投资组合选择模型及启发式算法研究
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摘要
本文分别基于随机性理论和模糊数的可能性理论对投资组合选择问题进行了深入的研究，并用遗传算法和粒子群算法这两种启发式算法进行求解。本文的主要研究内容概括如下：
1．考虑到现代证券投资组合理论在我国的实用性，从而基于卖空限制、交易费用限制和最小交易单位限制提出了具有投资限制的投资组合选择模型，并设计了一种改进的遗传算法求解我们所提出的整数规划模型。
2．研究了基于绝对偏差风险度量下的三种具有交易费用的投资组合模型，即具有交易费用的均值一绝对偏差投资组合模型、具有交易费用的均值一半绝对偏差投资组合模型和具有交易费用的均值一极大极小半绝对偏差投资组合模型，并分别对这三种不同风险度量下的模型进行实证比较研究，说明交易费用对投资组合选择有重要的影响，以及不同风险度量函数下所构建投资组合的优劣。
3．在假定证券的期望收益和风险具有可容许偏差的条件下，研究了具有交易费用和投资数量限制下的可容许有效投资组合问题。首先，提出了具有交易费用和投资数量限制的可容许有效投资组合模型。其次，设计了一种改进的粒子群算法求解上述问题。最后，通过一个实例说明我们所给出模型和算法的有效性，并比较了不同约束条件下可容许投资组合模型的上、下可容许有效边界。
4．在模糊数的可能性理论基础上研究了若干投资组合问题。首先，基于Carlsson和Fullér给出的上下可能性均值的定义提出了一种新的可能性均值的概念，即加权的可能性均值，并在此基础上提出了基于均值—方差效用函数的可能性投资组合模型。其次，以往关于具有交易费用的投资组合问题的研究大多是建立在不确定性为随机性的基础之上，从而基于可能性理论我们研究了存在交易费用的投资组合问题，建立了相应的模型，并给出了求解方法。最后，分别研究了存在融资和多约束(融资、流动性和投资数量约束下)条件下的可能性投资组合问题，建立了相应的模型。
5．研究了若干特殊隶属函数下的模糊可能性投资组合模型。分别基于Carlsson和Fullér的可能性均值、方差的定义与Zhang的上下可能性均值、方差定义的基础上，给出了收益率分别为三角模糊数、梯形模糊数和正态模糊数下的可能性投资组合模型，并对以上两种不同定义下的三种不同模糊数下的可能性投资组合模型进行了比较研究。
Based on stochastic theory and fuzzy possiblistic theory we study some portfolio selection problems, and we use two kinds of heuristics algorithms-genetic algorithm and particle swarm optimization for our proposed problems.
The primary contents will be generalized as follows:
1. Considering the practicality of moder portfolio selection theory, we propose a constrained portfolio selection model based on the no short sale, transaction costs and minimum lot constraints. Due to these complex constraints our proposed problem becomes a mixed integer optimization problem and traditional algorithms can't solve it efficiently. Thus, an improved genetic algorithm is designed to solve our proposed problem.
2. Based on the risk measurement of mean-absolute deviation we study three kinds of portfolio selection models with transaction costs, namely, mean-absolute deviation model with transaction costs, mean-semiabsolute deviation model with transaction costs, and mean-maxmin-semiabsolute deviation model with transaction costs, then an example of portfolio problem is given to compare three kinds of models, and comparative analysis is also given to show that transaction costs has great impact on the portfolio selection.
3. The admissible efficient portfolio model with transaction costs and quantity constrains are studied under the assumption that the expected return and risk of assets have admissible errors. Firstly, the admissible efficient portfolio model with transaction costs and quantity constrains is proposed. Secondly, traditional optimization algorithms fail to work efficiently for our proposed problem, so we designed an improved particle swarm optimization algorithm to solve it. At last, the effectiveness of the improved particle swarm optimization algorithm is demonstrated on a realistic portfolio selection problem.
4. Some fuzzy possibilistic portfolio selection problems are studied based on the possibilistic theory under the assumption that the returns of assets are fuzzy numbers. Firstly, we propose a new conception of possiblistic mean, namely, weighted possibilistic mean, which is the extension definition proposed by Carlsson and Fuller, then the possiblistic portfolio selection model with utility function is proposed. Secondly, based on the theory of possiblistic theory the possbilistic portfolio selection model with transaction costs is proposed, then the method for solving this problem is given. At last, the portfolio selection problem with borrowing constraints, and with multiple constraints (borrowing, liquditity and investing quantity constraints) are studied, respectively.
5. Study some possiblistic portfolio selection problems under returns of assets are some special fuzzy number. Regarding returns of assets as triangle fuzzy numbers, trapezoidal fuzzy numbers and normal fuzzy numbers respectively, some possiblistic portfolio selection models are proposed based on two kinds of definition of possiblistic mean and variance. Then, some comparative analyses are given to show the difference of these models under different definition.
