A small hydraulic machine is being planned for acquisition at a cost of $5,000. During its useful life of 7 years, it is expected to generate an annual profit of $3,400. Its annual maintenance cost is projected to be $300. The cost of space to house the machine and of the utilities is $900 per year

Question

A small hydraulic machine is being planned for acquisition at a cost of $5,000. During its useful life of 7 years, it is expected to generate an annual profit of $3,400. Its annual maintenance cost is projected to be $300. The cost of space to house the machine and of the utilities is $900 per year. Determine the payback period for the machine if the annual inflation rate is expected to be 5%.

Accepted Answer

This is Example 5.1, except that inflation is to be considered here. To evaluate the payback period, we need to determine the duration in which the $5,000 investment will be recovered.

The recurring expenditure comprises the space and utility cost of $900 and the maintenance cost of $300, both annual. Thus, the total operating cost is $900 + $300 = $1,200 per year. The net benefit from the investment in the machine is obtained by subtracting this total from the profit. This yields a net benefit of $3,400 – $1,200 = $2,200 per year, which needs to be modified for inflation.

The modification for inflation disturbs the uniform-series pattern of benefits, as seen in Example 15.1. Consequently, the payback-period formula applicable for uniform cash inflows cannot be used. Instead, the approach explained in Example 5.2 for non-uniform benefits where the [latex]cumulative[/latex] value of the net benefit is tracked is applicable. At the payback period the cumulative net benefit equals the $5,000 investment. The calculations can be streamlined by tabulating the data and the results, as follows.

Year	Net Benefit	Inflation Factor	Modified Net Benefit	Cumulative (MOD) Net Benefit
1	$2,200	[latex]1.05^{1}[/latex]	$2,095	$2,095
2	$2,200	[latex]1.05^{2}[/latex]	$1,995	$4,090
3	$2,200	[latex]1.05^{3}[/latex]	$1,900	$5,990
4	$2,200	[latex]1.05^{4}[/latex]
5	$2,200	[latex]1.05^{5}[/latex]
6	$2,200	[latex]1.05^{6}[/latex]
7	$2,200	[latex]1.05^{7}[/latex]

The inflation factors in the third column should be obvious. The modified net benefit is obtained by dividing the net benefit by the corresponding inflation factor. For example, for year 2, it is 2,200/1.05² = 1,995.

It is not essential to compute all the results for the last two columns of the table. To save time and effort, one should keep track of the cumulative value (last column) as calculations proceed. Once this value is equal to or greater than the investment ($5,000), the calculation ceases because the payback period becomes known by then.

The table shows that by the end of the third year, with cumulative net benefit of $5,990, the $5,000 investment is more than recovered. Hence the payback period is less than 3 years. The year-2 value of $4,090 suggests that it is greater than 2 years. Therefore, interpolate these two data, getting a payback period of 2.48 years, as shown as follows.

Payback period [latex]={2+(5,000-4,090)}/{(5,990-4,090)}[/latex]

[latex]={2+910}/{1,900}[/latex]

[latex]=2+0.48[/latex]

[latex]=2.48[/latex]

Should one interpolate to get the exact period for the payback? That depends on the accounting practice. If the bookkeeping follows year-end convention—usually so—then the payback period in this example is 3 years. However, if the revenues are realized round the year, and so accounted, then the exact answer of 2.48 years makes better sense.

In Example 5.1 where inflation was ignored, the payback period was determined to be 2.27 years. This example’s payback period of 2.48 years is longer since inflation delays the recovery of investment by diminishing the value or purchasing power of future benefits.

Question 15.3: A small hydraulic machine is being planned for acquisition a...

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