NPV or IRR? Why the two can rank your projects in opposite order

Two quotes sit on the desk of a bakery owner in Can Tho, and there is money for only one: an oven line at 400 million dong, or a delivery-van fleet at 1 billion. The accountant reports that the oven earns 26 per cent a year and the fleet 18. Choosing the oven seems obvious. Yet at a 12 per cent discount rate, picking the oven over the fleet leaves about 55.6 million dong of present value on the table. Both options and every figure below are invented by me for illustration. Only the arithmetic is real.

What question does each measure answer?

NPV answers "how much money does this project add?", and IRR answers "what percentage a year does it earn on the money put in?". NPV discounts every cash flow to today at the discount rate and subtracts the initial outlay; IRR is the discount rate at which that total is zero. In Capital budgeting techniques the accept rules are NPV above zero and IRR above the cost of capital.

The discount rate is the opportunity cost of capital: what you could earn elsewhere at the same risk, not the interest on a loan. Both measures rest on the time value of money. NPVs add up across projects; IRRs do not.

Why does IRR pick the oven while NPV picks the fleet?

Because IRR measures a rate and NPV measures scale, and the two projects differ in both scale and timing. The cash flows below are invented, after tax, at the end of each year, in million dong, with no resale value.

Results at a 12 per cent discount rate (assumed):

The oven needs little capital and pays early, so its rate is high. The fleet needs another 600 million and pays later, but pays more: 1,700 million of total inflows against 680 million. The last column of the first table is a "bridge" project: spend an extra 600 million to swap the oven for the fleet. Its IRR is 14.63 per cent, above 12, so the extra 600 million still earns more than you require. The NPV of the difference is about 55.6 million, the gap quoted in the opening paragraph.

At what discount rate does the ranking flip?

At 14.63 per cent. Below it the fleet has the higher NPV; above it the oven does. That figure is the IRR of the difference between the two projects, and it is called the crossover rate.

"The oven has the higher IRR" holds at every discount rate, whereas "who has the higher NPV" changes with the rate. In my reading, the crossover is also the price of the extra 600 million: if you could put that money to other work earning more than 14.63 per cent, the oven plus that other use beats the fleet; if you could only earn 12 per cent, the fleet wins.

What does payback leave out?

Everything after the payback point, and the time value of money. Payback ranks the oven (2.13 years) ahead of the fleet (3.38 years), yet once paid back the oven leaves only 280 million of surplus and the fleet 700 million (undiscounted). The library's framework lists the same two weaknesses, along with the strength that payback favours liquidity Capital budgeting techniques.

Graham and Harvey write that textbooks have lamented payback's shortcomings for decades and that the cut-off is usually arbitrary (Graham and Harvey, 2001, p. 200). They find no evidence that payback is rational for severely capital-constrained firms, and lean towards unsophisticated users as the explanation (Graham and Harvey, 2001, p. 200). They also cite McDonald (1998): for very uncertain investments, rules of thumb such as payback can approximate the optimal rule (Graham and Harvey, 2001, p. 200). I have not read McDonald. My inference is that a small business can use payback as a side filter, to see how long cash stays tied up, not to rank projects.

What do CFOs actually use?

Most use both NPV and IRR. The survey went out in February 1999 and drew 392 replies, a response rate of nearly 9 per cent (Graham and Harvey, 2001, p. 191). Table 2 gives the share who always or almost always use each technique: NPV 74.93 per cent, IRR 75.61, payback 56.74, discounted payback 29.45, and a hurdle rate 56.94 (Graham and Harvey, 2001, p. 198).

The detail that matters for a small firm: on a scale of 0 to 4, average use of NPV is 2.83 among small firms and 3.42 among large ones, where "large" means revenue above $1 billion (Graham and Harvey, 2001, p. 191; Graham and Harvey, 2001, p. 197). Small firms use payback (2.72) almost as often as NPV or IRR (Graham and Harvey, 2001, p. 200). The figures are self-reports: the authors say they cannot verify that beliefs match actions (Graham and Harvey, 2001, p. 197).

When does IRR give two answers, and what is the "reinvestment" assumption?

When cash flows change sign more than once, the IRR equation can have several solutions. The third invented example is a short lease on a central kitchen: pay 400 million today, collect 1,020 million in year 1, then pay 644 million in year 2 to restore the premises. The cash flows are −400, +1,020, −644 (million dong, invented).

NPV is zero at exactly two rates, 15 and 40 per cent. At 12 per cent it is −2.68 million. The rule "accept if IRR exceeds the cost of capital" sees both IRRs above 12 per cent and says yes, while NPV at 12 per cent is negative. NPV is positive only between 15 and 40 per cent (+2.78 million at 20), so the verdict depends on your real discount rate. This is why the library's framework says IRR "may have several solutions" Capital budgeting techniques.

Then the reinvestment assumption. IRR is the true rate of return only if interim cash is reinvested at that same rate, while NPV implicitly assumes reinvestment at the discount rate. This is a standard mathematical result. For the oven, 400 million earning 26.38 per cent a year for five years must grow to 1,289.37 million. The five inflows add up to only 680 million, and reach 1,289.37 million only if every one is reinvested at 26.38 per cent. Reinvested at 12 per cent they reach 917.75 million, equal to 18.07 per cent a year. The fleet's equivalents are 2,073.22 million and 15.70 per cent (this calculation is called MIRR). MIRR still ranks the oven above the fleet: correcting the reinvestment assumption does not remove the scale difference.

When is IRR perfectly fine?

For a single conventional project (outlay first, inflows after, one sign change) where the only question is accept or reject. Then an IRR above the discount rate is equivalent to a positive NPV, and a percentage is easier to present than a sum of money. For two mutually exclusive projects, use the IRR of the difference (14.63 per cent above) or compare NPVs directly.

What does NPV not tell you?

It does not say whether the forecast cash flows are right, nor whether the discount rate is right. In the invented example the ranking is sensitive to the forecast. Suppose the inflows of both options fall short by the same proportion. At about 8.5 per cent short the two NPVs are equal, and beyond that the oven wins. At 20 per cent short the oven's NPV is still 16.61 million while the fleet's is −58.88 million.

The discount rate has to be right for each project too: at about 18 per cent the fleet's NPV falls to zero. Graham and Harvey report that more than half of respondents would use the firm's overall discount rate for a project in an overseas market, whose risk likely differs from the firm's (Graham and Harvey, 2001, p. 232). For hard-to-reverse decisions, recompute NPV under each scenario (One number or several scenarios? shows how to build them). For a new idea, test feasibility first (checking feasibility before the plan).

Going deeper: what I read, and where the limits are

The only source I cite is Graham and Harvey (2001), Journal of Financial Economics, 60(2–3), pp. 187–243. I read the full text of the PDF laid out in the journal's printed pages, from Campbell Harvey's page at Duke, so the page numbers are print pages. Table 2 gives 75.61 per cent for IRR while the text says 75.7; I used the table. I have not read the studies they cite, such as McDonald (1998); what I say about it is the authors' account.

The survey reflects the beliefs of CFOs who replied in 1999. The authors say in an appendix that the sample is representative of the population surveyed (Graham and Harvey, 2001, p. 196); I did not check that part.

The library records the framework's origin as Dean (1951), applying the discounted-cash-flow logic of Fisher (1930) and Williams (1938). I have not read these three books, so I do not cite them. Every calculation in the article is standard arithmetic, computed by two independent methods.