What IRR and NPV actually measure
Net present value and internal rate of return both start from the same idea: a pound, dollar or euro that arrives next year is worth less than one in your hand today, because money in hand can earn a return between now and then. Discount every future cash flow back to today at a chosen rate, add them all together along with the upfront cost, and the total is the net present value, NPV, of the whole project. IRR asks a related but different question: at what single discount rate does that total come out to exactly zero? That rate is the project's own break-even rate of return, the growth rate the investment itself is implicitly delivering, independent of whatever rate you actually plan to use to judge it.
The two numbers are built from the same arithmetic and this calculator solves both from one set of inputs: an initial investment, entered as a positive amount because that is how a cost is normally quoted, and a year-by-year list of net cash flows afterwards, which can run negative in any year a project costs more than it returns.
The NPV formula, worked
CFt is the net cash flow in year t, r is the discount rate as a decimal, and t runs from year 0 (the investment, entered here as a positive figure and applied as −CF0) through to the final year n. Take this calculator's own starting numbers: invest 10,000, then get back 3,000, 4,000, 5,000 and 2,000 over the next four years, discounted at 10%. Divide each year's flow by 1.10 raised to that year's power, 2,727.27, 3,305.79, 3,756.57 and 1,366.03, add them to the −10,000 outlay, and the total comes to 1,155.66. A positive NPV means the discounted cash coming back is worth more than the outlay, at that rate; a negative one means the reverse.
IRR: the rate where NPV hits zero
IRR is simply the value of r that makes the NPV formula above equal zero for a given set of cash flows. There is no algebraic way to isolate r, the same reason the Interest Rate Calculator's loan-rate tab needs bisection rather than a formula, since r sits inside a power on every term from year 1 onward. This tool solves it the same way: it evaluates NPV at a sequence of rates across 0 to 100%, finds every point where the running total flips from positive to negative or back, and bisects each bracket, halving the gap between a rate that is too low and one too high, until the two sides agree to within a hundred-thousandth of a percentage point. On the calculator's own starting numbers that lands on an IRR of roughly 15.32%, and plugging 15.32% back into the NPV formula above returns a total within a rounding error of zero, which is exactly the definition being solved.
The NPV profile, and why the shape matters
The chart on this page plots NPV against the discount rate from 0 to 40%, the NPV profile. For a normal project, one big outlay followed only by positive returns, that line runs from a large positive value at a 0% discount rate down through zero and on into negative territory as the rate keeps rising, since a higher discount rate shrinks every future pound more than the last. Where the curve actually crosses zero is the IRR, marked automatically; your own discount rate is marked too, as a dashed vertical line, so you can see at a glance which side of the crossing it falls on. That single picture is doing more work than the headline percentage alone: it shows not just what the IRR is, but how sensitive the verdict is to the rate you chose, a curve that crosses steeply near your rate is a much more finely balanced decision than one that crosses a long way off.
The multiple-IRR problem
The tidy, one-crossing picture above assumes the cash flows only change sign once, negative then positive. Real projects sometimes do not: a mine that costs money to close down, a lease with a balloon payment, or a project needing a second round of investment partway through can all produce a cash-flow sequence that flips sign more than once, negative, then positive, then negative again. Descartes' rule of signs says the number of sign changes in the sequence is the maximum number of positive rates at which NPV can equal zero, though the true count can come in lower, by an even number, never higher. Two sign changes can mean two IRRs exist, or none at all; it is not something a single bisection run can be trusted to catch on its own.
This calculator does not assume the tidy case. It counts the sign changes in whatever cash flows you enter and, whenever there is at least one, scans the full 0 to 100% range in fine steps looking for every point NPV actually crosses zero, not just the first one it finds. A textbook example makes the point plainly: cash flows of −10,000, +25,000 and −15,600 change sign twice, and this calculator's scan finds NPV crossing zero at both 20% and 30%, two equally valid IRRs for the same project. Asking "is 20% or 30% the real IRR" is the wrong question; both are, and neither on its own tells you whether to go ahead. Whenever this calculator finds more than one crossing, it says so plainly and points at the profile chart, since the chart shows the whole shape a single percentage cannot.
IRR's reinvestment assumption, and MIRR's fix
Even a well-behaved, single-IRR project carries a quieter assumption worth knowing about. The IRR calculation implicitly assumes every positive cash flow gets reinvested, for however many years remain, at the IRR itself. For a modest IRR that is a reasonable-enough assumption; for a very high one, it is optimistic, since finding somewhere else to reinvest at 40% or 50% a year is not something most projects can promise. Modified internal rate of return, MIRR, replaces that assumption with an explicit one: negative flows are discounted back to year 0 at a finance rate, positive flows are compounded forward to the final year at a reinvestment rate, and one compound growth rate is solved between those two totals.
A textbook MIRR uses two separate rates for financing and reinvestment; this calculator keeps one input simple and uses your single discount rate for both. On the calculator's own starting numbers, an IRR of about 15.32% comes with an MIRR of about 13.05%, lower, because reinvesting the interim cash at 10% rather than at 15.32% is a more conservative, arguably more realistic, assumption. MIRR sitting below IRR is the normal pattern whenever IRR is comfortably above your reinvestment rate; the gap between the two is a rough measure of how much of the raw IRR figure is riding on that reinvestment assumption rather than the project's own cash flows.
Why NPV outranks IRR when comparing projects
For deciding whether a single project clears its own hurdle, IRR versus discount rate and NPV's sign at that rate agree; that is exactly what this calculator's own accept-or-reject verdict is built on. They can disagree, though, when ranking two mutually exclusive projects against each other, since IRR is a percentage and says nothing about scale, a project turning a small outlay into a slightly larger one can show a higher IRR than a much bigger project that generates far more actual pounds of value, and where cash flow timing differs between two options IRR's reinvestment assumption can distort the comparison further still. NPV, being a plain amount of money at your chosen rate rather than a percentage, does not carry either problem, which is the textbook reason NPV, not IRR, is the tool of choice whenever the question is which one project to pick rather than whether any one project clears its own bar.
Choosing your inputs and reading the results
Enter the initial investment as the positive amount actually spent; the calculator handles the sign. Each year afterwards takes whatever that year's net cash flow genuinely was, income minus any cost in that same year, and a negative row is entirely normal wherever a project has a costly year partway through rather than only at the very start. The discount rate should reflect your real cost of capital or the return genuinely available elsewhere, since both the accept-or-reject verdict and the MIRR figure are read directly against it; a rate plucked out of the air makes the verdict meaningless even though the arithmetic behind it stays correct. If the calculator reports that IRR is undefined, read the reason given rather than treating it as an error: cash flows that never change sign have no rate at which NPV can ever reach zero, and cash flows that do change sign but never cross inside the 0 to 100% window searched here usually mean the true rate is negative, the project loses money even with no discounting at all, which the NPV figure at your own rate already makes clear either way.
Questions people ask
Is a higher IRR always better?
Only when comparing projects of a genuinely similar size and shape. IRR is a percentage and says nothing about how much actual money is at stake, so a small project with a striking IRR can still create less value than a larger one with a lower IRR. For choosing between mutually exclusive projects, NPV is the more reliable figure; IRR is best read as the rate this specific project clears on its own, not as a ranking tool.
Why does this calculator sometimes say IRR is undefined?
Two different reasons produce that message. If every cash flow entered points the same way, all money going out or all money coming in, there is no discount rate at which the total can ever reach zero, so IRR genuinely does not exist. If the cash flows do change sign but the scan across 0 to 100% never finds a crossing, the true rate is either negative, this project loses money even undiscounted, or sits above 100%; either way the NPV figure at your own rate is shown alongside so you are never left with nothing to go on.
What does it mean if the calculator finds more than one IRR?
It means your cash flows change sign more than once, negative, then positive, then negative again is a common pattern, which by Descartes' rule of signs can produce more than one rate at which NPV equals zero. Both figures found are genuinely valid answers to "where does NPV cross zero", so treat the single-IRR accept-or-reject rule with caution here and read the NPV profile chart, and the NPV at your own rate, instead.
Why is MIRR usually lower than IRR?
IRR assumes every positive cash flow gets reinvested at the IRR itself for as long as the project runs, which is a generous assumption once the IRR is high. MIRR reinvests those same cash flows at your entered discount rate instead, a more conservative and often more realistic assumption, which is why MIRR typically comes in below IRR whenever IRR sits comfortably above that rate.
Can a project have a negative NPV but a positive IRR, or the other way round?
Not for a normal, single-sign-change project judged at its own IRR, since NPV is exactly zero there by definition; NPV is positive at rates below the IRR and negative above it. Once a cash-flow stream has more than one IRR, though, that clean relationship breaks down between the crossings, which is exactly why this calculator leans on the actual NPV figure at your chosen rate rather than a bare IRR-versus-rate comparison whenever more than one crossing is found.
Should I enter a negative number for a year that costs more than it returns?
Yes. Unlike a simple payback-period sum, IRR and NPV are built to handle a negative year mid-project, an expansion cost, a refit, or a second capital call, exactly as they would a positive one; just enter that year's genuine net figure, income minus cost, and let the calculator work from there.
How is this different from the Payback Period Calculator?
Payback period only asks how long the original outlay takes to come back and stops looking the moment it does; it ignores everything afterwards and treats every pound, near or distant, as worth the same. IRR and NPV weigh every cash flow across the whole life of the project and its timing properly. If a quick recovery-speed check is all you need, the simpler Payback Period Calculator answers that directly; reach for this page when the actual profitability question is the one that matters.
This calculator is an arithmetic and educational tool. It is informational only, not financial or investment advice, and none of its figures are a forecast, a recommendation, or a guarantee that any actual cash flow will behave as entered.