FRM Part 1 – Financial Markets and Products · FRM
Basis Risk in Financial Derivatives: The Whole Residual Risk
Basis risk is usually introduced as the risk that a hedge does not move perfectly with the thing it is hedging. That is true and it is not useful, because it describes the problem without telling you what you are actually exposed to.
The precise statement is sharper and it is the one the exam tests. When a hedge is placed today and lifted before the futures contract matures, the price the hedger ends up with is the futures price known today plus the basis on the day the hedge is lifted. The first term is certain. The second is not. Basis risk is not one risk among several that survive the hedge. After the hedge is in place, it is the entire remaining risk.
Key Takeaways
- Basis is the spot price of the asset being hedged minus the futures price of the contract used, and the sign convention must be fixed before anything else is said.
- Both a long and a short hedge settle at the initial futures price plus the closing basis, so after the hedge is placed the basis is the entire remaining risk.
- Basis risk has three sources: asset mismatch, timing mismatch, and uncertainty about the transaction date.
- A strengthening basis helps a short hedge and hurts a long hedge. Derive it rather than memorise it, because it reverses under the other sign convention.
- A cross hedge adds a second component that does not converge at maturity, because nothing forces two different assets to the same price.
- The minimum variance hedge ratio is the correlation times the ratio of standard deviations, and hedge effectiveness is the correlation squared.
What basis is, and the convention that matters
Basis is the spot price of the asset being hedged minus the futures price of the contract being used to hedge it.
bt = St − Ft
where S is the spot price of the asset actually held or needed, and F is the price of the futures contract used
Two details in that sentence do real work. The asset in the spot term is the asset the hedger cares about, which is not necessarily the asset the futures contract is written on. And the futures price is that of the specific contract chosen, which has a specific maturity. Change either choice and the basis changes.
The sign convention is worth stating once and holding to it. Some texts, particularly on interest rate futures, define basis as futures minus spot. Both conventions appear in practice and neither is wrong, but every statement about the basis strengthening or weakening reverses between them. This article uses spot minus futures throughout, which is the convention the FRM curriculum uses for commodity and general hedging problems.
The reason basis exists at all is that a futures price is a spot price carried forward. Storage, insurance, financing and any convenience yield from holding the physical asset all sit between the two. As the contract approaches delivery there is progressively less carrying left to price, so the two prices converge, and for a deliverable asset at the delivery location the basis goes to zero at maturity.
Where basis risk comes from
Basis risk has three sources, and separating them is worth doing because they are managed differently and only one of them is unavoidable.
| Source | What creates it | Example |
|---|---|---|
| Asset mismatch | The asset being hedged is not the asset underlying the futures contract | Hedging jet fuel with heating oil futures, because no liquid jet fuel contract is available |
| Timing mismatch | The hedge is lifted before the futures contract matures | A November exposure hedged with a December contract, closed out in November |
| Date uncertainty | The hedger does not know exactly when the asset will be bought or sold | A harvest whose completion date depends on weather |
The second and third are usually structural rather than chosen. A hedger picks the contract whose maturity is just past the exposure date, precisely because holding a contract into its delivery month brings delivery mechanics and thinning liquidity. The result is a hedge that is almost always lifted early, and therefore almost always exposed to a non-zero basis.
The first is a choice, and it is the one that dominates when it is present. A hedge whose futures contract is written on a different asset does not converge on the hedger’s spot price even at maturity, because there is nothing forcing two different assets to the same price. Section 5 treats it separately.
The equation that shows basis risk is the whole residual risk
Take a hedger who will buy an asset at a future date and hedges by going long futures today at price F1. At date 2 the asset is bought at spot S2 and the futures position is closed at F2.
Effective cost = S2 − (F2 − F1) = F1 + (S2 − F2) = F1 + b2
where F1 is known when the hedge is placed and b2 is the basis when it is lifted
The same algebra runs for a short hedge and lands on the same expression, with effective proceeds equal to F1 + b2. This is the single most important line in the topic. The spot price at date 2 has cancelled out completely. Where the market goes between the two dates does not affect the hedger at all. What affects the hedger is the basis on the day the hedge comes off, and nothing else.
A cable manufacturer will buy 500 tonnes of aluminium in late November. On 1 August the spot price is $2,450 per tonne and the December futures price is $2,510, so the current basis is $2,450 − $2,510 = −$60. The firm goes long December futures. On 20 November it buys the metal and closes the futures. Consider two outcomes in which the spot price ends up identical.
