A portfolio selection model based on possibility ... - Semantic Scholar
A portfolio selection model based on possibility theory using fuzzy two-stage algorithm Xue Deng, Rongjun Li
A portfolio selection model based on possibility theory using fuzzy two-stage algorithm 1
Xue Deng, 2 Rongjun Li * 1 Department of Mathematics School of Science South China University of Technology Guangzhou, Guangdong, 510640, China [email protected] 2 School of Business Administration South China University of Technology Guangzhou, Guangdong, 510640, China doi: 10.4156/jcit.vol5.issue6.14
Abstract The uncertainty of a financial market is traditionally dealt with probabilistic approaches. However, there are many non-probabilistic factors that affect the financial markets such that the return rate of risky assets may be regarded as fuzzy number, which is a powerful tool used to describe an uncertain environment with vagueness and ambiguity and some type of fuzziness. In this paper, the conventional probabilistic mean-variance model can be simplified as a bi-objective linear programming model based on possibility theory. Furthermore, the fuzzy two-stage algorithm is applied to solve our proposed bi-objective model. Finally, a numerical example of the portfolio selection problem is given to illustrate our proposed effective means and variances. The results of the example also show that the fuzzy two-stage algorithm solving the model is effective and feasible.
Keywords: Portfolio selection, Fuzzy number, Possibilistic mean, Possibilistic variance, Fuzzy two-stage algorithm
1. Introduction The mean-variance methodology for the portfolio selection problem, proposed originally by Markowitz [1,2,3], has played an important role in the development of modern portfolio selection theory. It combines probability with optimization techniques to model the behavior investment under uncertainty. The key principle of the mean-variance model is to use the expected return of the portfolio as the investment return and to use the variance of the expected return of the portfolio as the investment risk. Most of existing portfolio selection models are based on probability theory. The mean-variance portfolio selection problem has been studied by many researchers including Sharpe [4], Merton [5], Perold [6], Pang [7], Vörös [8] and Best [9], etc. Stein and Branke and Schmeck [10] discussed the efficient implementation quadratic programming that was specialized for large-scale mean-variance portfolio selection with a dense covariance matrix. The aim was to calculate the whole Pareto front of solution that represented the trade-off between maximizing expected return and minimizing variance of return. Yan and Miao and Li [11] discussed that variance was substituted by semi-variance in Markowitz‟s portfolio selection model. Moreover, one period portfolio selection is extended to multiperiod. And a hybrid genetic algorithm (GA), which makes use of the position displacement strategy of the particle swarm optimizer (PSO) as a mutation operation, was applied to solve the multi-variance model. There are many non-probabilistic factors that affect the financial market such that the return of risky asset is fuzzy uncertainty. And a number of empirical studies showed the limitations of using probabilistic approaches in characterizing the uncertainty of the financial market. Recently, a number of researchers investigated fuzzy portfolio selection problem. Bellman and Zadeh [12] proposed the basic fuzzy decision theory. Carlsson and Fullér [13] discussed some basic knowledge about possibilistic mean and variance of fuzzy numbers. Arenas and Rodríguez and Bilbao [14] took into
138
Journal of Convergence Information Technology Volume 5, Number 6, August 2010 account three criteria: return, risk and liquidity and used a fuzzy goal programming approach to solve the portfolio selection problem. Huang and Tzeng and Ong [15] discussed a new algorithm for the revised mean-variance model which is proposed to deal with the problem of uncertain portfolio selection. Wang and Zhu [16] gave an overview on the development of fuzzy portfolio selection using fuzzy approaches, quantitative analysis, qualitative analysis, the experts‟ knowledge and manager‟s subjective opinions. Ida [17] considered portfolio selection problem with interval and fuzzy objective function coefficients as a kind of multiple objective problems including uncertainties. And for this problem two kinds of efficient solutions are introduced: possibly efficient solution as an optimistic solution and necessarily efficient solution as a pessimistic solution. Abiyev and Menekay [18] presented fuzzy logic is utilized in the estimation of expected return and risk. Managers could extract useful information and estimate expected return by using not only statistical data, but also economical and financial behaviors of the companies and their business strategies. Bilbao and Pérez and Arenas and Rodríguez [19] discussed a portfolio selection problem using Sharpe‟s single index model in a soft framework. Estimations of subjective or imprecise future beta for every asset can be represented through fuzzy numbers on the basis of statistical data and the relevant knowledge of the financial analysis. Carlsson and Fullér and Heikkilä and Majlender [20] developed a methodology for valuing options on Research and Development (R&D) projects, when future cash flows are estimated by trapezoidal fuzzy numbers. In this paper, a new approach using fuzzy two-stage algorithm in the bi-objective portfolio selection model is presented based on possibility theory. We organize the paper as follows. In Section 2, we recall the notions of possibilistic means, variances and covariances of fuzzy numbers introduced by [13] and introduce the formulae of possibilistic means and variances of linear combination of fuzzy numbers. In Section 3, we propose the bi-objective possibilistic portfolio selection linear programming model. In Section 4, we present the ideas of fuzzy two-stage algorithm for solving multi-objective linear programming model. In Section 5, we use a numerical example of portfolio selection problem with triangular fuzzy numbers to illustrate our proposed effective means and variances and effective fuzzy two-stage algorithm in solving the model.
2. Possibilistic mean and variance and covariance Let us introduce some definitions, which we shall need in the following section. A fuzzy number A is a fuzzy set of the real line R with a normal, fuzzy convex and continuous membership function of bounded support. The family of fuzzy numbers is denoted by F . A level set of a fuzzy number
A is defined by [ A] {t R A(t ) } if 0 and [ A] cl{t R A(t ) 0} (the closure of the support of A ) if 0 . It is well known that if A is a fuzzy number then [ A] is a compact subset of
R for all [0,1] . Let
A and
B be fuzzy numbers with [ A] [a1 ( ), a2 ( )] and
[ B] [b1 ( ), b2 ( )] , for [0,1] . Next, we recall some notions introduced by Carlsson and Fullér [13], we define the possibilistic mean value of A by 1
M ( A) (a1 ( ) a2 ( ))d , 0
(1)
denote the possibilistic variance of A by 1 1 (2) (a2 ( ) a1 ( ))2 d , 20 and define the possibilistic covariance of A and B by 1 1 (3) Cov( A, B) (a2 ( ) a1 ( ))(b2 ( ) b1 ( ))d , 2 0 respectively. The standard deviation of A is defined by A Var ( A) . The following theorems show that the possibilistic mean value and variance of linear combinations of fuzzy numbers can easily be computed in a similar manner as in probability ([13]). Var ( A)
139
A portfolio selection model based on possibility theory using fuzzy two-stage algorithm Xue Deng, Rongjun Li Theorem 2.1. Let A and B be two fuzzy numbers and let , R , then Var ( A B ) 2Var ( A) 2Var ( B ) 2 Cov( A, B ).
