Chapter 25, Problem 7QP

Duration What is the duration of a bond with four years to maturity and a coupon of 8.6 percent paid annually if the bond sells at par?

Summary Introduction

To calculate: The duration of bond with the given maturity and coupon, and that are paid annually if bond sells at par.

Duration:

It is an object used in bond valuation to determine the sensitivity of the bond. High duration comes with high probability of interest rate risk while in lower duration there is low risk of interest rate fluctuation.

Explanation of Solution

Calculate coupon of the bond.

Given,

Par value of bond is $1,000.

Coupon rate is 8.6%.

Year to maturity is 4 years.

Formula to calculate coupon is,

$\text{Coupon}=\text{Par}\text{value}\times \text{Coupon}\text{rate}$

Substitute $1,000 for par value of bond, 8.6% for coupon rate and 4 years for year to maturity.

$\begin{array}{c}\text{Coupon}=\$1,000\times 8.6\%\\ =\$86\end{array}$

So, coupon is of $86.

Calculation of duration,

Year	Present value	Relative value	Duration
1	79.1	0.0791	0.0791
2	72.9	0.0729	0.1458
3	67.1	0.0671	0.2013
3	780.7	0.7807	3.1228
Total Present value	1,000	1	3.549

Table (1)

Working notes:

Calculation of present value for year 1,

$\begin{array}{c}\text{Present}\text{value}=\frac{\text{Coupon}}{\left(1+\text{Coupon}\text{rate}\right)}\\ =\frac{\$86}{\left(1+8.6\%\right)}\\ =\$79.1\end{array}$

Calculation of present value for year 2,

$\begin{array}{c}\text{Present}\text{value}=\frac{\text{Coupon}}{{\left(1+\text{Coupon}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{\$86}{{\left(1+8.6\%\right)}^2}\\ =\$72.9\end{array}$

Calculation of present value for year 3,

$\begin{array}{c}\text{Present}\text{value}=\frac{\text{Coupon}}{{\left(1+\text{Coupon}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{\$86}{{\left(1+8.6\%\right)}^3}\\ =\$67.1\end{array}$

Calculation of present value for year 4,

$\begin{array}{c}\text{Present}\text{value}=\frac{\text{Coupon}}{{\left(1+\text{Coupon}\text{rate}\right)}^{\text{Number}\text{of}\text{years}}}\\ =\frac{\$1,086}{{\left(1+8.6\%\right)}^4}\\ =\$780.7\end{array}$

Calculation of relative value of year 1,

$\begin{array}{c}\text{Relative}\text{value}=\frac{\text{Present}\text{value}}{\text{Par}\text{value}\text{of}\text{bond}}\\ =\frac{\$79.1}{\$1,000}\\ =0.0791\end{array}$

Calculation of relative value of year 2,

$\begin{array}{c}\text{Relative}\text{value}=\frac{\text{Present}\text{value}}{\text{Par}\text{value}\text{of}\text{bond}}\\ =\frac{\$72.9}{\$1,000}\\ =0.0729\end{array}$

Calculation of relative value of year 3,

$\begin{array}{c}\text{Relative}\text{value}=\frac{\text{Present}\text{value}}{\text{Par}\text{value}\text{of}\text{bond}}\\ =\frac{\$67.1}{\$1,000}\\ =0.0671\end{array}$

Calculation of relative value of year 4,

$\begin{array}{c}\text{Relative}\text{value}=\frac{\text{Present}\text{value}}{\text{Par}\text{value}\text{of}\text{bond}}\\ =\frac{\$780.7}{\$1,000}\\ =0.7807\end{array}$

Calculation of duration for year 1,

$\begin{array}{c}\text{Duration}=\text{Relative}\text{value}\times \text{Number of years}\\ =0.0791\times 1\\ =0.0791\end{array}$

Calculation of duration for year 2,

$\begin{array}{c}\text{Duration}=\text{Relative}\text{value}\times \text{Number of years}\\ =0.0729\times 2\\ =0.1458\end{array}$

Calculation of duration for year 3,

$\begin{array}{c}\text{Duration}=\text{Relative}\text{value}\times \text{Number of years}\\ =0.0671\times 3\\ =0.2013\end{array}$

Calculation of duration for year 4,

$\begin{array}{c}\text{Duration}=\text{Relative}\text{value}\times \text{Number of years}\\ =0.7807\times 4\\ =3.1228\end{array}$

Conclusion

So, the duration of the bond is 3.549 years.