A possibilistic approach to selecting portfolios with highest utility score

A possibilistic approach to selecting portfolios with highest utility score Christer Carlsson christer.carlsson@abo.fi Robert Fullér rfuller@abo.fi Péter Majlender peter.majlender@abo.fi Abstract The mean-variance methodology for the portfolio selection problem, originally proposed by Markowitz, has been one of the most important research fields in modern finance. In this paper we will assume that (i) each investor can assign a welfare, or utility, score to competing investment portfolios based on the expected return and risk of the portfolios; and (ii) the rates of return on securities are modelled by possibility distributions rather than probablity distributions. We will present an algorithm of complexity o(n 3 ) for finding an exact optimal solution (in the sense of utility scores) to the n-asset portfolio selection problem under possibility distributions. 1 A utility function for ranking portfolios The mean-variance methodology for the portfolio selection problem, originally proposed by Markowitz [4], has been one of the most important research fields in modern finance theory [7]. The key principle of the mean-variance model is to use the expected return of a portfolio as the investment return and to use the variance of the expected returns of the portfolio as the investment risk. Following [1] we shall asssume that each investor can assign a welfare, or utility, score to competing investment portfolios based on the expected return and risk of those portfolios. The utility score may be viewed as a means of ranking portfolios. Higher utility values are assigned to portfolios with more attractive riskreturn profiles. One reasonable function that is commonly employed by financial theorists assigns a risky portfolio P with a risky rate of return r P, an expected rate of return E(r P ) and a variance of the rate of return σ 2 (r P ) the following utility score [1]: U(P ) = E(r P ) 0.005 A σ 2 (r P ), (1) The final version of this paper appeared in: Fuzzy Sets and Systems, 131(2002) 13-21. 1

where A is an index of the investor s risk aversion (A 2.46 for an average investor in the U.S.A.). The factor of 0.005 is a scaling convention that allows us to express the expected return and standard deviation in equation (1) as percentages rather than decimals. Equation (1) is consistent with the notion that utility is enhanced by high expected returns and diminished by high risk. Because we can compare utility values to the rate offered on risk-free investments when choosing between a risky portfolio and a safe one, we may interpret a portfolio s utility value as its certainty equivalent rate of return to an investor. That is, the certainty equivalent rate of a portfolio is the rate that risk-free investments would need to offer with certainty to be considered as equally attractive as the risky portfolio. Now we can say that a portfolio is desirable only if its certainty equivalent return exceeds that of the risk-free alternative. In the meanvariance context, an optimal portfolio selection can be formulated as the following quadratic mathematical programming problem ( n U ) ( n r i x i = E ) ( n ) r i x i 0.005 A σ 2 r i x i max (2) subject to {x 1 + + x n = 1, x i 0, i = 1,..., n}, where n is the number of available securities, x i is the proportion invested in security (or asset) i, and r i denotes the risky rate of return on security i, i = 1,..., n. Denoting the rate of return on the risk-free asset by r f, a portfolio is desirable for the investor if and only if ( n ) U r i x i > r f. In this paper we will assume that the rates of return on securities are modelled by possibility distributions rather than probablity distributions. That is, the rate of return on the i-th security will be represented by a fuzzy number r i, and r i (t), t R, will be interpreted as the degree of possibility of the statement that t will be the rate of return on the i-th security. In our method we will consider only trapezoidal possiblity distributions, but our method can easily be generalized to the case of possibility distributions of type LR. In standard portfolio models uncertainty is equated with randomness, which actually combines both objectively observable and testable random events with subjective judgments of the decision maker into probability assessments. A purist on theory would accept the use of probability theory to deal with observable random events, but would frown upon the transformation of subjective judgments to probabilities. 2

The use of probabilities has another major drawback: the probabilities give an image of precision which is unmerited - we have found cases where the assignment of probabilities is based on very rough, subjective estimates and then the subsequent calculations are carried out with a precision of two decimal points. This shows that the routine use of probabilities is not a good choice. The actual meaning of the results of an analysis may be totally unclear - or results with serious errors may be accepted at face value. In standard portfolio theory the decision maker assigns utility values to consequences, which are the results of combinations of actions and random events. The choice of utility theory, which builds on a decision maker s relative preferences for artificial lotteries, is a way to anchor portfolio choices in the von Neumann- Morgenstern axiomatic utility theory. In practical applications the use of utility theory has proved to be problematic (which should be more serious than having axiomatic problems): (i) utility measures cannot be validated inter-subjectively, (ii) the consistency of utility measures cannot be validated across events or contexts for the same subject, (iii) utility measures show discontinuities in empirical tests (as shown by Tversky (cf. [5])), which should not happen with rational decision makers if the axiomatic foundation is correct, and (iv) utility measures are artificial and thus hard to use on an intuitive basis. As the combination of probability assessments with utility theory has these well- known limitations we have explored the use of possibility theory as a substituting conceptual framework. Let us introduce some definitions we shall need in the following section. A fuzzy number A is called trapezoidal with tolerance interval [a, b], left width α and right width β if its membership function has the following form 1 a t if a α t a, α 1 if a t b, A(t) = 1 t b if a t b + β, β 0 otherwise, and we use the notation A = (a, b, α, β). It can easily be shown that [A] γ = [a (1 γ)α, b + (1 γ)β], γ [0, 1]. where [A] γ denotes the γ-level set of A. Let [A] γ = [a 1 (γ), a 2 (γ)] and [B] γ = [b 1 (γ), b 2 (γ)] be fuzzy numbers and let λ R be a real number. Using the extension principle we can verify the following 3

Figure 1: Trapezoidal fuzzy number. rules for addition and scalar muliplication of fuzzy numbers [A + B] γ = [a 1 (γ) + b 1 (γ), a 2 (γ) + b 2 (γ)], [λa] γ = λ[a] γ. Let A F be a fuzzy number with [A] γ = [a 1 (γ), a 2 (γ)], γ [0, 1]. In [2] we introduced the (crisp) possibilistic mean (or expected) value and variance of A as E(A) = 1 0 γ(a 1 (γ) + a 2 (γ))dγ, σ 2 (A) = 1 2 1 0 γ ( a 2 (γ) a 1 (γ) ) 2 dγ. It is easy to see that if A = (a, b, α, β) is a trapezoidal fuzzy number then and E(A) = 1 0 γ[a (1 γ)α + b + (1 γ)β]dγ = a + b + β α 2 6. σ 2 (A) = [ (b a)2 (b a)(α + β) (α + β)2 b a + + = 4 6 24 2 + α + β ] 2 (α + β)2 +. 6 72 2 A possibilistic approach to portfolio selection problem Watada [6] proposed a fuzzy portfolio selection model where he used fuzzy numbers to represent the decision maker s aspiration levels for the expected rate of return and a certain degree of risk. Inuiguchi and Tanino [3] introduced a novel possibilistic programming approach to the portfolio selection problem: their approach, which prefers a distributive investment solution, is based on the minimax regret criterion (the regret which the decision maker is ready to undertake). In many important cases it might be easier to estimate the possibility distributions of rates of return on securities rather than the corresponding probability distributions. Consider now the portfolio selection problem with possibility distributions ( n U ) ( n r i x i = E ) ( n ) r i x i 0.005 A σ 2 r i x i max (3) subject to {x 1 + + x n = 1, x i 0, i = 1,..., n}. 4

where r i = (a i, b i, α i, β i ), i = 1,..., n are fuzzy numbers of trapezoidal form. It is easy to compute that ( n ) E r i x i = n [ 1 a i + b i + 1 ] 2 3 (β i α i ) x i, and ( n ) ( n [ σ 2 1 r i x i = b i a i + 1 ] ) 2 2 3 (α i + β i ) x i + 1 [ n ] 2 (α i + β i )x i. 72 Introducing the notations u i = 1 [ a i + b i + 1 ] 2 3 (β i α i ), v i = 0.005A 2 [ b i a i + 1 ] 3 (α i + β i ), w i = 0.005A 72 (α i + β i ), we shall represent the i-th asset by a triplet (v i, w i, u i ), where u i denotes its possibilistic expected value, and vi 2 + w2 i is its possibilistic variance multiplied by the constant 0.005 A. We will also assume that there are at least three distinguishable assets, with the meaning that if two assets have the same expected value and variance then they are considered indistinguishable (or identical in the framework of mean-variance analysis). That is, we assume that u i u j or vi 2 + w2 i v2 j + w2 j for i j. Then we will state the possibilistic portfolio selection problem (3) as u, x v, x 2 w, x 2 max; s.t.{x 1 + + x n = 1, x i 0, i = 1,..., n}. (4) The convex hull of {(v i, w i, u i ) : i = 1,..., n}, denoted by T, and defined by T = conv{(v i, w i, u i ) : i = 1,..., n} {( n n n ) = v i x i, w i x i, u i x i : n } x i = 1, x i 0, i = 1,..., n. is a convex polytope in R 3. Then (4) turns into the following three-dimensional nonlinear programming problem (v 2 0 + w 2 0 u 0 ) max; subject to (v 0, w 0, u 0 ) T, 5

