1. HANDBOOK OF THE ECONOMICS OF FINANCE VOLUME 1B

2. HANDBOOKS IN ECONOMICS 21 Series Editors KENNETH J. ARROW MICHAEL D. INTRILIGATOR Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo 3. HANDBOOK OF THE ECONOMICS OF FINANCE VOLUME 1B FINANCIAL MARKETS AND ASSET PRICING Edited by GEORGE M. CONSTANTINIDES University of Chicago MILTON HARRIS University of Chicago and REN E M. STULZ Ohio State University 2003 Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo 4. ELSEVIER B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands 2003 Elsevier B.V. All rights reserved. This work is protected under copyright by Elsevier, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. 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Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elseviers Science & Technology Rights Department, at the phone, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verication of diagnoses and drug dosages should be made. First edition 2003 Library of Congress Cataloging-in-Publication Data A catalog record from the Library of Congress has been applied for. British Library Cataloguing in Publication Data A catalogue record from the British Library has been applied for. ISBN: 0-444-50298-X (set, comprising vols. 1A & 1B) ISBN: 0-444-51362-0 (vol. 1A) ISBN: 0-444-51363-9 (vol. 1B) ISSN: 0169-7218 (Handbooks in Economics Series) The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands. 5. INTRODUCTION TO THE SERIES The aim of the Handbooks in Economics series is to produce Handbooks for various branches of economics, each of which is a denitive source, reference, and teaching supplement for use by professional researchers and advanced graduate students. Each Handbook provides self-contained surveys of the current state of a branch of economics in the form of chapters prepared by leading specialists on various aspects of this branch of economics. These surveys summarize not only received results but also newer developments, from recent journal articles and discussion papers. Some original material is also included, but the main goal is to provide comprehensive and accessible surveys. The Handbooks are intended to provide not only useful reference volumes for professional collections but also possible supplementary readings for advanced courses for graduate students in economics. KENNETH J. ARROW and MICHAEL D. INTRILIGATOR PUBLISHERS NOTE For a complete overview of the Handbooks in Economics Series, please refer to the listing at the end of this volume. 6. This Page Intentionally Left Blank 7. CONTENTS OF THE HANDBOOK VOLUME 1A CORPORATE FINANCE Chapter 1 Corporate Governance and Control MARCO BECHT, PATRICK BOLTON and AILSA R OELL Chapter 2 Agency, Information and Corporate Investment JEREMY C. STEIN Chapter 3 Corporate Investment Policy MICHAEL J. BRENNAN Chapter 4 Financing of Corporations STEWART C. MYERS Chapter 5 Investment Banking and Security Issuance JAY R. RITTER Chapter 6 Financial Innovation PETER TUFANO Chapter 7 Payout Policy FRANKLIN ALLEN and RONI MICHAELY Chapter 8 Financial Intermediation GARY GORTON and ANDREW WINTON Chapter 9 Market Microstructure HANS R. STOLL 8. viii Contents of the Handbook VOLUME 1B FINANCIAL MARKETS AND ASSET PRICING Chapter 10 Arbitrage, State Prices and Portfolio Theory PHILIP H. DYBVIG and STEPHEN A. ROSS Chapter 11 Intertemporal Asset-Pricing Theory DARRELL DUFFIE Chapter 12 Tests of Multi-Factor Pricing Models, Volatility, and Portfolio Performance WAYNE E. FERSON Chapter 13 Consumption-Based Asset Pricing JOHN Y. CAMPBELL Chapter 14 The Equity Premium in Retrospect RAJNISH MEHRA and EDWARD C. PRESCOTT Chapter 15 Anomalies and Market Efciency G. WILLIAM SCHWERT Chapter 16 Are Financial Assets Priced Locally or Globally? G. ANDREW KAROLYI and RENE M. STULZ Chapter 17 Microstructure and Asset Pricing DAVID EASLEY and MAUREEN OHARA Chapter 18 A Survey of Behavioral Finance NICHOLAS C. BARBERIS and RICHARD H. THALER Finance Optimization, and the Irreducibly Irrational Component of Human Behavior ROBERT J. SHILLER Chapter 19 Derivatives ROBERT E. WHALEY Chapter 20 Fixed Income Pricing QIANG DAI and KENNETH J. SINGLETON 9. PREFACE Financial economics applies the techniques of economic analysis to understand the savings and investment decisions by individuals, the investment, nancing and payout decisions by rms, the level and properties of interest rates and prices of nancial assets and derivatives, and the economic role of nancial intermediaries. Until the 1950s, nance was viewed primarily as the study of nancial institutional detail and was hardly accorded the status of a mainstream eld of economics. This perception was epitomized by the difculty Harry Markowitz had in receiving a PhD degree in the economics department at the University of Chicago for work that eventually would earn him a Nobel prize in economic science. This state of affairs changed in the second half of the 20th century with a revolution that took place from the 1950s to the early 1970s. At that time, key progress was made in understanding the nancial decisions of individuals and rms and their implications for the pricing of common stocks, debt, and interest rates. Harry Markowitz, William Sharpe, James Tobin, and others showed how individuals concerned about their expected future wealth and its variance make investment decisions. Their key results showing the benets of diversication, that wealth is optimally allocated across funds that are common across individuals, and that investors are rewarded for bearing risks that are not diversiable, are now the basis for much of the investment industry. Merton Miller and Franco Modigliani showed that the concept of arbitrage is a powerful tool to understand the implications of rm capital structures for rm value. In a world without frictions, they showed that a rms value is unrelated to its capital structure. Eugene Fama put forth the efcient markets hypothesis and led the way in its empirical investigation. Finally, Fischer Black, Robert Merton and Myron Scholes provided one of the most elegant theories in all of economics: the theory of how to price nancial derivatives in markets without frictions. Following the revolution brought about by these fathers of modern nance, the eld of nance has experienced tremendous progress. Along the way, it inuenced public policy throughout the world in a major way, played a crucial role in the growth of a new $100 trillion dollar derivatives industry, and affected how rms are managed everywhere. However, nance also evolved from being at best a junior partner in economics to being often a leader. Key concepts and theories rst developed in nance led to progress in other elds of economics. It is now common among economists to use theories of arbitrage, rational expectations, equilibrium, agency relations, and information asymmetries that were rst developed in nance. The committee for the 10. x Preface Alfred Nobel Memorial Prize in economic science eventually recognized this state of affairs. Markowitz, Merton, Miller, Modigliani, Scholes, Sharpe, and Tobin received Nobel prizes for contributions in nancial economics. This Handbook presents the state of the eld of nance fty years after this revolution in modern nance started. The surveys are written by leaders in nancial economics. They provide a comprehensive report on developments in both theory and empirical testing in nance at a level that, while rigorous, is nevertheless accessible to researchers not intimate with the eld and doctoral students in economics, nance and related elds. By summarizing the state of the art and pointing out as-yet unresolved questions, this Handbook should prove an invaluable resource to researchers planning to contribute to the eld and an excellent pedagogical tool for teaching doctoral students. The book is divided into two Volumes, corresponding to the traditional taxonomy of nance: corporate nance (1A) and nancial markets and asset pricing (1B). 1. Corporate nance Corporate nance is concerned with how businesses work, in particular, how they allocate capital (traditionally, the capital budgeting decision) and how they obtain capital (the nancing decision). Though managers play no independent role in the work of Miller and Modigliani, major contributions in nance since then have shown that managers maximize their own objectives. To understand the rms decisions, it is therefore necessary to understand the forces that lead managers to maximize the wealth of shareholders. For example, a number of researchers have emphasized the positive and negative roles of large shareholders in aligning incentives of managers and shareholders. The part of the Handbook devoted to corporate nance starts with an overview, entitled Corporate Governance and Control, by Marco Becht, Patrick Bolton, and Ailsa Roell (Chapter 1) of the framework in which managerial activities take place. Their broad survey covers everything about corporate governance, from its history and importance to theories and empirical evidence to cross-country comparisons. Following the survey of corporate governance in Chapter 1, two complementary essays discuss the investment decision. In Agency, Information and Corporate Investment, Jeremy Stein (Chapter 2) focuses on the effects of agency problems and asymmetric information on the allocation of capital, both across rms and within rms. This survey does not address the issue of how to value a proposed investment project, given information about the project. That topic is considered in Corporate Investment Policy by Michael Brennan in Chapter 3. Brennan draws out the implications of recent developments in asset pricing, including option pricing techniques and tax considerations, for evaluating investment projects. In Chapter 4, Financing of Corporations, the focus moves to the nancing decision. Stewart Myers provides an overview of the research that seeks to explain rms capital structure, that is, the types and proportions of securities rms use to nance their 11. Preface xi investments. Myers covers the traditional theories that attempt to explain proportions of debt and equity nancing as well as more recent theories that attempt to explain the characteristics of the securities issued. In assessing the different capital structure theories, he concludes that he does not expect that there will ever be one capital structure theory that applies to all rms. Rather, he believes that we will always use different theories to explain the behavior of different types of rms. In Chapter 5, Investment Banking and Security Issuance, Jay Ritter is concerned with how rms raise equity and the role of investment banks in that process. He examines both initial public offerings and seasoned equity offerings. A striking result discovered rst by Ritter is that rms that issue equity experience poor long-term stock returns afterwards. This result has led to a number of vigorous controversies that Ritter reviews in this chapter. Firms may also obtain capital by issuing securities other than equity and debt. A hallmark of the last thirty years has been the tremendous amount of nancial innovation that has taken place. Though some of the innovations zzled and others provided fodder to crooks, nancial innovation can enable rms to undertake protable projects that otherwise they would not be able to undertake. In Chapter 6, Financial Innovation, Peter Tufano delves deeper into the issues of security design and nancial innovation. He reviews the process of nancial innovation and explanations of the quantity of innovation. Investors do not purchase equity without expecting a return from their investment. In one of their classic papers, Miller and Modigliani show that, in the absence of frictions, dividend policy is irrelevant for rm value. Since then, a large literature has developed that identies when dividend policy matters and when it does not. Franklin Allen and Roni Michaely (Chapter 7) survey this literature in their essay entitled Payout Policy. Allen and Michaely consider the roles of taxes, asymmetric information, incomplete contracting and transaction costs in determining payouts to equity holders, both dividends and share repurchases. Chapter 8, Financial Intermediation, focuses more directly on the role nancial intermediaries play. Although some investment is funded directly through capital markets, according to Gary Gorton and Andrew Winton, the vast majority of external investment ows through nancial intermediaries. In Chapter 8, Gorton and Winton survey the literature on nancial intermediation with emphasis on banking. They explore why intermediaries exist, discuss banking crises, and examine why and how they are regulated. Exchanges on which securities are traded play a crucial role in intermediating between individuals who want to buy securities and others who want to sell them. In many ways, they are special types of corporations whose workings affect the value of nancial securities as well as the size of nancial markets. The Handbook contains two chapters that deal with the issues of how securities are traded. Market Microstructure, by Hans Stoll (Chapter 9), focuses on how exchanges perform their functions as nancial intermediaries and therefore is included in this part. Stoll examines explanations of the bid-ask spread, the empirical evidence for these explanations, and the implications for market design. Microstructure and Asset Pricing, 12. xii Preface by Maureen OHara and David Easley (Chapter 17), examines the implications of how securities trade for the properties of securities returns and is included in Volume 1B on Financial Markets and Asset Pricing. 2. Financial markets and asset pricing A central theme in nance and economics is the pursuit of an understanding of how the prices of nancial securities are determined in nancial markets. Currently, there is immense interest among academics, policy makers, and practitioners in whether these markets get prices right, fueled in part by the large daily volatility in prices and by the large increase in stock prices over most of the 1990s, followed by the sharp decrease in prices at the turn of the century. Our understanding of how securities are priced is far from complete. In the early 1960s, Eugene Fama from the University of Chicago established the foundations for the efcient markets view that nancial markets are highly effective in incorporating information into asset prices. This view led to a large body of empirical and theoretical work. Some of the chapters in this part of the Handbook review that body of work, but the efcient markets view has been challenged by the emergence of a new, controversial eld, behavioral nance, which seeks to show that psychological biases of individuals affect the pricing of securities. There is therefore divergence of opinion and critical reexamination of given doctrine. This is fertile ground for creative thinking and innovation. In Volume 1B of the Handbook, we invite the reader to partake in this intellectual odyssey. We present eleven original essays on the economics of nancial markets. The divergence of opinion and puzzles presented in these essays belies the incredible progress made by nancial economists over the second half of the 20th century that lay the foundations for future research. The modern quantitative approach to nance has its origins in neoclassical economics. In the opening essay titled Arbitrage, State Prices and Portfolio Theory (Chapter 10), Philip Dybvig and Stephen Ross illustrate a surprisingly large amount of the intuition and intellectual content of modern nance in the context of a single- period, perfect-markets neoclassical model. They discuss the fundamental theorems of asset pricing the consequences of the absence of arbitrage, optimal portfolio choice, the properties of efcient portfolios, aggregation, the capital asset-pricing model (CAPM), mutual fund separation, and the arbitrage pricing theory (APT). A number of these notions may be traced to the original contributions of Stephen Ross. In his essay titled Intertemporal Asset Pricing Theory (Chapter 11), Darrell Dufe provides a systematic development of the theory of intertemporal asset pricing, rst in a discrete-time setting and then in a continuous-time setting. As applications of the basic theory, Dufe also presents comprehensive treatments of the term structure of interest rates and xed-income pricing, derivative pricing, and the pricing of corporate securities with default modeled both as an endogenous and an exogenous process. 13. Preface xiii These applications are discussed in further detail in some of the subsequent essays. Dufes essay is comprehensive and authoritative and may serve as the basis of an entire 2nd-year PhD-level course on asset pricing. Historically, the empirically testable implications of asset-pricing theory have been couched in terms of the mean-variance efciency of a given portfolio, the validity of a multifactor pricing model with given factors, or the validity of a given stochastic discount factor. Furthermore, different methodologies have been developed and applied in the testing of these implications. In Tests of Multi-Factor Pricing Models, Volatility, and Portfolio Performance (Chapter 12), Wayne Ferson discusses the empirical methodologies applied in testing asset-pricing models. He points out that these three statements of the empirically testable implications are essentially equivalent and that the seemingly different empirical methodologies are equivalent as well. In his essay titled Consumption-Based Asset Pricing (Chapter 13), John Campbell begins by reviewing the salient features of the joint behavior of equity returns, aggregate dividends, the interest rate, and aggregate consumption in the USA. Features that challenge existing asset-pricing theory include, but are not limited to, the equity premium puzzle: the nding that 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 the representative-consumer, time-separable-preferences models examined by Grossman and Shiller (1981), Hansen and Singleton (1983), and Mehra and Prescott (1985). Campbell also examines data from other countries to see which features of the USA data are pervasive. He then proceeds to relate these ndings to recent developments in asset-pricing theory that relax various assumptions of the standard asset-pricing model. In a closely related essay titled The Equity Premium in Retrospect (Chapter 14), Rajnish Mehra and Edward Prescott the researchers who coined the term critically reexamine the data sources used to document the equity premium puzzle in the USA and other major industrial countries. They then proceed to relate these ndings to recent developments in asset-pricing theory by employing the methodological tool of calibration, as opposed to the standard empirical estimation of model parameters and the testing of over-identifying restrictions. Mehra and Prescott have different views than Campbell as to which assumptions of the standard asset-pricing model need to be relaxed in order to address the stylized empirical ndings. Why are these questions important? First and foremost, nancial markets play a central role in the allocation of investment capital and in the sharing of risk. Failure to answer these questions suggests that our understanding of the fundamental process of capital allocation is highly imperfect. Second, the basic economic paradigm employed in analyzing nancial markets is closely related to the paradigm employed in the study of business cycles and growth. Failure to explain the stylized facts of nancial markets calls into question the appropriateness of the related paradigms for the study of macro- economic issues. The above two essays convey correctly the status quo that the puzzle 14. xiv Preface is at the forefront of academic interest and that views regarding its resolution are divergent. Several goals are accomplished in William Schwerts comprehensive and incisive essay titled Anomalies and Market Efciency (Chapter 15). First, Schwert discusses cross-sectional and time-series regularities in asset returns, both at the aggregate and disaggregate level. These include the size, book-to-market, momentum, and dividend yield effects. Second, Schwert discusses differences in returns realized by different types of investors, including individual and institutional investors. Third, he evaluates the role of measurement issues in many of the papers that study anomalies, including the difcult issues associated with long-horizon return performance. Finally, Schwert discusses the implications of the anomalies literature for asset-pricing and corporate nance theories. In discussing the informational efciency of the market, Schwert points out that tests of market efciency are also joint tests of market efciency and a particular equilibrium asset-pricing model. In the essay titled Are Financial Assets Priced Locally or Globally? (Chapter 16), Andrew Karolyi and Rene Stulz discuss the theoretical implications of and empirical evidence concerning asset-pricing theory as it applies to international equities markets. They explain that country-risk premia are determined internationally, but the evidence is weak on whether international factors affect the cross-section of expected returns. A long-standing puzzle in international nance is that investors invest more heavily in domestic equities than predicted by the theory. Karolyi and Stulz argue that barriers to international investment only partly resolve the home-bias puzzle. They conclude that contagion the linkage of international markets may be far less prevalent than commonly assumed. At frequencies lower than the