Chapter 11, Problem 26P

(a):

To determine

Calculate the annual worth.

Explanation of Solution

Defender: Market value (MV) is $7,000. Upgrade cost (UC) is $22,000. Annual operating cost (AC) is $27,000. Salvage value (SV) is $21,000. Time period (n) is 3. Interest rate (i) is 10%.

Challenger: First cost (FC) is $65,000. Annual operating cost (AC) is $14,000. Salvage value (SV) is $123,000. Time period (n) is 3. Interest rate (i) is 10%.

Annual worth (AV) of the defender can be calculated as follows:

$\begin{array}{c}\text{AV}=-\left(\text{MV}+\text{UC}\right)\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)-\text{AC}+\text{SV}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ =-\left(7,000+22,000\right)\left(\frac{\text{0}\text{.1}{\left(1+\text{0}\text{.1}\right)}^{\text{3}}}{{\left(1+\text{0}\text{.1}\right)}^{\text{3}}-1}\right)-\text{27,000}+\text{12,000}\left(\frac{\text{0}\text{.1}}{{\left(1+\text{0}\text{.1}\right)}^{\text{3}}-1}\right)\\ =-29,000\left(\frac{\text{0}\text{.1}\left(1.331\right)}{1.331-1}\right)-\text{27,000}+\text{12,000}\left(\frac{\text{0}\text{.1}}{1.331-1}\right)\\ =-29,000\left(\frac{\text{0}\text{.1331}}{0.331}\right)-\text{27,000}+\text{12,000}\left(\frac{\text{0}\text{.1}}{0.331}\right)\\ =-29,000\left(0.402115\right)-\text{27,000}+\text{12,000}\left(0.302115\right)\\ =-11,661.34-27,000+3,625.38\\ =-35,035.96\end{array}$

The annual worth of the defender is -$35,035.96.

Annual worth (AV) of the challenger can be calculated as follows:

$\begin{array}{c}\text{AV}=-\text{FC}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)-\text{AC}+\text{SV}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ =-65,000\left(\frac{\text{0}\text{.1}{\left(1+\text{0}\text{.1}\right)}^{\text{3}}}{{\left(1+\text{0}\text{.1}\right)}^{\text{3}}-1}\right)-14,000+23,000\left(\frac{\text{0}\text{.1}}{{\left(1+\text{0}\text{.1}\right)}^{\text{3}}-1}\right)\\ =-65,000\left(\frac{\text{0}\text{.1}\left(1.331\right)}{1.331-1}\right)-14,000+23,000\left(\frac{\text{0}\text{.1}}{1.331-1}\right)\\ =-65,000\left(\frac{\text{0}\text{.1331}}{0.331}\right)-14,000+23,000\left(\frac{\text{0}\text{.1}}{0.331}\right)\\ =-65,000\left(0.402115\right)-14,000+23,000\left(0.302115\right)\\ =-26,137.48-14,000+6,948.65\\ =-33,188.83\end{array}$

The annual worth of the challenger is -$33,188.83. Since the annual worth of cost for challenger is lesser than the defender, replace the defender with the challenger.

(b):

To determine

Calculate the annual worth for 2 year.

Explanation of Solution

Annual worth (AV) of the challenger can be calculated as follows:

$\begin{array}{c}\text{AV}=-\text{FC}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)-\text{AC}+\text{SV}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ =-65,000\left(\frac{\text{0}\text{.1}{\left(1+\text{0}\text{.1}\right)}^{\text{2}}}{{\left(1+\text{0}\text{.1}\right)}^{\text{2}}-1}\right)-14,000+23,000\left(\frac{\text{0}\text{.1}}{{\left(1+\text{0}\text{.1}\right)}^{\text{2}}-1}\right)\\ =-65,000\left(\frac{\text{0}\text{.1}\left(1.21\right)}{1.21-1}\right)-14,000+23,000\left(\frac{\text{0}\text{.1}}{1.21-1}\right)\\ =-65,000\left(\frac{\text{0}\text{.121}}{0.21}\right)-14,000+23,000\left(\frac{\text{0}\text{.1}}{0.21}\right)\\ =-65,000\left(0.57619\right)-14,000+23,000\left(0.47619\right)\\ =-37,452.35-14,000+10,952.37\\ =-40,499.98\end{array}$

The annual worth of the challenger is -$40,499.98. Since the annual worth of 2 year cost for challenger is greater than the defender for 3 year cost, retain the defender.