Chapter M, Problem 9P

1.

To determine

Determine the required number of deposit and the amount of last deposit.

Explanation of Solution

Annuity: An annuity is referred as a sequence of payment of fixed amount of cash flows that occurs over the equal intervals of time.

Cash flow occurs during the first day of each time period is known as an annuity due, whereas cash flow occurs during the last day of each time period is known as an ordinary annuity.

FV_O represents Future Value of ordinary annuity = $40,000, and i represents interest rate for each of the stated time periods = 7% $\left(\frac{14\%}{2}\right)$ per semi-annum

Determine the number of deposits required.

$\begin{array}{c}{\text{FV}}_{\text{O}}=\text{Cash flow }\times \left({\text{f}}_{\text{on,i}}\right)\\ \text{\$40,000}=\text{\$4,000 }\times \left({\text{f}}_{\text{on=?,i=7\%}}\right)\\ \frac{\text{\$40,000}}{\text{\$4,000}}=\left({\text{f}}_{\text{on=?,i=7\%}}\right)\\ \text{10}\text{.000000}=\left({\text{f}}_{\text{on=?,i=7\%}}\right)\end{array}$

In the future value of an ordinary annuity of $1 table (at the end of the time value money module), it can be identified that the factor of 10.000000 is lies between 7 and 8 number of period at 7% column. This reveals that T Houser has to make 7 deposits of $4,000 each, and the 8^th deposit would be less than the amount of $4,000.

Now, to determine the amount of the last deposit, first calculate the future value of an annuity due of 7 deposits of $4,000 at 7%, using future value of annuity due formula.

$\begin{array}{l}{\text{FV}}_{\text{D}}=\text{cash flow }\times \left({\text{f}}_{\text{Dn=7,i=7\%}}\right)\\ {\text{FV}}_{\text{D}}=\text{cash flow }\times \left({\text{f}}_{\text{On+1=8,i=7\%}}-1\right)\\ {\text{FV}}_{\text{D}}=\$4,000\times \left(\text{10}\text{.259803}-1\right)\\ {\text{FV}}_{\text{D}}=\$4,000\times 9.\text{259803}\end{array}$

${\text{FV}}_{\text{D}}=\$37,039.21$

Here,

FV_D represents future value of annuity due.

Here, 10.259803 is taken from Future value of ordinary annuity of $1 table, where n = 8, i =7%. Factor of annuity due is calculated with the help of ordinary annuity table, as there is no separate table provided in this module for future value of annuity due.

Finally, determine the amount of required last deposit.

$\begin{array}{c}\text{Amount of last deposit = }\left[\begin{array}{l}\text{Fund need to accumulate}-\text{Future value}\\ \text{of annuity due of 7 deposits}\end{array}\right]\\ =\$40,000-\$37,039.21\\ =\$2,960.79\end{array}$

Therefore, the amount of last deposit will be $2,960.79.

Conclusion

Therefore, the required number of deposit of $4,000 each 6 months is 7 and the amount of last deposit is $2,960.79.

2.

To determine

Determine the required number of payments need to be made and the amount of last payment.

Explanation of Solution

Annuity: An annuity is referred as a sequence of payment of fixed amount of cash flows that occurs over the equal intervals of time.

Cash flow occurs during the first day of each time period is known as an annuity due, whereas cash flow occurs during the last day of each time period is known as an ordinary annuity.

PV_O represents Present Value of ordinary annuity = $20,000.

Determine the number of payments required.

$\begin{array}{c}{\text{PV}}_{\text{O}}=\text{Cash flow }\times \left({\text{p}}_{\text{on,i}}\right)\\ \text{\$20,000}=\text{\$4,000 }\times \left({\text{p}}_{\text{on=?,i=12\%}}\right)\\ \frac{\text{\$20,000}}{\text{\$4,000}}=\left({\text{p}}_{\text{on=?,i=12\%}}\right)\\ \text{5}\text{.000000}=\left({\text{p}}_{\text{on=?,i=12\%}}\right)\end{array}$

In the present value of an ordinary annuity of $1 table (at the end of the time value money module), it can be identified that the factor of 5.000000 is lies between 8 and 9 number of period at 12% column. This reveals that Person JC has to make 8 payments of $4,000 each year, and the 9^th payment would be less than the amount of $4,000.

Now, determine the amount of reduction in principal after 8^th payment, using present value of ordinary annuity formula (PV_O).

$\begin{array}{l}{\text{PV}}_{\text{O}}=\text{Cash flow }\times \left({\text{p}}_{\text{On=8,i=12\%}}\right)\\ {\text{PV}}_{\text{O}}=\text{\$4,000}\times 4.967640\\ {\text{PV}}_{\text{O}}=\$19,870.56\end{array}$

Here, 4.967640 is taken from Present value of ordinary annuity of $1 table, where n = 8, i =12%.

Now, $129.44 $\left(\$20,000-\$19,870.56\right)$ of the original principal is yet to be paid by Person JC. This $124.44 will accrue interest at 12% interest rate in 9^th year.

Now, determine the amount of required last payment.

$\begin{array}{l}\text{FV =PV}\times \left({\text{f}}_{\text{n,i}}\right)\\ \text{FV =\$129}\text{.44}\times \left({\text{f}}_{\text{n=9,i=12\%}}\right)\\ \text{FV =\$129}\text{.44}\times \text{2}\text{.773079}\\ \text{FV = \$358}\text{.95}\end{array}$

Hence, the amount of last payment will be $358.95.

This 2.773079 is taken from Future value of $1 table, where n = 9, i =12%.

Conclusion

Determine the required number of payment of $4,000 each year is 8 and the amount of last payment is $358.95.