Why a Futures Hedge Can Leave Residual Exposure
A futures hedge can reduce the effect of an adverse price movement without eliminating it. The hedge may use a futures contract on a similar, but not identical, asset; it may be closed before the contract expires; or the number of units covered by the futures position may differ from the exposure. In these cases, the spot price of the exposure and the futures price do not move in exactly the same way over the hedge period.
The remaining gain or loss is not necessarily evidence that the hedge was entered in the wrong direction. It can be the expected consequence of using a related futures contract as a proxy. To measure that residual, keep the cash-market position and the futures position separate until the closing date, then add their changes in value.
The word “corresponding” matters. If the futures contract covers a different grade, location, or related asset, \(S_t\) and \(F_t\) are prices for different things, expressed in compatible units. If the hedge is closed before expiry, they are prices observed on the hedge-closing date, not necessarily prices at futures delivery. Basis risk can arise in either situation.
For the calculations below, assume the hedge is opened at time 0 and closed at time 1. Ignore transaction costs, margin interest, and any interest earned on interim variation-margin cash flows. Under these assumptions, a long futures position on \(n\) units has gain \(n(F_1-F_0)\); a short position has gain \(n(F_0-F_1)\). These are futures gains or losses, not the initial value of the futures position.
Residual Exposure for a Matched-Quantity Hedge
Suppose a business will buy \(q\) units of an asset at time 1 and uses a long futures position on \(q\) corresponding units. A rise in the asset’s spot price increases the purchase cost, while a rise in the futures price produces a gain on the long futures position. The change in purchase cost net of the futures gain is
Thus, when the quantities match, the residual cost change equals the quantity hedged multiplied by the change in basis. A seller who owns \(q\) units and takes a short futures position has the same basis-change relationship for net sale proceeds: spot revenue plus futures gain changes by \(q(b_1-b_0)\).
Worked Example: A Buyer Closes a Cross-Hedge Before Delivery
A manufacturer expects to buy 4,000 units of a material in three months. It uses a related futures contract, with each contract covering 100 units, and buys 40 contracts. At the start, the material’s spot price is $72.00 per unit and the futures price is $68.40 per unit. Three months later, when the material is purchased and the futures contracts are closed, the spot price is $79.50 and the futures price is $75.10. Calculate the increase in purchase cost after the futures gain.
Method. Treat the cash purchase and the long futures position as separate cash flows. The contract quantity is \(40(100)=4{,}000\) units, so it matches the planned purchase. Assume the futures gain is not reinvested and ignore margin interest.
The increase in the spot purchase cost is
Compute the price difference carefully: \(79.50-72.00=7.50\), so the correct spot-cost increase is
The long futures gain per unit is \(75.10-68.40=\$6.70\). Across 4,000 units, the gain is
Therefore the net increase in purchase cost is \(30{,}000-26{,}800=\$3{,}200\). Check using basis: \(b_0=72.00-68.40=\$3.60\), \(b_1=79.50-75.10=\$4.40\), and \(4{,}000(4.40-3.60)=\$3{,}200\). The basis widened by $0.80 per unit, which accounts for the residual cost. The common mistake is to report only the $30,000 increase in spot cost and omit the futures gain.
Deriving the Basis Relationship
The matched-quantity result follows directly from adding the two sides of the hedge. This short derivation is useful because it fixes the signs for buyers and sellers.
A spot position of \(q\) units changes in value by \(q(S_1-S_0)\). For a buyer’s purchase-cost calculation, this is the increase in the amount paid.
A long position on \(q\) units gains \(q(F_1-F_0)\). Subtract that gain from the buyer’s increase in cost to obtain the net cost change.
Subtracting gives \(q[(S_1-F_1)-(S_0-F_0)]\), which is \(q(b_1-b_0)\) by the definition of basis.
For an asset owner, the spot value changes by \(q(S_1-S_0)\), and a short futures position gains \(q(F_0-F_1)\). Their sum is \(q[(S_1-F_1)-(S_0-F_0)]\), again \(q(b_1-b_0)\). The same expression appears, but it describes a change in net value or proceeds rather than an increase in a buyer’s cost.
Worked Example: A Seller Closes a Hedge Before Futures Expiry
A distributor plans to sell 2,500 units of a product in three months. Each futures contract covers 250 units, so it sells 10 contracts. At the hedge start, the spot price is $124.00 and the futures price is $121.20 per unit. At the sale date, before the futures contract expires, the spot price is $116.50 and the futures price is $114.90. Find the combined sale proceeds and futures gain, and compare the result with the initial spot value.
Method. Add the spot-sale proceeds to the gain on the short futures position. The futures position is short because a price decline reduces the distributor’s sale revenue.
Spot-sale proceeds are \(2{,}500(116.50)=\$291{,}250\). The short futures gain per unit is \(121.20-114.90=\$6.30\), giving a total futures gain of \(2{,}500(6.30)=\$15{,}750\). Combined proceeds are therefore \(291{,}250+15{,}750=\$307{,}000\), or \(\$307{,}000/2{,}500=\$122.80\) per unit.
The initial spot value was \(2{,}500(124.00)=\$310{,}000\), so combined proceeds are $3,000 below that value. Check the basis calculation: \(b_0=124.00-121.20=\$2.80\), while \(b_1=116.50-114.90=\$1.60\). The basis change is \(-\$1.20\) per unit, so \(2{,}500(-1.20)=-\$3{,}000\), matching the result. The common mistake is to assume the hedge guarantees the initial spot price; it does not when the basis changes.