Theorem 2.2. Let A1 , A2 ,
, An be n fuzzy numbers and let 0 , 1 , n
n
i 1
i 1
, n R , then
M (0 i Ai ) 0 i M ( Ai ) ,
and n
n
n
i 1
i 1
i j 1
Var (0 i Ai ) i 2Var ( Ai ) 2 i j Cov( Ai , Aj ) . In Theorems 2.1 and 2.2, the addition of fuzzy numbers and the multiplication by a scalar of fuzzy numbers are defined by the sup-min extension principle.
3. Bi-objective possibilistic efficient portfolio selection model In the conventional Markowitz‟s mean-variance model, the return rate of risky asset is denoted as a random variable. It is well known that the returns of risky assets are in a fuzzy uncertain economic environment and vary from time to time, the future states of returns and risk of risky assets cannot be predicted accurately. Fuzzy number is a more powerful tool used to describe an uncertain environment with vagueness and ambiguity. Based on these factors, we consider the portfolio selection problem under the assumption that the returns of assets are fuzzy numbers. For simplicity, we use the following notations: ri denotes the return rate of risky asset i , xi denotes the proportion of total investment funds devoted to the asset i , li and ui (ui li 0) are the lower and upper bounds of xi (i 1, 2, , n) , respectively. Suppose an investor starts with an existing portfolio and considers reallocating his/her wealth among n risky assets and a risk-free asset which is current deposit of bank, the lending interest rate is r0 , and the corresponding proportion of total investment is x0 . Let ri (ai , i , i ) (i 1, 2, , n) be triangular fuzzy numbers with center ai and left width i 0 and right width i 0 . A level set of ri can be denoted as
ri
[ai (1 ) i , ai (1 ) i ] [ i ( ), i ( )] , for all [0,1] , (i 1, 2, , n) . According to (1), (2) and (3), we can easily obtain 1 1 M (ri ) [ai ( ) i ( )]d ai ( i i ), 0 6 and 1 1 1 Var (ri ) [ i ( ) i ( )]2 d ( i i ) 2 , 2 0 24 and 1 1 1 Cov(ri , rj ) [ i ( ) i ( )][ j ( ) j ( )]d (i i )( j j ), 0 2 24 respectively. n
Then the return associated with the portfolio proportion ( x0 , x1 , x2 ,
, xn ) is r r0 x0 ri xi , and i 1
the corresponding possibilistic mean value and possibilistic variance of the return r are given by n n n 1 M (r ) M (r0 x0 ri xi ) r0 x0 xi M (ri ) r0 x0 xi (ai ( i i )) , 6 i 1 i 1 i 1 and
140
Journal of Convergence Information Technology Volume 5, Number 6, August 2010 n
n
n
i 1
i 1
Var (r ) Var (r0 x0 ri xi ) xi 2Var (ri ) 2 xi x j Cov(ri , rj )
i j 1
n
n
1 1 xi x j ( i i )( j j ) xi 2 (i i )2 2 24 i 24 i 1 j 1 2
1 n xi (i i ) . 24 i 1 Analogous to Markowitz‟s mean-variance methodology for the portfolio selection problem, the possibilistic mean value of the return is termed as measure of return and the possibilistic variance of the return is termed as measure of risk. Thus, the possibilistic mean-variance model of portfolio selection problem may be described by the following linear bi-objective optimal problem with equality and inequality constraints: n 1 max M (r ) r0 x0 xi (ai ( i i )) 6 i 1
1 n min Var (r ) xi (i i ) 24 i 1 s.t.
x0 x1 x2
2
xn 1,
(4)
0 x0 1, 0 li xi ui , i 1, 2, , n. Furthermore, the possibilistic mean-variance model (4) is equivalent to the following bi-objective linear programming: n 1 max M (r ) r0 x0 xi (ai ( i i )) 6 i 1 n
min
24Var (r ) xi ( i i ) i 1
s.t. x0 x1 x2
xn 1,
(5)
0 x0 1, 0 li xi ui , i 1, 2,
, n.
The possibilistic mean-variance model (5) is a simple bi-objective linear programming, and we can easily obtain the optimal solution of the possibilistic efficient portfolio model (5) by the following fuzzy two-stage algorithm.
4. Ideas and stages of fuzzy two-stage algorithm 4.1. Ideas of fuzzy two-stage algorithm Consider the following multi-objective linear programming model:
n
a x j 1
ij
j
bi , i 1, 2,
n
j 1
j 1
n
n
j 1
j 1
,Wr ( x )] [ c1 j x j , c2 j x j ,
min W ( x ) [W1 ( x ), W2 ( x ), s.t.
n
, Z q ( x )] [ c1 j x j , c2 j x j ,
max Z ( x ) [ Z1 ( x ), Z 2 ( x ),
n
, cqj x j ] j 1 n
, crj x j ] j 1
(P1)
, m,
0 l j x j u j , j 1, 2,
, n.
By the name of definition (two-stage), the process of solving the problem is divided into two stages. Its basic ideas are as follows: In Stage1, we try to obtain the total satisfactory optimal value (1) and optimal solution x (1) of equivalent problem for the objective set (which is the feasible solution of the original problem (P1)) by
141
A portfolio selection model based on possibility theory using fuzzy two-stage algorithm Xue Deng, Rongjun Li using the „max-min‟ fuzzy algorithm corresponding to logic „and‟. The corresponding mathematical model is Zimmermann minimum operator model. If the optimal solution of the inductive equivalent model is unique, then x (1) is our optimal solution, of course it is also the optimal solution of the original problem. Otherwise, we continue the second stage. In Stage2, we will try to construct a mathematical model maximizing average satisfaction degree for an objective set, and its corresponding additional condition is t (1) , t 1, 2, , q r , namely, every objective satisfaction degree is larger than or equal to the optimal satisfaction degree obtained in Stage1. Obviously, the solution of Stage2 is effective by the total complementation of arithmetic average operator. The constraints t (1) keep the balance among all the objective sets, and it is impossible to abandon any special objective.
4.2. Stages of fuzzy two-stage algorithm The mathematical description of two-stage algorithm is as follows: Stage1: According to the idea of Zimmermann, (P1) is equivalent to the following linear programming problem (P2). We can obtain the maximum satisfaction degree (1) of objective set and the feasible solution x (1) of the original problem by solving the following equivalent problem (P2): max Z ( x ) zk s.t. k , k 1, 2, zk zk w W ( x ) s s , s 1, 2, ws ws [0,1],
, q,
(P2)
, r,
x D.
where
zk max Z k ( x ), zk min Z k ( x ), k 1, 2,
, q,
ws min Ws ( x ), ws max Ws ( x ), s 1, 2,
, r,
xD
xD n
xD
xD
D { aij x j bi , i 1, 2, j 1
, m, 0 l j x j u j , j 1, 2,
, n.}.
O ( zk ( x); ws ( x)) and O ( zk ( x ); ws ( x )) are called the corresponding ideal solutions. In general, we do not know whether the solution of (P2) is unique, hence we can not determine whether x (1) is the optimal solution of the original problem. Effectiveness of x (1) must be verified by the following problem (P3). Stage2: Verifying the effectiveness of x (1) or looking for a new effective solution x ( 2) : q r 1 max [ k s ] q r k 1 s 1 Z ( x ) zk s.t. (1) k k , k 1, 2, zk zk
, q,
ws Ws ( x ) , s 1, 2, ws ws
, r,
(1) s
(P3)
k , s [0,1], x D. We can easily obtain the solution x ( 2) . And we can prove that x ( 2) is also the solution of linear programming (P2). Hence, if the solution of the linear programming (P2) is unique, then we must have x (2) x (1) . In this time, x (1) is the optimal solution to the original problem, and the work of Stage2 is only to check the fact. If the solution of (P2) is not unique, i.e. x (1) x (2) , in this situation, x (1) may
142
Journal of Convergence Information Technology Volume 5, Number 6, August 2010 be an effective solution or may be not an effective solution, but anyway, x ( 2) is always an effective solution. Thus, under any circumstance, the two-stage algorithm will insure to obtain an effective solution of the original problem in Stage2.
5. Numerical example In order to illustrate our proposed effective means and variances of the efficient portfolio in this paper, we considered a real portfolio selection example. In this example, we selected five stocks from Shanghai Stock Exchange, their returns ri , i 1, 2, ,5 are regarded as triangular fuzzy numbers. Based on the historical data, the future information, the advices of experts, the corporations‟ financial reports and the managers‟ subjective opinions, we can estimate their returns with the following possibility distributions: r1 (0.073,0.054,0.087), r2 ( 0 . 1 0 5 , 0 . 0 7 5 , 0 . 1 0 2 r)3, (0.138,0.096,0.123),
r4 ( 0 . 1 6 8 , 0 . 1 2 6 , 0 . 1 6 2 ) r,5 (0.208,0.168,0.213), and therefore, the level sets of ri , i 1, 2, ,5 are given by [r1 ] [0.019 0.054 , 0.160 0.087 ], [r2 ] [0.030 0.075 , 0.207 0.102 ], [r3 ] [0.042 0.096 , 0.261 0.123 ], [r4 ] [0.042 0.126 , 0.330 0.162 ], [r5 ] [0.040 0.168 , 0.421 0.213 ].
We selected a risk-free asset which is current deposit of bank with the lending interest rate r0 is 2%. The lower bound and upper bound vectors of x ( x0 , x1 , x2 , x3 , x4 , x5 ) are given by l (0,0.1,0.1,0.1,0.1,0.1) and u (1,0.7,0.7,0.8,0.8,0.9) , respectively. We use prime ( ' ) to denote matrix transposition and all non-primed vectors are column vectors for convenience. Thus, using model (5), we can obtain the bi-objective linear programming model based on the possibilistic mean-variance theory: max 0.02x0 0.0785 x1 0.1095 x2 0.1425 x3 0.1740 x4 0.2155 x5
min 0.1410x1 0.1770x2 0.2190x3 0.2880x4 0.3810x5 s.t. x0 x1 x2 x3 x4 x5 1, 0 x0 1, 0.1 x1 0.7,
(M1)
0.1 x2 0.7, 0.1 x3 0.8, 0.1 x4 0.8, 0.1 x5 0.9. By the fuzzy two-stage algorithm, we obtain the following ideal solutions: O ( z1 ( x ); w1 ( x)) (0.179750;0.120600), and O ( z1 ( x ); w1 ( x)) (0.082000;0.311100). Stage1: Model (1) is changed into the following Model (2):
143
A portfolio selection model based on possibility theory using fuzzy two-stage algorithm Xue Deng, Rongjun Li max s.t. (0.02x0 0.0785 x1 0.1095 x2 0.1425 x3 0.1740 x4 0.2155 x5 0.082) /(0.17975 0.082),
(0.3111 0.1410x1 0.1770x2 0.2190x3 0.2880x4 0.3810x5 ) /(0.3111 0.1206), x0 x1 x2 x3 x4 x5 1, 0 x0 1,
(M2)
0.1 x1 0.7, 0.1 x2 0.7, 0.1 x3 0.8, 0.1 x4 0.8, 0.1 x5 0.9,
[0,1]. We obtain the following optimal solution: (1) 0.516500, x (1) (0.079421, 0.1, 0.1, 0.520579, 0.1, 0.1), and the optimal mean and variance of return are M *(1) 0.133521, Var*(1) 0.001885 .
