FRM Exam Part I · How Do Firms Manage Financial Risk?

Minimum Variance Hedge Ratio and Basis Risk

Updated 11 October 2026 · Fact-checked

The minimum variance hedge ratio is h* = ρ × σS ÷ σF, the slope from regressing spot price changes on futures price changes. It sets the futures position that cuts the variance of the hedged position the most. Basis risk and cross-hedging make ρ fall below 1, so the hedge is imperfect.

Understand Hedge Ratio and Basis Risk

A hedge uses a futures position to offset price moves in an asset you own or will buy or sell. If the futures price moved exactly with your spot price, you would sell one unit of futures per unit of exposure and the risk would vanish. In practice the two prices do not move one for one.

The basis is the spot price minus the futures price: Basis = S − F. Basis risk is the uncertainty about the basis when you close the hedge. It arises when the hedge horizon differs from the futures expiry, or when the asset hedged is not the asset underlying the contract. If you hedge with a contract on a different asset, this is cross-hedging. For example, you may hedge jet fuel with heating oil futures.

Because of basis risk, a one-for-one hedge is not always the best. The minimum variance hedge ratio is the number of futures units per unit of exposure that minimises the variance of the hedged position. It depends on how volatile spot and futures are, and on how closely they move together. It equals the slope of a regression of ΔS on ΔF.

The formula is h* = ρ × σS ÷ σF. If correlation is 1 and volatilities are equal, h* = 1. If correlation is low, h* is smaller. The number of contracts is then N* = h* × QA ÷ QF, where QA is the exposure size and QF is the contract size.

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 gives effectiveness of 0.81. Even the best hedge leaves residual variance of σS² × (1 − ρ²).

Key formulas to remember

Basis: Basis = Spot price − Futures price; Sign convention follows Hull: spot minus futures. Check the question's definition.
Minimum variance hedge ratio: h* = ρ × σS ÷ σF; σS and σF are standard deviations of changes in spot and futures prices over the hedge life. ρ is their correlation. Equals the regression slope of ΔS on ΔF.
Number of futures contracts: N* = h* × QA ÷ QF; QA = size of exposure in units. QF = size of one futures contract in units. Round to the nearest whole contract.
Hedge effectiveness: Effectiveness = ρ² = h*² × σF² ÷ σS²; Proportion of the variance of the unhedged position eliminated.
Variance of hedged position (per unit of exposure): Var = σS² × (1 − ρ²); Valid at the optimal h*. Residual risk is its square root: σS × √(1 − ρ²).
Variance for any hedge ratio h: Var(h) = σS² + h² × σF² − 2 × h × ρ × σS × σF; Minimised at h = ρ × σS ÷ σF.

How to solve Hedge Ratio and Basis Risk questions

Use this order for any question on hedge ratios, basis risk or hedge effectiveness.

1Identify what you are hedging and what contract you use. Same asset means pure basis risk. Different asset means cross-hedging.
2List the inputs: σS, σF, ρ, or the regression slope. Note whether volatilities are of price changes over the same period.
3If the slope of ΔS on ΔF is given, h* equals that slope. Otherwise compute h* = ρ × σS ÷ σF.
4Compute contracts: N* = h* × QA ÷ QF. Use the correct sign: short futures to hedge a long exposure.
5Round to the nearest whole contract if the question asks for a number of contracts.
6For effectiveness, compute ρ². For residual risk, compute σS × √(1 − ρ²).
7For basis questions, compute the basis at both dates. Hedged outcome = S₂ + (F₁ − F₂) for a short hedger, which equals F₁ + b₂.
8Sanity check: h* should be near 1 for a good hedge and well below 1 when correlation is weak.

Quickest way: Shortcut: ratio of volatilities times correlation

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

Compute h* = ρ × σS ÷ σF in one line.
Multiply by exposure ÷ contract size to get contracts, then round.
Square ρ for effectiveness. Do not recompute variances.
Check the options: if only one is near your value, stop. Distractors often use σF ÷ σS or ρ² instead of ρ.
On a calculator, enter ρ, multiply by σS, divide by σF, and store the result for the contracts step.

Common mistakes in Hedge Ratio and Basis Risk

Inverting the volatility ratio and using σF ÷ σS.
The two symbols look alike and students recall 'ratio of volatilities' without the order.
Fix: Remember the regression logic: spot is the dependent variable, so the slope has σS on top. h* = ρ × σS ÷ σF.
Reporting hedge effectiveness as ρ instead of ρ².
Correlation is given directly, so students stop there.
Fix: Effectiveness is the variance reduction, which is R² = ρ². A ρ of 0.8 means 64% effectiveness.
Assuming the hedge ratio is always 1.
Textbook examples with identical assets teach one-for-one hedging.
Fix: Use 1 only when the optimal h* is 1. With cross-hedging or different volatilities, compute h* explicitly.
Using volatilities of price levels instead of price changes.
Data are shown as prices and students take their standard deviation directly.
Fix: The inputs are standard deviations of changes in S and F over the hedge period. Use what the question labels as such.
Forgetting to divide by the contract size, or mixing units.
Students compute h* and treat it as the number of contracts.
Fix: Use N* = h* × QA ÷ QF. Write down the units of QA and QF before dividing.
Believing a perfect hedge removes basis risk whenever the same asset is used.
Basis risk is linked only to different assets.
Fix: Basis risk also exists with the same asset if you close out before expiry or the expiry does not match your horizon.

Worked examples

Example 1

An airline will buy 2,000,000 gallons of jet fuel in three months and 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 the heating oil futures price is 0.040 per gallon over the hedge life. Correlation is 0.80. Find the minimum variance hedge ratio, the number of contracts and the hedge effectiveness.

Show the solution

h* = ρ × σS ÷ σF = 0.80 × 0.032 ÷ 0.040.
0.032 ÷ 0.040 = 0.80, so h* = 0.80 × 0.80 = 0.64.
N* = h* × QA ÷ QF = 0.64 × 2,000,000 ÷ 42,000.
0.64 × 2,000,000 = 1,280,000. Then 1,280,000 ÷ 42,000 = 30.48.
Round to 30 contracts. The airline buys (goes long) futures because it is hedging a future purchase.
Effectiveness = ρ² = 0.80² = 0.64.

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

Example 2

A fund manager has a long USD 10 million position and hedges with futures. The standard deviation of spot changes is 2.0% and of futures changes is 2.5%. Correlation is 0.90. Compute the minimum variance hedge ratio and the standard deviation of the hedged position in USD, using the exposure of USD 10 million.

Show the solution

h* = ρ × σS ÷ σF = 0.90 × 2.0 ÷ 2.5.
2.0 ÷ 2.5 = 0.80, so h* = 0.90 × 0.80 = 0.72.
Unhedged standard deviation = 2.0% × USD 10,000,000 = USD 200,000.
Residual standard deviation = σS × √(1 − ρ²) = 200,000 × √(1 − 0.81).
1 − 0.81 = 0.19 and √0.19 = 0.4359.
200,000 × 0.4359 = USD 87,178.

Answer: h* = 0.72; the hedged position has a standard deviation of about USD 87,178, down from USD 200,000.

Exam tips

Write the formula h* = ρ × σS ÷ σF first. Most errors come from getting σS and σF the wrong way round.
Expect questions that give a regression slope or R². The slope is h*, and R² is the hedge effectiveness.
When the answer asks for a number of contracts, check whether it wants a long or short position and round to the nearest whole contract.
For basis questions, compute S − F at both dates and note that a change in the basis is the source of hedge gain or loss.
Check for traps: unit mismatches, prices versus price changes, and distractor options that equal ρ² or σF ÷ σS.

Practice questions from How Do Firms Manage Financial Risk?

Hedge Ratio 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.

Hedge Ratio and Basis Risk: frequently asked questions

What is the minimum variance hedge ratio formula?

It is h* = ρ × σS ÷ σF. Here 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 is also the slope from regressing ΔS on ΔF.

What is basis risk in hedging?

Basis risk is the risk that the spot price minus the futures price changes unpredictably before you close the hedge. It occurs when the hedged asset differs from the contract's underlying, or when your horizon does not match the contract expiry. It stops a hedge from being perfect.

What is cross-hedging?

Cross-hedging means using a futures contract on a different asset from the one you are exposed to, such as heating oil futures for jet fuel. It works when prices are highly correlated. The lower the correlation, the lower the hedge effectiveness.

How do I calculate hedge effectiveness?

Hedge effectiveness is ρ², the R² of the regression of spot changes on futures changes. It is the proportion of the unhedged variance that the hedge removes. A correlation of 0.9 gives 81% effectiveness.