Electronic copy available at: http://ssrn.com/abstract=1634819 1 On the meaning of internal rates of return and why an internal rate of return is not an investment criterion By Michael J. Osborne School of Business, Management, and Economics Mantell Building, University of Sussex Brighton, East Sussex BN1 9RH, UK email: mo98@sussex.ac.uk First version July 2010 This version November 2011

Abstract All conceivable solutions to the internal rate of return equation are shown to have meaning as well as use. Internal rates of return are the units in which value is measured and the quantities of such units. This result implies a single internal rate of return cannot be an investment criterion. Moreover, a rate of return should not be considered in isolation; a rate is fully meaningful only when considered alongside its unorthodox partners. Key words: Capital budgeting; complex plane; internal rate of return; net present value JEL Classifications: C00, C60, E22, E40, G00, G1, G24, G30, G31, O16, O22 Electronic copy available at: http://ssrn.com/abstract=1634819 2 On the meaning of internal rates of return and why an internal rate of return is not an investment criterion 1.Introduction There is recent interest in the fact that the internal rate of return (IRR) equation produces multiple solutions for the rate of return, including solutions that are complex numbers.During the twentieth century only Dorfman (1981) makes explicit use of all possible IRRs, including the complex.In the current century, Hazen (2003), Osborne (2010) and Pierru (2010) employ all rates.These authors take different approaches, however.Hazen (2003) and Pierru (2010) take the conventional approach, using an individual rate as an investment criterion, while Dorfman (1981) and Osborne (2010) use all IRRs simultaneously as components of another financial concept. A question faced by any researcher employing all possible rates of return is what financial meaning is attributable to the unorthodox solutions, particularly the complex?This article extends the analysis in Osborne (2010) about use of all IRRs to the meaning of all IRRs. The conventional view about the meaning of an IRR is challenged.The conventional view is that, relative to the cost of capital, IRR is an investment criterion, although not a very good one because it possesses pitfalls.Numerous studies show practitioners continue to use IRR as an investment criterion despite five decades of education about its pitfalls.Such behavior suggests better arguments than the pitfalls are needed if practitioners are to be persuaded not to use IRR. This article argues for an alternative interpretation of IRR: the mark-up of an IRR over the cost of capital is the unit in which value is measured and the product of the mark-ups of all remaining IRRs over the cost of capital is the quantity of measurement units.Thus, total value, measured by the unit of measurement and the quantity of units, is determined by the entire set of internal rates of return, unorthodox as well as orthodox. The argument leads to several ancillary conclusions.First, a single rate of return is not very informative except in the simplest of circumstances.Interest rates make full sense only when all possible rates are considered together. Second, multiple-interest-rate analysis 3 supports arguments that the IRR pitfalls are spurious.Third, a better reason than the pitfalls for not using IRR as an investment criterion is that a single value of IRR cannot be an investment criterion.Practitioners may cease using IRR as an investment criterion if they can be persuaded it isnt one.Multiple-interest-rate analysis provides such a demonstration. 2.The pitfall of multiple rates of return This section contains a short explanation of the multiple-interest-rate problem.The ten-period cash flow in the equation below is taken from Brealey et al. (2011).The cash flow is a particular example of the oil-pump problem first described in general terms by Lorie and Savage (1955).An initial investment is followed by a series of inflows ending in final expenditure to dismantle equipment and clean up after the investment.

NPV = !3+1(1+ r)ii=19"!6.5(1+ r)10 = 0 The cash flow illustrates the IRR pitfall of multiple rates of return.The equation is a polynomial of order ten; therefore there are ten values of (1+r) solving the equation.Table 1 contains the ten solutions and the ten values of r implied by the solutions; Figure 1 plots the solutions. The narrow version of the multiple-interest-rate problem, the version most often found in the literature, is that there are two real solutions for (1+r).The solutions are 1.0350 and 1.1954, implying IRRs of 3.50% and 19.54%.Given that both real rates are feasible, several questions are posed. Does one rate apply, and, if so, which one?Do both rates apply, or neither? The wide version of the multiple-interest-rate problem concerns the fact that there are ten solutions in total, eight of which are unorthodox because they are complex numbers having the form a + b.iwhere i = !1 .Open questions exist about the financial use and meaning of all ten solutions, especially the unorthodox ones. This article discusses the meaning of all solutions in order to shed light on both versions of the multiple-interest-rate problem. 4 [ Table 1 about here ] [ Figure 1 about here ] 3.The capital budgeting literature concerning multiple interest rates Cannaday et al. (1986) contains a comprehensive survey of the multiple-interest-rate literature in the context of capital budgeting.Magni (2010) provides a more recent one.Much of the capital budgeting literature does not mention the existence of multiple IRRs.Of the literature mentioning multiple interest rates, most does not address all possible interest rates, including the complex; rather, it addresses the restricted set of real rates. There is a pattern to the development of ideas about multiple rates.A brief history is as follows.In the first stage, during the 1930s, researchers acknowledge the existence of n rates, but dismiss most of the solutions, especially the complex solutions, as having no economic significance.As a result, early research focuses on the restricted set of real rates, negative and positive. In the second stage, during the last half of the twentieth century, the focus progressively narrows.This trend has two features.First, there are attempts to limit the range of legitimate rates found along the real number line, and to isolate the most relevant, real, positive rate from the range.Second, there is literature showing how to restructure cash flows so as to force the production of only one, positive, real interest rate. In the third stage, after 2000, the trend reverses and there emerges an interest in making sense of the entire range of rates: real (positive and negative) and complex. This brief history is now described in more detail.Table 2 shows a chronological history of the literature sorted into categories.The categories are based on discussion in Magni (2010) with additions by the author. [ Table 2 about here ] Column 1 of Table 2 lists articles published between the 1930s and the 1960s.The authors examine possible solutions for the roots in the range zero to plus infinity along the real number line, implying a range of interest rates from minus 100% to plus infinity.Some 5 authors identify the restrictions on the structure of the cash flows necessary to guarantee a single, real-valued rate of return.From the beginning, the story is of restriction. It is likely that the lower bound of the permissible range of rates was imposed for the same reason that modern financial calculators and spreadsheets have their output from IRR calculations limited to solutions greater than minus 100%.The search for a root of the TVM equation is the search for the solutions of (1+r).At the point (1+r) = 0, any discounted value goes to infinity.Such a point is a barrier.Solutions on the real number line to the left-hand side of the barrier have traditionally been ignored.This neglect is possibly because the search for solutions is mathematically simpler if it is restricted to the right-hand side of the barrier.Another reason is doubt that values of r below minus 100% have financial meaning. Beginning in the 1950s, a second series of papers appears in which the authors further restrict the range of permissible solutions for the root (1+r) to between plus one and plus infinity, i.e. they rule out negative interest rates (column 2 of Table 2). The seminal paper of Lorie and Savage (1955) belongs to this group.They were first to point out the possibility of inconsistent ranking of investment projects by the NPV and IRR criteria.Lorie and Savage also introduce the oil-pump problem referred to earlier.The problem produces two real interest rates, both of which are positive and feasible, prompting discussion about which rate is most relevant, and why.The oil-pump problem is often quoted in the literature, perhaps because a genuine puzzle emerges in such a simple, unforced way from a realistic cash flow. Also in this group, Hirschleifer (1958) and Bailey (1959) follow in the tradition of Fisher (1907) by recommending analysis of the many, individual, two-period returns in a multi-period investment.This is multiple-interest-rate analysis of an entirely different kind to that studied here.Fishers approach divides and conquers the cash flow, thereby sidestepping the issue of multiple solutions addressed in the current study in which each and every solution applies to the entire cash flow. Some authors in this second category, e.g. Ramsey (1970), examine multi-period cash flows possessing several changes of sign and begin the practice of invoking Descartes law of signs to determine the number of real rates.1

