Arithmetic Series Under a government environment incentive plan companies installing air-pollution control equipment are entitled to a 50% grant to offset the purchase and maintenance costs.XYZ decides to avail of this incentive by acquiring equipment costing $50,000. The maintenance costs are

Question

Arithmetic Series

Under a government environment incentive plan companies installing air-pollution control equipment are entitled to a 50% grant to offset the purchase and maintenance costs. XYZ decides to avail of this incentive by acquiring equipment costing $50,000. The maintenance costs are estimated to be $4,000 in the first year, rising annually by $300. If the equipment’s useful life is 10 years, and annual interest rate is 6% compounded half-yearly, how much will the grant be?

Accepted Answer

If you can comprehend the problem, there is no need to draw the cashflow diagram. Otherwise, draw one. Note that the maintenance cost increases every year by the same amount, so the maintenance costs form an arithmetic series, for which an equation has been derived above. We need to determine the equivalent present cost, 50% of which will be paid by the government as a grant.

The present cost comprises the purchase price and the present value of all the maintenance costs over 10 years. The latter is the equivalent P-value for an arithmetic-series cashflow with A = $4,000 and G = $300. For determining it, we can use Equation (4.5a) or (4.5b). Let us opt for functional-notation-based Equation (4.5b). Since the compounding is half-yearly, [latex]i[/latex] is 3%, and for the same reason n is 20—number of compoundings over 10 years (if you have difficulty with these values of [latex]i[/latex] and n, reread the relevant sections of Chapter 3). Thus, the relevant data are: G = $300, [latex]i=3\%[/latex], and n = 20, for which the total present cost on the equipment[latex]^{16}[/latex].

[latex]P=G\frac{\left(1+i ight)^n-ni-1 }{i^2\left(1+i ight)^n } [/latex] (4.5a)

[latex]P=G\left\lgroup\frac{P}{G,i,n} ight group [/latex] (4.5b)

= Purchase cost + Present value of maintenance costs

[latex]=50,000+\left\{A({P}/{A},i,n)+G({P}/{G},i,n) ight\}[/latex]

[latex]=50,000+4,000({P}/{A},3\%,20) +300({P}/{G},3\%,20)[/latex]

[latex]=50,000+4,000 imes 14.877+300 imes 126.799[/latex]

[latex]=50,000+59.508+38.040[/latex]

[latex]=\$147,548[/latex]

Being 50% of the total present cost, the grant will be $73,774.

[latex]^{16}[/latex]See Fig. 4.7.

Question 4.4: Arithmetic Series Under a government environment incentive p...

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