Chapter 18, Problem 11P

To determine

Calculate the annual worth.

Explanation of Solution

Time period is indicated by (n). Interest rate is indicated by ‘i’.

The annual worth (AW) of interest rate 7% can be calculated as follows:

$\begin{array}{c}\text{AW}=-\text{Borrowing}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)-\text{MO}+\text{Selling price}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ =-12,000,000\left(\frac{\text{0}\text{.07}{\left(1+\text{0}\text{.07}\right)}^{\text{5}}}{{\left(1+\text{0}\text{.07}\right)}^{\text{5}}-1}\right)-3,100,000+2,000,000\left(\frac{\text{0}\text{.07}}{{\left(1+\text{0}\text{.07}\right)}^{\text{5}}-1}\right)\\ =-12,000,000\left(0.24389\right)-3,100,000+2,000,000\left(0.17389\right)\\ =-5,678,907\end{array}$

Therefore, the annual worth is -$5,678,907, which is less than the -$6,100,000. Thus, it is acceptable.

The annual worth (AW) of interest rate 10% can be calculated as follows:

$\begin{array}{c}\text{AW}=-\text{Borrowing}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)-\text{MO}+\text{Selling price}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ =-12,000,000\left(\frac{\text{0}\text{.1}{\left(1+\text{0}\text{.1}\right)}^{\text{5}}}{{\left(1+\text{0}\text{.1}\right)}^{\text{5}}-1}\right)-3,100,000+2,300,000\left(\frac{\text{0}\text{.1}}{{\left(1+\text{0}\text{.1}\right)}^{\text{5}}-1}\right)\\ =-12,000,000\left(0.2638\right)-3,100,000+2,300,000\left(0.1638\right)\\ =-5,888,836\end{array}$

Therefore, the annual worth is -$5,888,836, which is less than the -$6,100,000. Thus, it is acceptable.

The annual worth (AW) of interest rate 15% can be calculated as follows:

$\begin{array}{c}\text{AW}=-\text{Borrowing}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)-\text{MO}+\text{Selling price}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ =-12,000,000\left(\frac{\text{0}\text{.15}{\left(1+\text{0}\text{.15}\right)}^{\text{5}}}{{\left(1+\text{0}\text{.15}\right)}^{\text{5}}-1}\right)-3,100,000+2,500,000\left(\frac{\text{0}\text{.15}}{{\left(1+\text{0}\text{.15}\right)}^{\text{5}}-1}\right)\\ =-12,000,000\left(0.29832\right)-3,100,000+2,500,000\left(0.14832\right)\\ =-6,309,040\end{array}$

Therefore, the annual worth is -$6,309,040, which is greater than the -$6,100,000. Thus, it is not acceptable. Since the annual worth exceeds $6,100,000 at interest rate 15%, the decision is sensitive.