Sizing Futures Hedges For Known Exposures

Why Hedge Size Matters

A futures hedge is useful only to the extent that its size matches the exposure it is intended to offset. A buyer who will need a commodity later is hurt if its price rises, so a long futures position can gain when that price rises. A seller who will deliver a commodity later is hurt if its price falls, so a short futures position can gain when that price falls. In either case, too few contracts leave some exposure uncovered; too many create an exposure in the opposite direction.

This tutorial sizes a hedge when the underlying exposure and the futures contract refer to the same asset or quantity, and the hedge is closed when the exposure occurs. The calculations assume that the change in the relevant cash price equals the change in the futures price over the hedge period. This matching assumption isolates the contract-sizing calculation; it is not appropriate when the cash and futures prices move differently.

Definition: A long futures hedge uses long futures contracts to offset the risk of a future purchase or another position that loses value when the underlying price rises. A short futures hedge uses short futures contracts to offset the risk of a future sale or another position that loses value when the underlying price falls. The hedge ratio in contracts is the number of futures contracts held for the exposure being hedged.

Contract Count and Futures Gain or Loss

For a quantity exposure, divide the exposure quantity by the quantity covered by one contract. This gives the ideal contract count before any whole-contract constraint is applied. The exposure quantity and the contract size must use the same units.

N_{\mathrm{ideal}}=\frac{Q}{Q_c}

Here $Q$ is the quantity to be hedged and $Q_c$ is the quantity covered by one contract. Futures contracts are normally held in whole numbers, so a calculated non-integer count must be rounded to a feasible position. The nearest integer often gives the smallest quantity mismatch, but the appropriate choice can depend on the hedger’s objective.

If $N$ contracts are held and the futures price changes from $F_0$ to $F_1$, the aggregate futures gain for a long position is $NQ_c(F_1-F_0)$. A short position has the opposite gain or loss. These formulas calculate the total change over the period; daily marking to market allocates that change through the margin account but does not change the total price-change calculation.

\begin{aligned} \text{Long futures gain} &= NQ_c(F_1-F_0),\\ \text{Short futures gain} &= NQ_c(F_0-F_1). \end{aligned}

A positive result is a gain to the futures position, and a negative result is a loss. For a currency-denominated contract, use the contract’s stated unit and currency conventions. If the contract is quoted in index points rather than dollars per unit, use its multiplier to convert a point change into dollars.

Worked Example: Hedging a Known Material Purchase

A manufacturer expects to buy 18,000 kilograms of a material in four months. Each futures contract covers 3,000 kilograms. The manufacturer enters futures when the quoted price is $2.40 per kilogram and closes the position when the price is $2.58 per kilogram. Assume the cash price for the material changes by the same amount as the futures price. Find the hedge position, futures result, purchase cost, and combined net cost.

Method and direction. The manufacturer is a future buyer, so a price increase raises its purchase cost. It should take a long futures position, which gains when the futures price rises.

Contract count. The ideal count is $18{,}000/3{,}000=6$ contracts. Six contracts cover exactly $6(3{,}000)=18{,}000$ kilograms, so there is no quantity mismatch.

Futures gain. The price increase is $$2.58-$2.40=$0.18$ per kilogram. Therefore, the long position gains $6(3{,}000)($0.18)=$3{,}240$.

Purchase and net cost. At the assumed cash price of $2.58 per kilogram, the purchase costs $18{,}000($2.58)=$46{,}440$. Subtract the futures gain: $$46{,}440-$3{,}240=$43{,}200$. This equals $18{,}000($2.40)$, the purchase cost at the initial matched price.

Check and mistake avoided. The futures gain offsets the $0.18-per-kilogram increase on all 18,000 kilograms: $18{,}000($0.18)=$3{,}240$. A common error is to add the futures gain to the purchase cost; because the buyer’s futures position is long and gains here, the gain must be subtracted from the cost.

