What is an Investment Project s Implied Rate of Return?

ABACUS, Vol. 53,, No. 4,, 2017 2016 doi: 10.1111/abac.12093 GRAHAM BORNHOLT What is an Investment Project s Implied Rate of Return? How to measure a project s implied rate of return has long been an unresolved problem, except for some special cases. This paper derives return on present cost (ROPC) as the correct measure of an investment project s implied rate of return. The IRR is a biased measure except for projects classified as simple projects, and this bias is likely to be substantial in many real-world applications. Thus while net present values should be used to determine whether to accept/reject projects, I recommend that analysts use ROPC in place of the IRR as a measure of a project s true rate of return. Key words: Implied duration; IRR bias; Net present value; Project evaluation. When considering an investment project, two questions are of particular interest. First, by accepting this project, how much better off will the firm or investor be? Second, what is the project s implied rate of return? While the project s net present value (NPV) has long been established as the answer to the first question, to date there is no broadly acceptable answer to the second question. While some readers may expect that the internal rate of return (IRR) is also the implied rate of return, it is easy to show that this is only true for some special cases. Other readers may assume that a project s modified internal rate of return (MIRR) measures its implied rate of return. However, I show that MIRR fails badly as a measure of a project s implied rate of return. The problem of how to identify a project s implied rate of return is longstanding. According to a leading corporate finance text: Unfortunately, there is no wholly satisfactory way of defining the true rate of return of a long-lived asset... The internal rate of return is used frequently in finance. It can be a handy measure, but, as we shall see, it can also be a misleading measure (Brealey et al., 2006, p. 91). Problems with the use and misuse of the internal rate of return were identified in the 1950s by Lorie and Savage (1955) and by Hirshleifer (1958), and discussion has continued ever since. The basic problem with using IRR as the implied rate of return is that the IRR measure is the answer to a different question: what discount rate would make this project marginal (i.e., have NPV = 0)? In contrast, implied rate of return intuitively means an equivalent compound rate of return rather than the rate of return that GRAHAM BORNHOLT (g.bornholt@griffith.edu.au) is in the Finance discipline at the Griffith Business School, Griffith University, Gold Coast, Queensland. 513

ABACUS would have made the project marginal. Employing the concept of an equivalent compound rate of return, in this article I undertake a step-by-step derivation of the implied rate of return for investment projects from the simplest to the most general. This approach identifies return on present cost (ROPC) as the implied rate of return for all projects. I show that ROPC differs from IRR except for simple projects, and that IRR s bias will often be substantial in real-world applications. In the case of one mining project, for example, its IRR exceeded its implied rate of return by a substantial 7.8% per annum (30.2% versus 22.4%). Regarding the potential uses of the implied rate of return, an important caveat is in order. Net present values should continue to have primacy when determining whether or not projects are accepted because a high rate of return does not of itself imply a project has a high value. A project s worth depends not just on its rate of return but also on the project s duration and on the amount of funds invested. In the final section of the paper, an identity is derived that links together the four interrelated factors that characterize NPV. INVESTMENT PROJECTS AND PRESENT COST Since a firm cannot earn an implied rate of return on an investment unless the project involves funds being invested for a period of time, there is a need to define what is meant by an investment project and what it costs in cash flow terms. Let (C 0, C 1,...,C n ) represent a project s net cash flows, where n is the number of periods and where C t is the end-of-period t cash flow, t =0,1,, n. In order to differentiate between net cash inflows (C t > 0) and outflows (C t < 0) in the calculations that follow, define the positive part ( ) and the negative part (C t )ofc t,by ¼ C t if C t > 0, otherwise ¼ 0; C t ¼ C t if C t < 0, otherwise C t ¼ 0. Thus, project inflows are the nonzero s and outflows are the nonzero C t s. To cover situations where the decision to commence a project does not result in an immediate cost (C 0 = 0), a project is defined as an investment project if it has at least one cash outflow and at least one cash inflow and if its first cash outflow occurs before its first cash inflow. Henceforth in this paper, any references to projects refer only to investment projects. Since many corporate projects do not become cash flow positive in their first year, the cost of a project in net cash flow terms is frequently larger than its initial cost because the project may have a number of future cash outflows. 1 Since a firm would be indifferent between a series of outflows and the present value of these outflows, we can measure the cost of an arbitrary investment project by the negative of the present value of the project s outflows. Specifically, an investment project s cost from the firm s perspective is defined as the project s present cost (denoted PC) given by: 1 The word future in relation to a project is used in this paper to refer to periods following the initial period. That is, future periods have t 1. 514

