced by Attanasio, Banks, and Tanner (2002). Ait-Sahalia, Parker, and Yogo (2004) find that consumption of “luxury goods,” presumably enjoyed by stockholders, fits the equity premium with less risk aversion than that of normal goods. Vissing-Jorgensen (2002) is a good recent example of the large literature that actually estimates the first-order condition (29) in microdate, though only for a single asset over time rather than for the spread between stocks and bonds. Thus, we are a long way from a full estimate that accounts for the market as well as the size and value premia (say, the Fama–French 25) and other effects. Must we use micro data? While initially appealing, it’s not clear that the stockholder/non-stockholder distinction is vital. Are people who hold no stocks really not “marginal”? The costs of joining the stock market are trivial; just turn off your spam filter for a moment and that becomes obvious. Thus, people who do not invest at all choose not to do so in the face of trivial fixed costs. This choice must reflect the attractiveness of a price ratio relative to the consumer’s marginal rate of substitution; they really are “marginal” or closer to “marginal” than most theories assume. More formally, Heaton and Lucas (1996) examine a carefully calibrated portfolio model and find they need a very large transaction cost to generate the observed equity premium. Even non-stockholders are linked to the stock market in various ways. Most data on household asset holdings excludes defined-contribution pension plans, most of which contain stock market investments. Even employees with a defined-benefit plan should watch the stock market when making consumption plans, as employees of United Airlines recently found out to their dismay. Finally, while there are a lot of people with little stock holding, they also have little consumption and little effect on market prices. Aggregates weight by dollars, not people, and many more dollars of consumption are enjoyed by rich people who own stocks than the numbers of such people suggest. In sum, while there is nothing wrong with looking at stockholder data to see if their consumption really does line up better with stock returns, it is not so obvious that there is something

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terribly wrong with continuing to use aggregates, even though few households directly hold stock.

7. CHALLENGES FOR THE FUTURE Though this review may seem extensive and exhausting, it is clear at the end that work has barely begun. The challenge is straightforward: we need to understand what macroeconomic risks underlie the “factor risk premia,” the average returns on special portfolios that finance research uses to crystallize the cross section of assets. A current list might include the equity premium, and its variation over time underlying return forecastability and volatility, the value and size premiums, the momentum premium, and the time-varying term premia in bond foreign exchange markets. More premia will certainly emerge through time. On the empirical side, we are really only starting to understand how the simplest power utility models do and do not address these premiums, looking across data issues, horizons, time aggregation, and so forth. The success of ad-hoc macro factor and “production” models in explaining the Fama–French 25 is suggestive, but their performance still needs careful evaluation and they need connection to economic theory. The general equilibrium approach is a vast and largely unexplored new land. The papers covered here are like Columbus’ report that the land is there. The pressing challenge is to develop a general equilibrium model with an interesting cross section. The model needs to have multiple “firms”; it needs to generate the fact that low-price “value” firms have higher returns than high-price “growth firms”; it needs to generate the failure of the CAPM to account for these returns, and it needs to generate the comovement of value firms that underlies Fama and French’s factor model, all this with preference and technology specifications that are at least not wildly inconsistent with microeconomic investigation. The papers surveyed here, while path-breaking advances in that direction, do not come close to the full list of desiderata. Having said “macroeconomics,” “risk,” and “asset prices,” the reader will quickly spot a missing ingredient: money. In macroeconomics, monetary shocks and monetary frictions are considered by many to be an essential ingredient of business cycles. They should certainly matter at least for bond risk premia. (See Piazzesi (2005) for the state of the art on this question.) Coming from the other direction, there is now a lot of evidence for “liquidity” effects in bond and stock markets (see Cochrane (2005a) for a review), and perhaps both sorts of frictions are related.

References Abel, A. B. Asset prices under habit formation and catching up with the Joneses. American Economic Review 80 (1990): 38–42. Ait-Sahalia, Y., J. Parker, and M. Yogo. Luxury goods and the equity premium. Journal of Finance 59 (2004): 2959–3004. Alvarez, F., and U. J. Jermann. Using asset prices to measure the cost of business cycles. Journal of Political Economy 112 (2004): 1223–1256.

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315 Ang, A., M. Piazzesi, and M. Wei. What does the yield curve tell us about GDP growth? Journal of Econometrics (2004). Attanasio, O. P., J. Banks, and S. Tanner. Asset holding and consumption volatility. Journal of Political Economy 110 (2002): 771–792. Ball, R. Anomalies in relationships between securities’ yields and yield—surrogates. Journal of Financial Economics 6 (1978): 103–126. Bansal, R., R. F. Dittmar, and C. Lundblad. Consumption, dividends, and the cross-section of equity returns. Journal of Finance 60 (2005): 1639–1672. Bansal, R., and A. Yaron. Risks for the long run: A potential resolution of asset pricing puzzles. Journal of Finance 59(4) (2004): 1481–1509. Banz, R. W. The relationship between return and market value of common stocks. Journal of Financial Economics 9 (1981): 3–18. Basu, S. The relationship between earnings yield, market value, and return for NYSE common stocks: Further evidence. Journal of Financial Economics 12 (1983): 129–156. Belo, F. A pure production-based asset pricing model. Manuscript, University of Chicago (2005). Berk, J. B., R. C. Green, and V. Naik. Optimal investment, growth options and security returns. Journal of Finance 54 (1999): 1153–1607. Boldrin, M., L. J. Christiano, and J. Fisher. Habit persistence, asset returns, and the business cycle. American Economic Review 91 (2001): 149–166. Brainard, W. C., W. R. Nelson, and M. D. Shapiro. The consumption beta explains expected returns at long horizons. Manuscript, Economics Department. Yale University (1991). Brav, A., G. Constantinides, and C. Geczy. Asset pricing with heterogeneous consumers and limited participation: Empirical evidence. Journal of Political Economy 110 (2002): 793–824. Breeden, D. T. An intertemporal asset pricing model with stochastic consumption and investment opportunities. Journal of Financial Economics 7 (1979): 265–296. Breeden, D., M. Gibbons, and R. Litzenberger. Empirical tests of the consumption-oriented CAPM. Journal of Finance 44 (1989): 231–262. Brennan, M. J., Y. Xia, and A. Wang. Estimation and test of a simple model of intertemporal asset pricing. Journal of Finance 59 (2005): 1743–1776. Campbell, J. Y. Some lessons from the yield curve. Journal of Economic Perspectives 9 (1995): 129–152. Campbell, J. Y. Understanding risk and return. Journal of Political Economy 104 (1996): 298–345. Campbell, J. Y. Asset pricing at the millennium. Journal of Finance 55 (2000): 1515–1567. Campbell, J. Y. Consumption-based asset pricing. Chapter 13 in G. Constantinides, M. Harris, and R. Stulz, eds., Handbook of the Economics of Finance Vol. IB. North-Holland, Amsterdam, (2003): pages 803–887. . Campbell, J. Y., and J. H. Cochrane. By force of habit: A consumption-based explanation of aggregate stock market behavior. NBER Working paper 4995 (1995). Campbell, J. Y., and J. H. Cochrane. By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107 (1999): 205–251. Campbell, J. Y., and J. H. Cochrane. Explaining the poor performance of consumption based asset pricing models. Journal of Finance 55 (2000): 2863–2878. Campbell, J. Y., and R. J. Shiller. The dividend-price ratio and expectations of future dividends and discount factors. Review of Financial Studies 1 (1988): 195–228. Campbell, J. Y., and R. J. Shiller. Yield spreads and interest rate movements: A bird’s eye view. The Review of Economic Studies 58(3), Special Issue: The Econometrics of Financial Markets (1991): 495–514. Campbell, J. Y., R. J. Shiller, and K. L. Schoenholtz. Forward rates and future policy: Interpreting the term structure of interest rates. Brookings Papers on Economic Activity (1983): 173–223. Campbell, J. Y., and T. Vuolteenaho. Good beta, bad beta. American Economic Review 94 (2004): 1249–1275. Carhart, M. On persistence in mutual fund performance. Journal of Finance 52 (1997): 57–82. Chen, X., and S. Ludvigson. Land of addicts? An empirical investigation of habit-based asset pricing models. Manuscript, New York University (2004). Chetty, R., and A. Szeidl. Consumption commitments: Neoclassical foundations for habit formation. Manuscript, University of California at Berkeley (2004).

316

Chapter 7 • Financial Markets and the Real Economy Cochrane, J. H. The sensitivity of tests of the intertemporal allocation of consumption to near-rational alternatives. American Economic Review 79 (1989): 319–337. Cochrane, J. H. Explaining the variance of price-dividend ratios. Review of Financial Studies 5 (1991a): 243–280. Cochrane, J. H. Production-based asset pricing and the link between stock returns and economic fluctuations. Journal of Finance 46 (1991b): 207–234. Cochrane, J. H. Rethinking production under uncertainty. Manuscript, University of Chicago (1993). Cochrane, J. H. Permanent and transitory components of GNP and stock prices. Quarterly Journal of Economics 109 (1994): 241–266. Cochrane, J. H. A cross-sectional test of an investment-based asset pricing model. Journal of Political Economy 104 (1996): 572–621. Cochrane, J. H. Where is the market going? Uncertain facts and novel theories. Economic Perspectives, Federal Reserve Bank of Chicago 21(6) (November/December) (1997). Cochrane, J. H. New facts in finance. Economic Perspectives, Federal Reserve Bank of Chicago 23(3) (1999a): 36–58. Cochrane, J. H. Portfolio advice for a multifactor world. Economic Perspectives, Federal Reserve Bank of Chicago 23(3) (1999b): 59–78. Cochrane, J. H. Asset Pricing. Princeton University Press, Revised Edition, Princeton, NJ (2004). Cochrane, J. H. Liquidity trading and asset prices. NBER Reporter, National Bureau of Economic Research, www.nber.org/reporter ( 2005a). Cochrane, J. H. Financial markets and the real economy. Foundations and Trends in Finance 1 (2005b): 1–101. Cochrane, J. H. Financial markets and the real economy, in J. H. Cochrane, ed., Financial Markets and the Real Economy, Volume 18 of the International Library of Critical Writings in Financial Economics, Edward Elgar, London, (2006a): pages xi–lxix. Cochrane, J. H. The dog that did not bark: A defense of return predictability. Review of Financial Studies (2006b). Cochrane, J. H., and L. P. Hansen. Asset pricing explorations for macroeconomics. In O. Blanchard, and S. Fisher, eds., NBER Macroeconomics Annual (1992): pages 115–165. Cochrane, J. H., and M. Piazzesi. Bond risk premia. American Economic Review 95 (2005): 138–160. Cogley, T. Idiosyncratic risk and the equity premium: Evidence from the consumer expenditure survey. Journal of Monetary Economics 49 (2002): 309–334. Constantinides, G. Habit formation: A resolution of the equity premium puzzle. Journal of Political Economy 98 (1990): 519–543. Constantinides, G., and D. Duffie. Asset pricing with heterogeneous consumers. Journal of Political Economy 104 (1996): 219–240. Cooper, I., and R. Priestley. Stock return predictability in a production economy. Manuscript, Norwegian School of Management (2005). Craine, R. Rational bubbles: A test. Journal of Economic Dynamics and Control 17 (1993): 829–846. Daniel, K., and D. Marshall. Equity-premium and risk-free-rate puzzles at long horizons. Macroeconomic Dynamics 1 (1997): 452–484. Daniel, K., and S. Titman. Testing factor-model explanations of market anomalies. Manuscript, Northwestern University and University of Texas, Austin (2005). De Bondt, W. F. M., and R. Thaler. Does the stock market overreact? Journal of Finance 40 (1985): 793–805. De Santis, M. Interpreting aggregate stock market behavior: How far can the standard model go? Manuscript, University of California, Davis (2005). Eichenbaum, M., and L. P. Hansen. Estimating models with intertemporal substitution using aggregate time series data. Journal of Business and Economic Statistics 8 (1990): 53–69. Eichenbaum, M., L. P. Hansen, and K. Singleton. A time-series analysis of representative agent models of consumption and leisure choice under uncertainty. Quarterly Journal of Economics 103 (1988): 51–78. Engel, C. The forward discount anomaly and the risk premium: A survey of recent evidence. Journal of Empirical Finance 3 (1996): 123–192. Epstein, L. G., and S. E. Zin. Substitution, risk aversion and the temporal behavior of asset returns. Journal of Political Economy 99 (1991): 263–286.

John H. Cochrane

317 Estrella, A., and G. Hardouvelis. The term structure as a predictor of real economic activity. Journal of Finance 46 (1991): 555–576. Fama, E. F. Forward rates as predictors of future spot rates. Journal of Financial Economics 3 (1976): 361–377. Fama, E. F. Forward and spot exchange rates. Journal of Monetary Economics 14 (1984a): 319–338. Fama, E. F. The information in the term structure. Journal of Financial Economics 13 (1984b): 509–528. Fama, E. F. Efficient markets: II, Fiftieth Anniversary Invited Paper. Journal of Finance 46 (1991): 1575–1617. Fama, E. F., and D. M. James. Risk, return and equilibrium: Empirical tests. Journal of Political Economy 71 (1972): 607–636. Fama, E. F., and R. R. Bliss. The information in long-maturity forward rates. American Economic Review 77 (1987): 680–692. Fama, E. F., and K. R. French. Permanent and temporary components of stock prices. Journal of Political Economy 96 (1988a): 246–273. Fama, E. F., and K. R. French. Dividend yields and expected stock returns. Journal of Financial Economics 22 (1988b): 3–27. Fama, E. F., and K. R. French. Business conditions and expected returns on stocks and bonds. Journal of Financial Economics 25 (1989): 23–49. Fama, E. F., and K. R. French. The cross-section of expected stock returns. Journal of Finance 47 (1992): 427–465. Fama, E. F., and K. R. French. Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33 (1993): 3–56. Fama, E. F., and K. R. French. Multifactor explanations of asset-pricing anomalies. Journal of Finance 51 (1996): 55–84. Fama, E. F., and K. R. French. Size and book-to-market factors in earnings and returns. Journal of Finance 50 (1997a): 131–155. Fama, E. F., and K. R. French. Industry costs of equity. Journal of Financial Economics 43 ( 1997b): 153–193. Fama, E. F., and M. R. Gibbons. Inflation, real returns and capital investment. Journal of Monetary Economics 9 (1982): 297–323. Fama, E. F., and G. W. Schwert. Asset returns and inflation. Journal of Financial Economics 5 (1977): 115–146. Ferson, W. E., and G. Constantinides. Habit persistence and durability in aggregate consumption: Empirical tests. Journal of Financial Economics 29 (1991): 199–240. Ferson, W. E., and C. R. Harvey. Conditioning variables and the cross section of stock returns. Journal of Finance 54 (1999): 1325–1360. Gala, V. D. Investment and returns. Manuscript, University of Chicago (2006). Goetzmann, W. N., and P. Jorion. Testing the predictive power of dividend yields. Journal of Finance 48 (1993): 663–679. Gomes, J. F., L. Kogan, and L. Zhang. Equilibrium cross-section of returns. Journal of Political Economy 111 (2003): 693–732. Gourio, F. Operating leverage, stock market cyclicality and the cross-section of returns. Manuscript, University of Chicago (2004). Goyal, A., and I. Welch. Predicting the equity premium with dividend ratios. Management Science 49 (2003): 639–654. Goyal, A., and I. Welch. A comprehensive look at the empirical performance of equity premium prediction. Manuscript, Brown University, Revision of NBER Working Paper 10483 (2005). Grossman, S., A. Melino, and R. J. Shiller. Estimating the continuous-time consumption-based asset-pricing model. Journal of Business and Economic Statistics 5 (1987): 315–328. Grossman, S. J., and R. J. Shiller. The determinants of the variability of stock market prices. American Economic Review 71 (1981): 222–227. Grossman, S. J., and R. J. Shiller. Consumption correlatedness and risk measurement in economies with non-traded assets and heterogeneous information. Journal of Financial Economics 10 (1982): 195–210.

318

Chapter 7 • Financial Markets and the Real Economy Hall, R. E. Stochastic implications of the life cycle-permanent income hypothesis: Theory and evidence. Journal of Political Economy 86 (1978): 971–987. Hall, R. E. Intertemporal substitution in consumption. Journal of Political Economy 96 (1988): 339–357. Hall, R. E. The stock market and capital accumulation. American Economic Review 91 (2001): 1185–1202. Hamburger, M. J., and E. N. Platt. The expectations hypothesis and the efficiency of the Treasury bill market. Review of Economics and Statistics 57 (1975): 190–199. Hansen, L. P. Consumption, asset markets and macroeconomic fluctuations: A comment. Carnegie-Rochester Conference Series on Public Policy 17 (1982): 239–250. Hansen, L. P. Calculating asset prices in three example economies, in T. F. Bewley, ed., Advances in Econometrics, Fifth World Congress. Cambridge University Press, Cambridge, UK (1987). Hansen, L. P., and R. J. Hodrick. Forward exchange rates as optimal predictors of future spot rates: An econometric analysis. Journal of Political Economy 88 (1980): 829–853. Hansen, L. P., and T. J. Sargent. Exact linear rational expectations models: Specification and estimation, Federal Reserve Bank of Minneapolis Staff Report 71 (1981). Hansen, L. P., and K. J. Singleton. Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica 50 (1982): 1269–1288. Hansen, L. P., and K. J. Singleton. Stochastic consumption, risk aversion, and the temporal behavior of asset returns. Journal of Political Economy 91 (1983): 249–268. Hansen, L. P., and K. J. Singleton. Errata. Econometrica 52 (1984): 267–268. Hansen, L. P., and R. Jagannathan. Implications of security market data for models of dynamic economies. Journal of Political Economy 99 (1991): 225–262. Hansen, L. P., J. C. Heaton, J. Lee, and N. Roussanov. Intertemporal substitution and risk aversion. Manuscript, University of Chicago, Forthcoming in J. Heckman, ed., Handbook of Econometrics. North-Holland, Amsterdam (2006). Hansen, L. P., J. C. Heaton, and N. Li. Intangible risk? in C. Corrado, J. Haltiwanger, and D. Sichel, eds., Measuring Capital in the New Economy. University of Chicago Press, Chicago (2005): pages 111–152. Hansen, L. P., J. C. Heaton, and N. Li. Consumption strikes back? Measuring long-run risk. Manuscript. University of Chicago (2007). Heaton, J. C. The interaction between time-nonseparable preferences and time aggregation. Econometrica 61 (1993): 353–385. Heaton, J. C. An empirical investigation of asset pricing with temporally dependent preference. specifications. Econometrica 63 (1995): 681–717. Heaton, J. C., and D. J. Lucas. Evaluating the effects of incomplete markets on risk sharing and asset pricing. Journal of Political Economy 104 (1996): 443–487. Heaton, J. C., and D. J. Lucas. Portfolio choice and asset prices: The importance of entrepreneurial risk. Journal of Finance 55 (2000): 1163–1198. Hodrick, R. J. Dividend yields and expected stock returns: Alternative procedures for inference and measurement. Review of Financial Studies 5 (1992): 357–386. Jacobs, K., and K. Q. Wang. Idiosyncratic consumption risk and the cross-section of asset returns. Journal of Finance 59 (2004): 2211–2252. Jagannathan, R., and Z. Wang. The conditional CAPM and the cross-section of expected returns. Journal of Finance 51 (1996): 3–53. Jagannathan, R., and Y. Wang. Consumption risk and the cost of equity capital. NBER Working Paper 11026 (2005). Jagannathan, R., K. Kubota, and H. Takehara. Relationship between labor-income risk and average return: Empirical evidence from the Japanese stock market. Journal of Business 71 (1998): 319–347. Jegadeesh, N., and S. Titman. Returns to buying winners and selling losers: Implications for stock market efficiency. Journal of Finance 48 (1993): 65–91. Jermann, U. Asset pricing in production economies. Journal of Monetary Economics 41 (1998): 257–275. Jermann, U. The equity premium implied by production. Manuscript, University of Pennsylvania (2005). Kandel, S., and R. F. Stambaugh. Expectations and volatility of consumption and asset returns. Review of Financial Studies 3 (1990): 207–232.

John H. Cochrane

319 Kandel, S., and R. F. Stambaugh. Asset returns and intertemporal preferences. Journal of Monetary Economics 27 (1991): 39–71. Kandel, S., and R. F. Stambaugh. Portfolio inefficiency and the cross-section of expected returns. Journal of Finance 50 (1995): 157–184. Keim, D. B., and F. S. Robert. Predicting returns in stock and bond markets. Journal of Financial Economics 17 (1986): 357–390. Kocherlakota, N. R. Disentangling the coefficient of relative risk aversion from the elasticity of intertemporal substitution: An irrelevance result. Journal of Finance 45 (1990): 175–190. Kocherlakota, N. R. The equity premium: It’s still a puzzle. Journal of Economic Literature 34 (1996): 42–71. Kogan, L. Asset prices and real investment. Journal of Financial Economics 73 (2004): 411–431. Kreps, D. M., and E. L. Porteus. Temporal resolution of uncertainty and dynamic choice theory. Econometrica 46 (1978): 185–200. Kuhn, T. The Structure of Scientific Revolutions. University of Chicago Press, Chicago (third edition, 1996) (1962). Lakonishok, J., A. Shleifer, and R. W. Vishny. Contrarian investment, extrapolation, and risk. Journal of Finance 49 (1994): 1541–1578. Lamont, O. A. Earnings and expected returns. Journal of Finance 53 (1998): 1563–1587. Lamont, O. A. Investment plans and stock returns. Journal of Finance 55 (2000): 2719–2745. LeRoy, S. F., and R. D. Porter. The present-value relation: Tests based on implied variance bounds. Econometrica 49 (1981): 555–574. Lettau, M. Idiosyncratic risk and volatility bounds, or, can models with idiosyncratic risk solve the equity premium puzzle? Review of Economics and Statistics 84 (2002): 376–380. Lettau, M. Inspecting the mechanism: Closed-form solutions for asset prices in real business cycle models. Economic Journal 113 (2003): 550–575. Lettau, M., and S. Ludvigson. Consumption, aggregate wealth, and expected stock returns. Journal of Finance 56 (2001a): 815–849. Lettau, M., and S. Ludvigson. Resurrecting the (C)CAPM: A cross-sectional test when risk premia are timevarying. Journal of Political Economy 109 (2001b): 1238–1287. Lettau, M., and S. Ludvigson. Expected returns and expected dividend growth. Journal of Financial Economics (2004). Lewellen, J., and S. Nagel. The conditional CAPM does not explain asset-pricing anomalies. Manuscript, MIT (2004). Lewellen, J., S. Nagel, and J. Shanken. A skeptical appraisal of asset pricing tests. Manuscript, Dartmouth College, Stanford University, and Emory University (2006). Li, Q., M. Vassalou, and Y. Xing. Investment growth rates and the cross-section of equity returns. Manuscript, Columbia University (2003). Liew, J., and M. Vassalou. Can book-to-market, size and momentum be risk factors that predict economic growth? Journal of Financial Economics 57 (2000): 221–245. Lucas, R. E., Jr. Asset prices in an exchange economy. Econometrica 46 (1978): 1429–1446. Lustig, H., and S. Van Nieuwerburgh. Housing collateral, consumption insurance and risk premia: An empirical perspective. Journal of Finance (2004a). Lustig, H., and S. Van Nieuwerburgh. A theory of housing collateral, consumption insurance and risk premia. Manuscript, UCLA and NYU (2004b). Lustig, H., and A. Verdelhan. The cross-section of foreign currency risk premia and U. S. consumption growth risk. Manuscript, University of Chicago and UCLA (2004). Macaulay, F. R. Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States Since 1856. Publications of the National Bureau of Economic Research No. 33 (1938). Reprinted in Risk Classics Library, Risk Books (1999). Malloy, C., T. Moskowitz, and A. Vissing-Jorgenson. Job risk and asset returns. Manuscript, University of Chicago (2005). Mankiw, N. G., and S. Zeldes. The consumption of stockholders and non-stockholders. Journal of Financial Economics 29 (1991): 97–112. McCloskey, D. N. The rhetoric of economics. Journal of Economic Literature 21 (1983): 481–517.

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Chapter 7 • Financial Markets and the Real Economy Mehra, R., and E. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15 (1985): 145–161. Menzly, L. Influential observations in cross-sectional asset pricing tests. Manuscript, University of Chicago (2001). Menzly, L., T. Santos, and P. Veronesi. Understanding predictability. Journal of Political Economy 112 (2004): 1–47. Merton, R. C. An intertemporal capital asset pricing model. Econometrica 41 (1973): 867–887. Merz, M., and E. Yashiv. Labor and the market value of the firm. Manuscript, University of Bonn (2005). Nelson, C. R., and M. J. Kim. Predictable stock returns: The role of small sample bias. Journal of Finance 48 (1993): 641–661. Ogaki, M., and C. M. Reinhart. Measuring intertemporal substitution: The role of durable goods. Journal of Political Economy 106 (1998): 1078–1098. Pakos, M. Asset pricing with durable goods and non-homothetic preferences. Manuscript, University of Chicago (2004). Parker, J., and C. Julliard. Consumption risk and the cross-section of expected returns. Journal of Political Economy 113 (2005): 185–222. Pastor, L., and P. Veronesi. Rational IPO waves. Journal of Finance (2004). Petkova, R. Do the Fama–French factors proxy for innovations in predictive variables? Journal of Finance 61 (2006): 581–612. Piazzesi, M. Bond yields and the Federal Reserve. Journal of Political Economy 113 (2005): 311–344. Piazzesi, M., and M. Schneider. Equilibrium yield curves. Manuscript, University of Chicago and NYU, prepared for the 2006 Macroeconomics Annual (2006). Piazzesi, M., M. Schneider, and S. Tuzel. Housing, consumption, and asset pricing. Manuscript, University of Chicago, NYU, and UCLA (2004). Piazzesi, M., S. Martin, and T. Selale. Housing, consumption and asset pricing. Journal of Financial Economics 83 (2007): 531–569. Poterba, J., and L. H. Summers. Mean reversion in stock returns: Evidence and implications. Journal of Financial Economics 22 (1988): 27–60. Restoy, F., and P. Weil. Approximate equilibrium asset prices. NBER Working Paper 6611 (1998). Roll, R. The Behavior of Interest Rates, Basic Books, New York (1970). Roll, R. 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): 129–176. Roll, R., and S. A. Ross. On the cross-sectional relation between expected returns and betas. Journal of Finance 49 (1994): 101–121. Rozeff, M. S. Dividend yields are equity risk premiums. Journal of Portfolio Management 11 (1984): 68–75. Santos, T., and P. Veronesi. Labor income and predictable stock returns. Review of Fiancial Studies (2005). Sargent, T. J. Rational expectations and the term structure of interest rates. Journal of Money Credit and Banking 4 (1972): 74–97. Sargent, T. J. A note on maximum likelihood estimation of the rational expectations model of the term structure. Journal of Monetary Economics 5 (1978): 133–143. Schwert, G. W. Anomalies and market efficiency. Chapter 15 of G. Constantinides, M. Harris, and S. Stulz, eds., Handbook of the Economics of Finance. North-Holland, Amsterdam, (2003): 937–972. Shanken, J. On the estimation of beta pricing models. Review of Financial Studies 5 (1992): 1–34. Shiller, R. J. The volatility of long-term interest rates and expectations models of the term structure. Journal of Political Economy 87 (1979): 1190–1219. Shiller, R. J. Do stock prices move too much to be justified by subsequent changes in dividends? American Economic Review 71 (1981): 421–436. Shiller, R. J. Consumption, asset markets, and economic fluctuations. Carnegie-Rochester Conference on Public Policy 17 (1982): 203–238. Shiller, R. J. Stock prices and social dynamics. Brookings Papers on Economic Activity (1984): pages 457–510. Shiller, R. J., J. Y. Campbell, and K. L. Schoenholz. Forward rates and future policy: Interpreting the term structure of interest rates. Brookings Papers on Economic Activity (1983): pages 173–217.

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321 Stambaugh, R. F. The information in forward rates: Implications for models of the term structure. Journal of Financial Economics 21 (1988): 41–70. Stambaugh, R. F. Predictive regressions. Journal of Financial Economics 54 (1999): 375–421. Storesletten, K., C. I. Telmer, and A. Yaron. Cyclical dynamics of idiosyncratic labor market risk. Journal of Political Economy (2005). Sundaresan, S. M. Intertemporally dependent preferences and the volatility of consumption and wealth. Review of Financial Studies 2 (1989): 73–88. Tallarini, T. D., Jr. Risk-sensitive real business cycles. Journal of Monetary Economics 45 (2000): 507–532. Uhlig, H. Asset pricing with Epstein-Zin preferences. Manuscript, Humboldt University (2006). Vassalou, M. News related to future GDP growth as a risk factor in equity returns. Journal of Financial Economics 68 (2003): 47–73. Verdelhan, A. A habit-based explanation of the exchange rate risk premium. Manuscript, University of Chicago (2004). Vissing-Jorgensen, A. Limited asset market participation and the elasticity of intertemporal substitution. Journal of Political Economy 110 (2002): 825–853. Wachter, J. A consumption-based model of the term structure of interest rates. Manuscript, University of Pennsylvania (2004). Weil, P. The equity premium puzzle and the risk-free rate puzzle. Journal of Monetary Economics 24 (1989): 401–421. Yogo, M. A consumption-based explanation of expected stock returns. Journal of Finance (2006). Zhang, L. Anomalies. Manuscript, University of Rochester (2004). Zhang, L. The value premium. Journal of Finance 60 (2005): 67–104.

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APPENDIX This appendix gives a self-contained derivation of the discount factor under Epstein-Zin (1991) preferences.

Utility Index The consumer contemplates the purchase of ξ shares at price / pt with payoff xt+1 . / ∂ The maximum is achieved where Ut (ct − pt ξ, ct+1 + xt+1 ξ)/ / = 0. From the utility ∂ξ ξ=0 function 1 1−ρ % 1−ρ $ 1−γ 1−γ 1−ρ , (A1) Ut = (1 − β)ct + β Et Ut+1 we have ∂Ut ρ −ρ = Ut (1 − β)ct . ∂ct

(A2)

Then, the first order condition is %γ−ρ $ ∂Ut 1 1−γ 1−γ −γ ∂Ut+1 ρ 1−ρ Et Ut+1 pt = xt+1 . Et (1 − γ)Ut+1 Ut β ∂ct 1−ρ 1−γ ∂ct+1 Substituting from (A2) and canceling gives %γ−ρ $ ρ−γ −ρ ! 1−γ 1−γ −ρ ct pt = β Et Ut+1 Et Ut+1 ct+1 xt+1 . Thus, defining the discount factor from pt = E(mt+1 xt+1 ) gives ⎞ρ−γ

⎛

⎟ ⎜ Ut+1 ⎟ mt+1 = β ⎜ 1 ⎠ % 1−γ ⎝$ 1−γ Et Ut+1

ct+1 ct

−ρ .

(A3)

Market Return The utility function (A1) is linearly homogeneous. Thus, Ut =

∞ ∞ ∂Ut ∂Ut ct+j = Et ct+j ∂ct+j ∂ct+j j=0

Ut = Et ∂Ut /∂ct

j=0

∞ j=0

mt,t+j ct+j = Wt

(A4)

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John H. Cochrane

The final equality is the definition of total wealth—the value of the claim to consumption (including time t consumption). This is the heart of the idea—wealth reveals the utility index in (A3). We want an expression with the market return, not wealth itself, so we proceed as follows. Use the utility function (A1) to express the denominator of (A3) in terms of time t observables: 1 1−ρ 1 1 1−γ 1 1−γ 1−ρ 1−ρ 1−ρ Ut − (1 − β)ct = . Et Ut+1 β

(A5)

Now, substitute for Ut and Ut+1 from (A4), with (A2): Wt =

Ut Ut 1 1−ρ ρ = ρ = Ut ct . −ρ 1 − β ∂Ut /∂ct Ut (1 − β)ct

(Note with ρ = 1 the wealth-consumption ratio is constant: Wt /ct = 1/1 − β(Ut /ct )1−ρ .) Solving for Ut gives 1 −ρ Ut = Wt (1 − β)ct 1−ρ .

(A6)

Now, use (A5) and (A6) in (A3): ⎞ρ−γ

⎛ mt+1

⎟ ⎜ Ut+1 ⎟ =β⎜ 1 % 1−γ ⎠ ⎝$ 1−γ Et Ut+1

ct+1 ct

−ρ (A7)

.

Substituting into (A3) gives ⎛

−ρ ! β)ct+1 1−ρ 1

mt+1

⎞ρ−γ

⎜ ⎟ Wt+1 (1 − ⎟ =β⎜ 1 $ 1 ⎠ % 1−ρ ⎝ 1−ρ −ρ 1−ρ 1 Wt (1 − β)ct − (1 − β)ct β =β

ρ−γ 1+ 1−ρ

1−γ

= β 1−ρ 1−γ

= β 1−ρ

−ρ

Wt+1 ct+1 −ρ

1−ρ

ρ−γ 1−ρ

ct+1 ct

−ρ

Wt ct − ct ρ−γ ρ−γ −ρ 1−ρ +1 Wt+1 1−ρ ct+1 Wt − ct ct ρ−γ 1−γ 1−ρ −ρ 1−ρ Wt+1 ct+1 Wt − ct ct

ct+1 ct

−ρ

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Since this definition of wealth includes current consumption (dividend), the return on the wealth portfolio is W Rt+1 =

Wt+1 Wt − ct

so we have in the end mt+1 = β

1−γ 1−ρ

ρ−γ W 1−ρ Rt+1

ct+1 ct

−ρ

1−γ 1−ρ

.

If we define 1−γ γ−ρ , 1−θ = , 1−ρ 1−ρ then we can express the result as a combination of the standard consumption-based discount factor and the inverse of the market return: 1−θ −ρ θ 1 ct+1 . mt+1 = β ct RW θ=

t+1

Discount Factor in the ρ = 1 Case From (A1), let υ = ln U and let c now denote log consumption. Then we can write (A1) as 1 ln((1 − β)e(1−ρ)ct + βe(1−ρ)Qt ), 1−ρ 1 ln Et (e(1−γ)υt+1 ). Qt = 1−γ υt =

In the limit ρ = 1 (differentiating numerator and denominator), υt (1) = (1 − β)ct + βQt (1), where I use the notation υt (1), Qt (1) to remind ourselves that υt is a function of the preference parameter ρ, and results that only hold when ρ = 1. Next, assuming consumption and hence υt+1 (1) are log-normal and conditionally homoskedastic, we have 1 ln Et (e(1−γ)υt+1 (1) ) 1−γ 1 = (1 − β)ct + βEt [υt+1 (1)] + β(1 − γ)σ 2 [υt+1 (1)] , 2 ∞ (1 − γ) 2 1 σ [υt+1 (1)] . υt (1) = (1 − β) β j Et (ct+j ) + β 2 (1 − β)

υt (1) = (1 − β)ct + β

j=0

325

John H. Cochrane

The discount factor is, from (A7), ln mt+1 = ln(β) − ρ (ct+1 − ct ) + (ρ − γ) (vt+1 − Qt ) (Et+1 − Et ) ln mt+1 = −ρ (Et+1 − Et ) ct+1 + (ρ − γ)(Et+1 − Et )νt+1 In the case ρ = 1, with normal and homoskedastic consumption, we then have (Et+1 − Et ) ln mt+1 = − (Et+1 − Et )ct+1 + (1 − γ) (1 − β)(Et+1 − Et )

∞

β j(ct+1+j ).

j=0

It’s convenient to rewrite the discount factor in terms of consumption growth, as follows: W =

∞

β j ct+1+j = (ct+1 − ct ) + β(ct+2 − ct+1 ) + β 2 (ct+3 − ct+2 ) + · · · + ct

j=0

+ βct+1 + β 2 ct+2 + · · ·, W =

∞

β j Δct+1+j + ct + βW ,

j=0

W =

∞ 1 1 j β Δct+1+j + ct . 1−β 1−β j=0

Then, since (Et+1 − Et ) ct = 0, (Et+1 − Et ) ln mt+1 = − (Et+1 − Et) Δct+1 + (1 − γ)(Et+1 − Et )

∞

β jΔct+1+j

j=0

or (Et+1 − Et ) ln mt+1 = − γ (Et+1 − Et )(Δct+1 ) ∞ j + (1 − γ) (Et+1 − Et ) β Δct+1+j . j=1

Here we see the familiar consumption growth raised to the power γ, plus a newtermreflecting innovations in long-run consumption growth.

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Discussion: Financial Markets and the Real Economy1 Lars Peter Hansen Dept of Economics, University of Chicago

John Cochrane has done an admirable job of summarizing a rather extensive empirical literature. The work is so exhaustive that I will not even attempt to comment on it in a systematic way. There are many very nice aspects to his discussion, and what follows merely provides some minor amendments. To his credit, Cochrane considers again some of the early literature on consumptionbased asset pricing and compares quotes across papers in an attempt at intellectual history. This is interesting reading, but I would urge others to read the whole papers, not just quotes. Some important breakthroughs occurred prior to Mehra and Prescott (1985) and Hansen and Singleton (1983). While the Shiller (1982) paper that Cochrane features is a nice paper, I am personally a big fan of Grossman and Shiller (1980, 1981). These two joint papers really got researchers like Singleton and myself and others thinking of empirical implications of the consumption-based capital asset pricing model along with the earlier theoretical work of Rubinstein (1976), Lucas (1978), and Breeden (1979). It is unfortunate that only an abbreviated version, Grossman and Shiller (1981), was published because Grossman and Shiller (1980) was familiar to many people at the time. In this sense, the analogy to Columbus versus Erikson in the discovery of America is a bit misleading, although the important influence of Mehra and Prescott (1985) in subsequent research is undeniable. Given my Nordic origins, I have always been a bit partial to Erikson. It is interesting that the Shiller inequality that Cochrane refers to differs from the ones Cochrane uses to frame most of his discussion. Shiller deduced his inequality using information about the marginal distribution for consumption or more generally a stochastic discount factor along with the marginal distributions for separate returns. Hansen and Singleton (1982) used information from the joint distribution of stochastic discount factors and returns following in part Grossman and Shiller (1980). Hansen 1 I thank John Heaton and Ken Singleton for helpful comments. This material is based upon work supported by the National Science Foundation under Award Number SES0519372.

Lars Peter Hansen

327 and Jagannathan (1991) and the Hansen comment on Shiller used marginal information on stochastic discount factors in conjunction with information on the joint distribution of returns. This latter approach was motivated by an aim to produce a common set of diagnostics for a rich family of stochastic discount factor models. All three approaches are interesting and arguably serve different purposes. It is certainly true that the Shiller paper was a natural precursor to my work with Jagannathan. In Cochrane’s discussion of the Hansen and Singleton (1983) paper in his section of the equity-premium puzzle, it is not clear why Ken and I are even mentioned as part of the same discovery game. We focused on monthly postwar data and used a sample with a shorter span (but observed more frequently) for estimation and inference in contrast to both Grossman and Shiller (1980, 1981) and Mehra and Prescott (1985). For the postwar data sample we used, the mean returns could not be estimated with enough accuracy for reliable inference. A narrowly framed equity premium puzzle based on postwar data would have been much less dramatic and much easier to debunk. Perhaps we erred in focusing on such a short time period, but this choice is non-trival and has important consequences. It revolves in part around the following question: Did postwar investors presume the prewar volatility was germane when making investments? The whole point of Hansen and Singleton (1983) is to show that by exploiting conditioning information one can make non-trivial inferences with postwar data. Unfortunately, this led us to a related problem. While conditioning information could be helpful in identifying the intertemporal elasticity of substitution from asset market data and consumption, actual use of this information put us in a bind. You cannot simultaneously explain the conditional distribution of consumption as well as multiple returns. This bind was reflected in the conclusion of our paper, but certainly our prose did not match the elegance of Mehra and Prescott (1985). While our paper in the Journal of Political Economy exploited log-normality, our companion paper Hansen and Singleton (1982) (and awkwardly the errata in Hansen and Singleton (1984)) published in Econometrica found comparable results with multiple returns and conditioning information constructed as scaling factors using an estimation method that avoided log-normality. It is evident from our work that the heterogeneity in the risk exposure of returns including those we constructed through scaling posed a serious challenge to the power utility, representative consumer model. On the other hand, we were not as clever as Mehra and Prescott in describing and framing this as a puzzle. In contrast to Mehra and Prescott, statistical inference was front and center in our analysis and formally shaped how we looked at evidence, but this is only part of the difference in approaches. Although vast in its coverage, there is a missing link in Cochrane’s essay that is worth further consideration. Cochrane has separate discussions of the Fama and French (1995) empirical evidence based on portfolio constructed using ratios of book equity to market equity and Hall (2001)’s analysis of intangible capital. While I share Cochrane’s interest in Hall’s work, in Hansen et al. (2005), we view the Fama–French work as suggesting possibly important differences in the risk exposure of technologies that feature different mixes of tangible and intangible capital. If intangible capital is a primary source of

328


divergence in measures of book equity and market equity, then the Fama and French (1995) analysis suggests that the macroeconomic risk exposure of intangible capital may be fundamentally different from that of measured capital. This has potentially important modeling implications that are worth exploring further. Restoy and Weil (1998), Hansen et al. (1998), Tallarini (2000), and others feature the use of continuation values computed from consumption dynamics in conjunction with recursive utility. While Restoy and Weil (1998) focus on the role of consumption, they exploit its link to wealth and the return on the wealth portfolio. The link between continuation values and wealth becomes degenerate when the intertemporal elasticity of substitution is unity. This leads Restoy and Weil (1998) to exclude this case. Even with a unitary elasticity of substitution, however, continuation values still can be inferred from consumption dynamics by solving the utility recursion exactly or at least approximately. In fact, a unitary elasticity of substitution simplifies the calculation, as is evident from Cochrane’s discussion. By working with continuation values, Hansen et al. (2006) show that an approximation around ρ = 1, where ρ is the reciprocal of the intertemporal elasticity of substitution, is straightforward to compute for some alternative models of consumption dynamics. From Eq. (14) in Cochrane’s essay, the logarithm of the marginal utility of consumption is log mt+1 = −ρ(log ct+1 − log ct ) + (ρ − γ)log Ut+1 + πt , where πt is in the date t information set and Ut+1 is the continuation value for consumption at date t + 1. The term πt is inconsequential when characterizing the innovation to the logarithm of the marginal rate of substitution. Differentiating log mt+1 − πt with respect to ρ gives − log ct+1 + log ct + log Ut+1 + (ρ − γ)

d log Ut+1 . dρ

To localize around unity, we evaluate both log mt+1 − πt and its derivative at ρ = 1, scale the latter by ρ − 1, and add the terms: log mt+1 − πt ≈ − (log ct+1 − log ct ) + (1 − γ)log Ut+1 |ρ=1 d log Ut+1 + (ρ − 1) − log ct+1 + log ct + log Ut+1 |ρ=1 + (ρ − γ) |ρ=1 . dρ Hansen et al. (2006) compute continuation values and derivatives for log-normal consumption dynamics and for consumption dynamics that include some forms of stochastic volatility. These are analogous approximation formulas that characterize how asset values and local risk prices change as a function of the intertemporal substitution elasticity of investors. My final thought is a reflection about how explorations into alternative preferences have been or will be useful in macroeconomic analyses. Recently, Backus et al. (2004) wrote a useful summary on so-called exotic preferences and why they might or should

Lars Peter Hansen

329 be of interest to macroeconomists. In asset pricing, are exotic preferences merely a device to account for asset pricing facts, or do we aim for this evidence to be formally integrated into, say, macroeconomics models to be used in policy analysis? Similarly, what role will the asset pricing-based models with market imperfections have to play in constructing heterogenous agent models for use in addressing macroeconomic policy questions? It will be interesting to see how this empirically ambitious literature summarized by Cochrane will influence the construction of dynamic general equilibrium models. Will there be an analogous ambition that will pervade dynamic economic modelling more generally, or will asset pricing evidence be viewed in isolation? The jury is still out on such questions.

References Backus, D. K., B. R. Routledge, and S. E. Zin. Exotic preferences for macroeconomics. In M. Gertler, and K. Rogoff, eds., NBER Macroeconomics Annual (2004). Breeden, D. An intertemporal asset pricing model with stochastic consumption and investment opportunities. Journal of Financial Economics 7 (1979): 265–296. Fama, E. F., and K. R. French. Size and book-to-market factors in earnings and returns. Journal of Finance 50 (1995): 131–155. Grossman, S. J., and R. J. Shiller. Preliminary results on the determinants of the variability of stock market prices. University of Pennslyvania Press Philadelphia (1980). Grossman, S. J., and R. J. Shiller. The determinants of the variability of stock market prices. American Economic Review Papers and Proceedings 71 (1981): 222–227. Hall, R. E. The stock market and capital accumulation. American Economic Review 96 (2001): 222–227. Hansen, L. P., and R. Jagannathan. Implications of security market data for models of dynamic economies. Journal of Political Economy 99 (1991): 225–262. Hansen, L. P., and K. J. Singleton. Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica 50 (1982): 1269–1288. Hansen, L. P., and K. J. Singleton. Stochastic consumption, risk aversion, and the intertemporal behavior of asset returns. Journal of Political Economy 91 (1983): 249–268. Hansen, L. P., and K. J. Singleton. Errata. Econometrica 52 (1984): 267–268. Hansen, L. P., T. J. Sargent, and T. D. Tallarini. Robust permanent income and pricing. Review of Economic Studies 66 (1998): 873–907. Hansen, L. P., J. Heaton, and N. Li. Intangible risk? in C. Corrado, J. Haltiwanger, and D. Sichel, eds., Measuring Capital in the New Economy. University of Chicago Press, Chicago (2005), pages 111–152. Hansen, L. P., J. C. Heaton, J. Lee, and N. Rousanov. Intertemporal substitution and risk aversion. Forthcoming in the Handbook of Econometrics (2006). Lucas, R. E. Asset prices in an exchange economy. Econometrica 46 (1978): 1429–1445. Mehra, R. and E. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15 (1985): 145–161. Restoy, F., and P. Weil. Approximate equilibrium asset prices. NBER Working paper 6611 (1998). Rubinstein, M. The valuation of uncertain income streams and the pricing of options. Bell Journal of Economics 7 (1976): 407–425. Shiller, R. J. Consumption, asset prices and economic fluctuations. Carnegie-Rochester Conference on Public Policy 17 (1982): 203–238. Tallarini, T. D. Risk-sensitive real business cycles. Journal of Monetary Economics 45 (2000): 507–532.


CHAPTER 8

Understanding the Equity Risk Premium Puzzle George M. Constantinides∗ University of Chicago and NBER

1. Introduction 2. Habit Persistence 3. Limited Stock Market Participation and Per Capita Consumption 4. Incomplete Markets and Idiosyncratic Income Shocks 5. Concluding Remarks References

332 337 345 349 355 356

Abstract The unconditional mean of the aggregate equity risk premium is almost 6 percent per year even after adjusting downwards the sample mean premium for unanticipated events in the latter part of the 20th century. In this essay, I present theoretical and empirical research on three classes of generalizations of the standard neoclassical model and discuss their contribution toward a better understanding of the equity risk premium: preferences exhibiting habit persistence; borrowing constraints over the households’ life cycle that limit capital market participation and concentrate the stock market risk on the saving middle-aged households; and the recognition that idiosyncratic income shocks are persistent, uninsurable, and concentrated in economic recessions. JEL Classification: G12 Keywords: equity premium puzzle, habit persistence, borrowing constraints, limited participation, incomplete markets, idiosyncratic income shocks, lifecycle savings ∗I

thank John Cochrane, Hanno Lustig, Roberto Violi, and, particularly, Rajnish Mehra for their insightful comments and constructive criticism. I also thank Vito Gala and Lukasz Pomorski for their excellent research assistance and insightful comments. Naturally, I remain responsible for errors.


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332

Chapter 8 • Understanding the Equity Risk Premium Puzzle

1. INTRODUCTION The term “equity risk premium puzzle,” coined by Mehra and Prescott (1985), originally referred to the inability of standard neoclassical economic theory to reconcile the historically large realized premium of stock market return over the risk-free rate, with its low covariability with aggregate consumption growth. By now, the term encompasses a number of empirical regularities in the prices of capital assets that are at odds with the predictions of standard economic theory. The neoclassical economic model parsimoniously links the returns of all assets to per capita consumption growth through the Euler equations of consumption.1 According to the theory, the risk premia of financial assets are explained by their covariance with per capita consumption growth. However, per capita consumption growth covaries too little with the returns of most classes of financial assets, creating a whole class of asset pricing puzzles: the aggregate equity return, the long-term bond return, and the returns of various subclasses of financial assets are too large, too variable, and too predictable. Attempts to leverage the low covariability typically backfire, implying that the observed risk-free rate is too low and that its variance is too low. I discuss in some depth the aggregate equity puzzle because it exemplifies many of the problems that arise in attempting to explain the premium of any subclass of financial assets. The covariance of per capita consumption growth with the aggregate equity return is positive. The rational model explains why the aggregate equity premium is positive. However, the covariance is typically one order of magnitude lower than what is needed to explain the premium. Thus, the equity premium is a quantitative puzzle.2 The equity premium puzzle is robust. One may address the problem by testing the Euler equations of consumption or by calibrating the economy. Either way, it is a puzzle. In calibrating an exchange economy, the model cannot generate the first and second unconditional moments of the equity returns. In testing and rejecting the Euler equations of consumption, one abstracts from the market clearing conditions. Variations in the assumptions on the supply side of the economy do not resolve the puzzle. The challenge is a dual puzzle of the equity premium that is too high and the risk-free rate that is too low relative to the predictions of the model. In calibrating an economy, the strategy of increasing the risk aversion coefficient in order to lever the effect of the problematic low covariance of consumption growth with equity returns increases the predicted risk-free rate and aggravates the risk-free-rate puzzle. In testing the Euler equations of consumption, the rejections are strongest when the risk-free rate is included in the set of test assets. 1 See

Breeden (1979), Lucas (1978), Merton (1973), and Rubinstein (1976). and Constantinides (1991), Grossman and Shiller (1981), Hansen and Jagannathan (1991), Hansen and Singleton (1982), and many others tested and rejected the Euler equations of consumption. Mehra and Prescott (1985) calibrated an economy to match the process of consumption growth. They demonstrated that the unconditional mean annual premium of the aggregate equity return over the risk-free rate is, at most, 0.35 percent. This is too low, no matter how one estimates the unconditional mean equity premium. Weil (1989) stressed that the puzzle is a dual puzzle of the observed too high equity return and too low risk-free rate. In addition, Mehra (1998) shows that the Mehra–Prescott framework is unable to match the observed volatility of the U.S. stock market, expressed as a share of national income. 2 Ferson

George M. Constantinides

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Several generalizations of essential features of the model have been proposed to mitigate its poor performance. They include alternative assumptions on preferences,3 rare but disastrous market-wide events,4 distorted beliefs and learning,5 incomplete markets,6 market imperfections,7 and liquidity risk.8 They also include a better understanding of data problems such as limited participation of consumers in the stock market,9 temporal aggregation,10 and the survival bias of the U.S. capital market.11 The low covariance of the growth rate of aggregate consumption with equity returns is a major stumbling block in explaining the mean aggregate equity premium and the cross section of asset returns, in the context of a representative-consumer economy with time-separable preferences. Mankiw and Shapiro (1986) found that the market beta often explains asset returns better than the consumption beta does. Over the years, a number of different economic models have been proposed that effectively increase the covariance of equity returns with the growth rate of aggregate consumption, by proxying the growth rate of aggregate consumption with the stock market return in the Euler equations of consumption.12 Many of these generalizations contribute in part toward our better understanding of the economic mechanism that determines the pricing of assets. I refer the reader to the essays in this handbook; the textbooks by Campbell, Lo, and McKinlay (1997) and Cochrane (2001); and the articles by Campbell (2001, 2003), Cochrane (1997), Cochrane and Hansen (1992), Constantinides (2002), Kocherlakota (1996), and Mehra and Prescott (2003). My objective in this essay is to provide a better understanding of these issues when I generalize, one at a time, three of the key assumptions of the standard theory. First, 3 For example, Abel (1990), Bansal and Yaron (2004), Benartzi and Thaler (1995), Boldrin, Christiano, and Fisher (2001), Braun, Constantinides and Ferson (1993), Cagetti et al. (2002), Campbell and Cochrane (1999), Constantinides (1990), Epstein and Zin (1991), and Ferson and Constantinides (1991). 4 For example, Barro (2006), Danthine and Donaldson (1999), Mehra and Prescott (1988), and Rietz (1988). 5 Cecchetti, Lam and Mark (2000), Danthine and Donaldson (1999), Guidolin (2004), and Weitzman (2005). 6 For example, Bewley (1982), Brav, Constantinides, and Geczy (2002), Constantinides and Duffie (1996), De Santis (2004), Heaton and Lucas (1996), Jacobs and Wang (2004), Krebs (2002), Mankiw (1986), Mehra and Prescott (1985), and Storesletten, Telmer, and Yaron (2001, 2004). 7 For example, Aiyagari and Gertler (1991), Bansal and Coleman (1996), Constantinides, Donaldson, and Mehra (2002), Daniel and Marshall (1997), Danthine, Donaldson, and Mehra (1992), Guvenen (2005), He and Modest (1995), Heaton and Lucas (1996), and Parker and Julliard (2005). 8 See, Alvarez and Jermann (2001) and Lustig (2004). 9 For example, Attanasio, Banks, and Tanner (2002), Brav, Constantinides, and Geczy (2002), Brav and Geczy (1995), Mankiw and Zeldes (1991), and Vissing-Jorgensen (2002). 10 See Gabaix and Laibson (2001), Heaton (1995), and Lynch (1996). 11 See Brown, Goetzmann, and Ross (1995). However, Jorion and Goetzmann (1999, Table 6) found that the average real capital gain rate of a U.S. equities index exceeds the average rate of a global equities index that includes both markets that have and have not survived by merely one percent per year. 12 Friend and Blume (1975) explained the mean equity premium with low RRA coefficient by assuming a single-period economy in which the end-of-period consumption inevitably equals the end-of-period wealth. In the Epstein and Zin (1991) model, even though the preferences are defined over consumption alone, the stock market return enters directly in the Euler equations of consumption. Bakshi and Chen (1996) introduced a set of preferences defined over consumption and wealth—the spirit of capitalism—that also have the effect of introducing the stock market return in the Euler equations of consumption.

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I relax the assumption that preferences are time separable and introduce preferences that exhibit habit persistence. Second, I recognize that participation in the stock market by households is limited and that some aggregate consumption measures reflect the consumption of both participating and non-participating households in the stock market. Lastly, I relax the assumption that the market is complete and address the implications of incomplete consumption insurance. I begin with the standard assumptions of neoclassical economic theory, as adapted in finance. There are I consumers, i = 1, . . . , I, each consuming cti units of the consumption good in period t and having von Neumann–Morgenstern preferences % $ i ,... . E0 Ui c0i , . . . , cti , ct+1

(1.1)

There are J capital assets, j = 1, . . . , J , traded by the consumers in perfect markets. Prices, dividends, and income are denominated in units of the consumption good. The ex-dividend price of the jth asset at time t is Pj,t , its dividend is Dj,t , and its return between dates t and t + 1 is Rj,t+1 = Pj,t+1 + Dj,t+1 /Pj,t . The ith consumer’s budget constraint at time t is cti + yti +

J

i θj,t Pj,t ≤

j=i

J

i θj,t−1 Pj,t + Dj,t ,

(1.2)

j=i

i where yti is non-dividend income at time t and θj,t is the fraction of the jth asset held by the ith consumer after trading at time t. The ith consumer maximizes expected utility by choosing a consumption and i investment plan that satisfies the budget constraints. By definition, if cti , ct+1 is the optimal consumption plan at times t and t + 1, then the feasible consumption plan i i ct − δ, ct+1 + δRj,t+1 maximizes expected utility with respect to δ at δ = 0:

% $ i + δRj,t+1 , . . . ∂E0 Ui c0i , . . . , cti − δ, ct+1

∂δ

= 0.

(1.3)

δ=0

This variational argument leads to the following Euler equation of consumption between times t and t + 1: ∂Ui ∂Ui (1.4) Et − i + i Rj,t+1 = 0. ∂ct ∂ct+1 In the standard theory, preferences are typically specialized to be time separable as ∞

E0

t=0

i β ui ct , t

(1.5)

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with constant subjective discount factor β. There is no empirical justification for this specialization of preferences. In Section 2, I relax this assumption, introduce preferences that exhibit habit persistence, and explore its ramifications in understanding the observed equity premium. In any case, with time separable preferences as in Eq. (1.5), the Euler equation of consumption simplifies into the following equation: i i ∂ui ct+1 /∂ct+1 (1.6) Rj,t+1 = 1. Et β ∂ui cti /∂cti In the standard theory, the model is typically further specialized by assuming that the market is complete. There is little, if any, empirical justification for this assumption either. In Section 4, I relax this assumption, considering instead an incomplete market, and explore its implications in addressing the observed equity premium. In any case, under the assumption of market completeness, the equilibrium in this heterogeneous-consumer economy is isomorphic in its pricing implications to the equilibrium in a representative-consumer economy.13 For our purposes, market completeness implies that there exists a period utility function u (Ct ) of per capita consumption Ct ≡ I −1 Ii=1 cti , such that we may replace the I consumer-specific Euler equations (1.6) with the following Euler equation of consumption of the representative consumer: ∂u(Ct+1 )/∂Ct+1 (1.7) Rj,t+1 = 1. Et β ∂u(Ct )/∂Ct It is often assumed that the period utility function is of the power form, u (Ct ) = (1 − A)−1 (Ct )1−A , with constant relative risk aversion (RRA) coefficient A. With power utility, the Euler equation (1.7) further simplifies into the following:

−A Ct+1 Et β (1.8) Rj,t+1 = 1. Ct An advantage of the assumption that the period utility function is of the power form is that per capita consumption appears in the Euler equation only as per capita consumption growth, Ct+1 /Ct , which may be stationary even if the consumption level is non-stationary. In empirical work and calibration, per capita consumption growth in the Euler equation (1.8) is often taken to be the aggregate consumption growth, where the aggregate consumption is taken from the National Income and Product Accounts (NIPA). However, the aggregate consumption reported by NIPA is the sum total over all households, irrespective of whether these households are marginal in the equities and bond markets or not. In Section 3, I relax this assumption, recognizing instead the limited stock market participation by households and explore its ramifications in addressing 13 See

Section 4 for an elaboration of this.

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the observed equity premium. I pursue these issues in the context of a life cycle model in which young consumers are constrained from borrowing against their future labor income to invest in the stock market. Leaving aside the issue of limited capital market participation, I condition down the Euler equation (1.8) into unconditional form and linearize it under the assumption that consumption growth and asset returns are bivariate lognormally distributed: % $ % 1 $ ln E Ri,t+1 ≡ E ln Ri,t+1 + var ln Ri,t+1 2

1 2 Ct+1 Ct+1 − A × var ln = −ln β + A × E ln Ct 2 Ct

Ct+1 + A × cov ln Ri,t+1 , ln . Ct

(1.9)

% $ The term 1 A2 × var ln (Ct+1 /Ct ) reflects the decrease in the expected rate of return 2 $ % due to the precautionary demand for savings. The term A × cov ln Ri,t+1 , ln (Ct+1 /Ct ) reflects the “consumption-beta” risk premium. I take the difference of the linearized Euler equations for the market return and for the risk-free rate and find that the unconditional mean equity premium is equal to the product of the RRA coefficient of the representative consumer and the covariance of the market return with per capita consumption growth:14

$ % $ % Ct+1 . ln E RM,t+1 − ln E RF ,t+1 = A × cov ln RM,t+1 , ln Ct

(1.10)

I also find that the risk-free rate is equal to the sum of the subjective discount rate (−ln β) and the product of the RRA coefficient and the mean growth rate in consumption, less one-half the product of the squared RRA coefficient and the variance of consumption growth:

% $ Ct+1 1 Ct+1 − A2 × var ln . ln E RF ,t+1 = −ln β + A × E ln Ct 2 Ct

(1.11)

In this context, the elasticity of substitution in consumption is the increase$ in expected % consumption growth for a unit increase in the risk-free rate, ψ ≡ ∂ ln E Ct+1 /Ct /∂ $ % 15 ln E RF ,t+1 . With time-separable preferences and constant RRA coefficient, Eq. (1.11)

14 I

use the property that the risk-free rate RF ,t+1 is known at time t. The conditional covariance covt (ln RF ,t+1 , ln (Ct+1 /Ct )) is zero and, therefore, the unconditional covariance cov(ln RF ,t+1 , ln (Ct+1 /Ct )) is approximately zero. 15 This is a specialization of the general definition of the elasticity of substitution in consumption, ψ ≡ $ % ∂ ln Ct+j /Ct /∂ ln ut+j /ut .

337


implies that the product of the RRA coefficient and the elasticity equals one, RRA × ψ = 1. In terms of the elasticity, Eq. (1.11) may be written as

% $ 1 Ct+1 1 Ct+1 − . (1.12) × var ln ln E RF ,t+1 = −ln β + × E ln ψ Ct Ct 2ψ 2 With time-separable preferences and constant RRA coefficient, Eqs. (1.11) and (1.12) are interchangeable.16 Empirically, the covariance of per capita consumption growth with the market return is low and cannot generate the mean equity premium, as in Eq. (1.10), with a reasonable value of the RRA coefficient. If one assumes a sufficiently high value of the RRA coefficient in order to generate a realistic mean premium, the Euler equation of consumption with respect to the risk-free rate, Eq. (1.11), now generates a risk-free rate that is much higher than the historically observed risk-free rate. Thus, the puzzle is a dual puzzle that the observed mean equity premium is too high and the risk-free rate is too low.17 These are but two of the implications of the Euler equations (1.7) that are at odds with the data either in empirical work or calibration. In the following sections, I sequentially examine preferences that exhibit habit persistence, recognize the limited stock market participation by households, and relax the assumption of market completeness and explore their ramifications in better understanding the equity premium, the risk-free rate, and the Euler equations of consumption.

2. HABIT PERSISTENCE I explore the contribution of habit persistence in resolving the equity premium puzzle and related anomalies. Habit persistence has a long tradition in economic theory, dating back to Marshall (1920) and Duesenberry (1949). It is the property of preferences that an increase in consumption increases the marginal utility of consumption at adjacent dates relative to the marginal utility of consumption at distant ones. I model internal habit, xt , as an exponentially weighted sum of the representative consumer’s own past consumption, ct , as &t −at xt = e x0 + b ea(s−t) cs ds, (2.1) 0

where the parameter a is the persistence of habit. 16 With time-separable preferences that allow a variable RRA coefficient, u (C ) = (1 − A) −1 (C − K) 1−A , t t $ % % $ Eq.(1.11) is replaced % by ln E RF ,t+1 = −ln β + A × E ln (Ct+1 − K)/(Ct − K) − 12 A2 × $ var ln (Ct+1 − K)/(Ct − K) . It is still the case that A × ψ ≈ 1. However A no longer equals the RRA coefficient, and the product RRA × ψ need not equal one. Also, with time-non-separable preferences, such as habit-forming preferences, and state-non-separable preferences as in Epstein–Zin (1991), the product RRA × ψ need not equal one. With Epstein–Zin (1991) preferences, the risk-free rate is given by Eq. (1.12) in terms of the elasticity and not by Eq. (1.11) in terms of the RRA coefficient. 17 Note that the risk-free rate is decreasing in the RRA coefficient for a sufficiently high value of this coefficient. However, it takes an absurdly high value of the RRA coefficient to lower the risk-free rate in this way.

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Duesenberry (1949) introduced a model of external habit persistence in which the consumer does not take into account the effect of her own current consumption on her future preferences but rather takes into account the effect of other consumers’ consumption on her future preferences. I model external habit, Xt , as an exponentially weighted sum of per capita past consumption, Ct , of identical consumers as −a t

Xt = e

&t

X0 + b

ea (s−t) Cs ds.

(2.2)

0

In models with external habit, one typically assumes that there are a large number of identical consumers so that a consumer chooses her optimal consumption taking the path of per capita consumption as exogenous. Naturally, in equilibrium, ct = Ct . I switch from a discrete-time to a continuous-time framework and specialize the von Neumann–Morgenstern preferences stated in Eq. (1.1) as follows: &∞ e−ρt (1 − A)−1 (f (ct , xt , Xt ))1−A dt, (2.3) 0

where ρ ≡ −ln β is the subjective discount rate. More generally, I may define f (·) to be a function of internal and external habits at several different frequencies. The concavity parameter A has the interpretation as the RRA coefficient in the time-separable model, f (ct , xt , Xt ) = ct , but not generally so in the habit model even if f (·) is linear in its arguments. Building on earlier work by Ryder and Heal (1973) and Sundaresan (1989), I demonstrated in Constantinides (1990) that internal habit persistence can, in principle, reconcile the high mean equity premium with the low variance of consumption growth and with the low covariance of consumption growth with equity returns. In this model, habit is internal and linear: f (ct , xt , Xt ) = ct − xt .

(2.4)

From Eqs. (2.3) and (2.4), the period utility is u (ct , xt ) = (1 − A)−1 (ct − xt )1−A . I embed these preferences in a representative-consumer production economy and discuss their implications. There exists only one production good that is also the consumption good. This good may be consumed or invested without adjustment costs in two constant-returns-to-scale technologies. The technologies have constant returns to scale and rates of return over the period [t, t + dt] equal to rdt and μdt + σdw(t), respectively, where r, μ, and σ are constants and w(t) is a standard Brownian motion in R1 . I do not explicitly model the labor input to production because my focus is on the Euler equations of consumption and their implications for the equity premium puzzle. Since this is a representative-consumer economy, I interpret the optimal consumption and investment paths as the equilibrium paths and the shadow prices of assets as the equilibrium prices. In calibrating the model, I choose parameter values such that both technologies are always active. Then the rate of return of the risk-free technology determines the risk-free rate as r; and the parameters of the risky technology determine the

339


mean and standard deviation of the return on the stock market as μ and σ, respectively. I do not model the consumers’ labor income. Therefore, consumption equals dividends plus interest, net of new investment in production. One way to rationalize the assumption that consumers do not receive labor income is to view the capital invested in production as including human capital, the capitalized value of future labor income. Then the return on the risky technology is interpreted as the return on both the stock market and human capital. There is a single state variable in this economy, the habit-to-consumption ratio, xt /ct . In Constantinides (1990), I solved this model. The wealth, Wt , and consumption dynamics are xt

m2 σ 2 ct − xt r+a−b = exp n − t + mσw(t) , = x0 c0 − x0 2 W0 − r+a−b Wt −

(2.5)

where n≡

r − ρ (μ − r)2 (1 + A) + A 2A2 σ 2

(2.6)

m=

μ−r . Aσ 2

(2.7)

and

Consumption growth is dct = ct

n + b − (n + a)

xt ct

dt +

xt 1− ct

mσdw(t) ,

(2.8)

and the derived utility of wealth is V (Wt , xt , t) = θ × Wt −

1−A xt , r+a−b

(2.9)

where θ is a constant. The Arrow–Pratt relative risk aversion (RRA) coefficient is defined over atemporal gambles in terms of the derived utility of wealth as RRA = −

W VW W = VW

A xt /Wt 1− r+a−b

⎫ ⎧ 2 ⎪ ⎪ (1 (μ − A) − r) ⎪ ⎪ ρ − (1 − A) r − ⎨ 2 xt ⎬ 2Aσ =A 1+ × . ⎪ A (r + a) ct − xt ⎪ ⎪ ⎪ ⎭ ⎩

(2.10)

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The RRA coefficient is not given by the expression −ct ucc /uc = A/ (1 − xt /ct ).18 In the upcoming calibration, the RRA coefficient is very close to the parameter A, even though the expression A/ (1 − xt /ct ) is much higher than A. In calibrating the model, I choose the technology parameters r, μ, and σ such that the annual risk-free rate is 1 percent and the annual market return has mean 7 percent and standard deviation 16.5 percent, implying a mean annual equity premium of 6 percent. The goal is to choose the remaining parameters ρ, A, a, and b so that annual consumption growth rate has mean 1.8 percent and standard deviation 3.6 percent, while the RRA coefficient is kept at a plausibly low value.19 The order of magnitude of the subjective discount factor is not a controversial parameter, so I set ρ = 0.037 per year. Before I discuss the calibration of the habit model, I demonstrate that the timeseparable model fails to achieve the calibration goals. To see this, I set b = 0. If I want the model to imply that 100 percent (or 50 percent) of the wealth is invested in the stock market, I must set the RRA coefficient at 2.2 (or, 4.4). Then the model implies that the annual consumption growth has mean 3 percent (or, 1 percent) and standard deviation 16.5 percent (or, 8.2 percent). In order to generate the moments of the market return and the risk-free rate, the model requires much higher variability of consumption growth and covariability with the market return than what is observed in the data. With internal habit, several parameter combinations attain the calibration goal, as displayed in Constantinides (1990, Table 1). For example, with parameters A = 2.2, a = 0.5, and b = 0.405, the RRA coefficient has unconditional mean 2.88 percent, and the annual consumption growth has unconditional mean 1.8 percent and standard deviation 3.6 percent. On average, 79 percent of the wealth is invested in the stock market. Furthermore, consumption growth is substantially less variable than the growth in wealth. For example, with the above parameter values, the standard deviation of the instantaneous consumption growth is one-fifth of that of the growth in wealth. An intuition behind the internal habit model is that the RRA coefficient in Eq. (2.10) is increasing in the habit-to-consumption ratio and is, therefore, countercyclical. To see this, I first discuss a common property of economies with time-separable preferences and internal habit preferences. In an economic recession, equity returns are low and this event typically occurs when consumption is low and, therefore, the marginal utility of consumption is high. This effect alone is insufficiently strong to generate the order of magnitude of the observed premium if the RRA coefficient is constant, as in the standard model with time-separable preferences. In the habit model, the decrease in consumption increases the habit-to-consumption ratio, increases the RRA aversion coefficient, and raises the marginal utility of consumption much higher than in a model 18 If the utility function is time separable with period utility (1 − A) −1 c1−A , then V (W , t) = h W 1−A , where t t t h is a function of time and the state variables. In this case, the RRA coefficient exactly equals −cucc /uc because −Wt VW W /VW = A and −cucc /uc = A. If the utility function is time separable with period utility 1−A 1−A (1 − A)−1 ct − c and c is a constant, then V (Wt , t) = h Wt − c/r if the risk-free rate r is constant; as before, h is afunction of time and the state variables. In this is−W case, the RRA coefficient t VW W /V W = A/ 1 − c/rWt and is approximately equal to −cucc /uc = A/ 1 − c/ct , only if 1 − c/rWt / 1 − c/ct ≈ 1. 19 This exercise is the “reverse” of that in Mehra and Prescott (1985). In that paper, the authors start with a consumption process with a mean growth rate of 1.8 percent and standard deviation of 3.6 percent and seek reasonable parameter values to rationalize the observed equity premium.

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with time-separable preferences. It turns out that the combination of the two effects is sufficiently strong to generate the order of magnitude of the observed premium without an unreasonably high RRA coefficient. The same intuition drives the non-linear external habit model of Campbell and Cochrane (1999), which I discuss later. Another intuition behind the internal habit model may be gleaned by studying the relationship between the RRA coefficient and the elasticity of substitution in consumption. Recall our earlier discussion of the equity premium puzzle and the risk-free-rate puzzle through Eqs. (1.8) and (1.10). In the time-separable model with constant RRA coefficient, the product of the RRA coefficient and the elasticity, ψ, equals one. A high value of the RRA coefficient is needed to generate the observed equity premium. However, the high value of the RRA coefficient implies a low value for the elasticity and an unrealistically large value of the risk-free rate. By contrast, in the habit model, the product RRA × ψ can be substantially lower than one, as shown in Constantinides (1990, Table 1). For the particular choice of parameter values that I discussed earlier, the product RRA × ψ is about 0.26. Epstein and Zin (1991) put forth a model of preferences outside the class of von Neumann–Morgenstern preferences. Their model has the advantage that the RRA coefficient and the elasticity of substitution in consumption are neatly represented as two distinct parameters. Their model disentangles these two important economic concepts, with the former driving the equity premium and the latter driving the risk-free rate. However, the lesson derived from our earlier discussion of the habit model is that departure from von Neumann–Morgenstern preferences is not a necessary condition for disentangling these two important economic concepts. Even within the class of von Neumann–Morgenstern preferences, departure from time separability in the form of habit persistence suffices to disentangle the RRA coefficient and the elasticity of substitution in consumption. A limitation of the internal habit model is that it implies autocorrelations much higher than the observed ones at monthly, quarterly, and annual frequencies. The model also implies standard deviations somewhat higher than the observed ones. Whereas the model-implied instantaneous consumption growth is serially uncorrelated, the k-period -not. I define the k-period consumption growth rate ,+growth is *+consumption t+k t+2k cs ds and numerically calculate its standard deviation and cs ds as ln t t+k first-order autocorrelation for k = 1, 3, and 12 months and parameter values A = 2.2, a = 0.5, and b = 0.405. The results are reported in the following table. 1 Month Model-implied mean std dev

0.013

3 Months 0.026

12 Months 0.070

Ferson–Constantinides estimated std dev

0.008

0.022

0.031

Model-implied mean autocorrelation

0.383

0.531

0.693

−0.360

−0.310

0.410

Ferson–Constantinides estimated autocorr

In the same table, I also present the monthly estimates for the period 1959–1986, the seasonally adjusted quarterly estimates for the period 1948–1986, and the annual

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estimates for the period 1930–1986, as reported in Ferson and Constantinides (1991). It should be pointed out that the sample autocorrelation of the consumption growth rate varies widely over different samples. For example, the Ferson and Constantinides (1991) estimate of 0.41 autocorrelation of annual consumption growth for the period 1930–1986 contrasts with the Mehra and Prescott (1985) estimate of −0.14 for the period 1889–1978. Despite the wide range of these estimates, the internal habit model implies autocorrelations higher than the observed ones. Campbell and Cochrane (1999) argued that the annual endowment growth process (and, therefore, the equilibrium consumption growth process) should be close to i.i.d. However, an exchange economy with internal habit and i.i.d. annual endowment growth process yields a now endogenous risk-free rate that is excessively variable. This observation partly motivates the non-linear external habit model of Campbell and Cochrane (1999) that generates a constant endogenous risk-free rate when the annual endowment growth process is i.i.d. I discuss their model later. The calibration results of the internal habit model are reinforced by empirical evidence. Braun, Constantinides, and Ferson (1993) and Ferson and Constantinides (1991) tested the Euler equations of consumption in the special case of the linear internal habit model in which the habit depends only on the first lag of own consumption. They reported that the habit model performs better than the time-separable model and that the habit persistence parameter is economically and statistically significant. In what sense does the linear internal habit model fit the data better than the timeseparable model? Ferson and Constantinides (1991, Table 11) added a parameter L to the risk-free rate and re-estimated the parameters, including the parameter L, from the Euler equations. For the 5-asset system of Euler equations (Treasury bills, government bonds, and equity size deciles 1, 5, and 10), the estimated value of L is small and insignificantly different from zero. However, for the 2-asset system of Euler equations (Treasury bills and equity size decile 10), the estimated value of L is about 6 percent annually and significantly different from zero. More to the point, the estimated value of L is the same whether Ferson and Constantinides (1991) allow for habit in the Euler equations or set to zero the habit parameters. The fit to the data is better with habit than without. However, the improvement in the fit is not better in accommodating the mean equity premium. In both the time-separable and the linear internal habit models, they estimated an unexplained premium of about 6 percent annually. The interpretation of this little-noticed result is that habit persistence improves the fit of the model largely through its effect on moments other than the mean equity premium. Cecchetti, Lam, and Mark (1994), Cochrane and Hansen (1992), Gallant, Hansen, and Tauchen (1990), Hansen and Jagannathan (1991), and Heaton (1995) found that the linear internal habit model brings the moments of the marginal rate of substitution closer to the feasible Hansen and Jagannathan (1991) region than the time-separable model does. In exploring the differences between internal and external habit models, it is convenient to switch back to a discrete-time setup and state the preferences as ∞ (2.11) β t u (ct , xt , Xt ) . t=0

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I model internal habit, xt , as an exponentially weighted sum of the representative consumer’s own past consumption, ct , over S lags as xt = b

S

ϕs ct−s .

(2.12)

s=1

I model external habit, Xt , as an exponentially weighted sum of per capita past consumption, Ct , of identical consumers over S lags as Xt = b

S

ϕs Ct−s .

(2.13)

s=1

With external habit, u (ct , xt , Xt ) = u (ct , Xt ), the marginal utility of consumption at t is MUtEXT = β t

∂u (ct , Xt ) . ∂ct

(2.14)

By contrast, with internal habit, u (ct , xt , Xt ) = u (ct , xt ), the marginal utility of consumption at time t is S INT t ∂u (ct , xt ) t+s ∂u (ct+s , xt+s ) ∂xt+s + Et β . (2.15) MUt = β ∂ct ∂xt+s ∂ct s=1

The Euler equation of consumption for an asset with return Rt+1 is MUt = Et [MUt+1 × Rt+1 ].

(2.16)

With external habit, the contemporaneous consumption growth ct+1 /ct and its S lags appear in the Euler equations of consumption. By contrast, with internal habit, the S forward values of consumption growth, in addition to the contemporaneous consumption growth ct+1 /ct and its S lags appear in the Euler equations of consumption. Thus, external and internal habit models are empirically distinguishable.20 Turning to the empirical evidence, Bansal, Dittmar, and Lundblad (2005), Bansal and Yaron (2004), Hansen, Heaton, and Li (2005), Malloy, Moskowitz, and VissingJorgensen (2005), and Parker and Julliard (2005) reported that forward-looking consumption growth explains the cross section of stock returns better than contemporaneous consumption growth does. These results may be interpreted as evidence in 20 Hansen and Sargent (undated) and Cochrane (2001) pointed out that the pricing implications of internal and external habit models are identical, in the special case when habit is linear and the risk-free rate is constant. In the external habit model with period utility u (ct , Xt ) = u (ct − Xt ), the marginal utility is MUtEXT = β t u (ct − Xt ). If the Euler equation for the risk-free asset holds with external habit, $ % $ % −1 then u (ct − Xt ) = βRF Et u (ct+1 − Xt+1 ) . This implies that Et u (ct+1 − xt+1 ) = βRF u (ct − xt ) and MUtINT = β t u (ct − Xt ) − bβ t Ss=1 β s ϕs u (ct+s − Xt+1 ) = const. × β t u (ct − Xt ) = const. × MUtEXT . Therefore, if the Euler equations with external habit are satisfied, so are the Euler equations with internal habit.

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support of the internal habit models and against the external habit models. However, factors other than internal habit may be driving these results. They include measurement error, adjustment costs, non-separability of marginal utility of consumption from factors such as labor supply or housing stock, and Epstein–Zin (1991) preferences and behavioral factors. Ferson and Harvey (1992) tested the linear external habit model and reported positive results. Chen and Ludvigson (2005) treated the functional form of non-linear habit as an unknown and estimated it along with the other model parameters from the data in both the internal habit and external habit versions of the model. They concluded that habit is non-linear, is internal, and explains the cross section of stock returns better than most extant factor models do. Two specific non-linear habit models have received attention in the literature. The first, by Abel (1990), is a model of non-linear (multiplicative) internal and external D 1−D −1 habit, f (ct , xt , Xt ) = ct−1 Ct−1 ct , where D is a parameter. With external habit, D = 0, Abel (1990) found that the unconditional expected equity premium and risk-free rate are close to their sample values; however, the conditional expected values are too variable. With internal habit, D = 1, his results are inconclusive and oversensitive to the parameter values. This model is discussed in depth in Mehra’s essay in this handbook. Campbell and Cochrane (1999) modeled an exchange economy with i.i.d. annual endowment growth process and preferences in which the habit is external and is a nonlinear function of past changes in consumption rather than an exponentially weighted sum of past consumption levels, as in Eq. (2.2). The non-linear specification of the external habit is reverse-engineered such that the risk-free rate is either constant or has low variability. The model has in common with the Constantinides (1990) linear internal habit model the intuition that the RRA coefficient is an increasing function of the habitto-consumption ratio and is, therefore, countercyclical. The exogenous specification of the non-linear dependence of the external habit on past changes in consumption affords Campbell and Cochrane (1999) the flexibility to generate predictable long horizon returns and realistic behavior of several macroeconomic series along the business cycle. Their model hits the aggregate price-dividend ratio in a number of periods but misses it in the 1950s and 1990s. The major contribution of this model is that it sets a broad and ambitious research agenda that goes beyond the equity premium puzzle and addresses the broad interface between finance and macroeconomics. A drawback of the model is that the implied RRA coefficient is implausibly high and highly volatile: it takes the value of 80 in the steady state and a much higher value in economic recessions. Nevertheless, the model provides a flexible theoretical tool for exploring a broad set of issues, leaving concerns about the high risk aversion as a topic for future research. Menzly, Santos, and Veronesi (2004) developed a theoretical model that combines the Campbell and Cochrane preferences with an interesting cash flow process across industries. Dividend shares across industries vary over time in a stationary way, so no one industry dominates over time. The model explains two puzzling stylized empirical observations: first, dividend yield forecasts stock returns, but the strength of this relationship varies across subperiods and is absent in some subperiods; second, even though dividend growth is forecastable, dividend yield does not forecast dividend growth in


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simple regressions. The key to their model is that an increase in expected dividend growth decreases the dividend yield and increases the expected return. This weakens the commonly explored predictability of expected return by the dividend yield. Menzly, Santos, and Veronesi (2004) reported that their version of the non-linear external habit model helps explain the cross section of asset returns. Li (2001) reported that in both the linear and non-linear external habit models, the habit-to-consumption ratio has limited success in explaining the time series of the premia of stock and bond portfolios. Korniotis (2005) reported positive empirical results in testing factor pricing models that recognize non-linear external habit but only in conjunction with the assumption that the market is incomplete. High on the agenda for future research is the development of habit models that combine the low implied risk aversion of the Constantinides (1990) linear internal habit model with the flexibility of the nonlinear external habit model of Campbell and Cochrane (1999). On the empirical front, it is important to investigate the extent to which the empirical success of forward-looking consumption growth in explaining the cross section of asset returns is due to internal habit.

3. LIMITED STOCK MARKET PARTICIPATION AND PER CAPITA CONSUMPTION The low covariance of the growth rate of per capita consumption with equity returns is a major stumbling block in explaining the mean aggregate equity premium and the cross section of asset returns, in the context of a representative-consumer economy with time-separable preferences. The incorrect specification of per capita consumption contributes to the low correlation of the equity return with the growth rate of per capita consumption. Failure to adjust for the limited participation of households in the stock market is one source of this misspecification. Whereas we understood all along that many households whose consumption is counted in the NIPA-reported aggregate capital consumption do not hold equities, it took a paper by Mankiw and Zeldes (1991) to point out that the emperor has no clothes.21 Even though 52 percent of the U.S. adult population held stock either directly or indirectly in 1998, compared to 36 percent in 1989, stockholdings remain extremely concentrated in the hands of the wealthiest few.22 Furthermore, wealthy entrepreneurs may be infra-marginal in the stock market if their wealth is tied up in private equity. In Section 1, I pointed out that the unconditional mean equity premium is equal to the product of the RRA coefficient of the representative consumer and the covariance 21 Since then, several papers have studied the savings and portfolio composition of households, stratified by income, wealth, age, education, and nationality. See Blume and Zeldes (1993), Haliassos and Bertaut (1995), Heaton and Lucas (2000), Poterba (2001), and the collected essays in Guiso, Haliassos, and Jappelli (2001). 22 The percentages are only indicative of the extent of non-participation for two reasons. First, small stockholders may still be marginal. Second, given the wide variety of retirement plans, households have widely varying control on their investment in these plans.

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of per capita consumption growth with the market return. I now explicitly distinguish between the NIPA-reported aggregate consumption, CtNIPA , and the aggregate consumpI i tion, I × Ct ≡ i=1 ct , by the subset of households, i, i = 1, . . . , I, who are marginal in the equities market, identified as the stockholders. The NIPA-reported aggregate NIPA consumption growth, Ct+1 /CtNIPA , is a noisy measure of the stockholders’ aggregate consumption growth, Ct+1 /Ct . By explicitly recognizing the limited stock market participation of households, I may coax the theory to predict a higher value of the mean equity premium without increasing the RRA coefficient and without predicting an implausibly high value for the risk-free rate. Mankiw and Zeldes (1991) calculated per capita food consumption of a subset of households, designated as asset holders according to a criterion of asset holdings above some threshold. They found that the implied RRA coefficient decreases as the threshold is raised. Brav and Geczy (1995) confirmed their result by using per capita consumption of non-durables and services, reconstructed from the Consumer Expenditure Survey (CEX) database. Attanasio, Banks, and Tanner (2002), Brav, Constantinides, and Geczy (2002), and Vissing-Jorgensen (2002) found some evidence that per capita consumption growth can explain the equity premium with a relatively high value of the RRA coefficient, once we account for limited stock market participation. However, Brav, Constantinides, and Geczy (2002) pointed out that the statistical evidence is weak and the results are sensitive to the experimental design. My interpretation of these empirical results is that recognition of limited stock market participation is a step in the right direction, but this step alone is insufficient to explain asset returns. Constantinides, Donaldson, and Mehra (2002) introduced borrowing constraints in an overlapping-generations exchange economy and provided an endogenous explanation for the limited participation of young consumers in the stock market. They also demonstrated that limited participation in the borrowing constrained economy works in the right direction in increasing the equity premium, lowering the bond yield and lowering the variance of the bond and equity returns. In the Constantinides et al. (2002) economy, consumers live for three periods. In the first period, a period of human capital acquisition, consumers receive a relatively low endowment income. In the second period, consumers are employed and receive wage income subject to large uncertainty. In the third period, consumers retire and consume the assets accumulated in the second period. The key feature is that the bulk of the future income of the young consumers is derived from their wages forthcoming in their middle age, while the future income of the middle-aged consumers is derived primarily from their savings in equity and bonds. The young would like to invest in equity, given the observed large equity premium. However, they are unwilling to decrease their current consumption in order to save by investing in equity, because the bulk of their lifetime income is derived from their wages forthcoming in their middle age. They would like to borrow, but the borrowing constraint prevents them from doing so. Human capital alone does not collateralize major loans in modern economies for reasons of moral hazard and adverse selection. The model explains why many consumers do not participate in the stock market in the early phase of their life cycle.

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The future income of the middle-aged consumers is derived from their current savings in equity and bonds. Therefore, the risk of holding equity and bonds is concentrated in the hands of the middle-aged saving consumers. This concentration of risk generates the high equity premium and the demand for bonds, in addition to the demand for equity, by the middle-aged.23 The model recognizes and addresses simultaneously, at least in part, the equity premium, the limited participation in the stock market, and the demand for bonds. Constantinides et al. (2002) calibrated the model to match the following eight targets: the average share of income going to labor; the average share of income going to the labor of the young; the average share of income going to interest on government debt; the coefficient of variation of the 20-year wage income of the middle aged; the coefficient of variation of the 20-year aggregate income; the 20-year autocorrelation of the labor income; the 20-year autocorrelation of the aggregate income; and the 20-year cross correlation of the labor income and the aggregate income. Since the length of one period in this model is 20 years, for all securities (equity, bond, or consol), the annualized mean and standard deviation of the return are defined as the mean and standard deviation, respectively, of the annualized 20-year holding period return. Next I reproduce the first panel of Table 1 in Constantinides et al. (2002).24 Borrowing constrained Mean equity return Std dev of equity return Mean bond return Std dev of bond return Mean premium over bond Std dev of premium over bond Mean consol return Std dev of consol return Mean premium over consol

Borrowing unconstrained

8.40

10.20

23.00

42.00

5.10

9.00

15.40

27.60

3.40

1.10

18.40

31.60

3.70

9.90

19.10

27.60

4.70

0.30

Std dev of premium over consol

10.50

5.20

Corr of wages and dividend

−0.43

−0.43

Corr of wages and premium

−0.02

0.00

The borrowing constraint decreases the mean return of the 20-year or consol bond by about a factor of 2. This observation is robust to the calibration of the correlation and autocorrelation of the labor income of the middle-aged with the aggregate income. 23 See also the discussion in the related papers by Bertaut and Haliasos

(1997), Bodie, Merton, and Samuelson (1992), Jagannathan and Kocherlakota (1996), and Storesletten, Telmer, and Yaron (2001). 24 The RRA coefficient is set at 6; the coefficient of variation of the (20-year) aggregate income is set at 0.20; the coefficient of variation of the (2-year) wages is set at 0.25; the autocorrelation of aggregate income, the autocorrelation of wages, and the correlation of aggregate income and wages are all set at 0.1. The consol bond is in positive net supply, and the one-period (20-year) bond is in zero net supply.

