The value of a forward rate agreement
1. The idea
A forward rate agreement, or FRA, fixes today the interest rate for a future period on an amount, the notional, that itself never changes hands. One side receives a fixed rate \(R_K\) and pays the floating rate that will be set for the period, such as a three-month interbank rate; the other side does the opposite. Only the difference between the two rates on the notional is paid.
An FRA is valued as if the floating rate will turn out equal to today’s forward rate for the period, and its value is the present value of the payment that would then be made at the period’s end. FRA rates are quoted as simple yearly rates, applied over the period: over \(\Delta t\) years, a rate \(R\) grows money by \(1 + R\,\Delta t\). So the forward rate from zero rates, with continuous compounding, is first turned into one:
\[ R_F = \frac{e^{f\Delta t} - 1}{\Delta t}, \qquad V = (R_K - R_F)\, L\, \Delta t\, e^{-z_2 T_2}, \]
where \(f\) is the forward rate with continuous compounding, \(\Delta t = T_2 - T_1\) the period’s length in years, \(R_F\) the forward rate as a simple yearly rate, \(L\) the notional, and \(z_2\) the zero rate for the period’s end \(T_2\), all rates as decimals. \(V\) is the value to the side that receives fixed; the side that pays fixed has \(-V\). In practice an FRA is usually settled at the start of its period, with the payment discounted to that date; its value today is the same. Section 4 shows that payment. Some exercises say that the continuous rates may be used as they are; then use \(f\) in place of \(R_F\). Follow the convention the question states.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The value on a chart
In the question, the forward rate for the second year is \((0.035 \times 2 - 0.03 \times 1)/1 = 0.04\) with continuous compounding, which is \(e^{0.04} - 1\), or 4.0811% as a simple yearly rate. We receive 4.75%, so if the floating rate turns out at 4.0811%, we receive 4.75% − 4.0811% = 0.6689% of $2,000,000, about $13,378, at the end of year 2. Its present value, discounted with the two-year zero rate, is about \(\$13{,}378 \times e^{-0.035 \times 2} = \$12{,}474\).
Now move the fixed rate. The FRA is worth $0 to both sides at the forward rate, and what one side gains, the other loses.
4. One worked example, then your turn
In the worked example, we pay fixed for a period of three months.
Now one with hints, one step at a time.
Settled at the start of the period. Most FRAs are settled when their period starts, as soon as the floating rate for it, \(R_M\), a simple yearly rate, is set. The amount that would be due at the end, \((R_K - R_M)\,L\,\Delta t\) to the side that receives fixed, is paid at the start instead, discounted with that same rate for the period:
\[ \text{payment at the start} = \frac{(R_K - R_M)\, L\, \Delta t}{1 + R_M\, \Delta t}. \]
Then an FRA settled at the start of its period, with hints.
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 as the side that receives fixed, one as the side that pays fixed, and one for an FRA settled at the start of its period. 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: