Internal Rate of Return (IRR)
Reverse the question: which makes the present-value total exactly zero?
Reverse the NPV question
Which discount rate makes NPV exactly zero?
NPV starts with a chosen discount rate. Keep the same initial outlay and three inflows, then move the rate to find where the result reaches zero.
Substitute the current trial rate into NPV
Check the root using the solved IRR
- The slider is a trial rate; IRR is the separately solved root. Displayed values are rounded.
Current NPV is $221.03. Raise the discount rate to move toward zero. The curve crosses zero near 15.98%: that is the IRR.
This conventional cash-flow pattern has one IRR. Other patterns may have no solution or multiple roots; IRR does not promise the same realized return each year. The equation and limitations follow below.
What did you just see?
It summarizes a cash-flow series as an annualized break-even rate.
Cash-flow timing and solver limits
The initial outlay occurs at year 0; later rows occur at year end, with positive receipts and negative payments. Equal annual spacing makes the result an effective annual . Repeated sign changes may produce multiple roots. Numerical search has range and precision limits; no result does not prove no mathematical solution exists.
A higher rate can create less value
A costs US$100 now and pays US$130 in one year: IRR is 30.00%. B costs US$1,000 and pays US$1,200: IRR is 20.00%. At a common 10% required return, their NPVs are US$18.18 and US$90.91. A larger percentage on a small outlay need not create more money.
For mutually exclusive projects with comparable risk and adequate funding, NPV directly compares added value. With a binding capital budget, replication, alternative opportunities and feasible combinations also matter. Neither the bigger project nor the higher IRR wins unconditionally.
One payment stream can have two break-even rates
Pay 100 today, receive 230 in year one, then pay 132 in year two. At 10%, −100 + 230/1.10 − 132/1.10² = 0. Substituting 20% also gives zero: this stream has two IRRs. Picking the larger root arbitrarily is not a valid decision rule. Multiple sign changes permit multiple roots but do not guarantee them.
A conventional initial outlay followed only by receipts generally permits a useful IRR-versus-required-return comparison. For complex streams, inspect NPV at the relevant hurdle rate and the full curve. A failed numerical search does not prove no root exists; finding one number does not prove uniqueness. IRR is not a promise of the return earned each year.
Another example
- Enter an initial investment of 1,000 as the time-zero outflow.
- Enter inflows of 600 in year one and year two.
- IRR is about 13.07%, the rate at which these cash flows have an NPV near zero.
Show formal definition and formula
IRR is the discount rate at which cash flows have zero net present value.
IRR is the discount rate that makes the net present value of all cash flows equal zero.
Common mistakes
Do not overlook multiple roots or directly compare investments of very different sizes by IRR alone.
Frequently asked questions
Why can IRR have multiple answers?
When cash flows alternate signs, the NPV curve may cross zero more than once.
Can IRR be found without both inflows and outflows?
Generally no. At least one positive and one negative cash flow are needed.
Should I compare projects using IRR or NPV?
When scale or timing differs, NPV often better reflects value at a chosen discount rate.
You can now explore…
- What a zero crossing of NPV means
- When IRR may not be unique
Term reference4 terms
Open a term when you need a reminder. Underlined words in the article work too.