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The borrowing constraint goes a long way, albeit not all the way, toward resolving the risk-free-rate puzzle. If the young were able to borrow, they would do so and purchase equity; the borrowing activity of the young would raise the bond return, thereby exacerbating the risk-free rate puzzle. Second, the mean equity premium over the 20-year or consol bond is about 4 percent. This is satisfactory given that long-term bond returns typically command a premium over the short-term risk-free rate. This premium drastically decreases when the borrowing constraint is relaxed. If the young were able to borrow, the increase in the bond return would induce the middle-aged to shift their portfolio holdings in favor of bonds; the increase in the demand for equity by the young and the decrease in the demand for equity by the middle-aged work in opposite directions; on balance, the effect would be to increase the return on both equity and the bond while simultaneously shrinking the equity premium. Third, the correlation of the labor income of the middle-aged and the equity premium over the 20-year bond is much smaller in absolute value than the exogenously imposed correlation of the labor income of the middle-aged and the dividend. Thus, equity is attractive to the young because of the large mean equity premium and the low correlation of the premium with the wage income of the middle-aged, thereby corroborating another important dimension of the model. In equilibrium, it turns out that the correlation of the wage income of the middle-aged and the equity return is low.25 The young consumers would like to invest in equity because equity return has low correlation with their future consumption, if their future consumption is derived from their future wage income. However, the borrowing constraint prevents them from purchasing equity on margin. Furthermore, since the young consumers are relatively poor and have an incentive to smooth their intertemporal consumption, they are unwilling to decrease their current consumption in order to save by investing in equity. Therefore, the young choose not to participate in the equity market. Finally, the borrowing constraint lowers the standard deviation of the annualized, 20-year equity and bond returns. However, the standard deviation of the equity return and particularly the standard deviation of the bond return remain high. In related work, Guvenen (2005) calibrated a general equilibrium model with two classes of infinitely lived agents, one that is exogenously forbidden from investing in equities and another that holds the entire stock of equity. The model replicates many features of the macroeconomy and the real business cycle through the channel of unrealistically high variance of the consumption growth of the stockholders. Constantinides, Donaldson, and Mehra (2002, 2007) addressed yet another major misspecification of per capita consumption. They included inter vivos gifts and post mortem bequests in the definition of consumption thereby increasing the covariance of equity returns with the growth rate of consumption. In the context of an overlappinggenerations economy, they modeled the “consumption” of the old households as 25 The

low correlation of the wage income of the middle-aged and the equity return is a property of the equilibrium and obtains for a wide range of values of the assumed correlation of the wage income of the middle-aged and the dividend.


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consisting of the “joy of giving,” cB , in the form of inter vivos gifts and post mortem bequests, in addition to the direct consumption, cD , of durable goods, services, and the service flow from non-durables goods. The direct consumption of the ‘cohort, wealthy enough to be non-trivial investors in the capital markets, is dictated by the state of their health and their medical expenses rather than fluctuations of their wealth. The correlation between the direct consumption of the old and the stock market return is low, a prediction that is borne out empirically. Constantinides et al. (2007) captured this feature by assuming that direct consumption is held constant because of a binding constraint on direct consumption. The bequest equals the wealth of the old, net of the direct consumption of the old, and is a fortiori highly correlated with the stock market return. Thus, in an economy that recognizes bequests, the marginal rate of substitution with respect to the bequests of the old is better able to explain the mean equity premium and the risk-free rate than the marginal rate of substitution with respect to the direct consumption of the old. How does the OLG economy with bequests as in Constantinides et al. (2007) differ in its pricing implications from the OLG economy without bequests as in Constantinides et al. (2002), if the wage and dividend processes in the two economies are identical? With low risk aversion, the calibrated model produces realistic values for the mean equity premium and the risk-free rate, the variance of the equity premium, and the ratio of bequests to wealth. However, the variance of the risk-free rate is unrealistically high. Security prices tend to be substantially higher in an economy with bequests as compared to an otherwise identical one where bequests are absent. We are able to keep the prices low and generate reasonable returns and the equity premium by stipulating that a portion of the bequests skips a generation and is received by the young, rather than by the middle-aged. Whereas borrowing constraints may explain the non-participation of young consumers in the financial markets, we are still far from a satisfactory explanation of the limited participation of middle-aged consumers in the financial markets. A better understanding of limited participation may aid our understanding of the equity premium puzzle and the returns of financial assets. Ang, Bekaert, and Liu (2005) took a step in this direction by demonstrating that investors endowed with “disappointment aversion” preferences as in Gul (1991) may choose to refrain from participating in some financial markets for a wide range of preference parameters.

4. INCOMPLETE MARKETS AND IDIOSYNCRATIC INCOME SHOCKS In economic recessions, investors are exposed to the double hazard of stock market losses and job loss. Investment in equities not only fails to hedge the risk of job loss but accentuates its implications. Investors require a hefty equity premium in order to be induced to hold equities. The argument hinges on the fact that the market for job-loss insurance is incomplete. Moral hazard impedes the development of unemployment insurance markets

350


and extant unemployment compensation provides inadequate protection for investors wealthy enough to be significant players in the financial markets. For example, with jobloss insurance, an increase of 1 percent in the probability of job loss merely manifests itself as a 1 percent decrease in per capita consumption growth—a risk that is too small to generate the observed equity premium. Job loss is but one example of idiosyncratic income shocks that have the potential to generate a sizable equity premium. The observed correlation of per capita consumption growth with stock returns is low. Over the years, I have grown skeptical of how meaningful an economic construct per capita consumption is, and how hard we should push per capita consumption to explain returns. Per capita consumption is a meaningful economic concept only if the market is complete or all consumers have quadratic utility. Any time that we model the household sector by a representative consumer and proxy consumption with per capita (or, aggregate) consumption, we suppress the potentially major economic impact of uninsurable idiosyncratic income shocks. I begin to formalize these ideas with the observation that, in the competitive equilibrium of a complete market economy, heterogeneous households allocate their consumption in a way that they equalize, state by state, their marginal rate of substitution. Negishi (1960) pointed out that the same consumption allocation is attained when a social planner maximizes the judiciously weighted sum of the households’ utility functions. For this result to hold, the weight of each household’s utility function is chosen to be inversely proportional to the Lagrange multiplier of the budget constraint of the respective household. The claim follows from the observation that the households’ first-order conditions are identical to the first-order conditions of a social planner. Constantinides (1982) further showed that the social planner’s problem can be split into two subproblems.26 In the first subproblem, one constructs an increasing and concave utility function out of the social planner’s weighted sum of the households’ utility functions. In the second subproblem, the social planner maximizes the aboveconstructed utility function and ends up allocating consumption at each state equal to the total consumption allocated at that state in the heterogeneous-household economy. This implies that the equilibrium in a heterogeneous-household, full-information economy is isomorphic in its pricing implications to the equilibrium in a representative-household, full-information economy, if households have von Neumann–Morgenstern preferences. The strong assumption of market completeness is indirectly built into asset pricing models in finance and neoclassical macroeconomic models through the assumption of the existence of a representative household. Bewley (1982), Mankiw (1986), and Mehra and Prescott (1985) suggested the potential of enriching the asset pricing implications of the representative-household paradigm, by relaxing the assumption of complete markets.27 Constantinides and Duffie (1996) found that incomplete markets substantially enrich the implications of the representative-household model. Their main result is a proposition demonstrating, by 26 See

also the discussion in Mehra and Prescott (1980).

27 There is an extensive literature on the hypothesis of complete consumption insurance. See Altonji, Hayashi,

and Kotlikoff (1992), Attanasio and Davis (1997), Cochrane (1991), and Mace (1991).

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construction, the existence of household income processes, consistent with given aggregate income and dividend processes, such that equilibrium equity and bond price processes match the given equity and bond price processes. The starting point in the Constantinides and Duffie (1996) theory is the I × J Euler equations (1.4) of individual consumption by the I consumers for the J assets, specialized for preferences that imply constant relative risk aversion: $ −A % Et βgi,t+1 Rj,t+1 = 1, i = 1, . . . , I, j = 1, . . . , J. (4.1) i /cti and the RRA coeffiThe consumption growth of the ith consumer is gi,t+1 ≡ ct+1 cient is A. The point of departure from standard theory is the recognition that market incompleteness rules out the step of replacing the Euler equations (4.1) by the Euler equation (1.6) of the representative consumer. A stochastic discount factor (SDF), or pricing kernel, is defined as any random variable mt+1 with the following property: $ % j = 1, . . . , J. (4.2) Et mt+1 Rj,t+1 = 1, −A Therefore, Eq. (4.1) states that each consumer’s marginal rate of substitution, βgi,t+1 , is a valid SDF. I sum the I × J Euler equations (4.1) across households and obtain the expression I −A −1 j = 1, . . . , J. (4.3) Et β I gi,t+1 Rj,t+1 = 1, i=1

Equation (4.3) states that the weighted sum of the consumers’ marginal rate of substitution is a valid SDF also: I −A −1 mt+1 = β I gi,t+1 . (4.4) i=1

I expand Eq. (4.4) as a Taylor series up to cubic terms and obtain the following approximation for the SDF:

2 I A(A + 1) −1 gi,t+1 −A −1 I mt+1 = βgt+1 1 + 2 gt+1 i=1

3 I A(A + 1) (A + 2) −1 gi,t+1 − −1 . (4.5) I 6 gt+1 i=1

The term gt+1 ≡ I −1 Ii=1 gi,t+1 is the sample mean of the consumption growth rate %2 $ across consumers; I −1 Ii=1 gi,t+1 /gt+1 − 1 is the squared coefficient of variation %3 $ of the consumption growth rate across consumers; and I −1 Ii=1 gi,t+1 /gt+1 − 1 is a measure of the skewness of the cross-sectional variation of consumption growth.

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The theory requires that the idiosyncratic income shocks have three properties in order to explain the returns on financial assets. First, they must be uninsurable. If there exists a complete set of markets, the equilibrium of a heterogeneous-household, full-information economy is isomorphic in its pricing implications to the equilibrium of a representative-household, full-information economy and household consumption growth cannot do better than aggregate consumption growth in explaining the returns. To see this, note that, if a complete set of markets exists, then the heterogeneous households are able to equalize, state by state, their marginal rates of substitution. In particular for any state s at time t + 1, there exists a state-specific but consumer-independent parameter λs such that gi,t = gt = λs , i = 1, . . . , I. The SDF in Eq. (4.5) simplifies into −A mt+1 = βgt+1

(4.6)

or, equivalently, into28 mt+1 = β

Ct+1 Ct

−A .

(4.7)

Therefore, if there exists a complete set of markets, the equilibrium of a heterogeneoushousehold, full-information economy is isomorphic in its pricing implications to the equilibrium of a representative-household, full-information economy and household consumption growth cannot do better than aggregate consumption growth in explaining the returns. Second, the theory requires that the idiosyncratic income shocks be persistent. If the shocks are transient, then households can smooth their consumption by borrowing or by drawing down their savings.29 Third, the conditional variance, or some higher moment of the income shocks, must be countercyclical. In Eq. (4.5), the SDF is monotonically increasing in the conditional variance. If the conditional variance is countercyclical, then the SDF is countercyclical and covaries negatively with the market return, even though aggregate consumption has low covariance with the market return. In principle, this negative covariation gives rise to an equity premium that is absent in a complete market. Even if the conditional variance is not countercyclical, the cyclical or countercyclical behavior of some higher moment of the income shocks may generate a high equity premium. For example, in

28 When

there is a complete set of markets, Eq. (4.7) follows from the fact that, for any state s, i I −1 Ii=1 ct+1 Ct+1 = = λs = gt+1 . Ct I −1 Ii=1 cti

I expect that the SDF given by Eq. (4.7) is less susceptible to observation error than the SDF given by Eq. (4.6). 29 Aiyagari and Gertler (1991) and Heaton and Lucas (1996) found that consumers facing transient shocks come close to the complete-markets rule of complete risk sharing even with transaction costs and/or borrowing costs, provided that the supply of bonds is not restricted to an unrealistically low level.


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Eq. (4.5), the SDF is monotonically decreasing in the conditional skewness. If the conditional skewness is cyclical, then the SDF is countercyclical and covaries negatively with the market return, giving rise to an equity premium. A good example of a major uninsurable income shock is job loss. Job loss is uninsurable because unemployment compensation is inadequate. Layoffs have persistent implications on household income, even though the laid-off workers typically find another job quickly.30 Layoffs are countercyclical as they are more likely to occur in recessions. Recall that the main result in Constantinides and Duffie (1996) is a proposition demonstrating, by construction, the existence of household income processes, consistent with given aggregate income and dividend processes, such that equilibrium equity and bond price processes match the given equity and bond price processes. The proposition implies that the Euler equations of household (but not necessarily of per capita) consumption must hold. Furthermore, since the given price processes have embedded in them whatever predictability of returns by the price-dividend ratios, dividend growth rates, and other instruments that the researcher cares to ascribe to returns, the equilibrium price processes have this predictability built into them by construction. The first implication of the theory is an explanation of the countercyclical behavior of the equity risk premium: the risk premium is highest in a recession because the stock is a poor hedge against the uninsurable income shocks, such as job loss, that are more likely to arrive during a recession. The second implication is an explanation of the unconditional equity premium puzzle: even though per capita consumption growth is poorly correlated with stocks returns, investors require a hefty premium to hold stocks over short-term bonds because stocks perform poorly in recessions, when the investor is most likely to be laid off. In principle, I may directly test the I × J system of Euler equations (4.1) of household consumption. In the U.S., the best available disaggregated consumption data is provided by the Bureau of Labor Statistics’ Consumer Expenditure Survey (CEX) of quarterly consumption of selected households (not individual consumers). In practice, the direct test is difficult because household consumption data is reported with substantial error. Brav, Constantinides, and Geczy (2002) provided empirical evidence on the importance of uninsurable, idiosyncratic income risk on pricing. They put forth a series of candidate SDFs: the marginal rate of substitution of the representative consumer as in Eq. (4.6) or (4.7); the Taylor series expansion of the SDF up to quadratic terms; the Taylor series expansion of the SDF up to cubic terms, as in Eq. (4.5); and a loglinearized expansion of the SDF. They estimated the RRA coefficient and tested the set 30 The

empirical evidence is sensitive to the model specification. Heaton and Lucas (1996) modeled the income process as univariate and provided empirical evidence from the Panel Study on Income Dynamics (PSID) that the idiosyncratic income shocks are transitory. Storesletten, Telmer, and Yaron (2001) modeled the income process as bivariate and provided empirical evidence from the PSID that the idiosyncratic income shocks have a highly persistent component that becomes more volatile during economic contractions. Storesletten, Telmer, and Yaron (2007) corroborated the latter evidence by studying household consumption over the life cycle.

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of Euler equations of household consumption on the premium of the value-weighted and the equally weighted market portfolio return over the risk-free rate and on the premium of value stocks over growth stocks.31 Brav et al. (2002) did not reject the Euler equations of household consumption with RRA coefficient between two and four when the candidate SDF is the Taylor series expansion of the SDF up to cubic terms, as in Eq. (4.5). A RRA coefficient between two and four is economically plausible. They rejected the Euler equations of household consumption with any value of the RRA coefficient when the Taylor series expansion of the SDF does not include the cubic terms. This implies that, in addition to the mean and variance, the skewness of the cross-sectional distribution is important in explaining the equity premium. They also found that the log-linearized expansion of the SDF fails to explain the premia, possibly because the log-linearization downplays the effect of outliers. These results emphasize the role of the higher moments of the cross-sectional distribution in explaining the premia. Krebs (2002) provided a theoretical justification as to why it is possible that neither the variance nor the skewness, but higher moments of the cross-sectional distribution are important in explaining the equity premium. He extended the Constantinides and Duffie (1996) model by introducing rare idiosyncratic income shocks that drive consumption close to zero. In his model, the conditional variance and skewness of the idiosyncratic income shocks are nearly constant over time. Despite this, Krebs demonstrated that the original proposition of Constantinides and Duffie remains valid, that is, there exist household income processes, consistent with given aggregate income and dividend processes, such that equilibrium equity and bond price processes match the given equity and bond price processes. Essentially, he provided a theoretical justification as to why it may be hard to empirically detect the rare but catastrophic shocks in the low-order cross-sectional moments of household consumption growth. Jacobs and Wang (2004) expanded the set of test assets employed by Brav et al. (2002) to the set of the 25 Fama–French size and value portfolios and provided additional empirical evidence on the importance of uninsurable idiosyncratic income risk on pricing. They found that a two-factor asset pricing model, with the mean and crosssectional variance of the household consumption growth rate as factors, significantly outperforms the capital asset pricing model in explaining the cross section of asset returns. See also Semenov (2005) for a discussion of the role of higher moments. Korniotis (2005) investigated the potential incompleteness of the market across U.S. states while assuming complete consumption insurance within each state. He tested the Euler equation of consumption for each state, where state consumption is proxied by the annual statewide sales at retail establishments. He found that a factor pricing model that recognizes this incompleteness with the cross-sectional variance of the state consumption as a factor has modest success in explaining the cross section of asset 31 In related studies, Jacobs (1999) studied the PSID database on food consumption; Cogley (2002) and Vissing-Jorgensen (2002) studied the CEX database on broad measures of consumption; Jacobs and Wang (2004) studied the CEX database by constructing synthetic cohorts; and Ait-Sahalia, Parker, and Yogo (2004) proxied household consumption with the purchases of certain luxury goods.


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returns. He also found that a factor pricing model that simultaneously recognizes this incompleteness and non-linear external habit better explains the cross section of asset returns.

5. CONCLUDING REMARKS I examined the observed asset returns and concluded that the evidence does not support the case for abandoning the rational economic model. I argued that the standard model is greatly enhanced by relaxing some of its assumptions. In particular, I argued that we go a long way toward addressing market behavior by recognizing that consumers face uninsurable and idiosyncratic income shocks, for example, the loss of employment. The prospect of such events is higher in economic downturns, and this observation takes us a long way toward understanding both the unconditional moments of asset returns and their variation along the business cycle. I also argued that we should account for the limited stock market participation by households in constructing the per capita consumption growth rate by excluding from the measure of per capita consumption the households that are infra-marginal in the stock market. In this context, lifecycle considerations are important and often overlooked in finance. Borrowing constraints become important when placed in the context of the life cycle. The fictitious representative consumer that holds all the stock market and bond market wealth does not face credible borrowing constraints. Young consumers, however, do face credible borrowing constraints. I traced their impact on the equity premium, the demand for bonds—who holds bonds if the equity premium is so high?—and on the limited participation of consumers in the capital markets. Finally, I argued that relaxing the assumption of convenience that preferences are time separable drives a wedge between the preference properties of risk aversion and intertemporal elasticity of substitution, within the class of von Neumann–Morgenstern preferences. Further work along these lines may enhance our understanding of price behavior along the business cycle with credibly low risk aversion. Has the equity premium puzzle been resolved? I don’t think so. However, the pursuit of an explanation has helped us broaden the scope of our investigation in several important ways. We are now interested in understanding the mean, higher moments, covariability, and predictability of the return of different classes of financial assets. At the macro level, we study the short-term risk-free rate, the term premium of long-term bonds over the risk-free rate, and the aggregate equity premium of the stock market over the risk-free rate. At the micro level, we study the premium of individual stock returns and of classes of stocks, such as the small-capitalization versus large-capitalization stocks, the “value” versus “growth” stocks, and the historical record of losing versus winning stocks. Our pursuit has led us to study a broader class of preferences, beliefs and learning mechanisms, the evolution of state variables, market incompleteness, market imperfections, and notions of liquidity than those embedded in the standard neoclassical model.

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Our pursuit has also helped us gain a better understanding of data problems such as limited participation of consumers in the stock market, temporal aggregation, and survival biases. In my eclectic discussion of these issues, I hope to have convinced the reader that the integration of the notions of habit persistence, incomplete markets, the life cycle, borrowing constraints, and other sources of limited stock market participation are promising vantage points from which to study the prices of assets and their returns both theoretically and empirically.

References Abel, A. B. Asset prices under habit formation and catching up with the Joneses. American Economic Review Papers and Proceedings 80 (1990): 38–42. Ait-Sahalia, Y., J. A. Parker, and M. Yogo. Luxury goods and the equity premium. Journal of Finance 59 (2004): 2959–3004. Aiyagari, R. S., and M. Gertler. Asset returns with transactions costs and uninsured individual risk. Journal of Monetary Economics 27 (1991): 311–331. Altonji, J. G., F. Hayashi, and L. J. Kotlikoff. Is the extended family altruistically linked? American Economic Review 82 (1992): 1177–1198. Alvarez, F., and U. J. Jermann. Quantitative asset pricing implications of endogenous solvency constraints. Review of Financial Studies 14 (2001): 1117–1152. Ang, A., G. Bekaert, and J. Liu. Why stocks may disappoint. Journal of Financial Economics 76 (2005): 471–508. Attanasio, O. P., J. Banks, and S. Tanner. Asset holding and consumption volatility. Journal of Political Economy 110 (2002): 771–792. Attanasio, O. P., and S. J. Davis. Relative wage movements and the distribution of consumption. Journal of Political Economy 104 (1997): 1227–1262. Bakshi, G., and Z. Chen. The spirit of capitalism and stock market prices. American Economic Review 86 (1996): 133–157. Bansal, R., and J. W. Coleman. A monetary explanation of the equity premium, term premium, and risk-free rate puzzles. Journal of Political Economy 104 (1996): 1135–1171. Bansal, R., R. F. Dittmar, and C. T. Lundblad. Consumption, dividends and the cross-section of equity returns. Journal of Finance 60 (2005): 1639–1672. Bansal, R., and A. Yaron. Risks for the long run: A potential resolution of asset pricing puzzles. Journal of Finance 59 (2004): 1481–1509. Benartzi, S., and R. H. Thaler. Myopic loss aversion and the equity premium puzzle. Quarterly Journal of Economics 110 (1995): 73–92. Barro, R. J. Rare events and the equity premium. Quarterly Journal of Economics 121 (2006): 823–866. Bertaut, C. C., and M. Haliassos. Precautionary portfolio behavior from a life cycle perspective. Journal of Economic Dynamics and Control 21 (1997): 1511–1542. Bewley, T. F. Thoughts on tests of the intertemporal asset pricing model. Working paper, Northwestern University (1982). Blume, M. E., and S. P. Zeldes. The structure of stock ownership in the U.S. Working paper, University of Pennsylvania (1993). Bodie, Z., R. C. Merton, and W. F. Samuelson. Labor supply flexibility and portfolio choice in a life cycle model. Journal of Economic Dynamics and Control 16 (1992): 427–449. Boldrin, M., L. J. Christiano, and J. D. M. Fisher. Habit persistence, asset returns, and the business cycle. American Economic Review 91 (2001): 149–166. Braun, P. A., G. M. Constantinides, and W. E. Ferson. Time nonseparability in aggregate consumption: International evidence. European Economic Review 37 (1993): 897–920.


357

Brav, A., and C. C. Geczy. An empirical resurrection of the simple consumption CAPM with power utility. Working paper, University of Chicago (1995). Brav, A., G. M. Constantinides, and C. C. Geczy. Asset pricing with heterogeneous consumers and limited participation: Empirical evidence. Journal of Political Economy 110 (2002): 793–824. Breeden, D. T. An intertemporal asset pricing model with stochastic consumption and investment opportunities. Journal of Financial Economics 7 (1979): 265–296. Brown, S. J., W. N. Goetzmann, and S. Ross. Survival. Journal of Finance 50 (1995): 853–873. Cagetti, M., L. P. Hansen, T. Sargent, and N. Williams. Robustness and pricing with uncertain growth. Review of Financial Studies 15 (2002): 363–404. Campbell, J. Y. Asset pricing at the millennium. Journal of Finance 55 (2001): 1515–1567. Campbell, J. Y. Consumption-based asset pricing, in G. M. Constantinides, M. Harris, and R. Stulz, eds., Financial Markets and Asset Pricing: Handbook of the Economics of Finance, Volume 1B, Handbooks in Economics 21. North-Holland, Amsterdam, pp. 803–887 (2003). Campbell, J. Y., and J. H. Cochrane. By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107 (1999): 205–251. Campbell, J. Y., A. W. Lo, and A. C. MacKinlay. The Econometrics of Financial Markets. Princeton University Press, Princeton, NJ (1997). Cecchetti, S. G., P. Lam, and N. Mark. The equity premium and the risk free rate: Matching the moments. Journal of Monetary Economics 31 (1994): 21–46. Cecchetti, S. G., P. Lam, and N. Mark. Asset pricing with distorted beliefs: Are equity returns too good to be true? American Economic Review 90 (2000): 787–805. Chen, X., and S. C. Ludvigson. Land of addicts? An empirical investigation of habit-based asset pricing models. Working paper, New York University (2005). Cochrane, J. H. A simple test of consumption insurance. Journal of Political Economy 99 (1991): 957–976. Cochrane, J. H. Where is the market going? Uncertain facts and novel theories. Economic Perspectives (Federal Reserve Bank of Chicago) 21 (1997): 3–37. Cochrane, J. H. Asset Pricing. Princeton University Press, Princeton, NJ (2001). Cochrane, J. H., and L. P. Hansen. Asset pricing explorations for macroeconomics, in O. J. Blanchard, and S. Fischer, eds., NBER, Macroeconomics Annual. MIT Press, Cambridge, MA (1992). Cogley, T. Idiosyncratic risk and the equity premium: Evidence from the consumer expenditure survey. Journal of Monetary Economics 49 (2002): 309–334. Constantinides, G. M. Intertemporal asset pricing with heterogeneous consumers and without demand aggregation. Journal of Business 55 (1982): 253–267. Constantinides, G. M. Habit formation: A resolution of the equity premium puzzle. Journal of Political Economy 98 (1990): 519–543. Constantinides, G. M. Rational asset prices. Journal of Finance 57 (2002): 1567–1591. Constantinides, G. M., J. B. Donaldson, and R. Mehra. Junior can’t borrow: A new perspective on the equity premium puzzle. Quarterly Journal of Economics 117 (2002): 269–296. Constantinides, G. M., J. B. Donaldson, and R. Mehra. Junior is rich: Bequests as consumption. Economic Theory 32 (2007): 125–155. Constantinides, G. M., and D. Duffie. Asset pricing with heterogeneous consumers. Journal of Political Economy 104 (1996): 219–240. Daniel, K., and D. Marshall. The equity premium puzzle and the risk-free rate puzzle at long horizons. Macroeconomic Dynamics l (1997): 452–484. Danthine, J.-P., and J. B. Donaldson. Non-falsified expectations and general equilibrium asset pricing: The power of the peso. Economic Journal 109 (1999): 607–635. Danthine, J.-P., J. B. Donaldson, and R. Mehra. The equity premium and the allocation of income risk. Journal of Economic Dynamics and Control 16 (1992): 509–532. De Santis, M. Interpreting aggregate stock market behavior: How far can the standard model go? Working paper, UC Davis (2004). Duesenberry, J. S. Income, Saving, and the Theory of Consumer Behavior. Harvard University Press, Cambridge, MA (1949).

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Chapter 8 • Understanding the Equity Risk Premium Puzzle Epstein, L. G., and S. E. Zin. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: An empirical analysis. Journal of Political Economy 99 (1991): 263–286. Ferson, W. E., and G. M. Constantinides. Habit persistence and durability in aggregate consumption. Journal of Financial Economics 29 (1991): 199–240. Ferson, W. E., and C. R. Harvey. Seasonality and consumption-based asset pricing. Journal of Finance 47 (1992): 511–552. Friend, I., and M. E. Blume. The demand for risky assets. The American Economic Review 65 (1975): 900–922. Gabaix, X., and D. Laibson. The 6D bias and the equity premium puzzle, in B. Bernanke, and K. Rogoff, eds., NBER Macroeconomics Annual 2001. MIT Press, Cambridge, MA (2001). Gallant, A. R., L. P. Hansen, and G. Tauchen. Using conditional moments of asset payoffs to infer the volatility of intertemporal marginal rates of substitution. Journal of Econometrics 45 (1990): 141–179. Grossman, S. J., and R. J. Shiller. The determinants of the variability of stock market prices. American Economic Review Papers and Proceedings 71 (1981): 222–227. Guidolin, M. Pessimistic beliefs under rational learning: Quantitative implications for the equity premium puzzle. Working paper, University of Virginia (2004). Guiso, L., M. Haliassos, and T. Jappelli. Household Portfolios, MIT Press, Cambridge, MA (2001). Gul, F. A theory of disappointment aversion. Econometrica 59 (1991): 667–686. Guvenen, F. A parsimonious macroeconomic model for asset pricing: Habit formation or cross-sectional heterogeneity. Working paper, University of Rochester (2005). Haliassos, M., and C. C. Bertaut. Why do so few hold stocks? The Economic Journal 105 (1995): 1110–1129. Hansen, L. P., J. C. Heaton, and N. Li. Consumption strikes back? Measuring long run risk. Working paper, University of Chicago (2005). Hansen, L. P., and R. Jagannathan. Implications of security market data for models of dynamic economies. Journal of Political Economy 99 (1991): 225–262. Hansen, L. P., and K. J. Singleton. Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica 50 (1982): 1269–1288. He, H., and D. M. Modest. Market frictions and consumption-based asset pricing. Journal of Political Economy 103 (1995): 94–117. Heaton, J. C. An empirical investigation of asset pricing with temporally dependent preference specifications. Econometrica 63 (1995): 681–717. Heaton, J. C., and D. J. Lucas. Evaluating the effects of incomplete markets on risk sharing and asset pricing. Journal of Political Economy 104 (1996): 443–487. Heaton, J. C., and D. J. Lucas. Portfolio choice and asset prices: The importance of entrepreneurial risk. Journal of Finance 55 (2000): 1163. Jacobs, K. Incomplete markets and security prices: Do asset-pricing puzzles result from aggregation problems? Journal of Finance 54 (1999): 123–163. Jacobs, K., and K. Q. Wang. Idiosyncratic consumption risk and the cross-section of asset returns. Journal of Finance 59 (2004): 2211–2252. Jagannathan, R., and N. R. Kocherlakota. Why should older people invest less in stocks than younger people? Federal Bank of Minneapolis Quarterly Review 20 (1996): 11–23. Jorion, P., and W. N. Goetzmann. Global stock markets in the twentieth century. Journal of Finance 54 (1999): 953–980. Kocherlakota, N. R. The equity premium: It’s still a puzzle. Journal of Economic Literature 34 (1996): 42–71. Korniotis, G. M. External habit, incomplete markets, and the cross-section of expected returns. Working paper, University of Notre Dame (2005). Krebs, T. Testable implications of consumption-based asset pricing with incomplete markets. Journal of Mathematical Economics 40 (2002): 191–206. Li, Y. Expected returns and habit persistence. Review of Financial Studies 14 (2001): 861–899. Lucas, R., Jr. Asset prices in an exchange economy. Econometrica 46 (1978): 1429–1446. Lustig, H. The market price of aggregate risk and the wealth distribution. Working paper, UCLA (2004). Lynch, A. W. Decision frequency and synchronization across agents: Implications for aggregate consumption and equity returns. Journal of Finance 51 (1996): 1479–1497.


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Mace, B. J. Full insurance in the presence of aggregate uncertainty. Journal of Political Economy 99 (1991): 928–956. Malloy, C. J., T. J. Moskowitz, and A. Vissing-Jorgensen. Long-run stockholder consumption risk and asset returns. Working paper, University of Chicago (2005). Mankiw, N. G. The equity premium and the concentration of aggregate shocks. Journal of Financial Economics 17 (1986): 211–219. Mankiw, N. G., and M. D. Shapiro. Risk and return: Consumption beta versus market beta. Review of Economics and Statistics 68 (1986): 452–459. Mankiw, N. G., and S. P. Zeldes. The consumption of stockholders and nonstockholders. Journal of Financial Economics 29 (1991): 97–112. Marshall, A. Principles of Economics: An Introductory Volume. Macmillan, London (1920). Mehra, R. On the volatility of stock prices: An exercise in quantitative theory. International Journal of Systems Science 29 (1998): 1203–1211. Mehra, R., and E. C. Prescott. The equity premium: A puzzle. Working paper, Carnegie-Mellon University (1980). Mehra, R., and E. C. Precott. The equity premium: A puzzle. Journal of Monetary Economics 15 (1985): 145–161. Mehra, R., and E. C. Prescott. The equity premium: A solution? Journal of Monetary Economics 22 (1988): 133–136. Mehra, R., and E. C. Prescott. The equity premium in retrospect, in G. M. Constantinides, M. Harris, and R. Stulz, eds., Financial Markets and Asset Pricing: Handbook of the Economics of Finance, Volume 1B, Handbooks in Economics 21. North-Holland, Amsterdam (2003). Menzly, L., T. Santos, and P. Veronesi. Understanding predictability. Journal of Political Economy 112 (2004): 1–47. Merton, R. C. An intertemporal capital asset pricing model. Econometrica 41 (1973): 867–887. Negishi, T. Welfare economics and existence of an equilibrium for a competitive economy. Metroeconomica 12 (1960): 92–97. Parker, J. A., and C. Julliard. Consumption risk and the cross section of expected stock returns. Journal of Political Economy 113 (2005): 185–222. Poterba, J. M. The rise of the “equity culture”: U.S. stockownership patterns. 1989–1998, Working paper, MIT (2001). Rietz, T. A. The equity risk premium: A solution. Journal of Monetary Economics 22 (1988): 117–131. Rubinstein, M. The valuation of uncertain income streams and the pricing of options. Bell Journal of Economics 7 (1976): 407–425. Ryder, H. E., and G. M. Heal. Optimum growth with intertemporally dependent preferences. Review of Economic Studies 40 (1973): 1–43. Semenov, A. High-order consumption moments and asset pricing. Working paper, York University (2005). Storesletten, K., C. I. Telmer, and A. Yaron. Asset pricing with idiosyncratic risk and overlapping generations. Review of Economic Dynamics, forthcoming (2001). Storesletten, K., C. I. Telmer, and A. Yaron. Consumption and risk sharing over the life cycle. Journal of Monetary Economics 51 (2004): 609–633. Sundaresan, S. M. Intertemporally dependent preferences and the volatility of consumption and wealth. Review of Financial Studies 2 (1989): 73–89. Vissing-Jorgensen, A. Limited asset market participation and the elasticity of intertemporal substitution. Journal of Political Economy 110 (2002): 825–853. Weil, P. The equity premium puzzle and the risk-free rate puzzle. Journal of Monetary Economics 24 (1989): 401–421. Weitzman, M. L. A unified Bayesian theory of equity puzzles. Working paper, Harvard University (2005).

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Discussion: Understanding the Equity Risk Premium Puzzle Hanno Lustig UCLA and NBER

1. INTRODUCTION In his survey of the literature, George Constantinides lists several ingredients that help to resolve the equity premium puzzle: (1) missing or incomplete markets; (2) idiosyncratic risk depends on aggregate state; (3) borrowing constraints; (4) limited stock market participation; and (5) habits. I will limit my discussion to the first three mechanisms. To address these issues in a coherent way, I set up a version of the canonical “incomplete markets” model with a large number of agents. These agents face idiosyncratic and aggregate income risk. To really distinguish between aggregate and idiosyncratic risk, it is key to consider models with a large number of agents in which the income shocks are i.i.d. across agents (see Denhaan (2001) for a clear exposition). Each of these agents is endowed with a “labor income” tree. This tree cannot be traded. In addition, there is a Lucas tree that yields a constant fraction α of the aggregate endowment as its fruit in each period. There are two different trading environments I will consider for this economy. I start by allowing agents to trade a complete set of contingent claims (Section 2). In this Arrow–Debreu environment, agents can write contracts both on aggregate and idiosyncratic shocks. In Section 3, I close down the markets for idiosyncratic risk. Finally, Section 4 considers the interaction between market (in-)completeness and solvency constraints. Section 5 concludes. The mathematical derivations are in the Appendix. Building on recent work by Krueger and Lustig (2006), I argue that exogenously reducing the menu of traded assets does not necessarily help and might even be

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Hanno Lustig

counterproductive in helping us to understand the equity premium puzzle. As long as agents can trade a claim to aggregate consumption, Mehra and Prescott’s (1985) statement of the equity premium puzzle still applies.

1.1. Environment I denote the current aggregate shock by zt ∈ Z and the current idiosyncratic shock by yt ∈ Y . For simplicity, both Z and Y are assumed to be finite. Furthermore, let zt = (z0 , . . . , zt ) and y t = (y0 , . . . , yt ) denote the history of aggregate and idiosyncratic shocks. As shorthand notation, we use st = (yt , zt ) and st = (y t , zt ). We let the economy start at initial node z0 . Conditional on idiosyncratic shock y0 and thus s0 = (y0 , z0 ), the probability of a history st is given by πt (st |s0 ). We assume that these shocks follow a first-order Markov process with transition probabilities given by π(s |s).

1.2. Preferences and Endowments Consumers rank stochastic consumption streams {ct (st )} according to the following homothetic utility function: U (c)(s0 ) =

∞ t=0 st ≥s0

β t π(st |s0 )

ct (st )1−ρ , 1−ρ

where ρ is the coefficient of relative risk aversion and β ∈ (0, 1) is the constant time discount factor. The economy’s aggregate endowment process {et } depends only on the aggregate event history; we let et (zt ) denote the aggregate endowment at node zt . Finally, the growth rate of the aggregate endowment process λ(zt ) = et+1 (zt+1 )/et (zt ) are Markov in the aggregate state zt . If the aggregate shocks are i.i.d. over time, the log of aggregate consumption follows a random walk. Each agent draws a “labor income” share η(y ˆ t , zt ) as a fraction of the aggregate endowment in each period. Her labor income share depends only on the current individual and aggregate event. I denote the resulting individual labor income process by {ηt }, with ηt (st ) = η(yt , zt )et (zt ), where st = (st−1 , yt , zt ). Furthermore, I assume that the law of large numbers holds, so that π(st |s0 ) is not only a household’s individual probability of having income ηt (st ), but also the deterministic fraction of the population having that income. In addition, there is a Lucas tree that yields a constant share α of the total aggregate endowment, so that the total dividends of the tree are given by αet (zt ) in each period. The remaining fraction of the total endowment accrues to individuals as labor income, so that 1 − α denotes the labor income share. Therefore, by construction, the labor share

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of the aggregate endowment equals the sum over all individual labor income shares:

Πzt (yt )η(yt , zt ) = (1 − α),

yt ∈Y

for all zt , where Πzt (yt ) represents the distribution of idiosyncratic shocks yt , conditional on the aggregate shock zt . By the law of large numbers, the fraction of agents who draw y in state z only depends on z. At time 0, the agents are endowed with initial wealth θ0 . This wealth represents the value of an agent’s share of the Lucas tree producing the dividend flow in units of time 0 consumption, as well as the value of her labor endowment at date 0. We use Θ0 to denote the initial joint distribution of wealth and idiosyncratic shocks (θ0 , y0 ).

2. COMPLETE MARKETS To set the stage, we start by considering the complete markets environment: in each period, markets re-open and the households trade a complete menu of assets. In this case, labor income risk turns out to be irrelevant for asset pricing.1 We let at (st , yt+1 , zt+1 ) denote the quantity purchased of a security that pays off one unit of the consumption good if the shock in the next period is (yt+1 , zt+1 ). Its price today is given by qt (zt , yt+1 , zt+1 ). In addition, households trade shares in the Lucas tree. We use σt (st ) to denote the number of shares a household with history st = (y t , zt ) purchases today, and we let vt (zt ) denote the price of one share. An agent starting period t with initial wealth θt (st ) buys consumption commodities in the spot market and trades securities subject to the usual budget constraint: ct (st ) +

at (st , yt+1 , zt+1 )qt (zt , yt+1 , zt+1 ) + σt (st )vt (zt ) ≤ θt (st ).

zt+1

If the next period’s state is st+1 = (st , yt+1 , zt+1 ), her wealth is given by her labor income and the payoff from the contingent claim purchased in the previous period: θt+1 (st+1 ) = η(yt+1 , zt+1 )et+1 (zt+1 ) + at (st , yt+1 , zt+1 ) labor income contingent payoff + σt (st )[vt+1 (zt+1 ) + αet+1 (zt+1 )] . value of shares in Lucas tree 1 Strictly speaking, households do not have to actively trade all these claims. In the absence of information and commitment problems, they could simply hand over their endowment to a financial intermediary, in exchange for the “equilibrium” consumption process (see Atkeson and Lucas (1992) for a decentralization with financial intermediaries of constrained efficient allocations in the case of private information).

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In addition to the budget constraints, the households’ trading strategies are subject to a natural borrowing limit, one for each history st : t t AN t (s ) = η(s ) +

t+1 qt (zt , yt+1 , zt+1 )AN ). t+1 (s

yt+1 ,zt+1

This constraint prevents them from running Ponzi schemes. Since the Inada condition is satisfied, this natural borrowing limit never binds.

2.1. Equilibrium In equilibrium, agents line up their IMRS (intertemporal marginal rate of substitution) with the price of the Arrow security:

qt (z , yt+1 , zt+1 ) = π(s t

t+1

ct+1 (st , θ0 ) |st )β ct (st , θ0 )

−ρ .

Because of the homogeneity of the one-period utility kernel, a linear consumption sharing rule describes the equilibrium allocations; each agent consumes a fixed fraction of the aggregate endowment in all future states of the world: 1/ρ

ct (st , θ0 ) =

μ0

et (zt ), 1/ρ μ0 dΛ(μ0 )

where Λ is the cross-sectional distribution of initial weights μ0 . The weights are chosen such that the time-zero value of claim to {ct (st , θ0 ) − ηt (st )} equals net financial wealth θ0 . This being the case, the budget constraint is satisfied. We will refer to these weights as Pareto–Negishi weights. It is easy to verify that this consumption sharing rule satisfies the household firstorder condition, and that it clears the market by construction. Substituting for the consumption sharing rule in the household’s first-order condition produces the standard Breeden–Lucas stochastic discount factor (SDF):

mt,t+1

et+1 =β et

−ρ .

Hence, the stochastic discount factor is identical to that of this representative agent model (Rubinstein (1974) and Wilson (1968)). Constantinides (1982) extends this result to deal with preference heterogeneity, in addition to wealth heterogeneity.

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What does all this imply? Well, a representative agent who eats the aggregate e endowment process satisfies the Euler equation for any excess returns Rt+1 : −ρ et+1 e Rt+1 = 0, Et β et as well as the Euler equation for riskless bonds: −ρ et+1 Rtb = 1, Et β et where Rtb denotes the risk-free rate. This representative agent construct also works in incomplete market environments, as we will discuss in Section 3, as long as households can trade a claim to aggregate consumption.

2.2. Equity Premium Puzzle As shown by Hansen and Jagannathan (1991), we can restate the Sharpe ratio on equity as the product of the market price of consumption risk ρ std (Δ log et+1 ) and the correlation between log returns and log consumption growth: 0.45 =

E(Re ) e ), ρ std(Δ log et+1 ) corr(Δ log et+1 , log Rt+1 σ(Re )

where Re denotes the excess return on equity. The postwar standard deviation of nondurable aggregate consumption growth is only about 1.5 percentage points. So we obviously need a huge risk aversion coefficient of 30 to be able to match the observed Sharpe ratio on U.S. equity in postwar data. To make matters worse, the correlation between log consumption growth and the log return is only around 0.25. This, in essence, is Mehra and Prescott’s (1985) equity premium puzzle.

3. MISSING MARKETS Of course, households in the complete markets economy trade a very generous menu of assets. In particular, there are likely to be information and enforcement problems that prevent these contracts from being traded in reality. For example, if the idiosyncratic income shocks y are private information, these insurance contracts cannot simply be traded in decentralized markets. In fact, Cole and Kocherlakota (2001) show that, under some conditions, the presence of private information problems forces the closure of markets for idiosyncratic risk. That is why, in a second step, I close down the market for idiosyncratic shocks. Since the aggregation result relied on complete markets, it seems natural to expect that closing down the market for idiosyncratic income risk breaks this result. The work of Aiyagari and Gertler (1991), Telmer (1993), Lucas (1994), Heaton and Lucas (1996), Krusell and Smith (1997), and earlier work by Marcet and Singleton (1999) fit into this broad category.

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To examine the missing markets hypothesis, I close down the market for idiosyncratic labor income risk. To keep the analysis simple, I will still allow for trade in contingent claims on aggregate shocks. We call this economy the Arrow economy. The discussion that follows builds entirely on Krueger and Lustig (2006). Agents only trade claims on aggregate shocks: at (st , zt+1 ) denotes the quantity purchased of a security that pays off one unit of the consumption good if the shock in the next period is (yt+1 , zt+1 ). Its price today is given by qt (zt , zt+1 ). In addition to the budget constraints, the households’ trading strategies are subject to a modified version of the natural borrowing limits: t t A˜ N t (s ) = η(s ) +

qt (zt , zt+1 )

zt+1

π(yt+1 , zt+1 |st ) yt+1

π(zt+1 |zt )

t+1 ). A˜ N t+1 (s

3.1. Equilibrium As is easy to verify from the first-order conditions for optimality, in equilibrium, agents line up their expected marginal utility growth or expected IMRS (summing over y states tomorrow) with the price of the Arrow security: π(yt+1 , zt+1 |st ) ct+1 (st+1 , θ0 ) −ρ . qt (z , zt+1 ) = β π(zt+1 |zt ) ct (st , θ0 ) y t

t+1

Using the law of iterated expectations, this price can be restated as the cross-sectional average of the expected household IMRS (intertemporal marginal rate of substitution): qt (zt , zt+1 ) =

yt+1

−ρ ct+1 (st+1 , θ0 ) π(y t+1 , zt+1 |y0 , z0 ) β dΘ0 . ct (st , θ0 ) π(zt+1 |z0 )

(1)

Θ0 denotes the initial joint distribution of financial wealth and initial endowments. The standard incomplete markets model implies that the average IMRS, averaged across idiosyncratic states tomorrow, is a valid SDF. This averaging across idiosyncratic states is a key feature of the missing markets. So, does this help? Well, that depends on the cross-sectional distribution of consumption growth.

3.2. Mankiw’s Recipe for Generating Risk Premia How can we generate large risk premia in this environment? Mankiw (1986) provides a strikingly simple answer. If the cross-sectional variance of consumption growth is countercylical (henceforth CCV), then the incomplete markets SDF can provide a better match with the data: CCV increases the SDF more in recessions, thus increasing the price of aggregate risk.

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This result can easily be understood by going back to the state price in Eq. (1). Let σt2 denote the cross-sectional variance of consumption growth at time t. Using a secondorder Taylor expansion of the state price in Eq. (1) around the household’s average consumption growth rate, we can restate the SDF (approximately) as a function of the cross-sectional variance of consumption growth:2 ⎡

⎤

ρ(ρ + 1) 2 ⎥ rep ⎢ mt,t+1 = mt,t+1 ⎣1 + 2 σt+1 ⎦. 2 eet+1t The first term is the standard Breeden–Lucas SDF. The second part is the part contributed by the incompleteness, and it captures Mankiw’s effect. The stochastic discount factor increases in recessions or low aggregate consumption growth states if the second moment of the consumption growth distribution increases, provided that marginal utility is convex (ρ > −1).3 2 If somehow we can get the second moment of household consumption growth σt+1 to move around enough over the business cycle, then we might be able to increase the equity premium enough.

3.3. Constantinides and Duffie Constantinides and Duffie (1996) (henceforth CD) embed Mankiw’s mechanism in an equilibrium model. CD assume all the y shocks are permanent. There is no net financial wealth in this economy (α = 0). In equilibrium, household consumption simply equals the household labor endowment: ct (st , θ0 ) = η(y t )et (zt ). CD assume the consumption shares η are log-normal, and they choose the crosssectional standard deviation of Δ log η as follows: σt+1 =

2 ρ(ρ + 1)

et+1 log mt+1 + δ + ρ log , et

(2)

where mt+1 is a strictly positive SDF that essentially prices all assets, meaning that Et [mt+1 Rt+1 ] = 1

2 See

the Appendix for a derivation.

3 This effect is absorbed in the drift and only affects the risk-free rate in continuous-time diffusion models

Grossman and Shiller (1982)).

(see

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Hanno Lustig

0.7 0.6

std(var(␴t +1))

0.5 0.4 0.3 0.2 0.1 0 1

2

3

4 5 6 7 Risk Aversion ␳

8

9

10

2 FIGURE 1 Variance of cross-sectional standard deviation of consumption. Plots the implied std(σt+1 ) needed to match the postwar U.S. Sharpe ratio of 0.45 against the coefficient of risk aversion ρ.

for any return process Rt+1 . From Eq. (2), we can back out the volatility in the cross-sectional variance of consumption growth needed to match the maximum Sharpe ratio of 0.45.4 Our objective is to deliver a maximum Sharpe ratio of 0.45, the postwar Sharpe ratio for U.S. equity. So, how much action do we need in the cross-sectional distribution to match the Sharpe ratio on U.S. equity of 0.45? In order to give this mechanism the best shot at matching the data,5 I assume that the correlation between consump2 tion growth and the cross-sectional variance is −1: corr(Δ log (et+1 ), var(σt+1 )) = −1. The stdΔ log (et+1 ) is 0.015 (or 1.5 percentage points). This is the annual standard deviation of non-durable consumption growth in postwar U.S. data. Figure 1 plots 2 the standard deviation of σt+1 , the cross-sectional variance of consumption growth, needed to match the postwar U.S. Sharpe ratio of 0.45, against the coefficient of relative risk aversion, assuming the variance is perfectly negatively correlated with aggregate consumption growth. On average, the cross-sectional variance of household consumption growth is around 10 percentage points.6 This corresponds to a standard deviation of 1 percent; this is 4 See

the Appendix for a complete derivation. argument follows Cochrane (2001), p. 479. 6 The mean actually does not effect the excess return, only the risk-free rate and returns. 5 This

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the estimate inferred from the fanning out of consumption distributions as cohorts of consumers age by Deaton and Paxson (1998). When ρ = 2, a one standard deviation drop in aggregate consumption growth (of 1.5 percentage points) below its mean requires an increase of 40 percentage points (not percent) in the cross-sectional variance of consumption growth. That seems implausibly large. Next, we consider the case of ρ = 10 in the right corner of the graph. A one standard deviation drop in aggregate consumption growth (of 1.5 percentage points) still implies an 8-percentage-point increase in the cross-sectional variance of consumption growth. Hence, in a recession, the standard deviation of consumption growth would have to increase by 40 percent to explain the Sharpe ratio on equity. So, even if the shocks are not insurable, lots of variation in the cross-sectional dispersion of consumption growth is still needed.

3.3.1. Evidence This raises two main questions. First, is there any evidence in the data to support this mechanism quantitatively? In other words, is the correlation between the cross-sectional dispersion and returns strong enough? Cogley (2002) finds some evidence in favor of this mechanism, but he concludes that this effect is not strong enough. Brav et al. (2002) find more evidence in higher moments of the consumption growth distribution; these enter when we take a higher-order expansion of the average IMRS. Krueger and Lustig (2006) try to answer a different question: can the model endogenously generate CCV? The answer is no. Shutting down markets for idiosyncratic risk does not activate this mechanism endogenously.

3.4. Independence of Idiosyncratic Shocks from Aggregate Conditions To answer the question of whether incomplete market models endogenously activate the CCV mechanism, Krueger and Lustig (2006) assume that idiosyncratic shocks are independent of the aggregate shocks. Condition 1. Individual endowment shares η(yt , zt ) are functions of the current idiosyncratic state yt only, that is, η(yt , zt ) = η(yt ). Also, transition probabilities of the shocks can be decomposed as π(zt+1 , yt+1 |zt , yt ) = ϕ(yt+1 |yt )φ(zt+1 |zt ). That is, individual endowment shares and the transition probabilities of the idiosyncratic shocks are independent of the aggregate state of the economy z. In addition, we assume that the aggregate shocks are independent over time. Condition 2. Aggregate endowment growth is i.i.d.: φ(zt+1 |zt ) = φ(zt+1 ). The growth rate of aggregate endowment is uncorrelated over time, so that the logarithm of the aggregate endowment follows a random walk with drift. Under these

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Hanno Lustig

conditions, Krueger and Lustig (2006) show that the Breeden–Lucas consumptionCAPM also prices excess returns on the stock in the Arrow model:

Et

e β Rt+1

et+1 et

−ρ = 0,

where Rs denotes the return on a claim to aggregate consumption. The stochastic discount factor is given by −ρ et+1 κt . (3) mt,t+1 = β et Incomplete market models cannot endogenously activate the Mankiw (1986) mechanism unless it is built into the model, i.e., built into the idiosyncratic endowment process. This result is quite robust. When households trade only a bond and a stock, this result holds only if the aggregate shocks are i.i.d. (condition 2 needs to be satisfied). In equilibrium, these households only trade the stock (a claim to aggregate consumption) to smooth their consumption. By doing so, these households keep their wealth proportional to aggregate income in all states of the world. If the traded assets span the aggregate shocks, this result survives even if the aggregate shocks are not i.i.d. Even though markets are incomplete, the aggregation results we discussed in Section 1 survive. This result effectively means that Mehra and Prescott’s statement of the equity premium puzzle applies to a much larger class of models. Now, Storesletten et al. (2004) have documented a substantial increase in labor income risk faced by households in recessions. The data seem to violate condition 1. But, the result we derived tells us that we only need to worry about that part of idiosyncratic risk that is correlated with aggregate consumption growth. The rest is irrelevant for risk premia.

3.4.1. Borrowing Constraints This result is quite robust. Suppose we introduce tighter borrowing constraints that bind in equilibrium, instead of the natural borrowing constraints. The first type of constraint imposes a lower bound on the value of the asset portfolio at the end of the period today, at (st , zt+1 )qt (zt , zt+1 ) + σt (st )υt (zt ) ≥ Kt (st ), (4) zt+1

while the second type imposes state-by-state lower bounds on net wealth tomorrow, at (st , zt+1 ) + σt (st ) υt+1 (zt+1 ) + αet+1 (zt+1 ) ≥ Mt (st , zt+1 ) for all zt+1 . (5) As long as these borrowing constraints are consistent with a stationary equilibrium (that is, they do not become tighter or looser as the economy is growing), our irrelevance results survives. There is essentially no link between the extent of risk sharing and the equity risk premium.

370


Once we introduce an interaction between idiosyncratic and aggregate risk, the link between risk sharing and the price of aggregate risk is restored, but I show below that closing markets may hurt, rather than help, when agents have different marginal willingness to pay for a unit of aggregate consumption in different states of the world.

4. MISSING MARKETS AND STATE-DEPENDENT SOLVENCY CONSTRAINTS In a second step, we drop the independence condition in (1) and allow for interaction between the aggregate shocks and the idiosyncratic shocks. As it turns out, closing markets for idiosyncratic risk may end up reducing the volatility of the SDF. Suppose we consider tighter solvency constraints of type 2, as in Eq. (5), that actually bind in equilibrium. These constraints can be derived endogenously in environments characterized by limited commitment on the part of households. Alvarez and Jermann (2000) derive these solvency constraints to decentralize constrained efficient allocations when agents can walk away from their debts into autarchy, as in Kehoe and Levine (1993). Lustig (2006) considers a different environment in which agents cannot be excluded from trading. In that case, the solvency constraints are tighter and all borrowing is collateralized. When agents face binding solvency constraints, then the SDF is given by the highest (across households) expected IMRS: qt (z , zt+1 ) = maxst ,θ0 t

π(yt+1 , zt+1 |st ) ct+1 (st+1 , θ0 ) −ρ . β π(zt+1 |zt ) ct (st , θ0 ) y t+1

These Arrow prices are the only ones that prevent arbitrage opportunities in equilibrium from arising: since the price equals the highest average IMRS, this unconstrained household has no interest in shorting it; other households may want to, because they have a lower willingness to pay for consumption in that state of the world tomorrow, but they cannot because they face binding constraints.

4.1. Incomplete Markets What is the maximum effect we can get out of incomplete risk sharing? To answer this question, I tighten the solvency constraints such that agents end up in autarchy,7 and the equilibrium prices are given by qt (z , zt+1 ) = maxst t

π(yt+1 , zt+1 |st ) ηt+1 (st+1 ) −ρ . β π(zt+1 |zt ) ηt (st ) y t+1

7 See

Lettau (2002) for a similar argument.

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Hanno Lustig

Let us consider the cyclical calibration of Heaton and Lucas (1996) (see p. 455). Suppose ρ = 2 and β = 0.99. The maximum Sharpe ratio produced by this model if there is absolutely no risk sharing is only 0.22 (these numbers were produced by simulating 10,000 times). If we expand the menu of assets traded, while still keeping agents in autarchy, this maximum Sharpe ratio increases dramatically.

4.2. Complete Markets Now, suppose we revisit the original trading environment and let agents trade a complete menu of assets. In this case, the SDF is given by the maximum IMRS, state by state (Alvarez and Jermann (2000)) qt (zt , zt+1 ) = max(st+1 ,θ0 ) β

ct+1 (st+1 , θ0 ) ct (st , θ0 )

−ρ .

(6)

If we impose the same tight borrowing constraints, an autarchic equilibrium is obtained and the SDF is given by8 qt (zt , zt+1 ) = max(st+1 ,θ0 ) β

ηt+1 (st+1 ) ηt (st )

−ρ .

In this complete markets case, the Sharpe ratio is 0.64, three times higher than the Sharpe ratio in the incomplete markets case. The states prices are much more volatile in the complete markets case, because we are not averaging over different states. Figure 2 plots the autarchic Sharpe ratios for both trading environments, against the coefficient of relative risk aversion ρ. The dotted line is the incomplete markets Sharpe ratio; the full line is the complete markets one. In a complete markets model, Lustig (2006) shows that the wealth distribution dynamics impute enough volatility to the SDF in (6) to match risk premia at low levels of risk aversion, even when there is a substantial amount of equilibrium risk sharing, albeit at the cost of inducing too much risk-free rate volatility. In addition, Krueger et al. (2007) find some evidence in U.S. household consumption data that the consumption growth of unconstrained U.S. households (in the sense of ) is more correlated than aggregate consumption growth with stock returns. What is the intuition behind this? Well, for a household that faces binding constraints, in the incomplete markets world, the price of a unit of aggregate consumption tomorrow reflects the average value of the Lagrangian multiplier on the solvency constraint, averaged across different y states of the world tomorrow. This averaging actually may end up reducing the volatility of the SDF.

8 To

keep the analysis simple, we assume there is no outside financial wealth.

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Chapter 8 • Understanding the Equity Risk Premium Puzzle Autarchy Sharpe Ratios

1.8 1.6 1.4

Sharpe Ratio

1.2 1 0.8 0.6 0.4 0.2 0 1

1.5

2

2.5 3 3.5 Risk Aversion ␳

4

4.5

5

FIGURE 2 Complete vs. incomplete markets. This figure plots the maximum Sharpe ratio in autarchy for the incomplete markets economy (dotted line) and the complete markets economy (full line).

5. CONCLUSION In standard equilibrium models, closing down markets for idiosyncratic risk does not increase the price of aggregate risk, unless the risk increases in recessions. That part of idiosyncratic risk that does not increase in recessions does not matter for the equity premium. In addition, in models with state-dependent solvency constraints, closing down markets for idiosyncratic risk may reduce the volatility of the SDF, making it harder to close the gap between the theory and the data.

References Aiyagari, S. R., and M. Gertler. Asset returns with transaction costs and uninsured individual risk. Journal of Monetary Economics 27 (1991): 311–331. Alvarez, F., and U. Jermann. Efficiency, equilibrium, and asset pricing with risk of default. Econometrica 68(4) (2000): 775–798. Atkeson, A., and R. E. Lucas. On efficient distribution with private information. Review of Economic Studies 59 (1992): 427–453. Brav, A., G. M. Constantinides, and C. C. Geczy. Asset pricing with heterogenous consumers and limited participation: Empirical evidence. Journal of Political Economy 110(4) (2002): 793–824. Breeden, D. T. An intertemporal asset pricing model with stochastic consumption and investment opportunities. Journal of Financial Economics 7 (1979): 265–296. Cochrane, J. H. Asset Pricing. Princeton University Press, Princeton, NJ (2001).

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Cogley, T. Idiosyncratic risk and the equity premium: Evidence from the consumer expenditure survey. Journal of Monetary Economics 49 (2002): 309–334. Cole, H. L., and N. R. Kocherlakota. Efficient allocations with hidden income and hidden storage. Review of Economic Studies 68(3) (2001): 523–542. Constantinides, G. M. Intertemporal asset pricing with heterogeneous consumers and without demand aggregation. Journal of Business 55(2) (1982): 253–267. Constantinides, G. M., and D. Duffie. Asset pricing with heterogeneous consumers. Journal of Political Economy 104 (1996): 219–240. Deaton, A. S., and C. H. Paxson. Aging and inequality in income and health. American Economic Review 88(2) (1998): 248–253. Denhaan, W. J. Understanding equilibrium models with a small and a large number of agents. Journal of Economic Dynamics and Control 25 (2001): 721–746. Grossman, S., and R. Shiller. Consumption correlatedness and risk measurement in economies with nontraded assets and heterogeneous information. Journal of Financial Economics 10 (1982): 195–210. Hansen, L. P., and R. Jagannathan. Implications of security markets data for models of dynamic economies. Journal of Political Economy 99 (1991): 252–262. Heaton, J. C., and D. J. Lucas. Evaluating the effects of incomplete markets on risk sharing and asset pricing. Journal of Political Economy 104 (1996): 668–712. Kehoe, T., and D. Levine. Debt-constrained asset markets. Review of Economic Studies 60 (1993): 865–888. Krueger, D., and H. Lustig. The irrelevance of market incompleteness for the price of aggregate risk. Working paper, UCLA and University of Frankfurt (2006). Krueger, D., H. Lustig, and F. Perri. Evaluating asset pricing models with limited commitment using household consumption data. Journal of the European Economic Association (Papers and Proceedings), forthcoming (2007). Krusell, P., and A. Smith. Income and wealth heterogeneity, portfolio choice, and equilibrium asset returns. Macroeconomic Dynamics 1 (1997): 387–422. Lettau, M. Idisosyncratic risk and volatility bounds or can models with idiosyncratic risk solve the equity premium puzzle? Review of Economics and Statistics 84(2) (2002): 376–380. Lucas, D. Asset pricing with unidiversifiable income risk and short sales constraints: Deepening the equity premium puzzle. Journal of Monetary Economics 34 (1994): 325–341. Lucas, R. E. Asset prices in an exchange economy. Econometrica 46(6) (1978): 1429–1454. Lustig, H. The wealth distribution and aggregate risk. Mimeo, UCLA (2006). Mankiw, G. N. The equity premium and the concentration of aggregate shocks. Journal of Financial Economics 17 (1986): 211–219. Marcet, A., and K. Singleton. Equilibrium asset prices and savings of heterogeneous agents in the presence of incomplete markets and portfolio constraints. Macroeconomic Dynamics 3 (1999): 243–277. Mehra, R., and E. C. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15 (1985): 145–161. Rubinstein, M. An aggregation theorem for security markets. Journal of Financial Economics 1 (1974): 225–244. Storesletten, K., C. Telmer, and A. Yaron. Cyclical dynamics in idiosyncratic labor-market risk. Journal of Political Economy 112 (2004): 695–717. Telmer, C. Asset-pricing puzzles and incomplete markets. Journal of Finance 48 (1993): 1803–1832. Wilson, R. The theory of syndicates. Econometrica 36 (1968): 119–132.

A. SECOND-ORDER TAYLOR EXPANSION We use λ(st+1 , θ0 ) to denote household consumption growth: λ(st+1 , θ0 ) =

ct+1 (st+1 , θ0 ) , ct (st , θ0 )

374


and λa to denote the average consumption growth rate: a

t+1

λ (z

)=

y t+1

π(y t+1 , zt+1 |y0 , z0 ) t+1 λ(s , θ0 )dΘ0 . π(zt+1 |z0 )

Using a second-order Taylor approximation of the average IMRS in Eq. (1) around the average consumption growth rate, we can show that 1 −ρ−2 rep qt (zt , zt+1 ) = qt (zt , zt+1 ) + βρ(ρ + 1) π(zt+1 |zt )(λa (zt+1 )) var(c)(zt+1 ), 2 where the second cross-sectional moment of consumption growth is given by t+1

var(c)(z

)=

y t+1

π(y t+1 , zt+1 |y0 , z0 ) (λ(st+1 , θ0 ) − λa (zt+1 ))2 dΘ0 . π(zt+1 |z0 )

This implies we can restate the SDF (approximately) as 1 rep 2 mt,t+1 = mt,t+1 + βρ(ρ + 1)(λat+1 )−ρ−2 σt+1 . 2

B. CONSTANTINIDES AND DUFFIE So, how much action do we need in the cross-sectional distribution to match the Sharpe ratio? Staring from the equation for the cross-sectional standard deviation of household consumption growth in (2), note that 2 σt+1

2 = ρ(ρ + 1)

et+1 . log mt+1 + δ + ρ log et

Rearranging terms produces ρ2 (ρ + 1)2 et+1 2 2 var (log mt+1 ) = var(σt+1 ) + ρ var log 4 et ρ2 (ρ + 1) et+1 2 −2 var(σt+1 ) . cov log 2 et

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Hanno Lustig

We assume that

et+1 2 , var(σt+1 ) corr log et

= 1.

As a result, the relationship between the variance of the log SDF and the variance of the cross-sectional variance of consumption growth is given by var(log mt+1 ) =

ρ2 (ρ + 1)2 2 var(σt+1 ) + ρ2 var(Δlog (et+1 )) 4 2 + ρ2 (ρ + 1)std(Δ log (et+1 ))std(σt+1 ).


CHAPTER 9

Cash Flow Risk, Discounting Risk, and the Equity Premium Puzzle Gurdip Bakshi University of Maryland and

Zhiwu Chen∗ Yale University

1. Introduction 2. Economic Determinants of Equity Premium 2.1. Cash Flow Process 2.2. The Discounting Process 2.3. Dynamics of the Market Portfolio 2.4. Dynamics of the Equity Premium 3. Time-Series Data on S&P500 EPS, EPS Growth, and the Interest Rate 4. Implications of the Model for Equity Premium 4.1. How Large Is the Interest-Rate Risk Premium? 4.2. Maximum-Likelihood Estimation of the (Physical) Gt Process 4.3. Compensation for Cash Flow Risk and the Equity Premium 5. Concluding Remarks and Extensions Appendix References

379 381 381 382 383 385 387 389 389 391 392 396 398 400

∗ The

earnings data used in the paper are provided to us by I/B/E/S International Inc. We thank Doron Avramov, Charles Cao, Steve Heston, Ravi Jagannathan, Nengjiu Ju, Mark Loewenstein, Dilip Madan, Lior Menzly, and Harry Mamaysky for many constructive discussions on this topic. We are grateful to Rajnish Mehra for inviting us to contribute this chapter. The chapter has improved from the feedback of Rajnish Mehra and two anonymous referees. The computer codes used to implement the model are available from either of the authors.


377

378

Chapter 9 • Cash Flow Risk, Discounting Risk, and the Equity Premium Puzzle

Abstract This article investigates the impact of cash flow risk and discounting risk on the aggregate equity premium. Our approach is based on the idea that consumption is hard to measure empirically, so if we substitute out an empirically difficult-to-estimate marginal utility by a pricing kernel of observables, we can evaluate the empirical performance of an equilibrium asset pricing model in a different way. Once the pricing-kernel process is specified, we can endogenously solve for the equity premium, the price of the market portfolio, and the term structure of interest rates within the same underlying equilibrium. Embedded in the closed-form solution are compensations for cash flow risk and discounting risk. With the solution for the risk premium explicitly given, we then calibrate the model to evaluate its empirical performance. This approach allows us to avoid the impact of the unobservable consumption or market portfolio on inferences regarding the model’s performance. Our illustrative model is based on the assumption that aggregate dividend equals a fixed fraction of aggregate earnings plus noise, and the expected aggregate earnings growth follows a mean-reverting stochastic process. Moreover, the economy-wide pricing kernel is chosen to be consistent with (1) a constant market price of aggregate risk and (2) a mean-reverting interest rate process with constant volatility. Estimation results show that the framework can mimic the observed market equity premium. JEL Classification: G10, G12, D9 Keywords: equity premium puzzle, discounting risk, cash flow risk, risk premia, priced factors, risk compensation, actual earnings growth, expected earnings growth, interset rate, pricing kernel

Gurdip Bakshi and Zhiwu Chen

379

1. INTRODUCTION In their seminal contribution, Mehra and Prescott (1985) show that the observed equity premium on the S&P 500 market index is far too high given the stochastic properties of aggregate consumption and plausible assumptions about risk aversion. Furthermore, equity returns empirically covary little with aggregate consumption growth, implying also that the average equity premium can only be reconciled through an implausibly large coefficient of relative risk aversion. Table 1 in Mehra and Prescott (2003) documents that the average equity premium in the U.S. is 6.92 percent, while the real rate of interest is 1.14 percent, over the sample period of 1889–2000. Why have stocks delivered an average return of about 7 percent over risk-free bonds? Why is the observed real rate on Treasuries so low? Why is the systematic risk, as exemplified by the correlation between consumption growth and market-index return, so small? Collectively known as the equity premium puzzle, this set of questions has consumed financial economists over the past two decades and generated competing explanations ranging from (1) generalizations to state-dependent utility functions (Constantanides (1990), Epstein and Zin (1991), Benartzi and Thaler (1995), Bakshi and Chen (1996), Campbell and Cochrane (1999), and Barberis, Huang, and Santos (2001)), (2) the fear of catastrophic consumption drops (Reitz (1988)), (3) the presence of uninsurable and idiosyncratic income risk (Heaton and Lucas (1996) and Mankiw (1986)), (4) borrowing constraints (Constantinides, Donaldson, and Mehra (2002)), and (5) measurement errors and poor consumption growth proxies (Breeden, Gibbons, and Litzenberger (1989), Mankiw and Zeldes (1991), Ferson and Harvey (1992), and A¨ıt-Sahalia, Parker, and Yogo (2004)). Despite the substantial research efforts, there is controversy whether these explanations can completely explain all aspects of the equity premium puzzle (Mehra and Prescott (2003)), and the original puzzle remains unsolved. That is, under plausible parameterizations, existing models can only generate a small equity premium. This article expounds on a risk-based explanation without taking a stand on the precise parametric specification of the marginal utility function. Our approach is based on the idea that consumption is hard to measure empirically, so if we substitute out an empirically difficult-to-estimate marginal utility by a pricing-kernel function of observables, we can evaluate the empirical performance of an equilibrium asset pricing model in a different way. That is, once the pricing-kernel process is specified, we can endogenously solve for the equity premium, the current price of the market portfolio, and the term structure of interest rates within the same underlying equilibrium. Embedded in the closed-form solutions are compensations for cash flow risk and discounting risk. With these solutions for the risk premium, we can then calibrate the model to evaluate its empirical performance. This approach allows us to avoid the impact of unobservable consumption on inferences regarding an asset pricing model’s performance. We illustrate the potential of this modeling approach by using some simple assumptions. First, we posit that a fixed proportion of the market-portfolio earnings (plus some noise) will be paid out as dividends. This assumption allows us to directly link the stock price and the equity premium to the firm’s earnings, instead of dividends. This modeling feature is important because dividend-based stock valuation models have not succeeded

380


empirically, and investors are far more interested in the earnings of a stock rather than its dividends. Second, we assume some marginal utility function that is consistent with both a constant market price of aggregate risk and a single-factor Vasicek (1977) term structure of interest rates. It is further assumed that the market-portfolio earnings per share (EPS) obeys a proportional stochastic process, with its expected growth rate following a mean-reverting process (under the physical probability measure). Thus, in our equity valuation setting, there is an embedded stochastic term structure of interest rates, the expected EPS growth follows a stochastic process, the current market-index level depends on earnings (instead of dividends), and both cash flow risk and interest rate risk are priced. The rationale for our assumptions will be discussed in more details shortly. It is shown that risk aversion implicit in the pricing kernel introduces a wedge between the physical process and the risk-neutralized process of variables in the economy. Specifically, the working of risk aversion makes the risk-neutral drift of the interest rate process higher than its physical counterpart and leads to a heavier discounting of stochastic cash flow streams. This mechanism generates lower market valuations and a higher equity premium, even though this effect also raises bond yields. Risk aversion also affects the risk-neutralized cash flow process: the risk-neutral drifts for both the earnings and the expected earnings growth processes are lower than their counterpart under the physical probability measure. Such a mapping is suggestive of a positive compensation for both earnings risk and expected earnings-growth risk. Overall, the equity premium is a weighted sum of compensations for risks associated with interest rate, earnings, and expected earnings-growth shocks, with the weights dependent on the state-of-the-economy and the structural parameters. Our empirical implementation provides several insights on how discounting risk and cash flow risks are reflected and simultaneously priced in the S&P 500 index and default-free bonds. We find that the interest-rate risk premium is negative and it contributes to a 77.16 basis-point spread between the market-portfolio and the risk-free interest rate. Moreover, the compensation for expected earnings-growth risk is negligible, and the compensation for earnings risk is 6.53 percent. It is the risk premium for earnings uncertainty, and not expected earnings-growth uncertainty, that largely drives the equity premium. The total model-derived equity premium is 7.31 percent and quantitatively robust under perturbations to test design methods. Overall, our empirical exercise demonstrates that the signs of the risk premiums are consistent with economic theory and show promise in explaining the behavior of the average equity premium and the Treasury yield curve. We argue that replacing the marginal utility by a pricingkernel function of observables, and sensibly parameterizing the discounting structure and cash flows, is crucial to achieving a reasonable equity premium and improved performance. The purpose of this article is not to test whether a particularly parameterized economic model would be able to explain the observed equity premium under some reasonable set of parameter values. Rather, the goal is to show that given the unobservability of key economic variables (such as consumption and the market portfolio), an alternative approach to testing an economic model is to rely on its internal equilibrium

381


relations to substitute unobservable variables by functions of observable financial market variables. Then, a test on the resulting equilibrium relations amounts to a test on the economic model itself. Perhaps, another way to look at the results in this article is that it shows what basic properties an empirically successful pricing kernel must have in order to be consistent with the observed equity premium in the U.S. stock market. In what follows, Section 2 outlines assumptions and develops analytical expressions for the price of the market portfolio and the equity premium. Section 3 describes the data on S&P 500 earnings, equity premium, interest rates, and the panel of bond prices. Section 4 estimates the valuation model and discusses its implication for the equity premium. Concluding statements are provided in Section 5. The mathematical derivations for the price of the market portfolio and the equity premium are provided in the Appendix.

2. ECONOMIC DETERMINANTS OF EQUITY PREMIUM This section develops a framework to study the determinants of the time t price of the market portfolio, Pt , for each time t ≥ 0, and the instantaneous market-index risk premium μt − rt , for short interest rate rt . Consider a continuous-time, infinite-horizon economy whose underlying valuation standard is represented by some pricing-kernel process, denoted by Mt . Assume that the market portfolio entitles its holder to an infinite dividend stream {Dt : t ≥ 0}. Asset pricing models under the perfect-markets assumption imply &∞ Mu (1) Du du, Et Pt = Mt t

dMt dPt μt − rt = −Covt , /dt, (2) Mt P t where Et [·] is the time t conditional expectation operator with respect to the objective probability measure. All variables in Eqs. (1) and (2) are in nominal terms. In this framework, the instantaneous equity premium and the price of the market portfolio are determined endogenously and jointly within the same underlying risk-return equilibrium. The basic model outlined ahead is adopted from Bakshi and Chen (2005).

2.1. Cash Flow Process To explicitly solve Eqs. (1) and (2), assume that the market portfolio has a constant dividend-payout ratio (plus noise), α (with 1 ≥ α ≥ 0), that is, Dt dt = αYt dt + dZt ,

(3)

where Yt is the aggregate earnings per share (EPS) flow at t and hence Yt dt is the total EPS over the interval from t to t + dt, and dZt is the increment to a martingale process

382


with zero mean. The existence of dZt allows the market-portfolio dividends to randomly deviate from the fixed proportion of its EPS, and it makes Dt and Yt not perfectly substitutable. Although this temporary deviation could be correlated with recent earnings and past deviations, incorporating this feature, or the stochastic pay-out ratio feature, into the assumption would unnecessarily complicate the model (see Lintner (1956), Marsh and Merton (1987), Barsky and Delong (1993), and Menzly, Santos, and Venonesi (2004)). Under the objective probability measure, Yt is assumed to follow a process given below: dYt y = Gt dt + σy dWt , Yt g dGt = κg μ∗g − Gt dt + σg dWt ,

(4) (5)

for constants σy , κg , μ∗g , and σg . The long-run mean for both Gt and actual EPS growth dYt /Yt is μ∗g , and the speed at which Gt adjusts to μ∗g is reflected by κg . Further, 1/κg measures the duration of the firm’s business growth cycle. Volatility for both earnings growth and changes in Gt is time-invariant. The cash flow process parameterized in Eq. (4) offers enough flexibility to model the level of the market portfolio and the instantaneous equity premium (see also Bakshi and Chen (1997) and Longstaff and Piazzesi (2004)). First, both actual and expected earnings growth can take either positive or negative values, reflecting business cycles. Second, expected EPS growth Gt is mean-reverting and has both a permanent component (reflected by μ∗g ) and a transitory component, so that Gt can be high or low relative to its long-run mean μ∗g . Finally, since Yt is observable and Gt can be obtained from analyst estimates, we can learn about the equity premium based on readily identifiable and observable state variables.

2.2. The Discounting Process Turning to the pricing kernel, assume, as in Constantinides (1992), that Mt follows an Ito process satisfying dMt = −rt dt − σm dWtm , Mt

(6)

for a constant σm , where the instantaneous discounting rate, rt , follows the Ornstein– Uhlenbeck mean-reverting process: drt = κr μ∗r − rt dt + σr dWtr , (7) for constants κr , μ∗r , and σr . The pricing kernel can be interpreted in the context of the consumption-based −γ asset pricing model. Suppose Mt = Ct for coefficient of relative risk aversion γ and aggregate consumption Ct ; then Ito’s lemma impies dMt /Mt = −γ (dCt /Ct ) + 2 1 2 γ(1 + γ)(dCt /Ct ) . Thus, we can write risk-return equation (2) as μt − rt = γ Covt (dCt /Ct , dPt /Pt )/dt, and the equilibrium rt dt = γEt(dCt /Ct ) − 12 (γ)(1 + γ) Et(dCt /Ct )2 .

383


Thus, unlike the traditional approaches in Mehra and Prescott (1985) and Weil (1989), we independently model the interest-rate dynamics as specified in (7). Parameter κr measures the speed at which rt adjusts to its long-run mean μ∗r . The pricing kernel (6) leads to a single-factor Vasicek (1977) term structure of interest rates, that is, the τ-period bond price is B(t, τ) = exp(−ξ[τ] − ς[τ]rt ), where +τ +τ ς[τ] ≡ (1 − e−kr τ )/kr , and ξ(τ) ≡ − 12 σr2 0 ς 2 [u]du + (κr μr + Covt (dMt /Mt , drt )/dt) 0 ς[u]du. This approach provides interest-rate parameters that can be separately calibrated to the observed Treasury yield curve. Notice that shocks to expected growth, W g , may be correlated with both systematic shocks W m and interest-rate shocks W r , with their respective correlation coefficients denoted by ρg,m and ρg,r . In addition, the correlations of W y with W g , W m , and W r are respectively denoted by ρg,y , ρm,y , and ρr,y . Thus, both actual and expected EPS growth shocks are correlated with the pricing kernel and hence priced risk factors. The noise process dZt in Eq. (3) is, however, assumed to be uncorrelated with Gt , Mt , rt , and Yt , and not a priced risk factor.

2.3. Dynamics of the Market Portfolio Substituting assumptions (3)–(7) into (1) and (2), we can see that the conditional expectations in Pt must be a function of Gt , rt , and Yt . Applying Ito’s lemma to Pt and substituting the resulting expression into risk-return equation (2), we have the partial differential equation (PDE) for Pt (the details are given in the Appendix): ∂P ∂2 P ∂2 P 1 2 2 ∂2 P + (G − Π )Y σ σ Y σ σ Y σy Y + ρ + ρ y g,y y g r,y y r 2 ∂Y ∂Y∂G ∂Y∂r ∂Y 2 + ρg,r σg σr

∂2P ∂P 1 ∂2 P 1 ∂2 P + σr2 2 + κr (μr − r) + σg2 2 ∂G∂r 2 ∂r ∂r 2 ∂G

+ κg (μg − G)

∂P − rP + αY = 0, ∂G

(8)

subject to the transversality condition Pt < ∞. The transversality condition states that the stock price stays bounded for all combinations of the parameters governing cash flows, discounting, and their risk premiums. In the valuation equation PDE (8), we set μg ≡ μ∗g −

Πg , κg

(9)

μr ≡ μ∗r −

Πr , κr

(10)

384


which are, respectively, the long-run means of Gt and rt under the risk-neutral probability measure defined by the pricing kernel Mt . It can be shown that

dMt dYt . dt, , Mt Y t

. dMt , dGt dt, Πg ≡ −Covt Mt

. dMt Πr ≡ −Covt , drt dt Mt Πy ≡ −Covt

(11)

(12)

(13)

are the risk premia for the earnings shocks, expected earnings growth, and interest rate, respectively. Conjecture that the solution to the PDE (8) is of the form Pt = αYt

&∞ p[t, u; G, r]du,

(14)

0

where p[t, u; G, r] can be interpreted as the time t price of a claim that pays $1 at a future date t + u. Solving the resulting valuation equation and the associated Ricatti equations subject to the boundary condition p[t + u, 0; G, r] = 1 yields the claim price: p[t, u; G, r] = exp(ϕ[u] − [u]rt + ϑ[u]Gt ), where

1 σr2 1 − e−2κr u 2(1 − e−κr u ) − u + 2 κr2 2κr κr

−κr u κr μr + σy σr ρr,y 1−e u− κr κr

2 −2κ u 2 1 σg 1−e g −κg u − (1 − e ) u+ 2 κg2 2κg κg

κg μg + σy σg ρg,y 1 − e−κg u u− κg κg

σr σg ρg,r 1 1 − e−(κr +κg )u 1 −κr u −κg u ) − (1 − e )+ u − (1 − e , κr κg κr κg κr + κg

(15)

ϕ[u] ≡ − Πy u + − + + −

(16)

[u] ≡

1 − e−κr u , κr

(17)

ϑ[u] ≡

1 − e−κg u , κg

(18)

385


subject to the transversality condition that μr − μ g >

σr2 2κr2

−

σg2 σg σy ρg,y σr σy ρr,y σg σr ρg,r − − Πy + 2 + . κr κg κr κg 2κg

(19)

Thus, the model price for the market portfolio is the summed value of a continuum of claims that each pay at a future time an amount respectively determined by the earnings process. The presence of an integral in (14) should not hamper the applicability of the model, as the integral can be computed numerically. The valuation formula in (14) is not as simple to comprehend as the Gordon dividend growth model. Realize that the Gordon model is a special case in which both Gt and rt are constant over time: Gt = g and rt = r, for constants g and r. Consequently, both Mt and Yt follow a geometric Brownian motion. In this case, we obtain Pt = αYt /(r + Πy − g) provided r + Πy − g > 0. In our economic setting, valuation is more complex, as both discounting and cash flow forecasts have to be assessed at the same time.

2.4. Dynamics of the Equity Premium In deriving the valuation formula, we relied on a CAPM-like risk-return relation to arrive at the PDE in (8). In this sense, our model is consistent with and built upon developments in the risk-return literature. But, as seen, a risk-return equation alone is not sufficient to determine Pt since assumptions on the cash flow processes are also needed. Based on (2) and the pricing solution (14), we can show that the equity premium is

αYt dPt . dt + − rt , μt − rt ≡ Et Pt Pt

. dMt dPt dt, , = −Covt Mt P t

Yt ∂Pt 1 ∂Pt 1 ∂Pt + Πg + Πr , Pt ∂Yt Pt ∂Gt Pt ∂rt +∞ p[t, u; G, r] × ϑ[u]du 0 = Πy + Πg +∞ p[t, u; G, r]du 0

= Πy

+∞ − Πr

0

p[t, u; G, r] × [u]du , +∞ p[t, u; G, r]du 0

(20)

(21)

where p[t, u; G, r] is displayed in (15). Equation (20) shows that the equity premium is a weighted sum of the risk premiums for shocks respectively due to earnings, expected earnings growth, and interest rate, with weights equal to the sensitivity of the price with respect to the respective state variables.

386


+ ∞ Equation (21) follows from (20) since + ∞ (Yt /Pt )(∂Pt /∂Yt ) = 1, ∂Pt /∂Gt = αYt p[t, u; G, r] × ϑ[u]du, and ∂P /∂r = −αY t t t 0 p[t, u; G, r] × [u]du. Thus, the equi0 librium equity premium is a function of the time t interest rate, the expected EPS growth, the firm’s required risk premiums, and the structural parameters governing the cash flow and interest-rate processes. According to (21), μt − rt is independent of the current level of cash flows and is mean-reverting with the state of rt and Gt . The dynamics of the state variables under the equivalent martingale measure, Q, can facilitate our understanding of the nature of risk compensation in this economy. Based on (8), we may write the stock price as Pt = α

&∞

Q

Et

− + u rs ds Yu du, e t

(22)

t

where the processes for (Yt , Gt , rt ) under the Q-measure are dYt ˜ yt , = Gt − Πy dt + σy dW Yt ˜ gt , dGt = κg [μ∗g − Πg /κg ] − Gt dt + σg dW ˜ rt . drt = κr [μ∗r − Πr /κr ] − rt dt + σr dW

(23) (24) (25)

Economically, risk-averse investors seek to discount future cash flows more heavily under the equivalent martingale measure. For instance, we should expect Πr < 0, which makes the drift of the risk-neutral discounting process higher. Consistent with this effect, a higher long-run mean μr = μr − Πr /κr will simultaneously reduce the discount bond price and raise all Treasury yields. Thus, our decomposition in (20) shows that Πr < 0 can be expected to increase the overall equity premium, because ∂Pt /∂rt < 0. There is evidence from bond markets that the interest-rate risk premium is non-zero (see, for example, Duffee (2002)). A similar risk-aversion-based reasoning suggests that investors tend to be less optimistic about future cash flows under the equivalent martingale measure than under the physical probability measure. Intuitively, we have Πy > 0 and Πg > 0: the presence of both risk premiums decreases the drift of the (Yt , Gt ) process. The working of both of these forces reduces the present value of future cash flows and, thus, elevates the market risk premium. Thus, the earnings risk premium Πy , the expected earnings growth risk premium Πg , and the discounting risk premium receive positive compensation and contribute separately to the total equity premium. To explore the properties of the equity premium derived in (21), we turn to a comparative statics exercise and study how it responds to any structural parameter. In this example, κr = 0.23, μ∗r = 7.8 percent, σr = 0.012, κg = 1.44, μ∗g = 0.10, σg = 0.089, σy = 0.20, ρg,r = −0.05, ρg,y = 1, and α = 0.50. We fix the interestrate risk premium Πr = −0.002, the expected earnings growth risk premium Πg = 0.002, and the earnings risk premium Πy = 0.06. In all calculations, rt = 5.68 percent


387

and Gt = 7.48 percent, which are market-observed values as of July 16, 1998, and correspond to an S&P500 index level of 1174. Our numerical exercise shows that the equity premium is increasing in both Gt and μ∗g , but decreasing in both rt and μ∗r . Therefore, as expected, positive shocks to expected EPS growth tend to raise the equity premium, whereas positive shocks to interest rates depress it. However, the equity premium is much more sensitive to μ∗g (μ∗r ) than to Gt (rt ). Intuitively, these comparative static results hold because current expected EPS growth Gt may have a transitory component, whereas a change in μg is permanent. Lastly, the model equity premium increases with EPS growth volatility σy , the volatility of expected EPS growth σg , and the volatility of the interest rate σr . Risks as measured by these parameters raise the required compensation to shareholders. Modeling the EPS and the expected EPS processes explicitly indeed allows us to see how they affect the equity premium.

