CF · IRR
IRR Calculator
Find the rate that drives net present value to zero for any set of uneven cash flows — the same Newton iteration the BA II Plus runs behind the IRR key — with MIRR beside it, a warning when the flows admit more than one root, and a repayment table that shows the investment closing at exactly zero.
- Up to 24 cash-flow groups
- MIRR with your own rates
- Multiple-root warning
- NPV profile chart
The idea
What the internal rate of return really measures
Net present value asks how much a project is worth at a rate you supply. IRR turns the question round: it asks which rate would make the project worth exactly nothing. That rate is the return the project earns on the money still committed to it — not on the original outlay, and not on the total of the cash flows, but on the shrinking balance that remains unrecovered as the flows come in.
The repayment panel above is the honest way to see it. Charge the IRR as interest on the outstanding balance, subtract each cash flow when it arrives, and the balance closes on zero at the final period. Any other rate leaves money on the table or overdraws it. That is the whole definition, and it is why IRR behaves like the interest rate on a loan read from the lender's side.
0 = CF0 + CF1 ÷ (1 + r)¹ + CF2 ÷ (1 + r)² + … + CFn ÷ (1 + r)ⁿ
There is no algebraic solution for r beyond a handful of periods, which is why every calculator — the BA II Plus included — iterates towards it rather than evaluating a formula. See the NPV calculator for the same equation read the other way.
Step by step
How to calculate IRR by hand
You cannot solve for IRR directly, so by hand it becomes guess, measure, and close in. The example below is the one loaded above: £80,000 out today, then £20,000 for two years and £30,000 for three.
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Write the flows against their periods
t = 0 : −80,000 | t = 1–2 : +20,000 | t = 3–5 : +30,000
Two groups, not five rows. On the hardware that is CF1 = 20,000 with F1 = 2, then CF2 = 30,000 with F2 = 3.
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Guess a rate and price the flows at it
NPV at 15%
+4,307.56
Still positive, so 15% is too low — the project can afford to be charged more than that.
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Overshoot deliberately to bracket the answer
NPV at 20%
−5,559.41
Negative, so the root is trapped between 15% and 20%. One rate above zero and one below is all a solver needs to guarantee an answer.
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Interpolate between the two
r ≈ 15% + 5% × 4,307.56 ÷ (4,307.56 + 5,559.41)
17.1828%
Close, but a little high: NPV curves, and straight-line interpolation always cuts the corner. Repeat the step with the new bracket and the error collapses fast.
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Settle on the rate where the flows repay exactly
NPV at 17.063938%
17,084.68 + 14,594.32 + 18,700.44 + 15,974.55 + 13,646.01 = 80,000.00
IRR = 17.0639%
The discounted inflows come to the outlay to the penny, so NPV is zero and the rate is the IRR. The BA II Plus reaches the same figure by the same route, only faster.
The catch
When one project has two IRRs
IRR is the root of a polynomial of degree n, and a polynomial can have as many roots as it has changes of sign. A conventional project — pay once, receive thereafter — changes sign exactly once and has exactly one IRR. Add a later outflow and the guarantee is gone.
Take £1,000 out today, receive £5,000 next year, then pay £6,000 to close the site the year after. NPV is zero at 100% and again at 200%, and it is positive in between:
| Discount rate | NPV | Reading |
|---|---|---|
| 0% | −2,000.00 | Below zero |
| 50% | −333.33 | Below zero |
| 100% | 0.00 | First IRR |
| 150% | +40.00 | Above zero |
| 200% | 0.00 | Second IRR |
| 250% | −61.22 | Below zero |
Both 100% and 200% satisfy the equation, so neither is the IRR. A calculator will report whichever one its search happens to reach first, which is exactly why the results card above counts the sign changes and says so. MIRR for the same flows at a 10% finance and reinvestment rate is a single, unambiguous −3.9259% — and the sign tells you what the two positive roots hid: this is a project that loses money.
The fix
What MIRR changes, and why it is unique
MIRR does not solve a polynomial at all. It collapses the whole stream into two numbers first: every inflow is carried forward to the final period at a reinvestment rate you choose, and every outflow is pulled back to today at your finance rate. What is left is one payment out and one payment in, separated by n periods, and the growth rate between them is arithmetic rather than search.
MIRR = (TV of inflows ÷ PV of outflows) ^ (1 ÷ n) − 1
Because only one sign change survives the collapse, there is exactly one answer — always, for any stream. The price is that you have to state the reinvestment rate rather than let the maths invent one.
That is the real argument for it. Comparing two IRRs quietly credits each project with re-earning its own rate on interim cash, so a short project with a 40% IRR is being assumed to find another 40% home for every pound it returns. Set the reinvestment rate to something you could actually get and MIRR falls back towards the truth — which is why it is usually lower than IRR for a good project and higher for a bad one.
Reading the card
What each figure on the results card means
- Internal rate of return
- The rate per period at which the flows exactly repay the money still tied up. Compare it to your cost of capital, not to another project's IRR.
