One method for determining bond price is to discount all bond cash flows at the market discount rate. But since the yield curve is not flat, a more appropriate approach is one where each bond cash flow is discounted separately at the interest rate which corresponds to its maturity, i.e. the relevant spot interest rates.

## Spot interest rate

A spot interest rate (also called a **zero rate**) is the yield to maturity on a zero-coupon bond maturing on the date of each cash flow.

Following is the general expression for bond pricing using spot rates

\[ P=\frac{PMT}{\left(1+Z_1\right)^1}+\frac{PMT}{\left(1+Z_2\right)^2}+…+\frac{PMT+FV}{\left(1+Z_N\right)^N} \]Where Z_{1}, Z_{2} and Z_{n} are the relevant spot rates.

## Example

A bond has 3-years till maturity, and it pays 4% coupon rate. If the one-year, two-year and three-year spot rates at 3%, 4%, and 5% respectively, the value of the bond would be 97.4208 as shown below:

\[ P=\frac{4}{\left(1+3\%\right)^1}+\frac{4}{\left(1+4\%\right)^2}+\frac{4+100}{\left(1+5\%\right)^3}=97.4208 \]Since the price is less than 100, the bond is trading at a discount. The bond’s yield to maturity must be less than the coupon rate. Using the formula for bond price using a uniform rate, we can find out the bond’s yield to maturity.

\[ 97.4208=\frac{4}{\left(1+r\right)^1}+\frac{4}{\left(1+r\right)^2}+\frac{4+100}{\left(1+r\right)^3} \]It works out to 4.95%.