FRM Exam Part I · Futures Markets

Hedging with Futures and Basis Risk Explained

Updated 11 October 2026 · Fact-checked

Hedging with futures means taking a futures position that offsets a cash exposure: short futures if you will sell the asset, long futures if you will buy it. Basis risk is the uncertainty in spot minus futures. The minimum variance hedge ratio is h* = ρ × σS ÷ σF, and contracts N = h* × QA ÷ QF.

Understand Hedging with Futures and Basis Risk

A hedge uses futures to cut the risk of a price move in an asset you own or plan to trade. The futures gain or loss offsets the loss or gain on the cash position.

A short hedge means selling futures. You use it when you hold the asset or will sell it later, for example a producer or a fund holding bonds. If the price falls, the cash position loses and the short futures gains. A long hedge means buying futures. You use it when you will buy the asset later, for example an airline buying fuel. If the price rises, you pay more in the cash market but the long futures gains.

A perfect hedge is rare. The basis is spot price minus futures price (the sign convention used here). It is not constant. Its changes are basis risk. Basis risk arises because the hedge asset may differ from the exposed asset (cross hedging), the hedge may end at a date different from the futures delivery month, or you may close out before expiry. A strengthening basis (spot rises relative to futures) helps a short hedger and hurts a long hedger. A weakening basis does the opposite.

With a short hedge, the effective price you receive is F1 + b2, where F1 is the futures price when you open the hedge and b2 is the basis when you close it. With a long hedge, the effective price paid is F1 + b2 as well. The only unknown is b2, so basis risk is what remains.

In a cross hedge, hedging one-for-one is not optimal. The minimum variance hedge ratio picks the futures position that minimises the variance of the hedged position. It equals the slope of a regression of spot price changes on futures price changes. It rises with the correlation and with the ratio of spot volatility to futures volatility. Then you convert the ratio into a number of contracts using contract size.

Key formulas to remember

Basis: Basis = Spot price − Futures price; Use the sign convention stated in the question. Some texts define it the other way round.
Effective price with a hedge: Effective price = F1 + b2 (b2 = S2 − F2); F1 is the futures price at start. b2 is the basis when the hedge is closed. Same form for long and short hedges.
Minimum variance hedge ratio: h* = ρ × σS ÷ σF; ρ is the correlation between changes in spot and futures prices. σS and σF are standard deviations of those changes over the hedge period.
Number of contracts: N* = h* × QA ÷ QF; QA is the size of the exposure in units. QF is the size of one futures contract in the same units.
Hedge effectiveness: Effectiveness = ρ²; The share of variance of the spot change removed by the optimal hedge.
Hedge ratio as regression slope: h* = Cov(ΔS, ΔF) ÷ Var(ΔF); Equivalent to ρ × σS ÷ σF. Slope of ΔS regressed on ΔF.
Variance of hedged position (per unit): σ²hedged = σS² × (1 − ρ²); Residual variance when the optimal ratio is used.

How to solve Hedging with Futures and Basis Risk questions

Use this order for any hedging question. It covers direction, basis outcome, and sizing.

1Identify the exposure. Will you sell the asset later (short hedge) or buy it later (long hedge)? Or do you hold it now and fear a fall (short hedge)?
2Pick the futures side: short futures for a short hedge, long futures for a long hedge.
3If the question gives spot and futures prices at two dates, compute the futures gain or loss and the cash result, then the net outcome. Check it against F1 + b2.
4If the hedge uses a different asset or date, recognise basis risk and a cross hedge, and do not assume h = 1.
5Compute h* = ρ × σS ÷ σF. Make sure σS and σF are volatilities of price changes over the same period and in the same units.
6Compute N* = h* × QA ÷ QF. Use the same units for QA and QF.
7Round to the nearest whole contract unless told otherwise, and state long or short.
8Sanity check: h* should be positive when ρ is positive, and the result should be near QA ÷ QF if volatilities are similar.

Quickest way: Three-line hedge sizing

When to use it: Use it when the question gives ρ, σS, σF, exposure and contract size and asks for contracts or hedge ratio.

Write h* = ρ × σS ÷ σF and compute it first.
Compute QA ÷ QF as the one-for-one contract count.
Multiply the two numbers, round, and attach the direction (short if you sell the asset, long if you buy it).
If asked for effectiveness, just square ρ.

Common mistakes in Hedging with Futures and Basis Risk

Using the wrong side of the market, for example going long futures to protect a holding.
Students think of hedging as buying protection, like insurance.
Fix: Ask what hurts you. If a price fall hurts, you need a gain when price falls, so go short futures.
Swapping σS and σF in the hedge ratio.
The formula looks symmetric and the order is easy to forget.
Fix: Remember spot on top: h* = ρ × σS ÷ σF. It is the regression of spot changes on futures changes.
Using correlation of price levels or the volatility of returns in different units.
Questions give data in several forms.
Fix: Use correlation and standard deviations of changes over the hedge horizon, with both volatilities in the same terms.
Forgetting contract size and mixing units, such as barrels and thousands of barrels.
Time pressure and unit labels hidden in the stem.
Fix: Write QA and QF with units before dividing.
Saying basis risk disappears when you hedge.
Students assume futures lock in a price.
Fix: A hedge replaces price risk with basis risk. Only when the hedge ends at expiry in the same asset is basis risk close to zero.
Reversing which basis move helps which hedger.
The sign convention is confusing.
Fix: With basis = spot − futures, a strengthening basis helps a short hedger and hurts a long hedger. Check with the effective price F1 + b2: a higher b2 raises the price received.

Worked examples

Example 1

A refiner will buy 200,000 barrels of jet fuel in three months. No jet fuel futures exist, so it hedges with heating oil futures. The correlation between changes in jet fuel and heating oil futures prices is 0.85. The standard deviation of jet fuel price changes is 0.042 and that of futures price changes is 0.050 (both in USD per gallon). One heating oil contract covers 42,000 gallons. The exposure is 200,000 barrels, with 42 gallons per barrel. How many contracts should the refiner trade?

Show the solution

The refiner will buy fuel, so it needs a long hedge: buy futures.
Exposure in gallons: QA = 200,000 × 42 = 8,400,000 gallons.
Hedge ratio: h* = 0.85 × 0.042 ÷ 0.050 = 0.85 × 0.84 = 0.714.
Contracts: N* = 0.714 × 8,400,000 ÷ 42,000 = 0.714 × 200 = 142.8.
Round to the nearest whole number: 143.

Answer: Buy (go long) about 143 heating oil futures contracts.

Example 2

On 1 March an exporter expects to sell 50,000 units of a commodity on 1 June. The June futures price on 1 March is USD 80. On 1 June the exporter sells at the spot price of USD 76 and closes the futures at USD 77. What is the effective price per unit received, and what was the basis on 1 June?

Show the solution

The exporter sells later, so it uses a short hedge: sell futures on 1 March at 80.
Futures gain per unit: 80 − 77 = 3.
Cash sale price: 76. Effective price = 76 + 3 = 79.
Basis on 1 June: b2 = S2 − F2 = 76 − 77 = −1.
Check with the formula: F1 + b2 = 80 + (−1) = 79. This matches.

Answer: Effective price is USD 79 per unit, and the 1 June basis is −USD 1. Total proceeds are 50,000 × 79 = USD 3,950,000.

Exam tips

Read the direction of the exposure first. Many wrong options are the right size with the wrong side.
Check units. Contract size and exposure are often given in different units, such as barrels, gallons or tonnes.
If a question gives both ρ and the two volatilities, the answer is almost always h* = ρ × σS ÷ σF. Do not take h* = 1.
Know the sign convention of the basis in the question stem before judging whether a basis move helps or hurts.
Questions about effectiveness use ρ². Compute it before looking at options.

Practice questions from Futures Markets

Hedging with Futures and Basis Risk 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 Basis Risk: frequently asked questions

What is the minimum variance hedge ratio formula in FRM Part I?

It is h* = ρ × σS ÷ σF, where ρ is the correlation between spot and futures price changes. It is also the slope from regressing spot changes on futures changes. It minimises the variance of the hedged position.

How do I calculate the number of futures contracts for a hedge?

Multiply the hedge ratio by the exposure and divide by the size of one contract: N* = h* × QA ÷ QF. Keep both sizes in the same units. Round to the nearest whole contract.

What is basis risk in futures?

Basis risk is the risk that spot minus futures changes unpredictably over the hedge. It arises from asset mismatch, date mismatch or early close-out. It is the risk left after hedging.

When do I use a short hedge versus a long hedge?

Use a short hedge when you own or will sell the asset and fear a price fall. Use a long hedge when you will buy the asset and fear a price rise.