Teaching MIRR to Improve Comprehension of Investment Performance Evaluation Techniques

Transcription

1 Teaching MIRR to Improve Comprehension of Investment Performance Evaluation Techniques R. Brian Balyeat 1, Julie Cagle 2, and Phil Glasgo 3 ABSTRACT NPV and IRR are frequently used to evaluate investment performance, yet when they conflict many firms base capital budgeting decisions on IRR, though NPV is superior in specific cases. Teaching the MIRR technique should reinforce academia s preference of the NPV technique. The reinvestment assumptions of NPV and IRR are implicit and hidden from students, while the calculation of the MIRR technique forces explicit decisions regarding the investment and discounting of interim cash flows. Thus, by teaching the MIRR calculation students may gain a better understanding of the differences between the three techniques reinforcing the primacy of NPV. Key Words: NPV, IRR, MIRR Introduction Evaluating investment performance is fundamental to the finance discipline, and taught in both investment andfinancial management courses. 4 A puzzle related to performance evaluation techniques as used in practice is the similar frequency of use of the IRR technique relative to NPV for evaluating capital investments, even though most finance texts argue that NPV is superior in certain cases. 5 NPV is more consistent with wealth maximization when projects have unconventional cash flows, and in the cases of mutually exclusive projects with differences in scale (initial investment size) or timing of cash flows (whether larger cash flows occur later vs. early in project life). Brealy and Myers (2000, p. 108) point out that IRR is a derived figure without any simple economic interpretation and that it cannot be described as anything more than the discount rate that when applied to all cash flows makes NPV equal to zero. Hirshleifer(1958) concludes that anytime there are intermediate cash flows between investment and the termination of the project, the IRR rule is not generally correct. The MIRR (modified IRR) yields decisions identical to the NPV rule unless scale differences are present betweenmutually exclusive projects. Even with scale differencesbetween mutually exclusive projects, Shull (1992) shows an adjusted MIRR technique that leads to identical decisions to the NPV rule. 6 Should it concern finance academics that IRR is used so frequentlyin practice? Yes, given that IRR is not a valid measure of return for many projects, and that IRR may result in investment decisions that conflict with NPV and lead tosub-optimal investment decisions. Burns and Walker (1997)provide evidence 1 Department of Finance, Xavier University 2 Department of Finance, Xavier University 3 Department of Finance, Xavier University 4 See Phalippou (2008) for a discussion of the hazards of using IRR to measure performance in an investment context, particularly the case of private equity. Phalippou points out that the performance evaluation literature is largely found in corporate finance texts rather than investment texts. He also provides a MIRR calculation for measuring investment level and fund level performance. 5 See, for example, Ross, Westerfield, and Jordan (2011) pages 246 and Shull (1992) discusses an adjusted ORR method that is the MIRR technique adjusted so that it can lead to identical ranking decisions to NPV in the case of scale differences between mutually exclusive projects

2 on how capital budgeting decision criteria are used in practice, and more importantly,how the multiple decision techniques are treated when they conflict. The survey was sent to CFOs on the Fortune 500 industrials list. Forty-one percent of respondents indicated that IRR took priority in the case of a conflict, versus 29% for NPV and 2% for MIRR. This is evidence that IRR is being used by practitioners in ways which result insub-optimal investment decisions that are inconsistent with the recommendation by finance academics to prioritize NPV in the cases of conflict with IRR. Clearly, these results suggest academics need to do more to clarify the best use of capital budgeting decision criteria to students that will be future practitioners. This paper proceeds to make the case for teaching the MIRR decision rule not only because it is a superior measure of rate of return in some cases (e.g., when project cash flows change sign more than once) compared to IRR, but also because teaching MIRR reinforces NPV as the primary decision criteria for capital budgeting. When students calculate the MIRR for a project, they must explicitly consider how intermediate cash flows during the life of the project are treated. In so doing, the differences in the reinvestment assumptions between NPV, IRR, and MIRR can be highlighted. That the three techniques can lead to inconsistent investment decisions and ranking of projects can also be emphasized and will hopefully result in practitioners using NPV as the primary decision criteria over IRR in the case of conflict. The latter may be particularly important if there is a bias by practitioners toward rate of return techniques, like IRR, over NPV. Further, students and practitioners will become more aware that absent scale differences between projects, MIRR complements the NPV decision if properly calculated. 