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Fuzzy Inf. Eng. (2009)2:115-127 DOI 10.1007/s12543-009-0010-4 ORIGINAL ARTICLE Weighted Portfolio Selection Models Based on Possibility Theory Wei Chen Received: 19 March 2009/ Revised: 9 May 2009/ Accepted: 15 May 2009/ © Springer and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, we discuss portfolio selection problem in a fuzzy uncertain environment. Based on the Fuller’ ´ s and Zhang’s notations, we discuss some proper- ties of weighted lower and upper possibilistic means and variances as in probability theory. We further present two weighted possibilistic portfolio selection models with bounded constraint, which can be transformed to linear programming problems un- der the assumption that the returns of assets are trapezoidal fuzzy numbers. At last, a numerical example is given to illustrate our proposed eﬀective means and approaches. Keywords Portfolio selection · Weighted possibilistic mean · Weighted possibilis- tic variance · Linear programming 1. Introduction The mean-variance methodology for the portfolio selection problem, proposed origi- nally by Markowitz [1,2], 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. With the continuous eﬀort of var- ious researchers, Markowitz’s seminal work has been widely extended. But, most existing portfolio selection models (see [3,4,5,6]) are based on the probability distri- butions, in which uncertainty is equated with randomness. Though probability theory is one of the main techniques used for analyzing un- certainty in ﬁnance, the ﬁnancial market is also aﬀected by several non-probabilistic factors such as vagueness and ambiguity. Decision makers are commonly provided with information which is characterized by linguistic descriptions such as high risk, low proﬁt, high interest rate, etc. [7]. Fuzzy set theory proposed by Zadeh [8] in 1965, has become a helpful tool in integrating the experts’ knowledge and investors’ sub- jective opinions into a portfolio selection problem. Recently, a number of researchers Wei Chen () School of Information, Capital University of Economics and Business, Beijing 100070, P.R.China e-mail: cwcw2001@163.com 116 Wei Chen (2009) have focused on the fuzzy portfolio selection problem. Watada [9], Inuiguchi [10], Wang et al [11] and Ramaswamy [12] discussed portfolio selection by using fuzzy decision theory. Tanaka and Guo [13,14] proposed two kinds of portfolio selection models based on fuzzy probabilities and exponential possibility distributions, respec- tively. Carlsson and Fuller ´ [15] introduced the notions of lower and upper possibilistic mean values of a fuzzy number, then proposed a possibilistic approach to selecting portfolios with highest utility score in [16]. Ammar [17] solved the fuzzy portfolio optimization problem as a convex quadratic programming problem and provided an acceptable solution to it. Arenas Parra [18] proposed a fuzzy goal programming ap- proach to solve portfolio selection problem. Bilbao Terol [19] formulated a fuzzy compromise programming method to solve a portfolio selection