Probability and Risk: Foundations and Economic Implications of Probability-Dependent Risk Preferences

A large body of evidence has documented that risk preferences depend nonlinearly on outcome probabilities. We discuss the foundations and economic consequences of probability-dependent risk preferences and offer a practitioner's guide to understanding and modeling probability dependence. We argue that probability dependence provides a unifying framework for explaining many real-world phenomena, such as the equity premium puzzle, the long-shot bias in betting markets, and households' underdiversification and their willingness to buy small-scale insurance at exorbitant prices. Recent findings indicate that probability dependence is not just a feature of laboratory data, but is indeed manifest in financial, insurance, and betting markets. The neglect of probability dependence may prevent researchers from understanding and predicting important phenomena.

[1]  D. Prelec.  The Probability Weighting Function , 1998 .

[2]  R. Hodrick,et al.  The Cross-Section of Volatility and Expected Returns , 2006 .

[3]  Brian H. Boyer,et al.  Expected Idiosyncratic Skewness , 2009 .

[4]  J. Hey,et al.  INVESTIGATING GENERALIZATIONS OF EXPECTED UTILITY THEORY USING EXPERIMENTAL DATA , 1994, Experiments in Economics.

[5]  Atanu Saha,et al.  Expo-Power Utility: A ‘Flexible’ Form for Absolute and Relative Risk Aversion , 1993 .

[6]  Uzi Segal,et al.  SOME REMARKS ON QU|GGIN'S ANTICIPATED UTILITY* , 1986 .

[7]  J. Potters,et al.  An experiment on risk taking and evaluation periods , 1997 .

[8]  Uzi Segal,et al.  First order versus second order risk aversion , 1990 .

[9]  M. Yaari.  The Dual Theory of Choice under Risk , 1987 .

[10]  The Revealed Preference Implications of Reference Dependent Preferences , 2007 .

[11]  Jeffrey A. Dubin,et al.  A Microeconometric Analysis of Risk Aversion and the Decision to Self-Insure , 1994, Journal of Political Economy.

[12]  R. Schubert,et al.  Rationality on the rise: Why relative risk aversion increases with stake size , 2009 .

[13]  Robert Sugden,et al.  Disappointment and Dynamic Consistency in Choice under Uncertainty , 1986 .

[14]  The Nature of Risk Preferences: Evidence from Insurance Choices , 2013 .

[15]  Adrian Bruhin,et al.  Risk and Rationality: Uncovering Heterogeneity in Probability Distortion , 2009 .

[16]  M. Abdellaoui.  Parameter-Free Elicitation of Utility and Probability Weighting Functions , 2000 .

[17]  A. Tversky,et al.  Prospect theory: analysis of decision under risk , 1979 .

[18]  Uzi Segal,et al.  The Ellsberg Paradox and Risk Aversion: An Anticipated Utility Approach , 1987 .

[19]  Rachel T. A. Croson,et al.  Gender Differences in Preferences , 2009 .

[20]  John A List,et al.  A simple test of expected utility theory using professional traders. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[21]  David A. Chapman,et al.  First-Order Risk Aversion, Heterogeneity, and Asset Market Outcomes ∗ , 2009 .

[22]  L. J. Savage,et al.  The Utility Analysis of Choices Involving Risk , 1948, Journal of Political Economy.

[23]  John H. Kagel,et al.  Testing between alternative models of choice under uncertainty: Some initial results , 1990 .

[24]  Yoram Halevy.  Strotz Meets Allais: Diminishing Impatience and the Certainty Effect , 2008 .

[25]  Colin Camerer.  Individual Decision Making , 2020, The Handbook of Experimental Economics.

[26]  P. Wakker.  Prospect Theory: Frontmatter , 2010 .

[27]  Thierry Post,et al.  Loss Aversion with a State-Dependent Reference Point , 2011, Manag. Sci..

[28]  B. Jullien,et al.  Estimating Preferences under Risk: The Case of Racetrack Bettors , 2000, Journal of Political Economy.

[29]  M. Kenward,et al.  An Introduction to the Bootstrap , 2007 .

[30]  Alok Kumar.  Who Gambles in the Stock Market? , 2008 .