1 Descartes law of signs states that the number of changes of sign in the coefficients of a polynomial is greater then or equal to the number of real, positive roots (see Weisstein, 2003). The law also states that if the signs are reversed on all the coefficients attached to odd powers, then the number of changes of sign in the coefficients is 6 In the 1950s the search intensifies for a unique, real rate of return.In 1959, the first of a series of articles is published that invokes project truncation as a way to control the structure of the cash flow such that a unique, positive, real solution is guaranteed (column 3 of Table 2). Karmel (1959), in a response to Pitchford and Haggar (1958), shows that, if a project is terminable at any stage during its lifetime and provided that the scrap value is always non-negative, the marginal efficiency of the truncated project expected to have the highest marginal efficiency will be a unique value. Soper (1959) concurs: If the investor is aiming at a maximum rate of profit on his investment a maximum r then this will always cause him to discriminate between different lengths of life for the investment, choosing that length which includes the maximum number of consecutive yields which are still consistent with a discounting equation with one, and only one, positive root. A famous paper in this series is by Arrow and Levhari (1969) who revise the conclusion of the two earlier papers.The authors argue that choosing the truncation period to maximize net present value is superior to choosing the truncation period to maximize the rate of return.If, with a given constant rate of discount, we choose the truncation period so as to maximise the present value of the project, then the internal rate of return of the truncated project is unique.It is argued here that truncation is not an answer to the wide version of the multiple-interest-rate issue, only to the narrow version concerning multiple, real, positive solutions.An equation having a suitably truncated cash flow might yield only one positive, real interest rate, but there are still n solutions to the equation, therefore questions remain about the use and meaning of all solutions. The next stage in the search for a unique rate of return was the creation of the concept of modified internal rate of return, or MIRR (see column 4 of Table 2).Teichroew et al. (1965a,b) suggest the problem of multiple, real rates associated with non-simple projects is eliminated if different rates for project investment and project finance are used. Teichrow et

greater than or equal to the number of real, negative roots. A negative root in a financial polynomial implies the existence of an interest rate less than minus 100%. As indicated in the text, most financial calculators and spreadsheets will not calculate such interest rates and they are usually ignored. 2 The absolute values in Eq. (4) can be released selectively. The absolute value of |mi|=|ri-k|/(1+k) is retained if ri is complex which means |ri-k| is a positive, real number measuring a distance in the complex plane. The absolute value of |mi|=|ri-k|/(1+k) is released if ri is on the real number line which means the sign of a wholly real difference in interest rates is determined. This procedure determines the sign of the overall relationship between 7 al. do not discuss the concept of MIRR directly, but take a step towards it with their idea of separate borrowing and lending rates according to whether the project balance is positive or negative. Lin (1976) and Athanasopoulos (1978) begin the MIRR literature proper.They assert that the IRR method of appraisal assumes all returned cash flows are reinvested at the IRR, and that the NPV method assumes the returned cash flows are reinvested at the cost of capital.Researchers favoring use of MIRR assert that neither assumption is realistic.They suggest that if all project inflows are compounded forward to the terminal date at some reinvestment rate (preferably a more realistic rate than IRR), and all outflows are discounted back to the start date at some finance rate (possibly the cost of capital), then the terminal value relative to the start value is a monotonic function possessing one positive, real interest rate: MIRR. A number of commentators object to this approach: see Lohmann (1988), Keef and Roush (2001) and Eagle et al. (2008).They argue that the assumptions about reinvestment are fallacious, that neither the NPV nor the IRR criteria assume anything about how the incoming funds are reinvested.In the latter two articles the authors trace the assumptions back to a confusing discussion in Solomon (1956).Despite these objections to MIRR, articles continue to appear in support of it, e.g., Chang & Swales (1999) and Kierulff (2008). Like the truncation technique, the development of MIRR is an attempt to reconfigure the problem to give one positive, real interest rate as output.From the perspective of the current research the concept of MIRR is not helpful.The MIRR equation may yield only one positive, real MIRR, but the equation still provides n values for MIRR, and most of them are complex.Redefining how the relevant rate of interest is calculated does not answer the wide question about the use and meaning of every possible interest rate.In the 1980s several papers appear advocating criteria for choosing the correct rate of return from among multiple real solutions (column 5 in Table 2).Cannaday et al. (1986) suggest the following criterion: if the net future value function has a negative derivative at a relevant root, and the associated IRR is greater than minus one, then the IRR is appropriate.They admit that their approach does not always work: for some cash flows, more than one real, positive IRR satisfies the criterion. Continuing the theme of identifying the single, relevant rate of return, Zhang (2005) proposes a simple technique for accepting or rejecting a project having multiple, real IRRs. 8 The technique involves counting the number of real rates of return greater than the cost of capital; if the number is even then reject the project (because NPV must be negative) and if the number is odd then accept the project (because NPV must be positive).Zhang concludes that while the proposed method is not computationally easier than the NPV method, it serves as a simple way of retaining the use of the IRR without having conflicts between the two methods. Zhang demonstrates how the relevant IRR is identified: in the case of project acceptance it is the IRR above and closest to the cost of capital; in the case of project rejection it is the IRR below and closest to the cost of capital. The method is justified on the grounds that any investment decision based on it is supported by the NPV criterion.If NPV is the ultimate arbiter, however, why has the search for the IRR persisted over the years?Zhang concludes it is because practitioners find a criterion expressed as a rate of return desirable on the grounds it is intuitively appealing and easy to communicate. As detailed above, research on multiple rates prior to 2000 is characterized by gradual restriction, the restriction taking two forms.Either the restriction is on the structure of the cash flow in order to force one, real, positive rate of return, or it is on the range of permitted rates with an emphasis on the choice of one of them: the relevant rate. A new approach appears in the multiple-interest-rate literature after the millennium.The research opens up to consider all possible cash flows (no truncation) and all possible solutions, including the complex.Before reviewing these recent works we discuss the few twentieth century works mentioning all rates, including the complex (see the references with an asterisk in Table 2). Perhaps the earliest reference to all possible IRRs is in an exchange of views between economists in the 1930s.Boulding (1936a) contains a discussion of investment appraisal.In an appendix, he describes difficulties met when calculating IRR and presents a procedure for finding a solution.Wrights reply to Bouldings article (Wright, 1936) is interesting because, possibly for the first time in the academic literature, a researcher points out the existence of all n solutions in unequivocal terms by referring to the fundamental theorem of algebra. The theorem asserts that any algebraic equation of degree n has n solutions. Applied to Mr. Boulding's definition of the interest rate, it means that several different and reasonable values may be obtained for i (Wright, 1936) 9 In his reply to Wright, Boulding takes a stance that will, with rare exceptions, characterize most research on the subject of multiple IRRs for the next 70 years. Now it is true that an equation of the nth degree has n roots of one sort or another, and that therefore the general equation for the definition of a rate of interest can also have n solutions, where n is the number of years concerned. Nevertheless, in the type of payments series with which we are most likely to be concerned, it is extremely probable that all but one of these roots will be either negative or imaginary [complex], in which case they will have no economic significance.Boulding (1936b) Samuelson (1937) also discusses Boulding (1936a) and refers to the multiplicity of solutions but does not elaborate on the nature of the multiplicity. Thus, by the 1930s researchers were aware of the issue of multiple solutions to the time value of money equation.Since that time, Bouldings opinion, that the negative and complex solutions have no economic significance, has been the conventional view.This fact is probably why most of the works in the multiple-interest-rate literature surveyed by Cannaday et al. (1986) and Magni (2010) focus on the restricted set of real solutions.During the remainder of the twentieth century, authors who explicitly mention the possibility of negative and complex solutions include Hirschleifer (1958), Soper (1959), Feldstein and Flemming (1964), and Dorfman (1981).Apart from Dorfman, their comments are brief and negative. Soper (1959) echoes Boulding when he writes that some of these roots can be ignored as irrelevant; those which are less than zero or are complex.Hirschleifer (1958) and Feldstein and Flemming (1964) point out that some cash flows, including wholly positive or wholly negative cash flows, guarantee wholly complex solutions.Hirschleifer concludes that, because of such solutions, the idea that IRR represents a growth rate in any simple sense cannot be true.Feldstein and Flemmings sole comment is the examples given are rather peculiar. Dorfman (1981) is a significant exception to the multiple-interest-rate literature during the twentieth century, first because he discusses explicitly all possible solutions, and second because he employs them.He examines the case where the proceeds of an investment are serially reinvested in projects of exactly the same type, and the process is continued 10 indefinitely.The growth path of a dollar placed in such an investment is shown to depend simultaneously on all roots of the internal rate of return equation.In this sense, i.e. simultaneous employment of all IRRs as components of another financial concept, Dorfmans contribution is seminal.Moreover, researchers today benefit from the existence of sophisticated mathematical software to explore the complex plane, e.g. Maple, Mathcad, Mathematica and Matlab.The software appeared during the late 1980s.Dorfmans work was published nearly a decade before the software became widely available, making his work truly pioneering. This brief history now reaches the wider perspective of the multiple-interest-rate problem developing in the twenty-first century. First, there is research taking a skeptical view of unorthodox interest rates.Hartman and Schafrick (2004) and Magni (2010) discuss the complex solutions but adopt (different) procedures to eliminate such solutions from the analysis.The authors supply more sophisticated, extended rejections of the unorthodox solutions than Bouldings bald assertion of seventy years earlier. In contrast, there is research taking a positive view of the unorthodox solutions.Works by Hazen (2003), Osborne (2010), and Pierru (2010) employ all n rates of interest, including the highly negative and complex.These authors do so in different ways, however. Hazen sees each and every IRR as an investment criterion.The procedure he describes is roundabout, although it works for any IRR.He writes there is no need to discard unreasonable or extreme internal rates all are equally valid.Hazen employs the procedure from Lohmann (1988).The procedure uses an IRR to convert the original cash flow for a project into an alternative cash flow called an investment stream.Each IRR has an associated investment stream.The NPV of an investment stream is calculated using the cost of capital as discount rate.The original project is judged profitable if the investment stream is profitable.It does not matter which of the many IRRs is used to produce a decision.This roundabout method need only be applied once, and any IRR, with its associated investment stream, serves the purpose. Importantly, Hazen makes explicit use of complex-valued IRRs, although he considers only the real part of complex-valued IRRs and only the real part of complex-valued 11 investment streams.He demonstrates that the method gives invest/not-invest decisions for single projects consistent with decisions made using the NPV criterion.For Hazen the problem of multiple or non-existent internal rates of return universally regarded as a fatal flaw for the IRR method is not really a flaw at all, and can easily be dealt with conceptually and procedurally. Despite this result, Hazen urges use of NPV in preference to IRR because the NPV criterion is simpler.Moreover, his proposed IRR methodology still suffers from drawbacks.First, when comparing mutually exclusive projects, the ranking from his method can conflict with the ranking from the NPV criterion; this is the well-known pitfall of IRR introduced by Lorie and Savage (1955).Second, the method considers only the real parts of the complex solutions, ignoring any information contained in the imaginary components.Third, the method does not provide an interpretation of complex rates.In Hazens words: We are currently unaware of an economic interpretation of complex-valued rates of return or complex-valued investment streams, and without such an interpretation, it would be hard to justify any economic recommendation without resort to performance measures such as present value. Hazen (2003) Although it has drawbacks, Hazens contribution is important.The concept of an investment stream is employed in section 5.2 below, albeit with a different interpretation. Pierru (2010) also pursues meaning for complex interest rates. He examines them in the context of a portfolio of two assets. When a project involves the joint production of two outputs whose markets are subject to different risks, our approach allows the projects cash flows to be discounted at a single (but complex) rate.The single complex rate is interpreted to represent several different real rates at the same time.A difficulty is that the interpretation is confined to a narrow range of applications.Pierru acknowledges this when he writes we are aware of the apparently limited practical interest of the interpretations proposed . The approach to multiple IRRs taken by Osborne (2010) supplies the analytical starting point for the substance of the current article.The approach is summarized as follows.Eq. (1) is a base equation in which present value, P, depends on an initial investment, I0, and a series of cash flows ci discounted at the rate R. 1 20 2...(1 ) (1 ) (1 )nnc c cP IR R R= ! + + + ++ + +(1) 12 As R varies, so P varies.If the discount rate is chosen such that P is zero, then the rate is the internal rate of return, labeled r in Eq. (2).As discussed earlier, there are n possible values for the roots (1+r) solving Eq. (2), and these roots can be positive real numbers, negative real numbers or complex numbers. 1 20 20 ...(1 ) (1 ) (1 )nnc c cIr r r= ! + + + ++ + +(2) If the discount rate takes the value of the cost of capital, k, then P is labeled NPV.This is shown in Eq. (3). 1 20 2...(1 ) (1 ) (1 )nnc c cNPV Ik k k= ! + + + ++ + +(3) The relationship between the cost of capital and IRR can be expressed as (1 ) (1 )(1 ) r k m + = + +in which m is the interest rate marking up the cost of capital to the IRR.Assuming a single cost of capital, and given n values for IRR, there must be n mark-ups, i.e. (1 ) (1 )(1 )j jr k m + = + +for j from 1 to n.From the last equation we deduce ( ) / (1 )j jm r k k = ! + .Osborne (2010) employs this result to derive an equation for NPV per dollar invested.Eq. (4) shows all IRRs, including the unorthodox, serve as components of NPV.The equation states that the product of the mark-ups of all n IRRs over the cost of capital is equal to NPV scaled by the initial investment. 1 0njjNPVmI==!(4) Eq. (4) is a difference equation bridging the levels equations (2) and (3).The shift from Eq. (2) to Eq. (3) shows the increase in value of the dependent variable from zero to NPV is a function of the shift in the interest rate from IRR to the cost of capital.However, Eq. (4) shows the shift in value is not a function of the shift in the orthodox interest rate alone; rather, it depends on the shifts from every possible value of IRR to the single cost of capital. From one perspective, Osbornes analysis is in the spirit of Dorfman (1981): both analyses employ all interest rates simultaneously as components of another financial concept, rather than as individual rates of return per se.From another perspective, there is a difference between the two analyses.Dorfman employs all complex solutions in their raw form, i.e. in the form a + b.i, while the analysis in Osborne (2010) uses the absolute values of differences 13 between complex interest rates, which are real numbers equating (mathematically) to distances in the complex plane and (financially) to interest rate spreads.2 Unfortunately, while Eq. (4) incorporates all IRRs, thereby demonstrating use, the analysis has a shortcoming; it says nothing about meaning.Ideally, the mathematical relationship requires a meaningful story.Admitting this shortcoming, Osborne suggests it is a question for future work.This article addresses the shortcoming by providing a story. 4.Some groundwork about the meaning of IRRs: the standard value structure Before attributing meaning to the interest rates in Eq. (4) some preliminary remarks are made about value and its measurement.An initial assumption is that the price of an asset, by itself, conveys nothing about the value of the asset.An assets value becomes apparent when its price is compared with another price.The assets price can be compared with the price of another asset at the same moment in time, or compared with its own price at another moment in time. The relative value of two assets at a moment in time is the ratio of their individual prices.For example, ifP /P =$1.40/$1.00=1.4A B then asset A is 40% more valuable than asset B.The additional value is the difference in price relative to one of the prices.It is represented as follows:(PA! PB) PB= 0.4 .The additional value is always a pure number because the currency units cancel. The pure number representing additional value at a moment in time is not normally divided into specific units, although it could be.For example, the statement that asset A is 40% more valuable than asset B is the same as saying asset A is more valuable than asset B by 40 units of 0.01 or 1% each.The unit could be 5% in which case the additional value is 8 units of 0.05 each, i.e.(PA! PB) PB= 0.4 = 0.01!40 = 0.05!8Taking this view, additional value, or the difference between two prices relative to one of them, is the product of the unit of value and the number of units.This simple structure is called here the standard value structure.As outlined above, under the standard value

