IRR Calculator

Background.

Put $10,000 into a machine that brings back $3,000, $4,200 and $6,800 over the next three years and you receive $14,000 in total. That is not a 40% return, and it is not 40% spread over three years either, because most of the money arrives late and money that arrives late is worth less. The internal rate of return answers the question properly: what single annual rate, compounded, makes those three receipts exactly repay the $10,000? For this machine the answer is about 16.34%.

Put more formally, the IRR is the discount rate at which the net present value of the whole series is zero. That makes it a break-even figure. If the money tied up in the machine costs you less than 16.34% a year, whether as borrowing or as returns given up elsewhere, the cash flows more than cover it; if it costs more, they fall short. It judges the cash flows you expect rather than forecasting them, so any error in those figures passes straight into the rate.

Once a series has more than two entries there is no formula that can be rearranged to give the rate, so it is found by search. The calculator scans rates from −99% upward to locate where the net present value changes sign, then halves that interval repeatedly until the rate is pinned down.

One caution comes with the measure itself. A series that turns negative again later, such as a project with a clean-up bill at the end, can have two or more rates that each set the net present value to zero, and the calculator reports only the lowest. The number of sign changes in your series is shown so you can tell when that applies.

What is irr calculator?

The internal rate of return is the rate r that solves CF₀ + CF₁ ÷ (1 + r) + CF₂ ÷ (1 + r)² + … + CFₙ ÷ (1 + r)ⁿ = 0, where CFₜ is the cash flow at the end of period t and CF₀ is today's amount, usually the outlay. It is a rate per period: with yearly cash flows it is an annual rate, with monthly cash flows a monthly one. The IRR uses nothing but the cash flows themselves, which is what makes it internal, and that is also its limit: it says what the project earns on the money it has tied up, not how much value it adds in dollars. Net present value at your own required rate answers that second question.

How to use this calculator.

1. 01Type the cash flows as a comma-separated list, starting with period 0, today. Money you pay out is negative and money you receive is positive, so a purchase followed by three years of income looks like -10000, 3000, 4200, 6800.
2. 02Leave out thousands separators. Because commas separate the entries, -10,000, 3,000, 4,200, 6,800 is read as eight cash flows, -10, 0, 3, 0 and so on, and gives a meaningless 111.08%. The number of periods shown should match what you meant to enter.
3. 03Keep the periods equal in length and enter 0 for any period with no cash flow, so that a sum received in year 3 sits in the third place after the outlay.
4. 04Read the IRR as a rate per period of whatever length you used, and compare it with your own required rate of return.
5. 05Check the sign-change count. With more than one change, read the reported rate with care and look at the net present value at your required rate as well.

The formula.

The equation Σ CFₜ ÷ (1 + r)ᵗ = 0 is a polynomial in 1 ÷ (1 + r) whose degree is the number of periods, and beyond the simplest cases its roots can only be found numerically. The search runs in two stages. First the net present value is evaluated across a grid of rates, every 0.1 percentage point from −99% to +1,000% and then every 50 points up to +10,000%, stopping at the first pair of neighbouring rates where the value changes sign. Then bisection takes over: the net present value at the midpoint shows which half still holds the sign change, and the interval is halved again until it is narrower than 0.00000001 percentage points. Bisection cannot overshoot or wander off the way faster slope-following methods sometimes do; it only needs a sign change to start from. The net present value at the reported rate is shown as a check and should be within a fraction of a cent of zero. Descartes' rule of signs limits how many rates above −100% can solve the equation to the number of sign changes in the series, so one sign change means exactly one IRR. With more than one, the first root met on the way up from −99%, the lowest, is the one reported. A series needs at least two entries and at least one change between negative and positive; otherwise there is no rate to find.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Brealey, R.A., Myers, S.C., and Allen, F. — Principles of Corporate Finance, 13th edition. Chapter 5 develops IRR and documents the classic multiple-IRR "pump project" example (-1,600, +10,000, -10,000; roots at 25% and 400%).
2. [2]Microsoft Support — Go with the cash flow: calculate NPV and IRR in Excel. Defines IRR for periodic cash flows, documents the iterative search, and warns that different guesses can return different roots.
3. [3]CFA Institute — CFA Program Curriculum, Corporate Issuers topic area, covering IRR, the multiple-IRR problem, and its relationship to the net present value decision rule.
4. [4]Bodie, Z., Kane, A., and Marcus, A.J. — Investments, 12th edition. Discusses internal rate of return as a performance measure and its relationship to time-weighted return.