Static Portfolio Choice under Cumulative Prospect Theory

Bernard, Carole and Ghossoub, Mario (2009): Static Portfolio Choice under Cumulative Prospect Theory.

Abstract

We derive the optimal portfolio choice for an investor who behaves according to Cumulative Prospect Theory. The study is done in a one-period economy with one risk-free asset and one risky asset, and the reference point corresponds to the terminal wealth arising when the entire initial wealth is invested into the risk-free asset. When it exists, the optimal holding is a function of a generalized Omega measure of the distribution of the excess return on the risky asset over the risk-free rate. It conceptually resembles Merton’s optimal holding for a CRRA expected-utility maximizer. We derive some properties of the optimal holding and illustrate our results using a simple example where the excess return has a skew-normal distribution. In particular, we show how a Cumulative Prospect Theory investor is highly sensitive to the skewness of the excess return on the risky asset. In the model we adopt, with a piecewise-power value function with different shape parameters, loss aversion might be violated for reasons that are now well-understood in the literature. Nevertheless, we argue that this violation is acceptable.

• Item Type: MPRA Paper
• Language: English
• Keywords: Cumulative Prospect Theory, Portfolio Choice, Behavioral Finance, Omega Measure.
• Subjects: D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty; G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice ; Investment Decisions

References

Abdellaoui, M., H. Bleichrodt, and C. Paraschiv (2007): “Loss Aversion Under Prospect Thepry: A Parameter-Free Measurement,” Management Science, 53(10), 1659–1674.

Allais, M. (1953): “Le Comportement de l’Homme Rationnel Devant le Risque: Critique des Axiomes et Postulats de l’ Ecole Americaine,” Econometrica, 21(4), 503–546.

Barberis, N., and M. Huang (2008): “Stocks as Lotteries: The Implications of Probability Weighting for Security Prices,” American Economic Review, 98(5), 2066–2100.

Baucells, M., and F. Heukamp (2006): “Stochastic dominance and cumulative theory,” Management Science, 52(9), 1409–1423.

Benartzi, S., and R. Thaler (1995): “Myopic Loss Aversion and the Equity Premium Puzzle,” The Quarterly Journal of Economics, 110(1), 73–92.

Berkelaar, A., R. Kouwenberg, and T. Post (2004): “Optimal Portfolio Choice Under Loss Aversion,” The Review of Economics and Statistics, 86(4), 973–987.

Bernardo, A., and O. Ledoit (2000): “Gain, Loss and Asset Pricing,” Journal of Political Economy, 108(1), 144–172.

Camerer, C., G. Loewenstein, and M. Rabin (2004): Advances in Behavioral Economics. Princeton University Press.

Cascon, A., C. Keating, and W. Shadwick (2003): “The Omega Function,” The Finance Development Centre, Working Paper.

Davies, G. B., and S. E. Satchell (2007): “The Behavioural Components of Risk Aversion,” Journal of Mathematical Psychology, 51(1), 1–13.

De Giorgi, E., and T. Hens (2006): “Making Prospect Theory Fit for Finance,” Financial Markets and Portfolio Management, 20(3), 339–360.

De Giorgi, E., T. Hens, and H. Levy (2004): “Existence of CAPM Equilibria with Prospect Theory Preferences,” NCCR-FINRISK Working Paper, no. 85, Available at SSRN: http://ssrn.com/abstract=420184.

De Giorgi, E., T. Hens, and J. Mayer (2006): “A Behavioral Foundation of Reward-Risk Portfolio Selection and the Asset Allocation Puzzle,” EFA 2006 Zurich Meetings Paper, Available at SSRN: http://ssrn.com/abstract=899273.

De Giorgi, E., T. Hens, and M. Rieger (2008): “Financial Market Equilibria with Cumulative Prospect Theory,” Swiss Finance Institute Research Paper, no. 07-21, Available at SSRN: http://ssrn.com/abstract=985539.

Denneberg, D. (1994): Non-Additive Measure and Integral. Kluwer Academic Publishers.

Edwards, W. (1962): “Subjective Probabilities Inferred from Decisions,” Psychological Review, 69(2), 109–135.