2 The absolute values in Eq. (4) can be released selectively. The absolute value of |mi|=|ri-k|/(1+k) is retained if ri is complex which means |ri-k| is a positive, real number measuring a distance in the complex plane. The absolute value of |mi|=|ri-k|/(1+k) is released if ri is on the real number line which means the sign of a wholly real difference in interest rates is determined. This procedure determines the sign of the overall relationship between NPV and the spreads between real rates (ri-k). 14 structure, it is possible for the unit of value to vary and the number of units to adjust appropriately such that their product remains constant.This innocuous observation becomes important later in the analysis. The price of asset A today can be compared with its price yesterday or its likely price tomorrow.When time is included, the convention is to express the increase in value per unit time.The usual unit of time is the year and the rate of increase in value per year is variously known as effective annual rate (EAR), annual percentage rate (APR), compound annual rate (CAR), yield to maturity (YTM), or IRR, depending on context.3If AiPis the price of asset A at time i, and the price of asset A increases by 40% over four years, then the calculation is as follows:P4AP0A=1.4implies 1.4 = (1+r)4 in which r is the rate of interest per year.Therefore r = (1.4)1/4-1 = 0.087757 = 8.7757% per year.If the standard value structure applies to this situation, it must be possible to express the increment in price relative to the original price as a unit of value multiplied by a number of units.A candidate for the unit of value is the rate of interest per period, r.Under this assumption,(P4A! P0A) P0A= r. Xin which X is the number of units.It follows that 0.4 = 0.087757. X therefore X = 4.5580 units.The standard value structure has been imposed on this calculation; therefore it is not obviously valid to use r as the unit and the associated entity X as the number of units.This situation exists by assumption; it is not derived from deeper, more fundamental analysis.A derivation comes from the same ideas used by Osborne (2010) to produce the new expression for NPV per dollar invested, Eq. (4).The ideas are the special form of a polynomial, and the special relationship between the coefficients and roots of the special form. Any time-value-of-money polynomial can be rearranged into a special form. 111 0(1 ) (1 )nii nibz z=! + + =+ +" The parameters and the roots of this special form are linked in a particular way: the absolute value of the sum of the parameters is equal to the product of the absolute values of all possible interest rates solving the polynomial.

3 In Europe, in some circumstances, APR can have the same meaning as EAR in the US. 15 1 1n ni ji jb z= ==! " The polynomial in the example above is 44 0(1 )A AP r P = + .This equation is transformed into its special form. 404 4111 0(1 ) (1 )AAPPr r! "#$ %& '# + + =+ + The special form implies the special relationship 441 01Aj AjPrP=! =" in which rj represents every possible interest rate solving the polynomial. The left-hand side of the last equation is(P4A! P0A) P0A which is the increment in price relative to the original price; it has value 0.4.On the right-hand side, the orthodox interest rate is labeled r1 and is known to be 0.087757.Therefore the equation 44 012 0A Aj AjP Pr rP=!= " can be rewritten as0.4 0.087757 4.5580 = ! .4 A critical observation is that the last equation displays the standard value structure.On the left-hand side is the change in price relative to the original price.On the right-hand side, the orthodox interest rate, r1, is interpreted as the unit of value, while the product of the (n-1), unorthodox interest rates is interpreted as the number of units.In this way, the employment of r and its associated entity X emerge naturally from within the problem, and, moreover, meaning is attributed to the product of the (n-1), unorthodox interest rates. It is not clear, however, which of the four rates of return in the equation has the honor of being the unit of value.Mathematically the job could rotate among the four rates, the remaining three rates jointly taking on the role of measuring the quantity of units.The allocation of an orthodox rate to the role of measurement unit is arbitrary.It is demonstrated below that this fact is significant.

4 This result is easy to check using the original values for the roots in the complex plane.It is possible to visualize the result and calculate it with a hand calculator.The example has the same structure as a zero coupon bond.The roots of a zero coupon bond are distributed evenly around a circle of radius (1+r) where r is the orthodox interest rate, in this case 0.087757.Therefore the four roots (1+rj) are located at 1.087757, -1.087757, 1.087757i and -1.087757i where1 ! .The absolute values of the interest rates are the distances between the four roots and the point (1,0).The application of Pythagoras theorem establishes that the product of all four distances is 0.4, and the product of the three unorthodox distances is 4.5580. 16 The examples given so far in this section have simple structures.First, there are comparisons between the prices of two different assets, PA and PB, at the same moment in time.Second, there are comparisons between the prices of the same asset at two different moments in time, e.g. 0APand 4AP . The idea of the standard value structure is now applied to the concept of NPV.The analysis of NPV introduces two complications.First, the analysis is not restricted to comparisons of two prices, as in the earlier examples; instead, the analysis is of many cash flows distributed across time.Second, the analysis involves comparisons of values resulting from inputting two different interest rates.Input an IRR into the base equation, Eq. (1), and the result is zero.Input the cost of capital into the base equation and the result is NPV.Apparently, NPV results from a shift in the interest rate from the orthodox IRR to the cost of capital.However, as observed earlier, in reality things are not so simple.Eq. (4) demonstrates that NPV per dollar invested is the product of the simultaneous shifts in interest rate from every possible value of IRR to the cost of capital.The equation is repeated below for convenience. 1 0njjNPVmI==!in which(1 ) (1 )(1 )j jr k m + = + + (4) A critical observation is that Eq. (4) displays the standard value structure.Despite the complications introduced by many cash flows and the employment of differences between interest rates, similar reasoning about relative value applies.On the left-hand side of the equation is the increment in present value relative to the initial value of the investment.On the right-hand side of the equation is the product of the mark-ups of all IRRs over the cost of capital.One of the mark-ups, m1, is the mark-up of the orthodox IRR over the cost of capital; this mark-up is interpreted as the unit of value.The remaining mark-ups are the unorthodox values; under the standard value structure, the product of these mark-ups is interpreted as the quantity of units. Once again the possibility is noted that any of the n mark-ups could take the role of unit of value while the remaining, (n-1) mark-ups combine to give the quantity of units.The preceding discussion demonstrates how the standard value structure enables the attribution of meaning to all possible interest rates solving the TVM equation.The following section supports this interpretation with two deeper analyses. 17 5.On the interpretation of the mark-up of an IRR over the cost of capital as a unit of measurement and the product of all remaining mark-ups as the quantity of measurement units The question of interpretation is approached from two directions.The first route employs a concept from bond mathematics: Macaulay duration.The second route reinterprets the Hazen-Lohmann investment stream introduced earlier.As will be demonstrated, the two routes are complementary and mutually support the interpretation offered above. 5.1. A concept from bond mathematics gives meaning to all IRRs Earlier in this article there is a reference to the exchange of views between Wright (1936) and Boulding (1936a and 1936b) in which they discuss the implications of the fundamental theorem of algebra for the TVM equation.The fundamental theorem of algebra means the IRR equation factorizes into n factors implying n solutions for IRR.In order to factorize Eq. (1) it is first transformed.The equation is multiplied throughout by(1+ R)n and divided throughout by I0 to convert the equation to a more orthodox presentation in which the leading coefficient takes the value one.5Factorization of this transformed version of Eq. (1) is shown in Eq. (5). (1+ R)n PI0= (1+ R)n!1+ci I0(1+ R)ii=1n"#$%&'( =[(1+ R) !(1+r1)][(1+ R) !(1+r2)]...[(1+ R) !(1+rn)] (5) On the right-hand side of Eq. (5) is the product of all n factors of the equation.Each factor takes the form[(1 ) (1 )]jR r + ! + ; each factor contains a root (1+rj) and each root contains an IRR, rj.As we have seen, as R varies so P varies.If R takes the value of the orthodox IRR, here labeled r1, the right-hand side of Eq. (5) becomes zero and the left-hand side of the equation reduces to Eq. (2), which is the conventional IRR equation.1 20 21 1 10 ...(1 ) (1 ) (1 )nnc c cIr r r= ! + + + ++ + +(2)

