多期模糊投资组合优化模型及算法研究
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摘要
如何将投资者手中的财富按照其投资意图合理地分配到不同的资产中是广大从事金融研究学者和资产管理者所广泛关注的问题。在现实生活中，投资者的行为往往是多期的。继Markowitz单期均值-方差模型之后，已有不少学者在概率论框架下研究了具有随机不确定性的多期投资组合选择问题并且取得了一定的成果。而这些模型均只考虑了金融市场中的随机不确定性，忽略了资产收益的模糊不确定性。随着模糊集理论的广泛应用，一些学者开始尝试利用模糊集理论处理金融市场中的模糊不确定性。目前，对于模糊环境中多期组合选择的研究还停留在单期情形，而对于多期模糊投资组合选择问题的研究仍处在探索阶段。本文将综合运用模糊集理论、最优化方法以及智能优化方法来对多期模糊投资组合选择问题加以研究，进而构建多期模糊投资组合选择理论框架。
本文的主要研究工作以及创新点概括如下：
（1）提出了具有开环及闭环策略的基于可能性高阶矩的多期投资组合选择模型并设计了相应的求解算法。由于传统的大多数多期投资组合优化模型都是开环策略模型，并且它们通常只考虑资产收益和风险两个因素，而对高阶矩的研究鲜有涉及。事实上，已有大量研究表明高阶矩对投资决策的影响不容忽视。针对此问题，我们利用可能性理论来加以研究。首先在均值-方差的理论框架下，以可能性下半方差代替方差作为风险度量，构建了两个具有单一目标的开环多期投资组合优化模型。然后，以所提出的两个单目标模型为基础，增加考虑了偏度和峰度因素对投资决策的影响，分别利用可能性三阶矩和可能性四阶矩来度量其偏度和峰度，进而提出四个基于可能性矩的多目标投资组合优化模型，并设计了一个改进的遗传算法来对模型进行求解，同时结合我国证券市场上的真实数据给出了数值算例来阐述所提出模型的实用性。另外，为了进一步分析证券实际收益与其预期收益之间偏差对投资决策的影响，我们利用动态反馈控制理论提出了两个具有闭环策略的多期投资组合优化模型，并且通过应用实例对所提出的闭环策略模型与相应的开环策略模型进行对比分析。通过比较发现在考虑相同决策因素情形下依据闭环策略模型进行决策较开环策略模型更为有效。
（2）构建了可能性熵分散化度量指标，提出了一个基于可能性均值-半方差-熵的多期分散化投资组合选择模型，并设计了遗传模拟退火算法来对模型求解。在传统投资组合选择模型中通常利用比例熵来度量资产组合的分散化程度，而利用比例熵作为分散化程度的度量可能会导致极其分散的资产组合，这样得到的投资策略并非最优。为了克服上述问题，我们构建了一个新的可能性熵分散化度量指标，进而提出了一个带交易成本的可能性均值-半方差-熵多期投资组合模型，同时给出了一个基于比例熵的多期投资组合优化模型来进行对比分析。然后，设计了一个遗传模拟退火算法来对模型求解。最后，通过应用实例来阐述基于可能性熵分散化投资组合选择模型较基于比例熵分散化投资组合选择模型的优越性。
（3）分别提出了基于区间规划的多期投资组合选择模型和具有容许偏差的多期投资组合选择模型来研究新兴市场中信息不足情形以及具有容许偏差的投资组合优化问题，并给出了相应的求解算法。目前，有关新兴市场中信息严重不足情形下的投资组合选择问题的研究还处在单期情形，而对多期情形下的研究仍尚未涉及。本文利用区间规划的方法对该类问题加以研究，提出了四个基于区间规划的多期投资组合选择模型，借助表示区间数序关系的可能度定义将模型转化为清晰的非线性规划模型，进而设计基于可行解的粒子群方法来对其求解，并通过应用实例来阐述模型的实用性以及算法的有效性。此外，利用模糊测度理论来研究具有容许偏差的多期投资组合选择问题，提出了具有容许偏差多期投资组合选择模型，并设计了改进的微分进化算法来对模型进行求解。
（4）利用可信性理论研究了具有破产风险控制的多期投资组合问题，提出了两个基于可信性测度的多期破产风险控制模型，并设计了相应的混合智能算法对模型进行求解。针对模糊环境中的破产风险控制问题，目前还尚未涉及。本文利用可信性理论来加以研究，主要做了以下两个方面的工作：（i）基于可信性收益最大的多期破产控制问题研究。本文在均值-方差模型框架下，利用可信性理论对具有破产风险控制的多期模糊投资组合选择问题进行探讨，分别以可信性均值和方差来量化投资收益和风险，提出了一个自融资条件下的以可信性收益最大为目标的具有破产风险控制的多期投资组合选择模型，并且给出了一个新的混合遗传粒子群算法对模型进行求解。同时，通过一个应用实例来阐述了模型的实用性以及算法的有效性。（ii）基于均值-下方风险-熵的破产风险控制问题的研究。拓展了前面的基于可信性收益最大的破产风险模型，考虑了资产收益的模糊不确定性对投资决策选取的影响，以可信性下方风险代替可信性方差作为风险度量，以可信性熵作为资产收益模糊不确定程度的度量，提出了一个基于可信性均值-下方风险-熵的破产风险控制模型。由于所提出的模型为三目标规划问题，我们利用模糊多准则决策理论来将其转化成单目标规划模型，进而给出了一个混合智能算法对模型进行求解，并通过一个实例来说明算法的有效性。
How to reasonably allocate investors’ wealth among different assets based on their owninvetment intention is an extensively concerned problem for extensive financial researchersand asset managers. In real world, investors’ behaviors are usually multi-period. AfterMarkowitz’s single period mean-variance model, numerous researchers have proposed aseries of multi-period portfolio selection models under the framework of probability theory.All these models only consider the random uncertainty associated with financial market andneglect the fuzzy uncertainty on the return of asset. With the widely use of fuzzy set theory,people have realized that they could utilize the fuzzy set theory to handle the uncertainty infinancial markets. Up to now, researches on portfolio seletion in fuzzy environment are mainconcerned on portfolio selection in single period cases, and studies about multi-periodportfolio selection in fuzzy environment are still on exploratory stage. This thesis combinesfuzzy set theory, optimization method and intelligent optimization approaches to study themulti-period portfolio selection problem with fuzzy uncertainty. And then, we try to constructthe framework of multi-period fuzzy portfolio selection theory.