Separating Basis Risk from a Quantity Mismatch
The basis-only formula assumes the futures position covers exactly the same number of units as the spot exposure. If an asset owner holds \(q\) units but sells futures on only \(n\) units, the combined change in value is
The first term is the basis-change component for the full spot exposure. The second term is the price movement on the units left unmatched. This decomposition is an accounting identity; it does not require futures prices to move in a particular way. It also shows why a residual loss cannot automatically be attributed only to basis risk when the hedge quantity is imperfect.
Worked Example: Identifying Basis and Underhedging Components
An asset owner holds 6,000 units and sells futures on 5,200 units. At the start, the spot price is $54.00 and the futures price is $52.50. At the hedge close, the spot price is $51.60 and the futures price is $50.50. Calculate the combined change in value and divide it into basis and quantity-mismatch components. Ignore transaction costs and margin interest.
Method. The spot position changes by \(6{,}000(51.60-54.00)=6{,}000(-2.40)=-\$14{,}400\). The short futures position gains \(5{,}200(52.50-50.50)=5{,}200(2.00)=\$10{,}400\). Combined, the positions lose \(14{,}400-10{,}400=\$4{,}000\).
For the decomposition, the initial basis is \(54.00-52.50=\$1.50\), and the final basis is \(51.60-50.50=\$1.10\). Thus \(\Delta b=-\$0.40\), and the basis component is \(6{,}000(-0.40)=-\$2{,}400\). The unmatched quantity is \(6{,}000-5{,}200=800\) units, and the futures-price change is \(50.50-52.50=-\$2.00\). The quantity component is \(800(-2.00)=-\$1{,}600\). Together, \(-2{,}400-1{,}600=-\$4{,}000\), as calculated directly. The common mistake is to call the entire residual a basis loss and overlook the 800 units not covered by futures.
What Changes When the Hedge Ends Before Expiry?
A futures position used for a dated hedge is commonly closed when the cash-market transaction occurs, even if the contract has a later delivery date. The futures price at that close date is the relevant price for the hedge gain or loss. It is not valid to substitute a presumed delivery-date price for \(F_1\).
For a futures contract on the same deliverable asset, spot and futures prices generally converge at contract expiry under the delivery and market-access assumptions of the contract. That convergence does not imply they are equal on an earlier hedge-closing date. For a cross-hedge using a different asset, even expiry need not make the two prices equal. Therefore a calculation that sets \(b_1=0\) is justified only when the problem’s asset and timing conditions support that assumption.
| Hedge situation | What drives the residual | Key calculation |
|---|---|---|
| Matched units; hedge closed before expiry | Basis change over the hedge period | \(q(b_1-b_0)\) |
| Different asset or contract specification | Basis may not converge to zero, even at expiry | Use the stated spot and futures prices at the relevant dates |
| Futures quantity differs from spot exposure | Basis change and unmatched price exposure | \(q(b_1-b_0)+(q-n)(F_1-F_0)\) for a spot holding and short futures |
Common Exam Traps
- Reversing the basis definition. Here basis is \(S-F\). If a problem defines basis differently, the sign in the calculation must change accordingly. State the definition before using it.
- Using the wrong futures direction. A planned purchase exposed to rising prices generally uses long futures; an owner concerned about falling sale prices generally uses short futures.
- Assuming a perfect offset from similar price movements. A related futures price may move in the same direction as the spot price and still leave a residual because the changes differ.
- Ignoring contract units. Convert the number of contracts into the number of units covered before comparing it with the exposure.
- Assuming convergence on the hedge date. Convergence at expiry, when applicable, does not mean the spot and futures prices are equal when a hedge is closed earlier.
- Mixing gain and cost signs. For a buyer, report spot-cost increase minus long-futures gain. For an asset owner, add spot-value change and short-futures gain.
Check Your Understanding
For each problem, track the spot position and futures position separately. Ignore transaction costs and interest on margin cash flows.
- A buyer expects to purchase 1,800 units and buys 18 futures contracts, each covering 100 units. Spot prices move from $42.00 to $45.10, while futures prices move from $40.80 to $43.40. What is the increase in purchase cost after the futures gain? (Answer: $900)
- A seller owns 3,200 units and shorts 16 contracts, each covering 200 units. Initially \(S_0=\$88.00\) and \(F_0=\$86.50\); at the hedge close, \(S_1=\$84.20\) and \(F_1=\$83.60\). Find combined sale proceeds and the change from the initial spot value. (Answer: $278,720 proceeds, a decrease of $2,880 from the initial spot value)
- An owner holds 5,000 units and shorts futures on 4,500 units. The basis changes from $1.20 to $1.80, and the futures price rises by $1.20. Calculate the basis and quantity-mismatch components of the combined change in value. (Answer: basis component $3,000; mismatch component $600; total gain $3,600)
- A seller holds 2,400 units and shorts 12 contracts covering 200 units each. Initially the spot price is $75.00 and the futures price is $74.30. At the contract’s expiry, assume the same deliverable asset has a spot and futures price of $69.80. Find combined proceeds and compare them with the initial spot value. (Answer: $178,320 proceeds, a decrease of $1,680 from the initial spot value)