JOURNAL OF ECONOMICS AND FINANCE EDUCATION ∙ Volume 12∙ Number 1 ∙Summer 2013
Teaching MIRR to Improve Comprehension of
Investment Performance Evaluation Techniques
R. Brian Balyeat1, Julie Cagle2, and Phil Glasgo3
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.4A 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.5NPV 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
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 248.
6
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
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JOURNAL OF ECONOMICS AND FINANCE EDUCATION ∙ Volume 12∙ Number 1 ∙Summer 2013
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 1975. 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).
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In terms of “realistic reinvestment rate”, NPV received a composite average of 3.5 versus MIRR which
received 2.0.8Formal 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 1998-99 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.
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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 ai, we define MIRR with Lin’s latter approach and using Shull
(1992)notation:
MIRR=(TV/IB)(1/n) – 1.
(1)
where
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JOURNAL OF ECONOMICS AND FINANCE EDUCATION ∙ Volume 12∙ Number 1 ∙Summer 2013
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.
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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 1
0
1
2
|-------|-------| NPV= -18.68
-60 500 -500
IRR=16.2%, 617.13% 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
0
1
2
|-------|-------|
-60
0
50
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)2which is -8.71%. The NPV of the project is
-18.68 calculated using the original cash flows as -60 + 500/(1.10) - 500/(1.10)2 or using the rewritten
cash flows as -60 + 0/(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 2
0
1
2
|-------|-------| NPV= 1.32
-40 500 -500
IRR=9.61%, 1040.39% 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
0
1
2
|-------|-------|
-40
0
50
The project has a small NPV of 1.32 = -40 + 500/(1.10) - 500/(1.10)2 or -40 + 50/(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
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would stop after finding the first rate that works, it is likely some users of IRR would reject profitable
projects like the above.
Example 3
0
1
2
3
4
|-------|-------|-------|-------|
NPV= 21.33
IRR=354.47%,-133.13%MIRR= 14.80%
-65 400 -500 90 100
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
0
1
2
3
4
|-------|-------|-------|-------|
-65
0
-60 90 100
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=114.59. 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) + 100.
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 = -65 + 400/(1.10) – 500/(1.10)2 = 114.59. Also, from Equation (3), TV = 90(1.10) + 100 = 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 21.33 = -65+400/(1.10)-500/(1.10)2+90/(1.10)3+100/(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,
354.47%, one negative root, -133.13, 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 354.47% in this example is only achievable if the intermediate cash flows of 400 in year 1
and 90 in year 3 are reinvested at 354.47% and the cash flow of -500 at time 2 can be discounted at
354.47%. 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.9Also, 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.
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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.
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Biondi, Yuru. 2006. “The Double Emergence of the Modified Internal Rate of Return: The Neglected
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Burns, Richard M. and Joe Walker.1997. “Captial Budgeting Techniques Among the Fortune 500: A
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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%.
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JOURNAL OF ECONOMICS AND FINANCE EDUCATION ∙ Volume 12∙ Number 1 ∙Summer 2013
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.
1
2
A
Year
Cash
B
0
-65
C
1
400
D
2
-500
E
F
3
90
4
100
Because
Flowoutflows 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 -478.22 = -65 + -500/(1.10)2. The resulting cash flows can be seen in row three of the spreadsheet.
1
2
3
A
Year
Cash
B
0
-65
-478.22
C
1
400
400
D
E
2
-500
0
3
90
90
F
4
100
100
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 731.40 which is equivalent to 400*(1.10) 3+90*(1.01)+100. The resulting
final cash flows are in row four of the spreadsheet.
1
2
3
4
5
A
Year
Cash
Flow
MIRR
B
0
-65
-478.22
-478.22
11.21%
C
1
400
400
0
D
2
-500
0
0
E
3
90
90
0
F
4
100
100
731.40
The MIRR for the project is then calculated as the rate that equates the -478.22 initial outlay to an
inflow of 731.40 in four years. Thus, the MIRR is found from solving
731.40 = 478.22*(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%.
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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 478.22 in this example does not have any economic meaning.
Thus, the 11.21% MIRR based upon this base of 478.22 is difficult to interpret. It is unlikely a firm would
raise 478.22 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.
1
2
A
Year
Cash
B
0
-65
C
1
400
D
2
-500
E
3
90
F
4
100
Because
Flowcash 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) + -500. The resulting cash flows can be seen in row three of the spreadsheet.
1
2
3
A
Year
Cash
B
0
-65
-65
C
1
400
0
D
2
-500
-60
E
3
90
90
F
4
100
100
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 -114.59 =
-65 + -60/(1.10)2 and the resulting cash flows can be seen in row four.
1
2
3
4
A
Year
Cash
Flow
B
0
-65
-65
-114.59
C
1
400
0
0
D
2
-500
-60
0
E
3
90
90
90
F
4
100
100
100
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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) + 100. The
resulting final cash flows can be seen in row 5 of the spreadsheet.
1
2
3
4
5
6
A
Year
Cash
Flow
MIRR
B
0
-65
-65
-114.59
-114.59
14.80%
C
1
400
0
0
0
D
2
-500
-60
0
0
E
3
90
90
90
0
F
4
100
100
100
199
The MIRR for the project is then calculated as the rate that equates the -114.59 initial outlay to an
inflow of 199 in four years. Thus, the MIRR is found from solving
199.00
= 114.59*(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 114.59 now has an economic meaning. The
114.59 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 C2.
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