Question 4 of 7: Present Value of Ownership, and the Rate of Return on a Copier

Question 4: Present Value of Ownership, and the Rate of Return on a Copier (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Find. (a) the present value at 15 per cent of all the costs of owning and operating the computer for five years; (b) the unadjusted (accounting) rate of return on the copier purchase, and the same return after the incremental tax charge.

Approach. Part (a) is a straight discounted present-worth calculation — a single first cost, a five-year uniform series and a single recovery at the end. Part (b) is deliberately the other kind of calculation: the unadjusted rate of return is the accounting return, average annual income after depreciation divided by the investment, with no discounting at all; the adjusted figure is the same ratio after the incremental tax charge.

1. Part (a) — write the present value as three terms. Costs are positive in this statement, and the salvage is a recovery, so $$PV = P + A(P/A, i, n) - S(P/F, i, n)$$ with $P = 300{,}000$, $A = 100{,}000$, $S = 100{,}000$, $i = 0.15$ and $n = 5$, all amounts in Canadian dollars.
2. Evaluate the two interest factors at 15 per cent for five years. From the definitions, $$(P/A, 15\%, 5) = \frac{1-(1.15)^{-5}}{0.15} = 3.35216, \qquad (P/F, 15\%, 5) = (1.15)^{-5} = 0.49718$$ The salvage factor is worth noting on its own: at 15 per cent a dollar recovered five years out is worth only 49.7 cents today, so the residual value is worth barely half its face amount.
3. Substitute and total. Term by term, $$A(P/A) = 100{,}000 \times 3.35216 = 335{,}216$$ $$S(P/F) = 100{,}000 \times 0.49718 = 49{,}718$$ $$PV = 300{,}000 + 335{,}216 - 49{,}718$$ $$\boxed{\ PV = 585{,}500\ \text{CAD}\ }$$ (585,498 before rounding). Re-adding the three components confirms the total. The equivalent uniform annual cost is $585{,}498/3.35216 = 174{,}663$ CAD per year, which is the figure to compare against the annual cost of the manual alternative when the recommendation has to be made.
4. Note what the discounting is worth. The undiscounted arithmetic sum of the same cash flows is $300{,}000 + 5(100{,}000) - 100{,}000 = 700{,}000$ CAD; discounting at 15 per cent reduces the figure by 114,500 CAD, or 16.4 per cent. Reporting the undiscounted total would materially overstate the cost of ownership and, more importantly, would misrank this option against alternatives with different timing.

1. Part (b) — establish what the copier actually saves. The present arrangement costs the lease plus the per-page charge, $$C_{\text{lease}} = 6{,}500 + 0.02 \times 50{,}000 = 6{,}500 + 1{,}000 = 7{,}500\ \text{CAD per year}$$ Owning costs 1,500 CAD per year in paper and maintenance, so the gross annual saving before any charge for the machine itself is $$\Delta = 7{,}500 - 1{,}500 = 6{,}000\ \text{CAD per year}$$
2. Charge depreciation to get an accounting income. The unadjusted rate of return is an accounting measure, so the machine is written off over its economic life on a straight-line basis, $$d = \frac{P - S}{n} = \frac{10{,}000 - 2{,}000}{5} = 1{,}600\ \text{CAD per year}$$ and the average annual net income becomes $$I = \Delta - d = 6{,}000 - 1{,}600 = 4{,}400\ \text{CAD per year}$$
3. Divide by the investment for part (i). On the original investment, $$\text{unadjusted rate of return} = \frac{I}{P} = \frac{4{,}400}{10{,}000}$$ $$\boxed{\ \text{unadjusted rate of return} = 44\ \text{per cent}\ }$$ The project is plainly attractive: it also pays the machine back out of gross savings in $10{,}000/6{,}000 = 1.67$ years.
4. Deduct the incremental taxes for part (ii). The tax charge is an operating outflow of the project, so it reduces the same numerator, $$I_{\text{after tax}} = 4{,}400 - 1{,}000 = 3{,}400\ \text{CAD per year}$$ $$\boxed{\ \text{adjusted rate of return} = \frac{3{,}400}{10{,}000} = 34\ \text{per cent}\ }$$ Taxes take 22.7 per cent off the return but leave the decision unchanged.
5. Cross-check with a discounted rate of return. The accounting measures above ignore timing, so it is worth confirming the conclusion with the internal rate of return on the actual cash flows: an outlay of 10,000 now against five annual receipts of $6{,}000-1{,}000 = 5{,}000$ and a 2,000 salvage. Solving $-10{,}000 + 5{,}000(P/A,i,5) + 2{,}000(P/F,i,5) = 0$ gives $i = 43.1$ per cent after tax, and 54.4 per cent before tax. The two families of measure agree that this is a strongly positive project, which is the reassurance the cross-check is for.