Question

Suppose that the index model for stocks A and B is estimated from excess returns with the following results:

RA = 5.0% + 1.30RM + eA

RB = –2.0% + 1.6RM + eB

σM = 20%; R-squareA = 0.20; R-squareB = 0.12

Break down the variance of each stock to the systematic and firm-specific components. (Do not round intermediate calculations. Calculate using numbers in decimal form, not percentages. Round your answers to 4 decimal places.)

Answer 1

We can calculate the total standard deviation of the stocks using the R-squared equation provided below:

R-squared = Variance explained by External factor / Overall variance

Using the above equation, we can get following equation for stock A:

R-squared_A = (Beta_A² * (Standard Deviation of market)²) / (Total standard deviation_A)²

Beta of A = 1.3; Standard Deviation of Market = 20% or 0.20; R-squared of A = 0.20

=> 0.20 = (1.3 * 0.20)² / (Total Variance of A)

=> Total Variance of A = 1.3² * 0.20 = 0.3380 | Total Standard Dev of A = (0.338)^1/2 = 0.5814

Similarly, we can calculate Total Variance of B, where Beta of B = 1.6; R-squared of B = 0.12

=> 0.12 = (1.6 * 0.20)² / (Total Variance of B)

=> Total Variance of B = 0.1024 / 0.12 = 0.8533 | Total Standard Dev of B = (0.8533)^1/2 = 0.9238

Now, we will calculate Systematic risk for each stock, using which we can find the Firm-specific variance of the stocks.

Systematic risk or variance of A = Beta_A² * Variance of market = (1.3 * 0.20)² = 0.0676

Firm-specific variance of A = Total Variance of A - Systematic Variance of A = 0.3380 - 0.0676 = 0.2704

Systematic risk or variance of B = Beta_B² * Variance of market = (1.6 * 0.20)² = 0.1024

Firm-specific variance of B = Total Variance of B - Systematic Variance of B = 0.8533 - 0.1024 = 0.7509