Portfolio selection based on upper and lower exponential possibility distributions

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1 European Journal of Operational Research 114 (1999) 115±126 Theory and Methodology Portfolio selection based on upper and lower exponential possibility distributions Hideo Tanaka *, Peijun Guo Department of Industrial Engineering, Osaka Prefecture University, Gakuencho 1-1, Sakai, Osaka , Japan Received 24 June 1997; accepted 11 November 1997 Abstract In this paper, two kinds of possibility distributions, namely, upper and lower possibility distributions are identi ed to re ect experts' knowledge in portfolio selection problems. Portfolio selection models based on these two kinds of distributions are formulated by quadratic programg problems. It can be said that a portfolio return based on the lower possibility distribution has smaller possibility spread than the one on the upper possibility distribution. In addition, a possibility risk can be de ned as an interval given by the spreads of the portfolio returns from the upper and the lower possibility distributions to re ect the uncertainty in real investment problems. A numerical example of a portfolio selection problem is given to illustrate our proposed approaches. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy set; Investment analysis; Quadratic programg; Possibility distribution; Possibility portfolio 1. Introduction Possibility theory has been proposed by Zadeh [1] and advanced by Dubois and Prade [2] where fuzzy variables are associated with possibility distributions in the similar way that random variables are associated with probability distributions. Possibility distributions are represented as normal convex fuzzy sets, such as L±R fuzzy numbers, quadratic and exponential functions [3±6]. The theory of exponential possibility distributions has * Corresponding author. Tel.: ; fax: ; tanaka@ie.osakafu-u.ac.jp been proposed and applied to possibilistic data analysis [7±9]. As an application of possibility theory to portfolio analysis, possibility portfolio selection models were initially proposed in papers [10,11] where portfolio models are based on exponential possibility distributions rather than the mean-variance form in Markowitz's model. Although there are some similarities between Markowitz's model and possibility portfolio selection models, these two kinds of models analyze the security data in very di erent ways. Markowitz's model regards the portfolio selection as probability phenomena so that it imizes the variance of portfolio return subject to a given average return. On the contrary, possibility models /99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S ( 9 8 )

2 116 H. Tanaka, P. Guo / European Journal of Operational Research 114 (1999) 115±126 based on possibility distributions re ect portfolio experts' knowledge, which is characterized by the given possibility grades to security data. The basic assumption for using Markowitz's model is that the situation of stock markets in future can be correctly re ected by security data in the past, that is, the mean and covariance of securities in future is similar to the past one. It is hard to ensure this kind of assumption for the real ever-changing stock markets. On the other hand, possibility portfolio models integrate the past security data and experts' judgement to catch variations of stock markets more feasibly. Because experts' knowledge is very valuable for predicting the future state of stock markets, it is reasonable that possibility portfolio models are useful in the real investment world. This paper deals with portfolio selection problems based on two kinds of possibility distributions, namely, the upper and lower possibility distributions that are similar to rough sets to some extent to re ect two extreme opinions of experts. Using these two kinds of distributions, the corresponding portfolio selection models are formalized by quadratic optimization problems imizing spreads of possibility portfolio returns subject to the given center returns. It can be concluded that the portfolio return based on the lower possibility distribution has smaller possibility spread than the one based on the upper possibility distribution. A numerical example of a portfolio selection problem is given to illustrate our proposed approaches. 2. Markowitz's portfolio selection model Let us give a brief description of Markowitz's model. Assume that there are n securities denoted by S j (j ˆ 1;... ; n), the return of the security S j is denoted as x j and the proportion of total investment funds devoted to this security is denoted as r j. Thus, X n jˆ1 r j ˆ 1: 1 Since the return of the security x j (j ˆ 1;... ; n) vary from time to time, those are assumed to be random variables which can be represented by the pair of the average vector and covariance matrix. For instance, it is assumed that the observation data on returns of securities over m periods are given. At the discrete time i, the vector of n returns is denoted as x i ˆ x i1 ;... ; x in Š t. Thus, the total data over m periods are denoted as the following matrix. 0 1 x 11 x 12 x 1n x 21 x 22 x 2n B A : 2 x m1 x m2 x mn The average vector of returns over m periods is denoted as x 0 ˆ x 0 1 ;... ; x0 n Št and is written as 2 P m x 3 i1=m x 0 ˆ : 3 P m x in=m Also the covariance matrix Q ˆ q ij Š can be written as q ij ˆ Xm kˆ1 x ki x 0 i x kj x 0 j =m; i ˆ 1;... ; n j ˆ 1;... ; n : 4 Therefore random variables x j (j ˆ 1;... ; n) can be represented by the average vector x 0 and the covariance matrix Q, denoted as (x 0 ; Q). Now, the return associated with a portfolio r ˆ r 1 ;... ; r n Š t is given by z ˆ r t x: 5 The average and variance of z are given as E z ˆ E r t x ˆ r t Ex ˆ r t x 0 ; 6 V z ˆ V r t x ˆ r t Qr: 7 Since the variance of a portfolio return is regarded as the risk of investment, the best investment is one with the imum variance (7) subject to a given average return c. This leads to the following quadratic programg problem.

