Basis Risk And Cross-Hedging With Futures

Why a Futures Hedge Can Leave Residual Risk

A futures hedge can offset much of a price change without fixing the final cash flow exactly. The remaining uncertainty is basis risk: the spot price of the asset being hedged and the futures price may not move together perfectly. That can happen because the hedge is closed before the contract expires, or because the futures contract is written on a different but related asset.

The earlier material on sizing futures hedges addresses the number of contracts needed for a planned exposure. Here, the focus is on what the hedge leaves behind. In particular, a candidate must distinguish a futures position’s gain or loss from the change in the basis, and must not assume that a cross-hedge removes all price risk.

Definition: Let $S_t$ be the spot price per unit of the asset being hedged at time $t$, and let $F_t$ be the futures price per matching unit at that time. Define the basis by $b_t=S_t-F_t$. The basis changes by $\Delta b=b_T-b_0$ over a hedge period from time $0$ to time $T$. For a cross-hedge, $S_t$ is the price of the exposure asset and $F_t$ is the futures price on a different underlying asset.

This sign convention matters. A positive change in basis means $S_t-F_t$ has increased. For a seller who is short futures, an increase in the basis improves the hedged sale proceeds relative to a result calculated using an unchanged basis. For a buyer who is long futures, it increases the hedged purchase cost.

Basis Risk When the Hedge Closes Before Expiry

Suppose an asset owner plans to sell $Q$ units at time $T$, and shorts futures at time $0$. Assume the futures position represents the same asset and quantity, with no transaction costs. Ignore interest earned or paid on daily marking-to-market cash flows, so the cumulative futures gain is treated as $Q(F_0-F_T)$. The combined proceeds are:

$$ QS_T+Q(F_0-F_T)=Q(F_0+b_T). $$

The futures position offsets changes in the futures price, but the final proceeds still depend on the basis at the hedge date. If the basis is not known in advance, the final proceeds are not locked in. If the hedge is held to expiry and the contract is deliverable against the exact asset being sold, convergence may make the final basis zero under the contract’s delivery terms. That does not make the basis zero at an earlier close date, or for a different asset.

Worked Example: Closing a Short Hedge Before Expiry

A merchant plans to sell 1,000 units of a commodity in three months. The relevant futures contract represents 100 units, so the merchant shorts 10 contracts at a futures price of $52.00 per unit. At the sale date, before the futures contracts expire, the commodity’s spot price is $48.70 and the futures price is $49.10. Find the combined proceeds and compare them with the proceeds that would result if the initial basis had stayed unchanged. Ignore transaction costs and interest on marking-to-market cash flows.

Method. Calculate the spot-sale proceeds and the short-futures gain separately, then combine them. The position covers $10(100)=1{,}000$ units.

Spot proceeds. The sale brings in $1{,}000(48.70)=\$48{,}700$.

Futures gain. The short position gains $1{,}000(52.00-49.10)=\$2{,}900$.

Combined proceeds. The total is $\$48{,}700+\$2{,}900=\$51{,}600$.

Basis check. The initial basis was $b_0=51.40-52.00=-\$0.60$, where the initial spot price is $51.40. The closing basis is $b_T=48.70-49.10=-\$0.40$. The formula gives $1{,}000(52.00-0.40)=\$51{,}600$. If the initial basis had remained unchanged, the proceeds would have been $1{,}000(52.00-0.60)=\$51{,}400$. The actual basis increased by $0.20 per unit, adding $\$200$ to the proceeds. The calculation is consistent with both the cash flows and the basis formula.

Common mistake avoided. Treating the basis as unchanged would give $51,400, not the actual $51,600. The hedge reduces exposure to the futures-price movement; it does not make the closing basis equal to the initial basis.

Cross-Hedging and the Minimum-Variance Hedge Ratio

A cross-hedge uses futures on one asset to hedge price changes in another. The two prices may be related, but their changes will generally not match one-for-one. For example, differences in grade, location, or supply conditions can cause the exposure asset’s price to move differently from the futures price. A hedge ratio below or above one may be more appropriate than a one-for-one hedge.

Let $\Delta S$ and $\Delta F$ denote price changes over the same hedge horizon. Let $h$ be the number of futures-price units used per unit of exposure, with the futures position taken in the direction that offsets the exposure. The remaining price change per exposure unit is $\Delta S-h\Delta F$. The minimum-variance hedge ratio is:

h^*=\frac{\operatorname{Cov}(\Delta S,\Delta F)} {\operatorname{Var}(\Delta F)} =\rho\frac{\sigma_S}{\sigma_F}.

