Chapter M, Problem 17P

Comprehensive The following are three independent situations:

1. 1. K. Herrmann has decided to set up a scholarship fund for students. She is willing to deposit $5,000 in a trust fund at the end of each year for 10 years. She wants the trust fund to then pay annual scholarships at the end of each year for 30 years.
2. 2. Charles Jordy is planning to save for his retirement. He has decided that he can save $3,000 at the end of each year for the next 10 years, $5,000 at the end of each year for Years 11 through 20, and $10,000 at the end of each year for Years 21 through 30.
3. 3. Patricia Karpas has $200,000 in savings on the day she retires. She intends to spend $2,000 per month traveling around the world for the next 2 years, during which time her savings will earn 18%, compounded monthly. For the next 5 years, she intends to spend $6,000 every 6 months, during which time her savings will earn 12%, compounded semiannually. For the rest of her life expectancy of 15 years, she wants an annuity to cover her living costs. During this period, her savings will earn 10% compounded annually. Assume that all payments occur at the end of each period.

Required:

1. 1. In Situation 1, how much will the annual scholarships be if the fund can earn 6%? How much at 10%?
2. 2. In Situation 2,
   1. (a) How much will Charles have at the end of 30 years if his savings can earn 10%? How much at 6%?
   2. (b) If Charles expects to live for 20 years in retirement, how much can he withdraw from his savings at the end of each year if his savings earn 10%? How much at 6%?
   3. (c) How much would Charles need to invest today to have the same amount available at the time he retires as calculated in Situation 2(a) at 10%? How much at 6%?
3. 3. In Situation 3, how much will Patricia’s annuity be?

To determine

Determine the amount of annual scholarship that will be paid to the student at the end of each year for 30 years.

Explanation of Solution

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

Determine the amount of annual scholarship will be, if the fund earns 6% interest and compounded annually.

Calculate the future value of $5,000 to be deposited at the end of each year for 10 years.

$\begin{array}{c}{\text{FV}}_{\text{O}}=\text{Cash flow}\times \text{Future value of ordinary annuity at 6\% of 10 years}\\ =\text{\$5,000}\times \left({\text{f}}_{\text{On=10,i=6\%}}\right)\\ =\$5,000\times 13.180795\\ \text{=}\$65,903.98\end{array}$

Calculate amount of annual scholarship that will be paid to the student at the end of each year for 30 years.

$\begin{array}{c}{\text{FV}}_{\text{O}}=\text{Cash flow}\times \text{Future value of ordinary annuity at 6\% of 30 years}\\ \$65,903.98=\text{Cash flow}\times \left({\text{f}}_{\text{On=30,i=6\%}}\right)\\ \text{Cash flow}=\frac{\$65,903.98}{13.764831}\\ \text{=\$4,787}\text{.85}\end{array}$

Therefore, the amount of annual scholarship that will be paid to the student at the end of each year for 30 years is $4,787.85, if the fund earns 6% interest.

Determine the amount of annual scholarship will be, if the fund earns 10% interest and compounded annually.

Calculate the future value of $5,000 to be deposited at the end of each year for 10 years.

$\begin{array}{c}{\text{FV}}_{\text{O}}=\text{Cash flow}\times \text{Future value of ordinary annuity at 10\% of 10 years}\\ =\text{\$5,000}\times \left({\text{f}}_{\text{On=10,i=10\%}}\right)\\ =\$5,000\times 15.937425\\ \text{=}\$79,687.13\end{array}$

Calculate amount of annual scholarship that will be paid to the student at the end of each year for 30 years.

$\begin{array}{c}{\text{FV}}_{\text{O}}=\text{Cash flow}\times \text{Future value of ordinary annuity at 10\% of 30 years}\\ \$79,687.13=\text{Cash flow}\times \left({\text{f}}_{\text{On=30,i=10\%}}\right)\\ \text{Cash flow}=\frac{\$79,687.13}{9.426914}\\ \text{=\$8,453}\text{.15}\end{array}$

Therefore, the amount of annual scholarship that will be paid to the student at the end of each year for 30 years is $8,453.15, if the fund earns 10% interest.

To determine

Determine the amount will Person C have at the end of 30 years.

Explanation of Solution

Determine the amount will Person C have at the end of 30 years, if the savings can earn 10%.

Calculate the future value of $3,000 to be deposited at the end of each year for next 10 years.

$\begin{array}{c}{\text{FV}}_{\text{O}}=\text{Cash flow}\times \text{Future value of ordinary annuity at 10\% of 10 years}\\ =\text{\$3,000}\times \left({\text{f}}_{\text{On=10,i=10\%}}\right)\\ =\$3,000\times 15.937425\\ \text{=}\$47,812.28\end{array}$

Now, convert this $47,812.28 value of savings available in the savings account in year 10 to the future value at the end of year 30.

