Internal Rate of Return (IRR)

Internal Rate of Return (IRR) for Cash Flows

The Internal Rate of Return (IRR) is the discount rate that makes the net present value of a series of cash flows equal to zero. Formally, it satisfies Σ FC_t / (1+IRR)^t = 0, where FC_t is the cash flow in period t (the initial investment is conventionally negative). Because the equation is a polynomial of degree n, IRR is normally computed numerically (Newton-Raphson or bisection) rather than algebraically.

In project appraisal the decision rule is simple: accept the project if IRR > cost of capital (also called the hurdle rate or WACC). When comparing mutually exclusive projects, however, IRR can mislead because it ignores scale and reinvestment assumptions — many textbooks recommend pairing IRR with NPV. Non-conventional cash flows (multiple sign changes) may also produce multiple IRR roots; the MIRR (Modified IRR) resolves this by reinvesting positive flows at the cost of capital and discounting negative flows at the financing rate.

Applications

IRR is one of the most used metrics in corporate finance, project appraisal, real-estate underwriting, private equity and infrastructure concessions. Brazilian regulators such as CVM and exchanges like B3 (Bovespa) reference IRR in prospectuses and offering memoranda; multilateral lenders (BNDES, World Bank) use it to compare investment alternatives across sectors and currencies.

FAQ

What does a negative IRR mean? It indicates that, at every plausible discount rate, the project destroys value — the sum of discounted inflows never reaches the initial outlay. In practice, no rational investor should accept a project with a clearly negative IRR.

Why can the same cash flow have more than one IRR? When cash flows change sign more than once (e.g. large reinvestments mid-project), the NPV equation is a polynomial with multiple real roots; each root is mathematically a valid IRR. MIRR avoids this ambiguity.

IRR or NPV: which is better? NPV measures absolute value created and is the preferred theoretical criterion. IRR is intuitive (a percentage) and useful for ranking under a capital constraint, but should not be used in isolation for mutually exclusive projects with different scales.