An airline hedging 60,000 tonnes of jet fuel with 1,000-barrel crude futures gets a regression of jet fuel price changes on crude futures changes: slope 0.71, standard error 0.08, R-squared 0.64, 36 observations. Read it, set the hedge, and say how much risk it removes.

Buy 337 crude futures: 0.71 times 474,000 barrels of exposure is 336,540 barrels, 336.5 contracts of 1,000. The slope is the minimum-variance hedge ratio, and the R-squared of 0.64 says the hedge removes 64% of the variance of fuel cost, which cuts its standard deviation by 40%. The standard error of 0.08 puts the ratio between about 0.55 and 0.87, or 260 to 414 contracts, but the variance left barely changes across that range. The intercept does not enter the hedge.

Think of a tailor fitting a suit from a customer's measurements: three numbers matter, and the rest of the form is filing. The slope, 0.71, is the minimum-variance hedge ratioThe number of units of a futures contract per unit of exposure that makes the variance of the hedged position as small as possible. It equals the regression slope of exposure price changes on futures price changes.: on average, jet fuel moves 0.71 rupees for each rupee crude futures move, so each barrel of fuel needs 0.71 barrels of futures. The R-squared, 0.64, is the share of the variance of jet fuel changes that crude explains, which is the share the hedge can remove. The standard error, 0.08, says how precisely the slope is known. The intercept, 0.42 with a standard error of 1.10, is a drift in jet fuel that crude does not explain; it is statistically zero and plays no part in the hedge.

The airline buys fuel, so it loses when prices rise and needs a position that gains when they rise. It buys 337 crude futures: the exposure is 60,000 times 7.9, 474,000 barrels, the slope scales that to 336,540 barrels of crude, and at 1,000 barrels a contract that is 336.5. A one-for-one hedge of 474 contracts sounds safer and is not: because crude futures move more than jet fuel relative to their correlation, over-hedging adds crude risk back. The table puts the three choices side by side, using the correlation of 0.8 and the ratio of volatilities the slope implies.

The hedge removes 64% of the variance of fuel cost, which means the standard deviation falls to the square root of 0.36, 60% of its unhedged level, a 40% cut. Treasury should report the 40%, because budgets are discussed in rupees, not squared rupees. The slope's t-statistic is 0.71 over 0.08, about 8.9, so the relationship is not in doubt, but the 95% range of the slope runs from about 0.55 to 0.87, 260 to 414 contracts. That sounds like a wide range for a hedge; the figure shows why it matters less than it looks. The variance left is a U-shaped curve with a flat bottom, so a hedge ratio anywhere in that range leaves between 36% and about 39% of the variance.

Say the limits. The regression is on three years of monthly data, and the relationship between jet fuel and crude moves with refinery margins, which can widen sharply in a supply shock exactly when the airline most needs the hedge; 36% of the variance, the crack spread, stays unhedged. The hedge is set on monthly changes, while the airline buys fuel through a quarter, so the hedge should be layered across the purchase dates rather than closed at once. The conversion of 7.9 barrels a tonne is an average that varies with density. And futures are margined daily: a fall in crude means cash calls on the futures while the cheaper fuel saves money only later. The judgement: buy about 337 contracts, report a 40% cut in cost volatility, and say plainly that the crack spread is the risk left.