Central moments, stochastic dominance, moment rule, and diversification with an application

In this paper, we first develop some properties to state the relationships among central moments, stochastic dominance (SD), risk-seeking stochastic dominance (RSD), and integrals for the general utility functions and the polynomial utility functions of both risk averters and risk seekers. We then introduce the moment rule and establish some necessary and/or sufficient conditions between stochastic dominance and the moment rule for the general utility functions and the polynomial utility functions of both risk averters and risk seekers without imposing the same-location-scale-family condition. Thereafter, we apply the moment rules to develop some properties of portfolio diversification for the general utility functions and the polynomial utility functions for both risk averters and risk seekers. The findings in our paper enable academics and practitioners to draw preferences of both risk averters and risk seekers on their choices of portfolios or assets by using different moments. We illustrate this by using the moment rule tests to compare excess return of 49 industry portfolios from Kenneth French's online data library.