FRM Exam Part I · Pricing Financial Forwards and Futures

Hedging with Futures and the Optimal Hedge Ratio

Updated 11 October 2026 · Fact-checked

Hedging with futures means taking a futures position that offsets the price risk of an asset you hold or will trade. The optimal (minimum variance) hedge ratio is h* = ρ × σS ÷ σF. The number of contracts is h* × QA ÷ QF, where QA is your exposure and QF is the contract size.

Understand Hedging with Futures and Optimal Hedge Ratio

A futures hedge locks in a price by taking an opposite position in the futures market. If you own an asset, you short futures. If you will buy an asset later, you go long futures. Gains on the futures offset losses on the underlying.

The hedge is rarely perfect. Basis is spot price minus futures price. Hedging removes the price level risk but leaves basis risk, which is the uncertainty in the basis when you close the hedge. It arises when the asset hedged differs from the futures asset, when the hedge ends before delivery, or when the timing of the exposure is uncertain.

Because of basis risk, hedging one-for-one is not always best. The minimum variance hedge ratio picks the futures position that makes the variance of the hedged position as small as possible. It equals the slope of a regression of spot price changes on futures price changes. So h* = ρ × σS ÷ σF, where ρ is the correlation of spot and futures changes.

The hedge effectiveness is the R² of that regression, which is ρ². It is the share of spot variance removed by the hedge. The variance of the hedged position is σS² × (1 − ρ²) per unit of exposure.

For equity portfolios, you hedge with stock index futures. The portfolio's beta is the hedge ratio, since beta measures sensitivity to the index. To reduce beta to a target, you adjust the number of contracts. Tailing the hedge adjusts the contract count for daily settlement of futures, which matters for long hedges. It scales the position by the spot-to-futures price ratio.

Key formulas to remember

Minimum variance hedge ratio: h* = ρ × σS ÷ σF; σS and σF are standard deviations of spot and futures price changes over the hedge period. ρ is their correlation. Also equals the regression slope.
Number of contracts: N* = h* × QA ÷ QF; QA is the size of the position hedged (units of the asset). QF is the size of one futures contract (units).
Hedge effectiveness: Effectiveness = ρ²; The proportion of variance of the spot position eliminated by the optimal hedge.
Variance of hedged position: σ²hedged = σS² × (1 − ρ²) per unit; Applies when the optimal h* is used. Multiply by the squared size of the exposure for total variance.
Basis: Basis = S − F; Spot minus futures. Some texts use futures minus spot, so check the definition given in the question.
Tailed hedge (value form): N* = h* × VA ÷ VF; VA is the dollar value of the exposure. VF is the dollar value of one futures contract (futures price × contract size).
Contracts for changing portfolio beta: N = (β* − β) × P ÷ A; P is portfolio value, A is the value of one futures contract (index level × multiplier), β is current beta, β* is target beta. Positive means buy, negative means sell. To fully hedge, β* = 0 and N = −β × P ÷ A.

How to solve Hedging with Futures and Optimal Hedge Ratio questions

Use this approach for any question on futures hedges, hedge ratios or contract counts.

1Identify the exposure. Are you long the asset (short futures) or need to buy later (long futures)?
2Read what is given: σS, σF, ρ, or a regression slope. Note whether the figures are for price levels or changes.
3Compute h* = ρ × σS ÷ σF, or take the regression slope directly. For equity hedges, use beta instead.
4Find the units. Compute QA and QF in the same units, or VA and VF in the same currency.
5Compute N* = h* × QA ÷ QF. Apply tailing if the question gives spot and futures prices and asks for it.
6Round as the question asks. Contract counts are usually rounded to the nearest whole number.
7Check direction and size: short if you hold the asset, long if you must buy. Check effectiveness with ρ² if asked.

Quickest way: Three-line hedge check

When to use it: Use when the question gives ρ, σS and σF and asks for contracts or effectiveness.

Write h* = ρ × σS ÷ σF and compute it first. Sanity check: it should usually be near 1 for a good hedge.
Multiply by the ratio of exposure to contract size: N = h* × QA ÷ QF.
If asked for effectiveness, square ρ. If asked for residual variance, use σS² × (1 − ρ²).
For index hedges, skip h*. Use N = β × P ÷ A to hedge fully, or (β* − β) × P ÷ A to change beta.

