Bootstrapping Yield Curve

Bootstrapping the Yield Curve

Bootstrapping extracts spot rates from par bond prices iteratively. A par bond is priced at face value (price = 1), so its coupon rate equals its YTM.

Algorithm

For a par bond maturing in year $n$ with par rate $p_n$, and normalized face value = 1:

$1 = p_n \sum_{t=1}^{n-1} d_t + (1 + p_n) \cdot d_n$

Where $d_t = \frac{1}{(1+s_t)^t}$ is the discount factor at time $t$.

Solving for $d_n$ (the unknown discount factor for year $n$):

$d_n = \frac{1 - p_n \sum_{t=1}^{n-1} d_t}{1 + p_n}$

Then: $s_n = d_n^{-1/n} - 1$

Step-by-Step

1. Year 1: $s_1 = p_1$ (trivially, the 1-year par rate is the 1-year spot rate)
2. Year 2: Use $s_1$ to compute $d_1$, solve for $d_2$, then $s_2$
3. Year 3: Use $s_1, s_2$ to compute $d_1, d_2$, solve for $d_3$, then $s_3$
4. Continue iterating...