bkmsol_ch08

# B beta is the slope of the scl which is the measure

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b. Beta is the slope of the SCL, which is the measure of systematic risk. The SCL for Stock B is steeper; hence Stock B’s systematic risk is greater. c. The R 2 (or squared correlation coefficient) of the SCL is the ratio of the explained variance of the stock’s return to total variance, and the total variance is the sum of the explained variance plus the unexplained variance (the stock’s residual variance): ) (e σ σ β σ β R i 2 2 M 2 i 2 M 2 i 2 + = Since the explained variance for Stock B is greater than for Stock A (the explained variance is , which is greater since its beta is higher), and its residual variance σ 2 M 2 B σ β 2 (e B ) is smaller, its R 2 is higher than Stock A’s. d. Alpha is the intercept of the SCL with the expected return axis. Stock A has a small positive alpha whereas Stock B has a negative alpha; hence, Stock A’s alpha is larger. e. The correlation coefficient is simply the square root of R 2 , so Stock B’s correlation with the market is higher. 4. a. Firm-specific risk is measured by the residual standard deviation. Thus, stock A has more firm-specific risk: 10.3% > 9.1% b. Market risk is measured by beta, the slope coefficient of the regression. A has a larger beta coefficient: 1.2 > 0.8 c. R 2 measures the fraction of total variance of return explained by the market return. A’s R 2 is larger than B’s: 0.576 > 0.436 d. Rewriting the SCL equation in terms of total return (r) rather than excess return (R): r A – r f = α + β (r M – r f ) r A = α + r f (1 β) + β r M The intercept is now equal to: α + r f (1 β) = 1 + r f (l – 1.2) Since r f = 6%, the intercept would be: 1 – 1.2 = –0.2% 8-3

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5. The standard deviation of each stock can be derived from the following equation for R 2 : = σ σ β = 2 i 2 M 2 i 2 i R Explained variance Total variance Therefore: % 30 . 31 980 20 . 0 20 7 . 0 R A 2 2 2 A 2 M 2 A 2 A = σ = × = σ β = σ For stock B: % 28 . 69 800 , 4 12 . 0 20 2 . 1 B 2 2 2 B = σ = × = σ 6. The systematic risk for A is: 196 20 70 . 0 2 2 2 M 2 A = × = σ β The firm-specific risk of A (the residual variance) is the difference between A’s total risk and its systematic risk: 980 – 196 = 784 The systematic risk for B is: 576 20 20 . 1 2 2 2 M 2 B = × = σ β B’s firm-specific risk (residual variance) is: 4800 – 576 = 4224 7. The covariance between the returns of A and B is (since the residuals are assumed to be uncorrelated): 336 400 20 . 1 70 . 0 ) r , r ( Cov 2 M B A B A = × × = σ β β = The correlation coefficient between the returns of A and B is: 155 . 0 28 . 69 30 . 31 336 ) r , r ( Cov B A B A AB = × = σ σ = ρ 8-4
8. Note that the correlation is the square root of R 2 : 2 R = ρ Cov(r A ,r M ) = ρσ A σ M = 0.20 1/2 × 31.30 × 20 = 280 Cov(r B ,r M ) = ρσ B σ M = 0.12 1/2 × 69.28 × 20 = 480 9. For portfolio P we can compute: σ P = [(0.6 2 × 980) + (0.4 2 × 4800) + (2 × 0.4 × 0.6 × 336] 1/2 = [1282.08] 1/2 = 35.81% β P = (0.6 × 0.7) + (0.4 × 1.2) = 0.90 958.08 400) (0.90 1282.08 σ β σ ) (e σ 2 2 M 2 P 2 P P 2 = × = = Cov(r P ,r M ) = β P 2 M σ =0.90 × 400=360 This same result can also be attained using the covariances of the individual stocks with the market: Cov(r P ,r M ) = Cov(0.6r A + 0.4r B , r M ) = 0.6Cov(r A , r M ) + 0.4Cov(r B ,r M ) = (0.6 × 280) + (0.4 × 480) = 360 10. Note that the variance of T-bills is zero, and the covariance of T-bills with any asset is zero. Therefore, for portfolio Q: [ ] [ ] % 55 . 21 ) 360 3 . 0 5 . 0 2 ( ) 400 3 . 0 ( ) 08 . 282 , 1 5 . 0 ( ) r , r ( Cov w w 2 w w 2 / 1 2 2 2 / 1 M P M P 2 M 2 M 2 P 2 P Q = × × × + × + × = × × × + σ + σ = σ 75 . 0 0 ) 1 3 . 0 ( ) 90 . 0 5 . 0 ( w w M

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