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solna2 - Actsc 372 F08 Assignment 2 Solution(Total marks 30...

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Actsc 372 F08: Assignment 2 Solution (Total marks: 30) 1. (2 marks each) (a), (b) and (c) E ( A ) = 0 . 3(0 . 123 + 0 . 065 + 0 . 03) = 0 . 0726667 E ( B ) = 0 . 3( - 0 . 037 + 0 . 03 + 0 . 05) = 0 . 0143333 V ar ( A ) = E [ A - E ( A )] 2 = 0 . 00147089 s.d. ( A ) = 0 . 03835 V ar ( B ) = E [ B - E ( B )] 2 = 0 . 00138422 s.d. ( B ) = 0 . 0372 cov ( A, B ) = E [( A - E ( A ))( B - E ( B ))] = - . 00140855555 ρ = - 0 . 00140855555 0 . 03835 × 0 . 0372 = - 0 . 9871 2. (a) (3 marks) Solution 1: σ 2 P = x 2 1 σ 2 1 + (1 - x 1 ) 2 σ 2 2 + 2 x 1 (1 - x 1 ) · ρσ 1 σ 2 ∂σ 2 P x 1 = 2 x 1 σ 2 1 - 2(1 - x 1 ) σ 2 2 + (2 - 4 x 1 ) · ρσ 1 σ 2 = 0 x MIN 1 = σ 2 2 - ρσ 1 σ 2 σ 2 1 + σ 2 2 - 2 ρσ 1 σ 2 = 0 . 82 x MIN 2 = 1 - x MIN 1 = 0 . 18 μ x MIN = 0 . 82 μ 1 + 0 . 12 μ 2 = 9 . 08% σ 2 x MIN = [0 . 82(0 . 15)] 2 + 2(0 . 82)(0 . 18)(0 . 3)(0 . 15)(0 . 25) + [0 . 18(0 . 25)] 2 = 0 . 020475 σ x MIN = 14 . 309% Solution 2: Σ = 0 . 0225 0 . 01125 0 . 01125 0 . 0625 D = 0 . 0225(0 . 0625) - 0 . 01125 2 = 0 . 0012796875 Σ - 1 = 48 . 84004884 - 8 . 791208791 - 8 . 791208791 17 . 58241758 x MIN = 1 e T Σ - 1 e T Σ - 1 e = 1 48 . 84004884 40 . 04884005 8 . 791208789 = 0 . 82 0 . 18 μ x MIN = μ T x MIN = 9 . 08% σ 2 x MIN = ( x MIN ) T Σ - 1 x MIN = 0 . 020475 (b) i. (2 marks) x 1 μ 1 + (1 - x 1 ) μ 2 = 0 . 12 x 1 = 1 3 σ 2 P = 0 . 0352778 σ P = 18 . 78% 3
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ii. (3 marks) x 1 = - 2 / 3 , x 2 = 5 / 3 with σ P = 39 . 826%. This implies that to achieve such a high expected return of 18% requires to short-sell the bond fund. (c) i. (1 mark) z * = Σ - 1 μ - e T Σ - 1 μ e T Σ - 1 e Σ - 1 e = 2 . 67643 1 . 75824 - 4 . 43468 48 . 84004 40 . 04884 8 . 79121 = - 0 . 96 0 . 96 ii. (2 marks) μ x * = μ x MIN + τμ z * = 0 . 0908 + 0 . 0576 τ σ 2 x * = 0 . 020475 + 0 . 0576 τ 2 iii. (3 marks) 0 . 12 = 0 . 0908 + 0 . 0576 τ τ = 0 . 5069 0 . 18 = 0 . 0908 + 0 . 0576 τ τ = 1 . 5486 hence higher value τ implies less risk averse 3. (4 marks) σ z * = 0 implies ( z * ) T Σ z * = 0. Since Σ is positive definite, which leads to
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