The forward price of a share
1. The idea
A share bought today costs its spot price, \(S_0\). With a forward contract (Level 1) we can instead agree today to buy it on a later date, at today’s forward price: the delivery price at which a new forward costs nothing to enter. The seller of the forward can buy the share today with borrowed money, hold it until delivery and repay the loan with interest at the risk-free rate, the rate at which money can be borrowed and lent without risk. That interest is the cost of holding the share until delivery, its cost of carry. A forward price above the spot price plus the cost of carry would give the seller a riskless profit, an arbitrage, and one below it would give the buyer one, as the lesson on arbitrage shows. So the forward price is the spot price grown at the risk-free rate until delivery:
\[ F_0 = S_0\, e^{rT}, \]
where \(F_0\) is the forward price, \(S_0\) the spot price, \(r\) the risk-free rate a year, as a decimal, and \(T\) the time to delivery in years. The rate has continuous compounding: interest is added at every moment, so that $1 grows to \(e^{rT}\) dollars in \(T\) years, where \(e \approx 2.71828\); compounding in Level 3 shows where this comes from. This holds for a share that pays no dividends before delivery. The lessons on the forward price with dividends and the cost of carry add dividends, and storage costs for commodities such as oil and gold.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The forward price on a chart
In the question, the seller of the forward borrows $64 for nine months, buys the share and repays the loan with interest at delivery: \(\$64 \times e^{0.04 \times 0.75} = \$65.95\). So the forward price is $65.95, and the $1.95 of interest is the cost of carry.
Now move the time to delivery. The longer the wait, the more interest is added.
4. One worked example, then your turn
Now the same steps 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 delivery in less than a year and one for delivery in a year or more. 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: