Now that we have described NPV, we can introduce the internal rate of return or IRR. IRR is simply a specialized discount rate, namely the discount rate that yields a net present value of zero.

For example, if an investment will pay $10,000 in ﬁve years and the investor paid $8,000 for it at time zero (to), then the internal rate of return IRR is equal to 4.6 percent.

Since an IRR is simply a discount rate that yields a certain NPV (0), calculating an IRR involves trying many iterations until you arrive at the correct discount rate. As such, it is easier to use a ﬁnancial calculator or spreadsheet software to perform the iterations required. The important point is that the IRR is simply a speciﬁed discount rate in which the present value of the investment is equal to the amount paid for the investment. Said differently, the IRR is the discount rate when the NPV is equal to zero.

IRR is often used by investors to measure their return based on varying purchase prices. In our example above, if the investor paid $5,000 instead of $8,000, the IRR would be 14.9 percent (as compared to a return of 4.6 percent if the investor were to pay $8,000 for it). Investors will compare the projected IRR with their opportunity cost of capital and some will choose to invest if the IRR on a project is greater than their cost of capital — which is the same as determining whether the NPV is greater than zero.