多阶段均值—半方差模糊投资组合决策研究

张鹏++张卫国、

摘要： 考虑交易成本、借款限制、阀值约束和基数约束，提出多阶段均值—半方差模糊投资组合模型。在该模型中，收益水平被定义为可能的平均回报，风险水平被定义为回报的半方差。由于交易成本和基数约束，多阶段投资组合模型为具有路径依赖性的混合整数动态优化问题。文章提出了前向动态规划方法求解。最后，以一个具体的算例比较了不同的基数约束投资组合的最优投资策略。

关键词：多阶段模糊投资组合；均值—半方差；基数约束；交易成本；前向动态规划方法

中图分类号：F83248文献标志码：A文章编号：1009-055X（2014）05-0021-09

一、 引言

Markowitz[1]提出的单阶段均值—方差投资组合理论为现代投资组合的发展奠定了基础。虽然方差在投资组合决策中得到了广泛的应用，但其也有一些局限性[2， 3]。例如，在均值—方差模型中，同时去掉了高收益和低收益，而高收益正是投资者希望的。由于方差度量风险消除高低收益，牺牲了投资者获取高回报的可能。同时文献[4， 5]研究发现许多证券回报都是不对称分布的。为了克服均值—方差模型的这些局限性，人们运用下偏矩度量风险，这种方法只测量收益水平的下偏差，而半方差[2]则是最常用的下偏矩度量风险方法。

为了使Markowitz的模型更符合实际，人们在投资组合中限制资产的数量（基数约束）并规定了每个资产投资比例的上下限（阀值）。在过去几十年里Markowitz 基数约束模型被广泛地研究，尤其是从计算角度，如Anagnostopoulos 和 Mamanis [6]； Bertsimas 和 Shioda [7]； Fernández和Gómez [8]； Li 等 [9]； RuizTorrubiano 和 Suarez [10]； WoodsideOriakhi等[11]； Cesarone等[12]； Murray 和 Shek [13]； Cui 等 [14]； Le Thi 等 [15，16]； Deng 等[17]； Soleimani 等[18]； Sun等[19].这些研究分析了LAM （Limited Asset Markowitz）模型的计算复杂性，经典的Markowitz模型为凸二次规划模型，而LAM模型为0-1混合整数二次规划问题 （MIQP），该模型为NPhard 问题（见Bienstock [20]； Shaw [21] 的例子）。

以上模型假设投资为单阶段，但在现实生活中投资者可以在不同时段内重新分配自己的资产，所以投资决策应该是多阶段的。许多学者将单阶段的投资组合拓展到多阶段。Mossin [22]运用动态方法求出多阶段投资组合的最优投资策略。Hakansson [23] 分析了多阶段均值-方差投资组合有效前沿。Li，Chan和Ng [24] 用嵌入的方法把多阶段均值-安全首要投资组合模型转变为一个能用动态规划处理的问题，从而得到了最优投资策略及有效前沿的解析表达式。 使用同样的方法，Li和Ng [25] 研究了多阶段均值-方差投资组合模型，并得到了其有效前沿。Calafiore [26]考虑了具有金融资产分配序贯决策问题，并提出了具有线性控制的多阶段投资组合模型。 Zhu等[27] 提出了具有破产控制的多阶段均值-方差投资组合模型。Wei和Ye [28] 在随机市场情况下提出了具有破产控制的多阶段均值-方差投资组合模型。Güplnar和Rustem [29] 在随机情景树框架下构建多阶段均值-方差投资组合模型。Yu等 [30] 提出了具有破产控制的多阶段均值-绝对偏差投资组合模型。likyurt和zekici[31]在随机市场情况下提出了几种多阶段均值-方差投资组合模型。Yan和Li [32]和Yan等[33]用半方差代替方差，提出了多阶段均值—半方差投资组合模型。Plnar [34] 使用下方风险度量方法研究多阶段投资组合模型。考虑到线性的交易成本和投资组合的多样性及其偏度，Zhang等 [35，36] 和 Liu等 [37，38]分别提出了几种模糊多阶段投资组合模型，并分别运用遗传算法、混合智能算法和微分进化算法求解。

