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# Spot rates and bond pricing

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 Z1, Z2 and Zn 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%.

by Obaidullah Jan, ACA, CFA on Fri Feb 14 2020