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Describe how a plain vanilla interest rate swap's discount rates are computed.

FRM® Part I · Financial Markets and Products · Swaps · FRM1.FM.20.f

The idea

Swap valuation needs discount rates, and these are taken from a zero curve built by bootstrapping. Short maturities come from money market rates or overnight index swap rates, and longer zero rates are solved one maturity at a time from par swap rates. A par swap rate is the fixed rate that makes the swap worth zero, so a fixed bond paying that coupon is worth par. For an annual swap with par rate c and notional 1, the par condition is: Bond value = c × (DF1 + DF2 + … + DFn) + DFn = 1. Solving it for the last discount factor, using factors already found for earlier dates: DFn = (1 − c × sum of earlier DFs) ÷ (1 + c). The zero rate z for maturity T follows from the discount factor, for example with continuous compounding: DF = e^(−z × T). Collateralised swaps are generally discounted on an overnight index swap curve consistent with the collateral terms and market convention, rather than a LIBOR-style curve. The usual mistake is treating a par swap rate as the zero rate of the same maturity.

Formulas

Bootstrapping the last discount factor from a par swap rate

DFn = (1 − c × sum of earlier DFs) ÷ (1 + c)

c
the par swap rate per period
sum of earlier DFs
the discount factors already found for the earlier payment dates

From the par condition that a bond paying c is worth 1. A par swap rate is not the zero rate of the same maturity.

Discount factor from a continuously compounded zero rate

DF = e^(−z × T)

z
the continuously compounded zero rate
T
the maturity, in years

Use the exponential only for continuous rates; with annual compounding DF = 1 ÷ (1 + z) raised to T.

A practice question

From the question bank, with the reason each wrong option is wrong.

What is the par swap rate for a given maturity?

  1. The forward rate for the last period of the swap.Why not: The par rate is a weighted average of all the forward rates, not the last forward.
  2. The floating reference rate for the first payment period.Why not: The par rate is a fixed rate for the whole maturity, not the first reset rate.
  3. The fixed rate that makes the swap's value zero at initiation. - correct
  4. The average of the zero rates from today to maturity.Why not: Averaging the zero rates is not the par rate; the par rate depends on discount factors.

A par swap rate is the fixed rate that sets the present value of the fixed leg equal to that of the floating leg, so the swap is worth zero at initiation.

A harder one

The 6-month and 12-month discount factors are 0.9850 and 0.9680. An 18-month swap with semi-annual payments has a par rate of 4.0%. Which is the 18-month discount factor implied by the par swap rate?

  1. 0.8864Why not: Uses the annual 4.0% instead of the semi-annual 2.0% coupon at each date.
  2. 0.9804Why not: Discounts a single semi-annual period instead of solving from the par condition.
  3. 0.9609Why not: Leaves the final coupon out of the final cash flow, so divides by 1 instead of 1.02.
  4. 0.9421 - correct

Par condition: 1 = 0.02 x (0.9850 + 0.9680 + DF3) + DF3, so DF3 = (1 - 0.02 x 1.9530)/1.02 = 0.9421.

The hardest tier

Annual-pay par swap rates are 3.0% for 1 year, 4.0% for 2 years and 5.0% for 3 years. The 1-year zero rate equals the 1-year par rate. What is the 3-year zero rate (annual compounding), closest to?

  1. 4.00%Why not: Averages the three par rates instead of bootstrapping.
  2. 5.07% - correct
  3. 4.94%Why not: Reports a continuously compounded rate when annual compounding is required.
  4. 5.00%Why not: Treats the 3-year par swap rate as the 3-year zero rate.

DF1 = 1/1.03 = 0.9709; DF2 = (1 - 0.04 x DF1)/1.04 = 0.9242; DF3 = (1 - 0.05 x (DF1 + DF2))/1.05 = 0.8621. The 3-year zero rate = DF3^(-1/3) - 1 = 5.07%.

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