Semidefinite programming relaxation for portfolio selection with higher order moments

With the nor-normal distribution and nor-binomial utility function of investors, higher-order moments should be taken into consideration with portfolio selection. So far, studies on portfolio selection with higher-order moments are mainly conducted under the framework of mean-variance-skewness while kurtosis is seldom taken into account. But many literatures have pointed out kurtosis play more important role in portfolio selection than variance and skewness. So, in order to fill up the study gap, portfolio selection with higher-order moments will be studied, under the framework of mean-variance-skewness-kurtosis, in this paper. Firstly, the optimization model of portfolio selection with kurtosis minimization is set up. Then, semidefinite programming relaxation is suggested based on research results of Lasserre and Waki to resolve the difficulty of model solution from the higher order and non-convexity of objective function and the efficient frontier of portfolio selection model minimizing kurtosis is deduced theoretically. Finally, the theoretical efficient frontier and the effectiveness of semidefinite programming relaxation are verified through empirical analysis.