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Munich Personal RePEc ArchiveThe Reinvestment Rate AssumptionFallacy for IRR and NPV: A PedagogicalNoteMagni, Carlo Alberto and Martin, John D.University of Modena and Reggio Emilia, Baylor UniversityDecember 2017Online at https://mpra.ub.uni-muenchen.de/83889/MPRA Paper No. 83889, posted 11 Jan 2018 20:09 UTC

2017The Reinvestment RateAssumption Fallacy forIRR and NPVA PEDAGOGICAL NOTECARLO ALBERTO MAGNI AND JOHN D. MARTIN

Table of Contents1Introduction . 12The fallacy . 13Potential sources of the reinvestment rate fallacy . 2Early contributions to capital budgeting literature . 2Ranking Alternative Investments. 3The multiple IRR problem . 3The modified IRR . 3The scale effect . 4What’s the harm of assuming required reinvestment rates? . 5Is this really a problem of semantics? . 54Summary remarks. 5References . 7CARLO ALBERTO MAGNI AND JOHN D. MARTIN1

The Reinvestment Rate Assumption Fallacyfor IRR and NPV: A Pedagogical Note1IntroductionThe notion that the internal rate of return (IRR) and net present value (NPV) have reinvestment rateassumptions built into them has long been settled in the academic finance literature.1 Specifically, thereare no reinvestment rate assumptions built into, or implicit to, the computation and use of either theIRR or NPV. Once an investment’s cash flows are received they can be distributed to the firm’s creditorsor shareholders without any necessity to reinvest them. However, there persists the notion that IRR andNPV have implicit “reinvestment rate assumptions” embedded in them. For example, the followingstatement was taken from Investopedia:2. . . the traditional internal rate of return (IRR) assumes the cash flows from a project are reinvested at the IRR.Here are two more quotes taken from different websites, that focus on student users that make thesame point:The IRR rule assumes that intermediate cash flows from the project get reinvested at the IRR. Implicit is the assumption that thefirm has an infinite stream of projects yielding similar IRRs.34NPV and PI assume reinvestment at the discount rate. IRR assumes reinvestment at the internal rate of return.In this brief note, we first review the theoretical underpinnings of the rate of return assumption fallacy.Next, we offer two possible origins from the academic finance literature that may be responsible for thefallacy. Specifically, Ezra Solomon’s discussion of the ranking of mutually exclusive investments and JackHirshleifer’s discussion of the multiple IRR problem. We conclude that the reinvestment assumption is asufficient condition, not an implicit assumption, for solving the problems of conflicting ranking andmultiple IRRs.2The fallacyTo illustrate the fallacy of rate of return assumptions behind IRR and NPV we use a simple example.Here’s how it works. Consider a security market in which borrowing and lending rates (𝑟) are the same.Assume now that an investor has an opportunity to undertake a project with cash-flow stream𝑓 ( 𝐶0 , 𝑓1 , , 𝑓𝑛 ), where 𝑓𝑡 0 for each 𝑡 and 𝐶0 is the initial cash outlay required to finance theinvestment. The investor may then borrow an amount equal to the present value of the project’s futurecashflows, i.e., 𝑉0 𝑛𝑡 1 𝑓𝑡 (1 𝑟) 𝑡 . By taking these actions, the investor (i) realizes a cash inflowequal to 𝑉0 from the loan proceeds, (ii) pays 𝐶0 to finance the investment, and (iii) obligates theDudley (1972, p. 908) put it bluntly, “There is no such assumption implicit in the technique”. More recent papersinclude Rich and Rose (2014) and Walker et. al. g-91/reinvestment-assumptions-394-8295/.1CARLO ALBERTO MAGNI AND JOHN D. MARTIN1

