A Model Illustrating Multiple Interest Rate Analysis (MIRA)
Net present value (NPV) and internal rate of return (IRR) are criteria commonly employed by organizations to appraise investment projects. NPV is the present value of an investment's cash flows , for to , discounted at the cost of capital , where is the investment’s initial outlay at time zero (negative), and is the investment’s net cash flow at time (revenues minus costs; usually positive but sometimes negative). IRR is the value of the discount rate setting NPV to zero.
When deciding the viability of a single project, the criterion that its NPV be positive usually gives the same investment decision as the criterion that its IRR exceed the cost of capital . Exceptions exist when a project’s cash flows yield either no real-valued IRR (an anomalous situation) or multiple real-valued IRRs (an ambiguous situation). Additionally, when ranking multiple projects, the rank order determined by how far their IRRs exceed the cost of capital sometimes agrees with the rank order determined by how far their NPVs exceed zero, but sometimes it does not; projects’ cash flows can be such that the rank orders conflict (another ambiguity). Modern textbooks label these anomalies and ambiguities the "IRR pitfalls" (for example, see ). Given the pitfalls, conventional academic opinion is that NPV-based decisions are better than IRR-based decisions because the former possess "clarity and uniformity" .
Generations of business students have learned the academic preference for NPV; nevertheless, practitioners routinely ignore the advice. Research finds that managers in corporations , financial institutions , and government agencies  continue to use IRR.
The model described here employs the fundamental theorem of algebra to clarify the relationship between NPV and IRR, this clarification providing novel, cogent support to the academic preference for NPV. The fundamental theorem implies that an degree polynomial in solves for values of (see Related Link). Multiple interest rate analysis (MIRA) applies the fundamental theorem to the time value of money (TVM) polynomial. MIRA demonstrates that any TVM equation possesses a dual formula containing every interest rate (real and complex) solving its conventional counterpart, thereby employing interest rates that have been ignored by financial economists for centuries (see ).
The dual formula for NPV per dollar outlay, , is equal to the product of markups of every IRR over the cost of capital , that is, the product of for to , where is defined by the expression . This product conveniently divides into two parts, one familiar and the other unfamiliar. The familiar part is the markup, , based on a selected value of IRR, usually the one produced by a financial calculator or spreadsheet, here designated , meaning . The unfamiliar part is the product of the (previously ignored) unconventional markups, for to . MIRA demonstrates that this unfamiliar component contains valuable information; the product is equal to a statistic summarizing structure in the cash flows (duration, as defined by Macaulay ).
This result has implications. First, multiple solutions for IRR do not constitute a pitfall to be avoided; rather, all solutions convey information, and therefore they are to be embraced. Second, NPV per dollar outlay is a superior criterion to the markup of IRR over the cost of capital because it contains the additional information about project cash flow structure conveyed by the unconventional markups. Third, inconsistent rankings of multiple investment projects by the two criteria are wholly explained by differences in the projects' cash flow structures.
This Demonstration employs the dual NPV formula from . The formula is , in which is the markup of a selected IRR over the cost of capital. The selected IRR is usually the conventional one (assuming a real, positive IRR exists), though any one of the IRRs will do. The unfamiliar entity is , which is the product of the nonselected (usually unconventional) markups , for to , and .
MIRA demonstrates that the entity is equal to the duration of the adjusted set of cash flows , for to , duration being defined as their present-value, weighted-average term to maturity. This adjusted set is explained as follows. First, note that scaled cash flows determine the roots of the IRR equation, for to , and that this process reverses—the TVM polynomial's factors (each factor containing a root) expand to recover the TVM polynomial’s coefficients (cash flows). Second, note the earlier observation that the dual formula for NPV per dollar outlay, , divides into two parts: the familiar part and the unfamiliar part . This division is such that the expression for is comprised of unconventional roots, , for to , and the root containing the cost of capital. The factors containing these roots expand into a set of scaled cash flows slightly different from the original set. The two sets of cash flows and are highly correlated, possessing a similar structure, because they have roots in common and their roots differ only slightly: compared with . As a result, the duration of the set is close in value to the duration of the original set , but it is not the same: it is subtly different, and it is the duration of this adjusted set that enters the dual formula for NPV.
The model demonstrates that varying a cash flow or varying the cost of capital results in complex shifts in the roots and markups. Despite this complexity, the dual NPV formula exhibits a simple, invariant relationship between its three components: NPV per dollar outlay, the selected markup, and cash flow duration (as defined above).
Snapshot 1 illustrates the classic "oil-pump project" in which an outflow of cash today is followed by a stream of positive net cash flows ending with a negative cash flow in year ten, perhaps to cover cleanup costs . Two changes of sign in the sequence of cash flows results in two feasible IRRs of 3.5% and 19.54%. Conventional theory says that this situation is ambiguous because one IRR is higher than the assumed cost of capital and the other is lower. In contrast, MIRA sees no ambiguity, because the dual NPV formula incorporates all markups at once, including the two markups containing the feasible IRRs. Whichever markup of IRR over the cost of capital is selected as the IRR investment criterion, its companion markups combine with it, incorporating valuable information (about the structure of the project’s "selected-markup-adjusted-cash-flows") into the value of NPV per dollar outlay.
Snapshots 2 and 3 illustrate two projects whose NPVs and IRRs give conflicting rankings . The NPV per dollar outlay for project (5.9741 / 20=0.2987) exceeds the NPV per dollar outlay for project (5.6238 / 20=0.2812), despite the fact that the conventional markup of IRR over the cost of capital for [(0.1602–0.1) / 1.1=0.0547] is considerably less than that for [(0.2826–0.1) / 1.1=0.1660]. These inconsistent decisions are wholly explained by the duration of the adjusted cash flows for project exceeding that for (5.4620 compared with 1.6940, respectively).
It has long been known that cash flow structure is a factor in "anomalous or ambiguous" IRR-based decisions; for example,  describes offending cash flows as having "unusual time shapes and reversals." To date, however, the literature has lacked a fully satisfactory explanation. The explanation in this Demonstration is concise and subtle: concise because the unconventional interest rates collapse into a single, real-valued financial statistic: duration, capturing pattern in the adjusted cash flows; subtle because the route to the adjusted cash flows is not obvious, the route involving analysis in the complex plane.
 R. Brealey, S. Myers, and F. Allen, Principles of Corporate Finance, 11th ed., New York: McGraw-Hill, 2013.
 J. Lorie and L. Savage, “Three Problems in Rationing Capital,” The Journal of Business, 28(4), 1955 pp. 229–239. www.jstor.org/stable/2351049.
 J. Graham and C. Harvey, “The Theory and Practice of Corporate Finance: Evidence from the Field,” Journal of Financial Economics, 60(2–3), 2001 pp. 187–243. doi:10.1016/S0304-405X(01)00044-7.
 J. Cuthbert and M. Cuthbert, "Why IRR Is an Inadequate Indicator of Costs and Returns in Relation to PFI Schemes," Critical Perspectives on Accounting, 23(6), 2012 pp. 420–433. doi:10.1016/j.cpa.2012.05.001.
 M. J. Osborne, Multiple Interest Rate Analysis: Theory and Applications, Basingstoke, UK and New York: Palgrave Macmillan, 2014
 G. Hazen, “A New Perspective on Multiple Internal Rates of Return,” The Engineering Economist, 48(1), 2003 pp. 31–51. doi:10.1080/00137910308965050.
 F. Macaulay, Some Theoretical Problems Suggested by the Movement of Interest Rates, Bond Yields and Stock Prices in the United States since 1856, New York: National Bureau of Economic Research, 1938.