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A simple intuitive NPV-IRR consistent ranking
Accepted Manuscript Title: A simple intuitive NPV-IRR consistent ranking Authors: Moshe Ben-Horin, Yoram Kroll PII: DOI: Reference:
S1062-9769(17)30009-1 http://dx.doi.org/doi:10.1016/j.qref.2017.01.004 QUAECO 1000
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Received date: Revised date: Accepted date:
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Quarterly
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Please cite this article as: Ben-Horin, Moshe., & Kroll, Yoram., A simple intuitive NPV-IRR consistent ranking.Quarterly Review of Economics and Finance http://dx.doi.org/10.1016/j.qref.2017.01.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A simple intuitive NPV-IRR consistent ranking
Moshe Ben-Horin * and Yoram Kroll **
* Moshe Ben-Horin, Ono Academic College. ** Corresponding author: Yoram Kroll, Ono Academic College, and Ruppin Academic Center Israel. Tel.: +972-9-9523024; Mobile: +972-50-5718153, E-mail address: [email protected]
Highlights The NPV-IRR ranking conflict can be avoided by using a simple and intuitive cash flow .dominance rule Abstract Numerous recent studies have revisited the issue of the potential conflicting NPVIRR ranking of competing investment projects. Most have suggested procedures that resolve the conflict after performing an iso-NPV modification of at least one of the cash flows. However, none has provided a general sufficient condition that guarantees the absence of NPV-IRR ranking conflict. We define dominance between cash flow streams and show that if the streams are conventional, dominance of one stream over another ascertains no NPV-IRR ranking conflict. While dominance among original cash flows may be relatively rare, iso-NPV crossrisk adjustment and iso-NPV modification of one cash flow stream may easily
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reveal such dominance even if the original projects are subject to different risks. The resulting implication is a practical, simple and economically intuitive procedure that guarantees consistent NPV-IRR ranking, while minimizing the implicit or explicit distortions of the original competing cash flow streams and their IRRs.
Keywords: NPV, IRR, MIRR, AIRR, Cross-Risk Adjustment, Cash Flow Dominance JEL Classification: G31I.
I. Introduction Empirical surveys indicate that despite the well-known deficiencies of the Internal Rate of Return (IRR), it remains a prominent measure, along with the Net Present Value (NPV), for ranking the attractiveness of investment projects 1. The potential NPV-IRR ranking conflict of conventional cash flow streams in the context of capital budgeting has been dealt with in the literature for decades 2. Corporate
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See for example, Petry and Sprow; (1993) Payne, Heath and Gale (1999); Graham and Harvey (2002); Bierman and Smidt (2007) The conflict stems from flaws of the IRR measure as a rate of return on investment. Magni (2013), for example, lists 18 such flaws.
finance textbooks propose that in case of an NPV-IRR ranking conflict one should make the choice between the competing projects by comparing the IRR of the incremental cash flow with the hurdle rate (i.e., the cost of capital). This solution reduces the problem of ranking two competing projects, down to a question of accepting or rejecting one incremental cash flow stream. However, this solution is problematic. First, if the risk levels of the competing projects differ, one must discount each project’s cash flow using a different risk adjusted cost of capital. In such a case, one must apply a new cost of capital in the process of discounting the incremental cash flow stream. Estimating the appropriate new hurdle rate for discounting the incremental cash flow stream is neither trivial theoretically nor is it straightforward empirically. In addition, the incremental cash flow stream conceals information about the cash flow streams of the individual projects, and it is prone to have more than one sign variation in which case the incremental cash flow stream may have multiple IRRs or no real IRR even if the two original projects have conventional cash flow streams. In this paper, we present a simple general procedure by which the NPV-IRR potential ranking conflict of conventional investments3 disappears. The procedure is general because it does not merely deal with projects of different lengths and different sizes, but also with projects of different risks, a situation often neglected in the relevant literature. Generally, the procedure involves two steps. If the risks of the projects are equal, the first step is redundant:
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By definition, conventional cash flow streams have only one sign variation. In conventional investments, the sign variation is from negative (investment in the early period(s)) to positive (returns in later periods).
1. Cash flow streams A and B are first turned into equivalent-risk assets by applying a cross-risk adjustment (as defined below) to one of the cash flow streams, 2. An iso-NPV modification is performed on one of the cash flow streams, B for example, so as to make the cash flows of B equal to the cash flow of the competing project, A, at all points of time except one, such as at time t=04.
We show that applying the above procedure guarantees no conflict between the ranking of the resulting IRR and NPV values.
The rest of the paper is organized as follows: Section II presents a literature review, Section III presents our main claim, a proof and a discussion, Section IV presents a numerical example and the conclusions are presented in Section V.
II. Literature Review Recent studies re-examine the relationships between NPV and IRR. Hazen (2003) presented the concept of investment stream and used it to formulate a mathematical relationship between NPV and IRR. For a given cash flow stream, CFt in periods t = 1,2,…,n, Hazen defines the investment stream as follows: (1) ct (k ) (1 k )ct 1 (k ) CFt
for t 1,...., n 1
where k is the internal rate of return (k ≡ IRR), c0 (k ) CF0 and cn (k ) 0 . The present value of the investment stream, c, is:
4
4
In fact, the modification of B’s cash flows may be made equal to A’s cash flows at all points of time except one, arbitrary, point of time. In the example below, we demonstrate two such modifications.
n
(2) PV (c) ct (k )(1 r ) t t 0
where r is the cost of capital. Hazen then proves that the NPV is equal to: IRR r (3) NPV (r ) PV (c) 1 r
The economic intuition behind Hazen’s formulation is that the NPV is expressed as if it is that of a one-period project that has an initial investment equal to PV (c) , and a rate of return equal to IRR (of the original cash flow stream). The NPV is
IRR r equal to the one period discounted spread times the overall investment 1 r size, as measured by the present value of the investment stream, PV (c) . Magni (2010) presented the concept of average rate of return, AIRR. He showed that even when the periodic rates of return across the project’s life and the periodic costs of capital, vary (so that kt replaces k and rt replaces r), a similar relationship holds when the internal rate of return and the cost of capital in Eq. (3) are replaced by the weighted average rate of return as in Eq. (4): (4)
AIRR r NPV (r ) PV (c) 1 r n
n
t
t 1
t 1
j 1
Where AIRR ct 1ktt PV (c) , r ct 1 rt t PV (c) , and t (1 r j ) 1 n
for t=1,2,…,n, and PV (c) ct t where 0 1 . t 0
Equation (4) is similar to Equation (3) in that it too, expresses the NPV as if it is in a one period setting. Eqs. (3) and (4) may be used to resolve the NPV-IRR ranking problem, since, NPV(A) is greater than NPV(B) if and only if:
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(5)
AIRRA rA AIRRB rB PVA (c) 1 rA 1 rB
PVB (c)
Magni (2010) correctly claims that as the initial investment is c0 CF0 and the ending investment is equal to zero ( cn 0 ), then, any assumption about intermediate values of the stream at t=1,2…,n-1 will not change the present value of the project but will change the size of the terms on the right hand side of Eq.(4). Thus, one can always set PVA (c) PVB (c) and then compare projects by their average one period discounted premium. Osborne (2010) uses entirely different approach to formulate the relationship between IRR and NPV. He avoided the use of equation (3) that implicitly converts the multi-period cash flows to one-period modified cash flows. Instead, Osborne correctly claims that IRR is deficiently used for ranking comparisons relatively to NPV ranking, since to achieve correct ranking, one has to employ all the solutions (including the unreal ones) of the zero NPV polynomial equation. Namely, he considers all n real (positive or negative) as well as complex roots of the zero NPV polynomial function, where ranking projects by IRR disregard the other n-1 solutions. In other words, Osborne claims that the multiple IRR solutions are not a pit. On the contrary, he uses all real and complex solutions and obtains an exact formulation of the relationships between all the IRR values and NPV. Osborne claims that one may calculate the NPV as the real product between the initial investment outlay and the multiplicative of all mark-ups between all the IRR solutions and the cost of capital (see Osborne’s (2010) Equation 6 p. 236). However, Osborn is also aware of the fact that his analysis does not arm investors with a simple rule for a practical resolution of the NPV- IRR ranking conflict.
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Lin’s (1976) Modified IRR (MIRR) is a well-known measure that avoids potential multiple sign variations of a cash flow stream that may lead to either no real IRR or multiple IRR solutions5. The remedy of the MIRR is achieved by compounding all the positive interim cash flows to the end of the project’s life and discounting all the negative interim cash flows to the beginning of the project’s life. The MIRR thus transforms the original multi-period project into a one-“long”-period project by “emptying out” the interim cash flows and leaving only one negative initial cash flow and one positive ending cash flow6. Since the compounding and discounting of the cash flows are carried out at the cost of capital, the NPV of the project is preserved. Shull (1992), Kierulff (2008), (2010), Balyeat, Cagle and Glasgo (2013) claim that iso-NPV modifications of the cash flows that equate the initial investment of the competing projects resolve the NPV-IRR ranking conflict. However, their solution is limited to the case in which the two competing projects have the same risk level. Lin eliminates the ranking problem by modifying the project’s cash flow stream. Magni, on the other hand, keeps the same cash flow stream and eliminates the ranking problem by modifying the capital value stream. However, once the modified cash flows are forced to have the same initial investment the resulting cash flows and rates of return may lose their intuitive appeal, as they may be far different from the original cash flows and the original rates of return. In this paper we ask: Is there a general sufficient condition that guarantees no NPVIRR ranking conflict? We then define a simple dominance condition of one cash
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Ben-Horin and Kroll (2012) claim that the occurrence of multiple real as well as non-real IRRs is very rare in practice. Therefore, the problem of multiple IRRs is practically negligible. The MIRR is essentially the geometric mean (by the number of the project’s periods) of the rate of return on the “long” period of the project’s life.
flow over another and show that this dominance condition is a sufficient condition that guarantees the elimination of the NPV-IRR ranking conflict7. When comparing two projects for determining their wealth creation using rates of return, we perform a very simple iso-NPV modification to the cash flow of only one of the competing cash flows, and obtain economically intuitive rates of return that rank the competing projects in the same order as NPV ranking. The paper proceeds as follows: Section III presents our claim. Section IV provides numerical examples and demonstrates the simplicity and intuitive appeal of our suggested dominance procedure. Section V presents our conclusions. III. The main claim As mentioned above, the recent studies on the subject of resolving the NPV-IRR ranking conflict, specified or generated conditions under which the NPV-IRR ranking conflict is indeed resolved. We specify a general sufficient condition that ascertains no NPV-IRR ranking conflict.
Claim Assume that project A and project B, each has a single8 real IRR. Then, the following dominance relationship provides a sufficient condition that guarantees the elimination of the NPV-IRR ranking conflict: 1. A’s cash flow9 is equal to or higher than B’s cash flow at all points of time and it is strictly higher than B’s cash flow at least at one point of time. 2. The level of A’s cash flow risk is equal to or less than that of B’s cash flow risk at all points of time.
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The condition eliminates the possible ranking conflict of conventional investments. It also eliminates the possible conflict of ranking none conventional projects that have a single real IRR. In case there are more than one real IRR, then a preliminary modification of the cash flow is required. As we argued in a previous paper (2012), multiple rates of return are indeed very rare in practice. 8 If original cash flows are non-conventional, one may perform an iso-NPV cash flow modification and convert the cash flow into a conventional one prior to applying the procedure suggested in the claim. 9 For brevity, we refer to “cash flows” rather than “expected cash flows”. Clearly, investment decisions are based on expectations for future cash flows. Note that since we deal with investment projects, a higher cash flow at time t=0 means a lower initial investment amount.
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Proof To prove sufficiency we have to show that if A’s cash flow dominates B’s cash flow, then IRR( A) IRR( B) and NPV ( A) NPV ( B) . If indeed A’s cash flow dominates B’s cash flow, A’s cash flow is higher or equal to B’s cash flow at all points of time and thus necessarily IRR( A) IRR( B) . If, in addition, A’s cash flow risk is lower or equal to that of B’s cash flow risk at all point of time, then A’s cash flow is discounted using a lower or equal discount rate than that used to discount B’s cash flow, and therefore NPV ( A) NPV ( B) must hold as well. Clearly, in the absence of dominance, a conflict between NPV and IRR ranking may or may not exist. Discussion Two points are worth noting: First, no NPV-IRR ranking conflict, may exist even when dominance does not exist. Our claim asserts that dominance is a sufficient condition to ascertain the absence of a ranking conflict. Second, dominance between original cash flows is perhaps rare in practice. However, below we show that even when the original cash flow streams do not exhibit dominance relationship, one can easily apply an iso-NPV modification of only one of the competing cash flow streams to create dominance and obtain simple and economically intuitive rates of returns that are consistent with NPV ranking. Generating dominance between two projects with different level of risk The following two algorithmic steps may easily generate dominance between the cash flows of any two competing projects, A and B. The first step: equating the risk of the cash flow streams
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If the risks of the original cash flow streams are not equal, one may easily adjust the cash flow of one of the projects to account for the differential risk without affecting its NPV. Let the NPVs of the cash flows of projects A and B, be as in Eq.(6) and Eq. (7), respectively: 1
(6)
t NPV ( A) CFt (1 rj ( A)) t 0 j 0
1
(7)
t NPV ( B) CFt (1 rj ( B)) t 0 j 0
n
A
n
B
where CFt A denote the cash flows of project A, so that A (CF0A , CF1A ,..., CFnA ) is the cash flow stream of project A, and similarly CFt B denote the cash flows of project B so that B (CF0B , CF1B ,..., CFnB ) is the cash flow stream of project B, and r0 ( A) r0 ( B) 0 . We now obtain for each t a new cross-risk adjusted cash
flow of project B (which we denote B ), so that CFt B has the same risk as CFt A , while preserving B’s net present value. To achieve this, we set: t t (8) CFt B (1 r j ( B)) 1 CFt B (1 r j ( A)) 1 j 1 j 1
so that:
(9)
CFt
B
t 1 r j ( A) CFt 1 r ( B ) j 1 j B
1 r ( B) 1 r ( A), then B t
Suppose the following holds for a given t:
t
j
j 1
j
j 1
t
may be thought of as the original CFt B minus a risk premium that adjusts for the
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risk differential between that of CFt A and that of CFt B . Controlling for the future risk difference, we are now able to compare the alternative cash flow streams as if they had the same risk level, and discount the cash flows of both projects using the same costs of capital. In particular, let B* (CF0B , CF1B* ,..., CFnB* ) be B’s cash flow stream already modified for risk (i.e., first step has been accomplished). The second step: iso-NPV cash flow modification of one cash flow stream After controlling for the risk difference, we now turn B* into a modified A, here denoted as A where A [CF0B* PV (CF ), CF1A ,..., CFnA ] , and where n
PV (CF ) PV (CF A CF B ) (CFt A CFt B )(1 r ) 1 , is the present value
t 0
of the incremental project (A – B*), and where A can be viewed as the sum of two cash flow streams: A [CF0B* PV (CF ), CF1A ,..., CFnA ] (CF0B* , CF1B* ,..., CFnB* )
Note
[ PV (CF ), CF1A CF1B* , CF2A CF2B* ,..., CFnA CFnB* ] A can be described as the sum of B* and a modified incremental project, where
the incremental initial investment, CF0A CF0B , is replaced by the present value
of the incremental project, PV (CF ) PV (CF A CF B ) , which implies that its IRR is equal to its cost of capital. We denote this incremental cash flow by A : A [ PV (CF ), CF1A CF1B* , CF2A CF2B* ,..., CFnA CFnB* ]
and we may write: A B * A . In fact, this shows the link between our iso-NPV modification and the incremental IRR method and we will further demonstrate it below in Section IV. Note that the iso-NPV modification turns the comparison
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between A and B into a comparison between A and A, which implies that the comparison is between B * and B * A . In essence, the choice depends on A . Discussion The economic intuition behind A’s iso-NPV modification in this way is that an arbitrage opportunity clearly emerges in favor of one of the projects. The question that naturally arises is as follows: Why is it that only under the dominance conditions can we ascertain no NPV-IRR ranking conflict? The answer is rooted in the basic deficiency of IRR relative to NPV. The claim that NPV(A) > NPV(B) is basically equivalent to an arbitrage opportunity claim. Namely, the preference of project A over B by NPV implies that A creates more wealth than B and therefore A must possess a higher premium over its cost10. In this context, the claim that IRR(A) > IRR(B), is entirely different from the claim that NPV(A) > NPV(B) for two reasons: 1. A rate of return per se, implies nothing about wealth creation unless it relates to defined levels of invested capital its timing and risk. 2. The alternative costs of capital (including the appropriate risk premiums inherent in them) are not part of the IRR calculations, and thus by comparing projects by their IRRs, one essentially ignores the alternative investment opportunities. In light of these two reasons, the condition IRR(A) > IRR(B), does not necessarily lead to an arbitrage opportunity unless A dominates B. The only case where we can guarantee higher wealth creation of A over B based on the rate of return ranking, is when the cost of capital is indeed irrelevant or redundant with respect to the ranking
10
If the rights to the project are marketable, A’s price will be higher than B’s in a competitive market.
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task. We proved that it is guaranteed to be irrelevant if and only if one cash flow stream dominates the other, in which case the selection problem is trivial. The well-known Modified IRR (MIRR) generates an iso-NPV project with one outflow at t=0 and one inflow at t=n. The MIRR is simply the rate of return of an iso-NPV equivalent cash flow created by discounting all negative cash flows to time t=0, and compounding all positive cash flows to the end of the last period, i.e., t=n. Since the discounting and compounding are done at the cost of capital, the NPV is preserved, and the modified cash flows, consists of only one negative initial outlay and one positive ending cash flow. The iso-NPV modified cash flow resolves the potential problem of multiple real IRRs solution or no real IRR solution. However, as suggested by previous studies, [Shull (1992), Kierulff (2008), Balyeat, Cagle and Glasgow (2013)], to solve the potential NPV-IRR ranking conflict, one has to re-modify the equivalent cash flows and equate either the initial or the ending cash flows. As shown by Kierulff (2008) and others, for projects with equal risk levels, if the initial investment of the two competing modified cash flows are equated, then MIRR(A) > MIRR(B) if and only if NPV(A) > NPV(B). However, Kierulff’s MIRR modification, in which the initial outlays are equated, does not deal with the case of different risk levels of the two competing projects. To guarantee no NPV-IRR ranking conflict between projects with different risk levels, one has to apply our suggested cross-risk adjustment procedure prior to performing Kierulff’s modification to generate the MIRR. In doing this, one creates the dominance conditions that guarantee the elimination of NPV-IRR ranking conflict. Magni’s (2010) AIRR is computed without performing cash flows modifications. However, it may be viewed as if the original multi-period cash flow stream is 13
modified into a one period setting. Thus, under the AIRR paradigm, AIRR(A) > AIRR(B) is equivalent to NPV(A) > NPV(B) if the projects are of equal risk and if the capital streams c are PV-equivalent: PV A (c) PVB (c) . However, owing to Eq. (4), one may interpret a multi-period project as a one-period project whose beginning-of-period cash flow is PV(c) and the end-of-period cash flow is PV(c)(1+AIRR). From this point of view, when facing two projects A and B, we might consider the cash flow streams of the one-period modified projects
Am ( PV A (c), PV A (1 AIRR ( A))) and
B m ( PVB (c), PVB (1 AIRR ( B))) .
