A Note on Prospect Stochastic Dominance and Risk-return Analysis Tools

We consider the use of mean-variance analysis in investment decisions. Mean-variance analysis provides a simple tool to rank investments by comparing the expected returns and return variances of those investments. The rule itself is simple and elegant, when we compare two risky investments, if one has a higher mean return and lower variance of return than the other, then the former must be preferred to the latter. However, the rule does not tell us what to do if we compare two investments in which one of them has both a higher mean and a higher variance, a common occurrence in the real world. In addition, empirical studies show that people prefer a higher variance is actually not unusual when people face losses. This kind of phenomenon is hard to be explained within the standard expected utility framework, which assumes risk-averse everywhere on the outcome domain. This also implies that mean-variance analysis as a tool will have limitations when losses are involved since the expected utility framework lies at its foundation. In this paper, we make use of the S-shaped value function proposed in prospect theory which implies risk-seeking behaviour on the loss domain. We explore the connection between mean-variance analysis and the S-shaped value function by utilising the concept of ``prospect stochastic dominance''. Our results show that incorporating partial moments in mean-variance analysis enables us to apply the suitably modified tool to choices over investments with normally distributed returns by investors who have S-shaped value functions and so exhibit a preferences for investments that involve losses that have both a higher variance as well as a higher mean