Internal Rate of Return (IRR) Calculator

The internal rate of return is the discount rate at which the present value of a project's cash flows nets to zero. It is defined implicitly, as the root of the polynomial CF0 + CF1/(1+r) + CF2/(1+r)^2 + ... + CFn/(1+r)^n = 0, and no closed-form solution exists beyond low-order cases, so it must be found numerically - here by bisection, narrowing a bracket between negative 99 percent and 1000 percent until the net present value falls within a cent of zero. The rate is called internal because it depends only on the project's own cash flows and uses no external market rate as an input. Decision rules compare it against a hurdle: a corporate weighted average cost of capital typically in the 8 to 12 percent range, roughly 8 to 12 percent for stabilised real estate, 15 to 25 percent for value-add property, and 20 to 30 percent or higher for venture and private equity where illiquidity and failure rates demand a premium. Three limitations matter when reading the result. IRR implicitly assumes interim cash flows are reinvested at the IRR itself, which is unrealistic for high-return projects and is what the modified IRR corrects. Descartes' rule means a stream with more than one sign change can admit multiple mathematically valid rates, and this bisection returns only one of them. And IRR is scale-blind: a small project at 45 percent may create less total value than a large one at 14 percent, so it should be read alongside net present value, which is why the sensitivity table shows NPV at several discount rates.

The concept preceded the name. Irving Fisher analysed what he called the rate of return over cost in The Rate of Interest in 1907 and again in The Theory of Interest in 1930, defining it as the rate that equalises the present values of two competing investment options. John Maynard Keynes described essentially the same quantity as the marginal efficiency of capital in The General Theory of Employment, Interest and Money in 1936, and Kenneth Boulding worked through the arithmetic of a single investment in a 1935 paper. The method reached corporate practice through capital budgeting literature after the Second World War, most influentially with Joel Dean's Capital Budgeting in 1951, which argued that firms should rank projects by discounted rate of return rather than by accounting payback. Its weaknesses were catalogued quickly: James Lorie and Leonard Savage set out the multiple-root and mutually-exclusive-project problems in Three Problems in Rationing Capital in the Journal of Business in 1955, and the profession has favoured net present value for ranking decisions ever since. Practical adoption nonetheless came from computation. Solving the polynomial by hand meant interpolating between trial discount rates, so IRR stayed a specialist tool until financial calculators and then spreadsheet functions - Lotus 1-2-3 and later Excel, which uses an iterative Newton-style search from a seed guess - made it a keystroke.