How do you derive the minimum variance hedge ratio, and when does it differ from a naive 1:1 hedge?

The minimum variance hedge ratio (MVHR) minimizes the variance of the hedged portfolio rather than simply matching notional amounts. A naive 1:1 hedge only works perfectly when the hedging instrument moves in lockstep with the exposure.\n\nDerivation:\n\nThe hedged portfolio value change is:\n\ndV = dS - h x dF\n\nVariance of the hedged position:\n\nVar(dV) = sigma_S^2 - 2h x rho x sigma_S x sigma_F + h^2 x sigma_F^2\n\nTaking the derivative with respect to h and setting it to zero:\n\nh = rho x (sigma_S / sigma_F)\n\nwhere rho is the correlation between spot and futures price changes, sigma_S is the standard deviation of spot price changes, and sigma_F is the standard deviation of futures price changes.\n\nWhy 1:1 Fails:\n\nA 1:1 hedge assumes rho = 1 and sigma_S = sigma_F. In practice:\n- Cross-hedging (hedging jet fuel with crude oil futures) introduces basis risk\n- Different contract sizes create notional mismatches\n- Spot and futures volatilities often diverge\n\nWorked Example:\nBelmont Airlines wants to hedge 500,000 gallons of jet fuel. Crude oil futures (1,000 barrels per contract) are used as a cross-hedge.\n\n| Parameter | Value |\n|---|---|\n| sigma_S (jet fuel, $/gallon) | 0.035 |\n| sigma_F (crude oil, $/barrel) | 1.85 |\n| Correlation (rho) | 0.89 |\n\nh = 0.89 x (0.035 / 1.85) = 0.89 x 0.01892 = 0.01684\n\nThis means for each gallon of jet fuel exposure, hedge with 0.01684 barrels of crude oil.\n\nTotal barrels to hedge: 500,000 x 0.01684 = 8,420 barrels\nNumber of contracts: 8,420 / 1,000 = 8.42, rounded to 8 contracts\n\nA naive 1:1 approach (matching barrel-for-barrel) would over-hedge dramatically and actually increase portfolio variance.\n\nHedge Effectiveness:\n\nR-squared of the regression (rho^2 = 0.89^2 = 0.7921) tells us that the hedge eliminates approximately 79% of variance. The remaining 21% is unhedgeable basis risk.\n\nKey Exam Points:\n- MVHR is the slope of the OLS regression of spot price changes on futures price changes\n- Tailing the hedge adjusts for daily settlement (mark-to-market) effects\n- Over-hedging can increase risk just as much as under-hedging\n\nPractice MVHR calculations in our CFA Derivatives question bank.