Internal Rate of Return (IRR)
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
The root equation does not require literal reinvestment. To interpret IRR as a compound growth rate of terminal wealth, interim distributions must be reinvested at that rate. MIRR explicitly models financing and reinvestment rates.
Yes, when cash flows change sign more than once. A returned root is one solution, not necessarily the only solution or a useful investment ranking. Some streams have no admissible root; use an NPV profile to examine them.