IRR Calculator

Background.

An IRR calculator solves for the one number every capital-budgeting analysis eventually needs: the discount rate at which a cash-flow series' net present value is exactly zero. The internal rate of return is, in effect, the break-even annualized return baked into a project or investment's own cash flows — if your actual cost of capital is below the IRR, the project is expected to create value; if it is above the IRR, the project is expected to destroy value. Unlike net present value, which requires you to supply a discount rate and returns a dollar figure, IRR requires no rate at all — it solves for the rate itself, algebraically implied by the cash flows you already have.

There is no closed-form formula for IRR once a cash-flow series has more than two entries — it is a polynomial root-finding problem with no algebraic shortcut, which is exactly why every real IRR tool, from a financial calculator's IRR button to Excel's IRR() function, relies on a numerical search rather than a single-step equation. This calculator uses a bisection method: it scans a wide range of candidate discount rates from -99% to +10,000% for the point where net present value flips from negative to positive, then repeatedly narrows that bracket in half until the rate is pinned down to better than one part in ten billion. Bisection is the deliberate choice here over faster methods like Newton-Raphson because it is unconditionally convergent given any bracket with a sign change — it cannot diverge, overshoot into an invalid discount-rate domain, or oscillate the way Newton-Raphson occasionally can on unusual cash-flow shapes. If no such bracket exists anywhere in the searched range, the calculator throws a clear error rather than ever returning NaN or a fabricated root.

IRR has one well-documented mathematical quirk that this calculator addresses honestly rather than hiding: a cash-flow series that changes sign more than once — for example, an initial cost, followed by income, followed by a later cost, such as an oil well that must be capped and reclaimed at the end of its life — can have more than one mathematically valid internal rate of return. This is not a bug in any particular calculator; it is a property of the underlying polynomial, formally bounded by Descartes' rule of signs. The classic textbook illustration, cited in Brealey, Myers and Allen's Principles of Corporate Finance, is a cash-flow series of -1,600, +10,000, -10,000, which has exactly two valid IRRs: 25% and 400%. This calculator reports how many sign changes your series contains, and when that count exceeds one, it flags explicitly that more than one IRR may exist and that net present value — not IRR — is the more reliable decision tool for that cash-flow shape.

Enter your cash-flow series as a comma-separated list, starting with the period-0 outlay, exactly as you would for Quanta's NPV calculator. The tool returns the IRR as both a percentage and a decimal, the net present value at that computed rate (which should sit essentially at zero as a built-in sanity check on the solver itself), the number of sign changes in your series, and a count of parsed periods to help catch a dropped value.

What is irr calculator?

The internal rate of return (IRR) is the discount rate at which a series of cash flows produces a net present value of exactly zero. It is the implied, break-even annualized rate of return a project or investment generates purely from its own timing and size of cash flows, without reference to any externally chosen discount rate. Formally, IRR is the value r that solves the equation: the sum, for every period t, of the cash flow at time t divided by (1 plus r) raised to the power t, equals zero. Because this is a polynomial equation with degree equal to the number of periods, there is no general algebraic formula for r once there are more than two cash flows — practitioners and software alike rely on numerical root-finding methods such as bisection or Newton-Raphson. A cash-flow series with a single sign change (the ordinary "pay once up front, receive returns later" pattern) is guaranteed to have exactly one real, economically meaningful IRR. A series with more than one sign change can have more than one mathematically valid IRR, a well-known limitation formally bounded by Descartes' rule of signs and illustrated by the classic textbook "pump project" example.

How to use this calculator.

1. 01Enter your full cash-flow series as a comma-separated list. The first number is period 0 — your initial outlay, entered as a negative number if money is leaving your hands today, following the same convention as Quanta's NPV calculator.
2. 02Enter every subsequent cash flow in order: period 1, period 2, and so on, using consistent units and period length throughout.
3. 03Make sure the series includes at least one negative value and at least one positive value — IRR is undefined for a series that is all cost or all income.
4. 04Read the reported IRR. Compare it against your actual cost of capital or opportunity cost: an IRR above that threshold suggests value creation; an IRR below it suggests the opposite.
5. 05Check the NPV-at-IRR figure — it should be essentially zero. This is the calculator's own built-in confirmation that the numerical solver actually converged on a real root.
6. 06Check the sign-change count. If it is greater than 1, read the multiple-IRR FAQ below before relying on the reported rate as the sole decision criterion — consider running the NPV calculator at your actual discount rate instead.

The formula.

IRR is defined implicitly: it is whatever discount rate makes the net present value formula — the sum of every cash flow divided by (1 plus the rate) raised to its period index — equal exactly zero. Because this equation cannot generally be rearranged to isolate the rate algebraically once there are more than two cash flows, the calculator finds it numerically in two stages. First, it scans a wide grid of candidate rates from -99% up to +10,000%, evaluating net present value at each point, until it finds two adjacent candidates where the sign flips from negative to positive (or vice versa) — that pair brackets a real root. Second, it repeatedly bisects that bracket: it evaluates net present value at the midpoint, keeps whichever half still contains the sign change, and narrows the interval by half every iteration until the bracket width is smaller than one part in ten billion. This is the standard bisection root-finding method, chosen for its guaranteed convergence whenever a valid bracket exists, rather than a Newton-Raphson-style method that can occasionally diverge on unusual cash-flow shapes. All arithmetic inside the bisection loop runs in arbitrary-precision Decimal.js math. If the series contains more than one sign change, more than one bracket — and therefore more than one valid root — may exist; this calculator reports the first bracket found scanning upward from -99%, which is typically the lowest of the possible IRRs, and separately reports the total sign-change count so the possibility of additional roots is never hidden from the result.

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]U.S. Securities and Exchange Commission, Investor.gov — Internal Rate of Return (IRR) glossary entry.
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.