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Solutions+Problem+Set+9

Solutions+Problem+Set+9 - E 120 Problem Set 9 Solutions...

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E 120 : Problem Set 9 Solutions - Fall 2010 Problem 1 (a) Portfolio mean = E [ P ] = w 1 r 1 + w 2 r 2 = 0 . 2 × 0 . 05+0 . 8 × 0 . 2 = 0 . 17. Portfolio variance = V [ P ] = w 2 2 σ 2 2 = 0 . 8 2 × 0 . 15 2 = 0 . 0144. (b) Portfolio mean = E [ P ] = w 1 r 1 + w 2 r 2 = - 0 . 4 × 0 . 05 + 1 . 4 × 0 . 2 = 0 . 26. Portfolio variance = V [ P ] = w 2 2 σ 2 2 = 1 . 4 2 × 0 . 15 2 = 0 . 0441. Problem 2 In this case, the covariance matrix is given by V = 0 . 16 0 . 05 0 . 05 0 . 0625 . Hence, the portfolio variance is given by V ( P ) = x T V x = x 1 x 2 0 . 16 0 . 05 0 . 05 0 . 0625 x 1 x 2 = 0 . 16 x 2 1 + 0 . 1 x 1 x 2 + 0 . 0625 x 2 2 Using the constraint that x 2 = 1 - x 1 , we have V ( P )( x 1 ) = 0 . 16 x 2 1 + 0 . 1 x 1 (1 - x 1 ) + 0 . 0625(1 - x 1 ) 2 = 0 . 1225 x 2 1 - 0 . 025 x 1 + 0 . 0625 For V ( P )( x 1 ) to be minimized at x 1 , it is sufficient that V 0 ( P )( x 1 ) = 0 and V 00 ( P )( x 1 ) > 0, i.e. V 0 ( P )( x 1 ) = 0 . 245 x 1 - 0 . 025 = 0 = x 1 = 0 . 025 0 . 245 = 0 . 102 and V 00 ( P )( x 1 ) = 0 . 245 > 0 Hence, we have x 2 = 1 - 0 . 102 = 0 . 898, and the required expected return is given by E [ P ] = x 1 r 1 + x 2 r 2 = 0 . 102 × 0 . 2 + 0 . 898 × 0 . 1 = 0 . 1102 Problem 3 (a) Asset 1 has a higher mean return and a lower variance of return. It offers a higher expected return at lower risk. Hence, Asset 1 must be chosen over Asset 2.
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