Value Of An Existing Forward Contract

Why an Existing Forward Has Value

A forward contract is typically entered into with no payment at inception, but that does not mean it remains valueless. As market conditions change, a contract’s agreed delivery price may become more or less favorable than the price available on a new forward contract for the same asset and delivery date. The difference is an economic value today, even though cash may not be exchanged until settlement.

The key comparison is between the contract’s fixed delivery price, denoted by $K$, and the current no-arbitrage forward price, denoted by $F_t(T)$. Here $t$ is the valuation time and $T$ is the delivery time. The forward price $F_t(T)$ is the delivery price for a new forward contract entered at time $t$ and settling at time $T$; it is not the spot price at time $t$.

Definition: For a forward contract on one unit of an asset, let $K$ be the delivery price agreed at inception, $F_t(T)$ the current fair forward price for delivery at $T$, and $P(t,T)$ the time-$t$ discount factor for one unit payable at $T$. Under the no-arbitrage assumptions below, the value of the existing long forward is $P(t,T)\bigl(F_t(T)-K\bigr)$. The value of the existing short forward is the negative of this amount.

Model and Valuation Formula

Assume a frictionless market with no arbitrage, the same underlying asset and delivery date for the existing and comparison contracts, and no counterparty default risk or transaction costs. The discount factor must reflect the rate applicable from the valuation time $t$ to the delivery time $T$. Any income from the asset and any carrying costs relevant to the forward price must be included when determining $F_t(T)$. The formula does not, by itself, determine that forward price; it uses the current no-arbitrage forward price appropriate to the asset and contract.

Write $\tau=T-t$ for the remaining time to delivery. With a flat effective annual interest rate $i$, and when the cash flows are discounted at that rate, $P(t,T)=(1+i)^{-\tau}$. If the relevant rate is a constant force of interest $\delta$, then $P(t,T)=e^{-\delta\tau}$. With a term structure, use the discount factor specifically applicable to the remaining term, rather than assuming a single flat rate.

V_t^{\text{long}}=P(t,T)\bigl(F_t(T)-K\bigr), \qquad V_t^{\text{short}}=P(t,T)\bigl(K-F_t(T)\bigr)

These formulas value one unit of the underlying. For a contract covering $N$ units, multiply the one-unit value by $N$. A positive value is an asset to the party holding the position; a negative value is a liability. Thus, if $F_t(T)>K$, the existing long has a positive value: it can obtain the asset at a below-market forward price. The existing short has the opposite value.

Why the Formula Holds

A new forward contract with delivery price $F_t(T)$ has value zero when it is entered under the no-arbitrage model. Compare that new long forward with the existing long forward, which has delivery price $K$. At time $T$, the two contracts deliver the same asset, so their asset deliveries cancel when the contracts are compared. The cash difference between the long positions is $F_t(T)-K$: the existing long pays $K$, while the new long would pay $F_t(T)$.

The existing contract must therefore have the same value at time $t$ as a time-$T$ amount $F_t(T)-K$, discounted to $t$. This gives the long formula. Reversing both positions gives the short formula. The argument also shows why discounting is necessary: $F_t(T)-K$ is a difference in delivery-date cash amounts, not a value already measured at time $t$.

V_t^{\text{long}}=0+P(t,T)\bigl(F_t(T)-K\bigr)

Worked Valuations

Worked Example: A Long Forward with Continuous Income

A hypothetical investor holds a long forward on one share, with delivery price $K=\$108$. There is one year remaining. The share is currently worth $\$100$, its continuous income yield is $q=0.02$, and the continuously compounded risk-free rate is $r=0.12$. Assume the yield and rate are constant over the remaining year, and ignore transaction costs and default risk. Find the current contract value.

Method. First calculate the current no-arbitrage forward price using the continuous-yield model, then discount the difference from the agreed delivery price. For a constant force $r$ and proportional income yield $q$, the forward price is $S_t e^{(r-q)\tau}$, while the discount factor is $e^{-r\tau}$.

The remaining term is $\tau=1$ year, and the net carry rate is $r-q=0.12-0.02=0.10$. Therefore,

F_t(T)=100e^{(0.12-0.02)(1)} =100e^{0.10} =100(1.105170918\ldots) =110.5170918\ldots

The one-year discount factor is

P(t,T)=e^{-0.12(1)} =e^{-0.12} =0.8869204\ldots

The forward-price difference is

F_t(T)-K=110.5170918\ldots-108 =2.5170918\ldots

Thus the long contract’s value is

V_t^{\text{long}} =(2.5170918\ldots)(0.8869204\ldots) =2.232460\ldots

Rounded to the nearest cent, the existing long forward is worth $\$2.23$. The sign is positive because the investor can buy the share for $\$108$ under the existing contract, below the current fair forward price of approximately $\$110.52$. The calculation also avoids a common arithmetic error: $100e^{0.10}-108=2.5170918\ldots$, not $2.517018\ldots$.

Worked Example: A Forward with No Income and an Effective Annual Rate

A hypothetical fund entered a long forward on an asset when the delivery price was fixed at $K=\$260$. There are two years to delivery. The asset’s current spot price is $\$250$; assume it pays no income, has no carrying costs, and the effective annual risk-free rate is $i=0.04$ for both years. Find the current value of the long forward.