problem. Deng [20] proposed a new minimax model on an optimal portfolio selection with uncertainty of both randomness and estimation in inputs. Gupta [21] incorporated fuzzy set theory into a semi-absolute deviation portfolio selection model. Lacagnina and Pecorella [22] developed a multistage stochastic soft constraints fuzzy program with recourse in order to capture both uncertainty and imprecision as well as to solve a portfolio management problem. Huang [23,24] studied portfolio selection in a random fuzzy environment in which the security returns are stochastic variables with fuzzy infor- mation. Recently, Huang proposed mean-semivariance models in [25] and a new type of optimization model base on the new deﬁnition of risk for portfolio selection [26]. Zhang [27,28,29] discussed the portfolio selection problem based on the lower, upper and crisp possibilistic means and possibilistic variances introduced in [30]. Chen [31] discussed a possibilistic portfolio selection problem with investing constrains. In this paper, we will discuss a portfolio selection problem with bounded constraint based on the possibilistic theory. The rest of the paper is organized as follows. Some properties as in probability theory based on the Fuller’ ´ s and Zhang’s notations are discussed in Section 2. In Section 3, two weighted possibilistic portfolio models with bounded constraint are presented based on the weighted lower and upper possibilistic means and variances. To better illustrate the modelling idea of the paper, a numerical example is given in Section 4. Finally, we conclude this paper in Section 5. 2. Weighted Lower and Upper Possibilistic Means and Variances Let us introduce some deﬁnitions for the need of the following section. A fuzzy number A is a fuzzy set of the real lineR with a normal, fuzzy convex and continuous membership function of bounded support. The family of fuzzy numbers is denoted by F . Moreover, a function f :[0, 1] →R is said to be a weighting function if f is a non-negative, monotone increasing and satisﬁes the normalization condition f (γ)dγ = 1. Fuller ´ and Majlender [32] deﬁned the notations of the f− weighted lower and up- per possibilistic mean values of fuzzy number A as a (γ) f (Pos A a (γ) )dγ 1 1 M (A) = a (γ) f (γ)dγ = , f (Pos A a (γ) )dγ 0 Fuzzy Inf. Eng. (2009) 2: 115-127 117 a (γ) f (Pos A a (γ) )dγ 2 2 M (A) = a (γ) f (γ)dγ = , f (Pos A a (γ) )dγ where Pos denotes possibility, i.e. Pos A a (γ) =Π((−∞, a (γ)]) = γ, 1 1 Pos A a (γ) =Π([a (γ),∞)) = γ. 2 2 The following lemma can be easily obtained by using the deﬁnitions of weighted lower and upper possibilistic means. Lemma 1 Let A , A ,..., A be n fuzzy numbers, and let λ be a real number. Then 1 2 n ⎛ ⎞ n n ⎜ ⎟ ⎜ ⎟ L ⎜ ⎟ L ⎜ ⎟ M ⎜ A ⎟ = M (A ), i i f ⎝ ⎠ f i=1 i=1 ⎛ ⎞ n n ⎜ ⎟ ⎜ ⎟ U ⎜ ⎟ U ⎜ ⎟ M ⎜ A ⎟ = M (A ), i i ⎝ ⎠ f f i=1 i=1 ⎪ λM (A ) if λ 0, ⎨ i L f M (λA ) = f ⎪ λM (A ) ifλ< 0, λM (A ) if λ 0, U f ( ) M λA = i ⎪ ⎪ L λM (A ) ifλ< 0, where the addition and multiplication by a scalar of fuzzy numbers are deﬁned by the sup-min extension principle [8]. Theorem 1 