Portfolio Optimization With Higher-order Moments

Markowitz’s mean-variance model has the epoch-making significance，whichmakes finance get rid of the situation of purely descriptive study and the operationonly by experience，and settles the foundation for the development of modernportfolio theory. The necessary and sufficient condition, under which themean-variance model is consistent with the expected utility principle, is that thereturn rate of the risky asset obeys the normal distribution or the investor has theutility of quadratic function. However, unfortunately, the above condition has nopractical significance. A lot of empirical studies home and abroad have shown thatthe distribution of the return rate of the risky asset isn’t normal and the utilityfunction of investors isn’t the quadratic function. So the effect of higher-ordermoments should be considered in the decision of the portfolio optimization,otherwise suboptimal decision will follow. Many scholars home and abroad havemade the study of portfolio optimization with higher-order moments. These studieshave been conducted directly or indirectly. The direct method takes higher-ordermoments or the function consist of the first four moments as the objective functionin the problem of portfolio optimization, while the indirect method converts theproblem of portfolio optimization of investors’ utility maximization into theproblem of portfolio optimization with higher-order moments through taylor seriesexpansion of the expected utility function. Although the existing studies have beenadundant, there are still a lots of shortage needed be improved. This Dissertation,based on previous study results, intends to extend the study of portfolio optimizationwith higher-order moments in order to improve the existing study and makeportfolio optimization with higher-order moments truly become the method and toolthat could be referenced in the dicision of portfolio optimization.Firstly, from the view of direct method, the study of portfolio optimization withhigher moments is extended under the framework of the first four moments, thealgorithm of semidefinite programming relaxation is proposed in order to deal withthe difficulty of problem solution of portfolio optimization from non-convexity ofoptimization problem and the higher order of objective function. Taking theoptimization problem of kurtosis maximization as an example, the algorithm ofsemidefinite programming relaxation for the solution of the problem of portfoliooptimization with higher-order moments is put forward based on the research resultof Lasserre and Waki. The algorithm could convert the optimization problem thathas higher-order objective function into the optimization problem of linear matrixinequality using the theory of moment matrices, which could acquire global optimum solution at relatively fast convergence rate under the cond ition thatobjective function is higher-order polynomial and the optimization problem isnon-convex. Furthermore, the efficient frontier of portfolio optimization maximizingkurtosis is deduced theoretically, then the algorithm is applied and the deducedefficient frontier is verified and the validity of the algorithm of semidefniteprogramming relaxation is also proved in empirical analysis.Secondly, from the view of indirect method, the study of portfolio optimizationwith higher moments is extended where the scope of the application of indirectmethod is extended from exponential utility function to HARA utility function.Under the background of HARA utility function, the convergence conditions ofTaylor series to expected utility function is studied in order to make Taylor seriesbecomes the reasonable approximation for expected utility function and guaranteethe convergence of approximate solution to real solution. The sufficient conditionunder which Taylor series is convergent to the expected utility function isdemonstrated and the relationship between the selection of the expansion point ofTaylor series and the convergence of Taylor series is analyzed，through which thetheoretical basis and method instruction is provided in applying Taylor series forportfolio optimization with higher-order moments，and the ordinary practice thatsets utility function as exponential utility function for the guaranty of convergenceof Taylor series is avoided，therefore the scope of the application of indirect methodis extended from exponential utility function to HARA utility function.Thirdly, the problem that conditional coskewness matrix and cokurtosis matrixare difficult to estimate is solved, through which the study of portfolio optimizationwith higher moments is extended from static angle to dynamics angle. Based onprevious research results, a new model is proposed to describe the time-variantcharacteristics of the multivariate distrib ution of return of financial assets, in whichthe model of AR(1)-DCC(1,1)-GARCH(1,1) is proposed to describe theself-correlation of multivariate conditional expectation and the aggregation ofmultivariate conditional variance while the multivariate conditional skewed-tdistribution is proposed to describe the time-variant of skewed and fat-tailedcharacteristics of the multivariate return of financial assets. Methods of modelidentification, parameter estimation, model-testing of the new model and theestimation method of conditional coskewness matrix and cokurtosis matrix arealso proposed. Through applying the proposed model, the dynamic portfoliooptimization with higher order moments is studied and the optimization results ofdynamic and static optimization are compared in the empirical analysis.Finally, on the basis of the first four moments, portfolio optimization is studiedtaking the full information of the distribution of portfolio return rate into consideration. The approximate analytic expression of the distribution of the returnrate of portfolio is put forward based on Gram-Charlier expansion and the validityof the put-forward approximate model is analyzed. Then relative entropy is taken asthe measure index of the distance between the approximate distribution and goaldistribution and the model of portfolio optimization of minimizing relative entropyis established from the intention of taking the full information of distribution of thereturn rate of portfolio into consideration. Numerical example is followed toillustrate the theoretical analys is.The study of this dissertation ulteriorly improves the study of portfoliooptimization with higher-order moments,breaks the limitation of the existingresearch, and makes the study of portfolio optimization with higher-order momentsmore symmetrical and thorough, therefore has important theoretical value.Furthermore, this dissertation also enhances the application value of the portfoliooptimization, makes it truly become the referred tool by fund company, pensionfund and insurance fund etc., is helpful to the scientific decision of institutionalinvestors and the establishment of the rational investment idea, is ultimatelybeneficial to the prosperity and healthy development of financial market.