Common mistakes in Hedging with Futures and Optimal Hedge Ratio

Swapping σS and σF in the hedge ratio.
The formula is memorised as a pair of letters without thinking about the meaning.
Fix: Remember the ratio is the regression slope of spot on futures: spot volatility on top, futures volatility on the bottom.
Using the wrong sign or direction of the position.
Students focus on the number and forget whether the hedger is long or short the asset.
Fix: A holder of the asset shorts futures. Someone who needs to buy later goes long futures.
Treating hedge effectiveness as ρ instead of ρ².
Correlation is given, so it is tempting to quote it directly.
Fix: Effectiveness is the R² of the regression, so it equals ρ². A correlation of 0.8 means 64% of variance is removed.
Mixing units, such as tonnes in the exposure and barrels in the contract.
Exposure and contract size are given in different places or units.
Fix: Write both in the same unit before dividing. Check the final count is reasonable.
Using the portfolio beta formula with the wrong contract value.
Index futures value is index level times multiplier, which is easy to forget.
Fix: Compute A = index level × multiplier first. Then use N = β × P ÷ A.
Assuming a futures hedge removes all risk.
The hedge is described as locking in a price.
Fix: Basis risk remains. The hedge fixes the price only up to the uncertainty in the basis at close-out.

Worked examples

Example 1

An airline will buy 2,000,000 gallons of jet fuel in three months. It hedges with heating oil futures. Each contract covers 42,000 gallons. The standard deviation of the change in jet fuel price is 0.032 and of heating oil futures is 0.040 (same units), with correlation 0.8. How many contracts should it buy, and how effective is the hedge?

Show the solution

The airline needs to buy fuel, so it goes long futures.
h* = ρ × σS ÷ σF = 0.8 × 0.032 ÷ 0.040 = 0.64.
N* = h* × QA ÷ QF = 0.64 × 2,000,000 ÷ 42,000 = 1,280,000 ÷ 42,000 = 30.48.
Round to the nearest whole number: 30 contracts.
Effectiveness = ρ² = 0.8² = 0.64, so 64% of variance is removed.

Answer: Buy 30 heating oil futures contracts (30.48 rounded). The hedge is 64% effective.

Example 2

A fund holds a US equity portfolio worth $20,000,000 with beta 1.25. The S&P 500 futures price is 4,000 and the multiplier is $250. The fund wants to reduce beta to 0.50 using futures. How many contracts should it trade?

Show the solution

Value of one futures contract: A = 4,000 × 250 = $1,000,000.
Change in beta needed: β* − β = 0.50 − 1.25 = −0.75.
N = (β* − β) × P ÷ A = −0.75 × 20,000,000 ÷ 1,000,000.
N = −0.75 × 20 = −15.
A negative number means selling futures.

Answer: Sell 15 S&P 500 futures contracts.

Exam tips

Read whether σ figures are for price changes or price levels. The hedge ratio uses changes.
Questions often ask for effectiveness or residual variance after you find h*. Keep ρ visible on your scratch work.
For index hedges, compute the contract value (level × multiplier) first. This is the most common source of wrong answers.
Watch the wording on direction: buy or sell, long or short. Four options often differ only by sign.
Tailing questions give spot and futures prices. Remember the adjustment is h* × VA ÷ VF, which changes the count slightly from the untailed answer.

Practice questions from Pricing Financial Forwards and Futures

Hedging with Futures and Optimal Hedge Ratio in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Hedging with Futures and Optimal Hedge Ratio: frequently asked questions

What is the minimum variance hedge ratio formula?

h* = ρ × σS ÷ σF, where ρ is the correlation between spot and futures price changes. σS and σF are the standard deviations of those changes. It equals the slope of a regression of spot changes on futures changes.

What is basis risk in futures hedging?

Basis risk is the uncertainty about the spot minus futures difference when the hedge is closed. It comes from asset mismatch, timing mismatch and delivery month mismatch. It is the reason a futures hedge is not perfect.

What is tailing the hedge?

Tailing adjusts the number of futures contracts for daily settlement. Futures gains and losses are paid each day, so they can be reinvested or financed. The adjusted position is h* times the exposure value divided by the value of one futures contract, rather than units divided by units.

How do I hedge a stock portfolio with index futures?

Compute the value of one contract as index level times multiplier. To hedge fully, sell β × P ÷ A contracts. To move beta to a target β*, trade (β* − β) × P ÷ A contracts, selling if the result is negative.