Tollbridge Engineering: two machines, two lives, one budget
Tollbridge Engineering must choose between two mutually exclusive machines. Machine A costs $500,000, has a four-year life, no residual value and generates annual net cash inflows of $200,000. Machine B costs $2,000,000, has a six-year life, generates annual net cash inflows of $520,000 and would be sold for $150,000 at the end of year six. Tollbridge's cost of capital is 11% and it pays no tax. The finance director prefers Machine A: its internal rate of return is higher and it ties up less capital. The capital budget for the year is $2,000,000 and the company has other positive-NPV divisible projects available.
1. Calculate NPV, IRR and equivalent annual annuity for both machines and recommend which to buy.
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Machine A
- Annuity factor 11%, 4 years = 3.102
- PV of inflows = 200,000 x 3.102 = $620,400; NPV = $120,400
- IRR: 500,000/200,000 = 2.5, giving an IRR of approximately 21.8%
- EAA = 120,400/3.102 = $38,800 a year
Machine B
- Annuity factor 11%, 6 years = 4.230; discount factor year 6 = 0.535
- PV of inflows = 520,000 x 4.230 = $2,199,600
- PV of residual = 150,000 x 0.535 = $80,250
- Total PV = $2,279,850; NPV = $279,850
- IRR: approximately 15.6%
- EAA = 279,850/4.230 = $66,200 a year
Recommendation IRR ranks A first (21.8% against 15.6%) while NPV and EAA rank B first. Where projects are mutually exclusive and differ in scale, IRR is the wrong criterion: it measures a percentage return on a small outlay rather than the absolute wealth created, and it implicitly assumes reinvestment at the IRR itself. The lives also differ, so raw NPV is not comparable; the EAA converts each into an annual equivalent.
Buy Machine B. It creates $66,200 of value per year of service against $38,800, and it fits exactly within the $2,000,000 budget. The FD's preference is driven by a relative measure that cannot rank projects of different size.
2. How far could Machine B's annual cash inflow fall before your recommendation reverses?
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Sensitivity calculation The choice reverses when B's EAA falls to A's EAA of $38,800, a fall of 66,200 - 38,800 = $27,400 a year.
Because EAA = NPV/annuity factor and the annual inflow enters NPV as inflow x annuity factor, a $1 change in the annual inflow changes the EAA by exactly $1. So the inflow may fall by $27,400, from $520,000 to $492,600, a fall of 27,400/520,000 = 5.3%.
Interpretation
- The margin is thin. A 5% shortfall in demand, a small price reduction or a two-month delay in commissioning would wipe out B's advantage.
- The residual value is a second assumption of similar size: the $150,000 sale proceeds contribute $80,250 of present value. If the resale market is weak, part of the margin goes with it.
- Machine A, by contrast, needs a much larger proportionate fall in its $200,000 inflow (27,400/200,000 on a comparable basis, but starting from a lower EAA base) before it becomes loss-making, so A is the more robust choice if Tolldbridge is risk-averse rather than value-maximising.
Judgement The recommendation of B stands on the base-case numbers, but the appraisal should not stop there. Present the board with the sensitivity, a scenario where B's inflows fall 10%, and a consideration of whether the project can be staged or whether a shorter-life machine can be leased. Where the ranking turns on 5% of a forecast, the decision should rest on the quality of the market evidence and on flexibility, not on the point estimate alone.
3. The finance director argues that with only $2,000,000 available, Machine A's higher IRR makes it the better use of scarce capital. Evaluate this argument.
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Profitability index
- A: PV of inflows/PV of outlay = 620,400/500,000 = 1.241
- B: 2,279,850/2,000,000 = 1.140
Taken alone, this supports the FD. Under hard capital rationing, ranking by PI maximises NPV per dollar of the scarce resource, and A leaves $1,500,000 for other projects.
But the comparison must be made on the incremental outlay Choosing B rather than A means investing an extra $1,500,000 to obtain an extra NPV of 279,850 - 120,400 = $159,450, an incremental PI of 1.106. The question therefore becomes: can $1,500,000 deployed in the other available projects earn more than a 10.6% NPV uplift?
- If yes, A plus those projects beats B.
- If the other projects are merely 'positive NPV' and yield less than 10.6% per dollar, B wins despite its lower PI.
- If the budget is soft rationing (internal, not imposed by lenders), the constraint should be challenged altogether and both the capital and the better project funded.
Why IRR still does not settle it IRR is not a measure of capital efficiency in the sense the FD needs; it ignores the size of the outlay and cannot be added across projects. PI captures the rationing question properly, and the incremental PI above is the test that directly answers it.
Conclusion The FD has identified the right issue but used the wrong measure. Ask for the other projects' cash flows and PIs. On the evidence given, B is preferred; A is preferred only if the alternative uses of the freed $1.5m genuinely clear an incremental PI of 1.106.