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A cutting plane algorithm for MV portfolio selection model
Applied Mathematics and Computation 215 (2009) 1456–1462
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A cutting plane algorithm for MV portfolio selection model q Guohua Chen a,*, Xiaolian Liao a, Shouyang Wang b a b
Department of Mathematics, Hunan Institute of Humanities Science and Technology, Loudi 417000, China Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China
a r t i c l e
i n f o
Keywords: Possibility theory Portfolio selection Cutting plane algorithm
a b s t r a c t This paper deals with a portfolio selection problem with fuzzy return rates. A possibilistic mean variance (FMVC) portfolio selection model was proposed. The possibilistic programming problem can be transformed into a linear optimal problem with an additional quadratic constraint by possibilistic theory. For such problems there are no special standard algorithms. We propose a cutting plane algorithm to solve (FMVC). The nonlinear programming problem can be solved by sequence linear programming problem. A numerical example is given to illustrate the behavior of the proposed model and algorithm. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction In 1952, Markowitz [8–10] published his pioneering work which laid the foundation of modern portfolio analysis. Markowitz’s mean variance model has served as a basis for the development of modern financial theory over the past five decades. Traditionally, research has been undertaken on the assumption that future security returns can be correctly reflected by past performance and be represented by random variables.Since the security market is so complex, in many situations security returns cannot always be accurately predicted from historical data. They are beset with ambiguity and vagueness. To deal with this problem, researchers have made use of fuzzy set theory [18,16]. Assuming that the returns are fuzzy, a great deal of work has been dedicated to extrapolating traditional mean variance models: for example, Watada [14], Tanaka and Guo [11,12], Parra et al. [16], Carlsson et al. [4], Zhang and Nie [23], Bilbao-Terol et al. [1], Ostermark [20], Leon et al. [21], Ballestero and Romero [6], etc., Lacagnina and Pecorella [22] proposed a multistage stochastic soft constraints fuzzy model with recourse to solve a portfolio management problem. Shiang and Rong [26] constructed a pair of two-level mathematical programming models, based on which the upper bound and lower bound of the objective values were obtained. Bilbao et al. [34] proposed a new Fuzzy Compromise Programming approach is based on the obtaining of the minimum fuzzy distance to the fuzzy ideal solution of the portfolio selection problem. Enriqueta et al. [33] presented a fuzzy downside risk approach for managing portfolio problems in the framework of risk-return trade-off using interval-valued expectations. Zhang [24] discussed the portfolio selection problem for bounded assets based on the upper and lower possibilistic means and variances of fuzzy numbers. Huang [25] built a risk curve fuzzy portfolio selection model. Recent advances of portfolio selection model consider integration of various muti-criteria decision making models such as fuzzy AHP in Tiryaki and Ahlatcioglu [35] and expert systems such as Smimou et al. [36]. In this paper we propose an alternative approach whose main idea is based on Dyer and Wolsey paper [27]. We use cutting planes to reformulate and solve the problem using valid inequalities as cutting planes. The objective is to use these results to obtain an improved formulation of the problem. This work was motivated in part by the success of a series of papers q Supported by the National Natural Science Foundation of China under Grant No. 70221001. A project supported by Scientific Research Fund of Hunan Provincial Education Department (07C389). * Corresponding author. E-mail address: [email protected] (G. Chen).
0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.040
G. Chen et al. / Applied Mathematics and Computation 215 (2009) 1456–1462
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[28–30] for different problems. Claus and Westerland [31] proposed a new algorithm for solving convex nonlinear programming optimization problems. Diaz and Zabala [32] presented an approach based on integer programming formulations of the graph coloring problem. This paper is organized as follows. In Section 2, we introduce briefly possibilistic mean variance portfolio selection model with transaction costs. In Section 3, we introduce briefly the possibility theory and transform possibilistic mean portfolio selection model into a linear optimal problem with an additional quadratic constraint (FMVC). For such problems there are no special standard algorithms. In Section 4, we propose a cutting plane algorithm to solve (FMVC), In Section 5, an example is given to illustrate the proposed model. A few concluding remarks are finally given in Section 6. 2. Mean variance portfolio selection model with transaction costs 2.1. Mean variance approach A portfolio selection problem in the mean variance context can be written as (MV):
max
ðMVÞ
f ðxÞ ¼
n X
Eðr j Þxj
j¼1 n X n X
s:t
i¼1 n X
rij xi xj 6 w;
j¼1
xj ¼ 1;
ð1Þ
j¼1
0 6 xj 6 aj ;
j ¼ 1; . . . ; n;
where n is the number of risky securities, xi is the proportion invested in security i; Eðr i Þ is the expected returns of security i; rij is the covariance of the expected returns on security i and j; w is the risk tolerance factor of the investor. Obviously, the greater the factor w is, the more risk tolerance the investor has. The expected return and the risk of a portfolio are, respectively, given by
Rp ¼
n X
Eðr j Þxj
j¼1
and
r2p ¼
n X n X
rij xi xj :
t¼1 t¼1
The input data of the maximization problem are expected returns of securities and variance covariance matrix of the expected returns of securities in a portfolio. Transaction cost is one of the main sources of concern to portfolio managers. Arnott and Wanger [19] found that ignoring transaction costs would result in an inefficient portfolio. Yoshimoto’s empirical analysis [2] also drew the same conclusion. In this paper, we consider the proportional transaction costs. Assume the rate of transaction cost of asset jðj ¼ 1; . . . ; nÞ is cj , P thus the transaction cost of asset j is cj xj . The transaction cost of the portfolio x ¼ ðx1 ; x2 ; . . . ; xn Þ is nj¼1 cj xj . The actual exPn pected rate of return under transaction costs is therefore, given by rðxÞ ¼ i¼1 Eðr j Þxj cj ðxj Þ. Hence, our portfolio selection model with fixed transaction costs can be formulated as (MVC):
ðMVCÞ
max
f ðxÞ ¼
n X ðEðr j Þ cj Þxj j¼1
s:t
n X n X i¼1 n X
rij xi xj 6 w;
j¼1
xj ¼ 1;
ð2Þ
j¼1
0 6 xj 6 a j ;
j ¼ 1; . . . ; n;
where xj represents the proportion of the total amount of money devoted to security, aj represent the maximum proportion of the total amount of money devoted to security respectively. Let r j be the random variable representing the rate of return of security, Eðr j Þ is the expected returns of security j, In this model, the investor is trying to maximize the future value of his/her portfolio, which requires the risk that the variance his portfolio not to be greater than w.
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2.2. Possibilistic mean variance portfolio selection model In standard portfolio models uncertainty is equated with randomness, which actually combines both objectively observable and testable random events with subjective judgments of the decision maker into probability assessments. A purist on theory would accept the use of probability theory to deal with observable random events, but would frown upon the transformation of subjective judgments to probabilities. So in this paper we will assume that the rates of return on securities are modeled by possibility distributions rather than probability distributions. Applying possibilistic distribution may have two advantages [17]: (1) the knowledge of the expert can be easily introduced to the estimation of the return rates and (2) the reduced problem is more tractable than that of the stochastic programming approach. We consider the portfolio selection problem under the assumption that the rate of return on security is the trapezoidal fuzzy number
ðFMVCÞ
max
f ðxÞ ¼
n X ðEð~r j Þ cj Þxj j¼1
n X n X
s:t
r~ ij xi xj 6 w;
t¼1 t¼1 n X
xj ¼ 1;
ð3Þ
j¼1
0 6 xj 6 aj ;
j ¼ 1; . . . ; n;
~ ij ¼ cov ð~ri ; ~r j Þ is possibilistic covariance value of where Eð~rj Þ is possibilistic mean value of fuzzy return of security j; r fuzzy return of security i and security j. (FMVC) is a fuzzy optimal problem, in order to solve it, we must use possibility theory.
3. Possibility theory 3.1. Preliminary theory Possibility theory was proposed by Zadeh [15] and advanced by Dubois and Prade [5] where fuzzy variables are associated with possibility distributions in a similar way that random variables are associated with probability distributions in the probability theory. the possibility distribution function of a fuzzy variable is usually defined by the membership function ~ of R with membership functionla : R ! ½0; 1, of the corresponding fuzzy set. We call a fuzzy number any fuzzy subset a ~ be two fuzzy numbers with membership function l ðxÞ; l ðxÞ, respectively. Based on the concepts and techniques ~; b Let a a b of possibility theory founded by Zadeh [15], In this paper, we consider the trapezoidal fuzzy numbers which are fuzzy numbers fully determined by quadruples ~r ¼ ðr 1 ; r 2 ; r 3 ; r 4 Þ of crisp numbers such that r1 < r 2 < r 3 < r 4 i.e., whose membership functions can be denoted by:
8 xr 1 > > > r2 r1 > <1 lðxÞ ¼ xr4 > r r > > 3 4 > : 0
r1 6 x 6 r2 ; r2 6 x 6 r3 ; r3 6 x 6 r4 ; otherwise:
We mention that the trapezoidal fuzzy number is a triangular fuzzy number if r2 ¼ r 3 . (1) fuzzy numbers and operation. The sum of two trapezoidal fuzzy numbers is also a trapezoidal fuzzy number, the product of a trapezoidal fuzzy number and a scalar number is also a trapezoidal fuzzy number, and we can define the sum ~ ¼ ðb1 ; b2 ; b3 ; b4 Þ as ~ ¼ ða1 ; a2 ; a3 ; a4 Þ; b of a
~ ¼ ða1 þ b1 ; a2 þ b2 ; a3 þ b3 ; a4 þ b4 Þ; ~þb a ðka1 ; ka2 ; ka3 ; ka4 Þ k P 0; ~¼ ka ðka4 ; ka3 ; ka2 ; ka1 Þ k < 0: ~ is a crisp subset of R and is (2) possibilistic mean value, variance and covariance. The a-level set of a fuzzy numbers a ~ ¼ ða1 ; a2 ; a3 ; a4 Þ, then ½a ~a ¼ fxjla ðxÞ P a; x 2 Rg ¼ ½a1 þ aða2 a1 Þ; ~a ¼ fxjla ðxÞ P a; x 2 Rg, when a denoted by: ½a a4 aða4 a3 Þ, Carlsson and Fuller [3] introduced the notation of crisp possibilitic mean value of continuous possi~a ¼ ½aa ; aa . Then the crisp possibilistic bility distributions, which are consistent with the extension principle. Let ½a R R ~ as rða ~Þ ¼ 12 01 aðaa ~ as Eða ~Þ ¼ 01 aðaa þ aa Þda. Then the crisp possibilistic variance value of a mean value of a R1 2 a 1 a a a a ~ ~ ~ and b as cov ða ~; bÞ ¼ 2 0 aða a Þðb b ÞdaIt is easy to a Þ da. Then the crisp possibilistic covariance value of a ~ ¼ ðb1 ; b2 ; b3 ; b4 Þ is two trapezoidal fuzzy number then: ~ ¼ ða1 ; a2 ; a3 ; a4 Þ; b see that if a
G. Chen et al. / Applied Mathematics and Computation 215 (2009) 1456–1462
~Þ ¼ Eða
a2 þ a3 a1 þ a4 þ ; 3 6
rða~Þ ¼
ða3 a2 Þ2 ða4 a1 Þ2 ða3 a2 Þða4 a1 Þ ; þ þ 12 8 24
1459
~ ¼ ða3 a2 Þðb3 b2 Þ þ ða4 a1 Þðb4 b1 Þ þ ða3 a2 Þðb4 b1 Þ þ ðb3 b2 Þða4 a1 Þ : ~; bÞ cov ða 8 24 24 24