Method. With no income or carrying costs, the no-arbitrage forward price is the spot price accumulated at the effective annual rate for the remaining term. The discount factor for the same term is $v^2=(1.04)^{-2}$, where $v=1/(1+i)$.

The current fair forward price is

$$ F_t(T)=250(1.04)^2 =250(1.0816) =270.40 $$

The forward-price difference is $270.40-260=\$10.40$. The discount factor is

P(t,T)=\frac{1}{(1.04)^2} =\frac{1}{1.0816} =0.924556\ldots

Therefore,

V_t^{\text{long}} =10.40(0.924556\ldots) =9.615384\ldots

Rounded to the nearest cent, the contract is worth $\$9.62$ to the long. As a check, the same value can be found by comparing the spot price with the present value of the fixed delivery payment:

250-\frac{260}{1.0816} =250-240.384615\ldots =9.615384\ldots

The two methods agree. A frequent mistake is to use the original two-year term after the valuation date has advanced. The exponent and discount factor must both use the time remaining until delivery.

Worked Example: Valuing a Portfolio of Long and Short Forwards

A hypothetical trading account holds four long forward contracts and three short forward contracts on the same asset, all with the same delivery date. Each long has delivery price $K_L=\$51$, and each short has delivery price $K_S=\$53$. The current no-arbitrage forward price is $F_t(T)=\$50.40$, and the discount factor to delivery is $P(t,T)=0.96$. Each contract covers one unit. Find the account’s total current value.

Method. Value each side separately, keeping the long and short signs distinct, and then add the values. For one long, the price difference is $50.40-51=-\$0.60$, so

V_{t,L}^{(1)}=(50.40-51)(0.96) =(-0.60)(0.96) =-\$0.576

For four long contracts,

V_{t,L}^{(4)}=4(-0.576)=-\$2.304

For one short, the difference is $K_S-F_t(T)=53-50.40=\$2.60$. Its value is

V_{t,S}^{(1)}=(53-50.40)(0.96) =(2.60)(0.96) =\$2.496

For three short contracts, $V_{t,S}^{(3)}=3(2.496)=\$7.488$. The total account value is

V_t=-2.304+7.488=\$5.184

Rounded to the nearest cent, the portfolio is worth $\$5.18$. As a sign check, the short positions have delivery prices above the current fair forward price and are valuable; their combined positive value exceeds the loss on the long positions.

Common Traps and a Reliable Process

The formula is short, but exam errors often come from using the wrong price, reversing the position, or discounting over the wrong interval. Use the following sequence before substituting numbers.

Identify the position and its agreed delivery price.
Record whether the contract is long or short, and use its original fixed delivery price $K$, not a newly quoted price.

Find the current fair forward price.
Use the current no-arbitrage price for the same underlying and delivery date. Include the relevant income and carrying effects when calculating that price.

Use the discount factor to delivery.
Discount for the remaining term $T-t$, using the rate convention and term structure specified by the problem.

Check the sign and scale.
A long benefits when $F_t(T)>K$; a short benefits when $K>F_t(T)$. Multiply by the number of units or contracts only after valuing one unit.

For example, a wrong long-position calculation might use $P(t,T)(K-F_t(T))$. That is the short value, not the long value. If $F_t(T)=\$75$, $K=\$72$, and $P(t,T)=0.95$, the wrong calculation gives $0.95(72-75)=-\$2.85$. The correction for the long is

V_t^{\text{long}}=0.95(75-72)=\$2.85

The positive result matches the position’s economics: the long has the right to buy at $\$72$ when a new forward would require a delivery price of $\$75$. Conversely, the short’s value is $-\$2.85$.

Do not confuse the forward price with the spot price. The spot price is relevant when deriving a forward price from the asset and financing assumptions; the value formula compares the agreed delivery price with the current forward price for the remaining term. Nor should the undiscounted difference $F_t(T)-K$ be reported as the current value: it is a delivery-date amount and must be discounted.

Key takeaway: Value an existing long forward by subtracting its fixed delivery price from the current fair forward price, then discount that difference to the valuation date. Reverse the price difference for a short. Use only the remaining time to delivery.

Check Your Understanding

For each problem, assume no transaction costs or counterparty default risk. Round money answers to the nearest cent.

A long forward covers one unit of an asset with no income or carrying costs. One year remains, the spot price is $\$90$, the effective annual rate is $5\%$, and the contract delivery price is $\$94$. Find the current value. (Answer: $\$0.48$)
A current no-arbitrage forward price is $\$120$, the delivery price of an existing long is $\$115$, and the discount factor to delivery is $0.91$. Find the value of two identical long contracts, each covering one unit. (Answer: $\$9.10$)
A share has a continuous income yield of $2\%$. The continuously compounded risk-free rate is $6\%$, there are $1.5$ years to delivery, the spot price is $\$200$, and an existing long forward has delivery price $\$212$. Find its current value. (Answer: $\$0.34$)
Five short forward contracts each cover one unit and have delivery price $\$68$. The current fair forward price is $\$70.50$, and the discount factor to delivery is $0.93$. Find the total value of the short position. (Answer: $-\$11.63$)
An account holds three long forwards with delivery price $\$100$ and two short forwards with delivery price $\$106$, all on the same asset and delivery date. The current fair forward price is $\$104$, and the discount factor is $0.95$. Each contract covers one unit. Find the account’s total value. (Answer: $\$15.20$)

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