Hedging with futures: the minimum-variance hedge ratio and basis risk for FRM candidates
A futures hedge takes a futures position that should gain when the exposure you already hold loses. In practice the futures price rarely moves one for one with the price you care about, so the question becomes how many contracts to use and what risk is left over. This is a standard topic in FRM® Part I (Financial Markets and Products). The FRM page lists the other guides, and how to plan your FRM study helps with ordering them.
Basis and basis risk
The basis is the spot price of the asset being hedged minus the futures price of the contract used. Conventions on sign vary by textbook, so check the one in your reading. At expiry, if the contract is on the same asset in the same place, the basis converges to zero. Basis risk is the uncertainty in the basis when the hedge is closed. It arises when:
- the asset hedged differs from the asset underlying the futures (a cross hedge),
- the hedge ends on a date that is not the contract's expiry,
- the futures are for a different grade or location.
A perfect hedge removes all price risk only if the basis is known at the close. Otherwise the hedger swaps price risk for basis risk, which is typically smaller.
The minimum-variance hedge ratio
Let S be the change in the spot price over the hedge period and F the change in the futures price. The hedge ratio that minimises the variance of the hedged position is
h = rho x (sigma_S / sigma_F)
where rho is the correlation between the two changes and sigma_S and sigma_F are their standard deviations. It is the slope of a regression of spot changes on futures changes. If rho is 1 and the volatilities are equal, h is 1. The proportion of variance removed is rho^2, the hedge effectiveness.
The number of futures contracts is
N = h x (quantity hedged) / (size of one contract)
A sale (short) of futures hedges a long exposure, such as owning or needing to sell the asset later; a purchase (long) hedges a short exposure, such as a future purchase. Because contracts come in fixed sizes, N is rounded to the nearest whole number.
Worked example
An invented airline, Oxbridgeway Air, expects to burn 2 million gallons of jet fuel in three months. It buys heating oil futures, since no jet fuel contract is liquid. Contracts cover 42,000 gallons. From past data, the standard deviation of three-month fuel price changes is 0.040 per gallon (sigma_S) and that of futures changes is 0.050 (sigma_F), with correlation 0.92. These figures were checked in Python before writing.
- h = 0.92 x (0.040 / 0.050) = 0.736.
- N = 0.736 x 2,000,000 / 42,000 = 35.05, so 35 contracts (long, because the airline is buying fuel).
- Hedge effectiveness = 0.92^2 = 0.846, so about 85% of the variance is removed. Around 15% remains, the basis risk.
- The 35 contracts cover 1.47 million gallons, which is 0.735 of the exposure and matches h.
A naive one-for-one hedge would need about 48 contracts (2,000,000 / 42,000), over-hedging because the futures are more volatile than the spot price.
Tailing the hedge
When the futures are marked to market daily, some textbooks adjust the contract count by the ratio of spot value to futures value (tailing); the adjustment is small for short hedges and is often ignored in exam questions unless it is requested.
Usual mistakes
- Setting h = 1 by default instead of using the volatility ratio and correlation.
- Using price levels in place of price changes to estimate rho and the volatilities.
- Swapping the ratio so that h = rho x sigma_F / sigma_S.
- Taking the wrong side: a company that will buy the asset goes long futures.
- Believing a hedge locks in a price. It locks in a price plus or minus the basis at the end.
- Forgetting to round contracts to whole numbers.
How the exam typically tests it
Typically you compute h from given statistics, convert it to a contract count, work out the effectiveness as rho squared, or say which side of the market to trade. Questions on the variance of the hedged position follow the same logic as portfolio variance in Value at Risk explained. Charta's practice questions for this topic let you rehearse the arithmetic.
Common questions
What is the basis?
The spot price minus the futures price, in the convention most texts use. It converges towards zero at expiry for the same asset and location.
Why is the optimal hedge ratio often below 1?
Because h = rho x (sigma_S / sigma_F). If the correlation is less than 1, or the futures are more volatile than the spot, the minimum-variance position is smaller than the exposure.
Does a futures hedge remove all risk?
No. It removes the part of the price risk that moves with the futures. Basis risk remains, and so does the cash-flow risk from daily margin calls.
What does rho squared tell me?
It is the proportion of the spot price variance eliminated by the optimal hedge, the hedge effectiveness.
Sources
- GARP, the FRM programme: what the exam is and its two parts.
The exam and the syllabus can change. Check GARP's current details before you book.