3. TIME-SERIES DATA ON S&P500 EPS, EPS GROWTH, AND THE INTEREST RATE For the remainder of the chapter, we choose the S&P500 index as the proxy for the market portfolio. To explore whether the model equity premium derived in (21) is close to the sample equity premium requires three data inputs: • expected EPS growth Gt , • interest rate rt , • current EPS Yt , and the model parameters. For the S&P500 index, I/B/E/S did not start collecting analyst EPS estimates until January 1982. Thus, our focus is on the sample period from January 1982 to July 1998. Pastor and Stambaugh (2001) detect structural shifts in the equity premium, especially over the past two decades. Lettau, Ludvigson, and Watcher (2004) observe that the market price-to-earnings ratio rose sharply over this period and have argued in favor of the declining ex-ante equity risk premium explanation. The I/B/E/S U.S. History File contains mid-month observations on reported actual earnings per share and consensus analyst forecasts of future S&P500 earnings, plus the contemporaneous price. In implementation, I/B/E/S consensus analyst estimates for current-year S&P500 EPS (i.e., Fiscal Year 1, hereby FY1) is taken to be the proxy for Yt . In any given month, the FY1 estimate may contain actual quarterly EPS numbers for the past quarters of the fiscal year, with the EPS numbers for the remaining quarters being consensus analyst forecasts. Because firms’ earnings typically exhibit seasonalities, the total EPS over a fiscal year is a natural proxy for Yt . Analyst-expected EPS growth from the current fiscal year (FY1) to the next fiscal year (FY2) is the measure for Gt . This choice is reasonable since the year-over-year EPS growth has been the conventional calculation method in the industry. For instance, quarter-over-quarter and month-over-month (if available) EPS growth rates would not

388


be better proxies for Gt , as they would be subject to seasonal biases in earnings and revenue. Valuation formulas for the market index and the equity premium depend on the interest rate rt , for which there is no established benchmark. Empirically, movements in the 30-year Treasury yield are much more closely followed by stock market participants than the short-term rate, as the long-term yields often co-move strongly with S&P500 earnings yields. To be consistent with theory, however, we use the 3-month Treasury yield or those implied by the Kalman filter as candidates for rt in estimation and calibration. The 30-year Treasury yield is used in a robustness exercise. The source of monthly 3-month interest rate is DataStream International, Inc. To infer the interest-rate risk premium independent of the price observations on the market portfolio, we rely on a panel of Treasury yields. We choose Treasury securities with constant maturity of 6 months, 2 years, 5 years, and 10 years. The Treasury yield data is from the Federal Reserve Board. Table 1 shows that the average equity premium over the sample period is 8.76 percent and volatile. Although the average equity premium is somewhat higher than the 7 percent reported by Mehra and Prescott (1985, 2003), it is nonetheless of a similar order of magnitude. That the equity index provides a higher return relative to bonds is also a stylized feature over our shorter sample. The sample period is January 1982 to July 1998 with 199 monthly observations. The expected earnings-per-share growth for the S&P500 index, Gt , is the consensus earnings-per-share forecast for FY2 divided by FY1, minus 1. The price-to-earnings ratio, P/E, is the current S&P500 index level normalized by FY1 earnings per share. We report the average, the standard deviation, the maximum, and the minimum. The computation of the monthly equity premium is based on the 3-month interest rate. The earnings and price on S&P500 are collected from I/B/E/S, and the interest rates are from the Federal Reserve Board. Forward price-to-earnings ratio (the current price divided by FY1 earnings) has a sample average of 15.10, with a minimum price-to-earnings ratio of 7.28 and a maximum is 26.47. The average expected EPS growth for the S&P500 index is 10.13 percent and varies between 0.09 percent and 26.13 percent. Also reported are the average and standard deviation of the 3-month nominal interest rate, which are respectively 6.28 percent and 2.44 percent. TABLE 1 Equity Premium for S&P500 Index (January 1982 to July 1998) Average

Std.

Max.

Min.

Price-to-earnings ratio

15.10

4.13

26.47

7.28

Expected earnings growth

10.13%

5.31%

26.13%

0.09%

Interest rate

6.98%

2.13%

14.68%

Monthly equity premium

0.0073

0.040

0.162

5.68% −0.200

389


4. IMPLICATIONS OF THE MODEL FOR EQUITY PREMIUM The purpose of this section is twofold. First, we pursue a traditional risk-based explanation of the equity premium puzzle and present an estimation strategy aimed at recovering each of the three components of the equity premium in (21). That is, we estimate Πr , Πg , Πy , along with other model parameters, and judge empirical performance accordingly. Second, we quantitatively assess whether the risk premium parameterizations, interest-rate dynamics, and cash flow dynamics embedded in the valuation model are capable of generating a reasonably large equity premium. We conduct these tasks while simultaneously fitting the Treasury yield curve as close as possible. Hence, our approach circumvents the risk-free rate puzzle outlined in Weil (1989).

4.1. How Large Is the Interest-Rate Risk Premium? We first address the sign and magnitude of the interest-rate risk premium by using the Kalman filtering approach and a panel of Treasury bond yields. This approach (1) enables the estimation of the interest-rate risk premium jointly with the parameters of the interest-rate dynamics in (7) (i.e., κr , μ∗r , and σr ) and (2) allows us to test whether the interest-rate model is able to generate realistic yield curve movements. To implement this estimation procedure, we note that the transition equation for the instantaneous interest rate, rt , can be expressed as (e.g., Bergstrom (1984)) rt = μ∗r (1 − e−κr Δt ) + e−κr Δt rt−1 + ηt ,

(26)

where Et−1 [ηt ] = 0 and Et−1 [ηt2 ] = σr2 Δt, and ηt is a serially uncorrelated disturbance term that is distributed normal. Next, let Ψt = (Ψj,t , . . . , ΨJ,t ) be the month-t observed Treasury yields, where J denotes the number of yields employed in the estimation. As is standard from Babbs and Nowman (1999) and Chen and Scott (2003), the measurement equation describing observed Treasury yields is Ψt = Ut + Vt rt + υt ,

t = 1, . . . , T ,

(27)

where Ut is an N × 1 vector with ith element ξ[τi ]/τi , Vt is an N × 1 vector with ith element ς[τi ]/τi , and υt ∼ N (0, Ht ). The normality of υt and ηt allows us to implement a Kalman filter recursion based on the maximum-likelihood approach described in Harvey (1991). For this maximum-likelihood estimation, we select Treasury yields with maturity of 6 months, 2 years, 5 years, and 10 years and display the estimation results in Table 2. Panel A of this table shows that the interest-rate parameters are reasonable and the interest-rate risk premium is in line with economic theory.

390


TABLE 2 Interest-Rate Risk Premium Based on Kalman Filtering Estimation Panel A: Parameter estimates Parameter rt Process

κr

σr

μ∗r

Πr

Log-Lik

−0.0020

1804.93

0.2313

0.0128

0.0728

(0.0135)

(0.0008)

(0.0022)

(0.0005)

Panel B: Fitting errors for bonds 6 Months

2 Years

5 Years

10 Years

Median absolute pricing errors (bp)

37

25

35

50

Squared-root of mean squared errors (bp)

48

33

44

59

The reported parameters of the interest-rate process and the interest-rate risk premium are based on Kalman filtering. We specify the interest-rate process under the physical probability measure as drt = (κr μr − κr rt )dt + σr dWtr , and under the equivalent martingale measure as

˜ rt . drt = (κr μr − Πr − κr rt )dt + σr dW The estimation uses a monthly time series of Treasury yields with maturity of 6 months, 2 years, 5 years and 10 years. The asymptotic standard errors are in parentheses and based on the outer product of the log-likelihood function. The maximized log-likelihood function is reported as Log-Lik. Panel B reports the median absolute pricing errors (in bp) and the root-mean-squared pricing errors (in bp). Let us discuss these parameter estimates in turn. First, the long-run interest rate, μ∗r , is estimated at 7.28 percent and is of an order of magnitude similar to that reported in Babbs and Nowman (1999) and Chen and Scott (2003). Second, the estimated κr = 0.2313 implies a half-life of 2.99 years and indicates slow mean reversion of the interest-rate process. Third, the reported volatility of interest-rate changes, σr = 1.28 percent, suggests a relatively stable interest-rate process. Finally, the maximized loglikelihood value for the estimation is 1804.93, and the estimated parameters are several times larger than their standard errors, suggesting statistical significance. The estimated interest-rate risk premium, Πr , is, as we previously postulated, negative with a point estimate of −0.00201 (i.e., −20 basis points) and a standard error of 0.0005. Although the estimate appears quantitatively small, it can drive a substantial wedge between the risk-neutral and the physical interest-rate processes. To see this point more clearly, we compute μr = μ∗r − Πr /κr = 8.154 percent, which has the effect of raising the risk-neutral interest-rate drift by 86.9 basis points (hereafter, bp).

391


Intuitively, the risk factor Πr < 0 causes a heavier discounting of future cash flows and theoretically supports the presence of a positive equity premium as the partial derivative of Pt with respect to the interest rate is negative in (21). Bonds provide a hedge during periods of stock market declines, which justifies a negative interest-rate risk premium. We refer the reader to the related work of Buraschi and Jitsov (2005) on the inflation risk premium and Bakshi and Chen (1996b) on a general model of inflation and interest rates in a monetary economy. Goodness-of-fit statistics assessed in Panel B of Table 2 reveal that the interest-rate model provides reasonable fitting errors as measured by actual minus model-implied yield. Across the Treasury yield curve, the median absolute errors for 6-month, 2-year, 5-year, and 10-year yields are 37bp, 25bp, 35bp, and 50bp, respectively. In sum, the time series on the cross section of bond yields provide the desired flexibility in estimating the interest-rate risk premiums and the interest-rate parameters. Although there is scope for improvement, the pricing-kernel process can realistically mimic both the short and long ends of the yield curve through time.

4.2. Maximum-Likelihood Estimation of the (Physical) Gt Process The unavailability of contingent claims written directly on the Gt process precludes a joint estimation of the expected EPS growth processes in (5) and (24). We propose a two-step procedure to estimate Πg . First, we exploit the transition density function to estimate the structural parameters, Θg ≡ {κg , μ∗g , σg }, of the Gt process in (5). Second, taking Θg as given, we estimate Πg , along with other unknown parameters, based on the time series of S&P500 index (the criterion function is specified in Section 4.3), and consequently recover the risk-neutral Gt process in (24). Let {Gt : 1, . . . , T } be the monthly time series on expected earnings growth rate. The discrete equation corresponding to the Gt process in (5) is Gt = μ∗g + e−κg Gt−1 − μ∗g + ζt ,

(28)

where ζt is Gaussian mean zero and satisfies the condition E(ζt ζu ) = 0 for t = u, and E(ζt2 ) =

σg2 1 − e−κg . 2 κg

(29)

Guided by Nowman (1997), we construct the likelihood function as minus twice the logarithmic of the Gaussian likelihood function

max

κg ,μ∗g ,σg

T t=1

log

2 Gt − μ∗g − e−κg Gt−1 − μ∗g σg2 −κg 1−e + . 2 2 2κg σ /2κ (1 − e−κg ) g

g

(30)

392


Maximizing the log-likelihood function in (30) by the choice of Θg , we report the maximum-likelihood parameter estimates as (the standard errors are shown in parentheses) κg = 1.4401 μ∗g = 0.1024

(0.4411),

(31)

(0.0153),

(32)

σg = 0.0894

(0.0047),

(33)

with an average log-likelihood value of 2.29575. Several observations are relevant to our analysis. First, the point estimate of long-run expected earnings growth rate, μ∗g , is 10.04 percent, and close to the sample average documented in Table 1. Thus, analysts have been optimistic about the S&P500 index earnings growth. Second, the volatility of changes in the expected earnings-per-share growth, σg , is 8.94 percent, which is considerably more volatile than the interest-rate counterpart. Finally, according to the κg estimates, the S&P500 expected earnings growth rate is mean-reverting with a half-life, log(2)/κg , of 6 months. The duration of the expected earnings growth rate cycle is, thus, much shorter than the interestrate cycle and roughly consistent with stylized business cycle findings. Realizations of the physical Gt process are devoid of any information about the pricing measure, so the risk premium for expected earnings growth rate cannot be recovered through this estimation step.

4.3. Compensation for Cash Flow Risk and the Equity Premium To estimate the risk premium for expected EPS growth risk, Πg , and the risk premium for actual EPS growth, Πy , and assess their implications for the equity premium, we make several choices. First, to reduce the estimation burden, we preset ρg,y = 1, and ρ ≡ ρg,r = ρr,y . This assumption implies that the actual and expected EPS growth rates are subject to a common random shock in (4) and (5). Second, we set Θg and {κr , μ∗r , σr , Πr } to the values estimated in Section 4.2 and Table 2, respectively. Thus, we treat these parameter inputs as representing the true values. Substituting Θg and {κr , μ∗r , σr , Πr } into (14)–(19), we can see that five parameters, Θ ≡ {Πg , Πy , α, σy , ρ},

(34)

are still required to determine the price of the market portfolio, Pt , in (14). Observe that the valuation model for the market portfolio does not constitute a set of moment restrictions on asset prices; rather, it is an exact restriction on the price of the market portfolio in relation to the contemporaneous EPS, the expected EPS growth, and the interest rate. For this reason, the generalized method of moments and related econometric techniques may not be applicable.

393


Following the lead in fixed-income and option pricing, Θ is estimated using the time series of market prices. We use two estimation methods, one correcting, and the other not correcting, for the serial correlation of the model errors. Focusing on the first method, define from (14) the model price-to-earnings ratio as: &∞ Pt =α p[t, u; G, r]du, (35) pet ≡ Yt 0 and let pe ˜ t be the month-t observed price-to-earnings ratio. Our estimation procedure tries to find a Θ to solve / 0

2 T &∞ 01 1 (36) p[t, u; G, r]du − pe α RMSE ≡ min ˜t , Θ T 0 t=1

subject to the transversality condition in (19), μr − μg >

σr2 2κr2

−

σg2 σg σy ρg,y σr σy ρr,y σg σr ρg,r − − Πy + 2 + . κr κg κr κg 2κg

(37)

The transversality restriction ensures that Pt does not explode in each iteration of the minimization routine. The estimation method seeks to minimize the sum of squared errors between each observed price-to-earnings ratio and the model-determined priceto-earnings ratio. Fitting the price-to-earnings is desirable because Pt /Yt serves as a normalized price that is comparable across time periods. If the purpose would be to fit the observed price levels as closely as possible, the estimation procedure would then favor the higher price observations. The criterion function in (36) fails to account for the serial correlation of the model pricing errors. However, when we assume a first-order autoregressive process for the model error, the resulting estimates are similar. Hence, we omit them and focus on the least-squares method in (36). The optimized objective function value from (36), RMSE, is zero only if the obtained Θ estimate leads to a perfect fit of each market price-to-earnings by the model. In general, the average in-sample price-to-earnings pricing error will not be zero because the objective in (36) is to minimize the sum of squared errors, but not the average pricing errors. In our estimation approach, the estimated risk premiums and parameters reflect the historical valuation standards applied to the S&P500 index by investors. Panel A of Table 3 reports the parameter estimates of Θ when the 3-month Treasury rate is used as the proxy for rt . Consistent with how the market has priced the market portfolio in the past, the market-implied ρ is negative, with a ρ of −0.109. This mildly negative point estimate of ρ suggests that expected earnings growth rate is likely high when the interest rate is low, and vice versa. Estimation of the risk premiums is based on S&P500 index observations from January 1982 to July 1998 (199 observations). We minimize the distance between the

394


model price-to-earnings ratio and the market price-to-earnings ratio denoted by pe ˜ t: / 0

2 T &∞ 01 1 α RMSE ≡ min p[t, u; G, r]du − pe ˜t , Θ T 0 t=1

subject to the transversality condition μr − μ g >

σr2 2κr2

+

σg2 2κg2

+

σg σy ρg,y σr σy ρr,y σg σr ρg,r − − − Πy . κg κr κg κr

In this estimation, κr = 0.2313, σr = 0.0128, μ∗r = 0.0728, and λr = −0.00201, which are based on the results in Table 2, and ρg,y = 1, and ρ ≡ ρg,r = ρr,y . Parameters governing the dynamics of the expected earnings growth rate are fixed to κg = 1.4401, μ∗g = +∞ 0.1024, and σg = 0.089. We compute the model error t ≡ Yt (α 0 p[t, u; G, r]du − pe ˜ t) and report the average pricing errors and the average absolute pricing errors. The standard deviations are shown as Std(.). Each month we compute the model equity premium as +∞ μt − rt = Πy + Πg

0

p[t, u; G, r] × ϑ[u]du +∞ p[t, u; G, r]du 0

+∞

− Πr

0

p[t, u; G, r] × [u]du +∞ p[t, u; G, r]du 0

and report the sample average as Mean(μt − rt ), All calculations in Panel A are done using the 3-month Treasury rate as a proxy for the interest rate and repeated in Panel B using the 30-year Treasury rate. Another result worth emphasizing is that the dividend-payout ratio, α, is consistent with intuition: the estimated α = 0.41 does not depart substantially from the historical average payout ratios of 44.29 percent. Table 3 also provides the estimate of σy = 18.17 percent, with the conclusion that the cash flow process demonstrates high volatility. One central observation from Table 3 is that the market-implied expected-EPSgrowth risk premium, Πg = −0.145 percent, is surprisingly small relative to the marketimplied earnings risk premium, Πy = 6.531 + ∞the reported Πg , + ∞ percent. For example, implies that the sample average of Πg {( 0 p[t, u; G, r] × ϑ[u]du)/( 0 p[t, u; G, r]du)} is only 1bp. This finding indicates that accounting for the compensation for bearing expected-EPS-growth risk plays virtually no role in explaining the equity premium puzzle. If we accept the premise that the market fairly prices the S&P500 index and correctly reflects the market price of various risks, then our empirical findings have a straightforward interpretation: risk-averse agents may deem it unnecessary to “double-penalize” the physical drift of (Yt , Gt ) process. This may occur since Pt is homogenous of degree 1 in Yt and has a first-order impact on the stock price. Therefore, a large compensation in the form of Πy may make it unnecessary to require compensation for Gt risk.

395


TABLE 3 Estimation of Risk Premiums for Earnings Growth and Expected Earnings Growth Rate: Implications for Equity Premium Panel A: Estimation based on 3-month Treasury rate Πg 0.001450

Πy

α

σy

ρ

RMSE

Mean(t ) {Std(t )}

Mean(|t |) {Std(|t |)}

0.06531

0.4100

0.1817

−0.109

3.2293

−7.22% {23.98%}

18.30% {17.63}

Mean(μt − rt ) 7.312%

Panel B: Estimation based on 30-year Treasury yield Πg 0.001145

Πy

α

σy

ρ

RMSE

Mean(t ) {Std(t )}

Mean(|t |) {Std(|t |)}

0.06379

0.4744

0.1513

−0.074

3.1351

−7.62% {23.66%}

19.05% {15.92}

Mean(μt − rt ) 7.213%

˜ t ≡ Gt − λy . Therefore, we may write (23) To further explain our reasoning, define G ˜ t dt + σy dW˜ yt , where dG˜ t = (κg μ∗g − Πg − κg Πy − κg G˜ t )dt + and (24) as dYt /Yt = G ˜ gt . Thus, the presence of Πy reduces the level and drift +of∞the G˜ t process. σ g dW p[t, u; G, r] × [u]du)/ + ∞With Πr = −0.002, the sample average of −Πr {( 0 ( 0 p[t, u; G, r]du)} is 77.16 bp. This suggests that accounting for discounting risk can help alleviate the equity premium puzzle. Based on (21), the overall equity premium can, thus, be calculated as +∞ μt − rt = Πy + Πg

0

p[t, u; G, r] × ϑ[u]du +∞ p[t, u; G, r]du 0

+∞

− Πr

0

p[t, u; G, r] × [u]du , +∞ p[t, u; G, r]du 0

= 6.53% + 0.01% + 0.7716%, = 7.31%. The ability of this model to generate an equity premium of 7.31 percent is in sharp contrast with the exercise in Mehra and Prescott (1985) that a standard representative agent model calibrated to the per capita consumption data can generate, at most, a 0.40 percent, equity premium. Thus, the proper parameterization of both the discounting structure and the cash flow process is key to improving performance by an asset pricing model and to achieving a reasonable equity premium. Our exercise in Panel B of Table 3 demonstrates that the equity premium is virtually insensitive to the choice of the interest rate in the estimation procedure in (36). Another economic yardstick that can be applied is whether the estimated risk premiums and model parameters provide a “good enough” approximation of the

396


market’s implicit valuation process. In Table 3, we also present two percentage pricingerror measures, computed by dividing the market-to-model price difference by the market price: (1) the absolute percentage pricing error, and (2) the mean percentage pricing error. The mean pricing error reflects the average pricing performance, while the absolute pricing error reflects the magnitude of the pricing errors, as negative and positive errors do not cancel each other. According to the pricing-error measures, the model’s fit is reasonable: the average mean pricing error is −7.22 percent, with a standard deviation of 23.98 percent, and the absolute pricing error of the S&P500’s 18.30 percent. Given the negative sign of the average errors, the model price is, on average, higher than the market price. In summary, the class of models examined here is not only consistent with the average equity premium and the term structure of interest rates, but also mimics the time evolution of the S&P500 index. The latter dimension imposes a stringent restriction on the validity of the pricing framework and differentiates this chapter from other studies on the equity premium.

5. CONCLUDING REMARKS AND EXTENSIONS The equity premium puzzle advocated by Mehra and Prescott (1985) remains a fascinating problem awaiting new and novel answers. This paper investigated the impact of cash flow risk and discounting risk on the aggregate equity premium, the price of the market portfolio, and the default-free bond prices. Our theoretical approach is based on the observation that aggregate per capita consumption is hard to measure empirically. Thus, if we can replace the empirically difficult-to-estimate marginal utility by a pricing-kernel function of observables and then specify both the primitive process for discounting and the exogenous cash flow stream, we will have an equilibrium asset pricing model based on observable state variables. Once this is done, we can endogenously solve for the equity premium, the price of the market portfolio, and the term structure of interest rates within the same underlying equilibrium. Embedded in the closed-form solution for the market portfolio and the bond prices are compensations for cash flow risk and discounting risk. With the solution for the risk premium explicitly given, we can then estimate the model to evaluate its empirical performance. This approach allows us to avoid the impact of unobservable consumption on inferences regarding the model’s performance. Our illustrative model is based on the assumption that aggregate dividend equals a fixed fraction of aggregate earnings plus noise, and the expected aggregate earnings growth follows a mean-reverting stochastic process. Moreover, the economy-wide pricing kernel is chosen to be consistent with (1) a constant market price of aggregate risk and (2) a mean-reverting interest rate process with constant volatility. S&P500 index-based estimation results show that the framework is quantitatively useful in explaining the observed market equity premium. Specifically, we find that the interest-rate risk premium is negative and the cash flow risk premium is positive. Overall, disentangling the equity premium into its cash flow and discounting components produces an economically meaningful equity premium of 7.31 percent.


397

Our empirical results suggest three possible avenues for theoretical research. First, one can introduce richer cash flow dynamics and interest-rate dynamics that possess stochastic volatility. Having multidimensional structures for the state variables with priced volatility risks can lead to more realistic models for the market portfolio and the equity premium. Second, one can examine alternative risk premium specifications that allow for richer stochastic variation in the risk premiums. Third, the valuation model can be used to pin down the sources of market return predictability, as in Menzly, Santos, and Veronesi (2004). The equity premium puzzle occupies a special place in the theory of finance and economics, and more progress is needed to understand the spread of equities over bonds. Determining the factors that drive the equity premium over time, and across countries, will likely remain an active research agenda.

398


APPENDIX To derive the analytical solution to the market portfolio, we note from Eqs. (1) and (3) that Pt solves

&∞ Mu (38) Yu du, Et Pt = α Mt t since dZt is uncorrelated with dMt . We also require by the transversality condition that Pt < ∞ for all t, which is the condition that the price of the market portfolio remain bounded for all pricing kernel and cash flow processes. Inserting the pricing kernel process (6) into (38) and using the earnings process (4) and (5), we note, by the Markov property, that Pt can only be a function of Yt , rt , and Gt . Write P [Yt , Gt , rt ], where the interest-rate process is as specified in (7). Therefore, the dynamics of the market portfolio, by Ito’s lemma, is given by dPt =

∂P ∂P ∂P 1 ∂2 P 1 ∂2 P 1 ∂2 P 2 2 (dY ) + (dG) + (dr)2 + dY + dG + dr 2 2 2 2 ∂Y ∂Y 2 ∂G ∂G 2 ∂r ∂r +

∂2 P ∂2 P ∂2 P dYdG + dYdr + drdG. ∂Y ∂G ∂Y ∂r ∂G∂r

(39)

Substituting (39) into (2) implies that the instantaneous equity premium is μt − rt = −Covt = −Covt

dMt dPt . dt, , Mt P t

. dMt 1 ∂P 1 ∂P 1 ∂P , dt, dY + dG + dr Mt Pt ∂Y Pt ∂G Pt ∂r

(40)

where the instantaneous expected return is μt = Et[dPt /Pt ]/dt + (αYt /Pt ). Relying on (39) and taking expectations, we may obtain

dPt 1 1 ∂2 P 1 ∂P 1 1 ∂2 P 2 = E [(dY ) ] + [dY ] + Et [(dG)2 ] Et E t t Pt 2 Pt ∂Y 2 Pt ∂Y 2 Pt ∂G 2 +

1 ∂P 1 ∂P 1 1 ∂2 P Et [(dr)2 ] + Et [dG] + Et [dr] Pt ∂G 2 Pt ∂r 2 Pt ∂r

+

1 ∂2 P 1 ∂2 P 1 ∂2 P Et [dYdG] + Et [dYdr] + Et [drdG]. Pt ∂Y ∂G Pt ∂Y ∂r Pt ∂G∂r

(41)

399


Combining the expressions in (40) and (41) and using the definition of the instantaneous expected rate of return, we have 1 ∂2 P ∂P 1 ∂2 P ∂P Et [(dY )2 ] + Et [(dG)2 ] + Et [dY ] + Et [dG] 2 2 ∂Y ∂Y 2 ∂G 2 ∂G +

∂P ∂2 P 1 ∂2 P 2 E [(dr) ] + [dr] + E Et [dY dG] t t 2 ∂r 2 ∂r ∂Y ∂G

+

∂2 P ∂2 P Et [dY dr] + Et [drdG] − rP dt + αY dt ∂Y ∂r ∂G∂r = −Covt

dMt ∂P ∂P ∂P , dY + dG + dr . Mt ∂Y ∂G ∂r

(42)

Based on (42), now define the risk premium for the earnings shocks, expected earnings growth, and interest rate, respectively, as Πy ≡ −Covt

. dt,

. dMt dt, , dGt Mt

. dMt dt. , drt Mt

Πg ≡ −Covt

Πr ≡ −Covt

dMt dYt , Mt Y t

This immediately implies that 1 ∂2 P ∂P 1 ∂2 P ∂P Et [(dY )2 ] + Et [(dG)2 ] + Et [dY ] + Et [dG] 2 2 ∂Y ∂Y 2 ∂G 2 ∂G +

∂P ∂2 P 1 ∂2 P Et [(dr)2 ] + Et [dr] + Et [dY dG] 2 2 ∂r ∂r ∂Y ∂G

+

∂2 P ∂2 P Et [dY dr] + Et [drdG] − rP dt + αY dt ∂Y ∂r ∂G∂r =

∂P ∂P ∂P Y Πy dt + Πg dt + Πr dt. ∂Y ∂G ∂r

(43)

400


Simplifying this equation and using the dynamics for Yt , Gt and rt , leads to the following partial differential equation for Pt : 1 2 2 ∂2 P ∂P ∂2 P ∂2 P + (G − Πy )Y σy Y + ρg,y σy σg Y + ρr,y σy σr Y 2 2 ∂Y ∂Y ∂G ∂Y ∂r ∂Y + ρg,r σg σr

∂2 P ∂P 1 ∂2 P 1 ∂2 P + σr2 2 + κr (μr − r) + σg2 2 ∂G∂r 2 ∂r ∂r 2 ∂G

+ κg (μg − G)

∂P − rP + αY = 0 ∂G

(44)

and must be solved subject the restriction that Pt < ∞. In the valuation partial differential equation (44) we have set μg = μ∗g − (Πg /κg ) and μr ≡ μ∗r − (Πr /κr ). Consider the following candidate solution: &∞ p[t, ˆ u; Y , G, r]du. (45) Pt = α 0

Clearly, p[t ˆ + u, 0; Y , G, r] = Yt+u . Thus, we have the partial differential equation for p[t, ˆ u; Y , G, r] as (G − Πy )Y

∂2 pˆ ∂2 pˆ ∂pˆ + ρg,y σy σg Y + ρr,y σy σr Y ∂Y ∂Y ∂G ∂Y ∂r

+ ρg,r σg σr

∂2 pˆ ∂pˆ 1 2 ∂2 pˆ 1 ∂2 pˆ + σr2 2 + κr (μr − r) + σ ∂G∂r 2 ∂r ∂r 2 g ∂G 2

+ κg (μg − G)

∂pˆ ∂pˆ − r p¯ − = 0. ∂G ∂u

(46)

Suppose p[t, ˆ u; G, r] = Yt exp (ϕ[u] − [u]rt + ϑ[u]Gt ). Taking the required partial derivatives with respect to Yt , Gt , and rt and solving the valuation equations lead to a set of ordinary differential equations. Solving the ordinary differential equations subject to the boundary conditions ϕ[0] = 0, [0] = 0, and ϑ[0] = 0 yields (14) and (15). The transversality condition (19) ensures that the restriction ϕ[0] = 0 is satisfied.

References A¨ıt-Sahalia, Y., J. Parker, and M. Yogo. Luxury goods and the equity premium. Journal of Finance 59(6) (2004): 2959–3004. Babbs, S., and B. Nowman. Kalman filtering of generalized Vasicek term structure models. Journal of Financial and Quantitative Analysis 36 (1999): 115–130. Bakshi, G., and Z. Chen. The spirit of capitalism and stock market prices. American Economic Review 86(1) (1996a): 133–157. Bakshi, G., and Z. Chen. Inflation, asset prices and the term structure of interest rates in monetary economies. Review of Financial Studies 9(1) (1996b): 241–275.


401

Bakshi, G., and Z. Chen. An alternative valuation model for contingent claims. Journal of Financial Economics 44 (1997): 123–165. Bakshi, G., and Z. Chen. Stock valuation in dynamic economies. Journal of Financial Markets 8(2) (2005): 115–151. Barberis, N., M. Huang, and T. Santos. Prospect theory and asset prices. Quarterly Journal of Economics 116(1) (February) (2001): 1–53. Barsky, R., and J. Delong. Why does the stock market fluctuate? Quarterly Journal of Economics 108(2) (1993): 291–311. Benartzi, S., and R. Thaler. Myopic loss aversion and the equity premium puzzle. Quarterly Journal of Economics 110(1) (February) (1995): 73–92. Bergstrom, A. Continuous time stochastic models and issues of aggregation over time, in Z. Griliches, and M. Intriligator, eds., Handbook of Econometrics, Volume II. Elsevier Science, Amsterdam (1984). Breeden, D., M. Gibbons, and R. Litzenberger. Empirical test of the consumption-oriented CAPM. Journal of Finance 44(2) (1989): 231–262. Buraschi, A., and A. Jitsov. Time-varying inflation risk premia and the expectations hypothesis: A monetary model of the Treasury yield curve. Journal of Financial Economics 75(2) (2005): 429–490. Campbell, J., and J. Cochrane. By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107 (1999): 205–251. Chen, R., and L. Scott. Multi-factor Cox–Ingersoll–Ross models of the term structure: Estimates and tests from a Kalman filter. Mimeo, Rutgers University (2003). Constantinides, G. Habit formation: A resolution of the equity premium puzzle. Journal of Political Economy 98(3) (June) (1990): 519–543. Constantinides, G. A theory of the nominal term structure of interest rates. Review of Financial Studies 5 (1992): 531–552. Constantinides, G., J. Donaldson, and R. Mehra. Junior can’t borrow: A new perspective on the equity premium puzzle. Quarterly Journal of Economics 117(1) (2002): 269–296. Duffee, G. Term premia and interest rate forecasts in affine models. Journal of Finance 52(1) (2002): 405–443. Epstein, L., and S. Zin. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: An empirical analysis. Journal of Political Economy 99(2) (1991): 263–286. Ferson, W., and C. Harvey. Seasonality and consumption-based asset pricing. Journal of Finance 47(2) (1992): 511–552. Harvey, A. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press, New York (1991). Heaton, J., and D. Lucas. Evaluating the effects of incomplete markets on risk sharing and asset pricing. Journal of Political Economy 104(3) (1996): 443–487. Lettau, M., S. Ludvigson, and J. Wachter. The declining risk premium: What role does macroeconomic risk play? NBER #10270 (2004). Lintner, J. Distribution of incomes of corporations among dividends, retained earnings and taxes. American Economic Review 46 (1956): 97–113. Longstaff, F., and M. Piazzesi. Corporate earnings and the equity premium. Journal of Financial Economics 74(3) (2004): 401–421. Marsh, T., and R. Merton. Dividend behavior for the aggregate stock market. Journal of Business 60 (1987): 1–40. Mankiw, N. G. The equity premium and the concentration of aggregate shocks. Journal of Financial Economics 17(1) (September) (1986): 211–219. Mankiw, G., and S. Zeldes. The consumption of stockholders and nonstockholders. Journal of Financial Economics 29 (1991): 97–112. Mehra, R., and E. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15(2) (March) (1985): 145–161. Mehra, R., and E. Prescott. The equity premium in retrospect, in G. Constantinides, M. Harris, and R. Stulz, eds., Handbook of the Economics of Finance. North-Holland, Amsterdam (2003).

402

Chapter 9 • Cash Flow Risk, Discounting Risk, and the Equity Premium Puzzle Menzly, L., T. Santos, and P. Veronesi. Understanding predictability. Journal of Political Economy 112(1) (2004): 1–47. Nowman, K. Gaussian estimation of single-factor continuous time models of the term structure of interest rates. Journal of Finance 52(4) (1997): 1695–1706. Pastor, L., and R. Stambaugh. The equity premium and structural breaks. Journal of Finance 56 (2001): 1207–1239. Rietz, T. The equity risk premium: A solution. Journal of Monetary Economics 22(1) (July) (1988): 117–131. Vasicek, O. An equilibrium characterization of the term structure. Journal of Financial Economics 5 (1977): 177–188. Weil, P. The equity risk premium and the risk-free rate puzzle. Journal of Monetary Economics 24 (1989): 410–421.

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Vito D. Gala

Discussion: Cash Flow Risk, Discounting Risk, and the Equity Premium Puzzle Vito D. Gala∗ London Business School

1. DISCUSSION Aggregate consumption growth is far too smooth to generate the historically high equity premium with plausible assumptions about risk aversion. This is the equity premium puzzle originally documented in Mehra and Prescott (1985). Accounting for the historically low correlation between equity returns and aggregate consumption growth makes the puzzle worse: only with implausible large risk aversion might one reconcile the observed equity premium. In a growing economy with power utility investors, an implausible large coefficient of risk aversion is not a viable solution for the puzzle because it counterfactually leads to a high risk-free rate. Since the equity premium puzzle was identified by Mehra and Prescott (1985), considerable progress has been made in specifying investors’ marginal utility (see Cochrane (2006) for a detailed survey). In contrast, much less attention has been paid to the problem of modeling cash flows within an endowment equilibrium framework. Indeed, many papers in this literature constrain aggregate dividends to equal aggregate consumption. Important exceptions include Campbell (1986), Cecchetti et al. (1993), Campbell and Cochrane (1999), Abel (1999), Longstaff and Piazzesi (2004), Menzly et al. (2004), and Santos and Veronesi (2006). Bakshi and Chen (2005) also allow for aggregate dividends to differ from aggregate consumption. However, in contrast to the above-mentioned papers, their model only provides partial equilibrium results. Without any economic restriction between marginal utility and economic fundamentals, there is hardly any puzzle. The high equity premium ∗I

am grateful to Rajnish Mehra for his detailed comments.

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and the low risk-free rate are, literally speaking, no puzzles in their model: the dynamics for the pricing kernel, cash flow, and interest rate are specified exogenously. They develop a continuous-time, infinite horizon equity valuation model under the assumption of perfect markets. Aggregate dividend is a fixed proportion of aggregate earnings plus noise. The pricing kernel, M, is chosen to have a constant market price of aggregate risk and an interest rate dynamic consistent with a single-factor Vasicek (1977) model. As standard, the stock market value, P , equals the expected present value of future dividends, D: ∞ p(t, u; G, r)du, (1) Pt = αYt 0

where p can be interpreted as the time t price of a claim paying $1 at a future date t + u. Innovations to the interest rate, realized and expected earnings growth processes are correlated with innovations to the pricing kernel process. Thus, the equity premium results from a weighted average of the risk premia for the interest rate, earnings, and expected earnings shocks: dMt dPt P ,Y P ,G P ,r dt = Πy ξt + Πg ξt + Πr ξt , , (2) μt − rt = −Covt Mt P t where Π and ξ denote constant risk premia and time-varying price sensitivities to the respective state variables: earnings (Y ), expected earnings (G), and interest rate (r), respectively.

1.1. Calibration and Estimation Bakshi and Chen present an estimation strategy aimed at recovering each of the three components of the equity premium in (2), along with other model parameters, in order to assess whether the model can generate a reasonably large equity premium. First, they use a Kalman filtering approach based on a panel of Treasury bond yields with maturities of 1/2, 2, 5, and 10 years to estimate the interest rate risk premium jointly with the parameters of the interest rate dynamics. While the model can, on average, reasonably fit the yield curve as shown in the reported pricing errors, the choice of the Vasicek (1977) model comes with its own empirically counterfactual implications. The single-factor nature of the interest rate process makes it rather difficult to properly account for the observed movements and shapes in the yield curve such as hump and inverted hump shapes (see Piazzesi (2004)). Moreover, the model allows for negative nominal interest rates. However, under their current parameterization, nominal interest rates can be negative only with a probability of 0.006 percent.1 stationary distribution of the instantaneous interest rate is normal with mean μ∗r = 7.28 percent and ! standard deviation σr2 /2κr = (0.0128)2 /(2 × 0.2313) = 1.88 percent. Therefore, the probability that the

1 The

interest rate can reach negative values is Pr (r < 0) = 0.006 percent.

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Most importantly, the exogenous specification of the interest rate dynamics circumvents the risk-free rate puzzle outlined in Weil (1989). Although the three parameters controlling its law of motion, κr , μ∗r , and σr , are estimated to independently match the average yield curve, they provide a reasonable match with the historical mean and volatility of the short-term interest rate. The model yields mean and volatility of short-term interest rate equal to 7.28 percent and 1.88 percent versus 6.98 percent and 2.13 percent in historical data, respectively. The estimated interest rate risk premium Πr is only about −0.20 percent. In order to estimate the risk premia due to earnings (Πy ) and expected earnings shocks (Πg ), along with the other model parameters, they follow a two-step procedure. First, they directly estimate the parameters governing the dynamics of the expected earning growth process using analyst-expected EPS growth from I/B/E/S during the period January 1982 to July 1998. Second, taking the estimated parameters for the interest rate and expected growth rate dynamics as given, they estimate the risk premia and the other model parameters by minimizing the distance between the time series of the model implied and the actual price-earning ratios. Given the unavailability of traded contingent claims written directly on the expected earning process, they assume that earnings and expected earnings are perfectly correlated. This assumption allows them to identify the risk premium due to expected earnings shocks, Πg . Based on the estimated model parameters, they compute the model implied equity premium according to (2) as P ,Y P ,G P ,r + Πg × E ξt + Πr × E ξt E(μt − rt ) = Πy × E ξt = 6.53% × 1 + 0.14% × 0.07 − 0.20% × (−3.84) = 7.31%. Despite the high price sensitivity to interest rate, the low volatility of the short-term interest rate (σr = 1.28%) yields a small interest rate premium, which accounts for only about 10 percent of the overall equity premium. Similarly, despite the relatively high volatility of expected earnings growth (σg = 8.90%), the low implied risk premium and price sensitivity can only account for about 0.14 percent of model-implied equity premium. The linear homogeneity of the stock market value in aggregate earnings makes its price sensitivity identically equal to 1. As a result, the high volatility of the earnings growth (σy = 18.17%) generates a 6.53 percent earnings growth premium, which accounts for about 89 percent of the model-implied equity premium.

1.2. Where Is the Equity Premium Puzzle? With a volatility of 18.17 percent for the endowment process, there is no equity premium puzzle even in the equilibrium framework of Mehra and Prescott (1985). To see this, let us make the simplifying assumptions that Gt = g and rt = r. Then, Mt and Yt are geometric Brownian motions, and the stock market value is given by Pt =

αYt , r + Πy − g

(3)

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where Πy = −Cov(dM/M, dY /Y )/dt, and the implied equity premium can be computed as dM dP dt = Πy . (4) μ − r = −Cov , M P With power utility Πy = γ Cov(dY /Y , dY /Y ) /dt, and the equity premium equals μ − r = γσY2 .

(5)

Indeed, with i.i.d. endowment process and a representative agent with power utility over consumption, a coefficient of relative risk aversion γ = 0.088/0.1822 2.65 would be enough to generate a substantially high equity premium. Similarly, the high volatility of the endowment process would also circumvent the excess stock market volatility puzzle: dPt dPt αYt + dt − rdt = σ (6) = σy = 18.17%, σ Pt Pt Pt √ which is close to the reported historical value 4% × 12 = 13.86%. Therefore, the equity premium is not a puzzle in Bakshi and Chen since aggregate consumption growth volatility is substituted out by the volatility of aggregate earnings—about 19 times larger during the same sample period from 1982 to 1998— and the short-term interest rate is exogenously specified. The crucial assumption in their model is that aggregate dividends equal a fixed proportion of aggregate earnings.2 An important feature of this approach is that percentage changes in aggregate dividends are equal to percentage changes in earnings. Thus, the volatility and correlation properties of growth rates in dividends mirror those of aggregate earnings. With aggregate earnings being about 19 times more volatile than aggregate consumption, there is no equity premium puzzle. This approach is used in a number of recent papers such as Lee et al. (1999), Longstaff and Piazzesi (2004), Bakshi and Chen (2005), and others. Bakshi and Chen justify the computation of dividends as a constant proportion of aggregate earnings based on the idea that aggregate consumption is hard to measure. More generally, the rationale for the computation of dividends as a constant proportion of aggregate earnings stems from the well-known tendency of firms to artificially smooth their dividends over time, thus delinking reported dividends from actual corporate cash flows. Among others, Marsh and Merton (1987), DeAngelo et al. (1992), and Allen and Michaely (2002) document that firms tend to retain earnings during good times and pay dividends during bad times. In addition, when firms manage their dividends based on their signaling content or to resolve agency conflicts, the properties of reported dividends will most likely differ from the underlying firms’ cash flows. Furthermore, earnings-rather than dividends-based equity valuation models seem more 2 Since the zero-mean noise component is uncorrelated with the pricing kernel, it has no pricing implications. Hence, without loss of generality it can be omitted.

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407 appropriate in light of the empirical evidence that non-dividend-paying firms account for about one-quarter of the stock market value (see Fama and French (2001)). These considerations can provide a plausible motivation for using a measure of dividends that more closely reflects actual corporate cash flows. However, since the main results of the paper rest primarily on this assumption, it seems appropriate to address how quantitatively different the results would be under an alternative computation of aggregate dividends. For instance, a more appropriate measure of dividends would include cash dividends as well as share repurchases or corporate acquisitions (see Hall (2001)). Alternative measure of dividends affect stock market valuation not only through the volatility of their growth but also through their expected growth. Indeed, the timing of cash flows matters since it affects the “duration” of the stock market—price sensitivity to discount rate risk. Within a general equilibrium framework, the ideal measure for aggregate consumption would be stockholders’ consumption. As any agents, stockholders choose consumption streams, of which dividends represent only a portion. They hold the market portfolio; hence, the properties of their consumption stream directly affect asset prices. Mankiw and Zeldes (1991) report that 72.4 percent of the households in a representative sample from the 1984 Panel Study of Income Dynamics held no stocks at all. Even more surprisingly, only 47.7 percent of the households holding other liquid assets in excess of $100,000 held any equity. They also document a significant difference in the consumption pattern of stockholders versus non-stockholders. Specifically, the aggregate consumption of stockholders is more volatile and highly correlated with the equity risk premium than is the aggregate consumption of non-stockholders. It would be interesting to place the current valuation model within a general equilibrium framework where both the pricing kernel and the interest rate are endogenous. The historically high equity premium is not a puzzle per se. It only becomes a puzzle when compared to the properties of the endowment process and investors’ risk aversion.

References Abel, A. B. Risk premia and term premia in general equilibrium. Journal of Monetary Economics 43 (1999): 3–33. Allen, F., and R. Michaely. Payout policy. Working paper, The University of Pennsylvania, Wharton School of Business (2002). Bakshi, G., and Z. Chen. Stock valuation in dynamic economies. Journal of Financial Markets 8(2) (2005): 115–151. Campbell, J. Bond and stock returns in a simple exchange model. Quarterly Journal of Economics 101 (1986): 785–804. Campbell, J., and J. Cochrane. By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107 (1999): 205–251. Cecchetti, S. G., P.-S. Lam, and N. C. Mark. The equity premium and the risk-free rate: Matching the moments. Journal of Monetary Economics 31 (1993): 21–46. Cochrane, J. H. Financial markets and the real economy. The International Library of Critical Writings in Financial Economics, Richard Roll, ed. (2006). DeAngelo, H., L. DeAngelo, and D. J. Skinner. Dividends and losses. Journal of Finance 47 (1992): 1837– 1863.

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Chapter 9 • Cash Flow Risk, Discounting Risk, and the Equity Premium Puzzle Fama, E. F., and K. R. French. Disappearing dividends: Changing firm characteristics or increased reluctance to pay? Journal of Financial Economics 60 (2001): 3–43. Hall, R. E. The stock market and capital accumulation. American Economic Review 91 (2001). Lee, C., J. Myers, and B. Swaminathan. What is the intrinsic value of the Dow? Journal of Finance 54 (1999): 1693–1741. Longstaff, F., and M. Piazzesi. Corporate earnings and the equity premium. Journal of Financial Economics 74 (2004): 401–421. Mankiw, N. G., and S. P. Zeldes. The consumption of stockholders and non-stockholders. Journal of Financial Economics 29 (1991): 97–112. Marsh, T. A., and R. C. Merton. Dividend behavior for the aggregate stock market. Journal of Business 60 (1987): 1–40. Mehra, R., and E. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15(2) (1985): 145–161. Menzly, L., T. Santos, and P. Veronesi. Understanding predictability. Journal of Political Economy 112 (2004): 1–47. Piazzesi, M. Affine term structure models. Handbook of Financial Econometrics, Yacine Ait-Sahalia and Lars Peter Hansen, eds. (2004). Santos, T., and P. Veronesi. Labor income and predictable stock returns. Review of Financial Studies 19 (2006). Vasicek, O. An equilibrium characterization of the term structure. Journal of Financial Economics 5 (1977): 177–188. Weil, P. The equity risk premium and the risk-free rate puzzle. Journal of Monetary Economics 24 (1989): 410–421.

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Lior Menzly

Discussion: Cash Flow Risk, Discounting Risk, and the Equity Premium Puzzle Lior Menzly Proxima Alfa Investment

Abstract The two-stage calibration approach has long been the workhorse in evaluating the ability of models to match the observed equity premium. The approach calls for calibrating the model’s parameters using data that are not linked to the equity premium and then assessing the ability of the model to match the observed premium. In the following discussion, I demonstrate that calibrating a model to fit the dividend yield series, volatility of prices, or any price to fundamental ratio is likely to produce a large equity premium in a purely mechanical way. Put differently, the practice of calibrating the model to fit dividend yield is no different than matching the equity premium directly, and hence both approaches are subject to Roll’s critique in the same way.

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1. INTRODUCTION In a seminal paper, Mehra and Prescott (1985) demonstrated that a CRRA representative agent model produces a minuscule equity premium. This result has generated a variety of richer models all claiming to match the observed equity premium upon calibration. However, Roll (1974) has pointed out that it is always possible to construct a univariate linear factor model that, ex-post, prices all assets. It should come as no surprise that a model with enough degrees of freedom can match the equity premium if calibrated to do so. Thus, it is not clear whether the ability of these models to match the observed premium is a result of their economic merits or a mere manifestation of the added degrees of freedom. The critical question is how to evaluate a model’s ability to address the equity premium puzzle while avoiding the overfitting problem. Most papers employ a twostep approach: they first calibrate the model’s parameters using data that are not linked to the equity premium and then assess the ability of the model to match the observed equity premium. The limitation is that utility parameters, such as risk aversion, can only be indirectly estimated from asset prices. In the special cases, where the risk aversion is tied to the intertemporal marginal rate of substitution, the risk aversion parameter can be inferred from the risk-free rate. In most cases, however, price information, such as price to fundamentals or volatility of stock returns, has to be used to calibrate risk aversion as well as other parameters. By calibrating a model to match some form of price data, one may inadvertently force the model to match the equity premium as well. In the following discussion, I demonstrate that calibrating a model to fit the dividend yield series, volatility of prices, or any price to fundamental ratio is likely to produce a large equity premium in a purely mechanical way. Specifically, the practice of calibrating a model to fit the historical dividend yield inadvertently forces the model to match the simple historical average of the premium, and hence a model implied measure cannot be treated as an independent or a new estimate of the equity premium. To demonstrate this point, I will use the model developed by Bakshi and Chen (2006). The tractability and the flexibility of their model provide a suitable ground for the investigation. However, the concern raised in this article is not specific to their model; it applies equally to a large part of the literature. Moreover, the discussion does not bear on Bakshi and Chen’s (2006) main contribution that growth risk plays virtually no role in explaining the equity premium puzzle. The discussion proceeds as follows: Section 2 outlines the model. Section 3 describes the calibration procedure and the main results. Section 4 explores the mechanical link between calibrating the model to fit different moments of prices and matching the equity premium.

2. THE MODEL 2.1. Pricing Kernel The pricing kernel follows a general Ito process satisfying dMt = −rt dt − σm dWtm , Mt

(1)

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where drt = κr (μr − rt )dt + σr dWtr . Three modeling choices are embedded in the pricing kernel. First, the interest rate process is independent of the risk aversion and its parameters that can be separately calibrated to the observed Treasury yield curve (which leads to a single-factor Vasicek (1977) model). Second, the premium for aggregate risk is constant. Third, the model does not specify a factor to test (e.g., consumption growth), which provides further flexibility with the caveat that if one is free to choose any pricing kernel, one can always find a pricing kernel that prices all assets (ex-post).

2.2. Cash Flow Process Dividends are a fixed fraction of the earnings (plus some noise): Dt dt = αYt dt + dZt . Earnings growth follows the following process: dYt y = Gt dt + σy dWt , Yt g dGt = κg (μg − Gt )dt + σg dWt .

2.3. The Model—Solutions The model provides a closed-form solution for the price-to-earnings ratio and the equity premium. Price-earnings (p/e) ratio: Pt =α pet = Yt

∞ (2)

p [t, u; G, r]du, 0

where p [t, u; G, r] = exp(ϕ[u] − ζ[u]rt + ϑ[u]Gt ) and ϕ[u] , ζ[u] and ϑ[u] are provided in Bakshi and Chen (2006). Equity premium: ∞ μt − rt = Πy + Πg

0

p [t, u; G, r] × ϑ[u]du ∞ p u; G, r]du 0 [t,

∞

− Πr

0

p [t, u; G, r] × ζ[u]du ∞ p u; G, r]du 0 [t,

, (3)

where Πy , Πg , and Πr are constant premiums for cash flows, expected earnings growth, and interest rate risks, respectively. The model decomposes the equity premium into three components: (1) cash flow risk; (2) interest rate risk; and (3) expected growth rate risk.

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3. CALIBRATION The calibration of the model is based on a two-stage approach. The first stage calibrates the model to fit forward-looking earnings growth and the forward-looking price-toearnings ratio. The second stage estimates the forward-looking equity premium at each point in time given the forward-looking earnings growth and price-to-earnings ratios. This two-stage approach measures the premium in a way that imposes theoretical restrictions and employs information that is only indirectly related to the average return of stocks.

3.1. Calibrating the Model The goal here is to obtain an estimate of the equity premium that is based on valuation (p/e ratio) and fundamental (earnings growth) measures without using returns. The interest rate risk premium is separately estimated using a Kalman filter on a panel of Treasury bond yields. Structural parameters of the growth rate are estimated from the time series of analyst-expected EPS growth from the current to the next fiscal year. Cash flow and growth premiums along with other unknown parameters are estimated by minimizing the sum of squared errors between the time series of the model’s implied p/e ratio and of the observed p/e ratio: T 1

t )2 . (pet (Θ; Gt , rt ) − pe RMSE = min Θ T t=1

where Θ = {Πg , Πy , α, σy , ρ}. In the process of calibrating the model, two simplifying assumptions are made: ρy,g = 1, ρ = ρr,g = ρr,y ; shocks to cash flow and expected growth rate are fully correlated.

3.2. Estimation Results Two main results are the outcome of the calibration: first, plugging estimated parameters into the equity premium equation (3) yields an equity premium of 7.31 percent, meaning that the model matches the observed equity premium. Second, recall that the equity premium can be decomposed into three components: (1) cash flow risk; (2) interest rate risk; and (3) expected growth rate risk. Perhaps the most surprising result is that accounting for the compensation for bearing growth risk plays virtually no role in explaining the equity premium puzzle.

4. TWO-STAGE PROCEDURE—AN EMPIRICAL CONCERN It should come as no surprise that the calibration produces an estimate of the equity premium that is close to the average return. The argument is based on the fact that there

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is an almost exact relationship among dividend yield, dividend growth rate, and realized return through the use of Campbell and Shiller’s log-linearization: ln(Rt+1 ) = a + ln(d/pt ) − ρ ln(d/pt+1 ) + ln(dt+1 /dt ) , where ρ = 1/(1 + d/p) and a is a constant. This relationship can be further extended to incorporate the p/e ratio instead of dividend yield (see Vuolteenaho (2002)). This simple log-linearization of returns implies that any model that produces a perfect fit for p/e ratio or dividend yield will also match the observed history of returns, and hence its average return (or any other moments). However, most models do not have sufficient degrees of freedom to match the history of p/e. It is more likely that a calibrated model will fit the first moment and to some extent the second moment of the p/e ratio. Figure 1 shows that this is the case in Bakshi and Chen (2006), by plotting the implied p/e ratio in their model relative to the actual value. The figure demonstrates that the calibration will attempt to match the level of the p/e ratio and some long-term variation in prices.

Model Implied pe Ratio 30

25

p/e Ratio

20

15

10 Model Actual 5 1975

1980

1985

1990

1995

2000

Year

FIGURE 1

Actual price-to-earnings ratio compared to modeled price-to-earnings ratio.

2005

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However, a mechanical fit for the equity premium may still arise when calibrating the model to match just the level or the variation of the p/e ratio. In the case of the levels, it is sufficient to look at the Gordon formula, which is a special case of this model, as an approximation for such an outcome: E(pe) ≈

1 ⇒ π ≈ 8%. r+π−g

Plugging the average p/e ratio, r, and g, as calibrated by Bakshi and Chen (2006), into the formula yields an “equity premium” of around 8 percent. Similar analysis also applies in the case of matching the second moment of prices. Since the cash flow process is fairly smooth, price volatility has to come from variation in the valuation of future cash flow. To produce such variation, the pricing kernel has to be sufficiently volatile and correlated with shocks to the future growth rate. These conditions, in turn, imply a higher equity premium. This point is demonstrated by the Hansen and Jagannathan bound, where the price kernel variance provides an upper limit to the Sharpe ratio of the market portfolio: E RM ≤ std(mt ) Rf . std RM

5. CONCLUSION Using Bakshi and Chen (2006), I demonstrate that calibrating a model to fit the dividend yield series, volatility of prices, or any price to fundamental ratio may produce a large equity premium in a purely mechanical way. Put differently, the practice of calibrating the model to fit dividend yield is no different than matching the equity premium directly, and hence both approaches are subject to Roll’s critique in the same way.

References Bakshi, G., and Z. Chen. Cash flow risk, discounting risk, and the equity premium puzzle (2006). Mehra, R., and E. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15(2) (1985): 145–161. Roll, R. A critique of the asset pricing theory’s tests. Journal of Financial Economics 4 (1977): 129–176. Vasicek, O. A. An equilibrium characterization of the term structure. Journal of Financial Economics 5 (1977): 177–88. Vuolteenaho, T. What drives firm-level stock returns? Journal of Finance 57 (2002): 233–264.

CHAPTER 10

Distribution Risk and Equity Returns∗ Jean-Pierre Danthine Swiss Finance Institute, University of Lausanne and CEPR and

John B. Donaldson, and Paolo Siconolfi Columbia University

1. Introduction 2. The Business Cycle and the Labor Market 2.1. The Stylized Facts of the Business Cycle 2.2. The Labor Market 3. The Model Economy 3.1. Workers 3.2. Shareholders 3.3. The Firm 3.4. Equilibrium 3.5. Numerical Procedures and Calibration 4. An Economy with Distribution Risk Only 5. Adding Aggregate Uncertainty 6. Comparative Dynamics and Welfare Assessment 6.1. Changes in the Correlation of Productivity and Distribution Shocks 6.2. Changes in Risk Aversion and the Conditional Mean of the Distribution Shock 6.3. Other Comparative Dynamic Tests 6.4. Welfare Considerations 6.5. Explaining the Market Value to National Income Ratio

417 418 418 421 423 423 424 425 427 429 430 432 436 437 438 440 441 442

∗ We thank Elmar Mertens, an anonymous referee, the volume editor, Rajnish Mehra, and participants at the IBF-University of Lausanne brown bag seminar for their comments. Danthine’s research is carried out within the National Center of Competence in Research “Financial Valuation and Risk Management.” The National Centers of Competence in Research are managed by the Swiss National Science Foundation on behalf of the federal authorities. Donaldson and Siconolfi were assisted by the Faculty Research Fund of the Graduate School of Business, Columbia University.


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7. 8. 9. 10. 11.

Technology-Driven Variations in Factor Shares Robustness An Alternative Interpretation of the Sharing Mechanism Related Literature Concluding Comments References

443 446 448 452 459 460

Abstract In this chapter, we explore a class of models designed to explain both the stylized U.S. business cycle facts and the fundamental stylized facts of the U.S. financial markets. To illustrate this perspective, we entertain the hypothesis that observed variations in factor income shares are the result of changes in the balance of power between workers and capital owners in labor relations. We show that this view implies that income share variations represent a risk factor of first-order importance for the owners of capital and, consequently, are a crucial determinant of the return to equity. When both risks are calibrated to observations, this distribution risk dominates in importance the usual systematic risk for the pricing of assets. We also show that distribution risks may originate in non-traded idiosyncratic income shocks. This chapter illustrates the benefits of applying a macroeconomic perspective to the study of financial return phenomena. JEL Classification: E32, G12 Keywords: Income shares, Distribution risk, equity premium, limited market participation

Jean-Pierre Danthine, John B. Donaldson, and Paolo Siconolfi

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1. INTRODUCTION The objective of this chapter is to illustrate the challenges and achievements of applying macroeconomic modeling techniques to the understanding of financial market phenomena. It adopts the philosophy that the same internally consistent model should be expected to explain as broad a class of economic events as possible.1 This perspective simply acknowledges the fact that it is the same agents, consumer-worker-investors and firms, whose actions give rise to the entire spectrum of economic activity, both real and financial. A full-fledged model should thus be able to explain successfully both macroeconomic events and asset return patterns alike. This is a tall order. Indeed, one message of this volume is that the quantitative magnitude of the equity premium alone has yet to be adequately explained in any generally accepted context. It therefore behooves us to focus on seeking to explain a fairly narrow joint set of principal macroeconomic and financial stylized facts. The macroeconomic phenomenon most directly influencing the equity premium is the business cycle. This is the empirical observation that the rate of growth of an economy at times exceeds its long-run average (periods of “expansion”) and at other times falls short of it (“recessions”). Either of these circumstances can be quite persistent. Aggregate profit levels, furthermore, are profoundly sensitive to these macroeconomic states, growing robustly in times of expansion and declining substantially during periods of recession. If equity prices represent, as conventional financial theory suggests, the capitalized value of future profit streams, it is clear that business cycle variations will give rise to changes in stock prices, and thus to variation in equity returns (risk). Since most firms are affected by the business cycle similarly, business cycle risk is de facto a principal source of systematic risk. In addition, the risk-free rate on shortterm government bonds, by indirectly reflecting the demand for loans by firms (both the government and private firms compete for the same pool of private savings), is also sensitive to the cycle.2 We thus explore a particular production model with the objective of assessing the extent to which it is able to replicate, simultaneously, the stylized facts of the business cycle and the financial markets. Our efforts are intended to be illustrative of a broad class of models, detailed in Section 10, with this same focus. In particular, we emphasize how labor market compensation arrangements can profoundly influence the ability of an otherwise standard stochastic dynamic general equilibrium business cycle model to replicate basic financial market phenomena. An outline of the chapter is as follows. Section 2 presents a brief overview of the modern characterization of the business cycle while also providing both the intuition and empirical evidence underlying the labor market characterization. In particular, we argue for an economy in which firm owners insure workers under arrangements that lead to variation in factor shares. Section 3 then proposes a simple model of risk sharing 1An

approach identified with the notion of “cross-model verification” by the editor of this volume. See Mehra (2002), footnote 5. 2 The actions of the monetary authority to stabilize prices will influence the risk-free rate over the cycle as well.

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that illustrates this idea. Factor shares represent incomes to firm owners and workers. We interpret factor share variations as distribution risk. Section 4 reports the outcome of solving the model numerically and displays the impact of distribution risk in an economy with no other source of shocks. In Section 5, we add aggregate uncertainty. We show that the financial properties of the model are, to a large extent, determined by the characteristics of the income share shock, while the macroeconomic properties follow from the properties assumed for the aggregate technology shock. Section 6 discusses the relative importance of various parameters. Section 7 contrasts our economy with one where factor share variations are purely technology-driven. In Section 8, we explore the ability of the model to explain data variations across a variety of time periods. Section 9 offers an alternative motivation—non-traded idiosyncratic income shocks—for distribution risk. While the number of studies seeking to explain the financial stylized facts is already very large (see Kocherlakota (1996) and Mehra and Prescott (2003) for excellent surveys), few have focused purely on distribution considerations alone. We review the related theoretical literature in Section 10. Section 11 concludes the paper.

2. THE BUSINESS CYCLE AND THE LABOR MARKET 2.1. The Stylized Facts of the Business Cycle As noted in the introduction, we conceive of the business cycle as represented by deviations from trend behavior of the principal macro-aggregates (output, investment, consumption, etc.). These fluctuations about trend occur irregularly and with varying amplitude. There is, in addition, substantial co-movement in the cyclical aspects of the various series. Consumption, investment, and total employment, for example, all tend to vary in tandem with output along their respective growth paths. This latter fact is also part and parcel of the business cycle.3 In order to formalize these notions, it is necessary to employ a precise statistical methodology that decomposes an arbitrary macroeconomic time series xt into its growth and cyclical components: g

xt = xct + xt , where the obvious notation applies. This division of a macroeconomic time series into its constituent components is inevitably somewhat arbitrary. In any reasonable decomposition of a macroeconomic time series, however, we would expect the growth g component xt to represent the bulk of the original series and to be fairly smooth (lowfrequency fluctuations), since the factors affecting growth—education, capital stock growth, and technological innovation, to name but a few—change only very slowly. We would correspondingly expect the residual cyclical component to be relatively small but much more variable. 3 For a much more detailed discussion of the statistical representation of the business cycle, see Chapter 1 of Cooley (1995).

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Most modern business cycle studies employ the Hodrick–Prescott (1997) filter g methodology to undertake the separation. This filter defines the growth component xt of a time series xt as the solution to min g {xt }

T T g g g 2 g g 2 xt − x t + λ xt+1 − xt − xt − xt−1 , t=1

t=1

where the first term measures how closely the growth component tracks the original series, and the second is a measure of the smoothness of that same component. The parameter λ weights these two considerations, with λ = 1600 being the typically chosen calibration for quarterly data. With this choice, fluctuations with a frequency of less than 8 years (32 quarters) are eliminated from the growth component. Historical qualitative studies find that business cycles have typically been of three to five years in duration. Under the λ = 1600 assignment, these variations are assigned exclusively to the cyclical component. If this procedure is applied to the real GDP time series for the U.S. economy, Figure 1 provides a graphical representation of the result. Following standard statistical practice, the series is first logged so that it is the series {log GDPt ≡ Yt } that is being decomposed. Differences of the original {Yt } series from the trend line (growth component g {Yt }) define the cyclical component. The business cycle stylized facts are a set of statistics computed from the cyclical components of various major macroeconomic time series constructed as per above. In particular, the standard deviations and correlations with output are typically reported with the former capturing the average amplitude of the cyclical fluctuations, and the latter their general co-movement with output. A typical representation 12000

GDP HP-filtered GDP

10000 8000 6000 4000 2000 0 1940 FIGURE 1

1950

1960

1970

1980

1990

2000

2010

Log of real GDP and its growth component.

(Source: National Income and Product Accounts (NIPA) published by Bureau of Economic Analysis (BEA); Quarterly GDP, 1949–2004, in billions of chained 2000 dollars.)

420


TABLE 1 The Stylized Facts of the U.S. Business Cycle (Quarterly data; all series logged and subject to the H-P filter detrending procedurei ) (a)

(b)

(c)

Outputii

1.76

1.00

1.00

Consumptioniii

1.29

0.73

0.85

Investmentiv

8.60

4.88

0.92

Hoursv

1.66

0.94

0.76

Productivityvi

1.18

0.67

0.42

Capital stockvii

0.63

0.36

0.04

(a) SD (standard deviation) of the indicated cyclical component. (b) SD of cyclical component/SD cyclical component of output. (c) Correlation of the indicated cyclical component with the cyclical component of output. i Statistics compiled in Hansen (1985) corresponding to the period 55:3 through 84:1; Cooley and Prescott (1995) give a nearly identical summary for the period 1954.1 through 1991.2. ii Real GDP; the modifier “real” indicating that the series has been converted—via a GDP deflator—to 1972 dollar equivalents. iii Real total consumption expenditures. iv Real gross private domestic investment. v Total hours worked by persons working in the non-agricultural industries. vi Productivity is defined as real GDP/hours. vii The capital stock series is composed of non-residential equipment and structures.

of these stylized facts for the principal macro-aggregate time series is presented in Table 1. It is worthwhile to highlight a number of important regularities. Most apparently, investment is much more volatile than output at cyclical frequencies (nearly five times as much) while consumption is considerably less so.4 Hours are nearly as volatile as output, and more volatile than wages. Of all the series, capital stock is the least volatile, not surprisingly, as it represents the accumulation of many past investments. In addition, all series except capital strongly co-move with output (we say they are procyclical) with this especially being the case for investment, hours, and consumption. Furthermore, these patterns apply to a wide selection of data periods.5 A successful model of the U.S. business must, at a minimum, replicate the patterns evident in Table 1. In what follows, we illustrate this process in the context of a particular business cycle cum financial sector macro model. 4 The absolute volatility of output and consumption has, however, declined considerably since 1983. See, e.g., Jaimovich and Rebelo (2006). 5 They are less robust when computed across countries, however. See Danthine and Donaldson (1992).


421

2.2. The Labor Market Our model highlights the role of the labor market in replicating financial return data. Broadly speaking, labor market institutions and arrangements might be expected, first and foremost, to influence equity cash flows and returns. Since wage payments represent a claim against the firm’s cash flow of priority exceeding that of interest and principal payments, they effectively constitute an additional form of leverage with the well-known consequence of influencing equity return risk. Under a purely competitive labor market and Cobb–Douglas technology, however, aggregate wage payments vary directly with output: Nt Wt = Nt MPNt = (1 − α)Yt = (1−α) (factor shares are constant; Wt denotes the period t wage, while (1 − α)KtαNt Kt and Wt designate period t capital and labor, respectively, with MP standing for marginal product). Such an arrangement is very accommodating to firms as their wage bills will then expand and contract exactly in tandem with output (revenues); that is, with their ability to pay. If, however, the wage bill were to vary less than output, falling proportionately less in recessions and increasing less during expansions, then the risk of the firm’s free cash flow and derived dividends is likely to increase substantially. This phenomenon would prevail if firm shareholders were willing to insure workers against the extremes of competitive income determination in exchange for paying a lower wage on average. With stockholders likely to be more highly diversified than workers—and thus better equipped to tolerate the risk—such a voluntary arrangement makes sense. Alternatively, it might be forced on shareholders and the firms representing them by labor unions, political actions, or strong social conventions. In any case, consumption variation of the shareholder class is likely to increase as a result, with the attendant greater preference for risk free assets vis-à-vis risky ones. Relative to a purely competitive scenario, a higher equity premium should result. Circumstantial evidence in favor of the risk sharing story is factor share variation. In all developed economies, capital and labor income shares display persistent variations over time. For the U.S. (see Figure 2), the variability of the (log) wage share over the postwar period is approximately 3.45 percent on an annual basis, with a first-order autocorrelation larger than 0.97. Such variations are incompatible with the hypothesis of Walrasian labor markets, at least under standard assumptions on the aggregate technology.6 In what follows, we entertain the hypothesis that the observed variations in income shares are the result of infrequent changes in the balance of power between workers and capital owners in labor relations. The overall context is one of non-competitive labor markets where political (partially via taxes on labor and capital) and social forces (notably via the action of trade unions) influence the sharing of value added between capital and labor. We show that this view implies that income share variations represent a risk factor of first-order importance for the owners of capital and, consequently, are a crucial determinant of the return to equity. We demonstrate that, when both risks 6 In this sense, these variations falsify most existing business cycle models, which typically assume a constant returns to scale technology. We pursue a purely technological explanation for income share variations in Section 7.

422

Chapter 10 • Distribution Risk and Equity Returns 0.64

0.62

0.6

0.58

0.56

0.54

19

59 19 .1 60 19 .1 61 19 .1 62 19 .1 63 19 .1 64 19 .1 65 19 .1 66 19 .1 67 19 .1 68 19 .1 69 19 .1 70 19 .1 71 19 .1 72 19 .1 73 19 .1 74 19 .1 75 19 .1 76 19 .1 77 19 .1 78 19 .1 79 19 .1 80 19 .1 80 19 .1 81 19 .1 82 19 .1 83 19 .1 84 19 .1 85 19 .1 86 19 .1 87 19 .1 88 19 .1 89 19 .1 90 19 .1 91 19 .1 92 19 .1 93 19 .1 94 19 .1 95 19 .1 96 19 .1 97 19 .1 98 19 .1 99 20 .1 00 .1

0.52

FIGURE 2

U.S. labor share 1959–2000.

(Source: National Income and Product Accounts (NIPA) published by Bureau of Economic Analysis (BEA); U.S. labor share is defined as compensation of employees (deflated by the consumption expenditures index)/real GDP.)

are calibrated to observations (including the statistical properties of income shares), what may be called distribution risk dominates in importance the usual systematic risk (arising from economy-wide productivity shocks) for the pricing of financial assets. We make our point in the standard, separable utility infinite-horizon, production economy paradigm characteristic of many studies of macroeconomic and financial equilibrium. In doing so, our objective is to emphasize the power of the modeling philosophy advocated in this chapter. Our economy is one with two classes of agents: shareholders and workers. It has two distinguishing features. First, relative to a standard business cycle model, our economy features an additional source of uncertainty, resulting in variations in the shares of income going to capital and labor. This risk is assumed to be uninsurable. Second, there is limited financial market participation: workers do not trade financial assets. Labor market arrangements, however, are such that the distribution of income risk in our economy is period-by-period Pareto optimal: the allocation of resources conforms to the maximization of a social welfare function under aggregate feasibility constraints although the welfare weights in the social welfare function are time-varying. Our specific modeling of labor relations is borrowed from Danthine and Donaldson (2002). The equivalent central planning formulation may, however, be interpreted as summarizing a broader class of labor market arrangements. Section 8 develops such an alternative interpretation. We study the extent to which these considerations allow the model to replicate the basic financial stylized facts. These include not only the mean equity and risk-free returns (and thus the equity premium; cf. Mehra and Prescott (1985)), but also their respective standard deviations and correlations with aggregate consumption growth. Everywhere the standard Lucas (1978) asset pricing methodology is employed. Consistent with our modeling philosophy, we also examine the extent to which the aggregate

423


variations attendant to the uncertainty in factor shares are consistent with the observed properties of the business cycle.

3. THE MODEL ECONOMY Imagine an economy with two agents, a worker and a rentier shareholder/bondholder. The worker’s utility is denoted by v(W ), where W stands for his consumption level; the shareholder’s utility is u(C), with C being his consumption level. The worker inelastically supplies one unit of labor. The capital stock is K, the investment I, the per-period rate of capital depreciation Ω, and the production function f (·)λt , with λt a stochastic shock to the technology. The aggregate constraints of this economy are standard: Ct + Wt + It ≤ f (Kt , 1)λt , Kt+1 = (1 − Ω)Kt + It ,

K0 given.

Now assume there is a benevolent central planner maximizing a weighted sum of the two agents’ utilities given these aggregate constraints with the distinguishing feature that the welfare weight μ is a random variable whose stochastic process is known but exogenously given to the central planner: max E0

{Ct ,It ,Wt }

∞

[μ˜ t v(Wt ) + u(Ct )]

.

(1)

t=0

This is the economy we will investigate. In effect, we interpret the observed variations in factor income shares as the result of a bargaining process between capitalists and workers. We will show that such an economy not only rationalizes the evolution of income shares but also possesses many features of the actual macroeconomy. Adding the hypothesis that only one class of agents—shareholders—prices the outstanding financial assets, we will show that the implied distribution risk is of first-order significance for the properties of the equity market. In the section that follows, we propose a specific decentralized interpretation for such an economy, borrowed from Danthine and Donaldson (2002).

3.1. Workers We postulate a continuum of workers distributed on [0, 1] who each supplies one unit of labor inelastically and consumes his wage. This extreme assumption forces any worker income smoothing activities to remain in the context of their employment relationship with the firm. Workers are viewed, somewhat paternalistically, as permanent members of the firm with which they have a lifetime employment relationship. The essence of the employment contract is that workers supply one unit of labor in exchange for a wage that is substantially less variable than their marginal productivity. Note that, by

424


the nature of the contract to be analyzed, a worker’s income risk cannot be decreased by his participation in the financial markets. Accordingly, workers solve ∞

max E0 β t v ctw w w c ,n { t t } t=0

subject to

ctw ≤ wt nwt ,

nwt ≤ 1.

(2)

In the above problem, v(·) denotes a representative worker’s period utility function, β his subjective discount factor, and ctw his period t consumption; wt is his period t wage and nwt his period t labor supply. Conforming to notational custom, in this and all other problem formulations, Et stands for the period t expectations operator under rational expectations.7 The solution to this problem is particularly simple: ctw = wt , nwt = 1. That is, workers consume their wages and work their full-time endowment.

3.2. Shareholders A continuum of shareholders indexed to the unit interval is also assumed. Shareholders are rentiers; they consume their dividend and interest income payments and trade securities in the financial markets. They own all the securities “traded” in the economy and do not supply labor services.8 Their problem is max E0

{ct ,zt ,bt }

∞

β t u(ct )

subject to

ct + qte zt+1 + qtb bt+1 ≤ qte + dt zt + bt .

(3)

t=0

In this decision problem, u(·) denotes the period utility function of the representative shareholder and ct his period t consumption. For simplicity, the shareholder’s subjective discount factor β coincides with that of the worker. The shareholder’s decision variables, zt and bt , denote, respectively, his period t stockholdings and bondholdings. The corresponding period t prices of these securities are qte and qtb , the former with an associated period t dividend payment dt . The bonds considered are one-period discount 7 We assume all relevant information is public knowledge. For ease of exposition, we specify the process on the relevant state variables, dF (·), when we spell out the problem of the firm. The expectations operators at the individual worker and shareholder levels apply to stochastic processes—on wages, dividends, and consumption—that are direct transformations of dF (·). 8 Note that this assumption together with the assumption that workers do not trade financial assets imply a perfect coincidence between the shares of income going to workers and shareholders, respectively, and the labor and capital factor shares. This is an important distinguishing feature of our economy.

425


bonds (paying one unit of consumption with certainty after one period). Postulating an artificial economy with a measure of one for both shareholders and workers is intended to reflect the empirical fact that approximately one-half of U.S. households own stock either directly or indirectly (as pension assets).9 Under standard concavity and differentiability assumptions, the necessary and sufficient first-order equations for problem (3) are e

+ dt+1 , u1 (ct ) qte = βEt u1 (ct+1 ) qt+1 u1 (ct ) qtb

= βEt {u1 (ct+1 )}.

(4) (5)

Equation (4) has the unique non-explosive solution:

qte = Et

∞ j=1

βj

u1 (ct+j ) dt+j . u1 (ct )

(6)

3.3. The Firm For notational simplicity, there is one firm that behaves competitively and lives forever. The capital structure of this firm is composed of one perfectly divisible share and b one-period default-free bonds.10 With homogeneous firm owners, the firm’s objective clearly is to maximize its predividend stock market value, dt + qte , on a period-by-period basis. The key decision variable is the level of investment it , given that a long-term labor contract defines the relation between the firm and its workers. The latter stipulates that in exchange for delivering one unit of labor per period for their lifetime, workers receive a wage income that corresponds to an optimal risk sharing arrangement with firm owners. The firm’s decision problem may thus be represented as 9 A counterpoint to this observation is the reality that only 10 percent of U.S. households own roughly 90 percent of these same equities (Survey of Consumer Finance Data as reported, e.g., in Guo (2001)). This wealthy 10 percent is more likely to be the subset of the population receiving substantially all their income in the form of dividends and capital gains. If we were to assign measures νs = 0.10 to shareholders and νw = 0.90 to workers (non-shareholders essentially), problem (1) would be reformulated as ∞ νw max E0 μ˜ t v(Wt )) + u(Ct ) {Ct ,It ,Wt } νs t=0

subject to νs Ct + νw Wt + It ≤ f (Kt , 1)λt , Kt+1 = (1 − Ω)Kt + It , K0 given.

(1)

This problem is not materially different from the one we study. If, however, as we will subsequently assert, capital income represents 30 percent of total income, shareholders under (1) would enjoy very large consumption, a feature that would make them less risk averse at the margin. 10 With our focus on multiperiod labor market arrangements and the presence of debt in the firm’s capital structure, it would be inconsistent to admit a one-period firm, as is commonly the case in this literature.

426


max dt +

{it ,nt }

qte

≡ dt + Et

u1 (ct+1 ) e β qt+1 + dt+1 u1 (ct )

(7)

subject to dt = f (kt , nt )λ˜t − nt wt − it − b + bqtb , kt+1 = (1 − Ω) kt + it ; k0 given, nt = 1, μ˜ t v1 (wt ) = u1 (ct ),

(8) (9) (10)

dF (λt+1 , μt+1 ; λt , μt ) and initial values λ0 , μ0 given. In Eq. (8), f (·) is the firm’s (increasing, concave, and differentiable) production technology, λt is an aggregate shock to productivity, nt denotes the period t level of hours (employment) engaged by the firm, kt is the period t capital stock (with depreciation rate Ω), and it its period t investment. Dividends, dt , are output less the aggregate wage bill (nt wt ), the net interest payment (b − bqtb ), and the level of investment. Equation (9) is the standard equation of motion on the firm’s capital stock. Constraint (10) summarizes the terms of the contract. It is one designed to effect optimal risk sharing between workers and firm owners on a period-by-period basis, that is, once μt has been determined. As it is the only non-standard element in this arrangement, we postulate that a new value μt is drawn at the end of each period (that is, at the end of t − 1 for μt ). This new value of μ applies for the next period and is taken as a given by both parties. We mean this to represent the hypothesis that the relative weight of the two classes of agents is determined in a process that, to some extent, escapes the economic sphere. We thus take the view that the low-frequency movements in the wage share are the outcome of interactions taking place at the social and political levels and that we do not attempt to model.11 When at the negotiating table, however, economic agents make sure that income risk is efficiently allocated, taking that reality as a given. We capture the uncertainties inherent in this process by postulating that μ˜ t follows an exogenously specified stochastic process. The joint conditional density of (λ˜ t , μ˜ t ) is given by dF (λt+1 , μt+1 ; λt , μt ). The parameter μt determines not only the average consumption shares going to the agents in this economy but also their relative variability. The calibration of the stochastic process governing the value of μt will be guided by the desire to replicate the variation in income shares observed in the U.S. economy. The necessary and sufficient first-order conditions for the firm’s problem are u1 (ct ) = βEt {u1 (ct+1 ) [f1 (kt+1 , nt+1 )λt+1 + (1 − Ω)]}, μt v1 (wt ) = u1(ct ), nt = 1.

11 Political

(11)

elections, in particular, may profoundly affect income sharing possibilities. Think of the impact of Ms. Thatcher on the U.K. economy!

427


3.4. Equilibrium Let capital letters denote equilibrium quantities; market clearing then requires that the following conditions be satisfied. In all cases, the variable θ indexes the continua of economic participants: nt = 1

ctw dθ =

0

1

1 0 1

nwt dθ =

1

1 dθ ≡ 1,

(12)

0

wt dθ ≡ Wt ,

0

ct dθ ≡ Ct ,

0

zt = b=

1 0 1

zt dθ = 1, b dθ = B,

0

k t ≡ Kt

and

it ≡ It .

In equilibrium, aggregate variables are related according to Yt = f (Kt , 1)λt = Ct + Wt + It .

(13)

Total consumption Ct + Wt will be later labeled T Ct . In this economy, workers are explicitly (though not implicitly) passive, and the shareholder’s activities have the effect only of pricing financial assets after receiving their dividend and bond income. Equilibrium is thus characterized by imposing the relevant market clearing conditions on the optimality conditions of the firm. This yields u1 (C(st )) = β u1 (C(st+1 ))[f (Kt+1 , 1)λt+1 + (1 − Ω)]dF (·), (14) μt v1 (W (st )) = u1 (C(st )),

(15)

where we explicitly recognize the dependence of all the equilibrium quantities on the economy’s state variables st ≡ (Kt , λt , μt ). More formally, we define equilibrium as follows: Definition: Equilibrium for the economy defined in Eqs. (2)–(7) is a triple of functions C(st ) , W (st ), and I (st ) that jointly satisfy Eqs. (13), (14), and (15). It may be observed in particular that Eqs. (14) and (15) are the necessary and sufficient first-order conditions for the functional equation J (st ) =

max {μt v [f (Kt , 1)λt − C(st ) − I (st )] + u(C(st )) + β J [(1 − Ω)Kt + I (st ), λt+1 , μt+1 ] dF (·)},

C(st ),I (st )

(16)

428


which, in turn, is the functional equation for the central planning formulation of our model: J (s0 ) = subject to

max

{C(st ),I (st ),W (st )}

E0

∞

β t [μt v (W (st )) + u(C(st ))]

(17)

t=0

C(st ) + I (st ) + W (st ) ≤ f (Kt , 1)λt , Kt+1 = (1 − Ω)Kt + It , K0 given, λ0 , μ0 , dF (·) specified a priori.

Representation (17), which is nothing else than problem (1), confirms that all mutually advantageous exchanges of risks between workers and shareholders are affected, but for the impossibility of their contracting ex ante against future idiosyncratic income variation as their respective bargaining powers wax and wane.12 In this economy, the standard Lucas (1978) asset pricing methodology is employed to value financial assets; in particular, given the shareholders’ equilibrium consumption function, C(st ), the equity and risk-free debt security prices are computed as the solution to the following equations, respectively: q e (st ) = β

u1 (C(st+1 )) e [q (st+1 ) + d(st+1 )]dF (·) u1 (C(st ))

(18)

and q (st ) = β b

u1 (C(st+1 )) dF (·), u1 (C(st ))

(19)

where d(st ), the aggregate dividend, satisfies d(st ) = f (Kt , 1)λt − W (st ) − I (st ) − B + q b (st )B.

(20)

12 Problem (16) is the recursive equivalent to sequence problem (17). It asserts that (17) may be approached from a period-by-period decision perspective. The expression J (st ) represents the state contingent aggregate welfare achievable under the optimal policy rules. With this identification, (16) asserts that the optimal time-invariant policy rules may be derived by maximizing utility today, μt v(Wt ) + u(Ct ) conditional on the understanding that the optimal policies will be followed in the future with the resultant expected welfare β∫ J (Kt+1 , λt+1 , μt+1 )dF (·). As such, (16) is an application of the principle of optimality and constitutes the dynamic programming formulation of problem (17). These solution methods are discussed at great length in Stokey, Lucas, and Prescott (1989).

429


Lastly, the period-by-period returns to the ownership of these securities are constructed according to e 1 + rt,t+1 =

q e (st+1 ) + d(st+1 ) . q e (st )

(21)

1 . q b (st )

(22)

and b = 1 + rt,t+1

3.5. Numerical Procedures and Calibration Using standard discrete state-space methodologies, we solve expression (16) by value function iteration (see Christiano (1990)) to obtain accurate approximations to the economy’s optimal policy functions C(st ) and I (st ). Once the values of capital stock that identify the economy’s long-run stationary distribution and the equilibrium consumption and dividend functions have been identified, solving (18) and (19) to obtain the equilibrium stock and bond prices amounts to solving a system of linear equations. All financial and business cycle statistics are computed on the basis of artificially constructed return series of 400,001 periods in length. In all cases, the shareholder’s period utility function is hypothesized to be logarithmic, u(C) = ln(C), while the period utility function of the representative worker is postulated as v(W ) = W 1−γ /(1 − γ), with γ assuming a variety of values.13 The production function common to all simulation runs is the customary f (K, 1) = LKtα , with L a scale parameter. The parameter α is typically calibrated to reproduce the observed share of capital in total value added. Estimates on this number vary: the most commonly used value is 0.36. Cooley and Prescott (1995) justify a higher value of 0.4 by the inclusion of a measure of imputed income for government capital. Gollin (2002) argues that this number is likely to be too high because of improperly accounting, as capital income, the labor income of the self-employed. The various adjustments he proposes to correct for this leads him to estimating the 1992 U.S. capital income share in a range [0.23–0.34] (instead of the 0.4 obtained for that particular year under a more standard calculation). In our economy, income shares are jointly determined by the value of the parameter α, the average value of μ, and the risk aversion parameter γ. Our basic scenario will be one where the capital share is 0.3 (α = 0.3) and where {μt } is such that the average wage share matches its competitive counterpart. We then study the impact of alternative values of μ that represent circumstances where workers are effectively 13 A

justification for these choices is as follows. Models of the type considered here afford the shareholder substantial opportunity to smooth out his consumption streams by manipulating the investment plan. By the risk sharing requirement, wages will end up being similarly smooth. As a result, the equilibrium consumption series runs the risk of being too smooth relative to the data. If the shareholder were postulated to be more risk-averse than log (say with a CRRA of two or three), this model shortcoming would only be aggravated. The assumption of log utility for the agent undertaking the economy’s investment decision is thus typical of this model class.

430


paying a premium for income insurance provided by firm owners and thus receive a lower average share of value added. Lastly, we make two distinct assumptions for the shock process on the share parameter. We first explore the implications of an economy with no technology shock (λt ≡ 1), where μt is assumed to follow a two-state Markov chain; that is,

dF (μt+1 ; μt ):

μ1 μ2

μ1 π 1−π

μ2 1−π . π

(23)

In this case, μ1 , μ2 , and π are chosen to yield a process on factor shares that reasonably corresponds to the data. We then complete the model by allowing for aggregate uncertainty as well. A joint stochastic process on the risk sharing parameter μ and the multiplicative technology shock parameter λ is assumed. Together they follow a four-state Markov chain with transition matrix:

dF (λt+1 , μt+1 ; μt , λt ):

(λ1 , μ1 ) ⎡ ψ (λ1 , μ1 ) (λ1 , μ2 ) ⎢ π + Δ (λ2 , μ1 ) ⎣ σ H (λ2 , μ2 )

(λ1 , μ2 ) π ψ −Δ H σ

(λ2 , μ1 ) σ H ψ −Δ π

(λ2 , μ2 ) ⎤ H σ ⎥. π+Δ ⎦ ψ (24)

This choice of matrix admits a high degree of flexibility. The parameters λ1 , μ1 , λ2 , μ2 , ψ, and σ principally determine the behavior of the technology shock, and their values are chosen so that the economy’s equilibrium output series, when subjected to the Hodrick–Prescott (HP) detrending procedures, matches the standard deviation of output for the U.S. economy as well as its first-order autocorrelation. The remaining parameters are selected to replicate the time-series properties of the wage share. The targeted income share data can be described as follows. Let Yt denote output in period t. For the period 1947.1–1998.1, the behavior of the wage share through time is well approximated by a first-order autoregressive process with corr[ln(W /Y )t , ln(W /Y )t−1 ] = 0.974, SD(ln(W /Y )t ) = 3.45 percent, and corr[ln (W /Y )t , YtHP ] = −0.053. The average value of the wage share in this data series is 0.57. As explained before, this value needs to be adjusted for mis-measurement problems. We use a benchmark value of 0.7. All statistics apply to U.S. data.

4. AN ECONOMY WITH DISTRIBUTION RISK ONLY We first study the properties of our economy in the absence of technology shocks, i.e., with distribution shocks exclusively. The corresponding transition matrix is (23). Table 2 presents a sampling of results for one possible (μ1 , μ2 ) pair; analogous results for the U.S. economy and for a classic pure real business cycle (RBC) study (Hansen (1985)) are provided for comparison purposes.

431


TABLE 2 The Benchmark Case: Pure Distribution Risk Panel A: Financial and wage share statistics U.S. Dataiii

(a) re

6.98

rf

0.80

rp

6.18

dt+1 /dt W /Y

— 57

Hansen’s (1985) model

(b)

(c)

(a)

(b)

(c)

(a)

(b)

18.2

0.23

4.11

0.52

0.36

5.32

17.66

0.35

0.12

4.10

0.37

0.06

3.00

3.42

−0.42

19.1

—

0.01

0.27

—

2.32

17.68

—

11.98

—

—

9.36

—

—

13.44

—

3.45

—

64

0

—

0.70

4.13

—

5.67

(d) corr(ln W /Y , Y HP ) corr(ln(W /Y )t , ln(W /Y )t−1 )

Risk sharing economyi

(d)

(d)

−0.053

0

−0.0073

0.97

0

0.92

corr(re , Ct+1 /Ct )

—

—

corr(re , T Ct+1 /T Ct )ii

0.06

0.78

(c)

0.69 −0.54

Panel B: Aggregates (e)

(f)

(e)

(f)

(e)

(f)

Output

1.76

—

1.79

—

0.07

—

Total consumption

1.29

0.85

0.54

0.88

0.75

0.176

—

—

—

9.78

0.025

0.92

5.78

0.99

2.72

−0.055

—

0.54

0.88

1.90

0.063

0.04

1.35

0.98

0.24

1.000

Shareholder consumption Investment Wages Capital stock i μ = 73, μ = 228, α 1 2 ii In the case of Hansen

— 8.60 — 0.63

= 0.30, β = 0.99, Ω = 0.025, γ = 3, corr(μt , μt−1 ) = 0.98, π = 0.98, B ≡ 0, L = 1.25. (1985), this is the figure for representative agent consumption. iii Data sources: Hansen (1985), Mehra and Prescott (1985), Mankiw and Zeldes (1991), Kocherlakota (1996), Mehra (1998), Campbell (1999) and Bansal (2004). (a) Expected values in percent, annual frequency. (b) Standard deviation in percent, annual frequency. (c) Correlation with growth rate of output, annual frequency. (d) Indicated correlations: wage share on a quarterly frequency; others annualized. (e) Standard deviation in percent. (f) Correlation with output.

432


The intention of this exercise is purely suggestive. The RBC literature tells us that, in the absence of aggregate shocks, it would be surprising to match the macro data adequately, and Table 2 confirms this message. Yet given this very artificial context, the financial results displayed in Table 2 are surprisingly representative. While still falling short of the analogous U.S. figure by a substantial margin, the 2.32 percent premium is extremely high relative to what is typically obtained for this class of models (see Kocherlakota (1996)).14 This improvement comes from both the equity and risk-free returns, with the former increasing and the latter decreasing by roughly the same amount relative to the mean 4 percent value typically observed for both securities in this class of stationary models when β = 0.99. With respect to the various return standard deviations, the match is also quite good: the risky return and the premium are only about 7 percent more volatile than what is observed in the data, and the risk-free rate only 40 percent less so. Dividend volatility and the correlation of equity return with consumption growth are also close to what is observed. The principal shortcoming is the negative correlation of the equity return with aggregate consumption growth. Nevertheless, the present exercise suggests that distribution risk may well be a first-order-importance determinant of financial returns. This message, however, needs to be confirmed in a more realistic economy.

5. ADDING AGGREGATE UNCERTAINTY We now add aggregate uncertainty to distribution risk. The corresponding probability transition matrix is described by (24). The results of this exercise are presented in Table 3, where Case 1 admits variation in both the share parameter μ and the technology shock λ (aggregate uncertainty), while Case 2 admits aggregate uncertainty alone. With plausible aggregate shocks the model economy may now be evaluated in the light of business cycle data. Case 2 reproduces the results routinely obtained in the business cycle literature. Case 1 demonstrates that distributional risk does not disturb the performance of the macroeconomy. The following differences are worth noticing, however: distribution risk adds some extra volatility in aggregate consumption and investment. It should be viewed positively, as consumption is excessively smooth in the standard model. For this benchmark calibration, there is excessive wage volatility, suggesting that there is insufficient income insurance. More interestingly, wages are acyclical in Case 1, while they are strongly pro-cyclical in the more standard Case 2. The cyclicality of wages has been the subject of much controversy since Dunlop (1938) and Tarshis (1938) provided evidence suggesting that wages are nearly acyclical. Some have seen this observation as a source of falsification of the standard real business cycle model. Our interpretation of the labor market provides a potential resolution of this puzzle. On the financial front, the performance of the complete model is extremely good, delivering an equity premium of 3.12 percent.15 By contrast, Case 2 confirms that the 14 If we shift the basis of comparison to a sample of non-U.S. countries, this particular result looks even better: Goetzmann and Jorion (1997) report that for a sample of non-U.S. countries, the premium averages 3.5 percent when adjusted to factor out the effects of war-related market interruptions. 15 Note that this is an unlevered equity return.

433


TABLE 3 Adding Aggregate Shocks Panel A: Financial and wage share statistics Case 1i

Case 2ii

(a)

(b)

(c)

(a)

(b)

(c)

re

5.83

22.11

0.48

4.21

4.78

0.68

rf

2.71

4.40

0.09

4.04

1.68

0.13

rp

3.12

21.88

—

0.17

4.25

—

dt+1 /dt

—

16.57

—

—

3.90

—

W /Y

70

5.69

—

71

2.17

—

(d)

(d)

−0.07

−0.04

corr(ln(W /Y )t , ln(W /Y )t−1 )

0.92

0.89


0.69

0.79


0.06

0.79

corr(ln W /Y , Y HP )

Panel B: Macro-aggregates (e)

(f)

(e)

—

(f)

Output

1.77

1.77

—

Total consumption

1.16

0.620

0.87

0.95

Shareholder consumption

11.35

0.620

2.75

0.95

Investment

6.77

0.850

5.41

0.98

Wages

1.97

−0.003

0.69

0.95

Capital stock

0.59

0.060

0.48

0.06

Columns (a)–(f): same interpretation as in Table 2. Both cases: α = 0.30, β = 0.99, Ω = 0.025, γ = 3. i Case 1: λ = 1.056, λ = 0.944, μ = 73, μ = 228, ψ = 0.98376, π = 0.00133, σ = 0.00533, 1 2 1 2 H = 0.00967, Δ = 0.02, L = 1.25, yielding ρλt ,μt = −0.61, ρλt ,λt+1 = ρμt ,μt+1 = 0.97. ii Case 2: μ = μ = 150.5 = (73 + 228)/2. 1 2

standard real business cycle model is entirely prone to exhibiting the equity premium puzzle, delivering a very low premium of 0.17 percent.16 As in the pure distribution risk economy, the volatility of equity returns and of the premium is somewhat in excess of what is observed while the volatility of the 16 In a purely competitive model with only technology shocks—e.g., Hansen (1985)—we would anticipate even a much smaller value. The 0.17 percent is attributable to the large technology shocks (relative to Hansen (1985)) necessary to get the output variation to a level that corresponds to the data. That such large shocks are required is due to the high average level of risk aversion in the economy since the fraction of income to the more risk-averse workers (γ = 3) is 70 percent.

434


risk-free return is slightly too low. It is clear that these volatilities are, to a large extent, attributable to the share variations. It may be argued that this model is falsified by the excessive dividend volatility relative to what is observed. Before accepting this conclusion, one should, however, consider that the 11.98 percent figure reported in Table 2 for the U.S. economy corresponds to the distributed dividends series, which is a smoothed-out series. In our artificial economy, there are no retained earnings and no dividend smoothing. Our dividend series is closer to a free-cash-flow series, which is, inevitably, more variable. The wage share statistics are of special interest since they bear on the specifics of our model. As to the volatility of the wage share itself, it may come as a surprise that, even in the absence of variation in μ, substantial share variation (W /Y ) is observed (Table 3, Case 2, SD(W /Y ) = 2.17 percent). In a more standard representative agent model with our production technology and competitively determined factor shares, the SD(W /Y ) ≡ 0, irrespective of the degree of output uncertainty. The result in Case 2 is obtained because the risk sharing contract (μv1 (Wt ) = u1 (Ct )) does not imply a linear relationship between the increase in output net of investment and the resulting marginal increases in worker and shareholder consumption. This is attributable to the differences in the two agents’ degrees of relative risk aversion. One sees from Table 3 that, at −0.07, the Corr(ln(W /Y ), Y HP ) is close to the observed −0.04 while the autocorrelation of ln(W /Y ) is a little on the low side (at 0.92 vs. 0.97 for the observations). The most significant difference on this score between cases 1 and 2 is the dramatic improvement in the correlation between the equity return and aggregate consumption growth: it is too high in the standard RBC model (at 0.79) and almost exactly correct in our benchmark case. All in all, the economy with both distribution risk and aggregate shocks performs extremely well over an unusually wide range of financial and macroeconomic statistics. Specifically, it does contribute a large equity premium for a production model that delivers macroeconomic statistics remarkably close to those characterizing the U.S. economy. It also appears that the financial characteristics of the model are largely determined by the income sharing mechanism, while the business cycle properties are, to a large extent, determined by the technological uncertainty. For this specification at least, the incorporation of either source of uncertainty in the same model reinforces the attractive features of the other. The intuition for our results is relatively straightforward. With a negative correlation between μ and λ, justified by the observed countercyclicity of the wage share (corr(μt , λt ) = −0.63 in the benchmark case), low-productivity shocks coincide roughly half the time with a high μt realization, that is, with situations where the bargaining power of capitalists is low. In these circumstances, the normally low payment to capital owners is further reduced by the above average income share going to labor, and vice versa in periods of high-productivity shocks. These events have the consequence of making capital fundamentally riskier. As a result, stocks are less attractive to investors, bonds more attractive, and the premium rises. It is of interest to observe that our results so far have been obtained in the absence of a reinforcing financial leverage effect: in all simulations, B = 0, so that the bond is being priced and its return determined in zero net supply. Table 4 reports the

435


TABLE 4 Adding Financial Leverage B=1 (a)

D = 3.5% E (b)

Various B D B=2 = 7% E (a) (b)

(a)

D = 11% E (b) 26.86

B=3

re

5.99

23.46

6.16

25.09

6.36

rf

2.71

4.40

2.71

4.40

2.71

4.40

rp

3.27

23.25

3.45

24.82

3.64

26.69

dt+1 dt

—

18.86

—

22.31

—

27.54

(a) Average values; (b) standard deviations.

results obtained for the benchmark case modified to include various (modest) levels of financial leverage. Only the financial results are reported, as Modigliani–Miller obtains, and the real variables are unchanged relative to the benchmark. Note that the properties of the risk-free return are unchanged as well: while financial leverage modifies the properties of dividends, it does not alter the consumption of shareholders whose total income (interest + dividends) is unaltered. The equity premium increases from 3.12 percent to 3.64 percent for B = 3. The most significant impact of financial leverage is on the standard deviation of dividend growth and of equity returns which both increase to very high levels for the higher values of B. It is clear that the import of financial leverage pales alongside the role played by operating leverage. Note, for instance, that Case 1 of Table 6 (with a parametrization identical to the one considered here but with γ = 1 and B = 0) produces an equity premium of 7.78 percent as opposed to the premium of 3.64 percent obtained with financial leverage for an almost identical level of dividend volatility. We do not explore financial leverage ratios above B = 3 (or a debt-equity ratio of 11 percent) because, beyond this debt level, dividends are occasionally negative and they become excessively volatile.17 A final insight is forthcoming if one writes down the representative agent model directly analogous to the risk sharing construct as presented in problem (17): ∞ β t [u(C(st ))] (25) max E0 {C(st ),I (st )}

subject to

t=0

C(st ) + I (st ) ≤ f (Kt , 1)λt , Kt+1 = (1 − Ω)Kt + It , K0 given, λ0 , μ0 , dF (·) specified a priori. 17 As modest as the financial leverage effect appears to be in the distribution risk model, it nevertheless overshadows the analogous phenomenon in Hansen (1985). In that model, as the D/E ratio rises from zero to 0.52, the increase in the premium is a next-to-negligible 0.0011 percent.

436


Problems (17) and (25) (similarly parameterized; i.e., for both models α = 0.30, β = 0.99, γ rep.agent = γ shareholder = 1 and the same stochastic process on {λt }) have identical steady-state capital stocks, the same aggregate consumption, etc. It thus follows that Ct = f (Kt , 1)λt − Wt − It rep.agent

∼ Ct

− Wt .

(26)

Equation (26) effectively means that the wage bill serves a role similar to a postulated external habit. Relative to the representative agent construct, the wage bill is the postulated habit, and shareholder consumption in Problem (17) is nothing more than the representative agent consumption in excess of the “habit.”18 There are a number of dimensions along which the risk sharing “wage bill habit” replicates the complex non-linear habit evolution process found in the seminal paper by Campbell and Cochrane (1999). One of them is the cyclical pattern in risk aversion obtained in both models. The effective underlying mechanism is very different, however. In Campbell and Cochrane (1999), an external moving habit is postulated to express the desire of investors to maintain their relative societal consumption position (“keeping up with the Jones”; see Abel (1990)). For the risk sharing model, this feature need not be an hypothesized appendage, but rather arises entirely endogenously. For both models, the asset pricer’s consumption and his habit covary positively with output, although in the risk sharing case this also requires either (1) low risk aversion on the part of workers (cf. Table 5, cases with γ = 1, and γ = 2), or (2) non-negative correlation between μ and λ (Table 4, leftmost case). In addition the “wage bill habit” is less variable than the analogous representative agent consumption series. In the following section, the sensitivity of our results to parameter changes is examined as a step to gaining a better understanding of why they are observed in the first place.

6. COMPARATIVE DYNAMICS AND WELFARE ASSESSMENT The message of the two previous sections has been that distributional considerations— the relative shares of income going to capital and labor—are highly influential in determining the financial characteristics of equilibrium in a way that is in general harmony with business cycle phenomena. In this section, we explore how these results are influenced by the various model parameters, in particular by γ, μ, Eμ , and corr(λt , μt ), corr(μt , μt+1 ). 18 One can also view the wage bill as akin to a lump sum consumption tax on firm owners, since it does not directly enter their intertemporal optimality condition. We thank our referee for this observation.

437


TABLE 5 Comparative Dynamics: Changes in ρλμ Panel A: Financial and wage share statistics ρλμ = 0

ρλμ = −0.96

ρλμ = −1

(a)

(b)

(c)

(a)

(b)

(c)

(a)

re

5.43

18.42

0.21

6.20

24.93

0.68

6.24

25.26

0.71

rf

2.92

3.74

0.03

2.31

3.98

0.14

2.26

3.93

0.15

rp

2.51

18.34

—

3.89

24.85

—

3.98

25.20

—

dt+1 /dt

—

14.10

—

—

18.46

—

—

18.72

—

W /Y

70

4.73

—

70

6.33

—

70

6.41

—

(d)

(b)

(d)

(d)

−0.04

−0.11

−0.11


0.92

0.91

0.91


0.70

0.68

0.68

−0.06

0.36

0.44


corr(re , T Ct+1 /T Ct )

(c)

Panel B: Aggregates (e)

(f)

(e)

(f)

(e)

(f)

Output

1.78

—

1.78

—

1.78

—

Total consumption

1.15

0.75

0.57

0.260

0.43

0.16


10.17

0.26

12.41

0.960

12.56

0.99

Investment

5.96

0.87

8.37

0.960

12.56

0.99

Wages

2.02

0.33

1.51

−0.710

1.44

−0.93

Capital stock

0.53

0.07

0.72

0.097

0.74

0.10

Columns (a)–(f), same as Tables 1 and 2; all parameters conform to those underlying Case 1 of Table 3 except • ρλμ = 0: ψ = 0.97237, π = 0.01262, σ = 0.01487, H = 0.00013, Δ = 0.03. • ρλμ = −0.96: ψ = 0.98499, π = 0.00001, σ = 0.00051, H = 0.01449, Δ = 0.02. ψ = 0.985, π = 0.01262, σ = 0.01487, H = 0.015, Δ = 0. • ρλμ = −1:

6.1. Changes in the Correlation of Productivity and Distribution Shocks Intuitively, one expects that an important determinant of our results is the correlation between the two shocks. In particular, this correlation significantly determines the inherent riskiness, for the capital owners, of the labor market arrangements characteristic of this economy. The negative correlation adopted in our benchmark (meaning that capitalists’ weight in the social welfare function is low precisely when the economy is less productive) is motivated by the observation that the wage share is countercyclical, and indeed this countercyclical property is reflected in the results of Table 3. Table 5 confirms our intuition. The first case considered is one where ρλμ = 0; indeed, in that case, the

438


premium decreases to (a still respectable) 2.52 percent. This lower premium is explained by a decrease in the volatility of shareholder’s consumption and dividend growth. The wage share remains countercyclical (a result explained by the mechanism discussed in the previous subsection for the case where there are no distribution shocks), but the correlation falls in absolute value. And the correlation between equity returns and aggregate consumption becomes counterfactually negative. Conversely, a ρλμ more negative than in the benchmark case produces an increase in shareholders’ consumption volatility, dividend volatility, and higher premia. It is worth noticing that the volatility of aggregate consumption and investment (but not GDP) are affected as well. Although the correlation between aggregate consumption and equity returns is excessive, it has the attractive consequence of producing a remarkable 3.89 percent equity premium.

6.2. Changes in Risk Aversion and the Conditional Mean Distribution Shock For our choice of functional forms, the income sharing condition, Eq. (10), reduces to μ−1 Ct = Wt , γ

γ > 0.

For a fixed μ, a smaller value of γ implies a larger value of Wt , provided Wt > 1, which is the case for all scenarios considered in this chapter. This implies that as workers become less risk-averse, the share of income going to workers increases. As a consequence, shareholder consumption growth volatility increases while simultaneously being restricted to a region of greater utility function curvature: the operating leverage effect is thus stronger. This reduces the demand for risky stocks, increases the demand for risk-free assets, and increases the premium. This intuition is explored in Table 6, where we report the financial and aggregate statistics for scenarios of varying γ, centered on our benchmark of γ = 3. Confirming our intuition, the fraction of income going to workers is greatest when they are less risk-averse. In fact, the mean consumption of the firm owners in the γ = 1 case (0.065) is less than one-seventh its value when γ = 4 (0.4725). As a result, shareholders’ consumption growth (dividend growth) is much more volatile, thus increasing the premium to a level of 5.74 percent in the γ = 2 case and even an excessive 7.78 percent in the γ = 1 case. Another direct consequence of this observation is the increased volatility of the return on equity and the premium, which now lies at the upper bounds of acceptability. We observe that corr(ln(W /Y )t , YtHP ) and corr(ln(W /Y )t , ln(W /Y )t−1 ) are largely unaffected by the changes in γ, while the correlation between equity returns and aggregate consumption turns more positive, as desired to match the observations when γ decreases. On the macroeconomic side, there are no major effects of varying the parameter γ. The most significant impact is on the properties of wages. As workers’ risk aversion decreases, wages become more highly pro-cyclical and their volatility falls. The latter may be seen as somewhat of a curiosum: as the workers become less

439


TABLE 6 Comparative Dynamics: Changes in Workers’ Risk Aversion γ Panel A: Financial and wage share statistics γ=1

γ=2

γ=4

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

(c)

re

8.32

38.59

0.34

7.24

31.85

0.40

4.97

14.77

0.54

rf

0.54

8.58

0.08

1.50

6.80

0.08

3.42

2.90

0.10

rp

7.78

38.77

—

5.74

31.79

—

1.54

14.50

—

dt+1 /dt

—

27.75

—

—

23.24

—

—

11.34

—

W /Y

77

2.29

—

74

3.79

—

65

6.67

—

(d)

(d)

−0.06

−0.06

−0.06


0.86

0.90

0.93


0.63

0.65

0.72

corr(re , T Ct+1 /T Ct )

0.29

0.16

0.10


(d)

Panel B: Aggregates

Output Total consumption

(e)

(f)

(e)

(f)

(e)

(f)

1.78

—

1.77

—

1.77

—

0.83

0.92

0.94

0.79

1.21

0.60

17.71

0.47

15.29

0.54

8.03

0.72

Investment

5.78

0.97

6.19

0.92

6.78

0.84

Wages

0.81

0.72

1.29

0.26

2.24

−0.12

Capital stock

0.51

0.09

0.54

0.07

0.60

0.05


Columns (a)–(f), same as Tables 2 and 3; all parameters except γ conform to those underlying Case 1 of Table 3.

risk-averse, their consumption volatility decreases. Both results are easy to explain, however. When γ decreases, shareholders’ bargaining power diminishes and their share of income falls. Workers’ consumption (W ) then makes up an increasing fraction of aggregate consumption, and the properties of the two series (W and T C) naturally converge. Note that these results further refute the assertion that a reasonable replication of the financial data requires the assumption of an excessive level of risk aversion. In our model construct, what appear to be key are the relative income shares and their respective volatilities.

440


6.3. Other Comparative Dynamic Tests In this section, we briefly report on other tests for which we do not provide full tabular results. As just argued, from the perspective of financial return characteristics, the main impact of the parameter γ is to help determine the relative income shares. This is even more obviously the case for the average μ, and variation in this quantity can be expected to have the same effect (with an increase in μ going in the same direction as a reduction in γ). This intuition is confirmed in simulations. As for the case of changes in the risk aversion parameter, the size of the wage share is of overwhelming importance. With an increase in Eμ (workers become more successful as bargainers on average), the share of income to labor rises. As a further benefit to the negotiation success, the volatility of their wages also diminishes (in percentage terms). On the shareholders’ side, the reversed consequences together reinforce one another to push up the premium: not only does the fundamental income uncertainty more fully reside with the shareholder, but also the income share reduction forces the risk to be borne by shareholders on the more concave portion of their period utility surface: relative to a representative agent economy there is a larger habit. As a result, shareholders behave in a more risk-averse fashion and attempt to reduce their holdings of the risky security and increase their holdings of the risk-free one. This increases the premium, though at the price of excessive return volatility. For the benchmark Case 1 of Table 3, EC = 0.30 corresponding to Eμ = 150; when Eμ = 75 (same μ1 /μ2 ratio), EC = 0.45; if Eμ = 250, EC = 0.21. In the first case, the premium is 2.05 percent; in the second, it is 4.12 percent. We also analyzed the effects of the time-series characteristics of the share process on financial equilibrium and, in particular, its autocorrelation. The results of this exercise do not entirely conform to intuition. One may expect that the more persistent the variations in μ, the harder it would be for shareholders to bear the distribution risk and the higher the premium they would require for holding the risky asset. This is not systematically the case, and the passage from ρμμ = 0.97 to ρμμ = 0.5 in fact increases the premium from 3.12 percent to 4.91 percent (the reference is once again Case 1 of Table 3, where ρμμ = 0.97). The explanation for this is as follows: while more persistent μ shocks may be expected to “hurt more,” it is also a fact that they admit greater opportunity for behavioral adjustments (changes in savings and investment). In a sense, under the less persistent μ process, shareholders are less able to tailor their investment responses and they must simply endure the greater consumption volatility demonstrated in the statistics. The behavioral adjustments present in the case of ρμμ = 0.5, however, have far-reaching counterfactual consequences at the macro level. They lead to a categorical rejection of this parametrization. First, when the income sharing parameter becomes less persistent, its impact on macroeconomic volatility becomes much smaller and, for technology shocks of the same size as in the benchmark, output becomes excessively smooth. Both shareholders’ and total consumption relative volatility become excessive. More significatively, the autocorrelation of the (log) wage share becomes much too low (0.68), and the return to equity becomes strongly negatively correlated with consumption growth, two observations that disqualify this calibration.

441


6.4. Welfare Considerations So far we have imposed on shareholders and workers a specific arrangement by which workers do not participate in financial markets but obtain income insurance from their employers. Besides having significant descriptive power, this form of arrangement has a long-standing tradition in the labor market literature (under the heading “implicit contracts,” see, e.g., Rosen (1994)). While such an arrangement can be viewed as resulting from the same non-economic forces that also lead to variations in the respective bargaining power of the two parties, one could equally well argue that, apart for these variations in μ, the underlying arrangement has been willingly agreed upon and is socially optimal, for example, because of the presence of significant fixed costs to financial market participation. If the latter interpretation is adopted, one would then want to argue that shareholders should be no worse off under the chosen arrangement than under the Walrasian alternative. This would imply that they should receive an appropriate premium as a compensation for the income insurance they provide. In that spirit it is natural to ask: what is the average value of μ (i.e., the average income share, maintaining the same relative μ1 − μ2 variation) for which firm owners are as well off in the risk sharing economy as in the corresponding Walrasian equilibrium? The latter is defined as the solution to the functional equation: J (Kt , λt ) = max{u(f (Kt , 1)λt − It − Wt ) + β J (Kt+1 , λt+1 )dF (λt+1 , λt )}, It

(27)

Kt+1 = (1 − Ω)Kt + It ; It ≥ 0, with {Wt } given. Walrasian equilibrium, in particular, is captured not only by regarding the wage sequence as exogenous from the firm owner perspective, but also by imposing on the first-order condition to problem (27) the requirement that Wt = (1 − α) f (Kt , 1)λt .

(28)

Not surprisingly, incentive compatibility requires that the portion of output extracted by workers be somewhat smaller than what underlies Tables 2–5. Under problem (27) and (28), firm owners experience a smaller income uncertainty and receive 30 percent of total output. In order to agree to an income sharing scheme with variable shares, their average share of income must increase. Our computations show that shareholders are as well off under the defined labor arrangement scheme (including variable income shares, i.e., distribution risk) if their share of income is increased by a little over 1 percent of GDP, i.e., is equal to 31 percent instead of 30 percent. This is obtained for (μ1 , μ2 ) = (61, 190). With this higher share of income to firm owners, and under our benchmark calibration for the other parameters, the equity premium is 2.80 percent. The entire set of statistics is reported in Table 7.

442

Chapter 10

•

Distribution Risk and Equity Returns

TABLE 7 Incentive Compatible Case Panel A: Financial and wage share statisticsi μ1 = 61, μ2 = 190 (a)

(b)

(c)

re

5.65

20.70

0.49

rf

2.85

4.12

0.08

rp

2.80

20.51

—

dt+1 /dt

—

15.58

—

W /Y

69

5.99

—

(d) −0.06

corr(ln(W /Y ), Y HP ) corr(ln(W /Y )t , ln(W /Y )t − 1 )

0.92


0.69


0.05 Panel B: Aggregates (e)

(f)

Output

1.77

—

Total consumption

1.18

0.60

10.77

0.64

Shareholder consumption Investment

6.83

0.84

Wages

2.10

−0.04

Capital stock

0.60

0.06

i (a)–(f)

as in Table 2. Parameters: α = 0.30, β = 0.99, Ω = 0.025, γ = 3, corr(μt , μt−1 ) = 0.98; other parameters as in Table 3.

6.5. Explaining the Market Value to National Income Ratio In a recent article, Mehra (1998) has reminded us that the 1985 Mehra–Prescott asset pricing model is also unable to match the observed volatility of the q e /NI ratio (the value of market equity as captured by the S&P500 as a fraction of national income) for the U.S. economy. For the sample period 1929–1993, the value of this ratio is in the range [0.45, 1.90]; for the postwar period 1946–1993, the corresponding range is [0.48, 1.33]. It is striking that the standard Mehra–Prescott 1985 model, with logarithmic utility for the representative agent, generates a range of [0.54, 0.69] for the same ratio. We observe (Table 8) that the distribution risk model is able to generate vastly more variation in this important ratio. The benchmark case produces a multiple

443


TABLE 8 Properties of the q e /Y Ratio μ1 = 36, μ2 = 114

Benchmark

μ1 = 121, μ2 = 380

E(W /Y )

0.66

0.70

0.73

SD(q e /Y )

1.01

0.87

0.69

max(q e /Y )

14.85

3.43

2.53

min(q e /Y )

1.71

1.00

0.66

E(q e /Y )

3.11

2.07

1.46

All parameters other than μ1 , μ2 are the same across all three cases and coincide with the benchmark, Case 1 of Table 3.

between the minimum and the maximum values of this ratio, at 3.4, that stands in between the value observed in the postwar period, 2.77, and the one obtained for the entire Mehra sample, 4.22. It is also worth noting that the risk sharing model suggests an alternative explanation to that of McGrattan and Prescott (2005) for the steep rise in the q e /NI ratio observed since the 1960s in the U.S. Whereas those authors attribute the phenomenon to less oppressive capital taxation, our model suggests that the observed reduction in the labor share could be responsible as well. The (modest) reduction in the labor share from 0.73 for the case μ1 = 121; μ2 = 380 to 0.70 for our benchmark, for example, produces an increase of 41 percent in the mean value of the ratio q e /NI. This interpretation of the facts receives strong support from Boldrin and Peralta-Alva (2005).

7. TECHNOLOGY-DRIVEN VARIATIONS IN FACTOR SHARES In this section, we briefly entertain the hypothesis that the observed variations in factor shares are the result of time variations in the parameters of the aggregate production function. Let us assume a representative agent with utility of consumption u(Ct ) = ln Ct and an aggregate technology given by α

f (Kt , 1) = LKt t , where αt varies stochastically through time according to a known Markovian process. For consistent comparisons, we have assumed that nt ≡ 1 and that all investment is financed out of the capital share of income: d(Kt , αt ) = αt f (Kt , 1) − I (Kt , αt ).

(29)

444


Such a model formulation can be summarized by the recursive functional equation: Jˆ (Kt , αt ) = max {u(C(Kt , αt )) + β Jˆ (Kt+1 , αt+1 )dH (αt+1 , αt )} I (Kt ,αt )

subject to α

C(Kt , αt ) + I (Kt , αt ) ≤ LKt t , Kt+1 = I (Kt , αt ) + (1 − Ω)Kt ; K0 given, dH (αt+1 ; αt ) specified, Markovian.

(30)

This formulation differs from the earlier one in several essential ways. First, the consumption of the representative agent equals the sum total of his wage and dividend income, a fact that suggests it will be nearly impossible—while maintaining plausible output and consumption variations—to achieve substantial MRS variability. Second, for this formulation, the share of income to capital cannot be specified independently of the average level of capital stock: a lower Eα (to match a higher worker share) is coincident with lower equilibrium capital stock, output, etc. All variants of the risk sharing model display the same steady-state level of capital stock, since α is unchanging. These remarks together suggest that it will be very difficult to achieve the same results as in the earlier formulation where factor shares (over a limited range) can be expressed substantially independently of the capital stock process, an assertion that is borne out in the results of Table 9. In this case the process on α˜ t follows a two-state Markov process. It is possible to parameterize this process so as to replicate simultaneously the de-trended output variability and the properties of the labor share as per Case 1, Table 3. This is accomplished by using the same transition matrix as in the cases underlying Tables 2–4, while assigning the role of μt to αt . That is, we chose to assume no other shocks and set λt ≡ 1, thus isolating the consequences of a variable αt . This also implies that the process governing the evolution of α˜t coincides exactly with the process governing the evolution of μ˜t in the comparison cases. One may thus choose to compare the results in Table 9 with those of Table 3, where there are no aggregate shocks, or with those of Tables 3 and following, where macro shocks are present and macroeconomic volatility is comparable. For full consistency, u(Ct ) = ln Ct is also maintained. As is clearly seen, there is no measurable premium, and the return standard deviations are quite low relative to previous cases. The business cycle properties of the model, including the properties of the wage share, are, however, quite acceptable (except for the very high correlation of wages with output and the low relative volatility of consumption). These results underline the fact that factor share variations are not an additional source of risk in a representative agent model: what the agent does not get in the form of wages, he receives in the form of dividend payments. By contrast, income share variations are (highly) relevant in economies with two separate agent classes, as is the case in the other models proposed herein. This latter remark suggests that it would be more appropriate to compare the equilibrium characteristics of the risk sharing construct with one in which shareholders

445


TABLE 9 Technology-Induced Variations in Income Shares (a)

(b)

(c)

re

4.111

1.12

0.33

rf

4.111

1.04

0.19

rp

0.000

0.24

—

dt+1 /dt

—

11.09

—

Wt /Yt

69

1.82

—

(d) −0.03

corr(ln(W /Y )t , YtHP ) corr(ln(W /Y )t , ln(W /Y )t−1 )

0.99


0.97 (e)

(f)

Output

1.78

—

Consumption

0.45

0.20

Investment

8.65

0.97

Wages

1.06

0.99

Capital stock

0.73

0.14

(a)–(f) as in prior tables. α1 = 0.322, α2 = 0.285, β = 0.99, Ω = 0.025, ψ = 0.97237, π = 0.0162, σ = 0.01487, H = 0.00013, Δ = 0.002.

solve

J˘ (Kt , αt ) = max {u(C(Kt , αt )) + β J˘ (Kt+1 , αt+1 )dH (αt+1 , αt )} I (Kt ,αt )

subject to

˘ tαt , C(Kt , αt ) + I (Kt , αt ) ≤ αt LK Kt+1 = I (Kt , αt ) + (1 − Ω)Kt ; K0 given, dH (αt+1 ; αt ) specified, Markovian,

(31)

˘ tαt. As in the risk sharing and workers receive the competitive wage payment (1 − αt )LK model, shareholders consume their dividends and finance investment exclusively out of capital’s income share. Unfortunately, this formulation does not afford an entirely natural comparison for other reasons: unlike in either formulation (30) or in the risk sharing benchmark, wage payments are no longer a “tax” exclusively on shareholder consumption but on capital income as well. As a result, the steady-state capital stock is much smaller: when calibrated identically to the benchmark model for all mutual parameters, the steadystate capital stock for Problem (31) is 5.28;19 for the benchmark model and the model 19 That

is, β = 0.99, L˘ = 1.25, Ω = 0.025, u(Ct ) = ln Ct , Eαt = 0.30, with α1 = 0.312, α2 = 0.288.

446


implicit in problem (30), the corresponding value is 29.47. In addition, the technology share parameter αt now assumes two roles, one as a share disturbance and the other as a standard aggregate productivity shock where the two exhibit perfect positive correlation.20 Neither source of uncertainty gives rise to significant premia, and it would be surprising if they were to do so in combination. Uncertainties of this type are too easily smoothed out by manipulating investment appropriately. This latter claim is bore out in the respective standard deviations of shareholder consumption growth. For the benchmark model the figure is 16.57 percent (which equals dividend growth), while for Problem (31) and L˘ = 1.25 (given the steady-state capital stock of 5.28), it is 1.85 percent. If the economy is scaled up to give a comxmon steady-state capital stock level (L˘ = 3.3), the standard deviation of shareholder consumption growth is 0.93 percent. These formulations yield a premia in the range of 0.10 percent, which is the order of magnitude generated by standard real business cycle models (e.g., Hansen (1985)).

8. ROBUSTNESS The stylized facts of the U.S. business cycle reported in Table 1 (leftmost column) are based on quantity data for the period 1955:3 through 1984:4. It is natural to explore the relevance of these numbers for both more recent and more comprehensive historical periods. Table 10 presents the most salient macroeconomic and financial stylized facts for a variety of postwar time periods. Let us first consider macro aggregates (Panel B). For all data periods, the pattern of relative volatilities and correlations with output thought to characterize the business cycle is preserved: investment is more volatile than output, which is, in term, more volatile than total consumption. In all cases, investment is more highly correlated with output than consumption. Comparing the results for the period 1983:1–2004:4 (rightmost column) with those of the other periods, we see evidence suggesting that macroeconomic volatility has declined dramatically post 1983, a period generally believed to be characterized by more effective monetary policy implementation. In particular, all series are roughly 40–60 percent as volatile as in their benchmark counterparts (leftmost column). Panel A contains the most relevant financial stylized facts. Here the story is similar. For time periods encompassing more recent events, return volatilities are less across all the series, while mean equity returns and the premium are all higher than for the benchmark period. It is not clear whether these changes reflect any profound change in the nature of financial market risks or are purely an artifact of the late 1990s’ stock market boom. Is a model with systematic and distributional risk features capable of replicating, for various reasonable parameterizations, the wide range of macroeconomic and financial phenomena characteristic of the various data periods requires identifying α1 = λ1 and α2 = λ2 in matrix (25), with ψ = 0.985, π = 0, σ = 0, H = 0.015, and Δ = 0. With these values, ραα = 0.97, ρλλ = 0.97, and ραλ = 1.00 provided (λ0 , α0 ) = (λ1 , α1 ).

20 This

447


TABLE 10 Stylized Macroeconomic and Financial Facts: Various Postwar Time Periods 1955:1 1984:4 (Table 2)

1954:1 1991:2

1947:1 1983:4

1947:1 2004:4

1983:1 2004:4

Panel A: Financial statisticsi (a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

re

6.98

18.2

7.5

13.8

7.6

13.4

8.7

14.2

10.7

15.4

rf

0.80

1.2

2.4

0.3

2.5

0.9

2.4

2.1

1.8

rp

6.18

6.3

13.4

7.3

13.4

7.8

14.0

8.6

15.0

5.67 19.1

Panel B: Aggregatesii (c)

(d)

(c)

(d)

(c)

(d)

(c)

(d)

(c)

(d)

Output

1.76

—

1.72

—

1.88

—

1.56

—

0.97

—

Total consumption

1.29

0.85

1.27

0.83

1.22

0.68

1.11

0.77

0.75

0.75

Investment

8.60

0.92

8.24

0.91

5.41

0.75

4.84

0.89

3.69

0.91

i Financial data from the benchmark period is sourced as per the notes in Table 2. For all other time periods, the financial statistics were computed using annual data for the period indicated. ii Macrodata from the benchmark (leftmost) period is from Hansen (1985). For the period 1954:1–1991:2, data is provided by Cooley and Prescott (1995). For all other historical periods, the data is drawn from Jaimovich and Rebelo (2006). (a) Expected values in percent, annual frequency. (b) Standard deviation in percent, annual frequency. (c) Standard deviation in percent, de-trended data. (d) Correlation with output.

considered in Table 10? In particular, consider the period 1983:1–2004:4, where macro volatility is low and equity returns high. Within the model framework of this chapter, these phenomena required a reduction in systematic risk (effected by a reduced SD of the productivity disturbance) in conjunction with a lower worker CRRA. The latter change results in smaller income transfers from workers to firm owners (less insurance), so that the share of income to firm owners is reduced. The full set of statistics for a representative case of this type is presented in Table 11. Basically, the results are as anticipated. With less variation in the productivity disturbance, macro volatilities are reduced across the board. Investment and total consumption are, relatively, a bit too smooth, while the share statistics remain entirely satisfactory. This is in the context of a 5.51 percent premium. Return volatilities are clearly too high. While it is possible to exhibit an 8 percent premium (see Table 6), return volatilities exhibit an even greater magnitude. As such, these results suggest that the financial phenomena of the very late 20th to early 21st centuries is anomalous, and attributable, most likely, to events not captured by our model construct. This observation is consonant with the observed pronounced decline in the share of income to labor during this same period.

448


TABLE 11 Lower Systematic Risk: Representative Casei,ii Panel A: Financial and wage share statistics (a)

(b)

(c)

re

7.02

30.21

0.27

rf

1.51

7.15

0.07

rp

5.51

30.42

—

dt+1 /dt

—

22.12

—

W /Y

77

1.80

—

(d) −0.04

corr(ln W /Y , Y HP ) corr(ln(W /Y )t , ln(W /Y )t−1 )

0.92


0.66

corr(re , T Ct+1 /T Ct

)ii

0.16 Panel B: Aggregates (e)

(f)

Output

0.95

—

Total consumption

0.48

0.84


15.08

0.45

Investment

3.15

0.95

Wages

0.51

0.56

Capital stock

0.28

0.09

i (a)–(f) ii All

as in Table 2. parameters as in Table 3 except λ1 = 1.03, λ2 = 0.97, γ = −1.

9. AN ALTERNATIVE INTERPRETATION OF THE SHARING MECHANISM The variable sharing parameter μ˜ can also be rationalized in an alternative manner that differs significantly from the motivation heretofore proposed (where variation in μ˜ was postulated to reflect changes in the relative bargaining strengths of workers and firm owners). In particular, we will demonstrate that variable sharing will also arise endogenously in purely competitive contexts where workers receive an idiosyncratic income shock against which they are unable to insure due to restricted financial market participation. Let us denote this income shock by ξt when received in period t, and let us suppose that ξt and λt are both highly autocorrelated and cross-correlated. To make clear the intuition underlying the development to follow, we first review the story underlying the analogous complete markets case.


449

Suppose, as we assume, that workers are more risk-averse than shareholders. They will thus wish to sell contingent commodities (or state claims) that pay off in their high future income states, and purchase claims that pay off in their low future income states. Since, with a competitive labor market and Cobb–Douglas technology, low aggregate income states for workers are also typically low income states for shareholders, it is very likely that the equilibrium price of claims paying in low income states will be higher than their high income state counterparts. On balance, the claims that workers sell will have less aggregate value than the claims they purchase. On net, they thus willingly transfer positive amounts of current consumption to the shareholders in the context of their trading activities in exchange for a more stable future consumption stream. It is in this sense that the shareholders are “compensated” for providing income insurance to workers in a competitive equilibrium context. Under competitive labor markets and a reasonable model calibration, workers’ wage income will be relatively smooth, especially as compared with firm owner income. This fact suggests that workers will mostly seek to insure against variation in their idiosyncratic income component. In any event, under market completeness, for all states, (k, ξ, λ), v1 (C(k, ξ, λ)) = θˆ u1(C(k, ξ, λ)) for some constant θˆ after all trades have been executed. Suppose now that workers (for reasons of implicit moral hazard, etc.) cannot trade claims that distinguish among (ξt , λt ) realizations. The only mechanism open to workers for future consumption stabilization is to save via the acquisition of positive quantities of (kt , t) indexed claims from shareholders, at a net cost to themselves. These claims pay one unit of the consumption good if the particular (kt , t) state is observed irrespective of what (ξt , λt ) is simultaneously realized. In a sense, they represent (kt , t) conditional discount bonds. Consider two workers with identical preferences and initial wealth. Such trading restrictions mean that the period tˆ wealth of a worker who has experienced a favorable sequence of high idiosyncratic income stocks ξt in periods t < tˆ, and the period tˆ wealth of an otherwise identical worker who has experienced a sequence of low ξt shocks, t < tˆ, will be the same. The former would have enjoyed greater consumption and higher welfare than the latter, but this does not translate into the potential for higher wealth accumulation. If that were the case, it would imply that the high idiosyncratic income worker could sell contingent commodities that pay in high future ξt income states and use the proceeds to buy more (kt , t)-based claims—something prohibited by the market structure. By accumulating (kt , t)-based contingent commodities, workers can partially insure against low-wage income states (ceteris paribus, low capital states are low-wage states) but not specifically against low ξt or low λt states. By selling capital-based contingent commodities to workers, shareholders insure workers but only partially so, as these securities ignore (ξt , λt ) differences. In equilibrium the ratio u1 (C(kt , ξt , λt ))/ v1 (C(kt , ξt , λt )) = θ(ξt , λt ) will thus vary, though generally to a lesser extent than in the absence of any claims trading whatsoever. It is this quantity θ(ξt , λt ) that we identify with μ˜ t . The formalism behind these ideas is presented below, where first the generalized planning problem and, second, an overview of its competitive decentralization are

450


discussed.21 Problem (32) represents the planning problem: ∞ t max E β u(Ct ) + γ(ξt , λt )v(C t ) {Ct ,C t ,It }

(32)

t=0

subject to Ct + C t + It ≤ f (Kt )λt + ξt , Kt+1 = (1 − Ω) Kt + It , ¯ λ) ¯ given. K0 = K, (ξ0 , λ0 ) = (ξ, In formulation (32), γ(ξt , λt ) is the variable sharing parameter, which is postulated to be a one-to-one function of the economy’s two sources of uncertainty—the worker’s endowment shock and the firm’s technology shock—and C t denotes period t worker consumption; otherwise, the notation is the same as previously. The planner’s decision variables may be written as functions of the economy’s capital stock Kt and the history, up to and including the present, of the economy’s exogenous shocks {(ξt , λt )}Tt=0 ≡ zT ; we also identify (zt−1 , (ξt , λt )) = zt . Hence, Ct = C(Kt , zt ), C t = C(Kt , zt ) and It = I (Kt , zt ). In the decentralized economy corresponding to Problem (31), shareholders are presumed to have access to a complete set of contingent commodities markets. They therefore solve the following problem: ∞ t t max E β u(C (Kt , z )) (33) {Ct , It , nt }

t=0

subject to ∞ t=0

P (Kt , zt ) C(Kt , zt ) + I (Kt , zt ) + W (Kt , zt ) n(Kt , zt ) − λt f (Kt ) ≤ 0, (34) (Kt , zt ) Kt+1 (zt ) = (1 − Ω)Kt (zt−1 ) + I Kt , zt , 0 ≤ n Kt , zt ≤ 1,

K0 , (ξ0 , λ0 ) given. Since the owner has access to complete markets, he is confronted with one overall budget constraint (34). Let θˆ denote the multplier on this constraint. The relevant differential optimality condition is thus β t π(zt )u1 (C(Kt , zt )) = θˆ P (K t , zt ),

(35)

21 A more detailed development, including the recursive counterpart, is provided in a special appendix available at http://www.hec.unil.ch/jdanthine/. We present the non-recursive valuation equilibrium representation, as the underlying mechanism is more transparent in this case.

451


where π(zt ) denotes the probability of the indicated sequence of events, and P (Kt , zt ) is the period t present value (relative to t = 0) price of consumption (output). As noted, the worker is presumed to be highly constrained in his trading activities. Neither can he trade contingent commodities indexed by ξt , described earlier, nor commodities indexed by λt . Workers may thus transfer wealth across different realizations of (Kt , t), but not across realizations (Kt , t, ξˆt , λˆ t ) and (Kt , t, ξt , λ t ). As a result, in each period, workers face multiple budget constraints, each one indexed by a possible realization of (ξt , λt ). These constraints allow the worker to trade contingent commodities only across future (ξt , λt ) states that coincide with the particular (ξ, λ) experienced today. In every period t, workers thus solve max E

{nwt+j , C t+j }

∞

β

t+j

v C Kt+j , zt+j

(36)

j=1

subject to ∞

C Kt+j , zt+j − ξt − W(Kt+j , zt+j ) nw (Kt+j , zt+j ) = 0, P Kt+j , zt+j

j=1 Kt+j

(37) ∀ (ξt , λt ) ∈ Ξ × Γ, 0 ≤ nw (Kt+j , zt+j ) ≤ 1. Making specific the remarks above, there is one constraint of type (37) for every possible (ξt , λt ) that is feasible going forward. Let θ(ξt , λt ) denote the multiplier identified with the constraint indexed by (ξt , λt ). The relevant differential optimality condition from the workers’ problem is β t π(zt ) v (C(Kt , zt )) = θ(ξt , λt ) P (Kt , zt ).

(38)

In equilibrium, v (C (Kt , zt )) =

θ(ξt , λt )

u (C(Kt , zt )). θˆ

(39)

ˆ Define μ˜ t = γ(ξt , λt ) = θ/θ(ξ t , λt ), and the identification is complete (constraints are binding and multipliers are strictly positive). Because of an absence of market access, workers cannot insure against all eventualities. Across different future states, the marginal benefit to relaxing the relevant budget constraints will no longer be in fixed ratio but will change with the (ξt , λt ) state, a fact mirrored by the variable γ(ξt , λt ) ratio. Under this identification, roughly speaking, μ˜ t ≡ γ(ξ˜t , λ˜ t ) will be highly persistent when ξ˜t and λ˜ t are. Furthermore, ceteris paribus, we may identify a high level of ξt with

452


a high value of μt , and vice versa. More generally, whenever one agent is constrained in his options to trade contingent commodities (or state claims), then there will be incomplete insurance, which can be captured by a variable μ˜ parameter as per our initial model identification. The above equivalence allows us to connect our modeling effort to an important strand in the literature associated most closely with Constantinides and Duffie (1996). As emphasized in Constantinides and Duffie (1996), a reasonable replication of the financial stylized facts requires (1) uninsurable income shocks that are (2) highly persistent. By the identifications in this section, the assumed properties on the μ˜ t process may be interpreted as fully satisfying these conditions.

10. RELATED LITERATURE The central advantage of production economies for the understanding of the pattern of financial returns is the added discipline they present to the exercise. Since the actions of the same economic agents give rise to both macroeconomic and financial phenomena, it is a minimum expression of consistency that the same model be expected to replicate the financial and macroeconomic stylized facts, at least along a limited set of dimensions. This has been our perspective. In this section, we discuss other theoretical contributions with significant labor market features and their implications for financial return data. Related comments focusing on empirical contributions to the literature may be found in the essay by John Cochrane in this volume. Matching financial data in a production setting requires that the capital owner display a strong desire to smooth his consumption intertemporally (provoked by, e.g., a habit formation feature) while simultaneously acting in a context that makes it difficult to reallocate labor or capital to that same end. These latter restrictions essentially substitute for some form of market incompleteness: in either case, agents are prevented from smoothing their consumption across states and dates. In most models, it is the degree of restrictiveness in the labor market that ultimately holds sway vis-à-vis financial characteristics. There are four models, in particular, that we review; principal comparative output data is provided as available. In all cases, notation is harmonized to be consistent with that adopted in this paper. A first paper to emphasize the influence of labor market phenomena on equilibrium financial returns was Danthine et al. (1992). It proposed a model with shareholders, primary and secondary workers. These latter groups hold no securities (limited participation incompleteness). The primary workers are assumed to have a permanent, full employment association with the firm. Their compensation is governed by a risk sharing arrangement identical to the one proposed in this paper. At the other extreme, the secondary workers’ employment prospects are governed by a pure Walrasian mechanism, one that otherwise would lead to substantial income variation. In order to moderate this wage income variability, primary worker wages are postulated to be subject to a wage floor augmented by unemployment compensation (the wage floor is above the market clearing wage in some states) financed by a tax on

453


TABLE 12 Model Results: Danthine et al. (1992)i Financial and aggregate statistics (a)ii

(b)

re

4.56

0.84

rf

3.98

0.80

rp

0.58

0.06

diii t

—

5.36 (d)


0.06 (e)

(f)

Output

1.76

0.69

Total consumption

0.34

0.98


5.36

0.99

Investment

6.08

0.99

Wagesiv

0.22

0.10

Capital stock

0.54

0.03

i The reported statistics are drawn from Tables 3 and 4 in Danthine et al. (1992). ii (a), (b), (d)–(f) as in Table 2. iii The reported volatility is for the dividend annualized, not its growth rate. iv Wages are equivalent to total worker compensation.

corporate profits.22 As a result of these latter arrangements, all workers in the model experience income volatility less than what would occur under a full Walrasian scenario. Whether directly—via wage insurance—or indirectly—via the unemployment tax—the net effect of worker income stabilization is to shift income risk onto the shareholders. The principle model results are presented in Table 12. While the model is able to replicate the stylized business cycle facts very well and produces a premium substantially in excess of what is obtainable under a Hansen (1985) construct, the premium obviously falls significantly short of what is observed. Security return volatilities are also much too low. In effect, variable equilibrium labor supply in the secondary sector in conjunction with shareholder control over investment together provide too much opportunity for shareholder consumption smoothing. Indeed, shareholder consumption volatility is about half the level of the benchmark case of this paper (Table 3, Case (1)); otherwise, the macro series are very similar. In a sense, the current model is a simplified version of Danthine et al. (1992) where all workers are subject to the primary worker income determination mechanism, augmented 22 Thus, the model generates unemployment among the secondary workers as both worker types supply labor inelastically.

454


with an extra source of risk affecting the mechanism of income sharing itself. This second source of uncertainty is fundamental to its superior results along the financial dimensions.23 Boldrin and Horvath (1995) propose a contracting mechanism that is similar to Danthine et al. (1992). In equilibrium, it also has the consequence that employees supply resources to firm owners in high-income states and receive payments from them in lowoutput ones.24 In their setup, profits and hours both display high levels of variability in line with their respective empirical counterparts. As they do not present data on the pattern of financial returns characteristic of their model, it is difficult to directly compare their results with the other literature. By the nature of their model formulation, however, it is likely that their results would be similar to those in Danthine et al. (1992). Subsequent to Danthine et al. (1992), the literature approached the same set of issues more from the perspective of modifying shareholder preferences in order that they act in a more risk-averse fashion and less from the “operating leverage” perspective of worker income insurance. The first paper in this tradition was Jermann (1998), which postulates a representative agent style model with habit formation (leading to high MRS volatility) in conjunction with capital adjustment costs that make it difficult to smooth consumption via investment variation. The inability of the agent to smooth is strengthened by a fixed labor supply assumption. With these features, his model is able to explain the business cycle stylized facts in conjunction with the mean premium quite well, but at a cost of excessive risk-free rate volatility. See Table 13. Boldrin et al. (2001) demonstrate, however, that the high premium in Jermann (1998) is significantly reduced if a Hansen (1985)-style labor-leisure choice mechanism is introduced even while retaining the same adjustment cost specification. Thus modified, Jermann’s (1998) model also has the counterfactual feature that hours and output are negatively correlated. In this modified model, there are two opportunities for the representative agent (and therefore the representative shareholder) to smooth his consumption stream—by adjusting his hours and investment (though at a cost)—and, taken together, these are very effective consumption smoothing devices. As a result, the premium declines to 0.30 percent. The results in Jermann (1998) are thus not extremely robust. Jermann (1998) is nevertheless, important for establishing the basic modeling perspective for finance cum production models: make the security holders extremely desirous of smoothing their consumption in tandem with technological impediments to doing so. Boldrin et al. (2001) also review a number of possible model features and ultimately explore one with two sectors—one producing consumption and the other capital goods—where the allocation of capital and labor to each sector must be chosen one period in advance of knowing the respective technology shocks. This has the

23 It is likely also true, however, that the introduction of variable labor into the present model construct would reduce the premium. 24 Much of the added generality in Boldrin and Horvath (1995) comes from their contracting mechanism, which admits fixing wage and employment schedules many periods in advance. Due to computational problems, however, they ultimately focus one one-period-ahead contracts.

455


TABLE 13 Model Results: Jermann (1998)i Financial and aggregate statistics (a)ii

(b)

re

7.00

19.86

rf

0.82

11.64

rp

6.18

—

—

8.44

dt+1 /dt

(e) Output

1.76

Consumption

0.86

Investment

4.64

i The ii (a),

reported statistics are drawn from Tables 1 and 2 in Jermann (1998). (b), (e) as in Table 2.

consequence of reducing the ability of shifts in either factor of production to be used to smooth consumption significantly. It is the restrictions on labor market flows between sectors subsequent to shock realizations, in particular, that they view as most crucial to their results. In conjunction with standard habit formation preferences, these authors can explain the mean equity premium although investment volatility is a bit too low and the risk-free rate again displays excessive volatility, so much so that its standard deviation substantially exceeds that of the return on equity (see Table 14). We note that the excessive risk-free rate volatility of Jermann (1998) and Boldrin et al. (2001) is not a general consequence of the distributional risk perspective. Danthine and Donaldson (2002) revisit the original question posed in Danthine et al. (1992): to what extent can operating leverage cum income share variation simultaneously explain the business cycle and financial market stylized facts? It is an exploration that is accomplished in a slightly more abstract setting than in Danthine et al. (1992), whereby the latter’s elaborate labor market setup (temporary and permanent classes of workers, etc.) is summarized by a “net” risk sharing mechanism nearly identical to the one considered here. The present paper may be viewed as decomposing the general results in Danthine and Donaldson (2002) into the distributional and aggregate shock-related components. With several additional features, such as costs of adjusting the capital stock, Danthine and Donaldson (2002) also achieve an excellent and broad-based fit to the data (Table 15). Generally speaking, Table 15 replicates the one presented in Table 3 except that returns seem to conform to the data slightly less satisfactorily. This is attributable to the slightly lower W /Y ratio, which results in a more modest operating leverage effect. Our final theoretical comments concern Guvenen (2005). He assumes a perspective that may be viewed as providing an alternative macro interpretation for the variable risk sharing feature of the present model. Rather than assuming workers and

456


TABLE 14 Model Results: Boldrin et al. (2001)i Financial and aggregate statistics (a)ii

(b)

re

7.83

18.4

rf

1.20

24.6

rp

6.63

—

(e)

(f)

1.97

—

Output Total consumption

1.36

0.76

Investment

4.71

0.96

Hours

1.58

0.78

i This

data is drawn from Tables 1 and 2 in Boldrin et al. (2001). (b), (e), (f) as in Table 2.

ii (a),

TABLE 15 Model Results: Danthine and Donaldson (2002)i Financial and aggregate statistics (a)ii

(b)

(c)

re

5.92

22.20

0.26

rf

2.46

4.05

0.02

rp

3.46

22.34

—

—

16.72

—

dt+1 /dt

(d) W /Y

Output Total consumption Shareholder consumption

0.69

4.83

(e)

(f)

1.77 1.45

0.96

11.94

0.38

Investment

3.05

0.93

Capital stock

0.27

−0.005

i The ii (a),

−0.022

reported statistics are drawn from Table 4 in Danthine and Donaldson (2002). (b), (c), (e), (f) as in Table 1.


457

shareholders interacting in an uncertain bargaining context, Guvenen (2005) presumes that the population is divided into two groups with unequal financial market access.25 Shareholders participate in both stock and bond markets while non-shareholders trade only bonds. Both groups supply labor inelastically to the firm, and non-stockholders are modeled as being more risk-averse.26 With bond trading being their only mechanism for consumption smoothing, non-stockholders bid up bond prices, resulting in a low risk-free rate. In equilibrium, stockholders end up insuring non-stockholders by increasing their debt holding exactly when a low productivity realization reduces both agents’ income, and vice versa. As a result, bond market events act to create a high level of volatility of shareholder consumption, volatility against which they can insure only via management of the capital stock. Although the effective extent of income insurance provided by shareholders to non-shareholders is not as great as in the present model, the fundamental idea is the same. Guvenen (2005) also goes on to show that the consumption of non-shareholders serves a role similar to that of a slow-moving habit in his equilibrium asset pricing equation, a feature also present in our distributional risk sharing formulation. We note that these results seem to be more favorable vis-à-vis “distribution risk” along the dimension of the return volatilities, but less so with regard to the business cycle stylized facts. In particular, investment is insufficiently volatile. The substance of these theoretical contributions, broadly speaking, is as follows: (1) labor market arrangements have substantial impact on the volatility of profits and shareholder income. (2) In contexts where shareholders have limited ability to hedge this added income risk, its consequences for the equilibrium pricing of financial claims are profound and generally go in the direction of enhancing the models abilities to simultaneously replicate the stylized facts of the business cycle and financial markets. (3) Since the magnitude of the equity premium responds directly to low-frequency income shocks, it is convenient—in the sense of allowing for a superior replication of financial data within a simple context—to have alternative sources of income variation beyond that arising from business cycle co-movements. Our risk sharing mechanism is one such source. (4) A reasonable representation of the financial stylized facts requires income shocks that cannot be insured (smoothed). This may take the form of technological restrictions, as in many of the papers detailed in the present section, or various forms of market incompleteness. Our distribution risk perspective entails aspects of both these perspectives.27 25 The

stockholder/non-stockholder apportionment is exogenous in Guvenen (2005). (especially shareholders) were confronted by a labor-leisure choice (allowing the shareholders another dimension of consumption smoothing), the ability of Guvenen’s (2005) model to replicate the financial stylized facts would be substantially compromised. The same may be said of the present paper. A mechanism such as in Dow (1995), where labor decisions must be taken in advance, is a possible mechanism for preserving labor supply inflexibility. 27 All the models considered in this chapter are traditional consumption-based asset pricing models. Another class of models relies on pure production relationships to determine asset prices and returns. This line of work is most closely identified with Cochrane (1991) and is reviewed in his volume chapter. More recent contributions in the same vein include Gomes, Yaron, and Zhang (2002) and, most especially, Jermann (2005). 26 We suspect that if either agent

458


TABLE 16 Model Results: Guvenen (2005)i Financial and aggregate statistics (a)ii

(b)

(c)

re

5.30

14.10

—

rf

1.98

5.73

—

rp

3.32

14.70

0.30

(e) Output

2.4

Total consumption

2.3


4.6

Non-shareholder consumption

1.1

Investment

2.7

Capital stock

—

i These statistics are principally from Tables 2 and 11 of Guvenen (2003), which is the antecedent of Guvenen (2005). The latter version is considerably abbreviated, however, and lacks macro statistics. It is for this reason that we make the indicated choice. ii (a), (b), (e) as in Table 2; (c) is correlation with output.

The focus of research in empirical finance is to explain the cross section of security returns, where the notion of cross section is in reference to sets of specifically constructed portfolios rather than individual issues.28 Curiously, there are to date few studies that include labor market explanatory variables of any sort in the first stages of the Fama and MacBeth (1972) style regressions, which constitute the fundamental technique employed in these exercises. There are two exceptions to this general rule: Jagannathan and Wang (1996) and Santos and Veronesi (2004). Jagannathan and Wang (1996) include the growth rate in per capita labor income as an explanatory variable, a fact that allows their model to outperform the standard CAPM. Such a variable is completely consistent with the model presented in this paper: an above-average value of μ¯ t in a particular period is consistent with a simultaneous high growth rate in per worker labor income, and vice versa. Santos and Veronesi (2004) focus rather on the predictability of stock returns. Their labor market variable is the economy-wide labor income to total consumption ratio, a quantity that is perfectly positively correlated with μt in our model. What is of particular interest to us is the intuition provided by Santos and Veronesi (2004), all of which, by construction, applies to the “distributional risk” construct. They argue that the share of income due to wages, as with all other principal sectors of the economy is a stationary process. The significance of this fact for asset pricing is twofold: (1) if the share of income to labor is high and likely to remain so, investors’ MRS variability will be 28 The

famous 25 Fama and French (1993) portfolios are the principal case in point.


459

relatively insensitive to events in the stock market, and thus the market risk premium is likely to be small, and vice versa. We note, however, that this ignores the operating leverage effect: a higher share of income to labor suggests a higher fundamental riskiness in the equity cash flow, something, per se, likely to increase the premium. For most all cases presented in this chapter, the latter effect dominates the Santos and Veronesi (2004) intuition. (2) If the share of income from wages is above average, it is likely to decline with the consequence that future dividend growth is likely to exceed consumption growth, leading to high asset prices and returns. The time variation in the asset risk premium suggested by these comments, however, is fully a feature of the distributional risk model. Regressing stock returns on lagged values of this variable leads to statistically significant coefficients and adjusted R2 that exceed what would be obtained using, e.g., the lagged dividend-price ratio as the explanatory variable. We suspect that a similarly good fit could be obtained using data generated by our model. Including this ratio as an explanatory variable also allows the model to outperform the standard CAPM in explaining the returns to the 25 Fama and French (1993) portfolios. Given the results obtained here, none of this is surprising.

11. CONCLUDING COMMENTS The principal message of this chapter is that asset return patterns may have less to do with business cycle systematic risk factors and more to do with distributional effects. What appears to be key is that the insurance mechanism demands payments from the firm (shareholders) to workers precisely at times when shareholder cash flow is already low. We showed that this mechanism acts as a standard habit formation mechanism but one that arises endogenously via shareholder-worker interactions. Not only is the model able to replicate simultaneously a broad class of financial and business cycle stylized facts, but it is also able to do so with very low CRRAs for both shareholders and workers, a fact that reinforces the significance of the insurance mechanism per se. Both the equity premium and security return volatilities, in particular, match the data well. Of additional interest is the equity/output ratio, q e /Y ; not only is this ratio of an average level typical of the economy, but its variation is also characteristic of what is observed. Financial leverage effects are also magnified to a degree unmatched in more standard representative agent paradigms. The latest trends, for the U.S. at least, point in the direction of a falling labor share. Simultaneously, competitive pressures under the force of globalization appear to render traditional internal-to-the-firm insurance mechanisms, which are at the heart of the present paper, harder and harder to maintain (the plight of American automobile manufacturers is a dramatic case in point). On those counts, our model leads us to anticipate a future decrease in equity premia. While our viewpoint is entirely model-specific, this prediction confirms those made recently by several researchers (Jagannathan, McGrattan and Scherbina (2001), McGrattan, and Prescott (2000), Fama and French (2002)).

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Our literature review together with the present contribution illustrates two distinct perspectives on the simultaneous replication of the business cycle and financial stylized facts. The first, as embodied in Jermann (1998) and Boldrin et al. (2001), focuses on representative agent economies in which the agent has simultaneously a strong desire to consume (habit formation) in conjunction with severe technological constraints to doing so (cost of adjusting capital and fixed labor supply in the case of Jermann (1998) and restrictions on shifting capital and labor intertemporally and across sectors in the case of Boldrin et al. (2001)). These features are required because the bulk of the representative agents’ consumption, in either case, comes from wage income, which, if the model is correctly calibrated to the business cycle stylized facts, is relatively smooth. Capital income must therefore be highly variable and of a pattern that tends to destabilize consumption. The technological constraints make the smoothing of capital income especially difficult. These features, per se, are nevertheless insufficient for an adequate replication of the financial stylized facts: the asset holding class must also display very high marginal risk aversion, something that habit formation preferences provide. In heterogeneous agent models with distributional risk, however, neither high levels of effective risk aversion for the shareholding class nor complex technological restrictions are required. From the perspective of asset pricing, the provision of income insurance by the shareholding class to the working class is sufficient (it must be added that a somehow-restricted labor supply may be required as well). We note that income insurance is fundamentally a constraint on prices (in particular, wages) rather than quantities. Our equivalence in Section 9 goes a long way toward harmonizing these perspectives by observing that partial income insurance is essentially equivalent to particular types of trading restrictions. Trading restrictions prevent outputs from being transferred across various dates and states, something that technological constraints equivalently accomplish. In addition, Section 5 demonstrates that labor income, relative to shareholder consumption, behaves much the same as a slow moving habit. In this sense, distributional risk serves a dual role.

References Abel, A. Asset prices under habit formation and catching up with the Jones. American Economic Review, Papers and Proceedings 80 (1990): 38–42. Bansal, R. Long run risk and risk compensation in equity markets. Working paper, Duke University Fuqua School of Business (2004). Boldrin, M., L. Christiano, and J. Fisher. Habit persistence, asset returns, and the business cycle. American Economic Review 91 (2001): 149–166. Boldrin, M., and M. Horvath. Labor contracts and the business cycle. Journal of Political Economy 103 (1995): 972–1004. Boldrin, M., and A. Peralta-Alva. What did (really) happen to the U.S. stock market: Accounting for the last 50 years. Mimeo, U., of Miami (2005). Campbell, J. Asset prices, consumption and the business cycle. Chapter 9 in J. B. Taylor, and M. Woodford, eds., Handbook of Macroeconomics. Elsevier, North-Holland, Amsterdam (1999): 1231–1303.


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Campbell, J. Y., and J. Cochrane. By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107 (1999): 205–251. Christiano, L. Solving the stochastic growth model by quadratic approximation and value-function iteration. Journal of Business and Economic Statistics 8 (1990): 23–26. Cochrane, J. Production based asset pricing and the link between stock returns and economic fluctuations. Journal of Finance 46 (1991): 207–234. Constantinides, G., and D. Duffie. Asset pricing with heterogeneous consumers. Journal of Political Economy 104 (1996): 219–240. Cooley, T. F., and E. C. Prescott. Economic growth and business cycles. Chapter 1 in T. F. Cooley, ed., Frontiers of Business Cycle Research, Princeton University Press, Princeton, NJ (1995). Danthine, J. P., and J. B. Donaldson. Methodological and empirical issues in real business cycle theory. European Economic Review 69 (1992): 41–64. Danthine, J. P., and J. B. Donaldson. Labor relations and asset returns. Review of Economic Studies 69 (2002): 41–64. Danthine, J. P., J. B. Donaldson, and R. Mehra. The equity premium and the allocation of income risk. Journal of Economic Dynamics and Control 16 (1992): 509–532. Dow, J. P. Real business cycles and labor markets with imperfectly flexible wages. European Economic Review 39 (1995): 1683–1696. Dunlop, J. The movement of real and money wage rates. Economic Journal 48 (1938): 413–434. Fama, E., and K. French. Common risk factors in the returns to stocks and bonds. Journal of Financial Economics 33 (1993): 3–56. Fama, E., and K. French. The equity premium. Journal of Finance 57 (2002): 637–659. Fama, E., and J. D. Macbeth. Risk, return and equilibrium: empirical tests. Journal of Political Economy 81 (1972): 607–636. Goetzman, W., and P. Jorion. A century of global stock markets. NBER Working Paper 5901 (1997). Gollin, D. Getting income shares right. Journal of Political Economy 110 (2002): 458–474. Gomes, J. F., A. Yaron, and L. Zhang, Asset pricing implications of firm’s financing decisions. Mimeo, University of Pennsylvania (2002). Guo, H. A simple model of limited stock market participation, Federal Reserve Bank of St. Louis Review, (May/June) (2001): 37–47. Guvenen, F. A parsimonious macroeconomic model for asset pricing. Habit formation or cross-sectional heterogeneity. Rochester Center for Economic Research Working Paper No. 499 (2005). Hansen, G. Indivisible labor and the business cycle. Journal of Monetary Economics 16 (1985): 309–327. Hodrick, R. J., and E. C. Prescott. Post-war U.S. business cycles: An empirical investigation. Journal of Money, Credit, and Banking 29 (1997): 1–16. Jagannathan, R. E., and Z. Wang. The conditional CAPM and the cross-section of expected returns. Journal of Finance 51 (1996): 3–53. Jagannathan, R., E. McGrattan, and A. Scherbinal, The declining U.S. equity premium. NBER Working Papers 8172, National Bureau of Economic Research, Inc (2001). Jaimovich, N., and S. Rebelo. Can news about the future drive the business cycle. Mimeo, Northwestern University (2006). Jermann, U. The equity premium implied by production. University of Pennsylvania (2005). Jermann, U. J. Asset pricing in production economies. Journal of Monetary Economics 41 (1998): 257–275. Kocherlakota, N. The equity premium: It’s still a puzzle. Journal of Economic Literature 34 (1996): 42–71. Lucas, R. E., Jr. Asset prices in an exchange economy. Econometrica 46 (1978): 1429–1444. Mankiw, G., and S. Zeldes. The consumption of stockholders and non stockholders. Journal of Financial Economics 29 (1991): 97–112. McGrattan, E. R., and E. C. Prescott. Is the stock market overvalued? Federal Reserve Bank of Minneapolis Quarterly Review (2000).

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Chapter 10 • Distribution Risk and Equity Returns McGrattan, E. R., and E. C. Prescott. Taxes, regulations, and the value of U.S. and U.K. corporations. Federal Reserve Bank of Minneapolis Staff Report (2005). Mehra, R. On the volatility of stock prices: An exercise in quantitative theory. International Journal of Systems Science 29 (1998): 1203–1211. Mehra, R. Finance. Journal of Economic Dynamics and Control 26 (2002): 1069–1074. Mehra, R., and E. C. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 22 (1985): 145–161. Mehra, R., and E. C. Prescott. The equity premium puzzle in retrospect. G. M. Constantinides, M. Harris, and R. Stulz, eds., Handbook of the Economics of Finance. North-Holland, Amsterdam (2003). Rosen, S. Implicit Contract Theory, Elgar (1994). Santos, J., and P. Veronesi. Labor income and predictable stock returns. Review of Financial Studies 19 (2006): 1–44. Stokey, N., and R. E. Lucas, Jr. with E. C. Prescott. Recursive Methods in Economic Dynamics. Harvard University Press, Cambridge, MA (1989). Tarshis, L. Changes in real and money wage rates. Economic Journal 49 (1938): 150–154.

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Urban J. Jermann

Discussion: Distribution Risk and Equity Returns Urban J. Jermann Wharton School of the University of Pennsylvania

This paper studies the returns to aggregate capital with shocks to the relative income shares of capital and labor. Such distribution risk is introduced into a basic version of a real business cycle model with endogenous capital accumulation. Distribution risk is assumed to originate in shifts in the balance of power between workers and capital owners. The model with distributional shocks and standard productivity shocks is shown to explain a number of asset pricing regularities that more standard models cannot. Specifically, it can generate an equity premium of several percentage points. There are two main mechanisms at work in the model: (1) limited asset market participation and (2) distributional shocks. First, it is assumed that there are two types of agents: workers and asset holders. Workers work and consume their income, but do not participate in asset markets. Asset holders live from the returns of their investments. This assumption of limited asset market participation is important because the pricing kernel in this model is a function of the consumption of the asset holders only. With this assumption, the model liberates itself from the tyranny of the smoothness of aggregate consumption which plagues representative agent models. The second key assumption is that wage rates are not determined in competitive markets but through some bargaining procedure. Shifts in the balance of power in this bargaining procedure are assumed to generate shocks to wages. These shocks have a powerful effect on the model’s equity premium. For more clarity, consider the constraint introduced into the model that effectively links wages wt , consumption of the asset holders ct , and the exogenously specified distributional shock μt : μt v (wt ) = u (ct ). Given the functional forms assumed in the paper, this can be written as wt = (ct μt )1/γ ,

(1)

464


with γ the curvature parameter in the workers’ utility function. Thus, wages rates, as well as labor income (hours worked are assumed constant), are determined by the distributional shock and by the consumption of the asset holders. A positive μt shock increases the wage rate and reduces the (unlevered) dividend dt distributed to shareholders: dt = outputt − it − wt n, where it is investment and n is hours worked. Because asset holders’ consumption equals (unlevered) dividends in equilibrium: ct = dt , μt shocks have a very powerful effect on the equity premium. As shown in Eq. (1) for dividends, the impact of the distributional shock also depends on the investment response. Because μt does not directly affect the incentives to invest at the margin—indeed it acts like a lump sum tax on firms’ profits—investment does not react much to this shock and to changes in the wage rate. For this reason, shocks to wage rates have such a powerful effect on dividends, and thus on consumption of the asset holders. The same model driven exclusively by productivity shocks generates a much smaller equity premium because investment movements smooth out dividends. For instance, a positive productivity shock increases the incentives to invest so that dividends respond much less positively than if investment had remained unchanged. Through its lucid analysis and its focus on variable-factor income shares, this paper makes a very nice contribution to the literature on the equity premium. The analysis of how distributional shocks and limited asset market participation affect asset prices in the basic RBC model with endogenous capital accumulation is very instructive. This paper extends the important work on related models by the first two of the authors here. The distributional shocks are in the line of research that has considered operational leverage such as, for instance, Abel (1994) and Cecchetti et al. (1993). The potential importance of limited stock market participation has been suggested by Mankiw and Zeldes (1991). In addition to being very useful in teaching the reader about these channels, the paper’s analysis also contains a potentially important message. Indeed, the assumption of constant factor income shares made by the overwhelming majority of RBC studies is not harmless. This assumption, in fact, hurts the model’s ability to explain asset prices. As such, the paper here suggests that deviations from constant factor shares deserve more attention. In the rest of this report, I will consider a few more specific points. 1. The quantitative success of the model in generating a large equity premium with log-utility and endogenous capital accumulation is impressive. For that reason it would be useful to further examine quantitative model implications and compare these to relevant empirical counterparts. Indeed, one advantage of studying limited participation and distributional risk in generating a large

465

Urban J. Jermann

equity premium—as opposed to, say, some exotic preference specification—is that existing empirical evidence can be compared to some key model implications. Concerning the crucial consumption of asset holders, I am somewhat concerned that the model’s implication may be hard to support empirically. In the paper, the consumption volatility of the asset holders as well as its correlation with dividends are key in generating the strong quantitative results. Given that (unlevered) dividends are equal in equilibrium to the asset holders’ consumption, the model-implied correlation of 1 is obviously not going to be found empirically. Concerning volatility, the benchmark model (Table 3, Case 1) reports consumption volatility of equity holders at 11.35 percent and 1.16 percent for aggregate consumption. That is, consumption of asset holders is approximately 10 times as volatile as aggregate consumption. The study by Mankiw and Zeldes (1991), for instance, reports shareholders’ consumption to be 1.5 times as volatile as total consumption. In addition to the consumption of asset holders, more attention to wage data would also be useful. Given the assumption of constant hours worked, obviously, the model cannot replicate empirical wage and labor income simultaneously. However, because of the well-known difficulties in measuring labor income shares, presenting additional evidence on wage data would be helpful. 2. I believe the economic message of the paper could be strengthened by a more structural specification of the source of distributional risk. The wage-setting equation through which distributional shocks enter the model is added as a constraint to the firm problem. Unfortunately, this equation doesn’t lend itself easily to a realistic economic interpretation from the point of view of a decentralized economy. It is essentially postulated as a reduced-form representation of the outcome of some contracting problem. Section 8 of the paper presents an alternative setup without wage bargaining that can generate an equation that is isomorphic to the bargaining equation assumed in the model. While the model in Section 8 can provide further support for the use of the proposed structure, it also highlights the danger of interpreting the positive quantitative results of the paper as support for the idea that non-competitive labor markets play an important role in the U.S. economy. 3. Despite the paper’s strong quantitative findings for the equity premium, I have some concerns about the ability of distributional shocks to do well for other important asset prices. Indeed, while equity risk premiums have been very large, risk premiums for long-term government bonds have been historically very small. Given that distributional shocks are inherently stationary, the pricing kernel here will have a tendency to generate large risk premiums for long-term bonds. To illustrate this point, assume that the log of the marginal utility of consumption has an AR(1) representation

ln Mt + 1 = ln β + ρ ln Mt + εt+1 ,

with εt + 1 ∼ N 0, σ 2 .

466


With some algebra, it is easy to see that the holding premium for a k-period discount bond equals Rt + 1,k ln E = σ 2 1 − ρ(k − 1) . f Rt Assuming stationary marginal utility, |ρ| < 1, then Rt + 1,k lim ln E = σ2. f k→∞ Rt Assuming a pricing kernel that is consistent with a U.S. historical Sharpe ratio for aggregate equity of (6.18/19.1) = 0.32 (which is the value implied by Table 1 in the paper), then, if ρ isn’t too far from 1, the risk premium for a long-term discount bound would be approximately at least 0.322 = 10.24 percent.1 Of course, the model here will not have such gigantic term premiums because the Sharpe ratio implied by its equity returns is only about (3.12/21.88) = 0.14 (see Table 3, Case 1). Nevertheless, even with this reduced ability to generate risk premiums, premiums for long-term bonds are likely to be on the large side relative to their historical counterparts. This problem could be somewhat alleviated by allowing for non-stationary productivity shocks. However, to the extent that the inherently stationary distributional shocks are responsible for most of the volatility of the pricing kernel, this problem is unlikely to entirely disappear in this class of models.

References Abel, A. Exact solutions for expected rates of return under markov regime switching: Implications for the equity premium puzzle. Journal of Money, Credit and Banking 26 (1994): 345–361. Cecchetti, S. G., P. Lam, and N. C. Mark. The equity premium and the risk free rate: Matching the moments. Journal of Monetary Economics 31 (1994): 21–46. Mankiw, N., and S. P. Zeldes. The consumption of stockholders and nonstockholders. Journal of Financial Economics 29 (1991): 97–112.

p

p

p

arbitrage requires E(Rt + 1 Mt + 1 /Mt ) = 0, then (ERt + 1 /StdRt + 1 )2 ≤ var(Mt + 1 /Mt )/E 2 (Mt + 1 /Mt ) ≈ var(Mt + 1 /Mt ). For the process here var(Mt + 1 /Mt ) ≈ var(ln[Mt + 1 /Mt ]) ≈ 2σ 2 /(1 + ρ) ≈ σ 2 , for ρ close to 1.

1 No

CHAPTER 11

The Worldwide Equity Premium: A Smaller Puzzle Elroy Dimson, Paul Marsh, and Mike Staunton∗ London Business School

1. Introduction 2. Prior Estimates of the Equity Premium 2.1. Expert Opinion 3. Long-Run International Data 3.1. The DMS Global Database: Composition and Start date 3.2. The DMS Global Database: General Methodology and Guiding Principles 4. Long-Run Historical Rates of Return 4.1. Extremes of History 4.2. The Long-Run Perspective 5. New Global Evidence on the Equity Premium 5.1. The Equity Premium Around the World 5.2. A Smaller Risk Premium 5.3. Survivorship of Markets 5.4. Survivorship Bias Is Negligible 6. Decomposing the Historical Equity Premium 6.1. Unanticipated Success 6.2. Decomposition of the Equity Premium 6.3. From the Past to the Future 7. Conclusion References Appendix 1: Decomposition of the Equity Premium Appendix 2: Data Sources for the DMS Database

469 471 472 474 475 477 479 480 483 486 487 489 490 492 493 493 495 497 500 501 505 507

∗ We

are grateful to Rajnish Mehra and an anonymous referee, participants at over 40 seminars, and the 37 individuals who contributed the dataset described in Appendix 2.


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468

Chapter 11 • The Worldwide Equity Premium: A Smaller Puzzle

Abstract We use a new database of long-run stock, bond, bill, inflation, and currency returns to estimate the equity risk premium for 17 countries and a world index over a 106-year interval. Taking U.S. Treasury bills (government bonds) as the risk-free asset, the annualized equity premium for the world index was 4.7 percent (4.0 percent). We report the historical equity premium for each market in local currency and U.S. dollars, and decompose the premium into dividend growth, multiple expansion, the dividend yield, and changes in the real exchange rate. We infer that investors expect a premium on the world index of around 3–3½ percent on a geometric mean basis, or approximately 4½–5 percent on an arithmetic basis. JEL Classification: G12, G15, G23, G31, N20 Keywords: Equity risk premium; long run returns; survivor bias; financial history; stocks, bonds, bills, inflation.

Elroy Dimson, Paul Marsh, and Mike Staunton

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1. INTRODUCTION In their seminal paper on the equity premium puzzle, Mehra and Prescott (1985) showed that the historical equity premium in the U.S.—measured as the excess return on stocks relative to the return on relatively risk-free Treasury bills—was much larger than could be justified as a risk premium on the basis of standard theory. Using the accepted neoclassical paradigms of financial economics, combined with estimates of the mean, variance, and autocorrelation of annual consumption growth in the U.S. economy and plausible estimates of the coefficient of risk aversion and time preference, they argued that stocks should provide at most a 0.35 percent annual risk premium over bills. Even by stretching the parameter estimates, they concluded that the premium should be no more than 1 percent (Mehra and Prescott (2003)). This contrasted starkly with their historical mean annual equity premium estimate of 6.2 percent. The equity premium puzzle is thus a quantitative puzzle about the magnitude, rather than the sign, of the risk premium. Ironically, since Mehra and Prescott wrote their paper, this puzzle has grown yet more quantitatively puzzling. Over the 27 years from the end of the period they examined to the date of completing this contribution, namely over 1979–2005, the mean annual U.S. equity premium relative to bills using Mehra– Prescott’s definition and data sources was 8.1 percent. Logically, there are two possible resolutions to the puzzle: either the standard models are wrong, or else the historical premium is misleading and we should expect a lower premium in the future. Over the last two decades, researchers have tried to resolve the puzzle by generalizing and adapting the Mehra–Prescott (1985) model. Their efforts have focused on alternative assumptions about preferences, including risk aversion, state separability, leisure, habit formation, and precautionary saving; incomplete markets and uninsurable income shocks; modified probability distributions to admit rare, disastrous events; market imperfections, such as borrowing constraints and transactions costs; models of limited participation of consumers in the stock market, and behavioral explanations. There are several excellent surveys of this work, including Kocherlakota (1996), Cochrane (1997), Mehra and Prescott (2003), and most recently, Mehra and Prescott (2008). While some of these models have the potential to resolve the puzzle, as Cochrane (1997) points out, the most promising of them involve “deep modifications to the standard models” and “every quantitatively successful current story . . . . still requires astonishingly high risk aversion.” This leads us back to the second possible resolution to the puzzle, namely that the historical premium may be misleading. Perhaps U.S. equity investors simply enjoyed good fortune and the 20th century for them represented the “triumph of the optimists” (Dimson, Marsh, and Staunton (2002)). As Cochrane (1997) puts it, maybe it was simply “100 years of good luck”—the opposite of the old joke about Soviet agriculture being the result of “100 years of bad luck.” This good-luck story may also be accentuated by country selection bias, making the historical data even more misleading. To illustrate this, consider the parallel with selection bias in the choice of stocks, and the task facing a researcher who wished to estimate the required risk premium and expected return on the common stock of Microsoft.

