Swap rates: the fixed rate that makes floating worth par
An interest rate swap exchanges fixed for floating on a notional that never moves. At the start it is worth nothing to either side, which pins the fixed rate; afterwards, the annuity turns every basis point of rate difference into dollars.
- Compute the par swap rate from a strip of discount factors
- Bootstrap the next discount factor and zero rate from a par swap rate
- Value an existing swap to the fixed payer, and its DV01 through the annuity
- Choose the right side of a swap to fix a floating-rate loan
The intuition
A two-year swap: you pay a fixed rate each year and receive the floating rate on the same notional. The floating side is easy to value: a note that resets to the market rate every year is always worth par on a reset date, so the floating coupons alone are worth 1 minus the present value of the final principal, 1 − DF₂. The fixed side is the rate times the sum of the discount factors, the annuity. At the start the two must be equal, so the par swap rate is (1 − DF₂) ÷ (DF₁ + DF₂): with one-year and two-year zeros of 3% and 4%, about 3.98%.
Once rates move, the swap is no longer free. If you locked in paying 3.5% and the market rate is now 3.98%, you save 48 basis points a year for two years, and the annuity turns that into dollars: 0.0048 × 1.895 × $100 million is about $910,000 of value to you. The annuity times the notional times one basis point is the swap's DV01, the same number a bond desk uses, which is why swaps are the standard tool for hedging bonds without buying or selling them.
Annuity = Σ DFₜ. Par swap rate = (1 − DFₙ) ÷ annuity. Bootstrap: DFₙ = (1 − S × Σ DFₜ for t < n) ÷ (1 + S). Value to the fixed payer = (par − fixed) × annuity × notional. DV01 ≈ annuity × notional × 0.0001.
Why it works
- The conventions here: a plain vanilla swap with annual fixed payments; one curve for discounting and projecting, so the floating leg is worth par. Zero rates rise or fall linearly with maturity; discount factors are printed to five decimals and every answer comes from the printed factors. Value to the fixed payer = (par − fixed) × annuity × notional. DV01 is the annuity approximation, holding the discount factors fixed.
- Why the floating leg is worth 1 − DFₙ. Add a notional exchange at the end. A floating-rate note is worth par at every reset, so floating coupons plus the final principal are worth 1 today; take away the principal, worth DFₙ, and the coupons alone are 1 − DFₙ.
- The swap rate is an average of forwards. Setting the fixed leg equal to the floating leg makes the swap rate a discount-factor-weighted average of the one-year forward rates along the curve. On an upward-sloping curve it sits above the short zero and below the long forward.
- Bootstrapping. Par means S × (annuity) + DFₙ = 1. With the earlier discount factors known, DFₙ is the only unknown, and its zero rate is DFₙ^(−1/n) − 1. Start at the shortest maturity and pull each factor up from the ones before it.
- Which side fixes a loan. A borrower paying floating on a loan pays fixed and receives floating on the swap; the floating receipts cover the loan and a fixed cost is left, plus the loan's credit spread. What it gives up is the benefit if rates fall.
- What the DV01 leaves out. When rates rise the discount factors fall and the annuity shrinks, so each extra basis point is worth slightly less: the swap's convexity. For a few basis points the annuity DV01 is fine; for a large move, revalue the curve.
| Discount factors: 1 ÷ 1.03 and 1 ÷ 1.04² | 0.97087, 0.92456 |
| Annuity: 0.97087 + 0.92456 | 1.89543 |
| Floating leg: 1 − 0.92456 | 0.07544 |
| Par swap rate: 0.07544 ÷ 1.89543 | 3.980% |
| Value to a payer of 3.5% fixed: (3.980% − 3.5%) × 1.89543 × $100M | about $910,000, an asset |
| DV01: 1.89543 × $100M × 0.0001 | $18,954 a basis point |
The formulas
The present value of one unit a year.
The fixed rate that makes the fixed leg worth the floating leg.
Bootstrap: the next discount factor from the par rate and the earlier factors.
The rate difference, every year, discounted.
One basis point, every year, on the notional, discounted.
Worked example
Add the discount factors for the annuity, take 1 minus the last one for the floating leg, divide. The follow-up asks why the floating leg is worth exactly that.
See it move
Same swap. Change where the curve starts and how it slopes, the maturity, the notional, and the fixed rate you locked in against today's par rate.
- Raise the one-year zero. The par swap rate rises, and the annuity and the DV01 fall, because every discount factor falls.
- Steepen the zero curve. The swap rate rises and the annuity falls.
- Lengthen the swap and watch the swap rate: it rises on an upward-sloping curve and falls on a downward one. The annuity and the DV01 always rise, because more years are added.
- Raise your fixed rate above par. The value falls one DV01 per basis point; nothing else moves. Raise the notional and the value and the DV01 scale with it.
Run it backwards
The earlier discount factors and the par swap rate for the final year are given. What is the final discount factor, and its zero rate?
Par means S × (sum of all discount factors) + DFₙ = 1. The earlier factors are known, so DFₙ = (1 − S × their sum) ÷ (1 + S). The zero rate is DFₙ raised to −1/n, minus one.
That is bootstrapping: each maturity's discount factor is pulled up from the ones before it. Start with the shortest instrument and work out the curve, and the whole zero curve comes from a strip of par rates.
Traps
Say it in the interview
“How is a swap rate set, and what is an existing swap worth?”
Check yourself
5 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- Floating leg = 1 − DFₙ; fixed leg = rate × annuity; par rate = (1 − DFₙ) ÷ annuity.
- Bootstrap the next discount factor from the par rate and the earlier factors.
- Value to the fixed payer = (par − fixed) × annuity × notional; DV01 = annuity × notional × 0.0001.
- A floating borrower pays fixed and receives floating.