Fixing a rate today for a loan that starts in two years: the formula, the check against the spot curve, and why the gap over today's rate is not a fee.
A forward-starting swap rate is the discount factor at the start date minus the discount factor at the end date, divided by the sum of the discount factors on the fixed payment dates in between. On an illustrative upward-sloping curve where today's five-year swap rate is 3.39 per cent, a five-year swap starting in two years fixes at 3.72 per cent, 33 basis points higher. That gap is not a fee for waiting: it is the rate the curve already implies for those years.
Worked in full in Derivatives by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The question usually comes from a treasurer with a loan that does not exist yet: a refinancing in two years, a construction facility that converts to a term loan, a bond issue planned for next spring. A forward-starting payer swap fixes the rate today for a period that begins later. The first quote is a shock, because it is above the spot swap rate everyone watches, and the instinct is to read the difference as the bank's margin.
| Year | Zero rate | Discount factor | One-year forward rate |
|---|---|---|---|
| 1 | 3.00% | 0.9709 | 3.00% |
| 2 | 3.10% | 0.9408 | 3.20% |
| 3 | 3.20% | 0.9098 | 3.40% |
| 4 | 3.30% | 0.8782 | 3.60% |
| 5 | 3.40% | 0.8461 | 3.80% |
| 6 | 3.48% | 0.8144 | 3.88% |
| 7 | 3.55% | 0.7833 | 3.97% |
The hedge: a payer swap on £50,000,000, starting in two years, running five years, annual fixed payments. Single-curve pricing and no credit or funding charge, so that the rate shown is the mid-market rate before any dealer spread.
Discount factor for year t: DFt = 1 ÷ (1 + zt)t. For year 2: 1 ÷ 1.0312 = 0.9408. For year 7: 0.7833.
Forward swap rate = (DFstart − DFend) ÷ Σ DFt over the fixed payment dates, years 3 to 7.
Numerator: 0.9408 − 0.7833 = 0.1574. Annuity: 0.9098 + 0.8782 + 0.8461 + 0.8144 + 0.7833 = 4.2319.
Forward swap rate = 0.1574 ÷ 4.2319 = 3.720%.
In Excel, with discount factors for years 1 to 7 in B2:B8: =(B3-B8)/SUM(B4:B8). For a spot-starting swap the numerator starts from 1, so today's five-year rate is =(1-B6)/SUM(B2:B6) = 3.387%.
The formula is the same one that gives the fixed rate on an ordinary swap, with the first discount factor moved from today to the start date. The floating leg of a swap from year 2 to year 7 is worth DF2 − DF7 per unit of notional; the fixed rate is whatever makes the fixed leg worth the same.
The forward swap rate is a weighted average of the one-year forward rates for years 3 to 7, each weighted by its discount factor: 21.5, 20.8, 20.0, 19.2 and 18.5 per cent of the annuity on 3.40, 3.60, 3.80, 3.88 and 3.97 per cent. The weighted average is 3.720 per cent, the same number. Today's five-year swap averages the forwards for years 1 to 5, which include the two cheapest years on the curve, 3.00 and 3.20 per cent. The forward-starting swap drops those two years and adds years 6 and 7.
A consistency check that dealers use: a seven-year swap is a two-year swap followed by the two-into-five forward swap, weighted by their annuities. (3.098% × 1.9116 + 3.720% × 4.2319) ÷ 6.1435 = 3.527 per cent, which is the spot seven-year rate. If a forward quote fails that test by more than the spread, ask why.
| Start | Fixed rate | Above spot five-year, bp |
|---|---|---|
| Today | 3.39% | 0.0 |
| In 1 year | 3.56% | 17.7 |
| In 2 years | 3.72% | 33.3 |
On £50,000,000 the PV01 of the forward swap, the change in value for one basis point, is £21,159, so the 33 basis point gap is worth £705,595 in present value, or £166,733 a year of extra fixed interest over the five years. That is the figure a board will ask about. It is not a cost in the sense of money lost: if the curve's forwards turn out right, floating would have cost the same.
| Curve, zero rates years 1 to 7 | Spot 5y | 2y forward 5y | Difference, bp |
|---|---|---|---|
| Flat at 3.30% | 3.30% | 3.30% | 0 |
| Base case, 3.00% to 3.55% | 3.39% | 3.72% | 33 |
| Steeper, 2.80% to 3.90% | 3.57% | 4.24% | 67 |
| Inverted, 4.00% to 3.40% | 3.49% | 3.24% | −25 |
The sign of the gap is the slope of the curve. On an inverted curve the forward-starting swap is cheaper than the spot one. Either way, a treasurer who waits to fix "until the loan starts" is not saving a premium: he is betting that the rate at the start date will come in below the forward the curve already implies.
What the hedge does when the start date arrives, assuming for simplicity a flat curve at the new five-year rate:
| Five-year rate in 2 years | Value of the payer swap, £ | Net effect |
|---|---|---|
| 3.00% | −1,648,928 | Loan cheaper, swap pays away the gain |
| 3.72% | 0 | Break-even |
| 4.00% | 623,029 | Swap offsets the dearer loan |
| 4.50% | 1,711,867 | Swap offsets the dearer loan |
In every row the all-in fixed rate is 3.72 per cent plus the loan margin. The swap value is the price of certainty, and on an exposure that does not yet exist it is also a collateral question: a mark of minus £1,648,928 under a CSA is cash posted before a single pound has been borrowed.
The common mistake is comparing the forward quote with today's five-year rate and calling the 33 basis points the cost of the delay, or the dealer's margin. Neither is right. The dealer's charge is the spread between the quote and the mid rate computed above, typically a few basis points plus a credit charge. The rest is the curve. The same confusion appears in FX, where forward points are mistaken for a hedging cost; it is set out in how to calculate an FX forward rate. Check a quote by rebuilding the mid from the curve, then look at what is left.
Forward swap rate = (DFstart − DFend) ÷ Σ DF on the payment dates: 3.72 per cent here against 3.39 per cent spot. The difference is the slope of the curve, priced, not a premium paid. The book's swap, its year-by-year cash flows at the curve's own forwards and the three real cost lines of a hedge are in the free workbook for this case.
Because on an upward-sloping curve the years it covers have higher implied forward rates. A two-into-five swap averages the one-year forwards for years 3 to 7, here 3.40 to 3.97 per cent, while the spot five-year swap includes the cheaper years 1 and 2. The result is 3.72 against 3.39 per cent. On an inverted curve the forward rate is lower.
No. Like an ordinary swap it is struck at the rate that gives it zero value at inception, so nothing is paid upfront apart from the dealer's spread built into the rate. Its value then moves with rates: on 50 million, if the five-year rate is 4.50 per cent at the start date the payer swap is worth about 1,711,867, and at 3.00 per cent about minus 1,648,928.
Put the discount factors for each annual payment date in a column, say years 1 to 7 in B2:B8. For a swap starting in year 2 and ending in year 7, use =(B3-B8)/SUM(B4:B8). With the illustrative curve that returns 3.720 per cent. Replace B3 with 1 for a swap that starts today.
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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