Question 3 of 8: Break-even Analysis and Present Value of an Equipment Decision

Question 3: Break-even Analysis and Present Value of an Equipment Decision (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. Part (a) is a single-product cost-volume-profit model valid over the output range 1,500 to 2,500 units; part (b) is a deterministic present-worth calculation over an eight-year life at 10 percent.

Find. The break-even output at each of the two prices and the effect of the price increase; and the present value of all expenditures on the machine over its eight-year life.

Approach. Part (a): set revenue equal to total cost and solve for volume, then repeat at the higher price and test the answer against the range over which the cost model was stated to hold. Part (b): discount the operating annuity with the series present-worth factor, discount the salvage receipt with the single-payment present-worth factor, and add them to the first cost with the correct sign.

1. Part (a) — write the break-even condition. Break-even is the output at which contribution exactly absorbs the fixed cost, $pQ=F+vQ$, so $$Q_{BE}=\frac{F}{p-v}$$ where $p-v$ is the contribution margin per unit. At the stated price the margin is $20-10=10$ dollars per unit.
2. Evaluate at the base price. Substituting the given data, $$Q_{BE}=\frac{25{,}000}{20-10}=\boxed{2{,}500\ \text{units per period}}$$ corresponding to break-even revenue of $$2{,}500\times\$20=\$50{,}000$$ This result sits exactly on the upper limit of the range over which the cost model was stated, so it is admissible but only just; at any higher volume the fixed cost of $25,000 would have to be re-examined for a step increase.
3. Re-evaluate at the increased price. Raising the price to $25 widens the contribution margin to 15 dollars per unit while leaving both cost parameters untouched, so $$Q_{BE}^{\prime}=\frac{25{,}000}{25-10}=1{,}666.7\;\Rightarrow\;\boxed{1{,}667\ \text{units per period}}$$ at a break-even revenue of about $41,667. The break-even output falls by 833 units, one third of its former value, because the margin rose by one half and break-even volume varies inversely with margin.
4. Interpret the shift, and note the range restriction. The firm now covers its fixed cost 833 units sooner, so every unit of the old 2,500-unit break-even volume beyond 1,667 becomes profit, and the margin of safety at any given sales level widens. Two qualifications belong in the answer. The revised break-even of 1,667 units lies below the 1,500-unit floor of the stated validity range only if output falls below 1,500, which it does not; but at 1,667 units the firm is operating near the bottom of the range where the $25,000 fixed cost was measured, so the linear model should be re-fitted before the result is used for a pricing decision. More importantly, the calculation assumes the quantity sold is unaffected by the 25 percent price rise. If demand is elastic, the lower break-even volume may still be harder to reach than the higher one was, and that trade-off, not the arithmetic, is the real decision.

1. Part (b) — assemble the three cash-flow components. The present value of net expenditure is the first cost, plus the present worth of the operating annuity, less the present worth of the salvage receipt: $$PV=P+C\,(P/A,i,n)-S\,(P/F,i,n)$$ Costs are taken as positive because the question asks for the present value of expenditures; the salvage value is a receipt and therefore carries the opposite sign.
2. Evaluate the two interest factors at 10 percent for eight years. The series present-worth factor and the single-payment present-worth factor are $$(P/A,10\%,8)=\frac{1-(1.10)^{-8}}{0.10}=5.3349,\qquad (P/F,10\%,8)=(1.10)^{-8}=0.46651$$ These are the standard tabulated values; recomputing them from the compound-interest definition rather than reading them off a table avoids the interpolation errors that creep in when a table stops at a different life.
3. Discount each component. The operating annuity is worth $$3{,}000\times5.3349=\$16{,}004.78$$ today, and the salvage receipt is worth $$4{,}000\times0.46651=\$1{,}866.03$$ today. The salvage is worth less than half its face value because it is eight years away at 10 percent, which is why salvage rarely decides an equipment choice.
4. Combine to obtain the present value of expenditures. Adding the first cost and the discounted operating costs and subtracting the discounted salvage, $$PV=24{,}000+16{,}004.78-1{,}866.03=\boxed{\$38{,}138.75}$$ It is worth converting this to an equivalent annual cost for comparison with alternatives of different lives: multiplying by the capital-recovery factor $(A/P,10\%,8)=0.187444$ gives $7,148.88 per year. Any competing machine whose equivalent annual cost is below that figure is the better buy at 10 percent.