WHAT IS AN INVESTMENT PROJECT S S IMPLIED RATE OF RETURN? PC ¼ C t ; (1) where k is the discount rate used in the calculation of the project s NPV. Note that present cost arises naturally as the difference between net present value and the present value of the project s inflows since, using C t ¼ þ C t, NPV ¼ þ ¼ t PC: Þ That is, NPV is the surplus over present cost that is provided by the project s inflows. C t DERIVATION AND INTERPRETATION There is broad agreement about the rate of return for projects that have a particularly simple structure. Projects with only two nonzero cash flows, an outflow at the start and an inflow in period n (C 0 < 0 and C n > 0) have returns given by r = (C n / C 0 ) 1/n 1. The rate of return for such a project (called a bipole project henceforth in this paper) satisfies PC ¼ C 0 ¼ C n Þ n (2) This equation can be regarded as justifying the interpretation of r as the project s implied or equivalent compound rate of return. Specifically, if the investment cost was to be invested at the compound rate r per period for n periods then it would grow in value to equal the future inflow C n (since PC (1+r) n = C n from (2)). It is important to note that this rate of return is an equivalent compound rate of return because there is no expectation that the market value of the project in the intervening periods will grow every period at the compound rate r. For example, the purchase of a five-year zero-coupon bond for $713 that pays $1,000 on maturity has an implied return of 7% per annum but the value of the bond after one year (and hence the unrealized return for that year) will depend on market interest rates at that time. Equation (2) means that investing a bipole project s cost at its implied rate of return would be just sufficient to reproduce its cash inflow. This interpretation can be generalized to projects with multiple inflows. For example, suppose Project A costs $300 at time zero and returns $120 for period one and $288 for period two. We can say that Project A has an implied rate of return of 20% per period because the $300 515

ABACUS cost invested at a rate of 20% per period would be just sufficient to reproduce all of the project s cash inflows (since $100 of the $300 cost invested for one period would grow to $120 while the remaining $200 cost invested for two periods would grow to $288). This example suggests that the implied rate of return for a project is the rate of return that would be just sufficient to reproduce all of the project s cash inflows from investing the project s cost at that rate of return. That is, a project s implied rate of return is the rate of return on the project s cost. Given that present cost is an arbitrary project s cash flow cost from the firm s perspective, I define a project s implied rate of return as the project s return on present cost, determined as follows. The rate r is a project s return on present cost (ROPC) if investing the project s present cost at time zero at a compound rate of r per period would be just sufficient to reproduce all of the project s cash inflows. Now to reproduce for some t 1, we would need to invest = at time zero for t periods since = ð1 þ rþ t ¼. Thus for the present cost of the investment project to be just sufficient to reproduce all of the project s inflows means that r is defined by PC ¼ C t ¼ ; (3) where k is the discount rate used in the calculation of the project s NPV. Equation (3) means that the project s cost can be partitioned into components (the nonzero = s), each of which could reproduce one of the project s cash inflows if invested at time zero at the common rate r for the relevant number of periods. Note that equation (3) also means that ROPC can be conveniently calculated as the IRR of a transformed sequence of cash flows that retains the original sequence of inflows but which has only one outflow ( PC), positioned as the transformed sequence s initial cash flow. Given a particular value for the NPV discount rate k, it is easy to show that (3) implies that each investment project has a unique ROPC. For bipole projects, equation (3) simplifies to equation (2), meaning that ROPC =(C n / C 0 ) 1/n 1 for this special case. The question arises: under what circumstances will IRR coincide with ROPC? Now the IRR of a project is defined by the equation C t ð1 þ IRR ¼ 0: (4) Rearranging equation (4), using C t ¼ þ C t, yields C t ð1 þ IRR ¼ ð1 þ IRR: (5) Comparing (5) with (3), we can see that IRR = ROPC if there are no future outflows (C t ¼ 0 for all t 1). Call investment projects simple if they have no future outflows, and call all other investment projects non-simple. Since IRR = ROPC for 516

