Given the zero-coupon yield curve, we can easily derive spot rates since zero-coupon bonds are, by definition, already expressed in terms of spot rates. However, if you have coupon bond data, you can bootstrap the spot rate curve by following these steps:
1. Start with the shortest maturity bond:
Assume that you have the bond price and yield data for a set of coupon-paying bonds. First, take the bond with the shortest maturity (i.e., the bond that matures first) and find the associated spot rate. This is simply the yield-to-maturity of the bond.
2. Calculate the discounted cash flows (DCF) for the subsequent bonds:
For each bond with a longer maturity, use the spot rates you derived in the previous steps to discount the bond’s cash flows up to the last known spot rate.
3. Determine the spot rate for the bond’s remaining cash flows:
Using the bond’s clean price and the discounted cash flows (DCF) calculated in the previous step, derive the spot rate for the bond’s remaining cash flows. This is done by setting the DCF equal to the bond’s clean price and solving for the spot rate.
4. Repeat steps 2 and 3 for all bonds in the set.
Here’s an example:
Assume that you have the clean prices and yields for the following coupon-paying bonds:
| Maturity (Years) | Coupon Rate | Clean Price | Yield-to-Maturity |
|------------------|-------------|-------------|-------------------|
| 1 | 4% | 100 | 4.00% |
| 2 | 4.5% | 101 | 4.25% |
| 3 | 5% | 103 | 4.46% |We begin by calculating the spot rate for the 1-year bond:
s1 = 4.0%
Calculating the DCF for the 2-year bond based on the 1-year spot rate:
DCF of the 2-year bond when it matures after 1 year:
$\frac{4.5\% * 100}{1 + s_1} = \frac{4.5}{1.04} = 4.33$
Now, we derive the 2-year spot rate, s2, using the equation:
$101 = 4.33 + \frac{(4.5 + 100)}{(1 + s_2)^2}$
Solving for s2, we get:
s2 = 4.49%
Repeating the process for the 3-year bond, we first calculate the DCF for the first two years:
DCF1: $\frac{5\%*100}{1+s_1} = 4.81$
DCF2: $\frac{5\%*100}{(1+s_2)^2} = \frac{5}{(1.0449)^2} = 4.56$
Now, we derive the 3-year spot rate, s3, using the equation:
$103 = 4.81 + 4.56 + \frac{(5 + 100)}{(1 + s_3)^3}$
Solving for s3, we get:
s3 = 4.63%
Therefore, the bootstrapped spot rate curve is:
s1 = 4.00%
s2 = 4.49%
s3 = 4.63%