Why Dividends Change a Forward Price
A forward contract on a share requires the buyer to pay an agreed delivery price and receive the share at a future date. A buyer who instead purchases the share today receives any dividends paid while holding it. The forward buyer does not own the share before delivery and therefore does not receive those interim payments. To compare the two strategies fairly, the value of dividends paid before delivery must be allowed for in the forward price.
This tutorial considers known discrete cash dividends: fixed dollar amounts paid on specified dates. It does not treat dividends as a continuous yield. The pricing result is aligned with the SOA Exam FM (Financial Mathematics) syllabus.
Times \(t_j\) and \(T\) are measured in years. If a payment occurs halfway through a year, its present value uses the fractional power \(v^{0.5}\), not a full-year discount factor. The formula also assumes that the contract delivers a share and that its holder does not receive the pre-delivery dividends.
Why the Dividend Adjustment Works
The key is to compare the forward with a strategy that will deliver the same share at time \(T\). Buying a share today costs \(S_0\), but the owner also receives the specified dividends. At the valuation date, those future dividend payments have present value \(\sum_j D_jv^{t_j}\). Thus, the share stripped of its pre-delivery dividends has a net cost of \(S_0-\sum_jD_jv^{t_j}\).
Discount each known payment from its own payment date: \(D_jv^{t_j}\). Add these present values.
Subtract the dividend present value from \(S_0\). The resulting amount is the net cost associated with receiving the share at \(T\), with the interim dividends separated out.
Under the assumed financing rate, the net cost at time \(T\) is \(\left(S_0-\sum_jD_jv^{t_j}\right)(1+i)^T\). This is the fair delivery price.
This argument gives a short derivation of the formula. If the forward price were higher than the cost of acquiring and financing the dividend-adjusted share, a trader could use the cheaper strategy to deliver a share into the forward. If it were lower, the reverse comparison would favor the forward. In either case, the imbalance would create an arbitrage under the stated assumptions. Therefore, the fair price is the accumulated dividend-adjusted cost.
An equivalent form can make timing easier to see. Accumulate the no-dividend forward price \(S_0(1+i)^T\), then subtract each dividend accumulated from its payment date to delivery:
The two forms agree because \(D_jv^{t_j}(1+i)^T=D_j(1+i)^{T-t_j}\). The present-value form is often convenient when a problem gives dividend discount factors; the accumulated-dividend form makes it especially clear that earlier dividends receive more time to accumulate.
Pricing a Forward with Known Dividends
Worked Example: Two Dividends Before Delivery
A share is worth $100 today. It will pay a known dividend of $4 at the end of year 1 and $5 at the end of year 2. A forward delivers one share at the end of year 3. Assume a flat effective annual financing rate of 10%, compounded annually, and no taxes or transaction costs. Find the fair forward price.
Method. Discount each dividend to today, subtract their total present value from the share price, and accumulate the remainder for three years.
The discount factor is \(v=1/1.10\). The present values of the payments are $$ 4v^1=\frac{4}{1.10}=3.6363636\ldots, \qquad 5v^2=\frac{5}{1.10^2}=\frac{5}{1.21}=4.1322314\ldots. $$
Their total present value is \(3.6363636\ldots+4.1322314\ldots=7.7685950\ldots\). The dividend-adjusted share cost is therefore \(100-7.7685950\ldots=92.2314050\ldots\). Accumulating it to year 3 gives $$ F_{0,3}=92.2314050\ldots(1.10)^3 =92.2314050\ldots(1.331) =122.76. $$
Check. The no-dividend forward price would be \(100(1.10)^3=133.10\). The year-1 dividend grows for two years to \(4(1.10)^2=4.84\), while the year-2 dividend grows for one year to \(5(1.10)=5.50\). Thus \(133.10-4.84-5.50=122.76\), agreeing with the result. The fair forward price is $122.76.
Common mistake avoided: Subtracting the cash dividends without accounting for when they are paid. For example, \(133.10-4-5=124.10\) is incorrect: each dividend must be accumulated from its payment date to delivery before it is subtracted from the no-dividend forward price.
Check Each Payment Against the Delivery Date
A dividend belongs in the adjustment only if it is paid before delivery under the contract’s timing convention. A dividend paid after delivery is received by whoever owns the share after the forward settles, so it does not reduce the forward price. If a problem places a payment exactly on the delivery date, check the stated entitlement and settlement convention rather than silently treating it as a pre-delivery payment.
This timing rule is important even when a dividend amount is given alongside the other payments. The formula does not include every dividend mentioned in a scenario; it includes only the payments the share owner receives before the forward buyer takes delivery.
Worked Example: Excluding a Dividend Paid After Delivery
A share is worth $250 today. It pays $5 at the end of year 1, $7 at the end of year 3, and $12 immediately after delivery at the end of year 4. A forward delivers one share at the end of year 4. Assume a flat effective annual financing rate of 5%, compounded annually, with no taxes or transaction costs. Find the fair forward price.
