基于半定规划松弛的高阶投资组合优化研究

Portfolio Optimization with Higher Order Moments Based on Semidefi nite Programming Relaxation

以最小化峰度为例研究了具有高阶目标函数的投资组合优化问题。针对目标函数的高阶性与非凸性所带来的投资组合优化模型求解困难,根据Lasserre和Waki的研究成果,提出高阶投资组合优化模型的半定规划松弛算法;并从理论上推导得到最小化峰度的投资组合优化模型的有效前沿。最后通过实证分析,验证了理论推导得到的有效前沿,进而说明了半定规划松弛算法求解高阶投资组合优化问题的有效性。

With the nor-normal distribution of returns and the nor-binomial utility function of investors, higher-order moments should be taken into consideration in the process of optimizing investment portfolio. The current study on investment portfolio optimization with higher-order moments is mainly conducted under the framework of mean-varianee-skewness while kurtosis is seldom taken into aeeount. The phenomenon that kurtosis is seldom considered in portfolio optimization is mainly attributed to the problem that portfolio optimization under the framework of mean-varianee-skewness-kurtosis is difficult to be achieved for non-convexity problems. This issue is particularly true when kurtosis is placed in the objective function. The higher order of objeetive function results in moreproblem-solving difficulty. The existing methods for portfolio optimization considering kurtosis are PGP approaeh and Taylor series approximation to utility function. However, the global optimum solution couldn＇t be achieved through these two methods. Thus, in order to fill up the study gap the problem of portfolio optimization under the framework of mean-varianee-skewness-kurtosis will be studied in this paper. First, the model of portfolio optimization under the framework ofmean-varianee-skewness-kurtosis is established. In this model, kurtosis minimization is the objective function and the first three moments are taken as restrictions. Based on Lagrangian Multiplier Method, the problem of portfolio optimization is converted into the problem of minimizing Lagrangian function. Through the function can help deduce the efficient frontier describing the relationship between kurtosis and the first three moments of optimal portfolio return.The obtained efficient frontier reflects the relationship between the fourth moment and the first three moments of optimal portfolio return as a degenerated parabola. Second, the algorithm of semidefinite programming relaxation is proposed to solve portfolio optimization. The algorithm, originally proposed by Lasserre （2001） and Waki （2004） , can convert the optimization problem by using kurtosis as an objective function （fourth-order polynomial） into the optimization problem of linear matrix inequality by using the theory of moment matrices. This conversion can help acquire global optimum solution at a relatively fast convergence rate under the condition that objective function is higher-order polynomial and the optimization problem is non-convex. Finally, an empirical analysis is conducted based on the transaction data of Shanghai Stock Exchange. The algorithm of semidefinite programming relaxation is applied to resolving the problem of portfolio optimization in minimizing kurtosis. Based on the analysis result, the relationship between the fourth moment and the first three moments of optimal portfolio return as a degenerated parabola is verified and the validity of the algorithm of semidefinite programming relaxation is also proved.