WHAT IS AN INVESTMENT PROJECT S S IMPLIED RATE OF RETURN? all simple projects, it is legitimate to treat the IRR of a simple project as the project s implied rate of return. This means that, for example, the common interpretation of an annual coupon bond s yield-to-maturity as its implied rate of return is correct. On the other hand, the IRR of a non-simple project should not be interpreted as its implied rate of return. Non-simple investment projects are those with at least one future outflow (C j < 0 for some j 1). Non-simple projects are common among corporate projects because it would be unusual for such projects to become cash flow positive in their first year. It is more typical for major industrial and resource projects to be cash flow negative for the first few periods before becoming cash flow positive. Similarly, it is rare for new ventures and business startups to become cash flow positive in the first year. An illustrative example of a non-simple project is Project B in Table 1. If the NPV discount rate is 10% then B s present cost is $596.78 and its ROPC is 26.5%. In contrast, project B s IRR is a much larger 43.2%. The difference between the two rates is due to how cash outflows totaling $710 are discounted in the left-hand sides of IRR equation (5) and ROPC equation (3). The left-hand side of (5) equals $372.62, while the left-hand side of (3) equals the present cost of $596.78. Consider the corresponding simple projects with these values as costs and with the same inflows as Project B. In Table 1, Project C is the IRR-equivalent simple project and Project D is the ROPC-equivalent simple project. Project C has a larger NPV than Project B ($655.01 versus $430.85), and would be strongly preferred over Project B (because the same inflows are received for lower cost) even though they have the same IRR. In contrast, the firm would be indifferent in its preference toward Project B or the ROPC-equivalent Project D because both projects have the same NPV. The reason why IRR fails to be interpretable as Project B s implied rate of return is now clear. While Project B s IRR of 43.2% could be considered the return on TABLE 1 RATES OF RETURN Project 1 t =0 t =1 t =2 t =3 t =4 t =5 PC 1 NPV 1 ROPC % IRR % B 10 100 600 500 500 500 596.78 430.85 26.5 43.2 C 372.62 0 0 500 500 500 372.62 655.01 43.2 43.2 D 596.78 0 0 500 500 500 596.78 430.85 26.5 26.5 E 100 400 400 430.58 66.94 7.1 100 F 100 500 600 317.1 100 100 554.55 309.96 30.6 41.6 G 100 500 100 100 300 808.2 554.55 309.96 22.1 25.2 This table provides cash flows and resulting descriptive measures for illustrative projects B to G. PC is present cost, NPV is net present value, ROPC is the return on present cost, and IRR is the internal rate of return. 1 The NPV discount rate of each project is 10% per annum. 517

