Average internal rate of return for risky projects

Abstract

The average internal rate of return (AIRR) fixes many deficiencies associated with the traditional internal rate of return (IRR), including apparent inconsistency with net present value (NPV). The AIRR approach breaks down project NPV into scale (the capital invested) and economic efficiency (the AIRR), and maintains NPV consistency for accept/reject decisions. Here we examine extensions of the AIRR to risky capital asset projects, a domain where the IRR appears intractable. We show that one can uniquely break down a risky NPV into a risk-sensitive project scale and a risk-sensitive extended AIRR, representing risky project efficiency, so that consistency with NPV for accept/reject decisions is maintained in the certainty-equivalent sense, in direct analogy to the deterministic case. This novel breakdown gives managerial insight by helping determine a risky project’s locus of uncertainty, be it the project scale, or economic efficiency, or both. In this way, risky features of competing projects can be explored in more detail, leading to insights substantiating the NPV ranking. We also show that under risk neutrality, the expected AIRR is equal to the AIRR of the expected cash flow, a property that notoriously fails for the stochastic IRR.

Notes

1 The use of liquid assets (cash withdrawals from bank accounts or sales of marketable securities) for financing corporate projects is frequent, especially for small-scale projects or for large-scale projects undertaken by firms which own a substantial amount of liquid assets.

2 The AIRR function is the function $i_A\left(x\right)=r+\frac{PV(\mathit{F}|r)(1+r)}{x}.$ The book AIRR is the value taken on by the function for $x=\mathrm{PV}\left(\mathit{C}|r\right)$ when $\mathit{C}$ is the vector of (pro forma) book values.

3 This problem is equivalent to the problem of computing the IRR of a portfolio of projects, either as the IRR of the portfolio of the cash-flow streams of the constituent assets or as the weighted average of the IRRs of the constituent assets.

4 The case of deterministic $\mathit{C}$ is a relevant one, for it is not rare that the capital of a project consists of fixed assets only (i.e., no working capital) which are depreciated according to a prespecified depreciation schedule.

5 This approach cannot account for sensitivity to times at which uncertainties are resolved, the long standing “temporal risk” problem. See Smith (1998) for a discussion and resolution. In this paper, we assume temporal risk issues do not arise.

6 See Baucells and Bodily (2018) on use of present wealth as opposed to future wealth with utility functions.

Additional information

Notes on contributors

Gordon Hazen

Gordon B. Hazen ([email protected]) is Emeritus Professor of Industrial Engineering and Management Sciences at Northwestern University. He received his B.S. in mathematics from the University of North Dakota, his M.S. in statistics from Purdue University, and his Ph.D. in industrial engineering from Purdue University. His research interests include decision analysis, utility and preference theory, and medical decision-making. He has taught courses in probability, decision analysis, financial decision analysis, and stochastic models. He is a member of the editorial board of the journal Decision Analysis, and has served as Area Editor for Operations Research, associate editor for Naval Research Logistics and for Management Science, and recently on the editorial board of Medical Decision Making. He has received publication awards from the Decision Analysis Society of INFORMS, and multiple Meritorious Service Awards for editorial work for Operations Research.

Carlo Alberto Magni

Carlo Alberto Magni ([email protected]) is an Associate Professor at the Department of Economics “Marco Biagi” and the School of Doctorate E4E (Engineering for Economics - Economics for Engineering) of the University of Modena and Reggio Emilia. He received his B.S. in business and economics from the University of Modena and Reggio Emilia, his M.S. in Business Administration from the University of Turin, and his PhD from the University of Trieste. His teaching activities include engineering economics and financial management, principles and models for managerial decisions, mathematics for investment and credit, and calculus. His research areas include engineering economics, corporate finance, managerial accounting, and financial mathematics. He has written more than 100 papers and published in more than 30 different journals. In 2011, he won the “Eugene L. Award”', granted by the Engineering Economy Division of the American Society for Engineering Education (ASEE). Since 2013, he serves as area editor for The Engineering Economist. He is the author of the book “Investment Decisions and the Logic of Valuation. Linking Finance, Accounting, and Engineering”, recently published by Springer Nature.