Forward rates from zero rates
1. The idea
A zero rate is the rate on money lent from today until one date. Two zero rates for different dates also fix a rate for the period between those dates. Compare two ways to lend money until the later date:
- Lend it from today until the later date.
- Lend it until the earlier date, then lend it again until the later date, at a rate agreed today.
Both must end with the same amount, or one of them would be an arbitrage. The rate agreed today that makes them match is the forward rate. The rate that is actually on offer at the earlier date may turn out different.
\[ f = \frac{z_2 T_2 - z_1 T_1}{T_2 - T_1}, \]
where \(z_1\) and \(z_2\) are the zero rates for the dates \(T_1\) and \(T_2\), in years, with continuous compounding, as decimals, and \(f\) is the forward rate for the period from \(T_1\) to \(T_2\), with continuous compounding too. When the zero rates rise with the date, the forward rate is above both of them; when they fall, it is below both.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The forward rate on a chart
In the question, money lent for 12 months grows by the factor \(e^{0.03 \times 1} = e^{0.03}\), and money lent for 6 months by \(e^{0.026 \times 0.5} = e^{0.013}\). Over the 6 months between them, money grows by the factor \(e^{0.03 - 0.013} = e^{0.017}\), a return of \(\ln(e^{0.017}) = 0.017\) with continuous compounding, where \(\ln\) is the natural logarithm. Over 0.5 years, that is \(0.017/0.5 = 0.034\) a year: a forward rate of 3.40%, above both zero rates, as they rise.
Now move the zero rate for 12 months. Above 2.6% the zero rates rise and the forward rate is above both; below it they fall and the forward rate is below both.
4. One worked example, then your turn
The worked example is a one-year period, from year 3 to year 4.
Now the same steps with hints, one step at a time.
And one on your own, with falling zero rates.
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 one-year period and one for a period of three or six months. 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: