Internal Rate of Return (IRR)
The discount rate that makes the NPV of a cash flow stream exactly zero. Solved numerically via Newton-Raphson with bisection fallback.
When to use: Use when you want a single rate-of-return number for a project — easier to interpret than NPV for non-finance audiences. Compare IRR to your hurdle rate: IRR > hurdle = accept, IRR < hurdle = reject.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| IRR | Internal Rate of Return | Discount rate that makes NPV equal to zero | % |
| CashFlows | Cash Flows | Sequence of cash flows starting at period 0 (negative for outflows, positive for inflows) | $ |
Real-Life Examples
Example 1: 4-Year Project
Invest $1,000 today, receive $400/year for 4 years. What rate of return does this earn?
Given
Step-by-Step
The project earns 21.86% per year — well above a typical 10% hurdle. Accept.
Example 2: Mixed Cash Flows
Invest $10,000, then receive $3,000, $4,000, $5,000, $2,000.
Given
Step-by-Step
IRR of 15.32% — a healthy return. Compare to your cost of capital to decide.
Frequently Asked Questions
Yes — when cash flows change sign more than once (e.g. a project with a mid-life capital outlay), Descartes' rule of signs allows multiple positive real roots. In practice this is rare for typical investment-then-returns shapes; when it happens, prefer NPV or MIRR for unambiguous decisions.
IRR assumes intermediate cash flows are reinvested at the IRR itself. For high-IRR projects this is unrealistic — most companies cannot reinvest at 25-30% indefinitely. MIRR fixes this by letting you specify a separate, more realistic reinvestment rate.
Either (a) all cash flows have the same sign — no root exists; (b) cash flows are too irregular and the iteration fails to converge; or (c) numerical precision issues with extreme values. Verify your cash flows include at least one negative and one positive entry.
NPV is preferred for accept/reject decisions on individual projects and for ranking projects of different sizes. IRR is intuitive for communicating returns and useful when comparing projects with similar scale and timing. For projects with non-standard cash flows, both should be checked together.