Chapter 25, Problem 11QP

Summary Introduction

To calculate: The duration of liability after borrowing and lending market interest rate of 7.6 percent.

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

Calculation of duration

Year	Present value ($)	Relative value	Duration
7	17,965.3360	0.2781	1.9472
8	16,696.4094	0.2584	2.0672
9	15,517.1091	0.2402	2.1618
10	14,421.1051	0.2233	2.2320
Total present value	64,599.9596	1	8.4082

Table (1)

Working notes:

Calculation of present value for year 7,

$\begin{array}{c}\text{Present}\text{value}=\frac{\text{Future}\text{Value}}{{\left(1+\text{Rate}\text{of}\text{interest}\right)}^{\text{Number}\text{of }\text{Years}}}\\ =\frac{\$30,000}{{\left(1+7.6\%\right)}^7}\\ =\$17,965.3360\end{array}$

Calculation of present value for year 8,

$\begin{array}{c}\text{Present}\text{value}=\frac{\text{Future}\text{Value}}{{\left(1+\text{Rate}\text{of}\text{interest}\right)}^{\text{Number}\text{of }\text{Years}}}\\ =\frac{\$30,000}{{\left(1+7.6\%\right)}^8}\\ =\$16,696.4094\end{array}$

Calculation of present value for year 9,

$\begin{array}{c}\text{Present}\text{value}=\frac{\text{Future}\text{Value}}{{\left(1+\text{Rate}\text{of}\text{interest}\right)}^{\text{Number}\text{of }\text{Years}}}\\ =\frac{\$30,000}{{\left(1+7.6\%\right)}^9}\\ =\$15,517.1091\end{array}$

Calculation of present value for year 10,

$\begin{array}{c}\text{Present}\text{value}=\frac{\text{Future}\text{Value}}{{\left(1+\text{Rate}\text{of}\text{interest}\right)}^{\text{Number}\text{of }\text{Years}}}\\ =\frac{\$30,000}{{\left(1+7.6\%\right)}^{10}}\\ =\$14,421.1051\end{array}$

Calculation of relative value for year 7,

$\begin{array}{c}\text{Relative}\text{Value}=\frac{\text{Present}\text{value}}{\text{Sum}\text{of}\text{present}\text{value}\text{of}\text{total}\text{years}}\\ =\frac{\$17,965.3360}{\text{\$64,599}\text{.9596}}\\ =0.2781\end{array}$

Calculation of relative value for year 8,

$\begin{array}{c}\text{Relative}\text{Value}=\frac{\text{Present}\text{value}}{\text{Sum}\text{of}\text{present}\text{value}\text{of}\text{total}\text{years}}\\ =\frac{\$16,696.4094}{\text{\$64,599}\text{.9596}}\\ =0.2584\end{array}$

Calculation of relative value for year 9,

$\begin{array}{c}\text{Relative}\text{Value}=\frac{\text{Present}\text{value}}{\text{Sum}\text{of}\text{present}\text{value}\text{of}\text{total}\text{years}}\\ =\frac{\$15,517.1091}{\text{\$64,599}\text{.9596}}\\ =0.2402\end{array}$

Calculation of relative value for year 9,

$\begin{array}{c}\text{Relative}\text{Value}=\frac{\text{Present}\text{value}}{\text{Sum}\text{of}\text{present}\text{value}\text{of}\text{total}\text{years}}\\ =\frac{\$14,421.1051}{\text{\$64,599}\text{.9596}}\\ =0.2233\end{array}$

Calculation of duration year 7,

$\begin{array}{c}\text{Duration}=\text{Relative value}\times \text{Number of years}\\ =0.2781\times 7\\ =1.9472\end{array}$

Calculation of duration year 8,

$\begin{array}{c}\text{Duration}=\text{Relative value}\times \text{Number of years}\\ =0.2584\times 8\\ =2.0672\end{array}$

Calculation of duration year 9,

$\begin{array}{c}\text{Duration}=\text{Relative value}\times \text{Number of years}\\ =0.2402\times 9\\ =2.1618\end{array}$

Calculation of duration year 10,

$\begin{array}{c}\text{Duration}=\text{Relative value}\times \text{Number of years}\\ =0.2232\times 10\\ =2.2320\end{array}$

Conclusion

So duration of bond is 8.4082 years.