$\begin{array}{c}\text{FV}=\text{PV}\times \text{Future value of \$1 at 10\% of 20 years}\\ \text{=PV}\times \left({\text{f}}_{\text{n=20, i=10\%}}\right)\\ =\$47,812.28\times 6.727500\\ =\$321,657.11\end{array}$

Calculate the future value of $5,000 to be deposited at the end of each year for Years 11 through 20.

$\begin{array}{c}{\text{FV}}_{\text{O}}=\text{Cash flow}\times \text{Future value of ordinary annuity at 10\% of 10 years}\\ =\text{\$5,000}\times \left({\text{f}}_{\text{On=10,i=10\%}}\right)\\ =\$5,000\times 15.937425\\ \text{=}\$79,687.13\end{array}$

Now, convert this $79,687.13 value of savings available in the savings account in year 20 (earned from Year 11 through 20) to the future value at the end of year 30.

$\begin{array}{c}\text{FV}=\text{PV}\times \text{Future value of \$1 at 10\% of 10 years}\\ \text{=PV}\times \left({\text{f}}_{\text{n=10, i=10\%}}\right)\\ =\$79,687.13\times 2.593742\\ =\$206,687.86\end{array}$

Calculate the future value of $10,000 to be deposited at the end of each year for Years 21 through 30.

$\begin{array}{c}{\text{FV}}_{\text{O}}=\text{Cash flow}\times \text{Future value of ordinary annuity at 10\% of 10 years}\\ =\text{\$10,000}\times \left({\text{f}}_{\text{On=10,i=10\%}}\right)\\ =\$10,000\times 15.937425\\ \text{=}\$159,374.25\end{array}$

Finally, calculate the amount will Person C have at the end of 30 years, if the savings can earn 10%.

Deposit period	Present value
$3,000 deposited each year for first 10 years	$321,657.11
$5,000 deposited each year for next 10 years (Year 11 through 20)	$206,687.86
$10,000 deposited each year for last 10 years (Year 21 through 30)	$159,374.25
Amount available in savings account at the end of year 30	$687,719.22

Therefore, the amount will Person C have at the end of 30 years, if the savings can earn 10% is $687,719.22.

Determine the amount will Person C have at the end of 30 years, if the savings can earn 6%.

Calculate the future value of $3,000 to be deposited at the end of each year for next 10 years.

$\begin{array}{c}{\text{FV}}_{\text{O}}=\text{Cash flow}\times \text{Future value of ordinary annuity at 6\% of 10 years}\\ =\text{\$3,000}\times \left({\text{f}}_{\text{On=10,i=6\%}}\right)\\ =\$3,000\times 13.180795\\ \text{=}\$39,542.39\end{array}$

Now, convert this $39,542.39 value of savings available in the savings account in year 10 to the future value at the end of year 30.

$\begin{array}{c}\text{FV}=\text{PV}\times \text{Future value of \$1 at 6\% of 20 years}\\ \text{=PV}\times \left({\text{f}}_{\text{n=20, i=6\%}}\right)\\ =\$39,542.39\times 3.207135\\ =\$126,817.78\end{array}$

Calculate the future value of $5,000 to be deposited at the end of each year for Years 11 through 20.

$\begin{array}{c}{\text{FV}}_{\text{O}}=\text{Cash flow}\times \text{Future value of ordinary annuity at 6\% of 10 years}\\ =\text{\$5,000}\times \left({\text{f}}_{\text{On=10,i=6\%}}\right)\\ =\$5,000\times 13.180795\\ \text{=}\$65,903.98\end{array}$

Now, convert this $65,903.98 value of savings available in the savings account in year 20 (earned from Year 11 through 20) to the future value at the end of year 30.

$\begin{array}{c}\text{FV}=\text{PV}\times \text{Future value of \$1 at 6\% of 10 years}\\ \text{=PV}\times \left({\text{f}}_{\text{n=10, i=6\%}}\right)\\ =\$65,903.98\times 1.790848\\ =\$118,024.01\end{array}$

Calculate the future value of $10,000 to be deposited at the end of each year for Years 21 through 30.

$\begin{array}{c}{\text{FV}}_{\text{O}}=\text{Cash flow}\times \text{Future value of ordinary annuity at 6\% of 10 years}\\ =\text{\$10,000}\times \left({\text{f}}_{\text{On=10,i=6\%}}\right)\\ =\$10,000\times 13.180795\\ \text{=}\$131,807.95\end{array}$

Finally, calculate the amount will Person C have at the end of 30 years, if the savings can earn 6%.

Deposit period	Present value
$3,000 deposited each year for first 10 years	$126,817.78
$5,000 deposited each year for next 10 years (Year 11 through 20)	$118,024.01
$10,000 deposited each year for last 10 years (Year 21 through 30)	$131,807.95
Amount available in savings account at the end of year 30	$376,649.74

Therefore, the amount will Person C have at the end of 30 years, if the savings can earn 6% is $376,649.74.

