Pricing a forward rate agreement from two deposit rates, settling it against the fixing, and showing why the discounted payment locks the borrower's rate exactly.
An FRA settles in cash at the start of the interest period: notional times the difference between the reference rate fixing and the contract rate, times the days in the period over 360, discounted back at the fixing for the same period. On an illustrative 50,000,000 euro 3x9 FRA struck at 2.39 per cent, a six-month Euribor fixing of 2.85 per cent makes the seller pay the buyer 114,626. Undiscounted, the figure would be 116,278, and the borrower who bought the FRA would be overpaid.
Worked in full in Derivatives by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A company will draw a 50,000,000 six-month loan in three months' time at six-month Euribor plus 1.50 per cent. It wants to fix the rate today, so it buys a 3x9 FRA: it agrees to pay a fixed rate on a notional 50,000,000 for the period from month 3 to month 9 and to receive the Euribor fixing. No notional changes hands; only the difference is settled. All rates are illustrative.
| Input | Value |
|---|---|
| Notional | 50,000,000 |
| Deposit rate, 91 days (today to month 3) | 2.10% |
| Deposit rate, 273 days (today to month 9) | 2.30% |
| FRA period | 182 days |
| Day count | actual/360 |
| Loan margin | 1.50% |
| Six-month Euribor fixing at month 3 | 2.85% |
The FRA rate is the forward rate that makes borrowing for 91 days and rolling for 182 cost the same as borrowing for 273 days. If it did not, a dealer could lock in a riskless profit.
F = [(1 + rlong × dlong/360) / (1 + rshort × dshort/360) − 1] × 360 / (dlong − dshort)
= [1.017442 / 1.005308 − 1] × 360 / 182 = 0.012069 × 1.978 = 2.3873%
Excel: =((1+2.3%*273/360)/(1+2.1%*91/360)-1)*360/182
The contract is struck at 2.39 per cent. A dealer would add a few basis points of spread; it is left out here so the arithmetic is visible.
Three months later six-month Euribor fixes at 2.85 per cent, 46 basis points above the contract rate. The interest difference for the period is:
Difference at period end = 50,000,000 × (2.85% − 2.39%) × 182 / 360 = 116,277.78
Settlement = 116,277.78 / (1 + 2.85% × 182 / 360) = 116,277.78 / 1.014408 = 114,626
Excel: =50000000*(2.85%-2.39%)*182/360/(1+2.85%*182/360)
The discount uses the fixing itself, not the contract rate, because the fixing is the rate at which the recipient can invest the payment until the end of the period, when the loan interest falls due.
The loan costs 50,000,000 × (2.85% + 1.50%) × 182 / 360 = 1,099,583 of interest. The FRA payment of 114,626, invested at 2.85 per cent for 182 days, grows to 116,278. Net cost: 983,306, or 3.89 per cent a year, which is the contract rate of 2.39 per cent plus the 1.50 per cent margin to the basis point. The discounting is what makes the hedge exact.
| Fixing | FRA settlement to buyer | Unhedged interest | Hedged net interest |
|---|---|---|---|
| 1.85% | −135,235 | 846,806 | 983,306 |
| 2.10% | −72,535 | 910,000 | 983,306 |
| 2.39% | 0 | 983,306 | 983,306 |
| 2.60% | 52,395 | 1,036,389 | 983,306 |
| 2.85% | 114,626 | 1,099,583 | 983,306 |
| 3.10% | 176,703 | 1,162,778 | 983,306 |
| 3.35% | 238,625 | 1,225,972 | 983,306 |
Every row ends at 983,306. Below 2.39 per cent the company pays on the FRA and would have been better unhedged, by 136,500 at a 1.85 per cent fixing. That is not a cost of the hedge; it is the certainty it bought. Whether a treasury report shows the 135,235 paid as a loss or the fixed 983,306 as the outcome decides how the hedge is judged, and only the second describes what the company set out to do.
If, just after the trade, the forward rate for the same six-month period rises to 2.60 per cent, the FRA is worth the settlement it would produce at that rate, 52,395 at the settlement date, discounted to today over the 91 days to settlement. With a discount factor of 0.994720, kept at its original level for illustration, that is 52,118. Later in the life of the trade the same method applies with the shorter time to settlement. Each basis point on the fixing moves the settlement by about 2,492.
An FRA hedges one period. A borrower with a five-year floating loan needs a strip of FRAs or, more simply, a swap; the two are priced from the same forward rates, as worked in how to value an interest rate swap.
The book's own hedges, a forward, a swap and a collar, are reproduced line by line in the free workbook for this case.
Because it is paid at the start of the interest period, while the interest it compensates is paid at the end. The difference of 116,278 at the end of the period is worth 116,278 / 1.014408 = 114,626 on the settlement date. Paying the undiscounted amount would overpay the buyer by 1,652, the interest the buyer would earn on the payment over the 182 days.
The seller pays the buyer. The buyer of an FRA is the party that pays the fixed contract rate, typically a borrower hedging a future floating-rate period. With the fixing at 2.85 per cent against a contract rate of 2.39 per cent the seller pays 114,626; at a fixing of 1.85 per cent the buyer would pay 135,235.
From the two deposit or zero rates spanning the period: F = ((1 + r_long x d_long/360) / (1 + r_short x d_short/360) - 1) x 360 / (d_long - d_short). With 2.10 per cent for 91 days and 2.30 per cent for 273 days, the 3x9 rate is 2.3873 per cent, quoted here as 2.39 per cent.
This article is one calculation from Derivatives. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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