Applied Mathematics and Computation 217 (2010) 949–951

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

A chance-constrained portfolio selection model with risk constraints

Xiang Li a, Zhongfeng Qin b,*, Lixing Yang a

a The State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, Chinab School of Economics and Management, Beihang University, Beijing 100191, China

a r t i c l e i n f o

Keywords:Fuzzy portfolio selectionFuzzy chance-constrained programmingRisk constraint

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.06.035

* Corresponding author.E-mail addresses: xiang-li04@mail.tsinghua.edu.c

a b s t r a c t

The paper by Huang [Fuzzy chance-constrained portfolio selection, Applied Mathematicsand Computation 177 (2006) 500–507] proposes a fuzzy chance-constrained portfolioselection model and presents a numerical example to illustrate the proposed model. In thisnote, we will show that Huang’s model produces optimal portfolio investing in only onesecurity when candidate security returns are independent to each other no matter howmany independent securities are in the market. The reason for concentrative solution isthat Huang’s model does not consider the investment risk. To avoid concentrative invest-ment, a risk constraint is added to the fuzzy chance-constrained portfolio selection model.In addition, we point out that the result of the numerical example is inaccurate.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

Let xi denote the investment proportion in security i, ni the fuzzy return for the ith security, i = 1,2, . . . ,n, respectively, athe predetermined confidence level the investor accepts. Consider the following fuzzy chance-constrained portfolio selectionmodel [2]:

max �f

s:t: Cr n1x1 þ n2x2 þ � � � þ nnxn P �f� �

P a;x1 þ x2 þ � � � þ xn ¼ 1;xi P 0; i ¼ 1;2; . . . ;n;

8>>><>>>:

ð1Þ

where max �f is the a-optimistic value defined as ðn1x1 þ n2x2 þ � � � þ nnxnÞsupðaÞ ¼ sup �f jCrfn1x1 þ n2x2 þ � � � þ�

nnxn P �fgP ag, which means the maximal investment return the investor can obtain at confidence level a. In order toillustrate the proposed model, Huang [2] proposed the following example.

Example 1.1. Assume that there are 10 securities. Among them, returns of the first seven ones are triangular fuzzy variablesni = (ai,bi,ci), i = 1,2, . . . ,7, respectively. The fuzzy returns of the other three ones take the membership functions li(ri), i = 8, 9,10, where ri are real numbers. The data set is given in Table 1, where the last two columns are the 0.9-optimistic value andabsolute deviation E[jni � E[ni]j] for the fuzzy returns, respectively, which will be used in the following examples.

Suppose that the investor accepts 0.90 as the safe confidence level. A run of the hybrid intelligent algorithm [2] showsthat among 10 securities, in order to gain the maximum investment return at credibility not less than 0.90, the investorshould assign his money according to Table 2 [2].

. All rights reserved.

n (X. Li), qin@buaa.edu.cn, qzf05@mails.tsinghua.edu.cn (Z. Qin).

Table 1Fuzzy returns of 10 securities (units per stock).

Security i Fuzzy returns 0.9-optimistic value E[jni � E[ni]j]

1 (�0.2,2.1,2.5) 0.26 0.482 (�0.1,1.9,3.0) 0.30 0.453 (�0.4,3.0,4.0) 0.28 0.734 (�0.1,2.0,2.5) 0.32 0.445 (�0.6,3.6,4.0) 0.24 0.876 (�0.2,2.8,3.5) 0.40 0.637 (�0.2,3.2,4.0) 0.48 0.728 1

1þ r�4:42ð Þ2

0.40 1.57

9 11þð5r�4:4Þ2

0.48 0.16

10 e�ðr�2:0Þ2 0.73 0.44

Table 2Allocation of money to 10 securities.

Security i 1 2 3 4 5 6 7 8 9 10

Allocation of money 0.000 0.000 0.000 0.000 0.000 0.000 0.053 0.143 0.164 0.640

950 X. Li et al. / Applied Mathematics and Computation 217 (2010) 949–951

2. Remarks

In model (1), the objective function is to maximize the investment return (n1x1 + n2x2 + � � � + nnxn)sup(a). Since Li and Liu [3]has proved that

ðn1x1 þ n2x2 þ � � � þ nnxnÞsupðaÞ ¼ ðn1ÞsupðaÞx1 þ ðn2ÞsupðaÞx2 þ � � � þ ðnnÞsupðaÞxn;

model (1) is essentially a linear programming model:

max ðn1ÞsupðaÞx1 þ ðn2ÞsupðaÞx2 þ � � � þ ðnnÞsupðaÞxn

s:t: x1 þ x2 þ � � � þ xn ¼ 1;xi P 0; i ¼ 1;2; . . . ;n:

8><>:

It is easy to prove that the optimal portfolio of model (1) invests in only one security with maximal a-optimistic value,which contradicts with the traditional portfolio selection theory. The reason is that it does not consider the investment risk,which leads to the above concentrative portfolio.

