The forward price with dividends
1. The idea
The buyer of a forward does not receive the dividends paid before delivery; whoever holds the share does. So the seller of the forward, who buys the share today and holds it until delivery, collects the dividends, and they pay back part of the cost of carry. That lowers the forward price: subtract the dividends’ value today, their present value, from the spot price, then grow the rest at the risk-free rate until delivery:
\[ \begin{aligned} I &= D\, e^{-rt},\\ F_0 &= (S_0 - I)\, e^{rT}, \end{aligned} \]
where \(F_0\) is the forward price, \(S_0\) the spot price, \(D\) the dividend paid at time \(t\), \(I\) its present value, the amount that grows to \(D\) by time \(t\) at the risk-free rate (the factor \(e^{-rt}\) that turns \(D\) into \(I\) is a discount factor), \(T\) the time to delivery, both times in years, and \(r\) the risk-free rate with continuous compounding, as a decimal. With several dividends, \(I\) is the sum of their present values.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The dividend on a chart
In the question, the seller of the forward can buy the share for $50, collect the $2 dividend after 3 months and deliver the share after 9. The dividend’s present value is \(I = 2 \times e^{-0.06 \times 0.25} = 1.9702\). Grow the rest until delivery: \((50 - 1.9702) \times e^{0.06 \times 0.75} = \$50.24\).
Now move the size of the dividend. Each dollar of dividend lowers the forward price by a little more than a dollar, as the dividend could earn interest until delivery.
4. One worked example, then your turn
Now a share with two dividends, with hints, one step at a time.
And one on your own.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer for a share with one dividend and one for a share with two. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: