Construction efficient portfolios Possibilistic mean Var portfolio selection model.

135 where E ~ denotes fuzzy mean operator. We can see that if 4 3 2 1 , , , ~ r r r r r = is a trapezoidal fuzzy number then 6 3 ~ ~ 4 1 3 2 1 3 4 4 1 2 1 r r r r d r r r r r r r E + + + = − − + − + = ∫ λ λ λ λ . 6.16

6.3.3 Construction efficient portfolios

Let j x the proportional of the total amount of money devoted to security j, j M 1 and j M 2 represent the minimum and maximum proportion respectively of the total amount of money devoted to security j . The trapezoidal fuzzy number of ji r is 4 3 2 1 , , , ~ ji ji ji ji ji r r r r r = where 4 3 2 1 ji ji ji ji r r r r ≤ . In addition, we denote the VaR i level by the fuzzy number trapezoidal 4 3 2 1 , , , ~ i i i i i b b b b b = , q i , 1 = . Using this approach we see that the model given by 6.5-6.8 reduces itself to the form from the following theorem . Theorem 6.1 The possibilistic mean VaR portfolio selection for the vector mean VaR efficient portfolio model 6.5-6.8 is ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ − ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∑ ∑ ∑ ∑ = = = = ∈ n j j jq n j j jq n j j j n j j j R x x c x r E x c x r E n 1 1 1 1 1 1 ~ ~ ,..., ~ ~ max 6.17 i n j i j ji b x r Pos to subject β ≤ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∑ = 1 ~ ~ , q i , 1 = , 6.18 ∑ = = n j j x 1 1 , 6.19 n j M x M j j j , 1 , 2 1 = ≤ ≤ . 6.20 In the following using Section 6.3 we obtain the efficient portfolios given by the Theorem 6.1. Theorem 6.2. If q i i , 1 , = λ , then an efficient portfolio for possibilistic model is an optimal solution of the following problem: 136 ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∑ ∑ ∑ = = = ∈ n j j ji n j j ji q i i R x x c x r E n 1 1 1 ~ ~ max λ 6.21 i n j i j ji b x r Pos to subject β ≤ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∑ = 1 ~ ~ , q i , 1 = , 6.22 ∑ = = n j j x 1 1 , 6.23 n j M x M j j j , 1 , 2 1 = ≤ ≤ . 6.24 Using the fact that rate of return on security , 1 n j j = is given by trapezoidal fuzzy number, then we get the following results. Theorem 6.3 Let rate of return on security , 1 n j j = by the trapezoidal fuzzy number 4 3 2 1 , , , ~ ji ji ji ji ji r r r r r = where 4 3 2 1 ji ji ji ji r r r r ≤ and addition 4 3 2 1 , , , ~ i i i i i b b b b b = is trapezoidal fuzzy number for VaR level and i λ ,with q i , 1 = . Then using the possibilistic mean VaR portfolio selection model an efficient portfolio is an optimal solution for the following problem: ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − + + + ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = ∈ n j j ji n j j ji n j j ji n j j ji n j j ji q i i R x x c x r x r x r x r n 1 1 4 1 1 1 3 1 2 1 6 3 max λ 6.25 : . to ubject s 1 1 3 2 1 4 1 ≥ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − + ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − ∑ ∑ = = n j i j ji i n j i j ji i b x r b x r β β , q i , 1 = , 6.26 ∑ = = n j j x 1 1 , 6.27 n j M x M j j j , 1 , 2 1 = ≤ ≤ . 6.28 Proof : Really, from the equation 6.16, we have 137 6 3 ~ ~ 1 4 1 1 1 3 1 2 1 ∑ ∑ ∑ ∑ ∑ = = = = = + + + = ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ n j j ji n j j ji n j j ji n j j ji n j j ji x r x r x r x r x r E , p i , 1 = . From Lemma 6.1, we have that, for any q i , 1 = , i n j i j ji b x r Pos β ≤ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∑ =1 ~ ~ , is equivalent with 1 1 3 2 1 4 1 ≥ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − + ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − ∑ ∑ = = n j i j ji i n j i j ji i b x r b x r β β . Furthermore, from 6.25-6.28 given by Theorem 6.2, we get that this problem is of the following form : ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − + + + ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = ∈ n j j ji n j j ji n j j ji n j j ji n j j ji q i i R x x c x r x r x r x r n 1 1 4 1 1 1 3 1 2 1 6 3 max λ 6.29 to subject 1 1 3 2 1 4 1 ≥ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − + ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − ∑ ∑ = = n j i j ji i n j i j ji i b x r b x r β β , q i , 1 = 6.30 ∑ = = n j j x 1 1 , 6.31 n j M x M j j j , 1 , 2 1 = ≤ ≤ . 6.32 This completes the proof. □ Problem 6.29-6.32 is a standard multi-objective linear programming problem. For optimal solution we can used some algorithms of multiobjective programming Slowinski and Teghem [175] and White [209]. 138

6.4 A Weighted possibilistic mean value approach