Internal Rate of Return Appraisal of Renewable Energy Projects with Cases from Samso > Present Worth > Internal Rate of Return The internal rate of return of a cash flow is the interest rate implied by a cash flow stream. It is the yield of the investment when considering both positive and negative cash flows. If the investment is to be financed through a bank loan, then the bank loan's rate of interest must be less than the internal rate of return in order for the investment to be acceptable. Table of contentsIRR External Links IRR The internal rate of return (IRR) of a cash flow is equivalent to the interest rate of a bank account with an overdraft facility (cash credit). Assume that cash inflows are deposited and cash outflows withdrawn from the bank account. Then the internal rate of return is the interest rate that would just pay the cost of having the account. Example (bank loan) Take the cash flow stream (-10, 1, 1, 11) related to a bank account. It corresponds to taking out a loan of a principal of 10 euros, followed by payments of 10% interest for three years of 1 euro. The principal is paid back after the end of the third year (1 + 10 = 11). The balance of the account at the end of the period, the net future value, is formally NFV = -10 * (1 + i)^{3} + 1 * (1 + i)^{2} + 1 * (1 + i) + 11 = -10 * (1 + 0.1)^{3} + 1 * (1 + 0.1)^{2} + 1 * (1 + 0.1) + 11 = 0 The interest rate i = 0.1 results in NFV = 0, because the loan is exactly repaid at the end of the period, and the account is empty. The IRR of a cash flow stream is that interest rate of a bank account, which would result in a final balance (net future value) of zero. Equivalently, it results in a net present value of zero, since it is proportional to the net future value. Stated more precisely: Given the cash flow stream (x_{0}, x_{1}, ..., x_{K}), then the internal rate of return is a number i = IRR that satisfies the equation NPV = 0, that is 0 = x_{0} + x_{1}/(1+i) + x_{2}/(1+i)^{2} + ... + x_{K}/(1+i)^{K} The consequence of this definition is that we have to solve a Kth order polynomial equation in order to find IRR. It is in general difficult to find the solution, but it is possible with a computer program (for instance the Excel function IRR). Another problem is that the equation may have more than one solution. Many cash flows have, however, an initial negative outflow followed by positive inflows. In that case the IRR is well-defined. Notice that the IRR is determined entirely by the cash flows of the stream. There is no reference to the external market rates, which is why it is called internal rate of return. The higher IRR, the more acceptable the investment. If the IRR is higher than the prevailing market rate of interest, the investment is considered better than what is available externally in the financial market. If a project is to be financed by a loan, then the loan's interest rate must be lower than the IRR in order for the project to be economically acceptable. The lower bound for the IRR is the minimum rate of return that will be attractive to the investor, known as the minimum attractive rate of return (MARR). In some situations the MARR is the available interest rate in the external market, for instance the rate that can be achieved by investing in secure bonds. External Links Wikipedia Internal rate of return