Why Known Distributions Change a Forward Price
A forward contract fixes a payment today for delivery of an asset at a later date. If the asset will make known cash distributions before delivery, a forward buyer does not receive those distributions: the buyer receives the asset only at the delivery date. As a result, the fair forward price is not generally the price that would apply to an otherwise identical asset with no interim income.
The key is to separate two prices. A prepaid forward requires payment now for delivery of the asset at the future date. A standard forward requires no payment initially; its agreed delivery payment is made when the asset is delivered. Known distributions reduce the prepaid price, and financing that prepaid amount to the delivery date gives the forward price.
These formulas assume the distributions and their payment dates are known, the asset and the matching risk-free investments can be traded, and there are no transaction costs, taxes, or restrictions that prevent the replication. The discount factors must reflect the applicable risk-free term structure. If rates are constant at effective annual rate \(i\) and times are whole years, then \(D(0,t)=(1+i)^{-t}\). The formulas do not apply unchanged when distributions are uncertain or when the stated amount or entitlement depends on contractual details.
Why the Formula Holds
The prepaid price follows from a replication argument. Buying the asset costs \(S_0\), and the buyer receives each known distribution. To remove those interim receipts, offset each one with a risk-free obligation to pay \(C_j\) at time \(t_j\). The present value of that obligation is \(C_jD(0,t_j)\). The combined position costs the spot price less the present value of the distributions, has no net cash flow at the distribution dates, and delivers the asset at \(T\).
Pay \(S_0\) now and receive the asset, including its known cash distributions before \(T\).
For each distribution \(C_j\), arrange a risk-free payment of \(C_j\) at \(t_j\). Its current cost is \(C_jD(0,t_j)\), and it cancels the distribution received from the asset.
The net initial cost is \(S_0-\sum C_jD(0,t_j)\); there are no interim net cash flows, and the asset is held at \(T\). This is the prepaid forward.
A payment of \(F_{0,T}\) due at \(T\) has present value \(F_{0,T}D(0,T)\). Equating it to the prepaid cost gives \(F_{0,T}D(0,T)=P^{\mathrm{pre}}_{0,T}\), hence \(F_{0,T}=P^{\mathrm{pre}}_{0,T}/D(0,T)\).
The final step also explains the financing interpretation. The prepaid amount is paid now; the fair forward payment is that amount accumulated to time \(T\). With a constant annual effective rate, this means multiplying the prepaid price by \((1+i)^T\).
Worked Calculations
Worked Example: Discount Factors for Known Distributions
An asset costs $250. It will pay certain cash distributions of $6 at time 1 and $8 at time 2. Delivery is at time 3. The risk-free discount factors are \(D(0,1)=0.98\), \(D(0,2)=0.95\), and \(D(0,3)=0.90\). Find the prepaid forward price and the fair forward price, then check the forward payment's present value.
Method. Discount each distribution to time 0, subtract their present values from the spot price, and divide the prepaid price by the delivery-date discount factor.
The present value of the time-1 distribution is \(6(0.98)=5.88\). The present value of the time-2 distribution is \(8(0.95)=7.60\). Thus, the total present value of distributions is \(5.88+7.60=13.48\).
The fair forward price is
Check. The present value of the forward payment is \(262.80(0.90)=236.52\), exactly the prepaid price. The common mistake of subtracting the undiscounted distributions would give \((250-6-8)/0.90=262.22\), which fails to account for when the distributions are received.
Worked Example: Constant Effective Annual Interest
A financial asset costs $600 and is certain to pay $10 at the end of year 1 and $15 at the end of year 3. Find its prepaid forward price and fair forward price for delivery at the end of year 4. Assume a constant effective annual risk-free rate of 4%.
Method. Discount each distribution to time 0 at 4%, subtract their present values from $600, and accumulate the prepaid amount four years to obtain the forward price.
The present value of the year-1 payment is \(10/(1.04)=9.6153846\). The present value of the year-3 payment is \(15/(1.04)^3=15/1.124864\approx13.3349454\). Therefore,
At the same rate, \(D(0,4)=(1.04)^{-4}\), so the forward price is
The prepaid forward price is $577.05, and the fair forward price is $675.07. Rounding is to the nearest cent only at the final step. A useful check is to discount the unrounded forward price by \((1.04)^{-4}\); this recovers approximately $577.04967. The method avoids treating the two distributions as if they occurred at the delivery date.
Worked Example: Check Using the Present Value of the Forward Payment
An asset costs $500 and will make certain distributions of $7 at time 1 and $11 at time 2. Delivery is at time 3. The discount factors are \(D(0,1)=0.96\), \(D(0,2)=0.91\), and \(D(0,3)=0.86\). Find both prices and verify the forward price by present-value comparison.
Method. Use the given term-structure discount factors directly; do not assume a single annual interest rate.
The distributions have present values \(7(0.96)=6.72\) and \(11(0.91)=10.01\). Their total present value is \(6.72+10.01=16.73\). Thus,
The forward price is
To the nearest cent, the prepaid forward price is $483.27 and the forward price is $561.94. As a check, the present value of the forward payment is \(561.9418605(0.86)=483.27\). Using one flat annual rate here would be unjustified: the supplied discount factors need not correspond to a constant rate.
Common Traps and Interpretation
A prepaid price and a forward price are different quantities. The prepaid price is paid at time 0; the forward price is paid at delivery. Their connection is determined by the discount factor to the delivery date. A calculation can subtract the distributions correctly and still produce the wrong answer if it reports the prepaid price as the forward price.
- Subtracting cash amounts without discounting. A payment at time \(t_j\) affects today's value by \(C_jD(0,t_j)\), not by the full amount \(C_j\).
- Using the wrong time for financing. The prepaid amount is accumulated to \(T\), so use \(D(0,T)\) or its reciprocal, not a discount factor for an income date.
- Mixing up present and future values. The correct relation is \(F_{0,T}D(0,T)=P^{\mathrm{pre}}_{0,T}\). Dividing by \(D(0,T)\) converts a present value to a future payment.
- Using a flat rate when only a term structure is given. Apply each provided discount factor to its matching cash-flow date.
- Ignoring the distribution entitlement convention. The formula as stated includes known distributions paid before delivery that belong to the asset holder. If a payment occurs exactly at delivery, its treatment must be specified rather than assumed.
Check Your Understanding
For each problem, identify the prepaid price first, then convert it to the forward price. Answers are rounded to the nearest cent.
- An asset costs $320, pays a known $8 at the end of year 1, and is delivered at the end of year 2. The effective annual risk-free rate is 5%. Find the prepaid and forward prices. (Answer: prepaid $312.38; forward $344.40)
- An asset costs $900 and pays $12 at time 1 and $20 at time 2. Delivery is at time 3, with discount factors \(D(0,1)=0.97\), \(D(0,2)=0.93\), and \(D(0,3)=0.88\). Find both prices. (Answer: prepaid $869.76; forward $988.36)
- An asset costs $1,500 and pays $25 at the end of year 1 and $30 at the end of year 2. Delivery is at the end of year 4, and the effective annual risk-free rate is 3%. Find both prices. (Answer: prepaid $1,447.45; forward $1,629.12)
- An asset costs $780, pays $9 at time 0.5 and $14 at time 1.5, and is delivered at time 2. The discount factors are \(D(0,0.5)=0.985\), \(D(0,1.5)=0.955\), and \(D(0,2)=0.93\). Find both prices. (Answer: prepaid $757.77; forward $814.80)
- An asset costs $600 and pays known amounts of $5 at time 1 and $10 at time 2. Delivery is at time 3. Given \(D(0,1)=0.96\), \(D(0,2)=0.91\), and \(D(0,3)=0.86\), find both prices and check the present value of the forward payment. (Answer: prepaid $586.10; forward $681.51; present value of forward payment $586.10)