Question

Suppose the index model for stocks 1 and 2 has the following results
??1 =2%+0.8???? +??1,??2 =?1%+1.1???? +??2,???? =20%.
The standard deviations of ??1 and ??2 are 20% and 25% 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

R1 = 2% + 0.8Rm + e1,

Beta for stock1, b1 = 0.8 ( from the index model above)

R2 = -1% + 1.1Rm + e2

Beta for stock 2, b2 = 1.1

standard deviation of market , m = 20%

standard deviation of e1, s1 = 20%

standard deviation of e2, s2= 25%

1)

variance of stock 1, v1 = (b1*m)² +(s1)² =  (0.8*20)² +(20)² = 656

Standard deviation of stock 1 , S1= (v1)^(1/2) = (656)^(1/2) = 25.61249% = 25.61% ( rounding offf to 2 decimal places)

variance of stock 2, v2 = (b2*m)² +(s2)² =  (1.1*20)² +(25)² = 1109

Standard deviation of stock 2 , S2= (v2)^(1/2) = (1109)^(1/2) = 33.301651% = 33.30% ( rounding offf to 2 decimal places)

2)

covariance between stock 1 and market = b1/(m)² = 0.8/(20)² = 0.0002

covariance between stock 1 and market = b1/(m)² = 1.1/(20)² = 0.00275

3)

covariance between stock 1 and stock 2 = b1*b2*(m)² = 0.8*1.1*(20)² = 352

correlation coefficient = Covariance/(S1*S2) = 352/(25.61249*33.301651) = 0.412691094 OR 0.41 (rounding off to 2 decimal places)

4) variance of stock 1, v1 = (b1*m)² +(s1)² =  (0.8*20)² +(20)² = 656

where systematic risk =(b1*m)² = (0.8*20)² = 256

Firm-specific risk = (s1)² = (20)² = 400

for stock 2

variance of stock 2, v2 = (b2*m)² +(s2)² =  (1.1*20)² +(25)² = 1109

where systematic risk =(b2*m)² = (1.1*20)² = 484

Firm-specific risk = (s2)² = (25)² = 625