Chapter 8 - Chapter 8 Answer Key Problem Sets 5-12 CFA...

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Chapter 8 – Answer Key: Problem Sets: 5-12 CFA Problems: 2 & 5 Problem Sets: 5. a. To optimize this portfolio one would need: n = 60 estimates of means n = 60 estimates of variances 770 , 1 2 n n 2 = estimates of covariances Therefore, in total: 890 , 1 2 n 3 n 2 = + estimates b. In a single index model: r i r f = α i + β i (r M r f ) + e i Equivalently, using excess returns: R i = α i + β i R M + e i The variance of the rate of return on each stock can be decomposed into the components: (l) The variance due to the common market factor: 2 M 2 i σ β (2) The variance due to firm specific unanticipated events: ) e ( i 2 σ In this model: σ β β = j i j i ) r , r ( Cov The number of parameter estimates is: n = 60 estimates of the mean E(r i ) n = 60 estimates of the sensitivity coefficient β i n = 60 estimates of the firm-specific variance σ 2 (e i ) 1 estimate of the market mean E(r M ) 1 estimate of the market variance 2 M σ Therefore, in total, 182 estimates. Thus, the single index model reduces the total number of required parameter estimates from 1,890 to 182. In general, the number of parameter estimates is reduced from: ) 2 n 3 ( to 2 n 3 n 2 + +
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6. a. The standard deviation of each individual stock is given by: 2 / 1 i 2 2 M 2 i i )] e ( [ σ + σ β = σ Since β A = 0.8, β B = 1.2, σ (e A ) = 30%, σ (e B ) = 40%, and σ M = 22%, we get: σ A = (0.8 2 × 22 2 + 30 2 ) 1/2 = 34.78% σ B = (1.2 2 × 22 2 + 40 2 ) 1/2 = 47.93% b.
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