Discount factors bootstrapped from four par rates, both legs valued, the annuity shortcut, and why projecting today's fixing gets 43 per cent of the value wrong.
An interest rate swap is valued as the present value of the floating leg minus the present value of the fixed leg, both discounted with factors bootstrapped from today's swap curve. A company paying 4.00 per cent fixed on 100,000,000 with four years left, when the four-year par swap rate has risen to 4.70 per cent, holds a swap worth 2,503,825 in its favour. The shortcut gives the same number: the 0.70 per cent rate gap times the annuity of 3.5769 times the notional.
Worked in full in Derivatives by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A year ago a borrower swapped a floating-rate loan to fixed for five years at 4.00 per cent, paying annually. Rates have risen since. The swap is valued today, on a payment date, so no interest has accrued and four annual periods remain. All rates are illustrative, and the same curve is used to project and to discount, which keeps the arithmetic visible; a dealer would discount on an overnight index curve and project on the curve of the floating index.
| Input | Value |
|---|---|
| Notional | 100,000,000 |
| Fixed rate paid | 4.00% |
| Remaining term | 4 years, annual |
| Par swap rate, 1 year | 4.40% |
| Par swap rate, 2 years | 4.55% |
| Par swap rate, 3 years | 4.65% |
| Par swap rate, 4 years | 4.70% |
A par swap is worth zero, so each par rate pins down one more discount factor, given the ones before it:
DFn = (1 − Sn × (DF1 + … + DFn−1)) / (1 + Sn)
DF1 = 1 / 1.0440 = 0.957854
DF2 = (1 − 0.0455 × 0.957854) / 1.0455 = 0.914794
Excel, with par rates in B2:B5 and factors in C2:C5: =(1-B3*SUM($C$2:C2))/(1+B3), filled down.
| Year | Par rate | Discount factor | Zero rate | One-year forward |
|---|---|---|---|---|
| 1 | 4.40% | 0.957854 | 4.4000% | 4.4000% |
| 2 | 4.55% | 0.914794 | 4.5534% | 4.7071% |
| 3 | 4.65% | 0.872357 | 4.6571% | 4.8647% |
| 4 | 4.70% | 0.831886 | 4.7090% | 4.8650% |
The sum of the four factors, the annuity, is 3.576892: the present value of 1 a year for four years on this curve.
PV fixed leg = N × fixed rate × annuity = 100,000,000 × 4.00% × 3.576892 = 14,307,569
PV floating leg = N × (1 − DF4) = 100,000,000 × (1 − 0.831886) = 16,811,393
Value to the fixed payer = 16,811,393.18 − 14,307,568.67 = 2,503,825
Shortcut: (4.70% − 4.00%) × 3.576892 × 100,000,000 = 2,503,825
The floating leg formula works because a floating-rate note resets to par: paying the floating coupons and the notional at maturity is worth the notional today, so the coupons alone are worth the notional minus the discounted notional. The shortcut works because the four-year par rate is by definition the fixed rate that would make the swap worth zero today; the old swap is worth the difference in rate, paid as an annuity.
A treasurer who wants to see where the value sits can project each floating payment at its forward rate and discount the net settlement.
| Year | Forward rate | Floating received | Fixed paid | Net | Present value |
|---|---|---|---|---|---|
| 1 | 4.4000% | 4,400,000 | 4,000,000 | 400,000 | 383,142 |
| 2 | 4.7071% | 4,707,061 | 4,000,000 | 707,061 | 646,815 |
| 3 | 4.8647% | 4,864,665 | 4,000,000 | 864,665 | 754,297 |
| 4 | 4.8650% | 4,864,987 | 4,000,000 | 864,987 | 719,571 |
| Total | 2,503,825 |
Most of the value sits in the later years, because the forwards there are highest.
The swap's sensitivity is the annuity times the notional: a DV01 of about 35,769 for each basis point of parallel shift.
| Shift | 4-year par rate | Swap value |
|---|---|---|
| −100 bp | 3.70% | −1,098,621 |
| −50 bp | 4.20% | 723,815 |
| 0 | 4.70% | 2,503,825 |
| +50 bp | 5.20% | 4,242,597 |
| +100 bp | 5.70% | 5,941,285 |
The steps are not equal: the gain from +50 basis points is a little smaller than the loss from −50, because higher rates shrink the annuity that the rate gap is multiplied by. A 100 basis point fall turns the asset into a liability of 1,098,621, which, before thresholds and minimum transfer amounts, is the collateral a counterparty would call under a margin agreement. How that call compares across hedge tenors is worked in how much more margin a five-year hedge calls.
The frequent shortcut is to assume the floating rate stays at today's one-year rate, 4.40 per cent, for the life of the swap, and discount the 400,000 a year. That gives 1,430,757, which understates the value by 1,073,068, or 42.9 per cent, because it ignores the upward slope of the curve: the market already expects floating payments of nearly 4.87 per cent in years three and four. The forward curve, not the current fixing, is what a swap is worth against. The value is also before any credit adjustment; the counterparty risk on a swap in the money is priced in how to calculate CVA on an interest rate swap.
Both yield curves in the free workbook for this case are bootstrapped from their par rates the same way, and the swap is run at the curve's own implied forwards.
For a swap valued on a payment date, the value to the fixed payer is the current par swap rate for the remaining term minus the contract fixed rate, times the annuity (the sum of discount factors), times the notional. In the example (4.70 - 4.00 per cent) x 3.5769 x 100,000,000 = 2,503,825.
A floating-rate note that resets to the market rate is worth par on each reset date. Its coupons are therefore worth par minus the present value of the final repayment of principal. With a four-year discount factor of 0.831886, the floating coupons on 100,000,000 are worth 16,811,393.
Approximately the annuity times the notional times 0.0001, the DV01. Here 3.576892 x 100,000,000 x 0.0001 is about 35,769 per basis point. The relationship is not exactly linear: a 50 basis point rise adds 1,738,773 while a 50 basis point fall removes 1,780,009.
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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