THOSE WHO WORK REGULARLY with investment projects know that, if a project requires significant net cash infusions after the initial investment, multiple rates of return can potentially occur, and there has been considerable interest in the financial literature in simple methods of ruling out this possibility. Table 1 shows, for example, the projected yearly after-tax cash flows to an investor in the 50% tax bracket from one partnership unit in an FHA-insured Section 8 housing project for the elderly in Atlanta, Ga. Investment is required in years 0, 2, 3, 4, and 20. The large outflow in year 20 reflects the tax consequences of recapture of accelerated depreciation. Many oil-well projects also have three investment periods. Do the three separated intervals of negative cash flow mean that the project has multiple rates of return? Comparing projects An equivalent difficulty can easily arise when two mutually exclusive projects are being compared for possible adoption. The difference between the present values of two projects is precisely the present value of the stream of period-byperiod cash-flow differences between them. Of course, this present value varies according to the interest rate considered. An internal rate of return of the stream of differences is a interest rate at which the two projects have equal present value. If there is no such breakeven rate, that is, if the difference stream has no internal rate of return, then one project dominates the other, being preferable at all interest rates. If there are several breakeven rates, then preference between the projects switches several times as the interest rate varies. The cash-flow differences can easily change sign frequently-for example, if one project is subject to wider swings or less frequent payments than another, as in Table 2. (We keep the numbers in our examples small, for clarity, but they could, of course, denote millions of dollars.) Is there more than one interest rate at which Projects A and B are equally desirable? (For answers, see Section 6.7.)