Spot Rate from Zero-Coupon Bond
Inverts the zero-coupon bond price formula to recover the annualized spot rate. Each spot rate is a single-cash-flow discount rate; the curve of spot rates is the term structure of interest rates.
When to use: Use to back out a spot rate from an observed zero-coupon Treasury price (STRIPS), or as the first step in bootstrapping a full spot curve from a series of par bonds. Spot rates are the inputs to discount any future single cash flow. The returned rate is nominal annual compounded m times per year — feed it into the Forward Rate formula with the same m to keep the convention consistent.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| Spot | Spot Rate | Annualized zero-coupon spot rate as a decimal | % |
| F | Face Value | Par value paid at maturity | $ |
| P | Bond Price | Market price of the bond per face value unit | $ |
| N | Years to Maturity | Years remaining until the bond matures | years |
| m | Coupons per Year | Number of coupons paid per year (e.g. 2 for semi-annual) | integer |
Real-Life Examples
Example 1: 10-Year STRIPS at $610.27
A $1,000-face 10-year zero trades at $610.27. Semi-annual compounding (m = 2).
Given
Step-by-Step
The 10-year spot rate is 5%. Inverting the zero price recovers the same yield used to price it — and gives the discount rate for any single $1 cash flow received exactly 10 years from now.
Frequently Asked Questions
Because the term structure isn't flat — long rates differ from short rates. A 1-year cash flow gets discounted at the 1-year spot rate; a 10-year cash flow at the 10-year spot rate. The shape of the spot curve (upward, downward, humped) reflects market expectations and risk premia.
The procedure of solving for spot rates one tenor at a time from a series of par-bond yields. Use the 6-month rate to derive the 6-month spot, then the 1-year par bond to back out the 1-year spot given the known 6-month spot, and so on. Builds the full curve from observable bond yields.
YTM is a single discount rate that prices the whole bond; spot rates are tenor-specific rates that discount each cash flow individually. For a coupon bond, YTM is approximately a duration-weighted average of the spot rates the cash flows actually face.