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It would be foolish to extrapolate from Microsoft’s stellar past performance. Its success and survival makes it non-typical of companies as a whole. Moreover, in its core business, Microsoft has a market share above 50 percent. Since, by definition, no competitor can equal this accomplishment, we should not extrapolate expected returns from this one example of success. The past performance of individual stocks is anyway largely uninformative about their future returns, but when there is ex post selection bias based on past success, historical mean returns will provide an upward biased estimate of future expected returns. That is one reason why equity premium projections are usually based on the performance of the entire market, including unsuccessful as well as successful stocks.1 For similar reasons, we should also be uncomfortable about extrapolating from a stock market that has survived and been successful, and gained a market share of above 50 percent. Organized trading in marketable securities began in Amsterdam in 1602 and London in 1698, but did not commence in New York until 1792. Since then, the U.S. share of the global stock market as measured by the percentage of overall world equity market capitalization has risen to around 50 percent (see Dimson, Marsh, and Staunton (2004)). This reflects the superior performance of the U.S. economy, as evidenced by a large volume of initial public offerings (IPOs) and seasoned equity offerings (SEOs) that enlarged the U.S. equity market, and the substantial returns from U.S. common stocks after they had gained a listing. No other market can rival this long-term accomplishment. Mehra and Prescott’s initial focus on the U.S. and the ready availability of U.S. data have ensured that much of the subsequent research prompted by their paper has investigated the premium within the context of the U.S. market. The theoretical work usually starts with the assumption that the equity premium is of the magnitude that has been observed historically in the U.S., and seeks to show why the Mehra–Prescott observations are not (quite so much of) a puzzle. Some empirical work has looked beyond the U.S., including Jorion and Goetzmann (1999) and Mehra and Prescott (2003). However, researchers have hitherto been hampered by the paucity of long-run equity returns data for other countries. Most research seeking to resolve the equity premium puzzle has thus focused on empirical evidence for the U.S. In emphasizing the U.S.—a country that must be a relative outlier—this body of work may be starting from the wrong set of beliefs about the past. The historically measured equity premium could also be misleading if the risk premium has been non-stationary. This could have arisen if, over the measurement interval, there have been changes in risk, or the risk attitude of investors, or investors’ diversification opportunities. If, for example, these have caused a reduction in the risk premium, this fall in the discount rate will have led to repricing of stocks, thus adding to the magnitude of historical returns. The historical mean equity premium will 1 Another key reason is that equilibrium asset pricing theories such as the CAPM or CCAPM assign a special role to the value-weighted market portfolio. However, our argument for looking beyond the U.S. is not dependent on the assumption that the market portfolio should necessarily be the world portfolio. Instead, we are simply pointing out that if one selects a country that is known after the event to have been unusually successful, then its past equity returns are likely to be an upward biased estimate of future returns.


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then overstate the prospective risk premium, not only because the premium has fallen over time, but also because historical returns are inflated by past repricings that were triggered by a reduction in the risk premium. In this paper, we therefore revisit two fundamental questions: how large has the equity premium been historically, and how big is it likely to be in the future? To answer these questions, we extend our horizon beyond just the U.S. and use a new source of long-run returns, the Dimson–Marsh–Staunton (2006) database, to examine capital market history in 17 countries over the 106-year period from 1900 to 2005. Initially, we use the DMS database to estimate the historical equity premium around the world on the assumption that the premium was stationary. We then analyze the components of the premium to provide insights into the impact on historical returns of (1) luck and (2) repricing resulting from changes in the underlying risk premium. This then enables us to make inferences about the likely future long-run premium. Our paper is organized as follows. The next section reviews previous estimates and beliefs about the size of the equity premium. Section 3 describes the new DMS global database and explains why it represents a significant advance over previous data. Section 4 utilizes the database to present summary data on long-run returns and to illustrate why we need long-run histories to estimate premiums with any precision—even if the underlying processes are non-stationary. Section 5 presents new evidence on the historical equity premium around the world, assuming stationarity. Section 6 decomposes historical equity premiums into several elements, documenting the contribution of each to historical returns. Section 7 uses this decomposition to infer expectations of the equity premium, discusses why these are lower than the historical realizations, and provides a summary and conclusion. There are two appendices, one formalizing the methodology behind our decomposition, and the other documenting our data sources.

2. PRIOR ESTIMATES OF THE EQUITY PREMIUM Prior estimates of the historical equity premium draw heavily on the U.S., with most researchers and textbooks citing just the American experience. The most widely cited source is Ibbotson Associates, whose U.S. database starts in 1926. At the turn of the millennium, Ibbotson’s estimate of the U.S. arithmetic mean equity premium from 1926–1999 was 9.2 percent. In addition, before the DMS database became available, researchers such as Mehra and Prescott (2003), Siegel (2002), and Jorion and Goetzmann (1999) used the Barclays Capital (1999) and Credit Suisse First Boston (CSFB) (1999) data for the U.K. In 1999, both Barclays and CSFB were using identical U.K. equity and Treasury bill indexes that started in 1919 and gave rise to an arithmetic mean equity premium of 8.8 percent. In recent years, a growing appreciation of the equity premium puzzle made academics and practitioners increasingly concerned that these widely cited estimates were too high. This distrust proved justified for the historical numbers for the U.K., which were wrong. The former Barclays/CSFB index was retrospectively constructed, and from 1919–1935, was based on a sample of 30 stocks chosen from the largest companies

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(and sectors) in 1935. As we show in Dimson, Marsh, and Staunton (2000), the index thereby suffered from ex post bias. It represented a potential investment strategy only for investors with perfect foresight in 1919 about which companies were destined to survive (survivorship bias). Even more seriously, it incorporated hindsight on which stocks and sectors were destined in 1919 subsequently to perform well and grow large (success bias).2 After correcting for this ex post selection bias, the arithmetic mean equity premium from 1919–35 fell from 10.6 percent to 5.2 percent. The returns on this index were also flattered by the choice of start date. By starting in 1919, it captured the post–World War I recovery, while omitting wartime losses and the lower prewar returns. Adding in these earlier years gave an arithmetic mean U.K. equity premium over the entire 20th century of 6.6 percent, some 2¼ percent lower than might have been inferred from the earlier, incorrect data for 1919–1999. The data used by Ibbotson Associates to compute the historical U.S. equity premium is of higher quality and does not suffer from the problems that afflicted the old U.K. indexes. Those believing that the premium is “too good to be true” have therefore pointed their finger of suspicion mainly at success bias—a choice of market that was influenced by that country’s record of success. Bodie (2002) argued that high U.S. and U.K. premiums are likely to be anomalous, and underlined the need for comparative international evidence. He pointed out that long-run studies are almost always of U.S. or U.K. premiums: “There were 36 active stock markets in 1900, so why do we only look at two? I can tell you—because many of the others don’t have a 100-year history, for a variety of reasons.” There are indeed relatively few studies extending beyond the U.S. and the U.K. markets. Mehra and Prescott (2003) report comparative premiums for France, Japan, and Germany. They find a similar pattern to the U.S., but their premiums are based on post-1970 data and periods of 30 years or less. Ibbotson Associates (now owned by Morningstar) computes equity premiums for 16 countries, but only from 1970. Siegel (2002) reports premiums for Germany and Japan since 1926, finding magnitudes similar to those in the U.S. market. Jorion and Goetzmann (1999) provide the most comprehensive long-run global study by assembling a database of capital gain indexes for 39 markets, 11 of which started as early as 1921. However, they were able to identify only four markets, apart from the U.S. and the U.K., with pre-1970 dividend information. They concluded that, “the high equity premium obtained for U.S. equities appears to be the exception rather than the rule.” But in the absence of reliable dividend information, this assertion must be treated with caution. We therefore return to this question using comprehensive total returns data in Section 5.

2.1. Expert Opinion The equity premium has thus been a source of controversy, even among experts. Welch (2000) studied the opinions of 226 financial economists who were asked to forecast 2 After becoming aware of our research, Barclays Capital (but not CSFB) corrected their pre-1955 estimates of U.K. equity returns for bias and extended their index series back to 1900.

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10

Arithmetic mean equity risk premium 8.9

Late 1998 August 2001

8.5

8

7.1

6

5.5

4

3.4

2 0

Ibbotson (1926–1997)

FIGURE 1

Key finance textbooks

Welch 30-year Welch 30-year premium premium

Welch 1-year premium

Estimated arithmetic equity premiums relative to bills, 1998 and 2001.

the average annual equity premium over the next 30 years. Their forecasts ranged from 1 percent to 15 percent, with a mean and median of 7 percent. No clear consensus emerged: the cross-sectional dispersion of the forecasts was as large as the standard error of the mean historical equity premium. Most respondents to the Welch survey would have viewed the Ibbotson Associates Yearbook as the definitive study of the historical U.S. equity premium. At that time, the most recent Yearbook was the 1998 edition, covering 1926–1997. The first bar of Figure 1 shows that the arithmetic mean equity premium based on the Yearbook data was 8.9 percent per annum.3 The second bar shows that the key finance textbooks were on average suggesting a slightly lower premium of 8.5 percent. This may have been based on earlier, slightly lower, Ibbotson estimates, or perhaps the authors were shading the estimates down. The Welch survey mean is in turn lower than the textbook figures, but since the respondents claimed to lower their forecasts when the equity market rises, this may reflect the market’s strong performance in the 1990s. At the time of this survey, academics’ forecasts of the long-run premium thus seemed strongly influenced by the historical record. Certainly, leading textbooks advocated the use of the historical mean, including Bodie, Kane, and Marcus (1999) and Brealey and Myers (2000). The latter states, “Many financial managers and economists believe that long-run historical returns are the best measure available.” This was supported by researchers such as Goyal and Welch (2007) who could not identify a single predictive variable that would have been of robust use for forecasting the equity premium, and recommended “assuming that the equity premium is ‘like it always has been’.” Even Mehra and Prescott (2003) state, “. . . over the long horizon the equity premium 3 This is the arithmetic mean of the one-year geometric risk premiums. The arithmetic mean of the one-year arithmetic risk premiums, i.e., the average annual difference between the equity return and the Treasury bill return, was slightly higher at 9.1 percent.

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is likely to be similar to what it has been in the past and the returns to investment in equity will continue to dominate that in T-bills for investors with a long planning horizon.” The survey and textbook figures shown in the second and third bars of Figure 1 indicate what was being taught at the end of the 1990s in the world’s top business schools and economics departments. But by 2001, longer-term estimates were gaining publicity. Our own estimate (Dimson, Marsh, and Staunton (2000)) of the U.S. arithmetic mean premium over the entire 20th century of 7.7 percent was 1.2 percent lower than Ibbotson’s estimate of 8.9 percent for 1926–1997. In August 2001, Welch (2001) updated his survey, receiving 510 responses. Respondents had revised their estimates downward by an average of 1.6 percent. They now estimated an equity premium averaging 5.5 percent over a 30-year horizon, and 3.4 percent over a one-year horizon (see Figure 1). Those taking part for the first time estimated the same mean premiums as those who had participated in the earlier survey. While respondents to the earlier survey had indicated that, on average, a bear market would raise their equity premium forecast, Welch reports that “this is in contrast with the observed findings: it appears as if the recent bear market correlates with lower equity premium forecasts, not higher equity premium forecasts.” The academic consensus now appears to be lower still (e.g., see Jagannathan, McGrattan, and Scherbina (2000) and Fama and French (2002)). Investment practitioners typically agree (see Arnott and Ryan (2001) and Arnott and Bernstein (2002)), and the latest editions of many textbooks have reduced their equity premium estimates (for a summary of textbook prescriptions, see Fernandez (2004)). Meanwhile, surveys by Graham and Harvey (2005) indicate that U.S. CFOs have reduced their forecasts of the equity premium from 4.65 percent in September 2000 to 2.93 percent by September 2005. Yet predictions of the long-term premium should not be so sensitive to short-term market fluctuations. Over this period, the long-run historical mean premium—which just a few years earlier had been the anchor of beliefs—has fallen only modestly, as adding in the years 2000–2005 reduces the long-run mean by just 0.4 percent, despite the bear market of 2000–2002. The sharp lowering of the consensus view about the future premium must therefore reflect more than this, such as new ways of interpreting the past, new approaches to forecasting the premium, or new facts about global longterm performance, such as evidence that the U.S. premium was higher than in most other countries.

3. LONG-RUN INTERNATIONAL DATA We have seen that previous research has been hampered by the quality and availability of long-run global data. The main problems were the short time series available and hence the focus on recent data, the absence of dividends, ex post selection bias, and emphasizing data that is “easy” to access. Historically, the most widely used database for international stock market research has been the Morgan Stanley Capital International (MSCI) index series, but the MSCI


475

data files start only in 1970. This provides a rather short history for estimating equity premiums and spans a period when equities mostly performed well, so premiums inevitably appear large. Researchers interested in longer-term data have found no shortage of earlier stock price indexes, but, as is apparent in Jorion and Goetzmann (1999), they have encountered problems over dividend availability. We show in Section 6 that this is a serious drawback, because the contribution of dividends to equity returns is of the same order of magnitude as the equity premium itself and since there have been considerable cross-country differences in average dividend yield. The absence of dividends makes it hard to generate meaningful estimates of equity premiums. Even for countries where long-run total returns series were available, we have seen that they sometimes suffered from ex post selection bias, as had been the case in the U.K. Finally, the data sources that predated the DMS database often suffered from “easy data” bias. This refers to the tendency of researchers to use data that is easy to obtain, excludes traumatic intervals such as wars and their aftermath, and typically relates to more recent periods. Dimson, Marsh, and Staunton (2002) identify the most widely cited prior data source for each of 16 countries and show that equity returns over the periods covered are higher than the 1900–2000 returns from the DMS database by an average of 3 percent per year. Easy data bias almost certainly led researchers to believe that equity returns over the 20th century were higher than was really the case.

3.1. The DMS Global Database: Composition and Start Date These deficiencies in existing data provided the motivation for the DMS global database. This contains annual returns on stocks, bonds, bills, inflation, and currencies for 17 countries from 1900–2005 and is described in Dimson, Marsh, and Staunton (2006a, 2006b). The countries include the U.S. and Canada, seven markets from what is now the Euro currency area, the U.K. and four other European markets that have not embraced the Euro, two Asia-Pacific markets, and one African market. Together, they made up 91 percent of total world equity market capitalization at the start of 2006, and we estimate that they constituted 90 percent by value at the start of our period in 1900 (see Section 5 for more details). The DMS database also includes four “world” indexes based on the countries included in the DMS dataset. There is, first, a World equity index: a 17-country index denominated in a common currency, here taken as U.S. dollars, in which each country is weighted by its starting-year equity market capitalization or, in years before capitalizations were available, by its GDP. Second, there is an analogous 16-country worldwide equity index that excludes the U.S. (“World ex-U.S.”). Third and fourth, we compute a World bond index and a World ex-U.S. bond index, both of which are constructed in the same way, but with each country weighted by its GDP. The DMS series all commence in 1900, and this common start date aids international comparisons. The choice of start date was dictated by data availability and quality. At first sight, it appears feasible to start earlier. Jorion and Goetzmann (1999) note

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that, by 1900, stock exchanges existed in at least 33 of today’s nations, with markets in 7 countries dating back another 100 years to 1800. An earlier start date would in principle be desirable, as a very long series of stationary returns is needed to estimate the equity premium with any precision. Even with non-stationary returns, a long time series is still helpful,4 and it would anyway be interesting to compare 19th-century premiums with those from later years. Indeed, some researchers report very low premiums for the 19th century. Mehra and Prescott (2003) report a U.S. equity premium of zero over 1802–1862, based on Schwert’s (1990) equity series and Siegel’s (2002) risk-free rate estimates, while Hwang and Song (2008) claim there was no U.K. equity premium puzzle in the 19th century, since bonds outperformed stocks. These inferences, however, are unreliable due to the poor quality of 19th-century data. The equity series used by Hwang and Song omits dividends and, before 1871, suffers from ex post bias and poor coverage. From 1871–1913, they use a broader index (Grossman (2002)), but this has problems with capital changes, omitted data, and stocks disappearing. Within the range of likely assumptions about these disappearances, Grossman shows that he can obtain a 1913 end value of anywhere between 400 and 1700 (1871 = 100). Mehra and Prescott (2003) list similar weaknesses in Schwert’s 1802–1871 U.S. data, such as the lack of dividends, tiny number of stocks, frequent reliance on single sectors, and likelihood of ex post bias. These flaws undermine the reliability of equity premium estimates for the 19th century. Unfortunately, better 19th-century U.K. equity indexes do not exist, and, until recently, Schwert’s series was the only source of pre-1871 U.S. data. However, most recently, Goetzmann and Ibbotson (2008) employ a new NYSE database for 1815– 1925 (see Goetzmann, Ibbotson, and Peng (2001)) to estimate the 19th-century U.S. equity premium. But they highlight two problems. First, dividend data is absent pre1825 and incomplete from 1825–1871. Equity returns for 1825–1871 are thus estimated in two ways based on different assumptions about dividends, producing two widely divergent estimates of the mean annual return, namely, 6.1 percent and 11.5 percent, which are then averaged. Second, since Treasury bills or their equivalents did not yet exist, the risk-free rate proves even more problematic and has to be estimated from risky bonds. These two factors make it hard to judge the efficacy of their 19th-century equity premium estimates. Returning to the question of the start date for the DMS database, it is clear that, even for the U.S., the world’s best-documented capital market, pre-1871 data is still problematic. Wilson and Jones (2002) observe that after 1871, U.S. equity returns are of higher quality; but while a few other DMS countries also have acceptable series over this period, most, including the U.K., have no suitable data prior to 1900. Before then, there are virtually no stock indexes to use as a starting point, and creating new 19th-century indexes would be a major task, requiring hand collection of stock data 4 P´ astor and Stambaugh (2001) show that a long return history is useful in estimating the current equity premium even if the historical distribution has experienced structural breaks. The long series helps not only if the timing of breaks is uncertain but also if one believes that large shifts in the premium are unlikely or that the premium is associated, in part, with volatility.


477

from archives.5 For practical purposes, 1900 is thus the earliest plausible common start date for a comparative international database.

3.2. The DMS Global Database: General Methodology and Guiding Principles The DMS database comprises annual returns and is based on the best-quality capital appreciation and income series available for each country, drawing on previous studies and other sources. Where possible, data was taken from peer-reviewed academic papers or highly rated professional studies. From the end point of these studies, the returns series are linked into the best, most comprehensive, commercial returns indexes available. The DMS database is updated annually (see Dimson, Marsh, and Staunton (2006a, 2006b)). Appendix 2 lists the data sources used for each country. To span the entire period from 1900, we link multiple index series. The best index is chosen for each period, switching when feasible to better alternatives, as they become available. Other factors equal, we have chosen equity indexes that afford the broadest coverage of their market. The evolution of the U.S. equity series illustrates these principles. From 1900–1925, we use the capitalization weighted Cowles Index of all NYSE stocks (as modified by Wilson and Jones (2002)); from 1926–1961, we use the capitalization weighted CRSP Index of all NYSE stocks; from 1962–1970, we employ the extended CRSP Index, which over this period also includes Amex stocks; and from 1971 on, we utilize the Dow Jones Wilshire 5000 Index, which contains over 7,000 U.S. stocks, including those listed on Nasdaq. The creation of the DMS database was in large part an investigative and assembly operation. Most of the series needed already existed, but some were long forgotten, unpublished, or came from research in progress. In other cases, the task was to estimate total returns by linking dividends to existing capital gains indexes. But for several countries, there were periods for which no adequate series existed. For example, U.K. indexes were of poor quality before 1962, and far from comprehensive thereafter. To remedy this, we compiled an index spanning the entire U.K. equity market for 1955– 2005 (Dimson and Marsh (2001)), while for 1900–1955, we built a 100-stock index by painstaking data collection from archives. Similarly, we used archive data to span missing subperiods for Canada, Ireland, Norway, Switzerland, and South Africa. Virtually all of the DMS countries experienced trading breaks at some point in their history, often in wartime. Jorion and Goetzmann (1999) provide a list and discuss the origins of these interruptions. In assembling our database, we needed to span these gaps. The U.K. and European exchanges, and even the NYSE, closed at the start of World War I, but typically reopened 4–6 months later. Similarly, the Danish, 5 The Dow Jones Industrial Average was, we believe, the first index ever published. It began in 1884 with 11 constituents. Charles Dow had neither computer nor calculator, hence his limited coverage. While today, computation is trivial, creating indexes more than 100 years after the event poses a major data challenge. While it is often fairly easy to identify hard-copy sources of stock prices, the real problems lie in identifying (1) the full population, including births, name changes, and deaths and their outcome, and (2) data on dividends, capital changes, shares outstanding, and so on. Archive sources tend to be poorer, or non-existent, the further back one goes in time.

478


Norwegian, Belgian, Dutch, and French markets were closed for short periods when Germany invaded in 1940, and even the Swiss market closed from May to July 1940 for mobilization. There were other temporary closures, notably in Japan after the Great Tokyo Earthquake of 1923. These relatively brief breaks were easy to bridge.6 But three longer stock exchange closures proved more difficult: Germany and Japan from toward the end of World War II, and Spain during the Civil War. We were able to bridge these gaps,7 but as markets were closed or prices were controlled, the end-year index levels recorded for Germany for 1943–1947, Japan for 1945, and Spain for 1936–1938 cannot be regarded as market-determined values. This needs to be borne in mind when reviewing arithmetic means, standard deviations, and other statistics relating to annual returns computed using these values. Over each of these stock exchange closures, more reliance can be placed on the starting and ending values than on the intermediate index levels. We are therefore still able to compute changes in investors’ wealth and geometric mean returns over periods spanning these closures. Finally, there was one unbridgeable discontinuity, namely, bond and bill (but not equity) returns in Germany during the hyperinflation of 1922–1923, when German bond and bill investors suffered a total loss of −100 percent. This episode serves as a stark reminder that, under extreme circumstances, bonds and bills can become riskier than equities. When reporting equity premiums for Germany, whether relative to bonds or bills, we thus have no alternative but to exclude the years 1922–1923. All DMS index returns are computed as the arithmetic average of the individual security returns, and not as geometric averages (an inappropriate method encountered in certain older indexes); and all the DMS security returns include reinvested gross (pretax) income as well as capital gains. Income reinvestment is especially important, since, as we saw above, many early equity indexes measure just capital gains and ignore dividends, thus introducing a serious downward bias. Similarly, many early bond indexes record only yields, ignoring price movements. Virtually all DMS equity indexes are capitalization weighted and are calculated from year-end stock prices, but in the early years, for a few countries, we were forced to use equally weighted indexes or indexes based on average- or mid-December prices (see Appendix 2).

6 Since the DMS database records annual returns, trading breaks pose problems only when they span a calendar year boundary. For example, at the start of World War I, the NYSE was closed from 31 July until 11 December 1914, so it was still possible to calculate equity and bond returns for 1914. However, the London Stock Exchange closed in July 1914 and did not reopen until 5 January 1915, so prices for the latter date were used as the closing prices for 1914 and the opening prices for 1915. A similar approach was adopted for French returns during the closure of the Paris Exchange from June 1940 until April 1941. 7 Wartime share dealing in Germany and Japan was subject to strict controls. In Germany, stock prices were effectively fixed after January 1943; the market closed in 1944 with the Allied invasion and did not reopen until July 1948. Both Gielen (1944) and Ronge (2002) provide data that bridges the gap between 1943 and 1948. In Japan, stock market trading was suspended in August 1945, and although it did not officially reopen until May 1949, over-the-counter trading resumed in May 1946. The Oriental Economist Index provides relevant stock return data. In Spain, trading was suspended during the Civil War from July 1936 to April 1939, and the Madrid exchange remained closed through February 1940; over the closure we assume a zero change in nominal stock prices and zero dividends.


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Our guiding principle was to avoid survivorship, success, look-ahead, or any other form of ex post selection bias. The criterion was that each index should follow an investment policy that was specifiable in advance, so that an investor could have replicated the performance of the index (before dealing costs) using information that would have been available at the time. The DMS database and its world indexes do, however, suffer from survivorship bias, in the sense that all 17 countries have a full 106-year history. In 1900, an investor could not have known which markets were destined to survive. Certainly, in some markets that existed in 1900, such as Russia and China, domestic equity and bond investors later experienced total losses. In Section 5, we assess the likely impact of this survivorship bias on our worldwide equity premium estimates. The DMS inflation rates are derived from each country’s consumer price index (CPI), although for Canada (1900–1910), Japan (1900), and Spain (1900–1914) the wholesale price index is used, as no CPI was available. The exchange rates are year-end rates from The Financial Times (1907–2005) and The Investors’ Review (1899–1906). Where appropriate, market or unofficial rates are substituted for official rates during wartime or the aftermath of World War II. DMS bill returns are in general Treasury bill returns, but where these instruments did not exist, we used the closest equivalent, namely, a measure of the short-term interest rate with the lowest possible credit risk. The DMS bond indexes are based on government bonds. They are usually equally weighted, with constituents chosen to fall within the desired maturity range. For the U.S. and U.K., they are designed to have a maturity of 20 years, although from 1900–1955, the U.K. bond index is based on perpetuals, since there were no 20-year bonds in 1900, and perpetuals dominated the market in terms of liquidity until the 1950s. For all other countries, 20-year bonds are targeted, but where these are not available, either perpetuals (usually for earlier periods) or shorter maturity bonds are used. Further details are given in Appendix 2. In summary, the DMS database is more comprehensive and accurate than the data sources used in previous research, and it spans a longer period. This allows us to set the U.S. equity premium alongside comparable 106-year premiums for 16 other countries and the world indexes, thereby helping us to put the U.S. experience in perspective.

4. LONG-RUN HISTORICAL RATES OF RETURN In this section, we use the DMS dataset to examine real equity market returns around the world. In Table 1, we compare U.S. returns with those in 16 other countries, and long-run returns with recent performance, to help show why we need long time series when analyzing equity returns. The second column of Table 1 reports annualized real returns over the early years of the 21st century, from 2000–2005, the most recent 6-year period at the time of writing. It shows that real equity returns were negative in 7 of the 17 countries and that the return on the world index was −1.25 percent. Equities underperformed bonds and bills (not shown here) in 12 of the 17 countries. Inferring the expected equity premium from

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TABLE 1 Real Equity Returns in 17 Countries, 1900–2005 Annualized returns (percent p.a.) 2000 to 1990 to 2005 1999

Country Belgium

Properties of annual (percent) real returns, 1900–2005

1900 to 2005

Arithmetic mean

Standard Standard Excess Serial error deviation Skewness kurtosis correlation

3.99

9.13

2.40

4.58

2.15

22.10

0.95

2.33

Italy

−0.73

6.42

2.46

6.49

2.82

29.07

0.76

2.43

0.03

Germany

−4.08

9.89

3.09

8.21

3.16

32.53

1.47

5.65

−0.12

France

−1.64

12.53

3.60

6.08

2.25

23.16

0.41

−0.27

0.19

2.48

12.16

3.74

5.90

2.12

21.88

0.80

2.17

0.32

10.91

8.25

4.28

7.08

2.62

26.96

2.37

11.69

−0.06

Switzerland

1.11

13.95

4.48

6.28

1.92

19.73

0.42

0.38

0.18

Japan

0.64

−5.23

4.51

9.26

2.92

30.05

0.49

2.36

0.19

Ireland

5.14

11.79

4.79

7.02

2.15

22.10

0.60

0.81

−0.04

World ex-U.S. (USD)

0.11

3.41

5.23

7.02

1.92

19.79

0.58

1.41

0.25

Denmark

9.41

7.52

5.25

6.91

1.97

20.26

1.83

6.71

−0.13

Netherlands

−5.41

17.79

5.26

7.22

2.07

21.29

1.06

3.18

0.09

United Kingdom

−1.34

11.16

5.50

7.36

1.94

19.96

0.66

3.69

−0.06

World (USD)

−1.25

7.87

5.75

7.16

1.67

17.23

0.13

1.05

0.15

4.32

8.28

6.24

7.56

1.63

16.77

0.09

−0.13

0.16

United States

−2.74

14.24

6.52

8.50

1.96

20.19

−0.14

−0.35

0.00

South Africa

11.05

4.61

7.25

9.46

2.19

22.57

0.94

2.58

0.05

7.78

8.98

7.70

9.21

1.71

17.64

−0.25

0.06

−0.02

−0.70

15.02

7.80

10.07

2.20

22.62

0.55

0.92

0.11

Spain Norway

Canada

Australia Sweden

0.23

returns over such a short period would be nonsense: investors cannot have required or expected a negative return for assuming risk. This was simply a disappointing period for equities. It would be just as misleading to project the future equity premium from data for the previous decade. The third column of Table 1 shows that, with the exception of one country, namely, Japan, which we discuss later, real equity returns between 1990 and 1999 were typically high. Over this period, U.S. equity investors achieved a total real return of 14.2 percent per annum, increasing their initial stake fivefold. This was a golden age for stocks, and golden ages are, by definition, non-typical, providing a poor basis for future projections.

4.1. Extremes of History While the 1990s and early 2000s were non-typical, they were not unique. The top panel of Table 2 highlights other noteworthy episodes of world political and economic history

481


TABLE 2 Real Equity Returns in Key Markets over Selected Periods Real rate of return (percent) over the period Period

Description

U.S.

U.K.

France

Germany

Japan

−50

−66

66

World

World ex-U.S.

Selected Episodes −18

−36

−20

−21

1914–18

World War I

1919–28

Post–WWI recovery

372

234

171

18

30

209

107

1929–31

Wall Street Crash

−60

−31

−44

−59

11

−54

−47

1939–48

World War II

24

34

−41

−88

−96

−13

−47

1949–59

Post–WWII recovery

426

212

269

4094

1565

517

670

1973–74

Oil shock/recession

−52

−71

−35

−26

−49

−47

−37

1980–89

Expansionary 80s

184

319

318

272

431

255

326

1990–99

90’s tech boom

279

188

226

157

−42

113

40

2000–02

Internet “bust”

−42

−40

−46

−57

−49

−44

−46

Periods with Highest Returns 1-year periods 2-year periods 5-year periods

Return Period

Return Period

Return

57

97

66

155

121

70

79

1933

1975

1954

1949

1952

1933

1933

90

107

123

186

245

92

134

1927–28

1958–59

1927–28

1958–59

1951–52

1932–33

1985–86

233

176

310

652

576

174

268

1924–28

1921–25

1982–86

1949–53

1948–52

1985–89

1985–89

Return

−38

−57

−40

−91

−86

−35

−41

Period

1931

1974

1945

1948

1946

1931

1946

Return

−53

−71

−54

−90

−95

−47

−52

1930–31

1973–74

1944–45

1947–48

1945–46

1973–74

1946–47

Period

Periods with Lowest Returns 1-year periods 2-year periods 5-year periods

Period

Return Period

−45

−63

−78

−93

−98

−50

−56

1916–20

1970–74

1943–47

1944–48

1943–47

1916–20

1944–48

Longest Runs of Negative Real Returns Longest

Return

runs over

Period

106 years

Number of Years

−7

−4

−8

−8

−1

−9

−11

1905–20

1900–21

1900–52

1900–54

1900–50

1901–20

1928–50

16

22

53

55

51

20

23

since 1900. It shows real equity returns over the five worst episodes for equity investors, and over four “golden ages” for the world indexes and the world’s five largest markets. These five markets are of interest not just because of their economic importance, but also because they experienced the most extreme returns out of all 17 countries in our database.

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The five worst episodes for equity investors comprise the two world wars and the three great bear markets—the Wall Street Crash and Great Depression, the first oil shock and recession of 1973–1974, and the 2000–2002 bear market after the internet bubble. While the world wars were in aggregate negative for equities, there were relative winners and losers, corresponding to each country’s fortunes in war. Thus in World War I, German equities performed the worst (−66 percent), while Japanese stocks fared the best (+66 percent), as Japan was a net gainer from the war. In World War II and its aftermath,8 Japanese and German equities were decimated (−96 percent and −88 percent, respectively), while both U.S. and U.K. equities enjoyed small positive real returns. Table 2 shows that the world wars were less damaging to world equities than the peacetime bear markets. From 1929–1931, during the Wall Street Crash and ensuing Great Depression, the world index fell by 54 percent in real, U.S. dollar terms, compared with 20 percent during World War I and 13 percent in World War II. For the U.S., Germany, and the world index this was the most savage of the three great bear markets, and from 1929–1931 the losses in real terms were 60 percent, 59 percent, and 54 percent, respectively. From peak to trough, the falls were even greater. Table 2 records calendar year returns, but the U.S. equity market did not start falling until September 1929, reaching its nadir in June 1932, 79 percent (in real terms) below its 1929 peak. British and Japanese investors, in contrast, suffered greater losses in 1973–1974 than during the 1930s. This was the time of the first OPEC oil squeeze after the 1973 October War in the Middle East, which drove the world into deep recession. Over 1973–1974, the real returns on U.K., U.S., Japanese, and world equities were −71 percent, −52 percent, −49 percent, and −47 percent, respectively. The last row of the top panel of Table 2 shows that the world equity index fell by almost as much (44 percent in real terms) in the bear market of 2000–2002, which followed the late 1990s internet bubble. Table 2 shows the returns over calendar years, and from the start of 2000 until the trough of the bear market in March 2003, the real returns on U.S., U.K., Japanese, and German equities were even lower, at −47 percent, −44 percent, −53 percent, and −65 percent, respectively. The top panel of Table 2 also summarizes real returns over four “golden ages” for equity investors. The 1990s, which we highlighted in Table 1 as a recent period of exceptional performance, was the most muted of the four, with the world index showing a real return of 113 percent. While the 1990s was an especially strong period for 8 To measure the full impact of World War II on German and Japanese equity returns, it is necessary to extend the period through to 1948 to include the aftermath of the war. This is because, as noted above, stock prices in Germany were effectively fixed after January 1943, and the exchanges closed in 1944 with the Allied invasion and did not reopen until July 1948, when prices could finally reflect the destruction from the war. Meanwhile, German inflation from 1943–1948 was 55 percent. In Japan, the stock market closed in 1944, but over-the-counter trading resumed from 1946 onwards. The sharp negative real returns recorded in 1945, 1946, and 1947 thus reflect the hyperinflation that raged from 1945 onward (inflation from 1945–1948 was 5,588 percent), the resumption of trading at market-determined prices in 1946, and the breakup of the zaibatsu industrial cartels and the distribution of their shares to the workforce.


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the U.S. market (279 percent real return), the world index was held back by Japan.9 The world index rose by appreciably more during the 1980s (255 percent in real terms) and the two post–world war recovery periods (209 percent in the decade after World War I and 517 percent from 1949–1959). During the latter period, a number of equity markets enjoyed quite staggering returns. For example, Table 2 shows that during these nascent years of the German and Japanese “economic miracles,” their equity markets rose in real terms by 4094 percent (i.e., 40.4 percent p.a.) and 1565 percent (29.1 percent p.a.), respectively. The second and third panels of Table 2 show the returns for, and dates of, the one-, two-, and five-year periods during which each country and the world indexes experienced their highest and lowest returns. The picture that emerges reinforces the discussion above: in nearly all cases, the best and worst periods are drawn from, and are subsets of, the episodes listed in the top panel. Note that the spreads between worst and best are wide. One-year real returns range from −35 percent to +70 percent (world), −38 percent to +57 percent (U.S.), −91 percent to +155 percent (Germany), and −86 percent to 121 percent (Japan). Five-year real returns extend from −50 percent to +174 percent (world), −45 percent to +233 percent (U.S.), −93 percent to +652 percent (Germany), and −98 percent to 576 percent (Japan). Finally, the bottom panel of Table 2 reports the longest period over which each country (or world index) has experienced a cumulative negative real return. It shows that for the U.S., the longest such period was the 16 years from 1905–1920, when the cumulative return was −7 percent. This reconfirms Siegel’s (2002) observation that U.S. investors have historically always enjoyed a positive real return as long as they have held shares for at least 20 years. However, Table 2 shows that investors in other countries have not been so fortunate, with Japan, France, and Germany suffering extended periods lasting over half a century during which cumulative equity returns remained negative in real terms. Dimson, Marsh, and Staunton (2004) report that three-quarters of the DMS countries experienced intervals of negative real stock market returns lasting for more than two decades.

4.2. The Long-Run Perspective The statistics presented in Tables 1 and 2 and the discussion in the previous section serve to emphasize the volatility of stock markets and the substantial variation in yearto-year and period-to-period returns. Clearly, because of this volatility, we need to examine intervals that are much longer than five years or a decade when estimating means or equity premiums. The fourth column of Table 1 (shown in boldface) illustrates the perspective that longer periods of history can bring by displaying real equity 9 Table 2 shows that Japan experienced a real return of −42 percent during the 1990s (equivalent to an annualized real return of −5.2 percent p.a. as shown in the third column of Table 1). At the start of the 1990s, the Japanese stock market was the largest in the world by market capitalization, with a 40.4 percent weighting in the world index, compared with 32.2 percent for the U.S. market. Japan’s poor performance, coupled with its high weighting in the world index, and even higher weighting (60 percent) in the world ex-U.S., naturally had a depressing effect on the returns on the world and world-ex U.S. indexes (see Table 2 and column 2 of Table 1).

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returns over the 106-year period 1900–2005. Clearly, these 106-year returns contrast favorably with the disappointing returns over 2000–2005 (second column), but they are much lower than the returns in the 1990s (third column). The remaining columns of Table 1 present formal statistics on the distribution of annual real returns over 1900–2005, and again, they emphasize how volatile stock markets were over this period. The arithmetic means of the 106 one-year real returns are shown in the fifth column. These exceed the geometric means (fourth column) by approximately half the variance of the annual returns. The standard deviation column shows that the U.S., U.K., Swiss, and Danish equity markets all had volatilities of around 20 percent. While this represents an appreciable level of volatility, these countries are at the lower end of the risk spectrum, with only Australia and Canada having lower standard deviations. The highest volatility markets were Italy, Japan, and Germany, with volatilities close to, or above, 30 percent. These high levels of volatility imply that the arithmetic means are estimated with large standard errors (see column six). We return to this issue when we discuss the precision of equity premium estimates at the end of this section. The skewness and excess kurtosis columns in Table 1 show that returns were positively skewed except in the U.S., and in most countries, they were noticeably more fat-tailed than would be expected if they were normally distributed.10 Finally, the serial correlation column shows that to a good approximation, returns are serially independent. The average serial correlation coefficient was 0.07, and only two out of 17 coefficients were significant at the 95 percent level—only slightly higher than the proportion that would be expected from chance. The fourth column of Table 1 shows that the 106-year annualized real return on U.S. equities was 6.5 percent. The equivalent real return on non-U.S. equities—from the perspective of a U.S. investor, and as measured by the world index excluding the U.S.— was lower at 5.2 percent. This lends initial support to the concern about success bias from focusing solely on the U.S. market. At the same time, the gap is not large, and it is also clear from Table 1 that the stock markets of several other countries performed even better than the U.S. market. Table 1 shows real returns in local currency terms, however, rather than equity premiums, and we defer presenting comprehensive comparisons of the latter until Section 5.

10 The

average coefficients of skewness and excess kurtosis for the 17 countries were 0.76 and 2.60. This is consistent with our expectation that the distribution of annual stock returns would be lognormal, rather than normal, and hence positively skewed. But when we examine the distribution of log returns (i.e., the natural logarithm of one plus the annual return), we find average skewness and kurtosis of −0.48 and 3.25, i.e., the skewness switches from positive to negative, and the distributions appear even more leptokurtic. This finding is heavily influenced by the extreme negative returns for Germany in 1948 and Japan in 1946. As noted in Section 3, German returns from 1943–1948 and Japanese returns from 1945–1946 must be treated with caution, as although the total return over these periods is correct, the values for individual years cannot be regarded as market-determined. The values recorded for Germany in 1948 and Japan in 1946 thus almost certainly include accumulated losses from previous years. Excluding Germany and Japan, the coefficients of skewness and kurtosis based on log returns were −0.20 and 1.40, which are much closer to the values we would expect if annual returns were lognormally distributed.

485


Annualized percentage historical risk premiums 60 Annual 40

Rolling

20 0 220 240 260 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 FIGURE 2 Annual and rolling 10-year U.S. equity premiums relative to bills, 1900–2005.

2000

However, to reinforce the importance of focusing on long-run data, we briefly preview the equity premium data for the U.S. market. The bars in Figure 2 show the year-by-year historical U.S. equity premium calculated relative to the return on Treasury bills over 1900–2005.11 The lowest premium was −45 percent in 1931, when equities earned −44 percent and Treasury bills 1 percent; the highest was 57 percent in 1933, when equities earned 57.6 percent and bills 0.3 percent. Over the entire 106-year interval, the mean annual excess return over Treasury bills was 7.4 percent, while the standard deviation was 19.6 percent. On average, therefore, this confirms that U.S. investors received a positive, and large, reward for exposure to equity market risk. Because the range of year-to-year excess returns is very broad, it could be misleading to label these as “risk premiums.” As noted above, investors cannot have expected, let alone required, a negative risk premium from investing in equities. Many low and all negative premiums must therefore reflect unpleasant surprises. Nor could investors have required premiums as high as the 57 percent achieved in 1933. Such numbers are quite implausible as a required reward for risk, and the high realizations must therefore reflect pleasant surprises. To avoid confusion, it is helpful to refer to a return in excess of the risk-free rate, measured over a period in the past, simply as an excess return or as the “historical” equity premium (rather than equity premium). When looking to the future, it is helpful to refer to the “expected” or “prospective” equity premium. 11 For

convenience, we estimate the equity premium from the arithmetic difference between the logarithmic return on equities and the logarithmic return on the riskless asset. Equivalently, we define 1 + Equity Premium to be equal to 1 + Equity Return divided by 1 + Riskless Return. Defined this way, the equity premium is a ratio and therefore has no units of measurement. It is identical if computed from nominal or real returns, or if computed from dollar or euro returns.

486


The 10-year excess returns were sometimes negative, most recently in the 1970s and early 1980s. Figure 2 also reveals several cases of double-digit 10-year premiums. Clearly, a decade is too brief for good and bad luck to cancel out, or for drawing inferences about investor expectations. Indeed, even with over a century of data, market fluctuations have an impact. Taking the U.K. as an illustration, the arithmetic mean annual excess return from 1900–1949 was only 3.1 percent, compared to 8.8 percent from 1950–2005. As over a single year, all we are reporting is the excess return that was realized over a period in the past. To quantify the degree of precision in our estimates, we can compute standard errors. Assuming that each year’s excess return is serially independent,12 the √ standard error of the mean historical equity premium estimate is approximately σ/ T, where σ is the standard deviation of the annual excess returns, and T is the period length in years. Since we have seen that σ was close to 20 percent for the U.S. market, this implies that the standard error of the mean historical equity premium estimated over 10 years is 6.3 percent, while the standard error using 106 years of data remains quite high at approximately 2 percent. Since we saw in Table 1 that most countries had a standard deviation that exceeded that of the U.S. market, the standard error of the mean equity premium is typically larger in non-American markets. When estimating the historical equity premium, therefore, the case for using long-run data is clear. Stock returns are so volatile that it is hard to measure the mean historical premium with precision. Without long-run data, the task is impossible, and even with over a century of data, the standard error remains high—even if we assume that the underlying series is stationary.

5. NEW GLOBAL EVIDENCE ON THE EQUITY PREMIUM Figure 3 shows the annualized (geometric mean) historical equity premiums over the 106-year period from 1900–2005 for each of the 17 countries in the DMS database, as well as the world index and the world excluding the U.S. market. Countries are ranked by the equity premium relative to bills (or the nearest equivalent short-term instrument), displayed as bars. The line plot shows each country’s equity premium relative to bonds (long-term government bonds). Since the world indexes are computed here from the perspective of a U.S. (dollar) investor, the world equity premiums relative to bills are calculated with reference to the U.S. risk-free (Treasury bill) rate. The world equity premiums relative to bonds are calculated relative to the world bond indexes. Figure 3 shows that equities outperformed both bills and bonds in all 17 countries over this period and that, in general, the equity premium was large. The chart lends support to the concern about generalizing from the U.S. experience by showing that the U.S. equity premium relative to bills was 5.5 percent compared with 4.2 percent for the rest of the world. But while noteworthy, this difference is not that large, and Figure 3 12 We saw in Table 1 that this was a good approximation for real returns, and the same holds true for excess returns. For the U.S., the serial correlation of excess returns over 1900–2005 was 0.00, while the average across all 17 countries was 0.05. For excess returns defined relative to bonds rather than bills, the average serial correlation was 0.04.

487


Annualized percentage return 7

Equity premium vs. bills 5.5

5 4 3

6.2

Equity premium vs. bonds

6

2.8 2.9

3.1

3.4

3.6

3.8

4.1 4.2

6.8 6.6 6.7

7.1

5.7

4.7 4.4 4.5 4.5

2 1 0

Bel Den Nor Spa Swi Ger* Ire WxUS UK Can Neth Wld US Swe SAf Ita Jap Fra Aus *Germany omits 1922–1923.

FIGURE 3

Worldwide annualized equity premiums, 1900–2005.

shows that several countries had larger premiums than the U.S. For the world index (with its large U.S. weighting), the premium relative to bills was 4.7 percent. The U.K. equity premium was a little below the world average at 4.4 percent. Relative to long bonds, the story for the 17 countries is similar, although on average, the premiums were around 0.8 percent lower, reflecting the average term premium, i.e., the annualized amount by which bond returns exceeded bill returns. The annualized U.S. equity premium relative to bonds was 4.5 percent compared with 4.1 percent for the world on an ex-U.S. basis. Across all 17 countries, the equity premium relative to bonds averaged 4.0 percent, and for the world index it was also 4.0 percent.13 Thus, while U.S. and U.K. equities have performed well, both countries are toward the middle of the distribution of worldwide equity premiums, and even the U.S. is not hugely out of line compared to other markets.

5.1. The Equity Premium Around the World Table 3 provides more detail on the historical equity premiums. The left half of the table shows premiums relative to bills, while the right half shows premiums relative to government bonds. In each half of the table we show the annualized, or geometric mean, equity premium over the entire 106 years (i.e., the data plotted in Figure 3); the arithmetic mean of the 106 one-year premiums; the standard error of the arithmetic 13 Over the entire period, the annualized world equity risk premium relative to bills was 4.74 percent, compared with 5.51 percent for the U.S. Part of this difference, however, reflects the strength of the dollar. The world risk premium is computed here from the world equity index expressed in dollars, in order to reflect the perspective of a U.S.-based global investor. Since the currencies of most other countries depreciated against the dollar over the 20th century, this lowers our estimate of the world equity risk premium relative to the (weighted) average of the local-currency-based estimates for individual countries.

488


TABLE 3 Annualized Equity Premiums for 17 Countries, 1900–2005 Historical equity premium (percent) relative to bills

Historical equity premium (percent) relative to bonds

Country

Geometric mean

Arithmetic mean

Standard error

Standard deviation

Geometric mean

Arithmetic mean

Standard error

Standard deviation

Australia

7.08

8.49

1.65

17.00

6.22

7.81

1.83

18.80

Belgium

2.80

4.99

2.24

23.06

2.57

4.37

1.95

20.10

Canada

4.54

5.88

1.62

16.71

4.15

5.67

1.74

17.95

Denmark

2.87

4.51

1.93

19.85

2.07

3.27

1.57

16.18

France

6.79

9.27

2.35

24.19

3.86

6.03

2.16

22.29

Germany

3.83

9.07

3.28

33.49

5.28

8.35

2.69

27.41

Ireland

4.09

5.98

1.97

20.33

3.62

5.18

1.78

18.37

Italy

6.55

10.46

3.12

32.09

4.30

7.68

2.89

29.73

Japan

6.67

9.84

2.70

27.82

5.91

9.98

3.21

33.06

Netherlands

4.55

6.61

2.17

22.36

3.86

5.95

2.10

21.63

Norway

3.07

5.70

2.52

25.90

2.55

5.26

2.66

27.43

South Africa

6.20

8.25

2.15

22.09

5.35

7.03

1.88

19.32

Spain

3.40

5.46

2.08

21.45

2.32

4.21

1.96

20.20

Sweden

5.73

7.98

2.15

22.09

5.21

7.51

2.17

22.34

Switzerland

3.63

5.29

1.82

18.79

1.80

3.28

1.70

17.52

U.K.

4.43

6.14

1.93

19.84

4.06

5.29

1.61

16.60

U.S.

5.51

7.41

1.91

19.64

4.52

6.49

1.96

20.16

Average

4.81

7.14

2.21

22.75

3.98

6.08

2.11

21.71

World-ex U.S.

4.23

5.93

1.88

19.33

4.10

5.18

1.48

15.19

World

4.74

6.07

1.62

16.65

4.04

5.15

1.45

14.96

∗ Germany

omits 1922–1923.

mean; and the standard deviation of the 106 one-year premiums. The geometric mean is, of course, always less than the arithmetic mean, the difference being approximately one-half of the variance of the historical equity premium. Table 3 shows that the arithmetic mean annual equity premium relative to bills for the U.S. was 7.4 percent compared with 5.9 percent for the world excluding the U.S. market. This difference of 1.5 percent again lends support to the notion that it is dangerous to extrapolate from the U.S. experience because of ex post success bias. But again we should note that Table 3 shows that the U.S. was by no means the country with the largest arithmetic mean premium. Indeed, on a strict ranking of arithmetic mean premiums, it was eighth largest out of 17 countries. Care is needed, however, in comparing and interpreting long-run arithmetic mean equity premiums. For example, Table 3 shows that, relative to bills, Italy had the highest arithmetic equity premium at 10.5 percent, followed by Japan at 9.8 percent, France at


489

9.3 percent, and Germany at 9.1 percent. Yet these four countries had below-average equity returns (see Table 1). Table 3 shows that part of the explanation lies in the high historical volatilities in these four markets, 32 percent, 28 percent, 24 percent and 33 percent, respectively. As we saw above, much of this volatility arose during the first half of the 20th century, during, or in the aftermath of, the world wars. In all four cases, therefore, the long-run equity premium earned by investors (the geometric mean) was well below the arithmetic mean. But this is only part of the story, since Table 3 shows that these countries still had above-average geometric equity premiums, despite their below-average equity market returns. (Italy, Japan, and France had above-average premiums relative to bills, while Italy, Japan, and Germany had above-average premiums relative to bonds). The explanation, of course, lies in the very poor historical bill and/or bond returns in these four countries, and we return in Section 5.3 to the issue of poor equity returns coinciding with poor bill and bond returns. Table 3 shows that both the U.S. and U.K. equity premiums relative to bills had similar standard deviations of close to 20 percent per annum and that only four other countries had standard deviations that were as low, or lower, than this. As noted above, the relatively high standard deviations for the equity premiums for the 17 countries, ranging from 17–33 percent, indicate that, even with 106 years of data, the potential inaccuracy in historical equity premiums is still fairly high. Table 3 shows that the standard error of the equity premium relative to bills is 1.9 percent for the U.S., and the range runs from 1.6 percent (Canada) to 3.3 percent (Germany).

5.2. A Smaller Risk Premium By focusing on the world, rather than the U.S., and by extending the time span to 1900–2005, the equity premium puzzle has become quantitatively smaller. We saw in Section 2 that, before our new database became available in 2000, the most widely cited number for the U.S. arithmetic mean equity premium relative to bills was the Ibbotson (2000) estimate for 1926–1999 of 9.2 percent. Table 3 shows that by extending the time period backwards to include 1900–1925 and forwards to embrace 2000–2005, while switching to more comprehensive index series, the arithmetic mean equity premium shrinks to 7.4 percent. Table 3 also shows that the equivalent world equity premium over this same period was 6.1 percent. But while the puzzle has become smaller than it once was, 6.1 percent remains a large number. Indeed, Mehra and Prescott’s original article documented a premium of 6.2 percent, albeit for a different time period. As we noted in the introduction to this chapter, the equity premium, and hence the equity premium puzzle, continued to grow larger in the years after their paper was written. By extending the estimation period, and expanding our horizons to embrace the world, we have simply succeeded in reducing the puzzle back down to the magnitude documented in Mehra–Prescott’s original paper. If 6.2 percent was a puzzle, it follows that 6.1 percent is only a very slightly smaller puzzle. In terms of the empirical evidence, if we are to further shrink our estimate of the expected premium, two further possibilities remain. The first is that our world index is

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still upward biased because of survivorship bias in terms of the countries included. The second possibility relates to “good luck” and/or a systematic repricing of equities and their riskiness to investors over the last century. As we have seen, however, although the U.S. equity market has performed well, it was not a massive outlier. The challenge for the good luck/repricing hypothesis is thus to explain not just why the U.S. had “100 years of good luck,” but why the rest of the world was almost as fortunate. In the next subsection, we assess the possible impact of survivorship bias. Section 6 then addresses the issues of good luck and repricing.

5.3. Survivorship of Markets Several researchers, most notably Brown, Goetzmann, and Ross (1995) and Jorion and Goetzmann (1999), have suggested that survivorship bias may have led to overestimates of the historical equity premium. Li and Xu (2002) argue on theoretical grounds that this is unlikely to explain the equity premium puzzle, since, for survival models to succeed, the ex ante probability of long-term market survival has to be extremely small, which they claim contradicts the history of the world’s financial markets. In this section, we look at the empirical evidence on returns and survivorship, and reach the same conclusion as Li and Xu, namely that concerns over survivorship are overstated, especially with respect to true survivorship bias, namely, the impact of markets that failed to survive. In practice, however, the term “survivorship bias” is often used to also embrace ex post success bias as well as true survivorship bias. By comparing U.S. history with that of 16 other countries, we have already addressed the issue of success bias. While a legitimate concern, we are still left with a high historical 17-country world equity premium. Mehra (2003) has also noted that, with respect to its impact on the equity premium, success bias is partly mitigated by the tendency of successful markets to enjoy higher bond and bill returns, as well as higher equity returns; similarly, unsuccessful markets have tended to have lower real returns for both government securities and equities. In other words, there has been a positive correlation between real equity and real bill (or bond) returns.14 Among markets with high ex post equity premiums, there are naturally countries with excellent equity performance (like Australia); but there are also countries whose below-average equity returns nevertheless exceeded their disastrous bond returns (like Germany or Japan). Consequently, the cross-sectional dispersion of equity premiums is narrower than the cross-sectional dispersion of equity returns. Our equity premiums are, of course, measured relative to bills and bonds. In a number of countries, these yielded markedly negative real returns, often as a result of periods of very high or hyperinflation. Since these “risk-free” returns likely fell below investor expectations, the corresponding equity premiums for these countries are arguably overstated. Even this is not clear, however, as equity returns would presumably have 14 Over

the entire 106-year period, the cross-sectional correlation between the 17 real equity and 17 real bill (bond) returns was 0.63 (0.66). Measured over 106 individual years, the time-series correlations between real equity and real bill returns ranged from 0.01 in The Netherlands to 0.44 in Japan, with a 17-country mean correlation of 0.22, while the time-series correlations between real equity and real bond returns ranged from 0.11 in The Netherlands to 0.55 in the U.K., with a 17-country mean correlation of 0.37.


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been higher if economic conditions had not given rise to markedly negative real fixedincome returns. Depressed conditions were a particular feature of the first half of the 20th century, a period in which hyperinflations were relatively prevalent.15 Had economic conditions been better, it is possible that the equity premium could have been larger. Similarly, it could be argued that in the more successful economies, the ex post bill and bond returns may, over the long run, have exceeded investors’ expectations. We concluded earlier that provided a very long-run approach is taken, inferences from the U.S. do not appear to have given rise to very large overestimates of the historical world equity premium. It is still possible, however, that our world index overstates worldwide historical equity returns by omitting countries that failed to survive. The most frequently cited cases are those of Russia and China, whose equity markets experienced a compound rate of return of −100 percent.16 However, there are other stock markets, apart from Russia and China, which we have so far been unable to include in our sample due to data unavailability.17 At noted earlier, at the start date of our database in 1900, stock exchanges already existed in at least 33 of today’s nations. Our database includes 17 of these, and we would ideally like to assess their importance in terms of market capitalization relative to the countries for which we have no data. Unfortunately, the required data is not available. Such aggregate data was neither recorded nor even thought of in 1900.18 Rajan and Zingales (2003), however, do report a set of market capitalization to GDP ratios for 1913. By combining these with Maddison (1995) GDP data, coupled with some informed guesses for countries not covered by Rajan and Zingales, we can estimate approximate equity market capitalizations at that date. Based on these estimates, it is clear that the 17 DMS database countries dominated the early 20th-century world equity market. The largest omitted market is Russia, which we estimate in those days represented just under 5 percent of total world capitalization. Next is Austria-Hungary, which then incorporated Austria, Hungary, the Czech Republic, Slovakia, Slovenia, Croatia, Bosnia, and parts of modern-day Ukraine, Poland, and even Italy (Trieste), and which accounted for some 2 percent of world 15 In

our sample of countries over 1900–1949, the cross-sectional correlation between real equity and real bill (bond) returns was 0.68 (0.80). The time-series correlations between annual real equity and real bill (bond) returns had a 17-country mean of 0.31 (0.42). 16 It could be argued that the nationalization of corporations in Russia after the revolution of 1917 and in China after the communist victory in 1949 represented a redistribution of wealth, rather than a total loss. But this argument would not have been terribly persuasive to investors in Russian and Chinese equities at the time. It is possible, however, that some small proportion of equity value was salvaged in Russian and Chinese companies with large overseas assets, e.g., in Chinese stocks with major assets in Hong Kong and Formosa (now Taiwan). 17 We are endeavoring to assemble total return index series over 1900–2005 for countries such as New Zealand, Finland, and Austria; and we believe that, in principle, series for Argentina, India, Hong Kong, and other markets might also be compiled. 18 The few snippets of historical data that exist, e.g., Conant (1908), are expressed in terms of the nominal value of the shares outstanding rather than the total market value of the shares. Furthermore, figures are often given only for the total nominal value of all securities, rather than that of equities. For the U.S., U.K., and two other countries, we have meticulously constructed market capitalization data from archival sources relating to individual stocks. But for many of the other markets, it is possible that even the disaggregated archive source data may not have survived from the end of the 19th century to the present time.

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capitalization. Data described in Goetzmann, Ukhov, and Zhu (2007) suggests that the Chinese equity market accounted for 0.4 percent of world equity market capitalization in 1900. In addition, there was a group of Latin American markets, including Argentina, Brazil, Mexico, and Chile, that in total made up around 1½ percent of overall capitalization; and a number of small markets that total less than 1 percent.19 In addition to Russia and China, several other exchanges from 1900 did not survive World War II and ended in disaster, notably those in Czechoslovakia (now the Czech Republic and Slovakia), Hungary, and Poland (though these three countries were not independent states in 1900, being part of the Russian and the Austria-Hungary empires). We believe that the DMS database accounted for 90 percent of world equity capitalization at the start of the 20th century, and that omitted countries represented just 10 percent.

5.4. Survivorship Bias Is Negligible Our estimates of the equity premium are based on 17 surviving markets and, as noted earlier, ignore at least 16 non-surviving markets. To quantify the global impact of omitted markets, it is unnecessary to focus on individual markets as in Li and Xu (2002). We assume the annualized historical equity return for markets that survived for T years was Rsurvivors and that for markets missing from the DMS database, it was Romitted . Assume a proportion S of the worldwide equity market survived the entire period. Then the cumulative worldwide equity premium ERPworldwide is given by: (1 + ERPworldwide )T = [S(1 + Rsurvivors )T + (1 − S)(1 + Romitted )T ] /[(1 + Rriskfree )T ], (1) where Rriskfree is the risk-free interest rate for the reference country. An extreme assumption would be that all omitted markets became valueless, namely Romitted = −1; and that this outcome occurred for every omitted country in a single disastrous year, rather than building up gradually. The worldwide equity premium, incorporating omitted as well as surviving markets, would therefore be given by: (1 + ERPworldwide ) = S1/T (1 + Rsurvivors )/(1 + Rriskfree ) = S1/T (1 + ERPsurvivors ),

(2)

where ERPsurvivors is the historical equity premium for markets that survived. In our case, we estimate the proportion of the world equity market capitalization that survived was at least S = 0.9, and our time horizon is T = 106 years. To account for the omission of markets that existed in 1900 but did not survive, we must therefore adjust the ex post equity premium of the 17-country world index using a factor of 19 The

Latin American stock markets suffered several episodes of political and economic instability and hyperinflation; today, they account for some 1.15 percent of world market capitalization, which is roughly three-quarters of their weighting in 1913. The other markets, that in 1913 totaled less than 1 percent of world market capitalization, today account for some 2.3 percent of the world market; this group includes countries such as Egypt, Finland, Greece, Hong Kong (China), India, New Zealand, and Sri Lanka.


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S1/T = 0.91/106 = 0.999. The survivorship bias in the estimated equity premium is therefore the following: ERPsurvivors − ERPworldwide = (1 − S1/T )(1 + ERPsurvivors ) = (1 − 0.999)(1 + ERPsurvivors ) ≈ 0.001,

(3)

where the final approximation reflects the fact that ERPsurvivors is an order of magnitude below 1. We see that, at most, survivorship bias could give rise to an overstatement of the geometric mean risk premium on the world equity index by about one-tenth of a percentage point. If disappearance were a slower process, the index weighting of countries destined to disappear would have declined gradually and the impact of survivorship bias would have been even smaller. Similarly, if omitted markets did not all become valueless, the magnitude of survivorship bias would have been smaller still. While there is room for debate about the precise impact of the bias arising because some, but not all, equity markets experienced a total loss of value, the net impact on the worldwide geometric mean equity premium is no more than 0.1 percent. The impact on the arithmetic mean is similar.20 At worst, an adjustment for market survivorship appears to reduce the arithmetic mean world equity premium relative to bills from around 6.1 percent (see Table 3) to approximately 6.0 percent. Thus the equity premium puzzle has once again become smaller, but only slightly so.

6. DECOMPOSING THE HISTORICAL EQUITY PREMIUM The conventional view of the historical equity premium is that, at the start of each period, investors make an unbiased, albeit inaccurate, appraisal of the end-of-period value of the stock market. Consequently, the ex post premium, averaged over a sufficiently long interval, is expected to be a relatively accurate estimate of investors’ expectations. A key question is whether the historical premium may nevertheless be materially biased as a proxy for expectations because the past was in some sense unrepresentative. For instance, investors may have benefited from a century of exceptional earnings, or stock prices may have enjoyed a major, but non-sustainable, expansion in their valuation ratios. Our argument, which has some roots in Mehra and Prescott (1988), is that the historical equity premium may have beaten expectations not because of survivorship, but because of unanticipated success within the equity market. This analysis therefore draws on, and complements, Fama and French (2002), Ibbotson and Chen (2003), and Arnott and Bernstein (2003).

6.1. Unanticipated Success To examine whether history may have witnessed exceptional earnings and/or expanding valuation ratios, consider how the stock market’s past performance could, over multiple 20 It is duplicative to derive this formally. The intuition involves disappearance of 10 percent of the value of the market over a century, which represents a loss of value averaging 0.1 percent per year.

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decades, be below or above expectations. The 20th century opened with much promise, and only a pessimist would have believed that the next 50 years would involve widespread civil and international wars, the 1929 Crash, the Great Depression, episodes of hyperinflation, the spread of communism, the conflict in Korea, and the Cold War. During 1900–1949, the annualized real return on the world equity index was 3.5 percent, while for the world excluding the U.S. it was just 1.5 percent. By 1950, only the most rampant optimist would have dreamed that over the following half-century, the annualized real return on world equities would be 9.0 percent. Yet the second half of the 20th century was a period when many events turned out better than expected. There was no third world war, the Cuban missile crisis was defused, the Berlin Wall fell, the Cold War ended, productivity and efficiency accelerated, technology progressed, and governance became stockholder-driven. As noted by Fama and French (2002), among others, the 9.0 percent annualized real return on world equities from 1950 to 1999 probably exceeded expectations. In many countries, valuation ratios expanded, reflecting—at least in part—reduced investment risk. Over the course of the 20th century, the price-dividend ratio rose in all the DMS countries. Davis et al. (2000) and Siegel (2002) report that for the U.S. over the period since the 1920s, the aggregate stock market price-earnings and price-to-book ratios also rose, and Dimson, Nagel, and Quigley (2003) make similar observations for the U.K. In 1900, investors typically held a limited number of domestic securities from a few industries (Newlands (1997)). As the century evolved, new industries appeared, economic and political risk declined, closed- and open-ended funds appeared, liquidity and risk management improved, institutions invested globally, and finally, wealthier investors probably became more risk-tolerant. Yet even if their risk tolerance were unchanged, as equity risk became more diversifiable, the required risk premium is likely to have fallen. These trends must have driven stock prices higher, and it would be perverse to interpret higher valuation ratios as evidence of an increased risk premium. Furthermore, insofar as stock prices rose because of disappearing barriers to diversification, this phenomenon is non-repeatable and should not be extrapolated into the future. To unravel whether 20th century equity premiums were on balance influenced by exceptional earnings and expanding valuation ratios, in Appendix 1 we decompose long-term premiums into several elements.21 Appendix 1 shows that the historical equity premium is equal to the sum of the growth rate of real dividends, expansion in the pricedividend ratio, the mean dividend yield, and the change in the real exchange rate, less the risk-free real interest rate. Provided the summations and subtractions are geometric, this relationship is an identity. 21 In

outline, let Gdt be the growth rate of real dividends; GPDt be the rate at which the price/dividend ratio has expanded; Yt = Dt /Pt be the dividend yield, the ratio of aggregate dividends paid during period t divided by the aggregate stock price at the end of period t; Xt be the change in the real exchange rate; and Rft be the risk-free real interest rate. The geometric mean from period 1 through period t, denoted by boldface italic, is calculated like this for all variables: (1 + Yt ) = [(1 + Y1 )(1 + Y2 ) . . . (1 + Yt )]1/t . Appendix 1 shows that the equity risk premium is given by (1 + ERPt ) = (1 + Gdt )(1 + GPDt )(1 + Yt )(1 + Xt )/(1 + Rf t ), where boldface italic indicates a t-period geometric mean.

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6.2. Decomposition of the Equity Premium Table 4 reports these five components of the equity premium for each country. The first two columns show the growth rate of real dividends and the expansion in the pricedividend ratio. There is a widespread belief, largely based on the long-term record of the U.S. (Siegel (2002)), that nominal dividends can be expected to grow at a rate that exceeds inflation. In fact, only three countries have recorded real dividend growth since 1900 of more than 1 percent per year, and the average growth rate is −0.1 percent, i.e., the typical country has not benefited from dividends (or, in all likelihood, earnings) growing faster than inflation. Equally, there is the belief that superior stock market performance may be attributed to the expansion of valuation ratios. While there is some truth in this, it should not be overstated. Over the last 106 years, the price-dividend ratio of the average country grew by just 0.6 percent per year. Given the

TABLE 4 Decomposition of the Historical Equity Premium for 17 Countries, 1900–2005

Country

Real dividend growth rate

plus*

plus

plus

minus

equals

Expansion in the P/D ratio

Geometric mean dividend yield

Change in real exchange rate

U.S. real interest rate

Equity premium for U.S. investors

Australia

1.30

0.46

5.83

−0.24

0.96

6.42

Belgium

−1.57

0.08

3.95

0.62

0.96

2.05

0.72

0.98

4.46

−0.04

0.96

5.18

Denmark

−0.87

1.43

4.68

0.47

0.96

4.74

France

−0.74

0.42

3.93

−0.14

0.96

2.47

Germany

−1.54

0.97

3.69

0.23

0.96

2.35

Ireland

−0.25

0.38

4.66

0.25

0.96

4.05

Italy

−1.46

−0.08

4.05

0.10

0.96

1.58

Japan

−2.39

1.59

5.39

0.32

0.96

3.85

Netherlands

−0.16

0.41

5.00

0.27

0.96

4.54

Norway

−0.25

0.50

4.02

0.25

0.96

3.54

Canada

0.91

0.31

5.95

−0.80

0.96

5.38

−0.62

0.24

4.13

0.00

0.96

2.75

Sweden

2.88

0.67

4.09

−0.05

0.96

6.72

Switzerland

0.32

0.60

3.52

0.72

0.96

4.22

U.K.

0.61

0.18

4.68

−0.03

0.96

4.46

U.S.

1.32

0.75

4.36

0.00

0.96

5.51

South Africa Spain

−0.10

0.58

4.49

0.11

0.96

4.11

Std deviation

1.32

0.45

0.71

0.35

0.00

1.51

World (USD)

0.77

0.68

4.23

0.00

0.96

4.74

Average

*Note: Premiums are relative to bill returns. All summations and subtractions are geometric. Units are annualized percentage return.

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improved opportunities for stock market diversification, 0.6 percent seems a modest contribution to the historical equity premium. Each country’s real (local currency) capital gain is attributable to the joint impact of real dividend growth and expansion in the price-dividend ratio. Although the real capital gain is not reported explicitly in Table 4, note that only two countries achieved a real, local-currency capital gain of at least 2 percent per year: the U.S. (2.1 percent) and Sweden (3.6 percent). We should be cautious about extrapolating from these relatively large rates of capital appreciation to other markets around the world. The middle column of Table 4 is the geometric mean dividend yield over the 106year sample period. Averaged across all 17 countries, the mean dividend yield has been 4.5 percent, though it has been as large as 6.0 percent (in South Africa) and as low as 3.5 percent (in Switzerland). Interestingly, the countries whose mean dividend yield is closest to the cross-sectional average are Canada (4.5 percent) and the U.S. (4.4 percent). Drawing on Grullon and Michaely (2002) and Mauboussin (2006) to adjust for the impact of repurchases,22 which are more important in the U.S. than elsewhere, that country’s (adjusted) historical dividend yield rises to approximately 4.7 percent, which is just above the (unadjusted) 17-country average of 4.5 percent. To examine the equity premium from the perspective of a global investor located in a specific home country, such as the U.S., we convert from real, local-currency returns to real, common-currency returns. Taylor (2002) demonstrates that, over the very long term, exchange rate changes reflect purchasing power changes. It is unsurprising, then, to see that the annualized change in our 17 countries’ real exchange rate averages only 0.1 percent per year and that every country’s real exchange rate change was within the range ±1 percent. Note that, for the average country, the capital gain in real U.S. dollars (the sum of the second, third, and fifth columns) was just 0.6 percent per year (not reported in Table 4). Measured in real U.S. dollars, only two countries achieved a capital gain that exceeded 2 percent per year. Nine countries achieved a real U.S. dollar capital gain that was between zero and +2 percent; and six achieved between zero and −2 percent. The annualized real, local-currency returns were reported for all countries in Table 1; across all 17 countries, the average 106-year return is 5.0 percent. The real, USDdenominated returns (the sum of the second to the fifth columns in Table 4) average 5.1 percent. Deducting the U.S. risk-free interest rate of 0.96 percent in real terms, the equity premium for a U.S. investor buying stocks in each of the 17 markets is as listed on the right of Table 4: on average, the premium is 4.1 percent.

22 Since the 1980s, U.S. yields have been low relative to the past partly because, under prior tax rules, companies could return capital to shareholders more effectively on an after-tax basis by means of stock repurchases. From 1972–2000, Grullon and Michaely (2002) estimate that annual repurchases averaged 38.0 percent of cash dividends (57.5 percent from 1984–2000), while over 1977–2005, Mauboussin (2006) estimates the average to be 64.8 percent. Adding repurchases to the yield, the “adjusted dividend yield” for the U.S. rises from its raw historical average of 4.4 percent to 4.7 percent, whether we use the data from Grullon and Michaely (2002) or Mauboussin (2006). The impact of a similar adjustment to other countries’ dividend yield is smaller and often zero (see Rau and Vermaelen (2002)).


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The ex post equity premiums on the right of Table 4 vary cross-sectionally for two reasons: the expected reward for risk, and the impact of chance. In 1900, the expected premium for higher risk markets may have merited a high reward that was subsequently realized; if Australia, Canada, South Africa, and Sweden were such economies, they achieved relatively large ex post premiums of over 5 percent. The expected premium for safer markets may have been low; if these markets are typified by Belgium, France, Germany, Italy, and Spain, their ex post premiums were below 3 percent. However, this rationalization is not a credible explanation for historical performance. It is more likely that, in 1900, investors underestimated the probability of wars in Europe, not to mention the ultimate value of resource-rich economies like the U.S. and Canada. National returns thus probably had more to do with noise than with the expected premium in 1900, and averaging mitigates the impact of noise. In projecting the equity premium into the future, we therefore focus on the equally weighted worldwide average of 4.1 percent and on the market-capitalization weighted world index. The world index is shown in the bottom right corner of Table 4; from the point of view of a U.S.-based investor, the world equity premium was 4.7 percent.23

6.3. From the Past to the Future Over the long run, real returns accrued largely from dividend payments, but Dimson, Marsh, and Staunton (2000, 2002), Arnott and Ryan (2001), and Ritter (2005) highlight the time-series and cross-sectional variation of global equity premiums. Given the large standard errors of historical estimates, and the likelihood that risks and equity premiums are non-stationary, one cannot determine a precise, forward-looking expected premium. However, by considering separately each component of the historical equity premium, we can develop a framework for making inferences. We start by discussing the real dividend growth rate, followed by expansion in the price-dividend ratio, and then the average dividend yield. We also consider changes in the real exchange rate. The second column of Table 4 indicates that, over the last 106 years, real dividends in the average country fell by 0.1 percent per year; in the world index, they rose by +0.8 percent; and in the U.S., they rose by +1.3 percent. Siegel (2005) and Siegel and Schwartz (2006), among others, observe that these long-term dividend growth rates were not achieved by a cohort of common stocks. The growth is that of a portfolio whose composition evolved gradually; today it contains almost no stocks from 1900,

23 We also computed the premium from the viewpoint of investors in the other 16 countries (for example, with a Japanese investor’s premium based on every market’s local-currency return converted into yen); the 17-country average equity premium varied between 2.3 percent for Denmark and 9.2 percent for Italy, with an average across all 17 reference currencies of 4.8 percent. Similarly, we computed the world premium from the viewpoint of investors in the other 16 countries (again converting every market’s return into yen, and so on); the world equity premium varied between 2.9 percent for Denmark and 9.9 percent for Italy, with an average across all 17 reference currencies of 5.4 percent. This wide range of values is attributable mostly to differences in the annualized real risk-free rate between countries, rather than to exchange rate differences.

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and largely comprises companies that gained a listing subsequently.24 In large part, the long-term increase in index dividends reflects companies that not only gained a listing after 1900, but ceased to exist quite some years ago.25 So what real dividend growth can we anticipate for the future? The worldwide growth rate was 0.8 percent per year; relative pessimists might project real dividend growth that is zero or less (Arnott and Bernstein (2002)), while relative optimists might forecast indefinite real growth in excess of 1 percent (Ibbotson and Chen (2003)). The third column of Table 4 reports that, over the last 106 years, the price-dividend ratio in the average country expanded by +0.6 percent per year; in the world and U.S. indexes, it expanded by +0.7 percent and +0.8 percent, respectively. As discussed earlier, this expansion reflected, at least in part, the enhanced opportunity to reduce portfolio risk as institutions increased the scope for diversification both domestically and internationally. If investors’ risk tolerances are today similar to the past, we have already argued that the required risk premium is likely to have fallen and valuation ratios to have risen. There is no reason to expect the required risk premium to fall further over the long haul, so persistent multiple expansion seems unlikely. Without further expansion in the price-dividend ratio, this source of historical performance cannot contribute to forward-looking equity premiums. The fourth column of Table 4 shows that, over the last 106 years, the geometric mean dividend yield in the U.S. was 4.4 percent, compared with 4.5 percent for the average country and 4.2 percent for the world index. Contemporary dividend yields (i.e., yields at end-2005, at the conclusion of the 106-year period) are lower than the historical average, even when buybacks are incorporated (see footnote 22). Whether adjusted for stock repurchases or not, projected levels for the long-term, geometric mean dividend yield are unlikely to be as large as the worldwide historical average of 4.2 percent. To the extent that the current (end-2005) level of dividends is indicative, the mean yield is likely to be lower in the future by at least ½–1 percent. Over the long term, nominal exchange rates tend to follow fluctuations in relative purchasing power. The consensus forecast for changes over the long term in the real (inflation-adjusted) exchange rate is zero. While the fifth column of Table 4 indicates that, historically, Americans gained (and others lost) from the rising real value of the U.S. dollar, this pattern cannot be extrapolated. We may assume that, over the long term, the real exchange rate change is expected to average zero. The historical equity premium comprises the sum of the factors discussed in the preceding paragraphs, minus the real interest rate (see the penultimate column of Table 4). The final column of Table 4 reports the historical equity premiums for our 24 To illustrate how much the listed equity market has evolved, Dimson, Marsh, and Staunton (2002) report that almost two-thirds of the value of the U.S. market and half the value of the U.K. market was represented by railroad stocks at the end of 1899. 25 There can also be a spurious jump in measured dividends when indexes are chain-linked. As a dividend series switches from narrower to broader composition, or from pretax to net-of-tax dividend payments, this can give rise to a step in income that impacts dividend growth estimates and (in the opposite direction) changes in the price-dividend ratio. We experimented with making adjustments for this for the U.S. and U.K., but the impact on estimated long-term dividend growth from splicing index series was small, and we abandoned this idea.


499

17 countries; they have an average of a 4.1 percent premium, with a cross-sectional standard deviation of 1.5 percent. While forward-looking estimates cannot be precise, a long-term projection of the annualized equity premium might, at the very least, involve making an adjustment to the historical record for components of performance that cannot be regarded as persistent. First, the expected change in the real exchange rate may be assumed to be zero, which implies an upward bias of 0.1 percent in the cross-sectional average of the country equity premiums. Second, the historical expansion in the pricedividend ratio cannot be extrapolated and might be assumed to be zero, which implies an upward bias of 0.6 percent in the cross-sectional average. These two adjustments, alone, attenuate the average country equity premium from 4.1 percent to 3.4 percent. When the same adjustments are made to the world index, the world equity premium shrinks from 4.7 percent to 4.0 percent. We noted above that if current dividend levels are a guide to the future, then the prospective mean dividend yield on the world index is likely to be lower than the historical average by at least ½–1 percent. This suggests a current equity premium of approximately 3–3½ percent. Goyal and Welch (2007) conclude that for forecasting the equity risk premium one cannot do better than to project the historical average equity premium into the future, and Mehra (2003) contends that “over the long term, the equity premium is likely to be similar to what it has been in the past.” However, as Campbell and Thompson (2007) point out, this cannot be the full story. History suggests that some part of the historical premium represents equity investors’ good luck, and Fama and French (2002) say in relation to the period 1951–2000 that their “main message is that the unconditional expected equity premium . . . is probably far below the realized premium.” Jorion and Goetzmann (1999) justified estimating equity premiums from capital appreciation indexes, stating “to the extent that cross-sectional variations in [dividend return minus real interest rate] are small, this allows comparisons of equity premiums across countries.” They compared six markets with and without dividends, with similar conclusions, albeit over a sample period differing from the 1900–2005 interval used here. However, there is a cross-country standard deviation in dividend yields of 0.7 percent (see Table 4). If one computes the sum for each country of dividend yield plus dividend growth, the cross-sectional standard deviation is 1.6 percent. Our estimates of the equity premium avoid the inaccuracies that arise from the Jorion–Goetzmann approximation. The debate on the size of the equity premium is sometimes conducted in terms of the arithmetic mean.26 For a stationary series the arithmetic mean is straightforward to interpret, but as Lettau and van Nieuwerburgh (2007) highlight, the underlying parameters are unstable. This makes arithmetic means harder to interpret, which is why we undertake our decompositions using annualized returns. For those who focus on the

26 To illustrate, consider a hypothetical index that provides a zero equity premium over a two-period interval. Assume that, within this interval, it suffers from transient volatility; for instance, the single-period returns might be +900 percent and −90 percent. While the geometric mean return is zero, the arithmetic mean return is +405 percent.

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arithmetic mean equity premium, for the world index the latter is 1.3 percent larger than the geometric mean (see Table 3), and our forward-looking estimate of the arithmetic mean premium for the world index would be approximately 4½–5 percent. Twentieth-century financial history was a game of two halves. In the first half, markets were harsh on equity investors; but in the second half they were benevolent.27 As we show in Dimson, Marsh, and Staunton (2002), early in the century dividend yields were mostly high relative to interest rates, whereas more recently yields have generally been lower. Looking at the 1900–2005 period as a whole, the world equity market experienced dividend growth and price-dividend multiple expansion that contributed 0.8 percent and 0.7 percent per year, respectively, to long-run real returns and hence to the ex post equity premium. The remainder was contributed by the annualized dividend yield of 4.2 percent (for the world index) and a real exchange rate adjustment. This suggests that the equity premium expected by investors was lower than the realized premium. The fact that ex post equity premiums were enhanced by this rate of dividend growth and multiple expansion is the “triumph” experienced by 20th-century stock market investors.

7. CONCLUSION We have presented new evidence on the historical equity premium for 17 countries over 106 years. Our estimates, including those for the U.S. and U.K., are lower than frequently quoted historical averages. The differences arise from bias in previous index construction for the U.K. and, for both countries, our use of a longer time frame that incorporates the earlier part of the 20th century as well as the opening years of the new millennium. Prior views have been heavily influenced by the U.S. experience, yet we find that the U.S. equity premium is somewhat higher than the average for the other 16 countries. The historical equity premium, presented here as an annualized estimate (i.e., as a geometric mean), is equal to investors’ ex ante expectations plus the impact of luck. In particular, expanding multiples have underpinned past returns. In part, this reflects a general decline in the risk faced by investors as the scope for diversification has increased, and stocks have become more highly valued. In addition, past returns have also been enhanced during the second half of the 20th century by business conditions that improved on many dimensions. We cannot know today’s consensus expectation for the equity premium. However, after adjusting for non-repeatable factors that favored equities in the past, we infer that investors expect an equity premium (relative to bills) of around 3–3½ percent on a geometric mean basis and, by implication, an arithmetic mean premium for the world index of approximately 4½–5 percent. These estimates are lower than the historical 27 Averaged across all 17 countries, the real, local-currency annualized equity returns were 2.7 percent in the first half of the 20th century, versus 7.1 percent over the following 56 years. Note, however, that adverse stock market conditions also tended to impact the real returns from bonds and bills (see Section 5).


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premiums quoted in most textbooks or cited in surveys of finance academics. From a long-term historical and global perspective, the equity premium is smaller than was once thought. The equity premium survives as a puzzle, however, and we have no doubt that it will continue to intrigue finance scholars for the foreseeable future.

References Aleotti, A. Borsa e Industria 1861–1989: Cento Anni di Rapporti Difficili. Edizioni Comunita, Milan (1990). Annaert, J., F. Buelens, J. de Ceuster, L. Cuyvers, G. Devos, M. Gemis, H. Houtman-deSmedt, and J. Paredaens. Ontwerp van een Databank m.b.t. het Archief van de Beurs van Brussel. TEW Working paper 98:11, RUCA, Antwerp (1998). Arnott, R. D., and P. Bernstein. What risk premium is normal? Financial Analysts Journal 58(2) (2002): 64–85. Arnott, R. D., and W. J. Bernstein. Earnings growth: The 2 percent dilution, Financial Analysts Journal 58(5) (2003): 47–55. Arnott, R. D., and R. Ryan. The death of the risk premium: Consequences of the 1990s. Journal of Portfolio Management 27(3) (2001): 61–74. Barclays Capital. Equity-Gilt Study (1999). Ball, R., and J. Bowers. Shares, bonds, Treasury notes, property trusts and inflation: Historical returns and risks 1974–1985. Australian Journal of Management 11 (1986): 117–137. Ball, R., P. Brown, F. J. Finn, and R. R. Officer. Share Markets and Portfolio Theory, 2nd ed. University of Queensland Press, St. Lucia, Australia (1989). Bianchi, B. Appendice statistica: Il rendimento del consolidato dal 1862 al 1946, in F. Vicarelli, ed., Capitale Industriale e Capitale Finanziaro: Il Caso Italiano. Il Mulino, Bologna (1979). Biscaini, A. M., F. Cotula, and P. L. Ciocca. Italian financial structures: Long-term quantitative aspects (1879–1970), in G. Federico, ed., (1994), The Economic Development of Italy Since 1870. Elgar, Aldershot (1982). Bittlingmayer, G. Output, stock volatility and political uncertainty in a natural experiment: Germany 1880– 1940. Journal of Finance 53 (1998): 2243–2257. Bodie, Z. Longer time horizon “does not reduce risk.” Financial Times January 26 (2002). Bodie, Z., A. Kane, and A. J. Marcus. Investments, 4th ed. McGraw-Hill, New York (1999). Bowley, A. L. Wages and Income in the United Kingdom Since 1860. Cambridge University Press, Cambridge (1937). Brealey, R. A., R. Giammarino, E. Maynes, S. C. Myers, and A. J. Marcus. Fundamentals of Corporate Finance, 1st Canadian ed. McGraw-Hill, New York (1996). Brealey, R. A., and S. C. Myers. Principles of Corporate Finance, 6th ed. McGraw-Hill, New York (2000). Brown, S., W. Goetzmann, and S. Ross. Survival. Journal of Finance 50 (1995): 853–873. Campbell, J. Y., and S. B. Thompson. Predicting the equity premium out of sample: Can anything beat the historical average? Review of Financial Studies, forthcoming (2007). Christiansen, J., and B. Lystbaek. Afkast og Risiko pa Aktier og Obligationer 1915–1993. Finans Invest 3 (1994): 10–13. Cochrane, J. H. Where is the market going? Uncertain facts and novel theories. Economic Perspectives 21 (1997): 3–37. Credit Suisse First Boston. The CSFB Equity-Gilt Study (1999). Davis, J. L., E. F. Fama, and K. R. French. Characteristics, covariances and average returns: 1929–1997. Journal of Finance 55 (2000): 389–406. Dimson, E., and P. Marsh. The stability of U.K. risk measures and the problem of thin trading. Journal of Finance 38 (1983): 753–783. Dimson, E., and P. Marsh. U.K. financial market returns 1955–2000. Journal of Business 74 (2001): 1–31. Dimson, E., P. Marsh, and M. Staunton. The Millennium Book: A Century of Investment Returns. ABN AMRO/London Business School (2000).

502

Chapter 11 • The Worldwide Equity Premium: A Smaller Puzzle Dimson, E., P. Marsh, and M. Staunton. Triumph of the Optimists: 101 Years of Global Investment Returns. Princeton University Press, Princeton, NJ (2002). Dimson, E., P. Marsh, and M. Staunton. Irrational optimism. Financial Analysts Journal 60(1) (2004): 15–25. Dimson, E., P. Marsh, and M. Staunton. Global Investment Returns Yearbook 2006. ABN AMRO/London Business School (2006a). Dimson, E., P. Marsh, and M. Staunton. DMS Global Returns data module. Ibbotson Associates, Chicago (a subsidiary of Morningst Inc.) (2006b). Dimson, E., S. Nagel, and G. Quigley. Capturing the value premium in the United Kingdom. Financial Analysts Journal 59(6) (2003): 35–45. Eichholtz, P., K. Koedijk, and R. Otten. De Eeuw van Het Aandeel. Economisch Statistische Berichten 85(4238) (2000): 24–27. Fama, E., and K. French. Disappearing dividends: Changing firm characteristics or lower propensity to pay. Journal of Financial Economics 60 (2001): 3–43. Fama, E., and K. French. The equity premium. Journal of Finance 57 (2002): 637–659. Fernandez, P. Market risk premium: Required, historical and expected. Working paper, IESE Business School, Spain (2004). Firer, C., and H. McLeod. Equities, bonds, cash and inflation: Historical performance in South Africa 1925– 1998. The Investment Analysts Journal 50 (1999): 7–28. Frennberg, P., and B. Hansson. Computation of a monthly index for Swedish stock returns 1919–1989. Scandinavian Economic History Review 40 (1992a): 3–7. Frennberg, P., and B. Hansson. Swedish stocks, bonds, bills and inflation (1919–1990). Applied Financial Economics 2 (1992b): 79–86. Frennberg, P., and B. Hansson. Computation of a monthly index for Swedish stock returns 1919–1989: Update to end-1999. Unpublished note (2000). Fujino, S., and R. Akiyama. Security prices and interest rates: 1874–1975. The Documentation Center for Japanese Economic Statistics, Institute of Economic Research. Hitotsubashi University, Tokyo (1977). Gallais-Hamonno, G., and P. Arbulu. La rentabilité reelle des actifs boursiers de 1950 a` 1992. Economie et Statistique 280 (1995): 3–30. Gielen, G. Konnen Aktienkurse Noch Steigen? Langfristige Trendanalyse des Deutschen Aktienmarktes. Gabler, Wiesbaden (1994). Goetzmann, W. N., and R. G. Ibbotson. History and the equity risk premium, in R. Mehra, ed., Handbook of the Equity Risk Premium. Elsevier, Amsterdam (2008). Goetzmann, W. N., R. G. Ibbotson, and L. Peng. A new historical database for the NYSE 1815 to 1925: Performance and predictability. Journal of Financial Markets 4(1) (2001): 1–32. Goetzmann, W. N., A. Ukhov, and N. Zhu. China and the world financial markets 1870–1930: Modern lessons from historical globalization. The Economic History Review 60(2) (2007): 267–312. Gonzalez, A. S., and J. L. Suarez. Historical returns in the Spanish equity market. Journal of Investing 3(1) (1994): 66–71. Goyal, A., and I. Welch. A comprehensive look at the empirical performance of equity premium prediction. Review of Financial Studies, forthcoming (2007). Graham, J. R., and C. R. Harvey. The equity risk premium in September 2005: Evidence from the global CFO outlook survey. Working paper, Duke University (2005). Grossman, R. S. New indices of British equity prices, 1870–1913. Journal of Economic History 62 (2002): 121–146. Grullon, G., and R. Michaely. Dividends, share repurchases, and the substitution hypothesis. Journal of Finance 57 (2002): 1649–1684. Hamao, Y. A standard database for the analysis of Japanese security markets. Journal of Business 64 (1991): 87–101. Hamao, Y., and R. G. Ibbotson. Stocks, Bonds and Inflation 1989 Yearbook. Ibbotson Associates, Chicago (1989) (and subsequent annual supplements). Houston, W. R. The Annual Financial Review (Canadian). Houston’s Standard Publications, Toronto (1900– 1914).


