Internal Rate of Return (IRR)
The annualized rate at which a series of cash flows breaks even — the discount rate that makes net present value zero.
IRR is the lens by which investors compare projects with different shapes of cash flow. It answers: 'what rate of return is this project equivalent to?'
Mathematically, IRR is the rate r solving Σ Cₜ / (1+r)ᵗ = 0. There is no closed-form solution; calculators iterate.
Caveats: IRR is unreliable for cash-flow streams with multiple sign changes (it can have multiple solutions or none) and tends to overstate returns when the implied reinvestment rate is unrealistic.