What is IRR and how is it calculated?
The internal rate of return (IRR) is the discount rate at which a project's net present value is exactly zero: the rate r that solves 0 = CF0 + CF1 / (1 + r) + CF2 / (1 + r)^2 + ... + CFn / (1 + r)^n. Beyond two years there is no algebraic shortcut, so IRR is found by trial: try a rate, see whether NPV is above or below zero, and close in.
This calculator closes in by computer. It scans every rate from -99% to 1,000%, finds each point where NPV changes sign and narrows it by bisection until it is exact to many decimals. The default project, an outlay of $100,000 followed by $30,000, $40,000, $50,000 and $20,000, has an IRR of 15.32%. At any cost of capital below 15.32% its NPV is positive, and at any rate above it the NPV is negative.
How do you calculate IRR by hand with linear interpolation?
A written answer estimates IRR from two NPVs. Choose a lower rate L where NPV is positive and a higher rate H where it is negative, and join them with a straight line:
IRR is approximately L + NPV at L / (NPV at L - NPV at H) x (H - L)
Here is the default project at 10% and 20%, with factors from a 3-decimal present value table:
| Year | Cash flow | Factor 10% | PV at 10% | Factor 20% | PV at 20% |
|---|---|---|---|---|---|
| 0 | -$100,000 | 1.000 | -$100,000 | 1.000 | -$100,000 |
| 1 | $30,000 | 0.909 | $27,270 | 0.833 | $24,990 |
| 2 | $40,000 | 0.826 | $33,040 | 0.694 | $27,760 |
| 3 | $50,000 | 0.751 | $37,550 | 0.579 | $28,950 |
| 4 | $20,000 | 0.683 | $13,660 | 0.482 | $9,640 |
| NPV | $11,520 | -$8,660 |
IRR is approximately 10% + 11,520 / (11,520 + 8,660) x 10% = 10% + 0.5709 x 10% = 15.71%. The exact IRR is 15.32%, so the estimate is 0.39 percentage points high. The calculator repeats this sum for any two rates you set and prints the same two-column working under the result.
Why is the interpolated IRR higher than the exact IRR?
For a conventional project, one outflow followed by inflows, the NPV profile is a curve that falls steeply at low rates and flattens out at high ones. A straight line drawn between two points on that curve sits above it, so the line reaches zero at a higher rate than the curve does, and the interpolated IRR overstates the true one. The wider apart L and H are, the larger the overstatement:
| Higher rate H (L = 10%) | NPV at H, table factors | Interpolated IRR | Gap from 15.32% |
|---|---|---|---|
| 20% | -$8,660 | 15.71% | +0.39 points |
| 25% | -$16,600 | 16.15% | +0.82 points |
| 30% | -$23,500 | 16.58% | +1.26 points |
A pair of rates close to the IRR does much better: L at 15% and H at 16% gives 15.34%, only 0.02 points out. The second chart under the result plots this drift for your own cash flows. Treat the interpolated figure as a hand method and the exact one as the answer.
How do you calculate IRR in Excel or Google Sheets?
Put every cash flow in one column, year 0 first and negative, and use =IRR(B1:B5). Unlike the NPV function, IRR takes the year 0 value inside its range. For the default project it returns 15.32%. Microsoft's help page says the function starts from a guess of 10% unless you supply one, keeps going until the result is accurate within 0.00001 percent, and returns #NUM! if it has not settled after 20 tries. The values need at least one positive and one negative number, at regular intervals. Google Sheets offers IRR, XIRR and MIRR as well.
For cash flows on calendar dates use =XIRR(values, dates), which works on a 365-day year and returns the rate at which XNPV is zero, trying up to 100 times. If IRR or XIRR shows #NUM!, give it a different guess as the last argument, or check the NPV profile on this page: there may be no IRR at all, or more than one.
People also search for IRR from NPV. There is no formula that turns one into the other, but NPV at the IRR is zero, so =NPV(IRR(B1:B5), B2:B5) + B1 returns 0. That is a quick check that a range and its IRR belong together.
How do you find IRR on a BA II Plus or other financial calculator?