7 We first examinethe evidence on how alternative decision criteria are used in practice, followed by the academic perspective on the MIRR technique and how MIRR is calculated. Then, a discussion of reinvestment rate assumptions is provided, and a comparison of NPV, IRR, and MIRR decision criteria is made. We conclude than not only is MIRR a superior measure of rate of return to IRR, but that pedagogical emphasis on the MIRR criterion reinforces the primacy of the NPV technique. The Practice of Capital Budgeting The practice of capital budgeting has changed over time. Pike (1996) provides a longitudinal study of capital budgeting practice between 1975 and 1992 for 100 UK firms. In regards to evaluation techniques, he finds discounted cash flow methods are well established with 81% of firms reporting using IRR and 74% of firms reporting using NPV. The use of multiple techniques increased over time from one or two methods to four methods. The greatest growth for any one technique was with NPV with 42% of the sample introducing it since While these results are encouraging, Pike cautions that we know very little about how the discounted cash flow techniques are used in the decision making process. Burns and Walker (1997)include MIRR in their survey of Fortune 500 industrial CFOs to try and ascertain more about the how of capital budgeting practices. They find NPV and IRR dominate with more than 70% of firms using each, while MIRR was used by only 3% of respondents. The survey also asked about emphasis on each of the techniques and IRR received the greatest emphasis (48 of 100 points), followed by MIRR (45 of 100), and NPV (33 of 100). This suggests that while a small number of firms use the MIRR technique, of those that do, it receives considerable emphasis. MIRR was also indicated as a younger technique, with the only 50% of firms indicating that have used it for more than 10 years, versus 63% for IRR and 66% for NPV. Importantly, Burns and Walker (1997)also provide evidence on how the multiple techniques are treated when they conflict. Forty-one percentof respondents indicated that IRR took priority in the case of a conflict, versus 29% for NPV and 2% for MIRR. These results are significant because they suggest decisions are being made with priority placed on IRR instead of NPV when the rules conflict, which can lead to sub-optimal investment decisions. Similar to Pike (1996), the results of Burns and Walker (1997) indicate firms have increased their emphasis on IRR, MIRR, and NPV over the last 5-10 years, while payback, discounted payback, and average rate of return receive less emphasis. Interestingly, when asked about why a particular technique is used, ease of understanding is indicated more frequently for the MIRR (8.5) and NPV (4.3) techniques than IRR (3.5). The trend is similar across the three techniques for ease of computation and reliability. 7 See the section below MIRR Calculation for how we define the calculation of MIRR versus other authors and how our definition may result in a MIRR different from that obtained from the Microsoft Excel MIRR function or a financial calculator. Our definition is based on Lin s (1976) second definition for MIRR. Also see Shull (1992)

3 In terms of realistic reinvestment rate, NPV received a composite average of 3.5 versus MIRR which received Formal education was indicated as the dominant source of familiarity with the various techniques, suggesting academics have a significant role in regard to how these techniques are used in practice. Graham and Harvey (2001) surveyed 392 CFOs in regarding capital budgeting, among other topics, as well as descriptive information about their firms. NPV and IRR were indicated as the most frequently used of the capital budgeting techniques listed, in a list that also included payback, discounted payback, profitability index, accounting rate of return, and adjusted present value. Larger firms were also more likely to use NPV than small firms. The MIRR technique was not included in the survey. Ryan and Ryan (2002) survey 205 CFOs of Fortune 1000 companies. Use of seven different capital budgeting decision techniques including NPV, IRR, and MIRR was examined. While 96% of the firms reported use of NPV, 85.1% indicated they used it frequently. For IRR the usage rate was 92.1% with 76.7% of the firms using it frequently. By contrast, the MIRR usage rate was just under 50% and the technique is used frequently by fewer than 10% of respondents. In fact, MIRR ranked seventh out of seven for frequency of use. The size of the annual capital budget affected use of the various techniques. Firms with larger capital budgets were more likely to use NPV and IRR, whereas this was not the case for MIRR. MIRR was more frequently used by firms with mid-sized ($100 - $500 million) annual capital budgets than small (< $100 million) or large (>$500 million) budgets. The authors puzzle over the lack of use of MIRR and suggest it may gain acceptance over several decades as did other discounted cash flow techniques. Academic Perspective on MIRR The presumed reason for IRR s frequent use in the field is that managers prefer to make decisions based on returns rather than dollar amounts. Shull (1994, p. 162) argues that optimal investment decisions are not the sole objective behind rate of return methods. NPV already provides that, so rate of return methods are redundant for that reason alone. This suggests rate of return methods provide an advantage beyond NPV-consistent decisions, but exactly what this additional advantage is remains unclear. Finance texts now suggest MIRR as an alternative to IRR because it leads to decisions more consistent with wealth maximization for projects with nonconventional cash flows and mutually exclusive projects with different timing of cash flows. McDaniel, et al. (1988) credit Lin (1976) with early development of the term MIRR in a format similar to today s usage. Biondi(2006)traces the development of MIRR back to Duvillard in 1787, and the re-emergence in the 1950 s to Lorie and Savage (1955), and Solomon (1956), among others. Therefore, MIRR has been around quite some time and is covered in most finance texts. E.g., Brigham and Daves (2007)text indicates MIRR is a better measure than IRR of the project s true rate of return. However, neither Graham and Harvey (2001) nor Pike (1996) include MIRR in their surveys regarding the practice of capital budgeting, and when included in surveys, results indicate low use by practitioners. Kierluff(2008) suggests a lack of academic support has produced graduates relatively unaware of the power of MIRR. Kierluff(2008) describes MIRR as the more accurate measure of the attractiveness of an investment because the return depends not only on the investment itself, but also the return expected on the cash flows it generates. It is unclear why MIRR hasn t been embraced as the next best alternative to using NPV, as it is puzzling why IRR would be the primary decision criteria used in the case that multiple criteria conflict(burns and Walker, 1997). One issue may be that MIRR is not an internal rate of return in that a factor external to the project cash flows, the reinvestment rate, is used to determine the rate of return. Another contributing factor may be the confusion surrounding the academic debate regarding reinvestment rate assumptions. Carlson, Lawrence and Wort(1974) make this case: Note carefully that the most desirable solution to the reinvestment rate problem is not in selecting either the IRR or NPV methodology, depending on the situation. Rather, a much better solution is to explicitly select a consistent and accurate reinvestment rate for the alternatives under consideration (Solomon, 1969). 8 Although not advocated in this paper, the MIRR technique is sometimes calculated with two different rates for discounting and compounding. This may explain the different survey results for this item between NPV and MIRR

4 We address the issue of alternative reinvestment rate assumptions below, following the explanation of how to calculate MIRR. MIRR Calculation There is confusion about how to calculate MIRR with some ways to calculate it leading to decisions more consistent with NPV decisions than alternative calculations. This confusion may be inhibiting the adoption of MIRR technique. Ross, Westerfield, and Jordan (2011) describe three different possible MIRR calculations and note that detractors suggest the acronym should stand for meaningless internal rate of return. They point out that since MIRR is based on a modified set of cash flows it is no longer truly an internal rate of return, which is a legitimate criticism regarding the name of the technique. In the discount approach, all negative cash flows are discounted back to present, a reinvestment approach where all cash flows beyond the initial investment are compounded to the end of the project s life, and a combination approach where negative cash flows are discounted and positive cash flows are compounded. They also argue that it is irrelevant what is done with interim cash flows in that how cash flows are spent in the future does not affect their value today, yet their three approaches result in three different MIRRs. However, we believe how one deals with interim cash flows is key to understanding why NPV is a superior technique relative to IRR. Kierulff(2008), Brigham and Daves(2010), and Emery, Finnertyand Stowe (2007) and others describe a three step procedure for MIRR similar to Ross, Westerfield, and Jordan s combination approach. First, periodic cash flows must be determined for the project life. An issue occurs with simultaneously occurringinvestment funds (IF) and operating cash flows (OCF). McDaniel et al. (1988) argue that the flows should be separated and IF discounted at the marginal cost of capital because it measures the cost of meeting obligations to capital providers. Lin(1976) suggests two alternatives and both use the project s opportunity cost of capital as the relevant discount and compounding rate. Alternative oneis to net cash flows and net negative cash flows are discounted to time zero and net positive cash flows are compounded until the project s termination. This calculation is consistent with the Texas Instruments BA II Plus Professional and Microsoft Excel as long as the same reinvestment and discount rates are used and that rate is the cost of capital. See Ng s (2009) and Jones (2011) discussions of the inputs necessary to obtain the correct MIRR with the BAII Plus Professional calculator when the cash flows from the project have more than one sign change or when the cash flows start with multiple negative cash flows. With Lin s secondalternative, positive operating cash flows are invested and used to meet any subsequent cash outflows during the life of the project, resulting in a net cash flow. If positive operating cash flows are not available to offset subsequent cash flow, these funds must be obtained externally. Only net cash flows from external sources are discounted to time zero, and all other cash flows are compounded to the termination of the project. This is consistent with the notion that a firm would use internal sources of funds before going to external sources. Using Lin s second alternative for calculating the MIRR can yield different results from the first alternative and thus is not always consistent with the MIRR obtained either with the Texas Instruments BA II Plus Professional calculator or Microsoft Excel. Appendix I highlights the differences between Lin s two approaches in a Microsoft Excel Spreadsheet. Note differences only occur for those projects involving cash outflows that can be funded by previous cash inflows. When differences do occur, Lin s second alternative produces MIRRs greater (smaller) than those in Texas Instruments BA II Plus Professional calculator or Microsoft Excel when the project MIRR exceeds (is less than) the cost of capital. While Lin s two methods provide NPV-consistent accept and reject decisions for projects, the second approach is clearer to interpret according to Shull (1992, 1994). Shull(1992, p. 9) provides arguments for why Lin s latter approach, while resulting in the same investment decisions as the former approach, has interpretational advantages. The latter approach has an investment base that can be interpreted as the project s investment capital that could be invested in alternative opportunities and/or otherwise consumed. If the investment base has meaning, then the return calculated on this base is likewise meaningful. In turn, the terminal value considers all cash flows not in the project s investment base compounded at the cost of capital. Denoting a project s cash flows by a i, we define MIRR with Lin s latter approach and using Shull (1992)notation: MIRR=(TV/IB) (1/n) 1. (1) where

5 IB= The project s investment base ( and (2) TV=The project s terminal value (. (3) The variable n is the horizon period over which projects are evaluated, m is the last period with an unfunded(external) negative cash flow, and k is the project s cost of capital/opportunity cost of funds. TheIBcan be interpreted as the present value of the investment requirements funded by sources external to the project and the TVcan be thought of as the future value of all cash flows not considered in the investment base compounded to the project s termination. The MIRR equation can be rearranged into a more intuitive version as IB = TV / (1+MIRR) n (4) As with IRR, the MIRR is compared to the hurdle rate of the project cost of capital. As discussed in Shull (1992), the MIRR rule will provide the same investment decisions as the NPV rule when the same discount rate is used for both criteria. However, MIRR will not always rank mutually exclusive projects identically to NPV. An adjusted MIRR is needed for NPV-consistent rankings. Bernhard(1980) provides an incremental rate of return method, but its use may require many pair-wise comparisons. Shull (1992) provides two alternative adjusted MIRRs that do not require pair-wise comparisons, but may be more difficult to intuit and are likely beyond the scope of most