Two Packaging Lines: NPV, IRR, and the Scale Conflict
Halcyon Foods can invest in one of two packaging lines. Line X costs $100,000 and is expected to generate $40,000 per year for four years. Line Y costs $250,000 and is expected to generate $95,000 per year for four years. Both are depreciated straight-line to zero, have no salvage value, and carry the same business risk. Halcyon's cost of capital is 10%. The finance director prefers Line X because its internal rate of return is higher; the operations director prefers Line Y because it creates more absolute value. The board has a capital budget of $250,000 for this decision and cannot invest in partial projects or repeat either line. The company uses NPV for value creation but also reports IRR and payback to the board.
1. Calculate the NPV, approximate IRR, payback period, and profitability index for each line.
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Annuity factor at 10% for 4 years = [1 − 1/(1.10)^4]/0.10 = 3.1699.
NPV(X) = 40,000 × 3.1699 − 100,000 = 126,795 − 100,000 = $26,795. NPV(Y) = 95,000 × 3.1699 − 250,000 = 301,137 − 250,000 = $51,137.
IRR(X): factor = 100,000/40,000 = 2.500. At 20% the factor is 2.5887; at 22% it is 2.4936. Interpolating: 20 + [(2.5887 − 2.500)/(2.5887 − 2.4936)] × 2 ≈ 21.9%. IRR(Y): factor = 250,000/95,000 = 2.6316. At 18% the factor is 2.6901; at 20% it is 2.5887. Interpolating: 18 + [(2.6901 − 2.6316)/(2.6901 − 2.5887)] × 2 ≈ 19.2%.
Payback(X) = 100/40 = 2.50 years. Payback(Y) = 250/95 = 2.63 years. Profitability index(X) = 126,795/100,000 = 1.268. PI(Y) = 301,137/250,000 = 1.205.
2. Which line should the board choose under the $250,000 capital budget? Explain the conflict between NPV and IRR and support your answer with an incremental analysis.
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Choose Line Y. It has the higher NPV: $51,137 versus $26,795, a difference of $24,342 in favour of Y. NPV measures the addition to shareholder wealth, which is the appropriate objective when the projects are mutually exclusive and the budget can accommodate Y.
The IRR conflict is a scale effect. X earns a higher return per dollar invested, but it invests only $100,000. Y earns a slightly lower percentage return on a much larger base, so it creates more absolute value. IRR ranks projects by percentage return, not by wealth created.
Incremental analysis confirms the choice: Y requires an extra $150,000 and produces an extra $55,000 per year for four years. Incremental NPV at 10% = 55,000 × 3.1699 − 150,000 = 174,345 − 150,000 = $24,345 > 0. The incremental IRR is the rate where 55,000 × annuity factor = 150,000, factor = 2.7273, about 17.3%. Because the cost of capital is 10%, below 17.3%, the extra investment in Y is worthwhile. If the budget were only $100,000, X would be chosen; the profitability index also favours X per dollar, but the board cannot use that ranking when projects are indivisible and Y is affordable.
3. The finance director proposes using IRR alone because it “avoids the need to estimate the cost of capital.” Evaluate this proposal.
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The proposal is not sound. IRR cannot rank mutually exclusive projects of different scale, as this case shows: X has the higher IRR but Y has the higher NPV. IRR also assumes that interim cash flows are reinvested at the IRR, which may not be realistic, and it can give multiple or no solutions when cash flows change sign more than once. It still requires a hurdle rate-the cost of capital-to decide whether a project is acceptable, so it does not avoid estimating the cost of capital.
The board should use NPV as the primary rule and report IRR and payback as supplementary information. A sensitivity check around the 10% cost of capital is useful: the crossover rate is about 17.3%, so Y remains superior for any discount rate below that. If the cost of capital were above 17.3%, the incremental investment in Y would destroy value and X would be preferred. Payback also ignores cash flows after the payback date and the time value of money, so it should not drive the decision.