Question

You are considering making additional investments, and have gathered data about
two risky stocks:

Stock: CVX
Expected Return: 11.5%
Beta: 1.10
Firm-Specific Standard Deviation: 24%

Stock: Forsythe Inc.
Expected Return: 16%
Beta: .75
Firm-Specific Standard Deviation: 28%

Assumptions: The expected market return (as measured by the S&P 500 Index) is 10%. The
standard deviation of the S&P 500 Index is 15%.

a. Calculate the standard deviation of CVX.
b. Calculate the standard deviation of Forsythe.
c. Calculate the covariance between the two stocks.
d. Calculate the correlation between the two stocks.
e. How does the single-index model improve on the traditional Markowitz approach to
creating portfolios?

Answer 1

a) Standard deviation of CVX = (beta of CVX^2 * market variance + Firm Specific S.D of CVX^2) ^0.5

= (1.1^2*0.15^2+0.24^2)^0.5 = 0.291247 or 29.1247%

b)

Standard deviation of Forsythe = (beta of Forsythe^2 * market variance + Firm Specific S.D of Forsythe^2) ^0.5

= (0.75^2*0.15^2+0.28^2)^0.5 = 0.301755 or 30.1755%

c)

covariance between the two stocks. = beta of CVX*beta of Forsythe * market variance

= 1.1*0.75*0.15^2

= 0.0185625

d) correlation between the two stocks. = covariance between two stocks/ (SD of CVX*SD of Forsythe)

=0.0185625/(0.291247*0.301755)

=0.2112

e) The Single Index model reduces the requirement of input variables to many fold and only (3N+2) inputs are required for creating a portfolio of N stocks as compared to the requirement of (N^2+3N)/2 inputs in case of Markowitz model , thereby simplifying the process