Chapter B, Problem 18E

Practical applications of the time value of money

1. a. How much would you have to deposit today if you wanted to have $60,000 in four years? Annual interest rate is 9%.
2. b. Assume that you are saving up for a trip around the world when you graduate in two years. If you can earn 8% on your investments, how much would you have to deposit today to have $15,000 when you graduate?
3. c. Would you rather have $463 now or $1,000 ten years from now? Assume that you can earn 9% on your investments.
4. d. Assume that a college parking sticker today costs $90. If the cost of parking is increasing at the rate of 5% per year, how much will the college parking sticker cost in eight years?
5. e. Assume that the average price of a new home is $158,500. If the cost of a new home is increasing at a rate of 10% per year, how much will a new home cost in eight years?
6. f. An investment will pay you $10,000 in 10 years and it will also pay you $400 at the end of each of the next 10 years (years 1 thru 10). If the annual interest rate is 6%, how much would you be willing to pay today for this type of investment?
7. g. A college student is reported in the newspaper as having won $10,000,000 in the Kansas State Lottery. However, as is often the custom with lotteries, she does not actually receive the entire $10 million now. Instead she will receive $500,000 at the end of the year for each of the next 20 years. If the annual interest rate is 6%, what is the present value (today’s amount) that she won? (Ignore taxes.)

To determine

Calculate the amount that is to be deposited today to receive $60,000 in four years.

Explanation of Solution

Present Value:

Present value refers to the current value of future sum of money in lump sum or in instalments with a stated rate of interest.

$\begin{array}{c}\text{Present value = }\left(\begin{array}{l}\text{Present value factor}\\ \text{for 4 years at 9\%}\end{array}\right)\text{×Amount expected to receive}\\ \text{=0}\text{.7084}\times \text{\$60,000}\\ \text{=\$42,504}\end{array}$

Therefore, the present value of the amount that is to be deposited today to receive $60,000 in four years is $42,504.

To determine

Calculate the amount that is to be deposited today to receive $15,000 while graduating.

Explanation of Solution

Present Value:

Present value refers to the current value of future sum of money in lump sum or in instalments with a stated rate of interest.

$\begin{array}{c}\text{Present value = }\left(\begin{array}{l}\text{Present value factor}\\ \text{for 2 years at 8\%}\end{array}\right)\text{×Amount expected to receive}\\ \text{=0}\text{.8573}\times \text{\$15,000}\\ \text{=\$12,859}\text{.60s}\end{array}$

Therefore, the present value of the amount that is to be deposited today to receive $15,000 while graduating is $12,859.60.

To determine

Identify whether $463 currently or $1,000 will be received after 10 years from now.

Explanation of Solution

Future value:

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

Present Value:

Present value refers to the current value of future sum of money in lump sum or in instalments with a stated rate of interest.

Calculate the future value of $463 at 9% for 10 years:

$\begin{array}{c}\text{Future value=}\frac{\text{Amount deposited}}{\left(\begin{array}{l}\text{ Present value factor}\\ \text{of 10 years at 9\% interest}\end{array}\right)}\\ \text{=}\frac{\text{\$463}}{\text{0}\text{.4224}}\\ \text{=\$1,096}\text{.12}\end{array}$

Therefore, the future value of $463 is $1,096.12.

Calculate the present value of $1,000 will be received after 10 years from now:

$\begin{array}{c}\text{Present value=Amount deposited}\times \left(\begin{array}{l}\text{Present value factor of }\\ \text{10 years at 9\% interest}\end{array}\right)\\ \text{ =\$1,000}\times \text{0}\text{.4224}\\ \text{= \$422}\text{.40s}\end{array}$

Therefore, the present value of $1,000 will be received after 10 years from now is $422.40

To determine

Calculate the cost of college parking sticker in eight years.

Explanation of Solution

Future value:

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

Calculate the cost of college parking sticker in eight years:

$\begin{array}{c}\text{Future value=}\frac{\text{Amount deposited}}{\left(\begin{array}{l}\text{ Present value factor}\\ \text{of 8 years at 5\% interest}\end{array}\right)}\\ \text{=}\frac{\text{\$90}}{\text{0}\text{.6768}}\\ \text{=\$132}\text{.98}\end{array}$

Therefore, the cost of college parking sticker in eight years is $132.98.

To determine

Calculate the cost of new home in eight years.

Explanation of Solution

Future value:

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

Calculate the cost of new home in eight years.

$\begin{array}{c}\text{Future value=}\frac{\text{Amount deposited}}{\left(\begin{array}{l}\text{ Present value factor}\\ \text{of 8 years at 5\% interest}\end{array}\right)}\\ \text{=}\frac{\text{\$158,500}}{\text{0}\text{.4665}}\\ \text{=\$339,764}\text{.20}\end{array}$

Therefore, the cost of new home in eight years is $339,764.20.

To determine

Calculate the amount that will be paid today for given type of investment in this situation.

Explanation of Solution

Present Value:

Present value refers to the current value of future sum of money in lump sum or in instalments with a stated rate of interest.

Annuity:

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

Calculate the amount that will be paid today for given type of investment in this situation by using a present value of a lump sum part:

$\begin{array}{c}\text{Present value = }\left(\begin{array}{l}\text{Present value factor }\\ \text{of 10 years at 6\% interest }\end{array}\right)\text{× Amount to be received}\\ \text{=0}\text{.5584}\times \text{\$10,000}\\ \text{=5,584}\end{array}$

Calculate the amount that will be paid today for given type of investment in this situation by using a present value of an annuity part:

$\begin{array}{c}\text{Present value = }\left(\begin{array}{l}\text{Present value of annuity factor }\\ \text{of 10 years at 6\% interest }\end{array}\right)\text{× Amount to be received}\\ \text{=7}\text{.3601}\times \text{\$400}\\ \text{=\$2,944}\end{array}$

Therefore, the amount that will be paid today for given type of investment in this situation is $\left[\$5,584+\$2,944\right]$ $8,528.

To determine

Calculate the present value for the given transaction.

Explanation of Solution

Present Value:

Present value refers to the current value of future sum of money in lump sum or in instalments with a stated rate of interest.

Annuity:

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

Calculate the present value:

$\begin{array}{c}\text{Present value = }\left(\begin{array}{l}\text{Present value of annuity factor }\\ \text{of 20 years at 6\% interest }\end{array}\right)\text{× Amount to be received}\\ \text{=11}\text{.4699}\times \text{\$500,000}\\ \text{=\$5,734,950}\end{array}$

Therefore, the present value is $5,734,950.