1．考虑到现代证券投资组合理论在我国的实用性，从而基于卖空限制、交易费用限制和最小交易单位限制提出了具有投资限制的投资组合选择模型，并设计了一种改进的遗传算法求解我们所提出的整数规划模型。
2．研究了基于绝对偏差风险度量下的三种具有交易费用的投资组合模型，即具有交易费用的均值一绝对偏差投资组合模型、具有交易费用的均值一半绝对偏差投资组合模型和具有交易费用的均值一极大极小半绝对偏差投资组合模型，并分别对这三种不同风险度量下的模型进行实证比较研究，说明交易费用对投资组合选择有重要的影响，以及不同风险度量函数下所构建投资组合的优劣。
3．在假定证券的期望收益和风险具有可容许偏差的条件下，研究了具有交易费用和投资数量限制下的可容许有效投资组合问题。首先，提出了具有交易费用和投资数量限制的可容许有效投资组合模型。其次，设计了一种改进的粒子群算法求解上述问题。最后，通过一个实例说明我们所给出模型和算法的有效性，并比较了不同约束条件下可容许投资组合模型的上、下可容许有效边界。
4．在模糊数的可能性理论基础上研究了若干投资组合问题。首先，基于Carlsson和Fullér给出的上下可能性均值的定义提出了一种新的可能性均值的概念，即加权的可能性均值，并在此基础上提出了基于均值—方差效用函数的可能性投资组合模型。其次，以往关于具有交易费用的投资组合问题的研究大多是建立在不确定性为随机性的基础之上，从而基于可能性理论我们研究了存在交易费用的投资组合问题，建立了相应的模型，并给出了求解方法。最后，分别研究了存在融资和多约束(融资、流动性和投资数量约束下)条件下的可能性投资组合问题，建立了相应的模型。
5．研究了若干特殊隶属函数下的模糊可能性投资组合模型。分别基于Carlsson和Fullér的可能性均值、方差的定义与Zhang的上下可能性均值、方差定义的基础上，给出了收益率分别为三角模糊数、梯形模糊数和正态模糊数下的可能性投资组合模型，并对以上两种不同定义下的三种不同模糊数下的可能性投资组合模型进行了比较研究。
Based on stochastic theory and fuzzy possiblistic theory we study some portfolio selection problems, and we use two kinds of heuristics algorithms-genetic algorithm and particle swarm optimization for our proposed problems.
The primary contents will be generalized as follows:
1. Considering the practicality of moder portfolio selection theory, we propose a constrained portfolio selection model based on the no short sale, transaction costs and minimum lot constraints. Due to these complex constraints our proposed problem becomes a mixed integer optimization problem and traditional algorithms can't solve it efficiently. Thus, an improved genetic algorithm is designed to solve our proposed problem.
2. Based on the risk measurement of mean-absolute deviation we study three kinds of portfolio selection models with transaction costs, namely, mean-absolute deviation model with transaction costs, mean-semiabsolute deviation model with transaction costs, and mean-maxmin-semiabsolute deviation model with transaction costs, then an example of portfolio problem is given to compare three kinds of models, and comparative analysis is also given to show that transaction costs has great impact on the portfolio selection.
3. The admissible efficient portfolio model with transaction costs and quantity constrains are studied under the assumption that the expected return and risk of assets have admissible errors. Firstly, the admissible efficient portfolio model with transaction costs and quantity constrains is proposed. Secondly, traditional optimization algorithms fail to work efficiently for our proposed problem, so we designed an improved particle swarm optimization algorithm to solve it. At last, the effectiveness of the improved particle swarm optimization algorithm is demonstrated on a realistic portfolio selection problem.
4. Some fuzzy possibilistic portfolio selection problems are studied based on the possibilistic theory under the assumption that the returns of assets are fuzzy numbers. Firstly, we propose a new conception of possiblistic mean, namely, weighted possibilistic mean, which is the extension definition proposed by Carlsson and Fuller, then the possiblistic portfolio selection model with utility function is proposed. Secondly, based on the theory of possiblistic theory the possbilistic portfolio selection model with transaction costs is proposed, then the method for solving this problem is given. At last, the portfolio selection problem with borrowing constraints, and with multiple constraints (borrowing, liquditity and investing quantity constraints) are studied, respectively.
5. Study some possiblistic portfolio selection problems under returns of assets are some special fuzzy number. Regarding returns of assets as triangle fuzzy numbers, trapezoidal fuzzy numbers and normal fuzzy numbers respectively, some possiblistic portfolio selection models are proposed based on two kinds of definition of possiblistic mean and variance. Then, some comparative analyses are given to show the difference of these models under different definition.
引文
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