IRR Calculator

The calculation itself

IRR is the discount rate at which NPV equals zero — the point where a project stops adding value and starts destroying it.

There is no closed form. The equation is a polynomial in the discount factor, and beyond a couple of periods it has to be solved numerically. This one uses bisection: bracket a sign change and halve repeatedly. It is slower than Newton's method and cannot diverge, which is the right trade for a function that is known to misbehave.

Two consequences follow directly from the maths, and most calculators hide both. A series that never changes sign has no root at all. A series that changes sign several times can have several.

In practice

The same project as the NPV page: $1,000 out, then $500, $400, $300 and $100 back.

The IRR is 14.49%. Check it the other way — NPV evaluated at 14.4888% comes to zero to nine decimal places, which is the definition satisfied rather than a coincidence.

Against a 10% cost of capital that leaves 4.49 points of margin. That margin is not profit; it is the room the forecast has to be wrong in before the project stops paying for its capital.

Below the IRR, NPV is positive. Above it, negative. The two measures are the same statement read along different axes, which is why they only disagree when scale or timing enters.

When IRR has no answer

All-positive cash flows have no IRR. Neither do all-negative ones. There is no rate that makes a series of same-signed amounts sum to zero, and returning a number for it would mean inventing one.

Multiple sign changes — a project needing reinvestment partway through, a mine with closure costs — can produce several rates that all satisfy the equation. None is more correct than the others.

Descartes' rule of signs sets the bound: the number of sign changes limits how many positive roots there can be. This calculator counts them and says so, because a single IRR quoted on a multi-sign series is a number without a meaning.

In both cases NPV at your actual cost of capital is well defined and answers the question you were really asking.

Where the figure deceives

IRR is a rate, so it ignores size. Sixty percent on a $10,000 project beats twenty percent on a $10 million one by this measure and is worth a tiny fraction as much money. Where IRR and NPV rank projects differently, NPV is the one to follow.

It also assumes interim cash flows are reinvested at the IRR itself. On a project returning 45%, that assumption is doing enormous unstated work — modified IRR exists precisely to replace it with something defensible, and the two diverge more the higher the IRR.

And it is easy to game. Shortening a project raises its IRR while lowering its NPV, which is one reason IRR is popular in places where results are reported annually.

Acting on it

Quote IRR alongside NPV rather than instead of it, and use the margin over your cost of capital as the headline rather than the raw rate.

Check the sign-change count before trusting a single figure. More than one and the number is not well defined.

For very high IRRs, look at modified IRR too. The gap between them is a measure of how much of the return depends on reinvesting at a rate you probably cannot get.