IRR calculator: internal rate of return

Enter what the project costs up front, what it pays back each year, how many years it runs, and the rate you have to beat. It returns the IRR, the net present value at that hurdle rate, and a one-line verdict on whether the project clears it.

The numbers it opens on are the first worked example below: $10,000 out at the start and $3,000 back at the end of each of five years. That series earns 15.24 percent, so against a 10 percent cost of capital it is worth doing, and the NPV is $1,372.36.

The hurdle rate is an input rather than a fixed number because there is no universal one. It is what your money costs, plus whatever the risk of this particular project is worth to you.

IRR is defined by an equation rather than computed by one:

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

Money going out is negative and money coming back is positive, so time zero normally carries the minus sign. Each later cash flow is divided by $(1 + IRR)^t$, which makes the whole thing a polynomial in $IRR$ of degree $n$, and past the fourth degree a polynomial has no general root formula. So in practice every IRR you have ever seen was found by searching: guess a rate, price the series at it, adjust, repeat.

This page searches with Newton's method and falls back on bisection. The guard that verifies the published figures uses bisection alone, so two different searches have to land on the same rate before a number ships.

One shape does have a closed form. With a single amount out and a single amount back, the equation collapses to a plain growth rate, which is what the CAGR calculator works out. The third example below is exactly that shape.

An IRR on its own is a number with nothing to compare it to. Put it beside the rate your money costs and it becomes a decision.

Discount the five payments of $3,000 at 10 percent and they are worth $11,372.36 today, against the $10,000 paid out at the start, so the NPV is $1,372.36 and the project clears the bar. Raise the discount rate and that surplus shrinks, because every future payment is being divided by a bigger number. Raise it to the IRR, which is 15.24 percent for this series, and the surplus is gone. Above the IRR the NPV is negative and the project is destroying value at that cost of capital.

So the rule fits in a line: accept when the IRR is above your cost of capital, reject when it is below. The net present value calculator states the same decision in money rather than in rate, which is the better unit whenever you are choosing between projects rather than judging one.

Three things break IRR, and all three are about what a single rate cannot carry.

Size. A rate says nothing about how much money is involved. A 40 percent return on a small outlay can be worth less cash than a 12 percent return on a large one, and an IRR ranking will put them the wrong way round. When projects differ in size, compare NPV.

Reinvestment. The arithmetic behind IRR treats every payment as if it were put back to work at the IRR itself until the project ends. If the project earns 30 percent and everything else you own earns 6 percent, the IRR flatters it. The modified IRR, MIRR, exists for this: it discounts the outflows and compounds the inflows at rates you pick rather than at the answer.

Multiple roots. A polynomial of degree $n$ has up to $n$ roots. A series that turns negative again after turning positive, a mine that must be restored or a plant needing a mid-life rebuild, can genuinely have several rates that set NPV to zero, and each one is a real IRR. There is no honest way to choose between them, so this calculator returns no rate at all in that case and leaves you with the NPV at the cost of capital you actually face.