Eeckhoudt, L., C. Gollier, and H. Schlesinger (2005): Economic and Financial Decisions under Risk. Princeton University Press.

Ellsberg, D. (1961): “Risk, Ambiguity and the Savage Axioms,” Quarterly Journal of Economics, 75(4), 643–669.

Fellner, W. (1961): “Distortion of Subjective Probabilities as a Reaction to Uncertainty,” Quarterly Journal of Economics, 75(4), 670–689.

Fishburn, P. (1988): Nonlinear Preference and Utility Theory. The Johns Hopkins University Press.

Genton, M. (2004): Skew-El liptical Distributions and Their Applications: A Journey Beyond Normality. Chapman and Hall/CRC.

Gollier, C. (1996): “Optimum Insurance of Approximate Losses,” Journal of Risk and Insurance, 63(3), 369–380.

Gollier, C. (2001): The Economics of Risk and Time. The MIT Press.

Gomes, F. (2005): “Portfolio Choice and Trading Volume with Loss-Averse Investors,” Journal of Business, 78(2), 675–706.

Handa, J. (1977): “Risk, Probabilities and a New Theory of Cardinal Utility,” Journal of Political Economy, 85(1), 97–122.

Jarrow, R., and F. Zhao (2006): “Downside Loss Aversion and Portfolio Management,” Management Science, 52(4), 558–566.

Jin, H., and X. Y. Zhou (2008): “Behavioral Portfolio Selection in Continous Time,” Mathematical Finance, 18(3), 385–426.

Kahneman, D., and A. Tversky (1979): “Prospect Theory: An Analysis of Decision Under Risk,” Econometrica, 47(2), 263–291.

Keating, C., and W. Shadwick (2002): “A Universal Performance Measure,” Journal of Performance Measurement, 6(3), 59–84.

Kobberling, V., and P. Wakker (2005): “An Index of Loss Aversion,” Journal of Economic Theory, 122(1), 119–131.

Merton, R. (1969): “Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case,” The Review of Economics and Statistics, 51(3), 247–257.

Neilson, W. S. (2002): “Comparative Risk Sensitivity with Reference-Dependent Preferences,” The Journal of Risk and Uncertainty, 24(2), 131– 142.

Quiggin, J. (1982): “A Theory of Anticipated Utility,” Journal of Economic Behavior, 3(4), 323–343.

Schmeidler, D. (1986): “Integral Representation without Additivity,” Proceedings of the American Mathematical Society, 97(2), 255–261.

Schmeidler, D. (1989): “Subjective Probability and Expected Utility without Additivity,” Econometrica, 57(3), 571–587.

Schmidt, U., and H. Zank (2005): “What is Loss Aversion?,” The Journal of Risk and Uncertainty, 30(2), 157–167.

Schmidt, U., and H. Zank (2007): “Linear Cumulative Prospect Theory with Applications to Portfolio Selection and Insurance Demand,” Decisions in Economics and Finance, 30, 1–18.

Starmer, C. (2000): “Developments in Non-Expected Utility Theory: The Hunt for a Descriptive Theory of Choice under Risk,” Journal of Economic Literature, 38(2), 332–382.

Tversky, A., and D. Kahneman (1992): “Advances in Prospect Theory: Cumulative Representation of Uncertainty,” The Journal of Risk and Uncertainty, 5(4), 297–323.

Wakker, P. (1994): “Separating Marginal Utility and Probabilistic Risk Aversion,” Theory and Decision, 36(1), 1–44.

Wakker, P., and A. Tversky (1993): “An Axiomatization of Cumulative Prospect Theory,” The Journal of Risk and Uncertainty, 7(7), 147–176.

Wang, S., and V. Young (1998): “Ordering Risks: Expected Utility Theory versus Yaari’s Dual Theory of Risk,” Insurance: Mathematics and Economics, 22, 145–161.

Yaari, M. (1987): “The Dual Theory of Choice under Risk,” Econometrica, 55(1), 95–115.

Zank, H. (2009): “On Probabilities and Loss Aversion,” Theory and Decision - Forthecoming.