undergraduate finance classes. If rates of return provide something beyond NPV-consistent rankings, these modified MIRRsshould be used; otherwise, NPV can be used for the optimal ranking of projects. The Reinvestment Rate Assumption The correct reinvestment rate is an issue of considerable debate. In terms of the reinvestment rate, McDaniel, et al. (1988), Brigham and Daves(2010), and Emery, Finnerty and Stowe (2007), and Hirshleifer(1958) all advocate the project cost of capital as the discount rate. The rationale provide by McDaniel, et al. (1988)is that if firms take all positive NPV projects, then positive cash flows generated by projects reduce the need for external financing and save the cost of capital that would be required on these funds. Kierulff(2008) points out that NPV and IRR also make reinvestment rate assumptions. When the project cost of capital differs from the firm s cost of capital, NPV assumes that future projects of similar risk to the project under consideration will be found. Thus, projects will have a discount rate more than or less than the firm s cost of capital depending on the risk of the individual project. More commonly recognized is the IRR reinvestment assumption that noninvestment cash flows from the project will be reinvested at the same rate of return as the IRR of the project under consideration. For positive NPV projects this means the NPV reinvestment assumption is more conservative than that of the IRR. The issue of capital rationing is important in the reinvestment rate assumption. Carlson, Lawrence and Wort(1974) point out that the implicit reinvestment assumption comes not from the decision criteria, but from the objective of the investor. If the investor s objective is the maximize shareholder wealth, then a reinvestment assumption is included in the calculation of discounted cash flow criteria. Bacon (1977) indicates that absent capital rationing, the correct reinvestment rate is the firm s cost of capital since positive cash flows save the firm the external cost of raising funds. However, if capital is severely rationed, the rate of return on future marginal investments should be used. Bacon acknowledges the complexity of estimating this future reinvestment rate. Brealy and Myers (2000) point out that most capital rationing is soft, or self-imposed. The limit on the capital budget is an aid to financial control adopted by management. Further, Brealy and Meyers (2000) argue hard rationing would be rare for corporations in the U.S. Gitman and Mercurio(1982) provide survey evidence consistent with this. While two-thirds of Fortune 1000 firms indicated they were confronted with capital rationing, the dominant cause was a debt constraint imposed by internal management. Zhang (1997) suggests this control is related to managerial incentives to shirk. The competition created internally among managers when capital is limited reduces their likelihood of under-reporting project quality. Given that hard rationing is uncommon, the project cost of capital is the appropriate discount rate in most cases and we will proceed with this assumption. This is consistent with Shull (1992), Bernhard (1989), McDaniel, et al. (1988) and others

6 Comparing NPV, IRR, and MIRR NPV is calculated as the present value of all cash flows for the project discounted at the project s cost of capital. IRR is calculated as the discount rate that makes NPV equal to zero. To compare these two techniques with MIRR, we will use three mathematical examples. Note that these examples involve nonconventional cash flows so multiple IRRs can result. For each example, the project cost of capital is 10%. Example NPV= IRR=16.2%, % MIRR= -8.71% In Example 1, we allow for investment funds beyond the initiation of the project that can be fully funded from prior positive operating cash flows. The positive operating cash flow of 500 at year 1 when reinvested at 10 percent, provides a future value in year 2 of 500*(1.1)=550. Thus, the additional investment of 500 in year 2 need not be funded externally. The 550 generated from the 500 in year 1 offsets the entire 500 required in year 2 and nets an extra 50 in year 2. After using all available cash inflows to fund subsequent cash outflows, the cash flow stream of the project can be rewritten as Thus, the project requires an investment of 60 that only nets 50. For this project, the investment base (IB) is 60 and the terminal value (TV) is 50. The MIRR per Equation (4) is the solution to 60 = 50/(1+MIRR) 2 which is -8.71%. The NPV of the project is calculated using the original cash flows as /(1.10) - 500/(1.10) 2 or using the rewritten cash flows as /(1.10) + 50/(1.10) 2. The NPV is the same whether the original or rewritten cash flows are used. This is because the rewritten cash flows are simply the original cash flows where some or all of the cash flows are