The IRR formula finds the discount rate that makes a project’s net present value equal to zero. In other words, internal rate of return is the break-even compound return implied by the project’s own cash-flow amounts and timing.
0 = Σ[t=0 to n] CF_t ÷ (1 + IRR)^t
— Signed net cash flow at time t, including the initial investment
— Periodic discount rate that sets NPV to zero
— Number of periods after the valuation date
— Final cash-flow period
IRR uses the same discounted-cash-flow structure as NPV, but the unknown changes. In an NPV problem, you know the discount rate and solve for value. In an IRR problem, you set value to zero and solve for the rate.
The time-zero project cost is usually negative, while later net inflows are positive. At the IRR, the present value of the inflows exactly equals the present value of the outflows. If a conventional project’s IRR exceeds its required return, its NPV at that required return will be positive.
Most multi-period IRR equations cannot be rearranged into a simple closed-form solution. A financial calculator or spreadsheet searches numerically for a root. By hand, you can test discount rates until NPV changes sign and then interpolate for an estimate.
First, place every cash flow on a timeline and keep its sign. Second, calculate NPV at a trial rate. If a conventional project’s NPV is positive, try a higher rate; if NPV is negative, try a lower rate. Once two nearby rates produce opposite signs, the IRR lies between them. Repeat with narrower intervals or use interpolation.
Estimated IRR = r_low + [NPV_low ÷ (NPV_low − NPV_high)] × (r_high − r_low)
— Lower trial rate, normally with a positive NPV
— Higher trial rate, normally with a negative NPV
— NPV calculated at the lower rate
— NPV calculated at the higher rate
Interpolation assumes NPV changes roughly in a straight line between the two trial rates. The actual NPV curve is not linear, so interpolation is an approximation. Narrower brackets produce a better estimate.
A manufacturer can install an inspection system for $146,500 today. Expected year-end net cash inflows are $42,800, $49,600, $55,900, and $61,700 over four years. The cash-flow signs change once, from the initial outflow to later inflows, so this is a conventional pattern.
| Trial | Calculation | Result |
|---|---|---|
| NPV at 12% | −$146,500 + $42,800/1.12 + $49,600/1.12² + $55,900/1.12³ + $61,700/1.12⁴ | $10,255.08 |
| NPV at 15% | −$146,500 + $42,800/1.15 + $49,600/1.15² + $55,900/1.15³ + $61,700/1.15⁴ | $254.45 |
| NPV at 16% | −$146,500 + $42,800/1.16 + $49,600/1.16² + $55,900/1.16³ + $61,700/1.16⁴ | −$2,853.44 |
| Interpolated IRR | 15% + [$254.45 ÷ ($254.45 − −$2,853.44)] × 1% | 15.0819% |
| Numerical IRR | Rate that makes NPV approximately $0 | 15.0806% |
At approximately 15.0806%, the discounted inflows equal the $146,500 outlay. If the company requires 11%, the project passes the IRR rule because 15.0806% exceeds 11%. Its NPV at 11% would also be positive for this conventional cash-flow pattern.
Do not subtract the required return from IRR and call the difference value. A four-percentage-point spread does not reveal how many dollars the project adds. Calculate NPV at the required return to measure dollar value.
Place equally spaced cash flows in chronological order, including the initial negative amount, and apply IRR to the full range. Unlike the spreadsheet NPV function, IRR needs the time-zero cash flow inside its range. The optional guess gives the solver a starting point; changing it can produce a different root when the cash-flow pattern permits multiple solutions.
For actual calendar dates that are not equally spaced, use XIRR with a matching date range. A standard IRR calculation assumes each row is one equal period apart. Monthly cash flows produce a monthly IRR; convert it to an effective annual rate before comparing it with an annual hurdle rate.
Effective annual IRR = (1 + periodic IRR)^periods-per-year − 1
— IRR for one cash-flow interval, such as one month
— 12 for monthly periods, 4 for quarterly periods
Multiple IRRs. If cash-flow signs change more than once, the NPV equation can have more than one real solution. For example, a project with an initial cost, years of inflows, and a large cleanup cost may produce two IRRs or none that is economically useful. Inspect the NPV profile or use NPV at the required return.
Different project scale. A small project can have a higher percentage return but add fewer dollars than a larger project. For mutually exclusive choices, the highest IRR need not have the highest NPV.
Reinvestment interpretation. IRR is a project yield measure, but comparing projects solely through IRR can hide assumptions about intermediate cash flows. Modified IRR can state separate financing and reinvestment rates when a course or analysis calls for it.
Period mismatch. A monthly IRR is not directly comparable with an annual required return. Convert rates to the same effective period.
Omitted cash flows. Working-capital investment, terminal recovery, tax effects, and disposal costs can materially change the root. Use incremental cash flows and show the timeline.
Keep the initial outlay negative and later inflows positive. Do not discount the initial outlay for one period. Do not enter cumulative cash flows in the solver; enter each period’s actual net cash flow. Finally, check the answer by substituting the calculated IRR back into the NPV equation. A correctly calculated IRR should return an NPV very close to zero, allowing for rounding.
An NPV profile plots project NPV against discount rates. For the conventional inspection-system cash flows, the profile slopes downward: higher discount rates reduce the present value of positive future inflows. The point where the curve crosses zero is the 15.0806% IRR. This visual connects the two metrics instead of treating them as unrelated formulas.
At any required return below the crossing, this conventional project has positive NPV. At a required return above the crossing, it has negative NPV. That clean relationship can fail when cash-flow signs change more than once because the profile may cross zero several times.
Suppose one project returns cash early while another produces larger cash flows later. Their NPV profiles can cross. One project may have the higher IRR, while the other has the higher NPV at the company’s actual required return. If the projects are mutually exclusive, accepting the higher-IRR project automatically can sacrifice dollar value. Calculate incremental cash flows or follow the course’s NPV comparison method.
Project length also matters. IRR is periodic: an IRR from annual cash flows is annual, while an IRR from monthly cash flows is monthly. The solver does not infer calendar units from the numbers. The analyst supplies meaning through the timeline.
IRR is the rate that makes NPV zero. Use a solver for precision, verify the result in the NPV equation, and prefer NPV when cash flows are nonconventional or projects conflict.
Set the sum of every signed cash flow divided by one plus IRR raised to its period equal to zero, then solve for IRR.
Calculate NPV at trial rates until one result is positive and another is negative, then narrow the interval or interpolate. The estimate should be checked by recomputing NPV.
An IRR is acceptable under the basic rule when it exceeds the project’s risk-appropriate required return. There is no universal good percentage.
IRR is defined as the discount rate at which the present value of inflows equals the present value of outflows. Their signed sum is therefore zero.
Yes. Multiple sign changes in the cash-flow sequence can create multiple mathematical roots. An NPV profile and NPV at the required return are safer guides in that case.
IRR incorporates the timing of multiple cash flows and is an annualized compound rate when periods are annual. Simple ROI compares total gain with cost and may ignore timing.
By the FinanceBrain Team · Last verified July 12, 2026 · How we produce and verify articles