The Mean-Variance Rule for Investors with Reverse S-Shaped Utility

Our paper contributes to the literature by extending the theory of the mean-variance (MV) rules for both risk averters and risk lovers to the MV rule for investors with reverse S-shaped utility. To do so, we first introduce the definition of the MV rule for investors with reverse S-shaped utility. We then set up the conjecture for the preferences of investors with reverse S-shaped utility on different prospects by using the MV rule for investors with reverse S-shaped utility that they could get a higher expected utility for the preferred asset under some conditions. Thereafter, we look for the conditions that the conjecture could hold and construct a theorem for this purpose to show that when the negative parts of the assets follow one type (say, the first type) of location-scale family or the linear combination of location-scale families and when the positive parts of the assets follow another type (say, the second type) of location-scale family or the linear combination of location-scale families, then the preferences of the assets by using the MV rule for investors with reverse S-shaped utility is the same as those by using an expected utility for the investors with reverse S-shaped utility. We then extend the MV theory for investors with reverse S-shaped utility to develop some properties of portfolio diversification by the MV rule for investors with reverse S-shaped utility. The theory developed in our paper enables academics and practitioners to apply the theory developed in this paper to analyze some important empirical issues and draw inferences on the preferences for investors with reverse S-shaped utility