Internal Rate of Return (IRR)

Solving for the Discount Rate That Zeroes Out an Investment’s NPV

IRR (Internal Rate of Return) is the annual return an investment’s own cash flows imply — specifically, the discount rate at which the investment’s net present value works out to exactly zero. Enter the upfront investment and each period’s cash flow, and this calculator finds that rate.

Unlike the Payback Period Calculator, which simply asks how long until the money comes back, IRR accounts for the time value of money: a dollar returned next year is worth less than a dollar returned today, and IRR is the single rate that captures that trade-off across every cash flow at once. It’s directly comparable to a savings account’s interest rate or another investment’s own IRR, which is what makes it one of the most widely used capital-budgeting metrics for comparing different projects or investments.

The Formula

There’s no simple closed-form formula for IRR — it has to be solved numerically, the same way the Annual Percentage Rate (APR) Calculator solves for an effective interest rate. IRR is the rate r that satisfies:

$$0 = -(\vA{\text{Initial Investment}}) + \sum \left[\frac{\vB{\text{Cash Flow in period } t}}{(1+\vC{r})^{t}}\right]$$

This calculator searches for that rate using bisection — repeatedly narrowing a range of possible rates until the resulting net present value converges on zero.

Worked Example

A $10,000 investment returning $3,000, $4,000, $5,000, and $2,000 over four years:

1. Total cash returned: $3,000 + $4,000 + $5,000 + $2,000 = $14,000.
2. Net profit: $14,000 − $10,000 = $4,000.
3. Solving for the rate at which those four cash flows, discounted back to today, exactly equal the $10,000 invested gives an IRR of ≈ 15.32%.

Key Factors to Consider

• IRR assumes intermediate cash flows are reinvested at the IRR itself, an assumption that can be unrealistic for a high IRR. This is a well-known theoretical limitation of IRR — a related metric called Modified IRR (MIRR) addresses it by assuming reinvestment at a more realistic rate, though this calculator computes the standard, more widely reported IRR.
• This calculator assumes the well-behaved case: one upfront outlay followed by cash flows that don’t turn negative again. Under that pattern, exactly one real IRR exists — a more complex cash flow pattern with multiple sign changes (money going out again partway through) can mathematically produce more than one IRR or none at all, a genuinely tricky edge case standard IRR calculations aren’t built to handle cleanly.
• IRR and NPV can occasionally disagree about which of two investments is better. When comparing two mutually exclusive investments of different sizes or timing, the one with the higher IRR isn’t always the one that creates more actual value — NPV (Net Present Value) is often considered the more reliable metric for a direct dollar-value comparison between two specific investment choices.
• IRR is a percentage return; the total dollar amount at stake still matters separately. A small investment with an excellent IRR might create less total value than a larger investment with a more modest IRR — IRR describes efficiency of return, not the absolute size of the payoff.

Common Mistakes

• Comparing IRR alone across investments of very different sizes. A tiny investment can post a spectacular IRR while creating almost no real value — always look at IRR alongside the actual dollar amounts involved, not as a standalone ranking.
• Assuming reinvestment happens at the IRR itself. The math implicitly assumes every cash flow gets reinvested at the same rate the investment is earning — unrealistic for a high IRR, and part of why MIRR exists as a more conservative alternative.
• Applying standard IRR to a cash flow pattern with multiple sign changes. Money going out again partway through a project (a follow-on investment, for instance) can produce more than one mathematically valid IRR or none at all — this calculator’s bisection search assumes the simpler, single-sign-change case.
• Using IRR alone to choose between two mutually exclusive investments. The option with the higher IRR doesn’t always create more actual dollar value — NPV is generally the more reliable metric for a direct value comparison between two specific choices.

Useful to Know

• Want a direct dollar-value comparison between investment options rather than a percentage rate? The Net Present Value (NPV) Calculator calculator answers that more reliable comparison question directly.
• Just want to know how long until an investment’s cash flows repay the initial cost, ignoring the time value of money? The Payback Period Calculator calculator answers that simpler question.
• Need to bring a single future cash flow back to today’s dollars using a chosen discount rate instead of solving for one? The Present Value Calculator calculator does that directly.