5 The leading coefficient is the coefficient on the highest power of the variable. 18 If, in Eq. (5), R takes the value of the cost of capital, k, then the equation reduces to Eq. (3) for NPV. 1 20 2...(1 ) (1 ) (1 )nnc c cNPV Ik k k= ! + + + ++ + +(3) So far, the analysis is familiar.An unusual step is now taken with the transformation of Eq. (5) into something less familiar.On the right-hand side of the equation the orthodox root (1+r1) is replaced by the cost of capital (1+k) while all other roots remain the same.If this revised set of factors on the right-hand side of Eq. (5) is expanded, the coefficients on the left-hand side of the equation take new values because a different set of roots means a different equation.Eq. (6) is the transformed Eq. (5) in which the cost of capital replaces the orthodox root, but all else on the right-hand side of the equation remains the same, and the elements on the left-hand side carry an asterisk to show they, too, have changed. (1+ R)n P*I0* = (1+ R)n!1+ci*I0*(1+ R)ii=1n!"#$$%&''=[(1+ R) !(1+k)][(1+ R) !(1+r2)]...[(1+ R) !(1+rn)](6) If the variable R takes the value of the cost of capital then the first factor on the right-hand side of Eq. (6) is zero, therefore the entire right-hand side collapses to zero; it follows that P* is also zero, and the left-hand side becomes Eq. (7).Eq. (7) is a hybrid cash flow that is neither the IRR equation nor the NPV equation; it is hybrid because it is comprised of roots from each of these two levels equations.Eq. (7) has the (n-1), unorthodox roots from the IRR equation (2) and the single, orthodox root from the NPV equation, Eq. (3). * ** 100 ...(1 ) (1 )nnc cIk k= ! + + ++ + (7) Eq. (7) is the IRR equation with a single amendment. The amendment is that an IRR, in this instance the orthodox IRR, is shifted in a true, other-things-being-equal manner to the level of the cost of capital.The shift bears comparison with the shift from Eq. (2) to Eq. (3) but with a twist.The next paragraph is a short digression demonstrating how this hybrid equation can be interpreted. Various descriptive statistics exist to summarize a cash flow.One such statistic from bond mathematics is Macaulay duration.Macaulay duration is so-called partly because it was 19 first proposed by Macaulay (1938) as an approximation to the interest elasticity of the price of a bond and partly because it is defined as the weighted average maturity (or duration) of the cash flows.Each weight is the present value of the cash flow at a given maturity relative to the price of the bond.If Eq. (8) is the equation for the price of a bond then Eq. (9) expresses the bonds Macaulay duration. 1 (1 )niiicpr==+!(8) 1 11(1 )n nii ii icMD wi ip r= =! "= =# $+% &' 'where 11niiw==!(9)The digression now ends and the analysis resumes.Eq. (10) shows the result of applying the formula for Macaulay duration to the hybrid cash flow in Eq. (7).Macaulay duration, MD, is in bold to denote its association with the hybrid cash flow. * * *1 2* * 2 *0 01 1 1.1 .2 ... .(1 ) (1 ) (1 )MDnnoc c cnI k I k I k= + + ++ + + (10) It is asserted here that the multiple-interest-rate analysis in Osborne (2005) can be used to prove that Macaulay duration of the hybrid cash flow expressed by Eq. (10) is identical to the product of the (n-1), unorthodox mark-ups on the right-hand side of Eq. (4).Macaulay duration is, by definition, a number of periods.Thus, the product of the (n-1), unorthodox mark-ups in Eq. (4) is a number of periods. It follows that Eq. (4) can be rewritten as Eq. (11). NPVI0= MD.m1(11) Earlier it was suggested Eq. (4) displays the standard value structure.Eq. (11) is a rewritten version of Eq. (4) and it displays the structure more clearly.On the left-hand side is the increment in present value per dollar invested.On the right-hand side are two elements: the first element is a number of periods (MD); and the second element is the orthodox mark-up per dollar per period (m1).Thus, a concept from the bond market supports the earlier contention that the product of the (n !1) , unorthodox mark-ups of the IRR over the cost of capital represents a quantity of measurement units, the unit being the orthodox mark-up. If both sides of Eq. (11) are multiplied by I0 the relationship is stark and simple.Eq. (11a) demonstrates the increment in present value (NPV) is equal to the number of dollars 20 invested (I0) multiplied by a number of periods (MD) multiplied by the increment in value per dollar per period (m1). NPV = I0.MD.m1(11a) The composite element 1. MDmis comprised of the mark-ups of all IRRs over the cost of capital.If the cost of capital shifts, every mark-up shifts simultaneously.The element 1. MDmis, therefore, a single composite variable that should be considered as a whole.This last observation departs from the conventional interpretation of the relationship between NPV and the cost of capital. Conventionally, in the capital budgeting literature, the relationship between NPV and the discount rate is depicted in a graph.The discount rate is the independent variable on the horizontal axis and NPV is the dependent variable on the vertical axis.Typically, as the discount rate moves to the right, away from zero along the horizontal axis, NPV declines on the vertical.When the discount rate reaches the value of the cost of capital applicable to the project (however determined), NPV takes the value reflecting the projects worth.If the discount rate rises further to the level of the orthodox IRR then NPV becomes zero. Eq. (4) supplies a wholly different interpretation of the graph.The interpretation begins when the discount rate is equal to IRR.For a typical cash flow, as the discount rate moves to the left away from IRR, NPV increases from zero.But it is not the orthodox IRR alone from which the discount rate departs; the discount rate moves relative to every possible IRR.The differences between the discount rate and all IRRs change simultaneously as the discount rate shifts.Eq. (4) shows that when the discount rate arrives at the cost of capital the differences between the cost of capital and all IRRs combine to determine the value of NPV per dollar invested in the project. Thus, the independent variable in the typical textbook graph is a partial variable, making the conventional graph misleading.The conventional graph fails to convey information about additional action going on elsewhere in the complex plane. This additional action is also affected by movements in the discount rate and is, therefore, part of the independent variable; the additional action determines the number of periods (or times) an invested dollar is marked up, i.e. the additional action determines the quantity of measurement units. 21 In the next section the preceding argument is bolstered by analysis of the depreciation schedule. 5.2. Information from the depreciation schedule gives meaning to all IRRs In this section the analysis is about what happens to the cash flows in the IRR equation as time unfolds from one period to the next, i.e. we consider the depreciation schedule.The objective is to use the depreciation schedule to determine the total number of times an invested dollar would be marked up during the course of such an investment and to show it accords with the analysis in the previous section.To simplify the analysis, a fourth order version of the IRR equation is employed, Eq. (2a). 3 1 2 40 2 3 40(1 ) (1 ) (1 ) (1 )c c c cIr r r r= ! + + + ++ + + +(2a) This exercise is hypothetical.It is a what-if exercise.It is not assumed that the depreciation schedule for the IRR equation unfolds in reality.The current analysis concerns the number of times an invested dollar would be marked up in the event the depreciation is enacted using the IRR. [ Table 3 about here. ] Table 3 contains the depreciation schedule for Eq. (2a).Col. 2 shows the cash flows in each period.Col. 3 shows the number of dollars outstanding at the start of each period.By the fourth period there are zero dollars outstanding because the mark-up is IRR, which, by definition, reduces the entire cash flow to zero, i.e. the final element at the foot of Col. 3 is the IRR equation, Eq. (2a).Col. 4 shows the number of dollars marked up by the rate (1+r) at the start of each period; the information in Col. 4 is extracted from Col. 3.Col. 5 contains the elements of Col. 4 with the signs reversed and transposed to the appropriate timeframe.The sequence of dollar amounts in Col. 5 is the Hazen-Lohmann investment stream mentioned earlier during the discussion of Hazen (2003). The elements of the investment stream are the raw components of the objective, namely, the total number of times an invested dollar is marked up.As things stand, the elements of the investment stream in Col. 5 cannot be added because each element is a dollar 22 amount occurring in a different time period.The time value of money must be taken into account therefore a suitable discount rate is required.The cost of capital k is chosen because of the analysis in Hazen (2003).Hazen simplifies the complicated presentation of Lohmann (1988) to show that, in general, the following statement holds true: the present value of any cash flow stream discounted at rate k is equal to the product of two elements: (a) the present value of the associated investment stream created at the IRR but discounted at rate k; and (b) the element(r !k) / (1+k) .Critically, Hazen extends Lohmanns result, showing it holds true for all IRRs, i.e. any one of the IRRs can be employed to create elements (a) and (b). In this article, the Hazen-Lohmann result is given a different interpretation from that in Hazen (2003).The result is examined element by element.First, the present value of the cash flow stream in Col. 2 at discount rate k is NPV.Second, the elements of the investment stream in Col. 5 discounted at rate k are in Col. 6.The present value of the investment stream is the simple sum of the elements in Col. 6.This sum is here labeled Y.Third, if r is the orthodox IRR (r1) and the relationship between r1 and k is given by the expression (1+ r1) = (1+ k)(1+ m1) , then the element (r1! k) / (1+ k)is the mark-up of orthodox IRR over the cost of capital, earlier labeled m1.Thus, the Hazen-Lohmann result demonstrates 1. NPV Y m = .When NPV is scaled by the initial investment, the result is 10 0NPV YmI I= .Comparison of the last equation with Eq. (4) shows Y/I0 is equal to the product of the mark-ups of all unorthodox IRRs over the cost of capital. Given this last result, it is worth emphasizing the meaning of the entity Y/I0.Each element of the investment stream in Col. 5 of Table 3 contains the number of dollars marked up in each period during the life of the investment; Col. 6 contains their discounted counterparts.Therefore Y is the present value of the total number of dollars marked up, i.e. it is the number of times a dollar is marked up.Scaling Y by the initial investment gives the entity Y/I0, which is the number of times an invested dollar is marked up.It follows that the product of the mark-ups of the (n-1), unorthodox IRRs over the cost of capital is equal to the number of times an invested dollar is marked up, i.e. it is a number, or quantity, of mark-ups.Thus, analysis of the depreciation schedule in this section adds weight to the argument from bond mathematics in the previous section. 23 Two arguments from different sources demonstrate that the product of the mark-ups of all unorthodox IRRs over the cost of capital represents a quantity of the measurement unit, the measurement unit itself being the mark-up of the orthodox IRR over the cost of capital.All mark-ups work together, simultaneously producing the entire increment in value that is NPV. As observed earlier, given the structure of Eq. (4), any one of the mark-ups mj could be designated the unit of measurement and the product of the remaining (n-1) values of mj would comprise the number of measurement units.Mathematically, the choice of mark-up as measurement unit is immaterial.The two functions of measurement unit and quantity of units can rotate through the entire group of mark-ups and the analysis still works.Total value (NPV) is invariant to the state of the rotation within the group.As noted earlier, Hazen (2003) demonstrates that Lohmanns result applies to all IRRs and sees the possibility of rotation but his interpretation is different.Hazen views each IRR as an investment criterion in its own right; he does not associate the investment stream with the (n-1) remaining IRRs; concomitantly, he does not interpret the entire cluster of n rates as a single, financial entity.6

Practically, from the full set of mark-ups, it makes sense to designate an appropriate mark-up as measurement unit.By appropriate is meant wholly real, and involving an IRR having the order of magnitude of a conventional rate of interest.From this sense of what is appropriate as a measurement unit flows the distinction between orthodox and unorthodox mark-ups employed in the analysis so far.The distinction is a convenience to connect more readily with conventional analysis, but it is not mathematically necessary. 6.On the meaning of IRRThe interpretation offered here about the meaning of the entire cluster of interest rates has implications for the meaning of an IRR, conventionally viewed.As demonstrated above, the size of the measurement unit and the number of units are inextricably related; they are simultaneously determined, and they impact the value of an investment together.One implication of this result is that the conventional interpretation of a single IRR cannot be sustained: IRR, or the mark-up of IRR over the cost of capital, cannot be an investment criterion because the rate fails to convey all information about the return to a project.The net

6 Magni (2010) also employs the Hazen-Lohmann result but, ironically, does so in order to avoid use of complex IRRs. 24 return per dollar invested depends on how often an invested dollar is marked up, as well as on the mark-up itself.Information about how often an invested dollar is marked up is embedded in the (n-1), unorthodox mark-ups. For emphasis this point is restated in other words.The argument of this article is not that IRR is a poor investment criterion, as maintained in most expositions of capital budgeting.An archetypical example is found in Brealey et al. (2011): Many firms use internal rate of return as a criterion in preference to net present value. We think this is a pity. Although, properly stated, the two criteria are formally equivalent, the internal rate of return rule contains several pitfalls The argument advanced in this article is that a single value of IRR, or a single mark-up of an IRR over the cost of capital, by itself, cannot be an investment criterion.A single mark-up speaks of the value added to a single dollar at a moment in time.However, a mark-up says nothing about the number of dollars and the number of moments.A mark-up is vital information about a project but it is not full information. Multiple-interest-rate analysis has many implications.There is no need for concern about the existence of multiple IRRs, real or complex, or for undue concern about identifying one true, real IRR.All IRRs are determined simultaneously, every IRR has a part to play in the NPV cluster, all IRRs possess meaning, and all IRRs should be considered together, as a group. 7.The NPV-versus-IRR debate reviewed Academic researchers have long argued that NPV is a better investment criterion than IRR largely on the grounds that IRR has pitfalls.As observed earlier, despite this situation, numerous studies of capital budgeting practice at different times and in different places show many practitioners continue to use IRR.See, for example, the much-cited study of US data by Graham and Harvey (2001) and the more recent study of European data by Brounen et al. (2004). The most common argument proposed to explain this behavior is that practitioners would rather compare rates of return than compare dollar amounts.In this article additional reasons are proposed: first, practitioners continue to employ IRR as an investment criterion because, in conventional accounts, IRR is endorsed as an investment criterion; second, 25 practitioners are not persuaded by conventional arguments about the pitfalls of IRR that argue it is a poor criterion. The arguments in Osborne (2010) about use of multiple interest rates demonstrate that skepticism about the pitfalls is justified.In Brealey et al. (2011) the first pitfall is that an IRR, by itself, gives no indication of whether a project involves lending or borrowing.This is true but unimportant.Nobody takes an investment decision on the basis of a single statistic. Any investment proposal always contains supplementary information clarifying the nature of the investment.The second pitfall is that multiple IRRs exist.Eq. (4) makes this criticism moot. The third pitfall is that the rank order of a group of mutually exclusive projects given by the NPV criterion can conflict with the order given by IRR.This pitfall is not true within the multiple-interest-rate approach embedded in Eq. (4) because the ranking by NPV per dollar invested is always identical to the ranking by all possible IRRs.In short, three of the four pitfalls are not valid.7 The arguments in this article about meaning of multiple interest rates support this skepticism.Moreover, the same arguments recommend removing endorsement of IRR as an investment criterion.A better reason for preferring NPV is recognition that a single IRR, or a single mark-up of IRR over the cost of capital, cannot be an investment criterion.It is not a matter of degree (one is better than the other); it is a matter of kind (one is and the other is not). 8.Conclusion Since the time of Fisher (1907), the IRR, or the margin of IRR over the cost of capital, has been characterized in the financial literature as an investment criterion to rival the criterion of net present value.There has been much debate about their relative merits. Osborne (2010) shows NPV per dollar is composed of the mark-ups of every possible IRR over the cost of capital; every mark-up being simultaneously determined.This finding establishes use for all IRRs.