daily frequency, asset-pricing theory generally ignores the role of the microstructure of nancial markets. In their essay titled Microstructure and Asset Pricing (Chapter 17), David Easley and Maureen OHara survey the theoretical and empirical literature linking microstructure factors to long-run returns, and focus on why stock prices might be expected to reect premia related to liquidity or informational asymmetries. They show that asset-pricing dynamics may be better understood by recognizing the role played by microstructure factors and the linkages of microstructure and fundamental economic variables. All the models that are discussed in the essays by Campbell, Mehra and Prescott, Schwert, Karolyi and Stulz, and Easley and OHara are variations of the neoclassical asset-pricing model. The model is rational, in that investors process information rationally and have unambiguously dened preferences over consumption. Naturally, the model allows for market incompleteness, market imperfections, informational asymmetries, and learning. The model also allows for differences among assets for liquidity, transaction costs, tax status, and other institutional factors. Many of these variations are explored in the above essays. In their essay titled A Survey of Behavioral Finance (Chapter 18), Nicholas Barberis and Richard Thaler provide a counterpoint to the rational model by providing explanations of the cross-sectional and time-series regularities in asset returns by 15. Preface xv relying on economic models that are less than fully rational. These include cultural and psychological factors and tap into the rich and burgeoning literature on behavioral economics and nance. Robert Shiller, who is, along with Richard Thaler, one of the founders of behavioral nance, provides his personal perspective on behavioral nance in his statement titled Finance, Optimization and the Irreducibly Irrational Component of Human Behavior. One of the towering achievements in nance in the second half of the 20th century is the celebrated option-pricing theory of Black and Scholes (1973) and Merton (1973). The model has had a profound inuence on the course of economic thought. In his essay titled Derivatives (Chapter 19), Robert Whaley provides comprehensive coverage of the topic. Following a historical overview of futures and options, he proceeds to derive the implications of the law of one price and then the BlackScholes Merton theory. He concludes with a systematic coverage of the empirical evidence and a discussion of the social costs and benets associated with the introduction of derivatives. Whaleys thorough and insightful essay provides an easy entry to an important topic that many economists nd intimidating. In their essay titled Fixed-Income Pricing (Chapter 20), Qiang Dai and Ken Singleton survey the literature on xed-income pricing models, including term structure models, xed-income derivatives, and models of defaultable securities. They point out that this literature is vast, with both the academic and practitioner communities having proposed a wide variety of models. In guiding the reader through these models, they explain that different applications call for different models based on the trade-offs of complexity, exibility, tractability, and data availability the art of modeling. The Dai and Singleton essay, combined with Dufes earlier essay, provides an insightful and authoritative introduction to the world of xed-income pricing models at the advanced MBA and PhD levels. We hope that the contributions represented by these essays communicate the excitement of nancial economics to beginners and specialists alike and stimulate further research. We thank Rodolfo Martell for his help in processing the papers for publication. GEORGE M. CONSTANTINIDES University of Chicago, Chicago MILTON HARRIS University of Chicago, Chicago RENE STULZ Ohio State University, Columbus References Black, F., and M.S. Scholes (1973), The pricing of options and corporate liabilities, Journal of Political Economy 81:637654. 16. xvi Preface Grossman, S.J., and R.J. Shiller (1981), The determinants of the variability of stock market prices, American Economic Review Papers and Proceedings 71:222227. Hansen, L.P., and K.J. Singleton (1982), Generalized instrumental variables estimation of nonlinear rational expectations models, Econometrica 50:12691288. Mehra, R., and E.C. Prescott (1985), The equity premium: a puzzle, Journal of Monetary Economics 15:145161. Merton, R.C. (1973), Theory of rational option pricing, Bell Journal of Economics and Management Science 4:141183. 17. CONTENTS OF VOLUME 1B Introduction to the Series v Contents of the Handbook vii Preface ix FINANCIAL MARKETS AND ASSET PRICING Chapter 10 Arbitrage, State Prices and Portfolio Theory PHILIP H. DYBVIG and STEPHEN A. ROSS 605 Abstract 606 Keywords 606 1. Introduction 607 2. Portfolio problems 607 3. Absence of arbitrage and preference-free results 612 3.1. Fundamental theorem of asset pricing 614 3.2. Pricing rule representation theorem 616 4. Various analyses: ArrowDebreu world 618 4.1. Optimal portfolio choice 619 4.2. Efcient portfolios 619 4.3. Aggregation 620 4.4. Asset pricing 621 4.5. Payoff distribution pricing 622 5. Capital asset pricing model (CAPM) 624 6. Mutual fund separation theory 629 6.1. Preference approach 629 6.2. Beliefs 631 7. Arbitrage pricing theory (APT) 633 8. Conclusion 634 References 634 Chapter 11 Intertemporal Asset Pricing Theory DARRELL DUFFIE 639 Abstract 641 18. xviii Contents of Volume 1B Keywords 641 1. Introduction 642 2. Basic theory 642 2.1. Setup 643 2.2. Arbitrage, state prices, and martingales 644 2.3. Individual agent optimality 646 2.4. Habit and recursive utilities 647 2.5. Equilibrium and Pareto optimality 649 2.6. Equilibrium asset pricing 651 2.7. Breedens consumption-based CAPM 653 2.8. Arbitrage and martingale measures 654 2.9. Valuation of redundant securities 656 2.10. American exercise policies and valuation 657 3. Continuous-time modeling 661 3.1. Trading gains for Brownian prices 662 3.2. Martingale trading gains 663 3.3. The BlackScholes option-pricing formula 665 3.4. Itos Formula 668 3.5. Arbitrage modeling 670 3.6. Numeraire invariance 670 3.7. State prices and doubling strategies 671 3.8. Equivalent martingale measures 672 3.9. Girsanov and market prices of risk 672 3.10. BlackScholes again 676 3.11. Complete markets 677 3.12. Optimal trading and consumption 678 3.13. Martingale solution to Mertons problem 682 4. Term-structure models 686 4.1. One-factor models 687 4.2. Term-structure derivatives 691 4.3. Fundamental solution 693 4.4. Multifactor term-structure models 695 4.5. Afne models 696 4.6. The HJM model of forward rates 699 5. Derivative pricing 702 5.1. Forward and futures prices 702 5.2. Options and stochastic volatility 705 5.3. Option valuation by transform analysis 708 6. Corporate securities 711 6.1. Endogenous default timing 712 6.2. Example: Brownian dividend growth 713 6.3. Taxes, bankruptcy costs, capital structure 717 6.4. Intensity-based modeling of default 719 19. Contents of Volume 1B xix 6.5. Zero-recovery bond pricing 721 6.6. Pricing with recovery at default 722 6.7. Default-adjusted short rate 724 References 725 Chapter 12 Tests of Multifactor Pricing Models, Volatility Bounds and Portfolio Performance WAYNE E. FERSON 743 Abstract 745 Keywords 745 1. Introduction 746 2. Multifactor asset-pricing models: Review and integration 748 2.1. The stochastic discount factor representation 748 2.2. Expected risk premiums 750 2.3. Return predictability 751 2.4. Consumption-based asset-pricing models 753 2.5. Multi-beta pricing models 754 2.6. Mean-variance efciency with conditioning information 760 2.7. Choosing the factors 765 3. Modern variance bounds 768 3.1. The HansenJagannathan bounds 768 3.2. Variance bounds with conditioning information 770 3.3. The HansenJagannathan distance 773 4. Methodology and tests of multifactor asset-pricing models 774 4.1. The Generalized Method of Moments approach 774 4.2. Cross-sectional regression methods 775 4.3. Multivariate regression and beta-pricing models 781 5. Conditional performance evaluation 785 5.1. Stochastic discount factor formulation 787 5.2. Beta-pricing formulation 788 5.3. Using portfolio weights 790 5.4. Conditional market-timing models 792 5.5. Empirical evidence on conditional performance 793 6. Conclusions 794 References 795 Chapter 13 Consumption-Based Asset Pricing JOHN Y. CAMPBELL 803 Abstract 804 Keywords 804 1. Introduction 805 20. xx Contents of Volume 1B 2. International stock market data 810 3. The equity premium puzzle 816 3.1. The stochastic discount factor 816 3.2. Consumption-based asset pricing with power utility 819 3.3. The risk-free rate puzzle 824 3.4. Bond returns and the equity-premium and risk-free rate puzzles 827 3.5. Separating risk aversion and intertemporal substitution 828 4. The dynamics of asset returns and consumption 832 4.1. Time-variation in conditional expectations 832 4.2. A loglinear asset-pricing framework 836 4.3. The equity volatility puzzle 840 4.4. Implications for the equity premium puzzle 845 4.5. What does the stock market forecast? 849 4.6. Changing volatility in stock returns 857 4.7. What does the bond market forecast? 859 5. Cyclical variation in the price of risk 866 5.1. Habit formation 866 5.2. Models with heterogeneous agents 873 5.3. Irrational expectations 876 6. Some implications for macroeconomics 879 References 881 Chapter 14 The Equity Premium in Retrospect RAJNISH MEHRA and EDWARD C. PRESCOTT 889 Abstract 890 Keywords 890 1. Introduction 891 2. The equity premium: history 891 2.1. Facts 891 2.2. Data sources 892 2.3. Estimates of the equity premium 894 2.4. Variation in the equity premium over time 897 3. Is the equity premium due to a premium for bearing non-diversiable risk? 899 3.1. Standard preferences 902 3.2. Estimating the equity risk premium versus estimating the risk aversion parameter 912 3.3. Alternative preference structures 913 3.4. Idiosyncratic and uninsurable income risk 918 3.5. Models incorporating a disaster state and survivorship bias 920 4. Is the equity premium due to borrowing constraints, a liquidity premium or taxes? 921 4.1. Borrowing constraints 921 4.2. Liquidity premium 924 21. Contents of Volume 1B xxi 4.3. Taxes and regulation 924 5. An equity premium in the future? 