What a Forward Payoff Measures
A forward contract fixes a price today for a transaction at a specified future date. At maturity, the market price may be above or below that agreed delivery price. The difference determines the position's payoff: the long side is obligated to buy, while the short side is obligated to sell. Correctly identifying which price is paid and which price reflects the asset's market value is the key to getting the sign right.
This tutorial focuses on the payoff at maturity, not on how the delivery price is chosen or how an existing forward is valued before maturity. A delivery price can be specified for the problems here without assuming anything about how it was set.
The formulas describe the economic value of the forward exchange at maturity. Under physical delivery, the long pays \(QK\) and receives an asset worth \(QS_T\); the difference between those amounts is the long's payoff. Under cash settlement, the parties instead exchange the corresponding net amount. The payoff formula is the same in either case.
If the contract is quoted per unit, first find the per-unit difference and then multiply by the contract quantity. If a contract specifies a multiplier, such as a dollar amount per index point, include that multiplier in the effective quantity. The payoff is positive for the position that benefits from the difference between the maturity spot price and the delivery price.
Why the Payoff Formula Holds
For one unit, a long forward requires a payment of \(K\) at maturity in exchange for an asset whose market value at that time is \(S_T\). Comparing the value received with the amount paid gives \(S_T-K\). For \(Q\) units, both amounts are multiplied by \(Q\), giving \(Q(S_T-K)\). A short forward makes the opposite exchange: it receives \(K\) and gives up an asset worth \(S_T\). Its payoff is therefore \(Q(K-S_T)\).
The two payoffs must sum to zero when they refer to opposite sides of the same contract and ignore separate costs: one party's forward payoff is the other party's loss. This is a useful check on both the sign and the arithmetic.
| Maturity price comparison | Long forward | Short forward |
|---|---|---|
| \(S_T>K\) | Positive payoff | Negative payoff |
| \(S_T=K\) | Zero payoff | Zero payoff |
| \(S_T<K\) | Negative payoff | Positive payoff |
Worked Example: Physical Delivery of a Commodity
A buyer takes a long forward on 3 contracts, each covering 100 kilograms of a hypothetical metal, at a delivery price of $48.00 per kilogram. At maturity, the spot price is $53.25 per kilogram. Find the long's payoff and the short's payoff. Assume no separate initial payment.
Method. Use the maturity spot price minus the delivery price for the long, then multiply by the total number of kilograms.
The total quantity is \(3 \times 100=300\) kilograms. The per-kilogram difference is \(53.25-48.00=\$5.25\). Therefore,
The short payoff is the negative of the long payoff:
Under physical delivery, the long pays \(300(48.00)=\$14{,}400.00\) and receives metal worth \(300(53.25)=\$15{,}975.00\). The difference is \(\$15{,}975.00-\$14{,}400.00=\$1{,}575.00\), confirming the long payoff. The short has the equal and opposite result. The common mistake this method avoids is calculating only the cash paid for the metal and overlooking the value of what the long receives.
Payoff Is Not Always the Same as Profit
A contractual payoff records what the forward itself delivers at maturity. It does not automatically include any fee or other amount paid when the position is entered. If the position holder pays an initial cost, the holder's total economic result depends on how that cost is treated.
For a terminal-date comparison, suppose a cost \(C_0\) is paid at time 0 and the effective annual interest rate is \(i\). If that amount could otherwise have earned interest at rate \(i\), its accumulated value at time \(T\) is \(C_0(1+i)^T\). With no other cash flows, terminal profit is the forward payoff minus this accumulated cost. This convention assumes annual effective compounding and that the initial cost is paid by the position holder; it is not part of the forward's contractual payoff.
If an initial amount is received rather than paid, it is not a cost: its accumulated value is added to the terminal payoff. Be clear about the requested measure. Simply subtracting the amount originally paid gives a nominal, undiscounted comparison of dollars at different dates; accumulating the cost gives a comparison in maturity-date dollars under the stated interest assumption.
Worked Example: An Index Forward With an Entry Fee
An investor is long 4 hypothetical index forward contracts. Each contract has a multiplier of $25 per index point, and the delivery price is 1,260 points. At maturity, the index is at 1,195 points. The investor also paid a separate $800 entry fee at time 0. The contract matures in 2 years, and the effective annual interest rate for accumulating the fee is 5%. Find the contractual payoff and the terminal profit.
Method. First calculate the forward payoff without the fee. Then accumulate the fee for 2 years and deduct that maturity-date cost.
There are \(4 \times \$25=\$100\) of exposure per index point. The long's index-point difference is \(1{,}195-1{,}260=-65\). Thus the contractual payoff is
The separate entry fee accumulates to \(800(1.05)^2\). Since \(1.05^2=1.1025\), the accumulated fee is \(800(1.1025)=\$882.00\). The terminal profit is therefore
The contractual payoff remains \(-\$6{,}500.00\); the fee does not change that payoff. Terminal profit is \(-\$7{,}382.00\) under the stated interest assumption. As a sign check, the index finishes below the delivery price, so the long must have a loss. The method avoids the mistake of treating the entry fee as if it were included in the forward settlement.
Combining Forward Positions
When positions have the same underlying asset and maturity, calculate each payoff using its own delivery price, quantity, and direction, then add the results. A long position and a short position do not necessarily cancel: they offset only to the extent that their quantities and contract terms match. If all terms match and the quantities are equal, their forward payoffs sum to zero.
For positions with the same delivery price and maturity, it can be helpful to combine long and short quantities into a net quantity. Keep the direction explicit: a positive net quantity is long, and a negative net quantity is short. If delivery prices differ, calculate the payoff of each position separately before summing.
Worked Example: Netting Long and Short Contracts
A trader holds 3 long forward contracts, each covering 200 units of a hypothetical asset, and 2 short forward contracts, each covering 250 units of the same asset. All contracts have the same 1-year maturity and delivery price of $73.50 per unit. At maturity, the spot price is $75.00. The trader paid a separate $40 fee at time 0. The effective annual interest rate is 5%. Find the combined contractual payoff and the terminal profit after the fee.
Method. Compute the long and short payoff separately, add them, and then deduct the accumulated fee.
The long quantity is \(3 \times 200=600\) units, so its payoff is
The short quantity is \(2 \times 250=500\) units, so its payoff is
The combined contractual payoff is \(\$900.00-\$750.00=\$150.00\). Equivalently, the trader is net long \(600-500=100\) units, and \(100(75.00-73.50)=\$150.00\). The fee accumulates for one year to \(40(1.05)=\$42.00\). Therefore, terminal profit is \(\$150.00-\$42.00=\$108.00\).
The check is consistent: the spot price is above the delivery price, and the net position is long, so the combined forward payoff is positive. The fee reduces terminal profit but does not alter the contractual payoff.
Common Sign and Measurement Traps
The most frequent error is reversing the price difference for one side. A long agrees to pay \(K\) for an asset worth \(S_T\), so its payoff is \(Q(S_T-K)\), not \(Q(K-S_T)\). A short has the opposite payoff. A quick direction check often catches an error before any multiplication is needed.
Other checks help prevent errors in exam-style problems:
- Identify the side before subtracting. Long means \(S_T-K\); short means \(K-S_T\).
- Apply the full contract quantity. If a contract has a multiplier, include it once for each contract.
- Keep payoff and profit separate. A separately stated initial cost affects profit, not the contractual maturity payoff.
- Check the signs across the two parties. Opposite positions in the same contract must have equal and opposite payoffs.
These calculations are aligned with the SOA Exam FM (Financial Mathematics) syllabus because they require careful interpretation of a derivative position, its direction, contract size, and cash-flow timing. The central discipline is to state the position and the relevant prices before applying the formula.
Check Your Understanding
For each problem, find the requested maturity payoff or terminal profit. Treat any stated entry fee as separate from the contractual payoff.
- A long forward covers 150 units at a delivery price of $32.00 per unit. The maturity spot price is $35.60. Find the long payoff. (Answer: $540.00)
- A trader is short 5 contracts, each covering 40 units, at a delivery price of $18.75 per unit. The maturity spot price is $17.90. Find the combined short payoff. (Answer: $170.00)
- An investor is long 2 index forward contracts, each with a multiplier of $10 per index point. The delivery price is 4,200 points and the maturity index is 4,115 points. Find the contractual payoff. (Answer: -$1,700.00)
- A trader is long 4 contracts of 75 units each and short 3 contracts of 100 units each on the same asset, with the same delivery price and maturity. The delivery price is $29.00, and the maturity spot price is $31.00. The trader paid a separate $120 fee at time 0. Maturity is in 1 year, and the effective annual interest rate is 2.5%. Find the combined contractual payoff and terminal profit. (Answer: contractual payoff $0.00; terminal profit -$123.00)
- An investor is short 2 contracts, each covering 500 units, at a delivery price of $64.20 per unit. At maturity, the spot price is $66.10. The investor also paid a separate $250 fee 2 years earlier. The effective annual interest rate is 4%. Find the contractual payoff and terminal profit. (Answer: contractual payoff -$1,900.00; terminal profit -$2,170.40)