obviously holds by Lemma 1. Theorem 1 Let A , A ,..., A be n fuzzy numbers and let λ ,λ ,λ ,...,λ be n+ 1 1 2 n 0 1 2 n real numbers. Then ⎛ ⎞ n n ⎜ ⎟ ⎜ ⎟ L ⎜ ⎟ L ⎜ ⎟ M λ + λ A = λ + |λ|M (s (λ ) A ), ⎜ ⎟ 0 i i 0 i i i f ⎝ ⎠ f i=1 i=1 ⎛ ⎞ n n ⎜ ⎟ ⎜ ⎟ U ⎜ ⎟ U ⎜ ⎟ M λ + λ A = λ + |λ|M (s (λ ) A ), ⎜ ⎟ 0 i i 0 i i i f ⎝ ⎠ f i=1 i=1 where s(x) is a sign function of x∈R. Especially, if λ ,λ ,...,λ be n nonnegative real numbers, then 1 2 n n n L L M ( λ A +λ ) = λ + λ M (λ A ), i i 0 0 i i i f f i=1 i=1 n n U U M ( λ A +λ ) = λ + λ M (λ A ). i i 0 0 i i i f f i=1 i=1 118 Wei Chen (2009) Corresponding with the weighted lower and upper possibilistic means, Zhang [33] introduced weighted lower and upper possibilistic variances and covariances of fuzzy numbers as follows L L Var (A)= M (A)− a (γ) f (γ)dγ f f 1 2 (M (A)− a (γ)) f (Pos A a (γ) )dγ 1 1 = , f (Pos A a (γ) )dγ U U Var (A)= M (A)− a (γ) f (γ)dγ f f (M (A)− a (γ)) f (Pos A a (γ) )dγ 2 2 = , f (Pos A a (γ) )dγ L L L Cov (A) = (M (A)− a (γ))(M (B)− b (γ)) f (γ)dγ, 1 1 f f f U U U Cov (A) = (M (A)− a (γ))(M (B)− b (γ)) f (γ)dγ, 2 2 f f f γ γ where [A] = a (γ), a (γ) ,[B] = b (γ), b (γ) . 1 2 1 2 According to the above deﬁnitions we can get the following two Lemmas easily. Lemma 2 Let A be a fuzzy number, and let θ∈R be a real numbers. Then L L U U Var (A+θ) = Var (A), Var (A+θ) = Var (A). f f f f Lemma 3 Let A, B be two fuzzy numbers, and let λ∈R be a real number. Then L L U U Cov (A, B) = Cov (B, A), Cov (A, B) = Cov (B, A), f f f f L L U U Cov (A, A) = Var (A), Cov (A, A) = Var (A), f f f f L U Cov (A,λ) = 0, Cov (A,λ) = 0. f f The following theorems show some important properties of the weighted lower and upper possibilistic variances and covariances. Theorem 2 Let A, B be two fuzzy numbers, and let λ∈R be a real numbers. Then L L L L Var (A+ B) = Var (A)+ Var (B)+ 2Cov (A, B), f f f f U U U U Var (A+ B) = Var (A)+ Var (B)+ 2Cov (A, B), f f f f 2 L λ Var (A) if λ 0, L f Var (λA) = ⎪ 2 U λ Var (A) ifλ< 0, 2 U ⎪ λ Var (A) if λ 0, U f Var (λA) = f ⎪ 2 L λ Var (A) ifλ< 0. γ γ Proof Suppose [A] = a (γ), a (γ) and [B] = b (γ), b (γ) , then 1 2 1 2 [A+ B] = a (γ)+ b (γ), a (γ)+ b (γ) . 1 1 2 2 Fuzzy Inf. Eng. (2009) 2: 115-127 119 From deﬁnitions of weighted lower and upper possibilistc variances, we get that L L Var (A+ B)= M (A+ B)− a (γ)− b (γ) f (γ))dγ 1 1 f f L L = (M (A)− a (γ))+ (M (B)− b (γ)) f (γ))dγ 1 1 0 f f L L L = Var (A)+ Var (B)+ 2Cov (A, B) f f f and U U Var (A+ B)= M (A+ B)− a (γ)− b (γ) f (γ))dγ 2 2 f 0 f U U = (M (A)− a (γ))+ (M (B)− b (γ)) f (γ))dγ 2 2 f f U U U = Var (A)+ Var (B)+ 2Cov (A, B). f f f Moreover, from Lemma 1 and λa (γ),λa (γ) , ifλ 0, ⎨ 1 2 γ γ [λA] = λ[A] = λ a (γ), a (γ) = 1 2 ⎪ λa (γ),λa (γ) , ifλ< 0, 2 1 we get that for λ 0 L L Var (λA)= M (λA)−λa (γ) f (γ))dγ f f = λ(M (A)− a (γ)) f (γ))dγ 0 f 2 L = λ Var (A) and forλ< 0 L L Var (λA)= M (λA)−λa (γ) f (γ))dγ f 0 f = λ(M (A)− a (γ)) f (γ))dγ 2 U = λ Var (A). Similarly, we easily prove that 2 U ⎪ λ Var (A), ifλ 0, U f Var (λA) = f ⎪ 2 L λ Var (A), ifλ< 0. This completes the proof. Theorem 3 Let A, B be two fuzzy numbers, and letλ,μ,θ∈R be three real numbers. Then L 2 L 2 L L Var (λA+μB+θ) = λ Var (s(λ)A)+μ Var (s(μ)B)+ 2|λμ|Cov (s(λ)A, s(μ)B) f f f f U 2 U 2 U U Var (λA+μB+θ) = λ Var (s(λ)A)+μ Var (s(μ)B)+ 2|λμ|Cov (s(λ)A, s(μ)B) f f f f where s(x) is a sign function of x∈R. Proof Supposeλ< 0,μ < 0, then from Lemma 2, Lemma 3 and Theorem 2, we get 120 Wei Chen (2009) L L L L Var (λA+μB+θ)= Var (λA)+ Var (μB)+ 2Cov (λA,μB) f f f f 2 U 2 U U = λ Var (A)+μ Var (B)+ 2λμCov (A, B) f f f 2 L 2 L L = λ Var (−A)+μ Var (−B)+ 2λμCov (−A,−B) f f f 2 L 2 L L = λ Var (s(λ)A)+μ Var (s(μ)B)+ 2|λμ|Cov (s(λ)A, s(μ)B), f f f where 1, if x > 0, s(x) = ⎪ 0, if x = 0, −1, if x < 0. Similar reasoning holds for the case λ 0,μ 0. Now suppose thatλ> 0,μ< 0, then we get L L L L Var (λA+μB+θ)= Var (λA)+ Var (μB)+ 2Cov (λA,μB) f f f f 2 L 2 U L = λ Var (A)+μ Var (B)− 2λμCov (A,−B) f f f 2 L 2 L L = λ Var (A)+μ Var (−B)− 2λμCov (A,−B) f f f 2 L 2 L L Var (s(λ)A)+μ Var (s(μ)B)+ 2|λμ|Cov (s(λ)A, s(μ)B). = λ f f f Similar reasoning holds for the caseλ< 0,μ> 0, which ends the proof. From Theorem 3, we easily get the following conclusion. Theorem 4 Let A , A ,..., A be n fuzzy numbers, and let λ ,λ ,...,λ be n real 1 2 n 1 2 n numbers. Then ⎛ ⎞ n n n ⎜ ⎟ ⎜ ⎟ L ⎜ ⎟ 2 L L ⎜ ⎟ Var λ A = λ Var (A )+ 2 |λλ |Cov (s(λ )A, s(λ )A ), ⎜ ⎟ i i i i j i i j j f ⎝ ⎠ i f f i=1 i=1 i< j=1 ⎛ ⎞ n n n ⎜ ⎟ ⎜ ⎟ U ⎜ ⎟ 2 U U ⎜ ⎟ Var λ A = λ Var (A )+ 2 |λλ |Cov (s(λ )A, s(λ )A ), ⎜ ⎟ i i i i j i i j j f ⎝ ⎠ i f f i=1 i=1 i< j=1 where s(x) is a sign function of x∈R. 3. Model Formulation In this section, we formulate weighted lower and upper possibilistic portfolio selec- tion models. Let x be the proportion invested in asset j, and r be the random return j j rate of asset j, j = 1, 2,..., n. In order to describe conveniently, we introduce the following notations: x = (x , x ,..., x ) , 1 2 n r = (r , r ,..., r ) , 1 2 n L L L L M = (M (r ), M (r ),..., M (r )) , 1 2 n f f f f U U U U M = (M (r ), M (r ),..., M (r )) , 1 2 n f f f f L L Cov = (Cov (r, r )) , i j n×n f f U U Cov = (Cov (r, r )) , i j n×n f f e = (1, 1,..., 1) , Fuzzy Inf. Eng. (2009) 2: 115-127 121 L U L U M and M are weighted lower and upper possibilistic mean vector; Cov and Cov f f f f are weighted lower and upper possibilistic convariance matrix. It is well-known that the ﬁnancial market is aﬀected by many non-probabilistic factors. In a fuzzy uncertain economic environment, the future states of returns and risks of risky assets cannot be predicted accurately. Possibility theory has been pro- posed by Zadeh [8] and advanced by Dubois and Prade [34]. In this theory, fuzzy variables are associated with possibility distributions. In many important cases, it might be easier to estimate the possibility distributions of rates of return on risky as- sets, rather than the corresponding probability distributions. Based on these facts, we discuss the portfolio selection problem under the assumption that the returns of assets r, i = 1,..., n are n fuzzy numbers. From Theorem 1, the weighted lower and upper possibilistic means of the return associated with the