FRM Exam Part I · Using Futures for Hedging

Minimum Variance Hedge Ratio for FRM Part I

Updated 11 October 2026 · Fact-checked

The minimum variance hedge ratio is h* = ρ × σS ÷ σF, where ρ is the correlation between spot and futures price changes. It is the slope of a regression of spot changes on futures changes. Multiply h* by exposure ÷ contract size to get the optimal number of contracts.

Understand Minimum Variance Hedge Ratio

A hedge removes price risk only if the futures price moves one-for-one with the asset you hold. In practice it often does not. The hedged asset may differ from the futures underlying (cross hedging), or the maturity may not match. The leftover risk is basis risk.

So the right question is not how many futures offset the exposure in size, but how many minimise the variance of the hedged position. Suppose you are long the spot asset and short h futures per unit of spot. Over the hedge period, the change in value per unit of spot is ΔS − h × ΔF. Its variance is σS² + h² σF² − 2 h ρ σS σF.

Differentiate with respect to h and set to zero: 2 h σF² − 2 ρ σS σF = 0. This gives h* = ρ σS ÷ σF. It is also the slope coefficient from regressing ΔS on ΔF, because slope = Cov(ΔS, ΔF) ÷ Var(ΔF).

Read the formula in two parts. Correlation scales the hedge down when futures track spot poorly. The volatility ratio adjusts for futures that move more or less than spot. If ρ = 1 and σS = σF, then h* = 1, the textbook one-to-one hedge.

Hedge effectiveness is the share of spot variance removed by the hedge. It equals ρ², which is the R² of the regression. A correlation of 0.9 removes 81% of the variance, not 90%. The remaining variance is σS² × (1 − ρ²).

Key formulas to remember

Minimum variance hedge ratio: h* = ρ × σS ÷ σF = Cov(ΔS, ΔF) ÷ Var(ΔF); σS and σF are standard deviations of changes in spot and futures prices over the hedge period. h* is the slope of the regression of ΔS on ΔF.
Optimal number of contracts: N* = h* × QA ÷ QF; QA is the size of the position being hedged (units of the asset). QF is the size of one futures contract in the same units.
Hedge effectiveness: Effectiveness = ρ² = R² = 1 − Var(hedged) ÷ Var(unhedged); It is the proportion of variance eliminated.
Variance of hedged position (per unit): Var = σS² + h² σF² − 2 h ρ σS σF; At h = h*, this reduces to σS² × (1 − ρ²).
Minimum hedged standard deviation: σhedged = σS × √(1 − ρ²); Per unit of spot exposure. Multiply by the quantity hedged for the total.
Number of contracts using values (tailing form): N* = h* × VA ÷ VF; VA is the value of the position and VF is the value of one futures contract (futures price × contract size). Use it when the question gives values rather than units.

How to solve Minimum Variance Hedge Ratio questions

Use this order for any question on the minimum variance hedge ratio, number of contracts or hedge effectiveness.

1Identify what is given: σS, σF and ρ, or Cov(ΔS, ΔF) and σF, or regression output.
2Check that the volatilities are for the same horizon and are for price changes, not levels. If one is a variance, take the square root where needed.
3Compute h* = ρ × σS ÷ σF, or Cov ÷ σF² if covariance is given.
4Convert to contracts: N* = h* × QA ÷ QF. Use units if given, or values if given.
5Round to the nearest whole contract, as futures trade in whole numbers.
6Decide the direction: short futures hedges a long spot position, long futures hedges a short position (for example, a future purchase).
7If asked for effectiveness, compute ρ², then the hedged variance σS² (1 − ρ²) if needed.

Quickest way: Three-line hedge calculation

When to use it: Use for standard numeric MCQs where ρ, σS and σF are given directly.

Write h* = ρ × σS ÷ σF and plug in. Sanity check: h* should be near ρ when volatilities are similar.
Multiply by exposure ÷ contract size to get N*. Round at the end only.
For effectiveness, square ρ. For hedged risk, use σS × √(1 − ρ²).
Eliminate options with the wrong direction or those that forgot the volatility ratio or used ρ alone.

Common mistakes in Minimum Variance Hedge Ratio

Using h* = ρ × σF ÷ σS (ratio inverted).
The order of the volatilities is easy to swap from memory.
Fix: Remember it is the slope of ΔS on ΔF, so spot volatility is on top. If futures are more volatile than spot, you need fewer than ρ per unit.
Reporting effectiveness as ρ instead of ρ².
Correlation feels like a percentage of risk removed.
Fix: Effectiveness is variance reduction, so it is ρ². ρ = 0.8 gives 64%.
Computing the number of contracts without dividing by contract size.
Students stop after h* and treat it as the contract count.
Fix: h* is per unit of exposure. Always compute N* = h* × QA ÷ QF.
Plugging in variances as if they were standard deviations.
Questions sometimes give variance or covariance to test attention.
Fix: Take square roots of variances. If covariance is given, use h* = Cov ÷ σF².
Using the wrong hedge direction.
Focus on the number hides the long or short logic.
Fix: A holder of the asset shorts futures. Someone who will buy the asset later goes long futures.
Assuming a minimum variance hedge removes all risk when ρ is high but below 1.
A high ρ looks like a near-perfect hedge.
Fix: Residual variance is σS² (1 − ρ²). Basis risk always remains unless ρ = 1.

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 covering 42,000 gallons. The standard deviation of the change in jet fuel price over the period is 0.032 per gallon, the standard deviation of the change in the futures price is 0.040, and the correlation is 0.80. What are the minimum variance hedge ratio, the optimal number of contracts and the hedge effectiveness?

Show the solution

h* = ρ × σS ÷ σF = 0.80 × 0.032 ÷ 0.040 = 0.80 × 0.8 = 0.64.
N* = h* × QA ÷ QF = 0.64 × 2,000,000 ÷ 42,000 = 1,280,000 ÷ 42,000 = 30.476.
Round to the nearest whole contract: 30.
The airline will buy fuel, so it goes long futures.
Effectiveness = ρ² = 0.80² = 0.64, so 64% of the variance is removed.

Answer: h* = 0.64; go long about 30 contracts; hedge effectiveness = 64%.

Example 2

A fund holds a USD 15,000,000 bond portfolio and hedges with a futures contract whose value is USD 120,000 per contract. The variance of spot changes is 0.0009 and the variance of futures changes is 0.0016 (same units, same horizon). Their covariance is 0.00108. Find the optimal number of contracts and the standard deviation of the hedged position per unit of spot as a fraction of the unhedged standard deviation.

Show the solution

Standard deviations: σS = √0.0009 = 0.03 and σF = √0.0016 = 0.04.
Correlation: ρ = Cov ÷ (σS σF) = 0.00108 ÷ (0.03 × 0.04) = 0.00108 ÷ 0.0012 = 0.90.
h* = Cov ÷ σF² = 0.00108 ÷ 0.0016 = 0.675. Check: 0.90 × 0.03 ÷ 0.04 = 0.675.
N* = h* × VA ÷ VF = 0.675 × 15,000,000 ÷ 120,000 = 0.675 × 125 = 84.375.
Round to 84 contracts. The fund holds the bonds, so it shorts futures.
Hedged standard deviation ÷ unhedged = √(1 − ρ²) = √(1 − 0.81) = √0.19 = 0.4359.

Answer: Short about 84 contracts (h* = 0.675). The hedged standard deviation is about 43.6% of the unhedged standard deviation, with 81% of variance removed.

Exam tips

Check whether the question gives variances, covariance or standard deviations before using the formula. Questions often supply covariance to test h* = Cov ÷ Var(F).
Effectiveness questions are about ρ² or R². If the stem gives R² from a regression, you can read effectiveness straight from it, and h* is the regression slope.
Watch the units: contract size in units, futures value in currency. Use the matching version of N*.
Round the number of contracts only at the end, and to the nearest whole number unless told otherwise.
Expect conceptual options too: h* is below 1 when ρ is low or futures are more volatile than spot, and it can exceed 1 when spot is more volatile than futures and ρ is high.

Practice questions from Using Futures for Hedging

Minimum Variance Hedge Ratio: frequently asked questions

What is the minimum variance hedge ratio formula?

It is h* = ρ × σS ÷ σF. ρ is the correlation between changes in spot and futures prices, σS is the standard deviation of spot changes and σF is that of futures changes. It equals the slope of a regression of spot changes on futures changes.

How do I calculate the optimal number of futures contracts?

Multiply h* by the size of the exposure and divide by the size of one futures contract: N* = h* × QA ÷ QF. If values are given, use position value and futures contract value. Round to a whole number.

What is the hedge effectiveness formula?

Hedge effectiveness is ρ², the R² of the regression of spot changes on futures changes. It is the fraction of the variance of the unhedged position that the hedge removes. With ρ = 0.9, effectiveness is 81%.

What is cross hedging and why does it change the hedge ratio?

Cross hedging uses futures on a different asset from the one you hold, such as heating oil for jet fuel. Correlation is below 1 and volatilities differ, so the best ratio is not 1. The minimum variance formula adjusts for both.