Subjective Skewness of Return as an Explanation of the Optimal Choice between Gambles in Cumulative Prospect Theory

Abstract

Given that the expected return and variance of return of two gambles are equal  the hypothesis that the gamble with the greater  positive skewness of return will be chosen by an expected utility maximiser is appealing. However the hypothesis is  not, in general, correct. Brockett and Garven (1998) and Brocket and Kahane (1992) demonstrate this both theoretically and by constructing counter examples.A particularly revealing example is the following one constructed by Brockett and Kahane.  Gamble A has the two outcomes 2.45 and 7.49 with probabilities 0.5141 and 0.4859 respectively. Gamble B has the three outcomes 0, 4.947 and 10 with probabilities 0.12096, 0.750085 and 0.128955 respectively. Even though gamble A exhibits  lower expected return,  a higher variance and lower  positive skewness than gamble B it is preferred to gamble B by an expected utility maximiser on the basis of any standard utility function  such as power, log or exponential.  Consequently in this  example of theirs the expected utility maximiser exhibits an aversion to higher expected return and higher skewness and a preference for higher variance. As noted by Brockett and Kahane these results cannot be dismissed as decision makers “trading” variance for mean or skewness or having a strange idiosyncratic utility function.