Chapter 9, Problem 45P

a.

Summary Introduction

To calculate: The semi-annual payment of the investment.

Introduction:

Annuity:

When payments are made or received in a series at equivalent intervals, they are termed as an annuity. Such payments can be made weekly, monthly, quarterly, or annually.

Answer to Problem 45P

The semi-annual payment of the investment is $8,886.56

Explanation of Solution

The calculation of the semi-annual payment of the investment is shown below.

$\begin{array}{c}\text{Annuity}=\frac{\text{Future Value}}{\left(\frac{{\left(1+\frac{\text{Interest Rate}}{\text{Number of Payment in a Year }\left(\text{n}\right)}\right)}^{\left(\text{Number of Payments}\right)}-1}{\frac{\text{Interest Rate}}{\text{Number of Payment in a Year }\left(\text{n}\right)}}\right)}\\ =\frac{\text{\$250,000}}{\left(\frac{{\left(1+\frac{\text{10\%}}{2}\right)}^{\left(18\right)}-1}{\frac{10\%}{2}}\right)}\\ =\frac{\$250,000}{\left(\frac{2.40661923-1}{5\%}\right)}\\ =\frac{\$250,000}{28.1323846}\\ =\$8,886.56\end{array}$

b.

Summary Introduction

To calculate: The revised semiannual payment after a change in rate from 10% to 12%.

Introduction:

Annuity:

When payments are made or received in a series at equivalent intervals, they are termed as an annuity. Such payments can be made weekly, monthly, quarterly, or annually.

Answer to Problem 45P

The revised semiannual payment after a change in rate from 10% to 12% is $7,609.48.

Explanation of Solution

The calculation of the revised semiannual payment is shown below.

$\begin{array}{c}\text{Annuity}=\frac{\text{Additional Amount}}{\left(\frac{{\left(1+\frac{\text{Interest Rate}}{\text{Number of Payment in a Year }\left(\text{n}\right)}\right)}^{\left(\text{Number of Payments}\right)}-1}{\frac{\text{Interest Rate}}{\text{Number of Payment in a Year }\left(\text{n}\right)}}\right)}\\ =\frac{\text{\$128,371}\text{.56}}{\left(\frac{{\left(1+\frac{\text{12\%}}{2}\right)}^{\left(12\right)}-1}{\frac{12\%}{2}}\right)}\\ =\frac{\text{\$128,371}\text{.56}}{\left(\frac{2.01219647-1}{6\%}\right)}\\ =\frac{\text{\$128,371}\text{.56}}{16.86994117}\\ =\$7,609.48\end{array}$

Working notes:

1. Requirement of additional amount:

The calculation of the requirement of additional amount is shown below.

$\begin{array}{c}\text{Additional Amount}=\text{Required Future Value}-\text{Future Value at the End of 9th Year}\\ =\$250,000-\$121,628.44\\ =\$128,371.56\end{array}$

2. Future value of the payment made at the beginning of the 4th year:

The calculation of the future value of the payment made at the beginning of the 4th year is shown below.

$\begin{array}{c}{\text{Future Value}}_{\left(\text{At 4th Year}\right)}=\text{Annuity}\times \left(\frac{{\left(1+\frac{\text{Interest Rate}}{\text{Number of Payment in a Year }\left(\text{n}\right)}\right)}^{\left(\text{Number of Payments}\right)}-1}{\frac{\text{Interest Rate}}{\text{Number of Payment in a Year }\left(\text{n}\right)}}\right)\\ =\$8,886.56\times \left(\frac{{\left(1+\frac{\text{10\%}}{\text{2}}\right)}^{\left(\text{6}\right)}-1}{\frac{\text{10\%}}{\text{2}}}\right)\\ =\$8,886.56\times \left(\frac{1.34009564-1}{\text{5\%}}\right)\\ =\$8,886.56\times 6.8019128\\ =\$60,445.61\end{array}$

3. Future value of the payment from the beginning of the 4th year to the end of the 9th year:

The calculation of the future value of the payment from the beginning of the 4th year to the end of the 9th year is shown below.

$\begin{array}{c}\text{Future Value}={\text{Future Value}}_{\left(\text{At 4th Year}\right)}\times {\left(1+\frac{\text{Interest Rate}}{\text{Number of Payment in a Year }\left(\text{n}\right)}\right)}^{\left(\text{Number of Payments}\right)}\\ =\$60,445.61\times {\left(1+\frac{\text{12\%}}{\text{2}}\right)}^{\left(6\times 2\right)}\\ =\$60,445.61\times {\left(1+6\%\right)}^{12}\\ =\$60,445.61\times 2.01219647\\ =\$121,628.44\end{array}$