3 H. Tanaka, P. Guo / European Journal of Operational Research 114 (1999) 115± r r t Qr r t x 0 ˆ c; X n r i P 0: r i ˆ 1; 8 x i is revised to be 1. Taking the transformation y ˆ x a, the possibility distribution with a zero center vector is obtained as, P A y ˆ exp y t D 1 A y ˆ 0; D A e : 11 In what follows, the matrices D A in the upper and the lower possibility distributions are denoted as D u and D l, respectively. 3. Possibility distributions in portfolio selection problems Let us begin with the given data x i ; h i i ˆ 1;... ; m where x i ˆ x i1 ;... ; x in Š t is a vector of returns of n securities S j j ˆ 1;... ; n at the ith period and h i is an associated possibility grade given by expert knowledge to re ect a similarity degree between the future state of stock markets and the state of the ith sample. These grades h i i ˆ 1;... ; m are assumed to be expressed by a possibility distribution A de ned as P A x ˆ expf x a t D 1 A x a g ˆ a; D A e ; 9 where a ˆ a 1 ; a 2 ;... ; a n Š t is a center vector and D A is a symmetric positive de nite matrix, denoted as D A > 0. Given the data, the problem is to detere an exponential possibility distribution (9), i.e., a center vector a and a symmetric positive de nite matrix D A. According to two di erent viewpoints, two kinds of possibility distributions of A, namely, the upper and the lower possibility distributions are introduced in this paper. The upper and the lower possibility distributions denoted as P u and P l, respectively, should satisfy the inequality P u x P P l x with considering some similarities between our proposed methods and rough sets. From the formulation (9), it is obvious that the vector x with the highest possibility grade should be closest to the center vector a among all x i i ˆ 1;... ; m. Thus, the center vector a can be approximately estimated as 3.1. Identi cation of upper and lower possibility distributions by integrated model The upper and the lower distributions are used to re ect two kinds of distributions from the upper and the lower directions. In order to detere the matrix D u in the upper distribution, the following assumptions are given: (1) P u y i P h i ; i ˆ 1;... ; m (the constraint conditions), (2) imize P u y 1 P u y m (the objective function). The problem can be described as detering D u that imizes the objective function (2) subject to the constraint conditions (1). It means to obtain the smallest distribution among P u y i P h i i ˆ 1;... ; m in the sense of imizing P u y 1 P u y m. Fig. 1 illustrates the basic idea of the identi cation method. a ˆ x i ; 10 where x i denotes the vector with grade h i ˆ max kˆ1;...;m h k. The associated possibility grade of Fig. 1. Upper possibility distribution.

