IRRs

Internal rate of return (IRR) is a financial metric used to evaluate the profitability of potential investments. It is the discount rate that makes the net present value (NPV) of the project's cash flows equal to zero. In practice, one solves for r in the equation NPV = Σ CF_t / (1+r)^t = 0, where CF_t are the cash flows at time t, and CF_0 is the initial investment (often negative).

Calculation is typically done by numerical methods; there is no closed-form solution in general. If the project has conventional cash flows (one initial outlay followed by all positive inflows), IRR is unique. If cash flows alternate between positive and negative, multiple IRRs may exist, complicating interpretation.

Use: compare the IRR to a required rate of return or cost of capital. For independent projects, an IRR higher than the hurdle rate suggests acceptance. However, IRR can conflict with NPV when projects differ in scale or duration, and it assumes reinvestment of interim cash flows at the IRR, which may be unrealistic. Therefore, IRR should be considered alongside NPV and other metrics. To address reinvestment assumptions, the modified internal rate of return (MIRR) uses specified reinvestment and financing rates.

Limitations: susceptibility to multiple IRRs; misleading comparisons across projects with different sizes or lifespans; reliance on estimated cash flows; sensitive to timing.

IRR remains widely used due to its intuitive interpretation as a percentage return; it is common in capital budgeting, project appraisal, and financial analysis, but it is not a stand-alone decision rule.