Mean-risk analysis with enhanced behavioral content

We study a mean-risk model derived from a behavioral theory of Disappointment with multiple reference points. One distinguishing feature of the risk measure is that it is based on mutual deviations of outcomes, not deviations from a specific target. We prove necessary and sufficient conditions for strict first and second order stochastic dominance, and show that the model is, in addition, a Convex Risk Measure. The model allows for richer, and behaviorally more plausible, risk preference patterns than competing models with equal degrees of freedom, including Expected Utility (EU), Mean–Variance (M-V), Mean-Gini (M-G), and models based on non-additive probability weighting, such as Dual Theory (DT). In asset allocation, the model allows a decision-maker to abstain from diversifying in a positive expected value risky asset if its performance does not meet a certain threshold, and gradually invest beyond this threshold, which appears more acceptable than the extreme solutions provided by either EU and M-V (always diversify) or DT and M-G (always plunge). In asset trading, the model provides no-trade intervals, like DT and M-G, in some, but not all, situations. An illustrative application to portfolio selection is presented. The model can provide an improved criterion for mean-risk analysis by injecting a new level of behavioral realism and flexibility, while maintaining key normative properties.

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

[2]  Wing-Keung Wong,et al.  Gains from diversification on convex combinations: A majorization and stochastic dominance approach , 2010, Eur. J. Oper. Res..

[3]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[4]  James S. Dyer,et al.  A standard measure of risk and risk-value models , 1996 .

[5]  Maria Grazia Speranza,et al.  Conditional value at risk and related linear programming models for portfolio optimization , 2007, Ann. Oper. Res..

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

[7]  K. Arrow.  Essays in the theory of risk-bearing , 1958 .

[8]  J. Dyer,et al.  Relative risk—value models , 1997 .

[9]  Shlomo Yitzhaki,et al.  On an Extension of the Gini Inequality Index , 1983 .

[10]  Wlodzimierz Ogryczak,et al.  Dual Stochastic Dominance and Related Mean-Risk Models , 2002, SIAM J. Optim..

[11]  H. Levy.  Stochastic dominance and expected utility: survey and analysis , 1992 .

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

[13]  W. Ogryczak,et al.  LP solvable models for portfolio optimization: a classification and computational comparison , 2003 .

[14]  L. Eeckhoudt,et al.  Optimal insurance without expected utility: The dual theory and the linearity of insurance contracts , 1995 .

[15]  Enrico G. De Giorgi,et al.  Reward-Risk Portfolio Selection and Stochastic Dominance , 2005 .

[16]  P. Delquié,et al.  Disappointment without prior expectation: a unifying perspective on decision under risk , 2006 .

[17]  Josef Hadar,et al.  Rules for Ordering Uncertain Prospects , 1969 .

[18]  S. Werlang,et al.  Uncertainty Aversion, Risk Aversion, and the Optimal Choice of Portfolio , 1992 .

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

[20]  M. Rothschild,et al.  Increasing risk: I. A definition , 1970 .

[21]  S. Yitzhaki.  Stochastic Dominance, Mean Variance, and Gini's Mean Difference , 1982 .

[22]  Alain Chateauneuf,et al.  The no-trade interval of Dow and Werlang: Some clarifications , 2010, Math. Soc. Sci..

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

[24]  David E. Bell,et al.  Disappointment in Decision Making Under Uncertainty , 1985, Oper. Res..

[25]  Douglas W. Mitchell,et al.  Risk-value models: Restrictions and applications , 2003, Eur. J. Oper. Res..

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

[27]  Alexander Schied,et al.  Convex measures of risk and trading constraints , 2002, Finance Stochastics.

[28]  B. Stone.  A GENERAL CLASS OF THREE-PARAMETER RISK MEASURES , 1973 .

[29]  D. E. Bell.  Risk, return, and utility , 1995 .

[30]  Melvyn Sim,et al.  Satisficing Measures for Analysis of Risky Positions , 2009, Manag. Sci..

[31]  A. Tversky,et al.  Prospect theory: an analysis of decision under risk — Source link , 2007 .

[32]  Wlodzimierz Ogryczak,et al.  From stochastic dominance to mean-risk models: Semideviations as risk measures , 1999, Eur. J. Oper. Res..

[33]  M. Machina.  "Expected Utility" Analysis without the Independence Axiom , 1982 .

[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]  A. Ruszczynski,et al.  Portfolio optimization with stochastic dominance constraints , 2006 .

[36]  P. Schoemaker,et al.  Probability Versus Certainty Equivalence Methods in Utility Measurement: Are they Equivalent? , 1985 .

[37]  R. Rockafellar,et al.  Generalized Deviations in Risk Analysis , 2004 .

[38]  I. Gilboa,et al.  Maxmin Expected Utility with Non-Unique Prior , 1989 .

[39]  Rakesh K. Sarin,et al.  Risk-value models , 1993 .

[40]  M. Allais.  Le comportement de l'homme rationnel devant le risque : critique des postulats et axiomes de l'ecole americaine , 1953 .

[41]  Stephen E. Satchell,et al.  An Extended Family of Financial-Risk Measures , 1998 .

[42]  Wing-Keung Wong,et al.  Stochastic dominance and mean-variance measures of profit and loss for business planning and investment , 2007, Eur. J. Oper. Res..

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

[44]  P. Fishburn.  Mean-Risk Analysis with Risk Associated with Below-Target Returns , 1977 .

[45]  Wing-Keung Wong,et al.  Stochastic Dominance and Risk Measure: A Decision-Theoretic Foundation for VAR and C-Var , 2006, Eur. J. Oper. Res..

[46]  A. Rustichini,et al.  Ambiguity Aversion, Robustness, and the Variational Representation of Preferences , 2006 .

[47]  John A. Weymark,et al.  A single-parameter generalization of the Gini indices of inequality , 1980 .

[48]  Wlodzimierz Ogryczak,et al.  On Extending the LP Computable Risk Measures to Account Downside Risk , 2005, Comput. Optim. Appl..