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# When short sales are allowed problem 18 the managers

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When short sales are allowed (Problem 18), the manager’s Sharpe measure is higher (0.3662). The reduction in the Sharpe measure is the cost of the short sale restriction. The characteristics of the optimal risky portfolio are: β P = w M + w A × β A = (1 – 0.0931) + (0.0931 × 0.9) = 0.99 E(R P ) = α P + β P E(R M ) = (0.0931 × 2.8%) + (0.99 × 8%) = 8.18% 54 . 535 ) 03 . 969 , 1 0931 . 0 ( ) 23 99 . 0 ( ) e ( 2 2 P 2 2 M 2 P 2 P = × + × = σ + σ β = σ % 14 . 23 P = σ With A = 2.8, the optimal position in this portfolio is: 5455 . 0 54 . 535 8 . 2 01 . 0 18 . 8 y = × × = The final positions in each asset are: Bills 1 – 0.5455 = 45.45% M 0.5455 × (1 0.0931) = 49.47% A 0.5455 × 0.0931 × 0.3352 = 1.70% C 0.5455 × 0.0931 × 0.6648 = 3.38% 100.00% 8-11

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b. The mean and variance of the optimized complete portfolios in the unconstrained and short-sales constrained cases, and for the passive strategy are: E(R C ) 2 C σ Unconstrained 0.5685 × 8.42 = 4.79 0.5685 2 × 528.94 = 170.95 Constrained 0.5455 × 8.18 = 4.46 0.5455 2 × 535.54 = 159.36 Passive 0.5401 × 8.00 = 4.32 0.5401 2 × 529.00 = 154.31 The utility levels below are computed using the formula: 2 C C A 005 . 0 ) r ( E σ Unconstrained 8 + 4.79 – (0.005 × 2.8 × 170.95) = 10.40 Constrained 8 + 4.46 – (0.005 × 2.8 × 159.36) = 10.23 Passive 8 + 4.32 – (0.005 × 2.8 × 154.31) = 10.16 20. All alphas are reduced to 0.3 times their values in the original case. Therefore, the relative weights of each security in the active portfolio are unchanged, but the alpha of the active portfolio is only 0.3 times its previous value: 0.3 × 16.90% = 5.07% The investor will take a smaller position in the active portfolio. The optimal risky portfolio has a proportion w * in the active portfolio as follows: 01537 . 0 23 / 8 6 . 809 , 21 / 07 . 5 / ) r r ( E ) e ( / w 2 2 M f M 2 0 = = σ σ α = The negative position is justified for the reason given earlier. The adjustment for beta is: 0151 . 0 )] 01537 . 0 ( ) 08 . 2 1 [( 1 01537 . 0 w ) 1 ( 1 w * w 0 0 = × + = β + = Since w* is negative, the result is a positive position in stocks with positive alphas and a negative position in stocks with negative alphas. The position in the index portfolio is: 1 – (–0.0151) = 1.0151 To calculate Sharpe’s measure for the optimal risky portfolio we compute the information ratio for the active portfolio and Sharpe’s measure for the market portfolio. The information ratio of the active portfolio is 0.3 times its previous value: A = α / σ (e)= –5.07/147.68 = –0.0343 and A 2 =0.00118 8-12
8-13 Hence, the square of Sharpe’s measure of the optimized risky portfolio is: S 2 = S 2 M + A 2 = (8/23) 2 + 0.00118 = 0.1222 S = 0.3495 Compare this to the market’s Sharpe measure: S M = 8/23 = 0.3478 The difference is: 0.0017 Note that the reduction of the forecast alphas by a factor of 0.3 reduced the squared information ratio and the improvement in the squared Sharpe ratio by a factor of: 0.3 2 = 0.09
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When short sales are allowed Problem 18 the managers Sharpe...

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