[31]  Byoung-Hyoun Hwang,et al.  Initial Public Offerings as Lotteries: Skewness Preference and First-Day Returns , 2012, Manag. Sci..

[32]  Todd Mitton,et al.  Equilibrium Underdiversification and the Preference for Skewness , 2004 .

[33]  Peter P. Wakker,et al.  The Midweight Method to Measure Attitudes Toward Risk and Ambiguity , 2009, Manag. Sci..

[34]  Bryan R. Routledge,et al.  Exotic Preferences for Macroeconomists , 2004, NBER Macroeconomics Annual.

[35]  Mark J. Machina,et al.  Temporal risk and the nature of induced preferences , 1984 .

[36]  V. Crawford.  Equilibrium without independence , 1990 .

[37]  M. Allais.  L'Extension des Theories de l'Equilibre Economique General et du Rendement Social au Cas du Risque , 1953 .

[38]  Prospect Theory and Asset Prices , 1999 .

[39]  H. J. Einhorn,et al.  Expression theory and the preference reversal phenomena. , 1987 .

[40]  Ulrich Hoffrage,et al.  An Expected Utility Approach to Probabilistic Insurance: A Comment on Wakker, Thaler and Tversky (1997) , 1999 .

[41]  Peter P. Wakker,et al.  On the Intuition of Rank-Dependent Utility , 2000 .

[42]  H. Stott.  Cumulative prospect theory's functional menagerie , 2006 .

[43]  Uzi Segal,et al.  Two Stage Lotteries Without the Reduction Axiom , 1990 .

[44]  J. Quiggin.  A theory of anticipated utility , 1982 .

[45]  Charles A. Holt,et al.  Risk Aversion and Incentive Effects: New Data without Order Effects , 2005 .

[46]  R. Schubert,et al.  Gender, Financial Risk, and Probability Weights , 2006 .

[47]  J. Kagel,et al.  Animals' Choices over Uncertain Outcomes: Some Initial Experimental Results , 1985 .

[48]  A. Tversky,et al.  Who accepts Savage's axiom? , 1974 .

[49]  Peter P. Wakker,et al.  An index of loss aversion , 2005, J. Econ. Theory.

[50]  M. Rabin,et al.  Reference-Dependent Risk Attitudes , 2007 .

[51]  S. Legg,et al.  Dynamic Portfolio Choice and Asset Pricing with Narrow Framing and Probability Weighting , 2012 .

[52]  Justin R. Sydnor.  Over)insuring Modest Risks , 2010 .

[53]  A. Tversky,et al.  Advances in prospect theory: Cumulative representation of uncertainty , 1992 .

[54]  Colin Camerer.  Recent Tests of Generalizations of Expected Utility Theory , 1992 .

[55]  A. Tversky,et al.  Prospect Theory : An Analysis of Decision under Risk Author ( s ) : , 2007 .

[56]  Charles Bellemare,et al.  Myopic Loss Aversion: Information Feedback vs. Investment Flexibility , 2004 .

[57]  Steven J. Kachelmeier,et al.  Examining Risk Preferences under High Monetary Incentives: Experimental Evidence from the People's Republic of China , 1992 .

[58]  P. Wakker.  Prospect Theory: For Risk and Ambiguity , 2010 .

[59]  Valery Polkovnichenko.  Household Portfolio Diversification: A Case for Rank-Dependent Preferences , 2005 .

[60]  B. V. Praag,et al.  A parametric analysis of prospect theory’s functionals for the general population , 2009, SSRN Electronic Journal.

[61]  K. MacCrimmon.  DESCRIPTIVE AND NORMATIVE IMPLICATIONS OF THE DECISION THEORY POSTULATES. A SUMMARY OF EXPERIMENTAL RESULTS WITH BUSINESS EXECUTIVES , 1966 .

[62]  Feng Zhao,et al.  Probability Weighting Functions Implied in Options Prices , 2012 .

[63]  Uday S. Karmarkar,et al.  Subjectively weighted utility and the Allais Paradox , 1979 .

[64]  P. Schoemaker.  The Expected Utility Model: Its Variants, Purposes, Evidence and Limitations , 1982 .

[65]  John Conlisk,et al.  Three Variants on the Allais Example , 1989 .

