Chapter 17, Problem 60P

(a):

To determine

Calculate the annual worth.

Explanation of Solution

Time period is denoted by ‘n’ and after tax return is denoted by ‘i’. The remaining time period is denoted by ‘rn’. The annual value (AW) of defender can be calculated as follows:

$\begin{array}{c}{\text{AW}}_{\text{Defender}}=\left(\begin{array}{l}-\text{Market value}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{rn}}}{{\left(1+\text{i}\right)}^{\text{rn}}-1}\right)+\text{Salvage value}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{rn}}-1}\right)\\ -\text{OE}+\left(\left(\text{OE}+\frac{\text{First cost}-\text{Salvage value}}{\text{n}}\right)\times \text{Effective tax rate}\right)\end{array}\right)\\ =\left(\begin{array}{l}-\text{15,000}\left(\frac{\text{0}\text{.06}{\left(1+\text{0}\text{.06}\right)}^{\text{5}}}{{\left(1+\text{0}\text{.06}\right)}^{\text{5}}-1}\right)+\text{2,000}\left(\frac{\text{0}\text{.06}}{{\left(1+\text{0}\text{.06}\right)}^{\text{5}}-1}\right)\\ -\text{1,200}+\left(\left(\text{1,200}+\frac{28,000-2,000}{\text{10}}\right)\times \text{0}\text{.06}\right)\end{array}\right)\\ =\left(-\text{15,000}\left(0.2374\right)+\text{2,000}\left(0.1774\right)-\text{1,200}+228\right)\\ =-4,178\end{array}$

The annual worth of the defender is -$4,178.

The annual worth (AW) of challenger can be calculated as follows:

$\begin{array}{c}{\text{AW}}_{\text{Challenger}}=\left(\begin{array}{l}\left(-{\text{First cost}}_{\text{Challenger}}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\right)+{\text{Salvage}}_{\text{Challenger}}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)-{\text{OE}}_{\text{Challenger}}\\ +\left({\text{OE}}_{\text{Challenger}}+\frac{{\text{First cost}}_{\text{Challenger}}-{\text{Salvage}}_{\text{Challenger}}}{\text{n}}\right)\times \text{Effective tax rate}\end{array}\right)\\ =\left(\begin{array}{l}\left(-\text{15,000}\right)\left(\frac{\text{0}\text{.06}{\left(1+\text{0}\text{.06}\right)}^{\text{8}}}{{\left(1+\text{0}\text{.06}\right)}^{\text{8}}-1}\right)\\ +3,000\left(\frac{\text{0}\text{.1}}{{\left(1+\text{0}\text{.06}\right)}^{\text{8}}-1}\right)-\text{1,500}+\left(\text{1,500}+\frac{15,000-3,000}{\text{8}}\right)\times \text{0}\text{.06}\end{array}\right)\\ =\left(\left(-\text{15,000}\right)\left(0.16104\right)+3,000\left(0.10104\right)-\text{1,500}+180\right)\\ =-3,432\end{array}$

The annual worth of the challenger is -$3,432. Since the annual worth of the challenger is greater than the defender, select the challenger.

(b):

To determine

Calculate the annual worth through spreadsheet.

Explanation of Solution

The annual worth of the defender and challenger can be calculated using spreadsheet as follows:

(c):

To determine

Calculate the new annual worth.

Explanation of Solution

The annual value (AW) of defender can be calculated as follows:

$\begin{array}{c}{\text{AW}}_{\text{Defender}}=\left(-\text{Market value}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{rn}}}{{\left(1+\text{i}\right)}^{\text{rn}}-1}\right)+\text{Salvage value}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{rn}}-1}\right)-\text{OE}\right)\\ =\left(-\text{15,000}\left(\frac{\text{0}\text{.12}{\left(1+\text{0}\text{.12}\right)}^{\text{5}}}{{\left(1+\text{0}\text{.12}\right)}^{\text{5}}-1}\right)+\text{2,000}\left(\frac{\text{0}\text{.12}}{{\left(1+\text{0}\text{.12}\right)}^{\text{5}}-1}\right)-\text{1,200}\right)\\ =\left(-\text{15,000}\left(0.2774\right)+\text{2,000}\left(0.1574\right)-\text{1,200}\right)\\ =-5,046\end{array}$

The annual worth of the defender is -$5,046.

The spreadsheet function to calculate the annual worth of the defender is given below:

= -PMT(12%,5,-15000,2000) – 1200

The above function gives the value of -$5046.

The annual worth (AW) of challenger can be calculated as follows:

$\begin{array}{c}{\text{AW}}_{\text{Challenger}}=\left(\left(-{\text{First cost}}_{\text{Challenger}}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\right)+{\text{Salvage}}_{\text{Challenger}}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)-{\text{OE}}_{\text{Challenger}}\right)\\ =\left(\left(-\text{15,000}\right)\left(\frac{\text{0}\text{.12}{\left(1+\text{0}\text{.12}\right)}^{\text{8}}}{{\left(1+\text{0}\text{.12}\right)}^{\text{8}}-1}\right)+3,000\left(\frac{\text{0}\text{.1}}{{\left(1+\text{0}\text{.12}\right)}^{\text{8}}-1}\right)-\text{1,500}\right)\\ =\left(\left(-\text{15,000}\right)\left(0.16104\right)+3,000\left(0.10104\right)-\text{1,500}\right)\\ =-4,276\end{array}$

The annual worth of the challenger is -$4,276. Since the annual worth of the challenger is greater than the defender, select the challenger.

The spreadsheet function to calculate the annual worth of the defender is given below:

= -PMT(12%,8,-15000,3000) – 1500

The above function gives the value of -$4,276.