NPV vs IRR: What to Do When the Two Rules Rank the Same Projects Differently

They answer different questions

NPV asks: given that money next year is worth less than money today, and given a rate r that reflects what capital costs us, how much surplus does this project create in today's money? You supply r, and the answer comes back as an amount. IRR asks the inverse: at what rate r would the surplus be exactly zero? You supply nothing, and the answer comes back as a percentage. That is why IRR feels attractive — it needs no assumption — and why the feeling is misleading: the assumption has not disappeared, it has just moved to the moment you compare the IRR against a hurdle rate.

For a single project with one sign change in its cash flows, the two rules always agree: NPV is positive at r exactly when IRR exceeds r. The trouble starts the moment you have to choose between projects, because a percentage carries no information about size. Fifty percent of $10,000 and thirty percent of $100,000 are not comparable quantities, and a ranking rule that treats them as comparable will pick the smaller one every time. Capital budgeting is about how much value you end the year with, not how efficiently a small sum performed.

Scale and timing: the two ways they split

Scale first. The small refit turns $10,000 into $15,000 in a year — an IRR of 50 percent and an NPV of $3,889 at 8 percent. The large refit turns $100,000 into $130,000 — 30 percent, and $20,370. IRR ranks them small-then-large; NPV ranks them large-then-small. To see which ranking is right, look at the difference between the two projects: committing the extra $90,000 returns an extra $115,000, an incremental return of 27.8 percent. Anything above your 8 percent cost of capital is worth doing, so the extra $90,000 is worth committing, and the large project wins. The 50 percent figure was never available on more than $10,000.

Timing does the same damage with identical outlays. Quick payback returns $120,000 after one year on $100,000: a 20 percent IRR and $11,111 of value. The durable asset returns $20,000 a year for ten years on the same $100,000: a 15.1 percent IRR and $34,202 of value. The rate flatters the short project because a high rate over one year compounds into very little; the money rewards the long one because it keeps producing. The two NPV lines cross at a discount rate of 13.7 percent — below that, the durable asset is worth more; above it, the quick one is. Both IRRs sit above 13.7 percent, which is exactly why the ranking conflict exists at an 8 percent cost of capital.

When IRR has two answers, or none

IRR is the root of a polynomial, so it inherits that polynomial's behaviour: a cash-flow series with k sign changes can have up to k real roots. One sign change, one IRR. Two sign changes — the shape of any project that must be dismantled, cleaned up or restocked at the end — and there can be two. Take −$100,000 to build, +$230,000 in year one, −$132,000 in year two for restoration. Discount that series at 10 percent and the NPV is exactly zero. Discount it at 20 percent and the NPV is exactly zero again. Both are internal rates of return; both are useless as a decision variable, because there is no sense in which the project returns 10 percent and also 20 percent.

NPV has no such difficulty. Discount the same mine at your actual 8 percent cost of capital and it is worth −$206: marginally value-destroying, so do not build it. At 15 percent it is worth +$189, which is the polynomial telling you that this series only has positive value for rates strictly between its two roots — a fact with no business meaning at all. Series with no sign change at all, such as a grant that only ever pays out, have no IRR whatsoever, and a spreadsheet will return an error or, worse, whichever root sits nearest your starting guess. Where a rate is genuinely wanted for reporting, use a modified internal rate of return, which fixes an explicit reinvestment rate and therefore always has exactly one solution. But rank on NPV, and rank on the same discount rate for every project you compare.

Frequently asked questions

If I can only report one number, which should it be?

NPV, with the discount rate stated next to it. A net present value of $34,202 at 8 percent is a complete statement: it names the assumption and the amount of value created under it. An IRR of 15.1 percent names neither the size of the project nor the rate you are comparing against, and it silently assumes intermediate cash flows can be reinvested at 15.1 percent, which is usually false. Report IRR alongside if your audience expects it, but never rank on it.

What is the crossover rate, and how do I find it?

It is the discount rate at which two projects have the same NPV, and you find it by subtracting one cash-flow series from the other and taking the IRR of the difference. For the two refits, the difference is −$90,000 then +$115,000, whose IRR is 27.8 percent: below that rate the large project has the higher NPV. For the quick-versus-durable pair it is 13.7 percent. The crossover rate is the honest way to state a ranking conflict, because it tells you exactly which assumption about the cost of capital would flip your decision.

My spreadsheet returned one IRR for the mine. Is it wrong?

It is not wrong, it is incomplete. Spreadsheet IRR functions search numerically from a starting guess and stop at the first root they reach, so the same cash flows will return 10 percent or 20 percent depending on the guess you pass. Neither result announces that the other exists. The reliable check costs nothing: plot NPV against the discount rate across a range and look at how many times the curve crosses zero. If it crosses more than once, discard the rate entirely and decide on NPV at your actual cost of capital.