either discounted or compounded at the project s cost of capital. This discounting and compounding at the project s cost of capital is entirely consistent with the NPV methodology. However, the IRR methodology discounts the cash flows at a rate that is not necessarily the project s cost of capital and thus is inconsistent with the NPV methodology. Both the NPV and the MIRR indicate that the project should be rejected. However, the IRRs for the project are 16.2% and 617.3%. Both of the IRRs are greater than the project cost of capital of 10%. Some users of IRR might accept the project based on the rule to accept when IRR exceeds the cost of capital, which would result in a different investment decision than the NPV and MIRR criteria. Example NPV= 1.32 IRR=9.61%, % MIRR=11.8% The only difference between Example 1 and Example 2 is that the project s initial cost has been reduced from 60 to 40. As before, the 500 received in year 1 is invested at 10% to entirely offset the required outlay in year 2 of 500. Thus, the cash flow stream of the project can be rewritten as The project has a small NPV of 1.32 = /(1.10) - 500/(1.10) 2 or /(1.10) 2. The IB for this project is 40 and the TVis 50. The MIRR is the solution to 40 = 50/(1+MIRR) 2 which is 11.8%. Both NPV and MIRR would indicate the project should be accepted in this example, while the IRR technique shows rates both above and below the hurdle rate of 10%. Since most calculators and spreadsheet programs

7 would stop after finding the first rate that works, it is likely some users of IRR would reject profitable projects like the above. Example NPV= IRR=354.47%, %MIRR= 14.80% Example 3 considers the case where an investment cash flow in year 2 cannot be fully funded from the prior reinvested operating cash flow. The positive operating cash flow of 400 at year 1 when reinvested at 10 percent, provides a future value in year 2 of 400*(1.1)=440. Since this is insufficient to fund the additional investment of 500 in year 2, additional external funds of 60 are required at the end of year 2. The cash flow stream of the project can be rewritten as The IB for this project includes the initial cost of 65 and the discount value of the extra 60 of external financing needed in the second year. Thus, IB=65+60/(1.10) 2 = The TV includes the 90 in year 3 compounded forward one year at the project s cost of capital of 10% and the 100 at year 4. Thus, thetv=199 = 90(1.10) It is worth noting that the IB and TV of this project can be calculated directly from Equations (2) and (3). For this project, the last period with a negative cash flow is period 2. Thus, m = 2. From Equation (2), IB = /(1.10) 500/(1.10) 2 = Also, from Equation (3), TV = 90(1.10) = 199. While calculating the IB and TV in this manner is simpler, the intuition gained from directly funding future negative cash flows with prior positive cash flows is lost. The NPV for the project is = /(1.10)-500/(1.10) 2 +90/(1.10) /(1.10) 4 and the MIRR is 14.80% = (199/114.59) 1/4-1. Because the investment horizon of the project is four years, there are four potential IRRs. By Descarte s Rule of Signs, because the project involves a quartic equation with three sign changes, there are either 1 or 3 positive roots and 1 negative root. In this case, there is one positive root, %, one negative root, , and two complex roots. How to interpret multiple IRRs especially when one root is negative and two roots are complex is unclear at best. The NPV and MIRR of the project would indicate the project is moderately profitable, while the one positive IRR would indicate it is an exceptionally good project. The IRR of % in this example is only achievable if the intermediate cash flows of 400 in year 1 and 90 in year 3 are reinvested at % and the cash flow of -500 at time 2 can be discounted at %. Neither of these possibilities is realistic. However, the 14.80% MIRR is achievable if the positive cash flows in years 1 and 3 can be reinvested at 10% and the negative cash flow in year 2 can be raised at a cost of 10%. This example dramatically reinforces why the MIRR is a more realistic measure of the return of a project than the IRR. As examples 1 and 2 indicate, the IRR technique does not lead to the same investment decisions as the NPV technique, while MIRR matches the NPV investment decisions. 9 Also, the magnitude of the IRR (large and positive) in examples where NPV is negative or moderate illustrates IRR s flaw as a rate of return measure and the primacy of the NPV technique. MIRR is the better measure of the expected return on a project. Furthermore, the calculation of MIRR highlights the importance of intermediate project cash flows, how they are used, and at what rate they are reinvested or discounted in these techniques. The user of MIRR must not only choose a reinvestment rate and use it to determine terminal value, but must calculate the investment base as the present value of net external funds necessary to support the project. Conclusion Finance texts are consistent in pointing out concerns about the IRR technique. In many cases, it is simply not a valid measure of return. When there are multiple IRRsormutually exclusive projects with differences in scale or timing, the IRR can lead to suboptimal investment decisions. Yet, in practice IRR is given priority when multiple decision techniques conflict (Burns and Walker, 1997). Given the proclivity 9 See Shull (1992) pages 9-10 for a proof

