** Introduction of Internal Rate of Return **

The traditional internal rate of return (IRR) method of project selection has been shown to be inferior to the NPV method due to various problems of the IRR. For instance, the basic IRR method cannot rank mutually exclusive projects, the project with higher IRR potentially having lower net present value (NPV); and the IRR rule cannot readily be used in case of multiple real IRRs, which can occur when there are multiple changes in the sign of the cash flows. Thus NPV is preferred strongly by academicians as the criterion for investment decisions. Yet despite the clear advantages of the NPV rule, surveys indicate that many business practitioners prefer IRR over NPV. This is because business practitioners find the IRR method easier to understand than the NPV because the IRR can be calculated (though not used) without having to estimate the cost of capital, and the IRR, expressed as a percentage rate of return, Address correspondence to Duo Zhang, Department of Economics and Finance, University of Missouri-Rolla, 1870 Miner Circle, Rolla, MO 65409, USA. E-mail: zhangduo@umr.edu 328 D. Zhang is more intuitively appealing to evaluate investments and to communicate profitability. The preference for the IRR among practitioners regardless of its apparent drawbacks has motivated numerous efforts to improve and modify the IRR method to methodologies that are more consistent with the NPV than is the traditional IRR method. The modified IRR (MIRR), which assumes that cash flows from all projects are reinvested at the cost of capital rather than the project's own IRR, was proposed by Lin and Beaves .

It indicates relative profitability of projects better than the IRR but it cannot correctly rank order projects with different sizes. Shull and Hajdasinski adjust and improve the MIRR so the adjusted MIRR approach can rank order projects with different sizes in an NPV-consistent manner. Mc- Daniel et al. compute the marginal return on invested capital (MRIC). The MRIC is reasonable because it separates cash flows into external capital funds required and internal operating cash flows. Then (1 + IRR)T is computed as the ratio between the terminal value of the internal operating cash flows and the present value of the externally financed funding. Hazen shows that the traditional IRR method with multiple internal rates of returns in fact is consistent with the NPV in accepting or rejecting the cash flow stream, regardless of which IRR is used, given that one properly identifies the underlying investment stream as net investment or net borrowing. Hartman and Schafrick present a method of identifying the relevant IRR from among a set of real IRRs and using only the relevant IRR in decision making. All of these methods are NPV-consistent; however, except for that of Hartman and Schafrick, they either deviate from the traditional understanding of the IRRs among practitioners or, in the case of Hazen , involve transforming cash flow streams in a complicated way. Also, Hartman and Schafrick require a certain amount of computation to divide theNPV function into monotonic regions. Since the IRR is preferred by practitioners to the NPV due in part to its simplicity of interpretation, any complicated improvement on the IRR technique may not be appealing. Only a simple and communicable approach can be useful for practitioners.

The straightforward technique to be discussed in the present article can justify the usage of the IRR in an easily interpretable and communicable way in the business world. In the spirit of the "relevant internal rate of return" approach of Hartman and Schafrick , the present article proposes a way, to be referred to as the IRR parity technique, of using the traditional IRRs in capital budgeting. Unlike the method of Hartman and Schafrick, however, it is not necessary to divide the NPV function into monotonic regions to determine which internal rate of return is "relevant." All we need to do is to solve for all the real IRRs and then count the number of real IRRs that are greater than The IRR Parity Technique 329 the cost of capital. The parity (odd or even) of this number dictates the accept/reject decision. The decision between mutually exclusive projects can be made in an NPV-consistent manner using a sequence of incremental accept/reject decisions, as is known in the literature. This article begins with a theorem relating IRR parity and the NPV. Then the theorem is applied to the project acceptance problem and the project selection problem. Next the IRR parity technique is compared with the "relevant IRR" technique of Hartman and Schafrick. Finally, the article ends with a brief summary.