Then, AIRR ( A) AIRR ( B) is equivalent to NPV(A) > NPV(B) if either the initial capital of A m and B m are equal (i.e., PV A (c) PVB (c) ) or their ending cash flows are equal (i.e., PV A (1 AIRR ( A)) PVB (1 AIRR ( B)) ). If these conditions are met, one of the cash flow streams clearly dominates the other as per our general claim. If the projects risks are different, Magni suggests to consider the one period discounted risk premiums (see Eq. 5). Following an iso-NPV re-modification of the cash flow stream in order to equate the risk and the initial or ending cash flow values, the AIRR and MIRR provide projects ranking that are consistent with NPV ranking. Then both adhere to our wealth maximization dominance rule. Resolving the NPV- IRR ranking conflict by MIRR may lead to modifying at least one of the cash flow streams in a way that could be remote from the original cash flow, and as such may seem unintuitive, mainly to practitioners. Thus, the goal of minimizing the iso-NPV modifications seems plausible and appealing. Accordingly, our proposed method is consistent with wealth maximization as well as with AIRR and MIRR, but is potentially much more intuitive than MIRR . As for AIRR, it requires modifying the invested capital 14
stream, and it is not clear how to do it. In particular, if one makes use of the oneperiod projects A m and B m , as explained above, it is not clear how one should change the initial invested capital or the ending cash flows of A m and B m .
IV. A Numerical Example The example below shows that following a risk adjustment and following the equating modification of the initial, (or ending) cash flow, the well-known MIRR resolve to the NPV-IRR ranking conflict as it is based on a modified cash flow stream that is a special case of our dominance criterion. As noted above, the AIRR measure, does not require any cash flow modification, but may nonetheless be viewed as an IRR measure of a modified one period cash flow stream. Viewed through this lens one may interpret it as a special case of our dominance condition. INSERT TABLE 1 HERE The cash flows of projects A and B and their NPV and IRR values are presented in Table 1 lines 2 and 3, respectively. We assume that project B is riskier than project A, and accordingly, the cost of capital of project A is 5% and that of B is 8%. We denote project B after its cash flows were cross-risk adjusted to A’s level of risk, as B. Line 4 of Table 1 shows B ’s cash flows11. For example, B ’s cross-risk adjusted cash flow in year 3 was calculated as follows: 32.6456 1.05 1.083 30 .
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For convenience, B’s cross-risk adjusted cash flows were chosen as round numbers. The original cash flows were therefore “grossed up” by reversing the cross-risk adjustment procedure. For example, B’s original cash flow in year 3 was calculated as follows: 30 1.08 1.053 32.6456 . The result is that B’s cross-risk adjusted cash flow in year 3 is 30.
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The ranking conflict of the original cash flow streams is evident: IRR(A)=17.72% < IRR(B)=42.30%, while NPV(A)=38.44 > NPV(B)=31.00. Line 5 shows project A’s invested capital at the end of each year of the project’s life, calculated using Equation (1).12 The present value of A’s invested capital is given in line 6 and is calculated as follows: 4
(10)
PV A (c) t 0
ct ( A) 317.43 (1 r ) t
Note that the cash flow stream of A in line 1 is iso-NPV modified in line 7 to become the cash flow of a one-period investment project with an initial investment of -317.4 at t=0 and a single return of 373.67 at t=1. The modification retains the original NPV of 38.44 and the one period yield of 17.72%. Line 8 presents the invested capital of the original project B and line 9 shows that the present value (using r=8%) of the invested capital is equal to 97.61. Line 10 presents the invested capital of B. While Bs invested capital is lower than that of B’s at all points of time (except at t=0 and t=4), the present value of the invested capital stream (line 11) is identical to that of line 9. This is due to the lower discount rate used in calculating the present values (8% was used for line 8 and 5% was used for line 10). Line 12 presents the one period modified cash flows of the cross-risk adjusted B project that retain the original net present value and rate of return measures:
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The invested capital as defined in Equation (1), employs a simplifying assumption that the IRR is the rate of return in each of the projects periods, hence we obtain AIRR=IRR. This is tantamount to assuming certain "deprecation" values across periods. When the rates of returns are not equal across periods, the AIRR is generally not equal to the project’s IRR [see Magni (2010)]. Different periodic depreciation as well as different periodic costs of capital, yield different present values of invested capital. By Magni's approach, one may modify the cash flows to equate the present value of the invested capital of the two competing projects and if the cost of capital is the same for the two competing projects, then NPV(A) > NPV(B) if and only if AIRR(A) > AIRR(B). As demonstrated in this study, for equal risk alternatives, Magni’s implied modifications of capitals can be reframed in terms of modifications of cash flows so as to be consistent with our dominance conditions.
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NPV(B)=NPV(B)=31 and IRR(B)=38.34%. Lines 7 and 12 demonstrate that the ranking conflict is yet not resolved even though project B’s cash flows were crossrisk adjusted. The one period modified cash flows of the competing projects in lines 7 and 12 differ at two time points (-317.43 versus -97.61 at t=0 and 373.67 versus 135.03 at t=1). Therefore, the single period cash flows in lines 7 and 12 do not satisfy the dominance condition, and indeed the ranking problem persists. To assure the absence of ranking conflict, we must equate the single period cash flows at either t=0 or at t=1 thereby creating dominance. In line 13 and 14 we use isoNPV re-modifications of the one period modified cash flow of B in order to equate its initial investment to that of A’s (as shown in line 13) or to equate the ending one period modified cash flow of B to that of A’s (as shown in line 14). The result is that A’s one period cash flows dominate B ’s one period cross-risk adjusted cash flows by both NPV and IRR: They both signal the preference of A over B and thus over B:
NPV ( A) 38.44 31.00 NPV ( B ) NPV ( B) and (by using the cash flows in line 13):
IRR( A) AIRR ( A) 17.72% 15.25% MAIRR ( B ) Where MAIRR ( B ) is the AIRR ( B ) following the iso-NPV rescaling of the initial invested capital in B* to become equal to that of A’s. One can also solve the ranking conflict by using the cash flows in line 14 rather than those of line 13 to obtain: AIRR ( A) 17.72% 15.02% MAIRR ( B )
Note that the choice to equate the beginning cash flows of the projects or the ending cash flows is arbitrary. Indeed, line 14 of Table 1 presents B*’s cash flow after it was iso-NPV modified so that its ending (rather than the beginning) cash flow is 17
equal to that of A. The IRR ranking once again does not conflict the NPV ranking. The resulting IRR is now 15.02% and it generates equal NPV to that of our previous calculation as it now applies to a higher invested capital. Namely,
0.1502 0.05 0.1525 0.05 324.88 317.67 31 1.05 1.05 As Magni (2010) (2013) showed, the AIRR approach does not require the modification of the cash flow stream. Indeed, the AIRR of the cash flow stream of B may be calculated as follows:
AIRR ( B ) 0.05
30
20 5 30 25 2 3 1.05 1.05 1.05 1.05 4 1.05 0.1525 317.43
In this equation, the cost of capital (5%) is that of project A and of the cross-risk adjusted project B, and the denominator (317.43) is equal to the present value of A’s invested capital. Although the AIRR does not require the modification of a project’s cash flow after it has been risk-adjusted, it may be viewed as if the cash flow stream has been modified by compressing the project's multi-period cash flow stream into an equivalent single-period iso-NPV investment. Ranking projects by AIRR necessarily leads to correct value creation ranking if the original cash flows are of equal risk or the cash flows were cross-risk adjusted prior to calculating the AIRR and, in addition, the present value of the invested capital of the competing projects are equated prior to calculating the AIRR.
