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===Internal rate of return===
{{Main|Internal rate of return}}
The '''[[internal rate of return]]''' (IRR) is the [[Interest rate|discount rate]] that gives a [[net present value]] (NPV) of zero. It is a widely used measure of investment efficiency. To maximize return, sort projects in order of IRR.

Many projects have a simple cash flow structure, with a negative cash flow at the start, and subsequent cash flows are positive. In such a case, if the IRR is greater than the cost of capital, the NPV is positive, so for non-mutually exclusive projects in an unconstrained environment, applying this criterion will result in the same decision as the NPV method.

An example of a project with cash flows which do not conform to this pattern is a loan, consisting of a positive cash flow at the beginning, followed by negative cash flows later. The greater the IRR of the loan, the higher the rate the borrower must pay, so clearly, a lower IRR is preferable in this case. Any such loan with IRR less than the cost of capital has a positive NPV.

Excluding such cases, for investment projects, where the pattern of cash flows is such that the higher the IRR, the higher the NPV, for mutually exclusive projects, the decision rule of taking the project with the highest IRR will maximize the return, but it may select a project with a lower NPV.

In some cases, several solutions to the equation NPV = 0 may exist, meaning there is more than one possible IRR. The IRR exists and is unique if one or more years of net investment (negative cash flow) are followed by years of net revenues.  But if the signs of the cash flows change more than once, there may be several IRRs.  The IRR equation generally cannot be solved analytically but only via iterations.

IRR is the return on capital invested, over the sub-period it is invested. It may be impossible to reinvest intermediate cash flows at the same rate as the IRR. Accordingly, a measure called [[Modified Internal Rate of Return]] (MIRR) is designed to overcome this issue, by simulating reinvestment of cash flows at a second rate of return.

Despite a strong academic preference for maximizing the value of the firm according to NPV, surveys indicate that executives prefer to maximize returns{{Citation needed|date=July 2009}}.