Stage2: In order to check the effectiveness of x (1) , we need to solve the following Model (3): max =
1 2
2 s.t. 0.516500 1 (0.02x0 0.0785 x1 0.1095 x2 0.1425 x3 0.1740 x4 0.2155 x5 0.082) /(0.17975 0.082), 0.516500 2 (0.3111 0.1410x1 0.1770x2 0.2190x3 0.2880x4 0.3810x5 ) /(0.3111 0.1206), x0 x1 x2 x3 x4 x5 1,
(M3)
0 x0 1, 0.1 x1 0.7, 0.1 x2 0.7, 0.1 x3 0.8, 0.1 x4 0.8, 0.1 x5 0.9,
1 , 2 [0,1].
Then we can obtain the following optimal solution: 1 0.516500, 2 0.516500, 0.516500 (1) , x (2) (0.079421, 0.1, 0.1, 0.520579, 0.1, 0.1), M *(2) 0.133521, Var*(2) 0.001885 .
It is obvious that x (1) x (2) which means that the effective solution is unique. That is, the solution x obtained in Stage1 is effective and optimal for the original problem (M1). (1)
6. Conclusions In today‟s extremely competitive business environment, investors have already invested in various assets to keep their competitive advantage. In practical situations, regarding the expected return and the risk as two objective functions, we can propose a bi-objective linear programming model based on possibilistic mean and variance with triangular fuzzy numbers of return rate. Naturally, the fuzzy twostage algorithm can be utilized to solve our proposed model. The obtained results from the numerical example show that our proposed model is reasonable, and the two-stage algorithm is effective, and the optimal solution must be found for the original problem. Our proposed possibilistic approaches to selecting portfolios can better describe an uncertain decision problem with vagueness and ambiguity.
144
Journal of Convergence Information Technology Volume 5, Number 6, August 2010
7. Acknowledgements This research was supported by Guangdong Science and Technology Department Soft Science Sponsor Item, China (No. 2008B070800012) and Student Research Planning, South China University of Technology, China (No. Y1070180). The authors are also grateful to a referee for his/her very helpful comments and suggestions.
8. References [1] H. Markowitz, “Portfolio selection”, Journal of Finance, vol. 7, pp. 77¬-91, 1952. [2] H. Markowitz, Portfolio Selection: Efficient Diversification of Investment, Wiley, New York, 1959. [3] H. Markowitz, Analysis in Portfolio Choice and Capital Markets, Basil Blackwell, Oxford, 1987. [4] W. F. Sharpe, Portfolio Theory and Capital Markets, McGraw-Hill, New York, 1970. [5] R.C. Merton, “An analytic derivation of the efficient frontier”, Journal of Finance and Quantitative Analysis, vol. 9, pp. 1851-1872, 1972. [6] A.F. Perold, “Large-scale portfolio optimization”, Management Science, vol. 30, pp. 1143-1160, 1984. [7] J.S. Pang, “A new efficient algorithm for a class of portfolio selection problems”, Operational Research, vol. 28, pp. 754-767, 1980. [8] J. Vörös, “Portfolio analysis-An analytic derivation of the efficient portfolio frontier”, European Journal of Operational Research, vol. 203, pp. 294-300, 1986. [9] M.J. Best, J. Hlouskova, “The efficient frontier for bounded assets”, Mathematical Methods of Operations Research, vol. 52, pp. 195-212, 2000. [10] M. Stein, J. Branke, H. Schmeck, “Efficient implementation of an active set algorithm for largescale portfolio selection”, Computers & Operations Research, vol. 35, pp. 3945-3961, 2008. [11] W. Yan, R. Miao, S.R. Li, “Multi-period semi-variance portfolio selection: Model and numerical solution”, Applied Mathematics and Computation, vol. 194, pp. 128-134, 2007. [12] R. Bellman, L.A. Zadeh, “Decision making in a fuzzy environment”, Management Science, vol. 17, pp. 141-164, 1970. [13] C.Carlsson, R. Fullér, “On possibilistic mean and variance of fuzzy numbers”, Fuzzy sets and Systems, vol. 122, pp. 315-326, 2001. [14] M. Arenas, A. Bilbao, M.V. Rodríguez, “A fuzzy goal programming approach to portfolio selection”, European Journal of Operational Research, vol. 133, pp. 287-297, 2001. [15] J.J. Huang, G.H. Tzeng, C.S. Ong, “A novel algorithm for uncertain portfolio selection”, Applied Mathematics and Computation, vol. 173, pp. 350-359, 2006. [16] S.Y. Wang, S.S. Zhu, “On fuzzy portfolio selection problems”, Fuzzy Optimization and Decision Making, vol. 1, pp. 361-377, 2002. [17] M. Ida, “Solutions for the portfolio selection problem with interval and fuzzy coefficients”, Reliable Computing, vol. 10, pp. 389-400, 2004. [18] R.H. Abiyev, M. Menekay, “Fuzzy portfolio selection using genetic algorithm”, Soft Comput vol. 11, pp. 1157-1163, 2007. [19] A. Bilbao, B. Pérez, M. Arenas, M.V. Rodríguez, “Fuzzy compromise programming for portfolio selection”, Applied Mathematics and Computation, vol. 173, pp. 251-264, 2006. [20] C.Carlsson, R. Fullér, M. Heikkilä, P. Majlender, “A fuzzy approach to R&D project portfolio selection”, International Journal of Approximate Reasoning, vol. 44, pp. 93-105, 2007. [21] H.J. Zimmermann, “Description and optimization of fuzzy systems”, International Journal of General Systems, vol. 2, pp. 209-215, 1976. [22] R.J. Li, E. Stanly Lee, “Fuzzy approaches to multicriteria de novo programs”, Mathematical Analysis and Applications, vol. 153, pp. 97-111, 1990. [23] A. Gundel, S. Weber, “Utility maximization under a shortfall risk constraint”, Journal of mathematical Economics, vol. 44, pp. 1126-1151, 2008. [24] M.R. Haley, “A simple nonparametric approach to low-dimension, shortfall-based portfolio selection”, Finance Research Letters, vol. 5, pp. 183-190, 2008.
