Bootstrapping Spot Rates

Bootstrapping spot rates is a forward substitution method that allows investors to determine zero-coupon rates using the par yield curve. The par curve shows the yields to maturity on government bonds with coupon payments, priced at par, over a range of maturities.

Bootstrapping involves obtaining spot rates (zero-coupon rates) for one year, then using the one-year spot rate to determine the 2-year spot rate, and so on.

Spot rates obtained through bootstrapping are known as implied spot rates.

Example: Bootstrapping Spot Rates

Consider the following annual par rates for a coupon paying bond.

$$ \textbf{Annual Par-Rates} \\ \begin{array}{c|c} \textbf{Year} & \textbf{Par Rate} \\ \hline 1 & 2.00\% \\ \hline 2 & 2.60\% \\ \hline 3 & 2.90\% \\ \hline 4 & 3.80\% \end{array} $$

The one-year implied spot rate is 2%, as it is simply the one-year par yield.

The two-year implied spot rate is determined as follows:

$$ \begin{align*} 1 &=\frac{0.026}{1.02}+\frac{(1+0.026)}{\left(1+r\left(2\right)\right)^2} \\ r (2) & =2.61\% \end{align*} $$

We have bootstrapped the 2-year spot rate.

Similarly, the three-year spot rate can be bootstrapped by solving the equation:

$$ \begin{align*} 1 &=\frac{0.029}{1.02}+\frac{0.029}{{1.0261}^2}+\frac{1.029}{\left(1+r\left(3\right)\right)^3} \\ r (3) & = 2.91\% \end{align*} $$

The four-year spot rate is determined as:

$$ \begin{align*} 1 &=\frac{0.038}{1.02}+\frac{0.038}{{1.0261}^2}+\frac{0.038}{{1.0291}^3}+\frac{1.038}{\left(1+r\left(4\right)\right)^4} \\ r (4) & = 3.87\% \end{align*} $$

The zero-coupon rates are shown in the following table:

$$ \begin{array}{c|c|c} \textbf{Year} & \textbf{Par Rate} & \textbf{Zero-coupon rate} \\ & & \textbf{(Implied Spot rate)} \\ \hline 1 & 2.00\% & 2.00\% \\ \hline 2 & 2.60\% & 2.61\% \\ \hline 3 & 2.90\% & 2.91\% \\ \hline 4 & 3.80\% & 3.87\% \end{array} $$

Question

Determine the three-year implied spot rate given the following par rates.

$$ \begin{array}{c|c} \textbf{Year} & \textbf{Rate} \\ \hline 1 & 5.00\% \\ \hline 2 & 5.25\% \\ \hline 3 & 5.50\% \end{array} $$

1. 5.25%.
2. 5.52%.
3. 16.59%.

Solution

The correct answer is B.

The one-year implied spot rate is 5%, as it is simply the one-year par yield.

The two-year implied spot rate is determined as follows:

$$ \begin{align*} 1&=\frac{0.0525}{1.05}+\frac{(1.0525)}{\left(1+r\left(2\right)\right)^2} \\ r (2)& =5.26\% \end{align*} $$

We have bootstrapped the 2-year spot rate.

Similarly, the three-year spot rate can be bootstrapped by solving the equation:

$$ \begin{align*} 1&=\frac{0.055}{1.05}+\frac{0.055}{{1.0526}^2}+\frac{1.055}{\left(1+r\left(3\right)\right)^3} \\ r (3)& = 5.52\% \end{align*} $$

Reading 28: The Term Structure and Interest Rate Dynamics

LOS 28(b) Describe how zero-coupon rates (spot rates) may be obtained from the par curve by bootstrapping.

Why Bootstrapping Spot Rates Matters in CFA Level II

Bootstrapping spot rates is an important CFA Level II Fixed Income concept. Candidates may be tested on deriving zero-coupon rates from par yields, valuing bonds with spot curves, and interpreting term structure relationships.

To strengthen your preparation, explore the CFA Level II study program with practice questions, mock exams, and guided lessons.