Long Two, Short Three

Question

sigma_X = 2, sigma_Y = 1, correlation rho = 0.5. You're long 2 units of X and short 3 units of Y. Compute the portfolio variance operatornameVar(2X - 3Y).

Accepted Answer

Apply the bilinear formula with a = 2, b = -3: operatornameVar(2X - 3Y) = 4(4) + 9(1) + 2(2)(-3)(0.5)(2)(1) = 16 + 9 - 12 = mathbf13, so sigma_p = sqrt13 approx 3.61. (Simulation: 12.999. ✓) The trap is the cross term's sign: shorting flips b negative, so positive correlation now reduces variance, the short leg hedges the long leg. Compare across correlations: - rho = 0: variance = 25, no hedging benefit. - rho = 0.5: variance = 13, about half the risk for free. - rho = 1: variance = 16 + 9 - 24 = 1, and the slick check: with perfect correlation, sigma_p = |asigma_X + bsigma_Y| = |4 - 3| = 1, standard deviations combine like signed scalars. Two extensions interviewers reach for: 1. Minimum-variance hedge ratio. Hedging one unit of X with h units of Y: minimize operatornameVar(X - hY) over h, giving h^ = operatornameCov(X,Y)/operatornameVar(Y) = rho,sigma_X/sigma_Y = 1 here. The OLS slope formula again, hedging is regression. 2. Variance is a quadratic form. With more assets, sigma_p^2 = w^top Sigma w; this two-asset drill is the 2times 2 case, and being fluent in it is the entry ticket to every risk, sizing, and hedging conversation. Every risk report on a desk is this formula eval

Long Two, Short Three

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