模糊投资组合优化研究
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摘要
证券组合投资理论是现代金融学的重要部分,也是当今科学研究的难点和热点之一。其核心问题是如何在风险环境下对资源进行分配和利用。本文应用模糊数学和最优化原理来研究证券投资组合选择问题,试图为投资决策分析建立一种新的理论分析框架.全文内容共分为八章。
在第一章中,我们简要地介绍了本文的学术背景及意义,介绍了部分典型的投资组合选择模型以及模糊投资组合选择的国内外的研究动态.
在第二章中,我们研究了模糊均值-方差投资组合模型.第2.1节建立了模糊环境下投资组合选择的均值方差模型,利用隶属函数将模糊规划问题转化为带二次约束的线性优化问题,针对这类问题没有标准解法,引进割平面法把非线性规划问题近似转化为一系列线性规划问题求解.第2.2节我们考虑收益率为模糊数的投资组合选择问题,利用模糊约束简化方差约束,建立了投资组合选择的模糊线性规划模型,然后引进模糊期望把模糊线性规划问题化为普通参数线性规划问题.第2.3节我们考虑了预期收益率为模糊数的投资组合选择问题,沿用第二节的方法,建立了投资组合选择的模糊线性规划模型,然后利用模糊数学知识把模糊线性规划问题转化为多目标线性规划问题,我们利用模糊两阶段法对其求解.第2.4节我们利用模糊约束将均值-方差投资组合模型转化为模糊线性规划模型,用区间数来描述证券的期望收益率和风险损失率,建立了区间数模糊证券投资组合模型,然后利用区间数知识把区间规划问题转化为参数线性规划问题对该模型进行求解,最后给出了一个箅例来阐述方法的有效性.
在第三章中,我们探讨了收益率为模糊数的投资组合模型,提出了可能性均值安全第一投资组合模型。进一步,我们建立了带可能性约束的数学规划模型。利用模糊序关系,该模型可以转化为线性规划问题得到解决。
在第四章中,我们利用第三章的方法考虑方差因素建立了可能性均值-方差安全第一投资组合模型,利用模糊序及模糊数的可能性均值方差把我们的模型转化为二次规划问题,由于这类问题没有固定的解法,我们利用割平面法对其进行求解.
在第五章中,我们研究了模糊均值-绝对偏差投资组合模型。第5.1节我们对多目标证券组合投资模型进行了研究,模型以绝对偏差和代替方差.以换手率刻画流动性,该模型是一多目标线性优化问题,我们采用两阶段模糊算法对模型进行了求解,箅例给出了该模型的一个实例的最优解。第5.2节我们研究了带模糊流动性的收益率为模糊数的均值-绝对偏差投资组合模型,利用模糊数的均值把我们的模型进行转化求解.
在第六章中,我们考虑了模糊均值-β投资组合模型。第6.1节建立了多目标均值-β投资组合模型并利用模糊两阶段方法对模型进行了求解,第6.2节讨论了模糊环境下均值-β投资组合模型,利用隶属函数知识把模糊规划问题转化为参数线性规划问题,根据投资者的主观意愿选定参数可得满意的投资组合,第6.3节讨论了区间模糊数均值-β投资组合模型,引进区间数比较的满意度,利用区间数知识对模型进行了转化求解.
在第七章中,我们研究了模糊均值-熵投资组合模型。第7.1节我们建立了基于信息熵的证券投资组合模型,该模型是一多目标线性优化问题,我们采用两阶段模糊算法对模型进行了求解,箅例给出了该模型的一个实例的最优解.第7.2节讨论了区间模糊均值-熵投资组合模型,利用区间数满意度把区间数不等式转化为清晰数不等式,得到模型的参数规划,根据投资者的偏好决定参数值可对模型进行求解.
在第八章中,单位风险收益最大化的组合投资决策模型的分式规划解法
我们将预期收益率表示为模糊数,给出一个折衷考虑风险最小化和收益最大化的单目标决策方法。首先,以单位风险收益最大化为决策目标建立了投资组合的分式规划模型,考虑到分式规划问题的求解难度,我们利用遗传算法研究模型求解并给出算法步骤与数值箅例.
Portfolio theory has been an important part of modern financial theory and one of difficult and hot issues in scientific research nowadays. Its central problem is how to allocate and utilize capital assets under risk. The authors apply fuzzy mathematics and optimization theory to study portfolio selection problems systemically and deeply, we try to establish a new framework for investment analysis.This dissertation consists of eight chapters.
