The internal rate of return or return on an investment is the discount rate that equals the present value of the expected cash outflows with the present value of the expected income. From the mathematical point of view, it is represented by the rate r, in such a way that where at is the cash flow for period t, whether it is income or net cash outflows, n is the last period where a cash flow is expected. Cash and Σ: indicates the sum of discounted cash flows at the end of periods zero to n. If the outlay or initial cost of cash occurs at time O, the equation can be expressed as
In this form, r is the rate that discounts the series of future cash flows (A j up to an n) to equal the initial outlay at the time OA or. We implicitly assume that the cash income received from the investment is reinvested to achieve the same recovery rate as r.
To exemplify the use of the equation, let’s assume that we have an investment opportunity that requires a cash outlay at time O of $ 18,000, and is expected to p1oppose cash income of $ 5,600 at the end of each of the next five years. . The problem can be expressed as
The search for the internal rate of return, r, involves an iterative procedure that uses the: present values. Fortunately, computer programs and advanced calculators can do this for us. However, if you are curious about a manual method, let’s look again at our example. The series of cash flows is represented by an equal series of cash flows of $ 5,600, which will be received at the end of each of the next five years… We want to determine the discount factor that, when multiplied by $ 5,600, equals the cash outlay of $ 18,000 at time O. Suppose we start with three discount rates -14%, 16%, and 18% AND we calculate the value present of the series of cash flows. With the different discount factors shown in table B at the end of the book
In this way, the internal rate of return necessary to equalize the present value of cash income with the present value of expenses is approximately 16.8%. When, as we see above, the series of cash flows is an equal series and the initial outlay occurs at the moment or, in reality, there is no need for a trial and error method. We simply divide the initial outlay between the cash flow and look for the nearest discount factor. In our example, we divide $ 18,000 between $ 5,600, to obtain 3,214. The nearest discount factor in the five- year line
the interpolation gives only an approximation of the exact percentage; the relationship between the two discount rates is not linear with respect to the present value.