（企业管理专业论文）基于高阶矩的投资组合优化研究.pdf

国内图书分类号：f830.591 学校代码：10213 国际图书分类号：336.7 密级：公开 管理学博士学位论文管理学博士学位论文 基于高阶矩的投资组合优化研究 博 士 研 究 生 ： 彭胜志 导 师 ： 王福胜教授 申请学位 ： 管理学博士 学科 ： 企业管理 所 在 单 位 ： 管理学院 答 辩 日 期 ： 2012 年 7 月 授予学位单位 ： 哈尔滨工业大学 classified index: f830.591 u.d.c: 336.7 dissertation for the doctoral degree in management portfolio optimization with higher-order moments candidate： peng sheng-zhi supervisor： prof.wang fu-sheng academic degree applied for： doctor of management speciality： enterprise management affiliation： school of management date of defence： july, 2012 degree-conferring-institution： harbin institute of technology 摘 要 - i - 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 哈尔滨工业大学管理学博士学位论文 - ii - 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。 关键词：投资组合优化；高阶矩；非凸性；收敛性；时变性；完全分布信息 abstract - iii - abstract markowitzs mean-variance model has the epoch-making significance，which makes finance get rid of the situation of purely descriptive study and the operation only by experience，and settles the foundation for the development of modern portfolio theory. the necessary and sufficient condition, under which the mean-variance model is consistent with the expected utility principle, is that the return rate of the risky asset obeys the normal distribution or the investor has the utility of quadratic function. however, unfortunately, the above condition has no practical significance. a lot of empirical studies home and abroad have shown that the distribution of the return rate of the risky asset isnt normal and the utility function of investors isnt the quadratic function. so the effect of higher-order moments should be considered in the decision of the portfolio optimization, otherwise suboptimal decision will follow. many scholars home and abroad have made the study of portfolio optimization with higher-order moments. these studies have been conducted directly or indirectly. the direct method takes higher-order moments or the function consist of the first four moments as the objective function in the problem of portfolio optimization, while the indirect method converts the problem of portfolio optimization of investors utility maximization into the problem of portfolio optimization with higher-order moments through taylor series expansion of the expected utility function. although the existing studies have been adundant, there are still a lots of shortage needed be improved. this dissertation, based on previous study results, intends to extend the study of portfolio optimization with higher-order moments in order to improve the existing study and make portfolio optimization with higher-order moments truly become the method and tool that could be referenced in the dicision of portfolio optimization. firstly, from the view of direct method, the study of portfolio optimization with higher moments is extended under the framework of the first four moments, the algorithm of semidefinite programming relaxation is proposed in order to deal with the difficulty of problem solution of portfolio optimization from non-convexity of optimization problem and the higher order of objective function. taking the optimization problem of kurtosis maximization as an example, the algorithm of semidefinite programming relaxation for the solution of the problem of portfolio optimization with higher-order moments is put forward based on the research result of lasserre and waki. the algorithm could convert the optimization problem that has higher-order objective function into the optimization problem of linear matrix inequality using the theory of moment matrices, which could acquire global 哈尔滨工业大学管理学博士学位论文 - iv - optimum solution at relatively fast convergence rate under the condition that objective function is higher-order polynomial and the optimization problem is non-convex. furthermore, the efficient frontier of portfolio optimization maximizing kurtosis is deduced theoretically, then the algorithm is applied and the deduced efficient frontier is verified and the validity of the algorithm of semidefi nite programming relaxation is also proved in empirical analysis. secondly, from the view of indirect method, the study of portfolio optimization with higher moments is extended where the scope of the application of indirect method is extended from exponential utility function to hara utility function. under the background of hara utility function, the convergence conditions of taylor series to expected utility function is studied in order to make taylor series becomes the reasonable approximation for expected utility function and guarantee the convergence of approximate solution to real solution. the sufficient condition under which taylor series is convergent to the expected utility function is demonstrated and the relationship