In Chapter 1, the academic background and importance are briefly addressed, and the same time the developments of some classical portfolio selection model and fuzzy portfolio selection models are also introduced.
In Chapter 2, Fuzzy Mean-Variance portfolio selection model are studied. In sec-tion 2.1, Fuzzy set was applied to portfolio selection model, the fuzzy mean variance portfolio selection model was proposed(FMV),and the fuzzy programming problem can be transformed into a linear optimal problem (PMV)with an additional quadratic con-straint by fuzzy maths theory. For such problems there are no special standard algo-rithms. A cutting plane algorithm was proposed to solve (PMV), and then the nonlin-ear programming problem can be solved by sequence linear programming problem. A numerical example was given to illustrate the behavior of the proposed model and al-gorithm. In section 2.2,a portfolio selection problem with possibilistic return rates was considered. An efficient way is given to transform an traditional optimal problem with variance constraint into a fuzzy linear problem, a fuzzy linear programming problem is converted into an ordinary parameter linear programming problem by introducing possibilistic mean value and possibilistic variance, a numerical example is given to il-lustrate the behavior of the proposed model. In section 2.3, a portfolio selection problem with fuzzy expected return rates was considered. A efficient way is given to transform an traditional optimal problem with variance constraint into a fuzzy linear problem, a fuzzy linear programming problem is converted into an multi-objective parameter lin-ear programming problem by the knowledge of fuzzy math,Fuzzy two-stage algorithm was applied to solve it. a numerical example was given to illustrate the behavior of the proposed model. In section 2.4, a interval number fuzzy portfolio selection model is proposed in a method, which Markowitz portfolio selection model is converted into fuzzy linear programming model with fuzzy constraint and profit rates and risk rates of security is described by interval fuzzy number, then a interval programming problem is converted into a parameter linear programming problem by the knowledge of interval math. Finally, a numerical example is given to illustrate the validity of the method.
In chapter 3, a portfolio selection problem with fuzzy return rates is dealt. A pos-sibilistic mean safety-first portfolio selection model was proposed. Specially, by fuzzy order, we present a mathematical programming model with possibilistic constraint. The possibilistic programming problem can be solved by transforming it into a linear pro-gramming problem.
In chapter 4, fuzzy mean-variance, safety-first portfolio selection model is pro-posed when we consider the factor of variance, at the same time, the model is converted into an quadratic programming one with possibilistic mean of fuzzy order and fuzzy number. For there no special standard algorithms to such problems, we introduce a cutting plane algorithm to solve them.
In chapter 5, Fuzzy Mean absolute deviation portfolio selection is studied. In sec-tion 5.1, we study multi-objective portfolio selection(MADL), In model,the risk was taken as the sum of the absolute deviation of the risk assets instead of covariance. liquid-ity was depicted by turnover.Problem(MADL)is a multi-objective linear optimal prob-lem, Fuzzy two-stage algorithm was applied to solve it.The optimum solution of an example about this portfolio model was given with this new algorithm. In section 5.2 the thesis dealt with a mean absolute deviation portfolio selection problem with fuzzy return rates under fuzzy liquidity constraint, a new possibilistic programming approach based on possibilistic mean and fuzzy liquidity has been proposed, the problem can be reduced to a linear programming by possibility theory. A numerical example of portfolio selection problem is given to illustrate our proposed approach.
In chapter 6, Fuzzy Meanβportfolio selection is studied. In section 6.1, mean-βportfolio selection model was presented, and Fuzzy two-stage algorithm was applied to solve it. In section 6.2, mean-βportfolio selection model under the fuzzy constraint was discussed, the fuzzy programming problem was turned into parameter linear pro-gramming by means the knowledge of membership function, thus satisfied portfolio selection model can be obtained on the basis of the investors subjective wish to select the parameter. In section 6.3,interval fuzzy number mean-βportfolio selection model was discussed, interval number satisfied level was taken, by means of the knowledge of fuzzy numbers we obtained the solution of the model.
In chapter 7, the portfolio selection based on entropy is given, problem is a multi-objective linear optimal problem, Fuzzy two-stage algorithm is applied to solve it.The optimum solution of an example about this portfolio model is given with this new al-gorithm. Later, interval fuzzy entropy portfolio selection model was considered, and parameter programming of the model was obtained after interval number inequality was transformed into clear number inequality using interval number satisfied level.Thus the model can be solved according to risk preference of investor.