4 118 H. Tanaka, P. Guo / European Journal of Operational Research 114 (1999) 115±126 More detailed, the assumption (1) can be written as follows: P u y i P h i () y t i D 1 u y i 6 ln h i ; i ˆ 1;... ; m: 12 The objective function can be rewritten as follows: max X m y t i D 1 u y i: 13 Thus, the problem for obtaining D u becomes the following optimization problem. max D u X m y t i D 1 u y i y t i D 1 u y i 6 ln h i ; i ˆ 1;... ; m; D u > 0: 14 Fig. 2. Lower possibility distribution. Similarly, in order to detere the matrix D l in the lower distribution, the following assumptions are given: (1) P l y i 6 h i ; i ˆ 1;... ; m; (the constraint conditions), (2) maximize P l y 1 P l y m (the objective function). Likewise, the assumptions (1) and (2) can be converted into the following optimization problem to obtain the distribution matrix D l. D l X m y t i D 1 l y i y t i D 1 l y i P ln h i ; i ˆ 1;... ; m; D l > 0: 15 It means to obtain the largest distribution among P l y i 6 h i ; i ˆ 1;... ; m in the sense of maximizing P l y 1 P l y m. Fig. 2 shows the basic idea of the lower distribution. From the optimization problems (14) and (15), we know that the upper possibility distribution is obtained as the smallest possibility distribution subject to P l y i P h i. On the contrary, the lower possibility distribution is obtained as the largest possibility distribution subject to P l y i 6 h i. If we solve these two optimization problems separately, it can not be ensured that P u y P P l y holds for an arbitrary y. Let us consider the following model, which integrates Eqs. (14) and (15) to nd out D l and D u, simultaneously with the condition that D u D l is a semi-positive de nite matrix. D u ;D l X m y t i D 1 l y i Xm y t i D 1 u y i y t i D 1 u y i 6 ln h i ; y t i D 1 l y i P ln h i ; i ˆ 1;... ; m; D u D l P 0; D l > 0: 16 In this case, P u y and P l y are similar to rough set concept shown in Fig. 3 because D u D l P 0 ensures P u y P P l y. It is obvious that Eq. (16) Fig. 3. Graphic explanation of upper and lower distributions.

5 H. Tanaka, P. Guo / European Journal of Operational Research 114 (1999) 115± is a nonlinear optimization problem which is dif- cult to be solved. In order to solve the problem (16) easily let us rstly consider a simple linear programg problem without the conditions D u D l P 0 and D l > 0 in this problem. If the obtained matrix D u and D l cannot satisfy the conditions D u D l P 0 and D l > 0, we introduce the following auxiliary conditions to the constraint conditions of the problem (16) to obtain positive de nite matrices D l and D u such that D u D l P 0 holds by the linear programg. y t i D 1 u y i P e for i 2 E; 17 y t i D 1 u y j ˆ 0 for all i 6ˆ j and i; j 2 E 18 y t i D 1 l D 1 u y i P 0 for i 2 E; 19 y t i D 1 l y j ˆ 0 for all i 6ˆ j and i; j 2 E; 20 where E is the index set of n selected independent vectors fy 1 ;... ; y n g among y i i ˆ 1;... ; m with considering the condition m n, and e is a small positive number. It should be noted that the center vector y i ˆ 0 is not included in fy 1 ;... ; y n g. The equalities (18) and (20) are called the orthogonal conditions. It is proved afterwards that the constraint conditions (17)±(20) can ensure that D l > 0; D u > 0 and D u P D l hold. Thus, the following LP problem is formed. D u ;D l X m l y i Xm u y i u y i 6 ln h i for i ˆ 1;... ; m; l y i P ln h i for i ˆ 1;... ; m; u y i P e for i 2 E; y i t D 1 l D 1 u y i P 0 for i 2 E; l y j ˆ 0 for all i 6ˆ j and i; j 2 E; u y j ˆ 0 for all i 6ˆ j and i; j 2 E: 21 Theorem 1. The matrices D u and D l obtained from Eqs. (17)±(20) satisfy the condition D u D l P 0; D l > 0 and D u > 0. Proof. Because fy 1 ;... ; y n g are independent vectors in the n-dimensional space, an arbitrary vector z can be represented as z ˆ k 1 y 1 k 2 y 2 k n y n ; 22 where k i is a real number. Thus, using Eqs. (17) and (18), we have for z 6ˆ 0 z t D 1 u z ˆ k 1y 1 k 2 y 2 k n y n t D 1 u k 1 y 1 k 2 y 2 k n y n ˆ Xn k 2 i y i t D 1 u y i > 0: 23 It means that D u > 0. It follows from Eq. (19) that l y i P u y i > 0: 24 Thus, using Eqs. (20) and (24), we have z t D 1 l z ˆ k 1 y 1 k 2 y 2 k n y n t D 1 l k 1 y 1 k 2 y 2 k n y n ˆ Xn k 2 i y i t D 1 l y i > 0; 25 which means that D l > 0. Similarly, z t D 1 l D 1 l D 1 u z ˆ k 1y 1 k 2 y 2 k n y n t D 1 u k 1y 1 k 2 y 2 k n y n ˆ Xn k 2 i y i t D 1 l which means that D u D l P 0. D 1 u y i P 0; 26 Theorem 2. An optimal solution of Eq. (21) always exists. Proof. Let us consider a n n matrix K by which a set of linearly independent vectors fy 1 ;... ; y n g is transferred into a set of orthonormal vectors fz 1 ;... ; z n g where z it ˆ 0;... ; 0; 1; 0;... ; 0Š t. Thus, K y 1 ;... ; y n Š ˆ I; h 27 where I is the identical matrix. If we take positive de nite matrices D 1 u ˆ q 1 K t K; 28

6 120 H. Tanaka, P. Guo / European Journal of Operational Research 114 (1999) 115±126 and D 1 l ˆ q 2 K t K; 29 where q 1 6 q 2, it is obvious that D 1 u and D 1 l satisfy Eqs. (18) and (20) and D 1 l D 1 u ˆ q 2 q 1 K t K. The constraint condition of Eq. (21) becomes q 1 Ky i t Ky i 6 ln h i for i ˆ 1;... ; m; q 2 Ky i t Ky i P ln h i for i ˆ 1;... ; m; q 2 ˆ max ;...;m 1;i6ˆi ln h i= Ky i t Ky i ; 32 and a very small positive value for e, the obtained D u and D l satisfy all of constraint conditions. It means that there is an admissible set in the constraint conditions of Eq. (21). It should be noted that the center vector y i ˆ 0 is omitted in the rst two inequalities of Eq. (30), because Ky i t Ky i ˆ ln 1 ˆ 0. Thus, we consider i ˆ 1;... ; m 1 without y i ˆ 0 in detering q 1 and q 2 in Eqs. (31) and (32). h q 1 Ky i t Ky i P e; for i 2 E; q 2 q 1 Ky i t Ky i P 0; for i 2 E; q 1 Ky i t Ky j ˆ 0; for all i 6ˆ j and i; j 2 E; q 2 Ky i t Ky j ˆ 0 for all i 6ˆ j and i; j 2 E: If we take q 1 ˆ ;...;m 1;i6ˆi ln h i= Ky i t Ky i ; Here, orthogonal conditions are added to constraint conditions that can con ne the matrices D u and D l to positive de nite matrices and D u D l to a semi-positive de nite matrix. However, since there are many orthogonal conditions among independent vectors, it is very hard to select appropriate ones. To cope with this di culty, we use principle component analysis (PCA) to rotate the given data (y i ; h i ) to obtain a positive de nite matrix easily. Fig. 4 illustrates the rotation of orthogonal axes. The data can be transformed by linear transformation T. Columns of T are eigenvectors of the matrix R ˆ r ij, where rij is de ned as Fig. 4. Illustration of the linear transform by principle component analysis.

7 H. Tanaka, P. Guo / European Journal of Operational Research 114 (1999) 115± r ij ˆ ( ), X m Xm kˆ1 x ki a i x kj a j h k kˆ1 h k : 33 Without loss of generality, we assume that rank R ˆ n. It should be noted that T t T ˆ I. Using the linear transformation, the data y can be transformed into fz ˆ T t yg. Then we have P A z ˆ expf z t T t D 1 A Tzg: 34 According to the feature of PCA, T t D 1 A T is assumed to be a diagonal matrix as follows: 0 1 c 1 0 T t D 1 A T ˆ C A ˆ : B : A : 0 c n 35 Denote C A as C u and C l for the upper and the lower possibility distribution cases, respectively and denote c uj and c lj j ˆ 1;... ; n as the diagonal elements in C u and C l, respectively. The integrated model can be rewritten as follows. C l ;C u X m z t i C lz i Xm z t i C uz i z t i C lz i P ln h i ; z t i C uz i 6 ln h i ; i ˆ 1;... ; m; c uj P e; c lj P c uj ; j ˆ 1;... ; n; 36 where the condition c lj P c uj P e > 0 makes the matrix D u D l semi-positive de nite and matrices D u and D l positive. Thus, we have D u ˆ TC 1 u Tt ; 37 D l ˆ TC 1 l T t : This identi cation procedure is called as PCA method. In what follows, PCA method is used. It is simpler than the method based on orthogonal