What is the IRR formula?

IRR is the rate that solves 0 = Σ CFt ÷ (1 + IRR)t. It has to be found by trial and error or a numerical method.

How do I decide using IRR?

Accept a project if its IRR is above your required return (hurdle rate) and reject it if it is below.

Can a project have more than one IRR?

Yes, when the cash flows change sign more than once. Each sign change can add a root, so use NPV or MIRR to decide.

Does IRR match Excel's IRR function?

Yes, for normal cash flows (one sign change) the result matches Excel's IRR and numpy-financial's irr.

IRR Formula and Calculator

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Internal Rate of Return (IRR) Calculator

The internal rate of return is the discount rate at which the net present value of a set of cash flows is exactly zero. It is the project’s own compound rate of return, and it is compared with a hurdle rate: if the IRR is higher than the return you require, the project is attractive.

Formula

0 = NPV = Σ_t=0ⁿ CF_t(1 + IRR)^t

Where

IRR: the discount rate that makes NPV exactly zero
CF_t: net cash flow in period t (CF₀ is usually negative)
n: number of periods after time 0

Calculator

Cash flows, starting with time 0

Enter the initial investment as a negative number.

Result and step-by-step solution

Internal rate of return

16.3406%

Accept the project if the IRR exceeds your required return (hurdle rate).

Write the NPV equation
0 = −10,000.00 + 3,000.00(1 + IRR) + 4,200.00(1 + IRR)² + 6,800.00(1 + IRR)³
Try rates until NPV changes sign
Rate NPV
5% 2,540.76
10% 1,307.29
15% 255.61
20% −648.15
25% −1,430.40
NPV falls as the rate rises; it crosses zero between the rates where the sign flips.
Solve numerically
Bisection narrows the interval until NPV = 0 to 15 significant digits: IRR = 16.3406%
Check
NPV at 16.3406% = 0.00 ✓

Rate	NPV
5%	2,540.76
10%	1,307.29
15%	255.61
20%	−648.15
25%	−1,430.40

How to use the formula

There is no algebraic formula for IRR once there are more than a few cash flows, so it is found numerically. The calculator shows NPV at several trial rates so you can see where it changes sign, then narrows the interval by bisection until NPV equals zero to machine precision, and finally checks the answer by recomputing NPV at the IRR. Textbooks do the same by trial and error with linear interpolation.

In the example, 10,000 invested today returns 3,000, 4,200 and 6,800 over three years, an IRR of 16.34%. If the cash flows change sign more than once (for example, a large clean-up cost at the end), there can be several IRRs. The calculator lists them and reports the one closest to zero, the same convention numpy-financial uses; in that situation rely on NPV or the MIRR.

Frequently asked questions

What is the IRR formula?: IRR is the rate that solves 0 = Σ CF_t ÷ (1 + IRR)^t. It has to be found by trial and error or a numerical method.
How do I decide using IRR?: Accept a project if its IRR is above your required return (hurdle rate) and reject it if it is below.
Can a project have more than one IRR?: Yes, when the cash flows change sign more than once. Each sign change can add a root, so use NPV or MIRR to decide.
Does IRR match Excel’s IRR function?: Yes, for normal cash flows (one sign change) the result matches Excel’s IRR and numpy-financial’s irr.

Last reviewed: September 26, 2026. Calculations run in your browser and were checked against numpy-financial, SciPy and published spreadsheet examples.