基于高阶矩的投资组合优化研究
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摘要
马克维茨的均值—方差模型具有重要的意义，它使金融学摆脱了以往纯粹的描述性研究和单凭经验操作的状态，为现代投资组合理论的发展奠定了基础。均值—方差模型与期望效用原则具有一致性的充分必要条件为投资者的效用函数是二次函数或者风险资产的收益率服从正态分布。然而遗憾地是，该充分必要条件并不具有现实意义。国内外众多实证研究已经表明，风险资产收益率并不服从正态分布而投资者的效用函数也不是二次的。鉴于此，投资者在进行投资组合优化决策时应考虑高阶矩的影响，否则便会产生次优决策。目前，国内外很多学者已经对基于高阶矩的投资组合优化问题进行了研究，这些研究主要可以分为直接法和间接法。所谓直接法是将投资组合收益率的高阶矩或将包括高阶矩在内的各阶矩构成的函数直接作为目标函数的优化方法；而所谓间接法是通过期望效用函数的泰勒级数展开将最大化投资者期望效用的投资组合优化问题转化为基于高阶矩的投资组合优化问题的方法。虽然现有的研究已经比较系统深入，但仍存在诸多需要完善之处。本文将以前人研究成果为基础，对基于高阶矩的投资组合优化问题进行扩展研究，使得基于高阶矩的投资组合优化研究更加系统完善，并使其真正成为投资者进行投资组合优化决策时可供参考的方法和工具。
首先，在直接法下，对均值—方差—偏度—峰度框架下的投资组合优化问题进行扩展研究，提出求解高阶投资组合优化问题的半定规划松弛算法，解决了问题的高阶性和非凸性所带来的模型求解困难问题。以最小化峰度的投资组合优化问题为例，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，该算法利用矩矩阵理论，将以高阶多项式为目标函数的优化问题转化为线性矩阵不等式优化问题，其优点在于能够在目标函数为高阶且非凸的条件下，获得全局最优解且具有较快的收敛速度。此外，还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，并且通过在实证分析中运用半定规划松弛算法，验证了理论推导得到的有效前沿，同时也从另一方面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。
其次，在间接法下，对基于高阶矩的投资组合优化问题进行扩展研究，将间接法的适用范围从指数型效用函数扩展为HARA效用函数。在HARA效用函数的背景下，研究泰勒级数对效用函数的收敛条件，使得泰勒级数成为期望效用函数的合理近似，从而保证投资组合优化问题的近似解收敛于真实解。通过论证期望效用函数的泰勒级数收敛于实际期望效用函数的充分条件以及分析泰勒级数展开点的选择与泰勒级数收敛性之间的关系，为如何在HARA效用函数的背景下合理运用泰勒级数展开来研究基于高阶矩的投资组合优化问题提供理论依据和方法指导，从而避免目前为了保证收敛性而指定效用函数为指数型效用函数的做法，将间接法下基于高阶矩的投资组合优化研究从指数型效用函数扩展到HARA效用函数范围。
再次，解决了基于高阶矩的动态投资组合优化研究中所遇到的条件协偏度阵和条件协峰度阵难于估计的问题，实现了基于高阶矩的投资组合优化研究由静态向动态方向的扩展。在前人研究成果的基础上，构建了多元风险资产收益率分布时变模型，其中所建立的AR(1)-DCC(1,1)-GARCH(1,1)模型反映了多元条件下条件期望自相关性和条件方差聚集性，所建立的多元条件有偏学生t分布反映了多元条件下有偏厚尾性的时变特征，并提出模型识别、参数估计、模型检验方法和条件协偏度矩阵、条件协峰度矩阵的估计方法；然后利用所构建的模型研究高阶动态投资组合优化问题，构建了高阶动态投资组合优化模型并提出模型求解方法，并通过实证分析对高阶动静态投资组合优化结果进行比较。
最后，以前四阶矩为基础，进一步深化基于高阶矩的投资组合优化研究，提出考虑投资组合收益率完全分布信息的投资组合优化方法，将基于高阶矩的投资组合优化研究从只考虑前四阶矩向考虑投资组合收益率完全分布信息方向扩展。为了获得投资组合收益率分布的近似解析式，以投资组合收益率的前四阶矩为基础，提出基于Gram-Charlier展开的投资组合收益率分布近似模型，并对基于Gram-Charlier展开的投资组合收益率分布近似模型的有效性进行分析；然后根据所提出的基于Gram-Charlier展开的收益率分布近似模型进行投资组合优化研究，将相对熵作为投资组合收益率近似分布与目标分布之间距离的量化指标，从而构建最小化相对熵的投资组合优化模型并提出模型的求解方法并通过算例对理论分析进行说明。
上述研究进一步地完善了基于高阶矩的投资组合优化研究，在继承现有研究成果的基础上，突破现有研究的局限性，使得基于高阶矩的投资组合优化研究更加系统深入，因而具有重要的理论价值。此外，本文的研究在弥补目前基于高阶矩的投资组合优化问题研究不足的同时，也提高其实际应用价值，使其能够真正成为基金公司、养老基金和保险基金等诸多机构投资者的投资组合优化实践中可供参考的工具，这不仅有助于提高机构投资者的科学决策水平，而且有助于证券市场中理性投资理念的建立，并最终有利于金融市场的繁荣稳定和健康发展。
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.