project’s future cash flows to repay the loan. The resulting net cash-flow vector then is (𝑉0 𝐶0 , 0,0, ,0) (𝑁𝑃𝑉, 0,0, ,0) . Of course, the project is worth undertaking if and only if 𝑁𝑃𝑉 𝑛𝑡 0 𝑓𝑡 (1 𝑟) 𝑡 0. Note that since all the project’s future cash flows are converted to their presentvalue equivalent via the loan, and the project’s future cash flows are committed to repaying the loan,there are no future cash flows available to reinvest!That NPV analysis does not assume reinvestment can be demonstrated in several other ways,abstracting from any consideration of borrowing. Most notably, consider the price 𝑉0 of a portfoliotraded in the market replicating the project’s cash flow. Shareholders may undertake the project byinvesting 𝐶0 or buy a replicating portfolio by investing 𝑉0 . The sequence of prospective cash flows is thesame from both alternatives, so acceptance only depends on the difference between the project’s costand the price of the portfolio (i.e., the NPV), not on reinvestments of cash flows.5As for IRR, assuming 𝑓 ( 𝐶0 , 𝑓1 , , 𝑓𝑛 ) is a conventional cash flow stream (outflows precedinginflows), the NPV function is monotonically decreasing so that 𝑁𝑃𝑉 0 if and only if 𝐼𝑅𝑅 𝑟. As thecondition 𝑁𝑃𝑉 0 has been determined with no reinvestment consideration, the condition 𝐼𝑅𝑅 𝑟 isnot tied to reinvestment consideration as well. The IRR is the (assumed constant) rate of return on theinvested capital, period by period. The condition that 𝐼𝑅𝑅 𝑟 only means that if, in any given period,an investor invests an amount of capital equal to the capital that remains invested in the project, thenthe rate of return earned with the former would be greater than the rate earned with the latter.3Potential sources of the reinvestment rate fallacyHow is it that many analysts continue to refer to the IRR as a required “reinvestment rate” for futurecash flows when using the IRR as a project evaluation tool, and the cost of capital as the required“reinvestment rate” when using NPV? The answer may lie in some of the early writings regarding thedifficulties encountered when(i) ranking mutually exclusive investment opportunities (where IRR and NPV rankings are oftenin conflict), and(ii) multiple IRRs arise in some nonconventional projects.Early contributions to capital budgeting literatureIn the 1950s the finance literature devoted to the analysis of mutually exclusive investment projects andthe analysis of multiple IRRs both incorporated consideration for reinvestment rates. The discussion ofreinvestment rates in this context, we believe, may well be the source of the confusion aboutreinvestment rates and project IRRs and NPVs.For example, it seemed natural to consider reinvesting cash flows as one way to eliminate interim cashflows which were the source of the ranking conflicts between NPV and IRR or to overcome thedifficulties encountered in comparing mutually exclusive investment proposals with different initialinvestments and/or different investment lives and/or different cash-flow patterns, and for projectswhose cash flows have multiple IRRs. Even more so considering that, in 1950s, modern finance was notyet fully established, and the concept of terminal value seemed still common, as opposed to the idea of5The rate 𝑟 is the expected rate of return of the replicating portfolio.CARLO ALBERTO MAGNI AND JOHN D. MARTIN2

