Question

1.i) Suppose that there are many different companies whose stocks have the same beta, say 1. Can you form a portfolio to diversify the risk to get a lower beta?

ii) Single factor model. Suppose that the single factor model for stocks A and B is estimated from excess returns as follows

R^e_A = 3% + 0.7R^e_M + ε_A

R^e_B = −2% + 1.2R^e_M + ε_B

where, σ_M = 20% , R-squared of A is 20% and R-squared of B is 12%.

(a) What is the standard deviation of each stock?

(b) Break down the variance of each stock into its systematic and firm-specific components.

(c) Calculate information ratio of each stock.

(d) What are covariance and the correlation coefficient between the two stocks?

(e) What is the covariance between each stock and the market index? (f) What is the standard deviation and market beta of portfolio P with weights of 0.60 in A and 0.40 in B ?

Answer 1

1.(i)

There are many different companies, whose stocks have the same beta say 1.

Portfolio Beta = Sum of (Ws* B_s)

Where, W_s is weight of stock s and Sum(W_s) = 1

B_s is Beta of stock s

Since all stocks have same beta therefore, any portfolio we form have a beta of 1.

Hence, we can’t form a portfolio to diversify the risk to get a lower beta.

(ii)

R^e_A= 3%+ 0.7 R^e_M + e_A

Beta of Stock A = B_A = 0.7

R^e_B= -2%+ 1.2 R^e_M + e_B

Beta of Stock B = B_B = 1.2

sigma_M = 20%

Variance of Market index = (sigma_M )²= 0.20² = 0.0400

a)

R-square_A = 20% = Proportion of variance due to market / total variance of the stock A = (B_A x sigma_M )² / sigma_A²

Sigma_A² = (B_A x sigma_M )² / R-square_A = (0.7 x 0.20)²/ 0.20 = 0.098

Standard deviation of A = sigma_A = 0.3130 = 31.30%

R-square_B = 12% = Proportion of variance due to market / total variance of the stock B = (B_B x sigma_M )² / sigma_B²

Sigma_B² = (B_A x sigma_M )² / R-square_B = (1.2 x 0.20)²/ 0.12 = 0.48

Standard deviation of B = sigma_B = 0.6928 = 69.28%

b)

sigma_A ²= (B_A x sigma_M)² + sigma²(e_A) = Systematic component + Firm specific component

Systematic component = (B_A x sigma_M)² = (0.7 x 0.20)² = 0.0196

Firm specific component = sigma²(e_A) = sigma_A ²- (B_A x sigma_M)² = 0.098 - 0.0196 = 0.0784

Sigma_B ²= (B_B x sigma_M)² + sigma²(e_B) = Systematic component + Firm specific component

Systematic component = (B_B x sigma_M)² = (1.2 x 0.20)² = 0.0576

Firm specific component = sigma²(e_A) = sigma_A ²- (B_A x sigma_M)² = 0.48 - 0.0182 = 0.4224

c)

Information ratio = alpha / Std dev

Information ratio of A = 3% / 0.3130 = 0.0958

Information ratio of B = -2% / 0.6928 = -0.0289

d)

Covariance = Product of betas x Variance of market index

Cov (R_A , R_B) = B_A x B_B x sigma_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) / (sigma_A x sigma_B) = 0.0336/ (0.3130 x 0.6928) = 0.1549

e)

Covariance between stock and the market index

Covariance = stock beta x Variance of market index

Cov (A,M) = B_A x sigma_M² = 0.7 x 0.20² = 0.028

Cov (B,M) = B_B x sigma_M² = 1.2 x 0.20² = 0.048

f)

W_A =0.60

W_B = 0.40

a) Standard deviation of portfolio = √{ W_A²* sigma_A ² + W_B²* sigma_B ² + 2* W_A* W_B* Cov (R_A , R_B)}

=√{0.60²*0.098 + 0.40²*0.48 + 2*0.60*0.40*0.0336}

Standard deviation of portfolio P = 0.3076 = 30.76%

Beta of portfolio P= W_A* B_A + W_B* B_B

= 0.60*0.7 + 0.40*1.2 = 0.90