7 The fourth pitfall is that a non-constant yield curve means the cost of capital varies along the yield curve therefore it is difficult to compare an IRR with the many values of k applying at different times. This pitfall is also not valid.The associated analysis, however, also employing multiple interest rate analysis, is deferred to another paper. 26 The analysis in this article proposes meaning for IRR different from the conventional.The mark-up of an IRR over the cost of capital is the unit of value in which the worth of an investment is measured.The product of all other mark-ups of IRR over the cost of capital measures the quantity of such units.The entire constellation of mark-ups works as a single cluster having the standard value structure.NPV per dollar invested is one manifestation of the structure. For a last word about investment appraisal we revert almost to the beginning of the story, namely the seminal work by Lorie and Savage (1955) cited earlier in which the multiple-interest-rate puzzle is first discussed at length.The authors introduce the much-cited example of an investment in an oil-pump involving two cash outlays, one at the outset of the project, and one at the end when it is wound up, with positive cash flows between.As demonstrated by the numerical example at the beginning of this article, the results are two, real internal rates of return and a number of complex rates, plus the puzzling question from Lorie and Savage about which of the two, real rates is relevant as an investment criterion.On the basis of the argument in this article, the answer is neither.Change the question to which real rate serves as a unit of measurement? and the answer is either will do.The first real rate is a short ruler; the product of the (n-1), other rates providing the large number of short rulers necessary to give the length of the projects NPV.The second real rate is a long ruler; the product of the (n-1), other rates providing the small number of long rulers necessary to give the same length of NPV.A rate, or ruler, by itself, is not a measure of a projects worth.A rate, or ruler, in combination with its (n-1) partners, is such a measure.NPV and all IRRs are inextricably tied together in a tight and meaningful bundle.In this way, a long-standing and much-cited puzzle in corporate finance textbooks is solved. The argument that all interest rates should be considered together has profound implications for topics other than investment appraisal.This is because the time value of money equation has broad application across economics and finance.By the same arguments that a single IRR, by itself, cannot be a measure of the worth of an investment project, YTM cannot be a measure of the worth of a bond, EAR or APR cannot be a measure of the cost of a consumer loan, and CAR cannot be a measure of return on a financial investment.These concepts, however, are fundamental components of genuine measures of worth in their 27 respective areas of analysis.Such challenging thoughts mean that many applications of multiple-interest-rate analysis remain open to exploration. Acknowledgements I am grateful to Ephraim Clark and Yacine Belghitar for several stimulating conversations on capital budgeting viewed through multiple-interest-rate spectacles, and to Qi Tang for help with the mathematics. References Arrow, K. & Levhari, D. (1969) Uniqueness of the internal rate of return with variable life of investment, The Economic Journal, 79 (315), 560-566. Athanasopoulos, P. (1978) A note on the modified internal rate of return and investment criterion, The Engineering Economist, 23 (2), 131-133. Auckamp, D & Eckardt, W. (1976) A sufficient condition for a unique non-negative internal rate of return: Comment, Journal of Financial and Quantitative Analysis, 11 (2), 329-332. Bailey, M. (1959) Formal criteria for investment decisions, Journal of Political Economy, 67 (5), 476-488. Bernhard, R. (1977) Unrecovered investment, uniqueness of the internal rate of return, and the question of project acceptability, Journal of Financial and Quantitative Analysis, 12 (1), 33-38. Bernhard, R. (1979) A more general sufficient condition for a unique internal rate of return, Journal of Financial and Quantitative Analysis, 14 (2), 337-341. Bernhard, R. (1980) A simplification and extension of the Bernhard-De Faro sufficient condition for a unique non-negative internal rate of return, Journal of Financial and Quantitative Analysis, 15 (1), 201-209. 28 Boulding, K. (1936a) Time and investment, Economica, 3 (10), 196-220. Boulding, K. (1936b) Time and investment: A reply, Economica, 3 (12), 440-442. Brealey, R., Myers, S. & Allen, F. (2011) Principles of Corporate Finance, 10th edition, McGraw-Hill. Brounen, D., De Jong, A. & Koedijk, K. (2004) Corporate finance in Europe: confronting theory with practice, Financial Management, 33 (4), 71-101. Cannaday, R., Colwell, P. & Paley, H. (1986) Relevant and irrelevant internal rates of return, The Engineering Economist, 32 (1), 17-38. Chang, C. & Swales, G. (1999) A pedagogical note on modified internal rate of return, Financial Practice and Education, 9 (2), 132-137. De Faro, C. (1978) A sufficient condition for a unique non-negative internal rate of return: Further comment, Journal of Financial and Quantitative Analysis, 13 (3), 577-584. Dorfman, R. (1981) The meaning of internal rates of return, The Journal of Finance, 36 (5), 1011-1021. Eagle, D., Kiefer, D. & Grinder, B. (2008) MIRR vs. IRR: Exploring the logic of the incremental reinvestment assumption, Journal of International Finance and Economics, 8 (4), 69-75. Eatwell, J. (1975) A note on the truncation theorem, Kyklos, 28 (4), 870-875. Feldstein, M. & Flemming, J. (1964) The problem of time-stream evaluation: present value versus internal rate of return rules, Bulletin of the Oxford University Institute of Economics and Statistics, 26 (1), 79-85. Fisher, I. (1907) The Rate of Interest, Macmillan, NY, republished Martino Publishing, 2009. 29 Fisher, I. (1930) The Theory of Interest, Macmillan, NY, republished in the Library of Economics & Liberty, http://www.econlib.org/library/YPDBooks/Fisher/fshToI.html. Flemming, J & Wright, F. (1971) Uniqueness of the internal rate of return: A generalization, The Economic Journal, 81 (322), 256-263. Graham, J. & Harvey, C. (2001) The theory and practice of corporate finance: Evidence from the field, Journal of Financial Economics, 60 (2-3), 187-243 Gronchi, S. (1986) On investment criteria based on the internal rate of return, Oxford Economic Papers, 38 (1), 174-180. Hajdasinski, M. (1987) On relevant and irrelevant rates of return, The Engineering Economist, 32 (4), 347-353. Hartman, J. & Schafrick, I. (2004) The relevant internal rate of return, The Engineering Economist, 49 (2), 139-158. Hazen, G. (2003) A new perspective on multiple internal rates of return, The Engineering Economist, 48 (1), 31-51. Hazen, G. (2009) An extension of the internal rate of return to stochastic cash flows, Management Science, 55 (6), 1030-1034. Hirschleifer, J. (1958) On the theory of optimal investment decision, Journal of Political Economy, 66 (4), 329-352. Jean, W. (1968) On multiple rates of return, The Journal of Finance, 23 (1), 187-191. Kaplan, S. (1965) A note on a method for precisely determining the uniqueness or non-uniqueness of the internal rate of return for a proposed investment, The Journal of Industrial Engineering, 16 (Jan/Feb), 389-392. Karmel, P. (1959) The marginal efficiency of capital, The Economic Record, 35 (72), 429-434. 