Prepaid Forwards And Discrete Income Dates

Introduction

An asset that pays cash income before a forward contract expires is not economically equivalent to an otherwise identical asset that pays nothing. The holder of the asset receives that income; the holder of a forward contract generally does not receive it before taking delivery. A fair forward price must account for this difference.

The key distinction is between paying now for delivery later and agreeing to pay at delivery. The first is a prepaid forward; the second is an ordinary forward. For known, discrete income payments, the prepaid-forward price equals spot less the present value of the income the asset holder receives before delivery. Accumulating that amount to the delivery date gives the ordinary forward delivery price.

Definition: A prepaid forward price for delivery at time $T$ is the amount paid at time 0 for the right to receive the asset at $T$, with no further payment at delivery. An ordinary forward delivery price, denoted here by $K_0(T)$, is the amount paid at $T$ under a forward contract entered at time 0 with no initial payment. For a fairly priced ordinary forward, the initial contract value is zero.

Pricing With Known Discrete Income

Assume the asset has current spot price $S_0$, and pays known cash amounts $C_j$ at times $t_j$ before delivery. Let $D(0,t)$ be the risk-free discount factor from time $t$ to time 0. Assume the payments are certain, the asset and payments can be traded or replicated, and borrowing and lending use the same risk-free rates. Ignore taxes, transaction costs, and trading restrictions. The income included in the calculation must be income to the holder of the asset before it is delivered.

Theorem: Under these assumptions, the prepaid-forward price is $$ P_0(T)=S_0-\sum_{j} C_jD(0,t_j), $$ where the sum covers the income payments to which the asset holder is entitled before delivery. The fair ordinary forward delivery price is $$ K_0(T)=\frac{P_0(T)}{D(0,T)}. $$

With a constant effective annual interest rate $i$, the discount factor is $D(0,t)=v^t=(1+i)^{-t}$, where $v=1/(1+i)$. Thus the formulas can also be written $$ P_0(T)=S_0-\sum_j C_jv^{t_j}, \qquad K_0(T)=P_0(T)(1+i)^T. $$

The payments need not occur at whole-year intervals. If $i$ is an effective annual rate and time is measured in years, a payment at $t=0.5$ is discounted by $v^{0.5}$, not by $v$. If rates vary over time, use the appropriate discount factors rather than assuming one constant rate.

Why the Income Is Subtracted

The price adjustment follows from replication. Buy the asset now, then borrow by issuing risk-free claims that require payments equal to the asset’s known income. The amount received from issuing those claims is the present value of the income. At each income date, the asset’s cash payment offsets the corresponding borrowing payment. At delivery, the replicating position still owns the asset and has no remaining income-related payment.

\begin{aligned} \text{Initial cost} &= S_0-\sum_j C_jD(0,t_j)\\ &= P_0(T). \end{aligned}

This portfolio and a prepaid forward both require the same initial payment and deliver the asset at $T$, with no intervening net cash flows. If their prices differed, an investor could buy the cheaper position and sell the more expensive one. Therefore, in a no-arbitrage market, their prices must agree.

An ordinary forward postpones payment until $T$. The amount $P_0(T)$ paid now is equivalent to $P_0(T)/D(0,T)$ paid at $T$. That is why the fair delivery price is $K_0(T)=P_0(T)/D(0,T)$. Under a constant effective annual rate, this is simply $P_0(T)(1+i)^T$.

Identify the delivery date and income entitlement.
List each known payment the asset holder receives before delivery. Check the contract’s treatment of a payment dated exactly at delivery.

Find the present value of the income.
Discount each payment from its own payment date to time 0 using the applicable risk-free discount factor.

Calculate the prepaid-forward price.
Subtract the present value of the entitled income from spot.

Convert to an ordinary forward delivery price.
Divide the prepaid price by $D(0,T)$, or accumulate it to $T$ when using a constant effective annual rate.