ABACUS $372.62, the problem is that $372.62 does not have a useful interpretation as project B s cost because this $372.62 is a function of the project s inflows (since the left-hand side of (5) uses IRR to discount future outflows and IRR is a function of the project s inflows). It is the internal aspect of the internal rate of return that destroys its use as the implied rate of return for projects other than simple projects. In contrast, $372.62 is the cost of Project C, and 43.2% is C s ROPC. IRR works as the implied return for simple projects like C and D because it equals the return on present cost for these projects. It is important to realize that the failure of the IRR as a measure of the implied rate of return occurs much more frequently than the circumstances that give rise to multiple IRRs. For example, Project B has a conventional pattern of cash flows (all outflows occur before the first inflow) and so has a unique IRR. This means that the failure of the IRR as a measure of the implied rate of return is not restricted to unconventional projects. Unconventional projects can, however, produce extreme cases of IRR failure. For example, Project E in Table 1 has a negative NPV and a negative ROPC but an IRR of 100%. The non-simple projects in Table 1 all have internal rates of return that are larger than their respective implied rates of return. These are examples of a general result concerning IRR bias: Result 1. If the IRR of a non-simple project is greater than (less than) the NPV discount rate then this IRR is greater than (less than) the project s ROPC. Proof: See Appendix A. Thus if the IRR of a non-simple project exceeds the project s NPV discount rate then we know that the IRR over-estimates the project s implied rate of return. The Rosemont Copper mining project provides a real-world example of this overestimation in practice. A feasibility study of the Rosemont Copper Project produced the estimated aftertax cash flows listed in Table 2. 2 The study s conclusions were based on an NPV discount rate of 5% (see pp. 1 13). With this discount rate, Rosemont Copper has a net present value of $2,544 million, a present cost of $726 million, an ROPC of 22.4%, and an IRR of 30.2%. In this instance, the project s IRR overestimates the implied rate of return by a substantial 7.8% per annum. A larger NPV discount rate would lower the present cost and thus raise the implied rate of return. Raising the discount rate to 10%, for example, would lower the present cost to $651 million and raise the ROPC to 24.0%. Even with this change, IRR still overestimates the implied rate of return by 6.2% per annum. Dudley (1972) identified a common misconception about the IRR known as the reinvestment fallacy (see also Keane, 1979). The misconception is to believe that the IRR implicitly assumes that project inflows will be reinvested at the IRR rate. According to surveys by Keefe and Roush (2001) and Rich and Rose (2014), many introductory finance and accounting texts continue to include this error. Thus it is 2 From August 2007 NI 43 101 Technical Report, pp. 1 112, downloaded from http://www. augustaresource.com/rosemont-copper/technical-reports/default.aspx. 518

WHAT IS AN INVESTMENT PROJECT S S IMPLIED RATE OF RETURN? TABLE 2 ROSEMONT COPPER PROJECT Year 0 1 2 3 4 5 6 7 8 Cash flow ($000) 0 77,898 355,037 381,602 107,189 397,993 334,706 384,268 335,332 Year 9 10 11 12 13 14 15 16 17 Cash flow ($000) 346,024 356,427 335,108 309,388 297,716 309,131 313,789 322,738 326,366 Year 18 19 20 21 22 23 24 25 Cash flow ($000) 327,862 329,006 320,405 318,541 116,229 40,552 0 2,384 IRR = 30.2% For k = 5%: NPV ($000) = 2,544,423 PC ($000) = 725,860 ROPC = 22.4% MIRR 1 = 11.5% For k =10%: NPV ($000) = 1,312,182 PC ($000) = 650,939 ROPC = 24.0% MIRR 1 = 15.0% 1 Table 2 reports estimated annual cash flows (in $000 s) for each year for the Rosemont Copper Project, and resulting summary measures. Assumes the discount rate and the reinvestment rate are the same. 519