a present value equal to the price of a portfolio replicating the project’s cash flows. 6 Significantcontributions to this discussion came from Solomon (1956) and Hirshleifer (1958) and both suggestedthat incorporating consideration of reinvestment rates for interim project cash flows might provebeneficial.Ranking Alternative InvestmentsSolomon (1956) provided an important contribution to the analysis of mutually exclusive investmentalternatives in which he raised the question as to implicit reinvestment rates for IRR and NPV.7 In short,Solomon believed that both NPV and IRR have implicit reinvestment rates, the former at the cost ofcapital, the latter at the IRR itself. According to the author, the cause of the conflict lies in the differentreinvestment assumptions of IRR and NPV. To solve the conflict, he suggested that the analyst make an“explicit” and common assumption regarding the rate at which funds can be reinvested up to theinvestment’s terminal date to arrive at an appropriate ranking: “If a common assumption is adopted,both approaches will always rank projects identically” (p. 126).The multiple IRR problemHirshleifer (1958) provided an early and important discussion of the problem of multiple IRRs that canarise in projects that have non-conventional investment cash flows (see Section III B. of his paper).While he suggested that the IRR assumes reinvestment at the IRR itself, he criticized this implicitassumption, deeming it unrealistic. Consistent with Solomon, Hirshleifer recommended making anexplicit reinvestment rate estimate for the interim cash flows, which would turn the project into acourse of action with a unique IRR (later called Modified Internal Rate of Return). This unique IRR iscompatible with the NPV of the entire course of action, in the sense that NPV is positive if and only if theMIRR is greater than the cost of capital.The modified IRRSolomon (1956) and Hirshleifer (1958) may then be considered forerunners of the modified internal rateof return (MIRR) approach. If multiple IRRs occur only in projects with interim cash flows, thenSolomon’s and Hirshleifer’s arguments may be used to solve the multiple-IRR problem: making an NPVinvariant modification of the project such that the modified project has no interim cash flows, andconsequently the project’s IRR will be unique. Among the infinitely many ways to adjust a cash-flowstream, one may consider the explicit reinvestment of interim cash flows up to the terminal. However,the assumption of reinvestment of interim cash flows at some rate is introduced only for making themultiple IRR problem (and the ranking problem) disappear, not because any reinvestment option isimplicit in the IRR procedure, as seen above.Note, however, that by modifying the cash-flow stream, the meaning of the resulting rate (MIRR)changes. Further, the MIRR does not measure the project’s rate of return because it takes account of6Which in turn, is equal to the price that the project would have if it were traded in the security market.Renshaw (1957) summarized Solomon’s position on reinvestment rates as follows: “The contribution of theSolomon article was to point out that the apparent conflict between these two ranking procedures was due todiffering implicit assumptions about reinvestment rates (the present-value approach assumes reinvestment ofintermediate cash receipts at the discounting rate, while the internal rate of return approach assumesreinvestment at the internal rate) and to suggest that the conflict could be eliminated by making an explicitassumption about the expected return from reinvestment.” p. 193 A key observation here is the statementregarding “implicit assumptions regarding reinvestment rates”.7CARLO ALBERTO MAGNI AND JOHN D. MARTIN3

the reinvestments of the interim cash flows, which have nothing to do with the original investmentopportunity. The MIRR is simply the internal rate of return of a course of action which includes theproject being analyzed and other projects that are associated with the reinvestment of interim cashflows. Ross et. al., 2011, p. 250 describes the MIRR as follows: “it’s a rate of return on a modified set ofcash flows, not the project’s actual cash flows”. Therefore, the MIRR is the IRR of a portfolio of projects,not the original investment. The key point here is that a project’s rate of return should not be affectedby the choice of a reinvestment rate for its cash flows. Brealey, Myers and Allen (2011) put it verysuccinctly “The prospective return on another independent investment should never be allowed toinfluence the investment decision” (p. 141).The MIRR also solves the problem of conflicting ranking (if the initial outlays are the same): Thecomparison of any two MIRRs will result in the same ranking as the comparison of the two associatedNPVs. In other words, explicitly incorporating consideration for the reinvestment rate is sufficient tomake IRR and NPV consistent and to ensure existence and uniqueness of IRR. As a result, scholars andpractitioners have mistaken a sufficient condition for an implicit assumption.Scale effectsKeane (1979) supplied an enlightening view that contributes to dispel the misconception of thereinvestment assumption. He attributed the problem of ranking projects as evidence of the “scaleeffect”. That is, the conflict between NPV and IRR ranking is due to difference in scale of the projectsunder consideration, not to different reinvestment assumptions. By “scale” one should not merely referto initial outlays, but to the total units of capital outstanding and capital length of the project (see Keane1979, pp. 53-54). While NPV considers project scale as well as economic efficiency, IRR only measureseconomic efficiency, so it is not adequate for ranking projects.It is worth noting that the conflict between the notion of rate of return and NPV disappears if explicitreinvestment assumptions are made on competing projects (if the initial outlays are the same),8 becausethis changes the project scale in a way that makes the IRR ranking and the NPV ranking identical (as longas the initial outlays are equal).9 In other words, reinvestment assumptions can be used to solve theproblem of conflicting ranking.10It is worth noting that, while not mentioned by Keane (1979), the “scale effect” is also the essentialreason why reinvestment assumption solves the multiple-IRR problem: indeed, it can be shown that, ifinterim cash flows are reinvested, then the resulting modified project possesses outstanding capitalwhich has the same sign in each period (i.e., a higher scale), thereby ensuring unique a IRR.118The NPV must be computed on the modified cash-flow streams.It may be proved that a higher IRR is associated with a higher scale. Therefore, the project with higher IRR alsohas a higher value created.10The role of the scale effect and the relation of scale to rate of return is analyzed in Magni et al. 2017, where areconciliation between NPV ranking and rate-of-return ranking is supplied with no need of assuming anyreinvestment. See also Keefe and Rousch (2001) for a relatively recent paper on the inexistence of reinvestmentassumptions.11Multiple IRRs may not arise if capital outstanding has the same sign in each period.9CARLO ALBERTO MAGNI AND JOHN D. MARTIN4