portfolio (x , x ,..., x ) are given by 1 2 n n n L L L L M (r x) = M (x r ) = x M (r ) = M x, i i i i f f f f i=1 i=1 n n U U U U M (r x) = M (x r ) = x M (r ) = M x. i i i i f f f f i=1 i=1 From Theorem 4, the weighted lower and upper possibilistic variances of the return associated with the portfolio (x , x ,..., x ) are given by 1 2 n n n L 2 L L L Var (r x) = x Var (r )+ 2 x x Cov (r, r ) = x Cov x, i i j i j f i f f f i=1 i< j=1 n n U 2 U U U Var (r x) = x Var (r )+ 2 x x Cov (r, r ) = x Cov x. i i j i j f f f f i=1 i< j=1 Analogous to Markowitz’s mean-variance methodology for the portfolio selection problem, the weighted lower and upper possibilistic means correspond to the return, while the weighted lower and upper possibilistic variances correspond to the risk. Moreover, in order to decrease a risk, there exist usually the limits on maximum holdings in a realistic portfolio management problem. So the lower and upper bounds constraints on each asset would be useful for the investors to select portfolios in real world. Therefore, the weighted lower and upper possibilistic portfolio selection models can be formulated as min x Cov x s.t. M x μ, (1) e x 1, l x u, 122 Wei Chen (2009) and min x Cov x s.t. M x μ. (2) e x 1, l x u, where l = (l , l ,..., l ) , u = (u , u ,..., u ) are lower and upper bound constraint 1 2 n 1 2 n vectors. The weighted lower and upper possibilistic eﬃcient portfolios for all possible μ construct the lower and upper possibilistic eﬃcient frontiers. Solving (1) and (2) for all possible μ, the weighted lower and upper possibilistic eﬃcient frontier is derived explicitly. It should be noted that we used weighted lower and upper possibilistic means, variances and covariances to replace the probabilistic means, variances and covari- ances in Markowitz’s mean-variance model, respectively. Moreover, according to the deﬁnitions of lower and upper variances, we get that the weighted lower possibilistic variance Var (A) describes the weighted average of lower possibility of the squared deviations between the left limit and the lower possibilistic mean of level sets of A, and the weighted upper possibilistic variance Var (A) describes the weighted av- erage of upper possibility of the squared deviations between the right limit and the upper possibilistic mean of level sets of A. Therefore, based on the above facts, we conclude that the weighted lower possibilistic variance dosen’t necessarily imply a lower risk, and the weighted upper one dosen’t not necessarily imply upper risk. In the following, we assume r = (a , b ,α ,β )(j=1,. . . ,n) to be n trapezoidal j j j j j numbers, thus a γ− level sets of r is [r ] = a − (1−γ)α , b + (1−γ)β ,∀γ ∈ j j j j j j [0, 1]. Moreover, we assume weighting function to be f (γ) = (m + 1)γ . By simple computation, we can obtain M (r ) = a − , i i m+ 2 M (r ) = b + , i i m+ 2 ⎡ ⎤ ⎢ ⎥ ⎢ m+ 1 m+ 1 ⎥ L ⎢ ⎥ 2 ⎢ ⎥ Var (r ) = ⎢ − ⎥ α , f ⎣ ⎦ i m+ 3 m+ 2 ⎡ ⎤ ⎢ ⎥ ⎢ m+ 1 m+ 1 ⎥ U ⎢ ⎥ 2 ⎢ ⎥ Var (r ) = − β , ⎢ ⎥ f ⎣ ⎦ i m+ 3 m+ 2 ⎡ ⎤ ⎢ ⎥ m+ 1 m+ 1 ⎢ ⎥ L ⎢ ⎥ ⎢ ⎥ Cov (r, r ) = ⎢ − ⎥ αα , i j i j f ⎣ ⎦ m+ 3 m+ 2 ⎡ ⎤ ⎢ ⎥ ⎢ m+ 1 m+ 1 ⎥ U ⎢ ⎥ ⎢ ⎥ Cov (r ) = ⎢ − ⎥ ββ . i i j ⎣ ⎦ m+ 3 m+ 2 Therefore, the weighted lower and upper possibilistic mean-variance model (1) Fuzzy Inf. Eng. (2009) 2: 115-127 123 and (2) can be formulated as ⎡ ⎤ ⎡ ⎤ n n ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ m+ 1 m+ 1 ⎥ ⎢ ⎥ ⎢ ⎥ 2 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ min ⎢ − ⎥ x α + 2 x x αα ⎢ i j i j⎥ ⎣ ⎦ i i ⎣ ⎦ m+ 3 m+ 2 i=1 i j=1 (3) s.t. x a − μ, i i m+ 2 i=1 x + x +...+ x 1, 1 2 n l x u, i = 1, 2,..., n i i i and ⎡ ⎤ ⎡ ⎤ 2 n n ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ m+ 1 m+ 1 ⎥ ⎢ ⎥ ⎢ ⎥ 2 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ min − x β + 2 x x ββ ⎢ ⎥ ⎢ i j i j⎥ ⎣ ⎦ i i ⎣ ⎦ m+ 3 m+ 2 i=1 i j=1 s.t. x b + μ, (4) i i m+ 2 i=1 x + x +...+ x 1, 1 2 n l x u, i = 1, 2,..., n, i i i respectively. Further, model (3) and (4) are equivalent to the following linear programming problems: min xα i i i=1 s.t. x a − μ, (5) i i m+ 2 i=1 x + x +...+ x 1, 1 2 n l x u, i = 1, 2,..., n i i i and min xβ i i i=1 s.t. x b + μ, (6) i i m+ 2 i=1 x + x +...+ x 1, 1 2 n l x u, i = 1, 2,..., n. i i i Model (5) and (6) are two standard linear programming problems. One can use several algorithms of linear programming to solve them eﬃciently, for example, the simplex method. It should be noted that weighted lower and upper possibilistic mean-variance mod- els will change when the possibility distributions of the returns change. For example, 124 Wei Chen (2009) if r = (a , b ,α ) is a symmetric trapezoidal fuzzy number, that is α = β , then j j j j j j the objective function in the weighted lower possibilistic mean-variance model (5) is equal to that in the upper possibilistic mean-variance model (6). Moreover, if r = (a ,α ,β ) is a triangular fuzzy number, that is a = b , both of two models are j j j j j j still linear programming problems. 4. Numerical Example In this section, we consider a numerical example in [31], ﬁve stocks from Shenzhen Stock Exchange was selected, and their possibility distributions of the return as fol- lows, r = (0.073, 0.093, 0.054, 0.087), r = (0.085, 0.115, 0.075, 0.102), 1 2 r = (0.108, 0.138, 0.096, 0.123), r = (0.128, 0.168, 0.126, 0.162), 3 4 r = (0.158, 0.208, 0.168, 0.213). Here, we assume that the weighted function f (γ) = 3γ . Then, the lower and upper possibilistic portfolio models are, respectively, given by min 0.054x + 0.075x + 0.096x + 0.126x + 0.166x 1 2 3 4 5 s.t. 0.0595x + 0.0663x + 0.084x + 0.0965x + 0.116x μ, 1 2 3 4 5 (7) x + x + x + x + x 1, 1 2 3 4 5 l x u, i = 1, 2,..., 5 i i i and min 0.087x + 0.102x + 0.123x + 0.162x + 0.213x 1 2 3 4 5 s.t. 0.1147x + 0.1405x + 0.1688x + 0.2085x + 0.2612x μ, 1 2 3 4 5 (8) x + x + x + x + x 1, 1 2 3 4 5 l x u, i = 1, 2,..., 5, i i i where (l , l , l , l , l ) = (0, 0.1, 0, 0, 0.2), and (u , u , u , u , u ) = (0.5, 0.5, 0.4, 0.8, 1 2 3 4 5 1 2 3 4 5 0.8). Since model (7) and (8) are simple linear programming problems, we can solve them easily by software LINGO. At last, we obtain weighted lower and upper pos- sibilistic eﬃcient portfolios as shown in Tables 1-2. All eﬃcient