Valuing a plain vanilla interest rate swap: the bond view
A plain vanilla interest rate swap exchanges fixed payments for floating payments on the same notional. Nothing but interest is swapped, and the notional itself is never exchanged. Valuing one is a standard FRM® Part I exercise, and the neat way to do it is to treat each side as a bond. For the exam's structure see how the FRM exam is structured, and for other topics see the FRM page.
The bond view
Take the party who pays fixed and receives floating. Its position is:
Swap value = PV of floating leg - PV of fixed leg
Imagine that both legs also exchange the notional at the end. This changes nothing, because the two notional payments cancel. The fixed leg then looks like a fixed-rate bond paying coupon K x notional each year plus the notional at maturity. The floating leg looks like a floating-rate bond.
Right after a reset date, a floating-rate bond that pays the reference rate is worth its par value, because the next coupon is set at the current market rate. So the floating leg is worth the notional, and:
Swap value (fixed payer) = notional - (K x notional x sum of discount factors + notional x final discount factor)
For the fixed receiver the sign flips. This assumes the curve used for discounting and for projecting floating rates is the same single curve. In practice, markets have used different curves for discounting and projection since the financial crisis, so treat this as the textbook version.
The par swap rate
The par swap rate is the fixed rate K that makes the swap worth zero at the start. Setting the value to zero gives:
K = (1 - final discount factor) / (sum of discount factors)
where the sum runs over every fixed payment date. It is the rate at which the fixed bond is worth exactly par.
Worked example
An invented company, Tarn Holdings, enters a three-year annual swap with notional 50 million, paying fixed and receiving floating. The invented discount factors are 0.9709, 0.9392 and 0.9050 for years 1, 2 and 3. Computed and asserted in Python:
- Sum of discount factors = 2.8151.
- Par swap rate = (1 - 0.9050) / 2.8151 = 3.375% (3.3747%).
- Fixed leg at that rate: 0.033747 x 2.8151 + 0.9050 = 1.0000 per unit of notional, so it equals the floating leg.
- Cross-check: the implied forward rates are 3.00%, 3.38% and 3.78%. Discounting them gives a floating-leg present value of 0.0950 per unit, equal to 1 - 0.9050.
- Initial swap value = 0 (up to rounding).
A year later, two years remain and rates have risen. New discount factors are 0.9615 and 0.9246, so the annuity is 1.8861 and the new par rate is (1 - 0.9246) / 1.8861 = 4.00%.
- Fixed leg value per unit: 0.033747 x 1.8861 + 0.9246 = 0.98825.
- Swap value to the fixed payer = (1 - 0.98825) x 50 million = about 0.588 million.
- Check: (4.00% - 3.375%) x 1.8861 x 50 million gives the same 0.588 million.
The fixed payer gains because it locked in a rate below today's market rate. The counterparty receiving fixed is worse off by the same amount.
Why a swap starts near zero
At inception the fixed rate is set at the par swap rate, so the present value of what you expect to pay equals that of what you expect to receive. No cash is usually exchanged up front. The value then drifts as rates move: a fixed payer gains when rates rise and loses when they fall. A swap's interest-rate sensitivity resembles that of a long position in a floating bond and a short position in a fixed bond, which links to duration and convexity.
Usual mistakes
- Valuing the floating leg at par on a date that is not a reset date. Between resets, the floating bond is worth (notional plus the known next coupon) x the discount factor to the next reset.
- Swapping the sign for payer and receiver.
- Discounting with a flat yield when the question gives a set of discount factors.
- Summing discount factors only up to the penultimate payment, or adding the final notional twice.
- Confusing the par swap rate with the floating rate for the next period, or assuming the swap's market value equals its credit exposure. Exposure and counterparty risk are separate topics.
How the exam typically tests it
Typically you compute a par swap rate from discount factors or spot rates, or value an existing swap as the difference between two bonds. Charta's practice questions for this topic let you rehearse these steps.
Common questions
How do you value a plain vanilla swap?
Value the fixed leg as a fixed-rate bond and the floating leg as a floating-rate bond, then take the difference. The floating bond is worth par just after a reset.
What is the par swap rate?
It is the fixed rate that sets the swap's initial value to zero: one minus the final discount factor, divided by the sum of the discount factors.
Why is a swap worth about zero at inception?
The fixed rate is chosen so that the expected payments on both legs have equal present value, so neither side pays to enter.
Does the notional get exchanged?
In an interest rate swap on a single currency, no. It is added to both legs only as a device to apply bond formulas.
Sources
- GARP, the FRM programme: what the exam is and its two parts.
The exam and the syllabus can change. Check GARP's current details before you book.