927 Appendix A 928 Appendix B. The original analysis of the equity premium puzzle 930 B.1. The economy, asset prices and returns 930 References 935 Chapter 15 Anomalies and Market Efciency G. WILLIAM SCHWERT 939 Abstract 941 Keywords 941 1. Introduction 942 2. Selected empirical regularities 943 2.1. Predictable differences in returns across assets 943 2.2. Predictable differences in returns through time 951 3. Returns to different types of investors 956 3.1. Individual investors 956 3.2. Institutional investors 958 3.3. Limits to arbitrage 961 4. Long-run returns 961 4.1. Returns to rms issuing equity 962 4.2. Returns to bidder rms 964 5. Implications for asset pricing 966 5.1. The search for risk factors 966 5.2. Conditional asset pricing 967 5.3. Excess volatility 967 5.4. The role of behavioral nance 967 6. Implications for corporate nance 968 6.1. Firm size and liquidity 968 6.2. Book-to-market effects 968 6.3. Slow reaction to corporate nancial policy 969 7. Conclusions 970 References 970 Chapter 16 Are Financial Assets Priced Locally or Globally? G. ANDREW KAROLYI and RENE M. STULZ 975 Abstract 976 Keywords 976 1. Introduction 977 2. The perfect nancial markets model 978 2.1. Identical consumption-opportunity sets across countries 979 22. xxii Contents of Volume 1B 2.2. Different consumption-opportunity sets across countries 982 2.3. A general approach 988 2.4. Empirical evidence on asset pricing using perfect market models 992 3. Home bias 997 4. Flows, spillovers, and contagion 1004 4.1. Flows and returns 1007 4.2. Correlations, spillovers, and contagion 1010 5. Conclusion 1014 References 1014 Chapter 17 Microstructure and Asset Pricing DAVID EASLEY and MAUREEN OHARA 1021 Abstract 1022 Keywords 1022 1. Introduction 1023 2. Equilibrium asset pricing 1024 3. Asset pricing in the short-run 1025 3.1. The mechanics of pricing behavior 1026 3.2. The adjustment of prices to information 1029 3.3. Statistical and structural models of microstructure data 1031 3.4. Volume and price movements 1033 4. Asset pricing in the long-run 1035 4.1. Liquidity 1036 4.2. Information 1041 5. Linking microstructure and asset pricing: puzzles for researchers 1044 References 1047 Chapter 18 A Survey of Behavioral Finance NICHOLAS BARBERIS and RICHARD THALER 1053 Abstract 1054 Keywords 1054 1. Introduction 1055 2. Limits to arbitrage 1056 2.1. Market efciency 1056 2.2. Theory 1058 2.3. Evidence 1061 3. Psychology 1065 3.1. Beliefs 1065 3.2. Preferences 1069 4. Application: The aggregate stock market 1075 4.1. The equity premium puzzle 1078 23. Contents of Volume 1B xxiii 4.2. The volatility puzzle 1083 5. Application: The cross-section of average returns 1087 5.1. Belief-based models 1092 5.2. Belief-based models with institutional frictions 1095 5.3. Preferences 1097 6. Application: Closed-end funds and comovement 1098 6.1. Closed-end funds 1098 6.2. Comovement 1099 7. Application: Investor behavior 1101 7.1. Insufcient diversication 1101 7.2. Naive diversication 1103 7.3. Excessive trading 1103 7.4. The selling decision 1104 7.5. The buying decision 1105 8. Application: Corporate nance 1106 8.1. Security issuance, capital structure and investment 1106 8.2. Dividends 1109 8.3. Models of managerial irrationality 1111 9. Conclusion 1113 Appendix A 1115 References 1116 Finance, Optimization, and the Irreducibly Irrational Component of Human Behavior ROBERT J. SHILLER 1125 Chapter 19 Derivatives ROBERT E. WHALEY 1129 Abstract 1131 Keywords 1131 1. Introduction 1132 2. Background 1133 3. No-arbitrage pricing relations 1139 3.1. Carrying costs 1140 3.2. Valuing forward/futures using the no-arbitrage principle 1141 3.3. Valuing options using the no-arbitrage principle 1143 4. Option valuation 1148 4.1. The BlackScholes/Merton option valuation theory 1149 4.2. Analytical formulas 1151 4.3. Approximation methods 1157 4.4. Generalizations 1164 24. xxiv Contents of Volume 1B 5. Studies of no-arbitrage price relations 1166 5.1. Forward/futures prices 1167 5.2. Option prices 1169 5.3. Summary and analysis 1173 6. Studies of option valuation models 1173 6.1. Pricing errors/implied volatility anomalies 1174 6.2. Trading simulations 1176 6.3. Informational content of implied volatility 1179 6.4. Summary and analysis 1181 7. Social costs/benets of derivatives trading 1189 7.1. Contract introductions 1189 7.2. Contract expirations 1193 7.3. Market synchronization 1194 7.4. Summary and analysis 1197 8. Summary 1198 References 1199 Chapter 20 Fixed-Income Pricing QIANG DAI and KENNETH J. SINGLETON 1207 Abstract 1208 Keywords 1208 1. Introduction 1209 2. Fixed-income pricing in a diffusion setting 1210 2.1. The term structure 1210 2.2. Fixed-income securities with deterministic payoffs 1211 2.3. Fixed-income securities with state-dependent payoffs 1212 2.4. Fixed-income securities with stopping times 1213 3. Dynamic term-structure models for default-free bonds 1215 3.1. One-factor dynamic term-structure models 1215 3.2. Multi-factor dynamic term-structure models 1218 4. Dynamic term-structure models with jump diffusions 1222 5. Dynamic term-structure models with regime shifts 1223 6. Dynamic term-structure models with rating migrations 1225 6.1. Fractional recovery of market value 1225 6.2. Fractional recovery of par, payable at maturity 1228 6.3. Fractional recovery of par, payable at default 1229 6.4. Pricing defaultable coupon bonds 1229 6.5. Pricing Eurodollar swaps 1230 7. Pricing of xed-income derivatives 1231 7.1. Derivatives pricing using dynamic term-structure models 1231 7.2. Derivatives pricing using forward-rate models 1232 7.3. Defaultable forward-rate models with rating migrations 1234 25. Contents of Volume 1B xxv 7.4. The LIBOR market model 1237 7.5. The swaption market model 1241 References 1242 Subject Index I-1 26. This Page Intentionally Left Blank 27. Chapter 10 ARBITRAGE, STATE PRICES AND PORTFOLIO THEORY PHILIP H. DYBVIG Washington University in Saint Louis STEPHEN A. ROSS MIT Contents Abstract 606 Keywords 606 1. Introduction 607 2. Portfolio problems 607 3. Absence of arbitrage and preference-free results 612 3.1. Fundamental theorem of asset pricing 614 3.2. Pricing rule representation theorem 616 4. Various analyses: ArrowDebreu world 618 4.1. Optimal portfolio choice 619 4.2. Efcient portfolios 619 4.3. Aggregation 620 4.4. Asset pricing 621 4.5. Payoff distribution pricing 622 5. Capital asset pricing model (CAPM) 624 6. Mutual fund separation theory 629 6.1. Preference approach 629 6.2. Beliefs 631 7. Arbitrage pricing theory (APT) 633 8. Conclusion 634 References 634 Handbook of the Economics of Finance, Edited by G.M. Constantinides, M. Harris and R. Stulz 2003 Elsevier B.V. All rights reserved 28. 606 P.H. Dybvig and S.A. Ross Abstract Neoclassical nancial models provide the foundation for our understanding of nance. This chapter introduces the main ideas of neoclassical nance in a single-period context that avoids the technical difculties of continuous-time models, but preserves the principal intuitions of the subject. The starting point of the analysis is the formulation of standard portfolio choice problems. A central conceptual result is the Fundamental Theorem of Asset Pricing, which asserts the equivalence of absence of arbitrage, the existence of a positive linear pricing rule, and the existence of an optimum for some agent who prefers more to less. A related conceptual result is the Pricing Rule Representation Theorem, which asserts that a positive linear pricing rule can be represented as using state prices, risk-neutral expectations, or a state-price density. Different equivalent representations are useful in different contexts. Many applied results can be derived from the rst-order conditions of the portfolio choice problem. The rst-order conditions say that marginal utility in each state is proportional to a consistent state-price density, where the constant of proportionality is determined by the budget constraint. If markets are complete, the implicit state- price density is uniquely determined by investment opportunities and must be the same as viewed by all agents, thus simplifying the choice problem. Solving rst-order conditions for quantities gives us optimal portfolio choice, solving them for prices gives us asset pricing models, solving them for utilities gives us preferences, and solving them for probabilities gives us beliefs. We look at two popular asset pricing models, the CAPM and the APT, as well as complete-markets pricing. In the case of the CAPM, the rst-order conditions link nicely to the traditional measures of portfolio performance. Further conceptual results include aggregation and mutual fund separation theory, both of which are useful for understanding equilibrium and asset pricing. Keywords arbitrage, arbitrage pricing theory, investments, portfolio choice, asset pricing, complete markets, mean-variance analysis, performance measurement, mutual fund separation, aggregation JEL classication: G11, G12 29. Ch. 10: Arbitrage, State Prices and Portfolio Theory 607 1. Introduction The modern quantitative approach to nance has its original roots in neoclassical economics. Neoclassical economics studies an idealized world in which markets work smoothly without impediments such as transaction costs, taxes, asymmetry of information, or indivisibilities. This chapter considers what we learn from single- period neoclassical models in nance. While dynamic models are becoming more and more common, single-period models contain a surprisingly large amount of the intuition and intellectual content of modern nance, and are also commonly used by investment practitioners for the construction of optimal portfolios and communication of investment results. Focusing on a single period is also consistent with an important theme. While general equilibrium theory seeks great generality and abstraction, nance has work to be done and seeks specic models with strong assumptions and denite implications that can be tested and implemented in practice. 2. Portfolio problems In our analysis, there are two points of time, 0 and 1, with an interval of time in between during which nothing happens. At time zero, our champion (the agent) is making decisions that will affect the allocation of consumption between nonrandom consumption, c0, at time 0, and random consumption {cw} across states w = 1, 2, . . . , W revealed at time 1. At time 0 and in each state at time 1, there is a single consumption good, and therefore consumption at time 0 or in a state at time 1 is a real number. This abstraction of a single good is obviously not true in any literal sense, but this is not a problem, and indeed any useful theoretical model is much simpler than reality. The abstraction does, however, face us with the question of how to interpret our simple model (in this case with a single good) in a practical context that is more complex (has multiple goods). In using a single-good model, there are two usual practices: either use nominal values and measure consumption in dollars, or use real values and measure consumption in ination-adjusted dollars. Depending on the context, one or the other can make the most sense. Following the usual practice from general equilibrium theory of thinking of units of consumption at various times and in different states of nature as different goods, a typical consumption vector is C {c0, c1, . . . , cW }, where the real number c0 denotes consumption of the single good at time zero, and the vector c {c1, . . . , cW } of real numbers c1, . . . , cW denotes random consumption of the single good in each state 1, . . . , W at time 1. If this were a typical exercise in general equilibrium theory, we would have a price vector for consumption across goods. For example, we might have the following choice problem, which is named after two great pioneers of general equilibrium theory, Kenneth Arrow and Gerard Debreu: 30. 