Variance of a blend as a covariance with itself

Question

Two factors have variances \mathrm{Var}(X) = 3 and \mathrm{Var}(Y) = 2, with covariance \mathrm{Cov}(X, Y) = 1. Consider the blend W = 2X - 3Y. What is the variance of W?

Accepted Answer

The variance of a blend is just its covariance with itself, so the same expand-term-by-term method applies. Squaring the two coefficients gives the variance weights, and the cross term carries twice the product of the coefficients times the covariance: \mathrm{Var}(aX + bY) = a^2\,\mathrm{Var}(X) + b^2\,\mathrm{Var}(Y) + 2ab\,\mathrm{Cov}(X, Y). In words: square each weight for its own variance, then add twice the product of the weights times the covariance. Apply it with a = 2 and b = -3: \mathrm{Var}(W) = 2^2(3) + (-3)^2(2) + 2(2)(-3)(1) = 12 + 18 - 12 = 18. The standard deviation is \sqrt{…

Variance of a blend as a covariance with itself

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