張鵬++張衛(wèi)國、

摘要： 考慮交易成本、借款限制、閥值約束和基數(shù)約束，提出多階段均值—半方差模糊投資組合模型。在該模型中，收益水平被定義為可能的平均回報，風(fēng)險水平被定義為回報的半方差。由于交易成本和基數(shù)約束，多階段投資組合模型為具有路徑依賴性的混合整數(shù)動態(tài)優(yōu)化問題。文章提出了前向動態(tài)規(guī)劃方法求解。最后，以一個具體的算例比較了不同的基數(shù)約束投資組合的最優(yōu)投資策略。

關(guān)鍵詞：多階段模糊投資組合；均值—半方差；基數(shù)約束；交易成本；前向動態(tài)規(guī)劃方法

中圖分類號：F83248文獻(xiàn)標(biāo)志碼：A文章編號：1009-055X（2014）05-0021-09

一、 引言

Markowitz[1]提出的單階段均值—方差投資組合理論為現(xiàn)代投資組合的發(fā)展奠定了基礎(chǔ)。雖然方差在投資組合決策中得到了廣泛的應(yīng)用，但其也有一些局限性[2， 3]。例如，在均值—方差模型中，同時去掉了高收益和低收益，而高收益正是投資者希望的。由于方差度量風(fēng)險消除高低收益，犧牲了投資者獲取高回報的可能。同時文獻(xiàn)[4， 5]研究發(fā)現(xiàn)許多證券回報都是不對稱分布的。為了克服均值—方差模型的這些局限性，人們運用下偏矩度量風(fēng)險，這種方法只測量收益水平的下偏差，而半方差[2]則是最常用的下偏矩度量風(fēng)險方法。

為了使Markowitz的模型更符合實際，人們在投資組合中限制資產(chǎn)的數(shù)量（基數(shù)約束）并規(guī)定了每個資產(chǎn)投資比例的上下限（閥值）。在過去幾十年里Markowitz 基數(shù)約束模型被廣泛地研究，尤其是從計算角度，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].這些研究分析了LAM （Limited Asset Markowitz）模型的計算復(fù)雜性，經(jīng)典的Markowitz模型為凸二次規(guī)劃模型，而LAM模型為0-1混合整數(shù)二次規(guī)劃問題 （MIQP），該模型為NPhard 問題（見Bienstock [20]； Shaw [21] 的例子）。

以上模型假設(shè)投資為單階段，但在現(xiàn)實生活中投資者可以在不同時段內(nèi)重新分配自己的資產(chǎn)，所以投資決策應(yīng)該是多階段的。許多學(xué)者將單階段的投資組合拓展到多階段。Mossin [22]運用動態(tài)方法求出多階段投資組合的最優(yōu)投資策略。Hakansson [23] 分析了多階段均值-方差投資組合有效前沿。Li，Chan和Ng [24] 用嵌入的方法把多階段均值-安全首要投資組合模型轉(zhuǎn)變?yōu)橐粋€能用動態(tài)規(guī)劃處理的問題，從而得到了最優(yōu)投資策略及有效前沿的解析表達(dá)式。 使用同樣的方法，Li和Ng [25] 研究了多階段均值-方差投資組合模型，并得到了其有效前沿。Calafiore [26]考慮了具有金融資產(chǎn)分配序貫決策問題，并提出了具有線性控制的多階段投資組合模型。 Zhu等[27] 提出了具有破產(chǎn)控制的多階段均值-方差投資組合模型。Wei和Ye [28] 在隨機(jī)市場情況下提出了具有破產(chǎn)控制的多階段均值-方差投資組合模型。Güplnar和Rustem [29] 在隨機(jī)情景樹框架下構(gòu)建多階段均值-方差投資組合模型。Yu等 [30] 提出了具有破產(chǎn)控制的多階段均值-絕對偏差投資組合模型。likyurt和zekici[31]在隨機(jī)市場情況下提出了幾種多階段均值-方差投資組合模型。Yan和Li [32]和Yan等[33]用半方差代替方差，提出了多階段均值—半方差投資組合模型。Plnar [34] 使用下方風(fēng)險度量方法研究多階段投資組合模型?？紤]到線性的交易成本和投資組合的多樣性及其偏度，Zhang等 [35，36] 和 Liu等 [37，38]分別提出了幾種模糊多階段投資組合模型，并分別運用遺傳算法、混合智能算法和微分進(jìn)化算法求解。

在實際投資過程中有許多非概率因素影響投資，因此，風(fēng)險資產(chǎn)的收益為模糊不確定。近來，許多學(xué)者研究了模糊投資組合。Watada[39]和León等[40] 使用模糊決策理論研究投資組合。Tanaka和Guo [41， 42]分別提出了模糊概率和指數(shù)可能性兩種投資組合模型。Inuiguchi和Tanino [43]使用模糊規(guī)劃方法研究了極小極大后悔投資組合模型。Wang和Zhu [44]， Lai等[45] and Giove等[46] 構(gòu)建了區(qū)間規(guī)劃投資組合模型。Zhang和Nie [47] ，Zhang等[48] 假設(shè)期望收益和風(fēng)險具有可容許誤差，提出了的可容許有效投資組合，并得到不允許賣空情況下模型的有效前沿。Dubois和Prade [49]定義了模糊數(shù)的區(qū)間期望，認(rèn)為它們是確定的隨機(jī)集合，也提出模糊數(shù)的期望滿足可加性。Carlsson和Fullér [50] 提出了模糊數(shù)的上下可能性均值的一些性質(zhì)。Huang [51， 52， 53] 提出了均值-方差、均值-半方差和均值-風(fēng)險曲線的模糊投資組合模型。Zhang 等[54]， Zhang [55]， Zhang和Xiao [56] 提出了上下可能性均值和方差投資組合模型。Li等[57， 58]提出了均值-方差和均值-方差-偏度模糊投資組合模型。Carlsson等[59] 假設(shè)收益為梯形模糊數(shù)，提出了具有最高效用的模糊投資組合模型。

雖然模糊過程分析法在單階段模糊投資組合已有較多應(yīng)用，但很少有文章將這一方法運用于多階段模糊投資組合中?？紤]交易成本、閥值約束和基數(shù)約束，本文提出了一個具有風(fēng)險控制的多階段模糊投資組合模型，該模型為具有路徑依賴性的混合整數(shù)動態(tài)優(yōu)化問題，并提出了前向動態(tài)規(guī)劃方法求解。華 南 理 工 大 學(xué) 學(xué) 報（社 會 科 學(xué) 版）

第5期張鵬 等：多階段均值—半方差模糊投資組合決策研究

二、 可能性均值和方差

從表1和表2可得：當(dāng)投資組合所含資產(chǎn)的數(shù)量增大時，其終期財富也增大。

六、結(jié)論

本文討論了模糊環(huán)境下的多階段投資組合問題，在該模型中收益、風(fēng)險資產(chǎn)的風(fēng)險均為梯形模糊變量。運用模糊分析方法處理不精確數(shù)據(jù)，提出多階段模糊投資組合最優(yōu)化模型。由于該模型為模糊規(guī)劃問題，所以運用模糊決策方法將其轉(zhuǎn)化為顯示模型。多階段投資組合模型為具有路徑依賴性的混合整數(shù)動態(tài)最優(yōu)化問題。提出前向動態(tài)規(guī)劃方法求出模型的最優(yōu)投資策略。通過實證研究驗證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

（責(zé)任編輯：余樹華）

[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

（責(zé)任編輯：余樹華）

[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.

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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

（責(zé)任編輯：余樹華）