608 P.H. Dybvig and S.A. Ross Problem 1: ArrowDebreu Problem. Choose consumptions C {c0, c1, . . . , cW } to maximize utility of consumption U(C) subject to the budget constraint c0 + W w = 1 pwcw = W. (1) Here, U() is the utility function that represents preferences, p is the price vector, and W is wealth, which might be replaced by the market value of an endowment. We are taking consumption at time 0 to be the numeraire, and pw is the price of the Arrow Debreu security which is a claim to one unit of consumption at time 1 in state w. The rst-order condition for Problem 1 is the existence of a positive Lagrangian multiplier l (the marginal utility of wealth) such that U0(c0) = l, and for all w = 1, . . . , W, Uw(cw) = lpw. This is the usual result from neoclassical economics that the gradient of the utility function is proportional to prices. Specializing to the leading case in nance of time- separable von NeumannMorgenstern preferences, named after John von Neumann and Oscar Morgenstern (1944), two great pioneers of utility theory, we have that U(C) = v(c0) + W w = 1 pwu(cw). We will take v and u to be differentiable, strictly increasing (more is preferred to less), and strictly concave (risk averse). Here, pw is the probability of state w. In this case, the rst-order condition is the existence of l such that v (c0) = l, (2) and for all w = 1, 2, . . . , n, pwu (cw) = lpw, (3) or equivalently u (cw) = lw, (4) where w pw/pw is the state-price density (also called the stochastic discount factor or pricing kernel), which is a measure of priced relative scarcity in state of nature w. Therefore, the marginal utility of consumption in a state is proportional to the relative scarcity. There is a solution if the problem is feasible, prices and probabilities are positive, the von NeumannMorgenstern utility function is increasing and strictly concave, and there is satised the Inada condition limc u (c) = 0.1 There are 1 Proving the existence of a solution requires more assumptions in continuous-state models. 31. Ch. 10: Arbitrage, State Prices and Portfolio Theory 609 different motivations of von NeumannMorgenstern preferences in the literature and the probabilities may be objective or subjective. What is important for us is that the von NeumannMorgenstern utility function represents preferences in the sense that expected utility is higher for more preferred consumption patterns.2 Using von NeumannMorgenstern preferences has been popular in part because of axiomatic derivations of the theory [see, for example, Herstein and Milnor (1953) or Luce and Raiffa (1957, Chapter 2)]. There is also a large literature on alternatives and extensions to von NeumannMorgenstern preferences. For single-period models, see Knight (1921), Bewley (1988), Machina (1982), Blume, Brandenburger and Dekel (1991) and Fishburn (1988). There is an even richer set of models in multiple periods, for example, time-separable von NeumannMorgenstern (the traditional standard), habit formation [e.g., Duesenberry (1949), Pollak (1970), Abel (1990), Constantinides (1991) and Dybvig (1995)], local substitutability over time [Hindy and Huang (1992)], interpersonal dependence [Duesenberry (1949) and Abel (1990)], preference for resolution of uncertainty [Kreps and Porteus(1978)], time preference dependent on consumption [Bergman (1985)], and general recursive utility [Epstein and Zin (1989)]. Recently, there have also been some attempts to revive the age-old idea of studying nancial situations using psychological theories [like prospect theory, Kahneman and Tversky (1979)]. Unfortunately, these models do not translate well to nancial markets. For example, in prospect theory framing matters, that is, the observed phenomenon of an agent making different decisions when facing identical decision problems described differently. However, this is an alien concept for nancial economists and when they proxy for it in models they substitute something more familiar [for example, some history dependence as in Barberis, Huang and Santos (2001)]. Another problem with the psychological theories is that they tend to be isolated stories rather than a general specication, and they are often hard to generalize. For example, prospect theory says that agents put extra weight on very unlikely outcomes, but it is not at all clear what this means in a model with a continuum of states. This literature also has problems with using ex post explanations (positive correlations of returns are underreaction and negative correlations are overreactions) and a lack of clarity of how much is going on that cannot be explained by traditional models (and much of it can). In actual nancial markets, ArrowDebreu securities do not trade directly, even if they can be constructed indirectly using a portfolio of securities. A security is characterized by its cash ows. This description would not be adequate for analysis of taxes, since different sources of cash ow might have very different tax treatment, but we are looking at models without taxes. For an asset like a common stock or a bond, the cash ow might be negative at time 0, from payment of the price, and positive or zero in each state at time 1, the positive amount coming from any repayment of 2 Later, when we look at multiple-agent results, we will also make the neoclassical assumption of identical beliefs, which is probably most naturally motivated by common objective beliefs. 32. 610 P.H. Dybvig and S.A. Ross principal, dividends, coupons, or proceeds from sale of the asset. For a futures contract, the cash ow would be 0 at time 0, and the cash ow in different states at time 1 could be positive, negative, or zero, depending on news about the value of the underlying commodity. In general, we think of the negative of the initial cash ow as the price of a security. We denote by P = {P1, . . . , PN } the vector of prices of the N securities 1, . . . , N, and we denote by X the payoff matrix. We have that Pn is the price we pay for one unit of security n and Xwn is the payoff per unit of security n at time 1 in the single state of nature w. With the choice of a portfolio of assets, our choice problem might become Problem 2: First Portfolio Choice Problem. Choose portfolio holdings Q {Q1, . . . , QN } and consumptions C {c0, . . . , cW } to maximize utility of consumption U(C) subject to portfolio payoffs c {c1, . . . , cw} = X Q and budget constraint c0 + P Q = W. Here, Q is the vector of portfolio weights. Time 0 consumption is the numeraire, and wealth W is now chosen in time 0 consumption units and the entire endowment is received at time 0. In the budget constraint, the term P Q is the cost of the portfolio holding, which is the sum across securities n of the price Pn times the number of shares or other unit Qn. The matrix product X Q says that the consumption in state w is cw = n XwnQn, i.e., the sum across securities n of the payoff Xwn of security n in state w, times the number of shares or other units Qn of security n our champion is holding. The rst-order condition for Problem 2 is the existence of a vector of shadow prices p and a Lagrangian multiplier l such that pwu (cw) = lpw, (5) where P = pX. (6) The rst equation is the same as in the ArrowDebreu model, with an implicit shadow price vector in place of the given ArrowDebreu prices. The second equation is a pricing equation that says the prices of all assets must be consistent with the shadow prices of the states. For the ArrowDebreu model itself, the state-space tableau X is I, the identity matrix, and the price vector P is p, the vector of ArrowDebreu state prices. For the ArrowDebreu model, the pricing equation determines the shadow prices as equal to the state prices. Even if the assets are not the ArrowDebreu securities, Problem 2 may be essentially equivalent to the ArrowDebreu model in Problem 1. In economic terms, the important feature of the ArrowDebreu problem is that all payoff patterns are spanned, i.e., each potential payoff pattern can be generated at some price by some portfolio of assets. Linear algebra tells us that all payoff patterns can be generated if the payoff matrix X 33. Ch. 10: Arbitrage, State Prices and Portfolio Theory 611 has full row rank. If X has full row rank, p is determined (or over-determined) by Equation (6). If p is uniquely determined by the pricing equation (and therefore also all ArrowDebreu assets can be purchased as portfolios of assets in the economy), we say that markets are complete, and for all practical purposes we are in an Arrow Debreu world. For the choice problem to have a solution for any agent who prefers more to less, we also need for the price of each payoff pattern to be unique (the law of one price) and positive, or else there would be arbitrage (i.e., a money pump or a free lunch). If there is no arbitrage, then there is at least one vector of positive state prices p solving the pricing equation (6). There is an arbitrage if the vector of state prices is overdetermined or if all consistent vectors of state prices assign a negative or zero price to some state. The notion of absence of arbitrage is a central concept in nance, and we develop its implications more fully in the section on preference-free results. So far, we have been stating portfolio problems in prices and quantities, as we would in general equilibrium theory. However, it is also common to describe assets in terms of rates of return, which are relative price changes (often expressed as percentages). The return to security n, which is the relative change in total value (including any dividends, splits, warrant issues, coupons, stock issues, and the like as well as change in the price). There is not an absolute standard of what is meant by return, in different contexts this can be the rate of return, one plus the rate of return, or the difference between two rates of return. It is necessary to gure which is intended by asking or from context. Using the notation above, the rate of return in state w is rwn = (Xwn Pn)/Pn.3 Often, consumption at the outset is suppressed, and we specialize to von Neumann Morgenstern expected utility. In this case, we have the following common form of portfolio problem. Problem 3: Portfolio Problem using Returns. Choose portfolio proportions q {q1, . . . , qN } and consumptions c {c1, . . . , cW } to maximize expected utility of consumption W w = 1 pq u(cw) subject to the consumption equation c = Wq (1 + r) and the budget constraint q 1 = 1. Here, p = {p1, . . . , pW } is a vector of state probabilities, u() is the von Neumann Morgenstern utility function, and 1 is a vector of 1s. The dimensionality of 1 is determined implicitly from the context; here the dimensionality is the number of assets. The rst-order condition for an optimum is the existence of shadow state-price density vector and shadow marginal utility of wealth l such that u (cw) = lw (7) 3 One unfortunate thing about returns is that they are not dened for contracts (like futures) that have zero price. However, this can be nessed formally by bundling a futures with a bond or other asset in dening the securities and unbundling them when interpreting the results. Bundling and unbundling does not change the underlying economics due to the linearity of consumptions and constraints in the portfolio choice problem. 34. 612 P.H. Dybvig and S.A. Ross and 1 = E[(1 + r)]. (8) These equations say that the state-price density is consistent with the marginal valuation by the agent and with pricing in the market. As our nal typical problem, let us consider a mean-variance optimization. This optimization is predicated on the assumption that investors care only about mean and variance (typically preferring more mean and less variance), so we have a utility function V(m, v) in mean m and variance v. For this problem, suppose there is a risk- free asset paying a return r (although the market-level implications of mean-variance analysis can also be derived in a general model without a risky asset). In this case, portfolio proportions in the risky assets are unconstrained (need not sum to 1) because the slack can be taken up by the risk-free asset. We denote by m the vector of mean risky asset returns and by s the covariance matrix of risky returns. Then our champion solves the following choice problem. Problem 4: Mean-variance optimization. Choose portfolio proportions q {q1, . . . , qN } to maximize the mean-variance utility function V(r + (m r1) q, q Sq). The rst-order condition for the problem is m r1 = lSq, (9) where q is the optimal vector of portfolio proportions and l is twice the marginal rate of substitution Vv (m, v)/Vm(m, v), evaluated at m = r + (m r1) q and v = q Sq. The rst-order condition (9) says that mean excess return for each asset is proportional to the marginal contribution of volatility to the agents optimal portfolio. We have seen a few of the typical types of portfolio problem. There are a lot of variations. The problem might be stated in terms of excess returns (rate of return less a risk-free rate) or total return (one plus the rate of return). Or, we might constrain portfolio holdings to be positive (no short sales) or we might require consumption to be nonnegative (limited liability). Many other variations adapt the basic portfolio problem to handle institutional features not present in a neoclassical formulation, such as transaction costs, bidask spreads, or taxes. These extensions are very interesting, but beyond the scope of what we are doing here, which is to explore the neoclassical foundations. 3. Absence of arbitrage and preference-free results Before considering specic solutions and applications, let us consider some general results that are useful for thinking about portfolio choice. These results are preference- free in the sense that they do not depend on any specic assumptions about preferences 35. Ch. 10: Arbitrage, State Prices and Portfolio Theory 613 but only depend on an assumption that agents prefer more to less. Central to this section is the notion of an arbitrage, which is a money pump or a free lunch. If there is arbitrage, linearity of the neoclassical problem implies that any candidate optimum can be dominated by adding the arbitrage. As a result, no agent who prefers more to less would have an optimum if there exists arbitrage. Furthermore, this seemingly weak assumption is enough to obtain two useful theorems. The Fundamental Theorem of Asset Pricing says that the following are equivalent: absence of arbitrage, existence of a consistent positive linear pricing rule, and existence of an optimum for some hypothetical agent who prefers more to less. The Pricing Rule Representation Theorem gives different equivalent forms for the consistent positive linear pricing rule, using state prices, risk-neutral probabilities (martingale valuation), state-price density (or stochastic discount factor or pricing kernel), or an abstract positive linear operator. The results in this section are from Cox and Ross (1975), Ross (1976c, 1978b) and Dybvig and Ross (1987). The results have been formalized in continuous time by Harrison and Kreps (1979) and Harrison and Pliska (1981). Occasionally, the theorems in this section can be applied directly to obtain an interesting result. For example, linearity of the pricing rule is enough to derive put- call parity without constructing the arbitrage. More often, the results in this section help to answer conceptual questions. For example, an option pricing formula that is derived using absence of arbitrage is always consistent with equilibrium, as can be seen from the Fundamental Theorem. By the Fundamental Theorem, absence of arbitrage implies there is an optimum for some hypothetical agent who prefers more to less; we can therefore construct an equilibrium in the single-agent pure exchange economy in which this agent is endowed with the optimal holding. By construction the equilibrium in this economy will have the desired pricing, and therefore any no-arbitrage pricing result is consistent with some equilibrium. In this section, we will work in the context of Problem 2. An arbitrage is a change in the portfolio that makes all agents who prefer more to less better off. We make all such agents better off if we increase consumption sometime, and in some state of nature, and we never decrease consumption. By combining the two constraints in Problem 2, we can write the consumption C associated with any portfolio choice Q using the stacked matrix equation C = W 0 + P X Q. The rst row, W P Q, is consumption at time 0, which is wealth W less the cost of our portfolio. The remaining rows, X Q, give the random consumption across states at time 1. Now, when we move from the portfolio choice Q to the portfolio choice Q + h, the initial wealth term cancels and the change in consumption can now be written as DC = P X h. 36. 614 P.H. Dybvig and S.A. Ross This will be an arbitrage if DC is never negative and is positive in at least one component, which we will write as4 DC > 0 or P X h > 0. Some authors describe taxonomies of different types of arbitrage, having perhaps a negative price today and zero payoff tomorrow, a zero price today and a nonnegative but not identically zero payoff tomorrow, or a negative price today and a positive payoff tomorrow. These are all examples of arbitrages that are subsumed by our general formula. The important thing is that there is an increase in consumption in some state of nature at some point of time and there is never any decrease in consumption. 3.1. Fundamental theorem of asset pricing Theorem 1: Fundamental Theorem of Asset Pricing. The following conditions on prices P and payoffs X are equivalent: (i) Absence of arbitrage: ( /h) P X h > 0 . (ii) Existence of a consistent positive linear pricing rule (positive state prices): (p 0)(P = p X ). (iii) Some agent with strictly increasing preferences U has an optimum in Problem 2. Proof: We prove the equivalence by showing (i) (ii), (ii) (iii), and (iii) (i). (i) (ii): This is the most subtle part, and it follows from a separation theorem or the duality theorem from linear programming. From the denition of absence of arbitrage, we have that the sets S1 P X h | h Rn and S2 x RW + 1 | x > 0 must be disjoint. Therefore, there is a separating hyperplane z such that z x = 0 for all x S1 and z x > 0 for all x S2. [See Karlin (1959), Theorem B3.5] Normalizing so that the rst component (the shadow price of time zero consumption) is 1, we will see that p dened by (1 p ) = z/z0 is the consistent linear pricing rule we seek. Constancy 4 We use the following terminology for vector inequalities: (x y) (i) (xi yi), (x > y) ((x y) & (i) (xi > yi)), and (x y) (i) (xi > yi). 37. Ch. 10: Arbitrage, State Prices and Portfolio Theory 615 of zx for x S1 implies that (1 p ) P X = 0, which is to say that P = p X , i.e., p is a consistent linear pricing rule. Furthermore, z x positive for x S2 implies z 0 and consequently p 0, and p is indeed the desired consistent positive linear pricing rule. (ii) (iii): This part is proven by construction. Let U(C) = (1 p ) C, then Q = 0 solves Problem 2. To see this, note that the objective function U(C) is constant and equal to W for all Q: U(C) = (1 p ) C = (1 p ) W 0 + P X Q = W + (P + p X ) Q = W. (The motivation of this construction is the observation that the existence of the consistent linear pricing rule with state prices p implies that all feasible consumptions satisfy (1 p ) C = W.) (iii) (i): This part is obvious, since any candidate optimum is dominated by adding the arbitrage, and therefore there can be no arbitrage if there is an optimum. More formally, adding an arbitrage implies the change of consumption DC > 0, which implies an increase in U(C). One feature of the proof that may seem strange is the degeneracy (linearity) of the utility function whose existence is constructed. This was all that was needed for this proof, but it could also be constructed to be strictly concave, additively separable over time, and of the von NeumannMorgenstern class for given probabilities. Assuming any of these restrictions on the class would make some parts of the theorem weaker [(iii) implies (i) and (ii)] at the same time that it makes other parts stronger [(i) or (ii) implies (iii)]. The point is that the theorem is still true if (iii) is replaced by a much more restrictive class that imposes on U any or all of strict concavity, some order of differentiability, additive separability over time, and a von NeumannMorgenstern form with or without specifying the probabilities in advance. All of these classes are restrictive enough to rule out arbitrage, and general enough to contain a utility function that admits an optimum when there is no arbitrage. The statement and proof of the theorem are a little more subtle if the state space is innite-dimensional. The separation theorem is topological in nature, so we must restrict our attention to a topologically relevant subset of the nonnegative random variables. Also, we may lose the separating hyperplane theorem because the interior of the positive orthant is empty in most of these spaces (unless we use the sup-norm topology, in which case the dual is very large and includes dual vectors that do not support state prices). However, with some denition of arbitrage in limits, the economic content of the Fundamental Theorem can be maintained. 