Finds the discount rate at which the net present value comes out exactly zero. It says what rate the investment effectively earned, so projects can be compared without having to settle on a discount rate first. It is solved by bisection.
This says what rate an investment effectively earned. It is defined as the discount rate that brings the net present value to exactly zero.
is the initial outlay, the cash received in year and the internal rate of return. The equation cannot be solved for directly, so it is bracketed by bisection. Raising the discount rate only ever lowers the left-hand side, which makes bracketing reliable.
Net present value demands that a discount rate be chosen up front, and the conclusion turns on that choice. The internal rate of return avoids the question: it hands back a rate, which can then be set against whatever return is being required.
An outlay of 1,000,000 brings in 300,000 a year for five years, a total of 1,500,000 and a net gain of 500,000. The internal rate of return is 15.24%. Discounting the five years at that rate brings the present value of the cash flow to exactly 1,000,000, leaving a net present value of zero.
A total that exceeds the outlay still gives a low rate if the money arrives far in the future. The same 1,500,000 is worth a different rate over five years than over ten.
Cash flows that turn negative part way through can admit more than one solution. Only cash flows of zero or above are handled here.