IRR Calculator
Find sign-changing internal rates of return for regular signed cash flows, including multiple-IRR warnings.
Content updated August 3, 2026
Equal end-of-period timing only. The solver scans for sign-changing roots from −99.99% through 1,000% per period; an even-multiplicity root that only touches zero or a root outside that range may be missed. IRR can be absent or ambiguous and is not a complete investment decision rule.
Internal rate of return is the periodic discount rate that makes the net present value of a cash-flow series equal to zero. Enter every signed amount from period 0 onward: investments and costs are normally negative, while receipts are normally positive.
A cash-flow pattern can have no IRR, one IRR, or more than one IRR. This calculator searches a documented rate range and returns every sign-changing root it finds instead of silently choosing the first answer, then converts each periodic result to an effective annual rate for the selected interval.
The IRR formula
0 = Σ[CFₜ ÷ (1+r)ᵗ]; effective annual IRR = (1+r)ᵏ−1CFₜ = signed cash flow at period t, beginning with t = 0 · r = periodic IRR greater than −100% · k = 12 for monthly, 4 for quarterly, or 1 for annual periods.
Worked example
For cash flows −100, 60, and 60 at annual intervals, the sign-changing IRR is about 13.0662%. For −100, 230, and −132, NPV equals zero at both 10% and 20%, so the result is explicitly reported as multiple IRRs.
Assumptions, rounding, and limitations
Assumptions
- Cash flows are signed amounts in one consistent currency unit, with the first amount at period 0.
- All later cash flows occur at the ends of equally spaced monthly, quarterly, or annual periods.
- Each effective annual result compounds the periodic root over 12, 4, or 1 periods.
- The solver reports detected sign-changing roots and does not choose a required return or reinvestment rate.
Rounding: Root brackets are refined with up to 100 bisection steps while calculations retain floating-point precision. Displayed rates use four decimal places and the residual NPV uses currency precision.
Limitations
- Supports 2–120 cash flows, each from -1 trillion to 1 trillion, and searches periodic rates from −99.99% through 1,000%.
- A repeated root that touches zero without changing sign can be missed, as can roots outside the search range or numerically ill-conditioned roots. Irregular dated cash flows require an XIRR-style method and are not supported.
- IRR can be absent or ambiguous and does not incorporate a chosen opportunity cost, risk adjustment, financing constraint, tax treatment, cash-flow uncertainty, or recommendation.
Sources
- Circular A-94: Guidelines and Discount Rates for Benefit-Cost Analysis — U.S. Office of Management and Budget
- OMB Circulars and A-94 materials — U.S. Office of Management and Budget
- Discount Rate Policy — U.S. Government Accountability Office
How the root search works
For each trial rate, the calculator discounts period t by (1 + r) raised to t and adds every signed cash flow. It scans rates in log(1 + r) space from −99.99% through 1,000% per period, brackets every detected sign change, and refines each bracket by bisection.
Monthly and quarterly IRRs are periodic rates. Their effective annual equivalents use compounding equivalence: (1 + periodic IRR) raised to 12 or 4, minus one. That annual conversion is not the same as multiplying the periodic percentage by the number of periods.
Why multiple IRRs need special treatment
OMB Circular A-94 defines IRR as the discount rate that sets NPV to zero and notes that a stream changing between negative and positive more than once can have multiple values. In that case, comparing a single IRR with a required return can be ambiguous.
The same circular says IRR does not generally provide an acceptable decision criterion on its own. Review NPV at a defensible discount rate, the actual cash-flow dates, reinvestment assumptions, scale, financing, taxes, uncertainty, and alternatives alongside the root calculation.
Frequently asked questions
▶What does IRR mean?
It is a periodic discount rate at which the net present value of the entered signed cash flows equals approximately zero.
▶Why can a project have multiple IRRs?
Multiple changes between negative and positive cash flows can produce more than one rate where NPV crosses zero. The calculator lists the sign-changing roots it detects rather than selecting one.
▶What happens if no IRR is found?
The series may have no real sign-changing root in the supported range, may only touch zero without crossing it, or may have a root outside the scanned range. NPV at chosen discount rates can still be evaluated separately.
▶When does each cash flow occur?
The first entered amount is period 0. Every later amount occurs at the end of the next equally spaced month, quarter, or year.
▶How is a monthly IRR annualized?
The effective annual rate is (1 + monthly IRR)^12 − 1. For example, 1% per month is about 12.6825% effective annually.
Related calculators
Discount a signed series of monthly, quarterly, or annual cash flows to period zero.
Open calculator →Calculate return on investment as a total percentage and an annualised rate.
Open calculator →Work out the compound annual growth rate between a starting and ending value.
Open calculator →Disclaimer: IRR Calculator results are estimates for general information and education only, and are not financial, tax, legal or medical advice. Verify important decisions with a qualified professional.