Optimal Hedge Ratio Calculator (minimum-variance h* = ρ σS/σF & number of futures)

Understanding the optimal hedge ratio calculator — concept and principle

When the futures contract you can trade does not match your exposure exactly, a one-for-one hedge is not the best choice. This calculator finds the minimum-variance hedge ratio, the number of futures contracts to trade and how much risk the hedge removes. It is used by airlines, commodity producers and risk managers for cross hedges.

0.7798 × 2,000,000 gallons ÷ 42,000 per contract ≈ 37.13, so about 37 contracts minimise the hedged variance.

How it is calculated

Optimal hedge ratio: h* = ρ × σS ÷ σF.
ρ: correlation between changes in the spot price and the futures price; σS and σF: their standard deviations over the hedge horizon.
Optimal number of contracts: N* = h* × QA ÷ QF, where QA is the size of the position and QF the units per contract.
Hedge effectiveness = ρ², the share of spot-price variance removed.
Risk left = √(1 − ρ²), the fraction of the standard deviation that remains.

The principle

A hedger holds the spot exposure and takes the opposite position in h units of futures per unit of exposure. The variance of the hedged position is σS² + h²σF² − 2hρσSσF. Setting the derivative with respect to h to zero gives h* = ρσS ÷ σF. If the futures price moves more than the spot price, you need fewer contracts; if the two are only partly correlated, you also hedge less, because extra futures add their own risk.

Worked example

Default inputs (classic jet fuel hedged with heating oil futures): ρ = 0.928, σS = 0.0263, σF = 0.0313, position 2,000,000 gallons, 42,000 gallons per contract.

h* = 0.928 × 0.0263 ÷ 0.0313 ≈ 0.78.
Contracts = 0.7798 × 2,000,000 ÷ 42,000 ≈ 37.13, rounded to 37.
Units covered by futures = 0.7798 × 2,000,000 ≈ 1,559,514 gallons.
Hedge effectiveness = 0.928² ≈ 86.1%.
Risk left vs no hedge = √(1 − 0.861) ≈ 37.3%.

Each point is the calculator re-run with only “Standard deviation of futures price changes σF” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Standard deviation of futures price changes σF	Optimal hedge ratio h*
0.00783	3.119
0.0157	1.56
0.0235	1.04
0.0313	0.78
0.0391	0.624
0.0469	0.52
0.0626	0.39

How to read the result

A hedge ratio below 1 is normal for cross hedges. The effectiveness tells you how much variance remains even with the best hedge: here about 14% of the variance and 37% of the standard deviation. Estimates come from past data and change over time, so many hedgers re-estimate and rebalance regularly.

Common mistakes

Using price levels instead of price changes to estimate ρ and σ.
Using a different time horizon for the statistics than for the hedge.
Forgetting the contract size when converting the ratio into contracts.
Ignoring daily settlement, which may call for a "tailed" hedge based on values.

Keep in mind: Hedge ratios estimated from history may not hold in future markets. This is an educational tool, not trading or risk-management advice.

Frequently asked questions

What is a minimum variance hedge ratio?

The number of futures per unit of exposure that makes the variance of the hedged position as small as possible.

Why not hedge one-for-one?

If futures and spot do not move together perfectly, a full hedge can add risk instead of removing it.

How do I estimate correlation?

Use historical changes in spot and futures prices over the same intervals as your hedge horizon.

Sources: Hull, J. C., Options, Futures, and Other Derivatives, Chapter 3 (cross hedging, Example 3.3) · Ederington, L. H. (1979). The Hedging Performance of the New Futures Markets. Journal of Finance, 34(1).

Checked 9 Nov 2026. E. M. Aldrich, Econ 236 lecture notes “Hedging with Futures” (2016): optimal hedge ratio h* = ρσ_S/σ_F, optimal number of contracts N* = h*Q_S/Q_F, example ρ = 0.928, σ_S = 0.0263, σ_F = 0.0313 → h* ≈ 0.78 and about 37 contracts.

Your numbers