Chapter 9, Problem 10P

How much would you have to invest today to receive

a. $\text{\$15,000}$ in 8 years at 10 percent?

b. $\text{\$20,000}$ in 12 years at 13 percent?

c. $\text{\$6,000}$ each year for 10 years at 9 percent?

d. $\text{\$50,000}$ each year for 50 years at 7 percent?

Summary Introduction

To calculate: The amount required today for an investment to become $15,000 in 8 years at 10%.

Introduction

Present value:

The current value of an investment or an asset is termed as its present value. It is calculated by discounting the future value of the investment or asset.

Answer to Problem 10P

$6,997.61 is required today for the investment to become $15,000 in 8 years at 10%.

Explanation of Solution

The calculation of the present value is as follows.

$\begin{array}{c}\text{Present Value}=\frac{\text{Future Value}}{{\left(1+\text{Interest Rate}\right)}^{\text{Time Period}}}\\ =\frac{\$15,000}{{\left(1+10\%\right)}^8}\\ =\frac{\$15,000}{{\left(1.10\right)}^8}\\ =\frac{\$15,000}{2.14358881}\\ =\$6,997.61\end{array}$

Summary Introduction

To calculate: The amount required today for an investment to become $20,000 in 12 years at 13%.

Introduction

Present value:

The current value of an investment or an asset is termed as its present value. It is calculated by discounting the future value of the investment or asset.

Answer to Problem 10P

$4,614.12 is required today for the investment to become $20,000 in 12 years at 13%.

Explanation of Solution

The calculation of the present value is as follows.

$\begin{array}{c}\text{Present Value}=\frac{\text{Future Value}}{{\left(1+\text{Interest Rate}\right)}^{\text{Time Period}}}\\ =\frac{\$20,000}{{\left(1+13\%\right)}^{12}}\\ =\frac{\$20,000}{{\left(1.13\right)}^{12}}\\ =\frac{\$20,000}{4.33452310}\\ =\$4,614.12\end{array}$

Summary Introduction

To calculate: The amount required today for an investment to become $6,000 each year for 10 years at 9%.

Introduction

Present value:

The current value of an investment or an asset is termed as its present value. It is calculated by discounting the future value of the investment or asset.

Annuity:

When a payment is made or received in a series of equivalent intervals, it is termed as an annuity. Such payments can be made weekly, monthly, quarterly, or annually.

Answer to Problem 10P

$38,505.95 is required today for the investment to become $6,000 each year for 10 years at 9%.

Explanation of Solution

The calculation of the present value is as follows.

$\begin{array}{c}\text{Present Value}=\text{Annuity}\times \frac{1-\frac{1}{{\left(1+\text{Interest Rate}\right)}^{\text{Time Period}}}}{\text{Interest Rate}}\\ =\$6,000\times \frac{1-\frac{1}{{\left(1+9\%\right)}^{10}}}{9\%}\\ =\$6,000\times \frac{1-\frac{1}{{\left(1.09\right)}^{10}}}{0.09}\\ =\$6,000\times \frac{1-0.42241081}{0.09}\\ =\$6,000\times 6.41765767\\ =\$38,505.95\end{array}$

Summary Introduction

To calculate: The amount required today for an investment to become $50,000 each year for 50 years at 7%.

Introduction

Present value:

The current value of an investment or an asset is termed as its present value. It is calculated by discounting the future value of the investment or asset.

Annuity:

When a payment is made or received in a series of equivalent intervals, it is termed as an annuity. Such payments can be made weekly, monthly, quarterly, or annually.

Answer to Problem 10P

$38,505.95 is required today for an investment to become $50,000 each year for 50 years at 7%.

Explanation of Solution

The calculation of the present value is as follows.

$\begin{array}{c}\text{Present Value}=\text{Annuity}\times \frac{1-\frac{1}{{\left(1+\text{Interest Rate}\right)}^{\text{Time Period}}}}{\text{Interest Rate}}\\ =\$50,000\times \frac{1-\frac{1}{{\left(1+7\%\right)}^{50}}}{7\%}\\ =\$50,000\times \frac{1-\frac{1}{{\left(1.07\right)}^{50}}}{0.07}\\ =\$50,000\times \frac{1-0.03394776}{0.09}\\ =\$50,000\times 13.80074629\\ =\$690,037.31\end{array}$