or, equivalently, f(v 0, w 0, u 0 ) := v 2 0 + w 2 0 u 0 min; subject to (v 0, w 0, u 0 ) T, (5) where T is a compact and convex subset of R 3, and the implicit function g c (v 0, w 0 ) := v 2 0 + w 2 0 c, is strictly convex for any c R. This means that any optimal solution to (5) must be on the boundary of T. We will now present an algorithm for finding an optimal solution to (4) on the boundary of T. Note that T is a compact and convex polyhedron of R 3 and that any optimal solution to (5) must be on the boundary of T, which imply that any optimal solution can be obtained as a convex combination of at most 3 extreme points of T. In the algorithm by lifting the non-negativity conditions for investment proportions we shall calculate: (i) the (exact) solutions to all conceivable 3-asset problems with non-colinear assets, (ii) the (exact) solutions to all conceivable 2- assets problems with distinguishable assets, and (iii) the utility value of each asset. Then we compare the utility values of all feasible solutions (i.e. solutions with non-negative weights) and portfolios with the highest utility value will be chosen as optimal solutions to portfolio selection problem (5). Our algorithm will require O(n 3 ) steps, where n is the number of available securities. Consider three assets (v i, w i, u i ), i = 1, 2, 3, which are not colinear: (α 1, α 2, α 3 ) R 3, (α 1, α 2, α 3 ) 0, such that α 1 v 1 w 1 u 1 + α 2 v 2 w 2 u 2 (α 1 + α 2 ) v 3 w 3 u 3 = 0. Then the 3-asset optimal portfolio selection problem with not-necessarily nonnegative weights reads (v 1 x 1 +v 2 x 2 +v 3 x 3 ) 2 +(w 1 x 1 +w 2 x 2 +w 3 x 3 ) 2 (u 1 x 1 +u 2 x 2 +u 3 x 3 ) min (6) subject to x 1 + x 2 + x 3 = 1. Let us denote L(x, λ) = (v 1 x 1 + v 2 x 2 + v 3 x 3 ) 2 + (w 1 x 1 + w 2 x 2 + w 3 x 3 ) 2 (u 1 x 1 + u 2 x 2 + u 3 x 3 ) + λ(x 1 + x 2 + x 3 1), (7) 6

the Lagrange function of the constrained optimization problem (6). The Kuhn- Tucker necessity conditions are 2v 1 (v 1 x 1 + v 2 x 2 + v 3 x 3 ) + 2w 1 (w 1 x 1 + w 2 x 2 + w 3 x 3 ) u 1 + λ = 0, 2v 2 (v 1 x 1 + v 2 x 2 + v 3 x 3 ) + 2w 2 (w 1 x 1 + w 2 x 2 + w 3 x 3 ) u 2 + λ = 0, 2v 3 (v 1 x 1 + v 2 x 2 + v 3 x 3 ) + 2w 3 (w 1 x 1 + w 2 x 2 + w 3 x 3 ) u 3 + λ = 0, x 1 + x 2 + x 3 = 1, which lead us to the following linear equality system [ q 2 1 + r1 2 ] [ ] [ q 1 q 2 + r 1 r 2 x1 1/2(u1 u q 1 q 2 + r 1 r 2 q2 2 + = 3 ) q 1 v 3 r 1 w 3 r2 2 x 2 1/2(u 2 u 3 ) q 2 v 3 r 2 w 3 ], (8) where we used the notations q 1 = v 1 v 3, q 2 = v 2 v 3, r 1 = w 1 w 3 and r 2 = w 2 w 3. Now we prove that if (v i, w i, u i ), i = 1, 2, 3, are not colinear then equation (8) has a unique solution. Suppose that the solution to equation (8) is not unique, i.e. [ q 2 det 1 + r1 2 ] q 1 q 2 + r 1 r 2 q 1 q 2 + r 1 r 2 q2 2 + = 0. r2 2 That is, [ q 2 det 1 + r1 2 ] q 1 q 2 + r 1 r 2 q 1 q 2 + r 1 r 2 q2 2 + = (q r2 1 2 + r1)(q 2 2 2 + r2) 2 (q 1 q 2 + r 1 r 2 ) 2 2 ( [ ]) 2 = (q 1 r 2 q 2 r 1 ) 2 q1 r = det 1 = 0. q 2 r 2 Thus, the rows of ( q 1 r 1 q 2 r 2 ) are not linearly independent: (α1, α 2 ) 0 such that α 1 [q 1, r 1 ]+α 2 [q 2, r 2 ] = 0 α 1 [v 1 v 3, w 1 w 3 ]+α 2 [v 2 v 3, w 2 w 3 ] = 0. (9) We find that equation (8) turns into [ ] [ ] (q2+r 2 2) 2 α 2 2 α 1 α 2 x1 α 1 α 2 α1 2 x 2 = α 1 [ 1/2α1 (u 1 u 3 ) + α 2 (q 2 v 3 + r 2 v 3 ) 1/2α 1 (u 2 u 3 ) α 1 (q 2 v 3 + r 2 v 3 ) Multiplying both sides by [α 1, α 2 ] we get that u 1, u 2 and u 3 have to satisfy the equation [ 1 α1 2 2 α 1(u 1 u 3 ) 1 ] 2 α 2(u 2 u 3 ) = 0. ]. 7

If α 1 0, then we obtain α 1 (u 1 u 3 ) + α 2 (u 2 u 3 ) = 0, and from equation (9) it follows that v 1 v 2 v 3 α 1 w 1 + α 2 w 2 (α 1 + α 2 ) w 3 = 0, u 1 u 2 u 3 i.e. (v i, w i, u i ), i = 1, 2, 3, were colinear. If α 1 = 0, then α 2 0, and from equation (9) it follows that q 2 = r 2 = 0. Now we find that equation (8) turns into [ q 2 1 + r1 2 ] [ 0 x1 0 0 x 2 Multiplying both sides by [0, 1], we obtain We find that ] [ 1/2(u1 u = 3 ) q 1 v 3 r 1 v 3 ) 1/2(u 2 u 3 ) 1 2 (u 2 u 3 ) = 0. v 2 v 3 = w 2 w 3 = u 2 u 3 = 0, which means that (v i, w i, u i ), i = 1, 2, 3, were colinear. Which ends the proof. Using the general inversion formula [ ] 1 [ ] t1 t 2 1 t4 t = 2, t 3 t 4 t 1 t 4 t 2 t 3 t 3 t 1 ]. we find that the optimal solution to (8) is [ x 1 x 2 ] [ 1 q = 2 2 + ] r2 2 (q 1 q 2 + r 1 r 2 ) (q 1 r 2 q 2 r 1 ) 2 (q 1 q 2 + r 1 r 2 ) q1 2 + r2 1 [ ] 1/2(u1 u 3 ) q 1 v 3 r 1 v 3. 1/2(u 2 u 3 ) q 2 v 3 r 2 v 3 (10) We will now show that x = (x 1, x 2, 1 x 1 x 2) satisfies the Kuhn-Tucker sufficiency condition, i.e. L (x, λ) is a positive definite matrix at x = x in the subset defined by { y = (y1, y 2, y 3 ) R 3 : y 1 + y 2 + y 3 = 0 }. 8

Really, from (7) we get M := 1 v 2 L (x 1 2 + w2 1 v 1 v 2 + w 1 w 2 v 1 v 3 + w 1 w 3, λ) = v 1 v 2 + w 1 w 2 v2 2 + w2 2 v 2 v 3 + w 2 w 3 v 1 v 3 + w 1 w 3 v 2 v 3 + w 2 w 3 v3 2 + w2 3 and, therefore, the inequality v 1 v 1 = v 2 v 2 v 3 v 3 T w 1 w 1 + w 2 w 2 w 3 w 3 y T My = (v 1 y 1 + v 2 y 2 + v 3 y 3 ) 2 + (w 1 y 1 + w 2 y 2 + w 3 y 3 ) 2 0, (11) holds for any y R 3. So M is a positive semidefinite matrix. If y T My = 0 for some y = (y 1, y 2, y 3 ) 0, y 1 + y 2 + y 3 = 0, then from (11) we find v 1 y 1 + v 2 y 2 + v 3 y 3 = 0, w 1 y 1 + w 2 y 2 + w 3 y 3 = 0, and we would get that v 1 v 2 v 3 [ ] [ ] det w 1 w 2 w 3 q1 q = det 2 q1 r = det 1 = 0, r 1 1 1 1 r 2 q 2 r 2 which would lead us to a contradiction with the non-colinearity condition. So L is positive definite. Thus x is the unique optimal solution to (6) and x is an optimal solution to (4) (with n = 3) if x 1 > 0, x 2 > 0 and x 3 > 0 (the Kuhn-Tucker regularity condition). The optimal value of (6) will be denoted by U. Consider now a 2-asset problem with two assets, say (v 1, w 1, u 1 ) and (v 2, w 2, u 2 ), such that (v 1, w 1, u 1 ) (v 2, w 2, u 2 ): (v 1 x 1 +v 2 x 2 ) 2 +(w 1 x 1 +w 2 x 2 ) 2 (u 1 x 1 +u 2 x 2 ) min; s.t. x 1 +x 2 = 1. (12) Let us denote L(x, λ) = (v 1 x 1 +v 2 x 2 ) 2 +(w 1 x 1 +w 2 x 2 ) 2 (u 1 x 1 +u 2 x 2 )+λ(x 1 +x 2 1), (13) the Lagrange function of the constrained optimization problem (6). The Kuhn- Tucker necessity conditions are 2v 1 (v 1 x 1 + v 2 x 2 ) + 2w 1 (w 1 x 1 + w 2 x 2 ) u 1 + λ = 0, 2v 2 (v 1 x 1 + v 2 x 2 ) + 2w 2 (w 1 x 1 + w 2 x 2 ) u 2 + λ = 0, T, x 1 + x 2 = 1, 9

