Covariance of two overlapping index weightings

Question

Two factors have variances \mathrm{Var}(X) = 5 and \mathrm{Var}(Y) = 3, with covariance \mathrm{Cov}(X, Y) = 1. Two indices are built as U = X + Y, \qquad V = X + 2Y. What is the covariance of U and V?

Accepted Answer

Expand the covariance of the two blends term by term, treating each as a product to multiply out. Constants come out front, and a same-variable piece becomes a variance. \mathrm{Cov}(X + Y,\; X + 2Y) = \mathrm{Cov}(X, X) + \mathrm{Cov}(X, 2Y) + \mathrm{Cov}(Y, X) + \mathrm{Cov}(Y, 2Y). Rewrite each piece: = \mathrm{Var}(X) + 2\,\mathrm{Cov}(X, Y) + \mathrm{Cov}(X, Y) + 2\,\mathrm{Var}(Y). Plug in. = 5 + 2(1) + 1 + 2(3) = 5 + 2 + 1 + 6 = 14. The two indices move together strongly, which makes sense: they share the same X and both load positively on Y. Takeaway. Two blends that share components…

Covariance of two overlapping index weightings

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