Question

5. Suppose the index model for stocks 1 and 2 has the following results ?1 = 0.5% + 0.95?? + ?1, ?2 = −1.2% + 1.25?? + ?2, ?? = 25%. The standard deviations of ?1 and ?2 are 20% and 22% respectively. (

1) Calculate the standard deviation of stock 1 and stock 2.

(2) What is the covariance between each stock and the market index.

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

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

Answer 1

1. The standard deviation of each stock can be derived from the following equation for R²:

R_i²= Explained variance / total variance

Therefore, variance of stock 1 = [(Beta of stock 1)^2 X market variance] / R₁²= [(0.95)^2 X (25%)^2] / (20%)^2 = 141.02%

Hence, standard deviation of stock 1 = (141.02%)^0.5 = 118.75%.

Therefore, variance of stock 2= [(Beta of stock 1)^2 X market variance] / R_A²= [(1.25)^2 X (25%)^2] / (22%)^2 = 201.77%

Hence, standard deviation of stock 2 = (201.77%)^0.5 = 142.05%.

2. The covariance between each stock and the market index is calculated as below:

For stock 1

Standard deviation of e1 X Standard deviation of stock 1 X Standard deviation of market index

= 20% X 118.75% X 25% = 5.94%.

For stock 2

22% X 142.05% X 25% = 7.81%.

3. The covariance between the returns of stocks 1 and 2 are as follows:

Beta of stock 1 X Beta of stock 2 X Variance of market index = 0.95 X 1.25 X (25%)^2 = 7.42%

The correlation coefficient between the returns of stocks 1and 2 is:

Covariance between the returns of stocks 1 and 2 / (Std. dev. of stock 1 X Std. dev. of stock 2)

= 7.42% / (118.75% X 142.05%) = 4.40%.

4. The systematic risk for stock 1 = (0.95)^2 X (25%)^2 = 5.64%.

The firm-specific risk of stock 1 (the residual variance) is the difference between 1’s total risk and its systematic risk:

141.02% - 5.64% = 135.38%.

The systematic risk for stock 2= (1.25)^2 X (25%)^2 = 9.77%.

The firm-specific risk of stock 2(the residual variance) is the difference between 2’s total risk and its systematic risk:

201.77% - 9.77% = 192%.