30 Keef, S. & Roush, M. (2001) Discounted cash flow methods and the fallacious reinvestment assumption: a review of recent texts, Accounting Education, 10 (1), 105-116. Kierulff, H. (2008) MIRR: A better measure, Business Horizons, 51 (4), 321-329. Lin, A. (1976) The modified internal rate of return and investment criterion, The Engineering Economist, 21 (4), 237-247. Lohmann, J. (1988) The IRR, NPV and the fallacy of the reinvestment rate assumptions, The Engineering Economist, 33 (4), 303-330. Lorie, J. & Savage, L. (1955) Three problems in capital budgeting, The Journal of Business, 28 (4), 229-239. Magni, C. (2010) Average internal rate of return and investment decisions: a new perspective, The Engineering Economist, 55 (2), 150-180. Norstrom, C. (1972) A sufficient condition for a unique non-negative internal rate of return, Journal of Financial and Quantitative Analysis, 7 (3), 1835-1839. Osborne, M. (2005) On the computation of a formula for the duration of a bond that yields precise results, Quarterly Review of Economics and Finance, 45 (1), 161-183. Osborne, M. (2010) A resolution to the NPV-IRR debate?, Quarterly Review of Economics and Finance, 50 (2), 234-239. Pierru, A. (2010) The simple meaning of complex rates of return, The Engineering Economist, 55 (2), 105-117. Pitchford, J. & Hagger, A. (1958) A note on the marginal efficiency of capital, The Economic Journal, 68 (271), 597-600. 31 Pratt, J. & Hammond, J. (1979) Evaluating and comparing projects: Simple detection of false alarms, The Journal of Finance, 34 (5), 1231-1242. Promislow, S. & Spring, D. (1996) Postulates for the internal rate of return of an investment project, Journal of Mathematical Economics, 26 (3), 325-361. Ramsey, J. (1970) The marginal efficiency of capital, the internal rate of return, and net present value: An analysis of investment criteria, Journal of Political Economy, 78 (5), 1017-1027. Ross, S., Spatt, C. & Dybvig, P. (1980) Present values and internal rates of return, Journal of Economic Theory, 23 (1), 66-81. Samuelson, P. (1937) Some aspects of the pure theory of capital, Quarterly Journal of Economics, 51 (3), 469-496. Sen, A. (1975) Minimal conditions for monotonicity of capital value, Journal of Economic Theory, 11 (3), 340-355. Solomon, E. (1956) The arithmetic of capital budgeting decisions, The Journal of Business, 29 (2), 124-129. Soper, C. (1959) The marginal efficiency of capital: A further note, The Economic Journal, 69 (273), 174-177. Teichroew, D., Robichek, A. & Montalbano, M. (1965a) Mathematical analysis of rates of return under certainty, Management Science, 11 (3), 395-403. Teichroew, D., Robichek, A. & Montalbano, M. (1965b) An analysis of criteria for investment and financing decisions under certainty, Management Science, 12 (3), 151-179. Wright, C. (1936) A note on Time and Investment, Economica, 3 (12), 436-440. Zhang, D. (2005) A different perspective on using multiple internal rates of return: the IRR parity technique, The Engineering Economist, 50 (4), 327-335. 32 Table 1 The ten values of (1+ri) solving the equation for the ten-period cash flow The ten implied values of the internal rates of return denoted by |ri| -1.01893786364476 + 0.33494777328123i 2.0465 -1.01893786364476 - 0.33494777328123i 2.0465 -0.62079688298250 + 0.87465557973728i 1.8417 -0.62079688298250 - 0.87465557973728i 1.8417 0.02342568291221 + 1.07229648735530i 1.4504 0.02342568291221 - 1.07229648735530i1.4504 0.66774985716627 + 0.83922632328577i 0.9026 0.66774985716627 - 0.83922632328577i 0.9026 1.19542150168974 0.1954 1.035030244741150.0350

Table 2: A history of multiple-rate analysis in the capital budgeting literature (built from material in Magni (2010) with additions by the author). Decade 1 Types of project having a real-valued IRR between 2 Types of project having a real-valued IRR between 3 Project truncation to obtain one real-valued IRR 4 MIRR based on different lending & borrowing rates 5 Choosing the relevant IRR from real alternatives 6 All IRRs considered; each IRR classed as an investment criterion 7 All IRRs employed as components of another concept 1930-39Fisher (1930) Wright (1936)* Boulding (1936)* Samuelson (1937) 1940-49 1950-59 Pitchford et al. (1958) Lorie et al. (1955) Solomon (1956) Hirschleifer (1958)* Bailey (1959) Karmel (1959) Soper (1959)* Wright (1959) 1960-69 Kaplan (1965) Feldstein et al. (1964)* Teichroew et al. (1965a) Teichroew et al. (1965b) Jean (1968) Arrow et al. (1969) 1970-79Ramsey (1970) Auckamp et al. (1976) De Faro (1978) Bernhard (1977, 1979) Pratt et al. (1979) Flemming et al.(1971) Norstrom (1972) Eatwell (1975) Sen (1975) Lin (1976) Athanasopoulos (1978) 1980-89Bernhard (1980)Ross et al. (1980) Gronchi (1986) Lohmann (1988)Cannaday et al. (1986) Hajdasinski (1987) Dorfman (1981) 1990-99Promislow et al. (1996) Chang et al. (1999) 2000-10Keef et al. (2001) Eagle et al. (2008) Kierulff (2008) Zhang (2005) Hazen (2003) Hartman et al. (2004)* Hazen (2009) Magni (2010)* Pierru (2010) Osborne (2010) - The papers with an asterisk (*) are examples of papers mentioning the possibility of complex roots but presuming them economically irrelevant. - The literature is not easily divided into neat categories. Whatever the criteria, some papers straddle several categories. For example, Lorie and Savage (1955) anticipate in a footnote the work of Cannaday et al. (1986); Gronchi (1986) discusses both project truncation and different lending and borrowing rates; Teichroew et al. (1965) assume different lending and borrowing rates but do not go as far as recommending MIRR. (!1,") (0, !) Table 3: The depreciation schedule for a four-period cash flow showing the cash flow and investment streams Col. 1 Col. 2Col. 3Col. 4 (drawn from Col. 3)Col. 5Col. 6 PeriodCash flow stream The number of $ outstanding at the start of each period The number of $ marked up by (1+r) at the start of each period Negative of Col. 4 = the investment stream Col. 5: the investment stream discounted by the cost of capital 0 -I0-I00I0I0 1 c1-I0 (1+r)+ c1-I0I0 (1+r)- c1I0 (1+r)/(1+k)- c1 /(1+k) 2 c2-I0 (1+r)2+ c1(1+r)+c2-I0 (1+r)+ c1I0 (1+r)2- c1(1+r)-c2I0 (1+r)2/(1+k)2- c1(1+r)/(1+k)2 -c2 /(1+k)2 3 c3-I0 (1+r)3+ c1(1+r)2+c2 (1+r) +c3 -I0 (1+r)2+ c1(1+r)+c2I0 (1+r)3- c1(1+r)2 -c2 (1+r)-c3 I0 (1+r)3/(1+k)3-c1(1+r)2/(1+k)3-c2(1+r)/(1+k)3-c3 /(1+k)3 4c4-I0 (1+r)4+c1(1+r)3+c2 (1+r)2 +c3 (1+r)+c4 =0 -I0 (1+r)3+ c1(1+r)2 +c2 (1+r)+c3 The final element above is the IRR equation, Eq. (2a). NPV of the investment stream = sum of all elements above = Y Figure 1. Solutions to the oil-pump cash flow Ten values of (1+ri) are plotted in the complex plane. The plot comes from entering the equation for the ten-period cash flow into www.wolframalpha.com and asking for its roots. The lengths of the grey lines radiating from the origin (0,0) represent the absolute values of (1+ri). In this example, there are two real values of (1+ri) plotted close together on the real axis on the right-hand side of the figure at 1.0350 and 1.1954. The remaining eight values of (1+ri) have an imaginary component therefore they reside off the real number line.If lines were to be drawn from all ten values of the roots (1+ri) to a focus at the point (1,0) their lengths would represent the ten values of |ri| implied by the roots, i.e. |(1+ri)-1| = |ri|. These values are in Table 1.