In: Finance
Suppose that the index model for stocks A and B is estimated from excess returns with the following results:
RA = 3% + 0.7RM+eA
RB = -2%+1.2RM+eB
R2A= 0.2 R2B = 0.12σM = 20%
For portfolio P with investment proportions of 0.60 in A and 0.40 in B,
a. What is the standard deviation of each stock?
b. Break down the variance of each stock to the systematic and firm-specific components.
c. What is the covariance between each stock and the market index?
For portfolio Q with investment proportions of 0.50 in P, 0.30 in the market
index, and 0.20 in T-bills.
d. What is the standard deviation of each stock?
e. Break down the variance of each stock to the systematic and firm-specific components.
f. What is the covariance between each stock and the market index?
RA = 3% + 0.7RM + eA
RB = -2% + 1.2RM + eB
σM = 20%; R-squareA = 0.20; R-squareB = 0.12
The Formula for R-Squared Is
R2 = Systematic Variation/Total Variation
Systematic Risk = β⋅σmarket ⇒ Systematic Variance = (Systematic Risk)2
(a) 1. Stock A
Systematic Variance = (0.7*20%)^2 = 1.96%
Total Variance = Systematic Variance / R2 = 1.96% / 0.2 = 9.8%
(Standard deviation)A = SQRT(Total Variance) = 31.30%
2. Stock B
Systematic Variance = (1.2*20%)^2 = 5.76%
Total Variance = Systematic Variance / R2 = 1.96% / 0.2 = 48%
(Standard deviation)A = SQRT(Total Variance) = 69.28%
(b) Total Variance = Systematic Variance+Unsystematic Variance
Unsystematic Variance = Total Variance - Systematic Variance
Stock | Total Variance | Systematic Variance | Unsystematic Variance |
A | 9.80% | 1.96% | 7.84% |
B | 48.00% | 5.76% | 42.24% |
(c) β = Covar(Ri, Rm) / σm^2
1. Stock A : Covar(Ra, Rm) = βa*σm^2 = 0.7*(20%)^2 = 2.80%
2. Stock B : Covar(Rb, Rm) = βb*σm^2 = 1.2*(20%)^2 = 4.80%
(d) Standard deviation of stocks A & B are 31.30% and 69.28% respectively.
An index fund essentially tracks the market index benchmark, and hence its standard deviation is equal to the standard deviation of the market returns itself. Therefore, Standard deviation of market index is 20%.
A T-bill has fixed returns and is treated as a risk-free asset, hence the standard deviation of T-bill returns is zero.
(e) An index find is assumed to be well-diversified just like the market index. Hence, its unsystematic variance is considered to zero and its total risk = β*σm
Stock | Total Variance | Systematic Variance | Unsystematic Variance |
A | 9.80% | 1.96% | 7.84% |
B | 48.00% | 5.76% | 42.24% |
Market Index | 4% | 4% | 0.00% |
T-bill | 0% | 0% | 0.00% |
(f) β = Covar(Ri, Rm) / σm^2
1. Stock A : Covar(Ra, Rm) = βa*σm^2 = 0.7*(20%)^2 = 2.80%
2. Stock B : Covar(Rb, Rm) = βb*σm^2 = 1.2*(20%)^2 = 4.80%
3. Market Index: Covar(Rmi, Rm) = βmi*σm^2 = 1*(20%)^2 = 4.00%
4. T-bill: Covar(Rt, Rm) = βt*σm^2 = 0*(20%)^2 = 0%