variance of returns

From an old mock exam:

Hello all, I’ve been trying to do this question all day and am pretty stumped…I just don’t know how to apply the variance of the portfolio formula / covariance / correlation formulas to solve this problem. Does anyone know a way I can solve this please? :slight_smile:

An investor’s portfolio currently has an expected return of 11% with a variance of 0.0081. She is considering replacing 20% of the portfolio with a security that has an expected return of 12% and a standard deviation of 0.07. If the covariance between the returns on the exiting portfolio and the returns on the added security is 0.0058, the variance of returns on the new portfolio will be closest to:

  • A. 0.00545
  • B. 0.00724
  • C. 0.00984

Answer: B

use the formula:

(w1^2)Var(1) +(w2^2)Var(2) +2(w1)(w2)Cov(1,2)

Note that the risk of asset 1 is already quoted as variance and the risk of asset 2 is quoted only as standard deviation.

Var(new portfolio) = 0.8^2 *var (existing) +0.2^2 *var(new) + 2 * 0.2 * 0.8 * cov(existing, new)

= 0.8^2 * 0.0081 + 0.2^2 * (0.07^2) + 2 * 0.8 * 0.2 * 0.0058

= 0.007236

They gave you the standard deviation of the new security, which is why we square the 0.07.

Ahh I think I was over thinking this one, thanks!