Valuing An Existing Forward On A No-Income Asset

Why an Existing Forward Has a Value

A forward contract fixes a delivery payment when the contract is made, but market conditions can change before delivery. If the asset’s spot price or the market interest rate changes, the contract’s fixed payment may become more or less attractive than the terms available on a new forward. The value of the existing contract measures that difference at the valuation date.

This tutorial considers a forward on a no-income asset: an asset that pays no income between the valuation date and delivery. Assume the asset can be bought and held, the contract has no default risk, and there are no transaction costs, storage costs, or other carrying costs. Assume also that the market discount factor used to value the delivery payment is available for the same term and applies consistently to the contract. These assumptions matter; if the asset generates income or incurs costs, the valuation requires additional adjustments.

Definition: Let $S_t$ be the asset’s spot price at valuation time $t$, $K$ the fixed delivery price in an existing forward, and $D(t,T)$ the market discount factor from delivery time $T$ to time $t$. The value of the long forward at time $t$ is $V_t=S_t-KD(t,T)$. The value of the short forward is $-V_t$.

The contract’s delivery price $K$ is the amount specified when that contract was entered. It does not change just because the market changes. The fair forward price is the delivery price that would make a new forward have zero value at the valuation date. Under the assumptions above, it is $S_t/D(t,T)$. Thus, the value of an existing forward and the fair price of a new forward answer different questions.

Deriving the Valuation Formula

The key idea is to compare the forward with a position that has the same payoff at delivery. A long forward gives its holder the asset at time $T$ in exchange for paying $K$. Alternatively, the holder can buy the asset now and borrow the present value of $K$, so that the loan repayment is exactly $K$ at delivery.

Construct the comparison position.
Buy one unit of the asset now for $S_t$, and borrow $KD(t,T)$. With the applicable discount factor, that borrowing grows to $K$ by time $T$.

Compare delivery-date cash flows.
The comparison position holds the asset and owes $K$, so its net delivery-date payoff is the asset minus $K$. This is the same payoff as a long forward.

Compare current values.
The comparison position costs $S_t-KD(t,T)$. Since positions with the same future payoffs have the same value under the stated assumptions, this is the long forward’s value.

\begin{aligned} \text{Current cost of comparison position} &= S_t-KD(t,T),\\ V_t&=S_t-KD(t,T). \end{aligned}

This argument also explains the sign. If the asset’s spot price exceeds the present value of the fixed payment, the long forward has positive value. If the present value of the payment exceeds the spot price, the long forward has negative value. The short has the opposite value because the two parties’ contract values sum to zero under this model.

Equivalent form: Define the current fair forward price by $F_{t,T}=S_t/D(t,T)$. Then $S_t=D(t,T)F_{t,T}$, so the existing long forward’s value can also be written as $V_t=D(t,T)(F_{t,T}-K)$. This form compares the current fair delivery price with the contract’s fixed delivery price, then discounts the difference.

If a constant effective annual interest rate $i$ applies over $T-t$ years, then $D(t,T)=(1+i)^{-(T-t)}$. In that case the value is $S_t-K(1+i)^{-(T-t)}$. Use the discount factor for the time remaining to delivery and the market conditions at the valuation date, not automatically the rate or forward price that applied when the contract was first made.

Worked Valuations

Worked Example: Recovering the Fixed Delivery Price

An existing long forward is on a no-income asset. At the valuation date, the spot price is $\$500$, the discount factor to delivery is $0.95$, and the forward’s value to the long is $\$25$. Find the contract’s fixed delivery price and verify the result.

Method. Use the existing-contract valuation formula and solve for $K$. The discount factor already reflects the market’s valuation of one dollar payable at delivery.

$$ V_t=S_t-KD(t,T) $$

Substitute the stated values:

\begin{aligned} 25&=500-0.95K,\\ 0.95K&=500-25=475,\\ K&=\frac{475}{0.95}=500. \end{aligned}

The fixed delivery price is $\$500.00$. Check by substituting it into the valuation formula:

$$ 500-(0.95)(500)=500-475=25. $$

The check reproduces the stated value. A common mistake is to divide the spot price by the discount factor and call the result the existing contract’s delivery price. That operation finds the fair delivery price of a new forward; it does not solve for the fixed $K$ in this contract.

Worked Example: Revaluing a Forward After One Year

A two-year forward was entered when the spot price of a no-income asset was $\$1{,}000$. At that time, the effective annual interest rate for the two-year term was $6\%$, so the contract was entered at the fair forward price. One year later, the spot price is $\$1{,}080$, and the current effective annual rate for the one year remaining is $5\%$. Find the value of the existing contract to the long.

