Portfolio spread when the correlation is given instead of covariance

Question

You hold one unit each of two assets with variances $\mathrm{Var}(A) = 25$ and $\mathrm{Var}(B) = 9$ . Their correlation is $-0.2$ . The combined value is

$P = A + B.$

What is the variance of the combined value?

Accepted Answer

The variance-of-a-sum rule needs a covariance, but here we are handed a correlation, so first convert. Covariance equals correlation times the two standard deviations: \mathrm{Cov}(A, B) = \rho \cdot \mathrm{SD}(A)\,\mathrm{SD}(B) = (-0.2)(5)(3) = -3. (The standard deviations are \sqrt{25} = 5 and \sqrt{9} = 3.) Now use the sum rule: \mathrm{Var}(A + B) = \mathrm{Var}(A) + \mathrm{Var}(B) + 2\,\mathrm{Cov}(A, B) = 25 + 9 + 2(-3) = 34 - 6 = 28. The standard deviation is \sqrt{28} \approx 5.29. Because the two assets are negatively correlated, they partly offset each other, so the combined vari…

Portfolio spread when the correlation is given instead of covariance

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