- MIRR
- The same idea with the reinvestment assumption made explicit. Unique for every stream, including the ones that give IRR several roots.
- Sign changes in the flows
- One change means one IRR and you can trust it. Two or more means several rates satisfy the equation and the single figure above is only the first one found.
- Terminal value and present value of the outflows
- The two numbers MIRR is built from — inflows grown to the last period, outflows discounted to today. Shown so the MIRR figure can be checked by hand rather than trusted.
- NPV at the finance rate
- The cross-check. IRR above the finance rate and NPV above zero are the same statement; if they ever disagree, a sign or a frequency is wrong.
- Payback
- How many periods until the undiscounted cash turns positive, interpolated inside the crossover period — 3.33 means a third of the way through the fourth.
Watch for
Five mistakes that account for most wrong answers
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Ranking projects by IRR
IRR is blind to size. 40% on £1,000 returns £400 of value; 12% on £1m returns far more. Rank by NPV and quote IRR alongside it.
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Entering the outlay as a positive number
With every flow the same sign there is no crossing and no IRR — that is the Error 7 the hardware reports. CF0 must be negative for an investment.
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Reading a periodic IRR as an annual one
Quarterly flows give a quarterly rate. Annualise it by compounding — 1.04⁴ − 1 = 16.99%, not 4 × 4% — using the interest conversion calculator.
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Skipping a period with no cash flow
A gap year still occupies a period. Enter it as amount 0 with the right frequency; leaving it out pulls every later flow forward and inflates the rate.
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Trusting a single IRR when the flows change sign twice
Several rates satisfy the equation and the calculator reports whichever it reaches first. Check the sign-change count on the results card, and switch to MIRR when it is above one.
Related work
When IRR is not the tool you want
IRR answers one narrow question well: what rate does this stream earn on the capital it ties up. Several neighbouring questions look similar and need a different worksheet.
- NPV calculator — when you have a required return already and want the answer in money rather than as a rate.
- TVM calculator — when the payments are level. A single rate on an even annuity is I/Y, and it comes back instantly.
- Bond calculator — yield to maturity is an IRR, but with coupon dates, day counts and accrued interest handled for you.
- Interest conversion calculator — to turn a periodic IRR into an effective annual rate, or the other way round.
- Amortization calculator — for the borrower's side of the same arithmetic, period by period.
FAQ
IRR questions, answered
What does IRR actually tell you?
IRR is the rate at which the project exactly repays itself. Charge that rate as interest on whatever money is still tied up, take each cash flow off the balance as it arrives, and the balance lands on precisely zero at the last period. The repayment table on this page shows that happening line by line. It is the same idea as the interest rate on a loan, read from the other side.
Why does my BA II Plus show Error 7 when I press IRR CPT?
Error 7 is an iteration failure: the calculator ran out of attempts without landing on zero. It nearly always means the flows never cross zero — every amount has the same sign, so there is no rate that balances them — or the sign changes so many times that the search wanders. Check that CF0 was entered with +/− and that the frequencies are right.
Can one set of cash flows have more than one IRR?
Yes. A stream that changes sign more than once can cross zero more than once, and every crossing is a legitimate IRR. A mine that costs money to open, earns for ten years, then costs money to close is the classic case. This page counts the sign changes and warns you when several roots are possible; when they are, MIRR is the figure to quote because it is always unique.
IRR or NPV — which should decide the project?
NPV decides. It is measured in money and it adds up across projects, so a positive NPV is a direct statement about how much better off you are. IRR is a rate, which makes it easy to compare against a hurdle but blind to size: 40% on £1,000 beats 12% on £1m by IRR and loses badly by NPV. Use IRR to communicate, NPV to choose.
What is MIRR and when should I use it instead?
MIRR carries every inflow forward to the last period at a reinvestment rate you choose, pulls every outflow back to today at your finance rate, and reports the single growth rate between the two. Because only one sign change survives that collapse there is exactly one answer, and the reinvestment assumption is yours rather than an accident of the arithmetic. Use it when the flows change sign repeatedly or when IRR comes out far above anything you could actually re-earn.
Does IRR really assume you reinvest at IRR?
Not as an assumption baked into the maths — IRR is just the root of the NPV equation. But comparing two IRRs does implicitly credit each project with re-earning its own rate on interim cash, which is why a short, very high-IRR project can look better than it is. MIRR removes the sleight of hand by making the reinvestment rate an input.
How does this calculator find the rate?
Newton–Raphson from several starting guesses: evaluate NPV, evaluate its slope, step to where the tangent hits zero, repeat. It converges in a handful of passes on ordinary flows. When the iteration diverges — which awkward streams can make it do — the solver falls back to bracketing a sign change and bisecting it, so an answer is either exact or honestly reported as absent, never a stray number from a diverged run.
How do I enter a period with no cash flow?
Enter the amount as 0 and give it the right frequency. A year of nothing is a group of amount 0 times 1; it holds the place so later flows land in the right period. Skipping the row instead pulls everything forward and quietly inflates the rate.
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