The Distribution of Risk Aversion

This paper develops a framework for deriving and inferring the distribution of relative risk aversion from financial markets. The theoretical constructions (i) rely on a fairly robust form of aggregating the marginal rate of substitution of individuals that are either long or short the market-index, and (ii) specifies a positive measure for the risk aversion coefficient capturing the feature that a proportion of the population possesses a distinct risk aversion. The implementation of the theoretical model reveals substantial heterogeneity in the coefficient of relative risk aversion. Our empirical approach supports the competitive markets paradigm that enforces positive skewness in the risk aversion distribution. The evidence also points to the presence of a risk aversion distribution that is characterized by heavy tails. We discuss the asset pricing implications of theory and empirical findings.

[1]  Miles S. Kimball,et al.  Preference Parameters and Behavioral Heterogeneity: An Experimental Approach in the Health and Retirement Survey , 1995 .

[2]  D. Madan,et al.  Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options , 2000 .

[3]  Liuren Wu,et al.  Asset Pricing under the Quadratic Class , 2002, Journal of Financial and Quantitative Analysis.

[4]  J. Lintner.  The Aggregation of Investor's Diverse Judgments and Preferences in Purely Competitive Security Markets , 1969, Journal of Financial and Quantitative Analysis.

[5]  Richard J. Zeckhauser,et al.  Proper risk aversion , 1987 .

[6]  Robert B. Wilson.  THE THEORY OF SYNDICATES , 1968 .

[7]  Nicolas P. B. Bollen,et al.  Does Net Buying Pressure Affect the Shape of Implied Volatility Functions? , 2002 .

[8]  Miles S. Kimball,et al.  Standard Risk Aversion , 1991 .

[9]  C. Gollier.  The economics of risk and time , 2001 .

[10]  Mark Rubinstein,et al.  An aggregation theorem for securities markets , 1974 .

[11]  S. Ross.  SOME STRONGER MEASURES OF RISK AVERSION IN THE SMALL AND THE LARGE WITH APPLICATIONS , 1981 .

[12]  G. D'Avolio,et al.  The Market for Borrowing Stock , 2002 .

[13]  Darrell Duffie,et al.  Securities Lending, Shorting, and Pricing , 2001 .

[14]  Ravi Jagannathan,et al.  Implications of Security Market Data for Models of Dynamic Economies , 1990, Journal of Political Economy.

[15]  Patrick J. Dennis,et al.  Risk-Neutral Skewness: Evidence from Stock Options , 2002 .

[16]  Ravi Jagannathan,et al.  Assessing Specification Errors in Stochastic Discount Factor Models , 1994 .

[17]  J. Jackwerth.  Recovering Risk Aversion from Option Prices and Realized Returns , 1998 .

[18]  S. Ross,et al.  AN INTERTEMPORAL GENERAL EQUILIBRIUM MODEL OF ASSET PRICES , 1985 .

[19]  P. Carr,et al.  Variance Risk Premia , 2007 .

[20]  R. C. Merton,et al.  AN INTERTEMPORAL CAPITAL ASSET PRICING MODEL , 1973 .

[21]  D. Duffie.  Dynamic Asset Pricing Theory , 1992 .

[22]  Markus K. Brunnermeier,et al.  Do Wealth Fluctuations Generate Time-Varying Risk Aversion? Micro-Evidence on Individuals' Asset Allocation , 2008 .

[23]  Darrell Duffie,et al.  Securities lending, shorting, and pricing , 2002 .

[24]  R. Mehra,et al.  THE EQUITY PREMIUM A Puzzle , 1985 .

[25]  Kenneth J. Arrow,et al.  Aspects of the Theory of Risk Bearing--Yrjo Jahnsson Lectures , 1969 .

[26]  Hayne E. Leland,et al.  Who Should Buy Portfolio Insurance , 1980 .

[27]  W. Feller,et al.  An Introduction to Probability Theory and Its Applications, Vol. II , 1972, The Mathematical Gazette.

[28]  Jordan Stoyanov,et al.  ESSENTIALS OF STOCHASTIC FINANCE , 2000 .

[29]  L. Hansen,et al.  Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models , 1982 .

[30]  Jun Pan.  The jump-risk premia implicit in options: evidence from an integrated time-series study $ , 2002 .

[31]  Liuren Wu,et al.  Dampened Power Law: Reconciling the Tail Behavior of Financial Security Returns , 2004 .

[32]  George M. Constantinides,et al.  Intertemporal Asset Pricing with Heterogeneous Consumers and without Demand Aggregation , 1982 .

[33]  Adam V. Reed,et al.  Stocks are Special Too: An Analysis of the Equity Lending Market , 2002 .

[34]  J. Pratt.  RISK AVERSION IN THE SMALL AND IN THE LARGE11This research was supported by the National Science Foundation (grant NSF-G24035). Reproduction in whole or in part is permitted for any purpose of the United States Government. , 1964 .

[35]  David M. Kreps,et al.  Speculative Investor Behavior in a Stock Market with Heterogeneous Expectations , 1978 .

[36]  Jiang Wang.  The Term Structure of Interest Rates in a Pure Exchange Economy with Heterogeneous Investors , 1995 .

[37]  A. Lo,et al.  Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test , 1987 .

[38]  Roy Radner,et al.  Existence of Equilibrium of Plans, Prices, and Price Expectations in a Sequence of Markets , 1972 .

[39]  Robert F. Engle,et al.  Risk and Volatility: Econometric Models and Financial Practice , 2004 .

[40]  A. Shiryaev.  Essentials of stochastic finance , 1999 .

[41]  A. Lo.  The Statistics of Sharpe Ratios , 2002 .

[42]  S. Ross.  The Economic Theory of Agency: The Principal's Problem , 1973 .

[43]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[44]  H. White,et al.  On More Robust Estimation of Skewness and Kurtosis: Simulation and Application to the S&P500 Index , 2003 .

[45]  Lennart Bondesson,et al.  Generalized Gamma Convolutions and Related Classes of Distributions and Densities , 1992 .

[46]  Larry G. Epstein,et al.  Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: An Empirical Analysis , 1991, Journal of Political Economy.

[47]  J. Pratt.  RISK AVERSION IN THE SMALL AND IN THE LARGE11This research was supported by the National Science Foundation (grant NSF-G24035). Reproduction in whole or in part is permitted for any purpose of the United States Government. , 1964 .

[48]  J. Caballe,et al.  Mixed Risk Aversion , 1995 .

[49]  E. Miller.  Risk, Uncertainty, and Divergence of Opinion , 1977 .

[50]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .

[51]  Mark Britten-Jones,et al.  Option Prices, Implied Price Processes, and Stochastic Volatility , 2000 .

[52]  Christian Gollier,et al.  Risk Vulnerability and the temper-ing E ect of Background Risk , 1996 .

[53]  Markus K. Brunnermeier,et al.  Do Wealth Fluctuations Generate Time-Varying Risk Aversion? Micro-Evidence on Individuals' Asset Allocation , 2005 .

[54]  M. Rubinstein.,et al.  Recovering Probability Distributions from Option Prices , 1996 .

[55]  Andrea Buraschi,et al.  Model Uncertainty and Option Markets with Heterogeneous Beliefs , 2006 .

[56]  E. Fama,et al.  Some Properties of Symmetric Stable Distributions , 1968 .

[57]  Jonathan E. Ingersoll,et al.  Mean-Variance Theory in Complete Markets , 1982 .

[58]  P. Carr,et al.  The Variance Gamma Process and Option Pricing , 1998 .

[59]  Stephen Figlewski,et al.  Options, Short Sales, and Market Completeness , 1993 .

[60]  J. Ingersoll.  Theory of Financial Decision Making , 1987 .