Why a Matched Hedge Offsets the Price Change

The hedge formulas follow from multiplying the price change by the quantity exposed. Suppose a buyer needs $Q$ units, holds $N$ long contracts of $Q_c$ units each, and the cash and futures prices both rise by $\Delta$. The increase in the purchase cost is $Q\Delta$. The long futures gain is $NQ_c\Delta$. If the contract count exactly matches the exposure, $NQ_c=Q$, and the two amounts are equal.

\begin{aligned} \text{Increase in purchase cost} &= Q\Delta,\\ \text{Long futures gain} &= NQ_c\Delta,\\ NQ_c=Q \quad &\Longrightarrow \quad Q\Delta-NQ_c\Delta=0. \end{aligned}

Thus the futures gain offsets the price-driven increase in the buyer’s cost. For a seller, a price decline reduces sale revenue by $Q\Delta$ when $\Delta$ denotes the positive size of the decline. The matched short futures position gains $NQ_c\Delta$, offsetting that reduction. This justification depends on matching exposure quantity and price change; a contract count alone cannot guarantee an offset if those assumptions fail.

Worked Example: Rounding a Seller’s Contract Count

A grain cooperative expects to sell 52,000 bushels. Each futures contract covers 5,000 bushels. It chooses the nearest whole number of contracts to the ideal count and takes the position that reduces its exposure to a price decline. The futures price falls from $4.80 to $4.55 per bushel. Assume the cash price changes by the same amount. Calculate its futures gain, sale revenue, combined revenue, and residual quantity exposure.

Method and direction. The cooperative is a future seller, so a price decline reduces its sale revenue. It should take a short futures position.

Contract count. The ideal count is $52{,}000/5{,}000=10.4$. The nearest whole number is 10 contracts. These cover $10(5{,}000)=50{,}000$ bushels, leaving $52{,}000-50{,}000=2{,}000$ bushels unhedged. The hedge is an underhedge.

Futures gain. The futures price falls by $$4.80-$4.55=$0.25$ per bushel. The short position gains $10(5{,}000)($0.25)=$12{,}500$.

Sale and combined revenue. The cash sale revenue is $52{,}000($4.55)=$236{,}600$. Adding the short futures gain gives $$236{,}600+$12{,}500=$249{,}100$.

Check and mistake avoided. At the initial price, revenue on 52,000 bushels would be $52{,}000($4.80)=$249{,}600$. The 2,000 bushels not covered by futures lose $2{,}000($0.25)=$500$ of revenue, and $$249{,}600-$500=$249{,}100$, matching the combined result. Do not describe the 2,000-bushel difference as an overhedge: ten contracts cover 2,000 fewer bushels than the sale exposure.

Sizing a Dollar-Valued Exposure

Some contracts are quoted in index points, and their dollar value is determined by a contract multiplier. If the futures quote is $F_0$ index points and the multiplier is $M$ dollars per point, one contract has an initial notional value of $MF_0$ dollars. For a portfolio that has the same proportional sensitivity to the index, a first-pass contract count is the portfolio value divided by that one-contract notional value.

N_{\mathrm{ideal}}=\frac{V}{MF_0}

Here $V$ is the dollar value of the exposure, and $M$ is the dollar multiplier per index point. This formula is appropriate only under the stated same-sensitivity assumption. It is not a general rule for every dollar exposure: for example, a cash amount that does not change with the index is not an index-sensitive portfolio.

Worked Example: Hedging an Index Portfolio

An investor holds a hypothetical $1,200,000 portfolio whose value changes in the same proportion as a specified stock index. The index futures quote is 4,000 points, and each contract has a multiplier of $50 per point. The investor wants protection from a decline and shorts the ideal number of contracts. The futures price later falls to 3,920 points. Find the contract count, futures gain, and portfolio loss under the stated sensitivity assumption.

Method and direction. A fall in the index reduces the portfolio value, so the investor takes a short futures position.

Contract count. One contract’s initial notional value is $4{,}000($50)=$200{,}000$. The ideal count is $$1{,}200{,}000/$200{,}000=6$ contracts.

Futures gain. The index falls $4{,}000-3{,}920=80$ points. The short position gains $6($50)(80)=$24{,}000$.

