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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Par rates and the par curve · The Term Structure of Interest Rates: Spot, Par, and Forward Curves