portfolios contain security 2 and security 5. Table 1: weighted lower possibilistic eﬃcient portfolios μ x x x x x Risk 1 2 3 4 5 [0, 0.073) 0.5 0.3 0 0 0.2 0.0827 0.073 0.5 0.291 0.009 0 0.2 0.0829 0.074 0.5 0.2345 0.0655 0 0.2 0.0841 0.075 0.5 0.178 0.122 0 0.2 0.0853 0.080 0.3523 0.1 0.3477 0 0.2 0.0930 0.095 0 0.1 0.4 0.1656 0.3344 0.1223 0.105 0 0.1 0.1884 0 0.7116 0.1437 Fuzzy Inf. Eng. (2009) 2: 115-127 125 Table 2: weighted upper possibilistic eﬃcient portfolios μ x x x x x Risk 1 2 3 4 [0, 0.152) 0.5 0.3 0 0 0.2 0.1167 0.152 0.49 0.31 0 0 0.2 0.1169 0.155 0.3736 0.4264 0 0 0.2 0.1186 0.165 0.1503 0.5 0.1497 0 0.2 0.1251 0.190 0 0.1935 0.4 0.2065 0.2 0.1450 0.210 0 0.1 0.4 0.0412 0.4588 0.1628 0.240 0 0.1 0.0972 0.0028 0.8 0.1930 Table 1, representing the weighted lower possibilistic eﬃcient portfolio, shows that the eﬃcient portfolio is composed of three securities for μ in the interval [0, 0.073) and the minimal possibilistic risk is 0.0827. For all the weighted lower possibilistic eﬃcient portfolios, the portfolio with the minimal risk 0.0827 denotes (x , x , x , x , 1 2 3 4 x ) = (0.5, 0.3, 0, 0, 0.2) and portfolio with the maximal risk 0.1437 represents (x , x , 5 1 2 x , x , x ) = (0, 0.1, 0.1884, 0, 0.7116). 3 4 5 Table 2, representing the weighted upper possibilistic eﬃcient portfolio, shows that the eﬃcient portfolio is composed of three securities for μ in the interval [0, 0.152] and the minimal possibilistic risk is 0.1167. For all the weighted upper possibilistic eﬃcient portfolios, the portfolio with the minimal risk 0.1167 is (x , x , x , x , x ) = 1 2 3 4 5 (0.5, 0.3, 0, 0, 0.2) and portfolio with the maximal risk 0.1930 is (x , x , x , x , x ) = 1 2 3 4 5 (0, 0.1, 0.0972, 0.0028, 0.8). 5. Conclusions This paper deals with a portfolio selection problem with fuzzy returns. Based on the Fuller’ ´ s and Zhang’s notations, we have discussed some properties of weighted lower and upper possibilistic means and variances as in probability theory. We have proposed weighted lower and upper possibilistic mean-variance models for bounded assets, which can be transformed to the linear programming problems under the as- sumption that the returns of assets are trapezoidal fuzzy numbers. At last, we have found that our proposed approach is eﬀective based on the numerical results. Acknowledgments I thank the Editor-in-Chief and referees for their valuable comments and suggestions. This research is supported by the Social Science Research Project of Beijing Munic- ipal Education Commission of China (No. SM200910038005). References 1. Markowitz H (1952) Portfolio selection. Journal of Finance 7:77-91 2. Markowitz H (1959) Portfolio selection: Eﬃcient Diversiﬁcation of Investments. Wiley, New York 3. Sharpe WF (1970) Portfolio theory and capital markets. 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Fuzzy Information and Engineering – Taylor & Francis
Published: Jun 1, 2009
Keywords: Portfolio selection; Weighted possibilistic mean; Weighted possibilistic variance; Linear programming
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