The MIRR solution is presented in Table 2. INSERT TABLE 2 HERE
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Lines 2 to 4 of Table 2 simply replicate the corresponding lines of Table 1. Lines 5 to 7 present the modified cash flows (MCFs) and the respective MIRRs. Once again, the NPV-IRR ranking conflict is not resolved, as MCF(A) differs from MCF(B*) at two points in time (t=0 and t=4), and we obtain MIRR(A) = 13.23% < MIRR(B*) = 23.24% where NPV(A) = 38.44 > NPV(B*) = 31.00. To resolve the ranking conflict we perform a re-modification of B*’s cash flow (see line 8), and equate B* 's initial investment with that of A’s to be 109.07 while preserving B* 's NPV. The result is that the ending inflow of the re-modified MCF(B*) is only 170.25 compared with MCF(A), which equals 179.31. Only at this stage when the competing modified cash flows adhere to our dominance rule (compare lines 5 and 8) the MIRR rank the projects consistently with their value creation (i.e., MIRR(B*) is only 11.78% while MIRR(A) is 13.23%).
The simple dominance resolution of the NPV-IRR Ranking Conflict
In Table 3 line 4 we present the cash flow stream of A”. At points in time t = 1,2,3,4, the cash flows are equal to the difference CFt A CFt B* . At t=0, the initial investment of A” is equal to the present value of its own cash flow stream at times t = 1,2,3,4: 5 30 40 20 CF0A" 77.45 2 3 4 1.05 1.05 1.05 1.05
The NPV and IRR of the incremental cash flow stream A” are 0 and 5%, respectively. INSERT TABLE 3 HERE
19
Line 5 presents A s cash flow stream, which is identical to A’s cash flows at points of time t=1,2,3,4. A’s cash flow stream is equal to the sum of the cash flow streams of B* (line 3) and A (line 4) at all points of time: A [107.45,40,10,60,70] [30,20,5,30,25] [77.45,20,5,30,45]
The result is an iso-NPV modified cash flow of B*, with future cash flow stream identical to that of A’s but the initial investment (107.45) is higher than A’s (100). It follows that more initial capital must be invested in A to generate the same future cash flow stream as A’s. Thus, the NPV of A is lower than that of A by 7.45 and the IRR of A (14.71%) is lower than that of A (17.72%). Below we show that the one period discounted premium of A ’s IRR ( 14.71%) over the cost of capital (5%) is a weighted average of the one period discounted premiums of the IRRs of B* (38.34%) and A'' (5%) over the cost of capital. The weights are based on the present value of the invested capital at t=0 of A' (335.15), and the presents values of the invested capital of streams of A'' and B* (237.54 and 97.61 respectively) which sum up to 335.15. Namely,
( IRRA r ) ( IRRA" r ) ( IRRB* r ) PV (cA | r ) PV (cA" | r ) PV (cB* | r ) 1 r 1 r 1 r ( IRRA 0.05) (0.05 0.05) (0.3834 0.05) 335.15 237.54 97.61 1.05 1.05 1.05 Thus:
IRR A 0.05 (0.3834 0.05)
97.61 0.1471 14.71% 335.15
V. Conclusions NPV ranks investment projects by their amount of value creation. Comparing projects by their IRRs may be inconsistent with value creation, hence, NPV-IRR
21
ranking conflict is generally possible. However, if one of the competing alternatives has the same or higher expected cash flows at all points of time and the same or lower risk than the other alternative, then such ranking conflict is not possible. We define such a relationship between projects’ cash flow streams as “dominance” of the preferred cash flow over the other. In general, dominance relationships between original cash flow streams are very rare and conflicting NPVIRR ranking are frequent. However, dominance conditions may be easily generated by using a two-step iso-NPV modification that eliminates the conflict. First, one of the alternative streams is cross-risk adjusted using a procedure that equates its risk to the risk of the alternative stream. Second, an iso-NPV modification is performed on one of the cash flow streams, so that all the future cash flows from time t=1 to time t=n, are equated to the cash flows of the alternative project. When the competing projects have identical risk level (either originally or after performing the cross-risk adjustment), the AIRR as well as the MIRR resolve the ranking conflict, but as we demonstrated, they are, in fact only special cases that adhere to the proposed dominance rule. The MIRR, is often based on cash flows far remote from the original cash flows and thus may be quite different from the original cash flow's IRR. In addition, it is not independent of the cost of capital and thus cannot serve as hurdle rates for the cost of capital. Even if we disregard the fact that alternative rates of return, AIRR and MIRR, may be quite different from the original IRR, we are still left with a major deficiency that they could mislead investors to regard them as cutoff rates for the cost of capital. While our procedure also uses the cost of capital to modify one of the cash flow streams, the economic intuition behind the comparison of the rate of return of the
21
modified stream with the rate of return of the unmodified stream is straightforward. Considering the alternative rates of return in the market (i.e., the cost of capital), one of the cash flow streams emerges as generating the same future cash flows at the same or lower risk, but with lower initial investment than the competing alternative. Such dominance is intuitive and appealing and is well reflected in the higher rate of return of the dominating project.
Acknowledgments We wish to acknowledge the very helpful comments and suggestions of an anonymous referee. We also wish to thank the Ono Research Institute in Finance (ORIF) for its generous support, the seminar participants at Ono Academic College for their comments and suggestions on an earlier draft of the paper. Remaining errors are our responsibility.
References Balyeat, R.B., J. Cagle and P. Glasgo (2013), "Teaching MIRR to Improve Comprehension of Investment Performance Evaluation Techniques". Journal of Economics and Finance education 12, 39-50. Bierman, H. and S. Smidt, 2007, "The Capital Budgeting Decision", 9th Ed. New York, N.Y. Routledge. Ben-Horin, M. and Y. Kroll (2010) "IRR, NPV and PI Ranking: Reconciliation". Advances in Financial Education 8, 88-105. Ben-Horin, M., Kroll, Y. (2012). "The Limited Relevance of the Multiple IRRs". The Engineering Economist 57, 101 –118.
22
Graham and Harvey, 2002, "How Do CFOs Make Capital Budgeting and Capital Structure Decisions?", Journal of Applied Corporate Finance, 15, 8-22. Hazen, G.B. (2003) "A New Perspective on Multiple Internal Rates of Return". The Engineering Economist 48, 31–51. Lin, Steven A. 1976. “The Modified Internal Rate of Return and Investment Criterion.” Engineering Economist 21, 237-247 Kierulff, H. (2008), "MIRR: A Better Measure". Business horizon 51, 321-329. Magni, C.A. (2010), "Average Internal Rate of Return and Investment Decisions: A New Perspective". The Engineering Economist 55, 150–180. Magni, C.A. (2013), "The Internal Rate-of-Return approach and the AIRR paradigm: A refutation and a corroboration”, The Engineering Economist 58, 73–111. Petry, G. H., J. Sprow,(1993)"The theory and practice of finance in the 1990s", The Quarterly Review of Economics and Finance 33,359-381. Payne, J. D., W. C. Heath and I.R. Gale, (1999), “Comparative Financial practice in the USA and Canada: Capital Budgeting and Risk Assessment techniques”, Financial Practice and Education, 9, 16-24. Osborne, M. (2010) A resolution to the NPV-IRR debate? Quarterly Review of Economics and Finance, 50, 234-239. Shull, David, (1992), “Efficient Capital Project Selection Through a Yield-Based Capital Budgeting Technique", Engineering Economist 38 (1) (fall):1-18.
23
Table 1: Ranking projects A and B using Magni's AIRR. Project B’s cash flows are presented before (B) and after (B) they were cross-risk adjusted*. 1
Time (Years) t
0
1
2
3
4
IRR/AIRR
NPV
2
CF(A)
-100
40
-10
60
70
17.72%
38.44 (r=5%)
3
CF(B)
-30
20.57
-5.28
32.64
27.98
42.30%
31.00 (r=8%)
4
CF(B)
-30
20
-5
30
25
38.34%
31
5
Invested Capital in A
100
77.72
101.49
59.46
0
6
Present value of A’s invested capital (r=5%)
317.43
7
A's equivalent iso-NPV one-period cash flow
-317.43
373.67
17.72%
38.44
8
Invested Capital in B
30
22.12
36.76
19.66
0
9
Present value of B’s invested capital (r=8%)
97.61
10
Invested capital in B
30
21.5
34.75
18.07
0
11
Present value of B invested capital (r=5%)
97.61
12
B ‘s iso-NPV one-period cash flow
-97.61
135.03
38.34%
31
13
Iso NPV modification of the one-period equivalent cash flow of B by equating its initial invested capital to that of the one-period cash flow of A
-317.43
365.85
15.25%
31
14
Iso NPV modification of the one-period equivalent cash flow of B by equating its ending cash flow to that of the one-period equivalent cash flow of A
-324.88
373.67
15.02%
31
In this Table, we compute the standardized AIRR using the AIRR-consistent modification of cash flows which generates and
24
B m . The same results would be obtained using Magni’s (2013) Eqs. (29) and (30).
Am
Table 2: Ranking Projects A and B by Cross-Risk Adjusting and Re-modifying MIRR 1
Time (Years) t
0
1
2
3
4
IRR
2
CF(A)
-100
40
-10
60
70
17.72%
38.44 (r=5%)
3
CF(B)
-30
20.57
-5.28
32.64
27.98
42.30%
31.00 (r=8%)
4
CF(B)
-30
20
-5
30
25
38.34%
31.00 (r=5%)
5
MCF(A)
-109.07
0
0
0
179.31
13.23%
38.44 (r=5%)
6
MCF(B)
-34.54
0
0
0
89.15
26.76%
31.00 (r=8%)
7
MCF(B)
-34.54
0
0
0
79.65
23.24%
31.00 (r=5%)
8
Re-modification of MCF(B) that equates its initial outlay to that of MCF(A)
-109.07
0
0
0
170.25
11.78%
31 (r=5%)
25
MIRR
NPV
Table 3: Equating the Future Cash Flow Stream of B* with that of A 1
Time (Years) t
0
1
2
3
4
IRR
NPV
2
CF(A)
-100
40
-10
60
70
17.72%
38.44 (r=5%)
3
CF(B)
-30
20
-5
30
25
38.34%
31.00 (r=5%)
4
CF(A’’)
-77.45
20
-5
30
45
5.00%
0.00 (r=5%)
5
CF(A’) = CF(B*) + CF(A”)
-107.45
40
-10
60
70
14.71%
31.00 (r=5%)
26
S1062-9769(17)30009-1 http://dx.doi.org/doi:10.1016/j.qref.2017.01.004 QUAECO 1000
To appear in:
The
Received date: Revised date: Accepted date:
10-3-2016 7-12-2016 6-1-2017
Quarterly
Review
of
Economics
and
Finance
Please cite this article as: Ben-Horin, Moshe., & Kroll, Yoram., A simple intuitive NPV-IRR consistent ranking.Quarterly Review of Economics and Finance http://dx.doi.org/10.1016/j.qref.2017.01.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A simple intuitive NPV-IRR consistent ranking
Moshe Ben-Horin * and Yoram Kroll **
* Moshe Ben-Horin, Ono Academic College. ** Corresponding author: Yoram Kroll, Ono Academic College, and Ruppin Academic Center Israel. Tel.: +972-9-9523024; Mobile: +972-50-5718153, E-mail address: [email protected]
Highlights The NPV-IRR ranking conflict can be avoided by using a simple and intuitive cash flow .dominance rule Abstract Numerous recent studies have revisited the issue of the potential conflicting NPVIRR ranking of competing investment projects. Most have suggested procedures that resolve the conflict after performing an iso-NPV modification of at least one of the cash flows. However, none has provided a general sufficient condition that guarantees the absence of NPV-IRR ranking conflict. We define dominance between cash flow streams and show that if the streams are conventional, dominance of one stream over another ascertains no NPV-IRR ranking conflict. While dominance among original cash flows may be relatively rare, iso-NPV crossrisk adjustment and iso-NPV modification of one cash flow stream may easily
1
reveal such dominance even if the original projects are subject to different risks. The resulting implication is a practical, simple and economically intuitive procedure that guarantees consistent NPV-IRR ranking, while minimizing the implicit or explicit distortions of the original competing cash flow streams and their IRRs.
Keywords: NPV, IRR, MIRR, AIRR, Cross-Risk Adjustment, Cash Flow Dominance JEL Classification: G31I.
I. Introduction Empirical surveys indicate that despite the well-known deficiencies of the Internal Rate of Return (IRR), it remains a prominent measure, along with the Net Present Value (NPV), for ranking the attractiveness of investment projects 1. The potential NPV-IRR ranking conflict of conventional cash flow streams in the context of capital budgeting has been dealt with in the literature for decades 2. Corporate
1
2
2
See for example, Petry and Sprow; (1993) Payne, Heath and Gale (1999); Graham and Harvey (2002); Bierman and Smidt (2007) The conflict stems from flaws of the IRR measure as a rate of return on investment. Magni (2013), for example, lists 18 such flaws.
finance textbooks propose that in case of an NPV-IRR ranking conflict one should make the choice between the competing projects by comparing the IRR of the incremental cash flow with the hurdle rate (i.e., the cost of capital). This solution reduces the problem of ranking two competing projects, down to a question of accepting or rejecting one incremental cash flow stream. However, this solution is problematic. First, if the risk levels of the competing projects differ, one must discount each project’s cash flow using a different risk adjusted cost of capital. In such a case, one must apply a new cost of capital in the process of discounting the incremental cash flow stream. Estimating the appropriate new hurdle rate for discounting the incremental cash flow stream is neither trivial theoretically nor is it straightforward empirically. In addition, the incremental cash flow stream conceals information about the cash flow streams of the individual projects, and it is prone to have more than one sign variation in which case the incremental cash flow stream may have multiple IRRs or no real IRR even if the two original projects have conventional cash flow streams. In this paper, we present a simple general procedure by which the NPV-IRR potential ranking conflict of conventional investments3 disappears. The procedure is general because it does not merely deal with projects of different lengths and different sizes, but also with projects of different risks, a situation often neglected in the relevant literature. Generally, the procedure involves two steps. If the risks of the projects are equal, the first step is redundant:
3
3
By definition, conventional cash flow streams have only one sign variation. In conventional investments, the sign variation is from negative (investment in the early period(s)) to positive (returns in later periods).
1. Cash flow streams A and B are first turned into equivalent-risk assets by applying a cross-risk adjustment (as defined below) to one of the cash flow streams, 2. An iso-NPV modification is performed on one of the cash flow streams, B for example, so as to make the cash flows of B equal to the cash flow of the competing project, A, at all points of time except one, such as at time t=04.
We show that applying the above procedure guarantees no conflict between the ranking of the resulting IRR and NPV values.
The rest of the paper is organized as follows: Section II presents a literature review, Section III presents our main claim, a proof and a discussion, Section IV presents a numerical example and the conclusions are presented in Section V.
II. Literature Review Recent studies re-examine the relationships between NPV and IRR. Hazen (2003) presented the concept of investment stream and used it to formulate a mathematical relationship between NPV and IRR. For a given cash flow stream, CFt in periods t = 1,2,…,n, Hazen defines the investment stream as follows: (1) ct (k ) (1 k )ct 1 (k ) CFt
for t 1,...., n 1
where k is the internal rate of return (k ≡ IRR), c0 (k ) CF0 and cn (k ) 0 . The present value of the investment stream, c, is:
4
4
In fact, the modification of B’s cash flows may be made equal to A’s cash flows at all points of time except one, arbitrary, point of time. In the example below, we demonstrate two such modifications.
n
(2) PV (c) ct (k )(1 r ) t t 0
where r is the cost of capital. Hazen then proves that the NPV is equal to: IRR r (3) NPV (r ) PV (c) 1 r
The economic intuition behind Hazen’s formulation is that the NPV is expressed as if it is that of a one-period project that has an initial investment equal to PV (c) , and a rate of return equal to IRR (of the original cash flow stream). The NPV is
IRR r equal to the one period discounted spread times the overall investment 1 r size, as measured by the present value of the investment stream, PV (c) . Magni (2010) presented the concept of average rate of return, AIRR. He showed that even when the periodic rates of return across the project’s life and the periodic costs of capital, vary (so that kt replaces k and rt replaces r), a similar relationship holds when the internal rate of return and the cost of capital in Eq. (3) are replaced by the weighted average rate of return as in Eq. (4): (4)
AIRR r NPV (r ) PV (c) 1 r n
n
t
t 1
t 1
j 1
Where AIRR ct 1ktt PV (c) , r ct 1 rt t PV (c) , and t (1 r j ) 1 n
for t=1,2,…,n, and PV (c) ct t where 0 1 . t 0
Equation (4) is similar to Equation (3) in that it too, expresses the NPV as if it is in a one period setting. Eqs. (3) and (4) may be used to resolve the NPV-IRR ranking problem, since, NPV(A) is greater than NPV(B) if and only if:
5
(5)
AIRRA rA AIRRB rB PVA (c) 1 rA 1 rB
PVB (c)
Magni (2010) correctly claims that as the initial investment is c0 CF0 and the ending investment is equal to zero ( cn 0 ), then, any assumption about intermediate values of the stream at t=1,2…,n-1 will not change the present value of the project but will change the size of the terms on the right hand side of Eq.(4). Thus, one can always set PVA (c) PVB (c) and then compare projects by their average one period discounted premium. Osborne (2010) uses entirely different approach to formulate the relationship between IRR and NPV. He avoided the use of equation (3) that implicitly converts the multi-period cash flows to one-period modified cash flows. Instead, Osborne correctly claims that IRR is deficiently used for ranking comparisons relatively to NPV ranking, since to achieve correct ranking, one has to employ all the solutions (including the unreal ones) of the zero NPV polynomial equation. Namely, he considers all n real (positive or negative) as well as complex roots of the zero NPV polynomial function, where ranking projects by IRR disregard the other n-1 solutions. In other words, Osborne claims that the multiple IRR solutions are not a pit. On the contrary, he uses all real and complex solutions and obtains an exact formulation of the relationships between all the IRR values and NPV. Osborne claims that one may calculate the NPV as the real product between the initial investment outlay and the multiplicative of all mark-ups between all the IRR solutions and the cost of capital (see Osborne’s (2010) Equation 6 p. 236). However, Osborn is also aware of the fact that his analysis does not arm investors with a simple rule for a practical resolution of the NPV- IRR ranking conflict.
