Economics Letters 67 (2000) 349–351
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Anomalies in net present value calculations
*
James F. Oehmke
Department of Agricultural Economics and Liberty Hyde Bailey Scholar, 317 AgH, Michigan State University,
East Lansing, MI 48824-1069, USA
Received 1 March 1999; received in revised form 5 October 1999; accepted 29 November 1999
Abstract
The net present value measure of project worth may exhibit anomalous behavior if the annual net benefits of the project change sign more than once. This behavior may make net present value unsuitable for certain types of project-selection decisions. 2000 Elsevier Science S.A. All rights reserved.
Keywords: Rate of return; Net present value; Project evaluation
JEL classification: Q28; D92
Project evaluations collapse a time stream of net benefits into a single measure of project worth, such as the net present value (NPV) or internal rate of return (ROR). A well-known criticism of the ROR is that when a project’s net benefits switch sign more than once over time, there may be no, or multiple, RORs associated with the project. The criticism of the ROR is sufficient for many authors to recommend the NPV over the ROR (e.g. Sugden and Williams, 1978; Robison and Barry, 1996; Jenkins and Harberger, 1995; Belli, 1996).
This note shows that in exactly the cases which may give rise to multiple solutions for the ROR, the NPV measure also exhibits anomalous behavior. In particular, it is possible for the NPV to increase as the discount rate increases, or to decrease as the discount rate falls. Thus, the recommendation to use NPV rather than ROR as the basis for comparison is less straight forward than it seems.
The mathematical basis for this result is the equation
T
Bt
]]]
N5
O
t (1)(11r)
t50
where N represents NPV; T is the (possibly infinite) life of the project; B is the net benefit at time t,t
*Tel.:11-517-353-2981; fax: 11-517-432-1800. E-mail address: oehmke@pilot.msu.edu (J.F. Oehmke)
350 J.F. Oehmke / Economics Letters 67 (2000) 349 –351
and is assumed to be known; and r is either the opportunity cost of capital or the internal rate of return. The latter is calculated by setting N equal to 0 and solving for r. The NPV is calculated by setting r equal to a predetermined discount rate and solving for N. Since NPV and ROR are related by this equation, it is not surprising that net benefit streams which lead to idiosyncracies in the calculation of the ROR also lead to idiosyncracies in the NPV.
Differentiating Eq. (1) results in
When investment in the project occurs entirely at time 0, and net benefits are positive thereafter, then the sign of≠N /≠r is unambiguously negative: an increase in the discount rate decrease the NPV of the project. When the net benefits,hB : tt 51, 2, . . .j, take on both positive and negative values, then the sign of ≠N /≠r is indeterminate a priori. That is, an increase (or decrease) in the discount rate can increase or decrease the NPV. This result is in fact implicit in Hirshleifer’s (1958) Fig. 9. Hirshleifer uses his figure to discuss the issue of multiple RORs, but not NPV anomalies.
The result in this note shows that if a project has periodic reinvestment costs, then use of a lower discount rate may reduce the calculated NPV of the project. Consider a hypothetical project with payoffs of (2815, 900, 2100, 1200, 21200). This project has multiple RORs, equal to 4.5% and 12.3%. For discount rates between these two RORs, the NPV of the project is positive: for example, at 6% the NPV is 2.1 (not discounting the first time period). For discount rates greater than 12.3 or less than 4.5, the NPV is negative. For example, the NPV at 3% is 23.5. The intuition is that in period 4, the project may have cash for reinvestment, and a higher discount rate in this case will generate sufficient reinvestment income to raise the NPV.
The cases in which the NPV may exhibit anomalous behavior are exactly those cases in which there are multiple RORs. Let p(r) denote the polynomial on the right-hand side of Eq. (1). Then a necessary condition for anomalous behavior is that p(r) have at least two real roots. A sufficient condition is that
p(r) have (at least) two distinct real roots. With two distinct roots, Hirshleifer’s Fig. 9 (or its vertically
reversed image) applies: the graph of p(r) looks parabolic and crosses the axis at two distinct points. If there are no multiple real roots, then the necessary and sufficient conditions are equivalent. If there are multiple roots, then it is possible that there is a single numerical value for the ROR, and that the NPV
2
is always non-negative (or non-positive). For example, the polynomial p(r)5(r21) has a multiple root at 1; the NPV defined by N5p(r) is zero at r51 and positive elsewhere. This behavior is somewhat idiosyncratic in itself.
The number of different roots of a polynomial in any closed interval, [a, b], is characterized by Sturm’s theorem. The use of Sturm sequences provides a means for calculating the number of roots (Jacobson). In an applied sense, for a given project appraisal, spreadsheets such as QuattroPro can determine if there is more than one ROR. Symbolic manipulation programs such as Mathematica can provide numerical solutions for all the roots. If p(r) has more than one real root, then there are multiple RORs, and the NPV also exhibits anomalous behavior.
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to determine the range of discount rates for which the NPV is positive, analogously to sensitivity analysis.
Acknowledgements
I would like to thank Eric Crawford and an anonymous referee for helpful comments, and Robert Oehmke and Theresa Oehmke for mathematical information. Responsibility for all errors remains with the author. This research is supported in part by the Michigan Agricultural Experiment Station.
References
Belli, P., 1996. Handbook on Economic Analysis of Investment Operations. Operations Policy Department, The World Bank, Washington, DC.
Hirshleifer, J., 1958. On the Theory of Optimal Investment Decision. The Journal of Political Economy 66, 329–352. Jacobson, N., Basic Algebra 1, W.H. Freeman & Co., New York.
Jenkins, C., Harberger, A., 1995. Cost-Benefit Analysis of Investment Decisions, Harvard Institute for International Development, Cambridge, MA.
