IRR Calculator
Find the internal rate of return — the discount rate that drives net present value to zero — for any series of cash flows. Enter your investment and each year's cash flow to see the IRR, the NPV at your hurdle rate, and MIRR, along with a clear warning when the cash flows admit no IRR or more than one.
Negative values are allowed — enter a minus sign for a year that consumes cash rather than producing it.
Cash flow detail and IRR reconciliation
Every cash flow discounted at the calculated IRR appears in the last column. Because IRR is defined as the rate that makes net present value zero, those discounted figures must add up to about zero — the total row is the proof that the solver converged.
| Year | Cash flow | Cumulative cash flow | Discounted at IRR |
|---|
How the IRR calculator works
Internal rate of return is defined by an equation rather than by a formula you can rearrange. It is the discount rate that sets net present value to zero:
where CFt is the cash flow in year t and r is the IRR. Your initial investment enters as a negative CF0.
For anything beyond a four-year project this is a polynomial of degree n with n of 5 or more, and by the Abel-Ruffini theorem no general solution in radicals exists. Even for the cubic and quartic cases, where closed forms do exist, they are unwieldy enough that the equation is solved numerically in practice. This calculator uses bisection: it evaluates NPV at −99% and at +1000%, confirms the two results have opposite signs, then repeatedly halves the interval, keeping whichever half still contains the sign flip. After about 300 halvings the interval is narrower than any meaningful precision, and the rate at its centre is reported. Bisection is slower than the Newton method some tools use, but given a bracket where NPV has opposite signs at each end it is guaranteed to converge on a root; it cannot diverge or oscillate.
When no IRR exists
An IRR is a point where the NPV curve crosses zero, and plenty of cash flow series never cross it. If every entry is positive, every discounted term stays positive too — the discount factor (1 + r)t is positive at every rate above −100%, and a positive divided by a positive is a positive. The sum can never reach zero, so nothing sets it there. The mirror case applies if every entry is negative. That is not a limitation of the arithmetic; the quantity simply is not defined, and this page says so rather than showing a placeholder number.
A subtler failure is a curve that never crosses zero inside the searched range. Bisection needs a bracket: two rates whose NPVs have opposite signs. If NPV is the same sign at −99% and at +1000%, no root sits between them and the method has nothing to converge on. Rather than return the midpoint of a meaningless interval, the calculator reports that no internal rate of return exists in a reasonable range for these cash flows.
Why more than one IRR is possible
The NPV equation is a polynomial, and polynomials can have several roots. Descartes' rule of signs gives the ceiling: the number of positive real roots is at most the number of sign changes in the coefficient sequence, which here is the cash flow series in year order. A conventional project — money out once, then money in for years — changes sign exactly once and therefore has at most one IRR. But a project that pays out, then demands fresh capital for a rebuild or a cleanup, then pays out again, changes sign three times and can have up to three distinct rates that each drive NPV to zero, with no mathematical basis for calling any one of them the answer.
This calculator counts the sign changes on every recalculation and displays the count. When the count exceeds one and a rate is shown, a warning appears: multiple IRRs may exist, and the figure shown is only the first root the search happens to land on.
What NPV and MIRR add
Because IRR collapses timing and scale into one percentage, it discards information. NPV at a hurdle rate you choose keeps that information: it is stated in dollars, so a large project and a small one are no longer flattened onto the same scale, and it is defined for every cash flow series, including the ones that have no IRR at all.
MIRR addresses a different gap. The IRR equation implicitly assumes each interim inflow is reinvested at the IRR itself — a strong assumption when the IRR is 40% and your actual reinvestment options pay 5%. MIRR replaces the assumption with rates you state:
Positive flows are compounded forward to the final year, negative flows are discounted back to year 0, and the ratio is annualized over n years. Because it is computed directly rather than solved for, MIRR returns a single unambiguous value whenever the series contains at least one inflow and at least one outflow — including the sign patterns that give IRR several. Without both an inflow and an outflow the ratio is undefined and the calculator shows n/a.
Reading the reconciliation table
The cumulative column tracks the running total of undiscounted cash, showing the year the project turns cash-positive in nominal terms. The final column discounts each flow at the calculated IRR. Its total should sit within a rounding error of zero, because that is precisely what IRR means. If it did not, the rate above it would not be a root. It is a check you can run against any IRR figure, from any source.
Frequently asked questions
What does IRR actually measure?
IRR is the discount rate at which the net present value of a cash flow series equals zero. It expresses the return implied by the timing and size of the cash flows themselves, as a single annualized percentage. It measures the rate a project earns on the capital still tied up in it, per period, assuming every interim cash flow is reinvested at that same rate.
Why does this calculator sometimes say no IRR exists?
An IRR only exists if the NPV curve actually crosses zero. If every cash flow has the same sign, NPV can never reach zero, so there is no rate to find. The calculator also searches only between -99% and +1000%. If NPV has the same sign at both ends of that range, no root can be bracketed inside it, and the calculator says so rather than printing a meaningless number.
Can a set of cash flows have more than one IRR?
Yes. Descartes' rule of signs says the number of possible positive roots is at most the number of times the cash flow series changes sign. One sign change means at most one IRR. Two or more sign changes, which happen when a project needs new money after it starts paying out, can produce several rates that all set NPV to zero. This calculator counts the sign changes and warns you when more than one IRR is possible.
How is MIRR different from IRR?
IRR assumes interim cash flows are reinvested at the IRR itself, which is often unrealistic when the IRR is high. MIRR lets you state the rates explicitly: positive flows are compounded forward at a reinvestment rate, negative flows are discounted back at a finance rate, and the two are compared over the life of the project. MIRR also returns a single value even when the sign pattern would give IRR several.
Why should the discounted column sum to zero?
That column is the reconciliation. Discounting every cash flow at the IRR and adding the results reproduces the definition of IRR, which is the rate that makes NPV zero. If the total in that column is zero apart from rounding, the solver converged correctly. A total that is visibly far from zero would mean the reported rate is not a true root.
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This calculator is for educational and informational purposes only and does not constitute financial, investment, or tax advice. Estimates are based on the values you enter. Confirm all figures with a qualified professional before making decisions.