Corporate Finance & Investment Banking

NPV = Σ CFₜ / (1 + r)ᵗ − Initial Investment. Accept the project if NPV > 0; the discount rate r is the opportunity cost of capital.

IRR is the discount rate that makes NPV = 0: Σ CFₜ / (1 + IRR)ᵗ = Initial Investment. Accept if IRR > cost of capital (hurdle rate).

Number of years required for cumulative cash inflows to recover the initial outlay. Ignores time value of money and cash flows beyond the cutoff — a liquidity screen, not a value rule.

Non-conventional cash flows (sign changes more than once) can yield multiple IRRs by Descartes' rule, or no real IRR at all. Use NPV or MIRR when sign flips occur (e.g., abandonment costs at end of life).

The discount rate at which two projects' NPVs are equal; computed as the IRR of the incremental cash flows (Project A − Project B). Below crossover, the higher-NPV project differs from above crossover; explains NPV/IRR ranking conflicts.

MIRR reinvests interim cash flows at the cost of capital (not IRR) and discounts outflows at the financing rate, producing a single, more realistic return. MIRR = (FV of inflows / PV of outflows)^(1/n) − 1.