503

Huber, G. Evidence sur la performance relative des marchés obligataire et des actions en Suisse 1960–83. Working paper, Pictet & Cie, Geneva (1985). Hwang, S., and B. K. Song. Irrational exuberance in the long-run history of the U.K. stock market. Applied Economics, forthcoming (2008). Ibbotson Associates. Stocks, Bonds, Bills and Inflation Yearbook. Ibbotson Associates, Chicago (2000). Ibbotson, R., and P. Chen. Long-run stock returns: Participating in the real economy. Financial Analysts Journal 59 (2003): 88–98. Ibbotson, R. G., and R. Sinquefield. Stocks, bonds, bills and inflation: Year-by-year historical returns (1926–74). Journal of Business 49 (1976): 11–43. Jagannathan, R., E. McGrattan, and A. Scerbina. The declining U.S. equity premium. Federal Reserve Bank of Minneapolis Quarterly Review 24 (2000): 3–19. Jorion, P., and W. Goetzmann. Global stock markets in the Twentieth Century. Journal of Finance 54 (1999): 953–980. Kocherlakota, N. R. The equity premium: It’s still a puzzle. Journal of Economic Literature 34 (1996): 42–71. Kronenberger, F. Die Preisbewegung der Effekten in Deutschland, in Betriebs- und Finanzwirtschaftliche Forschungen. Heft 2, Berlin (1920). Laforest, P., and P. Sallee. Le pouvoir d’achat des actions, des obligations et de l’or de 1914 a` 1976. Economie et Statistique 86 (1997): 61–67. Lettau, M., and S. van Nieuwerburgh. Reconciling the return predictability evidence: In-sample forecasts, out-of-sample forecasts, and parameter instability. Review of Financial Studies, forthcoming (2007). Li, H., and Y. Xu. Survival bias and the equity premium puzzle. Journal of Finance 57 (2002): 1981–1995. Lamberton, D. McL. Share Price Indices in Australia. Law Book Company, Sydney (1958). Maddison, A. Monitoring the World Economy 1820–1992. Organization for Economic Cooperation and Development, Paris (1995). Mauboussin, M. J. Clear Thinking About Share Repurchases. Legg Mason Capital Management (2006). Meghen, P. J. Statistics in Ireland. Institute of Public Administration, Dublin (1970). Mehra, R. The equity premium: Why is it a puzzle? Financial Analysts Journal 59 (2003): 54–69. Mehra, R., and E. C. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15 (1985): 145–161. Mehra, R., and E. C. Prescott. The equity risk premium: A solution? Journal of Monetary Economics 22 (1988): 133–136. Mehra, R., and E. C. Prescott. The equity premium in retrospect, in G. M. Constantinides, M. Harris, and R. M. Stulz, eds., Handbook of the Economics of Finance: Volume 1B Financial Markets and Asset Pricing. Elsevier, Amsterdam (2003). Mehra, R., and E. C. Prescott. The equity premium ABCS in R. Mehra, ed. Handbook of the Equity Risk Premium. Elsevier, Amsterdam (2008). Mitchell, B. R. International Historical Statistics: Europe 1750–1993. Macmillan Press, London (1998). Newlands, J. Put Not Your Trust in Money. Association of Investment Trust Companies (1997). Nielsen, S., and O. Risager. Macroeconomic perspectives on stock and bond investments in Denmark Since the First World War, in T. M. Andersen, O. Risager, and S. H. E. Jensen, eds., Macroeconomic Perspectives on the Danish Economy. Macmillan Press, London (1999). Nielsen, S., and O. Risager. Stock returns and bond yields in Denmark 1922–99. Scandinavian Economic History Review 49(1) (2001): 63–82. Panetta, F., and R. Violi. Is there an equity premium puzzle in Italy? A look at asset returns, consumption and financial structure data over the last century. Temi di Discussione 353, Bank of Italy (1999). Panjer, H. H., and K.S. Tan. Report on Canadian Economic Statistics 1924–2001. Canadian Institute of Actuaries, Ottawa (2002). Parum, C. Historisk Afkast af Aktier og Obligationer i Danmark. Finans Invest 3 (1999a): 4–13. Parum, C. Estimation af Realkreditobligationsafkast i Danmark i Perioden 1925–1998. Finans Invest 7 (1999b): 12–15. Parum, C. An equally weighted index of Copenhagen stock prices 1900–1914. Unpublished note (2002). Pástor, L., and R. F. Stambaugh. The equity premium and structural breaks. Journal of Finance 56 (2001): 1207–1239.

504

Chapter 11 • The Worldwide Equity Premium: A Smaller Puzzle Rajan, R., and L. Zingales. The great reversals: The politics of financial development in the twentieth century. Journal of Financial Economics 69 (2003): 5–50. ¨ Rätzer, E. Die Pensionskasse aus Okonomischer Sicht. Paul Haupt, Berne (1983). Rau, R., and T. Vermaelen. Regulation, taxes and share repurchases in the U.K. Journal of Business 75 (2002): 245–282. Ritter, J. Economic growth and equity returns. Pacific-Basin Finance Journal 13(5) (2005): 489–503. Ronge, U. Die Langfristige Rendite Deutscher Standardaktien. Peter Lang, Frankfurt (2002). Sandez, M., and F. G. Benavides. La Bolsa de Madrid entre 1919 y 1936. Revista de la Bolsa de Madrid 87 (2000): 32–39. Schumann, C. G. W., and A. E. Scheurkogel. Industrial and commercial share price indices in South Africa. Bureau for Economic Research, Faculty of Commerce, University of Stellenbosch (1948). Schwartz, S. S., and W. T. Ziemba. The Japanese stock market 1949–1991, in W. T. Ziemba, W. Bailey, and Y. Hamao, eds. Japanese Financial Market Research. Elsevier Science, Amsterdam (1991). Schwert, G. W. Indexes of U.S. stock prices from 1802 to 1987. Journal of Business 63 (1990): 399–426. Siegel, J. J. Stocks for the Long Run, 3rd ed., McGraw-Hill, New York (2002). Siegel, J. J. The Future for Investors: Why the Tried and the True Triumph over the Bold and the New. Crown Business, New York (2005). Siegel, J. J., and J. D. Schwartz. The long-term returns on the original S&P 500 firms. Financial Analysts Journal 62(1) (2006): 18–31. Stehle, R., et al. Aktienrenditen national. Working paper, Humboldt University (2007). Downloaded from: http://www2.wiwi.hu-berlin.de/finance/daten. Stehle, R. Der size-effekt am Deutschen Aktienmarkt. Zeitschrift für Bankrecht und Bankwirtschaft 9(3) (1997): 237–260. Stehle, R., C. Wulff, and Y. Richter. Die Rendite deutscher Blue-chip-Aktien in der Nachkriegszeit— Rückberechnung des DAX für die Jahre 1948 bis 1954. Working paper, Humboldt University, Berlin (1999). Switzer, L. An equally weighted index of Montreal stock prices 1900–1914. Unpublished note (2000). Taylor, A. A century of purchasing power parity. Review of Economics and Statistics 84(1) (2002): 139–150. Thomas, W. A. The Stock Exchanges of Ireland. Francis Cairns Publications, Dublin, Ireland (1986). Timmermann, A. Udviklingen i de Danske Aktiekurser 1914–1990. Nationalokonomisk Tidsskrift 130 (1992): 473–482. Urquhart, M. C., and K. A. H. Buckley. Historical Statistics of Canada. Macmillan Company, Toronto (1965). Valbuena, S. F. Sources and methodology of Spanish Bolsa returns in the 20th century. Working paper, Universidad Complutense, Madrid (2000). Vandellos, J. A. El porvenir del cambio de la peseta. Instituto de Investigaciones Economicas, Banca Mas Sarda 1936 (reprinted 1974). Welch, I. Views of financial economists on the equity premium and other issues. Journal of Business 73 (2000): 501–537. Welch, I. The equity premium consensus forecast revisited. Cowles Foundation Discussion Paper Number 1325, Yale School of Management (2001). Whelan, S. From Canals to Computers: The Friends First Guide to Long-Term Investment Returns in Ireland. Friends First Asset Management, Dublin (1999). Whelan, S. Prudent Pension Planning. Hibernian Investment Managers, Dublin (2002). Wilson, J., and C. Jones. An analysis of the S&P 500 index and Cowles’s extensions: Price indexes and stock returns, 1870–1999. Journal of Business 75 (2002): 505–533. Wydler, D. Swiss stocks, bonds and inflation 1926–1987. Journal of Portfolio Management 15(2) (1989): 27–32. Wydler, D. The Performance of Shares and Bonds in Switzerland: An Empirical Study Covering the Years Since 1925. Pictet & Cie, Geneva (2001).

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APPENDIX 1: DECOMPOSITION OF THE EQUITY PREMIUM This appendix explains how we decompose the historical equity premium into five elements. These are, first, the average dividend yield over the sample period; next, the impact of real dividend growth, expansion of the price-dividend ratio, and the change in the real exchange rate; and finally, the risk-free interest rate that is used to compute the equity premium. Without loss of generality, the decomposition is in real (inflation adjusted) terms.

Capital Appreciation and Income We assume the dividend payment on the equity index portfolio is received at the end of period t and is equal to Dt , that the price at the end of period t − 1 is Pt−1 , and that inflation over period t runs at the rate It . Real dividends are dt = Dt /(1 + It )t , where the denominator measures the inflation rate from period 1 to period t, namely (1 + It )t = (1 + I1 )(1 + I2 ) · · · (1 + It ). The pricedividend ratio is PDt = Pt /Dt . The real capital gain over period t is given by: 1 + real gain t = (Pt /Pt−1 )/(1 + It ) ≡ [(Dt /Dt−1 )/(1 + It )] (PDt /PDt−1 ) = (dt /dt−1 )(PDt /PDt −1 ) = (1 + Gdt )(1 + GPDt ),

(A1)

where the growth rate of real dividends is Gdt = dt /dt−1 − 1, and the rate at which the price-dividend ratio has expanded is GPDt = PDt /PDt−1 − 1. As a proportion of the initial investment, real dividend income during period t is: real incomet = (Dt /Pt−1 )/(1 + It ) ≡ (Dt /Pt )(Pt /Pt−1 )/(1 + It ) = Yt (Pt /Pt−1 )/(1 + It ),

(A2)

where Yt = Dt /Pt is the dividend yield, defined as the ratio of aggregate dividends paid over period t divided by the aggregate stock price at the end of period t. Note that the terms to the right of Yt measure (one plus) the real capital gain over period t, as defined above.

Total Returns The real return is equal to the arithmetic sum of real capital gain (A1) and real income (A2), namely, 1 + real returnt ≡ [Dt /Pt−1 + (Pt /Pt−1 )]/(1 + It ) = (1 + Gdt )(1 + GPDt )(1 + Yt ).

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So far we have decomposed returns denominated in a single currency. If the assets are purchased in unhedged foreign currency, we assume that each period’s return is converted from foreign currency into home currency. The real return is then: 1 + real returnt = (1 + Gdt )(1 + GPDt )(1 + Yt )(1 + Xt ),

(A3)

where Xt is the increase in the inflation-adjusted value of the home currency relative to the foreign currency, namely the change in the real exchange rate.28

The Equity Premium Finally, we define the equity premium as the geometric difference between the real return defined in (A3) and the risk-free real interest rate, Rft . Hence the historical equity premium is: 1 + ERPt = (1 + real return t )/(1 + Rft ) = (1 + Gdt )(1 + GPDt )(1 + Yt )(1 + Xt )/(1 + Rft ).

(A4)

The historical equity premium is therefore equal to the sum of the real dividend growth rate, expansion in the price-dividend ratio, the dividend yield, and the change in the real exchange rate; less the risk-free real interest rate. All additions and subtractions are geometric. Consequently, the geometric mean equity premium from period 1 through period t may be decomposed as follows: (1 + ERPt ) = (1 + Gdt )(1 + GPDt )(1 + Yt )(1 + Xt )/(1 + Rf t ),

(A5)

where each term on the right-hand side of (A5) is the geometric mean of t single-period components. That is, (1 + Yt )t = (1 + Y1 )(1 + Y2 ) · · · (1 + Yt ), and so on. To sum up, the annualized historical equity premium may be decomposed geometrically into five elements. These are as follows: first, the mean growth rate in real dividends; second, the mean rate of expansion in the price-dividend multiple; third, the mean dividend yield; fourth, the mean change in the real exchange rate; and finally, the mean risk-free real interest rate. Finally, note that the reference country for the real exchange rate and the real interest rate must correspond. For example, the exchange rate may be relative to the U.S. dollar; and if so, the real interest rate should be the rate on the U.S. risk-free asset.

28 Obviously,

when the investment is in domestic securities, the change in the real exchange rate is Xt = 0.


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APPENDIX 2: DATA SOURCES FOR THE DMS DATABASE Section 3 outlined the general methodology and guiding principles underlying the construction of the DMS database (see also Dimson, Marsh, and Staunton (2002, 2006a, and 2006b)). This appendix describes the data sources used for each country. Australian equities are described in Officer’s chapter in Ball, Brown, Finn, and Officer (1989). Ball and Bowers (1986) provide a complementary, though brief, historical analysis. We are grateful to Bob Officer for making his database available to us. Officer compiled equity returns from a variety of indexes. The early period made use of data from Lamberton’s (1958) classic study. This is linked over the period 1958– 1974 to an accumulation index of 50 shares from the Australian Graduate School of Management (AGSM) and over 1975–1979 to the AGSM value-weighted accumulation index. Subsequently, we use the Australia All-Ordinary index. Bond returns are based on the yields on New South Wales government securities from the start of the century until 1914. For the period 1915–1949, the yields were on Commonwealth Government Securities of at least five years maturity. During 1950–1986, the basis is 10-year Commonwealth Government Bonds. From 1986, we use the JP Morgan Australian government bond index with maturity of over seven years. For 1900–1928, the short-term rate of interest is taken as the three-month time deposit rate. From 1929 onward, we use the Treasury bill rate. Inflation is based on the retail price index (1900–1948) and consumer price index (1949 onward). The switch in 1966 from Australian pounds to Australian dollars has been incorporated in the Exchange Rate index history. Belgium is being researched by Annaert, Buelens, de Ceuster, Cuyvers, Devos, Gemis, Houtman-deSmedt, and Paredaens (1998). We are grateful for access to their interim results for 1900–1928, which are subject to correction. From 1929, we use the National Bank of Belgium’s 80-share index. The market was closed from August 1944 to May 1945, and we take the closing level as the 1944 year-end value. For 1965–1979, we use the Banque Bruxelles Lambert 30-share index and from 1980 the Brussels Stock Exchange All Share Index. Up to 1956, bond returns are based on estimated prices for 4 percent government bonds. During the 1944–1945 closure, we take the last available value from 1944 as the year-end level. Over 1957–1967, the index is for bonds with a 5 to 20-year maturity; and during 1968–1985, for bonds with maturity over five years. Subsequent years use the JP Morgan Belgian government bond index with maturity of over five years. Short-term interest rates are represented over the period 1900–1926 by the central bank discount rate, followed during 1927–1956 by the commercial bill rate. From 1957 onward, we use the return on Treasury bills. Inflation is estimated for 1900–1913 using the consumer price index, and for 1914 we take the French inflation rate. Over 1915–1920 and 1941–1946, we interpolate the Belgian consumer price index from Mitchell (1998). From 1921, inflation is measured using the Institut National de Statistique’s consumer price index. Canadian stocks, bonds, bills, and inflation since 1924 are presented in Panjer and Tan (2002), with supplementary data kindly compiled for us by Lorne Switzer. For

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1900–1914, the annual index returns are based on Switzer’s (2000) equally-weighted Montreal index, adjusted for dividends. The equity series for 1915–1946 is taken from Urquhart and Buckley (1965). Houston (1900–1914) provides dividends for 1900 and hence the Canadian yield premium relative to the 1900 S&P, and Panjer and Tan (2002) estimate the Canadian yield relative to the 1924 S&P. To compute yearly total returns over 1900–1923, we interpolate the Canadian yield premium relative to the S&P. For the period 1947–1956, returns are for the TSE corporates, and from 1957 the TSE 300 total return index. The bond index for 1900–1923 is based on a 4 percent bond from Global Financial Data (GFD). For 1924–1936, we use the Government of Canada long bond index from Panjer and Tan (2002). Starting in 1936, the index is the Cansim index of bonds with maturity of over 10 years, switching in 2002 to the JP Morgan Canadian government bond index with maturity of over 10 years. For 1900–1933 the short-term rate is represented by U.S. Treasury bills or equivalent. From 1934 onward, the short-term rate is based on Canadian Treasury bills. Inflation is measured using the Canadian wholesale price index for 1900–1910. For 1911–1923, we switch to the Canadian consumer price index, and thereafter consumer price inflation is taken from Cansim. Danish stock market data has involved working with Claus Parum to extend his research back to 1900. We have also referred to the papers by Steen Nielsen and Ole Risager (1999, 2001) and Allan Timmermann (1992). Over the period 1900–1914, we use Parum’s (2002) equally weighted index of equity returns, which covers some 40 to 50 constituents each year. Thereafter, all the studies cited above are based on equity price indexes from Statistics Denmark, though we incorporate Parum’s adjustments for capital changes that are not incorporated into the published index numbers. For 1915– 2001, we use the data compiled in Parum (1999a, b and 2002), switching from 2002 to the Copenhagen KAX Index. Danish bond returns are estimated from yields on government bonds until 1924. For 1925–2001, our data is from Parum (1999a, b and 2002), who uses the return on mortgage bonds, a large and liquid asset class throughout the period, in contrast to more thinly traded government bonds, as described in Christiansen and Lystbaek (1994). From 2002, we use the JP Morgan Danish government bond index with maturity of over seven years. Short-term interest rates are represented by the central bank discount rate until 1975, and thereafter by the return on Treasury bills. France is documented by Laforest (1958) and then Laforest and Sallee (1977), for the first half of the 20th century, followed by Gallais-Hamonno and Arbulu (1995) for the period commencing in 1950. The common basis for equity returns in all the primary studies is the index series compiled by the Institut National de la Statistique et des Etudes Economiques (INSEE). The INSEE equity index is a weighted average of price relatives with about 300 constituents. Over the period from 1914–1918, we interpolate, assuming constant real returns. We use the SBF-250 from 1991 onward. The bond series for France, also compiled by INSEE, is based on consol yields. Over the period from 1914–1918, we interpolate, assuming constant nominal returns. We switch in 1950 to the Gallais-Hamonno and Arbulu (1995) series, which is the INSEE General Bonds Index, with coupons reinvested monthly as received. From 1993, we use the JP Morgan French government bond index with maturity of over 10 years. The short-term interest rate for France is based on the central bank discount rate until 1930. The rate is measured


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by the return on Treasury bills starting in 1931. To measure consumer price inflation, we use the consumption price index that is compiled by the Institut National de la Statistique et des Etudes Economiques, taken from Laforest (1958), Gallais-Hamonno and Arbulu (1995) and directly since 1981. German data was provided by George Bittlingmayer (1998) and Richard Stehle (1997); also see Stehle, Wulff, and Richter (1999) and Gregor Gielen (1994) and Ulrich Ronge (2002). We use Ronge’s reconstruction of the DAX 30-share index to provide nominal equity returns for 1900–1953. For August 1914–October 1918 Ronge uses OTC quotations for the 14 shares in his index that could be found in Kronenberger (1920). For 1954–1994, we use the Stehle (1997) comprehensive index, switching in 1995 to the CDAX attributed to Stehle/Hartmond-Reihe in Stehle (2007). For 1900–1923, German bond returns are based on the price of 3 percent perpetuals, which essentially lost all value during the 1922–1923 hyperinflation. For 1924–1935, the bond index is based on mortgage bonds, and for 1936–1951, it is based on 4.5 percent conversion (to 1943), 4.5 percent western zone (1946–1947), and 5 percent tax-free (from 1948) bonds. We use the REX performance index starting in 1968, switching in 1986 to the JP Morgan German government bond index with maturity of over seven years. The short-term rate of interest is represented by the discount rate on private bills through 1945. We assume rates of 2 percent during 1946–1950, 3 percent for 1951–1953, and use Treasury bills beginning in 1954. Inflation in Germany is from Gielen (1994), using consumer price level data from the Imperial Statistical Office (see Bittlingmayer (1998)). Inflation rates during 1922 and 1923 were inferred from exchange rates against the dollar. From 1993, we use the CPI from the Federal Statistical Office. Ireland was first studied by Shane Whelan (1999), who used Irish Central Statistical Office (CSO) data from 1934, and British data before that. Thomas (1986) provides some additional early data, but only in graphical form. We therefore created a new, market capitalization weighted index of Irish equity prices for 1900–1933 from original archive stock price and dividend sources (and this index has now been adopted by Whelan (2002)). For 1934–1983 we use the Irish CSO Price Index of Ordinary Stocks and Shares. Until 1987, we incorporate our estimates of U.K. dividend yields. From 1988, we use the Irish Stock Exchange Equity (ISEQ) total return index. The bond series for Ireland uses U.K. returns for 1900–1978. For 1979–1998, we use Whelan’s (1999) return on a 20-year representative Irish gilt, as estimated by Raida Stockbrokers, turning thereafter to the Datastream 10-year Irish government bond index. Short-term Irish interest rates again use U.K. Treasury bills for 1900–1969. From 1970, we use Irish Treasury bills. Up to the date of political independence from Britain, inflation is measured using Bowley’s (1937) cost of living index for 1900–1913 and the workingclass cost of living index for 1914–1922. For 1923–1952, we use Meghen’s (1970) Irish cost of living index, and from 1953, the Irish consumer price index. Italian data was provided by Fabio Panetta and Roberto Violi (1999). The equity data for 1900–1907 is from the Official List and supplementary sources, and this is extended through 1911 with data from Aleotti (1990). From 1912–1977, the share price and dividend series are based on the Bank of Italy index, which covers at least threequarters of the total market capitalization of the Italian equity market. Thereafter, the

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Bank of Italy’s index is calculated from the bank’s monthly share price database, which covers all listed shares. From 1999 onward, we use the Milan BCI performance index. The government bond returns over 1900–1944 are from Bianchi (1979). For the period 1945–1983, the index of total bond returns is based on a Treasury bond index with a coverage of over half, and often over three-quarters, of the value of all Treasury bonds in issue. Thereafter, the data are sourced from Panetta and Violi’s (1999) study. From 1988, we use the JP Morgan Italian government bond index with maturity of over three years. The short-term bank deposit rate to 1940 is from Biscaini, Cotula, and Ciocca (1982). Panetta and Violi estimate the values for the period 1941–1946, while for 1947– 1961 the figures are from the Bank of Italy’s Bollettino Economico. After that, the source is the Bank of Italy’s Bollettino Statistico. Japanese data of good quality are available from the Hamao (1991) database, and from the study by Schwartz and Ziemba (1991). We are grateful to Kenji Wada for facilitating provision of pre–World War I equity data. For 1900–1914, we use the Laspeyres price index for the Tokyo Stock Exchange (TSE), as published in Fujino and Akiyama (1977). Thereafter, share prices are represented by the Japan National Bank index for 1915–1932; the Oriental Economist Index from 1933 until September 1948 (although trading was suspended in August 1945, and no index values were published again until May 1946 when black market trading resumed in Tokyo); the Fisher index from September 1948 until the market officially reopened in May 1949; and the Nikkei-225 from May 1949 to 1951. During 1952–1970, we use the Japan Securities Research Institute total return index. From 1971, we use total returns from Hamao and Ibbotson (1989). Returns continue from 1995 with the TSE TOPIX index. Pre-1971 Japanese government bond index data is available from Global Financial Data. Until 1957, the returns are estimated from yield data. No yield information is available for the end of 1947, and the yield for 1946 is used instead. The data for 1948–1957 represent the yields on newly issued bonds. From 1957 through 1968, the bonds are those issued by Nippon Telephone and Telegraph. From 1971, we use the government bond index from Hamao and Ibbotson (1989), followed from 1995 by the JP Morgan Japanese government bond index with maturity of over 10 years. The short-term riskless rate is available from 1900. It is based on call money rates to 1959, and on Treasury bills thereafter. Inflation is measured by the wholesale price index for 1900, the retail price index for 1901–1946, and the consumer price index from 1947 onward. The Netherlands is based on work by Eichholtz, Koedijk, and Otten (2000). The equity returns over 1900–1918 are based on the Central Bureau of Statistics (CBS) general index of share prices, and historical yield data. For the period 1919–1951, returns are based on the 50-stock, CBS weighted arithmetic index. The exchange was closed from August 1944 to April 1946, so the end-year index levels are represented by the intra-year values that are closest to the turn of the year. During 1952–1980, returns are based on the CBS All Share index, with dividends estimated by the Dutch central bank. For 1981 onward, we use the CBS total return index, which went live in 1989 with retrospective estimation of the impact of income reinvestment, changing to the Amsterdam AMS All Share index from 2004. During 1900–1914, Dutch bond returns are represented by 2.5 percent and 3 percent consols. During 1915–1973, the Eichholtz– Koedijk–Otten bond index is based on a series of 3.5 percent bonds. From 1974, the


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index is the JP Morgan Netherlands government bond index with maturity of over seven years. For the riskless rate, during 1900–1940 we use the discount rate on three-month private bills. The rate is assumed unchanged when data was unavailable during August 1914 to December 1918, and from mid-May 1940 to the end of that year. From 1941 to date, we use the rate on Dutch Treasury bills. Inflation is measured using the consumer price index. No data was available between August 1944 and June 1945, and the index was interpolated for end-1944. Norway was introduced into the study through Thore Johnsen, Knut Kjær, and Bernt Ødegaard, who provided data and sources. Equity returns for 1900–1917 are derived from an equally weighted index based on all stocks listed in Statistisk Arbok and supplemented with those shares listed in Kierulf’s Handbook for which there was information on year-end prices and dividends. The index contained between 33–36 shares until the end of 1914, but this fell to 21 by the start of 1918. For the period 1918–1969, we use an all-share index including industrial, banking, and whaling/shipping shares calculated by Statistics Norway. From 1970, we use the Oslo Stock Exchange indexes. We first use the Industrial index, switching in 1983 to the General Index and then, from 1996, to the All Share index. During 1900–1992, Norwegian bond returns are based on Global Financial Data’s government bond yields. From 1993, the index is the Datastream government bond index with maturity of 10 years. For the riskless rate, during 1900–1971, we use the central bank discount rate, followed by money market rates until 1983. From 1984 to date, we use the rate on Norwegian Treasury bills. Inflation is measured using the consumer price index published by Statistics Norway. South African stocks, bonds, bills, and inflation since 1925 are presented in Firer and McLeod (1999), who, in turn, draw on earlier work going back to 1910 by Schumann and Scheurkogel (1948). These studies provide indexes for industrial and commercial companies in South Africa. However, mining and financial companies are of particular importance, especially early last century. We therefore create a market capitalization weighted index of mining and financial shares for 1900–1959, based on London price quotations. We blend our mining and financial indexes with the Firer and McLeod industrial index, by starting with a weighting of 5 percent in the industrial index at the start of 1910, with weights increasing to 25 percent by the start of 1950. From 1960–1978, we use the Rand Daily Mail Industrial Index and, from 1979, the Johannesburg Stock Exchange–Actuaries Equity Index. Up to 1924, bond returns are based on the yields for 4 percent government bonds. Subsequently, we use the bond returns from Firer and McLeod, based first on market yields together with a notional 20 year bond prior to 1980, followed by the JSE-Actuaries Fixed Interest Index (to 1985), the JSE-Actuaries All Bond Index (to 2000) and the BESA Government total return index from 2001 onward. Before 1925, short-term interest rates are represented by U.K. Treasury bills. Subsequently, we use the bill returns from Firer and McLeod, based on three-month fixed deposits (1925–1959), bankers’ acceptances (1960–1966), and thereafter negotiable certificates of deposits. Inflation is estimated prior to 1925 using the consumer price index and thereafter using the official price index from Central Statistical Services. The switch in 1961 from pounds to rand has been incorporated in the Exchange Rate index index history.

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Spanish stock returns are presented in Gonzalez and Suarez (1994) for the period commencing in 1941. Valbuena (2000) provides a longer-term perspective. Valbuena’s equity index for Spain over 1900–1918 is from Bolsa de Madrid. For 1919–1936, we use a total returns index from Valbuena (2000) that rectifies some problems in the Sandez and Benavides (2000) index. Trading was suspended during the Civil War from July 1936 to April 1939, and the Madrid exchange remained closed through February 1940. Over the closure we assume a zero change in nominal stock prices and zero dividends. During 1941–1985, we use the Gonzalez and Suarez (1994) data, subsequently linking this to the Bolsa de Madrid total return index. The bond series for 1900–1926 is based on the price of Spanish 4 percent traded in London through 1913 and in Madrid thereafter. For 1926–1957 and 1979–1987, it is based on Global Financial Data’s estimates for government bonds, with prices kept unaltered during the Civil War. A private bond index is used for 1958–1978. From 1988, we use the JP Morgan Spanish government bond index series with maturity of over three years. The short-term interest rate over 1900–1973 is the central bank discount rate. From 1974, we use the return on Treasury bills. Inflation during 1900–1914 is measured using the wholesale price index from Mitchell (1998). For 1915–1935, we use the consumer price index from Mitchell (1998); see also Vandellos (1936). During 1936–1940, we revert to the wholesale price index from Mitchell. For 1941–1985, we use the Spanish consumer price index from Gonzalez and Suarez (1994) and thereafter from the Instituto Nacional de Estadistica. Sweden is studied in a series of papers by Per Frennberg and Bjorn Hansson (1992a, 1992b, 2000), whose database on stocks, bonds, bills, and inflation covers the period 1919–1999. The Swedish stock market data we use starts at the end of 1900, and we assume that stock prices did not move over 1900; thereafter we use the index values of the Swedish Riksbank. Over the period 1900–1918, Swedish equity dividends are estimated from contemporaneous bond yields adjusted upwards by 1.33 percent (the mean yield premium over 1919–1936). From the start of 1919, the Swedish equity series is based on the share price index published in the journal Affarsvarlden, plus the dividend income estimated by Frennberg and Hansson (1992b). The government bond series uses data for 1900–1918 from The Economist. For 1919–1949, the returns are for perpetuals, and after that the series measures the return on a portfolio of bonds with an average maturity of 10 years. We use the JP Morgan Swedish government bond index with maturity of over five years from 2000. The short-term riskless rate of interest from 1900 is represented by the official discount rate of the Swedish Riksbank. Frennberg and Hansson (1992b) switch in 1980 to the return on short-term money market instruments, and from 1982 to Treasury bills. Inflation is represented by the Myrdal–Bouvin consumer price index before 1914, the cost of living index between 1914–1954, and the Swedish consumer price index for 1955 onward. Switzerland is investigated using the series spliced together by Daniel Wydler (1989, 2001) coupled with extra data kindly provided by Urs Walchli and Corina Steiner. We have created a new, equally weighted index of Swiss equity prices for 1900–1910. This uses the series of annual prices and dividend yields collected from Neue Zurcher Zeitung, with an average of 66 year-end stock prices over the period.


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Over 1911–1925, we use the index of 21 industrial shares from Statistiches Jahrbuch. The Swiss exchanges were closed during September 1914 to December 1915, so for end-1914 and end-1915, we use the index at the date closest to the year-end. For 1926– 1959, Ratzer (1983) estimates total returns. For 1960–1983, Huber (1985) computes the returns from index levels and dividends on the SBC index. Over 1984–1998, we use the Pictet return index, and then the Swiss All Share index. For Switzerland only, and solely for the period 1900–1915, we estimate bond returns from the short rate. We use the latter as a proxy for the yield on seven-year bonds, and infer the annual returns for this series. For 1915–1925, we use annual data from the Statistischen Bureau. The interval 1926–1959 employs Ratzer’s (1983) estimates based on redemption yields for new Swiss bond issues. The 1960–1980 period is represented by Huber’s (1985) bond index based on actual trading prices. From 1981, we use the Datastream 10-year Swiss government bond index. During 1900–1955, short-term rates are represented by the central bank discount rate, and for 1956–1979, by the return on three-month time deposits. From 1980 onward, we use the return on Treasury bills. Nominal returns are adjusted for inflation using movements in the Swiss consumer prices index. The United Kingdom is analyzed using index series described in Dimson and Marsh (2001) for the interval from 1955 to date, and in Dimson, Marsh, and Staunton (2002, 2006a) for the period 1900–1954. Because of biases and inaccuracies in prior index series, the last half-century is based on the fully representative record of equity prices maintained by London Business School and described in Dimson and Marsh (1983). The period up to the end of 1954 is based on an index of the returns from the 100 companies that, before each New Year, have the largest equity market capitalization. Share capital was checked against the annual Stock Exchange Official Yearbook up to 1955, to account for capital changes and corporate events. Before 1955, all cash flows are assumed to occur at the end of each year, including dividends, special dividends, returns of capital, and cash from acquisitions. Where companies are acquired for shares or merge, we base returns on the end-year share price of the acquirer or merged entity, taking account of the exchange ratio. Dividends were obtained from the Stock Exchange 10-Year Record published by Mathiesons. The U.K. bond index was compiled from original British government bond data. For the 1900–1954 period, the returns are based on 2½ percent consols, and for 1955–2000 the bond index measures the return on a portfolio comprising high-coupon government bonds with a mean maturity of 20 years. Throughout, Treasury bills are used to measure the short-term riskless rate of interest. Inflation is calculated using the retail price index and, before 1962, the index of retail prices. The United States was first researched in the Ibbotson and Sinquefield (1976) article and subsequent Ibbotson Associates updates. The broadest index of U.S. stock market returns is in Wilson and Jones (2002), and we use the latter for this study. Earlier sources are described in Goetzmann, Ibbotson, and Peng (2001). Our series, however, commences with the Wilson–Jones index data over 1900–1925. For 1926–1961, we use the University of Chicago’s Center for Research in Security Prices (CRSP) capitalization weighted index of all New York Stock Exchange stocks. For 1962–1970, we use the CRSP capitalization weighted index of NYSE, American, and Nasdaq stocks.

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From 1971 onward, we employ the Dow Jones Wilshire 5000 index. All indexes include reinvested dividends. The government bond series for 1900–1918 is based on 4 percent government bonds. Over 1919–1925, we use the Federal Reserve 10 to 15 year bond index. After that, bond returns are based on Ibbotson Associates’ long bond index. The bill index uses commercial bills during 1900–1918. From 1919 onward, the series is based on U.S. Treasury bills. Inflation is based on the consumer price index. The world is represented by an equity series that comprises a 17-country, commoncurrency (here taken as U.S. dollars) index. For each period, we take a market’s local-currency return and convert it to U.S. dollars. We therefore have the return that would have been received by a U.S. citizen who bought foreign currency at the start of the period, invested it in the foreign market throughout the period, liquidated his or her position, and converted the proceeds back at the end of the period into U.S. dollars. We assume that at the beginning of each period our investor bought a portfolio of 16 such positions in each of the foreign markets in this study, plus domestic equities, weighting each country by its size. We use GDP weights with start-decade rebalancing before 1968 due to a lack of reliable data on capitalizations prior to that date. Thereafter, we use country capitalizations from Morgan Stanley Capital International (MSCI). The above procedure results in an index expressed in U.S. dollars. To convert this to real terms, we then adjust by the U.S. inflation rate. This gives rise to a global index return denominated in real terms, from the point of view of our notional U.S. investor. Our 17-country world bond market index is constructed in the same way. This is again weighted by country size, to avoid giving, say, Belgium the same weight as the U.S. Equity capitalization weights are inappropriate here, so the bond index is GDP-weighted throughout. The short-term risk-free rate is taken as the return on U.S. Treasury bills. The inflation rate is as for the U.S.

CHAPTER 12

History and the Equity Risk Premium William N. Goetzmann and Roger G. Ibbotson

1. 2. 3. 4. 5. 6.

Introduction Historical Conception and Measurement of the Equity Risk Premium Stocks, Bonds, Bills, and Inflation History as Written by the Winners? The Equity Premium Over the Very Long Term Conclusion References

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JEL Classification: G12 Keywords: monopolistic advantage, insurable risk, uncertainty, stock price indices, capital-weighted, re-investment of dividends, Capital Asset Pricing Model, government bond data, Ibbotson Associates, Vanguard, Boston money market, ex ante measurement


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Chapter 12 • History and the Equity Risk Premium

1. INTRODUCTION The equity premium puzzle posed in Mehra and Prescott (1985) was motivated by historical evidence on the return of U.S. stocks in excess of the riskless rate. Much of our own research has focused on estimating the equity risk premium using long-term historical data, and examining how historical accidents may relate to the classic puzzle. While the equity premium is a fascinating topic for scholarship, it is also one of the most important economic topics in modern finance. The equity risk premium is widely used to forecast the growth of investment portfolios over the long term. It is also used as an input to the cost of capital in project choice, and employed as a factor in the expected rate of return to stocks. Given its prevalence in practice and its importance to academic thought, it is interesting to discover that the calculation of the equity risk premium is a fairly new phenomenon. Reliable data to estimate the historical premium of stocks over bonds was only collected in the mid 20th century, and precise econometric estimates of the equity premium only came after the development of the theory that uses it as a central input—the capital asset pricing model, or CAPM. Our own contributions to the literature on the equity risk premium—working both separately and together on a series of empirical research studies conducted with colleagues through the years—have been in several areas. First, research by Roger Ibbotson and Rex Sinquefield provided some of the first accurate calculations of the annual rate of return on U.S. asset classes over long investment horizons with specific measures of the equity and other risk premiums. These calculations have come into widespread academic and industrial use as inputs to research and investment decision-making through numerous works that Ibbotson has produced. Second, Will Goetzmann and co-authors Stephen Brown and Stephen Ross proposed and examined the hypothesis that the equity premium estimated from U.S. financial data alone is subject to a bias due to analysis of a winning market rather than losing ones. Third, both of us together with our co-author Liang Peng have constructed one of the most complete long-term databases of U.S. financial returns yet developed and have used it to study the variations in the equity risk premium through nearly 200 years. Finally, both of us have worked on other related concepts, including the predictability of the equity risk premium, alternate ways of measuring the equity risk premium, and on other applications of the concept. In this chapter, we first review the historical development of the idea of the equity risk premium in financial economics as the theory of compensation for investment risk, and how it developed in tandem with the empirical analysis of historical investment performance. Next we summarize some of our past findings about the historical equity risk premium and present further analysis on potential survival biases. Finally, we update our analysis of the historical performance of the New York Stock Exchange over the period 1792 to the present, and include a measure of the U.S. equity risk premium over more than two centuries. This last section is based upon indices constructed from individual stock and dividend data collected over a decade of research at the Yale School of Management and other research institutions.

William N. Goetzmann and Roger G. Ibbotson

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2. HISTORICAL CONCEPTION AND MEASUREMENT OF THE EQUITY RISK PREMIUM One of the earliest and most succinct expressions of the concept of the equity risk premium came from John Stuart Mill in his 1848 classic, Principles of Political Economy. Writing about a farmer considering investment in the land, Mill argues that . . . he will probably be willing to expend capital on it (for an immediate return) in any manner which will afford him a surplus profit, however small, beyond the value of the risk, and the interest which he must pay for the capital if borrowed, or can get for it elsewhere if it is his own.1 Mill thus separates profit into three parts: first, there is the interest that must be paid for the capital borrowed—in effect, the opportunity cost of money. This is equivalent to the riskless rate. The second component is the “value of the risk” associated with the investment. This is equivalent to the equity risk premium. Mill’s third component is a surplus profit, no matter how small. In modern parlance, the “alpha”—a portion of compensation expected to be small in a competitive market. Despite Mill’s early formulation of the idea, the concept of equity profit as compensation for risk did not develop quickly. The capital theorist Eugen von Böhm-Bawerk (1884) sought to explain interest rates through the intertemporal nature of the production process. He likewise recognized the logical spread between risky and riskless debt; however, he intentionally left aside the problem of measurement of entrepreneurial returns. Economists at the turn of the century tended to focus on the apparent paradox of profit and perfect competition rather than risk and return. Columbia University professor John Bates Clark, whose on-going debate with Bo¨ hm-Bawerk focused on interpretations of the intertemporal nature of production and consumption, asserted that returns in excess of the riskless rate were due to monopolistic advantage, rather than compensation for insurable risk. In his view, innovation led to a comparative advantage which was in turn rewarded by excess return.2 Chicago economist Frank Knight in turn challenged Clark’s formulation by asserting the central importance of risk in concepts of profit. In his 1921 work Risk, Uncertainty 1 Book 2, Chapter 16, The Principles of Political Economy, 1848. See also J.S. Mill, Essays on Some Unsettled Questions of Political Economy Essay IV: On Profits, and Interest.

The profits of stock are the surplus which remains to the capitalist after replacing his capital: and the ratio which that surplus bears to the capital itself is the rate of profit. The gross profits from capital, the gains returned to those who supply the funds for production, must suffice for these three purposes. They must afford a sufficient equivalent for abstinence, indemnity for risk, and remuneration for the labour and skill required for superintendence. It is somewhat unclear whether he is referring only to a return that covers a probability of expected loss instead of the equity risk premium’s increase of expected return to cover systematic risk. 2 Clark (1892). See a response to Clark by Hawley (1893).

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and Profit, he reviewed the role of risk in the economic theory of profit up to the 1920s and took exception to the lack of distinction in previous analyses between quantifiable and unquantifiable risk—the latter he termed uncertainty, but both of which he asserted should command an investment premium. Although widely regarded as an argument for consideration of “meta-uncertainty” in financial models, LeRoy and Singell (1987) point out that Knight’s distinction was more likely about insurable, vs. uninsurable risk—in modern parlance, complete and incomplete markets. Knight’s philosophical treatise did little, however, to clarify how the different roles of risk and uncertainty would affect prices and business ventures in a practical manner, and he was completely silent on the issue of how one might quantify the equity risk premium. As theorists debated the role of risk in the expected return to investment, empirical researchers in the early 20th century began to collect historical performance data from the markets. The earliest attempts to construct stock price indices were motivated by the need for a “barometer” of current market trends, or as an indicator of fluctuating macroeconomic conditions. Charles Henry Dow’s famous index of 30 stocks was not originally intended as a measure of long-term investment performance, but rather as a daily measure of the market. A number of macroeconomists began to create stock price indexes in the early 20th century. Mitchell (1910, 1916), Persons (1916, 1919), Cole and Frickey (1928) are among the various scholars who collected U.S. equity prices and constructed indices as a means to study the interaction between economic cycles and the financial markets. Smith and Horne (1934) and Bowley, Schwartz and Smith (1931) built similar indices for Great Britain. None of them addressed the obvious question—at least from our modern perspective—of long-term investment returns.3 Edgar Lawrence Smith’s 1924 book, Common Stocks as Long Term Investments, is the first significant attempt to advocate equity investing as a means to achieve higher investment returns. We should not underestimate the importance of his contribution to the study of the equity risk premium. Smith collected price and dividend data for U.S. stocks over the period 1837 to 1923, and computed a total return index, which he compared to a fixed rate of interest over the corresponding period. He also collected historical price and dividend data for stocks and corporate bonds over the period 1866 through 1923 from the Boston and New York Stock Exchanges. He formed stock and bond investment portfolios of 10 securities each as the basis for simulating investor performance over 4 different time periods. He studied the relative appreciation returns and income returns from both asset classes and documented fairly convincingly that over a variety of subperiods equities yielded higher income than bonds and also provided significant capital appreciation. Smith simulated the performance of these portfolios in a number of ways. The simplest was to treat the income and capital appreciation returns from the stock and bond portfolios separately and to show that stocks nearly always dominated in both measures. He came close to developing a total return measure for the equity premium by the mechanical process of taking the income return each year from stocks and “paying” 3 For an excellent discussion of the development of early equity indices, see Pierre-Cyrille Hautcoeur and Muriel Petit-Koñczyk (2005). For a complete list of indices developed before Cowles (1938), see Cowles’ own discussion and notes in his volume.


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out of it the amount generated by the bond portfolio and then re-investing the residual back into shares. The relative growth of the stock portfolio through this procedure can be interpreted as a measure of the equity premium—at least with respect to corporate bonds. Smith’s book was not only widely read by investors but also closely studied by scholars. It was immediately cited by Yale’s Irving Fisher as an argument for investing in a diversified portfolio of equities over bonds.4 Based on Smith’s findings, Fisher theorized that the trend toward investment in diversified portfolios of common stock was actually changing the equity premium in the 1920s. His views on the factors influencing the equity risk premium are worth quoting at length. Studies of various writers, especially Edgar Smith and Kenneth Van Strum have shown that in the long run stocks yield more than bonds. Economists have pointed out that the safety of bonds is largely illusory since every bondholder runs the risk of a fall in the purchasing power of money and this risk does not attach to the same degree to common stock, while the risks that do attach to them may be reduced, or insured against, by diversification. . . . It is in this way that investment trusts and investment council tend to diminish the risk to the common stock investor. This new movement has created a new demand for such stocks and raised their prices, at the same time it has tended to decrease the demand for, and to lower the price of, bonds.5 Smith’s empirical approach to measuring the relative performance of the two asset classes was widely imitated in later studies. In 1937, Brown University Professor Chelcie Bowland published a synthesis of research following Smith’s book and showed how the common stock investment strategy performed through the worst years of the depression.6 Bowland concluded, on considerable empirical evidence, that the theory of common stock investment survived the crash. An interesting feature of the studies cited in Bowland’s book is that none of them produced what we now think of as a measure of the equity premium—that is, the difference in total return between a portfolio of equities and the riskless rate over the same period. The most carefully crafted early empirical analysis of the long-term performance of the stock market was Common Stock Indices, by Alfred Cowles III, published in 1938. This ambitious study, undertaken before the advent of computers, but assisted by the invention of Holerith cards, collected individual stock prices (actually monthly highs and lows by stock) and dividends from 1872 to 1937 for the NYSE. Its stated goal was to “portray the average experience of those investing in this class of security in the United States from 1871 to 1937.”7 Two important features of the Cowles study were that he collected data on virtually all of the stocks on the New York Stock Exchange and that he capital-weighted 4 Fisher

(1925). (1930). 6 Bowland (1937). 7 Cowles (1938). 5 Fisher

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them, a procedure that allowed the index to simulate a passive buy and hold investment strategy. The one serious limitation of the Cowles study is that it relied on the average of high and low prices during the month as a proxy for end-of-month stock prices. This had a smoothing effect on the returns, downward biasing the volatility and muddying later econometric analysis of the data. Oddly enough, given such widespread interest in Edgar Smith’s earlier study, the Cowles analysis was silent on the relative performance of stocks and bonds. The first book to explicitly define model, and estimate an equity risk premium was John Burr Williams’ The Theory of Investment Value, also published in 1938. According to Williams, “The customary way to find the value of a risky security has always been to add a ‘premium for risk’.”8 He provides a table of “Interest Rates, Past Present and Future” which takes the riskless rate as the long-term government bond rate of 4 percent and the expected return to “Good stocks” as 5 1/2 percent.9 Williams estimated the forward equity premium from a dividend discount model, and he was careful to explain that historical (i.e., past) estimates provide a good forecast of the future, even when they deviate from present conditions. In sum, by the end of the 1930s, economists had developed a clear conception of the equity risk premium, a means to measure rates of return on investments, and had collected historical data extending back through American financial history for several decades. The first empirical study by Edgar Smith of the relative attractiveness of stocks vs. bonds is generally regarded as a major factor in the rush by retail investors into the stock market in the 1920s, and Irving Fisher is often taken to task for his theory that stock prices increased to new levels in the 1920s as a result of a decreasing equity risk premium. Alfred Cowles created the first relatively accurate long-term index of total return to investing in common stocks, and J. B. Williams provided the first numeric estimate of the forward-looking equity risk premium. Their combined work thus provided a valuable foundation for the next generation of financial research on the equity risk premium. The next major attempt to empirically quantify equity returns was undertaken at the University of Chicago. Beginning in 1960, the CRSP, the Chicago Center for Research on Security Prices, headed by economists Lawrence Fisher and James H. Lorie, systematically began to collect stock prices and dividends from U.S. capital market history. Fisher and Lorie published the results of their study of returns to U.S. stocks in 1964, as “Rates of Return on Investments in Common Stocks”10 and in 1977 as a volume including returns to U.S. government securities as well. 11 Like Cowles, they based their analysis of individual share prices and re-investment of dividends of U.S. stocks. 8 Williams

(1938). Of special note to those interested in financial history is that Williams solves algebraically for the discount rate on the common stock of a firm as a function of the discount rate for the all-equity firm and the firm debt—preceding Modigliani and Miller in arguing that “the investment value of an enterprise . . . in no way depends upon what the company’s capitalization is” (p. 2). 9 Ibid., p. 387. 10 Fisher and Lorie (1964). 11 Fisher and Lorie (1977).


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The theoretical developments in financial economics in the 1950s and 1960s made these empirical estimates of rates of return particularly interesting. In 1952, Harry Markowitz published his famous model of portfolio selection that explicitly linked investment return and risk. Markowitz proposed taking as inputs to his model the historical means, variances, and covariances of individual securities, although he regarded this as a method that could be improved upon with better forecasting tools. Yale economist James Tobin (1958) is credited with adding an important feature to the Markowitz model: a non-zero riskless rate, leading to an explicit expression for a risk premium. The Markowitz–Tobin model, as it is now applied, identifies an optimal portfolio of assets in expected return and standard deviation space by the point of tangency formed by a ray extending from the expected return of the riskless (zero standard deviation) asset to the continuous frontier of portfolios providing the highest return for each level of standard deviation. The difference between the return of the riskless asset and the expected return of the tangency portfolio in this model is the equity risk premium.12 In the Markowitz–Tobin framework, the size of the equity risk premium is an empirical question. Later scholars took a theoretical approach to its estimation. The Sharpe–Lintner–Mossin capital asset pricing model (CAPM) was independently developed in the 1960s in part as a means to identify the optimal portfolio of risky assets in the Markowitz–Tobin framework. As such, the CAPM takes an analytical approach to the equity risk premium. The theory endogenizes asset prices as a function of the risk aversion of the representative investor and the variance-covariance structure of the universe of assets. The shape of the investor’s utility function, parameterized by the coefficient of risk aversion for the market as a whole, is central to identification of the equity premium. In the framework of the CAPM, if the form of the utility function and the coefficient of risk aversion are both known, then knowledge of the variance-covariance of the universe of assets (or the variance of the portfolio of risky assets) is sufficient to identify the spread between risky and riskless asset portfolios. An important feature of the Markowitz–Tobin model and the CAPM is that they provide a theoretical foundation for estimating the magnitude of the equity risk premium directly from investor preferences. It was not until Mehra and Prescott (1985), however, that anyone attempted to compare the equity premium implied by preferences with the empirical measures provided by historical returns.

3. STOCKS, BONDS, BILLS, AND INFLATION In 1976, Ibbotson and Sinquefield published Stocks, Bonds, Bills and Inflation: Yearby Year Historical Returns (1926–1974). Stock market returns were calculated as total returns from the S&P500 index (which at the time did not include dividends) plus the 12 The Markowitz framework is a single-period model. As such, the arithmetic return and the geometric return are the same.

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income return for the S&P stocks. The authors also used CRSP government bond data to include U.S. Treasury bond and bill indexes. They also included a corporate bond index constructed from bond yields, as well as inflation rates. Besides displaying total returns across all yearly hold periods, the paper was unique in that it explicitly measured historical risk premiums, not only for equities, but also including the horizon (maturity) premium, the default premium, and real interest rate. All results were presented in real (inflation-adjusted) terms as well. In Ibbotson Associates Yearbooks, they later added the small stock premium, the value premium, as well as various other data. These historical premiums came to be used as the risk premium input into the CAPM model, as well as for other models. Later in 1976, Ibbotson and Sinquefield showed how historical data can be used to simulate probability distributions of future returns. They started with the then-current yield curve, with its implicit forward interest-rate structure. They added the various historical premiums using bootstrapping methods that retained the correlation structure among the asset classes. They used a historical equity risk premium that was measured during the previous half-century to be 6.3 percent geometrically relative to U.S. Treasury bills, but a lesser number relative to longer-term bonds that contained horizon risk premiums. Interestingly, 1976 was also the year that Vanguard launched its market index trust, a passive, value-weighted portfolio of large-cap U.S. stocks designed to match the performance of the S&P500. The excellent excess performance of stocks from 1926 to 1974 reported in Ibbotson and Sinquefield may have helped build demand for a passive portfolio that sought to capture these historical returns. Just as the empirical study by Edgar Lawrence Smith demonstrated to investors in the 1920s the superiority of stocks over bonds and motivated a move toward equities, indexation made a lot of sense to investors in the 1970s when they had five decades of historical performance upon which to base their expectations for the future. An important historical note is that the decade of the 1970s was one of the worst periods in U.S. capital market history for stocks. The crash of 1973–1974, the experience of double-digit inflation, and the erosion of capital were fresh in the public imagination in 1976. Equity returns had not exceeded debt returns over nearly the entire preceding decade. Thus, the historical equity premium appeared to most as a wildly optimistic forecast of the future expected return of stocks over T-bills. In fact it was not—the equity risk premium since 1976 has nearly matched the estimate made at the time. There are several ways in which one might estimate an expected risk premium used for forecasting. One way is to extrapolate historical risk premiums, as did Ibbotson and Sinquefield. Another is to use investor demand models based upon investor risk aversion, as did Mehra and Precott. A third way is to look at the type of returns that the corporate sector supplies. Diermeir, Ibbotson, and Siegel (1984) and later Ibbotson and Chen (2003) used this supply approach. They extrapolated the cash flows and earnings growth generated by companies themselves. These forecasts tend to give somewhat lower historical risk premiums, primarily because part of the total return of the stock market has come from price-to-earnings ratio expansion. This expansion


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is not predicated to continue on indefinitely, and is removed from the expected risk premium.

4. HISTORY AS WRITTEN BY THE WINNERS? A major conceptual problem with equating the ex post historical realization of the equity premium with its ex ante expectation is that history could simply have turned out better than people expected. Recall that in 1938, J. B. Williams calculated the market forecast of the equity risk premium as a mere 1 1/2 percent. He might have been shocked to see the realized premium over the following five decades exceed his forecast by a multiple of four. A more subtle problem is that historical records of successful markets tend to get preserved, but it is difficult to dig up the records of failed markets. To study these issues, Brown, Goetzmann, and Ross (1995) built a model of stock market performance that examined the ex post historical return conditional upon survival. Their analytical results showed that when a market is confronted with the continued possibility of failure (modeled statistically as an absorbing lower bound), the ex post realized growth can be substantially higher than the unconditional mean. The fact that the U.S. survived two world wars and periods of global political unrest that caused other major markets to fail would suggest that the mean return of the U.S. market is an overestimate of its true historical mean. It is important to point out that this view of the equity risk premium is quite different than the critique of the equity premium puzzle posed by University of Iowa economist Thomas Rietz. Rietz (1988) theorized that a high ex ante equity premium could be justified by a fear of a huge crash. An equity premium of 6 percent would thus reflect the potential for an event that may not have been realized in America’s financial history but that nevertheless was a possibility in people’s minds. One limitation of the Rietz critique is that it conflicts with historical expectations of stock returns—most notably the J. B. William’s forward-looking premium in 1938 of 1 1/2 percent. The survivorship story, on the other hand, is entirely consistent with low historical expectations of the future, although it would be unlikely to reduce the equity risk premium to such a low number. In order to estimate the potential scale of the survival bias, and to look for empirical evidence that the U.S. market was an unusual performer in the global economy, Jorion and Goetzmann (1999) collected capital appreciation and inflation data for a large sample of the world’s equity markets from 1921 forward. They found that the U.S. was the top performer out of 39 markets, in terms of real capital appreciation of stocks. Studying this effect subsequently with a smaller but richer sample, Dimson, Marsh, and Staunton (2002) collect total real returns for 12 countries over the entire century and find that the U.S. was in the top quartile of performers in real terms—inched out by Sweden and Australia over the first couple of decades of the century.13 The Jorion and Goetzmann 13 Dimson,

Marsh, and Staunton (2002).

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GDP-weighted index of the 44 non-U.S. equity markets had an arithmetic real return of 3.84 percent per year compared to the U.S. value of 5.48 percent, suggesting a survival bias on the order of 1.5 percent per annum. The geometric return values were closer (roughly 90 basis points) because the world market index was less volatile over the period due to diversification. Thus, the work found evidence that, while the equity premium estimate is affected by survival issues, the magnitude is insufficient to explain away the equity premium puzzle.

5. THE EQUITY PREMIUM OVER THE VERY LONG TERM One of the major issues with statistical estimation of the realized equity risk premium is that a very long time series of stationary returns is required to achieve a high degree of confidence in the estimate. The longer the data series, the more accurate the equity risk premium calculation, as long as the fundamental expectations have remained the same. In order to estimate the U.S. equity premium using total returns over the longest possible time period, we have collected the most complete data set of U.S. stock prices and dividends assembled to date.14 Working with economist Liang Peng, we gathered individual security data from U.S. financial periodicals on a monthly basis, beginning with the official list of the New York Stock Exchange in 1815 and collected available dividends from 1825 to 1872. Among other things, this direct data collection enabled us to collect month ending prices, avoiding the Cowles data problem of averaging high and low prices for the month. It also allowed us to avoid the heterogeneity problems of Ibbotson and Brinson (1987), Schwert (1990), and Goetzmann (1993), all of whom had to rely upon chained indices constructed by earlier researchers.15 Recently, Sylla, Wilson, and Wright (2000) completed the collection of weekly NYSE stock prices over the late 18th and early 19th centuries.16 Richard Sylla generously shared this data with us, and from it we are able to construct a complete index of capital appreciation returns for the New York Stock Exchange from 1792 (its inception) to the present. In this chapter, using the combined databases of Sylla, Wilson, and Wright (2000) and Goetzmann, Ibbotson, and Peng (2000), as well as annual bond yield data from Homer and Sylla’s The History of Interest Rates, we are able to estimate an equity premium for the New York Stock Exchange over its pre-1925 history. Capital appreciation returns in this study are based on a price-weighted index of all stocks trading in the year, using the last price observation in the year. Income returns

14 Goetzmann,

Ibbotson, and Peng (2000). NYSE database is available for download on the website of the International Center for Finance at the Yale School of Management. www.icf.yale.edu. 16 Sylla, Wilson, and Wright (2000). 15 The


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from 1825 to 1871 are constructed in two ways and then averaged. The first method is to sum all the dividends paid in that year and divide by the prices of all firms that traded the previous year. This understates the income return because some dividend data is missing. The second method focuses solely on the income return of firms that paid regular dividends and for which prices were collected—this likely overestimates returns because some stocks may not pay dividends. It is important to note that no previous researchers collected actual dividend data on U.S. stocks before 1872. All analyses before our own have been based upon econometric methods to fill in income returns. Thus, although our two methodologies sometimes diverge, they are at least based upon empirical observation. However, like previous analysts, in order to estimate the pre-1825 dividends, we employ a linear forecasting model using next year’s dividend and this year’s capital appreciation as forecasting variables.17 Our dividend returns from 1872 to 1925 are taken from Cowles (1938). The most problematic variable in our study is the riskless rate. In 1792, the U.S. was an emerging market and U.S. securities were considered far from riskless. Homer and Sylla report yields on U.S. government and U.S. municipal debt in the 18th and 19th centuries. For long stretches of time, the yields on municipals were less than the yields of treasuries, and this difference was not due to tax treatment. We took the minimum yield of the two series each year as a measure of a (nearly) riskless rate and the pre1798 data was set to the 1798 value. Of course, yields are not returns when the security is not held to maturity. Thus, using them to capture the riskless rate ignores the capital appreciation component of bond returns. Returns to shorter maturity instruments are available for periods of U.S. financial history before 1926. Mehra and Prescottt (1985) used commercial paper rates reported in the earlier edition of Homer and Sylla back to 1883. We report these in our analysis as well. Homer (1963) notes that these are quite volatile in the early years, perhaps reflecting the lack of an organized money market.18 Commercial paper rates from 1830 to 1857 are from the Boston money market, and later from the New York market. We econometrically estimated commercial paper rates from 1792 to 1834 by projecting them on bond yields.19 Other types of short term rates in the early 19th century are dubious due to regulation of interest rates. Inflation data is taken from Ibbotson and Brinson (1987). Table 1 presents the results of the analysis for the period 1792–1925. Note that the arithmetic equity premium measured by spread over bonds is about 3.8 percent and the geometric spread is 2.72 percent. A major issue, however is that the interest-rate series we have constructed is not riskless. Comparing the returns instead to inflation indicates an arithmetic real return of about 7 percent. The bond returns would appear to be anomalous with respect to expectations, not the equity returns. The low capital appreciation returns to stocks prompted us to test the hypothesis that for much of U.S. financial history investors expected their equity returns to come in the form of dividend payouts rather than capital appreciation. We found that vast majority of stock in our sample traded around par value, implying an expectation of payout. 17 The

R-square of this model was 0.17, suggesting that it has some explanatory power. p. 317. 19 The regression yielded an R-square of 0.22 with a t-statistic of 5. 18 Homer,

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TABLE 1 Summary Statistics for New York Stock Exchange Returns, U.S. Bond Yields, Call Money Rates, and Inflation, 1792–1925 Arithmetic return

Geometric return

Standard deviation

Stocks

7.93%

6.99%

14.64%

Cap ap

1.91%

—

—

Income

6.01%

—

—

Bonds

4.17%

4.16%

4.17%

Comm. paper

7.62%

7.57%

3.22%

Inflation

0.85%

0.61%

7.11%

TABLE 2 Summary Statistics for U.S. Stocks, Bonds, Bills, and Inflation, 1926–2004 Arithmetic return

Geometric return

Standard deviation

Stocks

12.39%

10.43%

20.31%

Cap ap

7.85%

—

—

Income

4.27%

—

—

LT govt. bonds

5.82%

5.44%

9.30%

T-bills

3.76%

3.72%

3.14%

Inflation

3.12%

3.04%

4.32%

Source: Stocks, Bonds, Bills and Inflation, 2005 Yearbook, Ibbotson Associates, Chicago.

If this were strictly the case, it would allow the ex ante measurement of the equity risk premium through financial history. Unfortunately, we cannot verify that expected returns carried no anticipated long-term price growth. The 1.91 percent appreciation we document in Table 1 exceeds the realized inflation over the period, and thus may have been ex ante expected. On the other hand, in our paper we show that through most of U.S. capital market history, dividend yields were higher than bond yields, suggesting that differential income flow was a major part of the compensation for equity risk during the early period. Table 2 presents summary statistics for the major U.S. asset classes over the period 1926–2004. The arithmetic average return to common stocks over the second period is 400 basis points per year higher than in the first. In real terms, however, this differential is slightly less dramatic: 7.08 percent to 9.27 percent. The differences of real geometric means for the two periods are within a percent of each other.


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6. CONCLUSION The concept of the equity risk premium is fundamental to modern financial theory and a basic building block in most forecasting models of long-term expected investment returns. A review of the economic literature after the turn of the 19th century suggests that the concept of the equity risk premium was not clearly formulated until the late 1930s. While the notion of return as a premium for risk above and beyond the pure time value of money dates at least to the work of John Stuart Mill, the basic technique of calculating the total return to investing in equities vs. debt developed relatively slowly. The historical development of the concept and measurement of the equity risk premium provides the context for our own research contributions. The first Ibbotson and Sinquefield study represented a culmination of research on the basic building blocks of expected returns for different asset classes. The notion of building up expected returns from blocks of risk premia was viewed by J. B. Williams as a natural approach. However, it is surprising how long it took for the basic empirical calculus of risk and return to come into use. The Ibbotson and Sinquefield numbers as reported in 1976 were striking evidence that common stock investment, so avidly proposed by Smith and Fisher five decades earlier, was, in fact, a wise course of action to take. A new generation of investors in the 1970s used these numbers as a guide to expectations of future returns to equity investment, and 25 years later they were not disappointed. History proved an accurate forecast. The sheer magnitude of the equity premium in U.S. capital markets over the 20th century has caused both scholars and practitioners to ask whether these returns were simply an accident of history or evidence of a different kind of attitude toward risk than seems justified by theoretical models. The importance of Mehra and Prescott (1985) is that it pointed out the apparent contradiction between the U.S. market experience and academic models of human behavior. A part of our own inquiry into the equity premium puzzle has been the question of whether the history we examine is an unusual path—one unlikely to be realized in the future. Most Americans who lived through a significant portion of the 20th century counted themselves fortunate compared to large sectors of the global populace who suffered catastrophic loss of savings as a result of the political tumult of two world wars and widespread redistribution of wealth. In light of America’s political and economic success in the 20th century, it is not surprising to find that its markets dominated as well. A test of this “survivor” story in our research finds some empirical support, but does not fully explain the high equity risk premium enjoyed through U.S. capital market history. The survival hypothesis suggests that the American experience may not be the best example on which to base future expectations—then again, maybe it is for those who plan to continue investing in the U.S. capital markets. Perhaps the positive American experience was actually due to our particular configuration of laws, political system, cultural mixture, and practical orientation. A longer look at the American financial experience affords a chance to test this proposition. Although the 20th century may be the American Century, the 19th century

528


was not. Europe’s financial markets were dominant through the First World War. American finance was parochial and limited. One important qualification to this, however, was the comparative freedom of American equity markets. Britain severely limited the issuance of corporate shares until the mid-19th century, and full corporate access to the capital markets did not exist until the British Companies Acts of the 1860s. By that time, American equity markets had been operating in New York, Philadelphia, and Boston for many decades. Indeed, the U.S. might have been the best market to study the early development of unfettered capitalism in the early 19th century. What do we find when we look at the 19th century numbers? First, the measurement of the equity risk premium in the U.S. before 1925 is nearly impossible due to largely to the fact that there was no meaningful riskless rate as a benchmark. Had the capital asset pricing model been transported back from the 1960s to the 1860s, the challenge would have been to determine Rf , not Rm . When we look at the premium of stocks over inflation, however, we find that the real rate of equity returns in the first 125 years of its history pretty nearly resembles the real rate of equity returns in the last 80 years. Through that trajectory of time, the U.S. went from being an emerging market to a capital importer to a capital exporter. Given what would seem to be major regime changes in the U.S. economy, it is extraordinary to find such stability in the rate of return on investment capital. Is that stability due to a “country factor” in modern asset pricing terminology? This will have to await further tests using data from global capital market history. Our research together and separately has focused to a large degree on measurement and interpretation of history. Despite decades of research on early capital markets, however, much remains to be done. Our understanding of the historical experience of investors is relatively limited once we step beyond a few well-studied markets. Basic information about investor returns is lacking and may never be recovered. Nevertheless, efforts to quantify the equity risk premium are well rewarded by insights into both the stability and dynamics of long-term investment performance.

References Böhm-Bawerk, E. von. Capital and Interest: A Critical History of Economical Theory. Macmillan and Co., London (1884). Bowland, C. The Common Stock Theory of Investment. The Ronald Press Company, New York (1937). Bowley, A. L., G. L. Schwartz, and K. C. Smith. A New Index of Securities. London and Cambridge Economic Service, London (1931). Brown, S. J., W. N. Goetzmann, and S. A. Ross. Survival. Journal of Finance 50 (July) (1995): 853–73. Clark, J. B. Insurance and business profit. The Quarterly Journal of Economics 7(1) (1892): 40–54. Cole, A. C., and E. Frickey. The course of stock prices, 1825–66. Review of Economic Statistics 10(3) (1928): 117–139. Cowles, A. Common Stock Indices, Cowles Commission for Research in Economics. Monograph number 3, Principia Press, Bloomington, IN, p. 2 (1938). Dimson, E., P. Marsh, and M. Staunton. The Millenium Book: A Century of Investment Returns, ABN-AMRO and London Business School, p. 55 (2002). Fisher, I. Stocks vs. bonds. American Review of Reviews, July (1925). Fisher, I. The Theory of Interest. The Macmillan Company, New York, pp. 220–221 (1930). Fisher, L., and J. H. Lorie. Rates of return on investments in common stocks. Journal of Business 37(1) (1964): 1–21.


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Fisher, and Lorie. A Half Century of Returns on Stocks and Bonds, University of Chicago Graduate School of Business (1977). Frickey. An index of industrial stock prices. Review of Economics and Statistics 3(8) (1921): 264–277. Goetzmann, W. Patterns in three centuries of stock-market prices. Journal of Business (66)2 (1993): 249–270. Goetzmann, W., R. G. Ibbotson, and L. Peng. A new historical database for the NYSE 1815 to 1925: Performance and predictability. The Journal of Financial Markets 4(1) (2000): 1–32. Hawley, F. B. The risk theory of profit. The Quarterly Journal of Economics 7(4) (1893): 459–479. Hautcoeur, P.-C., and M. Petit-Koñczyk. The development of the Paris Bourse in the interwar period. Working Paper, Université Paris I (Matisse) and DELTA (2005). Homer, S. A History of Interest Rates. Rutgers University Press, New Brunswick, NJ (1963). Homer, S., and R. Sylla. A History of Interest Rates, third edition. Rutgers University Press, New Brunswick, NJ (1991). Ibbotson, R. G., and G. P. Brinson. Investment Markets. McGraw-Hill, New York (1987). Ibbotson, R. G., and R. A. Sinquefield. Stocks, bonds, bills, and inflation: Year-by-year historical returns (1926–1974). Journal of Business 49(1) (1976a): 11–47. Ibbotson, R. G., and R. A. Sinquefield. Stocks, bonds, bills and inflation: Simulations of the future 1976–2000. Journal of Business 49(3) (1976b): 313–338. Ibbotson, R. G., J. J. Diermeier, and L. B. Siegel. The Demand for capital market returns: A new equilibrium theory, Financial Analyst Journal, January/February, (1984): 22–33. Ibbotson, R. G., and P. Chen. Long-run stock returns: participating in the real economy, Financial Analysts Journal, 59 (1) ( 2003): 88–98. Jorion, P., and W. N. Goetzmann. Global stock markets in the twentieth century. Journal of Finance 54(3) (1999): 953–80. Knight, F. Risk, Uncertainty and Profit. Hart, Shaftner and Marx, New York (1921). LeRoy, S. F., and L. D. Singell, Jr. Knight on risk and uncertainty. Journal of Political Economy 95(2) (1987): 394–406. Markowitz, H. Portfolio selection. The Journal of Finance 7(1) (1952): 77–91. Mehra, R., and E. C. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15(2) (1985): 145–161. Mill, J. S. The Principles of Political Economy. Longmans, Green and Co., London (1848). Mill, J. S. Essays on Some Unsettled Questions of Political Economy. History of Economic Thought Books, McMaster University Archive for the History of Economic Thought, edition 2, number mill 1874 (1874). Mitchell, W. C. The prices of American stocks: 1890–1909. Journal of Political Economy 18(5) (1910): 345–380. Mitchell, W. C. A critique of index numbers of the price of stocks. Journal of Political Economy 24(7) (1916): 625–693. Modigliani, F., and M. H. Miller. The cost of capital, corporation finance and the theory of investment. American Economic Review, 48 (June) (1958): 261–97. Persons, W. M. Construction of a barometer based upon annual data. American Economic Review 4(4) (1916): 739–769. Persons, W. M. Indices of General Business Conditions. Harvard University Committee on Economic Research, Cambridge (1919). Rietz, T. The equity premium: A solution? Journal of Monetary Economics 22(1) (1988): 133–136. Schwert, W. Indexes of U.S. stock prices from 1802 to 1987. Journal of Business 63(3) (1990): 399–426. Smith, E. L. Common Stocks as Long Term Investments. The Macmillan Company, New York (1924). Smith, K. C., and G. F. Horne. An Index Number of Securities. London and Cambridge Economic Service, London (1934). Sylla, R., J. Wilson, and R. E. Wright. Database of securities prices, major U.S. markets, 1790–1850. InterUniversity Consortium for Political and Social Research (2000). Tobin, J. Liquidity preference as behavior towards risk. The Review of Economic Studies 25 (1958): 65–86. Williams, J. B. The Theory of Investment Value. Harvard University Press, Cambridge, MA (1938).

Chapter 12 • Discussion of Chapters 11 and 12

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Discussion: “The Worldwide Equity Premium: A Smaller Puzzle” and “History and the Equity Risk Premium” Stephen F. LeRoy University of California, Santa Barbara

“The Equity Premium: A Puzzle” by Mehra and Prescott had been in circulation for several years before it was published 20 years ago. I had the dubious honor of being a referee for one of the several journals that considered and rejected the paper before it was published. I anticipated that the paper would not be to the taste of the editor. However, I found the paper very interesting, so I devoted a fair amount of effort to writing a referee report persuasive enough to turn him around. I did not succeed in doing so. Like many readers, the editor who sent the equity premium paper to me was put off by the strong assumptions that Mehra and Prescott adopted. Many readers were confident that the equity premium puzzle would be cleared up as soon as these assumptions were weakened. I had the opposite reaction: I could not see why the admittedly strong assumptions that Mehra and Prescott adopted would bias the outcome toward producing too low an equity premium. For example, in a representative agent setting, agents bear no idiosyncratic risk, and that is surely unrealistic. But why would ignoring uninsurable risk induce agents to hold more stock and fewer bonds than they would otherwise, as is required if the equity premium predicted by theory is to be biased downward? Neither stock nor bonds is a hedge against idiosyncratic risk, so I could see no reason why excluding idiosyncratic risk would bias the outcome. Whether or not this intuition is correct has been a major research focus in the last 20 years. Almost immediately after Mehra and Prescott’s paper was published, Mankiw (1986) argued against the point just made, suggesting that excluding idiosyncratic risk is likely to increase the equity premium. More recently, Constantinides and Duffie (1996) showed that if the correlations between insurable and uninsurable risk are specified appropriately, part of the equity premium puzzle can, in fact, be explained along these

Stephen F. LeRoy

531 lines. Going in the opposite direction, Levine and Zame (1996) proved that under some conditions the assumption of market completeness when markets are, in fact, incomplete does not materially affect the result. As these citations imply, the issue has not yet been fully resolved. More generally, over the past 20 years economists have become much more accepting of the representative agent setting that Mehra and Prescott assumed. In the earlier period, it seemed obvious that macroeconomic fluctuations represented failures of the price system, coordination problems, and the like. It is impossible to analyze such problems in a representative-agent setting. Most macroeconomists presumed that, whatever the merits of representative agent models in classroom exercises, not much could be learned from them about the real world. With the advent of real business cycle theory, however, it came to seem more and more natural to analyze macroeconomic problems in a representative-agent setting, to the point where doing so is now rarely questioned. It is interesting that in this case progress—if that is what we are talking about—consists of relying on models that in an earlier era would have seemed to be grossly oversimplified. There are, incidentally, other similar examples in economics: in the Keynesian era, it was commonly asserted that the competitiveness assumption of the classical and neoclassical economists was an unacceptable oversimplification. That argument is rarely heard now, although there is no shortage of papers that specify non-competitive settings. Many found Mehra and Prescott’s specification of two states to be equally unappealing, particularly in an empirical paper. However, I was in no position to take issue with that modeling decision since I had adopted a similar two-state specification in a paper investigating the connection between stock price volatility and risk aversion (LaCivita and LeRoy (1981), cited by Mehra and Prescott). The elegant derivation of the Black– Scholes option pricing model in a binomial setting by Cox et al. (1979) suggested that a great deal can be learned from even the simplest two-state model. In any case, it later became clear that the Mehra and Prescott conclusions could be generated in settings that did not require the two-state limitation (Hansen and Singleton (1983), Hansen and Jagannathan (1991), for example). I recall that my initial reaction to Mehra and Prescott’s paper was one of annoyance, not at them for adopting an unnecessarily restrictive modeling specification, but at myself for not having figured out how to apply the two-state specification of LaCivita and LeRoy to model a growing economy, as Mehra and Prescott did. Taking the growth rates of the aggregate endowment (instead of the levels, as in LaCivita and LeRoy) as the states was obviously part of the solution. But, then, marginal utilities of consumption must be evaluated at an infinity of consumption levels, rather than just two as when the states represent endowment levels. It seemed to me that the model would surely become intractable. Mehra and Prescott’s solution, obvious with hindsight, was to restrict attention to homothetic preferences, so as to generate a stochastic steady-state equilibrium. However, I simply didn’t see that point. In several respects, my reaction to Mehra and Prescott’s paper when I was refereeing it was less than satisfactory, even making generous allowance for the fact that economists’ understanding of equilibrium finance models was much less advanced then than now. Like many, I had the idea that restricting attention to exchange economies involved a major loss of generality. I recall writing to Prescott about this. He patiently

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explained that even in a production economy the relations between consumption and asset prices would be as he and Mehra specified them, so no loss of generality is involved in not modeling the production side of the economy. I got the distinct impression then that he had already dealt with that question more than once. I expect that he and Mehra have heard it many times over the past 20 years. Turning to Goetzmann and Ibbotson’s paper, it is very useful to have a historical perspective on the equity premium puzzle. I was surprised to learn from their paper how recent a topic it is: it would seem that the question of how returns on various investments compare would have engaged economists from the earliest days of capitalism. Apparently not. Partly, no doubt, the fact that economists did not ask this question reflects the fact that they did not have the tools—analysis of financial markets under risk and risk aversion—to anwer it. It is ironic that one interpretation of the equity premium puzzle is that it indicates that the tools we have now are still far from satisfactory. At least now we are asking the question. Another surprise is that it was Mill who gave the first (somewhat) clear statement of the determinants of the equity premium as a payment to those who bear risk. I think of Mill as the first of the classical economists to appreciate the importance of capital theory, which is as close as economists prior to the modern era came to financial economics. However, Mill’s extremely long and tedious discussion of capital theory was far from successful, even by the standards of the mid-19th century. One gets the feeling from reading Mill that he himself partially realized that he wasn’t getting at the root of the matter. Hence my surprise at learning that it was Mill, rather than economists of the next generation, who did make genuine progress with capital theory, who alone in the 19th century gave a clear attribution of the equity premium to risk. There is a major omission in Goetzmann and Ibbotson’s review of the evolution of the idea of the equity premium: consumption-based asset pricing (Lucas (1978), Breeden (1979)). The equity premium puzzle is not a puzzle primarily because it contradicts the market version of CAPM, as Goetzmann and Ibbotson’s discussion implies—that would imply only that preferences and return distributions do not conform to the restrictive specification needed to generate the market version of CAPM. The consumption-based version of CAPM, on the other hand, says that consumption beta (in a representative-agent setting, the covariance of asset returns with the marginal utility of consumption) measures a security’s risk and therefore governs its risk premium. In a two-date setting, the market version of CAPM and the consumption version are equivalent if agents have quadratic utility, since then the marginal utility of consumption is collinear to the aggregate endowment, which is the payoff on the market portfolio. In general, market CAPM is a special case of consumption CAPM. Consequently, rejection of consumption CAPM implies rejection of market CAPM, but the converse is not necessarily true. The general framework that Mehra and Prescott assumed embodies consumption CAPM, but not necessarily market CAPM. However, in their specific implementation, Mehra and Prescott assumed that each event has exactly two successor events. In that case, the market return necessarily lies in the efficient plane. This in turn implies the validity of conditional CAPM (that each event and its successor events look like miniature versions of the two-date CAPM model). Consumption CAPM, in

Stephen F. LeRoy

533 contrast to market CAPM, holds under very weak assumptions, and this is why the equity premium puzzle is so important: as weak as these assumptions appear to be, they appear to be sharply contradicted by the data. Dimson et al. have provided a useful service in assembling an international data set of variables related to the equity premium. A major purpose of their paper is to evaluate the quantitative validity of the observation of Brown et al. (1995), that focusing exclusively on U.S. data, as Mehra and Prescott did, runs the risk of “survival bias.” Just as average mutual fund returns are biased if the sample consists only of mutual funds that survived the entire sample period, the estimated equity premium is biased upward if attention is limited to the U.S., in which both debt and equity markets survived its entire history (unless one considers Confederate debt). It is far from clear that the analogy to mutual funds is appropriate. Upward bias in the estimated equity premium would result if equities have higher probability of default than debt markets, assuming that the differing default probabilities were incorporated in asset prices. In that case, conditioning on no default would induce a higher upward bias in equity returns than in bond returns. It is far from clear that this condition is satisfied. The major cataclysms of the 20th century, such as the Russian revolution and the world wars, resulted in huge losses for both equity and debt holders, while events like the German hyperinflation of the 1920s resulted in greater losses for debt holders than equity holders. As this argument suggests, Dimson et al. find that there is not much evidence of survival bias in their data set. A much stronger case can be made, however, that limiting attention to the U.S. induces what Dimson et al. call “success bias”: the stock of the U.S. economy, both metaphorically and literally, has been rising faster than expected throughout its entire history. Consider, for example, the post–World War II period. Even though after World War II the U.S. was the dominant economy in the world, investors were very concerned about the possibility that the economy would revert to the depressed conditions of the 1930s. In the 1950s and 1960s, investors clearly revised upward their point estimates of economic growth rates in the U.S., and also attached successively lower uncertainty to their estimates, implying a lower risk premium for equities. As Dimson et al. pointed out, the capital gains that resulted from the ensuing changes in expectations would induce an upward bias in the estimated equity premium. In light of this effect, I find surprising Dimson et al.’s finding that equity premia estimated from international data are only slightly lower than those based on U.S. data alone. A major conclusion that Dimson et al. stress is that, even with a century of data, small-sample problems persist. Surely they are correct about this, and it is an important point. Financial data are not produced by a random-number generator producing independent draws from a stationary population. Rather, they are produced by real countries with real histories. That being so, it is not surprising that even long data sets do not produce data that conform to finance theory. Perhaps the implication is that we should not interpret the equity premium puzzle as providing strong evidence against consumption-based asset pricing. Both of the papers under discussion make valuable contributions to the equity premium literature.

534


References Breeden, D. T. An intertemporal asset pricing model with stochastic consumption and investment opportunities. Journal of Financial Economics 7 (1979): 265–296. Brown, S. J., W. N. Goetzmann, and S. A. Ross. Survival. Journal of Finance 50 (July) (1995): 853–873. Constantinides, C., and D. Duffie. Asset pricing with heterogeneous consumers. Journal of Political Economy 104 (1996): 219–240. Cox, J. C., S. A. Ross, and M. Rubinstein. Option pricing: A simplified approach. Journal of Financial Economics 7 (1979): 229–263. Hansen, L. P., and R. Jagannathan. Implications of security market data for models of dynamic economies. Journal of Political Economy 99 (1991): 225–262. Hansen, L. P., and K. J. Singleton. Stochastic consumption, risk aversion, and the temporal behavior of asset returns. Econometrica 91 (1983): 249–265. LaCivita, C. J., and S. F. LeRoy. Risk aversion and the dispersion of asset prices. Journal of Business 54 (1981): 535–547. Levine, D. K., and W. R. Zame. Debt constraints and equilibrium in infinite horizon economies with incomplete markets. Journal of Mathematical Economics 26 (1996): 103–131. Lucas, R. E. Asset prices in an exchange economy. Econometrica 46 (1978): 1429–1445. Mankiw, N. G. The equity premium and the concentration of aggregate shocks. Journal of Financial Economics 17 (1986): 211–219.

CHAPTER 13

Can Heterogeneity, Undiversified Risk, and Trading Frictions Solve the Equity Premium Puzzle? John Heaton University of Chicago and NBER and

Deborah Lucas Northwestern University and NBER

1. Introduction 2. Labor Income as Background Risk 2.1. Calibrating the Income Process 2.2. Adding Trading Frictions 3. Entrepreneurial Income as Background Risk 4. Limited Participation and Limited Diversification 5. Conclusions References

537 539 544 547 552 555 556 556

Abstract Can the historical equity premium be explained as a rational equilibrium outcome when risk-averse agents with conventional preferences are faced with non-diversifiable sources of risk (e.g., from labor or entrepreneurial income), and when trading frictions prevent them from using financial assets to effectively self-insure against transitory shocks? Our research suggests that it is difficult to generate the historical equity premium in realistically parameterized models of this sort. Nevertheless, investigations of these factors clearly reveal the ingredients necessary for any consumption-based model to match returns data. Using simplified versions of some of our earlier models and HANDBOOK OF THE EQUITY RISK PREMIUM c 2008 by Elsevier B.V. All rights of reproduction in any form reserved. Copyright

535

536

Chapter 13 • Can Heterogeneity, Undiversified Risk, and Trading Frictions

other models in the literature, in this chapter we illustrate the promise and limitations of incomplete risk diversification and trading frictions as explanations for the equity premium puzzle. We also present new results on the likely importance of entrepreneurial income risk. JEL Classification: G10, G12, G28, N42, D9 Keywords: equity premium, idiosyncratic shocks, trading costs, entrepreneurial risk.

