Valuing Forwards With Discrete Income

Why Interim Distributions Change a Forward’s Value

An existing forward contract fixes a delivery payment at the time the contract is entered, but the value of that contract can change as market conditions change. If the underlying asset pays known cash distributions before delivery, a holder of the asset receives those payments; a holder of a forward does not receive them before the delivery date. The forward’s value must therefore account for the distributions that remain unpaid.

This tutorial considers a forward on an asset with known, deterministic cash distributions on specified dates before delivery. The central task is to value the contract at a later valuation date, not to calculate its original delivery price. The distinction matters: the delivery payment $K$ is fixed by the contract, while the forward’s current value depends on the current spot price and the present values of future cash flows.

Definition: At valuation date $s$, let $S_s$ be the spot price of the asset, $T$ the delivery date, and $K$ the fixed delivery payment in an existing forward. Suppose known cash distributions $C_j$ are payable at dates $u_j$, where $s<u_j<T$. Let $P(s,u)$ denote the time-$s$ price of a default-free zero-coupon payment of 1 at date $u$, under the applicable discount curve. The value of the long forward is $$ V_s=S_s-\sum_j C_jP(s,u_j)-KP(s,T). $$ The value to the short position is $-V_s$. Only distributions that are still to be paid after $s$ and before $T$ are included.

The formula assumes that the distribution amounts and payment dates are known, the asset can be bought or sold at the stated spot price, and the same relevant discount factors can be used to value the cash flows. It also assumes no transaction costs, taxes, or restrictions that prevent the replication described below. If a distribution is uncertain, this known-cash-flow formula does not directly apply without additional modeling assumptions.

Why the Present-Value Adjustment Works

A forward gives its long holder an asset at date $T$ in exchange for $K$. To see what that right is worth today, first consider a prepaid forward: an arrangement paid for at date $s$ that delivers the asset at $T$, with no further payment at delivery. To replicate that delivery, buy the asset now and make payments equal to its distributions as they arise. The asset provides the distributions, and the matching payments remove them from the replicating position, leaving the asset itself at $T$.

The cost of this replication is the spot price less the present value of the distributions. A regular long forward differs from this prepaid forward only because the long pays $K$ at $T$, rather than paying the prepaid cost at $s$. The present value of that future payment is $KP(s,T)$. Subtracting it gives the existing forward’s value.

\begin{aligned} \text{Prepaid forward value at }s &=S_s-\sum_j C_jP(s,u_j),\\ V_s &=\text{Prepaid forward value at }s-KP(s,T)\\ &=S_s-\sum_j C_jP(s,u_j)-KP(s,T). \end{aligned}

This reasoning also gives a useful sign check. A long forward benefits when the spot price rises, and is reduced in value by a larger fixed delivery payment or by larger distributions still to be paid. The short position has the opposite value because each contract’s gain is the other party’s loss.

Set the valuation date.
Use the spot price and discount factors as of the date the existing contract is being valued.

List remaining distributions.
Include only known payments after the valuation date and strictly before delivery. Do not include payments already made.

Discount each cash flow.
Apply the factor from the valuation date to that particular distribution date, and the factor to $T$ to the delivery payment.

Calculate and interpret the value.
Subtract the present values from spot for the long position; reverse the sign for the short position.

Worked Valuations

Worked Example: Flat Annual Effective Interest Rate

A long forward was entered earlier on an asset. At the current valuation date, the asset’s spot price is $\$120$. The contract requires the long to pay $\$100$ in three years. The asset will pay known cash distributions of $\$6$ at the end of year 1 and $\$8$ at the end of year 2. Assume a flat annual effective interest rate of $10\%$, with annual compounding. Find the current value of the long forward.

Method. Discount each remaining distribution and the fixed delivery payment to the valuation date, then subtract them from the spot price. This is an existing contract, so the delivery payment is the contract’s fixed $\$100$, not a newly calculated fair delivery price.

The annual discount factor is $v=1/1.10$. Thus the factors for years 1, 2, and 3 are:

v=\frac{1}{1.10}=0.9090909,\qquad v^2=\frac{1}{1.10^2}=0.8264463,\qquad v^3=\frac{1}{1.10^3}=0.7513148.

The present value of the year-1 distribution is $\$6(0.9090909)=\$5.4545454$. The present value of the year-2 distribution is $\$8(0.8264463)=\$6.6115704$. The present value of the delivery payment is $\$100(0.7513148)=\$75.1314800$. Therefore:

$$ V_0=120-5.4545454-6.6115704-75.1314800 =32.8024042. $$

Rounded to the nearest cent, the long forward is worth $\$32.80$. The short position is worth $-\$32.80$. As a check, the long value is positive because the spot price, after allowing for the distributions, exceeds the present value of the contract’s delivery payment.

Common mistake avoided: Subtracting the distributions’ face amounts, $\$6+\$8$, instead of their present values would give $\$120-\$14-\$75.1314800=\$30.87$. That understates the long’s value because the distributions are received in the future and must be discounted.

Worked Example: Discount Factors from a Term Structure

At a valuation date, an asset has spot price $\$250$. A long forward requires delivery of the asset in 2.5 years for a fixed $\$230$. Before delivery, the asset will pay known distributions of $\$10$ in 0.5 years and $\$14$ in 1.5 years. The relevant zero-coupon discount factors are $P(0,0.5)=0.97$, $P(0,1.5)=0.91$, and $P(0,2.5)=0.86$. Find the current value of the long.

Method. Use the supplied factor for each cash-flow date. Since rates need not be constant across maturities, one flat annual rate should not be imposed.

The present value of the first distribution is $\$10(0.97)=\$9.70$. The present value of the second is $\$14(0.91)=\$12.74$. The delivery payment has present value $\$230(0.86)=\$197.80$. Substitution gives:

V_0=250-9.70-12.74-197.80=\$29.76.

The long forward is worth $\$29.76$, and the short is worth $-\$29.76$. The arithmetic also checks directly: the total present value subtracted from spot is $\$9.70+\$12.74+\$197.80=\$220.24$, and $\$250-\$220.24=\$29.76$.

Common mistake avoided: Applying the maturity factor $0.86$ to both distributions would discount them for 2.5 years rather than for their own dates. Each cash flow must use its matching discount factor.

Worked Example: Valuing a Contract After a Distribution Has Been Paid

A long forward has a fixed delivery payment of $\$150$ at the end of year 4. At the end of year 1, the contract is revalued. The spot price is then $\$146$, and the annual effective interest rate from that date is $5\%$. The asset paid a $\$3$ distribution at the end of year 1, and will pay known distributions of $\$4$ at the end of year 2 and $\$6$ at the end of year 3. Find the value at the end of year 1.

Method. Treat the end of year 1 as the valuation date. The $\$3$ distribution has already been paid, so it is not a future cash flow from this date. The remaining times are one year to the $\$4$ payment, two years to the $\$6$ payment, and three years to delivery.

The required discount factors are:

v=\frac{1}{1.05}=0.9523810,\qquad v^2=\frac{1}{1.05^2}=0.9070295,\qquad v^3=\frac{1}{1.05^3}=0.8638376.

The present values at the end of year 1 are $\$4v=\$3.8095238$, $\$6v^2=\$5.4421769$, and $\$150v^3=\$129.5756$. Therefore:

V_1=146-3.8095238-5.4421769-129.5756 \approx \$7.17.

The long position is worth $\$7.17$ to the nearest cent, and the short position is worth $-\$7.17$. For a check, the total present value subtracted from spot is approximately $\$138.83$, leaving approximately $\$7.17$.

Common mistake avoided: Including the $\$3$ already paid would incorrectly reduce the value by $\$3$. Once the valuation date has passed that payment date, the distribution is no longer a future cash flow of the contract’s underlying asset.

Common Timing and Sign Errors

The formula is compact, but its inputs depend on reading the contract and timeline carefully. The most reliable approach is to draw or list the valuation date, each remaining distribution date, and the delivery date before calculating. In particular:

Do not confuse value with delivery price. $K$ remains fixed in an existing contract. The current value is not obtained by replacing $K$ with a newly quoted delivery price.
Do not use undiscounted distribution amounts. Subtract each distribution’s present value, not its face amount.
Do not include past payments. A distribution already paid before the valuation date does not reduce current value.
Match each factor to its date. In a term structure, a distribution and the delivery payment will generally have different discount factors.
Keep the position’s sign clear. The short forward’s value is the negative of the long’s value under the same contract terms.

A distribution payable exactly at delivery requires careful attention to the contract’s settlement convention and ownership rights. The formula above is stated for distributions paid strictly before delivery; do not include a payment at $T$ without first establishing who is entitled to receive it under the contract.

Key takeaway: For a long forward on an asset with known cash distributions before delivery, subtract the present value of every distribution still to be paid and the present value of the fixed delivery payment from the current spot price. Use the valuation-date discount factor appropriate to each payment date.

Check Your Understanding

For each problem, value the long forward unless the question asks for the short position. Round money to the nearest cent.

An asset has spot price $\$80$. A long forward requires $\$70$ in two years. It will pay a known $\$3$ distribution in one year. Assume an annual effective rate of $5\%$. What is the long forward’s value? (Answer: $\$13.65$)
The spot price is $\$180$. A forward’s fixed delivery payment is $\$160$ in two years. Known distributions of $\$5$ and $\$7$ are payable in 0.5 and 1.5 years. The matching discount factors are $0.98$, $0.94$, and $0.90$ for 0.5, 1.5, and 2 years. Find the long forward’s value. (Answer: $\$24.52$)
At the end of year 2, an existing forward has spot price $\$205$, a fixed delivery payment of $\$200$ at the end of year 4, and one remaining known distribution of $\$8$ at the end of year 3. The annual effective rate is $4\%$. Find the value of the long. (Answer: $\$12.40$)
An asset’s spot price is $\$300$. Known distributions of $\$12$ and $\$15$ will be paid at the ends of years 1 and 2. A three-year long forward is entered at a delivery payment $K$. The annual effective rate is $5\%$. What value of $K$ makes the forward’s value zero when entered? (Answer: $\$318.31$)
At a valuation date, the asset has spot price $\$500$. A forward requires $\$430$ in three years. A $\$18$ distribution was already paid before the valuation date; remaining distributions of $\$20$ and $\$30$ are due in one and two years. The discount factors for years 1, 2, and 3 are $0.96$, $0.92$, and $0.88$. Find the long’s value and the short’s value. (Answer: Long $\$74.80$; short $-\$74.80$)

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