Cumulative Prospect Theory

Portfolio Choice Under Cumulative Prospect Theory: An Analytical Treatment∗

Xue Dong He† and Xun Yu Zhou‡ September 14, 2010

Abstract 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 when the reference point is the risk-free return and when it is not (while the utility function is piece-wise 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. Key words: Portfolio choice, single period, cumulative prospect theory, reference point, loss aversion, S-shaped utility function, probability weighting, well-posedness

We are grateful for comments from seminar and conference participants at Cambridge, London School of Economics, King’s College, TU Berlin, Le s´ minaire Bachelier, National University of Singapore, Academia e Sinica, Chinese Academy of Sciences, Nankai, Princeton, UT Austin, Man’s Oxford Offsite Quant Forum, the 2007 Workshop on Mathematical Control Theory and Finance, the 2007 Financial Engineering and Risk Management Conference, the 2008 Oxford Behavioural Finance Conference, and the 4th Annual CARISMA Conference. He acknowledges a start-up fund of Columbia University. Zhou acknowledges financial support from the RGC Earmarked Grant CUHK418605, and a start-up fund...