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Portfolio Choice Under Cumulative Prospect Theory: An Analytical Treatment

  • Xue Dong He

    ()

    (Industrial Engineering and Operations Research Department, Columbia University, New York, New York 10027)

  • Xun Yu Zhou

    ()

    (Nomura Centre for Mathematical Finance, University of Oxford, Oxford OX1 3LB, United Kingdom; Oxford-Man Institute of Quantitative Finance, University of Oxford, Oxford OX2 6ED, United Kingdom; and Department of Systems Engineering and Engineering Management, Chinese University of Hong Kong, Shatin, Hong Kong)

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    We formulate and carry out an analytical treatment of a single-period portfolio choice model featuring a reference point in wealth, S-shaped utility (value) functions with loss aversion, and probability weighting under Kahneman and Tversky's cumulative prospect theory (CPT). We introduce a new measure of loss aversion for large payoffs, called the large-loss aversion degree (LLAD), and show that it is a critical determinant of the well-posedness of the model. The sensitivity of the CPT value function with respect to the stock allocation is then investigated, which, as a by-product, demonstrates that this function is neither concave nor convex. We finally derive optimal solutions explicitly for the cases in which the reference point is the risk-free return and those in which it is not (while the utility function is piecewise linear), and we employ these results to investigate comparative statics of optimal risky exposures with respect to the reference point, the LLAD, and the curvature of the probability weighting. This paper was accepted by Wei Xiong, finance.

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    File URL: http://dx.doi.org/10.1287/mnsc.1100.1269
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    Article provided by INFORMS in its journal Management Science.

    Volume (Year): 57 (2011)
    Issue (Month): 2 (February)
    Pages: 315-331

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    Handle: RePEc:inm:ormnsc:v:57:y:2011:i:2:p:315-331
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    1. Amos Tversky & Daniel Kahneman, 1979. "Prospect Theory: An Analysis of Decision under Risk," Levine's Working Paper Archive 7656, David K. Levine.
    2. Arjan B. Berkelaar & Roy Kouwenberg & Thierry Post, 2004. "Optimal Portfolio Choice under Loss Aversion," The Review of Economics and Statistics, MIT Press, vol. 86(4), pages 973-987, November.
    3. Kobberling, Veronika & Wakker, Peter P., 2005. "An index of loss aversion," Journal of Economic Theory, Elsevier, vol. 122(1), pages 119-131, May.
    4. Shlomo Benartzi & Richard H. Thaler, 1995. "Myopic Loss Aversion and the Equity Premium Puzzle," The Quarterly Journal of Economics, Oxford University Press, vol. 110(1), pages 73-92.
    5. N. Gregory Mankiw & Stephen P. Zeldes, 1990. "The Consumption of Stockholders and Non-Stockholders," NBER Working Papers 3402, National Bureau of Economic Research, Inc.
    6. Francisco J. Gomes, 2005. "Portfolio Choice and Trading Volume with Loss-Averse Investors," The Journal of Business, University of Chicago Press, vol. 78(2), pages 675-706, March.
    7. Nicholas Barberis & Ming Huang, 2007. "Stocks as Lotteries: The Implications of Probability Weighting for Security Prices," NBER Working Papers 12936, National Bureau of Economic Research, Inc.
    8. Marc Rieger & Mei Wang, 2006. "Cumulative prospect theory and the St. Petersburg paradox," Economic Theory, Society for the Advancement of Economic Theory (SAET), vol. 28(3), pages 665-679, 08.
    9. Tversky, Amos & Kahneman, Daniel, 1992. "Advances in Prospect Theory: Cumulative Representation of Uncertainty," Journal of Risk and Uncertainty, Springer, vol. 5(4), pages 297-323, October.
    10. Bernard, Carole & Ghossoub, Mario, 2009. "Static Portfolio Choice under Cumulative Prospect Theory," MPRA Paper 15446, University Library of Munich, Germany.
    11. George Wu & Richard Gonzalez, 1996. "Curvature of the Probability Weighting Function," Management Science, INFORMS, vol. 42(12), pages 1676-1690, December.
    12. Hanqing Jin & Xun Yu Zhou, 2008. "Behavioral Portfolio Selection In Continuous Time," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 385-426.
    13. Harry Markowitz, 1952. "The Utility of Wealth," Journal of Political Economy, University of Chicago Press, vol. 60, pages 151.
    14. Mohammed Abdellaoui, 2000. "Parameter-Free Elicitation of Utility and Probability Weighting Functions," Management Science, INFORMS, vol. 46(11), pages 1497-1512, November.
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