8 for the use, and misuse, of IRR in practice, more needs to done pedagogically to emphasize the superiority of the NPV and MIRR decision rules. MIRR is a more accurate measure of return in the case of projects with intermediate cash flows than IRR, is not subject to multiple rates of return, and provides investment decisions that are NPV-consistent with one exception. For mutually exclusive projects with different scales, MIRR will not provide the same ranking as NPV, thus the NPV rule or one of Shull s(1992)adjusted MIRR approaches must be used. Academics and finance texts should give clear preference of MIRR over IRR as it overcomes most of the problems of IRR, resulting in a more accurate rate of return in most cases. Furthermore, teaching the MIRR calculation allows reinforcement of the primacy of the NPV decision ruleby placing attentionon the treatment and investment of intermediate project cash flows. References Bacon, Peter The Evaluation of Mutually Exclusive Investments. Financial Management (Summer): Bernhard, Richard H Base Selection for Modified Rates of Turn and its Irrelevance for Optimal Project Choice. Engineering Economist 35(1) (Fall): Biondi, Yuru The Double Emergence of the Modified Internal Rate of Return: The Neglected Financial Work of Duvillard( ) in a Comparative Perspective. European Journal of the History of Economic Thought 13:3(September): Brealy, Richard and Steward Myers Fundamentals of Corporate Finance, 6 th edition, McGraw-Hill Higher Education. Brigham, Eugene F. and Philip R. Daves Intermediate Financial Management, 10 th edition, South- Western Cengage. Burns, Richard M. and Joe Walker Captial Budgeting Techniques Among the Fortune 500: A Rationale Approach. Managerial Finance 23(9):3-15. Carlson, C. Robert, Michael L. Lawrence, and Donald H. Wort Clarification of the Reinvestment Assumption in Capital Analysis. Journal of Business Research, 2(2) (April): Emery, Douglas R., John D. Finnerty, and John D. Stowe Corporate Financial Management, 3 rd edition, Pearson Prentice Hall. Gitman, Lawrence J. and Vincent A. Mercurio Cost of Capital Techniques Used by Major U.S. Firms: Survey and Analysis of Fortune s Financial Management (Winter): Graham, J. and C. Harvey The Theory and Practice of Corporate Finance: Evidence from the Field. Journal of Financial Economics 60: Hirshleifer, J On the Theory of Optimal Investment Decision. Journal of Political Economy LXVI:4. Jones, Steven T A Clarification Regarding the MIRR Shortcut Function in the Texas Instruments BAII Plus Professional. Journal of Financial Education 37 (Spring/Summer), Keane, S The Internal Rate of Return and the Reinvestment Fallacy. Abacus 15(1) (June): Kierulff, Herbert MIRR: A Better Measure. Business Horizons 51: Lin, Steven A The Modified Internal Rate of Return and Investment Criterion. Engineering Economist 21(Summer):

9 Lorie, H.J. and J.L. Savage Three Problems in Capital Rationing. Journal of Business 28(4): McDaniel, William R., William E. McCarty, and Kenneth A. Jessell Discounted Cash Flow with Explicit Reinvestment Rates: Tutorial and Extension. The Financial Review 23:3(August): Ng, Chee Error in the MIRR Estimate Using the Texas Instruments BAII Plus Professional. Journal of Financial Education35 (Fall): Phalippou, Ludovic The Hazards of Using IRR to Measure Performance: The Case of Private Equity. Journal of Performance Measurement 12(4) (Summer): Pike, Richard A Longitudinal Survey of Capital Budgeting Practices. Journal of Business Finance and Accounting 23(1) (January): Ross, Stephen, Randolph Westerfield, and Bradford Jordan Essentials ofcorporate Finance, 7 th ed., McGraw-Hill Irwin. Ryan, Patricia A. and Glenn P. Ryan Capital Budgeting Practices of the Fortune 1000: How Have Things Changed? Journal of Business and Management 8(4) (Fall): Shull, David Efficient Capital Project Selection Through a Yield-Based Capital Budgeting Technique. Engineering Economist 38 (1) (Fall):1-18. Shull, David Interpreting Rates of Return: A Modified Rate of Return Approach. Financial Practice and Education (Fall): Shull, David Overall Rates of Return: Investment Bases, Reinvestment Rates and Time Horizons. Engineering Economist 39(2) (Winter): Solomon, E The Arithmetic of Capital Budgeting Decisions. Journal of Business