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# bkmsol_ch10 - CHAPTER 10 INDEX MODELS l a To optimize this...

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CHAPTER 10: INDEX MODELS l. 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 + + 10-1

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2. 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. The expected rate of return on a portfolio is the weighted average of the expected returns of the individual securities: E(r P ) = w A E(r A ) + w B E(r B ) + w f r f where w A , w B , and w f are the portfolio weights for Stock A, Stock B, and T-bills, respectively. Substituting in the formula we get: E(r P ) = (0.30 × 13) + (0.45 × 18) + (0.25 × 8) = 14% The beta of a portfolio is similarly a weighted average of the betas of the individual securities: β P = w A β A + w B β B + w f β f The beta for T-bills ( β f ) is zero. The beta for the portfolio is therefore: β P = (0.30 × 0.8) + (0.45 × 1.2) + 0 = 0.78 The variance of this portfolio is: ) e ( P 2 2 M 2 P 2 P σ + σ β = σ where 2 M 2 P σ β is the systematic component and ) e ( P 2 σ is the nonsystematic component. Since the residuals (e i ) are uncorrelated, the non-systematic variance is: ) e ( w ) e ( w ) e ( w ) e ( f 2 2 f B 2 2 B A 2 2 A P 2 σ + σ + σ = σ = (0.30 2 × 30 2 ) + (0.45 2 × 40 2 ) + (0.25 2 × 0) = 405 where σ 2 (e A ) and σ 2 (e B ) are the firm-specific (nonsystematic) variances of Stocks A and B, and σ 2 (e f ), the nonsystematic variance of T-bills, is zero. The residual standard deviation of the portfolio is thus: σ (e P ) = (405) 1/2 = 20.12% The total variance of the portfolio is then: 47 . 699 405 ) 22 78 . 0 ( 2 2 2 P = + × = σ 10-2
The standard deviation is 26.45%.

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