What the internal rate of return actually measures
The IRR is the compound rate of return a project earns on the capital it has outstanding, period by period, until that capital comes back. Formally it is the discount rate that drives net present value to zero. Practically it answers the question a credit committee asks: what interest rate is this project effectively paying us?
That framing explains both why the IRR is the most quoted number in capital budgeting and why it is so often misused. It is a rate, so it is scale-free — a 40% IRR on a $50,000 project and a 12% IRR on a $50 million project are both real, but only one of them changes the company. It also comes from a polynomial, so it is guaranteed neither to exist nor to be unique.
The decision rule: accept when the IRR exceeds the return you require on capital of that risk, reject when it does not. That required return is normally your weighted average cost of capital, raised for a project riskier than the firm's existing assets. The hurdle-rate field sets exactly that comparison, and the calculator reports the spread in percentage points beside the net present value at the same rate.
The equation, and why it has no closed-form solution
Write the project as a signed cash-flow stream: CF₀ is negative because you pay out at the start, and CF₁ through CFₙ are whatever the project delivers. Discount each at an unknown rate and require the total to be zero. That is the whole formula.
Substitute x = 1/(1 + IRR) and it becomes a polynomial of degree n: CF₀ + CF₁x + CF₂x2 + … + CFₙxn = 0. Two consequences follow, and both matter in practice.
First, there is no algebraic formula for the root once n exceeds four. Every IRR you have ever seen was found by iteration — Excel's IRR takes a guess and runs Newton steps; this calculator scans the rate axis for sign changes and bisects each bracket. Bisection is slower but it cannot diverge, and it finds roots a single Newton start would miss.
Second, Descartes' rule of signs caps the number of positive roots at the number of sign changes in the stream. A conventional project — one outflow then nothing but inflows — changes sign once and has exactly one IRR. A project with a mid-life reinvestment or an end-of-life decommissioning cost changes sign twice or more and can carry two or three mathematically valid IRRs. This calculator counts your sign changes and lists every root it finds.
Worked example: $100,000 for five annual payments of $30,000
You spend $100,000 today on equipment that will generate $30,000 of net cash in each of the next five years. Because the inflows are level, you can solve this by hand with an annuity-factor table.
- Turn the problem into an annuity factor. You need the rate at which five payments of $30,000 are worth $100,000 today, so the factor you are looking for is 100,000 ÷ 30,000 = 3.333333.
- Bracket it in the table. The five-period factor at 15% is 3.352155 and at 16% it is 3.274292. Your 3.333333 sits between them, so the IRR is between 15% and 16% — nearer the 15% end.
- Interpolate. (3.3521551 − 3.3333333) ÷ (3.3521551 − 3.2742937) = 0.0188218 ÷ 0.0778614 = 0.24173, so linear interpolation gives 15% + 0.2417% = 15.2417%.
- Check the exact root. Solved to convergence, the answer is 15.238237%. Linear interpolation overshoots by 0.0035 percentage points because the annuity factor is a convex function of the rate, not a straight line. Over a one-point bracket that error is negligible; over a ten-point bracket it is not.
Now verify it the way the calculator's table does. At 15.238237% the discount factors are 0.8677675, 0.7530205, 0.6534467, 0.5670398 and 0.4920588. Multiply each by $30,000: $26,033.03 + $22,590.61 + $19,603.40 + $17,011.20 + $14,761.76 = $100,000.00 — the outlay back to the cent, which is exactly what “NPV equals zero” means.
Finally, apply the decision rule. At a 10% hurdle the project's NPV is $30,000 × 3.790787 − $100,000 = +$13,724, and the IRR beats the hurdle by 5.24 percentage points. Both rules say accept, which is what you expect from a conventional single-sign-change stream.
How to read the number you get
Start with the sign of the spread, not the size of the IRR. An IRR of 9% against an 8% cost of capital creates value; an IRR of 25% against a 30% required return on a venture-stage project destroys it. The level of an IRR tells you nothing until you pair it with the risk-matched return you could earn elsewhere.
Then check the three things an IRR quietly hides.
Scale. IRR is a percentage, so it cannot tell you how much money you make. Ranking mutually exclusive projects by IRR routinely picks the smaller one. Rank by NPV; use IRR only to confirm each candidate clears the hurdle.
Timing. A high rate applies only while the capital is deployed. A 40% IRR earned over eight months leaves you holding cash for the other sixteen, and the return you actually get on that cash is the market rate, not 40%.
The reinvestment assumption. Solving the IRR equation implicitly values every interim cash flow as if it compounds at the IRR itself until the project ends. The modified IRR replaces that with a rate you specify, and the gap between the two is the size of the assumption you were making.
The money multiple this calculator reports — total cash in divided by total cash out — is a useful cross-check. Two streams both returning 1.5× your money, one over five years and one over two, have identical multiples and very different IRRs, which is why private-equity investors read both.