[66]  Oliver Spalt.  Probability Weighting and Employee Stock Options , 2011, Journal of Financial and Quantitative Analysis.

[67]  R. Sobel,et al.  Chapter 8 – Unifying the Favorite-Longshot Bias with Other Market Anomalies , 2008 .

[68]  Colin Camerer,et al.  Violations of the betweenness axiom and nonlinearity in probability , 1994 .

[69]  S. Chew.  Axiomatic utility theories with the betweenness property , 1989 .

[70]  M. Machina.  Choice under Uncertainty: Problems Solved and Unsolved , 1987 .

[71]  Justin Wolfers,et al.  Nber Working Paper Series Explaining the Favorite-longshot Bias: Is It Risk-love or Misperceptions? , 2022 .

[72]  Ole Hagen,et al.  Towards a Positive Theory of Preferences under Risk , 1979 .

[73]  Ori Levy,et al.  Theories of choice under risk: Insights from financial markets , 2009 .

[74]  Mohammed Abdellaoui,et al.  Risk Preferences at Different Time Periods: An Experimental Investigation , 2011, Manag. Sci..

[75]  Adrian Bruhin,et al.  Viewing the future through a warped lens: Why uncertainty generates hyperbolic discounting , 2010 .

[76]  D. Prelec,et al.  Compound Invariant Weighting Functions in Prospect Theory , 2000 .

[77]  A. Tversky,et al.  Options traders exhibit subadditive decision weights , 1996 .

[78]  Prospect Theory by Peter P. Wakker , 2010 .

[79]  Colin Camerer.  Prospect Theory In The Wild: Evidence From The Field , 1998 .

[80]  John A. List,et al.  Do professional traders exhibit myopic loss aversion? An experimental analysis , 2005 .

[81]  Stocks as Lotteries: the Implications of Probability Weighting for Security Prices , 2007 .

[82]  William T. Harbaugh,et al.  The Fourfold Pattern of Risk Attitudes in Choice and Pricing Tasks , 2009 .

[83]  P. Cook,et al.  The Peculiar Scale Economies of Lotto , 1991 .

[84]  R. Thaler,et al.  Anomalies: Risk Aversion , 2001 .

[85]  Robert Sugden,et al.  Reference-dependent subjective expected utility , 2003, J. Econ. Theory.

[86]  R. Thaler,et al.  Myopic Loss Aversion and the Equity Premium Puzzle , 1993 .

[87]  Herbert Walther.  Normal-randomness expected utility, time preference and emotional distortions , 2003 .

[88]  Uzi Segal,et al.  Observing different orders of risk aversion , 1994 .

[89]  David W Harless,et al.  The predictive utility of generalized expected utility theories , 1994 .

[90]  Eddie Dekel.  An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom , 1986 .

[91]  R. Thaler,et al.  Individual Preferences, Monetary Gambles, and Stock Market Participation: A Case for Narrow Framing , 2006 .

[92]  R. Mehra.  The Equity Premium Puzzle: A Review , 2007 .

[93]  J. Kagel,et al.  Tests of 'fanning out' of indifference curves: Results from animal and human experiments , 1990 .

[94]  K. MacCrimmon,et al.  Utility Theory: Axioms Versus ‘Paradoxes’ , 1979 .

[95]  J. Ingersoll,et al.  Non-Monotonicity of the Tversky-Kahneman Probability-Weighting Function: A Cautionary Note , 2007 .

[96]  R. Sugden.  Alternatives to Expected Utility: Foundations , 2004 .

[97]  Richard Gonzalez,et al.  On the Shape of the Probability Weighting Function , 1999, Cognitive Psychology.

[98]  Faruk Gul.  A Theory of Disappointment Aversion , 1991 .

[99]  P. Wakker,et al.  Dynamic Choice and NonExpected Utility , 1998 .

[100]  M. Machina.  Payoff Kinks in Preferences over Lotteries , 2000 .

[101]  D. Cross,et al.  Subjective probability and delay. , 1991, Journal of the experimental analysis of behavior.

[102]  C. Starmer.  Developments in Non-expected Utility Theory: The Hunt for a Descriptive Theory of Choice under Risk , 2000 .

[103]  R. Schubert,et al.  Energy-Using Durables: The Role of Time Discounting in Investment Decisions , 2011 .