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Solving For The Internal Rate Of Return Directly

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2001 Annual Conference


Albuquerque, New Mexico

Publication Date

June 24, 2001

Start Date

June 24, 2001

End Date

June 27, 2001



Page Count


Page Numbers

6.883.1 - 6.883.11

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Wayne Cottrell

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ASession 1639

Solving for the Internal Rate of Return Directly

Wayne D. Cottrell, University of Utah


The trial-and-error and graphical methods for computing the internal rate of return (IRR) are traditionally taught in engineering economics and financial management courses. While developing an understanding for direct computation of the IRR may enhance the learning process, little attention is devoted to direct solution. This paper explores the direct solution of the IRR using the roots of polynomials. Direct computation is feasible in three cases: two cash flows during an N-year study period, three cash flows during a two-year period, and four cash flows during a three-year period with equal revenues in years 1-3. As the degree of the polynomial increases, directly computing the IRR becomes laborious. Sharing these points with engineering economics and management students may improve the understanding of both the IRR’s definition and its computational methods.


The rate of return is a percentage figure that indicates the relative yield of different uses of capital. Rate of return methods are used to compare investment alternatives. Since the rate of return is an interest rate, the interpretation is straightforward. The internal rate of return (IRR) is one of several that are either commonly used or have been proposed in the literature; the others include the minimum attractive rate of return (MARR) and the external rate of return (ERR),15 the accounting rate of return (ARR),14 the overall rate of return,3,4 the modified internal rate of return (MIRR),9 the effective rate of return,2 and the adjusted modified internal rate of return (ADJMIRR).13 Some of the less frequently used rates of return have similar bases.8 The MARR is often used as a benchmark to which another rate of return is compared.15 One definition of the IRR is “the rate that yields a present worth of zero by assuming that all cash flows are reinvested at the IRR11.” Another definition is “the interest rate earned on the unrecovered project balance of investment such that, when the project terminates, the unrecovered project balance will be zero10.” The two definitions are equivalent in that the IRR equates the present worth of project’s cash expenses with its cash receipts. For a balance between expenses and income to occur, all income is presumed to be reinvested at an interest rate equivalent to the IRR. That is, all returns on funds remain internally invested in the project, such that there is no unrecovered balance at the project’s termination. The IRR is the interest rate at which:

EW (receipts) = EW (expenses), (1)

Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition Copyright © 2001, American Society for Engineering Education

Cottrell, W. (2001, June), Solving For The Internal Rate Of Return Directly Paper presented at 2001 Annual Conference, Albuquerque, New Mexico.

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