In chapter 8, Fractional Utility function Portfolio Selection model is presented. About the largest return and the smallest risk of the portfolio selection problem, a utility function that balances the returns and risks is advanced, and then the portfolio selec-tion model is built based on the nonlinear fractional programming, in order to solve the model, an genetic algorithm is proposed,and the numerical example of it are given.
在第一章中,我们简要地介绍了本文的学术背景及意义,介绍了部分典型的投资组合选择模型以及模糊投资组合选择的国内外的研究动态.
在第二章中,我们研究了模糊均值-方差投资组合模型.第2.1节建立了模糊环境下投资组合选择的均值方差模型,利用隶属函数将模糊规划问题转化为带二次约束的线性优化问题,针对这类问题没有标准解法,引进割平面法把非线性规划问题近似转化为一系列线性规划问题求解.第2.2节我们考虑收益率为模糊数的投资组合选择问题,利用模糊约束简化方差约束,建立了投资组合选择的模糊线性规划模型,然后引进模糊期望把模糊线性规划问题化为普通参数线性规划问题.第2.3节我们考虑了预期收益率为模糊数的投资组合选择问题,沿用第二节的方法,建立了投资组合选择的模糊线性规划模型,然后利用模糊数学知识把模糊线性规划问题转化为多目标线性规划问题,我们利用模糊两阶段法对其求解.第2.4节我们利用模糊约束将均值-方差投资组合模型转化为模糊线性规划模型,用区间数来描述证券的期望收益率和风险损失率,建立了区间数模糊证券投资组合模型,然后利用区间数知识把区间规划问题转化为参数线性规划问题对该模型进行求解,最后给出了一个箅例来阐述方法的有效性.
在第三章中,我们探讨了收益率为模糊数的投资组合模型,提出了可能性均值安全第一投资组合模型。进一步,我们建立了带可能性约束的数学规划模型。利用模糊序关系,该模型可以转化为线性规划问题得到解决。
在第四章中,我们利用第三章的方法考虑方差因素建立了可能性均值-方差安全第一投资组合模型,利用模糊序及模糊数的可能性均值方差把我们的模型转化为二次规划问题,由于这类问题没有固定的解法,我们利用割平面法对其进行求解.
在第五章中,我们研究了模糊均值-绝对偏差投资组合模型。第5.1节我们对多目标证券组合投资模型进行了研究,模型以绝对偏差和代替方差.以换手率刻画流动性,该模型是一多目标线性优化问题,我们采用两阶段模糊算法对模型进行了求解,箅例给出了该模型的一个实例的最优解。第5.2节我们研究了带模糊流动性的收益率为模糊数的均值-绝对偏差投资组合模型,利用模糊数的均值把我们的模型进行转化求解.
在第六章中,我们考虑了模糊均值-β投资组合模型。第6.1节建立了多目标均值-β投资组合模型并利用模糊两阶段方法对模型进行了求解,第6.2节讨论了模糊环境下均值-β投资组合模型,利用隶属函数知识把模糊规划问题转化为参数线性规划问题,根据投资者的主观意愿选定参数可得满意的投资组合,第6.3节讨论了区间模糊数均值-β投资组合模型,引进区间数比较的满意度,利用区间数知识对模型进行了转化求解.
在第七章中,我们研究了模糊均值-熵投资组合模型。第7.1节我们建立了基于信息熵的证券投资组合模型,该模型是一多目标线性优化问题,我们采用两阶段模糊算法对模型进行了求解,箅例给出了该模型的一个实例的最优解.第7.2节讨论了区间模糊均值-熵投资组合模型,利用区间数满意度把区间数不等式转化为清晰数不等式,得到模型的参数规划,根据投资者的偏好决定参数值可对模型进行求解.
在第八章中,单位风险收益最大化的组合投资决策模型的分式规划解法
我们将预期收益率表示为模糊数,给出一个折衷考虑风险最小化和收益最大化的单目标决策方法。首先,以单位风险收益最大化为决策目标建立了投资组合的分式规划模型,考虑到分式规划问题的求解难度,我们利用遗传算法研究模型求解并给出算法步骤与数值箅例.
Portfolio theory has been an important part of modern financial theory and one of difficult and hot issues in scientific research nowadays. Its central problem is how to allocate and utilize capital assets under risk. The authors apply fuzzy mathematics and optimization theory to study portfolio selection problems systemically and deeply, we try to establish a new framework for investment analysis.This dissertation consists of eight chapters.
In Chapter 1, the academic background and importance are briefly addressed, and the same time the developments of some classical portfolio selection model and fuzzy portfolio selection models are also introduced.
In Chapter 2, Fuzzy Mean-Variance portfolio selection model are studied. In sec-tion 2.1, Fuzzy set was applied to portfolio selection model, the fuzzy mean variance portfolio selection model was proposed(FMV),and the fuzzy programming problem can be transformed into a linear optimal problem (PMV)with an additional quadratic con-straint by fuzzy maths theory. For such problems there are no special standard algo-rithms. A cutting plane algorithm was proposed to solve (PMV), and then the nonlin-ear programming problem can be solved by sequence linear programming problem. A numerical example was given to illustrate the behavior of the proposed model and al-gorithm. In section 2.2,a portfolio selection problem with possibilistic return rates was considered. An efficient way is given to transform an traditional optimal problem with variance constraint into a fuzzy linear problem, a fuzzy linear programming problem is converted into an ordinary parameter linear programming problem by introducing possibilistic mean value and possibilistic variance, a numerical example is given to il-lustrate the behavior of the proposed model. In section 2.3, a portfolio selection problem with fuzzy expected return rates was considered. A efficient way is given to transform an traditional optimal problem with variance constraint into a fuzzy linear problem, a fuzzy linear programming problem is converted into an multi-objective parameter lin-ear programming problem by the knowledge of fuzzy math,Fuzzy two-stage algorithm was applied to solve it. a numerical example was given to illustrate the behavior of the proposed model. In section 2.4, a interval number fuzzy portfolio selection model is proposed in a method, which Markowitz portfolio selection model is converted into fuzzy linear programming model with fuzzy constraint and profit rates and risk rates of security is described by interval fuzzy number, then a interval programming problem is converted into a parameter linear programming problem by the knowledge of interval math. Finally, a numerical example is given to illustrate the validity of the method.
In chapter 3, a portfolio selection problem with fuzzy return rates is dealt. A pos-sibilistic mean safety-first portfolio selection model was proposed. Specially, by fuzzy order, we present a mathematical programming model with possibilistic constraint. The possibilistic programming problem can be solved by transforming it into a linear pro-gramming problem.
In chapter 4, fuzzy mean-variance, safety-first portfolio selection model is pro-posed when we consider the factor of variance, at the same time, the model is converted into an quadratic programming one with possibilistic mean of fuzzy order and fuzzy number. For there no special standard algorithms to such problems, we introduce a cutting plane algorithm to solve them.
In chapter 5, Fuzzy Mean absolute deviation portfolio selection is studied. In sec-tion 5.1, we study multi-objective portfolio selection(MADL), In model,the risk was taken as the sum of the absolute deviation of the risk assets instead of covariance. liquid-ity was depicted by turnover.Problem(MADL)is a multi-objective linear optimal prob-lem, Fuzzy two-stage algorithm was applied to solve it.The optimum solution of an example about this portfolio model was given with this new algorithm. In section 5.2 the thesis dealt with a mean absolute deviation portfolio selection problem with fuzzy return rates under fuzzy liquidity constraint, a new possibilistic programming approach based on possibilistic mean and fuzzy liquidity has been proposed, the problem can be reduced to a linear programming by possibility theory. A numerical example of portfolio selection problem is given to illustrate our proposed approach.
In chapter 6, Fuzzy Meanβportfolio selection is studied. In section 6.1, mean-βportfolio selection model was presented, and Fuzzy two-stage algorithm was applied to solve it. In section 6.2, mean-βportfolio selection model under the fuzzy constraint was discussed, the fuzzy programming problem was turned into parameter linear pro-gramming by means the knowledge of membership function, thus satisfied portfolio selection model can be obtained on the basis of the investors subjective wish to select the parameter. In section 6.3,interval fuzzy number mean-βportfolio selection model was discussed, interval number satisfied level was taken, by means of the knowledge of fuzzy numbers we obtained the solution of the model.
In chapter 7, the portfolio selection based on entropy is given, problem is a multi-objective linear optimal problem, Fuzzy two-stage algorithm is applied to solve it.The optimum solution of an example about this portfolio model is given with this new al-gorithm. Later, interval fuzzy entropy portfolio selection model was considered, and parameter programming of the model was obtained after interval number inequality was transformed into clear number inequality using interval number satisfied level.Thus the model can be solved according to risk preference of investor.
In chapter 8, Fractional Utility function Portfolio Selection model is presented. About the largest return and the smallest risk of the portfolio selection problem, a utility function that balances the returns and risks is advanced, and then the portfolio selec-tion model is built based on the nonlinear fractional programming, in order to solve the model, an genetic algorithm is proposed,and the numerical example of it are given.
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