首先，在直接法下，对均值—方差—偏度—峰度框架下的投资组合优化问题进行扩展研究，提出求解高阶投资组合优化问题的半定规划松弛算法，解决了问题的高阶性和非凸性所带来的模型求解困难问题。以最小化峰度的投资组合优化问题为例，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，该算法利用矩矩阵理论，将以高阶多项式为目标函数的优化问题转化为线性矩阵不等式优化问题，其优点在于能够在目标函数为高阶且非凸的条件下，获得全局最优解且具有较快的收敛速度。此外，还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，并且通过在实证分析中运用半定规划松弛算法，验证了理论推导得到的有效前沿，同时也从另一方面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。
其次，在间接法下，对基于高阶矩的投资组合优化问题进行扩展研究，将间接法的适用范围从指数型效用函数扩展为HARA效用函数。在HARA效用函数的背景下，研究泰勒级数对效用函数的收敛条件，使得泰勒级数成为期望效用函数的合理近似，从而保证投资组合优化问题的近似解收敛于真实解。通过论证期望效用函数的泰勒级数收敛于实际期望效用函数的充分条件以及分析泰勒级数展开点的选择与泰勒级数收敛性之间的关系，为如何在HARA效用函数的背景下合理运用泰勒级数展开来研究基于高阶矩的投资组合优化问题提供理论依据和方法指导，从而避免目前为了保证收敛性而指定效用函数为指数型效用函数的做法，将间接法下基于高阶矩的投资组合优化研究从指数型效用函数扩展到HARA效用函数范围。
再次，解决了基于高阶矩的动态投资组合优化研究中所遇到的条件协偏度阵和条件协峰度阵难于估计的问题，实现了基于高阶矩的投资组合优化研究由静态向动态方向的扩展。在前人研究成果的基础上，构建了多元风险资产收益率分布时变模型，其中所建立的AR(1)-DCC(1,1)-GARCH(1,1)模型反映了多元条件下条件期望自相关性和条件方差聚集性，所建立的多元条件有偏学生t分布反映了多元条件下有偏厚尾性的时变特征，并提出模型识别、参数估计、模型检验方法和条件协偏度矩阵、条件协峰度矩阵的估计方法；然后利用所构建的模型研究高阶动态投资组合优化问题，构建了高阶动态投资组合优化模型并提出模型求解方法，并通过实证分析对高阶动静态投资组合优化结果进行比较。
最后，以前四阶矩为基础，进一步深化基于高阶矩的投资组合优化研究，提出考虑投资组合收益率完全分布信息的投资组合优化方法，将基于高阶矩的投资组合优化研究从只考虑前四阶矩向考虑投资组合收益率完全分布信息方向扩展。为了获得投资组合收益率分布的近似解析式，以投资组合收益率的前四阶矩为基础，提出基于Gram-Charlier展开的投资组合收益率分布近似模型，并对基于Gram-Charlier展开的投资组合收益率分布近似模型的有效性进行分析；然后根据所提出的基于Gram-Charlier展开的收益率分布近似模型进行投资组合优化研究，将相对熵作为投资组合收益率近似分布与目标分布之间距离的量化指标，从而构建最小化相对熵的投资组合优化模型并提出模型的求解方法并通过算例对理论分析进行说明。
上述研究进一步地完善了基于高阶矩的投资组合优化研究，在继承现有研究成果的基础上，突破现有研究的局限性，使得基于高阶矩的投资组合优化研究更加系统深入，因而具有重要的理论价值。此外，本文的研究在弥补目前基于高阶矩的投资组合优化问题研究不足的同时，也提高其实际应用价值，使其能够真正成为基金公司、养老基金和保险基金等诸多机构投资者的投资组合优化实践中可供参考的工具，这不仅有助于提高机构投资者的科学决策水平，而且有助于证券市场中理性投资理念的建立，并最终有利于金融市场的繁荣稳定和健康发展。
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.
引文
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37Chang C.,Pactwa T.,Prakash A. Selecting a Portfolio with Skewness:RecentEvidence from US,European, and Latin American Equity Markets[J].Journal ofBanking and Finance.2003,27:1375-1390
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40Yu L.,Wang S.Y.,Lai K.K. Neural Network-based Mean-variance-skewnessModel for Portfolio Selection[J]. Computer and Operations Research.2008,35(1):34-46
41Briec W.,Kerstens K., Lesourd J. Single Period Markowitz Portfolio Selection,Performance Gauging and Duality:A Variation on the Luenberger ShortageFunction[J]. Journal of Optimization Theory and Applications.2004,120(1):1-27
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44叶蕾,陈志娟,叶中行.有高阶矩的最优投资组合问题研究[J].运筹学学报.2007,增刊:6-11
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48余婧.均值—方差—近似偏度投资组合模型和实证分析[J].运筹学学报.2010,14(1):106-114
49Jaksa C., Vassilis P.,Fernando Z. Optimal portfolio allocation with highermoments[J]. Annals of Finance.2008,4(1):1-28
50Tauchen G.,Hussey R. Quadrature-based Methods for Obtaining ApproximateSolutions to Nonlinear Asset Pricing Models[J].Econometrica.1991,59(2):371-396
51Ang A., Bekaert G. International Asset Allocation with Regime Shifts [J].Review of Financial Studies.2002,15(4):1137-1387
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53Garlappi L.,Skoulakis G. A State Variable Decomposition Approach for SolvingPortfolio Choice Problem[R]. Working Paper.University of Texas atAustin.2008
54Hlawitschka W. The Empirical Nature of Taylor Series Approximations toExpected Utility[J].American Economic Review.1994,84(3):713-719
55Berenyi Z. Accounting for Illiquidity and Non-Normality of Returns in thePerformance Assessment[R].Working Paper.University of Munich.2001
56Loistl O. The Erroneous Approximation of Expected Utility by Means of aTaylor’s Series Expansion:Ana lytic and Computational Results[J]. AmericanEconomic Review.1976,66(5):904-910
57Lhabitant F.S. On the (ab)use of Taylor Series Approximations for PortfolioSelection. Portfolio Performance and Risk Management[R]. Working Paper.University of Lausanne.1998
58刘树人.指数效用函数下投资组合问题的有效集方法[J].数学理论与应用.2005,25(2):23-25
59初叶萍,张鹏.无风险资产的投资组合效用最大化的模型研究[J].武汉理工大学学报(信息与管理工程版).2006,28(8):118-121
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61常浩,荣喜民.随机参数和随机资金流环境下基于二次效用函数的投资组合优化[J].应用数学学报.2011,34(4):703-711
62傅俊辉,张卫国,陆倩,梅琴.考虑偏度风险和峰度风险的非线性期货套期保值模型[J].系统工程.2009,27(10):44-48
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64王茜,武伟伟.幂效用函数的最优投资组合研究[J].数学理论与应用.2010,30(4):42-44
65崔媛媛,王建琼,庄泓刚.参数不确定条件下考虑偏度的投资组合[J].系统工程理论与实践.2011,31(9):1628-1634
66Geweke J., Amisano G. Compound Markov Mixture Models with Application inFinance[R]. Working Paper. University of Iowa.2001
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