ABACUS important to address the issue of reinvestment so as to ensure that no similar misconception arises for the implied rate of return. A careful reading of the derivation of ROPC above equation (3) shows that ROPC does not involve any implicit or explicit assumptions about how a project s future inflows will be used or reinvested. 3 The magnitude and interpretation of a project s ROPC remains the same even if the project s inflows are not reinvested but are instead distributed to shareholders or are spent by individuals. ROPC is the implied rate of return in the equivalent compound sense that investing the project s cost at the ROPC rate would be just sufficient to reproduce all of the project s inflows. This interpretation does not imply that these inflows will then be reinvested. Moreover, while we can say that each dollar of the project s cost earns the ROPC rate of return, we must be careful not to mistakenly assume that this means that all of the project s cost is invested for the whole n periods of the project (as only C þ n = Þn is invested to the end of period n). 4 Assuming a reinvestment rate for the project s inflows moves the analysis from the real project to a longer-duration hypothetical project with a different NPV (unless the reinvestment rate is assumed to be k) or with a different rate of return (unless the reinvestment rate is assumed to be ROPC). Such assumptions are not necessary. THE FAILURE OF THE MIRR APPROACH In contrast to the implied rate of return approach, the modified internal rate of return (MIRR) approach explicitly assumes a reinvestment rate to apply to all but the last inflow of a project. In addition, it is common in the MIRR approach to use the NPV discount rate (k) as the reinvestment rate. The net effect of the MIRR method is to convert both simple and non-simple projects into an artificial bipole project. 5 The question naturally arises: for projects in general, can a project s MIRR be reliably interpreted as its implied rate of return? The answer to this question is no, as the following simple example clearly demonstrates. Suppose that Project H involves an investment of $100 now, followed by a single inflow of $200 in year one, and that the NPV discount rate is 10%. Since this simple project is a bipole project, the 3 4 5 In the unlikely circumstances that the firm has decided how a project s future inflows would be reinvested (and for how long), then the project should only be treated as an interim project. Instead, the relevant project to analyze would be the final project that results from combining the original project with the reinvestment projects. This comment is included because in the IRR case it appears that the reinvestment fallacy (in its modern form) may be the result of some users incorrectly assuming that all of the project s initial cost is invested until the end of the project. While there are a number of variants to MIRR proposed in the literature, they all reinvest inflows up to a common point (usually the end of a project). See, for example, Bernhard (1979). 520

WHAT IS AN INVESTMENT PROJECT S S IMPLIED RATE OF RETURN? various return measures are all equal (100% per annum). Now suppose Project H is improved by having a further cash inflow of $20 at the end of year two. The improved simple project is no longer a bipole project, and its implied rate of return increases to 109.5% (ROPC = IRR = 109.5%). However, the improved project s MIRR collapses to 54.9% per annum as a result of the MIRR approach assuming that the $200 inflow from year one will now be reinvested for one year at 10% per annum. Thus, the MIRR of 54.9% is not the rate of return of the improved project but is the rate of return of a hypothetical project (the bipole project that involves an investment of $100 now, followed by a single inflow of $240 in year two) that combines the actual project with an assumed reinvestment project. It is clearly a mistake to assume that a project s MIRR represents the project s implied rate of return. We saw in the previous section that whenever the IRR of a non-simple project exceeds its NPV discount rate then the IRR is an over-estimate of the project s true rate of return. In contrast, the MIRR often leads to substantial under-estimation of the project s rate of return in similar circumstances. For example, the Rosemont Copper project in Table 2 has an ROPC of 22.4% but an MIRR of only 11.5%. The MIRR s gross underestimation by 10.9% per annum in this case is the unsurprising outcome from combining a project with a return of 22.4% per annum with reinvestment projects assumed to earn 5% per annum. THE IMPLIED RATE OF RETURN AND CAPITAL BUDGETING It is well established in the literature that NPV is the appropriate measure for determining whether or not to accept a project. Of necessity, a ranking of available projects using the implied rate of return need not coincide with a ranking based on NPV. For this reason, NPV should continue to have primacy when deciding whether or not to accept a project. The worth of a project depends not just on its rate of return but also on the lengths of time over which this return is earned and on the amount of funds invested, as is directly observable from the NPV of bipole projects NPV ¼ C n Þ n þ C 0 ¼ C 0 Þ n þ C 0 Þ n ¼ PC 1 : Þ Þ n (6) In short, a project s implied rate of return is just one component of the project s worth. Nevertheless, a project s rate of return may have a secondary role in some circumstances. For example, suppose the firm has mutually exclusive projects with very 521

