A weighted possibilistic mean variance and covariance of fuzzy numbers

142 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − + + ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = ∈ 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 1 1 4 1 2 1 1 1 8 2 max λ subject to 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 = , ∑ = = n j j x 1 1 , n j M x M j j j , 1 , 2 1 = ≤ ≤ .

6.5 A weighted possibilistic mean variance and covariance of fuzzy numbers

The classical mean-variance portfolio selection problem uses the variance as the measure for risk, which puts the same weight on the down side and upside of the return. In this section, we study the “weighted” possibilistic mean-variance and covariance portfolio selection model. Definition 6.2 Fuller and Majlender, [53] Let F ∈ r~ be a fuzzy number with ] , [ ] ~ [ 2 1 λ λ λ r r r = , ] 1 , [ ∈ λ . The w -weighted possibilistic variance of r~ is ∫ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 1 2 1 2 2 ~ λ λ λ λ d w r r r Var w λ λ λ λ λ λ λ λ d w r r r r r r 2 2 2 1 1 2 2 1 2 2 1 2 1 ∫ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥⎦ ⎤ ⎢⎣ ⎡ + − + ⎥⎦ ⎤ ⎢⎣ ⎡ − + = where weighting function is non-decreasing and satisfies ∫ = 1 1 λ λ d w . 6.48 We note that the weighted possibilistic variance of r~ is defined as the expected value of the squared deviations between the arithmetic mean and the endpoints of its level sets, i.e. the lower possibility-weighted average of the squared distance between the left-hand endpoint and the arithmetic mean of the endpoints of its level sets plus the upper 143 possibility weighted average of the squared distance between the right-hand endpoint and the arithmetic mean of the endpoints their of its level sets. The standard deviation of r~ is defined by ~ ~ r Var r = σ 6.49 Let r~ fuzzy number and w be a weighting function, we define the weighted possibilistic variance of r~ by ∫ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 1 2 1 2 2 ~ λ λ λ λ d w r r r Var w and the weighted covariance of r~ and b ~ is defined as ∫ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = 1 1 2 1 2 2 . 2 , ~ λ λ λ λ λ λ d w b b r r b r Cov w . If ] 1 , [ , 2 ∈ = λ λ λ w ∫ − = 1 1 2 2 1 ~ λ λ λ λ d r r r Var w , 6.50 and ∫ − − = 1 1 2 1 2 2 2 1 , ~ λ λ λ λ λ λ d b b r r b r Cov w . 6.51 Let , , , ~ 4 3 2 1 r r r r r = and , , , ~ 4 3 2 1 b b b b b = be fuzzy numbers of trapezoidal form. Let ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − = − 1 1 1 2 2 1 γ λ γ λ w , where 1 ≥ γ , be a weighting function then the power-weighted variance and covariance r~ and b ~ are computed by λ λ λ λ γ γ d r r r Var w ∫ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = − 1 2 1 2 1 2 1 1 2 1 2 ~ ⎥⎦ ⎤ ⎢⎣ ⎡ − + + − + − + − − − = 3 18 1 2 8 1 2 1 2 1 4 1 2 2 4 3 4 3 1 2 2 1 2 γ γ γ γ r r r r r r r r 144 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + + − + − + − − − = 1 6 3 1 4 2 2 1 2 4 1 2 2 4 3 4 3 1 2 2 1 2 γ γ γ γ r r r r r r r r ∫ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − + − − + − − − = − 1 2 1 4 3 1 2 4 3 1 2 1 1 2 1 . 2 1 1 2 ~ , ~ λ λ λ λ γ γ d b b b b r r r r b r Cov w = ⎥⎦ ⎤ ⎢⎣ ⎡ − + + + − + + − + − − − − 1 6 3 1 4 2 1 2 4 1 2 4 3 4 3 4 3 4 3 1 2 1 2 1 2 γ γ γ γ b b r r r r b b r r b b r r Theorem 6.6 The mean-variance efficient portfolio model is ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∑ ∑ ∑ = = = ∈ n j j ji n j j ji w q i i R x x c x r E n 1 1 1 ~ ~ max λ 6.52 to subject i n j i j ji b x r Pos β ≤ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∑ = 1 ~ ~ , q i , 1 = , 6.53 ∑ = = n j j x 1 1 , 6.54 n j M x M j j j , 1 , 2 1 = ≤ ≤ . 6.55 In the next theorem we extend Theorem 6.3 to the case weighted possibility mean- variance approach with a special weighted λ w . Theorem 6.7 Let ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − − = − 1 1 1 2 2 1 γ λ γ λ w , 1 ≥ γ the weighted possibility mean variance of the trapezoidal 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 a trapezoidal number for VaR i level, q i , 1 = . For q i i , 1 , = λ , then the possibilistic mean variance portfolio selection model is 145 ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ + ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − + − ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = ∈ 1 4 4 1 2 1 2 4 1 2 max 1 4 1 3 1 1 2 2 1 1 1 2 1 γ γ γ γ λ n j j ji j n j ji j ji n j j ji n j n j j ji j ji q i i R x x r x r x r x r x r x r n + ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ − − ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ + − ∑ ∑ ∑ = = = n j j ji n j j ji n j j ji x c x r x r 1 2 1 4 1 3 1 6 12 1 2 γ γ 6.56 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 = , 6.57 ∑ = = n j j x 1 1 , 6.58 n j M x M j j j , 1 , 2 1 = ≤ ≤ . 6.59 Proof: The proof is the on the line of Theorem 6.5. □ 146 C HAPTER 7 ATZBERGER’S EXTENSION OF MARKOWITZ PORTFOLIO SELECTION

7.1 Introduction