Three Prices and Values That Must Not Be Confused
A forward contract fixes terms today for an exchange at a future date. Once the contract is in place, its delivery price stays fixed, even as market conditions change. A new forward contract entered into later may have a different delivery price. The existing contract can therefore have a positive or negative value before it matures.
The key distinction is between the delivery price in an existing contract, the market forward price for a new contract, and the current value of the existing contract. The market forward price is not, by itself, the value of an existing contract; the two prices must be compared, and the difference must be valued at the appropriate discount factor.
Here, \(K\) is fixed when the contract is made. By contrast, \(F_t(T)\) is observed at the later valuation time and is the delivery price that makes a new contract have zero value then. A contract that began at zero value can gain or lose value as \(F_t(T)\) changes.
Why the Value Is a Discounted Price Difference
The formula follows by comparing the existing long contract with a new short contract having the same delivery date and terms. The new short contract has zero value at the market forward price. Combining it with the existing long cancels the asset delivery: the existing long receives one unit of the asset and pays \(K\), while the new short delivers that unit and receives \(F_t(T)\). The combined position therefore receives \(F_t(T)-K\) at time \(T\).
Under the stated assumptions, the value at time \(t\) of that certain terminal cash difference is its amount multiplied by the discount factor. Since the new short position has zero value, the value of the combined position is just the value of the existing long position. Thus:
This reasoning also establishes the sign. If the market forward price is above the existing delivery price, the long can buy at \(K\) under the old contract when a new buyer would pay \(F_t(T)\); the long position has positive value. If the market forward price is below \(K\), the long is committed to a less favorable purchase price and has negative value.
When the annual effective interest rate \(i\) is constant and \(T-t=n\) whole years, the discount factor is \(D(t,T)=v^n=1/(1+i)^n\). If discounting uses a term structure, use the discount factor for the specific period from \(t\) to \(T\); do not substitute an unrelated annual rate. A rate quoted for some other period must first be converted consistently.
Worked Example: Valuing a Long Forward with a Flat Annual Rate
A hypothetical investor holds a long forward that requires purchasing a specified asset for \(K=\$11{,}800\) in two years. The current market forward price for a new contract with the same terms is \(F_0(2)=\$12{,}500\). Assume a constant annual effective interest rate of \(i=5\%\), annual compounding, and no default or transaction-cost effects. Find the value of the existing long position.
Method. Compare the existing delivery price with the current market forward price, then discount the difference over the two years remaining.
The market forward price exceeds the old delivery price by:
The two-year discount factor is:
Therefore:
Rounded to the nearest cent, the existing long forward is worth \$634.92. This is positive because the long can buy at \$11,800 while the delivery price for a new matching contract is \$12,500. The value is less than \$700 because the difference is received at the future delivery date, not today.
Check. The short position has value \(-\$634.92\), so the long and short values sum to zero.
Common mistake avoided. Treating the \$700 price difference as a value today ignores the two-year delay. The correct value discounts that difference by \(1/(1.05)^2\).
Interpreting Long and Short Values
The delivery-price comparison has opposite implications for the two sides of a forward. A lower locked purchase price benefits the long; a higher locked sale price benefits the short. Both parties use the same discount factor, but their values have opposite signs.
| Comparison | Long value | Short value |
|---|---|---|
| \(F_t(T)>K\) | Positive | Negative |
| \(F_t(T)=K\) | Zero | Zero |
| \(F_t(T)<K\) | Negative | Positive |
These signs describe the value of the contract, not the cash amount either party has already paid. At initiation, the delivery price is normally set to the market forward price for a new contract, so \(F_t(T)=K\) and the initial value is zero. At a later valuation time, the original \(K\) remains unchanged while the market quote may move.
Worked Example: A Long Forward with Negative Value
A hypothetical manufacturer has a long forward to buy a specified input for \(K=\$11{,}250\) in six months. The current market forward price for a new contract with the same delivery date and terms is \(F_t(T)=\$10{,}800\). The current term structure gives \(D(t,T)=0.93\). Assume no default or transaction-cost effects. Find the long and short values and determine which side benefits from the market change.
Method. Use the current market quote for a new contract, not the original locked price, to measure the price difference.
For the long:
The corresponding short value is:
The long position is worth \(-\$418.50\), while the short position is worth \(+\$418.50\). The short benefits because the existing contract lets that party sell for \$11,250, above the current delivery price of \$10,800 for a new matching contract.
