Chapter 13, Problem 32P

(a):

To determine

Calculate the payback period.

Explanation of Solution

Investment (C) is $245,000. Expense (AE) is $38,000 per year. Income (AI) is $92,000. MARR (i) is 15%. Salvage value (SV) is 15% of the first cost.

Payback period (n) can be calculated as follows:

$\begin{array}{c}\left(\text{AI}-\text{AE}\right)\left(\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}\right)+\frac{\text{SV}}{{\left(1+\text{i}\right)}^{\text{n}}}=\text{C}\\ \left(92,000-38,000\right)\left(\frac{{\left(1+\text{0}\text{.15}\right)}^{\text{n}}-1}{\text{0}\text{.15}{\left(1+\text{0}\text{.15}\right)}^{\text{n}}}\right)+\frac{\text{0}\text{.15}\times \text{245,000}}{{\left(1+\text{0}\text{.15}\right)}^{\text{n}}}=245,000\\ 54,000\left(\frac{{\left(1+\text{0}\text{.15}\right)}^{\text{n}}-1}{\text{0}\text{.15}{\left(1+\text{0}\text{.15}\right)}^{\text{n}}}\right)+\frac{36,750}{{\left(1+\text{i}\right)}^{\text{n}}}=245,000\\ 54,000\left(\frac{{\left(1+\text{0}\text{.15}\right)}^{\text{n}}-1}{\text{0}\text{.15}{\left(1+\text{0}\text{.15}\right)}^{\text{n}}}\right)+\frac{36,750}{{\left(1+\text{i}\right)}^{\text{n}}}=245,000\end{array}$

Substitute n as 7 by trial and error method to verify the time period in the above calculation.

$\begin{array}{c}-54,000\left(\frac{{\left(1+\text{0}\text{.15}\right)}^{\text{7}}-1}{\text{0}\text{.15}{\left(1+\text{0}\text{.15}\right)}^{\text{7}}}\right)+\frac{36,750}{{\left(1+\text{0}\text{.15}\right)}^{\text{7}}}=-245,000\\ -54,000\left(\frac{2.6600-1}{\text{0}\text{.15}\left(2.6600\right)}\right)+\frac{36,750}{2.6600}=-245,000\\ -54,000\left(\frac{1.6600}{0.399}\right)+\frac{36,750}{2.6600}=-245,000\\ -54,000\left(4.1604\right)+13,815.79=-245,000\\ -224,661.6+13,815.79=-245,000\\ -210,845.81<-245,000\end{array}$

Since the calculated present value of the cost is less than the initial cost, increase the payback period to 8.

$\begin{array}{c}-54,000\left(\frac{{\left(1+\text{0}\text{.15}\right)}^{\text{8}}-1}{\text{0}\text{.15}{\left(1+\text{0}\text{.15}\right)}^{\text{8}}}\right)+\frac{36,750}{{\left(1+\text{0}\text{.15}\right)}^8}=-245,000\\ -54,000\left(\frac{3.05902-1}{\text{0}\text{.15}\left(3.05902\right)}\right)+\frac{36,750}{3.05902}=-245,000\\ -54,000\left(\frac{2.05902}{0.45885}\right)+12,013.65=-245,000\\ -54,000\times 4.4873+12,013.65=-245,000\\ -242,314.2+12,013.65=-245,000\\ -230,300.55\approx -245,000\end{array}$

Since the calculated value is nearly equal to the initial cost, it is verified that the payback period is 8 years.

(b):

To determine

Calculate the payback period using spreadsheet.

Explanation of Solution

The spreadsheet function for payback period is given below:

= NPER(15%,54000,-245000,0.15*245000)

The above function gives the value of 7.4 years.