On the BA II Plus, cash flows go into the Cash Flow worksheet just as they do for NPV: CF, then 2nd and CLR WORK, then CFo and each Cnn with its frequency Fnn. The Texas Instruments guidebook then finds IRR in two keystrokes: press IRR, then CPT. With the default flows the display reads 15.32. No discount rate is needed, because the IRR is the rate being solved for. The worksheet takes up to 24 cash flows after CFo, and a repeated amount needs only one entry with its frequency.
If the calculator stops with an error instead of a rate, the cash flows may have no change of sign or more than one, which this page detects and explains. HP's 10bII+ and 12C and Sharp's EL-738 also solve IRR from stored cash flows, under different key names; the cash-flow chapter of each manual gives the exact sequence, which this page does not repeat.
What happens when there is no IRR, or more than one?
An IRR exists only where NPV crosses zero. If every cash flow has the same sign, it never does. A conventional project has one change of sign and one IRR. Each extra change of sign, such as a large clean-up cost in the final year, can add another crossing, and the number of IRRs can never exceed the number of sign changes.
An example in euros: pay €10,000 today, receive €23,000 in year 1 and pay €13,200 in year 2. NPV is zero at both 10% and 20%. It is -€200 at 0%, +€18.90 at 15% and -€48.00 at 25%, so the project adds value only for a cost of capital between those two rates. Neither 10% nor 20% is the return on the project. Enter these three rows above and the calculator lists both rates, counts the two sign changes and draws the hump-shaped profile.
What is MIRR and how is it calculated?
The modified internal rate of return assumes the cash a project releases is reinvested at the cost of capital, not at the IRR, and it always produces a single answer. With one rate re for both finance and reinvestment: MIRR = (PV of inflows / PV of outflows)^(1/n) x (1 + re) - 1, with both present values taken at re and n the project life in years.
For the default project at 10%, the inflows are worth $111,556.59 today against an outflow of $100,000, so MIRR = (1.115566)^(1/4) x 1.10 - 1 = 13.05%. That lands between the 10% cost of capital and the 15.32% IRR, as it usually does, because IRR quietly assumes each receipt earns 15.32% again. =MIRR(B1:B5, 10%, 10%) gives the same figure; the spreadsheet version also accepts separate finance and reinvestment rates. For the two-IRR euro example, MIRR at a 15% cost of capital is 15.05%.
NPV and IRR: what if they rank projects differently?
For a single project, NPV and IRR agree: an IRR above the cost of capital goes with a positive NPV. They can disagree when you have to pick one project out of two. An example in pounds, at a 10% cost of capital:
| Project | Cash flows | NPV at 10% | IRR |
|---|---|---|---|
| A | -£100,000, then £30,000, £40,000, £50,000, £20,000 | £11,556.59 | 15.32% |
| B | -£20,000, then £10,000 a year for 3 years | £4,868.52 | 23.38% |
B has the higher IRR because it is small and quick; A adds £6,688.07 more value. The two NPV profiles cross at 13.69%, which is the IRR of the difference between the projects, so at any cost of capital below that A has the larger NPV. When only one can go ahead, finance theory sides with the higher NPV, because NPV measures money added while IRR is a rate earned on very different sums.
Common IRR mistakes, and what this calculator leaves out
- Reading IRR as a yearly return on all the money. It is a break-even discount rate, and it assumes released cash earns the same rate again.
- Ranking projects of different sizes by IRR. Compare NPVs, or work out the IRR of the difference between them.
- Interpolating with both rates on one side of the IRR. The formula then extrapolates and the error grows; the calculator flags it.
- Ignoring extra sign changes. A second outflow late in the project can create a second IRR.
- Calling a monthly IRR a yearly one. Annualize it: (1 + monthly rate)^12 - 1, so 1% a month is 12.68% a year.
The calculator works only on the cash flows you enter, at year ends, before tax and inflation. It searches from -99% to 1,000%, so a rate outside that range is not reported, and two IRRs lying extremely close together may show as one. It does not judge risk or the quality of the forecasts.
IRR glossary
Internal rate of return (IRR): the discount rate that makes NPV zero. Linear interpolation: estimating the IRR from the NPVs at two rates as if the profile were a straight line. NPV profile: a chart of NPV against the discount rate. Conventional cash flows: an outflow followed only by inflows, with one sign change. MIRR: a single-answer rate of return that assumes reinvestment at the cost of capital. Cost of capital: the return the providers of finance require; the hurdle an IRR is compared with. Incremental IRR: the IRR of the difference between two projects, where their NPV profiles cross.
Frequently asked questions
How do you calculate IRR by hand?
Work out NPV at a rate where it is positive and at a rate where it is negative, then interpolate: IRR is about L + NPV at L / (NPV at L - NPV at H) x (H - L). For the default project, NPVs of $11,520 at 10% and -$8,660 at 20% give about 15.71%, against an exact 15.32%.
Why is my interpolated IRR different from Excel's IRR?
Excel solves for the exact rate, while interpolation draws a straight line across a curve, which overstates the IRR of a conventional project. Rounded table factors move it a little further. Choosing two rates closer to the IRR shrinks the difference.
Can a project have more than one IRR?
Yes, when its cash flows change sign more than once. Paying 10,000 today, receiving 23,000 next year and paying 13,200 the year after gives IRRs of both 10% and 20%. In that case use NPV at the cost of capital or MIRR, which each give one answer.
What does it mean when IRR cannot be calculated?
Usually that NPV never reaches zero, most often because every cash flow has the same sign, for example when the year 0 investment was entered as a positive number. In a spreadsheet it can also mean the function needs a different starting guess.
What is the difference between IRR and MIRR?
IRR assumes cash released by the project is reinvested at the IRR itself. MIRR assumes it is reinvested at the cost of capital, so it is usually lower and always single-valued. For the default project, IRR is 15.32% and MIRR at 10% is 13.05%.
How are NPV and IRR related?
IRR is the rate at which NPV equals zero. If the cost of capital is below the IRR, NPV is positive, and if it is above, NPV is negative. When ranking competing projects the two can point different ways, and NPV is the measure of value added.
Which rates should I use for the interpolation?
One rate that gives a positive NPV and one that gives a negative NPV, as close to the IRR as you can judge. Start from the cost of capital; if NPV there is positive, try a higher rate until it turns negative. Rates far apart still work but make the estimate rougher.
How do I turn a monthly IRR into an annual one?
Compound it rather than multiplying: annual rate = (1 + monthly rate)^12 - 1. A monthly IRR of 1% is 12.68% a year, not 12%. XIRR in a spreadsheet already returns an annual rate from dated cash flows.
What is the difference between IRR and payback?
IRR uses every cash flow and the time value of money to give a rate of return. Payback counts only how long it takes to recover the outlay and ignores everything after that point. The default project pays back in 2.6 years and has an IRR of 15.32%.
Does the IRR change with currency?
No. IRR is a percentage, so the same cash flows give the same rate in dollars, pounds, euros or dirhams. Switching currency here only changes how amounts are formatted, and nothing is converted.
Courses related to IRR and NPV
ACCA SL Financial Management-FM Online Course for Exam Preparation
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See what the course coversSources and review
Last reviewed 13 November 2026 by the NIFM Academy editorial team. Next review by 13 November 2027.
- ACCA, Financial Management (FM) formulae sheet and maths tables (accaglobal.com, file S23_FM_Formulae.pdf, the version linked from the FM exam formulae and maths tables page), present value and annuity tables with factors printed to 3 decimal places for 1% to 20% and 1 to 15 periods, used for the 10% and 20% factors in the worked table. Checked 14 November 2026.
- Microsoft Support, IRR function, IRR(values, [guess]); guess defaults to 10%; accurate within 0.00001 percent; #NUM! after 20 tries; needs at least one positive and one negative value at regular intervals. Checked 13 November 2026.
- Microsoft Support, XIRR function, XIRR(values, dates, [guess]); 365-day year; the rate at which XNPV is zero; up to 100 tries. Checked 13 November 2026.
- Microsoft Support, MIRR function, MIRR(values, finance_rate, reinvest_rate); needs at least one positive and one negative value. Checked 13 November 2026.
- Texas Instruments, BA II PLUS guidebook (online edition), Cash Flow worksheet, IRR computed with the IRR key then CPT after entering CFo, Cnn and Fnn; up to 24 cash flows after CFo. Checked 13 November 2026.