FRM Part I · FRM Exam Part I · Multivariate Random Variables
A portfolio holds 60% in asset X and 40% in asset Y. The returns are jointly normal with standard deviations of 10% for X and 20% for Y, and a correlation of 0.25. What is the portfolio standard deviation?
The portfolio variance is 0.0036 + 0.0064 + 0.0024 = 0.0124, which includes twice the weighted covariance term. Its square root is about 11.14%. This is below the 14% weighted average of volatilities because correlation is below one.
- A11.14%Correct
- B10.00%
- C10.58%
- D14.00%
Explanation
Variance = 0.6^2*0.10^2 + 0.4^2*0.20^2 + 2*0.6*0.4*0.25*0.10*0.20 = 0.0036 + 0.0064 + 0.0024 = 0.0124. The square root is 11.14%. Ignoring the covariance gives 10.00%. Leaving out the factor of 2 on the covariance gives 10.58%. The weighted average of the standard deviations gives 14.00%, which ignores diversification.
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