To determine

Determine the amount that can be withdrawn by Person C at the end of each year, if he expects to live for 20 years in retirement.

Explanation of Solution

Determine the amount that can be withdrawn by Person C at the end of each year, if he expects to live for 20 years in retirement, if the savings can earn 10%.

$678,719.22 is the amount will be available in the savings account on the date of Person C’s retirement, which is calculated in part 2(a).

$\begin{array}{c}{\text{PV}}_{\text{O}}=\text{Cash flow}\times \text{Present value of ordinary annuity at 10\% of 20 years}\\ \$687,719.22=\text{Cash flow}\times \left({\text{P}}_{\text{On=20,i=10\%}}\right)\\ \$687,719.22=\text{Cash flow}\times \text{8}\text{.513564}\\ \text{Cash flow}\text{=}\$80,779.24\end{array}$

Therefore, an amount of $80,779.25 can be withdrawn by Person C at the end of each year, if he expects to live for 20 years in retirement and if the savings can earn 10%.

Determine the amount that can be withdrawn by Person C at the end of each year, if he expects to live for 20 years in retirement, if the savings can earn 6%.

$376,649.74 is the amount will be available in the savings account on the date of Person C’s retirement, which is calculated in part 2(a).

$\begin{array}{c}{\text{PV}}_{\text{O}}=\text{Cash flow}\times \text{Present value of ordinary annuity at 6\% of 20 years}\\ \$376,649.74=\text{Cash flow}\times \left({\text{P}}_{\text{On=20,i=6\%}}\right)\\ \$376,649.74=\text{Cash flow}\times 11.469921\\ \text{Cash flow}\text{=}\$32,838.04\end{array}$

Therefore, an amount of $32,838.04 can be withdrawn by Person C at the end of each year, if he expects to live for 20 years in retirement and if the savings can earn 6%.

To determine

Determine the amount to be invested by Person C today to have the same amount available at the time he retires as calculated in situation 2 (a).

Explanation of Solution

Determine the amount to be invested by Person C today to have the same amount available on his retirement as calculated in situation 2 (a).

$\begin{array}{c}\text{PV}=\text{Future value}\times \text{Present value of \$1 at 10\% of 30 years}\\ =\$687,719.22\times \left({\text{P}}_{\text{n=30, i=10\%}}\right)\\ =\$687,719.22\times \text{0}\text{.057309}\\ \text{=\$39,412}\text{.50}\end{array}$

Therefore, amount to be invested by Person C today to have the same amount available at the time he retires as calculated in situation 2 (a) is $39,412.50.

Determine the amount to be invested by Person C today to have the same amount available on his retirement as calculated in situation 2 (b).

$\begin{array}{c}\text{PV}=\text{Future value}\times \text{Present value of \$1 at 6\% of 30 years}\\ =\$376,649.74\times \left({\text{P}}_{\text{n=30, i=6\%}}\right)\\ =\$376,649.74\times \text{0}\text{.174110}\\ \text{= \$65,578}\text{.49}\end{array}$

Therefore, amount to be invested by Person C today to have the same amount available at the time he retires as calculated in situation 2 (b) is $65,578.49.

To determine

Determine the annuity that Person P wants to cover the last 15 years of life expectancy.

Explanation of Solution

First calculate value of savings available at the end of year 2.

$\begin{array}{c}\begin{array}{l}\text{Value of savings at}\\ \text{the end of Year 2}\end{array}\}\text{ = }\left(\text{PV}\times \left({\text{f}}_{n=2\times 12,i=\frac{18\%}{12}}\right)\right)-\left(\text{C}\times \left({\text{f}}_{On=2\times 12,i=\frac{18\%}{12}}\right)\right)\\ =\left(\$200,000\times 1.429503\right)-\left(\$2,000\times 28.633521\right)\\ =\$285,900.60-\$57,267.04\\ =\$228,633.56\end{array}$

Next, calculate value of savings available at the end of year 7.

$\begin{array}{c}\begin{array}{l}\text{Value of savings at}\\ \text{the end of Year 7}\end{array}\}\text{ = }\left(\text{PV}\times \left({\text{f}}_{n=5\times 2,i=\frac{12\%}{2}}\right)\right)-\left(\text{C}\times \left({\text{f}}_{On=5\times 2,i=\frac{12\%}{2}}\right)\right)\\ =\left(\$228,633.56\times 1.790848\right)-\left(\$6,000\times 13.180795\right)\\ =\$409,447.95-\$79,084.77\\ =\$330,363.18\end{array}$

Now calculate amount of annuity that Person P wants to cover the last 15 years of her life expectancy.

$\begin{array}{c}\text{Annuity=}\frac{{\text{PV}}_{\text{O}}}{{\text{P}}_{\text{On=15, i=10\%}}}\\ =\frac{\$330,363.18}{7.60608}\\ =\$43,434.09\end{array}$

Therefore, the amount of annuity that Person P wants to cover the last 15 years of life expectancy is $43,434.09.