3.2. Model formulation By possibility theory, possibilistic mean variance portfolio selection model can be transformed into linear optimal problem with an additional quadratic constraint
max
ðFMVCÞ
f ðxÞ ¼
n X r 12 þ r13
3
j¼1
s:t
n X
þ
r 11 þ r 14 cj Þxj 6
n X
ðr i3 r i2 Þðr j3 r j2 Þ ðr i4 r i1 Þðr j4 r j1 Þ þ 8 24 i¼1 j¼1 ðr i3 r i2 Þðr j4 r j1 Þ ðr j3 r j2 Þðr i4 r i1 Þ xi xj 6 w; þ þ 24 24
n X
xj ¼ 1;
ð4Þ
j¼1
0 6 xj 6 a j ;
j ¼ 1; . . . ; n:
4. Cutting plane method Problem (FMVC) is a linear optimal problem with an additional quadratic constraint. For such problems there are no special standard algorithms. Of course, one could treat this problem with general methods of nonlinear optimization, but this would lead to inefficient algorithms. In this paper, we propose a cutting plane algorithm to solve (FMVC), the cutting plane method for solving convex programming problems first introduced by Kelley [13] and Cheney and Goldstein [7]. P P ðr r Þðr r Þ ðr r Þðr r Þ ðr r Þðr r Þ ðr r Þðr r Þ Let gðxÞ ¼ w ni¼1 nj¼1 i3 i2 8 j3 j2 þ i4 i124 j4 j1 þ i3 i224 j4 j1 þ j3 j224 i4 i1 xi xj , then g(x) is a concave function on Rn , Let
G ¼ fx : gðxÞ P 0g and
T ¼ fx :
n X
xj ¼ 1; 0 6 xj 6 aj ; j ¼ 1; . . . ; ng
j¼1
Step 1. Solve the linear programs:
ðFMVC0Þ
max
f ðxÞ ¼
s:t
x 2 T:
n X r 12 þ r 13 j¼1
3
þ
r11 þ r 14 cj Þxj 6 ð5Þ
Let x0 be the optimal solution, If x0 is contained in the set
G ¼ fx : x 2 T; gðxÞ P 0g stop, an optimum of (FMVC) has been reached. Otherwise let k ¼ 0 and go to Step 2. Step 2. Solve the linear programs:
ðFMVCkÞ
max
f ðxÞ ¼
n X r12 þ r 13 j¼1
s:t
3
þ
r 11 þ r14 cj cj Þxj 6
g~ðxk Þ ¼ gðxk Þ þ Dgðxk ÞT ðx xk Þ P 0; x 2 T:
ð6Þ
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Step 3. Let xkþ1 be the optimal solution of the preceding linear program. If xkþ1 2 G, stop. otherwise set k ¼ k þ 1 and return to Step 2. Denoted by Sk the feasible set of linear program solved in Step 2 of iteration k. Theses sets are nested – that is,
Sk Sk1 S0 : Theorem 1. Let g be closed concave function on the compact convex set T Rn such that at every point x 2 T the sets of subgradient DgðxÞ are nonempty and there exists a K such that
supfknk : n 2 DgðxÞ; x 2 Tg 6 k: Further assume that G, the feasible set of (FMVC), is nonempty and contained in T. Let
Sk ¼ Sk1
\ fx : g~k ðx; xk Þ P 0g;
where S0 ¼ T. If xkþ1 2 Sk is such that
f ðxkþ1 Þ ¼ cT xkþ1 ¼ maxfcT x : x 2 Sk g; then the sequence fxk g contains a subsequence that converges to an optimal solution of (FMVC). Proof. First we observe from
Sk Sk1 S0 ; that ff ðxk Þg is monotone decreasing. Hence if fxk g contains a subsequence that converges to a point x 2 G,then ff ðxk Þgconverges to ff ðx Þg and x solve (MVC). Suppose now that xk does not have a subsequence converging to a point in G. Then there exists an a > 0 such that
gðxh Þ 6 a for h ¼ 0; 1; . . . ; k. If xkþ1 maximizing cT x on Sk , then xkþ1 2 T and
gðxh Þ þ Dgðxh ÞT ðxkþ1 xh Þ P 0;
h ¼ 0; 1; . . . ; k
from the last two relations and the Schwarz inequality, it follows that
a 6 gðxh Þ 6 Dgðxh ÞT ðxkþ1 xh Þ 6 Kkxkþ1 xh k: Hence for every subsequence fkp g of indices we have
kxkp xkq k P
a K
;
q < p;
that is, fxk g does not have a cauchy subsequence, which contradicts that fxk g T is bounded. h 5. Numerical example In this section, we will give a numerical example to illustrate the proposed mean variance portfolio selection model. Consider a 3-securities problem: c1 ¼ 0:003; c2 ¼ 0:004; c3 ¼ 0:005 expected rates are ~r1 ¼ ð0:12; 0:15; 0:21; 0:24Þ; ~r2 ¼ ð0:12; 0:16; 0:22; 0:26Þ; ~r3 ¼ ð0:20; 0:28; 0:38; 0:4Þ:w ¼ 0:0025; aj ¼ 0:5, covariance matrix is
0
1 0:0017 0:0024 0:0037 B C 0:0041 A @ 0:0024 0:002 0:0037 0:0041 0:0046 from (FMVC) we obtained:
max
f ðxÞ ¼ 0:177x1 þ 0:186x2 þ 0:315x3
s:t
0:0017x21 þ 0:0048x1 x2 þ 0:0074x1 x3 þ 0:002x22 þ 0:0082x2 x3 þ 0:0047x23 6 0:0025; x1 þ x2 þ x3 ¼ 1; 0 6 xj 6 0:5;
ð7Þ
j ¼ 1; 2; 3:
Let
T ¼ fx : x 2 R3 ; x1 þ x2 þ x3 ¼ 1; 0 6 x1 6 0:5; 0 6 x2 6 0:5; 0 6 x3 6 0:5g: Applying the cutting plane algorithm, we first solve the linear program of maximizing f(x) subject to x 2 T. the optimal solution is at x0 ¼ ð0; 0:5; 0:5Þ, where f ðx0 Þ ¼ 0:2502; gðx0 Þ ¼ 0:0012 < 0, we construct the linear constraint
G. Chen et al. / Applied Mathematics and Computation 215 (2009) 1456–1462
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Table 1 w and return and variance. w
x
Return
Variance
0.0025 0.0028 0.003 0.0032
(0.461538,0.5,0.038462) (0.384615,0.5,0.115385) (0.26921,0.5,0.23079) (0.192308,0.5,0.307692)
0.1868085 0.197423 0.213346 0.223962
0.00226506 0.00253681 0.002925591 0.003168786
gðx0 Þ þ Dgðx0 ÞT ðx x0 Þ P 0 or
0:0061x1 0:0061x2 0:0087x3 þ 0:0062 P 0: Solving the linear program:
ðFMVC1Þ
max s:t
f ðxÞ ¼ 0:177x1 þ 0:186x2 þ 0:315x3 0:0061x1 0:0061x2 0:0087x3 þ 0:0051 P 0; x1 þ x2 þ x3 ¼ 1; 0 6 xj 6 0:5;
ð8Þ
j ¼ 1; 2; 3;
we find the optimal solution at x1 ¼ ð0:461358; 0:5; 0:038462Þ, where f ðx1 Þ ¼ 0:186808; gðx1 Þ ¼ 0:00023494 > 0, stop. the return of the portfolio is 0.186808, the risk (variance) of the portfolio is 0.00226506. When w ¼ 0:0028, we find the optimal solution at x1 ¼ ð0:384615; 0:5; 0:115385Þ, where f ðx1 Þ ¼ 0:197423; gðx1 Þ ¼ 0:00026319 > 0, stop. The return of the portfolio is 0.197423, the risk (variance) of the portfolio is 0.00253681. When w ¼ 0:003, we find the optimal solution at x1 ¼ ð0:269231; 0:5; 0:230769Þ, where f ðx1 Þ ¼ 0:213346; gðx1 Þ ¼ 0:000074409 > 0, stop. The return of the portfolio is 0.213346, the risk (variance) of the portfolio is 0.002925591. When w ¼ 0:0032, we find the optimal solution at x1 ¼ ð0:192308; 0:5; 0:307692Þ, where f ðx1 Þ ¼ 0:223962; gðx1 Þ ¼ 0:00031214 > 0, stop. The return of the portfolio is 0.223962, the risk (variance) of the portfolio is 0.003168786. The example shows that optimal possibility return is increasing with increasing of w, as Table 1 shows. Through choosing the values of the parameter w according to the investor’s frame of mind, the investor may get a favorite investment strategy. 6. Conclusion In this paper, we consider trapezoidal possibility distribution as the possibility distribution of the rates of returns on the securities and propose a possibilistic mean variance portfolio selection model (FMVC). Problem (FMVC) is a linear optimal problem with an additional quadratic constraint. For such problems there are no special standard algorithms. We propose a cutting plane algorithm to solve (FMVC), The problem can be solved by solving a sequence linear programming problem. The possibilistic programming problem can be solved by transforming it into a linear programming problem based on the possibilistic theory and cutting plane algorithm. A numerical example is given to illustrate the proposed method can be used efficiently to solve portfolio selection problem. Acknowledgements We thank the Editor-in-Chief and referees for their penetrating remarks and suggestions concerning earlier versions of this paper. References [1] A. Bilbao-Terol, B. Perez-Gladish, M. Arenas-Parra, M.V. Rodriguez-Uria, Fuzzy compromise programming for portfolio selection, Applied Mathematics and Computation 173 (2006) 251–264. [2] A. Yoshimoto, The mean–variance approach to portfolio optimization subject to transaction costs, Journal of the Operational Research Society of Japan 39 (1996) 99–117. [3] C. Carlsson, R. Fuller, On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems 122 (2001) 315–326. [4] C. Carlsson, R. Fuller, P. Majlender, A possibilistic approach to selecting portfolios with highest utility score, Fuzzy Sets and Systems 131 (2002) 13–21. [5] D. Dubois, H. Prade, Possibility Theory, Plenum Press, New York, 1998. [6] E. Ballestero, C. Romero, Portfolio selection: a compromise programming solution, Journal of Operational Research Society 86 (1996) 347–1377. [7] E.W. Cheney, A.A. Goldstein, Newton’s method for convex programming and Tchebycheff approximation, Numerische Mathematik 1 (1959) 253–268. [8] H. Markowitz, Portfolio selection, Journal of Finance 7 (1952) 77–91. [9] H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, Wiley, New York, 1959. [10] H. Markowitz, Mean Variance Analysis in Portfolio Choice and Capital Markets, Basil Blackwell, New York, 1987. [11] H. Tanaka, P. Guo, Portfolio selection based on upper and lower exponential possibility distributions, European Journal of Operational research 114 (1999) 115–126.
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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A cutting plane algorithm for MV portfolio selection model q Guohua Chen a,*, Xiaolian Liao a, Shouyang Wang b a b
Department of Mathematics, Hunan Institute of Humanities Science and Technology, Loudi 417000, China Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China
a r t i c l e
i n f o
Keywords: Possibility theory Portfolio selection Cutting plane algorithm
a b s t r a c t This paper deals with a portfolio selection problem with fuzzy return rates. A possibilistic mean variance (FMVC) portfolio selection model was proposed. The possibilistic programming problem can be transformed into a linear optimal problem with an additional quadratic constraint by possibilistic theory. For such problems there are no special standard algorithms. We propose a cutting plane algorithm to solve (FMVC). The nonlinear programming problem can be solved by sequence linear programming problem. A numerical example is given to illustrate the behavior of the proposed model and algorithm. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction In 1952, Markowitz [8–10] published his pioneering work which laid the foundation of modern portfolio analysis. Markowitz’s mean variance model has served as a basis for the development of modern financial theory over the past five decades. Traditionally, research has been undertaken on the assumption that future security returns can be correctly reflected by past performance and be represented by random variables.Since the security market is so complex, in many situations security returns cannot always be accurately predicted from historical data. They are beset with ambiguity and vagueness. To deal with this problem, researchers have made use of fuzzy set theory [18,16]. Assuming that the returns are fuzzy, a great deal of work has been dedicated to extrapolating traditional mean variance models: for example, Watada [14], Tanaka and Guo [11,12], Parra et al. [16], Carlsson et al. [4], Zhang and Nie [23], Bilbao-Terol et al. [1], Ostermark [20], Leon et al. [21], Ballestero and Romero [6], etc., Lacagnina and Pecorella [22] proposed a multistage stochastic soft constraints fuzzy model with recourse to solve a portfolio management problem. Shiang and Rong [26] constructed a pair of two-level mathematical programming models, based on which the upper bound and lower bound of the objective values were obtained. Bilbao et al. [34] proposed a new Fuzzy Compromise Programming approach is based on the obtaining of the minimum fuzzy distance to the fuzzy ideal solution of the portfolio selection problem. Enriqueta et al. [33] presented a fuzzy downside risk approach for managing portfolio problems in the framework of risk-return trade-off using interval-valued expectations. Zhang [24] discussed the portfolio selection problem for bounded assets based on the upper and lower possibilistic means and variances of fuzzy numbers. Huang [25] built a risk curve fuzzy portfolio selection model. Recent advances of portfolio selection model consider integration of various muti-criteria decision making models such as fuzzy AHP in Tiryaki and Ahlatcioglu [35] and expert systems such as Smimou et al. [36]. In this paper we propose an alternative approach whose main idea is based on Dyer and Wolsey paper [27]. We use cutting planes to reformulate and solve the problem using valid inequalities as cutting planes. The objective is to use these results to obtain an improved formulation of the problem. This work was motivated in part by the success of a series of papers q Supported by the National Natural Science Foundation of China under Grant No. 70221001. A project supported by Scientific Research Fund of Hunan Provincial Education Department (07C389). * Corresponding author. E-mail address: [email protected] (G. Chen).
0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.040
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[28–30] for different problems. Claus and Westerland [31] proposed a new algorithm for solving convex nonlinear programming optimization problems. Diaz and Zabala [32] presented an approach based on integer programming formulations of the graph coloring problem. This paper is organized as follows. In Section 2, we introduce briefly possibilistic mean variance portfolio selection model with transaction costs. In Section 3, we introduce briefly the possibility theory and transform possibilistic mean portfolio selection model into a linear optimal problem with an additional quadratic constraint (FMVC). For such problems there are no special standard algorithms. In Section 4, we propose a cutting plane algorithm to solve (FMVC), In Section 5, an example is given to illustrate the proposed model. A few concluding remarks are finally given in Section 6. 2. Mean variance portfolio selection model with transaction costs 2.1. Mean variance approach A portfolio selection problem in the mean variance context can be written as (MV):
max
ðMVÞ
f ðxÞ ¼
n X
Eðr j Þxj
j¼1 n X n X
s:t
i¼1 n X
rij xi xj 6 w;
j¼1
xj ¼ 1;
ð1Þ
j¼1
0 6 xj 6 aj ;
j ¼ 1; . . . ; n;
where n is the number of risky securities, xi is the proportion invested in security i; Eðr i Þ is the expected returns of security i; rij is the covariance of the expected returns on security i and j; w is the risk tolerance factor of the investor. Obviously, the greater the factor w is, the more risk tolerance the investor has. The expected return and the risk of a portfolio are, respectively, given by
Rp ¼
n X
Eðr j Þxj
j¼1
and
r2p ¼
n X n X
rij xi xj :
t¼1 t¼1
The input data of the maximization problem are expected returns of securities and variance covariance matrix of the expected returns of securities in a portfolio. Transaction cost is one of the main sources of concern to portfolio managers. Arnott and Wanger [19] found that ignoring transaction costs would result in an inefficient portfolio. Yoshimoto’s empirical analysis [2] also drew the same conclusion. In this paper, we consider the proportional transaction costs. Assume the rate of transaction cost of asset jðj ¼ 1; . . . ; nÞ is cj , P thus the transaction cost of asset j is cj xj . The transaction cost of the portfolio x ¼ ðx1 ; x2 ; . . . ; xn Þ is nj¼1 cj xj . The actual exPn pected rate of return under transaction costs is therefore, given by rðxÞ ¼ i¼1 Eðr j Þxj cj ðxj Þ. Hence, our portfolio selection model with fixed transaction costs can be formulated as (MVC):
ðMVCÞ
max
f ðxÞ ¼
n X ðEðr j Þ cj Þxj j¼1
s:t
n X n X i¼1 n X
rij xi xj 6 w;
j¼1
xj ¼ 1;
ð2Þ
j¼1
0 6 xj 6 a j ;
j ¼ 1; . . . ; n;
where xj represents the proportion of the total amount of money devoted to security, aj represent the maximum proportion of the total amount of money devoted to security respectively. Let r j be the random variable representing the rate of return of security, Eðr j Þ is the expected returns of security j, In this model, the investor is trying to maximize the future value of his/her portfolio, which requires the risk that the variance his portfolio not to be greater than w.
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2.2. Possibilistic mean variance portfolio selection model In standard portfolio models uncertainty is equated with randomness, which actually combines both objectively observable and testable random events with subjective judgments of the decision maker into probability assessments. A purist on theory would accept the use of probability theory to deal with observable random events, but would frown upon the transformation of subjective judgments to probabilities. So in this paper we will assume that the rates of return on securities are modeled by possibility distributions rather than probability distributions. Applying possibilistic distribution may have two advantages [17]: (1) the knowledge of the expert can be easily introduced to the estimation of the return rates and (2) the reduced problem is more tractable than that of the stochastic programming approach. We consider the portfolio selection problem under the assumption that the rate of return on security is the trapezoidal fuzzy number
ðFMVCÞ
max
f ðxÞ ¼
n X ðEð~r j Þ cj Þxj j¼1
n X n X
s:t
r~ ij xi xj 6 w;
t¼1 t¼1 n X
xj ¼ 1;
ð3Þ
j¼1
0 6 xj 6 aj ;
j ¼ 1; . . . ; n;
~ ij ¼ cov ð~ri ; ~r j Þ is possibilistic covariance value of where Eð~rj Þ is possibilistic mean value of fuzzy return of security j; r fuzzy return of security i and security j. (FMVC) is a fuzzy optimal problem, in order to solve it, we must use possibility theory.
3. Possibility theory 3.1. Preliminary theory Possibility theory was proposed by Zadeh [15] and advanced by Dubois and Prade [5] where fuzzy variables are associated with possibility distributions in a similar way that random variables are associated with probability distributions in the probability theory. the possibility distribution function of a fuzzy variable is usually defined by the membership function ~ of R with membership functionla : R ! ½0; 1, of the corresponding fuzzy set. We call a fuzzy number any fuzzy subset a ~ be two fuzzy numbers with membership function l ðxÞ; l ðxÞ, respectively. Based on the concepts and techniques ~; b Let a a b of possibility theory founded by Zadeh [15], In this paper, we consider the trapezoidal fuzzy numbers which are fuzzy numbers fully determined by quadruples ~r ¼ ðr 1 ; r 2 ; r 3 ; r 4 Þ of crisp numbers such that r1 < r 2 < r 3 < r 4 i.e., whose membership functions can be denoted by:
8 xr 1 > > > r2 r1 > <1 lðxÞ ¼ xr4 > r r > > 3 4 > : 0
r1 6 x 6 r2 ; r2 6 x 6 r3 ; r3 6 x 6 r4 ; otherwise:
We mention that the trapezoidal fuzzy number is a triangular fuzzy number if r2 ¼ r 3 . (1) fuzzy numbers and operation. The sum of two trapezoidal fuzzy numbers is also a trapezoidal fuzzy number, the product of a trapezoidal fuzzy number and a scalar number is also a trapezoidal fuzzy number, and we can define the sum ~ ¼ ðb1 ; b2 ; b3 ; b4 Þ as ~ ¼ ða1 ; a2 ; a3 ; a4 Þ; b of a
~ ¼ ða1 þ b1 ; a2 þ b2 ; a3 þ b3 ; a4 þ b4 Þ; ~þb a ðka1 ; ka2 ; ka3 ; ka4 Þ k P 0; ~¼ ka ðka4 ; ka3 ; ka2 ; ka1 Þ k < 0: ~ is a crisp subset of R and is (2) possibilistic mean value, variance and covariance. The a-level set of a fuzzy numbers a ~ ¼ ða1 ; a2 ; a3 ; a4 Þ, then ½a ~a ¼ fxjla ðxÞ P a; x 2 Rg ¼ ½a1 þ aða2 a1 Þ; ~a ¼ fxjla ðxÞ P a; x 2 Rg, when a denoted by: ½a a4 aða4 a3 Þ, Carlsson and Fuller [3] introduced the notation of crisp possibilitic mean value of continuous possi~a ¼ ½aa ; aa . Then the crisp possibilistic bility distributions, which are consistent with the extension principle. Let ½a R R ~ as rða ~Þ ¼ 12 01 aðaa ~ as Eða ~Þ ¼ 01 aðaa þ aa Þda. Then the crisp possibilistic variance value of a mean value of a R1 2 a 1 a a a a ~ ~ ~ and b as cov ða ~; bÞ ¼ 2 0 aða a Þðb b ÞdaIt is easy to a Þ da. Then the crisp possibilistic covariance value of a ~ ¼ ðb1 ; b2 ; b3 ; b4 Þ is two trapezoidal fuzzy number then: ~ ¼ ða1 ; a2 ; a3 ; a4 Þ; b see that if a
G. Chen et al. / Applied Mathematics and Computation 215 (2009) 1456–1462
~Þ ¼ Eða
a2 þ a3 a1 þ a4 þ ; 3 6
rða~Þ ¼
ða3 a2 Þ2 ða4 a1 Þ2 ða3 a2 Þða4 a1 Þ ; þ þ 12 8 24
1459
~ ¼ ða3 a2 Þðb3 b2 Þ þ ða4 a1 Þðb4 b1 Þ þ ða3 a2 Þðb4 b1 Þ þ ðb3 b2 Þða4 a1 Þ : ~; bÞ cov ða 8 24 24 24
3.2. Model formulation By possibility theory, possibilistic mean variance portfolio selection model can be transformed into linear optimal problem with an additional quadratic constraint
max
ðFMVCÞ
f ðxÞ ¼
n X r 12 þ r13
3
j¼1
s:t
n X
þ
r 11 þ r 14 cj Þxj 6
n X
ðr i3 r i2 Þðr j3 r j2 Þ ðr i4 r i1 Þðr j4 r j1 Þ þ 8 24 i¼1 j¼1 ðr i3 r i2 Þðr j4 r j1 Þ ðr j3 r j2 Þðr i4 r i1 Þ xi xj 6 w; þ þ 24 24
n X
xj ¼ 1;
ð4Þ
j¼1
0 6 xj 6 a j ;
j ¼ 1; . . . ; n:
4. Cutting plane method Problem (FMVC) is a linear optimal problem with an additional quadratic constraint. For such problems there are no special standard algorithms. Of course, one could treat this problem with general methods of nonlinear optimization, but this would lead to inefficient algorithms. In this paper, we propose a cutting plane algorithm to solve (FMVC), the cutting plane method for solving convex programming problems first introduced by Kelley [13] and Cheney and Goldstein [7]. P P ðr r Þðr r Þ ðr r Þðr r Þ ðr r Þðr r Þ ðr r Þðr r Þ Let gðxÞ ¼ w ni¼1 nj¼1 i3 i2 8 j3 j2 þ i4 i124 j4 j1 þ i3 i224 j4 j1 þ j3 j224 i4 i1 xi xj , then g(x) is a concave function on Rn , Let
G ¼ fx : gðxÞ P 0g and
T ¼ fx :
n X
xj ¼ 1; 0 6 xj 6 aj ; j ¼ 1; . . . ; ng
j¼1
Step 1. Solve the linear programs:
ðFMVC0Þ
max
f ðxÞ ¼
s:t
x 2 T:
n X r 12 þ r 13 j¼1
3
þ
r11 þ r 14 cj Þxj 6 ð5Þ
Let x0 be the optimal solution, If x0 is contained in the set
G ¼ fx : x 2 T; gðxÞ P 0g stop, an optimum of (FMVC) has been reached. Otherwise let k ¼ 0 and go to Step 2. Step 2. Solve the linear programs:
ðFMVCkÞ
max
f ðxÞ ¼
n X r12 þ r 13 j¼1
s:t
3
þ
r 11 þ r14 cj cj Þxj 6
g~ðxk Þ ¼ gðxk Þ þ Dgðxk ÞT ðx xk Þ P 0; x 2 T:
ð6Þ
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Step 3. Let xkþ1 be the optimal solution of the preceding linear program. If xkþ1 2 G, stop. otherwise set k ¼ k þ 1 and return to Step 2. Denoted by Sk the feasible set of linear program solved in Step 2 of iteration k. Theses sets are nested – that is,
Sk Sk1 S0 : Theorem 1. Let g be closed concave function on the compact convex set T Rn such that at every point x 2 T the sets of subgradient DgðxÞ are nonempty and there exists a K such that
supfknk : n 2 DgðxÞ; x 2 Tg 6 k: Further assume that G, the feasible set of (FMVC), is nonempty and contained in T. Let
Sk ¼ Sk1
\ fx : g~k ðx; xk Þ P 0g;
where S0 ¼ T. If xkþ1 2 Sk is such that
f ðxkþ1 Þ ¼ cT xkþ1 ¼ maxfcT x : x 2 Sk g; then the sequence fxk g contains a subsequence that converges to an optimal solution of (FMVC). Proof. First we observe from
Sk Sk1 S0 ; that ff ðxk Þg is monotone decreasing. Hence if fxk g contains a subsequence that converges to a point x 2 G,then ff ðxk Þgconverges to ff ðx Þg and x solve (MVC). Suppose now that xk does not have a subsequence converging to a point in G. Then there exists an a > 0 such that
gðxh Þ 6 a for h ¼ 0; 1; . . . ; k. If xkþ1 maximizing cT x on Sk , then xkþ1 2 T and
gðxh Þ þ Dgðxh ÞT ðxkþ1 xh Þ P 0;
h ¼ 0; 1; . . . ; k
from the last two relations and the Schwarz inequality, it follows that
a 6 gðxh Þ 6 Dgðxh ÞT ðxkþ1 xh Þ 6 Kkxkþ1 xh k: Hence for every subsequence fkp g of indices we have
kxkp xkq k P
a K
;
q < p;
that is, fxk g does not have a cauchy subsequence, which contradicts that fxk g T is bounded. h 5. Numerical example In this section, we will give a numerical example to illustrate the proposed mean variance portfolio selection model. Consider a 3-securities problem: c1 ¼ 0:003; c2 ¼ 0:004; c3 ¼ 0:005 expected rates are ~r1 ¼ ð0:12; 0:15; 0:21; 0:24Þ; ~r2 ¼ ð0:12; 0:16; 0:22; 0:26Þ; ~r3 ¼ ð0:20; 0:28; 0:38; 0:4Þ:w ¼ 0:0025; aj ¼ 0:5, covariance matrix is
0
1 0:0017 0:0024 0:0037 B C 0:0041 A @ 0:0024 0:002 0:0037 0:0041 0:0046 from (FMVC) we obtained:
max
f ðxÞ ¼ 0:177x1 þ 0:186x2 þ 0:315x3
s:t
0:0017x21 þ 0:0048x1 x2 þ 0:0074x1 x3 þ 0:002x22 þ 0:0082x2 x3 þ 0:0047x23 6 0:0025; x1 þ x2 þ x3 ¼ 1; 0 6 xj 6 0:5;
ð7Þ
j ¼ 1; 2; 3:
Let
T ¼ fx : x 2 R3 ; x1 þ x2 þ x3 ¼ 1; 0 6 x1 6 0:5; 0 6 x2 6 0:5; 0 6 x3 6 0:5g: Applying the cutting plane algorithm, we first solve the linear program of maximizing f(x) subject to x 2 T. the optimal solution is at x0 ¼ ð0; 0:5; 0:5Þ, where f ðx0 Þ ¼ 0:2502; gðx0 Þ ¼ 0:0012 < 0, we construct the linear constraint
G. Chen et al. / Applied Mathematics and Computation 215 (2009) 1456–1462
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Table 1 w and return and variance. w
x
Return
Variance
0.0025 0.0028 0.003 0.0032
(0.461538,0.5,0.038462) (0.384615,0.5,0.115385) (0.26921,0.5,0.23079) (0.192308,0.5,0.307692)
0.1868085 0.197423 0.213346 0.223962
0.00226506 0.00253681 0.002925591 0.003168786
gðx0 Þ þ Dgðx0 ÞT ðx x0 Þ P 0 or
0:0061x1 0:0061x2 0:0087x3 þ 0:0062 P 0: Solving the linear program:
ðFMVC1Þ
max s:t
f ðxÞ ¼ 0:177x1 þ 0:186x2 þ 0:315x3 0:0061x1 0:0061x2 0:0087x3 þ 0:0051 P 0; x1 þ x2 þ x3 ¼ 1; 0 6 xj 6 0:5;
ð8Þ
j ¼ 1; 2; 3;
we find the optimal solution at x1 ¼ ð0:461358; 0:5; 0:038462Þ, where f ðx1 Þ ¼ 0:186808; gðx1 Þ ¼ 0:00023494 > 0, stop. the return of the portfolio is 0.186808, the risk (variance) of the portfolio is 0.00226506. When w ¼ 0:0028, we find the optimal solution at x1 ¼ ð0:384615; 0:5; 0:115385Þ, where f ðx1 Þ ¼ 0:197423; gðx1 Þ ¼ 0:00026319 > 0, stop. The return of the portfolio is 0.197423, the risk (variance) of the portfolio is 0.00253681. When w ¼ 0:003, we find the optimal solution at x1 ¼ ð0:269231; 0:5; 0:230769Þ, where f ðx1 Þ ¼ 0:213346; gðx1 Þ ¼ 0:000074409 > 0, stop. The return of the portfolio is 0.213346, the risk (variance) of the portfolio is 0.002925591. When w ¼ 0:0032, we find the optimal solution at x1 ¼ ð0:192308; 0:5; 0:307692Þ, where f ðx1 Þ ¼ 0:223962; gðx1 Þ ¼ 0:00031214 > 0, stop. The return of the portfolio is 0.223962, the risk (variance) of the portfolio is 0.003168786. The example shows that optimal possibility return is increasing with increasing of w, as Table 1 shows. Through choosing the values of the parameter w according to the investor’s frame of mind, the investor may get a favorite investment strategy. 6. Conclusion In this paper, we consider trapezoidal possibility distribution as the possibility distribution of the rates of returns on the securities and propose a possibilistic mean variance portfolio selection model (FMVC). Problem (FMVC) is a linear optimal problem with an additional quadratic constraint. For such problems there are no special standard algorithms. We propose a cutting plane algorithm to solve (FMVC), The problem can be solved by solving a sequence linear programming problem. The possibilistic programming problem can be solved by transforming it into a linear programming problem based on the possibilistic theory and cutting plane algorithm. A numerical example is given to illustrate the proposed method can be used efficiently to solve portfolio selection problem. Acknowledgements We thank the Editor-in-Chief and referees for their penetrating remarks and suggestions concerning earlier versions of this paper. References [1] A. Bilbao-Terol, B. Perez-Gladish, M. Arenas-Parra, M.V. Rodriguez-Uria, Fuzzy compromise programming for portfolio selection, Applied Mathematics and Computation 173 (2006) 251–264. [2] A. Yoshimoto, The mean–variance approach to portfolio optimization subject to transaction costs, Journal of the Operational Research Society of Japan 39 (1996) 99–117. [3] C. Carlsson, R. Fuller, On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems 122 (2001) 315–326. [4] C. Carlsson, R. Fuller, P. Majlender, A possibilistic approach to selecting portfolios with highest utility score, Fuzzy Sets and Systems 131 (2002) 13–21. [5] D. Dubois, H. Prade, Possibility Theory, Plenum Press, New York, 1998. [6] E. Ballestero, C. Romero, Portfolio selection: a compromise programming solution, Journal of Operational Research Society 86 (1996) 347–1377. [7] E.W. Cheney, A.A. Goldstein, Newton’s method for convex programming and Tchebycheff approximation, Numerische Mathematik 1 (1959) 253–268. [8] H. Markowitz, Portfolio selection, Journal of Finance 7 (1952) 77–91. [9] H. Markowitz, Portfolio Selection: Efficient Diversification of Investments, Wiley, New York, 1959. [10] H. Markowitz, Mean Variance Analysis in Portfolio Choice and Capital Markets, Basil Blackwell, New York, 1987. [11] H. Tanaka, P. Guo, Portfolio selection based on upper and lower exponential possibility distributions, European Journal of Operational research 114 (1999) 115–126.
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