NPV vs IRR: a value against a rate

Both measures start from the same cash flows and the same discounting. What differs is which side of the equation you hold fixed.

NPV fixes the rate and solves for a value:

$$NPV = \sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}$$

You supply $r$, the return you demand on money tied up at this risk, and the answer comes back as an amount in today's money.

IRR fixes the value at zero and solves for the rate:

$$0 = \sum_{t=0}^{n} \frac{CF_t}{(1+IRR)^t}$$

Nothing goes in but the cash flows. What comes out is the rate at which the project exactly breaks even in present value terms.

That relationship is why the two agree as often as they do, and it is worth stating with its condition attached. Take the ordinary shape, money out first and money in afterwards. Discount below the IRR and the NPV is positive, discount above it and the NPV is negative, and at the IRR it is zero. Accept or reject then lands the same way whichever measure you compute.

Turn the series around, money in first and out afterwards, and the picture inverts. That shape is a borrowing rather than an investment, its NPV rises as the discount rate rises, and the rule becomes accept when your cost of capital is above the IRR instead of below it. A rate quoted without the shape of the series behind it does not tell you which of those two cases you are in. The net present value calculator and the IRR calculator run the same arithmetic on the same series and simply report different ends of it.

A percentage says nothing about how much money is at stake. A high rate on a small outlay ranks above a moderate rate on a large one, while the large one may put far more cash in your hands. NPV carries size by construction, because it is denominated in money rather than in percent. Scale every cash flow in a project and the NPV scales with them while the IRR does not move at all, which is the same fact seen from the other side.

Rates also do not add up. Two NPVs can be summed to value a whole programme of independent work. Two IRRs cannot be averaged into anything meaningful unless the outlays and the lives happen to match, which they rarely do.

Then there is the reinvestment story, which deserves care because it is repeated everywhere. The usual version says IRR assumes every interim receipt goes straight back to work at the IRR, while NPV assumes the same at the discount rate. Neither assumption is in the arithmetic. Both measures are present value sums, and discounting states an equivalence between money at two dates rather than a forecast about where a receipt ends up. What is true is narrower and still worth knowing: read an IRR as the compound return you will actually earn across the whole life of the project and you have made that assumption yourself, which is exactly the reading a rate far above your alternatives invites. The modified internal rate of return makes the assumption explicit instead of implied, with outflows discounted at your cost of capital and inflows compounded at a reinvestment rate you name, both chosen rather than inherited from the answer.

IRR is the root of a polynomial. Substitute $x = 1/(1+r)$ into the equation above and it becomes an ordinary polynomial of degree $n$ in $x$, which can have up to $n$ roots. Descartes' rule of signs bounds how many of them are positive: at most the number of sign changes in the cash flow series, and short of that only by an even number. One sign change therefore leaves exactly one positive root, so there is exactly one rate above minus one hundred percent and the question never arises.

Series that change sign more than once are ordinary. A mine that must be restored at the end of its life, a plant needing a mid-life rebuild, a job that takes payment in advance, spends it, then collects the balance at handover: in each of them the direction of the money turns over twice rather than once. Two sign changes allow two rates that each set the value to zero, both genuine, with nothing in the mathematics to rank them. Pick whichever root a solver happens to reach first and you have published an arbitrary number as the answer. The same shape can instead leave no real rate at all, when the roots come out complex, and a series that never turns negative has none either, since there is nothing to bring to zero.

There is also no formula to fall back on. The one shape with a closed form worth writing down is a single amount out and a single amount back, and that is just a compound annual growth rate. Past a handful of periods the polynomial cannot be solved by radicals at all, so every IRR you have ever read was found by a solver iterating towards it.

NPV has none of this trouble. Hand it any series and any rate and it returns exactly one number, because it evaluates an expression rather than searching for a root. That is why the IRR calculator on this site reports a rate when the cash flows change sign exactly once, and gives the value at your cost of capital in every other case.

For a single project judged on its own merits, with one sign change in its cash flows and money going out before it comes back, the two measures agree and the choice between them is a matter of taste. That covers a good share of ordinary decisions, which is why the argument only surfaces once you are choosing between things.

What it depends on comes down to three questions. Are the projects mutually exclusive, so that taking one rules out the other? Do they differ in size? Do they differ in timing or in length, one paying back early and one paying back late? Any yes, and the two rankings can split, because IRR is silent on all three.

The split has a location. The crossover rate is the discount rate at which two projects have equal NPV, and for a pair whose profiles cross once it sits below both IRRs. Above the crossover the project with the higher IRR also has the higher NPV, so the rankings agree. Below it they part, and the higher rate belongs to the project worth less. IRR does not misrank at random, then: it ranks the pair the way NPV would if your discount rate sat above the crossover, whatever rate you actually face.

When they split, follow the NPV. The reason is not convention. NPV measures what a decision adds to your wealth at the rate you actually pay for money, and a rate cannot be spent. One qualification belongs here: where capital is genuinely rationed, so that you cannot take every project worth taking, what maximises value is NPV per unit of the scarce resource rather than NPV alone. That is still an NPV rule, and it is still not an IRR rule.

IRR earns its keep even so. It states a return without a discount rate having to be agreed first, which helps when the hurdle is arguable, and it measures headroom: a project whose IRR sits six points above the rate you are discounting at can absorb six points of increase in that rate before the decision turns over. Compute both, decide on the NPV, quote the IRR.