在实际投资过程中有许多非概率因素影响投资，因此，风险资产的收益为模糊不确定。近来，许多学者研究了模糊投资组合。Watada[39]和León等[40] 使用模糊决策理论研究投资组合。Tanaka和Guo [41， 42]分别提出了模糊概率和指数可能性两种投资组合模型。Inuiguchi和Tanino [43]使用模糊规划方法研究了极小极大后悔投资组合模型。Wang和Zhu [44]， Lai等[45] and Giove等[46] 构建了区间规划投资组合模型。Zhang和Nie [47] ，Zhang等[48] 假设期望收益和风险具有可容许误差，提出了的可容许有效投资组合，并得到不允许卖空情况下模型的有效前沿。Dubois和Prade [49]定义了模糊数的区间期望，认为它们是确定的随机集合，也提出模糊数的期望满足可加性。Carlsson和Fullér [50] 提出了模糊数的上下可能性均值的一些性质。Huang [51， 52， 53] 提出了均值-方差、均值-半方差和均值-风险曲线的模糊投资组合模型。Zhang 等[54]， Zhang [55]， Zhang和Xiao [56] 提出了上下可能性均值和方差投资组合模型。Li等[57， 58]提出了均值-方差和均值-方差-偏度模糊投资组合模型。Carlsson等[59] 假设收益为梯形模糊数，提出了具有最高效用的模糊投资组合模型。

虽然模糊过程分析法在单阶段模糊投资组合已有较多应用，但很少有文章将这一方法运用于多阶段模糊投资组合中。考虑交易成本、阀值约束和基数约束，本文提出了一个具有风险控制的多阶段模糊投资组合模型，该模型为具有路径依赖性的混合整数动态优化问题，并提出了前向动态规划方法求解。华 南 理 工 大 学 学 报（社 会 科 学 版）

第5期张鹏 等：多阶段均值—半方差模糊投资组合决策研究

二、 可能性均值和方差

从表1和表2可得：当投资组合所含资产的数量增大时，其终期财富也增大。

六、结论

本文讨论了模糊环境下的多阶段投资组合问题，在该模型中收益、风险资产的风险均为梯形模糊变量。运用模糊分析方法处理不精确数据，提出多阶段模糊投资组合最优化模型。由于该模型为模糊规划问题，所以运用模糊决策方法将其转化为显示模型。多阶段投资组合模型为具有路径依赖性的混合整数动态最优化问题。提出前向动态规划方法求出模型的最优投资策略。通过实证研究验证了模型和算法的有效性。

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The Possibilistic Multiperiod Meansemivariance Portfolio Selection

ZZHENG Peng1，ZHANG Weiguo2

（1.School of Management， Wuhan University of Science and Technology， Wuhan 430081， Hubei， China.

2.School of Business Administration， South China University of Technology， Guangzhou 510641， Guangdong， China）

Abstract： This paper discusses a multiperiod portfolio selection problem in fuzzy environment. A possibilistic mean semivariance model for multiperiod portfolio selection is presented by taking into account the transaction costs， borrowing constraints， threshold constraints and cardinality constraints. In the proposed model， the return level is quantified by the possibilistic mean of return， and the risk level is characterized by the possibilistic semivariance of return. Because of the transaction costs and cardinality constraints， the multiperiod portfolio selection is the mix integer dynamic optimization problem with path dependence. Furthermore， the forward dynamic programming method is designed to obtain the optimal portfolio strategy. Finally， the comparison analysis of the different cardinality constraints is provided by a numerical example to illustrate the efficiency of the proposed approaches and the designed algorithm.

Keywords：multiperiod fuzzy portfolio selection； mean semivariance； cardinality constraint； transaction costs； the forward dynamic programming method

（责任编辑：余树华）

[60]A.Saeidifar， E. Pasha， The possibilistic moments of fuzzy numbers and their applications[J]， Journal of Computational and Applied Mathematics 2 （2009）1028–1042.

[61]Xue Deng， Rongjun Li， A portfolio selection model with borrowing constraint based on possibility theory [J]， Applied Soft Computing 12 （2012）754–758.

[62]S.J. Sadjadi， S.M. Seyedhosseini， Kh. Hassanlou， Fuzzy multi period portfolio selection with different rates for borrowing and Lending [J]， Applied Soft Computing 11 （2011）3821–3826.

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[65]D. Bertsimas， D. Pachamanova， Robust multiperiod portfolio management in the presence of transaction costs [J]， Computers and Operations Research 35（2008）3–17.

