Lesson 5 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 5 of 6 · 12 min

Spot rates from forward rates, and pricing with forwards

Chaining forward rates rebuilds the spot curve, because a spot rate is the geometric average of the forward rates that cover its period, so bonds can be priced with either set of rates.

In short

  • The 0y1y rate is the 1-year spot rate; the others in a 1-year forward curve are 1y1y, 2y1y, 3y1y…
  • Multiply growth factors along the forward curve: (1+ZN)N=(1+0y1y)(1+1y1y)⋯(1+(N−1)y1y)(1+Z_N)^N = (1+0y1y)(1+1y1y)\cdots(1+(N-1)y1y).
  • So a spot rate is the geometric mean of the forward rates up to its maturity, not their arithmetic average.
  • Forwards of different tenors chain too, as long as they are back to back: 0y1y × 1y2y × 3y3y gives the 6-year spot rate.
  • To price with forwards, discount each cash flow by the product of all forward factors up to its date. The price equals the spot-rate price.

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