Chapter 11, Problem 37P

(a):

To determine

Calculate the annual worth.

Explanation of Solution

Defender: The first cost (FC) is $160,000. The annual operating cost (AC) is $7,000. The salvage value for the first year (SV1) is $50,000 and for the second year (SV2) is $40,000. The interest rate is 12% compounded monthly. Thus, the effective interest rate (i) is 1%$\left(\frac{0.12}{12}\right)$ .

Challenger: The first cost (FC) is $210,000. The annual operating cost (AC) is $5,000. The salvage values for first year (SV1) is $100,000, second year (SV2) is $7,000, and third year (SV3) is $45,000. The interest rate is 12% compounded monthly. Thus, the effective interest rate (i) is 1%$\left(\frac{0.12}{12}\right)$.

The annual worth (AV) is calculated using the following formula:

$\text{AV}=-\text{FC}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)-\text{AC}+\text{SVn}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)$        (1)

Substitute the respective values in Equation (1) to calculate the annual worth (AV) of defender in 12-month period.

$\begin{array}{c}\text{AV}=-160,000\left(\frac{0.01{\left(1+\text{0}\text{.01}\right)}^{12}}{{\left(1+\text{0}\text{.01}\right)}^{12}-1}\right)-7,000+50,000\left(\frac{0.01}{{\left(1+\text{0}\text{.01}\right)}^{12}-1}\right)\\ =-160,000\left(\frac{0.01\left(1.12683\right)}{1.12683-1}\right)-7,000+50,000\left(\frac{0.01}{1.12683-1}\right)\\ =-160,000\left(\frac{0.01127}{0.12683}\right)-7,000+50,000\left(\frac{0.01}{0.12683}\right)\\ =-160,000\left(0.08885\right)-7,000+50,000\left(0.07885\right)\\ =-14,216-7,000+3,942.5\\ =-17,274\end{array}$

The annual worth of the defender for 12 months is -$17,2754.

Substitute the respective values in Equation (1) to calculate the annual worth (AV) of challenger in 12-month period.

$\begin{array}{c}\text{AV}=-210,000\left(\frac{0.01{\left(1+\text{0}\text{.01}\right)}^{12}}{{\left(1+\text{0}\text{.01}\right)}^{12}-1}\right)-5,000+100,000\left(\frac{0.01}{{\left(1+\text{0}\text{.01}\right)}^{12}-1}\right)\\ =-210,000\left(\frac{0.01\left(1.12683\right)}{1.12683-1}\right)-5,000+100,000\left(\frac{0.01}{1.12683-1}\right)\\ =-210,000\left(\frac{0.01127}{0.12683}\right)-5,000+100,000\left(\frac{0.01}{0.12683}\right)\\ =-210,000\left(0.08885\right)-5,000+100,000\left(0.07885\right)\\ =-18,658.5-5,000+7,885\\ =-15,774\end{array}$

The annual worth of the challenger for 12 months is -$15,774.

The challenger can be replaced if the annual cost of challenger is lower than the defender. Thus, the firm can replace the defender with challenger.

(b):

To determine

Calculate the annual worth for 2 years.

Explanation of Solution

Table 1 shows the annual worth of the defender and challenger for 2 years (24 months), which were obtained using Equation (1).

Table 1

Project	AV
Defender	-13,048
Challenger	-12,290

The challenger can be replaced if the annual cost of challenger is lower than the defender. Thus, the firm can replace the defender with challenger.

(c):

To determine

Calculate the annual worth for 3 years.

Explanation of Solution

Table 1 shows the annual worth of the defender and challenger for 3 years (36 months), which were obtained using Equation (1).

Table 1

Project	AV
Defender	-14,292
Challenger	-10,349

The challenger can be replaced if the annual cost of challenger is lower than the defender. Thus, the firm can replace the defender with challenger.