Lesson 3 of 6 · All lessons in this module
  1. Spot rates and the spot curve
  2. Pricing a bond with spot ratesVideo lesson
  3. Par rates and the par curveVideo lesson
  4. Implied forward rates from spot ratesVideo lesson
  5. Spot rates from forward rates, and pricing with forwardsVideo lesson
  6. Comparing the spot, par and forward curves

Lesson 3 of 6 · 13 min

Par rates and the par curve

A par rate is the coupon rate that would make a bond priced off the spot curve worth exactly par, so it is that bond's coupon rate and its YTM at once.

In short

  • A par rate is the YTM (and coupon rate) at which a bond priced with spot rates is worth 100% of par.
  • Solve 100=PMT1+Z1+⋯+PMT+100(1+ZN)N100 = \frac{PMT}{1+Z_1} + \dots + \frac{PMT+100}{(1+Z_N)^N} for PMT; PMT ÷ 100 is the par rate.
  • Shortcut with discount factors: PMT=(100−100 DFN)/(DF1+⋯+DFN)PMT = (100 - 100\,DF_N)/(DF_1 + \dots + DF_N).
  • The 1-year par rate equals the 1-year spot rate. Longer par rates depend on all spot rates up to maturity.
  • Par curves describe hypothetical bonds priced at par, which avoids tax and trading distortions of premium and discount bonds. The daily US Treasury yield curve is a par curve.
  • Between coupon dates, set the flat price (not the full price) equal to 100.

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