which leads us to the following linear equation [(v 1 v 2 ) 2 + (w 1 w 2 ) 2 ] x 1 = 1 2 (u 1 u 2 ) (v 1 v 2 )v 2 (w 1 w 2 )w 2. (14) If (v 1 v 2 ) 2 + (w 1 w 2 ) 2 0 then we find that x = (x 1, 1 x 1 ), where [ ] x 1 1 1 = (v 1 v 2 ) 2 + (w 1 w 2 ) 2 2 (u 1 u 2 ) (v 1 v 2 )v 2 (w 1 w 2 )w 2, (15) is the unique solution to equation (14). If v 1 = v 2 and w 1 = w 2 then from (14) we find u 1 = u 2, which would contradict the initial assumption that the two assets are not identical. It can easily be seen that L (x, λ) is a positive definite matrix in the subset defined by { y = (y1, y 2 ) R 2 : y 1 + y 2 = 0 }. So, x is the unique optimal solution to (12), and if x > 0 then x is an optimal solution to (4) with n = 2. 3 An algorithm In this Section we provide an algorithm for finding an optimal solution to the n- asset possibilistic portfolio selection problem (4). The algorithm will terminate in o(n 3 ) steps. Step 1 Let c := + and x c := [0,..., 0]. Step 2 Choose three points from the bag {(v i, w i, u i ) : i = 1,..., n} which have not been considered yet. If there are no such points then go to Step 9, otherwise denote these three points by (v j, w j, u j ), (v k, w k, u k ) and (v l, w l, u l ). Let (v 1, w 1, u 1 ) := (v j, w j, u j ), (v 2, w 2, u 2 ) := (v k, w k, u k ) and (v 3, w 3, u 3 ) := (v l, w l, u l ). Step 3 If [ ] [ ] q1 r det 1 v1 v = det 3 w 1 w 3 = 0, q 2 r 2 v 2 v 3 w 2 w 3 then go to Step 2, otherwise go to Step 4. Step 4 Compute the first two component, [x 1, x 2 ], of the optimal solution to (6) using equation (10). Step 5 If [x 1, x 2, 1 x 2 x 1 ] > 0 then go to Step 6, otherwise go to Step 2. 10

Step 6 If U < c then go to Step 7, otherwise go to Step 2. Step 7 Let c = U, where U is the optimal value of (6), and let Step 8 Go to Step 2. j-th {}}{ k-th {}}{ l-th {}}{ x c = [0,..., x 1, 0,..., 0, x 2, 0,..., 0 x 3, 0,..., 0]. Step 9 Choose two points from the bag {(v i, w i, u i ) : i = 1,..., n} which have not been considered yet. If there are no such points then go to Step 16, otherwise denote these two points by (v j, w j, u j ) and (v k, w k, u k ). Let (v 1, w 1, u 1 ) := (v j, w j, u j ) and (v 2, w 2, u 2 ) := (v k, w k, u k ). Step 10 If (v 1 v 2 ) 2 + (w 1 w 2 ) 2 0 then go to Step 9, otherwise go to Step 11. Step 11 Compute the first component, x 1, of the optimal solution to (12) using equation (15). Step 12 If [x 1, x 2 ] = [x 1, 1 x 1 ] > 0 then go to Step 13, otherwise go to Step 9. Step 13 If U < c then go to Step 14, otherwise go to Step 9. Step 14 Let c = U, where U is the optimal value of (12), and let Step 15 Go to Step 9. j-th {}}{ k-th {}}{ x c = [0,..., x 1, 0,..., 0, x 2, 0,..., 0]. Step 16 Choose a point from the bag {(v i, w i, u i ) : i = 1,..., n} which has not been considered yet. If there is no such points then go to Step 20, otherwise denote this point by (v i, w i, u i ). Step 17 If v 2 i + w2 i u i < c then go to Step 18, otherwise go to Step 16. Step 18 Let c = v 2 i + w2 i u i and let Step 19 Go to Step 16. i-th {}}{ x c = [0,..., 0, 1, 0,..., 0]. Step 20 x c is an optimal solution and c is the optimal value of the original portfolio selection problem (4). 11

4 Example We shall illustrate the proposed algorithm by a simple example. Consider a 3-asset problem with A = 2.46 and with the following possibility distributions and, therefore, r 1 = ( 10.5, 70.0, 4.0, 100.0), r 2 = ( 8.1, 35.0, 4.4, 54.0), r 3 = ( 5.0, 28.0, 11.0, 85.0) (v 1, w 1, u 1 ) = (6.386, 1.359, 45.750), (v 2, w 2, u 2 ) = (3.469, 0.763, 21.717), (v 3, w 3, u 3 ) = (3.604, 1.255, 23.833). It should be noted that the first asset may yield negative rates of return with degree of possibility one. Usually, the support of fuzzy numbers representing the possibility distributions of rates of return can not contain any return that is less than -100%, because one can never lose more money than the original investment. First consider the 3-asset problem with (v 1, w 1, u 1 ), (v 2, w 2, u 2 ) and (v 3, w 3, u 3 ). Since [ ] [ ] q1 r det 1 2.782 0.105 = det = 1.352 0, q 2 r 2 0.135 0.491 we get [ x 1 x 2 ] 1 = 1.352 2 [ ] [ 0.259 0.427 0.800 0.427 7.751 0.044 ] = [ 0.124 0.373 ], and, since, we get (Step 7) [x 1, x 2, x 3] = [0.124, 0.373, 0.503] > 0. U := 9.386 and x := [0.124, 0.373, 0.503]. Thus [0.124, 0.373, 0.503] is a qualified candidate for an optimal solution to (3). Let us consider all conceivable 2-asset problems (1, 2), (1, 3) and (2, 3), where the numbers stand for the corresponding assets chosen from the bag {(v 1, w 1, u 1 ), (v 2, w 2, u 2 ), (v 3, w 3, u 3 )}. Here we are searching for optimal solutions on the edges of the triangle generated by the assets. 12

Select (1,2). Since we get (v 1 v 2 ) 2 + (w 1 w 2 ) 2 = 8.864 0, U := 9.336 and [x 1, x 2] = [0.163, 0.837] > 0. Thus [0.163, 0.837, 0] is a qualified candidate for an optimal solution to (3). Select (1,3). Since we get (v 1 v 3 ) 2 + (w 1 w 3 ) 2 = 7.751 0, U := 9.352 and [x 1, x 3] = [0.103, 0.897] > 0. Thus [0.103, 0, 0.897] is a qualified candidate for an optimal solution to (3). Select (2,3). Since we get (v 2 v 3 ) 2 + (w 2 w 3 ) 2 = 0.259 0, U := 9.277 and [x 2, x 3] = [0.171, 0.829] > 0. Thus [0, 0.171, 0.829] is a qualified candidate for an optimal solution to (3). Finally, we compute the utility values of all the three vertexes of the triangle generated by the three assets: v 2 1 + w 2 1 u 1 = 3.122, and [1, 0, 0] is the corresponding feasible solution to (3). v 2 2 + w 2 2 u 2 = 9.101, and [0, 1, 0] is the corresponding feasible solution to (3). v 2 3 + w 2 3 u 3 = 9.269, and [0, 0, 1] is the corresponding feasible solution to (3). Comparing the utility values of all feasible solutions we find that the only solution to the 3-asset problem is x = [0.124, 0.373, 0.503] with a utility value of 9.386. The optimal risky portfolio will be preferred to the risk-free investment (by an investor whose degree of risk-aversion is equal to 2.46) if r f < 9.386%. 13

5 Summary In this paper we have considered portfolio selection problems under possibility distributions and have presented an algorithm for finding an exact (i.e. not approximate) optimal solution to these problems. First we have proved that the boundary of the set of feasible solutions (which is a convex polytope) must contain all optimal solutions to the problem. Then we have considered all possible sides, edges and vertexes that could be generated from the given triplets and computed the optimal portfolios of (i) three assets that could generate a side, and (ii) two assets that could generate an edge of the convex hull of all assets. Then we have compared the utility values of all feasible solutions (i.e. solutions with non-negative weights) and portfolios with highest utility value have been chosen as optimal solutions to portfolio selection problem. References [1] Z. Bodie, A. Kane and A.J. Marcus, Investments (Irwin, Times Mirror Higher Education Group, Boston, 1996). [2] C. Carlsson and R. Fullér, On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems, 122(2001) 315-326. [3] M. Inuiguchi and T. Tanino, Portfolio selection under independent possibilistic information, Fuzzy Sets and Systems, 115(2000) 83-92. [4] H. Markowitz, Portfolio selection, Journal of Finance, 7(1952) 77-91. [5] A.Tversky, Intransitivity of Preferences, Psychological Review, 76(1969) 31-45. [6] J. Watada, Fuzzy portfolio selection and its applications to decision making, Tatra Mountains Math. Publ., 13(1997) 219-248. [7] Y. Xia, B. Liu, S. Wang and K.K. Lai, A model for portfolio selection with order of expected returns, Computers & Operations Research, 27(2000) 409-422. 6 Citations [A12] Christer Carlsson, Robert Fullér and Péter Majlender, A possibilistic approach to selecting portfolios with highest utility score, FUZZY SETS AND SYSTEMS, 131(2002) 13-21. [MR1920826] 14

in journals A12-c26 Xiaoxia Huang, Risk curve and fuzzy portfolio selection, Computers and Mathematics with Applications, 55(2008) 1102-1112. 2008 http://dx.doi.org/10.1016/j.camwa.2007.06.019 Carlsson et al. [A12] found the optimum portfolio by use of their own denition of mean and variance of fuzzy numbers [A14]. (page 1102) A12-c25 Pankaj Gupta, Mukesh Kumar Mehlawat, Anand Saxena, Asset portfolio optimization using fuzzy mathematical programming, Information Sciences, 178(2008), pp. 1734-1755. 2008 http://dx.doi.org/10.1016/j.ins.2007.10.025 Carlsson et al. [A12] introduced a possibilistic approach for selecting portfolios with the highest utility value under the assumption that the returns of assets are trapezoidal fuzzy numbers. (page 1735) A12-c24 Huang, X., A new perspective for optimal portfolio selection with random fuzzy returns, INFORMATION SCIENCES, 177 (23), pp. 5404-5414. 2007 http://dx.doi.org/10.1016/j.ins.2007.06.003 A12-c23 Zhang, W.-G., Wang, Y.-L., Notes on possibilistic variances of fuzzy numbers, APPLIED MATHEMATICS LETTERS, 20 (11), pp. 1167-1173. 2007 http://dx.doi.org/10.1016/j.aml.2007.03.002 A12-c22 Huang, X., Portfolio selection with fuzzy returns, JOURNAL OF IN- TELLIGENT AND FUZZY SYSYTEMS, 18 (4), pp. 383-390. 2007 A12-c21 Huang XX, Two new models for portfolio selection with stochastic returns taking fuzzy information, EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 180 (1): 396-405 JUL 1 2007 http://dx.doi.org/10.1016/j.ejor.2006.04.010 15

A12-c20 Zhang, W.-G., Wang, Y.-L., Chen, Z.-P., Nie, Z.-K., Possibilistic meanvariance models and efficient frontiers for portfolio selection problem, IN- FORMATION SCIENCES, 177 (13), pp. 2787-2801. 2007 http://dx.doi.org/10.1016/j.ins.2007.01.030 A12-c19 Zhang, W.-G., Possibilistic mean-standard deviation models to portfolio selection for bounded assets, APPLIED MATHEMATICS AND COMPU- TATION 189 (2), pp. 1614-1623. 2007 A12-c18 Smimou, K., Bector, C.R., Jacoby, G., A subjective assessment of approximate probabilities with a portfolio application, RESEARCH IN IN- TERNATIONAL BUSINESS AND FINANCE, 21 (2), pp. 134-160. 2007 A12-c17 Vercher E, Bermudez JD, Segura JV, Fuzzy portfolio optimization under downside risk measures, FUZZY SETS AND SYSTEMS 158 (7): 769-782 APR 1 2007 http://dx.doi.org/10.1016/j.fss.2006.10.026 We shall illustrate the above results by a simple example from [A12] with three assets whose returns are the following trapezoidal fuzzy numbers: (page 774) A12-c16 Lin PC, Chen JS, FuzzyTree crossover for multi-valued stock valuation, INFORMATION SCIENCES 177 (5): 1193-1203 MAR 1 2007 http://dx.doi.org/10.1016/j.ins.2006.08.017 A12-c15 Guohua Chen, Shou Chen, Yong Fang, Shouyang Wang, A Possibilistic Mean VaR Model for Portfolio Selection, ADVANCED MODELING AND OPTIMIZATION, Volume 8, Number 1, pp. 99-107. 2006 http://www.ici.ro/camo/journal/vol8/v8a8.pdf A12-c14 Huang XX, Fuzzy chance-constrained portfolio selection, APPLIED MATHEMATICS AND COMPUTATION 177 (2): 500-507 JUN 15 2006 http://dx.doi.org/10.1016/j.amc.2005.11.027 A12-c13 Zdenek Zmeskal, Value at risk methodology of internationalindex portfolio under soft conditions (fuzzy-stochastic approach), INTERNATIONAL REVIEW OF FINANCIAL ANALYSIS, 14(2005) 263-275. 2005 http://dx.doi.org/10.1016/j.irfa.2004.06.011 16