The main researches and contributions of this thesis can be summarized as follows:
（1）We propose some multi-period portfolio selection models based on possiblistichigher moments with open-loop and closed-loop policies. Meanwhile, we designcorresponding algorithms to solve these proposed models. Since most of traditionalmulti-period portfolio optimization models are open-loop policy models, they usuallyconsider two main factors, namely, return and risk. Few researches have considered theinfluence of higher moments on the multi-period portfolio selection. Actually, numerousstudies have shown that the effect of higher moments on portfolio decision-making cannot beneglected. For this problem, we use possibility theory to investigate it. Under the frameworkof mean-variance, we substitute possbilistic variance by possbilistic semi-variance as riskmeasure and propose two single-objective models with open-loop policies. Then, on the basisof the two models, we consider the influence of skewness and kurtosis on portfoliodecision-making. We characterize the skewness and kurtosis of portfolio by third possiblisitcmoment and fourth possiblisitc moment, respectively. After that, we propose fourmulti-objective portfolio optimizaiton models based on possibilistic higher moments tosimulate investors’ investment behavior. We design an improved genetic algorithm to solvethese proposed model. Meanwhile, numerical examples by collecting real data in Chinesesecurity markets are given to demonstrate the application of the proposed models. Additionally, to make further analysis about the deviation between the real return of securityand its expected return on portfolio decision-making, we propose two multi-period portfoliooptimization models by using dynamic feedback control theory. Besides, comparsion analysisis provided by an application example to highlight the advatages of closed-loop policy modelsover the corresponding open-loop policy models.
（2）We construct a possiblistic entropy to measure the diversifcation degree of portfolio.Then, we propose a multi-period diversified portfolio selection model based on possiblisticmean-semivariance-entropy. A genetic simulated annealing algorithm is designed for solution.In traditiondal portfolio selection models, researchers were accustomed to employ theproportion entropy to measure the diversification degree of portfolio. However, usingproportion entropy may lead to an extremely diversified portfolio, which is not an optimalinvestment strategy. To overcome aforementioned shortcomings, we construct a novelpossiblistic entropy to measure the diversification degree of portfolio and then we present apossiblistic mean-semivariance-entropy model with transaction cost. Meantime, we stillconstruct a multi-period portofolio optimization model based on proportion entropy formaking a comparsion analysis. Then, a genetic simulated annealing algorithm is designed forsolving the proposed model. Finally, application examples are given to demonstrate theadvantages of diversified portflio model based on possiblistic entropy over the proportionentropy model.
（3）We propose a multi-period portfolio seleciton model by using interval programmingand a multi-period portfolio seleciton model with admissible deviations to investigate themulti-period portfolio optimization problems in emerging markets with information severeshortage and admissible deviations, respectively. Solution algorithms are also given to solvethe proposed models. So far, researches about emerging markets, in which the historicalinformation is severe shortage, are all concerned about single period portfolio selection. Thereare few studies on multi-period portfolio selection. Thus, the purpose of this thesis is to usethe interval programming approach to investigate above-mentioned problem. We propose fourmulti-period portfolio selection models by using interval programming. Then, we transformthe proposed four interval programming models into corresponding crisp form of nonlinearprogramming problems by using the definition of possibility degree, which measures theorder relations of interval numbers. Finally, a feasibility-based particle swarm optimization(PSO) algorithm is designed to for solution. Application examples are also given todemonstrate the application of these models and the effectiveness of the designed algorithm. Besides, to make further analysis investors’ decision-making mind, we also use the fuzzymeasure theory to investigate multi-period portfolio selection problem with admissibledeviations and give an improved differential evolution aglorithm to solve the proposedmodels.
（4）We investigate multi-period portfolio selection problem with bankruptcy control byusing credibility theory and propose two multi-period bankruptcy control model based oncredibilitic measure. Meanwhile, we design corresponding hybrid intelligent algorithms tosolve the two proposed models. Until now, few researches have concerned on the multi-periodbankruptcy control problem in fuzzy environment. This thesis uses the credibility theory tostudy above-mentioned problem. The main contents of this topic can be summarized asfollows:（i）Research on multi-period bankrutcy control model with the objective ofmaximizing credibilitic return. Under the framework of mean-variance model, we usecredibility theory to study multi-period fuzzy portfolio selection problem with bankruptcycontrol. We qualify the investment return and risk by credibilitic mean and variance,respectively. Then, we propose a self-finacing bankruptcy control model for multi-periodportfolio seletion with the objective to maximize the credibilitic return. Meanwhile, we designa novel hybrid gentic algorithm with particle swarm algorithm for solution. After that, we givean empirical analysis to illustrate the application of the proposed model and demonstrate theeffectiveness of the designed algorithm.（ii）Researches on bankruptcy control problem basedon credibilitic return-lower side risk-entropy. This thesis extends the aforementioned riskcontrol model with maximizing credibilitic return. We consider the influence of the fuzzyuncertainty associated with the return of asset on portfolio decision-making. We substitutecredibilitic variance with the credibilitic lower side risk as the risk measure of asset andqualify the fuzzy uncertainty on the return of asset by credibilitic entropy. A credibiliticmean-lower side risk-entropy portfolio optimization model with bankruptcy control isproposed. Since the proposed model is a tri-objective optimization problem, we employ fuzzymutli-criteria decision-making theory to transform it into a corresponding single objetiveprogramming problem. A hybrid intelligent algorithm is designed to for solution and anumerical example is given to demonstrate the effectiveness of the designed algorithm.
本文的主要研究工作以及创新点概括如下：
（1）提出了具有开环及闭环策略的基于可能性高阶矩的多期投资组合选择模型并设计了相应的求解算法。由于传统的大多数多期投资组合优化模型都是开环策略模型，并且它们通常只考虑资产收益和风险两个因素，而对高阶矩的研究鲜有涉及。事实上，已有大量研究表明高阶矩对投资决策的影响不容忽视。针对此问题，我们利用可能性理论来加以研究。首先在均值-方差的理论框架下，以可能性下半方差代替方差作为风险度量，构建了两个具有单一目标的开环多期投资组合优化模型。然后，以所提出的两个单目标模型为基础，增加考虑了偏度和峰度因素对投资决策的影响，分别利用可能性三阶矩和可能性四阶矩来度量其偏度和峰度，进而提出四个基于可能性矩的多目标投资组合优化模型，并设计了一个改进的遗传算法来对模型进行求解，同时结合我国证券市场上的真实数据给出了数值算例来阐述所提出模型的实用性。另外，为了进一步分析证券实际收益与其预期收益之间偏差对投资决策的影响，我们利用动态反馈控制理论提出了两个具有闭环策略的多期投资组合优化模型，并且通过应用实例对所提出的闭环策略模型与相应的开环策略模型进行对比分析。通过比较发现在考虑相同决策因素情形下依据闭环策略模型进行决策较开环策略模型更为有效。
（2）构建了可能性熵分散化度量指标，提出了一个基于可能性均值-半方差-熵的多期分散化投资组合选择模型，并设计了遗传模拟退火算法来对模型求解。在传统投资组合选择模型中通常利用比例熵来度量资产组合的分散化程度，而利用比例熵作为分散化程度的度量可能会导致极其分散的资产组合，这样得到的投资策略并非最优。为了克服上述问题，我们构建了一个新的可能性熵分散化度量指标，进而提出了一个带交易成本的可能性均值-半方差-熵多期投资组合模型，同时给出了一个基于比例熵的多期投资组合优化模型来进行对比分析。然后，设计了一个遗传模拟退火算法来对模型求解。最后，通过应用实例来阐述基于可能性熵分散化投资组合选择模型较基于比例熵分散化投资组合选择模型的优越性。
（3）分别提出了基于区间规划的多期投资组合选择模型和具有容许偏差的多期投资组合选择模型来研究新兴市场中信息不足情形以及具有容许偏差的投资组合优化问题，并给出了相应的求解算法。目前，有关新兴市场中信息严重不足情形下的投资组合选择问题的研究还处在单期情形，而对多期情形下的研究仍尚未涉及。本文利用区间规划的方法对该类问题加以研究，提出了四个基于区间规划的多期投资组合选择模型，借助表示区间数序关系的可能度定义将模型转化为清晰的非线性规划模型，进而设计基于可行解的粒子群方法来对其求解，并通过应用实例来阐述模型的实用性以及算法的有效性。此外，利用模糊测度理论来研究具有容许偏差的多期投资组合选择问题，提出了具有容许偏差多期投资组合选择模型，并设计了改进的微分进化算法来对模型进行求解。
（4）利用可信性理论研究了具有破产风险控制的多期投资组合问题，提出了两个基于可信性测度的多期破产风险控制模型，并设计了相应的混合智能算法对模型进行求解。针对模糊环境中的破产风险控制问题，目前还尚未涉及。本文利用可信性理论来加以研究，主要做了以下两个方面的工作：（i）基于可信性收益最大的多期破产控制问题研究。本文在均值-方差模型框架下，利用可信性理论对具有破产风险控制的多期模糊投资组合选择问题进行探讨，分别以可信性均值和方差来量化投资收益和风险，提出了一个自融资条件下的以可信性收益最大为目标的具有破产风险控制的多期投资组合选择模型，并且给出了一个新的混合遗传粒子群算法对模型进行求解。同时，通过一个应用实例来阐述了模型的实用性以及算法的有效性。（ii）基于均值-下方风险-熵的破产风险控制问题的研究。拓展了前面的基于可信性收益最大的破产风险模型，考虑了资产收益的模糊不确定性对投资决策选取的影响，以可信性下方风险代替可信性方差作为风险度量，以可信性熵作为资产收益模糊不确定程度的度量，提出了一个基于可信性均值-下方风险-熵的破产风险控制模型。由于所提出的模型为三目标规划问题，我们利用模糊多准则决策理论来将其转化成单目标规划模型，进而给出了一个混合智能算法对模型进行求解，并通过一个实例来说明算法的有效性。
How to reasonably allocate investors’ wealth among different assets based on their owninvetment intention is an extensively concerned problem for extensive financial researchersand asset managers. In real world, investors’ behaviors are usually multi-period. AfterMarkowitz’s single period mean-variance model, numerous researchers have proposed aseries of multi-period portfolio selection models under the framework of probability theory.All these models only consider the random uncertainty associated with financial market andneglect the fuzzy uncertainty on the return of asset. With the widely use of fuzzy set theory,people have realized that they could utilize the fuzzy set theory to handle the uncertainty infinancial markets. Up to now, researches on portfolio seletion in fuzzy environment are mainconcerned on portfolio selection in single period cases, and studies about multi-periodportfolio selection in fuzzy environment are still on exploratory stage. This thesis combinesfuzzy set theory, optimization method and intelligent optimization approaches to study themulti-period portfolio selection problem with fuzzy uncertainty. And then, we try to constructthe framework of multi-period fuzzy portfolio selection theory.
The main researches and contributions of this thesis can be summarized as follows:
（1）We propose some multi-period portfolio selection models based on possiblistichigher moments with open-loop and closed-loop policies. Meanwhile, we designcorresponding algorithms to solve these proposed models. Since most of traditionalmulti-period portfolio optimization models are open-loop policy models, they usuallyconsider two main factors, namely, return and risk. Few researches have considered theinfluence of higher moments on the multi-period portfolio selection. Actually, numerousstudies have shown that the effect of higher moments on portfolio decision-making cannot beneglected. For this problem, we use possibility theory to investigate it. Under the frameworkof mean-variance, we substitute possbilistic variance by possbilistic semi-variance as riskmeasure and propose two single-objective models with open-loop policies. Then, on the basisof the two models, we consider the influence of skewness and kurtosis on portfoliodecision-making. We characterize the skewness and kurtosis of portfolio by third possiblisitcmoment and fourth possiblisitc moment, respectively. After that, we propose fourmulti-objective portfolio optimizaiton models based on possibilistic higher moments tosimulate investors’ investment behavior. We design an improved genetic algorithm to solvethese proposed model. Meanwhile, numerical examples by collecting real data in Chinesesecurity markets are given to demonstrate the application of the proposed models. Additionally, to make further analysis about the deviation between the real return of securityand its expected return on portfolio decision-making, we propose two multi-period portfoliooptimization models by using dynamic feedback control theory. Besides, comparsion analysisis provided by an application example to highlight the advatages of closed-loop policy modelsover the corresponding open-loop policy models.