Here, $\rho$ is the correlation between the two price changes, while $\sigma_S$ and $\sigma_F$ are their standard deviations. These statistics must refer to the same observation interval and a horizon relevant to the planned hedge. In practice, they are estimated from historical paired price changes; the resulting ratio is an estimate, not a guarantee.

Model assumptions: The minimum-variance formula applies when the price changes are measured on a consistent basis, their variances are finite, and a linear hedge is used. It chooses $h$ to minimize the variance of $\Delta S-h\Delta F$. It does not guarantee a profit, eliminate all risk, or ensure that past relationships persist.

Why the Minimum-Variance Formula Holds

For a given $h$, the variance of the residual price change is:

\begin{aligned} \operatorname{Var}(\Delta S-h\Delta F) &=\operatorname{Var}(\Delta S)+h^2\operatorname{Var}(\Delta F) -2h\operatorname{Cov}(\Delta S,\Delta F). \end{aligned}

The first term does not depend on $h$. Differentiate the expression with respect to $h$, set the derivative to zero, and solve:

2h\operatorname{Var}(\Delta F)-2\operatorname{Cov}(\Delta S,\Delta F)=0 \quad\Longrightarrow\quad h^*=\frac{\operatorname{Cov}(\Delta S,\Delta F)} {\operatorname{Var}(\Delta F)}.

When $\operatorname{Var}(\Delta F)>0$, the coefficient of $h^2$ is positive, so this stationary point minimizes the residual variance. Substituting $\operatorname{Cov}(\Delta S,\Delta F)=\rho\sigma_S\sigma_F$ gives the correlation-and-volatility form of the ratio. At this optimal ratio, the residual variance is $\sigma_S^2(1-\rho^2)$. Thus a perfect correlation can eliminate this modeled price-change risk, but a correlation with absolute value below one leaves residual variance.

Worked Example: Calculating a Cross-Hedge Size

A food processor expects to sell 50,000 units of a specialty ingredient in two months. It considers shorting futures on a related ingredient. For two-month price changes, assume the estimated standard deviations are $2.40 per unit for the specialty ingredient and $3.00 per unit for the futures underlying, with correlation $0.80$. Each futures contract covers 1,000 units. Find the minimum-variance number of contracts and the residual standard deviation of the aggregate price change under this model.

Method. First calculate the minimum-variance hedge ratio, then convert the exposure into contracts. For a planned sale, the futures position is short.

Hedge ratio. The ratio is $h^*=0.80(2.40/3.00)=0.64$. This means hedging 0.64 futures units per unit of the exposure asset.

Contract count. The target is $0.64(50{,}000)=32{,}000$ futures units. Dividing by 1,000 units per contract gives $32$ contracts to short.

Residual variance per exposure unit. The covariance is $0.80(2.40)(3.00)=5.76$. Therefore:

\begin{aligned} \operatorname{Var}(\Delta S-0.64\Delta F) &=2.40^2+0.64^2(3.00^2)-2(0.64)(5.76)\\ &=5.76+3.6864-7.3728\\ &=2.0736. \end{aligned}

The residual standard deviation is $\sqrt{2.0736}=\$1.44$ per unit. Across 50,000 units, the model’s aggregate residual standard deviation is $50{,}000(1.44)=\$72{,}000$. Without a hedge, the corresponding standard deviation would be $50{,}000(2.40)=\$120{,}000$. As a check, the variance reduction is $1-(1-0.80^2)=0.64$, or 64%, consistent with the optimal-ratio result.

Common mistake avoided. Using the correlation, 0.80, as the hedge ratio would ignore the different price-change volatilities. The correct ratio is 0.64, not 0.80.

Applying a Cross-Hedge and Reading the Residual

A hedge ratio describes a target hedge based on an estimated relationship. Once the hedge date arrives, the actual spot and futures price changes determine the result. The ratio does not turn the cross-hedge into a matched-asset hedge. In particular, a cross-hedger cannot generally rewrite the final cash flow as a futures price plus the basis for one identical deliverable asset.

If the exposure is $Q$ units and each contract covers $M$ units, a target ratio $h$ gives approximately $hQ/M$ contracts. Contract counts must be whole numbers, so rounding can add a small amount of residual exposure. The position direction also depends on the underlying cash flow: a seller typically shorts futures, while a buyer typically goes long.

Worked Example: A Buyer’s Cross-Hedge at the Hedge Date

A manufacturer will buy 12,000 units of a specialty material. It hedges with futures on a related material; each contract covers 500 units. Its estimated minimum-variance ratio is $0.75$, and it goes long the corresponding number of contracts. At the hedge date, the specialty material costs $33.20 per unit, while the futures price has risen from $28.00 to $31.50. Find the net cost after the futures gain and the net cost per unit. Ignore transaction costs and interest on marking-to-market cash flows.

Method. Apply the ratio to the exposure to find the futures quantity, then subtract the long-futures gain from the spot purchase cost.

Futures quantity and contracts. The target futures quantity is $0.75(12{,}000)=9{,}000$ units. This is $9{,}000/500=18$ contracts to buy.

Spot purchase cost. The manufacturer pays $12{,}000(33.20)=\$398{,}400$.

Futures gain. The futures price rises by $31.50-28.00=\$3.50$ per unit. The gain is $9{,}000(3.50)=\$31{,}500$.

Net cost. After applying the gain, the net cost is $\$398{,}400-\$31{,}500=\$366{,}900$, or $\$366{,}900/12{,}000=\$30.575$ per unit. Without the hedge, the price increase from $30.00 to $33.20 would add $\$38{,}400$ to the original purchase cost. The futures gain offsets $31,500 of that increase, leaving a $6,900 increase, or $0.575 per unit. These figures check the net-cost calculation.

Common mistake avoided. Subtracting the futures price change from every unit of the specialty material would count a hedge of 12,000 units, not the 9,000-unit futures position actually taken. The hedge ratio must be applied before calculating the gain.

Basis Risk, Cross-Hedge Risk, and Exam Traps

There are two closely related sources of residual risk. Basis risk arises when the spot and futures prices relevant to a hedge do not maintain the assumed relationship through the hedge date. It is especially visible when a hedge is closed before expiry, because convergence at expiry does not apply at that earlier date. Cross-hedge risk arises when the hedged asset differs from the futures underlying, so even a well-chosen hedge ratio leaves price changes that do not offset.

In a simple model using a single matched asset, a seller’s proceeds depend on $F_0+b_T$, not necessarily on $F_0+b_0$. For a cross-hedge, both the hedge ratio and the relationship between asset-price and futures-price changes matter. A large correlation alone is not enough to determine the number of contracts; the two standard deviations matter too.

Check the basis sign. With $b_t=S_t-F_t$, do not reverse the subtraction. Write the definition before interpreting whether the basis has increased.
Check the hedge date. Closing before expiry means the closing basis can remain nonzero, even if a deliverable contract is expected to converge at expiry.
Match units and direction. Convert the target futures quantity into contracts using the contract size, then choose a short position for a planned sale or a long position for a planned purchase.
Keep the model’s limits in view. The minimum-variance ratio depends on estimated price-change statistics. It reduces modeled variance; it does not promise a particular realized outcome.

Key takeaway: A futures hedge offsets futures-price movements, but the remaining result can depend on basis movements, differences between the hedged asset and the futures underlying, and contract-size rounding. Calculate the residual rather than assuming it is zero.

Check Your Understanding

For each problem, use the stated assumptions and distinguish the exposure price from the futures price.

A seller has 3,000 units to sell and shorts 30 futures contracts, each covering 100 units, at $41.20 per unit. At the hedge date, the spot price is $40.10 and the futures price is $39.70. What are the combined proceeds, ignoring costs and interest on marking-to-market cash flows? (Answer: $124,800)
A seller has a 10,000-unit exposure and uses futures with a 400-unit contract size. The price-change standard deviations are $4.00 for the exposure asset and $2.50 for the futures underlying, with correlation $0.75$. Find the minimum-variance ratio and the target number of contracts. (Answer: $h^*=1.20$; short 30 contracts)
A buyer needs 8,000 units and goes long futures at $62.00 per unit. Each contract covers 200 units. At the hedge date, the spot price is $68.00 and the futures price is $66.50. Find the net cost after the futures gain, ignoring costs and interest on marking-to-market cash flows. (Answer: $508,000, or $63.50 per unit)
For an optimal linear cross-hedge with correlation $0.60$, what proportion of the exposure’s price-change variance remains in the minimum-variance model? (Answer: $1-0.60^2=0.64$, or 64%)
A seller has 2,500 units to sell. Initially, the spot price is $18.05 and the futures price is $18.40. The seller shorts futures for 2,500 units. At the hedge date, the spot price is $18.10 and the futures price is $17.90. Find the combined proceeds and the change in basis. (Answer: $46,500; the basis changes from $-\$0.35$ to $+\$0.20$, an increase of $0.55 per unit)

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