A New NPV-Consistent Return Criterion for Investment Decisions

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Prof. Weber introduces the “selective IRR,” which leads to investment decisions that are consistent with the familiar net present value rule. The results are detailed in a forthcoming article in the Journal of Mathematical Economics.
NPV – the cornerstone for investment decisions. One of the most popular rules of investment, familiar to virtually anyone who ever attended business or engineering school, is to accept a project if and only if its net present value (NPV) is positive. The NPV measures the current value of the project, which is obtained by discounting all the future project cash flows to the present at the external interest rate r which represents the cost of capital. While the rule is widely known and useful in practice, it says nothing about the return on the investor’s capital.
Duplicitous IRR. Consider a project which requires an investment of 5 dollars now, pays out 16 dollars in one year, and requires another investment of 12 dollars in two years. The NPV of the corresponding cash-flow stream (-5, 16, -12) is positive only if r is between 20% and 100%. Hence, by the NPV criterion, if the investor’s cost of capital is 10% he should not accept the project (because the NPV is about -0.37 < 0), whereas if r is 25%, then he should accept the project (because the NPV is equal to 0.12 > 0). But what would be the return on the invested capital in either case? Many students have also learned about the internal rate of return (IRR), which corresponds to an interest rate i at which the NPV becomes zero. For the cash-flow stream (-5, 16, -12) there are two possible IRR-values: 20% or 100%. For either the NPV vanishes. But which one is the correct return on investment? The school wisdom typically ends at this point. The student is taught that because the IRR is not unique, it cannot be used to measure the return of a project, nor is it useful to make investment decisions.
The rate-of-return dilemma. Having rejected the IRR as possible investment criterion, the investor is left in a somewhat miserable situation, in a world where everybody compares investment yields and where relative returns obviously matter, but where – according to school wisdom – actual investment decisions should be taken using the NPV criterion. Yet the NPV is oblivious to relative returns, which may be seen as follows: an NPV of 10 can result after one period from investing 10 at a 100% return or from investing 1000 at a 1% return. Every rational investor would agree that doubling 10 in one period would be preferable to earning a meager 1% return on the bigger investment of 1000 because the 1% yield is substantially lower than what the investor could obtain elsewhere. Yet, the NPV rule would not be able to tell the difference between the two projects. The dilemma is that everybody needs and likes returns to evaluate investments, but there is always a fear that it might lead to decisions that are inconsistent with the NPV rule.
A new NPV-consistent rate of return.In a research paper, recently accepted for publication by the Journal of Mathematical Economics, Prof. Weber introduces the “selective IRR” as a new investment criterion. The selective IRR leads to NPV-consistent decisions and retains the features of the familiar IRR, which has been already termed the “marginal efficiency of capital” by John Maynard Keynes in the 1930s. The selective IRR, denoted by i(r), depends on a given external return benchmark r, which reflects the cost of capital. This dependence is much in the same vein as the NPV, which also depends on r as seen earlier for the cash-flow stream (-5, 16, -12). Stated in simple terms, the selective IRR determines a unique return from the set of possible IRRs or plus/minus infinity. The selective IRR investment rule is to invest in the project if and only if i(r) exceeds r, which Prof. Weber demonstrated to be equivalent to the NPV-rule. For the earlier example cash-flow stream of (-5, 16, -12) the selective IRR i(r) equals minus infinity if r < 20%, i(r) = 20% if r = 20%, and i(r) = 100% if r exceeds 20%. It is clear that these values for the selective IRR lead to the same investment decisions as the NPV rule. That is, i(r) exceeds r if and only if the NPV is positive, or equivalently, the excess return i(r)-r is positive exactly when the NPV is positive. Thus, an investor with a cost of capital of r = 10% would report the return of a project with the cash-flow stream (-5, 16, -12) as minus infinity (and therefore completely unacceptable), whereas an investor with a cost of capital of r = 25% would report the return of the same cash-flow stream as 100% which is very attractive indeed.
Ease of computation. For a given cost of capital r, the selective IRR, i(r), is obtained as follows: if the NPV at r is positive then i(r) is the next larger possible IRR-value if it exists or else infinity; otherwise, i(r) is the next smaller possible IRR-value if it exists or else minus infinity.
An addition to textbooks. Using the selective IRR instead of the standard IRR, the dilemma is resolved. The selective IRR always exists and is always unique. When finite it also coincides with one of the possible IRR values. Most importantly, by comparing the selective IRR with the cost of capital, an investor will be led to exactly the same decisions as under the NPV rule, which has been the cornerstone of investing since the seminal work by Irving Fisher in 1930.
References
Fisher, I. (1930) The Theory of Interest. Macmillan, New York.
Keynes, J.M. (1936) The General Theory of Employment, Interest, and Money. Macmillan, New York.
Weber, T.A. (2014) “On the (Non-)Equivalence of IRR and NPV,” Journal of Mathematical Economics, in press. [ http://dx.doi.org/10.1016/j.jmateco.2014.03.006 ]