If ρ = 1.0 and σ_S = σ_F, the minimum-variance hedge ratio is:

1.0. h^* = ρ × σ_S / σ_F = 1.0 × 1.0 = 1.0. Perfect correlation and equal volatilities make the naive one-for-one hedge optimal. In every other case the ratio differs from unity.

An exporter has spot exposure with σ_S = 0.020. The hedge instrument has σ_F = 0.040 and correlation ρ = 0.90 with the spot. The minimum-variance hedge ratio is closest to:

0.450. h^* = 0.90 × (0.020 / 0.040) = 0.90 × 0.5 = 0.45. The futures are twice as volatile as the spot, so fewer contracts are needed to offset the dollar shock. Picking a 1:1 hedge would over-hedge and inject futures noise into the portfolio.

Hedging & Basis Riskadvanced

Minimum Variance Hedge Ratio

The minimum-variance hedge ratio $h^*$ is the proportion of the spot exposure to be hedged with futures that minimises the variance of the hedged position. It equals $h^* = \rho\,\sigma_S / \sigma_F$ , where $\rho$ is the correlation between the change in the spot price and the change in the futures price, and $\sigma_S$ , $\sigma_F$ are their standard deviations. The result is the slope of the regression $\Delta S = a + h\,\Delta F + \varepsilon$ , and the regression $R^2 = \rho^2$ measures hedge effectiveness.

Try it yourself

Minimum-variance hedge ratio

You hold an asset and short futures to offset its price moves. The hedge ratio h is how many units of futures you short per unit of exposure. The variance of the hedged position is a parabola in h, minimised at h* = ρ · σ_spot / σ_futures. The deeper the correlation, the more variance the hedge removes — exactly ρ² of it.

Hedge effectiveness 64%. At h* the hedge removes 64% of the unhedged variance, leaving 0.52 of residual (basis) risk you cannot diversify away.

Hedge ratio h*0.96

Optimal contracts N* (exact)19.2

N* rounded to whole contracts19

Hedge effectiveness ρ²64%

Unhedged variance σ_spot²1.44

Minimum variance at h*0.52

Correlation ρ0.8

σ_spot (spot change std dev)1.2

σ_futures (futures change std dev)1

Exposure Q_A (units to hedge)10,000

Contract size Q_F (units per contract)500

h* = ρ · σ_spot / σ_futures · N* = h* · Q_A / Q_F · Var(h) = σ_spot² + h²·σ_futures² − 2·h·ρ·σ_spot·σ_futuresσ_spot and σ_futures are standard deviations of price changes over the hedge horizon. Effectiveness ρ² is the share of spot variance the hedge removes; the rest is basis risk. Contracts are discrete, so N* is rounded to whole contracts in practice.

Why it matters

A naive hedger sells one futures contract per unit of spot exposure. That is only optimal when the spot and the futures change by the same amount on average. If futures are twice as volatile as the spot, half as many contracts will offset the average shock. If correlation is below one, the additional uncertainty from the futures leg means the variance-minimising ratio is further reduced. The formula bundles both adjustments into a single number that can be estimated by a simple OLS regression.

Before you read on — recall

If $\rho = 1.0$ and $\sigma_S = \sigma_F$ , the minimum-variance hedge ratio is:

Formulas

Minimum-variance hedge ratio

h^* = \rho_{SF} \times \frac{\sigma_{\Delta S}}{\sigma_{\Delta F}}

Equivalently

h^* = \mathrm{Cov}(\Delta S, \Delta F) / \mathrm{Var}(\Delta F)

, the OLS slope from regressing spot changes on futures changes. Source: Hull (2022) §3.5.

Optimal number of futures contracts

N^* = h^* \times \frac{Q_A}{Q_F}

Q_A

is the size of the position being hedged and

Q_F

is the size of one futures contract, both in the same units. Round to the nearest whole contract.

Hedge effectiveness

R^2 = \rho^2

Fraction of variance in the hedged-position changes removed by hedging at

h^*

. Reported in Ederington (1979) as the standard measure of hedge quality.

