Chapter 14, Problem 29P

(a):

To determine

Calculate the present worth without considering the inflation.

Explanation of Solution

Table-1 shows the cash flow of different alternatives.

Table-1

Alternative	A	B
First cost (C)	-31,000	-48,000
Annual cost (AM) per year	-28,000	-19,000
Salvage value (SV)	5,000	7,000
Time period (n)	5	5

The interest rate (i) is 10%, and the inflation rate (f) is 3%.

The present worth (PW) for Alternative A can be calculated as follows:

$\begin{array}{c}\text{PW}=\text{C}+\text{AM}\left(\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}\right)+\text{SV}\left(\frac{1}{{\left(1+\text{i}\right)}^{\text{n}}}\right)\\ =-31,000-28,000\left(\frac{{\left(1+\text{0}\text{.1}\right)}^{\text{5}}-1}{\text{0}\text{.1}{\left(1+\text{0}\text{.1}\right)}^{\text{5}}}\right)+5,000\left(\frac{1}{{\left(1+\text{0}\text{.1}\right)}^{\text{5}}}\right)\\ =-31,000-28,000\left(3.7908\right)+5,000\left(0.6209\right)\\ =-31,000-106,143.4+3,104.5\\ =-134,038.9\end{array}$

The present worth is -$134,038.9. The present worth for Alternate A can be calculated using a spreadsheet as follows:

= - PV(10%,5,-28000,5000) – 31000

The above spreadsheet function gives the value of -$134,37.42.

The present worth (PW) for Alternative B can be calculated as follows:

$\begin{array}{c}\text{PW}=\text{C}+\text{AM}\left(\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}\right)+\text{SV}\left(\frac{1}{{\left(1+\text{i}\right)}^{\text{n}}}\right)\\ =-48,000-19,000\left(\frac{{\left(1+\text{0}\text{.1}\right)}^{\text{5}}-1}{\text{0}\text{.1}{\left(1+\text{0}\text{.1}\right)}^{\text{5}}}\right)+7,000\left(\frac{1}{{\left(1+\text{0}\text{.1}\right)}^{\text{5}}}\right)\\ =-48,000-19,000\left(3.7908\right)+7,000\left(0.6209\right)\\ =-48,000-72,025.2+4,346.3\\ =-115,\mathrm{678.9.}\end{array}$

The present worth is -$115,678.9. Since the present worth of the cost is lower for Alternate B, select Alternate B. The present worth for alternate B can be calculated using a spreadsheet as follows:

= - PV(10%,5,-19000,7000) - 48000

The above spreadsheet function gives the value of -$115,678.5.

(b):

To determine

Calculate the present worth with considering the inflation.

Explanation of Solution

The real interest rate (if) can be calculated as follows:

$\begin{array}{c}\text{if}=\text{i}+\text{f}+\text{i}\times \text{f}\\ =0.1+0.03+0.1\times 0.03\\ =0.13+0.003\\ =0.133\end{array}$

The real interest rate is 13.3%.

The present worth (PW) for Alternative A can be calculated as follows:

$\begin{array}{c}\text{PW}=\text{C}+\text{AM}\left(\frac{{\left(1+\text{if}\right)}^{\text{n}}-1}{\text{if}{\left(1+\text{if}\right)}^{\text{n}}}\right)+\text{SV}\left(\frac{1}{{\left(1+\text{if}\right)}^{\text{n}}}\right)\\ =-31,000-28,000\left(\frac{{\left(1+\text{0}\text{.133}\right)}^{\text{5}}-1}{\text{0}\text{.133}{\left(1+\text{0}\text{.133}\right)}^{\text{5}}}\right)+5,000\left(\frac{1}{{\left(1+\text{0}\text{.133}\right)}^{\text{5}}}\right)\\ =-31,000-28,000\left(3.4916\right)+5,000\left(0.5356\right)\\ =-31,000-97,764.8+2,678\\ =-126,086.8\end{array}$

The present worth is -$126,086.8. The present worth for Alternate A can be calculated using a spreadsheet as follows:

= - PV(13.3%,5,-28000,5000) - 31000

The above spreadsheet function gives the value of -$126,87.79.

The present worth (PW) for Alternative B can be calculated as follows:

$\begin{array}{c}\text{PW}=\text{C}+\text{AM}\left(\frac{{\left(1+\text{if}\right)}^{\text{n}}-1}{\text{if}{\left(1+\text{if}\right)}^{\text{n}}}\right)+\text{SV}\left(\frac{1}{{\left(1+\text{if}\right)}^{\text{n}}}\right)\\ =-48,000-19,000\left(\frac{{\left(1+\text{0}\text{.133}\right)}^{\text{5}}-1}{\text{0}\text{.133}{\left(1+\text{0}\text{.133}\right)}^{\text{5}}}\right)+7,000\left(\frac{1}{{\left(1+\text{0}\text{.133}\right)}^{\text{5}}}\right)\\ =-48,000-19,000\left(3.4916\right)+7,000\left(0.5356\right)\\ =-48,000-66,340.4+3,749.2\\ =-110,591.2\end{array}$

The present worth is -$115,678.9. Since the present worth of the cost is lower for Alternate B, select Alternate B. The present worth for Alternate B can be calculated using a spreadsheet as follows:

= - PV(13.3%,5,-19000,7000) - 48000

The above spreadsheet function gives the value of -$110,591.83.

(c):

To determine

Calculate the interest rate that makes both the alternates equal.

Explanation of Solution

The interest rate that makes both alternates’ present worth equal can be calculated using the spreadsheet as follows: