Chapter 11, Problem 42P

(a):

To determine

Calculate the market value of the machine.

Explanation of Solution

Defender: The annual cost of the defender for first year (ACD1) is $40,000, and it increases (ACDI) by $2,000 per year. The time period (n) is 4 years. The interest rate (i) is 12%.

Challenger: The market value (MVC) is $150,000. The annual value for the first year (ACC1) is $10,000, and it increases (ACI) by $500 per year. The salvage value (SVC) is 50,000. The time period (n1) is 10 years.

The market value of defender (MVD) is calculated as follows:

$\begin{array}{c}\left(\begin{array}{l}-\text{MVD}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ -\left(\text{AC1}+\text{ACI}\left(\frac{1}{\text{i}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{n1}}-1}\right)\right)\end{array}\right)=\left(\begin{array}{l}-\text{MVC}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n1}}}{{\left(1+\text{i}\right)}^{\text{n1}}-1}\right)\\ -\left(\text{AC1}+\text{ACI}\left(\frac{1}{\text{i}}-\frac{\text{n}}{{\left(1+\text{i}\right)}^{\text{n1}}-1}\right)\right)+\text{SVC}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n1}}-1}\right)\end{array}\right)\\ \left(\begin{array}{l}-\text{MVD}\left(\frac{\text{0}\text{.12}{\left(1+\text{0}\text{.12}\right)}^4}{{\left(1+\text{0}\text{.12}\right)}^{\text{4}}-1}\right)\\ -\left(\text{40,000}+\text{2,000}\left(\frac{1}{\text{0}\text{.12}}-\frac{\text{4}}{{\left(1+\text{0}\text{.12}\right)}^{\text{4}}-1}\right)\right)\end{array}\right)=\left(\begin{array}{l}-\text{150,000}\left(\frac{\text{0}\text{.12}{\left(1+\text{0}\text{.12}\right)}^{10}}{{\left(1+\text{0}\text{.12}\right)}^{\text{10}}-1}\right)\\ -\left(\text{10,000}+\text{500}\left(\frac{1}{\text{0}\text{.12}}-\frac{\text{5}}{{\left(1+\text{0}\text{.12}\right)}^{\text{10}}-1}\right)\right)\\ +\text{50,000}\left(\frac{\text{0}\text{.12}}{{\left(1+\text{0}\text{.12}\right)}^{\text{10}}-1}\right)\end{array}\right)\\ \left(\begin{array}{l}-\text{MVD}\left(0.32923\right)\\ -\left(\text{40,000}+\text{2,000}\left(1.3589\right)\right)\end{array}\right)=\left(\begin{array}{l}-\text{150,000}\left(0.17698\right)\\ -\left(\text{10,000}+\text{500}\left(3.5847\right)\right)+\text{50,000}\left(0.05698\right)\end{array}\right)\\ \left(-\text{MVD}\left(0.32923\right)-\left(\text{40,000}+2,717.8\right)\right)=\left(-26,547-\left(\text{10,000}+1,792.35\right)+2,849\right)\\ -\text{MVD}\left(0.32923\right)=42,717.8-35,490.35\\ -\text{MVD}=\frac{7,227.45}{0.32923}\\ \text{MVD}=-21,952.59\end{array}$

The market value of the defender is -$21,952.59. The negative sign indicates that the owner of the machine should pay the amount in order to purchase the challenger.

(b):

To determine

Calculate the annual value using spreadsheet.

Explanation of Solution

The market value of the defender (MVD) is calculated as follows:

The market value of the defender is $21,950, which is shown in cell B3. The positive sign indicates that the given amount has to be paid in order to buy machine B.