- Outcome A: spot is $2,780 and the December future is $2,820, so b2 = −$40.
- Effective cost in A = 2,780 − (2,820 − 2,510) = 2,780 − 310 = $2,470 per tonne. Check against the formula: F1 + b2 = 2,510 − 40 = $2,470.
- Outcome B: spot is again $2,780, but the December future is $2,795, so b2 = −$15.
- Effective cost in B = 2,780 − (2,795 − 2,510) = 2,780 − 285 = $2,495 per tonne. Check: 2,510 − 15 = $2,495.
- Across 500 tonnes the two outcomes differ by 500 × 25 = $12,500, on identical spot prices.
Answer: the spot price was the same in both outcomes and the hedger paid different prices. The difference came entirely from the basis on the day the hedge was lifted. This is what it means to say the hedge converts price risk into basis risk rather than eliminating risk.
Strengthening and weakening, and who each one helps
A basis strengthens when it increases, meaning the spot price rises relative to the futures price. It weakens when it falls. With the convention fixed and the equation in hand, the consequences follow mechanically rather than needing to be memorised.
| Basis move | Short hedge (will sell the asset) | Long hedge (will buy the asset) |
|---|---|---|
| Strengthens, b2 rises | Helps. Proceeds F1 + b2 are higher | Hurts. Cost F1 + b2 is higher |
| Weakens, b2 falls | Hurts. Proceeds are lower | Helps. Cost is lower |
In the worked example above the basis moved from −$60 to −$40 in one outcome and to −$15 in the other. Both are strengthenings, since both are increases, and the stronger one produced the higher cost for a long hedge. That is the table working exactly as written, and it is a useful check that the convention has been applied consistently.
Reading “strengthening” as “good”. It is directional, not favourable, and which side it favours depends entirely on whether the hedger is short or long the futures. The word describes what the number did, not what happened to the hedger. Candidates who memorise a good and bad mapping rather than deriving it from F1 + b2 reverse it under exam pressure, and they reverse it again if a question uses the futures minus spot convention.
Cross hedging, where the basis stops converging
Everything so far assumed the futures contract is written on the asset being hedged, so that convergence at maturity is guaranteed. A cross hedge breaks that assumption, and it breaks the guarantee with it.
Write S* for the spot price of the asset actually held and S for the spot price of the asset underlying the futures. The basis then splits into two pieces.
S*2 − F2 = (S2 − F2) + (S*2 − S2)
the first bracket is the ordinary basis, which converges to zero at maturity; the second is the spread between the two assets, which does not
This is why a cross hedge is a different proposition rather than a slightly worse version of the same thing. Holding the contract to maturity removes the first component and leaves the second untouched. An airline hedging jet fuel with heating oil futures still carries the jet fuel to heating oil spread on the day it lifts the hedge, and that spread has its own drivers: refinery economics, seasonal demand for one product but not the other, and regional supply disruptions that hit one grade harder than the other.
The practical consequence is that a cross hedge needs the relationship between the two assets to be examined directly, not assumed. Which brings the discussion to how much of the hedging instrument to hold.
The minimum variance hedge ratio
A one for one hedge is only optimal when the hedged asset and the futures move one for one. Where they do not, the ratio that minimises the variance of the hedged position is the one that accounts for both the correlation between them and their relative volatilities.
h* = ρ × (σΔS / σΔF)
ρ is the correlation between the change in spot and the change in futures, and the two sigmas are the standard deviations of those changes
The number of contracts follows from the hedge ratio, the size of the exposure and the contract size.
N* = h* × (QA / QF)
QA is the units of exposure and QF is the units per futures contract
An airline expects to buy 2,000,000 gallons of jet fuel and hedges with heating oil futures of 42,000 gallons each. Monthly changes in the jet fuel spot price have a standard deviation of 0.032, monthly changes in the heating oil futures price have a standard deviation of 0.040, and their correlation is 0.85.
- h* = 0.85 × (0.032 / 0.040) = 0.85 × 0.80 = 0.68.
- Units of futures required = 0.68 × 2,000,000 = 1,360,000 gallons.
- N* = 1,360,000 / 42,000 = 32.4, so the airline takes 32 contracts.
- Hedge effectiveness is ρ² = 0.85² = 0.7225.
Answer: 32 contracts, removing about 72% of the variance of the unhedged position and leaving about 28% in place. The residual is basis risk, and no choice of hedge ratio removes it, because h* minimises that variance rather than eliminating it.