conditions. Theorem 3. In the linear programg (LP) problem (36), matrices C u and C l always exist. Proof. Let us take C u ˆ qi and C l ˆ pi in Eq. (36). Thus the constraint conditions of Eq. (36) can be written as pz t i z i P ln h i ; i ˆ 1;... ; m; qz t i z i 6 ln h i ; i ˆ 1;... ; m; q P e; p P q: 38 If we take p ˆ max ;;m 1;i6ˆi ln h i=z t i z i ; q ˆ ;;m 1;i6ˆi ln h i=z t i z i and e 6 q, inequalities (38) can hold. Therefore, there is an admissible set in the constraint conditions of the LP problem (36). It should be noted that the vector z i ˆ 0 is omitted, because z t i z i ˆ ln 1 ˆ 0 in Eq. (38). Thus, we consider i ˆ 1;... ; m 1 without z i ˆ 0 in detering the values for p and q. h This theorem implies that using PCA method we always can obtain the matrices D u and D l in the upper and lower possibility distributions Veri cation of our proposed methods Assume that the given data y i ; h i ; i ˆ 1;... ; m; are obtained from an exponential possibility distribution Y ˆ 0; A r e where the center vector is zero. In other words, the following equations hold. P Y y i ˆ expf y t i A r 1 y i g ˆ h i ; for i ˆ 1;... ; m: 39 Let us consider the following optimization problem for nding out the upper possibility matrix A u and the lower possibility matrix A l from the above given data. A l ;A u J A l ; A u ˆ Xm y t i A 1 l y i P ln h i ; y t i A 1 u y i 6 ln h i ; where i ˆ 1;... ; m. y t i A 1 l y i Xm y t i A 1 u y i 40 Theorem 4. The optimal solutions of A u and A l in the problem (40) are A r. Proof. The optimization problem (40) can be separated into the following two optimization problems:

8 122 H. Tanaka, P. Guo / European Journal of Operational Research 114 (1999) 115±126 max A u and A l J 1 A u ˆ Xm y t i A 1 u y i y i t A 1 u y i 6 ln h i ; i ˆ 1;... ; m: J 2 A l ˆ Xm y t i A 1 l y i y t i A 1 l y i P ln h i ; i ˆ 1;... ; m: Since data y i ; h i i ˆ 1;... ; m are obtained from the exponential possibility distribution 0; A r e, the data y i ; h i i ˆ 1;... ; m satisfy the Eq. (39). Therefore, A r is an admissible solution of Eqs. (41) and (42). Assume that there is another matrix A 0 such as J 1 A 0 > J 1 A r in Eq. (41). Then, for some i, y t i A 0 1 y i > y t i A 0 1 y i ˆ ln h i ; 43 which shows that A 0 is not admissible. Thus, A r is the optimal solution of Eq. (41). In the same way, we can prove that the optimal solution of Eq. (42) is also A r. Therefore, both A u and A l are A r. h This theorem means that our methods for detering an exponential possibility distribution can obtain the actual matrix A r if the given data are governed by an exponential possibility distribution with a distribution matrix A r. Moreover, the upper and the lower possibility distributions are equal to A r. 4. Possibility portfolio selection models The portfolio return can be written as z ˆ r t x: 44 Because x is governed by a possibility distribution a; D A e, z is a possibility variable Z. According to the extension principle, the possibility distribution of a portfolio return Z is de ned by P Z z ˆ max expf x a t D 1 A x a g: 45 fxjzˆr t xg Solving the optimization problem (45), the possibility distribution of a portfolio return Z can be obtained as P Z z ˆ expf z r t a 2 r t D A r 1 g ˆ r t a; r t D A r e ; 46 where r t a is the center value and r t D A r is the spread of a portfolio return Z. Given the lower and the upper possibility distributions, the corresponding portfolio selection models are given as follows: Portfolio selection model based on upper possibility distributions. r r t D u r r t a ˆ c; X n r i P 0: r i ˆ 1; 47 Portfolio selection model based on lower possibility distributions. r t D l r r r t a ˆ c; X n 48 r i ˆ 1; r i P 0; where c is an expected center value of possibility portfolio return. It is straightforward that models (47) and (48) are quadratic programg problems imizing the spread of a possibility portfolio return Z. Theorem 5. The spread of the possibility portfolio return based on the lower possibility distribution is not larger than the one based on the upper possibility distribution. Proof. Suppose that the optimal solutions obtained from the problems (47) and (48) are denoted as r u and r l ; respectively, with considering the same center value. According to the feature of the upper and lower possibility distributions, i.e. D u D l P 0, the following inequality holds. r t u D ur u P rt u D lr u : 49 Because r l is the optimal solution of Eq. (48), we have

9 H. Tanaka, P. Guo / European Journal of Operational Research 114 (1999) 115± r t u D lr u P rt l D l r l : As a result, r t u D ur u P rt l D l r l ; which proves the theorem. h The nondoated solutions with considering two objective functions, i.e., the spread and the center of a possibility portfolio in the possibility portfolio selection models (47) and (48) can form e cient frontiers. De nition 1. E cient frontiers from the upper and lower possibility portfolio selection models are called possibility e cient frontier I and possibility e cient frontier II, respectively. De nition 2. Two spreads of possibility portfolio returns from the upper and lower possibility distributions with the same given center value form an interval. This interval is called a possibility risk interval. The possibility risk interval is used to re ect the uncertainty in portfolio selection problems. 5. Numerical example In order to illustrate the proposed approaches let us consider an example shown in Table 1 introduced by Markowitz [12]. In this example, since we can consider that the recent sample is more similar to the future state, it is assumed that the possibility grade h i can be obtained as h i ˆ 0:2 0:7 t 1 =17 t ˆ 1;... ; 18 : 52 In Table 1, Nos. 1±9 are American Tobacco, AT&T, United States Steel, General Motors, Atchison&Topeka&Santa Fe, Coca-Cola, Borden, Firestone and Sharon Steel, respectively. The return of the security S j during a year is de ned as x t;j ˆ p t 1;j d t;j p t;j =p t;j ; 53 where p t;j is the closing price of the security S j (j ˆ 1;... ; 9) in the year t and d t;j is the dividend of this security in the same year. The center vector, and upper and lower possibility distribution matrices were obtained by Eqs. (10), (36) and (37) with e ˆ 0:001 as shown in Table 2. Using models (47) and (48), we obtained two possibility e cient frontiers shown in Fig. 5. It can Table 1 Returns on nine securities and possibility grades h i Year #1 #2 #3 #4 #5 #6 #7 #8 #9 Am.T A.T.&T. U.S.S. G.M. A.T.&S. C.C Bdn. Frstn. S.S (1) )0.305 )0.173 )0.318 )0.477 )0.457 )0.065 )0.319 )0.4 ) (2) (3) ) )0.424 ) )0.093 ) (4) ) )0.043 )0.189 )0.077 )0.051 )0.09 ) (5) )0.28 )0.183 )0.171 ) ) )0.194 ) (6) ) ) (7) (8) (9) (10) )0.088 )0.046 )0.078 )0.272 )0.037 ) ) (11) )0.127 ) ) (12) ) ) ) (13) ) (14) ) ) (15) ) ) (16) (17) ) )0.072 ) )0.084 ) (18)

10 124 H. Tanaka, P. Guo / European Journal of Operational Research 114 (1999) 115±126 Table 2 Two possibility distributions a ˆ [0.154, 0.176, 0.908, 0.715, 0.469, 0.077, 0.112, 0.756, 0.185] D u ˆ 52:242 39:936 0:146 52:592 26:514 61: :506 67: :276 39:936 91:779 21:404 14:957 93: : :639 73:068 56:717 0:146 21: : :493 89: : : : :681 52:592 14: : :694 62:849 47: :92 100:631 43:709 26:514 93:979 89:657 62: : :189 70:492 81:383 78:534 61: : :189 47: : : : : : : : : :92 70: : : : :171 67:634 73: : :631 81: : : : : :276 56: :681 43:709 78: : : : :757 D 1 ˆ 51:951 39:730 0:992 53:331 27:108 61: :395 66: :576 39:730 91:583 22:443 15:762 94: : :615 72:127 56:543 0:992 22: : :498 87: : : : :174 53:331 15: : :979 64:849 47: :868 96:485 44:235 27:108 94:476 87:167 64: : :85 70:628 83:794 79:083 61: : :954 47: :85 559: : : :76 149: : : :868 70: : : : :328 66:677 72: :912 96:485 83: : : : : :576 56: :174 44:235 79: :76 238: : :418 Fig. 5. Possibility e cient frontiers I and II.

11 H. Tanaka, P. Guo / European Journal of Operational Research 114 (1999) 115± be said from Fig. 5 that the spread of the possibility portfolio return from Eq. (47) is always larger than that from Eq. (48). This fact stems from the concept of the lower and the upper possibility distributions. They can be regarded as two extreme opinions playing a reference role for an investor. The corresponding risk with c ˆ 0.3 is an interval value, i.e., [ , ], which re ect the uncertainty in real investment problems. Figs. 6 and 7 show the securities selected by the possibility portfolio selection models (47) and (48) Fig. 6. Possibility portfolio based on the upper possibility distribution with c ˆ 0.3 (1,2,3,8: 0%, 4: 23.3%, 5: 11.1%, 6: 16.0%, 7: 31.2%, 9: 18.4%). in the case of c ˆ 0.3, respectively. The result shows that the number of the obtained securities from Eq. (48) is more than the one from Eq. (47). It implies that the portfolio from model (48) tends to take more distributive investment than the one from Eq. (47). 6. Conclusions Markowitz's model basically assumes that the situation of stock markets in future can be characterized by the past security data. It is di cult to ensure this kind of assumption in real investment problems. On the other hand, possibility portfolio models try to predict the state of stock markets in future by integrating the past security data and experts' judgement, which can gain an insight into a change of stock markets. This paper proposes an integrated model for obtaining two kinds of possibility distributions, i.e., the upper and the lower possibility distributions to re ect the di erent viewpoints of experts in portfolio selection problems. The corresponding portfolio selection problems are converted into quadratic programg problems. It can be said that portfolio returns based on lower possibility distributions have smaller possibility spreads than those based on upper possibility distributions. In addition, the investment risk can be described as an interval value to re ect the uncertainty in real investment problems. Because portfolio experts' knowledge is characterized by the upper and lower possibility distributions, the obtained portfolio would re ect portfolio experts' judgement. References Fig. 7. Possibility portfolio based on the lower possibility distribution with c ˆ 0.3 (1,3,8: 0%, 2: 10.1%, 4: 20.1%, 5: 15.2%, 6: 14.3%, 7: 25.0%, 9: 15.3%). [1] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3±28. [2] D. Dubois, H. Prade, Possibility Theory, Plenum Press, New York, [3] H. Tanaka, Fuzzy data analysis by possibilistic linear models, Fuzzy Sets and Systems 24 (1987) 363±375. [4] H. Tanaka, I. Hayashi, J. Watada, Possibilistic linear regression analysis for fuzzy data, European Journal of Operational Research 40 (1989) 389±396. [5] H. Tanaka, H. Ishibuchi, Identi cation of possibilistic linear systems by quadratic membership functions of

12 126 H. Tanaka, P. Guo / European Journal of Operational Research 114 (1999) 115±126 fuzzy parameters, Fuzzy Sets and Systems 41 (1991) 145±160. [6] H. Tanaka, H. Ishibuchi, Evidence theory of exponential possibility distributions, International Journal of Approximate Reasoning 8 (1993) 123±140. [7] H. Tanaka, P. Guo, H.-J. Zimmermann, Possibility distributions of fuzzy variables in fuzzy linear programg problems, Proceedings of Seventh International Fuzzy Systems Association World Congress, vol. 3, 1997, pp. 48±52. [8] H. Tanaka, P. Guo, Identi cation methods for exponential possibility distributions, Proceedings of Sixth IEEE International Conference on Fuzzy Systems, vol. 2, 1997, pp. 687±692. [9] H. Tanaka, H. Ishibuchi, S. Yoshikawa, Exponential possibility regression analysis, Fuzzy Sets and Systems 69 (1995) 305±318. [10] H. Tanaka, P. Guo, Portfolio selection method based on possibility distributions, Proceedings of Fourteenth International Conference on Production Research, vol. 2, 1997, pp. 1538±1541. [11] H. Tanaka, H. Nakayama, A. Yanagimoto, Possibility portfolio selection, Proceedings of Fourth IEEE International Conference on Fuzzy Systems, 1995, pp. 813±818. [12] H.M. Markowitz, Portfolio Selection: E cient Diversi cation of Investments, Wiley, New York, 1959.