38. 616 P.H. Dybvig and S.A. Ross 3.2. Pricing rule representation theorem Depending on the context, there are different useful ways of representing the pricing rule. For some abstract applications (like proving putcall parity), it is easiest to use a general abstract representation as a linear operator L(c) such that c > 0 L(c) > 0. For asset pricing applications, it is often useful to use either the state-price representation we used in the Fundamental Theorem, L(c) = w pwcw, or risk-neutral probabilities, L(c) = (1 + r )1 E [cw] = (1 + r )1 w p wcw. The intuition behind the risk-neutral representation (or martingale representation5 ) is that the price is the expected discounted value computed using a shadow risk-free rate (equal to the actual risk-free rate if there is one) and articial risk-neutral probabilities p that assign positive probability to the same states as do the true probabilities. Risk- neutral pricing says that all investments are fair gambles once we have adjusted for time preference by discounting and for risk preference by adjusting the probabilities. The nal representation using the state-price density (or stochastic discount factor) to write L(c) = E[wcw] = w pwwcw. The state price density simplies rst- order conditions of portfolio choice problems because the state-price density measures priced scarcity of consumption. The state-price density is also handy for continuous- state models in which individual states have zero state probabilities and state prices but there exists a well-dened positive ratio of the two. Theorem 2: Pricing Rule Representation Theorem. The consistent positive linear pricing rule can be represented equivalently using (i) an abstract linear functional L(c) that is positive: (c > 0) (L(c) > 0) (ii) positive state prices p 0: L(c) = W w = 1 pwcw (iii) positive risk-neutral probabilities p 0 summing to 1 with associated shadow risk-free rate r : L(c) = (1 + r )1 E [cw] (1 + r )1 w p wcw (iv) positive state-price densities 0: L(c) = E[c] w pwwcw. Proof: (i) (ii): This is the known form of a linear operator in RW ; p 0 follows from the positivity of L. (ii) (iii): Note rst that the shadow risk-free rate must price the riskless asset c = 1: W w = 1 pw1 = (1 + r )1 E [1], which implies (since E [1] = 1) that r = 1/p 1 1. Then, matching coefcients in W w = 1 pwcw = (1 + r )1 w p wcw, 5 The reason for calling the term martingale representation is that using the risk-neutral probabilities makes the discounted price process a martingale, which is a stochastic process that does not increase or decrease on average. 39. Ch. 10: Arbitrage, State Prices and Portfolio Theory 617 we have that p = p/1 p, which sums to 1 as required and inherits positivity from p. (iii) (iv): Simply let w = (1 + r )1 p w (which is the same as pw/pw). (iv) (i): immediate. Perhaps what is most remarkable about the Fundamental Theorem and the Representation Theorem is that neither probabilities nor preferences appear in the determination of the pricing operator, beyond the initial identication of which states have nonzero probability and the assumption that more is preferred to less. It is this observation that empowers the theory of derivative asset pricing, and is, for example, the reason why the BlackScholes option price does not depend on the mean return on the underlying stock. Preferences and beliefs are, however, in the background: in equilibrium, they would inuence the price vector P and/or the payoff matrix X (or the mean return process for the BlackScholes stock). Although the focus of this chapter is on the single-period model, we should note that the various representations have natural multiperiod extensions. The abstract linear functional and state prices have essentially the same form, noting that cash ows now extend across time as well as states of nature and that there are also conditional versions of the formula at each date and contingency. In some models, the information set is generated by the sample path of security prices; in this case the state of nature is a sample path through the tree of potential security prices. For the state-price density in multiple periods, there is in general a state-price-density process { t} whose relatives can be used for valuation. For example, the value at time s of receiving subsequent cash ows cs+1, cs+2 . . . ct is given by t t = s + 1 Es t s ct , (10) where Es[] denotes expectation conditional on information available at time s. Basically, this follows from iterated expectations and dening t as a cumulative product of single-period s. Similarly, we can write risk-neutral valuation as Ps = E s Pt (1 + r s )(1 + r s + 1) . . . (1 + r t ) . (11) Note that unless the risk-free rate is nonrandom, we cannot take the discount factors out of the expectation.6 This is because of the way that the law of iterated expectations works. For example, consider the value V0 at time 0 of the cash ow in time 2. V0 = (1 + r 1 )1 E 0 [V1] = (1 + r 1 )1 E 0 [(1 + r 2 )1 E 1 [c2]] = (1 + r 1 )1 E 0 [(1 + r 2 )1 c2]. (12) 6 It would be possible to treat the whole time period from s to t as a single period and apply the pricing result to that large period in which case the discounting would be at the appropriate (t s)-period rate. The problem with this is that the risk-neutral probabilities would be different for each pair of dates, which is unnecessarily cumbersome. 40. 618 P.H. Dybvig and S.A. Ross Now, (1 + r 1 )1 is outside the expectation (as could be (1 + r s + 1)1 in Equation (11), but (1 + r 2 )1 cannot come outside the expectation unless it is nonrandom.7 So, it is best to remember that when interest rates are stochastic, discounting for risk-neutral valuation should use the rolled-over spot rate, within the expectation. 4. Various analyses: ArrowDebreu world The portfolio problem is the starting point of a lot of types of analysis in nance. Here are some implications that can be drawn from portfolio problems (usually through the rst-order conditions): optimal portfolio choice (asset allocation or stock selection) portfolio efciency aggregation and market-level implications asset pricing and performance measurement payoff distribution pricing recovery or estimation of preferences inference of expectations We can think of many of these distinctions as a question of what we are solving for when we look at the rst-order conditions. In optimal portfolio choice and its aggregation, we are solving for the portfolio choice given the preferences and beliefs about returns. In asset pricing, we are computing the prices (or restrictions on expected returns) given preferences, beliefs about payoffs, and the optimal choice (which is itself often derived using an aggregation result). In recovery, we derive preferences from beliefs and idealized observations about portfolio choice, e.g. at all wealth levels. Estimation of preferences is similar, but works with noisy observations of demand at a nite set of data points and uses a restriction in the functional form or smoothing in the statistical procedure to identify preferences. And, inference of expectations derives probability beliefs from preferences, prices, and the (observed) optimal demand. In this section, we illustrate the various analyses in the case of an ArrowDebreu world. Analysis of the complete-markets model has been developed by many people over a period of time. Some of the more important works include some of the original work on competitive equilibrium such as Arrow and Debreu (1954), Debreu (1959) and Arrow and Hahn (1971), as well as some early work specic to security markets such as Arrow (1964), Rubinstein (1976), Ross (1976b), Banz and Miller (1978) and Breeden and Litzenberger (1978). There are also a lot of papers set in 7 In the special case in which c2 is uncorrelated with (1 + r 2)1 (or in multiple periods if cash ows are all independent of shadow interest rate moves), we can take the expected discount factor outside the expectation. In this case, we can use the multiperiod riskfree discount bond rate for discounting a simple expected nal. However, in general, it is best to remember the general formula (11) with the rates in the denominator inside the expectation. 41. Ch. 10: Arbitrage, State Prices and Portfolio Theory 619 multiple periods that contributed to the nance of complete markets; although not strictly within the scope of this chapter, we mention just a few here: Black and Scholes (1973), Merton (1971, 1973), Cox, Ross and Rubinstein (1979) and Breeden (1979). 4.1. Optimal portfolio choice The optimal portfolio choice is the choice of consumptions (c0, c1, . . . , cW ) and Lagrange multiplier l to solve the budget constraint (1) and the rst-order conditions (2) and (3). If the inverse I() of u () and the inverse J() of v () are both known analytically, then nding the optimum can be done using a one-dimensional monotone search for l such that J(l) + W w = 1 pwI(lpw/pw) = W. In some special cases, we can solve the optimization analytically. For logarithmic utility, v(c) = log(c) and u(c) = d log(c) for some d > 0, optimal consumption is given by c0 = W/(1 + d) and cw = pwWd/((1 + d) pw) (for w = 1, . . . , W). The portfolio choice can also be solved analytically for quadratic utility. 4.2. Efcient portfolios Efcient portfolios are the ones that are chosen by some agent in a given class of utility functions. For the ArrowDebreu problem, we might take the class of utility functions to be the class of differentiable, increasing and strictly concave time-separable von NeumannMorgenstern utility functions U(c) = v(c0) + W w = 1 pwu(cw).8 Since u() is increasing and strictly concave, (cw > cw ) u (cw) < u (cw). Consequently, the rst- order condition (4) implies that (cw > cw ) ( w < w ). Since the state-price density w pw/pw is a measure of priced social scarcity in state w, this says that we consume less in states in which consumption is more expensive. This necessary condition for efciency is also sufcient; if consumption reverses the order across states of the state- price density, then it is easy to construct a utility function that satises the rst-order conditions. Formally, Theorem 3: ArrowDebreu Portfolio Efciency. Consider a complete-markets world (in which agents solve Problem 1) in which state prices and probabilities are all strictly positive, and let U be the class of differentiable, increasing and strictly concave time-separable von NeumannMorgenstern utility functions of the form U(c) = v(c0) + W w = 1 pwu(cw). Then there exists a utility function in the class U that chooses the consumption vector c satisfying the budget constraint if and only if consumptions at time 1 are in the opposite order as the state-price densities, i.e., (w, w {1, . . . , W})((cw > cw ) ( w < w )). 