Spot rates from forward rates, and pricing with forwards

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: $(1+Z_N)^N = (1+0y1y)(1+1y1y)\cdots(1+(N-1)y1y)$.
• 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.