XXIX(2): Zhang, Guochang Moral Hazard in Corporate Investment and the Disciplinary Role of Voluntary Capital Rationing. Management Science 43(6) (June): APPENDIX Lin s MIRR Calculations Illustrated in an Excel Spreadsheet This appendix details how the same example can be worked using the two methods detailed in Lin (1976). A rationale via a numerical example will also be provided for the authors prefer one of these methods over the other. The appendix will use Example 3 from the Comparing NPV, IRR, and MIRR section of this paper. The tables below are designed to illustrate how the example would look when worked step-by-step in an Excel spreadsheet. For the example, an initial outlay of 65 will be required that will lead to subsequent cash flows of 400, -500, 90, and 100 over the next four years. The cost of capital for the project will be 10%

10 Using the first method in Lin (1976) negative cash flows after the initial investment are not funded using prior positive cash flows. The cash flows are entered in row two in columns B through F. Columns B through F are labeled as years 0 to 4. Because Flow outflows are not funded with previous cash inflows, the -500 cash flow in cell will need to be discounted back two yearsto the beginning of the project. With a discount rate of 10%, the result in cell B2 becomes = /(1.10) 2. The resulting cash flows can be seen in row three of the spreadsheet. 3 Flow Additionally, the positive cash flows in years 1,3, and 4 must be compounded to the end of the project. Thus, the entry in cell F4 becomes which is equivalent to 400*(1.10) 3 +90*(1.01)+100. The resulting final cash flows are in row four of the spreadsheet Flow MIRR 11.21% The MIRR for the project is then calculated as the rate that equates the initial outlay to an inflow of in four years. Thus, the MIRR is found from solving = *(1+MIRR) 4 MIRR = (731.40/478.22) 1/4-1 MIRR = 11.21% This is equivalent to using Excel s built in MIRR function on rows two, three, or four because the Excel calculation is equivalent to Lin s first method. Excels MIRR function requires three inputs. The syntax for the MIRR function in Excel is MIRR(Values, Finance_rate, Reinvest_rate) = MIRR(B4:F4,10%,10%) = 11.21%

11 As discussed in the MIRR Calculation section of this paper, Lin (1976) suggests using the project s cost of capital as both the relevant discount rate (Finance_rate in Excel) and the compounding rate (Reinvest_rate in Excel). Further, Shull (1992) notes that MIRR and NPV will give the same investment decision when the same discount rate is used for both criteria. Unfortunately, the final initial outlay of in this example does not have any economic meaning. Thus, the 11.21% MIRR based upon this base of is difficult to interpret. It is unlikely a firm would raise in capital to fund this project given the cash flow in year 1 of 400 can be used to offset most of the 500 expense in year 2. Using the second method in Lin (1976) the negative cash flows after the initial investment are funded using prior positive cash flows. The cash flows will be the same as in the prior example. The cash flows are entered in row two in columns B through F and these columns are labeled as years 0 to 4. Because Flow cash inflows can now be used to fund subsequent cash outflows, the 400 in year 1 will be used to partially offset the -500 in year 2. With a discount rate of 10%, the result in cell D2 becomes -60 = 400(1.10) The resulting cash flows can be seen in row three of the spreadsheet. 3 Flow Since there are no other cash outflows preceded by positive inflows, one continues as with the previous technique. First, cash outflows are discounted back to the beginning of the project. The cash flow of -60 in cell D3 is discounted back to the beginning of the project. The result in cell B3 becomes = /(1.10) 2 and the resulting cash flows can be seen in row four Flow

12 The final step is to compound all remaining cash inflows to the end of the project. Thus, the 90 in cell E4 must be compounded forward one year. The result in cell F4 becomes 199 = 90(1.10) The resulting final cash flows can be seen in row 5 of the spreadsheet Flow MIRR 14.80% The MIRR for the project is then calculated as the rate that equates the initial outlay to an inflow of 199 in four years. Thus, the MIRR is found from solving = *(1+MIRR) 4 MIRR = (199.00/114.59) 1/4-1 MIRR = 14.80% The advantage of this method is that the initial outlay of now has an economic meaning. The represents the investment base of the project. This is the project s required investment capital that could be invested in alternative opportunities and/or otherwise consumed. Using the MIRR syntax on the modified cash flows in rows three, four, or five, yields the 14.8% using Lin s second method. This is not true for the cash flows in row two because the cash flow in D2 is not fully funded by the cash flow in C