Step 1: Find the contract’s fixed delivery price. At entry, the fair forward price for two years was the spot price accumulated at $6\%$ annually. This determines $K$; the contract price remains fixed after entry.

\begin{aligned} K&=1{,}000(1.06)^2\\ &=1{,}000(1.1236)\\ &=\$1{,}123.60. \end{aligned}

Step 2: Discount the fixed payment using current conditions. There is one year left, so use the current $5\%$ annual effective rate for one year:

D(t,T)=\frac{1}{1.05}=0.952380952\ldots

The present value of the fixed payment is:

\frac{1{,}123.60}{1.05}=\$1{,}070.095238\ldots

Step 3: Subtract the present value from spot.

V_t=1{,}080-1{,}070.095238\ldots =\$9.904761\ldots

The existing long forward is worth $\$9.90$, rounded to the nearest cent. As a check, the current fair forward price for the remaining year is $1{,}080(1.05)=\$1{,}134$. Its difference from $K$ is $\$10.40$, and discounting that difference gives $10.40/1.05=\$9.904761\ldots$, the same value. Using the original $6\%$ rate to discount the remaining payment would ignore the changed market rate.

Worked Example: A Forward That Is Unfavorable to the Long

An existing long forward on a no-income asset requires payment of $\$2{,}700$ at delivery in one year. At the valuation date the spot price is $\$2{,}500$, and the one-year market discount factor is $0.96$. Find the value to the long and the value to the short.

Method. First discount the contract’s fixed payment, then subtract that present value from the spot price. The short’s value is the negative of the long’s value.

\begin{aligned} KD(t,T)&=2{,}700(0.96)=\$2{,}592,\\ V_t^{\text{long}}&=2{,}500-2{,}592=-\$92,\\ V_t^{\text{short}}&=-V_t^{\text{long}}=+\$92. \end{aligned}

The long position is worth $-\$92.00$ and the short position is worth $\$92.00$. The long’s fixed payment has a present value greater than the asset’s spot price, making the contract unfavorable to the long under current terms.

Check using the fair forward price: $F_{t,T}=2{,}500/0.96=\$2{,}604.166666\ldots$. The fair price is $\$95.833333\ldots$ below $K$. Discounting that difference gives $0.96(-95.833333\ldots)=-\$92$, as required. The short value has the opposite sign, so the two values sum to zero.

Common Exam Traps and Interpretation

The calculation is short, but the setup requires care. Identify the valuation date, the delivery date, the asset’s spot price on the valuation date, the contract’s unchanged delivery price, and the discount factor for the remaining term. Keep the perspective clear: $S_t-KD(t,T)$ is the value to the long; the short’s value is its negative.

Confusing fair price and contract value. $S_t/D(t,T)$ is the fair delivery price for a new forward. The existing contract’s value is $S_t-KD(t,T)$.
Discounting the wrong amount or for the wrong term. Discount the fixed delivery payment $K$ from the actual delivery date to the valuation date. If one year remains, do not use a two-year discount factor simply because the original contract had a two-year term.
Using the wrong rate date. The contract’s $K$ is set at entry, but its value today uses the current market discount factor for the time remaining.
Reversing the position’s sign. A positive value to the long is a negative value to the short, and vice versa.
Applying the no-income formula when its assumptions fail. If the asset pays income, or has relevant costs of carry, the comparison position’s cash flows change. The formula here is specifically for a no-income asset under the assumptions stated above.

For a quick reasonableness check, compare spot with the present value of the payment. If $S_t$ is greater, the long’s value must be positive; if $S_t$ is smaller, it must be negative. The equivalent fair-forward-price form provides a second check when the current fair price is easy to calculate.

Key takeaway: For an existing forward on a no-income asset, keep the fixed delivery price $K$, discount it using the market factor for the remaining term, and subtract that present value from spot to obtain the long’s value: $V_t=S_t-KD(t,T)$.

Check Your Understanding

For each problem, determine the requested value using the long-forward convention unless stated otherwise. Round dollar amounts to the nearest cent when needed.

An existing long forward has spot price $\$700$, delivery price $\$710$, and discount factor $0.97$. Find its value to the long. (Answer: $\$11.30$)
A no-income asset’s spot price is $\$1{,}250$. An existing long forward requires $\$1{,}300$ in two years. The current effective annual rate is $4\%$. Find the forward’s value. (Answer: $\$48.08$)
An existing short forward has delivery price $\$880$, spot price $\$900$, and discount factor $0.94$. Find its value to the short. (Answer: $-\$72.80$)
An existing long forward is worth $-\$30$. The spot price is $\$1{,}500$, and the discount factor to delivery is $0.90$. Find the fixed delivery price and verify it by substitution. (Answer: $\$1{,}700.00$)
For an existing long forward, the spot price is $\$3{,}200$, the delivery price is $\$3{,}600$, and the discount factor is $0.875$. Find both the value of the forward and the current fair delivery price for a new forward. (Answer: value $\$50.00$; fair delivery price $\$3{,}657.14$)

What You'll Learn