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CHAPTER 10LESSON 053 MIN READ

Hedge ratios and basis risk

How much of one thing to sell against another, and the risk that remains.

WHY IT MATTERS

A hedge ratio chooses how much of another instrument offsets your exposure. The mismatch that remains is basis risk.

A hedge is rarely a perfect offset. A refiner hedging jet fuel may have no liquid jet contract available and hedge with diesel; a producer of a regional grade hedges with a benchmark. The question is how many units of the hedge instrument to trade against one unit of exposure, and the standard answer is the minimum-variance hedge ratio.

h = cov(exposure changes, hedge changes) / var(hedge changes)
h
units of hedge instrument per unit of exposure
cov, var
computed on CHANGES, over a stated window
This is the slope of a regression of exposure changes on hedge changes. It minimises the variance of the hedged position, which is not the same as eliminating it.

What remains after the hedge is basis risk: the variability of the difference between the exposure and the hedge. A minimum-variance hedge minimises that residual variance and does not remove it, and the size of what is left is the honest measure of how good the hedge is. A hedge ratio quoted without its residual variance has told you the recipe and withheld the result.

Three practical points follow. The ratio is estimated, so it has a standard error, and a ratio of 0.85 with wide uncertainty is a different proposition from the same number tightly estimated. The ratio is not stable, because the relationship between a regional grade and a benchmark changes with logistics and quality economics, so it has to be re-estimated and the re-estimation window is a real choice. And the ratio depends on the horizon: the relationship over daily changes and over monthly changes can differ substantially, and the right one is the horizon over which you actually bear the exposure.

There is also a cash-flow dimension that the variance calculation ignores entirely. A minimum-variance hedge can be economically correct and still generate margin calls that the underlying exposure does not fund until later, which is the timing problem described in futures contracts. Minimising price variance and surviving the hedge are different objectives.

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Minimum-variance ratios and the basis they leave, with the residual reported alongside the ratio.