Method. Include the first two dividends because they are paid before delivery. Exclude the $12 payment because it occurs immediately after delivery.
The no-dividend forward price is $$ 250(1.05)^4=250(1.21550625)=303.8765625. $$ The year-1 dividend accumulates for three years, so its delivery-date value is $$ 5(1.05)^{4-1}=5(1.157625)=5.788125. $$ The year-3 dividend accumulates for one year, so its delivery-date value is $$ 7(1.05)^{4-3}=7(1.05)=7.35. $$
Subtracting these two adjustments gives $$ F_{0,4}=303.8765625-5.788125-7.35=290.7384375. $$ Rounded to the nearest cent, the fair forward price is $290.74.
Check. Using present values, the dividends to include are \(5/1.05\) and \(7/(1.05)^3\). Subtracting their sum from $250 and accumulating the remainder for four years gives the same price. Adding the $12 payment to the dividend adjustment would be a timing error: the forward buyer receives the share before that payment is made.
Common mistake avoided: Including a dividend merely because it is listed in the problem. The payment time, not just the payment amount, determines whether it affects the forward price.
Solving for an Unknown Dividend
The same relation can be rearranged if a fair forward price is given and a dividend amount is unknown. First calculate the no-dividend forward price, then subtract the delivery-date value of each known pre-delivery dividend. The difference between that amount and the quoted fair price must be the delivery-date value of the unknown dividend. This is a useful check that the dividend’s timing has been applied consistently.
Worked Example: Inferring a Dividend from a Fair Forward Price
A share is worth $200 today. A known dividend of $6 will be paid at the end of year 1, and an unknown cash dividend \(D\) will be paid at the end of year 2. A fair forward delivers the share at the end of year 3 for $216.51. Assume a flat effective annual financing rate of 5%, compounded annually, with no taxes or transaction costs. Find \(D\).
Method. Use the accumulated-dividend form and isolate the unknown payment’s delivery-date value.
Without dividends, the forward price would be $$ 200(1.05)^3=200(1.157625)=231.525. $$ The year-1 dividend grows for two years, so its value at delivery is $$ 6(1.05)^2=6(1.1025)=6.615. $$ The unknown year-2 dividend grows for one year, so the fair-price equation is $$ 216.51=231.525-6.615-1.05D. $$
Rearrange step by step: $$ 1.05D=231.525-6.615-216.51=8.40, \qquad D=\frac{8.40}{1.05}=8. $$ The unknown dividend is $8.00.
Check. Substitution gives \(231.525-6.615-8(1.05)=231.525-6.615-8.40=216.51\), exactly the stated fair forward price. The common error here is to subtract \(D\) as though it were paid at delivery; because it is paid one year earlier, its delivery-date adjustment is \(1.05D\).
Practical Checks and Exam Traps
A reliable calculation keeps three dates separate: the valuation date, each dividend payment date, and the delivery date. The dividend is discounted from its payment date to the valuation date in the prepaid-cost form, or accumulated from its payment date to delivery in the equivalent form. Both methods give the same answer when the periods are measured consistently.
- Use the right period. A dividend paid at \(t_j\) is accumulated for \(T-t_j\) years, not for \(T\) years.
- Do not subtract an undiscounted dividend from today’s share price. Convert each future payment to a present value before subtracting it from \(S_0\).
- Exclude payments after delivery. The forward buyer receives the share only at delivery and is not entitled to a later dividend under the assumptions here.
- Use the stated rate convention. This formula uses a flat effective annual rate and year-based times. For a fractional year, use the corresponding fractional power.
- Check both forms when useful. The prepaid-cost expression and the accumulated-dividend expression should give the same forward price.
These steps distinguish a discrete cash dividend from an income yield and make the timing adjustment explicit. The result is the fair delivery price for a newly initiated forward under the stated assumptions; it does not by itself give the value of an existing contract.
Check Your Understanding
For each problem, assume known cash dividends, a flat effective annual financing rate compounded annually, and no taxes or transaction costs. Round dollar answers to the nearest cent unless an exact amount is requested.
- A share is worth $90. It pays $3 at the end of year 1, and a forward delivers the share at the end of year 2. The financing rate is 4%. Find the fair forward price. (Answer: $94.22)
- A share is worth $160. It pays $4 at the end of year 1 and $6 at the end of year 2. A forward delivers at the end of year 3, and the financing rate is 5%. Find the fair forward price. (Answer: $174.51)
- A share is worth $75. It pays $2 at the end of year 1 and $5 at the end of year 2. A forward delivers at the end of year 3, and the financing rate is 8%. Find the fair forward price. (Answer: $86.75)
- A share is worth $300. It pays $8 at the end of year 1 and $10 at the end of year 3. A forward delivers at the end of year 4, and the financing rate is 4%. Find the fair forward price. (Answer: $331.56)
- A share is worth $100. An unknown dividend is paid at the end of year 1, and a forward delivers at the end of year 2 for a fair price of $115.50. The financing rate is 10%. Find the dividend. (Answer: $5.00)