Check. The two position values offset: \(-\$418.50+\$418.50=\$0\).
Common mistake avoided. Using \(K-F_t(T)\) for the long would produce a positive value. That reverses the long’s perspective: the long pays \(K\) and receives the asset, so the relevant difference is \(F_t(T)-K\).
Solving for an Unknown Contract Term
The same relationship can be rearranged when a problem gives the value and asks for an unknown price. For a long position:
These rearrangements use the long value. If the given value is for the short, first use \(V_t^L=-V_t^S\). The discount factor must be positive, so dividing by it does not change the sign of the value difference.
Worked Example: Recovering the Locked Delivery Price
A hypothetical firm holds a long forward with two years remaining. A new matching forward has market delivery price \(F_t(T)=\$15{,}200\), and the relevant discount factor is \(D(t,T)=0.90\). The existing long position is worth \(V_t^L=\$1{,}260\). Assume the contracts match in all relevant terms and there are no default or transaction-cost effects. Find the delivery price \(K\) locked in the existing contract.
Method. Rearrange the long-value equation to isolate \(K\).
Start with:
Divide by the discount factor:
Thus the market forward price exceeds the old delivery price by \$1,400, and:
The locked delivery price is \$13,800. Since it is below the current market forward price, the long has a positive value, as expected.
Check. Substitution gives \(0.90(\$15{,}200-\$13{,}800)=0.90(\$1{,}400)=\$1{,}260\), matching the stated value.
Common mistake avoided. Subtracting the value directly from the market forward price would give \$13,940. The price difference at delivery is \(V_t^L/D(t,T)\), not \(V_t^L\).
Assumptions and Exam Traps
The relationship applies when the existing position and the new market forward have the same underlying, delivery date, amount, and other relevant contractual terms. It values the forward position from the perspective of its holder, using the discount factor for the period remaining until delivery. It does not automatically include separate fees, collateral adjustments, counterparty default risk, or payments made before delivery. If the contracts differ, their value difference cannot be reduced to the formula above without accounting for those differences.
In particular, do not confuse the market forward price with an expectation of the asset’s future spot price. Here, \(F_t(T)\) means the delivery price available for a new contract at time \(t\); it is used to compare contract terms. Nor should the current market forward price be substituted for the original \(K\): the locked delivery price identifies the existing contract, while the current quote is used to value it.
Record \(K\) from the existing contract and determine whether the holder is long or short.
Use \(F_t(T)\) for a new contract with the same underlying, delivery date, and relevant terms.
For the long, calculate \(D(t,T)(F_t(T)-K)\). Reverse the sign for the short.
A long benefits when the market forward price exceeds \(K\); a short benefits when it is below \(K\).
A frequent exam trap is to report \(F_t(T)-K\) as the value without discounting. That difference is the net amount associated with the matched contracts at delivery; the value at the current time is \(D(t,T)(F_t(T)-K)\). Another is to reverse the sign for a long. Check the perspective explicitly: a long buys at \(K\), so a higher market forward price makes the existing purchase contract more valuable.
Check Your Understanding
For each problem, use the stated assumptions and report dollar values to the nearest cent where needed.
- A long forward has delivery price \(K=\$5{,}000\). A new matching forward has market price \(F_t(T)=\$5{,}200\), and \(D(t,T)=0.96\). Find the long value. (Answer: \(\$192.00\))
- A long forward has \(K=\$19{,}500\), with two years remaining. The annual effective interest rate is \(i=4\%\), compounded annually, and a new matching forward has price \(\$20{,}000\). Find the long value. (Answer: \(\$462.28\))
- A long forward has \(K=\$74{,}000\). The current matching market forward price is \(\$72{,}500\), and \(D(t,T)=0.97\). Find both the long and short values. (Answer: Long \(-\$1{,}455.00\); short \(+\$1{,}455.00\))
- A long forward is worth \(\$840\). The matching market forward price is \(\$9{,}800\), and \(D(t,T)=0.93\). Find the locked delivery price. (Answer: \(\$8{,}896.77\))
- A short forward has delivery price \(\$40{,}000\). A new matching forward has price \(\$38{,}500\), and \(D(t,T)=0.92\). Find the short’s value and state whether it is positive or negative. (Answer: \(+\$1{,}380.00\), positive)