in proceedings A12-c13 J. Zhang and W. Tang and C. Wang and R. Zhao, Fuzzy Dynamic Portfolio Selection for Survival, in: Advanced Intelligent Computing Theories and Applications. With Aspects of Theoretical and Methodological Issues, Lecture Notes in Computer Science, vol. 4681, Springer, pp. 34-45. 2007 http://dx.doi.org/10.1007/978-3-540-74171-8 5 With the introduction of fuzzy set theory by Zadeh [18] in 1965, researchers began to realize that they could employ fuzzy set theory to manage portfolio in another type of uncertain environment called fuzzy environment. Since then a lot of researchers began to study the portfolio selection problem, such as Carlsson et al [A12], Inuiguchi and Tanino [5], Léon et al [6] and Tanaka and Guo [15]. (page 34) A12-c12 Silva, Ricardo C.; Verdegay, Jose L.; Yamakami, Akebo, Two-phase method to solve fuzzy quadratic programming problems, IEEE International Fuzzy Systems Conference (FUZZ-IEEE 2007), 23-26 July 2007, pp.1-6. 2007 http://dx.doi.org/10.1109/fuzzy.2007.4295501 A12-c11 Wei Chen, Runtong Zhang, Wei-Guo Zhang and Yong-Ming Cai, A Fuzzy Portfolio Selection Methodology Under Investing Constraints, Fuzzy Information and Engineering, Proceedings of the Second International Conference of Fuzzy Information and Engineering (ICFIE), Advances in Soft Computing Series, Vol. 40, Springer, [ISBN 978-3-540-71440-8] pp. 564-572. 2007 http://dx.doi.org/10.1007/978-3-540-71441-5 61 A12-c10 Po-Chang Ko, Ping-Chen Lin, Yao-Te Tsai, A Nonlinear Stock Valuation Using a Hybrid Model of Genetic Algorithm and Cubic Spline, In: Proceedings of the Second International Conference on Innovative Computing, Information and Control, 2007 (ICICIC 07). 2007 http://dx.doi.org/10.1109/icicic.2007.58 A12-c9 Takashi Hasuike, Hiroaki Ishii, Portfolio Selection Problems Considering Fuzzy Returns of Future Scenarios, In: Proceedings of the Second International Conference on Innovative Computing, Information and Control, 2007 (ICICIC 07). 2007 17

http://dx.doi.org/10.1109/icicic.2007.457 A12-c8 Lan, Yuping; Lv, Xuanli; Zhang, Weiguo A Linear Programming Model of Fuzzy Portfolio Selection Problem, IEEE International Conference on Control and Automation, (ICCA 2007), May 30 2007-June 1 20 07, Guangzhou, China, [ISBN: 978-1-4244-0818-4], pp. 3116-3118. 2007 http://www.ieeexplore.ieee.org/iel5/4376306/4376307/04376935.pdf? A12-c7 Bermudez, Jose D.; Segura, Jose V.; Vercher, Enriqueta, A fuzzy ranking strategy for portfolio selection applied to the Spanish stock market, Fuzzy Systems Conference, 2007 ( FUZZ-IEEE 2007), [doi 10.1109/FUZZY.2007.4295466], 23-26 July 2007, London, UK, pp. 1-4. 2007 http://ieeexplore.ieee.org/xpls/abs all.jsp?arnumber=4295466 Some fuzzy approaches to the portfolio selection problem have also been considered (see, for instance, Carlsson et al. [A12], Leon et al. [12], Tanaka and Guo [17] and Watada [23]). A12-c6 Zhang WG, Chen QQ, Lan HL, A portfolio selection method based on possibility theory, LECTURE NOTES IN COMPUTER SCIENCE 4041: 367-374 2006 http://dx.doi.org/10.1007/11775096 34 Carlsson [A12] introduced a possibilistic approach to selecting portfolios with highest utility score. (page 368) A12-c5 Chen YJ, Liu YK, Chen JF, Fuzzy portfolio selection problems based on credibility theory, LECTURE NOTES IN ARTIFICIAL INTELLIGENCE 3930: 377-386 2006 http://dx.doi.org/10.1007/11739685 40 On the other hand, based on possibility theory [4] [18], a lot of researchers such as Carlsson, Fullér and Majlender [A12], Inuiguchi and Tanino [6], Tanaka, Guo and Türksen [16] and León, Liern and Vercher [8] have devoted their efforts to the fuzzy portfolio selection problem. (page 377) A12-c4 Huang, X., Credibility based fuzzy portfolio selection, IEEE International Conference on Fuzzy Systems, art. no. 1681709, pp. 159-163. 2006 http://dx.doi.org/10.1109/fuzzy.2006.1681709 18

Many scholars such as Watada [34], Tanaka and Guo [30], Tanaka, Guo and Turksen [31], Parra et al [27] and Carlsson et al [A12] have employed possibility measure to describe security returns and extended Markowitzs mean-variance modelling idea in different ways. (page 159) A12-c3 Chen, Y.-J., Liu, Y.-K. Portfolio selection in fuzzy environment 2005 International Conference on Machine Learning and Cybernetics, ICMLC 2005, pp. 2694-2699 2005 http://dx.doi.org/10.1109/icmlc.2005.1527400 A12-c2 Zhang, J.-P., Li, S.-M. Portfolio selection with quadratic utility function under fuzzy enviornment 2005 International Conference on Machine Learning and Cybernetics, ICMLC 2005, pp. 2529-2533 2005 http://dx.doi.org/10.1109/icmlc.2005.1527369 in books A12-c1 B. Liu, Uncertainty Theory, Series: Studies in Fuzziness and Soft Computing, Vol. 154, Springer, [ISBN: 978-3-540-73164-1]. 2007 19