Chapter B, Problem 1AE

Calculate present and future values. 10-15 min.

Presented next are four independent situations related to future and present values.

Requirement

1. Using the tables in the appendix, calculate the future or present value of each item as needed

1. a. $8,000 is deposited in the bank today for a period of six years. Calculate the value of the $8,000 at the end of six years assuming it earns 7 percent interest
2. b. How much must you invest today in order to receive $3,000 at the end of each year for the next four years assuming you can earn 12 percent interest?
3. c. $4,500 will be invested at the end of each year for a period of three years. Calculate the value of the investment at the end of three years assuming it earns 10 percent interest.
4. d. The company you work for wants to purchase a new piece of equipment that is estimated to cost $29,000 ten years from now. How much must they invest today in order to have the $29,000 necessary to purchase the equipment if they can earn 6 percent interest?

To determine

Calculate the present value and the future value for each item.

Explanation of Solution

Time value of money:

Time value of money refers to the concept that the value of money available at present worth more in the future due to its potential earning capacity.

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.

$\text{Present Value = }\frac{1}{{\left(1+\text{i}\right)}^{\text{n}}}\times \text{Amount}$

Future value:

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

a.

Calculate the future value for the following transaction.

$\begin{array}{c}\text{Future value = Principal amount}\times \text{ Future value factor}\\ \text{=\$8,000}\times \text{1}\text{.501(1)}\\ =\$12,008\end{array}$

Therefore, the value of the $8,000 at the end of 6 years is $12,008.

Working Note:

Calculate the future value factor:

$\begin{array}{c}\text{Future Value factor= }{\left(1+r\right)}^n\\ ={\left(1+0.07\right)}^{{}^6}\\ =1.501\end{array}$ (1)

b.

Calculate the present value of annuity for the following transaction.

$\begin{array}{c}\text{Present value of annuity = Principal amount×Present value of annuity factor}\\ \text{=\$3,000}\times 3.037(2)\\ \text{=\$9,111}\end{array}$

Therefore, the amount of $9,111 must be invested today in order to receive $3,000 at the end of each year for the next four years.

Working note:

Calculate the present value of annuity factor:

$\begin{array}{c}\text{Present Value of Annuity factor= }\frac{\text{1}-{\left(1+\text{i}\right)}^{-\text{n}}}{\text{i}}\\ =\frac{\text{1}-{\left(1+0.12\right)}^{-4}}{12\%}\\ =3.037\end{array}$ (2)

c.

Calculate the future value of annuity for the following transaction.

$\begin{array}{c}\text{Future value of annuity = Principal amount}\times \text{ Future value of annuity factor}\\ \text{=\$4,500}\times 3.310\text{(3)}\\ =\$14,895\end{array}$

Therefore, the value of the investment at the end of three years is $14,895.

Working Note:

Calculate the future value of annuity factor:

$\begin{array}{c}\text{Future Value factor of annuity= }\frac{{(1+r)}^n-1}{\text{r}}\\ =\frac{{(1+0.10)}^3-1}{10\%}\\ =3.310\end{array}$ (3)

d.

Calculate the present value for the following transaction.

$\begin{array}{c}\text{Present value = Principal amount×Present value}\\ \text{=\$29,000}\times 0.558(4)\\ \text{=\$16,182}\end{array}$

Therefore, the company must invest an amount of $16,182 at present in order to  have the $29,000 amount to purchase the equipment.

Working note:

Calculate the present value of annuity factor:

$\begin{array}{c}\text{Present Value Factor = }\frac{1}{{\left(1+\text{i}\right)}^{\text{n}}}\\ =\frac{1}{{\left(1+0.06\right)}^{10}}\\ =0.558\end{array}$ (4)