145
A portfolio selection model based on possibility theory using fuzzy two-stage algorithm 1
Xue Deng, 2 Rongjun Li * 1 Department of Mathematics School of Science South China University of Technology Guangzhou, Guangdong, 510640, China [email protected] 2 School of Business Administration South China University of Technology Guangzhou, Guangdong, 510640, China doi: 10.4156/jcit.vol5.issue6.14
Abstract The uncertainty of a financial market is traditionally dealt with probabilistic approaches. However, there are many non-probabilistic factors that affect the financial markets such that the return rate of risky assets may be regarded as fuzzy number, which is a powerful tool used to describe an uncertain environment with vagueness and ambiguity and some type of fuzziness. In this paper, the conventional probabilistic mean-variance model can be simplified as a bi-objective linear programming model based on possibility theory. Furthermore, the fuzzy two-stage algorithm is applied to solve our proposed bi-objective model. Finally, a numerical example of the portfolio selection problem is given to illustrate our proposed effective means and variances. The results of the example also show that the fuzzy two-stage algorithm solving the model is effective and feasible.
Keywords: Portfolio selection, Fuzzy number, Possibilistic mean, Possibilistic variance, Fuzzy two-stage algorithm
1. Introduction The mean-variance methodology for the portfolio selection problem, proposed originally by Markowitz [1,2,3], has played an important role in the development of modern portfolio selection theory. It combines probability with optimization techniques to model the behavior investment under uncertainty. The key principle of the mean-variance model is to use the expected return of the portfolio as the investment return and to use the variance of the expected return of the portfolio as the investment risk. Most of existing portfolio selection models are based on probability theory. The mean-variance portfolio selection problem has been studied by many researchers including Sharpe [4], Merton [5], Perold [6], Pang [7], Vörös [8] and Best [9], etc. Stein and Branke and Schmeck [10] discussed the efficient implementation quadratic programming that was specialized for large-scale mean-variance portfolio selection with a dense covariance matrix. The aim was to calculate the whole Pareto front of solution that represented the trade-off between maximizing expected return and minimizing variance of return. Yan and Miao and Li [11] discussed that variance was substituted by semi-variance in Markowitz‟s portfolio selection model. Moreover, one period portfolio selection is extended to multiperiod. And a hybrid genetic algorithm (GA), which makes use of the position displacement strategy of the particle swarm optimizer (PSO) as a mutation operation, was applied to solve the multi-variance model. There are many non-probabilistic factors that affect the financial market such that the return of risky asset is fuzzy uncertainty. And a number of empirical studies showed the limitations of using probabilistic approaches in characterizing the uncertainty of the financial market. Recently, a number of researchers investigated fuzzy portfolio selection problem. Bellman and Zadeh [12] proposed the basic fuzzy decision theory. Carlsson and Fullér [13] discussed some basic knowledge about possibilistic mean and variance of fuzzy numbers. Arenas and Rodríguez and Bilbao [14] took into
138
Journal of Convergence Information Technology Volume 5, Number 6, August 2010 account three criteria: return, risk and liquidity and used a fuzzy goal programming approach to solve the portfolio selection problem. Huang and Tzeng and Ong [15] discussed a new algorithm for the revised mean-variance model which is proposed to deal with the problem of uncertain portfolio selection. Wang and Zhu [16] gave an overview on the development of fuzzy portfolio selection using fuzzy approaches, quantitative analysis, qualitative analysis, the experts‟ knowledge and manager‟s subjective opinions. Ida [17] considered portfolio selection problem with interval and fuzzy objective function coefficients as a kind of multiple objective problems including uncertainties. And for this problem two kinds of efficient solutions are introduced: possibly efficient solution as an optimistic solution and necessarily efficient solution as a pessimistic solution. Abiyev and Menekay [18] presented fuzzy logic is utilized in the estimation of expected return and risk. Managers could extract useful information and estimate expected return by using not only statistical data, but also economical and financial behaviors of the companies and their business strategies. Bilbao and Pérez and Arenas and Rodríguez [19] discussed a portfolio selection problem using Sharpe‟s single index model in a soft framework. Estimations of subjective or imprecise future beta for every asset can be represented through fuzzy numbers on the basis of statistical data and the relevant knowledge of the financial analysis. Carlsson and Fullér and Heikkilä and Majlender [20] developed a methodology for valuing options on Research and Development (R&D) projects, when future cash flows are estimated by trapezoidal fuzzy numbers. In this paper, a new approach using fuzzy two-stage algorithm in the bi-objective portfolio selection model is presented based on possibility theory. We organize the paper as follows. In Section 2, we recall the notions of possibilistic means, variances and covariances of fuzzy numbers introduced by [13] and introduce the formulae of possibilistic means and variances of linear combination of fuzzy numbers. In Section 3, we propose the bi-objective possibilistic portfolio selection linear programming model. In Section 4, we present the ideas of fuzzy two-stage algorithm for solving multi-objective linear programming model. In Section 5, we use a numerical example of portfolio selection problem with triangular fuzzy numbers to illustrate our proposed effective means and variances and effective fuzzy two-stage algorithm in solving the model.
2. Possibilistic mean and variance and covariance Let us introduce some definitions, which we shall need in the following section. A fuzzy number A is a fuzzy set of the real line R with a normal, fuzzy convex and continuous membership function of bounded support. The family of fuzzy numbers is denoted by F . A level set of a fuzzy number
A is defined by [ A] {t R A(t ) } if 0 and [ A] cl{t R A(t ) 0} (the closure of the support of A ) if 0 . It is well known that if A is a fuzzy number then [ A] is a compact subset of
R for all [0,1] . Let
A and
B be fuzzy numbers with [ A] [a1 ( ), a2 ( )] and
[ B] [b1 ( ), b2 ( )] , for [0,1] . Next, we recall some notions introduced by Carlsson and Fullér [13], we define the possibilistic mean value of A by 1
M ( A) (a1 ( ) a2 ( ))d , 0
(1)
denote the possibilistic variance of A by 1 1 (2) (a2 ( ) a1 ( ))2 d , 20 and define the possibilistic covariance of A and B by 1 1 (3) Cov( A, B) (a2 ( ) a1 ( ))(b2 ( ) b1 ( ))d , 2 0 respectively. The standard deviation of A is defined by A Var ( A) . The following theorems show that the possibilistic mean value and variance of linear combinations of fuzzy numbers can easily be computed in a similar manner as in probability ([13]). Var ( A)
139
A portfolio selection model based on possibility theory using fuzzy two-stage algorithm Xue Deng, Rongjun Li Theorem 2.1. Let A and B be two fuzzy numbers and let , R , then Var ( A B ) 2Var ( A) 2Var ( B ) 2 Cov( A, B ).
Theorem 2.2. Let A1 , A2 ,
, An be n fuzzy numbers and let 0 , 1 , n
n
i 1
i 1
, n R , then
M (0 i Ai ) 0 i M ( Ai ) ,
and n
n
n
i 1
i 1
i j 1
Var (0 i Ai ) i 2Var ( Ai ) 2 i j Cov( Ai , Aj ) . In Theorems 2.1 and 2.2, the addition of fuzzy numbers and the multiplication by a scalar of fuzzy numbers are defined by the sup-min extension principle.
3. Bi-objective possibilistic efficient portfolio selection model In the conventional Markowitz‟s mean-variance model, the return rate of risky asset is denoted as a random variable. It is well known that the returns of risky assets are in a fuzzy uncertain economic environment and vary from time to time, the future states of returns and risk of risky assets cannot be predicted accurately. Fuzzy number is a more powerful tool used to describe an uncertain environment with vagueness and ambiguity. Based on these factors, we consider the portfolio selection problem under the assumption that the returns of assets are fuzzy numbers. For simplicity, we use the following notations: ri denotes the return rate of risky asset i , xi denotes the proportion of total investment funds devoted to the asset i , li and ui (ui li 0) are the lower and upper bounds of xi (i 1, 2, , n) , respectively. Suppose an investor starts with an existing portfolio and considers reallocating his/her wealth among n risky assets and a risk-free asset which is current deposit of bank, the lending interest rate is r0 , and the corresponding proportion of total investment is x0 . Let ri (ai , i , i ) (i 1, 2, , n) be triangular fuzzy numbers with center ai and left width i 0 and right width i 0 . A level set of ri can be denoted as
ri
[ai (1 ) i , ai (1 ) i ] [ i ( ), i ( )] , for all [0,1] , (i 1, 2, , n) . According to (1), (2) and (3), we can easily obtain 1 1 M (ri ) [ai ( ) i ( )]d ai ( i i ), 0 6 and 1 1 1 Var (ri ) [ i ( ) i ( )]2 d ( i i ) 2 , 2 0 24 and 1 1 1 Cov(ri , rj ) [ i ( ) i ( )][ j ( ) j ( )]d (i i )( j j ), 0 2 24 respectively. n
Then the return associated with the portfolio proportion ( x0 , x1 , x2 ,
, xn ) is r r0 x0 ri xi , and i 1
the corresponding possibilistic mean value and possibilistic variance of the return r are given by n n n 1 M (r ) M (r0 x0 ri xi ) r0 x0 xi M (ri ) r0 x0 xi (ai ( i i )) , 6 i 1 i 1 i 1 and
140
Journal of Convergence Information Technology Volume 5, Number 6, August 2010 n
n
n
i 1
i 1
Var (r ) Var (r0 x0 ri xi ) xi 2Var (ri ) 2 xi x j Cov(ri , rj )
i j 1
n
n
1 1 xi x j ( i i )( j j ) xi 2 (i i )2 2 24 i 24 i 1 j 1 2
1 n xi (i i ) . 24 i 1 Analogous to Markowitz‟s mean-variance methodology for the portfolio selection problem, the possibilistic mean value of the return is termed as measure of return and the possibilistic variance of the return is termed as measure of risk. Thus, the possibilistic mean-variance model of portfolio selection problem may be described by the following linear bi-objective optimal problem with equality and inequality constraints: n 1 max M (r ) r0 x0 xi (ai ( i i )) 6 i 1
1 n min Var (r ) xi (i i ) 24 i 1 s.t.
x0 x1 x2
2
xn 1,
(4)
0 x0 1, 0 li xi ui , i 1, 2, , n. Furthermore, the possibilistic mean-variance model (4) is equivalent to the following bi-objective linear programming: n 1 max M (r ) r0 x0 xi (ai ( i i )) 6 i 1 n
min
24Var (r ) xi ( i i ) i 1
s.t. x0 x1 x2
xn 1,
(5)
0 x0 1, 0 li xi ui , i 1, 2,
, n.
The possibilistic mean-variance model (5) is a simple bi-objective linear programming, and we can easily obtain the optimal solution of the possibilistic efficient portfolio model (5) by the following fuzzy two-stage algorithm.
4. Ideas and stages of fuzzy two-stage algorithm 4.1. Ideas of fuzzy two-stage algorithm Consider the following multi-objective linear programming model:
n
a x j 1
ij
j
bi , i 1, 2,
n
j 1
j 1
n
n
j 1
j 1
,Wr ( x )] [ c1 j x j , c2 j x j ,
min W ( x ) [W1 ( x ), W2 ( x ), s.t.
n
, Z q ( x )] [ c1 j x j , c2 j x j ,
max Z ( x ) [ Z1 ( x ), Z 2 ( x ),
n
, cqj x j ] j 1 n
, crj x j ] j 1
(P1)
, m,
0 l j x j u j , j 1, 2,
, n.
