Chapter 3, Problem 34PS

Prices and spot interest rates What spot interest rates are implied by the following Treasury bonds? Assume for simplicity that the bonds pay annual coupons. The price of a one-year strip is 97.56%, and the price of a four-year strip is 87.48%.

Summary Introduction

To determine: The spot interest rates implied by treasury bonds.

Explanation of Solution

Determine $r_1$:

$\begin{array}{c}\$97.56=\frac{\$100}{\left(1+r_1\right)}\\ \left(1+r_1\right)=\frac{\$100}{\$97.56}\\ =2.50\%\end{array}$

Hence, $r_1$ is 2.50%.

Determine$r_4$:

$\begin{array}{c}\$87.48=\frac{\$100}{\left(1+r_4\right)}\\ \left(1+r_4\right)=\frac{\$100}{\$87.48}\\ =3.40\%\end{array}$

Hence, $r_1$ is 3.40%.

Determine$r_5$:

$r_5$ can be solved by eliminating the 1-4 cash flow of 1.5 times 2% coupon bond and 3% coupon bond.

$\begin{array}{c}\left(1.5\times \$928.90\right)–\$974.30=\frac{\left[1.5\times \$1,020–\left(\$1,030\right)\right]}{{\left(1+r_5\right)}^5}\\ =0.036or3.6\%\end{array}$

Determine$r_3$:

$r_3$ can be solved by reducing the 3-year, 5% coupon bond and 2.5 times of reduced 5-year 2% coupon bond which eliminates the 2-year cash flow:

$\begin{array}{c}\$928.90=\frac{\$20}{1.025}+\frac{\$20}{{\left(1+r_2\right)}^2}+\frac{\$20}{{\left(1+r_3\right)}^3}+\frac{\$20}{{1.034}^4}+\frac{\$1,020}{{1.035955}^5}\mathrm{.....}\left(1\right)\\ \$1,054.20=\frac{\$50}{1.025}+\frac{\$50}{{\left(1+r_2\right)}^2}+\frac{\$1,050}{{\left(1+r_3\right)}^3} \\ \$1,005.419512=\frac{\$50}{{\left(1+r_2\right)}^2}+\frac{\$1,050}{\left(1+r_3\right)}\mathrm{...........}\left(2\right)\end{array}$

Solve equation 1 and 2

$\begin{array}{c}\$1,005.419512–\left(2.5\times \$37.030021\right)=\frac{\$1,050-\left(2.5\times \$20\right)}{{\left(1+r_3\right)}^3}\\ =\frac{\frac{\$912.844459}{\$1,000}}{{\left(1+r_3\right)}^3}\\ r_3=0.030863,or3.0863\%\end{array}$

Determine$r_2$:

$r_2$ can be solved by 3-year bond and $r_3$ and $r_1$

$\begin{array}{c}\$1,054.20=\frac{\$50}{1.025}+\frac{\$50}{{\left(1+r_2\right)}^2}+\frac{\$1,050}{{1.030863}^3}\\ \$46.932036=\frac{\$50}{{\left(1+r_2\right)}^2}\\ =0.0322\end{array}$

Hence, the spot rates are 2.50%, 3.22%, 3.09%, 3.40% and 3.60 for years 1 to 5.