Explain how the rate on an interest rate forward contract is fixed
The idea
An interest rate forward, or forward rate agreement, fixes today a rate that will apply to a future period on a notional principal. The fixed rate is set so that the contract has zero value at initiation, which means it must equal the implied forward rate from today's spot rates; any other rate would allow an arbitrage by borrowing for one period and lending for another. With annual compounding, the one-year rate one year forward is (1 + S2)² / (1 + S1) - 1. For money-market periods, the forward rate for days a to b is [(1 + Rb x b/360) / (1 + Ra x a/360) - 1] x 360 / (b - a). A long FRA pays the fixed rate and receives the floating rate, so it gains when rates rise and is used by a future borrower to lock in borrowing costs; a short FRA suits a future lender. The settlement payment is the rate difference on notional for the period, discounted at the reference rate.
Formulas
Forward rate from two money-market rates
F = [(1 + R_b × b ÷ 360) ÷ (1 + R_a × a ÷ 360) − 1] × 360 ÷ (b − a)
- F
- the forward rate
- R_b
- the rate for the longer period
- R_a
- the rate for the shorter period
- a
- the days to the start of the forward period
- b
- the days to its end
Money-market quotes use a 360-day year.
A practice question
The fixed rate in a forward rate agreement is set at initiation so that:
- the long party is certain to gainWhy not: No-arbitrage pricing gives neither side an advantage at the outset.
- the contract has zero value to both sides - correct
- it equals today's spot rate for the periodWhy not: The rate must match the implied forward rate, not the current spot rate.
At initiation the FRA rate equals the implied forward rate from the spot curve, so no money changes hands and neither party starts with value.
A harder one
The one-year spot rate is 3.0% and the two-year spot rate is 4.0%, both annually compounded. What is the implied one-year rate one year from now?
- 3.50%Why not: Averages the two spot rates, which gives something between them rather than the forward.
- 4.00%Why not: Uses the two-year spot rate, ignoring that the first year earns only 3%.
- 5.01% - correct
Forward = 1.04² / 1.03 - 1 = 1.0816 / 1.03 - 1 = 5.01%. Investing for two years at 4% must equal one year at 3% followed by one year at the forward rate.
The hardest tier
A long 3x6 FRA on 10 million notional has a fixed rate of 3.0%. At settlement the 90-day reference rate is 3.5% (360-day basis). What does the long receive at settlement?
- 12,500Why not: Correct interest difference, but not discounted back from the end of the 90-day period.
- 12,392 - correct
- 50,000Why not: Applies the full annual rate difference, omitting the 90/360 day-count fraction.
Interest difference = (0.035 - 0.030) x 90/360 x 10,000,000 = 12,500, paid at the start of the period, so discount it: 12,500 / (1 + 0.035 x 90/360) = 12,392.
In the app: 9 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.
Take the free diagnostic →