Present value of $1 per period — the annuity factors behind a hand-solved IRR
| Periods | 5% | 8% | 10% | 12% | 15% | 20% | 25% |
|---|---|---|---|---|---|---|---|
| 3 | 2.7232 | 2.5771 | 2.4869 | 2.4018 | 2.2832 | 2.1065 | 1.9520 |
| 5 | 4.3295 | 3.9927 | 3.7908 | 3.6048 | 3.3522 | 2.9906 | 2.6893 |
| 7 | 5.7864 | 5.2064 | 4.8684 | 4.5638 | 4.1604 | 3.6046 | 3.1611 |
| 10 | 7.7217 | 6.7101 | 6.1446 | 5.6502 | 5.0188 | 4.1925 | 3.5705 |
Each factor is (1 − (1 + r)⁻ⁿ) ÷ r evaluated at the rate and period count shown. The method only works when the cash flows are level; for uneven streams there is no shortcut and you need the numerical solver.
When your stream changes sign more than once
A project with an interim outflow — a mid-life overhaul, a working-capital rebuild, a mine reclamation cost in the final year — changes sign two or more times. Descartes' rule then permits as many positive IRRs as there are sign changes, and streams with two real IRRs are easy to construct: −$1,000 at t = 0, +$2,600 at t = 1, −$1,680 at t = 2 has NPV zero at both 20% and 40%.
Verify that yourself: at 20%, −1,000 + 2,600/1.2 − 1,680/1.44 = −1,000 + 2,166.67 − 1,166.67 = 0. At 40%, −1,000 + 2,600/1.4 − 1,680/1.96 = −1,000 + 1,857.14 − 857.14 = 0. Both roots are correct; neither is the return. Enter $1,000 as the initial investment and 2600, -1680 as the cash flow list and this calculator reports both roots, withholds the accept/reject verdict, and points you at the NPV instead.
When that happens, the IRR rule has no meaning and you fall back on net present value, which is single-valued at every discount rate, or on the modified IRR, which is single-valued by construction.
Mistakes that make an IRR wrong or misleading
- Mixing period lengths. Monthly flows give a monthly IRR. Compound it — a 2% monthly IRR is 26.8% a year, not 24%.
- Ranking mutually exclusive projects by IRR. The higher rate frequently belongs to the smaller project. Rank by NPV and let IRR act only as a screen.
- Leaving out the terminal cash flow. Salvage proceeds, working-capital release and decommissioning costs belong in the final period, and each moves the IRR materially.
- Mixing nominal cash flows with a real hurdle rate, or the reverse. Pick one convention for both and stay in it.
- Reading a lending-shaped stream backwards. If you receive cash first and pay later, a high IRR is bad news — it is the rate you are paying. Check the NPV, which never inverts.
- Treating an IRR as achieved rather than forecast. The number inherits every optimism in your projections, and it is more sensitive to the late years than payback is.
What this calculator assumes
Cash flows are assumed to arrive at the end of each equal-length period, with the outlay falling entirely at period 0. Every flow is nominal, discounted at a single constant rate, and taken as certain — the IRR contains no risk adjustment at all, which is why the risk has to live in the hurdle rate you compare it against.
The solver searches from −99% to +501% per period, so a stream whose only root lies outside that window reports no IRR. Rates at or below −100% are excluded because (1 + r) turns non-positive and the discount factors stop existing.
Taxes, depreciation and financing are not modelled. Build them into the cash flows first: an after-tax project stream is after-tax operating cash flow plus the depreciation tax shield, with interest excluded because the cost of debt already sits inside the discount rate.
Where IRR sits among the other decision rules
IRR is one of four rules a capital budgeting pack shows side by side, and each answers a different question. Net present value answers “how much value in today's dollars?” — the theoretically correct rule, additive across projects and single-valued at every discount rate, so make the NPV calculator your primary test. IRR answers “at what rate?”, which is why lenders, sponsors and boards who do not share your cost of capital keep asking for it. Payback answers “how long until the money is back?”; it ignores everything after the recovery date but captures a real liquidity constraint, and the payback period calculator and its discounted version cover both variants. Profitability index — present value of inflows divided by the outlay — answers “how much value per dollar committed?” and is the right rule when capital is rationed rather than merely priced.
In a well-behaved conventional project all four agree. When they disagree, the disagreement is information: the projects differ in scale, in timing, or in the shape of their sign pattern. Follow NPV, and use the others to understand why NPV lands where it does.
Key terms
- Hurdle rate
- The minimum return a project must earn to be funded. Normally the weighted average cost of capital, raised for projects riskier than the firm's existing business.
- NPV profile
- A plot of net present value against discount rate. Its horizontal-axis crossings are the IRRs, and its steepness shows how sensitive the project is to the cost of capital.
- Conventional cash flows
- One initial outflow followed by inflows only — a single sign change, and therefore exactly one IRR.
- Money multiple
- Total cash received divided by total cash invested, undiscounted. Often called MOIC. It ignores timing entirely, which is why it is quoted next to an IRR rather than instead of one.
- Reinvestment assumption
- The implicit assumption inside the IRR equation that interim cash flows compound at the IRR until the project ends.