Lesson 2 of 6 · All lessons in this module
  1. Swaps versus a series of forward rate agreements
  2. Pricing a swap: the par swap rateVideo lesson
  3. Why issuers and investors use swaps
  4. Swap price versus swap valueVideo lesson
  5. How swap values change: time, rates and credit exposure
  6. Integrated cases: a treasury desk's futures and swap hedges

Lesson 2 of 6 · 14 min

Pricing a swap: the par swap rate

The swap's price is the par swap rate: the single fixed rate at which the present value of the fixed payments equals the present value of the expected floating payments, with floating payments expected at the implied forward rates.

In short

  • Expected floating payments are the implied forward rates for each period; discount each one, and each fixed payment, with the zero rate for its date.
  • The par swap rate sNs_N solves ∑IFRi−1,1DFi=sN∑DFi\sum IFR_{i-1,1}DF_i = s_N\sum DF_i, so sN=∑IFRi−1,1DFi / ∑DFis_N = \sum IFR_{i-1,1}DF_i \,/\, \sum DF_i.
  • It equals the coupon of a par bond priced off the same zero curve: sN=(1−DFN)/∑DFis_N = (1 - DF_N)/\sum DF_i.
  • The swap rate is a multi-period breakeven rate: an internal rate of return on the implied forward rates through the swap's maturity.
  • The whole swap is worth zero at inception, but individual periods are not: some expected net flows are positive and some negative.

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