Worked Example: Two Income Payments Before Delivery

A hypothetical asset has spot price $S_0=\$150$. It pays known cash income of $\$6$ at $t=1$ and $\$9$ at $t=2$. A forward delivers the asset at $T=3$. The effective annual risk-free rate is $i=4\%$. Find the prepaid-forward price and the fair ordinary forward delivery price. Use the method of discounting the payments to time 0, then accumulating the prepaid price to delivery. This avoids treating the prepaid price as if it were already payable at $T$.

The discount factor is $v=1/1.04$. The present value of the first payment is $$ \frac{6}{1.04}=5.76923077. $$ The present value of the second payment is $$ \frac{9}{1.04^2}=\frac{9}{1.0816}=8.32100628. $$ Thus the total present value of income is $$ 5.76923077+8.32100628=14.09023705. $$

Subtract this from spot to obtain the prepaid-forward price: $$ P_0(3)=150-14.09023705=135.90976295. $$ The ordinary forward delivery price is the prepaid amount accumulated for three years: $$ K_0(3)=135.90976295(1.04)^3 =135.90976295(1.124864)=152.88. $$

Answer: The prepaid-forward price is $\$135.91$, and the ordinary forward delivery price is $\$152.88$. As a check, carrying spot to time 3 gives $150(1.04)^3=\$168.7296$. The income payments accumulated to time 3 total $6(1.04)^2+9(1.04)=6.4896+9.36=\$15.8496$. Their difference is $\$152.88$. The final money amounts are rounded to the nearest cent.

Payment Dates and Contract Entitlement

A payment date alone does not always settle whether income belongs in the adjustment. What matters is whether the asset holder receives the payment before the asset is transferred. If the payment date equals $T$, read the delivery convention carefully. If the forward buyer receives the asset only after the distribution has been paid to the seller, that payment is excluded from the buyer’s income and must be included in the price adjustment. If the buyer receives the asset with the right to that distribution, it is not subtracted as income retained by the seller.

Worked Example: A Distribution on the Delivery Date

A hypothetical asset has spot price $\$100$, and the effective annual risk-free rate is $4\%$. It pays $\$4$ at $t=1$ and $\$6$ at $t=2$. Delivery is at $T=2$, immediately after the distribution at $t=2$; the seller receives that distribution. Find the prepaid-forward price and ordinary forward delivery price. The method is to include both payments because the asset holder receives both before transfer. The common mistake is to exclude the time-2 payment merely because its date equals $T$.

The present value of the payments is $$ \frac{4}{1.04}+\frac{6}{1.04^2} =3.84615385+5.54733728 =9.39349113. $$ Therefore, $$ P_0(2)=100-9.39349113=90.60650887. $$ Accumulating to time 2 gives $$ K_0(2)=90.60650887(1.04)^2 =90.60650887(1.0816)=98.00. $$

A direct check gives the same result: $100(1.0816)-4(1.04)-6=108.16-4.16-6=\$98.00$. If the contract instead transferred the asset immediately before the time-2 distribution, the buyer would receive the $\$6$ payment. Only the time-1 payment would then be subtracted, and the fair delivery price would be $108.16-4.16=\$104.00$. The timing convention changes the answer by exactly the $\$6$ payment at delivery. Final money amounts are shown to the nearest cent.

Checking a Quoted Forward Price

The no-arbitrage formula also provides a way to check a quoted ordinary forward price. Suppose the quoted delivery price is greater than the fair price. An investor can borrow the spot price, buy the asset, and enter a short forward. The asset’s income reduces the borrowing needed by the delivery date. The investor then delivers the asset under the forward and uses the forward payment to repay the remaining debt. If the quote is too high, the difference is a positive, risk-free amount at delivery under the stated assumptions.

Worked Example: An Overpriced Ordinary Forward

A hypothetical asset is priced at $\$90$ and pays known income of $\$2$ at $t=1$. An ordinary forward delivers it at $T=2$. The effective annual risk-free rate is $4\%$, but the market delivery price is quoted as $\$96.00$. Determine whether the quote is consistent with no arbitrage and calculate the delivery-date arbitrage gain if it is not. The method is to compare the quote with the fair price and replicate the short-forward obligation. The common mistake is to compare the quote with spot instead of with spot carried forward after allowing for income.