Worked examples

Scenario

An airline plans to buy 2,000,000 gallons of jet fuel in one month. Heating oil futures cover 42,000 gallons each. From historical monthly data, $\sigma_{\Delta S} = 0.0263$ , $\sigma_{\Delta F} = 0.0313$ , and $\rho = 0.928$ .

Solution

Hedge ratio $h^* = 0.928 \times (0.0263 / 0.0313) = 0.7777$ . Optimal contracts $N^* = 0.7777 \times (2{,}000{,}000 / 42{,}000) = 37.03$ , so the airline buys 37 heating oil futures. Hedge effectiveness $R^2 = 0.928^2 = 0.861$ , meaning 86% of jet fuel price variance is removed. This worked example follows Hull (2022) Example 3.3.

Scenario

An Australian grain trader hedges 50,000 tonnes of barley using ASX Eastern Wheat futures (50 tonnes per contract). Regression of monthly barley price changes on wheat futures changes gives $h^* = 0.85$ .

Solution

Optimal contracts $N^* = 0.85 \times (50{,}000 / 50) = 850$ . The trader shorts 850 ASX wheat futures. With $h^*$ implying a correlation around 0.85 to 0.90 depending on volatilities, hedging effectiveness lies in the 70 to 80 percent range. The remaining variance is barley-specific basis risk.

Common mistakes

✗The hedge ratio is always one. $h^* = 1$ only when $\rho = 1$ and $\sigma_S = \sigma_F$ . In every other case, the optimal ratio differs from unity. The naive 1:1 hedge typically over-hedges or under-hedges and leaves more variance in the portfolio.
✗A high correlation is sufficient. Hedge quality also depends on the relative volatilities. Two assets with $\rho = 0.99$ but very different volatility scales still need a non-trivial $h^*$ to balance dollar exposures.
✗The ratio is constant over time. Volatilities and correlations vary with market conditions. Hull (2022) recommends **re-estimating $h^*$ ** when the regression window shifts substantially or when the market regime changes.

Revision bullets

•** $h^* = \rho\,\sigma_S / \sigma_F$ ** minimises hedged variance
•Equals the OLS slope of $\Delta S$ on $\Delta F$
•Optimal contracts $N^* = h^* \times Q_A / Q_F$
• $h^* = 1$ only when $\rho = 1$ and equal volatilities
•Hedge effectiveness $R^2 = \rho^2$
•Re-estimate when market regimes shift

Quick check

If $\rho = 1.0$ and $\sigma_S = \sigma_F$ , the minimum-variance hedge ratio is:

An exporter has spot exposure with $\sigma_S = 0.020$ . The hedge instrument has $\sigma_F = 0.040$ and correlation $\rho = 0.90$ with the spot. The minimum-variance hedge ratio is closest to:

Connected topics

Delta Hedge Cross Hedge Tailing

More in Hedging & Basis Risk

Hedge vs Spec Short Hedge Long Hedge Beta Hedge Change Beta Basis Basis Risk

In learning paths

Hedging & Risk Path

Sources

Hull (2022), §3.5, Example 3.3
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Derives $h^* = \rho\,\sigma_S / \sigma_F$ , links it to the OLS regression slope, and works through the airline jet-fuel example used in our case study.
Ederington (1979)
Ederington, Louis H. The Hedging Performance of the New Futures Markets. Journal of Finance, Vol. 34, No. 1, March 1979, pp. 157-170.
Foundational paper that estimated minimum-variance hedge ratios and reported hedge effectiveness as the regression $R^2$ , the standard benchmark since.
Johnson (1960)
Johnson, Leland L. The Theory of Hedging and Speculation in Commodity Futures. Review of Economic Studies, Vol. 27, No. 3, 1960, pp. 139-151.
Earliest derivation of the minimum-variance hedge ratio in a mean-variance utility framework, predating Ederington's empirical estimation.
CME Group, Equity Index Hedging
CME Group. Hedging and Risk Management for Equity Index Futures. CME Education Centre, accessed 2026.
Applies the minimum-variance framework to equity portfolios and discusses how to translate the ratio into contract counts.

How to cite this page

Dr. Phil's Quant Lab. (2026). Minimum Variance Hedge Ratio. Derivatives Atlas. https://phucnguyenvan.com/concept/minimum-variance-hedge-ratio