[66]N.Gulp？nar， B. Rustem， R. Settergren， Multistage stochastic meanvariance portfolio analysis with transaction cost [J]， Innovations， in Financial and Economic Networks 3（2003）46–63.

[67]E. Vercher， J. Bermudez， J. Segura， Fuzzy portfolio optimization under downside risk measures [J]， Fuzzy Sets and Systems 158 （2007）769-782.

The Possibilistic Multiperiod Meansemivariance Portfolio Selection

ZZHENG Peng1，ZHANG Weiguo2

（1.School of Management， Wuhan University of Science and Technology， Wuhan 430081， Hubei， China.

2.School of Business Administration， South China University of Technology， Guangzhou 510641， Guangdong， China）

Abstract： This paper discusses a multiperiod portfolio selection problem in fuzzy environment. A possibilistic mean semivariance model for multiperiod portfolio selection is presented by taking into account the transaction costs， borrowing constraints， threshold constraints and cardinality constraints. In the proposed model， the return level is quantified by the possibilistic mean of return， and the risk level is characterized by the possibilistic semivariance of return. Because of the transaction costs and cardinality constraints， the multiperiod portfolio selection is the mix integer dynamic optimization problem with path dependence. Furthermore， the forward dynamic programming method is designed to obtain the optimal portfolio strategy. Finally， the comparison analysis of the different cardinality constraints is provided by a numerical example to illustrate the efficiency of the proposed approaches and the designed algorithm.

Keywords：multiperiod fuzzy portfolio selection； mean semivariance； cardinality constraint； transaction costs； the forward dynamic programming method

（责任编辑：余树华）

[60]A.Saeidifar， E. Pasha， The possibilistic moments of fuzzy numbers and their applications[J]， Journal of Computational and Applied Mathematics 2 （2009）1028–1042.

[61]Xue Deng， Rongjun Li， A portfolio selection model with borrowing constraint based on possibility theory [J]， Applied Soft Computing 12 （2012）754–758.

[62]S.J. Sadjadi， S.M. Seyedhosseini， Kh. Hassanlou， Fuzzy multi period portfolio selection with different rates for borrowing and Lending [J]， Applied Soft Computing 11 （2011）3821–3826.

[63]R.D.Arnott， W.H.Wagner， The measurement and control of trading costs [J]， Financial Analysts Journal 6 （1990）73–80.

[64]A. Yoshimoto， The mean–variance approach to portfolio optimization subject to transaction costs [J]， Journal of the Operational Research Society of Japan 39 （1996）99–117.

[65]D. Bertsimas， D. Pachamanova， Robust multiperiod portfolio management in the presence of transaction costs [J]， Computers and Operations Research 35（2008）3–17.

[66]N.Gulp？nar， B. Rustem， R. Settergren， Multistage stochastic meanvariance portfolio analysis with transaction cost [J]， Innovations， in Financial and Economic Networks 3（2003）46–63.

[67]E. Vercher， J. Bermudez， J. Segura， Fuzzy portfolio optimization under downside risk measures [J]， Fuzzy Sets and Systems 158 （2007）769-782.

The Possibilistic Multiperiod Meansemivariance Portfolio Selection

ZZHENG Peng1，ZHANG Weiguo2

（1.School of Management， Wuhan University of Science and Technology， Wuhan 430081， Hubei， China.

2.School of Business Administration， South China University of Technology， Guangzhou 510641， Guangdong， China）

Abstract： This paper discusses a multiperiod portfolio selection problem in fuzzy environment. A possibilistic mean semivariance model for multiperiod portfolio selection is presented by taking into account the transaction costs， borrowing constraints， threshold constraints and cardinality constraints. In the proposed model， the return level is quantified by the possibilistic mean of return， and the risk level is characterized by the possibilistic semivariance of return. Because of the transaction costs and cardinality constraints， the multiperiod portfolio selection is the mix integer dynamic optimization problem with path dependence. Furthermore， the forward dynamic programming method is designed to obtain the optimal portfolio strategy. Finally， the comparison analysis of the different cardinality constraints is provided by a numerical example to illustrate the efficiency of the proposed approaches and the designed algorithm.

Keywords：multiperiod fuzzy portfolio selection； mean semivariance； cardinality constraint； transaction costs； the forward dynamic programming method

（责任编辑：余树华）