The par swap rate built from five forward rates, why it is a discount-weighted average and not a simple one, the net cash flows it implies, and what the dealer's spread is worth.
The fixed rate on a new swap is the par rate, the rate that makes the swap worth zero: (1 − final discount factor) ÷ sum of the discount factors. On one-year forward rates of 4.40, 4.60, 4.75, 4.85 and 4.90 per cent, a five-year annual swap's fixed rate is 4.6883 per cent. That is the discount-weighted average of the forwards, a little below their simple average of 4.70.
Worked in full in Derivatives by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A borrower swapping a floating loan to fixed agrees to pay one rate for the life of the swap against a floating rate that will change every year. The fixed rate is not a forecast and not a negotiated number. It is pinned down by the forward curve: any other rate would hand value to one side on the day the swap is signed.
Illustrative rates. To keep the arithmetic visible, the same curve projects the floating rate and discounts the cash flows, and both legs pay annually; a dealer would discount on an overnight index curve.
| Input | Value |
|---|---|
| Notional | 100,000,000 |
| Term and payments | 5 years, annual |
| One-year forward rate, year 1 | 4.40% |
| Year 2 | 4.60% |
| Year 3 | 4.75% |
| Year 4 | 4.85% |
| Year 5 | 4.90% |
| Dealer spread on the fixed rate | 2.5 bp |
DFn = DFn−1 ÷ (1 + fn), starting from DF0 = 1
DF1 = 1 ÷ 1.0440 = 0.957854; DF2 = 0.957854 ÷ 1.0460 = 0.915731; and so on
Excel, forwards in B2:B6: C2 =1/(1+B2), then C3 =C2/(1+B3) filled down
| Year | Forward | Discount factor | Forward × DF | Weight in the rate |
|---|---|---|---|---|
| 1 | 4.40% | 0.957854 | 0.042146 | 21.89% |
| 2 | 4.60% | 0.915731 | 0.042124 | 20.92% |
| 3 | 4.75% | 0.874206 | 0.041525 | 19.98% |
| 4 | 4.85% | 0.833768 | 0.040438 | 19.05% |
| 5 | 4.90% | 0.794822 | 0.038946 | 18.16% |
| Sum | 4.376381 | 0.205178 | 100% |
The fixed leg is worth N × K × the sum of the discount factors, the annuity. The floating leg, paying each year's forward, is worth N × Σ fi × DFi, which collapses to N × (1 − DF5) because each term fi × DFi equals DFi−1 − DFi.
K = (1 − DF5) ÷ Σ DFi
= (1 − 0.794822) ÷ 4.376381 = 0.205178 ÷ 4.376381 = 4.6883%
Excel: =(1-C6)/SUM(C2:C6)
The last column of the table shows why it is not 4.70. Each forward counts in proportion to its discount factor, so year 1 at 4.40 per cent carries 21.89 per cent of the weight and year 5 at 4.90 per cent only 18.16. On a rising curve the par rate sits below the simple average, here by 1.17 basis points.
| Year | Floating received | Fixed paid | Net to the payer | Present value |
|---|---|---|---|---|
| 1 | 4,400,000 | 4,688,303 | −288,303 | −276,153 |
| 2 | 4,600,000 | 4,688,303 | −88,303 | −80,862 |
| 3 | 4,750,000 | 4,688,303 | 61,697 | 53,936 |
| 4 | 4,850,000 | 4,688,303 | 161,697 | 134,818 |
| 5 | 4,900,000 | 4,688,303 | 211,697 | 168,261 |
| Total | 0 |
If the forwards are realised, the fixed payer pays more than floating in the first two years and receives it back later. That pattern, not a bet, is what a rising curve does to a new swap, and it is why a swap that is "in the money" at year 3 is not evidence that the hedge was a good call.
| Curve | Simple average of forwards | Par rate | Annuity |
|---|---|---|---|
| Base, rising 4.40 to 4.90 | 4.70% | 4.6883% | 4.3764 |
| Flat at 4.70 | 4.70% | 4.7000% | 4.3656 |
| Inverted, 4.90 down to 4.40 | 4.70% | 4.7113% | 4.3551 |
| Base plus 50 bp on every forward | 5.20% | 5.1871% | 4.3162 |
Three curves with the same average give three different rates: the shape matters, not just the level. Shorter swaps on the base curve fix at 4.4000 per cent for one year, 4.4978 for two, 4.5780 for three and 4.6413 for four.
Client rate: 4.6883% + 2.5 bp = 4.713%
Value to the dealer: 100,000,000 × 0.00025 × 4.376381 = 109,410, or 25,000 a year
The spread is invisible in the rate the client sees, and the same annuity converts it into money. The annuity is also the swap's sensitivity to rates, worked in the DV01 of an interest rate swap.
K = (1 − DFn) ÷ Σ DF: 4.6883 per cent on the illustrative curve, the forwards weighted by their discount factors. Once signed, the same factors value the swap as rates move, worked in valuing an interest rate swap after rates move. The book's workbooks bootstrap both of its curves from ten par rates and price every hedge from them; they are on the free companion page for this book.
Because each forward is weighted by its discount factor, and later payments are worth less today. On forwards rising from 4.40 to 4.90 per cent the weights fall from 21.89 to 18.16 per cent, so the higher later forwards count for less: the par rate is 4.6883 per cent against a simple average of 4.70. On a flat curve the two are equal.
Put the discount factors in a column, say C2:C6, and use =(1-C6)/SUM(C2:C6). Each factor is the previous one divided by (1 + that year's forward). On the worked curve the factors run from 0.957854 to 0.794822, the annuity is 4.376381 and the formula returns 4.6883 per cent.
Notional x spread x annuity. A 2.5 basis point spread added to the par rate on a 100,000,000 five-year swap is 25,000 a year, and 109,410 in present value at the annuity of 4.376381. The client sees a rate of 4.713 per cent and rarely sees the 109,410.
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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