Question

Using payback, ARR, and NPV with unequal cash flows

Hughes Manufacturing, Inc. has a manufacturing machine that needs attention. The company is considering two options. Option 1 is to refurbish the current machine at a cost of \(2,600,000. If refurbished, Hughes expects the machine to last another eight years and then have no residual value. Option 2 is to replace the machine at a cost of \)3,800,000. A new machine would last 10 years and have no residual value. Hughes expects the following net cash inflows from the two options:

Year	Refurbish current machine	Purchase new machine
1	\(1,760,000	\)2,970,000
2	440,000	490,000
3	360,000	410,000
4	280,000	330,000
5	200,000	250,000
6	200,000	250,000
7	200,000	250,000
8	200,000	250,000
9		250,000
10		250,000
Total	\(3,640,000	\)5,700,000

Hughes uses straight-line depreciation and requires an annual return of 10%.

Requirements

1. Compute the payback, the ARR, the NPV, and the profitability index of these two options.

2. Which option should Hughes choose? Why?

Short answer

1. Capital budgeting calculations:

Method	Refurbish current machine	Purchase new machine
Payback	3.86 years	3 years
ARR	10%	10%
NPV	$257,880	$581,520
Profitability index	1.10	1.15

2. The business entity must select the option of purchasing a new machine.

Step-by-step solution

Definition of Payback Period

A capital budgeting metric that determines the time period in which the investment will give back the cash invested or the investment/cash recovery period is known as the payback period.

Calculation of payback period, ARR, NPV, and profitability index

Calculation of payback period:

1. Refurbishment of current machine:

$\begin{array}{c}\mathrm{Payback}\mathrm{period}=\mathrm{Year}\mathrm{prior}\mathrm{to}\mathrm{full}\mathrm{recovery}+\left(\frac{\mathrm{Recovery}\mathrm{amount}\mathrm{in}\mathrm{last}\mathrm{year}}{\mathrm{Cash}\mathrm{flow}\mathrm{in}\mathrm{last}\mathrm{year}}\right)\\ =3+\left(\frac{\mathrm{\$}2,840,000-\mathrm{\$}2,600,000}{\mathrm{\$}280,000}\right)\\ =3+0.86\\ =3.86\end{array}$

2. Purchase of new machine:

$\begin{array}{c}\mathrm{Payback}\mathrm{period}=\mathrm{Year}\mathrm{prior}\mathrm{to}\mathrm{full}\mathrm{recovery}+\left(\frac{\mathrm{Recovery}\mathrm{amount}\mathrm{in}\mathrm{last}\mathrm{year}}{\mathrm{Cash}\mathrm{flow}\mathrm{in}\mathrm{last}\mathrm{year}}\right)\\ =2+\left(\frac{\mathrm{\$}3,870,000-\mathrm{\$}3,460,000}{\mathrm{\$}410,000}\right)\\ =2+1\\ =3\end{array}$

Working note:

Year	Refurbish current machine	Cumulative	Purchase new machine	Cumulative
1	$1,760,000	$1,760,000	$2,970,000	$2,970,000
2	440,000	$2,200,000	490,000	$3,460,000
3	360,000	$2,560,000	410,000	$3,870,000
4	280,000	$2,840,000	330,000
5	200,000		250,000
6	200,000		250,000
7	200,000		250,000
8	200,000		250,000
9			250,000
10			250,000
Total	$3,640,000		$5,700,000

Calculation of ARR:

1. Refurbishment of machine:

$\begin{array}{c}\mathrm{ARR}=\frac{\mathrm{Average}\mathrm{annual}\mathrm{operating}\mathrm{income}}{\mathrm{Average}\mathrm{investment}}\times 100\\ =\frac{\mathrm{\$}130,000}{\frac{\mathrm{\$}2,600,000+0}{2}}\times 100\\ =10\mathrm{\%}\end{array}$

Working note:

Particular	Amount $
Total net cash flows during the life of the project	$3,640,000
Less: Total depreciation during the life of the asset$\left(\$2,600,000-\$0\right)$	2,600,000
Total operating income during the operating life	$1,040,000
Asset operating life in years	8
Average annual operating income$\left(\frac{\$1,040,000}{8}\right)$	$130,000

2. Purchase a new machine:

$\begin{array}{c}\mathrm{ARR}=\frac{\mathrm{Net}\mathrm{cash}\mathrm{in}\mathrm{flows}}{\mathrm{Initial}\mathrm{investment}}\times 100\\ =\frac{\mathrm{\$}190,000}{\frac{\mathrm{\$}3,800,000+\mathrm{\$}0}{2}}\times 100\\ =10\mathrm{\%}\end{array}$

Working note:

Particular	Amount $
Total net cash flows during the life of the project	$5,700,000
Less: Total depreciation during the life of the asset$\left(\$3,800,000-\$0\right)$	3,800,000
Total operating income during the operating life	$1,900,000
Asset operating life in years	10
Average annual operating income$\left(\frac{\$1,900,000}{8}\right)$	$190,000

Calculation of NPV:

1. Refurbish of current machine:

Year	Refurbish current machine	X	Present value factor $\left(\frac{1}{{\left(1+r\right)}^n}\right)$	=	Present value
1	$1,760,000	X	0.909	=	$1,599,840
2	440,000	X	0.826	=	$363,440
3	360,000	X	0.751	=	$270,360
4	280,000	X	0.683	=	$191,240
5	200,000	X	0.621	=	$124,200
6	200,000	X	0.564	=	$112,800
7	200,000	X	0.513	=	$102,600
8	200,000	X	0.467	=	$93,400
Total present value of net cash inflow					$2,857,880
Less: initial investment					(2,600,000)
Net present value					$257,880

2. Purchase of new machine:

Year	Purchase new machine	X	Present value factor$\left(\frac{1}{{\left(1+r\right)}^n}\right)$	=	Present value
1	$2,970,000	X	0.909	=	$2,699,730
2	490,000	X	0.826	=	$404,740
3	410,000	X	0.751	=	$307,910
4	330,000	X	0.683	=	$225,390
5	250,000	X	0.621	=	$155,250
6	250,000	X	0.564	=	$141,000
7	250,000	X	0.513	=	$128,250
8	250,000	X	0.467	=	$116,750
9	250,000	X	0.424	=	$106,000
10	250,000	X	0.386	=	$96,500
Total present value net cash inflow					$4,381,520
Less: initial investment					(3,800,000)
Net present value					$581,520

Calculation of profitability index:

1. Refurbish current machine:

$\begin{array}{c}\mathrm{Profitability}\mathrm{index}=\frac{\mathrm{Total}\mathrm{present}\mathrm{value}\mathrm{of}\mathrm{net}\mathrm{cashflows}}{\mathrm{Initial}\mathrm{investment}}\\ =\frac{\mathrm{\$}2,857,880}{\mathrm{\$}2,600,000}\\ =1.10\end{array}$

2. Purchase of new machine:

$\begin{array}{c}\mathrm{Profitability}\mathrm{index}=\frac{\mathrm{Total}\mathrm{present}\mathrm{value}\mathrm{of}\mathrm{net}\mathrm{cashflows}}{\mathrm{Initial}\mathrm{investment}}\\ =\frac{\mathrm{\$}4,381,520}{\mathrm{\$}3,800,000}\\ =1.15\end{array}$

Appropriate option

The business entity must purchase a new machine because its payback period is lower, and NPV and profitability index are higher than the current machine’s refurbishing.