Chapter 11, Problem 15P

(a):

To determine

Calculate the economic service life.

Explanation of Solution

First cost (FC) is $180,000. Annual cost (AC) in year 1 and 2 is $84,000 after it increases by $5,000 per year. Salvage value (SV) is 25% to the first cost. MARR (i) is 15%. Time period (n) is 5 years.

Table -1 shows the total annual salvage value for different years as follows:

Table -1

Year	AC	SV
1	84,000	45,000
2	84,000	45,000
3	89,000	45,000
4	94,000	45,000
5	99,000	45,000

Total annual value (TA) of the cost can be calculated using the formula as follows:

${\text{TA}}_{\text{Year n}}=\left(\begin{array}{l}-\text{FC}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\\ -\left(\begin{array}{l}\frac{{\text{AC}}_1}{{\left(1+\text{i}\right)}^{\text{1}}}+\frac{{\text{AC}}_2}{{\left(1+\text{i}\right)}^2}+\frac{{\text{AC}}_3}{{\left(1+\text{i}\right)}^3}\\ +\frac{{\text{AC}}_4}{{\left(1+\text{i}\right)}^4}+\frac{{\text{AC}}_5}{{\left(1+\text{i}\right)}^{\text{5}}}\end{array}\right)\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)+\text{SV}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)\end{array}\right)$        (1)

The value of n1 in the first year is 1. Substitute the respective values in Equation (1) to calculate the total annual value in the year 1.

$\begin{array}{c}{\text{TA}}_{\text{1}}=\left(\begin{array}{l}-\text{180,000}\left(\frac{\text{0}\text{.15}{\left(1+\text{0}\text{.15}\right)}^1}{{\left(1+\text{0}\text{.15}\right)}^1-1}\right)\\ -\left(\begin{array}{l}\frac{84,000}{{\left(1+\text{0}\text{.15}\right)}^1}+\frac{0}{{\left(1+\text{0}\text{.15}\right)}^2}+\frac{0}{{\left(1+\text{0}\text{.15}\right)}^3}\\ +\frac{0}{{\left(1+\text{0}\text{.15}\right)}^4}+\frac{0}{{\left(1+\text{0}\text{.15}\right)}^5}\end{array}\right)\left(\frac{\text{0}\text{.15}{\left(1+\text{0}\text{.15}\right)}^1}{{\left(1+\text{0}\text{.15}\right)}^1-1}\right)+\text{45,000}\left(\frac{\text{0}\text{.15}}{{\left(1+\text{0}\text{.15}\right)}^{\text{1}}-1}\right)\end{array}\right)\\ =\left(-\text{180,000}\left(\frac{\text{0}\text{.15}\left(1.15\right)}{\left(1.15\right)-1}\right)-\left(\frac{84,000}{1.15}\right)\left(\frac{\text{0}\text{.15}\left(1.15\right)}{\left(1.15\right)-1}\right)+\text{45,000}\left(\frac{\text{0}\text{.15}}{1.15-1}\right)\right)\\ =\left(-\text{180,000}\left(\frac{0.1725}{0.15}\right)-\left(\frac{84,000}{1.15}\right)\left(\frac{0.1725}{0.15}\right)+\text{45,000}\left(\frac{\text{0}\text{.15}}{0.15}\right)\right)\\ =\left(-\text{180,000}\left(1.15\right)-\left(\frac{84,000}{1.15}\right)\left(1.15\right)+\text{45,000}\left(1\right)\right)\\ =\left(-207,000-84,000+\text{45,000}\right)\\ =-246,000\end{array}$

The total annual value in the year 1 is -$246,000.

Table -2 shows the total annual value for the different years that are obtained using Equation (1) as follows:

Table -2

Year	Total annual value
1	-246,000
2	-173,791
3	-1581,317
4	-141,190
5	-135,934

Economic service life occurs at the time period where the annual value of the cost is lower. The lower total annual value -$135,934 occurs at the year 5. Thus, economic service life is 5.

(b):

To determine

Calculate the economic service life using spreadsheet.

Explanation of Solution

Total annual value (TA) of the cost can be calculated using the spreadsheet as follows:

Thus, the lower value of the total annual value is -$135,934 and the economic service life is 5.