What’s the harm of assuming required reinvestment rates?To illustrate the type of problems that can arise when assuming implicit reinvestment rates consider thefollowing paraphrased conversation that occurred between the CFO and a financial analyst at a majorpublic firm (relayed to the authors via a phone conversation):“We have an investment opportunity that promises an extremely high IRR of 30%. However, we are worried aboutthe decision to undertake the investment since the firm does not expect to have investment opportunities toreinvest those cash flows to earn 30%.”The implication being that without the opportunity to reinvest the cash thrown off by the 30% project ata similar rate of return, the project’s IRR would not be realized. This type of reasoning can be quitemisleading and costly to the firm. Specifically, it might lead the firm to avoid very high rate of returnprojects (paradoxically, the higher the IRR of the project, the more problematic is the project’sacceptance).In fact, the cash flows the investment opportunity promises can simply be distributed as they arereceived to the firm’s creditors in the form of interest and/or principal payments or to stockholders viadividends or share repurchases, and the firm will realize the promised 30% return, regardless of thedestination of those cash flows, because the IRR is the return generated by one dollar of capital thatremains invested in the project. As such, the IRR does not have anything to say about the rate of returnof the capital that is divested from the project. Instead IRR speaks to the return earned on the capitalthat is invested in the project.Is this really a problem of semantics?To this point we have been using adjectives like implicit and explicit when referring to reinvestmentrates. Is it possible that some of the misunderstanding about reinvestment rates has something to dowith language? For example, when an analyst attempts to move project cash flows either forward orbackward in time in such a way as to not alter the project’s NPV, then the rate of interest that preservesthe investment’s NPV is the rate used to evaluate the investment (i.e., the cost of capital). Similarly, ifthe analyst wishes to preserve the project’s IRR, the rate of interest that should be used to move cashflows either forward or backward is the IRR of the project itself. Does this observation mean that theserates of interest are implied reinvestment rates? Asked somewhat differently, do interim cash flowshave to be reinvested to realize the promised NPV or IRR? The answer is, of course, no. Remember thatthe investment cash flows do not need to be reinvested to guarantee the NPV or IRR of the project. It iseasy to imagine that this relationship between interest rates and preserving NPV or IRR might bemisconstrued to mean these rates are “required” or “mandatory” when they are not. Reinvestment isnot implicit in the IRR and NPV; rather, it is a sufficient condition for solving ranking conflicts betweenNPV and IRR and the multiple-IRR problem.4Summary remarksDespite demonstrated theory to the contrary, many continue to believe that NPV and IRR have implicitreinvestment rates. The reinvestment rate for NPV being the cost of capital for the investment and forIRR it is the IRR itself. The notion that project cash flows must be reinvested to assure the investorrealizes the IRR as a rate of return (should cash flows materialize as expected) or that these cash flowsmust earn the cost of capital to earn the estimated NPV is simply not the case. Once received, theCARLO ALBERTO MAGNI AND JOHN D. MARTIN5

investment cash flows can be distributed to the firm’s creditors or shareholders, or simply held in shortterm investments awaiting reinvestment at some future date at a yet unknown rate of return.We offer a possible explanation for the persistence of the reinvestment rate assumption that derive outof some of the foundational work on the evaluation of investments. Specifically, the answer may lie insome of the early writings regarding the ra