8 A non-time-separable version would be of the form U(c) = W w = 1 pwu(c0, cw). 42. 620 P.H. Dybvig and S.A. Ross Proof: The only if part follows directly from the rst-order condition and concavity as noted in the paragraph above. For the if part, we are given a consumption vector with the appropriate ordering and we will construct a utility function that will choose it and satisfy the rst-order condition with l = 1. For this, choose v(c) = exp((c c0)) (so that v (c0) = 1 as required by Equation 2), and choose u (c) to be any strictly positive and strictly decreasing function satisfying u (cw) = w for all w = {1, 2, . . . , W}, for example, by connecting the dots (with appropriate treatment past the endpoints) in the graph of w as a function of cw. Integrating this function yields a utility function u() such that the von NeumannMorgenstern utility function satises the rst-order conditions, and by concavity this rst-order solution is a solution. Friendly warning. There are many notions of efciency in nance: Pareto efciency, informational efciency, and the portfolio efciency we have mentioned are three leading examples. A common mistake in heuristic arguments is to assume incorrectly that one sense of efciency necessarily implies another. 4.3. Aggregation Aggregation results typically show what features of individual portfolio choice are preserved at the market level. Many asset pricing results follow from aggregation and the rst-order conditions. The most common type of aggregation result is the efciency of the market portfolio. For most classes of preferences we consider, the efcient set is unchanged by rescaling wealth, and consequently the market portfolio is always efcient if and only if the efcient set is convex. This is because the market portfolio is a rescaled version of the individual portfolios. (If the portfolios are written in terms of proportions, no rescaling is needed). When the market portfolio is efcient, then we can invert the rst-order condition for the hypothetical agent who holds the market portfolio to obtain the pricing rule. In the ArrowDebreu world, the market portfolio is always efcient. This is because the ordering across states is preserved when we sum individual portfolio choices to form the market portfolio. Consider agents m = 1, . . . , M with felicity functions v1 (), . . . , vM () and u1 (), . . . , uM () and optimal consumptions C1 , . . . , CM . The following results are close relatives of standard results in general equilibrium theory. Theorem 4: Aggregation Theorem. In a pure exchange equilibrium in a complete market, (i) all agents order time 1 consumption in the same order across states, (ii) aggregate time 1 consumption is in the same order across states, (iii) equilibrium is Pareto optimal, and (iv) there is a time separable von NeumannMorgenstern utility function that would choose optimally aggregate consumption. Proof: (i) and (ii) Immediate, given Theorem 3. 43. Ch. 10: Arbitrage, State Prices and Portfolio Theory 621 (iii) Let lm be the Lagrangian multiplier at the optimum in the rst order condition in agent ms decision problem. Consider the problem of maximizing the linear social welfare function with weights lm , namely N n = 1 ln vn (c0) + W w = 1 un (cn w) . It is easy to verify from the rst-order conditions from the equilibrium consumptions that they solve this problem too. This is a concave optimization, so the rst-order conditions are sufcient, and since the welfare weights are positive the solution must be Pareto optimal (or else a Pareto improvement would increase the objective function). (iv) Dene vA (c) maxcn s N n = 1 ln vn (cn) to be the rst-period aggregate felicity function and dene uA (c) maxcn s W n = 1 ln un (cn) to be the second-period aggregate felicity function. Then the utility function vA (c0) + E[uA (cw)] is a time-separable von NeumannMorgenstern utility function that would choose the markets aggregate consumption, since the objective function is the same as for the social welfare problem described under the proof of (iii). There is a different perspective that gives an alternative proof of the existence of a represenatitive agent (iv). The existence of a representative agent follows from the convexity of the set of efcient portfolios derived earlier. The main condition we require to have this work is that the efcient set of portfolio proportions is the same at all wealth levels, which is true here and typically of the cases we consider. 4.4. Asset pricing Asset pricing gets its name from valuation of cash ows, although asset pricing formulas may be expressed in several different ways, for example as a formula explaining expected returns across assets or as a moment condition satised by returns that can be tested econometrically. Let vA () and uA () represent the preferences of the hypothetical agent who holds aggregate consumption, as guaranteed by the aggregation theorem 4. Then we can solve the rst-order conditions (2) and (3) to compute pw = pwuA (cA w)/vA (cA 0 ) and therefore the time-0 valuation of the time-1 cash ow vector {c1, . . . , cW } is L(c1, . . . , cW ) = W w = 1 pw uA (cA w) vA (cA 0 ) cw = E uA (cA w) vA (cA 0 ) cw . (13) This formula (with state-price density w = uA (cA w)/vA (cA 0 ) is the right one for pricing assets, but asset pricing equations are more often expressed as explanations of mean 44. 622 P.H. Dybvig and S.A. Ross returns across assets or as moment conditions satised by returns. Dening the rate of return (the relative value change) for some asset as rw (cw P)/P where cw is the asset value in state w and P is the assets price. Letting rf be the risk-free rate of return (or the riskless interest rate), which must be rf = 1 E[uA (cA w)/vA (cA 0 )] , (14) we have that Equation (13) implies E[rw] = rf + (1 + rf ) cov uA (cA w) vA (cA 0 ) , rw , (15) so that the risk premium (the excess of expected return over the risk-free rate) is proportional to covariance of return with the state-price density. This is the representation of asset pricing in terms of expected returns, and is also the so-called consumption-capital asset pricing model (CCAPM) that is more commonly studied in a multiperiod setting. Either of the pricing relations could be used as moment conditions in an asset-pricing test, but it is more common to use the moment condition 1 = E uA (cA w) vA (cA 0 ) (1 + rw) , (16) to test the CCAPM. This same equations characterize pricing for just about all the pricing models (perhaps with optimal consumption for some agent in place of aggregate consumption). Recall that the rst-order conditions are just about the same whether markets are complete or incomplete. The main difference is that the state prices are shadow prices (Lagrangian multipliers) when markets are incomplete, but actual asset prices in complete markets. Either way, the rst-order conditions are consistent with the same asset pricing equations. 4.5. Payoff distribution pricing For von NeumannMorgenstern preferences (expected utility theory) and more general Machina preferences, preferences depend only on distributions of returns and payoffs and do not depend on the specic states in which those returns are realized. Consider, for example, a simple example with three equally probable states, p1 = p2 = p3 = 1 3 . Suppose that an individual has to choose one of the following payoff vectors for consumption at time 1: c1 = (1, 2, 2), c2 = (2, 1, 2), and c3 = (2, 2, 1). These three consumption patterns have the same distribution of consumption, giving consumption of 1 with probability 1 3 and consumption of 2 with probability 2 3 . Therefore, an agent with von NeumannMorgenstern preferences or more general Machina preferences 45. Ch. 10: Arbitrage, State Prices and Portfolio Theory 623 would nd all these consumption vectors are equally attractive. However they do not all cost the same unless the state-price density (and in the example, the state price) is the same in all states. However, having the state-price density the same in all states is a risk-neutral world all consumption bundles priced at their expected value which is not very interesting since all risk-averse agents would choose a riskless investment.9 In general, we expect the state-price density to be highest of states of social scarcity, when the market is down or the economy is in recession, since buying consumption in states of scarcity is a form of insurance. Suppose that the state-price vector is p = (.3, .2, .4). Then the prices of the bundles can be computed as p ca = .3 1 + .2 2 + .4 2 = 1.5, p cb = .3 2 + .2 1 + .4 2 = 1.6, and p cc = .3 2 + .2 2 + .4 1 = 1.4. The cheapest consumption pattern is cc , which places the larger consumption in the cheap states and the smallest consumption in the most expensive state. This gives us a very useful cash-value measure of the inefciency of the other strategies. An agent will save 1.5 1.4 = 0.1 cash up front by choosing cc up front instead of ca or 1.6 1.4 = 0.2 cash up front by choosing cc instead of cb . Therefore, we can interpret 0.1 as a lower bound on the amount of inefciency in ca , since any agent would pay that amount to swap to cc and perhaps more to swap to something better. The only assumption we need for this result is that the agent has preferences (such as von NeumannMorgenstern preferences or Machina preferences) that care only about the distribution of consumption and not the identity of the particular states in which different parts of the distribution are realized. The general result is based on the deep theoretical insight that you should buy more when it is cheaper. This means that efcient consumption is decreasing in the state-price density. We can compute the (lower bound on the) inefciency of the portfolio by reording its consumption in reverse order as the state-price density and computing the decline in cost. The payoff distributional price of a consumption pattern is the price of getting the same distribution the cheapest possible way (in reverse order as the state-price density). There is a nice general formula for the distributional price. Let Fc() be the cumulative distribution function of consumption and let ic() be its inverse. Similarly, let F() be the cumulative distribution function of the state-price density and let i() be its inverse. Let c be the efcient consumption pattern with distribution function Fc(). Then the distributional price of the consumption pattern can be written as E[c ] = 1 z = 0 ic(z) i(1 z) dz. (17) In this expression, z has units of probability and labels the states ordered in reverse of the state-price density, i(1 z) is state-