[61]  J. K. Hunter,et al.  Measure Theory , 2007 .

[62]  Per Krusell,et al.  Income and Wealth Heterogeneity in the Macroeconomy , 1998, Journal of Political Economy.

[63]  David M. Kreps,et al.  Martingales and arbitrage in multiperiod securities markets , 1979 .

[64]  A. M. Mathai,et al.  Quadratic forms in random variables : theory and applications , 1992 .

[65]  Darrell Duffie,et al.  Asset Pricing with Heterogeneous Consumers , 1996, Journal of Political Economy.

[66]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[67]  Jérôme Detemple,et al.  Intertemporal Asset Pricing with Heterogeneous Beliefs , 1994 .

[68]  Wayne E. Ferson,et al.  Seasonality and Consumption-Based Asset Pricing , 1992 .

[69]  Suleyman Basak,et al.  An Equilibrium Model with Restricted Stock Market Participation , 1997 .

[70]  R. Lucas.  ASSET PRICES IN AN EXCHANGE ECONOMY , 1978 .

[71]  Stephen Gordon,et al.  A Preference Regime Model of Bull and Bear Markets , 2000 .

[72]  Benoit B. Mandelbrot,et al.  Fractals and Scaling in Finance , 1997 .

[73]  W. Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[74]  R. Jarrow.  Heterogeneous Expectations, Restrictions on Short Sales, and Equilibrium Asset Prices , 1980 .

[75]  Michael S. Gibson,et al.  Dynamic Estimation of Volatility Risk Premia and Investor Risk Aversion from Option-Implied and Realized Volatilities , 2007 .

[76]  Yisong S. Tian,et al.  The Model-Free Implied Volatility and Its Information Content , 2005 .

[77]  R. Bliss,et al.  Option-Implied Risk Aversion Estimates , 2004 .

[78]  J. Campbell,et al.  By Force of Habit: A Consumption‐Based Explanation of Aggregate Stock Market Behavior , 1995, Journal of Political Economy.

[79]  Fischer Black,et al.  Mean Reversion and Consumption Smoothing , 1989 .

[80]  M. Yor,et al.  The Fine Structure of Asset Retums : An Empirical Investigation ' , 2006 .

[81]  W. Andrew.  LO, . Long-Term Memory in Stock Market Prices, Econometrica, , . , 1991 .

[82]  D. Madan,et al.  Spanning and Derivative-Security Valuation , 2000 .

[83]  Joel M. Vanden.  Option Coskewness and Capital Asset Pricing , 2006 .

[84]  A. Lo,et al.  Nonparametric Risk Management and Implied Risk Aversion , 2000 .

[85]  Dilip B. Madan,et al.  Optimal positioning in derivative securities , 2001 .

[86]  G. Constantinides,et al.  Habit Persistence and Durability in Aggregate Consumption: Empirical Tests , 1991 .

[87]  Patrick L. Brockett,et al.  A class of utility functions containing all the common utility functions , 1987 .

[88]  P. Carr,et al.  Time-Changed Levy Processes and Option Pricing ⁄ , 2002 .

[89]  Jun Pan.  The Jump-Risk Premia Implicit in Options : Evidence from an Integrated Time-Series Study , 2001 .

[90]  Gurdip Bakshi,et al.  The Spirit of Capitalism and Stock-Market Prices , 1996 .

[91]  Stephen Figlewski,et al.  The Informational Effects of Restrictions on Short Sales: Some Empirical Evidence , 1981, Journal of Financial and Quantitative Analysis.

[92]  M. Blume,et al.  The Demand for Risky Assets , 1975 .

[93]  Yacine Ait-Sahalia,et al.  Luxury Goods and the Equity Premium , 2001 .

[94]  David S. Bates.  Post-'87 crash fears in the S&P 500 futures option market , 2000 .

[95]  Hayne E. Leland,et al.  On Equilibrium Asset Price Processes , 1993 .