ABACUS similar net present values and present costs but with different implied rates of return. Projects F and G in Table 1 have the same NPV and PC but Project F has a much-larger ROPC (30.6% versus 22.1%) because its early inflows are larger than those of Project G. A firm may prefer Project F to Project G because the firm will receive more cash earlier from F, allowing it to distribute the proceeds or to reinvest in further projects sooner than Project G s cash flows would allow. The above discussion highlights the need for a meaningful duration measure when making comparisons between projects because differences in project duration can be an important driver of NPV differences. One possible duration measure (M) is a version of Macaulay s duration based on ROPC 6 : M ¼ t : (7) From the definition of ROPC in equation (3), we can see that the denominator of M equals present cost and that M is the weighted average number of periods that the project s present cost is invested. For a standard coupon bond investment, M simplifies to Macaulay s bond duration measure (since ROPC = yield-to-maturity for annual bonds). Although Macaulay s measure does not directly link to NPV, we can define a related duration measure that does link to NPV in a meaningful way. A project s implied duration is the length of the project s equivalent bipole. A project s equivalent bipole is defined to be the bipole project with the same NPV, the same NPV discount rate, the same present cost, and the same implied rate of return as the project itself. Solving bipole equation (6) for n and replacing n with D to denote this implied duration measure yields: D ¼ ln ð NPV þ PCÞ ln ð PC Þ lnþ lnþ for k r; (8) where r is the implied rate of return (ROPC). A project s implied duration can be regarded as the effective centre of the project s inflows. For the k=rcase not included in (8), inspection of (3) shows that NPV will equal zero when k = r, and thus if k = r then k must equal a root of the IRR equation (5). If IRR* denotes this root then k = r only when k = IRR*. The duration measures M and D will be numerically close to each other for NPV discount rates close to IRR* because D M approaches zero if the discount rate k approaches IRR*. That is, 6 Past research articles describing project duration typically employ IRR as the discount rate in the duration measure. See, for example, Boardman et al. (1982). 522

WHAT IS AN INVESTMENT PROJECT S S IMPLIED RATE OF RETURN? lim ð k IRR* D M Þ ¼ 0: (9) (For the proof of this result, see Appendix B). For this reason, set D = M for the k = r case that is not included in (8). The great advantage of defining project duration as the implied duration D is that it means that we can link together the four interrelated factors that drive NPV in a fundamental NPV identity that is valid for all investment projects: " # Þ D NPV ¼ PC 1 : (10) Þ That is, a project s NPV is completely determined by its present cost (PC), its NPV discount rate (k), its implied rate of return (r = ROPC), and its implied duration (D). As an illustrative example of the use of this identity, compare projects B and F from Table 1. Applying (8), Project B s implied duration is 3.9 years while Project F s implied duration is 2.6 years. Project B s larger NPV ($430.85 versus $309.96) can be regarded as the result of earning a smaller rate of return (26.5% versus 30.6%) on a larger investment ($596.78 versus $554.55) for a considerably longer duration (3.9 years versus 2.6 years). In summary, equation (10) provides a useful framework for discussing salient differences between projects. FINAL COMMENTS AND CONCLUSION The implied rate of return is an intuitive concept that has until now lacked a reliable measure, except in some special cases. This paper proposes that a project s return on its present cost (ROPC) is the appropriate measure of its implied rate of return. ROPC is the implied rate of return in the sense that investing the project s cost at the ROPC rate would be just sufficient to reproduce all of the project s future inflows. A project s ROPC coincides with its IRR in the case of simple projects. However, many large corporate projects are non-simple projects because there are usually a number of periods of outflows before these projects become cash flow positive. Since the bias of the internal rate of return is likely to be substantial in many real-world applications, I recommend that ROPC replace IRR as the measure of an investment project s true rate of return. REFERENCES Bernhard, R. H. (1979), Modified Rates of Return for Investment Project Evaluation A Comparison and Critique, Engineering Economist, Vol. 24, pp. 161 7. Boardman, C. M., W. Reinhart, and S. Celec (1982), The Role of the Payback Period in the Theory and Applications of Duration to Capital Budgeting, Journal of Business Finance and Accounting, Vol. 9, No. 4, pp. 511 22. 523

