Chapter M, Problem 11E

1.

To determine

Determine the accumulated amount in the savings account on December 31, 2021.

Answer to Problem 11E

The accumulated amount in the savings account on December 31, 2021 is $64,041.28.

Explanation of Solution

Future Value: The future value is value of present amount compounded at an interest rate until a particular future date.

Person HD deposits $40,000 on January 1, 2016. Interest rate of 8% is compounded semiannually. Thus the interest rate per one half-year is 4%$\left(\frac{8\%peryear}{\text{12}}\times 6\text{months in a semiannual}\right)$. Number of time period from January 1, 2016 to December 31, 2021 is 12 semi annum.

Determine the future value as on January 1, 2016.

$\begin{array}{c}\text{FV=PV×}\left({\text{f}}_{\text{n=12,i=4\%}}\right)\\ =\$40,000\times 1.601032\\ =\$64,041.28\end{array}$

Hence, the amount of $40,000 deposited for 12 semi annum at 4% interest compounded semi-annually will have a future value of $64,041.28.

Note:

FV stands for Future value

PV stands for Present value

i stands for interest rate for each of the stated time periods

n stands for number of time periods

FV factor (Future value of $1: n = 12, i =4%) is taken from the table value (Table 1 at the end of the time value money module).

2.

To determine

Determine the accumulated amount in the fund on December 31, 2020, after the receipt of December 31 bonuses of 2020.

Answer to Problem 11E

Accumulated amount in the fund on December 31, 2020, after the receipt of December 31 bonuses of 2020 is $31,764.24.

Explanation of Solution

Person BJ deposits the bonus amount of $5,000 each year in his savings accounts, starts from December 31, 2016. Deposit will earn 12% interest per annum. Number of time period from December 31, 2016 to December 31, 2020 is 5 years.

Determine the future value ordinary annuity.

$\begin{array}{c}{\text{FV}}_{\text{O}}\text{=Cash flow×}\left({\text{f}}_{\text{on=5,i=12\%}}\right)\\ =\$5,000\times 6.352847\\ =\$31,764.24\end{array}$

Hence, the amount of $5,000 deposited each year for 5 years at 12% compound interest has a future value of $31,764.24.

Note:

FV_O stands for future value ordinary annuity

i stands for interest rate for each of the stated time periods

n stands for number of time periods

f_O stands for factor of annuity due

Future value of ordinary annuity of $1: n =5, i =12% is taken from the table value (Table 2 at the end of the time value money module).

3.

To determine

Determine the amount to be paid on January 1, 2019 by Person RS.

Answer to Problem 11E

The amount to be paid on January 1, 2016 by Person RS is $11,566.29.

Explanation of Solution

Person RS owes $30,000 on a non- interest bearing note due January 1, 2026. On January 1, 2016 he offers to pay the amount after discounting the note at 10% compounded annually.

Determine the present value as on January 1, 2016.

$\begin{array}{c}\text{PV=FV×}\left({\text{p}}_{\text{n=10,i=10\%}}\right)\\ =\$30,000\times 0.385543\\ =\$11,566.29\end{array}$

Hence, Person RS would have to pay $11,566.29, on January 1, 2016.

Note:

FV stands for Future value

PV stands for Present value

i stands for interest rate for each of the stated time periods

n stands for number of time periods

PV factor (Present value of $1: n = 10, i =10%) is taken from the table value (Table 3 at the end of the time value money module).

4.

To determine

Determine the cost of annuity for Person JS.

Answer to Problem 11E

The cost of annuity for Person JS is $50,303.06.

Explanation of Solution

Person JS will receive $6,000 each period on June 30 and December 31 for the next 6 years, from the annuity purchased on January 1, 2016. Interest rate is 12% per annum. Cash flow will be receivable in 6 months, and then the interest rate per one half-year is 6%$\left(\frac{12\%peryear}{\text{12}}\times 6\text{months in a semiannual}\right)$ .

Number of time period from June 30, 2016 to December 31, 2022 is 12 semi annum.

Determine the present value ordinary annuity (Cost of annuity).

$\begin{array}{c}{\text{PV}}_{\text{O}}\text{=Cash flow×}\left({\text{p}}_{\text{on=12,i=6\%}}\right)\\ =\$6,000\times 8.383844\\ =\$50,303.06\end{array}$

Hence, the cost of annuity is $50,303.06.

Note:

PV_O stands for present value ordinary annuity

i stands for interest rate for each of the stated time periods

n stands for number of time periods

P_O stands for factor of present value ordinary annuity

Present value of ordinary annuity of $1: n =12, i =6% is taken from the table value (Table 4 at the end of the time value money module).

5.

To determine

Determine the amount of equal annual contribution.

Answer to Problem 11E

Amount of equal annual contribution would be $4,467.20.

Explanation of Solution

n – 5 annual cash flows

I – Interest rate 10% compounded annually

Future value – $30,000

First cash flow starts on December 31, 2016, and December 31, 2020 is the last cash flow. Here, the cash flow occurs during the first day of each time period, hence it is an annuity due.

Determine the future value annuity due.

$\begin{array}{c}{\text{Future value}}_{\text{D}}\text{= Cash flow}\times \left({\text{f}}_{\text{Dn=5,i=10\%}}\right)\\ \$30,000=\text{Cash flow}\times \left({\text{f}}_{\text{On +1=6,i=10\%}}-1\right)\\ \$30,000=\text{Cash flow}\times \left(7.715610-1\right)\\ \$30,000=\text{Cash flow}\times \left(6.715610\right)\end{array}$

$\begin{array}{c}\text{Cash flow}=\frac{\$30,000}{6.715610}\\ =\$4,467.20\end{array}$

Hence, the 5 equal annual contributions are $4,467.20.

Note:

Future Value_D stands for future value annuity due

Future value of ordinary annuity of $1: n = 6, i =10% is taken from the table value (Table 2 at the end of the time value money module).

There is no separate table provided in this module for future value of annuity due. Thus, factor of annuity due is calculated with the help of ordinary annuity table.

6.

To determine

Determine the amount of equal annual withdrawals.

Answer to Problem 11E

Amount of equal annual withdrawals would be $2,525.68.

Explanation of Solution

n – 6 equal future annual withdrawals start from December 31, 2027.

I –Interest rate 10% compounded annually

Investment (Present value) – $11,000

Here, the cash flow occurs during the last day of each time period, hence it is an ordinary annuity.

Determine the present value ordinary annuity.

$\begin{array}{c}{\text{PV}}_{\text{O}}\text{= Cash flow}\times \left({\text{p}}_{\text{On=6,i=10\%}}\right)\\ \$11,000=\text{Cash flow}\times 4.355261\\ \text{Cash flow}=\frac{\$11,000}{4.355261}\\ \text{Cash flow}=\$2,525.68\end{array}$

Hence, the 6 equal annual withdrawals would be $2,525.68.

Note:

Present value of ordinary annuity of $1: n = 6, i =10% is taken from the table value (Table 4 at the end of the time value money module).