Portfolio loss and check. The proportional index decline is $80/4{,}000=0.02$, or 2%. The portfolio loss is $$1{,}200{,}000(0.02)=$24{,}000$. The futures gain equals that loss, so the combined value change is $-$24{,}000+$24{,}000=$0$ under the assumed exact sensitivity match. A common mistake is to treat 80 index points as $80 per contract; the $50 multiplier must be applied to every point of movement.

Choosing a Feasible Hedge

An ideal contract count is a sizing calculation, not always a position that can actually be held. Quantity-based futures contracts are indivisible, so a non-integer ideal count requires a choice between underhedging and overhedging. An underhedge leaves some original exposure uncovered. An overhedge covers more than the original quantity and leaves a net position that moves in the opposite direction for the excess quantity.

The arithmetic below is a useful check whenever contract counts are rounded:

State the exposure and its units.
Identify the quantity to buy or sell, or the dollar value and sensitivity of an index exposure.

Choose the position direction.
Use long futures for a future purchase; use short futures for a future sale. For another exposure, identify which price movement would create a loss and select the futures position that gains on that movement.

Calculate and make the count feasible.
Divide by the contract quantity, or by the contract’s dollar notional when the exposure has matching proportional sensitivity. If needed, choose a whole number of contracts.

Calculate the futures result and mismatch.
Apply the long or short price-change formula, then compare the total quantity covered with the exposure quantity.

A frequent exam trap is to identify the correct number of contracts but reverse their direction. A buyer whose cost rises with the underlying price needs a long position, not a short one. Another trap is to report a gain without stating whether it belongs to the long or short side. Always write the direction first, then apply the matching formula.

Key takeaway: First match the futures position’s direction to the risk, then size it using consistent units. Check $NQ_c$ against the exposure quantity and calculate the futures result using the actual price change. These steps make underhedges and overhedges visible instead of hiding them in a contract-count answer.

Correct hedge sizing is a practical application of futures mechanics and is aligned with the SOA Exam FM (Financial Mathematics) syllabus. The clean offsets shown here rely on a known exposure, matching underlying, and matched price movements; when those assumptions do not hold, the simple quantity calculation does not by itself establish an exact offset.

Check Your Understanding

For each problem, state the hedge direction and calculate the requested futures result or residual exposure. Unless stated otherwise, assume the cash and futures prices change by the same amount.

A food processor will buy 24,000 kilograms of an ingredient. Each contract covers 4,000 kilograms. It uses the exact contract count, and the futures price rises by $0.15 per kilogram. Calculate the number and direction of contracts and the futures gain. (Answer: 6 long contracts; $3,600 gain.)
A grower expects to sell 31,500 bushels. Each contract covers 5,000 bushels, and the grower chooses the nearest whole number of contracts. The futures price falls by $0.08 per bushel. Find the position, futures gain, and residual quantity exposure. (Answer: 6 short contracts; $2,400 gain; underhedged by 1,500 bushels.)
A buyer needs 45,000 units, with 8,000 units per contract, and chooses 6 long contracts. The futures price rises by $0.25 per unit. Calculate the futures gain and state whether the hedge is under- or over-sized. (Answer: $12,000 gain; overhedged by 3,000 units.)
A hypothetical $900,000 portfolio changes in the same proportion as an index currently quoted at 3,600. Each index futures contract has a $25-per-point multiplier. The investor shorts the ideal whole number of contracts; the index later falls to 3,540. Find the contract count and futures gain, and check it against the portfolio loss. (Answer: 10 short contracts; $15,000 futures gain; $15,000 portfolio loss.)
A buyer needs 17,000 barrels and takes 17 long contracts, each covering 1,000 barrels. The futures price falls from $72.40 to $71.90 per barrel. Calculate the futures result and, assuming an equal cash-price fall, the reduction in the physical purchase cost. State the combined effect on cost. (Answer: $8,500 futures loss; $8,500 lower purchase cost; no net change in cost under the matching-price assumption.)

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