Two cautions travel with this calculation. The hedge ratio is estimated from historical data, so it inherits whatever period was used, and a ratio fitted over a calm stretch can be badly wrong in a disrupted one. And a correlation of 0.85 sounds high until it is squared, at which point more than a quarter of the variance remains. Hedge effectiveness of 72% is a real reduction and it is not protection.
What cannot be done about it
Standard advice on managing basis risk runs to monitoring, dynamic adjustment, diversification of instruments and better technology. Each of these is worth doing and none of them removes the exposure, so it is worth being clear about what each one actually achieves.
Choosing a contract whose maturity sits closer to the exposure date genuinely shrinks the expected basis, because there is less carrying left to price. This is the single most effective lever available, and it is bounded by liquidity, since contracts far out or in unusual months trade thinly and the transaction cost of a poor fill can exceed the basis saved.
Choosing a contract on a closer asset removes the second component of a cross hedge basis. This too is bounded, because if a liquid contract on the right asset existed the hedger would already be using it.
Monitoring and dynamic rebalancing address a different problem: they keep the hedge ratio current as correlations and volatilities move. They do not reduce b2, which is set by the market on the day the hedge is lifted.
The honest framing is that a hedge is a trade of one exposure for a smaller one. Price risk is large and largely unpredictable. Basis risk is smaller and, for a well matched contract held near to maturity, considerably better behaved. That is the whole benefit, and a hedger who expects more than that will at some point be surprised by a hedge that behaved exactly as designed.
Which is also why basis risk sits early in the FRM curriculum rather than as a footnote to hedging. It is the first place a candidate meets the idea that a risk management action changes the shape of an exposure rather than removing it, and that idea runs through everything that follows.
The short version
- Basis is the spot price of the asset being hedged minus the futures price of the contract being used, and the sign convention must be fixed before anything else is said.
- Both a long and a short hedge settle at F1 + b2, so after the hedge is placed the basis at lift-off is the entire remaining risk.
- Three sources: asset mismatch, timing mismatch and uncertainty about the transaction date.
- A strengthening basis helps a short hedge and hurts a long hedge. Derive it from F1 + b2 rather than memorising it.
- A cross hedge adds a second component that does not converge at maturity, because nothing forces two different assets to the same price.
- The minimum variance hedge ratio is ρ × σΔS / σΔF, and hedge effectiveness is ρ², which is always lower than the correlation suggests.
Frequently Asked Questions
What exactly is basis risk?
Basis is the spot price of the asset being hedged minus the futures price of the contract used to hedge it. Basis risk is the uncertainty about what that difference will be on the day the hedge is lifted. Because both a long and a short hedge settle at the initial futures price plus the closing basis, the basis is not one residual risk among several after a hedge is placed. It is the entire remaining exposure.
Why does basis risk exist if the hedge is designed to remove price risk?
Because a futures price is a spot price carried forward, and the carrying costs between them do not disappear on a fixed schedule. The two prices converge only as delivery approaches. A hedge lifted before the contract matures keeps whatever basis is left, and hedgers almost always lift early to avoid delivery mechanics and thinning liquidity in the delivery month.
Does a strengthening basis help or hurt?
It depends on the direction of the hedge. A strengthening basis means the basis increases, so the spot price rises relative to the futures price. Since both hedges settle at the initial futures price plus the closing basis, a higher closing basis raises the proceeds for a short hedge and raises the cost for a long hedge. Derive it rather than memorise it, because the answer reverses under the futures minus spot convention.
How is cross hedging different?
A cross hedge uses a contract written on a different asset, which splits the basis into two parts: the ordinary basis, which still converges to zero at maturity, and the spread between the two assets, which does not. Holding the contract to delivery removes the first and leaves the second untouched, so a cross hedge is a different proposition rather than a slightly worse version of the same one.
How is the optimal hedge ratio calculated?
The minimum variance hedge ratio is the correlation between spot and futures price changes multiplied by the ratio of their standard deviations. The number of contracts is that ratio multiplied by the size of the exposure and divided by the contract size. The ratio minimises the variance of the hedged position; it does not eliminate it.
What does hedge effectiveness of 0.72 actually mean?
It means about 72% of the variance of the unhedged position has been removed and about 28% remains as basis risk. Hedge effectiveness is the square of the correlation, so a correlation that sounds high translates into a less impressive variance reduction. A correlation of 0.85 leaves more than a quarter of the variance in place.
Loading comments...
Add your Thoughts: