考虑背景风险因素的可能性投资组合选择模型研究
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摘要
投资决策的核心问题是如何在众多金融产品中选择最优的组合进行投资。经典的投资组合模型根据投资者的风险厌恶程度将财富按一定比例在风险资产之间进行分配，以达到分散风险，确保收益的目的。经典的投资组合模型认为投资者在投资中只承受金融风险，而在实际的投资环境中投资者还要应对风险性非金融风险的影响。比如说，劳动收入、健康状况、持有房产等因素导致的风险（背景风险）。这些背景风险在金融市场上不能通过资产组合配置来分散，而它们的存在很大程度影响了投资者在金融市场中的投资行为，进而使得投资组合问题复杂化。迄今为止，已有一些学者对考虑背景风险的投资组合问题进行了研究。而这些研究主要建立在随机理论基础之上，他们将资产组合的未来收益看成随机变量。
然而现实的金融市场中存在许多非随机因素的影响，尤其在一个模糊的投资环境中，风险资产的收益表示为模糊数。因此考虑模糊不确定性的投资组合选择问题也是学术界研究的重要领域之一。
本文结合以上两点，依据可能性理论对考虑背景风险因素的可能性投资组合选择问题进行了系统的分析和研究，建立了不同环境下考虑背景风险因素的可能性投资组合选择模型。并对考虑背景风险因素的投资组合及其可能性有效前沿进行深入剖析，加深人们对背景风险的认识。本文主要研究工作和创新包括下面四个方面：
一、构建了基于背景风险偏好度的可能性均值-方差模型以及含流动性约束的双目标可能性MV模型，讨论了不同背景风险偏好度对投资组合风险及可能性有效前沿的影响。在分析现有的关于背景风险研究的现状基础上，以可能性理论为基础，将风险资产和背景资产的收益率均视为模糊变量，提出了基于背景风险偏好度的可能性均值-方差模型以及含流动性约束的双目标可能性MV模型。模型在现有研究成果的基础上考虑了边界限制、交易费用、流动性等现实因素对投资策略的影响。依据模糊集理论将风险资产和背景资产的收益率视为LR-类模糊变量，进而给出了两模型的具体表达式。然后，通过对比分析展示了不同背景风险偏好度下的可能性投资组合的有效前沿，探讨了流动性约束对具有背景风险的可能性投资组合的影响。实证表明：背景风险偏好度对投资组合风险和可能性有效前沿都有一定的影响。当给定的期望收益值相同时，背景风险偏好度越接近于1，投资者越偏好投资风险，其所承受的总风险越小，投资组合的可能性有效前沿向左上方移动。
二、构建了具有VaR约束的模糊投资组合模型以及具有背景风险和风险价值的模糊投资组合模型，分析了置信水平和VaR直线的截距对最优策略的影响，进一步探讨了背景资产的均值和方差的变动对投资组合风险及可能性有效前沿的影响。首先，将随机不确定条件下的VaR约束推广到可信性测度下。并把可信性测度下的VaR约束和交易费用加入到模型中，分别建立了具有VaR约束的模糊投资组合模型以及具有背景风险和交易费用的模糊投资组合模型。其次，以风险资产和背景资产收益率服从钟形可能性分布的情况为例，分析了置信水平和VaR直线的截距对最优投资策略的影响，展示了背景资产的均值和方差不同取值下的投资组合风险及可能性有效前沿。实证表明：当具有背景风险和交易费用的模糊投资组合模型中的其它参数保持不变，背景资产的均值增大时，其可能性有效前沿向左上方移动，投资者所承受的总风险减小。保持模型其它参数值不变，背景资产收益率的方差增大时，其可能性有效前沿向右平移，投资者承受的总风险增加。
三、给出了两个模糊数乘积的可能性均值、可能性方差和可能性协方差，构建了具有背景风险的国际投资组合选择模型。以可能性理论为依据，推导了两个模糊数乘积的可能性均值、可能性方差和可能性协方差。以此为基础，将汇率风险与背景风险同时引入到投资组合模型中。考虑到汇率的浮动性和不确定性，我们将汇率设为一个模糊变量，建立了具有背景风险的国际投资组合选择模型。对比分析了汇率风险和背景风险对投资决策的影响，给出了不同情况下的可能性有效前沿。实证结果表明：当给定的单位投资价值相同时，具有背景风险和汇率风险的投资组合风险更大。如果忽略对它们的考虑，投资者在投资中将低估投资组合风险，使其蒙受损失。
四、构建了具有背景风险的可能性投资组合调整模型。已有的关于投资组合调整模型都认为投资者所面对的风险只有一种，即投资风险。这些研究没有考虑投资者的劳动收入、健康状况等背景风险的影响。针对这一点，我们在可能性投资组合基础上进一步地研究了具有背景风险的可能性投资组合调整模型，探讨了背景风险偏好度对投资组合调整策略的影响。实证显示：背景风险偏好度的变化影响着投资者的投资调整策略。当背景风险偏好度减小时，投资者所承受的背景风险增加，使得投资者在投资中的总风险增加，其可能性有效前沿向右下方移动。
The fundamental objective of investment decisions is to select the optimal investmentportfolio from a large number of financial assets. The classical portfolio selection modelsassume that investors are risk aversion and they could allocate their wealth among risky assetswith the purpose of dispersing the risk and making profit. In the classical models, they onlyconsider financial risk. However, in real world, there exist some background risks which areresulted from risky non-financial assets such as the risks resulted from labor income, healthstatus, real estate and so on. These background risks will greatly affect the investmentbehavior and make portfolio selection more complicated with the reason that background riskscannot be dispersed by adjustment in financial markets. Up to now, some researchers haveinvestigated the influence of background risks on portfolio selection. These literatures areproposed under the framework of probability theory, in which the return of risky asset isviewed as a random variable.
However, there exist many non-random factors in real financial market. Especially, in afuzzy uncertain economic environment, the return of risky assets is represented by a fuzzynumber. Therefore, how to select the optimal investment portfolio in a fuzzy uncertaintyenvironment has become an important research field.
Taking the above-mentioned two aspects into consideration, we aim to systematicallystudy the problem of portfolio selection with background risk factors based on possibilitytheory and propose some possibilistic portfolio selection models with background risk undervarious environments. In order to make it better understood, we will analysis the porposedpossibilistic portfolio selection models and their efficient frontiers. The main work andcontributions of this thesis can be summarized as the following four aspects:
First, we present a possibilitic mean-variance model on the basis of the preferencedegree of background risk and formulate a two-objective possibilistic portfolio model withliquidity constraint. We also discuss the influence of the different appetite degrees ofbackground risk and their efficient frontiers. We analyze the existing researches aboutbackground risk and regard the returns of risky assets and background assets as fuzzyvariables. Then, we propose a possibilitic mean-variance model based on the preferencedegree of background risk and a two-objective possibilistic portfolio model with liquidityconstraint, respectively. Assume that the returns of risk assets and background assets obeyLR-type possibility distribution, we propose two specific portfolio selection models based onfuzzy set theory. After that, we compare the efficient frontiers of the above-mentioned possibilistic portfolio model with various appetite degrees of background risk and analyze theinfluence of the liquidity constraint on the allocation of possibilistic portfolio withbackground risk. Empirical results indicate that the appetite degree of background risk affectthe portfolio risk and efficient frontiers. Suppose that the given expected return levels areidentical, as the value of background risk appetite degree approximates to1, the investors willprefer market risk. In this case, the investors will suffer less portfolio risk and thecorresponding efficient frontier of the portfolio will move to the upper left.
Second, we propose two fuzzy portfolio selection models, namely, a fuzzy portfolioselection model with VaR constraint and a fuzzy portfolio selection model with backgroundrisk and value at risk (VaR). In the proposed two models, we discuss the influence of theconfidence level and the intercept of the straight line of VaR on the optimal investmentstrategy. Furthermore, we study the influence of the background assets mean and variance onthe efficient frontier. Then, we extend the classical VaR risk measure by credibility measureand define a fuzzy VaR risk measure. Consider the defined fuzzy VaR risk measure, wedevelop a fuzzy portfolio selection model with VaR constraints. At the same time, weincorporate the transaction cost and background risk into above-mentioned model and presenta fuzzy portfolio selection model with background risk and transaction cost. Then, under theassumption that the expected returns of risky assets and background assets are bell-shapefuzzy variables, we analyze the influence of confidence level and the intercept of the straightline of VaR on the optimal investment strategy. Meanwhile, we demonstrate the effect of thevariations of background assets mean and variance on its efficient frontier. Keep the otherparameters in the model invariant, when the means of background assets increase, the efficientfrontier moves to the upper left and the investment risk decreases. On the other hand, when thevariance of returns on background asset decreases and the other parameters in the model areinvariant, the efficient frontier moves to the left and the investment risk decreases.
Third, we give the possibilistic mean, variance and covariance of the multiplication oftwo fuzzy numbers. Meanwhile, we formulate an international portfolio selection model withbackground risk. Based on possibilistic theory, we compute the possibilistic mean value,variance and covariance of the multiplication of two fuzzy numbers. On the basis ofafore-mentioned factors, we integrate both background risk and exchange rate risk intoportfolio selection model. Consider the variability and uncertainty of exchange rate, we viewboth the value of unit investment on risky asset and exchange rate as fuzzy numbers. Then, wepropose a possibilistic portfolio selection model with background risk and exchange rate risk.We comparatively analyze the influence of two risk abvoe on portfolio decision-making. Empirical results demonstrate that, when the given expected value is a invariant, the portfoliorisk with exchange rate risk and background risk is higher than the one without them. It meansthat if we neglect background risk and exchange rate risk, the investors will underestimateportfolio risk and suffer greater losses.
Finally, we propose a possibilistic portfolio adjusting model with background risk. Inmost existing literatures, they only consider the investment risk. However, they ignore thebackground risk generated by many factors such as labor income, proprietary income,investments in real estate, etc. Based on this fact, we further study a possibilistic portfolioadjusted model with background risk under the framework of possibility theory. We discussthe influence of the appetite degrees of background risk on the portfolio adjusting strategies.Empirical results indicate that the preference degree of background risk affects the optimaladjusting strategy. When the investors' appetite degree about background risk decreases, theportfolio risk will increase and the corresponding efficient frontier moves to the lower right.
然而现实的金融市场中存在许多非随机因素的影响，尤其在一个模糊的投资环境中，风险资产的收益表示为模糊数。因此考虑模糊不确定性的投资组合选择问题也是学术界研究的重要领域之一。
本文结合以上两点，依据可能性理论对考虑背景风险因素的可能性投资组合选择问题进行了系统的分析和研究，建立了不同环境下考虑背景风险因素的可能性投资组合选择模型。并对考虑背景风险因素的投资组合及其可能性有效前沿进行深入剖析，加深人们对背景风险的认识。本文主要研究工作和创新包括下面四个方面：
一、构建了基于背景风险偏好度的可能性均值-方差模型以及含流动性约束的双目标可能性MV模型，讨论了不同背景风险偏好度对投资组合风险及可能性有效前沿的影响。在分析现有的关于背景风险研究的现状基础上，以可能性理论为基础，将风险资产和背景资产的收益率均视为模糊变量，提出了基于背景风险偏好度的可能性均值-方差模型以及含流动性约束的双目标可能性MV模型。模型在现有研究成果的基础上考虑了边界限制、交易费用、流动性等现实因素对投资策略的影响。依据模糊集理论将风险资产和背景资产的收益率视为LR-类模糊变量，进而给出了两模型的具体表达式。然后，通过对比分析展示了不同背景风险偏好度下的可能性投资组合的有效前沿，探讨了流动性约束对具有背景风险的可能性投资组合的影响。实证表明：背景风险偏好度对投资组合风险和可能性有效前沿都有一定的影响。当给定的期望收益值相同时，背景风险偏好度越接近于1，投资者越偏好投资风险，其所承受的总风险越小，投资组合的可能性有效前沿向左上方移动。
二、构建了具有VaR约束的模糊投资组合模型以及具有背景风险和风险价值的模糊投资组合模型，分析了置信水平和VaR直线的截距对最优策略的影响，进一步探讨了背景资产的均值和方差的变动对投资组合风险及可能性有效前沿的影响。首先，将随机不确定条件下的VaR约束推广到可信性测度下。并把可信性测度下的VaR约束和交易费用加入到模型中，分别建立了具有VaR约束的模糊投资组合模型以及具有背景风险和交易费用的模糊投资组合模型。其次，以风险资产和背景资产收益率服从钟形可能性分布的情况为例，分析了置信水平和VaR直线的截距对最优投资策略的影响，展示了背景资产的均值和方差不同取值下的投资组合风险及可能性有效前沿。实证表明：当具有背景风险和交易费用的模糊投资组合模型中的其它参数保持不变，背景资产的均值增大时，其可能性有效前沿向左上方移动，投资者所承受的总风险减小。保持模型其它参数值不变，背景资产收益率的方差增大时，其可能性有效前沿向右平移，投资者承受的总风险增加。
三、给出了两个模糊数乘积的可能性均值、可能性方差和可能性协方差，构建了具有背景风险的国际投资组合选择模型。以可能性理论为依据，推导了两个模糊数乘积的可能性均值、可能性方差和可能性协方差。以此为基础，将汇率风险与背景风险同时引入到投资组合模型中。考虑到汇率的浮动性和不确定性，我们将汇率设为一个模糊变量，建立了具有背景风险的国际投资组合选择模型。对比分析了汇率风险和背景风险对投资决策的影响，给出了不同情况下的可能性有效前沿。实证结果表明：当给定的单位投资价值相同时，具有背景风险和汇率风险的投资组合风险更大。如果忽略对它们的考虑，投资者在投资中将低估投资组合风险，使其蒙受损失。
四、构建了具有背景风险的可能性投资组合调整模型。已有的关于投资组合调整模型都认为投资者所面对的风险只有一种，即投资风险。这些研究没有考虑投资者的劳动收入、健康状况等背景风险的影响。针对这一点，我们在可能性投资组合基础上进一步地研究了具有背景风险的可能性投资组合调整模型，探讨了背景风险偏好度对投资组合调整策略的影响。实证显示：背景风险偏好度的变化影响着投资者的投资调整策略。当背景风险偏好度减小时，投资者所承受的背景风险增加，使得投资者在投资中的总风险增加，其可能性有效前沿向右下方移动。
The fundamental objective of investment decisions is to select the optimal investmentportfolio from a large number of financial assets. The classical portfolio selection modelsassume that investors are risk aversion and they could allocate their wealth among risky assetswith the purpose of dispersing the risk and making profit. In the classical models, they onlyconsider financial risk. However, in real world, there exist some background risks which areresulted from risky non-financial assets such as the risks resulted from labor income, healthstatus, real estate and so on. These background risks will greatly affect the investmentbehavior and make portfolio selection more complicated with the reason that background riskscannot be dispersed by adjustment in financial markets. Up to now, some researchers haveinvestigated the influence of background risks on portfolio selection. These literatures areproposed under the framework of probability theory, in which the return of risky asset isviewed as a random variable.
However, there exist many non-random factors in real financial market. Especially, in afuzzy uncertain economic environment, the return of risky assets is represented by a fuzzynumber. Therefore, how to select the optimal investment portfolio in a fuzzy uncertaintyenvironment has become an important research field.
Taking the above-mentioned two aspects into consideration, we aim to systematicallystudy the problem of portfolio selection with background risk factors based on possibilitytheory and propose some possibilistic portfolio selection models with background risk undervarious environments. In order to make it better understood, we will analysis the porposedpossibilistic portfolio selection models and their efficient frontiers. The main work andcontributions of this thesis can be summarized as the following four aspects:
First, we present a possibilitic mean-variance model on the basis of the preferencedegree of background risk and formulate a two-objective possibilistic portfolio model withliquidity constraint. We also discuss the influence of the different appetite degrees ofbackground risk and their efficient frontiers. We analyze the existing researches aboutbackground risk and regard the returns of risky assets and background assets as fuzzyvariables. Then, we propose a possibilitic mean-variance model based on the preferencedegree of background risk and a two-objective possibilistic portfolio model with liquidityconstraint, respectively. Assume that the returns of risk assets and background assets obeyLR-type possibility distribution, we propose two specific portfolio selection models based onfuzzy set theory. After that, we compare the efficient frontiers of the above-mentioned possibilistic portfolio model with various appetite degrees of background risk and analyze theinfluence of the liquidity constraint on the allocation of possibilistic portfolio withbackground risk. Empirical results indicate that the appetite degree of background risk affectthe portfolio risk and efficient frontiers. Suppose that the given expected return levels areidentical, as the value of background risk appetite degree approximates to1, the investors willprefer market risk. In this case, the investors will suffer less portfolio risk and thecorresponding efficient frontier of the portfolio will move to the upper left.
Second, we propose two fuzzy portfolio selection models, namely, a fuzzy portfolioselection model with VaR constraint and a fuzzy portfolio selection model with backgroundrisk and value at risk (VaR). In the proposed two models, we discuss the influence of theconfidence level and the intercept of the straight line of VaR on the optimal investmentstrategy. Furthermore, we study the influence of the background assets mean and variance onthe efficient frontier. Then, we extend the classical VaR risk measure by credibility measureand define a fuzzy VaR risk measure. Consider the defined fuzzy VaR risk measure, wedevelop a fuzzy portfolio selection model with VaR constraints. At the same time, weincorporate the transaction cost and background risk into above-mentioned model and presenta fuzzy portfolio selection model with background risk and transaction cost. Then, under theassumption that the expected returns of risky assets and background assets are bell-shapefuzzy variables, we analyze the influence of confidence level and the intercept of the straightline of VaR on the optimal investment strategy. Meanwhile, we demonstrate the effect of thevariations of background assets mean and variance on its efficient frontier. Keep the otherparameters in the model invariant, when the means of background assets increase, the efficientfrontier moves to the upper left and the investment risk decreases. On the other hand, when thevariance of returns on background asset decreases and the other parameters in the model areinvariant, the efficient frontier moves to the left and the investment risk decreases.
Third, we give the possibilistic mean, variance and covariance of the multiplication oftwo fuzzy numbers. Meanwhile, we formulate an international portfolio selection model withbackground risk. Based on possibilistic theory, we compute the possibilistic mean value,variance and covariance of the multiplication of two fuzzy numbers. On the basis ofafore-mentioned factors, we integrate both background risk and exchange rate risk intoportfolio selection model. Consider the variability and uncertainty of exchange rate, we viewboth the value of unit investment on risky asset and exchange rate as fuzzy numbers. Then, wepropose a possibilistic portfolio selection model with background risk and exchange rate risk.We comparatively analyze the influence of two risk abvoe on portfolio decision-making. Empirical results demonstrate that, when the given expected value is a invariant, the portfoliorisk with exchange rate risk and background risk is higher than the one without them. It meansthat if we neglect background risk and exchange rate risk, the investors will underestimateportfolio risk and suffer greater losses.
Finally, we propose a possibilistic portfolio adjusting model with background risk. Inmost existing literatures, they only consider the investment risk. However, they ignore thebackground risk generated by many factors such as labor income, proprietary income,investments in real estate, etc. Based on this fact, we further study a possibilistic portfolioadjusted model with background risk under the framework of possibility theory. We discussthe influence of the appetite degrees of background risk on the portfolio adjusting strategies.Empirical results indicate that the preference degree of background risk affects the optimaladjusting strategy. When the investors' appetite degree about background risk decreases, theportfolio risk will increase and the corresponding efficient frontier moves to the lower right.
引文
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