Question

Suppose that the index model for stocks A and B is estimated from excess returns with the following results: Ra : 3% + 0.7 Rm + ea Rs--2% + 1.2 Rm + eb What is the standard deviation for each stock? Break down the variance of each stock to the systematic and firm-specific components What are the covariance and correlation coefficient between the two stocks? What is the covariance between each stock and the market index? Are the intercepts of the two regressions consistent with the CAPM? Interpret their values. a. b. c. d. e.

Answer 1

Since there are multiple sub parts to the question, I will answer the first four.

Part (a)

R_a = 3% + 0.7 R_m + e_a

Beta of A = B_a = 0.7

_m = 20% = 0.20

R²_a = 0.20 = Proportion of variance due to market / total variance of the stock A = (B_a x _m )² / _a²

Hence,  _a² = (B_a x _m )² / R²_a  = (0.7 x 0.2)² / 0.2 = 0.098

Hence, standard deviation of stock A, _a = = 0.3130 = 31.30%

Similarly,

_b² = (B_b x _m )² / R²_b  = (1.2 x 0.2)² / 0.12 = 0.4800

Hence, standard deviation of stock B, _b = = 0.6928 = 69.28%

Part (b)

For Stock A:

_a² = (B_a x _m )² + ²(e_a) = Systematic component + Firm specific component

Systematic component = (B_a x _m)² = (0.7 x 0.2)² = 0.0196

Firm specific component = ²(e_a) = _a² - (B_a x _m )² = 0.3130² - 0.0196 = 0.0784

Similarly for Stock B

Systematic component = (B_b x _m)² = (1.2 x 0.2)² = 0.0576

Firm specific component = ²(e_b) = _b² - (B_b x _m )² = 0.6928² - 0.0576 = 0.4224

Part (c)

Covariance = Product of betas x market index risk

Cov (R_a , R_b) = B_a x B_b x _m² = 0.7 x 1.2 x 0.20² = 0.0336

Correlation coefficient between the two stocks = Covariance / Product of standard deviations = Cov (R_a , R_b) / (_a x _b) = 0.0336 / (0.3130 x 0.6928) = 0.1549

Part (d)

Beta of a stock = Covariance between stock's return and market return / variance of the market return

Hence, Cov between stock's return and market return = Beta x Variance of the market return

Hence, Cov(R_a, R_m) = B_a x _m² = 0.7 x 0.20² = 0.0280

Cov(R_b, R_m) = B_b x _m² = 1.2 x 0.20² = 0.0480