ABACUS Brealey, R. A., S. Myers, and F. Allen (2006), The Principles of Corporate Finance, 8th ed, McGraw-Hill Irwin, New York. Dudley, C. L., Jr (1972), A Note on Reinvestment Assumptions in Choosing between Net Present Value and Internal Rate of Return, Journal of Finance, Vol. 27, No. 4, pp. 907 15. Hirshleifer, J. (1958), On the Theory of Optimal Investment Decision, Journal of Political Economy, Vol. 66, pp. 329 52. Keane, S. M. (1979), The Internal Rate of Return and the Reinvestment Fallacy, Abacus, Vol. 15, No. 1, pp. 48 55. Keefe, S. P. and M. L. Roush (2001), Discounted Cash Flow Methods and the Fallacious Reinvestment Assumption: A Review of Recent Texts, Accounting Education, Vol 10, No. 1, pp. 105 16. Lorie, J. H. and L. J. Savage (1955), Three Problems in Rationing Capital, Journal of Business, Vol. 28, pp. 228 39. Rich, S. P. and J. T. Rose (2014), Re-examining an Old Question: Does the IRR Method Implicitly Assume a Reinvestment Rate?, Journal of Financial Education, Vol. 40, Nos 1/2, pp. 152 66. APPENDIX A: PROOF OF RESULT 1 This appendix contains the proof of the result that if the IRR of a non-simple project is greater than (less than) the NPV discount rate then the IRR is greater than (less than) the project s ROPC. The result is a straightforward consequence of equations (3) and (5) applied to non-simple projects. Proof: All non-simple investment projects satisfy the following two conditions: C t 0 for all t; and C i < 0 for some i 1; and (A1) 0 for all t; and C þ j > 0 for some j 1: (A2) Consider first the greater than case: IRR > k C t ð1 þ IRR < ð1 þ IRR < C t using ða1 ð1 þ ROPC from 3 IRR > ROPC: using ða2þ Þ ðþand ðþ 5 The less than case follows in the same way by reversing the inequalities in the greater than case. 524

WHAT IS AN INVESTMENT PROJECT S S IMPLIED RATE OF RETURN? APPENDIX B: PROOF THAT D-M 0 IF K IRR* This appendix shows that lim k IRR* ðd M Þ ¼ 0; where D ¼ ln ð NPV þ PCÞ ln ð PC Þ ; and M ¼ lnþ lnþ t : Now (3) and (5) imply that if k approaches an IRR (denoted IRR* in case there are multiple IRRs) then either r = IRR* in the case of simple projects or r must also approach IRR*. For the limit of D, we can use the right-hand side of (3) to show that lim k IRR* D ¼ lim k IRR* 0 ln B @ C þ t ln lnþ lnþ 1 C A (A3) The right-hand side of (A3) can be evaluated using L Hopital s rule because both the numerator and the denominator of (A3) approach zero as k approaches IRR*. Let f(k) denote the numerator and g(k) denote the denominator of the right-hand side of (A3). The derivative of f(k) with respect to k is f ðkþ ¼ t þ1 t þ1 : dr dk (A4) The derivative of g(k) with respect to k is g ðkþ ¼ 1 Þ : dr dk 1 Þ (A5) For non-simple projects, as both k and r approach the constant value IRR* in the limit, it follows from (3) and (5) that lim k IRR* dr dk ¼ 0 (A6) For simple projects, r is independent of k (since r = IRR*) so dr/dk = 0 and thus (A6) holds for simple projects also. Applying L Hopital s rule and (A4), (A5) and (A6) yields: 525

ABACUS lim k IRR* D ¼ lim k IRR* fðkþ gk ð Þ ¼ 2 ¼ lim k IRR* 6 4 2 ¼ lim k IRR* 6 4 lim k IRR* t f ðkþ g ðkþ þ1 t þ1 t þ1 3 7 5 = 1 Þ 3 : dr 7 dk5 = 1 dr Þdk 1 Þ using ða6þ ¼ lim k IRR* ¼ lim k IRR* t t ¼ t ð1 þ IRR* ð1 þ IRR* since r IRR* ðorr ¼ IRR* Þ as k IRR* ¼ lim k IRR* M: Therefore lim k IRR* ðd MÞ ¼ 0: 526