between the selection of the expansion point of taylor series and the convergence of taylor series is analyzed，through which the theoretical basis and method instruction is provided in applying taylor series for portfolio optimization with higher-order moments，and the ordinary practice that sets utility function as exponential utility function for the guaranty of convergence of taylor series is avoided，therefore the scope of the application of indirect method is extended from exponential utility function to hara utility function. thirdly, the problem that conditional coskewness matrix and cokurtosis matrix are difficult to estimate is solved, through which the study of portfolio optimization with higher moments is extended from static angle to dynamics angle. based on previous research results, a new model is proposed to describe the time-variant characteristics of the multivariate distribution of return of financial assets, in which the model of ar(1)-dcc(1,1)-garch(1,1) is proposed to describe the self-correlation of multivariate conditional expectation and the aggregation of multivariate conditional variance while the multivariate conditional skewed-t distribution is proposed to describe the time-variant of skewed and fat-tailed characteristics of the multivariate return of financial assets. methods of model identification, parameter estimation, model-testing of the new model and the estimation method of conditional coskewness matrix and cokurtosis matrix are also proposed. through applying the proposed model, the dynamic portfolio optimization with higher order moments is studied and the optimization results of dynamic and static optimization are compared in the empirical analysis. finally, on the basis of the first four moments, portfolio optimization is studied taking the full information of the distribution of portfolio return rate into abstract - v - consideration. the approximate analytic expression of the distribution of the return rate of portfolio is put forward based on gram-charlier expansion and the validity of the put-forward approximate model is analyzed. then relative entropy is taken as the measure index of the distance between the approximate distribution and goal distribution and the model of portfolio optimization of minimizing relative entropy is established from the intention of taking the full information of distribution of the return rate of portfolio into consideration. numerical example is followed to illustrate the theoretical analysis. the study of this dissertation ulteriorly improves the study of portfolio optimization with higher-order moments ,breaks the limitation of the existing research, and makes the study of portfolio optimization with higher-order moments more symmetrical and thorough, therefore has important theoretical value. furthermore, this dissertation also enhances the application value of the portfolio optimization, makes it truly become the referred tool by fund company, pension fund and insurance fund etc., is helpful to the scientific decision of institutional investors and the establishment of the rational investment idea, is ultimately beneficial to the prosperity and healthy development of financial market. keywords: portfolio optimization, higher-order moments, non-convexity, convergence, time-variance, fully information of distribution 哈尔滨工业大学管理学博士学位论文 - vi - 目 录 摘 要 . i abstract . iii 第 1 章 绪 论 . 1 1.1 选题背景与问题提出 . 1 1.1.1 选题背景 . 1 1.1.2 问题提出 . 3 1.2 研究目的和研究意义 . 5 1.2.1 研究目的 . 5 1.2.2 研究意义 . 5 1.3 国内外研究现状 . 8 1.3.1 以高阶矩为目标函数的投资组合优化问题的研究现状 . 8 1.3.2 基于期望效用函数的投资组合优化问题的研究现状. 11 1.3.3 多元条件高阶矩建模及动态投资组合问题的研究现状 . 13 1.3.4 国内外研究现状评述. 15 1.4 研究内容 . 17 1.5 研究方法与技术路线 . 18 1.5.1 研究方法 . 18 1.5.2 技术路线 . 19 第 2 章 直接法下基于高阶矩的投资组合优化 . 20 2.1 高阶矩的含义与投资者的高阶矩偏好 . 20 2.1.1 高阶矩的含义 . 20 2.1.2 投资者的高阶矩偏好. 21 2.2 直接法下投资组合优化模型的构建与求解 . 24 2.2.1 基于高阶矩的投资组合优化模型的构建 . 24 2.2.2 投资组合收益率前四阶矩的代数表述 . 25 2.2.3 最小化峰度问题的求解方法 . 27 2.3 最小化峰度问题的有效前沿. 29 2.4 实证分析 . 31 2.4.1 样本数据分析 . 32 2.4.2 问题的求解 . 34 目 录 - vii - 2.4.3 峰度与偏度和方差之间的关系分析. 36 2.5 本章小结 . 40 第 3 章 间接法下基于高阶矩的投资组合优化 . 41 3.1 效用函数的设定 . 41 3.2 hara 效用函数的泰勒级数收敛性研究 . 42 3.2.1 泰勒级数展开的重新表述 . 43 3.2.2 泰勒级数的收敛条件分析 . 44 3.3 投资组合背景下的泰勒级数收敛性研究 . 47 3.3.1 基于期望效用的投资组合模型及其泰勒级数近似. 48 3.3.2 泰勒级数近似模型的求解方法 . 49 3.3.3 投资组合优化背景下的泰勒级数收敛条件分析 . 50 3.4 实证分析 . 55 3.5 本章小结 . 59 第 4 章 考虑高阶矩时变性的动态投资组合优化 . 60 4.1 多元风险资产收益率分布的动态模型 . 60 4.1.1 模型表达 . 61 4.1.2 建模过程 . 64 4.1.3 条件协偏度阵 t s和条件协峰度阵 t k的估计 . 67 4.1.4 实证分析 . 68 4.2 基于高阶矩的动态投资组合优化问题 . 74 4.2.1 高阶动态投资组合优化模型的建立与求解 . 75 4.2.2 实证分析 . 77 4.3 本章小结 . 82 第 5 章 考虑投资组合收益率完全分布信息的投资组合优化 . 83 5.1 基于 gram-charlier展开的收益率分布近似模型研究. 84 5.1.1 gram-charlier 的理论推导. 84 5.1.2 风险资产收益率概率密度函数的 gram-charlier 近似 . 86 5.1.3 gram-charlier 展开式近似收益率分布的有效性分析 . 88 5.2 基于投资组合收益率近似分布模型的投资组合优化研究 . 92 5.2.1 投资组合收益率分布近似模型的确定 . 92 5.