Optimal Hedge Ratio Calculator
Enter the spot and futures price volatilities, correlation coefficient, and position size to find the optimal hedge ratio (h*) that minimises variance. The calculator also tells you how many futures contracts to use, the hedging effectiveness (R squared), and the dollar amount covered. Results update instantly as you type.
What is the optimal hedge ratio?
The optimal hedge ratio (h*) is the proportion of a futures position relative to a spot position that minimises the variance of the hedged portfolio. It is derived from the minimum-variance model first formalised by Ederington (1979) and popularised in derivatives textbooks by John C. Hull. Rather than always hedging one-to-one, the optimal ratio accounts for the fact that spot and futures prices rarely move in perfect lockstep. A ratio below 1.0 means a smaller futures position is needed, while a ratio above 1.0 means the futures leg should exceed the spot exposure. Negative ratios arise when spot and futures prices move inversely, requiring a long futures position.
The formula and how it works
The formula is h* = rho x (sigma_S / sigma_F), where rho is the correlation coefficient between changes in the spot and futures prices, sigma_S is the standard deviation of spot price changes, and sigma_F is the standard deviation of futures price changes. From h* you can derive the number of contracts: N* = h* x (Q_A / Q_F), where Q_A is the value of the position to be hedged and Q_F is the notional value of one futures contract. Hedging effectiveness is R2 = rho squared, representing the fraction of variance that the minimum-variance hedge eliminates. The remaining fraction (1 - R2) is basis risk, the irreducible residual variance caused by imperfect correlation between the two instruments.
Inputs and how to measure them
Spot volatility and futures volatility are typically estimated from historical price data using an annualised standard deviation of log returns, though implied volatility can also be used for short-dated hedges. The correlation coefficient is the Pearson correlation of the paired return series over the same measurement window. Common lookback windows range from 20 to 252 trading days. Shorter windows capture recent regime shifts but are noisy; longer windows are smoother but may lag structural changes. The exposure value is the current mark-to-market value of the position in the chosen currency, and the contract size is the fixed notional per futures contract as stated in the exchange specifications.
Model assumptions and practical limitations
The minimum-variance model assumes that the volatilities and the correlation coefficient remain constant over the hedge horizon, that the relationship between spot and futures prices is linear, and that transaction costs and margin requirements are negligible. In practice, all three quantities change over time, so the hedge ratio should be recalculated periodically using updated estimates. Cross-hedging (using a futures contract on a related but different asset) introduces additional basis risk not captured by this model. For commodity hedgers, seasonality in the underlying can make a rolling estimate more appropriate than a fixed-window one. The model is a starting point for risk management decisions, not a substitute for judgment.
Optimal hedge ratio interpretation guide
| h* range | Classification | Futures position | Implication |
|---|---|---|---|
| h* > 1.02 | Over-hedge | Short > exposure | Spot is more volatile; increase futures notional |
| h* = 1.00 (+/-0.02) | Perfect hedge | Short = exposure | One futures unit offsets one spot unit |
| 0.5 to 0.98 | Partial hedge | Short < exposure | Futures only partially track spot movements |
| 0 to 0.49 | Under-hedge | Small short | Low correlation or low spot vol; minimal risk reduction |
| ~0 | Ineffective hedge | Negligible | Correlation near zero; hedge provides almost no benefit |
| h* < 0 | Negative hedge | Long futures | Inverse relationship; long futures needed to offset spot loss |
How to read the h* value and what hedging action it implies.
Frequently asked questions
What does an optimal hedge ratio of 0.75 mean?
A hedge ratio of 0.75 means you should take a futures position equal to 75% of the spot exposure to minimise the variance of the combined portfolio. If you hold USD 1 million of the spot asset, you need USD 750,000 notional in futures contracts (which you would then divide by the contract size to get the number of contracts). The other 25% of the exposure is deliberately left unhedged because hedging it would actually increase variance given the imperfect correlation.
Why is the optimal hedge ratio sometimes greater than 1?
h* exceeds 1 when the spot asset is more volatile than the futures contract relative to their correlation. Because the futures price moves less per unit of notional than the spot price, you need a larger futures position to offset each unit of spot exposure. This is common when hedging with a related but not identical instrument, for example hedging jet fuel with crude oil futures, where the correlation is high but the volatilities differ.
What is hedging effectiveness (R2) and why does it matter?
Hedging effectiveness equals the correlation coefficient squared (rho squared). It measures the fraction of the spot variance that the minimum-variance futures hedge eliminates. For example, a correlation of 0.90 gives R2 = 0.81, meaning 81% of variance is removed and 19% remains as unavoidable basis risk. A low R2 signals that the chosen futures contract is a poor hedging vehicle for this spot position, and you should consider whether a more correlated instrument exists.
How do I estimate the inputs from real price data?
Collect daily closing prices for both the spot asset and the futures contract over your chosen lookback window (typically 20 to 252 trading days). Compute log returns for each series. Sigma_S is the annualised standard deviation of spot log returns (daily standard deviation multiplied by the square root of 252); sigma_F is the same for futures. The correlation coefficient is the Pearson correlation of the two return series. Use the same time window for both series so the estimates are consistent.
Should I round the number of contracts up or down?
This calculator rounds to the nearest integer. In practice the direction of rounding depends on your risk preference: rounding up gives a slight over-hedge, which leaves you slightly net short; rounding down gives a slight under-hedge, leaving you with some unhedged exposure. For most institutional hedgers, rounding to the nearest whole number is standard. If the fractional part is significant (above 0.4 of a contract), some traders split the difference by entering a partial position through a different instrument or contract month.
Can the optimal hedge ratio be negative?
Yes. A negative h* occurs when the correlation between spot and futures price changes is negative, meaning the two prices tend to move in opposite directions. In that case, you need to buy (go long) futures rather than sell (go short) to offset losses in the spot position. Negative correlations arise in cross-hedges where the hedging instrument has an inverse economic relationship with the exposure, for example hedging a portfolio of airline stocks partly with crude oil futures given the negative correlation between airline profits and fuel costs.