Project IRR Calculator — Capital Budgeting

The internal rate of return is the root of a polynomial: the single discount rate at which the present value of a project's inflows exactly cancels its outflows, leaving NPV at zero. Because the equation has no closed-form solution beyond a few periods, it must be solved numerically, and this tool uses Newton-Raphson iteration, starting from a 10 percent guess and refining it using the slope of the NPV curve until successive estimates differ by less than one part in ten billion. IRR is then compared against a hurdle rate, usually the weighted average cost of capital plus a risk premium. Industrial corporates commonly set hurdles between 10 and 15 percent, infrastructure and utility projects between 6 and 9 percent, and private equity sponsors target 20 to 25 percent gross. Two further outputs guard against IRR's blind spots. Payback period shows how many years of undiscounted inflow are needed to recover the outlay, which is a crude liquidity test but a real constraint for a firm with limited cash. Discounted payback applies the same test to present values and is always longer. The profitability index, present value of inflows divided by the initial outlay, ranks projects when the capital budget is fixed. The limitations of IRR are structural rather than computational. If the sign of the cash flow sequence changes more than once, Descartes' rule of signs permits multiple real roots and the reported figure is only one of them. IRR also implicitly assumes interim receipts are reinvested at the IRR itself, which flatters high-return projects; modified IRR addresses this by specifying a separate reinvestment rate. And IRR is scale-blind, so a 40 percent return on a small outlay can rank ahead of a 15 percent return that creates far more absolute value.

The concept predates the name. Irving Fisher, in The Theory of Interest (1930), defined what he called the rate of return over cost as the rate that equates the present values of two competing investment streams. John Maynard Keynes gave the same quantity a different label in The General Theory of Employment, Interest and Money (1936), calling it the marginal efficiency of capital and defining it as the discount rate that makes the present value of a capital asset's expected yields equal to its supply price. Kenneth Boulding set out the mathematics of a single investment's internal rate in a 1935 article in the Quarterly Journal of Economics, and the ambiguity of the underlying polynomial was visible almost immediately. The method entered corporate practice with Joel Dean's Capital Budgeting in 1951, which advocated discounted measures over the payback and accounting-return rules then dominant. Its weaknesses were documented quickly: James Lorie and Leonard Savage, writing in the Journal of Business in 1955 on capital rationing, showed concretely that a project with non-conventional cash flows can have more than one internal rate of return, and Ezra Solomon set out the reinvestment-rate objection soon afterwards. That debate produced the modified IRR and cemented NPV as the theoretically preferred criterion. IRR nevertheless remained popular because a percentage is easy to communicate to a board, and the 2001 Graham and Harvey survey of 392 chief financial officers found it used always or almost always by about three-quarters of respondents, marginally ahead of NPV itself.