# 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 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.

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