By the name of definition (two-stage), the process of solving the problem is divided into two stages. Its basic ideas are as follows: In Stage1, we try to obtain the total satisfactory optimal value (1) and optimal solution x (1) of equivalent problem for the objective set (which is the feasible solution of the original problem (P1)) by
141
A portfolio selection model based on possibility theory using fuzzy two-stage algorithm Xue Deng, Rongjun Li using the „max-min‟ fuzzy algorithm corresponding to logic „and‟. The corresponding mathematical model is Zimmermann minimum operator model. If the optimal solution of the inductive equivalent model is unique, then x (1) is our optimal solution, of course it is also the optimal solution of the original problem. Otherwise, we continue the second stage. In Stage2, we will try to construct a mathematical model maximizing average satisfaction degree for an objective set, and its corresponding additional condition is t (1) , t 1, 2, , q r , namely, every objective satisfaction degree is larger than or equal to the optimal satisfaction degree obtained in Stage1. Obviously, the solution of Stage2 is effective by the total complementation of arithmetic average operator. The constraints t (1) keep the balance among all the objective sets, and it is impossible to abandon any special objective.
4.2. Stages of fuzzy two-stage algorithm The mathematical description of two-stage algorithm is as follows: Stage1: According to the idea of Zimmermann, (P1) is equivalent to the following linear programming problem (P2). We can obtain the maximum satisfaction degree (1) of objective set and the feasible solution x (1) of the original problem by solving the following equivalent problem (P2): max Z ( x ) zk s.t. k , k 1, 2, zk zk w W ( x ) s s , s 1, 2, ws ws [0,1],
, q,
(P2)
, r,
x D.
where
zk max Z k ( x ), zk min Z k ( x ), k 1, 2,
, q,
ws min Ws ( x ), ws max Ws ( x ), s 1, 2,
, r,
xD
xD n
xD
xD
D { aij x j bi , i 1, 2, j 1
, m, 0 l j x j u j , j 1, 2,
, n.}.
O ( zk ( x); ws ( x)) and O ( zk ( x ); ws ( x )) are called the corresponding ideal solutions. In general, we do not know whether the solution of (P2) is unique, hence we can not determine whether x (1) is the optimal solution of the original problem. Effectiveness of x (1) must be verified by the following problem (P3). Stage2: Verifying the effectiveness of x (1) or looking for a new effective solution x ( 2) : q r 1 max [ k s ] q r k 1 s 1 Z ( x ) zk s.t. (1) k k , k 1, 2, zk zk
, q,
ws Ws ( x ) , s 1, 2, ws ws
, r,
(1) s
(P3)
k , s [0,1], x D. We can easily obtain the solution x ( 2) . And we can prove that x ( 2) is also the solution of linear programming (P2). Hence, if the solution of the linear programming (P2) is unique, then we must have x (2) x (1) . In this time, x (1) is the optimal solution to the original problem, and the work of Stage2 is only to check the fact. If the solution of (P2) is not unique, i.e. x (1) x (2) , in this situation, x (1) may
142
Journal of Convergence Information Technology Volume 5, Number 6, August 2010 be an effective solution or may be not an effective solution, but anyway, x ( 2) is always an effective solution. Thus, under any circumstance, the two-stage algorithm will insure to obtain an effective solution of the original problem in Stage2.
5. Numerical example In order to illustrate our proposed effective means and variances of the efficient portfolio in this paper, we considered a real portfolio selection example. In this example, we selected five stocks from Shanghai Stock Exchange, their returns ri , i 1, 2, ,5 are regarded as triangular fuzzy numbers. Based on the historical data, the future information, the advices of experts, the corporations‟ financial reports and the managers‟ subjective opinions, we can estimate their returns with the following possibility distributions: r1 (0.073,0.054,0.087), r2 ( 0 . 1 0 5 , 0 . 0 7 5 , 0 . 1 0 2 r)3, (0.138,0.096,0.123),
r4 ( 0 . 1 6 8 , 0 . 1 2 6 , 0 . 1 6 2 ) r,5 (0.208,0.168,0.213), and therefore, the level sets of ri , i 1, 2, ,5 are given by [r1 ] [0.019 0.054 , 0.160 0.087 ], [r2 ] [0.030 0.075 , 0.207 0.102 ], [r3 ] [0.042 0.096 , 0.261 0.123 ], [r4 ] [0.042 0.126 , 0.330 0.162 ], [r5 ] [0.040 0.168 , 0.421 0.213 ].
We selected a risk-free asset which is current deposit of bank with the lending interest rate r0 is 2%. The lower bound and upper bound vectors of x ( x0 , x1 , x2 , x3 , x4 , x5 ) are given by l (0,0.1,0.1,0.1,0.1,0.1) and u (1,0.7,0.7,0.8,0.8,0.9) , respectively. We use prime ( ' ) to denote matrix transposition and all non-primed vectors are column vectors for convenience. Thus, using model (5), we can obtain the bi-objective linear programming model based on the possibilistic mean-variance theory: max 0.02x0 0.0785 x1 0.1095 x2 0.1425 x3 0.1740 x4 0.2155 x5
min 0.1410x1 0.1770x2 0.2190x3 0.2880x4 0.3810x5 s.t. x0 x1 x2 x3 x4 x5 1, 0 x0 1, 0.1 x1 0.7,
(M1)
0.1 x2 0.7, 0.1 x3 0.8, 0.1 x4 0.8, 0.1 x5 0.9. By the fuzzy two-stage algorithm, we obtain the following ideal solutions: O ( z1 ( x ); w1 ( x)) (0.179750;0.120600), and O ( z1 ( x ); w1 ( x)) (0.082000;0.311100). Stage1: Model (1) is changed into the following Model (2):
143
A portfolio selection model based on possibility theory using fuzzy two-stage algorithm Xue Deng, Rongjun Li max s.t. (0.02x0 0.0785 x1 0.1095 x2 0.1425 x3 0.1740 x4 0.2155 x5 0.082) /(0.17975 0.082),
(0.3111 0.1410x1 0.1770x2 0.2190x3 0.2880x4 0.3810x5 ) /(0.3111 0.1206), x0 x1 x2 x3 x4 x5 1, 0 x0 1,
(M2)
0.1 x1 0.7, 0.1 x2 0.7, 0.1 x3 0.8, 0.1 x4 0.8, 0.1 x5 0.9,
[0,1]. We obtain the following optimal solution: (1) 0.516500, x (1) (0.079421, 0.1, 0.1, 0.520579, 0.1, 0.1), and the optimal mean and variance of return are M *(1) 0.133521, Var*(1) 0.001885 .
Stage2: In order to check the effectiveness of x (1) , we need to solve the following Model (3): max =
1 2
2 s.t. 0.516500 1 (0.02x0 0.0785 x1 0.1095 x2 0.1425 x3 0.1740 x4 0.2155 x5 0.082) /(0.17975 0.082), 0.516500 2 (0.3111 0.1410x1 0.1770x2 0.2190x3 0.2880x4 0.3810x5 ) /(0.3111 0.1206), x0 x1 x2 x3 x4 x5 1,
(M3)
0 x0 1, 0.1 x1 0.7, 0.1 x2 0.7, 0.1 x3 0.8, 0.1 x4 0.8, 0.1 x5 0.9,
1 , 2 [0,1].
Then we can obtain the following optimal solution: 1 0.516500, 2 0.516500, 0.516500 (1) , x (2) (0.079421, 0.1, 0.1, 0.520579, 0.1, 0.1), M *(2) 0.133521, Var*(2) 0.001885 .
It is obvious that x (1) x (2) which means that the effective solution is unique. That is, the solution x obtained in Stage1 is effective and optimal for the original problem (M1). (1)
6. Conclusions In today‟s extremely competitive business environment, investors have already invested in various assets to keep their competitive advantage. In practical situations, regarding the expected return and the risk as two objective functions, we can propose a bi-objective linear programming model based on possibilistic mean and variance with triangular fuzzy numbers of return rate. Naturally, the fuzzy twostage algorithm can be utilized to solve our proposed model. The obtained results from the numerical example show that our proposed model is reasonable, and the two-stage algorithm is effective, and the optimal solution must be found for the original problem. Our proposed possibilistic approaches to selecting portfolios can better describe an uncertain decision problem with vagueness and ambiguity.
144
Journal of Convergence Information Technology Volume 5, Number 6, August 2010
7. Acknowledgements This research was supported by Guangdong Science and Technology Department Soft Science Sponsor Item, China (No. 2008B070800012) and Student Research Planning, South China University of Technology, China (No. Y1070180). The authors are also grateful to a referee for his/her very helpful comments and suggestions.
8. References [1] H. Markowitz, “Portfolio selection”, Journal of Finance, vol. 7, pp. 77¬-91, 1952. [2] H. Markowitz, Portfolio Selection: Efficient Diversification of Investment, Wiley, New York, 1959. [3] H. Markowitz, Analysis in Portfolio Choice and Capital Markets, Basil Blackwell, Oxford, 1987. [4] W. F. Sharpe, Portfolio Theory and Capital Markets, McGraw-Hill, New York, 1970. [5] R.C. Merton, “An analytic derivation of the efficient frontier”, Journal of Finance and Quantitative Analysis, vol. 9, pp. 1851-1872, 1972. [6] A.F. Perold, “Large-scale portfolio optimization”, Management Science, vol. 30, pp. 1143-1160, 1984. [7] J.S. Pang, “A new efficient algorithm for a class of portfolio selection problems”, Operational Research, vol. 28, pp. 754-767, 1980. [8] J. Vörös, “Portfolio analysis-An analytic derivation of the efficient portfolio frontier”, European Journal of Operational Research, vol. 203, pp. 294-300, 1986. [9] M.J. Best, J. Hlouskova, “The efficient frontier for bounded assets”, Mathematical Methods of Operations Research, vol. 52, pp. 195-212, 2000. [10] M. Stein, J. Branke, H. Schmeck, “Efficient implementation of an active set algorithm for largescale portfolio selection”, Computers & Operations Research, vol. 35, pp. 3945-3961, 2008. [11] W. Yan, R. Miao, S.R. Li, “Multi-period semi-variance portfolio selection: Model and numerical solution”, Applied Mathematics and Computation, vol. 194, pp. 128-134, 2007. [12] R. Bellman, L.A. Zadeh, “Decision making in a fuzzy environment”, Management Science, vol. 17, pp. 141-164, 1970. [13] C.Carlsson, R. Fullér, “On possibilistic mean and variance of fuzzy numbers”, Fuzzy sets and Systems, vol. 122, pp. 315-326, 2001. [14] M. Arenas, A. Bilbao, M.V. Rodríguez, “A fuzzy goal programming approach to portfolio selection”, European Journal of Operational Research, vol. 133, pp. 287-297, 2001. [15] J.J. Huang, G.H. Tzeng, C.S. Ong, “A novel algorithm for uncertain portfolio selection”, Applied Mathematics and Computation, vol. 173, pp. 350-359, 2006. [16] S.Y. Wang, S.S. Zhu, “On fuzzy portfolio selection problems”, Fuzzy Optimization and Decision Making, vol. 1, pp. 361-377, 2002. [17] M. Ida, “Solutions for the portfolio selection problem with interval and fuzzy coefficients”, Reliable Computing, vol. 10, pp. 389-400, 2004. [18] R.H. Abiyev, M. Menekay, “Fuzzy portfolio selection using genetic algorithm”, Soft Comput vol. 11, pp. 1157-1163, 2007. [19] A. Bilbao, B. Pérez, M. Arenas, M.V. Rodríguez, “Fuzzy compromise programming for portfolio selection”, Applied Mathematics and Computation, vol. 173, pp. 251-264, 2006. [20] C.Carlsson, R. Fullér, M. Heikkilä, P. Majlender, “A fuzzy approach to R&D project portfolio selection”, International Journal of Approximate Reasoning, vol. 44, pp. 93-105, 2007. [21] H.J. Zimmermann, “Description and optimization of fuzzy systems”, International Journal of General Systems, vol. 2, pp. 209-215, 1976. [22] R.J. Li, E. Stanly Lee, “Fuzzy approaches to multicriteria de novo programs”, Mathematical Analysis and Applications, vol. 153, pp. 97-111, 1990. [23] A. Gundel, S. Weber, “Utility maximization under a shortfall risk constraint”, Journal of mathematical Economics, vol. 44, pp. 1126-1151, 2008. [24] M.R. Haley, “A simple nonparametric approach to low-dimension, shortfall-based portfolio selection”, Finance Research Letters, vol. 5, pp. 183-190, 2008.
145
