94 528 8 2 01 42 8 y in contrast with a passive

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94 . 528 8 . 2 01 . 0 42 . 8 y = × × = In contrast, with a passive strategy: 5401 . 0 23 8 . 2 01 . 0 8 y 2 = × × = This is a difference of: 0.0284 The final positions of the complete portfolio are: Bills 1 – 0.5685 = 43.15% M 0.5685 × l.0486 = 59.61% A 0.5685 × (–0.0486) × (–0.6142) = 1.70% B 0.5685 × (–0.0486) × 1.1265 = – 3.11% C 0.5685 × (–0.0486) × (–1.2181) = 3.37% D 0.5685 × (–0.0486) × 1.7058 = – 4.71% 100.00% [sum is subject to rounding error] Note that M may include positive proportions of stocks A through D. 19. a. If a manager is not allowed to sell short he will not include stocks with negative alphas in his portfolio, so he will consider only A and C: α σ 2 (e) α σ 2 (e) α / σ 2 (e) Σα / σ 2 (e) A 1.6 3,364 0.000476 0.3352 C 3.4 3,600 0.000944 0.6648 0.001420 1.0000 The forecast for the active portfolio is: α = (0.3352 × 1.6) + (0.6648 × 3.4) = 2.80% β = (0.3352 × 1.3) + (0.6648 × 0.7) = 0.90 σ 2 (e) = (0.3352 2 × 3,364) + (0.6648 2 × 3,600) = 1,969.03 σ( e ) = 44.37% 8-10
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The weight in the active portfolio is: 0940 . 0 23 / 8 03 . 969 , 1 / 80 . 2 / ) R ( E ) e ( / w 2 2 M M 2 0 = = σ σ α = Adjusting for beta: 0931 . 0 ] 094 . 0 ) 90 . 0 1 [( 1 094 . 0 w ) 1 ( 1 w * w 0 0 = × + = β + = The information ratio of the active portfolio is: A = α / σ (e) =2.80/44.37 = 0.0631 Hence, the square of Sharpe’s measure is: S 2 = (8/23) 2 + 0.0631 2 = 0.1250 Therefore: S = 0.3535 The market’s Sharpe measure is: S M = 0.3478 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
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