John Heaton and Deborah Lucas

537

1. INTRODUCTION The equity premium puzzle is the inability of a standard representative consumer asset pricing model (e.g., Lucas (1978)), calibrated with aggregate data, to match the historical differential between average stock returns and the risk-free rate. In pursuing a solution to the puzzle, our research strategy has been to weaken one or more of the maintained assumptions in the standard model. In particular, we have focused on deviations from complete markets arising from a variety of factors. These include undiversifiable background risks such as labor income and private business income, trading frictions, and limited portfolio diversification, all of which can reduce tolerance for investment risk. We have also considered whether these deviations from the standard model become more important when agents’ preferences deviate from the constant relative risk aversion (CRRA) specification assumed by Mehra and Prescott (1985). The results of these investigations suggest the robustness of the puzzle to these important deviations from market completeness. Further, they suggest the ingredients necessary for any consumption-based model to match the data, which may be helpful in evaluating proposed solutions. Using calibrated models similar to our earlier work and others found in the literature, in this chapter we illustrate the promise and limitations of incomplete risk diversification and trading costs as explanations for the equity premium puzzle. In a consumption-based model, the critical determinants of the predicted equity premium are the variance of consumption risk, its correlation with stock returns, and the curvature of the assumed utility function. An increase in any of these factors increases investors’ distaste for stock market risk and tends to increase the predicted equity premium. Aggregate consumption growth exhibits low volatility and is only weakly correlated with stock returns. Hence, a moderately risk-averse individual faced with this consumption process will require only a small premium to assume stock market risk. By contrast, many individuals’ consumption appears to be much more volatile than in aggregate, due to factors such as incomplete diversification of income risk or poor portfolio diversification. In some cases, individual consumption may also be more highly correlated with stock returns than is aggregate consumption. It is therefore natural to ask whether differences between aggregate and individual consumption processes can help explain the puzzle. It should be emphasized that this is primarily an empirical question, not a theoretical one. As shown in Section 2, it is fairly easy to construct a risky consumption process that allows the standard model to match the equity premium (although with CRRA preferences it is difficult to simultaneous match the low volatility of the risk-free rate). The challenge is to reconcile the equity premium, and asset returns more generally, with consumption processes that are suggested by data. To directly test whether an otherwise standard consumption-based model could explain asset returns when calibrated with individual data, it would be convenient to study the statistical properties of individual consumption. But individual consumption, when it is measured at all, is measured with considerable error. Expenditures do not equate to consumption, for instance, because of the lumpiness of durable purchases. Aggregating across different categories of consumption also can be problematic. A more

538


insurmountable difficulty is the absence of a U.S. data set that includes a long time series of data on household consumption.1 Time-series data is available, by contrast, on individual income. A popular source is the Panel Survey of Income Dynamics (PSID), which tracks the various components of income for a representative sample of households over many decades. The data can be used to look for differences in income risk across households with different characteristics, such as between stockholders and non-stockholders. It also can be used to test the standard model, since the model is essentially a non-linear filter that maps income to consumption by solving consumers’ constrained optimization problems. Most of the research discussed in this chapter relies on this approach. Several complications arise when the exogenous driving process is income rather than consumption, as model predictions are quite sensitive to the assumed market structure and income process. The most critical assumptions are (1) the form of the budget and wealth constraints, particularly the severity of borrowing and short sales restrictions, (2) the persistence of the income process, (3) the minimum income realization, and (4) the correlation between individual income growth and the stock market. To briefly elaborate on these sensitivities: the structure of trading frictions and the persistence of income shocks determine the extent to which individuals can smooth income shocks through capital markets. Borrowing and short sales restrictions, and any trading costs, limit how much idiosyncratic income risk can be effectively smoothed by saving. More persistent income shocks similarly limit peoples’ ability to self-insure by borrowing and lending, since they represent a permanent change in wealth that can only be absorbed by consumption changes. The minimum income realization puts a floor on consumption. It thereby serves as a risk-free bond that tends to increase tolerance for investment risk. The higher is the floor on income and hence on consumption, the lower is the predicted equity premium. Finally, the predicted equity premium is very sensitive to the correlation between individual income shocks and stock returns. Without a significant positive correlation between the two, even a very risky individual income process does little to increase the predicted premium. The seemingly conflicting results in the literature on the success of a heterogeneous agent approach can be largely understood in terms of differences in these critical assumptions. Very persistent individual income processes, or very limited opportunities to smooth transitory shocks by transacting in financial securities, combined with a significant correlation between non-investment income and investment income, and a low floor rate on overall income, can generate a sizable equity premium (e.g., Mankiw (1986), Constantinides and Duffie (1996), Storesletten, Telmer, and Yaron (2007), Constantinides, Donaldson, Mehra (2002)). On the other hand, with moderate persistence of income shocks, some trading frictions, a non-zero floor on income, and weak correlation between investment and non-investment income, it is difficult to explain the equity premium in this class of models (e.g., Heaton and Lucas (1992, 1995, 1996), Lucas (1994), Telmer (1993)). 1 Food consumption in the PSID is an exception, but food may not be a reliable proxy for a broader consumption measure.


539

When taking heterogeneity into account, a salient feature of the data is that stocks are concentrated in the portfolios of high-wealth households. As first emphasized by Mankiw and Zeldes (1991), it is then the consumption process of stockholders that should be most relevant for asset prices. In fact, the sources of income risk for highwealth households look quite different than for the typical household, with labor income playing a smaller role and income from privately held businesses becoming significant. In Section 3, we discuss our findings on the implications of private business income as a background source of risk, and for the first time examine these effects in a calibrated model. An alternative to non-investment income as the cause of higher individual consumption risk is undiversified investment risk. That is, if individual stock portfolios are less diversified than in aggregate, it is the statistical properties of those portfolios that affect the returns investors demand. Lack of diversification can arise, for instance, if having managers or business owners hold high equity stakes in their businesses overcomes agency problems. Such undiversified investment income is particularly important to the wealthy households who may be the marginal participants in equity markets. As with non-investment income, the critical consideration is the statistical properties of the resulting consumption process of stockholders. Consumption financed primarily with income from poorly diversified portfolios has properties consistent with a high equity premium—a non-negligible probability of low consumption realizations, high variance, and high correlation with the stock market. As with other explanations based on consumption volatility arising from market imperfections, the deeper question is whether there is evidence of frictions large enough to induce such a costly lack of risk sharing. In remainder of this chapter we elaborate on these themes, using simple calibrated models to illustrate some of the theoretical points, and summarizing the related empirical evidence. In Section 2, we discuss the literature on labor income as a source of uninsurable risk affecting asset returns, with and without trading costs. In Section 3, we consider income risk from closely held businesses; in Section 4, the risk arising from incomplete portfolio diversification. Section 5 concludes the paper.

2. LABOR INCOME AS BACKGROUND RISK The first papers to consider undiversifiable income risk as an explanation of the equity premium puzzle focused on labor income. Labor income is a natural candidate because of its size (it accounts for about two-thirds of aggregate income) and the difficulty of insuring it due to problems of moral hazard and legal limits on selling it forward. Mankiw (1986) first illustrated the potential for labor income risk to resolve the equity premium puzzle with a stylized two-period model. He shows that if some fraction of the population is forced to bear a disproportionate share of aggregate shocks, the predicted equity premium can be made arbitrarily large. A basic question explored in two closely related papers (Lucas (1994) and Telmer (1993)) is whether Mankiw’s results continue to hold in an infinite horizon model similar in structure to Mehra and Prescott (1985), but with the addition of exogenous idiosyncratic labor income shocks.

540


The model that we use to illustrate some general conclusions, drawn from these analyses is the following: we assume that investors maximize expected utility at time t over a horizon T , generally taken to be infinite, and that the utility function is time additive, with period utility exhibiting constant relative risk aversion. The coefficient of relative risk aversion is γ, and the subjective discount factor is β:

Ut,i = Et

T

x

β U (ct+x,i )

1−γ where U (ct+x,i ) = ct+x,i −1 /(1 − γ).

(1)

x=0

Agent i invests an amount of funds pst st+1,i in stocks and pbt bt+1,i in risk-free, one-period bonds with face value bt+1,i and consumes ct,i at time t. The I types of agents are distinguished by their income processes and possibly their access to capital markets. The consumption and saving choice is subject to the flow wealth constraint: ct,i + pst st+1,i + pbt bt+1,i ≤ st,i dt + pst + bt,i + yt,i ,

(2)

where yt,i is risky, exogenously specified, non-tradable income, pst is the price of stock at the beginning of time t, st is the share of stocks, pbt is the price of a pure discount one-period bond per dollar of face value, and bt is the face value of the bond. The resulting Euler equation for asset of type j, j = s, b is Et

* −γ j ct+1,i / ct,i (1 + rt+1 ) ≤ 1/β ,

(3)

where rts is the return on stocks and rtb is the return on bonds between timest − 1 and t. The stock return, rts , consists of dividends and capital gains: rts = dt + pst /pst−1 . The risk-free rate, rtb , is 1/pbt−1 − 1. Equation (3) holds with equality for any unconstrained investor i. In that case, rearranging gives an expression for the equity premium: Et

* −γ s b ct+1,i / ct,i rt+1 − rt+1 =0 .

(4)

The ability to self-insure by saving is limited by exogenous borrowing and short sales constraints, which are used to control access to capital markets. By assumption, investors can borrow only a limited amount in the bond market and can short only a limited amount of stocks. When a borrowing or short sales constraint is binding, Eq. (3) becomes an inequality, and (4) only holds for unconstrained agents. In fact, asset prices are always determined by the common marginal rate of substitution of the unconstrained agents, since in equilibrium they must be content with their choices.

541


The presence of credit-constrained agents implies that the identity of the marginal investor changes over time. That is, an agent who receives a low income realization and has little or no savings is unable to borrow, and hence his preferences do not affect bond prices. If, however, in a subsequent period he receives a high income realization, he is likely to again participate in the bond market as a saver. Because the identity of market participants changes over time, the aggregate consumption process does not price assets. The changing identity of the marginal investor also precludes closed-form solutions to the model and complicates its numerical solution. Assuming an aggregate endowment of one share of stock paying a stochastic aggregate dividend, dt , risk-free bonds in zero net supply, and aggregate labor income Yt , the market clearing conditions at each time t are I

ct,i = dt + Yt ,

(5a)

bt,i = 0,

(5b)

st,i = 1.

(5c)

i=1 I i=1 I i=1

An equilibrium for this model is a sequence of stock and bond prices, and consumption and portfolio allocations for each agent, such that Eqs. (1)–(3) hold and the market clearing conditions (5) hold in all states and at all times. Notice that the background income process yt affects the asset prices defined by (3) only indirectly, through its effect on the variability of consumption and its correlation with financial returns. While this background risk could come from a variety of sources, including wages, restricted pension holdings, housing rents, and private businesses, in this section we emphasize labor income as the risk source. Under the assumption that consumption growth and returns are lognormally distributed conditional on information at time t, (4) can be written as2 s $ s % s 1 b + γ covt ln ct+1,i /ct,i , rt+1 . Et rt+1 − rt+1 = − vart rt+1 2

(6)

The equity premium increases with the covariance between stock returns and consumption growth, and the coefficient of relative risk aversion. Rough calibrations of (6) illustrate the original equity premium puzzle as well as its potential solution by appealing to higher consumption risk induced by market imperfections. Using Eq. (6) and historical statistics on stock returns and consumption growth, we can ask what value of γ implies an equity premium of 7 percent? Consistent with historical data, assume a standard deviation of stock returns of 18 percent, a standard deviation of aggregate consumption growth of 3.7 percent, and a correlation between aggregate consumption growth and stock returns of 0.3. This implies a risk aversion coefficient 2A

derivation can be found, for example, in Hansen and Singleton (1983).

542


of 43, well outside the range considered plausible. Conversely, one can ask about the implied equity premium for a given level of risk aversion. At γ = 10 (well above the 3 to 5 range for this parameter considered reasonable by Mehra and Prescott), the implied equity premium for these parameters is only 0.38 percent.3 If individual consumption volatility is considerably higher than aggregate consumption volatility, the consumption-based model can generate predictions in accord with the data. For example, setting consumption volatility to 12 percent and holding other parameters as above, (6) implies that a risk aversion coefficient of 13.3 produces the target risk premium of 7 percent. At γ = 10, the implied equity premium is a respectable 4.9 percent. Equation (6) can also be used to demonstrate the sensitivity of the predicted premium to the correlation between consumption growth and stock returns. In the Mehra and Prescott’s analysis, dividend volatility and consumption volatility are equated, inducing a fairly high correlation between returns and income. Under the extreme assumption of perfect correlation between returns and consumption growth, and holding other parameters as in the low consumption risk example, the implied equity premium increases from 0.38 percent to 5 percent for γ = 10. With consumption volatility of 12 percent, the predicted premium rises from 4.9 to 20 percent. On the other hand, if the correlation between aggregate consumption and stock returns is set close to zero, which empirical evidence suggests may be the case, the equity premium is negligible even if consumption variability is high. These calculations illustrate an important conclusion from these earlier papers—high consumption variability, in combination with a moderate to high correlation between consumption growth and stock returns, has the potential to resolve the equity premium puzzle. Before turning to more complicated calibration results, it is useful to demonstrate another key finding of the earlier investigations: in calibrated models in which labor income has a positive minimum realization, the capitalized value of that minimum serves as a bond that can significantly increase tolerance for stock market risk. Because of this, adding risky labor income can actually make it more difficult to resolve the equity premium puzzle. The effect of a floor on labor income can be illustrated most clearly in a decisiontheoretic portfolio choice model, rather than in an equilibrium asset pricing model. In Heaton and Lucas (2000a), we analyze a model that is the portfolio choice analogue of the model described by Eqs. (1)–(5), calibrated to U.S. data. We find that including a realistic amount of risky labor income results in portfolios with 100 percent stock investments, or when borrowing is permitted, a levered position in stocks. To see why this occurs, it is convenient to consider Merton’s (1971) closed-form solution for the share of wealth invested in stocks, w ∗ : ∗

w W =

E(rs ) − rf (W + y) γ var(rs )

(7)

3 By assuming volatile stock returns, rather than deriving the volatility of stock returns from a dividend process calibrated with historical data, these examples actually understate the severity of the equity premium puzzle.

543


In this model W is financial wealth, y is the constant flow of labor income, and rf is the risk-free rate. In the absence of labor income, this model generates reasonable portfolio shares for moderate levels of risk aversion. For instance, assuming an equity premium of 6 percent and a 15 percent annual standard deviation of stock returns, a portfolio with 50 percent in stocks implies a coefficient of relative risk aversion of 2.67. Now, consider labor income of the same magnitude as financial wealth, which roughly matches the mean ratio of non-housing, non-business wealth to permanent income in the PSID (Carroll and Samwick (1995)). Holding the risk aversion coefficient at 2.67, predicted stockholdings rise to 100 percent of financial wealth. For households with even less financial wealth relative to income, the implication is that they would borrow to buy stocks. Intuitively, the certain labor income is analogous to the fixed income stream from a bond, so it substitutes for bonds in the agent’s utility function. While the simple calculations based on Eqs. (6) and (7) are suggestive, they abstract from the effect of more carefully modeling the joint statistical processes governing dividends and labor income, and the possibility that the effects of borrowing and short sales constraints will alter the relation between income and consumption in a way that has a significant effect on required returns. Table 1, reproduced from Lucas (1994), confirms that the conclusions drawn from Eq. (6) can carry over to a more fully specified model of labor income and dividends. The table is generated from a calibrated version of the model described above, where two infinitely lived agents receive equal and offsetting idiosyncratic labor income shocks when aggregate output is low. The idiosyncratic shocks are structured so that the employed worker receives income 1.22 times that of the unemployed worker when aggregate income growth is low. The aggregate income process is identical to that in Mehra and Prescott (1985), with dividends equal a constant 30 percent of aggregate income, with labor income making up the balance. Table 1 shows that the resulting equity premium matches the historical average premium, even with risk aversion as low as 2.5, but only under the strong assumption of no access to capital markets so that consumption equals income. TABLE 1 Model Predictions of Asset Returns and Consumption Growth with Idiosyncratic Shocks and no Trading in Stocks or Bonds 1.0

γ

1.5

2.5

E(rs )

0.071

0.076

0.098

σ(rs )

0.035

0.055

0.100

E(rb )

0.034

0.020

−0.002

σ(rb )

0.029

0.015

0.046

− rb )

0.037

0.056

0.099

σ(rs − rb )

0.007

0.080

0.081

E(rs

E(C /C)

0.021

0.019

0.021

σ(C /C)

0.080

0.080

0.081

544


Table 1 also suggests that the risk-free rate puzzle—the observation that the standard model also predicts too high a risk-free rate—is resolved in this model. The reason is that with heterogeneous agents, the risk-free rate clears the market for agents who are not constrained. The price setters are then the subset of the population whose current income and desire to save are relatively high. This implies that the market clearing interest rate is lower than with a representative agent. A general finding in this literature is that binding borrowing constraints make it relatively easy to solve the risk-free rate puzzle, whereas the equity premium puzzle is much more robust to the addition of financial frictions. The results of Table 1 are reversed with the opening of capital markets. When agents are assumed to have access to trading in either the stock or bond market, a similarly calibrated model (in terms of the income process and preference parameters) fails to reproduce the historical equity premium. Even with quite limited market access, the predicted equity premium falls to levels comparable to those found by Mehra and Prescott (1985), and equilibrium individual consumption volatility can be indistinguishable from aggregate consumption volatility (Lucas (1994)). For instance, the predicted equity premium is close to zero when agents are allowed to trade but not short-sell stocks, and borrowing is prohibited. The reason is that agents effectively smooth transitory idiosyncratic shocks by accumulating savings (here in the form of stocks) when income is high and reducing savings when income is low. The floor on labor income, as discussed above, also increases risk tolerance. The general conclusion that emerges from these analyses is that if idiosyncratic income shocks translate to similarly volatile consumption shocks, the equity premium puzzle can be resolved using individual consumption data. Making a persuasive case for this explanation, however, requires finding evidence that income shocks are fairly permanent, as assumed by Mankiw (1986) and Constantinides and Duffie (1996), or establishing that the barriers to trading in financial markets are sufficient to discourage self-insuring transitory income shocks. Another important observation is that in these types of models, the only way to prevent self-insurance against transitory shocks is to restrict access to all financial markets, since agents readily substitute to trading in the low-cost market. These quantitative issues are addressed in Heaton and Lucas (1996), where we calibrate a related model that incorporates transaction costs and an empirically estimated income process.

2.1. Calibrating the Income Process The persistence and volatility of individual income processes are critical determinants of the predicted equity premium in this class of models. Hence, model predictions are sensitive to the calibration of these processes. A robust finding of the literature is that more permanent income shocks cannot be self-insured by trading in financial securities and hence can lead to a high predicted equity premium in heterogeneous agent models. This is implicit in Mankiw (1986), and established very generally in Constantinides and Duffie (1996). In consumption-based asset pricing models, the reason that permanent


545

income shocks imply much higher consumption risk than transitory shocks is that the former represent a much larger wealth shock. The connection between the persistence of shocks and their wealth effects can be demonstrated most easily using the standard consumption-savings model. Assuming log preferences and complete markets, consumption is a constant fraction of wealth each period, equal to the risk-free rate multiplied by wealth. Hence, the volatility of wealth translates into the volatility of consumption. Permanent shocks to income induce large innovations in wealth, and influence consumption one for one. For example, consider a permanent increase in income of $100 per year, and a risk-free rate of 3 percent. For an infinitely lived agent, wealth increases by $100/0.03. Each year, consumption increases by 0.03($100/0.03) = $100; the permanent income shock translates to a permanent consumption shock. By contrast, a transitory shock of the same magnitude adds only $100 to wealth, and consumption increases by only 3 percent of this, or $3 per year.4 For this reason, it seems preferable to focus on measuring and realistically calibrating wealth shocks, rather than trying to resolve the empirically complicated question of whether income shocks are predominantly permanent or transitory. A further reason to emphasize the size of wealth shocks, rather than the permanent-transitory distinction, is that with finitely lived individuals the difference between the two is not well-defined. In fact, agents with very short horizons (e.g., just another period or two to live) effectively have permanent shocks regardless of the statistical properties of the income process. Aside from the unconditional volatility of wealth shocks, the implied premium is affected by the conditional volatility in individual shocks, as emphasized by Constantinides and Duffie (1996). Income specifications exhibiting higher individual conditional volatility in the low aggregate state imply a higher equity premium than processes with the same unconditional volatility but no state dependency. A related observation is that income specifications with the possibility of a catastrophic state also are more successful in reproducing the historical equity premium (Freeman (2004), Rietz (1988)). There is a mechanical reason that a higher conditional volatility in recessions or a low probability catastrophic state increases the implied equity premium in this class of models. Either assumption allows a coincidence between very bad individual and aggregate shocks, without causing a counterfactually high correlation between individual income growth and stock returns, or exessive aggregate labor income volatility. That is, if the distribution of individual shocks simply shifted down in recessions and up in recoveries by a constant amount, it would induce a counterfactually high correlation between labor income and dividends, and too much labor income volatility. Increasing dispersion in the low aggregate state increases the predicted equity premium because it creates a very low income realization that is correlated with the low dividend state. In the case of symmetrically distributed conditional volatility, the higher dispersion also creates a very high individual income realization in the low aggregate state. Because of the non-linearity of marginal utility, however, the high individual income realizations have much less influence on the predicted premium than the low realizations. 4 See

also Kocherlakota (1996), page 62, for further elaboration of this point.

546


There are several practical considerations in calibrating the income process for heterogeneous agent models. Model solutions are complicated by the much higherdimensional state space needed to track the cross-sectional wealth distribution. To make the problem numerically tractable, it is imperative to keep the exogenous state space small enough to be manageable. A first-order autoregressive representation of the income growth process is conveniently parsimonious and fits the data reasonably well. It allows a very high degree of persistence, although it precludes permanent shocks. For example, in Heaton and Lucas (1996), we estimate the aggregate process as

Xta

0.1487 0.0557 0.0278 a = Xt−1 + −0.5016 0.9168 0.0121

0 0.1961 a ε + , 0.0536 t −0.2607

(8)

a where γta = Yta /Yt−1 is aggregate income growth, δta = Dta /Yta is the dividend share, $ a % a and Xt = log γt , log δta . In the calibrations reported ahead, several specifications are considered for the individual income dynamics. In the base case, the individual income dynamics, based on data from the PSID, are also represented by an AR(1) process: i log ηti = −3.35 + 0.529 log ηt−1 (9) + εit , i , and the coefficients are the sample means from individual housewhere ηti = Yti /Yt−1 i hold estimates. σ εt is the sample mean of the household standard deviation of εit . To calibrate the model, Eqs. (8) and (9) were discretized and represented as a first-order Markov chain with eight possible states.5 Recent work by Guvenen (2005) that revisits the decomposition of income risk between permanent and transitory shocks appears to support this specification. Using a variant of (9) that includes aggregate income growth on the right-hand side, we find little evidence of countercyclical volatility in individual income growth. Nevertheless, in order to test the conjecture that the countercyclical component exists but is not reflected in the relatively short time series used in these estimates, we also calibrated the heterogeneous agent model with twice the conditional variance in the low aggregate growth state as in the high aggregate growth state, preserving the unconditional variance. That calibration is consistent with the imputed conditional volatility estimates of Storlesletten, Telmer, and Yaron (2004). The calibration results suggest that conditional volatility of this magnitude is not sufficient in itself to generate a significant equity premium. In Section 3, we modify these income processes to represent an economy where entrepreneurs are the dominant participants in financial markets. The purpose is to see whether the greater income volatility and its higher correlation with the market imply a significantly larger equity premium than in previous calibrations based on labor income.

5 Appendices

A and B in Heaton and Lucas (1996) discuss the estimation of Eqs. (8) and (9) in detail.


547

2.2. Adding Trading Frictions Since the risk associated with measured labor income processes appears insufficient in itself to generate a significant equity premium, we turn to the question of whether individual risk becomes more important in the presence of trading frictions. Trading frictions that can be easily analyzed in this class of models fall into two broad categories: transactions costs associated with trading financial securities, and borrowing and short sales constraints. Both types of frictions influence equilibrium returns, through several distinct channels that are discussed in this subsection. While predicted returns are affected in the expected direction, frictions must be severe to match the historical equity premium.

2.2.1. Transactions Costs Trading costs in the form of quadratic or pseudo-proportional transactions costs6 have the potential to increase the predicted equity premium through both a direct effect of changing relative prices and an indirect effect of reducing risk sharing. Borrowing costs paid exclusively by the borrower increase the predicted equity premium through a direct effect that for a given level of borrowing, higher trading costs imply a lower market clearing interest rate. The indirect effect is that the reduction in trading volume due to trading costs increases equilibrium consumption volatility, which can in principle also increase the predicted equity premium. In Heaton and Lucas (1996), we decompose these two effects and conclude that the direct effect is likely to be quantitatively more important than the indirect effect. The relatively small indirect effect is related to the finding that trade shifts to the market with lower costs. This allows most labor income risk to be shared, unless trading costs are assumed to be large in all asset markets. The direct effect of trading costs depends critically on which counterparty is assumed to pay them. To maintain a given level of borrowing demand, a 1 percent increase in trading costs requires a 1 percent lower interest rate. Similarly, to maintain a given supply of loans requires an increase in interest rates that offsets any increase in trading costs. When transaction costs are evenly split between borrowers and lenders, these effects are largely offsetting and the net effect on the equilibrium interest rate is small in the parameterizations considered.7 Instead, the equilibrium is restored by a reduction in the amount transacted. If, however, borrowers directly pay trading costs but lenders do not, the equilibrium interest rate will be lower than if the transactions costs are split evenly across the two counterparties. The rate adjustment makes up for the wedge between the all-in cost of borrowing and lending, and restores equilibrium (in this case there is both

6 Psuedo-proportional costs are defined as quadratic in a region close to zero, and linear outside that region. This structure, as well as quadratic costs, have the advantage of differentiability, and so are easier to work with than strictly proportional or fixed costs, which involve discontinuities. 7 In general, the effect will depend on the relative elasticities of the demand and supply functions.

548


a rate and quantity effect). Since the risk-free rate is taken to be a lending rate from the perspective of households (generally the Treasury bill rate) and obtaining consumer credit involves various fees, such a cost asymmetry is plausible. A similar logic applies to the stock market—only asymmetric costs have a significant direct effect on equilibrium stock returns. These observations are consistent with the finding from calibration exercises that the only cost configuration that yields a sizable equity premium is when all trading costs are paid by borrowers in the bond market, but trading costs are split evenly between buyers and sellers in the stock market. The indirect effect arises because for a given interest rate, a borrower will take out a smaller loan when trading costs are higher. Similarly, for a given interest rate, a lender will extend less credit with higher trading costs. Consequently, trading costs reduce the total volume of trade, which implies reduced risk sharing. This can decrease the tolerance for investment risk and increase the equity premium. With reasonable trading costs, however, our calibrations suggest that the indirect effect is likely to be small, accounting for less than 1 percent of the equity premium. This is because with CRRA utility, when the coefficient of relative risk aversion is assumed to be high enough to generate a non-negligible equity premium, agents are willing to bear fairly large transactions costs to smooth consumption. The resulting consumption process, although less smooth than in the same model without trading frictions, is still very similar to that in the representative agent case. Fixed costs also can contribute to the indirect effect of a higher equity premium due to reduced risk sharing. Explicitly incorporating fixed costs is difficult because they create non-convexities that preclude use of methods generally employed to solve these models. One reason that fixed costs can affect returns is that they discourage small investors from participating in the stock market, which concentrates market risk on a subset of the population. The non-participation channel is explored in Section 4. The model described by Eqs. (1)–(5), when augmented with quadratic or pseudoproportional trading costs and calibrated with the income process described in Section 2.1, generates predictions consistent with the above descriptions. For example, in Heaton and Lucas (1996), we consider trading costs in the stock market, κ st+1,i , st,i ; Zt , where κ is assumed to be differentiable in its two arguments, and Zt is the state. Both sellers and buyers pay a symmetric cost. For the case of quadratic costs: 2 κ st+1,i , st,i ; Zt = kt st+1,i − st,i pst ,

(10)

where kt is a constant. Trading costs in the market for one-period bonds are denoted by ω bt+1,i ; Zt , where ω is also differentiable. In some specifications costs are symmetric between borrowers and lenders, and in others only the borrower pays the cost. The asymmetric quadratic specification is %2 $ ω bt+1,i , bt,i ; Zt = Ωt min 0, bt+1,i pbt ,

(11)

549


0.09 0.08 0.07

Return

0.06 0.05

Stock Return Bond Return Equity Premium Net Premium

0.04 0.03 0.02 0.01 0

0

FIGURE 1

0.2

0.4

0.6

0.8

1 ␻

1.2

1.4

1.6

1.8

2

Effect of transactions costs on returns. (From Heaton and Lucas (1996)).

where Ωt is a constant. For unconstrained agents, the first-order conditions (3) and (4) are replaced by −γ s pt + κ1 st+1,i , st,i ; Zt ct,i * −γ s pt+1 + dt+1 − κ2 st+2,i , st+1,i ; Zt+1 (12) = βEt ct+1,i and

* −γ −γ b . pt + ω1 bt+1,i ; Zt = βEt ct+1,i ct,i

(13)

The market clearing conditions and budget constraints are also adjusted to include trading costs.8 As discussed above, the results are sensitive to the size and incidence of trading costs and the severity of the borrowing constraint. Figure 1 (Figure 1 from Heaton and Lucas (1996)) illustrates the main result, that the direct effects are much larger than the indirect effect, for the case of symmetric quadratic costs in the stock market and one-sided borrowing costs. In the figure, the parameter ω (which also controls K) interacts with trading behavior to determine the level of transactions costs in the stock market. Borrowing costs are assumed to also increase 8 Market

clearing requires

I i=1

ct,i + κ st+1,i , st,i ; Zt + ω bt+1,i ; Zt = dt + Yt . The budget constraint

becomes ct,i + pst st+1,i + pbt bt+1,i + κ st+1,i , st,i ; Zt + ω bt+1,i ; Zt ≤ st,i dt + pst + bt,i + yt,i .

550


0.06 Stock Market Bond Market

Percentage Costs

0.05

0.04

0.03

0.02

0.01

0

0

FIGURE 2

0.2

0.4

0.6

0.8

1 ␻

1.2

1.4

1.6

1.8

2

Average transactions costs. (From Heaton and Lucas (1996)).

with ω.9 The line labeled “Net Premium” is the portion of the premium attributable to consumption risk, which is a measure of the indirect effect. As explained above, the indirect effect is small even when trading costs are high, because risk-averse agents tolerate paying substantial transaction costs to avoid consumption risk. Very large transactions costs are needed to generate a significant equity premium. To see this, Figure 2 (Figure 2 from Heaton and Lucas (1996)) presents the average stock and bond market transactions costs as functions of ω. Trading costs are not linear in ω because the cost function is non-linear and trading behavior is endogenous. At the high end of the range for the estimated equity premium, the average trading cost in the stock market is 5 percent of the quantity traded, and the marginal trading cost is 10 percent. These types of results lead us to the conclusion that moderate trading costs and realistic labor income risk are not sufficient to resolve the equity premium puzzle.

2.2.2. Borrowing and Short Sales Constraints While proportional or quadratic transactions costs alter the volume and terms of trade, they do not change asset market participation rates; everyone still participates in

9 The cost parameters are linked because agents substitute toward trading almost exclusively in the lower-cost market when the spread between the parameters gets too large.


551

securities markets, but trading volume falls. If such costs were the only frictions, the aggregate consumption growth process would still determine asset prices according to Eqs. (12) and (13). In contrast, with intermittently binding borrowing and short sales constraints, market participants’ identity can change over time. It is these constraints that preclude any simple aggregation over individual consumption processes. Like the indirect effect of transactions costs, borrowing and short sales constraints can increase consumption volatility and the predicted equity premium by reducing opportunities for risk sharing. For constrained agents, consumption equals income. For unconstrained agents, there are fewer trading partners. Borrowing and short sales constraints can be imposed exogenously or determined endogenously. The earlier literature on trading frictions imposed exogenous constraints, set to levels motivated by observed borrowing and stock trading behavior. For instance, it is commonly assumed that short sales of stock are prohibitively costly and that borrowing is limited to a modest fraction (e.g., 10 to 20 percent) of annual average labor income. Two main results emerge: (1) borrowing constraints can dramatically reduce the risk-free rate; and (2) if income shocks are persistent but not permanent, only very severe borrowing and short sales constraints stop consumption smoothing via asset markets. The first result, as explained earlier, is due to the fact that the constrained agents would be willing to pay a high interest rate to borrow, but the unconstrained agents whose preferences determine asset prices want to save. The dominance of savers in pricing drives down the risk-free rate. The second result is less general. It arises because the total value of financial wealth is high relative to the estimated size of uninsurable shocks, and the shocks are not too persistent. This allows a large portion of individual income shocks to be buffered by drawing down financial assets, even if borrowing is restricted to only a small fraction of average annual income (Aiyagari and Gertler (1991)). Recently, there has been a growing interest in endogenous borrowing constraints, where state-contingent limits on debt are derived from assumptions about the enforcement mechanism for debt repayment. An important question is whether endogenous constraints tend to be more restrictive, or to generate more correlated risk, and hence whether they might explain a higher equity premium. The results of these exercises have been mixed. Zhang (1997) assumes that the punishment for default is a permanent forfeiture of the right to participate in asset markets. Due to the severity of the punishment, the resulting borrowing constraint is effectively more lenient than assumed in most analyses with exogenously imposed constraints. Thus, Zhang does not find that these endogenous constraints resolve the equity premium puzzle. Lustig (2007) proposes a much more stringent rule for accessing capital markets. He assumes that agents can only borrow to the extent that they have enough financial collateral to ensure the promised debt repayment is made in full. This has the effect of precluding any borrowing by non-stockholders. In the model, the identity of non-stockholders, and hence those facing the constraint, changes over time. This induces another source of aggregate risk in the model.

552


3. ENTREPRENEURIAL INCOME AS BACKGROUND RISK The analyses discussed in the last section strongly suggest that although labor income is an important source of largely undiversifiable risk for most households, it is unlikely to explain the equity premium puzzle in long horizon models, even in the presence of trading frictions. Labor income shocks have a large transitory component that can be self-insured by saving. There is low correlation with stock market returns, at least at annual or shorter frequencies. Also problematic is that labor income, as well as other income sources such as unemployment insurance, welfare, and intrafamily transfers, tend to put a floor on income that serves as a safe bond, and thereby increases the propensity to take risk in the stock market. These observations motivate a search for other sources of background risk that might have more promising statistical properties. Entrepreneurial income from closely held businesses has properties that suggest it could be an important source of background risk that is more correlated with stock returns than is labor income. We present evidence for this in Heaton and Lucas (2000c) and augment and update it in Curcuru et al. (2007). The main considerations include (1) its higher correlation with stock returns, (2) its higher volatility, (3) the much higher average financial wealth of business owners than wage earners, and (4) the large portion of stock market value held by households that also have entrepreneurial income. Using aggregate annual data from the National Income and Product Accounts, and stock returns from CRSP for the period from 1947–2003, we find a correlation between aggregate wage income growth and stock returns of only 0.06, but a higher correlation between aggregate business income growth and stock returns of 0.11.10 The volatility of aggregate business income growth is also more than twice as high as for wage income growth; 4.52 percent vs. 2.06 percent annually. Using the Panel of Individual Tax Return Data for the 1979–1990 period, we estimate that individual non-farm proprietary income has a volatility of 64 percent annually, almost twice as high as the growth rate of real wage income of 35 percent annually. Entrepreneurs not only face riskier income streams, but they are more likely to influence asset market returns due to their much higher wealth and higher rates of stock market participation. Table 2 reports estimates from the 2001 Survey of Consumer Finances on the portfolio characteristics of business owners vs. non-owners. Business owners have an average net worth that is four times higher than non-business owners. The typical share of assets invested in their own businesses is 32.5 percent, suggesting significant undiversified risk exposure. Business owners also hold substantial wealth in liquid assets including stocks. We estimate that households with businesses worth in excess of $10,000 account for about one-third of total stockholding. In Heaton and Lucas (2000c), we use this evidence to motivate the estimation of a linear asset pricing model with proprietary income as a risk factor. We find that a model that includes aggregate proprietary income, a value-weighted stock index, and a credit spread outperforms a similar model with labor income in place of proprietary income. 10 For

individuals, the correlation between wages and the market is likely to vary over the life cycle.

553


TABLE 2 Mean Portfolio Characteristics of Business Owners vs. Non-Owners Owners

Non-owners

Liquid fin. assets/total assets

24.9

37.9

Stocks/liquid fin. assets

55.8

47.8

Bonds/liquid fin. assets

18.2

20.3

Cash/liquid fin. assets

26.1

31.9

Owner-occupied housing/total assets

34.3

54.2

6.8

5.7

Other real estate/total assets Business/total assets

32.5

—

Age

49.1

52.0

Education (years) Income Net worth

14.4

13.5

$169,693

$69,533

$1,298,065

$323,255

Tabulations are from the 2001 Survey of Consumer Finance and based on survey weights. Source: Curcuru et al. (2007).

When the Fama–French pricing factors HML and SMB are also included, proprietary income remains significant, while HML becomes insignificant. These results suggest that entrepreneurial income is a risk factor that influences asset prices. The analysis, however, does not directly address the equity premium puzzle. It has not been established, however, whether explicitly modeling the income process of entrepreneurs in a heterogeneous agent model such as the one described in Section 2.2 can generate a larger predicted premium than one calibrated with labor income. The only work we are aware of that addresses this question is Polkovnichenko (2001), who considers a model in which entrepreneurs and wage earners trade in financial securities to smooth consumption. He finds that the model can explain the larger holdings of stocks by entrepreneurs due to a higher precautionary demand for savings resulting from their more volatile income. The equilibrium risk premium, however, is similar to that in the representative agent case. The small price effect, as in the models discussed above, is because there is sufficient trading in financial securities to buffer individual stocks. His analysis, however, abstracts both from trading costs and idiosyncratic risk. To further explore whether entrepreneurial income risk might resolve the equity premium puzzle, we recalibrate the model described in Section 2. The calibration is designed to give entrepreneurial income risk its best chance to succeed in this class of models. For tractability we are limited to two agents, both of whom are taken to be entrepreneurs. Restricting attention to the portion of income going to entrepreneurs can be interpreted as reflecting their much higher asset market participation rate, and large share of aggregate financial wealth. We assume an aggregate income process split equally between the dividends from traded stocks and from aggregate entrepreneurial income. Consistent with the data, aggregate entrepreneurial income has twice the

554


volatility of aggregate labor income, and idiosyncratic entrepreneurial income shocks have twice the volatility of idiosyncratic labor income shocks. To maintain high correlation between entrepreneurial income and dividend income, we continue to assume that there is little variation in the share of dividend income in aggregate income. Since proprietary income is the only other source of income in the model, this specification has the effect of increasing dividend volatility. Lettau, Ludvigson, and Wachter (2005) argue for the empirical plausibility of this assumption about aggregate dividend volatility. We do not have direct evidence on the persistence of entrepreneurial income, but consider two cases for the AR(1) coefficient. In the first, as estimated for labor income, the coefficient is 0.53. We call this the “low persistence” case. In the second, reflecting the possibility of more persistent shocks, the autoregressive parameter is increased to 0.8. We call this the “high persistence” case. We also consider two cases for the conditional volatility of entrepreneurial income. In the first, called the “constant shock distribution case,” the volatility of idiosyncratic shocks is independent of the aggregate state. In the second, called the “cyclical shock distribution case,” the conditional volatility of entrepreneurial income is 25 percent higher in the low aggregate state as in the high aggregate state. Table 3 reports the predictions of the model for stock returns, bond returns, and the equity premium for the case of zero transactions costs. We continue to assume that β = 0.95 and γ = 1.5. As in the results calibrated to labor income, the predicted equity premium is not large. Even with the large shocks of these examples, agents in the model can readily trade to self-insure.

TABLE 3 Expected Stock Return, Bond Return, and Equity Premium with Risks Calibrated Using Entrepreneurial Income A. Constant shock distribution Persistence of idiosyncratic shocks Low %

High %

Expected stock return

8.1

8.1

Expected bond return

7.7

7.7

Equity premium

0.4

0.4

B. Cyclical shock distribution Persistence of idiosyncratic shocks Low %

High %

Expected stock return

8.2

8.1

Expected bond return

7.7

7.7

Equity premium

0.5

0.4


555

4. LIMITED PARTICIPATION AND LIMITED DIVERSIFICATION Limited stock market participation can increase the required equity premium by concentrating stock market risk on a subset of the population (e.g., Basak and Cuocco (1998), Constantinides, Donaldson, and Mehra (2002), Saito (1995), Polkovnichenko (2004) and Vissing-Jorgensen (2002)). The poor diversification of many individual portfolios is also a source of incomplete risk sharing. Both have the theoretical potential to explain the equity premium puzzle. As with other explanations based on market incompleteness, however, the unresolved question is whether these effects are large enough to make a quantitatively significant difference. In Heaton and Lucas (2000b), we consider both of these mechanisms in an overlapping generations (OG) model. We conclude that the increases in participation of the magnitude witnessed in the past two decades are unlikely to cause a significant reduction in the predicted equity premium going forward, but that improved portfolio diversification might explain a fall in the equity premium of several percentage points. To understand why limited participation may have little quantitative significance for the equity premium, it is useful to review basic facts about the distribution of wealth, and its dynamics over time. Calculations from the Fed’s Survey of Consumer Finances (SCF) suggest that despite increases in stock market participation in the last 15 years, wealth and stock holdings in the U.S. remain highly concentrated in dollar terms. For example, in 1989, the top 10 percent of the wealth distribution held 84 percent of the stock. This dropped slightly to 83 percent in 1995 and further to 76.6 percent in 2001. Although these high concentrations might suggest the potential for greater risk sharing, the low wealth levels of households not already holding stock means that risk sharing could be little improved by low-wealth households taking pure long positions in the market. Presumably risk could be better shared with levered or derivative positions, but such behavior is not observed, presumably because of trading costs or market imperfections. Gomes and Michaelides (2005) also reach the conclusion that participation rates do not explain the equity premium, in a richer calibrated life-cycle model with a small fixed cost for stock market participation, and an endogenous participation decision. Although raw changes in participation are unlikely to explain changes in expected returns, the way in which the wealthy participate, beyond the distinction between entrepreneurs and non-entrepreneurs, could be important. Tallies of stock market participation rates do not distinguish between diversified and undiversified investment strategies. In Heaton and Lucas (2000c), we show theoretically that changes in the level of diversification of the wealthy (substitution from individual stocks to diversified pension and mutual funds) is an important change in portfolio choice behavior that might explain some reduction in expected stock returns. The technical mechanism is that less diversified portfolios have fatter tails and, in particular, higher volatility, a nonnegligible probability of a catastrophic outcome, and greater skewness. The question remains, however, if greater diversification is valuable enough to justify a high equity premium, why greater diversification is not achieved.

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5. CONCLUSIONS We conclude that there is no simple resolution of the equity premium puzzle based on heterogeneity, undiversified risk, and trading frictions in models that are otherwise structurally similar to that in Mehra and Prescott (1986). Nevertheless, these factors influence economic behavior and are likely to play a significant role in explaining asset prices. Perhaps most importantly, the continuing concentration of asset market participation by wealthy and older households suggests that better information on the preferences, incentives and constraints for this group could improve our understanding of the equity premium, and asset prices more generally.

References Aiyagari, R., and M. Gertler. Asset returns with transactions costs and uninsured individual risk. Journal of Monetary Economics (1991): 311–331. Basak, S., and D. Cuocco. An equilibrium model with restricted stock market participation. Review of Financial Studies 11 (1998): 309–341. Carroll, C. D., and A. A. Samwick. The nature of precautionary wealth. NBER working paper #W 5193 (1995). Constantinides, G., and D. Duffie. Asset pricing with heterogeneous consumers. Journal of Political Economy 104 (1996): 219–240. Constantinides, G. M., J. B. Donaldson, and R. Mehra. Junior can’t borrow: A new perspective on the equity premium puzzle. Quarterly Journal of Economics 17 (2002): 269–296. Curcuru, S., J. Heaton, D. Lucas, and D. Moore. Heterogeneity and portfolio choice: Theory and evidence. Handbook of Financial Econometrics forthcoming (2007). Freeman, M. C. Can market incompleteness resolve asset pricing puzzles? Journal of Business Finance and Accounting 31(7–8) (2004): 928–949. Gomes, F., and A. Michaelides. Asset pricing with limited risk sharing and heterogeneous agents. Manuscript, London School of Economics (2005). Guvenen, F. Learning your earning: Are labor income shocks really very persistent? Manuscript, University of Rochester (2005). Hansen, L. P., and K. Singeton. Stochastic consumption, risk aversion, and the temporal behavior of asset returns. Journal of Political Economy 91 (1983): 249–265. Heaton, J., and D. Lucas. The effects of incomplete insurance markets and trading costs in a consumptionbased asset pricing model. Journal of Economic Dynamics and Control 16(3–4) (1992): 601–620. Heaton, J., and D. Lucas. The importance of investor heterogeneity and financial market imperfections for the behavior of asset prices. Carnegie Rochester Papers 42 (1995): 1–32. Heaton, J., and D. Lucas. Evaluating the effects of incomplete markets on risk sharing and asset pricing. Journal of Political Economy 104 (1996): 443–487. Heaton, J., and D. Lucas. Portfolio choice in the presence of background risk. Economic Journal 110 (460) (2000a): 1–26. Heaton, J., and D. Lucas. Stock prices and fundamentals. NBER Macroeconomics Annual (2000b). Heaton, J., and D. Lucas. Asset pricing and portfolio choice: The role of entrepreneurial risk. Journal of Finance 55(3) (2000c): 1163–1198. Kocherlakota, N. The equity premium: It’s still a puzzle. Journal of Economic Literature XXXIV (1996): 42–71. Lettau, M., S. Ludvigson, and J. Wachter. The declining equity premium: What role does macroeconomic risk play? Manuscript, New York University (2005). Lucas, D. Asset pricing with undiversifiable income risk and short sales constraints: Deepening the equity premium puzzle. Journal of Monetary Economics (1994).


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Lucas, R. E. Asset prices in an exchange economy. Econometrica 46(6) (1978): 1429–1445. Lustig, H. The market price of aggregate risk and the wealth distribution. Manuscript, UCLA (2007). Mankiw, N. G. The equity premium and the concentration of aggregate shocks. Journal of Financial Economics 17 (1986): 211–219. Mankiw, N. G., and S. Zeldes. The consumption of stockholders and non-stockholders. Journal of Financial Economics 29 (1991): 97–112. Mehra, R., and E. Prescott. The equity premium—a puzzle. Journal of Monetary Economics 15 (1985): 145–161. Merton, R. C. Optimum consumption and portfolio rules in a continuous time model. Journal of Economic Theory 3(4) (1971): 373–413. Polkovnichenko, V. Limited stock market participation and the equity premium, finance research letters, (2004) vol. 1, #1. Reitz, T. A. The equity premium: A solution. Journal of Monetary Economics 22 (1988): 117–131. Saito, M. Limited market participation and asset pricing. Manuscript, University of British Columbia (1995). Storesletten, K., C. Telmer, and A. Yaron. Asset pricing with idiosyncratic risk and overlapping generations. Forthcoming, Review of Economics Dynamics (2007). Storesletten, K., C. Telmer, and A. Yaron. Cyclical dynamics in idiosyncratic labor-market risk. Journal of Political Economy 112 (2004): 695–717. Telmer, C. I. Asset pricing puzzles and incomplete markets. Journal of Finance 48 (1993): 1803–1832. Vissing-Jørgensen, A. Limited asset market participation and the elasticity of intertemporal substitution. Journal of Political Economy 110 (2002): 835–853. Zhang, H. Endogenous borrowing constraints with incomplete markets. The Journal of Finance 52(5) (1997): 2187–2209.

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Discussion: Can Heterogeneity, Undiversified Risk, and Trading Frictions Solve the Equity Premium Puzzle? Kjetil Storesletten University of Oslo

1. INTRODUCTION Heaton and Lucas’ chapter reviews the role background risk plays for the equity premium. With background risk we understand risk idiosyncratic to the agent, such as labor income risk or entrepreneurial risk. Why are we obsessed with the role of background risk for explaining the equity premium? The obvious first answer is that it might work: individual consumption growth is orders of magnitude more volatile than aggregate consumption growth. But there is also a broader motivation. The study of heterogeneity in macroeconomics aims at explaining the evolution of the joint distribution of consumption, earnings, and wealth. The promise of this line of research is to provide tools to evaluate government policies and to deepen our understanding of inequality. To this end, macroeconomic facts such as the evolution of consumption inequality represent useful restrictions of microeconomic theories—any micro theory of inequality and markets should be consistent with these macro facts (see, e.g., Krueger and Perri (2006) and Storesletten et al. (2004) for examples of such an approach). Risk and individuals’ attitudes toward risk are at the heart of these theories, and the equity premium puzzle—the inability of a reasonably calibrated consumption-based asset pricing model to explain the premium—represents a major road block for this literature. In their chapter, Heaton and Lucas analyze labor income risk, transaction costs (in connection with background risk), entrepreneurial risk, and the role of limited stock market participation, and I will now review their findings.

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2. LABOR INCOME RISK Workers’ labor earnings are subject to idiosyncratic fluctuations. To evaluate the role of such risk, one must take a stand on the market structure for hedging such risk. Due to asymmetric information or limited enforcement, it seems plausible to assume that agents have only limited insurance against these shocks. A natural starting point is to follow Huggett (1993) and Aiyagari (1994) and assume no insurance markets for idiosyncratic risk, over and above that agents can save in aggregate assets such as stocks and bonds. This is the approach Heaton and Lucas pursue, and it represents in some sense the maximum kick one can get out of idiosyncratic risk. Solving such models require, in general, numerical techniques.1 The problem is that in absence the of complete markets (a representative agent), the standard aggregation results break down, so the whole wealth distribution enters as a state variable (see, e.g., R´ıos–Rull (1994)). To date, the most fruitful solution approach has been the one suggested by Krusell and Smith (1997, 1998). They show that by focusing on a restricted set of relevant moments of the wealth distribution, such as the average capital stock and the fraction of constrained agents, one can obtain a remarkably good prediction of future prices and the evolution of these relevant moments, a result they interpret as “approximate aggregation.”2 Now, let us review the results documented in Heaton and Lucas’ chapter. They consider an exogenous autoregressive process for labor income. Overall, they find very little effect of idiosyncratic labor income risk on asset prices in general and on the equity premium in particular. The main reason for the result is that the labor income process is not very persistent (the autocorrelation is just 0.6), so that agents are able to smooth these shocks quite well by simply using a bond to buffer the shocks. As is well known, with sufficiently loose constraints on borrowing and a sufficiently low interest rate, agents can insure transitory shocks arbitrarily well by borrowing and lending (Telmer (1993) and Levine and Zame (2002)). Once idiosyncratic risk has been aggregated away, one is back to the representative-agent economy (and the associated equity premium puzzle). However, the consensus view in labor economics is that agents are subject to substantial and highly persistent shocks to earnings. For example, Abowd and Card (1989) obtain this conclusion based on covariance structure of labor earnings.3 Indeed, when allowing more persistent shocks, the asset pricing implications can be substantial. 1 Two well-known exemptions are when preferences are either linear-quadratic or of the constant absolute risk aversion type. Alternatively, one could put additional restrictions on markets or on the income process. In particular, Constantinides and Duffie (1996) focus on a class of models with multiplicative unit-root shocks, which implies an autarky result, so the model can be solved analytically (see also Krebs (2003) and Storesletten et al. (2007a)). An important alternative branch of the literature focuses on constrained-efficient limited enforcement models that also allow analytical results (see, e.g., Alvarez and Jermann (2001) and Lustig (2004)). 2 Heaton and Lucas (1996) simplify their computational problem by considering economies with a particularly simple form of heterogeneity—there are only two types of agents in the economy. Since the distribution of wealth can be fully characterized by a one-state variable—the wealth of the first agent—it is feasible to solve this problem without relying on the Krusell–Smith approach. 3 Clearly, even though earnings were to follow a unit-root process, these fluctuations need not impact consumption allocations, provided that the fluctuations were predicted or fully or partly insurable.

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For example, Storesletten et al. (2007a) show, using a reasonable calibration, that idiosyncratic labor income risk can account for between one-third and one-sixth of the empirical equity premium (see also Storesletten et al. (2007b)).

3. TRANSACTION COSTS What about transaction costs? Could it be that with costs on each transaction, the nearcomplete insurance result above would vanish? Heaton and Lucas assume proportional transaction costs for stocks, but frictionless trading in bonds. Their main finding is that implausible costs would be required to account for the equity premium. The intuition for this result is that even with large motives to trade (transitory shocks), agents satisfy their trading needs using bonds. As for stocks, the agents buy and hold, but do not require a large premium for doing so. And it doesn’t help to consider more persistent labor income shocks, because this would reduce the motives for trading (because it is harder to “self-insure” against permanent risk), which implies an even smaller role for transaction costs (in fact, in Constantinides and Duffie (1996), agents do not trade at all even in the absence of constraints, so constraints on portfolio allocations would be irrelevant!).

4. CONCENTRATING AGGREGATE RISK ON FEWER HANDS The source of the equity premium puzzle is that all agents share equally the aggregate risk and that the aggregate consumption process does not fluctuate much. Clearly, one way out would be to concentrate the aggregate risk on fewer agents, so that the aggregate consumption process for the stock owners could be very volatile, even with quite smooth aggregate consumption. Heaton and Lucas consider two such channels for concentrating risk on fewer hands: limited stock market participation and entrepreneurial risk.

4.1. Entrepreneurial Risk Entrepreneurs are interesting because they hold large stakes in one or a few companies (due to, say, asymmetric information problems). Since entrepreneurs’ holdings amount to a large share of the aggregate U.S. wealth, and the firm-specific risk in these companies is likely to be correlated with the market portfolio, it is particularly important to investigate entrepreneurs’ income and consumption processes. Heaton and Lucas (2000) document some important facts about business owners. First, they have more volatile income growth, substantially larger than the volatility of workers’ individual earnings growth. Moreover, they own substantial amounts of publicly traded stocks, as well as large stakes in their own companies. However, it turns out that the aggregate wage income of entrepreneurs has a low correlation with the market return on equity. At this point, one might have high hopes for this channel of individual risk to deliver large asset-pricing implications.

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In order to evaluate the quantitative implications of this “entrepreneurial risk,” Heaton and Lucas (2000) focus on the labor income of entrepreneurs, emphasizing the larger volatility and higher correlation with the return on the market portfolio. So the entrepreneurs’ holdings in their own companies are treated as exogenous and nontradeable, and the income from these holdings are incorporated in the stochastic labor income. When calibrating the economy, Heaton and Lucas focus on relatively transient shocks (ρ = 0.8) and find a small effect. Does this paper close the discussion on the role of entrepreneurs? The key shortcoming of their argument is that entrepreneurs are assumed to have a fixed amount of capital invested in the firm, and hence a fixed exposure to risk. A more plausible and standard model of entrepreneurs would allow them to accumulate and de-accumulate wealth in their own company, maintaining the assumption that the return on this firmspecific capital is idiosyncratic albeit with a small aggregate component (as in Angeletos and Calvet (2006); see also see Krebs (2003)). The following argument shows that this channel could potentially yield a large equity premium. Consider the following version of a Constantinides–Duffie model with entrepreneurs. Each entrepreneur is born with one unit of capital, invested in a backyard storage technology with idiosyncratic return, but no labor income over and above this. Bonds and stocks are in zero net supply. Idiosyncratic returns are independent over time and across people. If the expected return on the backyard technology is sufficiently low, a no-trade result will apply: the agents will consume the same fraction of the gross return on their respective backyard storage and will reinvest the rest in their storage. However, their ownership of stocks and bonds will remain zero. With this autarky result in hand, it is clear that the model is observationally equivalent to the standard Constantinides–Duffie model, which can deliver a large equity premium (see Storesletten et al. (2007b) for an example).

4.2. Limited Participation What about limited stock market participation? Empirically, stock owners’ consumption has substantially larger conditional variance than that of individuals who do not hold stocks (Attanasio et al. (2002), Brav et al. (2002), Vissing-Jørgensen (2002)). An important and interesting paper by Fatih Guvenen (2005) documents that this channel has a surprisingly large potential for accounting for the empirical asset prices. Since this paper speaks so directly to the issue at hand, I think it’s useful to review it here. Guvenen (2005) examines a real business cycle model with heterogeneous agents. Agents differ in two respects—in their preferences (they have different intertemporal elasticity of substitution; henceforth EIS) and in their access to equity markets. A majority of agents (80 percent) are barred from the stock market and have a low EIS of 1/10. A minority (20 percent) have high EIS of 1/2 and have full access to stocks and bonds. Interestingly, a plausibly calibrated version of this simple model has a surprisingly good performance in terms of explaining financial statistics such as the equity premium, the Sharpe ratio, the covariance between the risky rate and the risk-free rate,

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the predictability of returns, etc. The intuition for this good performance is that it can be reinterpreted as a version of the external habit model of Campbell and Cochrane (1999), henceforth CC. In particular, Guvenen shows that the “external habit” of CC has approximately the same stochastic properties as the non-stockholders of Guvenen’s model, so that the pricing kernels are quantitatively very similar in the two models. The main channels for the large equity premium in his model are that, first, limited participation concentrates more aggregate risk on the stockholders. Consequently, their consumption growth is much more volatile than the aggregate consumption growth, and they require a premium for shouldering this risk. Second, the non-stockholders must smooth their consumption through the bond market. Since they have a lower EIS than the stockholders, they are more interested in smoothing consumption. In particular, in recessions (booms) the non-stockholders want to reduce wealth (accumulate wealth). This is picked up by the stockholders, further adding to the pro-cyclical variability of their consumption growth, and therefore increasing the premium they require to hold stocks. Interestingly, the resulting movements in the distribution of wealth (between stockholders and non-stockholders) are not extremely large in Guvenen’s model. Returning to Heaton and Lucas, their critique against stories for the equity premium based on limited participation is that, first, it is not clear where the participation constraint comes from. In fact, it turns out that the non-participants in Guvenen’s model are willing to pay substantial amounts to gain access to equity markets. A possible reply to this critique is that non-stockholders have labor income risk and also have their wealth tied up in housing, due to house ownership being tax-advantaged. Second, the required concentration of aggregate risk on stockholders might be implausibly large. Finally, at the end of the day, we are aiming for consumption-based models of asset pricing that are consistent with both asset pricing facts and business cycle facts. Any serious business cycle model must incorporate a theory of fluctuations in aggregate labor supply, since hours worked is very pro-cyclical. However, endogenous labor supply makes it harder to account for the equity premium. If one were to introduce endogenous labor supply in any of the asset pricing models based on idiosyncratic labor income risk (such as Constantinides and Duffie (1996), Heaton and Lucas (1996), Krusell and Smith (1997), Guvenen (2005), or Storesletten et al. (2007a)), individuals would exploit the opportunity to smooth the marginal utility of consumption by varying their labor supply. In particular, Uhlig (2005) considers a representative-agent model with large risk aversion and shows that such a model has two undesired properties. First, the aggregate labor supply becomes countercyclical. Second, the equity premium fall substantially.

5. CONCLUSION Bottom line, the view of Heaton and Lucas is that the equity premium puzzle cannot be resolved by introducing incomplete markets and heterogeneity. While I agree that heterogeneity alone is unlikely to account for the whole premium, it seems clear that heterogeneity delivers a significant share of the equity premium.

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References Abowd, J. M., and D. Card. On the covariance structure of earnings and hours changes. Econometrica 57 (1989): 411–45. Aiyagari, S. R., Uninsured idiosyncratic risk and aggregate saving. Quarterly Journal of Economics 109 (1994): 659–684. Alvarez, F., and U. Jermann. Quantitative asset pricing implications of endogeneous solvency constraints. Review of Financial Studies 14 (2001): 1117–1152. Angeletos, G.-M., and L. Calvet. Idiosyncratic production risk, growth and the business cycle. Journal of Monetary Economics 53(6) (2006): 1095–1116. Attanasio, O. P., J. Banks, and S. Tanner. Asset holding and consumption volatility. Journal of Political Economy 110(4) (2002): 771–792. Brav, A., G. M. Constantinides, and C. Geczy. Asset pricing with heterogeneous consumers and limited participation: Empirical evidence. Journal of Political Economy 110 (2002): 793–824. Campbell, J. Y., and J. Cochrane. By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107 (1999): 205–251. Constantinides, G. M., and D. Duffie. Asset pricing with heterogeneous consumers. Journal of Political Economy 104 (1996): 219–240. Guvenen, F. A parsimonious macroeconomic model for asset pricing: Habit formation or cross-sectional heterogeneity? Working paper, University of Rochester (2005). Heaton, J., and D. J. Lucas. Evaluating the effects of incomplete markets on risk sharing and asset pricing. Journal of Political Economy 104 (1996): 443–487. Heaton J., and D. J. Lucas. Portfolio choice and asset prices: The importance of entrepreneurial risk. Journal of Finance 55 (2000): 1163–1198. Huggett, M. The risk free rate in heterogeneous-agents, incomplete insurance economies. Journal of Economic Dynamics and Control 17 (1993): 953–969. Krebs, T. Human capital risk and economic growth. Quarterly Journal of Economics 118(2) (2003): 709–744. Krueger, D., and F. Perri. Does income inequality lead to consumption inequality? Evidence and theory, Review of Economic Studies 73(1) (2006): 163–193. Krusell, P., and A. A. Smith. Income and wealth heterogeneity, portfolio choice, and equilibrium asset returns. Macroeconomic Dynamics 1(2) (1997): 387–422. Krusell, P., and A. A. Smith. Income and wealth heterogeneity in the macroeconomy. Journal of Political Economy 106 (1998): 867–896. Levine, D. K., and W. R. Zame. Does market incompleteness matter? Econometrica 70(5) (2002): 1805–1839. Lustig, H. The market price of aggregate risk and the wealth distribution. Working paper, Stanford University (2004). R´ıos-Rull, J. V. On the quantitative importance of market completeness. Journal of Monetary Economics 34 (1994): 463–496. Storesletten, K., C. J. Telmer, and A. Yaron. Consumption and risk sharing over the life cycle. Journal of Monetary Economics 59(3) (2004): 609–633. Storesletten, K., C. J. Telmer, and A. Yaron. Asset pricing with idiosyncratic risk and overlapping generations (in press). Review of Economic Dynamics (2007a). Storesletten, K., C. J. Telmer, and A. Yaron. Asset prices and intergenerational risk sharing: The role of idiosyncratic earnings shocks. In Handbook of Investments: Equity Risk Premium, R. Mehra, ed. North Holland, Amsterdam (2007b). Telmer, C. I. Asset pricing puzzles and incomplete markets. Journal of Finance 48 (1993): 1803–1832. Uhlig, H. Explaining asset prices with external habits and wage rigidities in a DSGE model. Working paper, Humboldt University (2007). Vissing-Jørgensen, A. Limited asset market participation and the elasticity of intertemporal substitution. Journal of Political Economy 110(4) (2002): 825–853.


CHAPTER 14

Asset Prices and Intergenerational Risk Sharing: The Role of Idiosyncratic Earnings Shocks∗ Kjetil Storesletten University of Oslo and CEPR

Chris Telmer Tepper School of Business, Carnegie Mellon University and

Amir Yaron The Wharton School, the University of Pennsylvania, and NBER

1. Introduction 2. An Analytical Example of the Constantinides–Duffie Model 2.1. Calibration of the Constantinides–Duffie Economy 2.2. Model Implications 3. Incorporating the Life Cycle 3.1. Calibration 4. Quantitative Results 4.1. Asset Pricing Implications 4.2. Sensitivity Analysis 5. Conclusions References A. Calibration Appendix B. Asset Pricing

567 569 570 571 573 576 577 580 581 581 584 587 590

∗ This chapter builds on the work in Storesletten, Telmer, and Yaron (2006). We thank Rajnish Mehra and Stan Zin for detailed comments.


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Chapter 14 • Asset Prices and Intergenerational Risk Sharing: The Role of Idiosyncratic Earnings Shocks

Abstract In their seminal paper, Rajnish Mehra and Edward Prescott (1985) were the first among many subsequent authors to suggest that non-traded labor-market risk may provide a resolution to the equity premium puzzle. The most direct demonstration of this was Constantinides and Duffie (1996), who showed that, under certain conditions, crosssectionally uncorrelated unit root shocks that become more volatile during economic contractions can resolve the puzzle. We examine the robustness of this to life-cycle effects. Retired people, for instance, do not face labor-market risk. If we incorporate them, to what extent will the equity premium be resurrected? Our answer is “not very much.” Our model, with realistic life-cycle features, can still account for about 75 percent of the average equity premium and the Sharpe ratio observed on the U.S. stock market. JEL Classification: E2, G11, G12, E32 Keywords: idiosyncratic shocks, counter cyclical volatility (CCV), human capital, equity premium, Sharpe ratio, portfolio choice, life-cycle.

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1. INTRODUCTION This chapter analyzes the channels by which idiosyncratic earning shocks affect asset prices. The findings and framework are largely built on Storesletten, Telmer, and Yaron (2006). We show that the asset pricing effects of idiosyncratic shocks depend on the relative magnitude of human capital shocks to financial risk. The exposure to this dimension changes dramatically over the life cycle—making life-cycle consideration an important element of our framework. More specifically, we show that individuals’ portfolio choices are sensitive to idiosyncratic earning shocks when (1) they display persistence and countercyclical volatility (CCV)—features documented using earnings data and that (2) the shocks to human capital are large relative to financial risk when young. Young agents are more susceptible to human capital shocks since they hold little financial capital, because the shocks are persistent, they have a large impact on discounted earnings (i.e., human capital). As agents age, the exposure to financial (human) capital increases (declines). With that logic, agents with no labor income—retirees—can accommodate the largest exposure to aggregate risk. However, since they have less labor income than the middle-aged, they still prefer to reduce their exposure to financial risk somewhat. Moreover, with countercyclical volatility of earnings shocks, agents who are fully exposed to human capital risk—the youngest agents—would like the smallest exposure to aggregate risk. As a consequence, our model displays average portfolio rules that are hump-shaped in age. That is, agents choose to hold very little equity when young, levered equity positions when middle-aged, and intermediate equity positions when retired. We show that in equilibrium these portfolio choices manifest themselves into asset prices that depend on how much intergenerational risk sharing takes place across cohorts. Without idiosyncratic shocks and with constant aggregate wages (and thus return on human capital), the old would prefer to issue debt while the young would prefer to be levered in equity. Such intergenerational risk sharing would tend, ceteris paribus, to lower the equity premium. However, the presence of countercyclical idiosyncratic shocks dissuades the young from holding equity, which curtails intergenerational risk sharing. Following the work in Storesletten, Telmer, and Yaron (2006), we show that quantitatively the effects of idiosyncratic risk are significant. Idiosyncratic risk inhibits intergenerational risk sharing, imposing a disproportionate share of aggregate risk on the wealthy middle-aged cohorts who demand an equity premium for their exposure to this risk. We use a stationary overlapping-generations model to show how life-cycle portfolio choices interact with intergenerational risk sharing to accentuate the equity premium. For a risk aversion of 8, our model is able to account for about 75 percent of the average equity premium and the Sharpe ratio observed on the U.S. stock market. It is important to note that the driving force in our model is a concentration of aggregate risk and equity ownership on middle-aged and old agents. The model of Constantinides, Donaldson, and Mehra (2002) shares a similar feature. Where their model is driven by portfolio constraints, ours is driven by portfolio choices made in

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light of how non-tradeable and tradeable risks interact. We discuss these issues further ahead. An advantage of our model relates to risk-sharing behavior. U.S. data on income and consumption indicates that, while complete markets may not characterize the world, neither does a distinguishing feature of the Constantinides and Duffie (1996) framework: autarky.1 The cross-sectional standard deviation of U.S. consumption, for instance, is roughly 35 percent smaller than that of non-financial earnings.2 However, our framework shows that in spite of the autarky dimension and the lack of realistic life-cycle features, the Constantinides–Duffie model is still able to provide useful quantitative asset pricing results. Papers by Balduzzi and Yao (2007), Brav, Constantinides, and Geczy (2002), Cogley (2002), Ramchand (1999), and Sarkissian (2003) investigate the Constantinides–Duffie model’s consumption Euler equation restrictions directly and find mixed evidence. Our approach, emphasizing endogenous asset pricing, seems to be consistent with the channels put forth in their paper (e.g., persistent and time-varying idiosyncratic risk). The idea that market incompleteness may contribute to the equity premium is not new, and by and large, most of the quantitative findings have been “negative” in terms of the ability of the proposed models to generate a viable equity premium. The agents in these models tend to be “very efficient” in insuring themselves against the relatively transitory income shocks they face; thus, the resulting equity premium is essentially the one derived in Mehra and Prescott (1985) (e.g., Telmer (1993), Heaton and Lucas (1996)). To generate a sizeable equity premium, such models usually had to resort to large transaction costs and/or tight borrowing constraints. The distinguishing aspect of our paper, life-cycle, has also important implications for the ability to capture persistent (almost unit root) processes for individual earnings shocks while maintaining aggregate quantities that are characterized by relatively transitory processes. A number of studies have examined more specifically the quantitative implications of the Constantinides and Duffie (1996) model. The closest to our work is Krusell and Smith (1997) and Gomes and Michaelides (2004). The latter use a similar life-cycle model, with fixed-entry costs to the stock market and preference heterogeneity across stock and non-stockholders. They show that preference heterogeneity and not the fixed-entry costs is the important dimension for matching the relative size of stockholders. Others include Aiyagari (1994), Aiyagari and Gertler (1991), Alvarez and Jermann (2001), den Haan (1994), Guvenen (2005), Heaton and Lucas (1996), Huggett (1993), Lucas (1994), Mankiw (1986), Marcet and Singleton (1999), R´ıos-Rull (1994), Telmer (1993), Weil (1992), and Zhang (1997). The stationary OLG framework we develop owes much to previous work by R´ıos-Rull (1994), Huggett (1996) and Storesletten (2000). More recent 1 By “autarky” we mean a situation where there is no trade amongst living agents and they are forced to consume their endowments every period. 2 A large number of papers, including Altonji, Hayashi, and Kotlikoff (1991), Attanasio and Davis (1996), Attanasio and Weber (1992), Cochrane (1991), Deaton and Paxson (1994), Mace (1991), and Storesletten, Telmer, and Yaron (2004a), provide evidence suggestive of imperfect risk sharing. Altug and Miller (1990) find opposing evidence. The numerical value cited here is based upon evidence in both Deaton and Paxson (1994) and Storesletten, Telmer, and Yaron (2004a).

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examples using life-cycle economies to asses portfolio choice and equity returns include Olovsson (2004) and Benzoni, Collin-Dufresne, and Goldstein (2004). The remainder of this chapter is organized as follows. In Section 2, we formulate a version of the Constantinides and Duffie (1996) model, calibrate it, and examine its quantitative asset pricing properties. In Section 3, we introduce life-cycle savings by assuming that retirement income is equal to zero; that is, a model with trade with a more realistic distribution of human to financial wealth. In Section 4, we analyze the economic workings of this model using a class of computational experiments. Section 5 concludes the paper.

2. AN ANALYTICAL EXAMPLE OF THE CONSTANTINIDES–DUFFIE MODEL We begin with an analytical example of the Constantinides and Duffie (1996) model. There are two asset markets, a one-period riskless bond and an Arrow–Debreu security. Agents live for H ≤ ∞ periods and have standard CRRA preferences: max E

H t=1

1−γ

c β t it 1−γ

.

(1)

Labor income for agent i in period t is given by yit = Ct exp(zit ),

(2)

where Ct is aggregate consumption and zit is agent i’s share of aggregate consumption during period t. Aggregate consumption growth follows a two-state process and equals 1 + z in booms and 1 − z in recessions. The probability of remaining in an aggregate state is P . Individual’s share of consumption, zit , follows a unit root process with heteroskedastic innovations: zit = zi,t−1 + ηit ,

(3)

zi0 = 0,

σ2 ηit ∼ N − t , σt2 , 2

(4) (5)

where the time-varying conditional variance depends on consumption growth, zt , according to σt2 =

a − b · zt a + b · zt

in recessions, in booms,

(6)

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where the coefficient b is the sensitivity of the cross-sectional variance to aggregate growth rate. That is, b defines the heteroskedasticity of the process σt . For example, the Constantinides and Duffie intuition by which the cross-sectional variation rises during bad times entail a b-coefficient that is negative (i.e., the cross-sectional variance grows during negative aggregate growth). Agents trade two assets in zero net supply—a one-period bond and a one-period Arrow–Debreu security paying one unit of consumption in booms and zero in recessions. Following Constantinides and Duffie (1996), it is straightforward to show that the equilibrium consumption allocation is autarky, that is cit = yit . The pricing kernel for this economy satisfies the standard pricing Euler equation: Et [Mt+1 Ri,t+1 ] = 1,

(7)

ˆ t+1 /Ct )−ˆγ and γˆ and βˆ are, respectively, the risk aversion and discount where Mt+1 = β(C factor of the aggregate “mongrel” consumer (see Eqs. (17) and (18) in Constantinides and Duffie (1996)) and are given by γ (γ + 1) · b, 2 (γ + 1) γ a . βˆ = β · exp 2 γˆ = γ −

(8) (9)

Equilibrium prices in this economy are now very easy to compute and follow the standard pricing formulas using CRRA preferences with the adjusted time discount and risk aversion. In particular, as shown in detail in the appendix, in the simple case of i.i.d. aggregate shocks (namely, P = 1/2), the Sharpe ratio for the excess return on the risky asset, denoted rt+1 , is given by the familiar formula: (1 − z)−ˆγ − (1 + z)−ˆγ E(r) = ≈ z · γˆ = std(log(Ct+1 /Ct ))γˆ , std(r) (1 + z)−ˆγ + (1 − z)−ˆγ

(10)

where the approximation exploits that log(1 + z) ≈ z for small z. As is clear from the denominator of Eq. (10), the return on the risky Arrow–Debreu asset can have a large standard deviation. However, by combining the risky Arrow– Debreu asset with the bond, it is straightforward to construct a portfolio that looks like a an “equity stock,” i.e., an asset whose return has a standard deviation of about 10 percent. Note however, that the Sharpe ratio on the stock will be the same as that of the risky Arrow–Debreu asset—making the above computation informative.

2.1. Calibration of the Constantinides–Duffie Economy To quantify the implications of the above model, we need to specify the process for consumption growth as well as the countercyclical volatility. We now ask if values of a and b implied by labor market data help the model account for the equity premium. We


571

use estimates from Storesletten, Telmer, and Yaron (2004b), which are based on annual PSID data, 1969–1992. They show that (1) idiosyncratic shocks are highly persistent and that a unit root is plausible, (2) the conditional standard deviation of idiosyncratic shocks is large, averaging 17 percent, and (3) the conditional standard deviation is countercyclical, increasing by roughly 68 percent from expansion to contraction (from 12.5 to 21.1 percent). In Appendix A, we show that these estimates map into values a = 0.0143 and b = −0.1652. We use a stochastic process for zt , which is essentially the same as that in Mehra and Prescott (1985): a two-state Markov chain with mean, and a standard deviation of aggregate consumption growth of 0.018, 0.033, and a transition matrix in which P = 2/3, approximately matching an autocorrelation of −0.14. We choose the “effective” discount factor, β ∗, to match the average U.S. annual risk-free interest rate, 1.3 percent. We vary the risk aversion γ from 3 to 8.3 We also vary the degree of countercyclical variation in idiosyncratic income process. In addition to the countercyclical variability (CCV) estimated in Storesletten, Telmer, and Yaron (2004b), we also examine a timeinvariant process in which the variance of idiosyncratic risk is constant across recessions and expansions and is set to the unconditional volatility. Finally, we also examine as a benchmark the “complete markets” economy, one in which the idiosyncratic volatility is set to zero. Table 1 reports the results. In the column “SR%,” we report the percent Sharpe ratio (which is slightly different from Eq. (10) because P > 1/2). We set std(rt ), the standard deviation of the excess return on the risky security, to be 10 percent by changing the weights of the portfolio of the risky Arrow–Debreu asset and the bond. Table 1 clearly demonstrates that as risk aversion is increased, the Sharpe ratio rises. Not surprisingly, in order to maintain the risk-free rate at 1.3 percent, β needs to be lowered. It appears that in order to generate a large Sharpe ratio, one requires both a sizeable risk aversion (i.e., risk aversion of 8) and the presence of CCV. Furthermore, the presence of CCV seems to have a much more dramatic effect in the case of risk aversion of 8. In that case, the introduction of CCV raises the Sharpe ratio from 23 to 37 percent, relative to a modest rise in the Sharpe ratio of only 3 percent in the case of low risk aversion of 3.

2.2. Model Implications There are several lessons to be taken from the special structure of the Constantinides– Duffie economy described earlier. First, the examples have zero capital. By increasing aggregate wealth to some positive level, agents are more likely to be able to selfinsure some fraction of their shocks, even if these shocks are permanent. The reason for this is the following. If agents have some positive financial wealth in addition to their human capital (defined as discounted future labor earnings), then a permanent 3 Cogley (2002) formulates an asset pricing model with idiosyncratic risk (to individuals’ consumption) and uses the empirical time-varying cross-sectional moments of consumption growth from the Survey of Consumer Expenditures (CEX) to ask what level of risk aversion would be required to account for the empirical equity premium. Interestingly, this approach delivers a risk aversion of 8 (assuming a plausible level of measurement error).

572


TABLE 1 Asset Pricing Properties—No-Trade Economies Risk aversion

Risk-free rate

Equity premium

Mean

Std Dev

Mean

Std Dev

Sharpe ratio

U.S. data

1.30

1.88

6.85

16.64

41.17

U.S. data, unlevered

1.30

1.88

4.11

10.00

41.17

Models Without Trade (Constantinides–Duffie): Complete markets

3.0

1.30

—

0.87

10.0

8.7

No CCV

3.0

1.30

—

0.87

10.0

8.7

Estimated CCV

3.0

1.30

—

1.15

10.0

11.5

Complete markets

8.0

1.30

—

2.26

10.0

22.6

No CCV

8.0

1.30

—

2.26

10.0

22.6

Estimated CCV

8.0

1.30

—

3.72

10.0

37.2

“Models Without Trade” corresponds to a calibration of the Constantinides and Duffie (1996) model using the idiosyncratic risk estimates from Storesletten, Telmer, and Yaron (2004b), Table 1, and the aggregate consumption moments from Mehra and Prescott (1985). Details are given in Appendix A. Rows labeled “Complete Markets” are economies with no idiosyncratic volatility (i.e., σt = 0). Rows labeled “No CCV” represent a “homoskedastic” economy, i.e., an economy with idiosyncratic risk with time-invariant variance. This conditional variance is set equal to the average conditional variance across recessions and booms, Var (η) = 0.0114. The rows labeled “Estimated CCV” correspond to economies with idiosyncratic risk equal to a unit root version of the estimated process in Storesletten, Telmer, and Yardon (2004b). Further details are given in Appendix A. In all the models, K/Y = 0. U.S. sample moments are computed using non-overlapping annual returns, from end of January to end of January, 1956–1996. Estimates of means and standard deviations are qualitatively similar using annual data beginning from 1927, or a monthly series of overlapping annual returns. Equity data corresponds to the annual return on the CRSP value-weighted index, inclusive of distributions. Risk-free returns are based on the one-month U.S. Treasury bill. Nominal returns are deflated using the GDP deflator. All returns are expressed as annual percentages. Unlevered equity returns are computed using a debt to firm value ratio of 40 percent, which is taken from Graham (2000).

shock to earnings implies a less-than-proportional change in total wealth (as financial wealth is unaffected). Consequently, financial wealth helps agents to partially smooth even permanent shocks (see e.g., Storesletten, Telmer, and Yaron (2004a)). Since agents in this economy all have the same normalized “target wealth” (see Carroll (2004) for details), a positive amount of aggregate wealth introduces a motive for trade and an endogenous wealth distribution—a key feature of the calibrated economy in Section 3. An economy with trade and a non-trivial distribution of wealth is in sharp contrast to the Constantinides–Duffie economy, in which the wealth distribution is degenerate (all agents have zero financial wealth). Since agents are able to better smooth consumption in economies with positive wealth (and trade), it is intuitive that countercyclical risk should have a smaller impact on asset prices. Thus, the Sharpe ratio in the Constantinides–Duffie economy represents an “upper bound” on the Sharpe ratio in economies with trade.


573

Another important shortcoming of the Constantinides–Duffie economy presented is that agents receive labor income all periods of life. Therefore, asset prices (e.g., Eq. (10)) are derived from a “worker” with labor income who is bearing all of aggregate risk. A more realistic economy should have some retirement years with no labor income risk. However, since these retirees do not face any labor income risk, their attitude toward risk should be the same as a standard Mehra–Prescott representative agent. One might therefore suspect that the presence of retirees will reduce the price of risk in the model. To analyze the effect of retirement, Storesletten, Telmer, and Yaron (2006) extend the no-trade Constantinides–Duffie model to include retirees. The key insight of that analysis is that in order for the retirees to be content with no trade, one has to endow them with a “leveraged” claim to aggregate consumption. Consequently, in the autarkic equilibrium, retirees take over some of the aggregate risk from the workers. In contrast, in the complete markets case, the workers and the retirees are equally exposed to aggregate risk. In this sense, countercyclical income risk tends to mitigate the intergenerational risk sharing. Moreover, the equilibrium price of risk will, intuitively, lie in between the Constantinides–Duffie benchmark and that of the complete markets model. Overall, the analysis shows that the Constantinides–Duffie model is quite successful at accounting for significant component of the equity premium given a realistic parameterization of idiosyncratic risk. The model has several counterfactual features (such as excess volatility of the risk-free rate) that we discuss in more detail in Storesletten, Telmer, and Yaron (2006) and that, in principle, can be rectified with a richer process for σt . However, for the remainder of this chapter, we focus on the implications of the issues discussed above regarding age, retirement, and risk sharing by quantitatively analyzing a calibrated model that includes these features.

3. INCORPORATING THE LIFE CYCLE The Constantinides–Duffie framework is useful in that it serves as a quantitative frame of reference for what we are ultimately interested in: models in which idiosyncratic risk motivates trade between heterogeneous agents. Why do we view the no-trade model as being insufficient? First, partial risk sharing is an undeniable aspect of U.S. data on labor earnings, consumption, and labor supply (see Heathcote, Storesletten, and Violante (2005), Storesletten, Telmer, and Yaron (2004a)). Partial risk sharing is likely to mitigate the asset pricing implications of idiosyncratic labor-market risk, so incorporating it is important for quantitative questions such as ours. Second, our emphasis is on how idiosyncratic risk interacts with life-cycle economics. That is, whereas the Constantinides–Duffie framework restricts the distribution of idiosyncratic shocks, we seek to incorporate the distribution of what is being shocked: human capital. Having a realistic distribution of capital necessarily means that we must incorporate trade. Finally, in a life-cycle model with trade, we are able to incorporate certain aspects of reality into our calibration (e.g., the demographic structure), thereby making for a more robust

574


quantitative exercise. We proceed by summarizing the critical aspects of the model, which is more formally laid out in Storesletten, Telmer, and Yaron (2006). There are H overlapping generations of agents, indexed by h = 1, 2, . . . , H, with a continuum of agents in each generation. Preferences are U (c) = Et

H

h 1−γ β h cit+h /(1 − γ),

(11)

h=1

where cith is the consumption of the ith agent of age h at time t and β and γ denote the discount factor and risk aversion coefficients, respectively. Newborn agents have zero financial wealth. Retirees receive zero labor income. Non-tradeable endowments take the form of labor efficiency units that are inelastically supplied to firms in returns for a wage of wt per unit. These labor efficiency units are denoted nhit , for agent i of age h at date t. They are governed by the stochastic process log nhit = κh + zhi,t ,

(12)

where κh is used to characterize the cross-sectional distribution of mean income across ages, and the idiosyncratic component zhi,t , follows: zit = zi,t−1 + ηit , zi0 = 0,

ηit ∼ N −

σt2 2 ,σ 2 t

(13) (14)

.

(15)

The time-varying conditional variance depends on an aggregate shock, Zt , according to σt2 = σE2

if Z ≥ E(Z)

= σC2

if Z < E(Z).

Individual labor income is the product of labor supplied and the wage rate: yith = wt nhit .4 Firms are represented by an aggregate production technology to which agents rent capital and labor services. Labor is supplied inelastically and, in aggregate, is fixed at N.

4 Storesletten,

Telmer, and Yaron (2006) abstract from bequest motives. The main difference in the economic outcomes would be that with bequests wealth would not fall sharply during retirement. We abstracted from these effects in order to focus exclusively on the main point of the paper—the effect of changes in the ratio of human capital to total wealth during working age.

575


Denoting aggregate consumption, output, and capital as Yt , Ct , and Kt , respectively, the production technology is Yt = Zt Ktθ N 1−θ , Kt+1 = Yt − Ct + (1 − δt )Kt ,

(16) (17)

rt = θZt Kt1−θ N 1−θ − δt ,

(18)

θ)Zt Kt−θ N −θ ,

(19)

wt = (1 −

where rt is the return on capital (the risky asset), wt is the wage rate, θ is capital’s share of output, Zt is an aggregate shock, and δt is the depreciation rate on capital. The depreciation rate is stochastic: δt = δ + (1 − Zt )

s , Std(Zt )

(20)

where δ controls the average and s is, approximately, the standard deviation of rt .5 Turning back to the household sector, agents can trade in a riskless one-period bond and in ownership of aggregate capital. An agent’s decision problem is to maximize (11) subject to the following sequence of budget constraints. We omit the i and t notation and express things recursively. Budget constraints are

+ b h+1 q(μ, Z) ch + kh+1 ah

kH+1 b H+1

≤ ah + nh w(μ, Z), = kh r(μ, Z) + bh , ≥ 0, ≥ 0,

(21)

where ah denotes beginning-of-period wealth, kh and bh are beginning-of-period capital

and bond holdings, and kh+1 and b h+1 are end-of-period holdings. We do not impose any portfolio restrictions over and above restricting terminal wealth to be non-negative (the third and fourth restrictions). The competitive equilibrium for this economy follows what is by now standard in a production economy with heterogeneous agents (e.g., R´ıos-Rull (1994), Krusell and Smith (1997)). The specific details are given in Storesletten, Telmer, and Yaron (2006). In essence, equilibrium requires market clearing for bonds, rental rates for capital and labor being given by their respective marginal productivity, and an equilibrium law of motion for the cross-sectional distribution of wealth. The main difficulty is that the wealth distribution enters as a state variable. To solve for an approximate equilibrium (in lieu of a multidimensional distribution of wealth), we use the computational methods developed 5 Greenwood, Hercowitz, and Krusell (1997) have used a similar production technology in a business cycle context. Boldrin, Christiano, and Fisher (2001) have done so in an asset pricing context. We view that our technology is essentially a reduced-form representation of, for instance, Greenwood, Hercowitz, and Krusell (1997), Eq. (B3).

576


by Krusell and Smith (1997) and adapted to life-cycle economies in Storesletten, Telmer, and Yaron (2006).

3.1. Calibration The model time period is one year. The aggregate shock in Eq. (18) follows a first-order Markov chain with values Z ∈ {0.9725, 1.0275}. The unconditional probabilities are 0.5, and the transition probabilities are such that the probability of remaining in the current state is 2/3 (so that the expected duration of a “business cycle” is 6 years). Capital’s share of output, θ from Eq. (18), is set to 0.40, and the average annual depreciation rate, δ, is set to match the average risk-free rate of 1.3 percent. This results in δ = 0.164. The average wage rate, w, is set equal to (1 − θ)E(K)θ N −θ . The magnitude of the depreciation shocks in Eq. (20) is set so that the standard deviation of aggregate consumption growth is 3.3 percent. We choose this (as opposed to matching the variability of equity returns) because, just as in representative agent models, realistic properties for aggregate consumption are the primary disciplinary force on asset pricing models with heterogeneity. Equation (7) makes this clear. The resulting implications for the standard deviation of equity returns is reported in Table 3. The volatility of the theoretical equity premium is roughly 7 percent, 3 percentage points less than the U.S. sample value. The persistent component of hours worked, zhit , follows a unit root process with innovations governed by a four-state Markov chain, two states corresponding to an expansion and the other two a contraction. The conditional variances, σE2 and σC2 , are set to 0.0095 and 0.0467, respectively, which are taken from Storesletten, Telmer, and Yaron (2004b), Table 1, and then scaled down so that the unconditional variance matches that of the ρ = 0.92 process (a value of 0.0281). The parameters κh are chosen so as to match the PSID mean age profile in earnings. Young agents are born with zero assets and retired agents receive zero labor income. This serves as the primary motive for trade. It also results in a realistic life-cycle distribution of human to financial capital—younger agents hold most of the former, whereas older agents hold most of the latter—which, as we’ll see, plays an important role in portfolio choice. The demographic structure is calibrated to correspond to several simple properties of the U.S. work force. Agents are “born” at age 22, retire at age 65, and are dead by age 85. “Retirement” is defined as having one’s labor income drop to zero and having to finance consumption from an existing stock of assets. Retired agents comprise roughly 20 percent of the population. Risk aversion is set to target the Sharpe ratio (discussed below, in reference to Table 1), and the discount factor, β, is chosen so that the average capital to output ratio is 3.3. Table 2 illustrates the aggregate properties of our economy. The sample size for the U.S. data is chosen to be the same as that used by Mehra and Prescott (1985). As is discussed in the text, the production side of our economy is unrealistic. Aggregate consumption variability, however, matches the data, as does the variability

577


TABLE 2 Aggregate Moments—Economies with Trade Panel A: Population moments of growth rates, theoretical economy Std Dev

Autocorrelation

Correlation with output

Output

0.930

−0.307

1.000

Investment

1.214

−0.309

0.998

Consumption

0.038

−0.117

0.935

Panel B: Sample moments of growth Rates, U.S. economy, 1929–1982 Std Dev

Autocorrelation

Correlation with output

Output

0.062

0.561

1.000

Investment

0.358

0.225

0.143

Consumption

0.036

0.353

0.471

U.S. sample moments are based on annual NIPA data, 1929–1982. Theoretical moments are computed as sample averages of a long simulated time-series.

of the risky asset return (discussed above). Our model does not resolve the well-known problems with production-based asset pricing models. It is best viewed in the same way one views any endowment economy: a model with realistic properties of aggregate consumption that can be supported by some, potentially unrealistic, production technology. Alternatively, one can view our economy as featuring a linear technology— commonplace in the finance literature—where we are explicit about the implications of from the production side of the model—not very commonplace in the finance literature.

4. QUANTITATIVE RESULTS Ultimately, we are interested in the asset pricing implications of idiosyncratic labormarket risk. These implications are manifest in how the heterogeneous-agent portfolio rules interact in general equilibrium. Therefore, we begin by describing portfolio choice. Figure 1 graphs the portfolio rules in levels, and Figure 2 graphs the (age-specific average) share of stocks as a fraction of financial wealth. For expositional reasons, we show results from our benchmark economy—that which we call the “CCV economy”— as well as results from an analogous complete markets economy. We start with the complete markets economy. In this case, aggregate consumption growth equals that of individual consumption growth, cohort by cohort. However, the portfolio rules that support this allocation differ substantially across ages. Figure 2 shows that portfolio rules are hump-shaped, with the middle-aged workers holding the largest position in equities. The reasons are as follows. First, retirees have zero labor income. So, in order to replicate aggregate consumption risk, they hold diversified portfolios of stocks and bonds, the reason being that stock returns are much more volatile than aggregate consumption growth (10.9 percent versus 3.3 percent). Second,


Average Portfolio Holdings CM vs. CCV 15

CCV stocks

10 Portfolio Holding

578

CM stocks 5

0

CCV bonds CM bonds

25 20

30

40

50

60

70

80

90

Age

FIGURE 1

Quantity of bonds and stocks, by age.

the older workers hold relatively much of their financial wealth in stocks because, analogously to the well-known model of Bodie, Merton, and Samuelson (1992) (BMS) model, their labor income has bond-like properties. In BMS this means that labor income is deterministic. In our model, it just means that it’s a lot less volatile than stock returns, which bear the brunt of the depreciation-rate shocks from Eq. (20). The upshot, nevertheless, is the same; older workers hold more stocks than retirees because their labor income serves as a partial hedge against their stock portfolio. Finally, young workers have negative financial wealth and, as a result, the youngest agents in the complete market economy actually short-sell stocks! The result is a hump-shaped portfolio profile over the entire life cycle. What drives this, vis-à-vis BMS, is a combination of negative financial wealth and risky wages. Average wage rates are risky and perfectly correlated with stock returns. With negative financial wealth, it is as if the agent is levered in aggregate risk.6 Hence, by shorting stocks, 6 Consider, for example, a worker who maintains a large debt invested in bonds. If consumption equals wages net of the deterministic interest-rate payments, consumption will be more volatile than wages. With

579


Median Portfolio Shares CM vs. CCV: Avg. Across Cycles

1.5

CCV

Portfolio Shares

1

0.5 CM

0

FIGURE 2

30

40

50

Age

60

70

80

Bond and stock portfolio shares, by age.

young agents reduce their exposure to aggregate risk, thus implementing the complete markets allocation. Now consider the CCV economy. With CCV and incomplete markets, aggregate risk is no longer shared uniformly across different age cohorts. It gets shifted from the young to the old, because the young face the most human-capital risk and CCV affects only human-capital risk, not financial-capital risk. This implication is clearly borne out in Figure 1, where the stock profile of the CCV economy is, for the young, shifted to the right. All agents younger than 55, in the CCV economy, hold less stocks and more bonds than their complete markets counterparts. After age 55, the relationship is reversed. Similarly, Figure 2 shows that after 55, the share of stocks is higher in the CCV economy than in the complete markets economy. Further details are available in Storesletten, Telmer, and Yaron (2006). sufficiently large debt, the agent’s consumption will be more volatile than aggregate consumption, so she would want to reduce her exposure to aggregate risk. Consequently, the agent would short stocks as an insurance against aggregate risk: shorting stocks implies that in good (bad) times, when earnings growth is large (small), stock repayment is large (small).

580


The resulting hump-shaped pattern in equity ownership in the CCV economy (see Figure 2) is broadly consistent with U.S. data and has been the focus of recent work by Amerkis and Zeldes (2000) and Heaton and Lucas (2000). Brown (1990) shows that non-tradeable labor income can generate hump-shaped portfolio rules in age, and Amerkis and Zeldes (2000) discuss a similar phenomenon.

4.1. Asset Pricing Implications Table 3 reports the Sharpe ratio, the risk-free rate, and the first two moments of the risky return for this class of models. Each row describes a different economy. In the case of complete markets, there are no idiosyncratic shocks to earnings and only aggregate shocks are operative. In the case of “No CCV,” there are idiosyncratic shocks, but they are homoskedastic with respect to aggregate shocks. We think of our “benchmark economy” as one with a risk aversion coefficient of 8, which is close to the value that delivers the sample Sharpe ratio in the Constantinides–Duffie model (see Table 1). We also report moments for a more standard value of 3. The Sharpe ratio in our benchmark economy is about 33 percent.7 Adding countercyclical variation to idiosyncratic shocks increases the Sharpe ratio by about 6 percentage points (an increase of about one-fourth). This compares to the Constantinides–Duffie model’s value of 37 percent, slightly less than the sample estimate of 41 percent. TABLE 3 Asset Pricing Properties—Economies with Trade Risk-free rate

Risk aversion

β

K/Y

Complete markets

3

0.965

3.3

No CCV

3

0.947

3.3

Estimated CCV

3

0.948

3.3

0.0168

Complete markets

8

0.96

3.3

No CCV

8

0.809

3.3

Estimated CCV

8

0.801

3.3

Large CCV

8

0.797

3.3

σE2

Equity premium

Sharpe ratio

σC2

Mean

Mean

Std Dev

0

0

4.4

0.63

7.0

9.0

0.0114

0.0114

2.6

0.80

7.0

11.5

0.0059

2.3

0.86

6.7

12.8

0

0

4.7

2.48

9.3

26.8

0.0114

0.0114

2.6

7.6

26.1

0.0168

0.0059

1.3

2.32

7.1

32.6

0.0204

0.0023

1.6

2.51

6.6

38.0

19.8

“Models with Trade,” are described in Section 3. The calibration procedure is discussed in the text. All economies are calibrated so that aggregate consumption volatility is 3.3 percent. The “Homoskedastic Economy” is distinguished by the volatility of idiosyncratic shocks not varying with aggregate shocks. The idiosyncratic shocks are calibrated so the unit root economy has the same average volatility as that in an economy based on the estimates of Storesletten, Telmer, and Yardon (2004b). 7 We

focus on the Sharpe ratio because the volatility of the risky return varies across our different economies. As described above, this is because the shock variability, from Eq. (20), is chosen so that aggregate consumption variability remains constant across economies. In Table 3, for instance, we see that higher CCV is associated with lower volatility in equity returns. This is because, ceteris paribus, higher CCV results in higher aggregate consumption variability, which, then, necessitates us to reduce the variability of the depreciation shocks.


581

Life-cycle effects, thus, mitigate our model’s ability to account for the equity premium, vis-à-vis the Constantinides–Duffie framework. This is due to the existence of retirees who do not face labor-market risk. Nevertheless, our model still delivers a Sharpe ratio of 33 percent, which compares to the complete markets value of 27 percent and the Mehra–Prescott model’s value (with risk aversion of 8) of 23 percent. In addition, the existence of retirees also dampens the effect of CCV. CCV delivers an additional 14 percentage points to the Sharpe ratio in the Constantinides–Duffie model, but only 8 points in our model.8

4.2. Sensitivity Analysis Lowering the risk aversion to 3 obviously lowers the Sharpe ratio and also the contribution of CCV risk to the Sharpe ratio (over and above the Sharpe ratio with complete markets). However, the qualitative findings discussed above and documented in the figures (all of which pertain to economies with a risk aversion of 8) remain unchanged with a risk aversion of 3. The age portfolio profiles in Figure 2 do not change much for other parts of the wealth distribution, or change much over the business cycle. As all the figures thus far were for the average investor, Figure 3 documents portfolio holdings of the top 10 percent of the wealth distribution. The portfolio holdings in this figure show that the portfolio choice patterns underlying the model’s equity premium are relatively constant across the wealth distribution. Figures 4 and 5 provide the age profile of portfolio shares as a function of the aggregate state for both the complete market and the CCV economy, respectively. The upshot is that these portfolio profiles do not change in any significant manner across recessions and expansions, although, as expected, there is a slight increase in equity investment during expansions.

5. CONCLUSIONS This chapter asks whether idiosyncratic labor market risk is quantitatively important for asset prices. This question is not new and often the answer has been “no” (e.g., Telmer (1993), Heaton and Lucas (1994)). On the other hand, Constantinides and Duffie (1996) show that with permanent shocks to marginal utility and infinitely lived agents, the dynamic properties of idiosyncratic risk can crucially affect asset prices. A natural question to arise is whether a realistic calibration, which takes into account the fact that workers and retirees face differential labor risk, will lead to very different conclusions. The premise of our investigation is that idiosyncratic risk is naturally tied down to life-cycle effects. Workers face earnings shocks while retirees don’t, and the young have less financial assets to secure themselves against these shocks. This life-cycle structure 8 We have ignored frictions such as portfolio constraints and trading costs. It is quite likely that an important aspect of the overall impact of CCV involves its interaction with such frictions. See, for example, Alvarez and Jermann (2001), Gomes and Michaelides (2004), and Lustig (2004).


Portfolio Shares in Stocks: 90th Wealth-percentile,CCV

1

0.8 Portfolio Share in Stocks

582

0.6

0.4

0.2

0

20.2

30

40

50

60

70

80

Age

FIGURE 3

Portfolio shares for top 10 earnings percentile, by age.

of idiosyncratic labor risk provides prima facie a role for intergenerational risk sharing by which the old will help the young smooth shocks. This presents a challenge to the asset pricing story by Constantinides and Duffie (1996). Our investigation shows, however, that introducing idiosyncratic risk within a life-cycle context matters—that is, the equity premium can be sizeable. Moreover, the model delivers an age “hump-shaped” portfolio choice pattern—a feature consistent with the data. The risk premium in the model reflects both the countercyclical volatility risk emphasized by Constantinides and Duffie (1996), and a “concentration of aggregate risk” upon the middle-aged and old, alluded to by Mankiw (1986). These two risks are manifestations of two key features of the model. First, the life-cycle nature of the ratio of human capital to financial wealth and the fact only the former is affected by idiosyncratic risk clearly make such risks be more important for the young (as older agents have built a non-trivial financial wealth). It follows that the CCV effect is therefore less important for older agents, who therefore are more content to hold equity. The second feature stems from the fact that returns are more volatile than wages, making older agents, whose wealth is mostly in the form of financial wealth, be less tolerant to holding equity.

583


Median Portfolio Shares: Complete Markets, Conditional on Cycle

2 1.8 1.6

Portfolio Shares

1.4 Stocks in recessions

1.2 Stocks in booms

1 0.8 0.6 0.4 0.2 0

40

45

50

55

60

65

70

75

80

85

Age

FIGURE 4

Portfolio shares conditional on business cycle, by age.

The solid line conditions on aggregate expansions. The dashed line conditions on aggregate contractions.

These two offsetting effects imply that young agents hold zero equity, retired agents hold diversified portfolios of equity and bonds, and middle-aged agents hold levered equity, issuing bonds to both the young and the old—resulting in the hump shape age profile for equity holdings. In Constantinides, Donaldson, and Mehra (2002) (CDM), young agents are endowed with very small wealth and are barred from borrowing or shorting equity. The young therefore choose not to hold any assets. Consequently, the age profile of equity holdings is also hump-shaped. Thus, the equity premium in CDM crucially depends on the concentration of aggregate risk on the middle-aged agents. However, the reasons for why the young do not hold equity in CDM are fundamentally different from those in our model. In our model, the decision to avoid equity is driven by risk—avoidance of the countercyclical volatility risk. On the other hand, in the CDM framework, young agents view equity as a desirable investment, that is, any positive savings would have been channeled to equity. Which of these interpretations is more important, although it is quite plausible both can coexist, is something we leave for future research.

584


1.5

Median Portfolio Shares in Stocks: CCV, Conditional on Cycle

Portfolio Share in Stocks

Booms

Recessions

1

0.5

0

30

40

50

60

70

80

Age

FIGURE 5

Portfolio shares conditional on business cycle, by age.

The solid line conditions on aggregate expansions. The dashed line conditions on aggregate contractions.

References Aiyagari, S. R. Uninsured idiosyncratic risk and aggregate saving. Quarterly Journal of Economics 109 (1994): 659–684. Aiyagari, S. R., and M. Gertler. Asset returns with transactions costs and uninsured individual risk. Journal of Monetary Economics 27 (1991): 311–331. Altonji, J. G., F. Hayashi, and L. J. Kotlikoff. Risk sharing, altruism, and the factor structure of consumption. NBER Working paper number 3834 (1991). Altug, S., and R. A. Miller. Households choices in equilibrium. Econometrica 58 (1990): 543–570. Alvarez, F., and U. Jermann. Quantitative asset pricing implications of endogeneous solvency constraints. Review of Financial Studies 14 (2001): 1117–1152. Amerkis, J., and S. P. Zeldes. How do household portfolio shares vary with age? Unpublished manuscript, Columbia University (2000). Attanasio, O., and S. J. Davis. Relative wage movements and the distribution of consumption. Journal of Political Economy 104 (1996): 1227–1262. Attanasio, O., and G. Weber. Consumption growth and excess sensitivity to income: Evidence from U.S. micro data. Unpublished manuscript, Stanford University (1992).


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Balduzzi, P., and T. Yao. Testing heterogeneous-agent models: An alternative aggregation approach. Journal of Monetary Economics 54 (2007): 369–412. Benzoni, L., P. Collin-Dufresne, and R. Goldstein. Portfolio choice over the life-cycle in the presence of “trickle down” labor income. Working paper, University of Minnesota (2004). Bodie, Z., R. C. Merton, and W. F. Samuelson. Labor supply flexibility and portfolio choice in a life-cycle model. Journal of Economic Dynamics and Control 16 (1992): 427–449. Boldrin, M., L. J. Christiano, and J. D. Fisher. Habit persistence, asset returns and the business cycle. American Economic Review 91 (2001): 149–166. Brav, A., G. M. Constantinides, and C. Geczy. Asset pricing with heterogeneous consumers and limited participation: Empirical evidence. Journal of Political Economy 110 (2002): 793–824. Brown, D. P. Age clienteles induced by liquidity constraints. International Economic Review 31 (1990): 891–911. Carroll, C. Theoretical foundations of buffer stock saving. Working paper, Johns Hopkins University (2004). Cochrane, J. H. A simple test of consumption insurance. Journal of Political Economy 99 (1991): 957–976. Cogley, T. Idiosyncratic risk and the equity premium: Evidence from the consumer expenditure survey. Journal of Monetary Economics 29 (2002): 309–334. Constantinides, G. M., J. B. Donaldson, and R. Mehra. Junior can’t borrow: A new perspective on the equity premium puzzle. Quarterly Journal of Economics 117 (2002): 269–296. Constantinides, G. M., and D. Duffie. Asset pricing with heterogeneous consumers. Journal of Political Economy 104 (1996): 219–240. Deaton, A., and C. Paxson. Intertepmoral choice and inequality. Journal of Political Economy 102 (1994): 437–467. den Haan, W. Heterogeneity, aggregate uncertainty and the short term interest rate: A case study of two solution techniques. Working paper, University of California at San Diego (1994). Gomes, F., and A. Michaelides. Asset pricing with limited risk sharing and heterogeneous agents. Working paper, London Business School (2004). Graham, J. R. How big are the tax benefits of debt? Journal of Finance 55 (2000): 1901–1941. Greenwood, J., Z. Hercowitz, and P. Krusell. Long-run implications of investment-specific technological change. American Economic Review 87 (1997): 342–362. Guvenen, F. A parsimonious macroeconomic model for asset pricing: Habit formation or cross-sectional heterogeneity? Working paper, University of Rochester (2005). Heathcote, J., K. Storesletten, and G. L. Violante. Two views of inequality over the life-cycle. Journal of the European Economic Association (Papers and Proceedings) 3(2–3) (2005): 543–552. Heaton, J., and D. J. Lucas. The importance of investor heterogeneity and financial market imperfections for the behavior of asset prices. Carnegie Rochester Conference Series on Public Policy (1994). Heaton, J., and D. J. Lucas. Evaluating the effects of incomplete markets on risk sharing and asset pricing. Journal of Political Economy 104 (1996): 443–487. Heaton, J., and D. J. Lucas. Portfolio choice and asset prices; the importance of entrepreneurial risk. Journal of Finance 55 (2000): 1163–1198. Huggett, M. The risk-free rate in heterogeneous-agents, incomplete insurance economies. Journal of Economic Dynamics and Control 17 (1993): 953–969. Huggett, M. Wealth distribution in life-cycle economies. Journal of Monetary Economics 38 (1996): 469–494. Krusell, P., and A. A. Smith. Income and wealth heterogeneity, portfolio choice, and equilibrium asset returns. Macroeconomic Dynamics 1 (1997): 387–422. Lucas, D. J. Asset pricing with undiversifiable risk and short sales constraints: Deepening the equity premium puzzle. Journal of Monetary Economics 34 (1994): 325–341. Lustig, H. The market price of aggregate risk and the wealth distribution. Working paper, Stanford University (2004). Mace, B. J. Full insurance in the presence of aggregate uncertainty. Journal of Political Economy 99 (1991): 928–956. Mankiw, N. G. The equity premium and the concentration of aggregate shocks. Journal of Financial Economics 17 (1986): 211–219.

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Chapter 14 • Asset Prices and Intergenerational Risk Sharing: The Role of Idiosyncratic Earnings Shocks Marcet, A., and K. J. Singleton. Equilibrium assets prices and savings of heterogeneous agents in the presence of portfolio constraints. Macroeconomic Dynamics 3 (1999): 243–277. Mehra, R., and E. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15 (1985): 145–161. Olovsson, C. Social security and the equity premium puzzle. Working paper, IIES (2004). Ramchand, L. Asset pricing in open economies with incomplete markets: Implications for foreign currency returns. Journal of International Money and Finance 18 (1999): 871–890. R´ıos-Rull, J. V. On the quantitative importance of market completeness. Journal of Monetary Economics 34 (1994): 463–496. Sarkissian, S. Incomplete consumption risk sharing and currency risk premiums. Review of Financial Studies 16(3) (2003): 983–1005. Storesletten, K. Sustaining fiscal policy through immigration. Journal of Political Economy 108 (2000): 300–323. Storesletten, K., C. I. Telmer, and A. Yaron. Consumption and risk sharing over the life cycle. Journal of Monetary Economics 59(3) (2004a), 609–633. Storesletten, K., C. I. Telmer, and A. Yaron. Cyclical dynamics in idiosyncratic labor market risk. Journal of Political Economy 112(3) (2004b): 695–717. Storesletten, K., C. I. Telmer, and A. Yaron. Asset pricing with idiosyncratic risk and overlapping generations, (in press) Review of Economic Dynamics (2006). Telmer, C. I. Asset pricing puzzles and incomplete markets. Journal of Finance 48 (1993): 1803–1832. Weil, P. Equilibrium asset prices with undiversifiable labor income risk. Journal of Economic Dynamics and Control 16 (1992): 769–790. Zhang, H. Endogenous borrowing constraints with incomplete markets. Journal of Finance 52 (1997): 2187–2209.

587


A. CALIBRATION APPENDIX This appendix first describes the calibration of the no-trade (Constantinides and Duffie (1996)) economies in Section 2 and Table 1 and then goes on to describe the calibration of the economies with trade, presented in Section 3 and Table 2. It also demonstrates the sense in which our specification for countercyclical volatility—heteroskedasticity in the innovations to the idiosyncratic component of log income—is consistent with the approach used by previous authors (e.g., Heaton and Lucas (1996), Constantinides and Duffie (1996)). In each case, the cross-sectional variance that matters turns out to be the variance of the change in the log of an individual’s share of income and/or consumption.

A.1. Calibration of No-Trade Economies Aggregate consumption growth follows an i.i.d. two-state Markov chain, with a mean growth of 1.8 percent and standard deviation of 3.3 percent. This is essentially the process used in Mehra and Prescott (1985) with slightly more conservative volatility. The Constantinides and Duffie (1996) model is then “calibrated” via a re-interpretation of the preference parameters of the Mehra and Prescott (1985) representative agent. Recall that we use β and γ to denote an individual agent’s utility discount factor and risk aversion parameters, respectively. Constantinides and Duffie (1996) construct a representative agent (their Eq. (16)) whose rate of time preference and coefficient of relative risk aversion are (using our notation) − log β ∗ = − log(β) −

γ(γ + 1) a 2

(22)

and γ∗ = γ −

γ(γ + 1) b, 2

(23)

respectively. In these formulas, the parameters a and b relate the cross-sectional variance 2 , using in the change of the log of individual i’s share of aggregate consumption (yt+1 Constantinides–Duffie’s notation) to the growth rate of aggregate consumption: ci,t+1 /ct+1 ct+1 . = a + b log Var log ct cit /ct

(24)

All that we require, therefore, are the numerical values for a and b that are implied by our PSID-based estimates in Table 1 of Storesletten, Telmer, and Yaron (2004b). Our estimates are based on income, yit . Because the Constantinides–Duffie model is autarkic, we can interpret these estimates as pertaining to individual consumption, cit . Balduzzi and Yao (2005), Brav, Constantinides, and Geczy (2002), and Cogley (2002) take the alternative route and use microeconomic consumption data. While their results are generally supportive of the model, they each point out serious data problems associated with using consumption data. Income data is advantageous is this sense.

588


In addition, our objective is just as much relative as it is absolute. That is, consumption is endogenous in the model of Section 3, driven by risk sharing behavior and the exogenous process for idiosyncratic income risk. What Table 1 asks is, “what would the Constantinides–Duffie economy look like, were its agents to be endowed with idiosyncratic risk of a similar magnitude?” Also, “how does our model measure up, in spite of its non-degenerate (and more realistic) risk sharing technology?” Using income data seems appropriate in this context. Accordingly, for the remainder of this appendix, we set cit = yit . We need to establish the relationship between our specification for idiosyncratic shocks and the log shares of aggregate consumption in Eq. (24). Denote an individual i’s share at time t as γit , so that log γit ≡ log cit − log E˜ t cit , where the notation E˜ t (·) denotes the cross-sectional mean at date t, so that E˜ t cit is date t, per capita aggregate consumption. The empirical specification in Storesletten, Telmer, and Yaron (2004b) identifies an idiosyncratic shock as the residual from a log regression with year-dummy variables: zit = log cit − E˜ t log cit , which have a cross-sectional mean of zero, by construction, and a sample mean of zero, by least squares. The difference between our specification and the log-share specification is, therefore, log γit − zit = E˜ t log cit − log E˜ t cit = E˜ t log γit − log E˜ t γit . The share, γit , is defined so that its cross-sectional mean is always one. The second term is therefore zero. For the first term, note that in both our economy and the statistical model underlying our estimates, the cross-sectional distribution is lognormal, conditional on knowledge of current and past aggregate shocks. If some random variable x is lognormal and E(x) = 1, then E(log x) = −Var(log x)/2. As a result, 1 log γit − zit = − V˜ t (log γit ), 2 where V˜ t denotes the cross-sectional variance operator. Because lives are finite in our model, and because we interpret data as being generated by finite processes, this crosssectional variance will always be well defined, irrespective of whether or not the shocks are unit root processes.

589


The quantity of interest in Eq. (24) can now be written as log

ci,t+1 /ct+1 ≡ log γi,t+1 − log γit cit /ct 1 = zi,t+1 − zit − V˜ t+1 (log γi,t+1 ) − V˜ t (log γit ) . 2

(25)

The term in parentheses—the difference in the variances—does not vary in the cross section. Consequently, application of the cross-sectional variance operator to both sides of Eq. (25) implies ci,t+1 /ct+1 ˜ = V˜ t+1 zi,t+1 − zit . Vt+1 log cit /ct The process underlying our estimates is zi,t+1 − zit = (1 − ρ)zit + ηi,t+1 , where the variance of ηi,t+1 depends on the aggregate shock. For values of ρ close to one, the variance of changes in zit is approximately equal to the variance of ηi,t+1 . The left side of Eq. (24) is, therefore, approximately equal to the variance of innovations, ηi,t+1 : ci,t+1 /ct+1 ≈ V˜ t+1 ηi,t+1 . V˜ t+1 log cit /ct For unit root shocks—which we assume for most of Section 3—this holds exactly. The estimates of σE and σC in Storesletten, Telmer, and Yaron (2004b), Table 1, are therefore sufficient to calibrate the Constantinides–Duffie model. All that remains is to map our estimates into numerical values for a and b from Eq. (24). Since aggregate consumption growth is calibrated to be an i.i.d. process with a mean and standard deviation of 1.8 percent and 3.3 percent, respectively, aggregate consumption growth, the variable on the right-hand side of Eq. (24), takes on only two values, 5.1 percent and −1.5 percent. Computing the parameters a and b then simply involves two linear equations: σE2 = a + 0.051b, σC2 = a − 0.015b. Storesletten, Telmer, and Yaron’s (2004b) estimates are σE2 = 0.0156 and σC2 = 0.0445. These estimates, however, are associated with ρ = 0.952. For our unit root economies, we scale them down so as to maintain the same average unconditional variance (across age). This results in σE2 = 0.0059 and σC2 = 0.0168. The resulting values for a and b are a = 0.0143 and b = −0.1652.

590


B. ASSET PRICING Following the Euler Eq. (7), and recalling that P is the probability of remaining in a given state, the price of the risky security is βP (1 + z)−ˆγ if t is a boom, pt = (26) β(1 − P )(1 + z)−ˆγ if t is recession. Similarly, the price of the bond is qt = Et βMt+1 βP (1 + z)−ˆγ + (1 − P )(1 − z)−ˆγ if t is boom, = if t is recession. β (1 − P )(1 + z)−ˆγ + P (1 − z)−ˆγ

(27)

It then follows that the unconditional bond price is simply E (qt ) =

1 β (1 + z)−ˆγ + (1 − z)−ˆγ . 2

(28)

Consequently, the realized return on the excess risky asset, rt+1 , can be expressed as 1 − 1 if t + 1 is boom, rt+1 = pj 1 qj (29) − qj if t + 1 is recession. where j ∈ {boom, recession} denotes the aggregate state in period t. Given the expressions in Eqs. (27) and (29), it is easy to solve for (10) in the case of i.i.d. aggregate shocks (P = 1/2).

591

Darrell Duffie

Discussion: Asset Prices and Intergenerational Risk Sharing: The Role of Idiosyncratic Earnings Shocks Darrell Duffie Graduate School of Business, Stanford University

The chapter is an extremely clever exploration of the implications of uninsurable labor income risk for intergenerational risk sharing and for conditional mean asset returns. As usual, the holy grail is a realistic explanation of the equity premium puzzle pointed out by Mehra and Prescott (1985). The path explored in this paper is based on incomplete markets and persistent idiosyncratic labor income shocks, along the lines of Constantinides and Duffie (1996) (CD). In this chapter, however, the authors relax the CD assumption that all investors have the same sorts of labor income shocks. In particular, young agents tend to have more exposure than older agents to the risk of changes in labor income. Older agents, who tend to have accumulated significant financial wealth, are more exposed to financial risk. In this limited sense then, the spirit of this chapter might be described as “Constantinides and Duffie (1996) meets Constantinides, Donaldson, and Mehra (2002).” The reader need not be scared away by the potential for an ugly result of this marriage. Storesletten et al. (STY) have accomplished a great deal within the confines of a surprisingly elegant model. It is especially impressive that the authors are able to solve the model by allowing trade, through the introduction of a production sector. Agents are endowed with idiosyncratic amounts of labor, which is hired by firms as a factor of production. The model is then solved numerically, as opposed to the explicit solution of the CD model. As the authors of this chapter point out, the main thrust of the paper could be handled by explicit calculations within the CD no-trade model, extended only by calibration of the idiosyncratic labor income risk to the seniority of the agent. It is worth mentioning that the results of the CD model do not require no trade; the consumption levels of the agents that is taken as given in that model could in principle be the end result of trading the available securities. For example, STY report the interesting fact that the cross-sectional standard deviation of U.S. consumption is

592


35 percent smaller than that of U.S. non-financial earnings. This is a weak defense of the CD model, of course, in that the resulting distribution of consumption that is taken as given in that model is highly stylized and symmetric. The STY model is preferred on grounds of realism. In any case, STY report at the end of Section 3 that trade is not the key to explaining the risk premium, and, of course, why should it be? The thrust of the STY results is clear from their Figure 1, which shows that as investors age they absorb a greater and greater fraction of the total across agents of the covariances between the investors’ consumption growths and total market returns. Integenerational risk sharing is not efficient, because the markets that would allow more risk sharing are missing. The older, wealthier agents demand a significant risk premium to bear their high shares of stock market risk, because young workers already have plenty of labor income risk to deal with. It was somewhat surprising to me that, while the STY model is successful at capturing the implications of incomplete markets and idiosyncratic labor income risk for life-cycle portfolio behavior, the STY model is not a leap forward on the equity premium puzzle itself. The authors show that, at the same levels of risk aversion, the calibrated STY model actually shows lower Sharpe ratios for the equity market than those of the similarly calibrated CD model. I still prefer the STY model on grounds of realism. As a footnote, I will take this opportunity to point out that the appeal to the law of large numbers for a continuum of i.i.d. random variables in the CD and STY models, indeed the entire related literature, can now be improved by citing the recent results of Yeneng Sun.1 While the appeal in CD (1996) to an earlier foundation for the exact law of large numbers for a continuum of random variables is formally correct, that earlier foundation has some undesirable features, as explained by Sun, and can now be avoided.

References Constantinides, G. M., J. B. Donaldson, and R. Mehra. Junior can’t borrow: A new perspective on the equity premium puzzle. Quarterly Journal of Economics 117 (2002): 269–296. Constantinides, G. M., and D. Duffie. Asset pricing with heterogeneous consumers. Journal of Political Economy 104 (1996): 219–240. Mehra, R., and E. Prescott. The equity premium: A puzzle. Journal of Monetary Economics 15 (1985): 145–161. Storesletten, K., C. I. Telmer, and A. Yaron. Consumption and risk sharing over the life cycle. Journal of Monetary Economics 59(3) (2004a), 609–633. Storesletten, K., C. I. Telmer, and A. Yaron. Cyclical dynamics in idiosyncratic labor market risk. Journal of Political Economy 112(3) (2004b): 695–717. Storesletten, K., C. I. Telmer, and A. Yaron. Asset pricing with idiosyncratic risk and overlapping generations, (in press) Review of Economic Dynamics (2006). Sun, Y. The exact law of large numbers via Fubini extension and characterization of insurable risks. Journal of Economic Theory 12 (2006): 31–69.

1 See

Sun (2006).

Index

A Abel, A. 14, 16n, 24n, 39n, 56, 117, 121n, 122, 124, 126, 127, 147, 149, 152, 160, 161, 163, 164n, 277, 333n, 344, 403, 436, 464 “Accessibility” theory of framing 226 Agent heterogeneity 75–78 and intermediation cost 113–114 borrowing and lending 113 idiosyncratic risk 113 Aggregate consumption 170, 177, 196, 210, 333 growth rate 346, 352, 403, 537, 541, 551, 562, 569, 587 Aggregate dividends 170, 189, 249, 403, 406, 407, 505 Aggregate risk 76, 113, 564, 579, 582 Aggregate uncertainty distribution risk, addition to 432–436 Aggregation, in preference–based theories 41–44 Ait-Sahalia, Y. 72, 73, 313, 354n, 379 Aiyagari, S.R. 25n, 40n, 86n, 111, 333n, 352n, 364, 551, 559, 568 Akiyama, R. 510 Aleotti, A. 509 Allais, M. 47n Allen, F. 406 Altonji, J.G. 350n, 568n Altug, S. 568n Alvarez, F. 25n, 40n, 63n, 297, 333n, 370, 371, 559, 568, 581n Amerkis, J. 580 Ang, A. 221n, 249, 349 Angeletos, G.-M. 561 Annaert, J. 507 Arbulu, P. 508, 509 Arnott, R.D. 474, 493, 498 Arrow–Debreu models 35, 265, 569, 570, 571

Arrow–Pratt relative risk aversion (RRA) 339 Asset pricing 590 economies with trade 580 and equilibrium 172–173 implications 180–184, 580–581 and intergenerational risk sharing Constantinides–Duffie model 569–573 discussion 591–592 idiosyncratic earnings shocks role 565, 577–581 models 381 no-trade economies 572 theory 12–14 life cycle savings 88 and returns 30–34 Asset returns and equity premium alternative preference structures 41 features 147 Atkeson, A. 362 Attanasio, O. 25n, 40n, 313, 333n, 346, 350n, 561, 568n Auerbach, A.J. 14 Australia 7, 480, 497, 507 Australian Graduate School of Management (AGSM) 507 Autarky 568n Average transactions cost 550 Avramov, D. 377 Azeredo, F. xx, 1, 6, 23n, 37, 46n, 52n, 74

B Babbs, S. 389, 390 Backus, D.K. 47n, 50n, 62n, 124, 328 Bad times 258, 267, 276, 579n

593

594

Index Bakshi, G. 333n, 377, 379, 381, 382, 391, 403, 406, 410, 411, 413, 414 Balvers, R. 237 Balduzzi, P. 568, 587 Ball, R. 251n, 507 Banks, J. 25n, 40n, 104, 127, 313, 333n, 346, 561 Bansal, R. 24n, 25n, 37, 39n, 40n, 50, 51, 167, 168, 169, 172, 173, 175, 176, 177, 178, 179, 180, 183, 184, 185, 186, 187, 188, 189, 190, 191, 194, 195, 196, 198, 283, 284, 285, 333n, 343, 431 Banz, R.W. 251n Barberis, N. 40n, 62, 63n, 67, 68, 69, 70, 71, 75n, 77, 78, 199, 202, 203, 204, 208, 209, 210, 211, 212, 213, 214, 215, 217, 224, 225, 226, 230, 379 Barclays Capital 471 Barrillas, F. 40 Barro, R. 14, 39, 82, 83, 83n, 84, 85, 233, 333n Barsky, R. 382 Basak, S. 25n, 40n Basu, S. 251n Becker, G.S. 14 Behavioral models, of preference structure 61 generalized disappointment aversion 64–67 happiness maintenance preferences 62–64 prospect theories 67–71 Bekaert, G. 161, 221n, 222n, 349 Belgium 497, 507, 514 Belo, F. 237, 293 Benartzi, S. 24n, 39n, 67n, 202, 207, 208, 210, 215, 216, 333n, 379 Benavides, F.G. 512 Benchmark level of consumption aggregate movements 161–162 canonical asset 126 price 127–129 rate of return 129–131 closed-form expressions second moments 138–142 dividend-price ratio and rate of return for stocks 142–146 linear approximations 137–138 accuracy 153–156 equity premium 133 risk premium 131–132 term premium 132–133 log-normality 134–136 equity premium 133 risk premium 131–132 term premium 132–133

micro-economics foundations and aggregation 163 implication 162–163 preferences 120–126 rate of return special cases 146–153 Benefit pension plans 113 Bentley, S. xx Benzoni, L. 569 Bergstrom, A. 389 Berk, J.B. 298 Bernstein, P. 474, 498 Bernstein, W.J. 493 Bertaut, C. 224, 345n. 347n Bewley, T.F. 25n, 40n, 333n, 350 Bianchi, B. 510 Billingsley, P. 29 Biscaini, A.M. 510 Bittlingmayer, G. 509 Bivariate model for aggregate consumption and aggregate corporate earnings 195 Black–Scholes option pricing model 531 Bliss, R.R. 247, 278 Blume, M.E. 333n, 345n Bodie, Z. 347n, 472, 473, 578 Boldrin, M. 24n, 39n, 78, 80, 81, 295, 296, 297, 333n, 443, 454, 455, 456, 460, 575n Bond and stock portfolio shares 579 Bond prices 43, 351, 353, 354, 381, 383, 429, 541 compensation 396 Bonomo, M. 64n Book-to-market ratio 303 Borrowing constraints 87, 89, 110–113, 331 bond yield 113 consumption 112 economy, versions of 111 equity 113 and short sales constraints 550–551 Borrowing costs 547, 549–550 Boston money market 525 Bowers, J. 507 Bowland, C. 519 Bowley, A.L. 509, 518 Brainard, W.C. 280 Braun, P.A. 333n, 342 Brav, A. 25n, 40n, 311, 312, 333n, 346, 353, 354, 368, 561, 568, 587 Brealey, R.A. 473 Breeden, D. 25n, 286, 326, 332n, 379, 532 Breeden–Lucas consumption CAPM 369 Brennan, M.J. 304

595

Index Brinson, G.P. 524, 525 Brock, W.A. 14, 35 Brokerage charges 105 Brown, D.P. 580 Brown, P. 507 Brown, S. 25n, 39n, 85, 110, 333n, 490, 516, 533 Brownian motion process 75, 92, 385, 405 Buckley, K.A.H. 508 Buelens, F. 507 Buraschi, A. 391 Business cycle 417 and labor market 421–423 model 51, 296, 445, 446 modern characterization 418–420 growth component 422 statistical representation 418–419 stylized facts of U.S. 420 time variation expected returns, correlation 246–257 Business owners 552 vs. non-owners portfolio characteristics 553

C Cagetti, M. 333n Calibration 340, 412, 429–430, 576–577 of Constantinides–Duffie Economy 570–571 estimation results 404–405, 412 of model 412 of no-trade economies 587–589 Calvet, L. 561 Campanale, C. 81 Campbell, J. 24n, 25n, 39n, 40n, 52, 58, 59, 60, 61, 63, 66, 75, 77, 90, 91, 121n, 122, 151, 161, 225, 232, 237, 246, 247, 248, 250, 261, 265, 269, 273, 275, 276, 277, 278, 285, 295, 296, 298, 305, 306, 333, 341, 342, 344, 345, 403, 431, 436, 499, 562 Canada 475, 479, 484, 507–508 Canonical asset 126, 156 closed-form solutions 138–142 price 127–129 rate of return 129–131 Cao, C. 377 Capital Asset Pricing Model (CAPM) 12, 241, 244, 258, 260, 280, 289, 300, 302, 306–307, 325, 458, 516, 521, 528, 532–533 Capital-based contingent commodities 449 Carroll, C. 162, 572 Cash-flow beta model 186–188, 306 performance 189–190

Cash flow process 411 of equity premium 381–382 Cash flow risk 186, 189–190, 380, 392–396, 403, 409 and equity premium puzzle 377 Cecchetti, S. 85, 150, 333n, 342, 403, 464 Center for Research in Security Prices (CRSP) 4, 477, 513, 522, 552 Central Bureau of Statistics (CBS) 510 Chan, Y.L. 75, 162n Chapman, D. 75, 76, 77, 162n Chen, P. 493, 498 Chen, R. 389, 390 Chen, X. 278, 344 Chen, Z. 333n, 377, 379, 380, 381, 382, 391, 403, 406, 410, 411, 413, 414 Chetty, R. 280 Chew, S.H. 47n Christiano, L. 24n, 39n, 295, 296, 297, 333n, 429, 454, 455, 456, 460, 575n Christiansen, J. 508 Cicchetti, C. 205 Ciocca, P.L. 510 Clark, D. xx Clark, J.B. 517 Cochrane, J. 16, 24n, 25n, 39n, 40n, 50n, 51n, 52, 58, 59, 60, 61, 63n, 66, 75, 77, 90, 91, 121n, 122, 151, 161, 225, 237, 246, 247, 248, 249, 251, 256, 261, 266, 267, 275, 276, 280, 285, 286, 288, 290, 291, 292, 293, 295, 298, 309, 314, 331, 333, 341, 342, 343n, 344, 345, 350, 367n, 379, 403, 436, 457n, 469, 562, 568n Cogley, T. 311, 354n, 368, 568, 571n, 587 Cohen, A. 205 Cointegration parameter 190 Cole, A.C. 518 Cole, A.H. 3 Cole, H.L. 364 Coleman, J.W. 25n, 40n, 104, 333n Collin-Dufresne, P. 569 Common Stock Indices 519 Common Stocks as Long Term Investments 518 Complete markets 86, 93, 231, 278, 350, 362, 371–372, 573, 579, 581 equilibrium 363–364 equity premium puzzle 364 Conditional labor income model 306–307 Conditional mean distribution shock 438 and risk aversion, changes 438–439 Conditional one-factor model 286 Constant relative risk aversion (CRRA) 16, 30, 41, 50, 52, 410, 537, 548, 569, 570

596

Index features aggregation 41–44 equilibrium return stationarity 41 separating risk and time preferences 46–52 separating time and risk preferences 44–46 time-consistent planning 44 preferences 90n Constant shock distribution case 554 Constantinides, C. 530 Constantinides, G. xxi, 1, 15n, 24n, 25n, 37, 39n, 40, 43n, 55, 58, 78, 86, 87, 88n, 89, 90n, 91, 111, 112, 121n, 122n, 160, 237, 275, 277, 279, 307–308, 311, 331, 332n, 333, 338, 339, 340, 341, 342, 344, 345, 346, 347, 348, 349, 350, 351, 353, 354, 360, 363, 366, 368, 379, 382, 452, 538, 544, 545, 555, 561, 562, 566, 567, 568, 569, 570, 572, 581, 582, 583, 587, 591 Constantinides–Duffie model 307–308, 581, 588, 589 analytical example 569–573 calibration 570–571 implications 571–573 for entrepreneurs 561 and idiosyncratic risk 307–311 life cycle incorporation 573–577 calibration 576–577 mechanism 366–368, 374–375 evidence 368 Consumption benchmark levels closed-form results 117, 121–123 and factor models 286 Consumption-based asset pricing model 186, 241, 257–258, 537 Consumption-benchmark ratio 153 Consumption-beta risk premium 336 Consumption data 5–6 Consumption Expenditure Survey (CEX) 346 Consumption externalities 122 of keeping/catching up with Joneses 120 with leverage 150 in distorted beliefs 152–153 without leverage in rational expectations 149–150 Consumption growth process 51 Consumption growth rate, serial correlation in 23–25 Consumption models 267 intertemporal substitution 270 power utility 267–270 risk aversion 269–270 utility functions 270–273

Consumption shock 195 Consumption-to-wealth ratio 63 Consumption volatility 542 Contingent claims and macro markets value 184 Cooley, T.F. 420, 429, 447 Cooper, I. 249 Corporate equity 36, 102, 110 versus T-bills 102 value 109 Cotula, F. 510 Counter cyclical volatility (CCV) 365, 368, 567, 571, 577, 579–580, 581, 582 Countercyclical risk aversion 52, 60, 63, 75, 77 Cowles, A. 4, 518n, 519, 525 Cox, J.C. 14, 531 Craine, R. 249 Credit Suisse First Boston 471 Cross-sectional implications, in asset portfolios 185–190 cash flows in consumption risks measurement 186–188 dividends 188–189 performance 189–190 Cuban Missile Crisis 83, 494 Cuoco, D. 25n, 40n Curcuru, S. 552, 553 Cuyvers, L. 507 Cyclical shock distribution case 554 D Da Silva, A. 89, 90 Daniel, K. 25n, 40n, 197, 254, 280, 333n Danthine, J.-P. 16, 25n, 40n, 52, 53, 76, 78, 85, 90, 333n, 415, 420, 422, 423, 452, 453, 454, 455, 456 Data sources 3–6 1802–1871 3–4 1871–1926 4 1926-present 4–5 consumption data 5–6 for DMS database 507–514 DataStream International Inc. 388 Davis, J.L. 494 Davis, S.J. 88, 112, 350n, 568n De Bondt, W.F.M. 251 De Ceuster, J. 507 De Santis, M. 308, 333n DeAngelo, H. 406 DeAngelo, L. 406 Deaton, A. 368, 568n Decision-theoretic portfolio choice model 542 Dekel, E. 47n

597

Index DeLong, B.J. 40n, 382 Den Haan, W.J. 360, 568 Denmark 497n, 508 Devos, G. 507 Dimmock, S. 227 Dimson, E. 7, 40n, 467, 469, 470, 472, 474, 475, 477, 483, 494, 497, 498n, 500, 507, 513, 523 Disappointment-averse preferences see Generalized disappointment aversion (GDA) Disaster events 39 and survivorship bias 81–86 Discount factor 239, 240, 272, 324 Discounted payoffs 239 Discounting process, of equity premium 382–383 Discounting risk 377 and equity premium puzzle 377 Distorted beliefs of rates of return 151, 164 consumption externalities with leverage 152–153 standard utility function without leverage 152 Distribution risk and equity returns 415 aggregate uncertainty adding 432–436 business cycle 418–420 comparative dynamics and welfare assessment 436–443 economy with distribution risk 430–432 labor market 421–423 model economy 423 equilibrium 427–429 firm 425–426 numerical procedures and calibration 429–430 shareholders 424–425 workers 423–424 related literature 452–459 robustness 446–448 sharing mechanism, alternative interpretation of 448–452 technology-driven variation, in factor shares 443–446 Distribution tax rate 108 Dittmar, R.F. 168, 169, 185, 186, 188, 189, 190, 191, 237, 283, 343 Dividend-payout ratio 384, 394 Dividend-price ratio 120, 245, 275 and rate of return correlation of stocks 142–146 Dividend volatility 66, 71, 432, 542, 554 Dividends and consumption correlation 163–165, 190 Division of labor 255 mimicking portfolio theorem 243–244 DMS bond indexes 479

DMS global database composition and start date 475–477 methodology and guiding principles 477–479 DMS inflation rates 479 Donaldson, J. xxi, 1, 15n, 16, 25n, 37, 40n, 44, 47n, 52, 53, 76, 78, 85, 87, 89, 90, 91n, 101, 102, 111, 333n, 346, 348, 349, 379, 415, 420, 422, 423, 452, 453, 454, 455, 456, 538, 555, 567, 583, 591 Dow, C.H. 518 Dow, J.P. 457n Dow Jones Industrial Average 477n Dubin, J. 205 Duesenberry, J. 158, 337, 338 Duffee, G. 386 Duffie, D. 25n, 40n, 88n, 112, 279, 310, 333n, 350, 351, 353, 354, 366, 452, 530, 538, 544, 545, 562, 566, 568, 569, 570, 572, 581, 582, 587, 591 Dunlop, J. 432 Durables vs. non-durables 274–275 Dynamics of equity premium 385–387 Dynan, K. 162 E Earnings per share (EPS) growth 380, 387–388, 392, 405, 412 Economic determinants, of equity premium 381–387 cash flow process 381–382 discounting process 382–383 dynamics of equity premium 385–387 market portfolio dynamics 383–385 Economies with trade aggregate moments 577 asset pricing properties 580 Efficient Method of Moments (EMM) technique 169, 184 Eichenbaum, M. 273 Eichholtz, P. 510 Einav, L. 205 Elasticity of intertemporal substitution (EIS) 45, 47, 48, 65, 211, 274, 284 Elasticity of substitution in consumption 336, 341 Empirical equity premium 39 Engstrom, E. 161 Entrepreneurial income 552–554 Entrepreneurial risk 560–562 Epstein, L. 16n, 24n, 39, 46, 47, 48, 49, 50, 51, 52, 53, 54n, 65, 168, 170, 173, 195, 197, 203, 211, 216, 217, 218, 221, 222, 225n, 271, 272, 282, 322, 333n, 337n, 341, 344, 379

598

Index Epstein-Zin preference 49–50, 282–283 consumption dynamics 284 CRRA and EIS, variation 52–55 discount factor 323–324 market return 321–323 utility index 285, 322 Equilibrium asset pricing model 542 Equilibrium market phenomena quantitative predictions 232–233 Equilibrium return stationarity 41 Equity gains/losses 67 Equity market portfolio 168 Equity markets cross-sectional implications 185–190 intertemporal elasticity of substitution (IES) 167 long-run risks and risk compensation 170–184, 194–198 low-frequency component, in consumption 194–195 preferences 195–197 returns and long-run cash flows 197–198 volatility risks 175–176, 182–183 Equity premium puzzle 24, 51, 57, 158, 159, 242, 258, 261–266, 258, 364, 379, 403, 405–407, 409, 469, 523, 535, 558 analysis 29–34 and cash flow risk, discounting risk 377 complete markets 362–363 equilibrium 363–364 discussion calibration 412 model 410–411 two-stage procedure 412–414 dual puzzle 332 environment 361 future 266–267 habit persistence 337–345 heterogeneity 539 idiosyncratic income shocks 349–355 incomplete markets 349–355 limited stock market participation and per capita consumption 345–349 loss aversion and narrow framing 199, 203–207 missing markets 364–370 and state-dependent solvency constraints 370–372 preferences and endowments 361–362 trading frictions 538, 547 undiversified risk 539, 556 Equity premium-risk-free rate puzzle 259–260 Equity protection puzzle 230–232

Equity return 171n, 476, 526 and distribution risks 415 Equity return data 1802–1871 3 1871–1926 4 1926-present 4 Estimates, of equity premium 6–9 Estrella, A. 249 Euler equation 172–173, 332, 342, 351, 367, 540, 568, 570 Ex ante measurement, of equity risk premium 526 Expectations hypothesis 246, 247 Expected earnings growth 380, 382, 384, 392 Expected excess return see Risk premia Expected returns 527, 555 variation across assets 245–246 variation over time 244–245 Expected utility 47, 62, 76, 92, 203, 272, 334, 540 Expected utility (EU) agent vs. PT agent 230–232 External habit persistence 338

F Factor shares, technology-driven variations in 443–446 Falato, A. 62, 63, 64 Fama, E. 147, 150, 241, 243, 246, 247, 248, 249, 251, 252, 253, 255, 278, 286, 288, 289, 302, 327, 407, 458, 459, 474, 493, 494, 499 Fama–French 25 portfolios 190, 254–255, 274, 281, 287, 288, 304 Fama–French pricing factors 553 Fama–French three-factor model 244, 254–256, 287, 303 Fernandez, P. 474 Ferson, W. 24n, 39n, 275, 305, 332n, 333n, 342, 344, 379 Financial markets and real economy consumption models 267–268 power utility 267–270 division of labor 244 equity premium 257–260 future 266–267 puzzle 261–266 and factor models 286–290 facts cross section of returns 251–257 return forecasts 246–251 variation across assets 245–246 variation over time 244–245 finance 242–243

599

Index general equilibrium business cycle models 295–297 challenges 299–301 cross section of returns 298–299 intangible capital 297–298 market returns and macroeconomics 294–295 idiosyncratic risk 302, 307–314 labor income 302–307 in macroeconomics 242 mimicking portfolio theorem 243–244 new utility functions 270–273 Epstein and Zin 282–283 habits 275–278 long horizon 280–282 non-separabilities across goods 273–275 related models 278–280 production-based asset pricing 290, 290–294 cross-sectional tests 292 time-series tests 291–292 risk premia 239–242 Financial stylized facts 421, 446, 452 Finn, F.J. 507 Firer, C. 511 Firm, in model economy 425–426 Fisher, I. 519 Fisher, J. 24n, 39n, 195, 295, 296, 333n, 454, 455, 456, 460, 575n Fisher, L. 20, 520 Fixed costs 548 Forecasting variables 249–250 statistics 250–251 Foreign exchange forecast 247 France 472, 497, 508–509 Freeman, M.C. 545 French, K. 147, 150, 241, 247, 248, 249, 251, 252, 253, 255, 256, 286, 288, 289, 302, 326, 327, 407, 458n, 459, 474, 493, 494, 499 Frennberg, P. 512 Frickey, E. 518 Friend, I. 333n Fujino, S. 510

G Gabaix, X. 25n, 40n, 230, 233, 333n Gala, V.D. 37, 299, 331, 403 Gali, J. 122, 124 Gallais-Hamonno, G. 508, 509 Gallant, A.R. 342 Gallant, R.A. 168, 169, 172, 184 Garcia, R. 64n Garduno, H. 237

GDP time series for U.S. economy 419 Geczy, C. 25n, 40n, 311, 312, 333n, 346, 353, 354, 368, 561, 568, 587 Gemis, M. 507 General equilibrium model asset pricing 295–297 business cycle models 296–297 challenges 299–301 cross section of returns 298–299 intangible capital 297–298 market returns and macroeconomics 294–296 Generalized disappointment aversion (GDA) 64–67 Generalized expected utility (GEU) 46 Geometric risk premium 184 German 85, 482, 483, 509, 533 Gertler, M. 25n, 40n, 86n, 111, 333n, 352n, 364, 551, 568 Geweke, J. 92n Giannikos, C. 53, 74, 89n, 90 Gibbons, M. 246, 285, 379 Gielen, G. 478, 509 Gneezy, U. 206n Goetzmann, W. 25n, 39n, 40n, 85, 110, 251, 333n, 432n, 470, 471, 472, 475, 476, 477, 490, 492, 499, 513, 515, 516, 523, 524, 533 Goetzmann–Ibbotson, comments on 532 Gold standard period 8, 9, 105 Goldstein, D.G. 231 Goldstein, R. 569 Gollin, D. 429 Gomes, F. 42n, 91, 158, 162, 203n, 555, 568, 581n Gomes, J.F. 298, 457 Gonzalez, A.S. 512 Gordon, S. 52 Gourio, F. 237, 299 Government bond data 522 Goyal, A. 251, 473, 499 Graham, J.R. 474, 572 Great Depression 82, 84, 110, 482, 494 Green, R.C. 298 Greenwood, J. 575n Gregory, A.W. 124 Grenadier, S. 161 Grossman, R.S. 476 Grossman, S. 5, 18n, 36, 262, 263, 264, 310, 326, 327, 332n, 366n Grullon, G. 496 Guidolin, M. 333n Guirguis, H. 53 Guiso, L. 345n Gul, F. 47n, 62, 64, 65, 220, 223, 349

600

Index Guo, H. 425 Guvenen, F. 76, 78, 90, 91, 333n, 348, 455, 457, 458, 546, 562, 568 H Habit formation 62, 78–79, 160, 460 axiomatic treatment 61 Campbell and Cochrane mechanism 58–61 concept 55–57 in consumption 121–122 equity premium, relationship to 57–58 Habit persistence 295, 331, 337–345 marginal rate of substitution 276–277 model 275–278 Haliassos, M. 224, 345n, 347n Hall, R.E. 267, 269, 284, 297, 327, 407 Hamao, Y. 510 Hamburger, M.J. 247n Hansen, G. 78, 81, 420, 430, 431, 433n, 435n, 446, 447, 453, 454 Hansen, L. 18n, 23, 24, 36, 40, 50n, 74, 93, 167, 169, 179, 183, 186, 190, 195, 196, 197, 198, 237, 246, 247n, 261, 262, 263, 264, 265, 266, 267, 268, 269, 273, 277, 280, 283, 284, 289, 297, 300, 325, 326, 327, 332n, 333, 342, 343, 364, 531, 541n Hansen–Jagannathan calculation 309 Hansson, B. 512 Happiness maintenance preferences 62–64 Hardouvelis, G. 249 Harvey, A. 389 Harvey, C. 304, 344, 379, 474 Hassapis, C. 224 Hautcoeur, P.-C. 518n Hawley, F.B. 517n Hayashi, F. 350n, 568n He, H. 25n, 40n, 333n Heal, G.M. 338 Heathcote, J. 573 Heaton, J. 25n, 40n, 167, 190, 194, 224, 273, 275, 283, 284, 297, 306, 310, 311, 313, 333n, 342, 343, 345n, 352n, 353n, 364, 371, 379, 535, 538, 542, 544, 546, 547, 548, 549, 550, 552, 553, 555, 559n, 562, 568, 580, 581, 587 Hercowitz, Z. 575n Heston, S. 377 High-beta stocks 88n, 112n Historical equity premium, decomposition of 471, 493–495 capital appreciation and income 505 from past to future 497–500 total equity premium 506

total returns 505–506 unanticipated success 494 History, of equity risk premium 515, 516, 523–524 historical conception and measurement 517–521 literature 516 in long-term 524–526 Stocks, Bonds, Bills and Inflation 521–523 Hodrick, R. 222n, 246, 250, 419 Homer, S. 4, 525 Horne, G.F. 518 Horvath, M. 454 Houston, W.R. 508 Houtman-deSmedt, H. 507 Huang, H. 199, 202, 208, 209, 210, 215, 225, 226, 230, 231, 232, 233 Huang, M. 40n, 62, 63n, 67, 68, 69, 70, 71, 75, 77, 78, 199, 202, 203, 204, 210, 211, 212, 213, 214, 217, 224, 230, 379 Huber, G. 513 Huggett, M. 559, 568 Human capital 208, 255, 303, 567, 571 Hump-shaped portfolio profile 578, 582 “Hunger” 240 CPAM 241 Hwang, S. 476 I I/B/E/S International Inc. 377 Ibbotson, R.G. 40n, 476, 493, 498, 510, 513, 515, 516, 521, 522, 524, 525, 527 Ibbotson Associates 4, 150, 471, 513, 522 Idiosyncratic earnings shocks, role of asset pricing and intergenerational risk sharing 565, 591 Idiosyncratic income shocks 75, 88, 112, 544 and incomplete markets 352–355 Idiosyncratic risk 104, 113, 307 Constantinides and Duffie constructive existence theorem 307–310 empirical work 311–312 micro data 312–313 Idiosyncratic shocks 278, 307–308, 310, 361, 362, 364, 368–370, 543, 544, 554, 567, 571, 580, 588 Incomplete consumption insurance 334 Incomplete markets 40, 349–353, 360, 365, 370–371, 372, 579 Individual consumption volatility 542, 544 Inflationary expectations 105 Ingersoll, Jr., J.E. 14 Insurable risk 517 vs. uninsurable risk 518

601

Index Intangible corporate capital 109 Interest rate 268, 269, 279, 295, 305, 307, 378, 387–388, 405, 508 risk premium 389–391 of S&P500 index 387–388 in transactions costs 547–548 Intermediation cost, impact of 113–114 Internal habit models 56, 342, 345 Intertemporal elasticity of substitution (IES) 168, 170, 327, 355, 561 Intertemporal marginal rate of substitution (IMRS) 53, 123, 170, 363, 410 of consumption 107 T-bills 102–106 Intertemporal substitution elasticity 269, 284, 296 Investment plans 291 Investment return 290, 527 Ireland 509 Irish Central Statistical Office (CSO) 509 Irish Stock Exchange Equity (ISEQ) 509 Isen, A.M. 62 Italy 488, 509–510 J Jacobs, K. 312, 333n, 354 Jagannathan, R. 23, 24, 93, 235, 261, 262, 265, 277, 280, 281, 302, 326, 332n, 342, 347n, 364, 377, 459, 474, 531 Jaimovich, N. 420n, 447 Japan 478, 480, 482, 510 Jappelli, T. 345n Jealousy 121, 122n Jegadeesh, N. 256 Jermann, U. 25n, 40n, 58, 63n, 78, 79, 80, 81, 164, 292, 294, 295, 297, 333n, 370, 371, 454, 455, 457n, 460, 463, 559, 568, 581n Jitsov, A. 391 Johnsen, T. 44, 47n Johnson, E. 225, 228, 231 Jones, C. 476, 477, 513 Jorion, P. 85, 251, 333n, 432n, 470, 471, 472, 475, 477, 490, 499, 523 Julliard, C. 278, 281, 283, 284, 333n, 343 Ju, N. 377 K Kac, M. xx Kahneman, D. 62, 201, 203, 204, 205, 206n, 207 Kallenbrunner, G. 51, 81n Kalman filtering approach, 389, 390, 404 Kandel, S. 22n, 148n, 216, 217, 285, 288 Kane, A. 473

Keeping up with Joneses 56, 120, 122n Kehoe, T. 370 Keynesian era, 531 Khare, A. xx, 37 Kihlstrom, R. 47n Kiku, D. 167 Kim, M.J. 251 King, R.G. 56n Knetsch, A. 203n Knight, F. 517–518 Kocherlakota, N. 25n, 40n, 49n, 154, 261, 284, 333, 347n, 418, 431, 432, 364, 469, 546n Koedijk, K. 510 Kogan, L. 75, 161n, 291, 298 Koopmans, T.C. 47 Korean War period 105 Korniotis, G.M. 345, 354 Kotlikoff, L.J. 14, 350n, 568n Krebs, T. 40n, 333n, 354, 559n, 561 Kreps, D.M. 44, 46, 47, 195, 272 Krishnamurthy, A. 106 Kronenberger, F. 509 Krueger, D. 371, 360, 365, 368, 369, 558 Krusell, P. 364, 559, 562 568, 575, 576 Kubota, K. 303 Kuhn, T. 261 Kumar, A. 227 Kuznets, S. 5 Kydland, F. xx, 14, 15 L Labor income 302–307, 360–362 income process calibration 544–546 linear discount factor models 302–304 micro data 306 model 306–307 proprietary income 306 as risk 539–544, 559–560 VAR framework 305 Labor market 421–423, 581 LaCivita, C.J. 531 Laforest, P. 508 Laibson, D. 40n, 25n, 233, 333n Lakonishok, J. 255 Lam, P. 85, 150, 333n, 342, 403, 464 Lamont, O.A. 249 Large-scale monetary gambles, attitudes to 202, 216–218 Lee, C. 406 Lee, J. 195, 196, 197, 198, 273, 283, 284 Legal-regulatory system 108, 109 LeRoy, S.F. 249, 518, 530, 531 LeRoy, S.H. 16

602

Index Lettau, M. 63, 78, 117, 127, 249, 250, 273n, 279, 285, 303, 311, 370n, 387, 499, 554 Leverage equity claims 158, 163–165 and distorted beliefs 164–165 dividends and consumption correlation 163–165 Levine, D. 370, 530, 559 Lewellen, J. 255, 289 Li, H. 490, 492 Li, N. 273, 283, 285, 289, 292, 298, 343 Li, Q. 292 Li, Y. 345 Liew, J. 256 Life cycle incorporation, 573–577 for asset pricing 88 models, 336 Lim, S. 227 Limited stock market participation 90, 335–336, 555, 561–562 and per capita consumption 345–349 Linear approximations 137–138 accuracy 153–156 equity premium 138 risk premium 137 second moments 138–142 term premium 137 Linear internal habit model, 342, 344 Linearized unconditional variance 141 Lintner, J. 382 Liquidity 104 idiosyncratic risk 104 Lagrange’s multipliers 104 Litterman, R.B. 5 Litzenberger, R. 286, 379 Liu, J. 221, 349 Lo, A.W. 333 Lochstoer, L. 51, 81n Loewenstein, G. 205n Loewenstein, M. 377 Log-normality 134 equity premium 136 risk premium 135 term premium 135 Long run consumption risk 50, 195, 281 Long-run growth risks model 170–184 data and model implications 176–184 asset pricing implications 180–184 contingent claims and macro markets value 184 data and growth rate dynamics 176–179 preference parameters 179–180

long-run growth rate risks 171–174 uncertainty risks 174–176 preferences and environment 170–171 dividend process 171 equilibrium and asset prices 172–173 pricing of 173–174 Long-run historical rates of return 479–486 extremes of history 480–483 long-run perspective 483–486 Long-run international data 474–479 DMS global database composition and start date 475–477 methodology and guiding principles 477–479 Longstaff, F. 382, 403 Lorie, J.H. 520 Loss aversion and narrow framing (LANF) 71, 201, 203–207, 235 “accessibility” theory of framing 226 dynamic aspects 225–226 equilibrium market phenomena quantitative predictions, 232–233 equity protection puzzle 230–232 importance 220–223 modeling 207–212 prospect theory (PT) agent vs. EU agent 230–232 in stock market pricing and participation puzzles, 255–256 Lovallo, D. 205n Low-frequency component, in consumption 194–195 Lucas, D. 25n, 40n, 224, 306, 310, 311, 313, 325, 333n, 345n 352n, 353n, 364, 371, 379, 535, 538, 539, 542, 543, 544, 546, 547, 548, 549, 550, 552, 553, 555, 559n, 562, 568, 580, 581, 587 Lucas Jr, R. xx, 2, 14, 15, 30, 36, 63n, 64, 71, 78n, 119, 127, 147, 148, 159, 163, 306, 310, 311, 313, 325, 332n, 362, 422, 428, 532, 537 Lucas’ model 15 Ludvigson, S. 63, 117, 127, 249, 250, 277, 279, 285, 303, 344, 387, 554 Lundblad, C. 168, 169, 172, 179, 185, 186, 188, 189, 190, 191, 283, 343 Lustig, H. 279, 281, 331, 333n, 360, 365, 368, 369, 370, 371, 551, 559, 581n Luttmer, E.G.J. 25n, 40n Luxury consumption goods versus basic consumption goods 72 Lynch, A.W. 25n, 333n Lystbaek, B. 508

603

Index M Macaulay, F.R. 3, 247n Macbeth, J.D. 458 Mace, B.J. 350n, 568n MacKinlay, A.C. 333 Macro aggregates 418, 446 Macro variables and forecastability 249–250 Macroeconomic modeling techniques for financial markets 417 Macroeconomics for financial markets and real economy 242 Madan, D. 377 Maddison, A. 491 Magin, K. 40n Malloy, C. 50n, 306, 343 Mamaysky, H. 377 Mankiw, G. 379, 365, 369, 431 Mankiw, N. 14, 25n, 40n, 88n, 112, 216, 224, 313, 333, 345, 346, 350, 379, 407, 464, 465, 530, 538, 539, 539, 544, 568, 582 Mankiw’s recipe for risk premia generation, 365–366 Marcet, A. 364, 568 Marcus, A.J. 473 Marginal rate of substitution 103, 123, 272, 313, 328, 349, 350, 363, 410 Mark, N. 85, 150, 333n, 342, 403, 464 Market betas 246, 252, 254 Market imperfections 469, 539, 555 Market incompleteness and trading frictions 86 borrowing constraints 89 restricted participation 86–91 Market portfolio, 52n, 174, 391, 392, 396, 398, 414 dynamics 383–385 Markov process 15, 30, 261, 361 Markowitz, H. 521 Markowitz-Tobin model, 521 Marsh, P. 7, 40n, 467, 469, 470, 472, 474, 475, 477, 483, 497, 498n, 500, 507, 523, 513 Marsh, T. 382, 406 Marshall, A. 337 Marshall, D. 25n, 40n, 197, 222n, 280, 333n Mauboussin, M.J. 496 McCloskey, D.N. 261 McGrattan, E. 2n, 15n, 25n, 40n, 101, 102, 105, 106, 107, 108, 109, 110, 443, 474, 459 McLeod, H. 511 Meghen, P.J. 509 Mehra, C. xx

Mehra, R. 1, 2, 2n, 3n, 5n, 7, 8, 9, 13, 14, 15, 16, 17n, 19, 20n, 24, 25n, 29, 30, 33, 35, 35n, 36, 37, 39, 40, 41, 43n, 44n, 45, 46, 48, 49, 51, 52n, 53, 54, 56n, 58n, 59, 62, 63, 64, 65, 69, 74, 78, 82, 83, 87, 89, 90n, 91, 92, 101, 102, 105, 106, 111, 113, 114, 117, 119, 146, 147, 148, 149, 150, 154, 158, 159, 167, 199, 201, 209, 210, 230, 261, 262, 263, 264, 265, 325, 326, 331, 332, 333n, 340n, 342, 346, 348, 349, 350, 361, 364, 377, 379, 383, 388, 395, 396, 403, 405, 410, 417n, 418, 422, 431, 442, 452, 453, 454, 455, 469, 470, 471, 472, 473, 476, 490, 493, 499, 516, 521, 525, 527, 537, 538, 539, 543, 544, 555, 566, 567, 568, 571, 572, 576, 583, 587, 591 Mellon, C. xx Melino, A. 53, 54, 55, 55n, 62, 63, 263 Menzly, L. 249, 288, 298, 306, 344, 345, 377, 397, 382, 403, 409 Mertens, E. 415 Merton, R. 382, 305, 332n, 347n, 406, 578 Merz, M. 292 Michaelides, A. 42n, 91, 162, 555, 568, 581n Michaely, R. 406, 496 Mill, J.S. 517, 527 Miller, R.A. 568n Mimicking portfolio theorem 243–244 Missing markets Constantinides and Duffie 366–368 equilibrium, 365 idiosyncratic shocks 368–370 Mankiw’s receipe 365–366 Mitchell, B.R. 507, 512 Mitchell, W.C. 518 Model economy 423 equilibrium 427–429 firm 425–426 numerical procedures and calibration 429–430 shareholders 424–425 workers 423–424 Model uncertainty 40, 91–93 Modest, D.M. 25n, 40n, 333n Momentum factor 256, 257 Momentum spread puzzle 185 Monte Carlo experiments 177 Moore, D. 552, 553 Morgan Stanley Capital International (MSCI) index 474–475, 514 Moskowitz, T. 50n, 306, 343 Mossin, J. 2n Mulligan, C. 195 Multifactor models 241

604

Index Multiple goods 71 basic consumption goods 72 capital goods 454 durable goods 74 luxury consumption goods 72 non-durable goods 74 non-housing consumption 72 Myers, J. 406 Myers, S.C. 473

N Nagel, S. 255, 288, 494 Naik, V. 298 National de la Statistique et des Etudes Economiques (INSEE) 508, 509 National Income and Product Accounts (NIPA) 5, 114, 335, 552 capital returns 105 National income ratio, explaining market value to 442–443 Negishi, T. 350 Nelson, C.R. 251, 280 Nelson, W.R. 280 Neoclassical growth model 14, 15 Netherlands, The 510–511 New York Stock Exchange, 4, 513, 516, 518, 519, 524 Newlands, J. 494 Nielsen, S. 508 No-trade economies asset pricing properties, 572 calibration, 587–589 Non-diversifiable risk 11, 39, 102 Non-financial earnings, 568, 592 Non-rational expectations 164–165, 166 Non-risk-based explanations agent heterogeneity and intermediation costs, impact of 113–114 borrowing constraints 110–113 government regulations and rules, effect of 106–107 T-bills 102–106 taxes 107–110 Non-tradeable and tradeable risks, 568 Norway 511 Nowman, B. 389, 390 Nowman, K. 391 Numerical procedures and calibration, of model economy 429–430

O Odean, T. 226 Officer, R.R. 507 Olovsson, C. 569 One-step-ahead innovation 178 Opp, M. 237 OPEC oil squeeze 482 Otten, R. 510 Overland, J. 162 Overlapping generations model 87, 111, 555, 567, 574

P Pakos, M. 275 Palazzo, D. 237 Panel Survey of Income Dynamics (PSID) 538 Panetta, F. 509, 510 Panjer, H.H. 507, 508 Parameter uncertainty 53 Paredaens, J. 507 Pareto optimal 42, 422 Parker J. 72, 73, 233, 278, 281, 283, 284, 313, 333n, 343, 354, 379 Parum, C. 508 Pastor, L. 291, 387, 476n Paxson, C. 368, 568n Peng, L. 476, 513, 516, 524 Peralta-Alva, A. 443 Percentage pricing-error measures, 396 absolute pricing error, 396 mean pricing error, 396 Permanent income hypothesis 258 Perri, F. 371, 558 Persons, W.M. 518 “Peso problems” 85 Petit-Koñczyk, M. 518n Petkova, R. 306 Piazzesi, M. 71, 72n, 237, 247, 249, 274, 278, 283, 314, 382, 403, 404, 406 Platt, E.N. 247n Polkovnichenko, V. 75, 76, 77, 162, 553, 555 Pomorski, L. 331 Porter, R. 168, 249 Porteus, E.L. 44, 46, 47, 47n, 195, 206n, 248, 271, 345n Portfolio-based theorem and division of labor 243–244 Portfolio choice, 567 Portfolio returns 185 Post World War II 105, 533 Power utility 173, 267–270

605

Index Preferences 14–25, 44–52, 120–126, 170–171, 361–362 based theories 39 aggregation 41–44 equilibrium return stationarity 41 time-consistent planning 44 parameters 153, 179–180 Prescott, E.C. xx, 1, 2, 3n, 5n, 8, 13, 14, 15, 16, 19, 20n, 24, 25n, 29, 33, 35, 36, 37, 39, 40, 41, 43n, 44n, 45n, 46, 48, 49, 51, 52n, 53, 54, 56n, 60, 62, 63, 64, 65, 69, 74, 82, 83, 91, 92, 101, 102, 105, 106, 107, 108, 109, 110, 113, 114, 119, 146, 147, 148, 149, 150, 154, 158, 159, 201, 209, 210, 261, 262, 263, 264, 265, 325, 326, 332, 333, 340n, 342, 350, 361, 364, 379, 383, 388, 395, 396, 403, 405, 410, 418, 419, 420, 422, 428n, 429, 431, 442, 443, 447, 459, 469, 470, 471, 472, 473, 476, 493, 516, 521, 525, 527, 537, 539, 543, 544, 566, 568, 571, 572, 576, 587, 591 Price 41, 63, 153, 169, 195, 239, 245, 248, 290, 294, 334, 363, 371, 383, 414, 449, 478n, 518, 531 of canonical asset 127–129 Price-dividend ratios 175, 178–179, 248–249, 266, 279, 297, 494, 496, 505, 506 Price-to-earnings ratio, 388, 393, 394, 411, 413 Pricing kernel 67, 123, 124, 174, 351, 378, 382, 404, 410–411, 570 Priestley, R. 249 Principles of Political Economy, 517 Prior estimates, of equity premium 471–474 expert opinion 472–474 Production economies 78–81 consumption smoothing 79 and financial returns 452 Production-based asset pricing 78, 290, 292–294, 577 models 81 time-series tests 291–292 cross-sectional tests 292 Productivity and distribution shocks, correlation, 437–438 Profit, 11, 110, 169, 258, 454, 457, 517 Prospect theory (PT) 67–71, 203 vs. EU 230–232 Pseudo-proportional costs 547n

Q Q theory 249, 291–292, 294 Quadratic costs 547, 548, 549

Quantitative implications of equity premium 212–216 Quantity of bonds and stocks, 578 Quiggin, J. 62 Quigley, G. 494 R Rabin, M. 203, 205, 216 Rajan, R. 491 Ramchand, L. 568 Rank dependent expected utility preferences (RDEU) 75 Rate of return 39, 57, 140, 142 distorted beliefs 151–153 of canonical asset 129–131 Rational expectations, of rate of return consumption externalities with leverage 150 without leverage 149–150 standard utility function without leverage 148–149 Rätzer, E. 513 Rau, R. 496n Ravina, E. 162 Ravn, M. 55n Read, D. 205n Real Business Cycle Theory 14–15, 102, 531 Rebelo, S. 56n Recursive preferences 47, 195, 196–197 Redelmeier, D. 205n Reitz, T. 63n, 81, 82, 83, 85 Representative agent model 363, 395, 435, 444, 576 Restoy, F. 273, 327 “Retirement”, 576 Return forecasts variation over time 246–249 Return volatilities 85, 447, 450 Returns, cross section 251–257, 282, 298–299 Richter, Y. 509 Rietz, T. 24n, 233, 333n, 523 R´ıos-Rull, J.V. 559, 568, 575 Risager, O. 508 Risk aversion 168 parameter changes in 438 versus equity risk premium, in estimation 35–36 Risk compensation 173, 194, 378 in equity markets 194 low-frequency component, in consumption 194–195 preferences 195–197 returns and long-run cash flows 197–198

606

Index Risk free asset 68, 106, 212, 221, 421, 468 Risk free security, return on 1802–1871 4 1871–1926 4 1926-present 5 Risk preferences 44–46, 50, 65, 81, 197 Risk premia 131–132, 176, 239–242, 378 definition 132 discount factor 239, 240 for earnings shocks, 399 for expected earnings growth, 399 “hunger” 240 for interest rate, 399 linear approximations 137 under log-normality 135 Risk-based explanations, of equity premium 37 alternative preference structures 41 disaster events and survivorship bias 81–86 market incompleteness and trading frictions 86–91 model uncertainty 91–93 production economies 78–81 Risk-free rate puzzle 20–22, 266, 269, 348, 389, 405, 544 Riskless consumption sequences 47 Ritter, J. 497 Robustness 196, 446–448 Roll, R. 242, 247n, 288, 410 Roll’s theorem 243 Romer, C.D. 6n Ronge, U. 478n, 509 Ross, S. 14, 25n, 39n, 85, 110, 288, 333n, 441, 490, 516, 531, 533 Roussanov, N. 195, 196, 197, 198, 237, 273, 283, 284, 326, 328 Routledge, B. 47n, 62, 63n, 64, 65, 66, 67, 77, 328 Rozeff, M.S. 247 Rubinstein, M. 16, 326, 332n, 363, 531 Ryder, H.E. 338

S S&P500 index 387–388, 391, 392, 396, 521–522 time-series data 387–388 EPS growth 387–388 equity premium, 388 interest rate 387–388 Sah, R. 40n, 58n, 87, 91 Saito, M. 555 Sallee, P. 508 Samuelson, W.F. 347n, 578 Sandez, M. 512

Santos, T. 62, 63n, 67, 68, 69, 70, 71, 77, 164n, 202, 208, 209, 210, 215, 233, 249, 255, 298, 306, 307, 344, 345, 379, 382, 403 Sargent, T. 40, 179, 195, 247 Sarkissian, S. 568 Scerbina, A. 474 Scheinkman, J. 198n Scheurkogel, A.E. 511 Schmitt-Groh S. 55n Schneider, M. 71, 72n, 225n, 249, 274, 284 Schoenholtz, K.L. 247 Schumann, C.G.W. 511 Schwartz, A. 206n Schwartz, G.L. 518 Schwartz, J.D. 497 Schwartz, S.S. 510 Schwert, W. 3, 243, 246, 476, 524 Scott, A. 237 Scott, L. 389, 390 Second-order Taylor expansion, 373–374 Semenov, A. 354 Sensitivity analysis, 581 Shah, V. 1 Shanken, J. 255 Shapiro, M.D. 280, 333 Share process, time-series characteristics of 440 Shareholders 90, 91, 424–425, 452, 457, 464 Sharing mechanism, alternative interpretation of 448–452 Sharpe–Lintner–Mossin CAPM, 521 Sharpe ratio 183, 259, 277, 297, 310, 364, 369, 371, 414, 466, 567, 570, 580, 592 Sharpe, W.F. 231 Shaw, W.H. 5 Shefrin, H. 226 Shiller, R.J. 4, 5, 18n, 36, 63, 247, 248, 249, 250, 261, 262, 263, 264, 265, 310, 325, 326, 332n, 366n Shleifer, A. 255 Short-run risk component 174, 189 Siconolfi, P. 78, 90, 415 Siegel, J. 4, 471, 472, 476, 483, 494, 495, 497 Simulation model 275, 279 Singell Jr, L.D. 518 Singleton, K.J. 18n, 36, 74, 262, 263, 264, 266, 267, 268, 269, 273, 289, 325, 326, 332n, 364, 531, 541n, 568 Sinquefield, R. 513, 516, 521, 522, 527 Skinner, D.J. 406 Small monetary gambles, attitudes to 218–220 Smith, A. 364, 559, 562, 568, 575, 576 Smith, E.L. 518

607

Index Smith, K.C. 518 Smith, W.B. 3 Song, B.K. 476 South Africa 496, 511 Spain 478, 479, 512 St-Amour, P. 52 Stambaugh, R. 22n, 148n, 216, 217, 247, 251, 285, 288, 387, 476n Standard consumption-savings model 545 Standard errors 177, 264, 390, 486, 497 Standard first-order Taylor series 171 Standard model 119n, 561 Standard portfolio model 260 Standard preferences 14, 236 Standard theory 293, 351, 469 habit persistence, 335 incomplete market, 335 Standard utility function without leverage in distorted beliefs 152 in rational expectations 148–149 Statman, M. 226 Staunton, M. 7, 467, 469, 470, 472, 474, 475, 477, 483, 497, 498n, 500, 507, 513, 523 Staunton, R. 40n Steady-state capital stock 445, 446 Stehle, R. 509 Stephen, P.Z. 40n Stochastic discount factor (SDF) 17, 196, 197, 239, 278, 272, 286, 326, 351, 363–364, 369 Stochastic general equilibrium model 60 Stock market 25, 36, 91, 201, 206, 209, 215, 224, 294, 297, 313, 334, 345–349, 523, 552, 555 pricing and participation puzzle 235–236 Stocks, Bonds, Bills and Inflation: Year by Year Historical Returns (1926–1974), 521–523 Stokey, N. 428n Storesletten, K. 25n, 40n, 87n, 311, 333n, 347n, 353n, 369, 538, 558, 565, 567, 568, 567, 571, 572, 573, 574, 575, 576, 579, 580, 587, 588, 589 Suarez, J.L. 512 Success bias 472, 490, 533 Summers, L.H. 14, 248 Sundaresan, S.M. 55, 276, 338 Surplus consumption process 59–61 Surplus profit, 517 Survivorship bias 490, 492–493 Survivorship, of markets 490–492 Swaminathan, B. 406 Sweden 496, 497, 512 Switzer, L. 508

Switzerland 512–513 Sylla, R. 524 Symmetric cost 548 Systematic risk 175, 186 Szeidl, A. 280

T T-bills 102–106 as proxy, for intertemporally smooth consumption 102–106 government regulations and rules, effect of 106–107 liquidity 104 transaction balances 104–106 Takehara, H. 303 Tallarini, T.D. 296, 297, 328 Tan, K.S. 507, 508 Tangible corporate capital 109 Tanner, S. 25n, 40n, 313, 333n, 346, 561 Tarshis, L. 432 Taxes 107–110 distribution tax rate 108 marginal tax rates 107 Taylor, A. 496 Taylor expansion 241n Telmer, C. 25n, 40n, 87n, 333n, 347n, 353n, 311, 364, 369, 538, 558, 559, 565, 567, 568, 571, 572, 573, 574, 575, 576, 579, 580, 581, 587, 588, 589 Temporal aggregation problems 25 Term premium 132–133 definition 133 linear approximations 137 under log-normality 135 Tervsky, A. 62 Thaler, R. 24n, 39n, 67n, 202, 203, 204, 205, 206n, 207, 208, 210, 215, 216, 224, 225, 228, 251, 333n, 379 The Theory of Investment Value, 520 Thomas, W.A. 509 Thompson, S.B. 499 Timmermann, A. 508 Titman, S. 255, 256 Tobin, J. 521 Trading costs 537, 547–550 direct effect 547 Trading friction 39–40 borrowing and short sales constraints 550–551 transactions costs 547–550

608

Index Transaction balances 104–106 brokerage charges 105 high-yielding assets 104 high-yielding securities 104 inflationary expectations 105 record keeping 104 stocks and bonds 105 Transactions costs 547–550, 560 Treasury bills 36, 168, 342, 485, 508, 511–514 Treasury bond yields, 106, 389, 404, 412 Treasury Inflation Protected Securities (TIPS) 5 Treasury rate, 394, 397 Treasury securities 106, 388 Tuzel, S. 71, 72n, 249, 278 Tversky, A. 203, 204, 205, 205n, 206 Two-stage procedure, 412–414 U Uhlig, H. 78, 285 Ukhov, A. 492 Uncertainty risks 114, 174–176, 518 pricing of 175–176 Unconditional multifactor model 286 Unconditional variance rate of return of canonical asset 139–140 Uninsurable risk, 518, 530 United Kingdom, The 513 United States, The 513–514 capital market, history 520 Unlevered equity 127 Uribe, M. 55n Urquhart, M.C. 508 Utility functions 61n, 270–273 Epstein-Zin preference 282–283 Utility functions (continued) habits 275–278 long horizon 280–282 non-separabilities across goods 273–275 related models 278–280 Utility index 272, 274, 285, 322 V Valbuena, S.F. 512 Valuation model for equity premium, 389 cash flow risk compensation, 392–396 Gt process, maximum likelihood estimation, 391–392 interest-rate risk premium, 389–391 Value spread puzzle 185 Van Nieuwerburgh, S. 280, 499 Vanguard 522

Variable sharing parameter 448, 450 Variation, in equity premium 9–11 Vasicek, O. 380, 383, 404 Vassalou, M. 256, 289, 292 Verdelhan, A. 278, 282 Vermaelen, T. 496n Veronesi, P. 85, 164n, 249, 291, 298, 306, 344, 345, 382, 397, 403 Viceria, L. 237 Violante, G.L. 573 Violi, R. 331, 509, 510 Vishny, R.W. 255 Vissing-Jorgensen, A. 25n, 40n, 50n, 106, 224, 313, 333n, 343, 346, 354n, 555, 561 Volatility risk 168, 175 Von Böhm-Bawerk, E. 517 Von Neumann-Morgenstern preferences 334 Vuolteenaho, T. 306, 413 W Wachter, J. 117, 127, 161, 278, 387, 554 “Wage bill habit” 436 Wage payments, in labor market 421 Wage share statistics 434 Wall Street Crash 482 Walrasian equilibrium 441, 452 Wang, A. 304, 312, 333n, 354 Wang, Y. 280, 281 Wang, Z. 303 Weber, G. 568n Wei, M. 249 Weil, D. 162 Weil, P. 16n, 20, 48, 49, 57n, 162n, 184, 195, 261, 273, 327, 332n, 383, 389, 405, 568 Weitzman, M. 40, 91, 92, 93, 333n Welch, I. 251, 472, 473, 474, 499 Welfare considerations 461–462 Werner, J. 16 Whelan, S. 509 Wickens, M. 237 Willen, P. 88, 112 Williams, J.B. 520, 523 Wilson, J. 476, 477, 513, 524 Wilson, R. 363 Workers 423–424 and shareholders, in labor market 422, 449 World bond index 475 World equity index 475 Worldwide equity premium 467, 487–489 decomposition 493, 495–497 from past to future 497–500 unanticipated success 493–494

609

Index global evidence 486–493 smaller risk premium 489–490 survivorship bias 492–493 survivorship of markets 490–492 long-run historical rates of return 479–486 extremes of history 480–483 long-run perspective 483–486 long-run international data 474–479 DMS global database 475–479 prior estimates 471–474 Wright, R.E. 524 Wulff, C. 509 Wydler, D. 512 X Xia, Y. 304 Xing, Y. 292 Xiong, W. 203n Xu, Y. 490, 492 Y Yang, A.X. 53, 54, 55, 62, 63 Yao, T. 568, 587 Yaron, A. 24n, 25n, 39n, 40n, 50, 51, 87n, 117, 127, 167, 168, 169, 172, 173, 175, 176, 177,

178, 179, 180, 183, 184, 186, 187, 189, 190, 191, 194, 195, 196, 283, 284, 285, 311, 333n, 347n, 353n, 369, 457, 538, 558, 565, 567, 568n, 573, 571, 572, 573, 574, 575, 576, 579, 580, 587, 588, 589 Yashiv, E. 291, 292 Yogo, M. 50n, 72, 73, 74, 237, 274, 313, 354n, 379 Yu, J. 117

Z Zame, W.R. 530, 559 Zeckhauser, R.J. 14 Zeldes, S. 25n, 216, 224, 313, 333n, 345, 346, 379, 407, 431, 464, 465, 539, 580 Zhang, H. 551, 568 Zhang, L. 291, 298, 457 Zhu, N. 492 Ziemba, W.T. xxiii, 510 Zin, S. 16n, 24n, 39, 47n, 62n, 46, 47, 48, 49, 50, 51, 52, 53, 54n, 65, 62, 63n, 64, 65, 66, 67, 77, 124, 168, 170, 173, 195, 197, 203, 211, 216, 217, 218, 221, 222, 271, 272, 282, 321, 333n, 337n, 341, 344, 379, 565 Zingales, L. 491


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