In addition, it is clear that the computational result in Table 2 is inaccurate. The accurate optimal portfolio should investall the money to security 10 because it has the maximum 0.9-optimistic value 0.73.

3. A revised model

The reason for concentrative solution of model (1) is that it does not consider the investment risk. For each portfolio(x1,x2, . . . ,xn), Cai et al. [1] defined a risk measure for random returns as R(x1,x2, . . . ,xn) = max16i6nxiE[jni � E[ni]j], where Eis the expected value operator for random variables. If we define a similar risk measure for fuzzy returns and consider itto model (1), we get the following revised model:

max ðn1ÞsupðaÞx1 þ ðn2ÞsupðaÞx2 þ � � � þ ðnnÞsupðaÞxn

s:t: max16i6n

xiE jni � E½ni�j½ � 6 d;

x1 þ x2 þ � � � þ xn ¼ 1;xi P 0; i ¼ 1;2; . . . ;n;

8>>>><>>>>:

ð2Þ

where E is the expected value operator for fuzzy variables [4], and d is the maximum risk level the investor can tolerate.Essentially, model (2) is equivalent to the following linear programming model

max ðn1ÞsupðaÞx1 þ ðn2ÞsupðaÞx2 þ � � � þ ðnnÞsupðaÞxn

s:t: xi 6 d=E½jni � E½ni�j�; i ¼ 1;2; . . . ;n;

x1 þ x2 þ � � � þ xn ¼ 1;xi P 0; i ¼ 1;2; . . . ;n:

8>>><>>>:

ð3Þ

Table 3Allocation of money to 10 securities with different maximum risk levels.

Risk level 1 2 3 4 5 6 7 8 9 10

0.09 0.0000 0.0000 0.0000 0.0000 0.0000 0.0945 0.1252 0.0042 0.5730 0.20310.10 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.139 0.0000 0.6353 0.22570.20 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2743 0.0000 0.2743 0.45140.30 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1615 0.0000 0.1615 0.67700.40 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0486 0.0000 0.0486 0.90270.50 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000

X. Li et al. / Applied Mathematics and Computation 217 (2010) 949–951 951

Remark 3.1. In fact, there are several other risk measures for fuzzy returns such as variance, semivariance, absolutedeviation and so on. Hence, we may define several variations for the revised model by taking different risk measures.

Example 3.1. Let us reconsider Example 1.1 by using model (2). A run of the function fmincon in Matlab 7.1 shows that as theincreasing of the maximum risk level d, the allocation invested to security 10 increases linearly because it owns the maxi-mum 0.9-optimistic value. Especially, when d is larger than E[jn10 � E[n10]j] = 0.44, the optimal portfolio allocates all themoney to security 10 because in this case (0,0,0,0,0,0,0,0,0,1) is a feasible portfolio of model (2). The summarized resultsare shown in Table 3.

Acknowledgements

The authors are indebted to the anonymous referees for their detailed comments and suggestions. The work was partiallysupported by China Postdoctoral Science Foundation (No. 20090450024), the National Natural Science Foundation of China(No.70901006) and the State Key Laboratory of Rail Traffic Control and Safety (Nos. RCS2009ZT001, RCS2008ZZ001) of Bei-jing Jiaotong University. The research of Z. Qin was supported in part by the Fundamental Research Funds for the CentralUniversities (No. YWF-10-02-028).

References

[1] X. Cai, K. Teo, X. Yang, X. Zhou, Portfolio optimization under a minimax rule, Management Sciences 46 (7) (2000) 957–972.[2] X. Huang, Fuzzy chance-constrained portfolio selection, Applied Mathematics and Computation 177 (2006) 500–507.[3] X. Li, B. Liu, The independence of fuzzy variables with applications, International Journal of Natural Sciences and Technology 1 (1) (2006) 95–100.[4] B. Liu, Y. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems 10 (4) (2002) 445–450.