Chapter 7, Problem 5E

(a)

To determine

Calculate the present discounted value of $50,000, which is received 1 year from now.

Explanation of Solution

Present discounted value (x) of $50,000, which is received 1 year from now with 3% interest rate (R), is calculated as follows.

$\text{Present discounted value}=\frac{\text{Future value}}{{\left(\text{1}+\text{Rate of interest}\right)}^{\text{Time period}}}$        (1)

Substitute the respective values in Equation (1).

$\begin{array}{l}\text{x}=\frac{\text{\$50,000}}{{\left(\text{1}+0.03\right)}^1}\\ \text{x}=\frac{\text{\$50,000}}{{\left(1.03\right)}^1}\\ \text{x}=48,544\end{array}$

The present discounted value is $48,544.

(b)

To determine

Calculate the present discounted value of $50,000 received 10 year from now.

Explanation of Solution

Substitute the respective values in Equation (1) to calculate the present discounted value (x) of $50,000 received 10 years from now.

$\begin{array}{c}\text{x}=\frac{\text{\$50,000}}{{\left(\text{1}+0.03\right)}^{10}}\\ \text{x}=\frac{\text{\$50,000}}{{\left(1.03\right)}^{10}}\\ \text{x}=\frac{50,000}{1.3439}\\ \text{x}=37,205\end{array}$

The present discounted value is $37,205.

(c)

To determine

Calculate the present discounted value of $100 every year, forever, starting immediately.

Explanation of Solution

The present discounted value of $100 every year, forever, is calculated as follows:

$\begin{array}{c}\text{x}=100+\frac{100}{{\left(1+\text{R}\right)}^1}+\frac{100}{{\left(1+\text{R}\right)}^2}+\frac{100}{{\left(1+\text{R}\right)}^3}+\mathrm{...}\\ =100\times \left[1+\frac{100}{{\left(1+\text{R}\right)}^1}+\frac{100}{{\left(1+\text{R}\right)}^2}+\frac{100}{{\left(1+\text{R}\right)}^3}+\mathrm{...}\right]\\ =100\times \frac{1}{1-\frac{1}{1+\text{R}}}\\ =100\times \frac{1+\text{R}}{\text{R}}\\ =100\times \frac{1.03}{0.03}\\ =\$3,433\end{array}$

The present discounted value is $3,433.

(d)

To determine

Calculate the present discounted value of $100 every year, forever, starting 1 year from now.

Explanation of Solution

The present discounted value of $100 every year, forever, starting 1 year from now is calculated as follows:

$\begin{array}{c}\text{x}=\frac{1}{1+\text{R}}\times 100\times \frac{1+\text{R}}{\text{R}}\\ =\frac{1}{1+0.03}\times 100\times \frac{1.03}{0.03}\\ =\frac{1}{1.03}\times 100\times \frac{1.03}{0.03}\\ =0.9708\times 100\times 34.3333\\ =3,333\end{array}$

The present discounted value is $3,333.

(e)

To determine

Calculate the present discounted value of $100 every year for the next 50 year, starting immediately.

Explanation of Solution

The present discounted value of $100 every year, forever, is calculated as follows:

$\begin{array}{c}\text{x}=100\times \frac{1-{\left(\frac{1}{1+\text{R}}\right)}^{50}}{1-\left(\frac{1}{1+\text{R}}\right)}\\ =100\times \frac{1-{\left(\frac{1}{1+0.03}\right)}^{50}}{1-\left(\frac{1}{1+0.03}\right)}\\ =100\times \frac{1-{\left(\frac{1}{1.03}\right)}^{50}}{1-\frac{1}{\left(1.03\right)}}\\ =100\times \frac{1-0.2281}{1-0.9708}\\ =100\times \frac{0.7719}{0.0292}\\ =100\times 26.43\\ =2,643\end{array}$

The present discounted value is $2,643.