（2）We construct a possiblistic entropy to measure the diversifcation degree of portfolio.Then, we propose a multi-period diversified portfolio selection model based on possiblisticmean-semivariance-entropy. A genetic simulated annealing algorithm is designed for solution.In traditiondal portfolio selection models, researchers were accustomed to employ theproportion entropy to measure the diversification degree of portfolio. However, usingproportion entropy may lead to an extremely diversified portfolio, which is not an optimalinvestment strategy. To overcome aforementioned shortcomings, we construct a novelpossiblistic entropy to measure the diversification degree of portfolio and then we present apossiblistic mean-semivariance-entropy model with transaction cost. Meantime, we stillconstruct a multi-period portofolio optimization model based on proportion entropy formaking a comparsion analysis. Then, a genetic simulated annealing algorithm is designed forsolving the proposed model. Finally, application examples are given to demonstrate theadvantages of diversified portflio model based on possiblistic entropy over the proportionentropy model.
（3）We propose a multi-period portfolio seleciton model by using interval programmingand a multi-period portfolio seleciton model with admissible deviations to investigate themulti-period portfolio optimization problems in emerging markets with information severeshortage and admissible deviations, respectively. Solution algorithms are also given to solvethe proposed models. So far, researches about emerging markets, in which the historicalinformation is severe shortage, are all concerned about single period portfolio selection. Thereare few studies on multi-period portfolio selection. Thus, the purpose of this thesis is to usethe interval programming approach to investigate above-mentioned problem. We propose fourmulti-period portfolio selection models by using interval programming. Then, we transformthe proposed four interval programming models into corresponding crisp form of nonlinearprogramming problems by using the definition of possibility degree, which measures theorder relations of interval numbers. Finally, a feasibility-based particle swarm optimization(PSO) algorithm is designed to for solution. Application examples are also given todemonstrate the application of these models and the effectiveness of the designed algorithm. Besides, to make further analysis investors’ decision-making mind, we also use the fuzzymeasure theory to investigate multi-period portfolio selection problem with admissibledeviations and give an improved differential evolution aglorithm to solve the proposedmodels.
（4）We investigate multi-period portfolio selection problem with bankruptcy control byusing credibility theory and propose two multi-period bankruptcy control model based oncredibilitic measure. Meanwhile, we design corresponding hybrid intelligent algorithms tosolve the two proposed models. Until now, few researches have concerned on the multi-periodbankruptcy control problem in fuzzy environment. This thesis uses the credibility theory tostudy above-mentioned problem. The main contents of this topic can be summarized asfollows:（i）Research on multi-period bankrutcy control model with the objective ofmaximizing credibilitic return. Under the framework of mean-variance model, we usecredibility theory to study multi-period fuzzy portfolio selection problem with bankruptcycontrol. We qualify the investment return and risk by credibilitic mean and variance,respectively. Then, we propose a self-finacing bankruptcy control model for multi-periodportfolio seletion with the objective to maximize the credibilitic return. Meanwhile, we designa novel hybrid gentic algorithm with particle swarm algorithm for solution. After that, we givean empirical analysis to illustrate the application of the proposed model and demonstrate theeffectiveness of the designed algorithm.（ii）Researches on bankruptcy control problem basedon credibilitic return-lower side risk-entropy. This thesis extends the aforementioned riskcontrol model with maximizing credibilitic return. We consider the influence of the fuzzyuncertainty associated with the return of asset on portfolio decision-making. We substitutecredibilitic variance with the credibilitic lower side risk as the risk measure of asset andqualify the fuzzy uncertainty on the return of asset by credibilitic entropy. A credibiliticmean-lower side risk-entropy portfolio optimization model with bankruptcy control isproposed. Since the proposed model is a tri-objective optimization problem, we employ fuzzymutli-criteria decision-making theory to transform it into a corresponding single objetiveprogramming problem. A hybrid intelligent algorithm is designed to for solution and anumerical example is given to demonstrate the effectiveness of the designed algorithm.
引文
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