The fair delivery price is $$ K_0(2)=90(1.04)^2-2(1.04) =97.344-2.08=95.264. $$ The quote exceeds the fair price by $$ 96.00-95.264=0.736. $$

Arbitrage strategy. At time 0, borrow $\$90$, buy the asset, and agree to sell it forward for $\$96.00$ at time 2. At time 1, receive $\$2$ of income and use it to reduce the borrowing. By time 2, the original borrowing would have grown to $90(1.04)^2=\$97.344$. The $\$2$ repayment at time 1 reduces the debt at time 2 by $2(1.04)=\$2.08$, leaving a balance of $$ 97.344-2.08=95.264. $$ Deliver the asset under the forward and receive $\$96.00$. After repaying the $\$95.264$ debt, the remaining amount is $$ 96.00-95.264=0.736. $$

The quote is too high by $\$0.736$ at delivery, or $\$0.74$ to the nearest cent. The cash flows confirm the result: the bought asset supplies the delivery, its income offsets part of the loan, and the forward proceeds exceed the remaining debt. This is an arbitrage only under the assumptions stated, including the ability to borrow at the risk-free rate and trade without costs or restrictions.

Common Traps and Limits

The formula is compact, but its application depends on matching dates, cash flows, and units. In particular, the present value adjustment is made at time 0, while the ordinary forward delivery price is an amount at $T$.

Accumulating the wrong amount. The prepaid price is paid at time 0. Accumulate that price to $T$ to obtain the ordinary forward delivery price; do not add the present value of income to spot.
Discounting every payment for the full term. A payment at $t_j$ is discounted for $t_j$ years, not $T$ years. For example, with $T=3$, a payment at $t=1$ uses $D(0,1)$.
Ignoring entitlement at delivery. Include or exclude a payment at $T$ according to who receives it under the contract’s delivery convention.
Using expected income as if it were certain. The stated formula uses known, fixed payments. If income is uncertain, simply subtracting its expected present value does not in general establish a no-arbitrage price.
Mixing time units. With an effective annual rate, express payment times in years and use the corresponding fractional powers when needed.

Key takeaway: For known income received by the asset holder before delivery, subtract its present value from spot to find the prepaid-forward price. Divide by $D(0,T)$, or accumulate to $T$, to find the fair ordinary forward delivery price. Always determine who receives each payment before doing the calculation.

Check Your Understanding

For each problem, state whether the requested amount is a prepaid-forward price or an ordinary forward delivery price. Assume a constant effective annual risk-free rate and no taxes, transaction costs, or trading restrictions.

An asset has spot price $\$100$, pays $\$4$ at $t=1$, and is delivered at $T=2$. The effective annual rate is $5\%$. Find the prepaid-forward price and fair ordinary delivery price. (Answer: prepaid-forward price $=\$96.19$; ordinary delivery price $=\$106.05$)
An asset has spot price $\$250$, pays $\$8$ at $t=1$ and $\$10$ at $t=2$, and is delivered at $T=3$. The effective annual rate is $4\%$. Find the fair ordinary delivery price. (Answer: $\$262.16$)
The prepaid-forward price is $\$140$. The asset pays known income of $\$5$ at $t=1$, and the effective annual rate is $5\%$. What spot price is consistent with no arbitrage? (Answer: $\$144.76$)
An asset has spot price $\$90$, pays $\$2$ at $t=1$, and is delivered at $T=2$. The effective annual rate is $4\%$. A market forward quotes a delivery price of $\$96.00$. Find the fair price and the delivery-date gain from the arbitrage trade. (Answer: fair delivery price $=\$95.264$; gain $=\$0.736$, or $\$0.74$ to the nearest cent)
An asset has spot price $\$300$, pays $\$9$ at $t=1$ and $\$5$ at $t=2$, and is delivered at $T=2$. The effective annual rate is $3\%$. If delivery occurs after both payments have been received by the seller, find the fair ordinary forward delivery price. How would it change if the buyer received the time-2 payment? (Answer: $\$304.00$ if the seller receives both payments; $\$309.00$ if the buyer receives the time-2 payment)

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