6
Lin’s (1976) Modified IRR (MIRR) is a well-known measure that avoids potential multiple sign variations of a cash flow stream that may lead to either no real IRR or multiple IRR solutions5. The remedy of the MIRR is achieved by compounding all the positive interim cash flows to the end of the project’s life and discounting all the negative interim cash flows to the beginning of the project’s life. The MIRR thus transforms the original multi-period project into a one-“long”-period project by “emptying out” the interim cash flows and leaving only one negative initial cash flow and one positive ending cash flow6. Since the compounding and discounting of the cash flows are carried out at the cost of capital, the NPV of the project is preserved. Shull (1992), Kierulff (2008), (2010), Balyeat, Cagle and Glasgo (2013) claim that iso-NPV modifications of the cash flows that equate the initial investment of the competing projects resolve the NPV-IRR ranking conflict. However, their solution is limited to the case in which the two competing projects have the same risk level. Lin eliminates the ranking problem by modifying the project’s cash flow stream. Magni, on the other hand, keeps the same cash flow stream and eliminates the ranking problem by modifying the capital value stream. However, once the modified cash flows are forced to have the same initial investment the resulting cash flows and rates of return may lose their intuitive appeal, as they may be far different from the original cash flows and the original rates of return. In this paper we ask: Is there a general sufficient condition that guarantees no NPVIRR ranking conflict? We then define a simple dominance condition of one cash
5
6
7
Ben-Horin and Kroll (2012) claim that the occurrence of multiple real as well as non-real IRRs is very rare in practice. Therefore, the problem of multiple IRRs is practically negligible. The MIRR is essentially the geometric mean (by the number of the project’s periods) of the rate of return on the “long” period of the project’s life.
flow over another and show that this dominance condition is a sufficient condition that guarantees the elimination of the NPV-IRR ranking conflict7. When comparing two projects for determining their wealth creation using rates of return, we perform a very simple iso-NPV modification to the cash flow of only one of the competing cash flows, and obtain economically intuitive rates of return that rank the competing projects in the same order as NPV ranking. The paper proceeds as follows: Section III presents our claim. Section IV provides numerical examples and demonstrates the simplicity and intuitive appeal of our suggested dominance procedure. Section V presents our conclusions. III. The main claim As mentioned above, the recent studies on the subject of resolving the NPV-IRR ranking conflict, specified or generated conditions under which the NPV-IRR ranking conflict is indeed resolved. We specify a general sufficient condition that ascertains no NPV-IRR ranking conflict.
Claim Assume that project A and project B, each has a single8 real IRR. Then, the following dominance relationship provides a sufficient condition that guarantees the elimination of the NPV-IRR ranking conflict: 1. A’s cash flow9 is equal to or higher than B’s cash flow at all points of time and it is strictly higher than B’s cash flow at least at one point of time. 2. The level of A’s cash flow risk is equal to or less than that of B’s cash flow risk at all points of time.
7
The condition eliminates the possible ranking conflict of conventional investments. It also eliminates the possible conflict of ranking none conventional projects that have a single real IRR. In case there are more than one real IRR, then a preliminary modification of the cash flow is required. As we argued in a previous paper (2012), multiple rates of return are indeed very rare in practice. 8 If original cash flows are non-conventional, one may perform an iso-NPV cash flow modification and convert the cash flow into a conventional one prior to applying the procedure suggested in the claim. 9 For brevity, we refer to “cash flows” rather than “expected cash flows”. Clearly, investment decisions are based on expectations for future cash flows. Note that since we deal with investment projects, a higher cash flow at time t=0 means a lower initial investment amount.
8
Proof To prove sufficiency we have to show that if A’s cash flow dominates B’s cash flow, then IRR( A) IRR( B) and NPV ( A) NPV ( B) . If indeed A’s cash flow dominates B’s cash flow, A’s cash flow is higher or equal to B’s cash flow at all points of time and thus necessarily IRR( A) IRR( B) . If, in addition, A’s cash flow risk is lower or equal to that of B’s cash flow risk at all point of time, then A’s cash flow is discounted using a lower or equal discount rate than that used to discount B’s cash flow, and therefore NPV ( A) NPV ( B) must hold as well. Clearly, in the absence of dominance, a conflict between NPV and IRR ranking may or may not exist. Discussion Two points are worth noting: First, no NPV-IRR ranking conflict, may exist even when dominance does not exist. Our claim asserts that dominance is a sufficient condition to ascertain the absence of a ranking conflict. Second, dominance between original cash flows is perhaps rare in practice. However, below we show that even when the original cash flow streams do not exhibit dominance relationship, one can easily apply an iso-NPV modification of only one of the competing cash flow streams to create dominance and obtain simple and economically intuitive rates of returns that are consistent with NPV ranking. Generating dominance between two projects with different level of risk The following two algorithmic steps may easily generate dominance between the cash flows of any two competing projects, A and B. The first step: equating the risk of the cash flow streams
9
If the risks of the original cash flow streams are not equal, one may easily adjust the cash flow of one of the projects to account for the differential risk without affecting its NPV. Let the NPVs of the cash flows of projects A and B, be as in Eq.(6) and Eq. (7), respectively: 1
(6)
t NPV ( A) CFt (1 rj ( A)) t 0 j 0
1
(7)
t NPV ( B) CFt (1 rj ( B)) t 0 j 0
n
A
n
B
where CFt A denote the cash flows of project A, so that A (CF0A , CF1A ,..., CFnA ) is the cash flow stream of project A, and similarly CFt B denote the cash flows of project B so that B (CF0B , CF1B ,..., CFnB ) is the cash flow stream of project B, and r0 ( A) r0 ( B) 0 . We now obtain for each t a new cross-risk adjusted cash
flow of project B (which we denote B ), so that CFt B has the same risk as CFt A , while preserving B’s net present value. To achieve this, we set: t t (8) CFt B (1 r j ( B)) 1 CFt B (1 r j ( A)) 1 j 1 j 1
so that:
(9)
CFt
B
t 1 r j ( A) CFt 1 r ( B ) j 1 j B
1 r ( B) 1 r ( A), then B t
Suppose the following holds for a given t:
t
j
j 1
j
j 1
t
may be thought of as the original CFt B minus a risk premium that adjusts for the
11
risk differential between that of CFt A and that of CFt B . Controlling for the future risk difference, we are now able to compare the alternative cash flow streams as if they had the same risk level, and discount the cash flows of both projects using the same costs of capital. In particular, let B* (CF0B , CF1B* ,..., CFnB* ) be B’s cash flow stream already modified for risk (i.e., first step has been accomplished). The second step: iso-NPV cash flow modification of one cash flow stream After controlling for the risk difference, we now turn B* into a modified A, here denoted as A where A [CF0B* PV (CF ), CF1A ,..., CFnA ] , and where n
PV (CF ) PV (CF A CF B ) (CFt A CFt B )(1 r ) 1 , is the present value
t 0
of the incremental project (A – B*), and where A can be viewed as the sum of two cash flow streams: A [CF0B* PV (CF ), CF1A ,..., CFnA ] (CF0B* , CF1B* ,..., CFnB* )
Note
[ PV (CF ), CF1A CF1B* , CF2A CF2B* ,..., CFnA CFnB* ] A can be described as the sum of B* and a modified incremental project, where
the incremental initial investment, CF0A CF0B , is replaced by the present value
of the incremental project, PV (CF ) PV (CF A CF B ) , which implies that its IRR is equal to its cost of capital. We denote this incremental cash flow by A : A [ PV (CF ), CF1A CF1B* , CF2A CF2B* ,..., CFnA CFnB* ]
and we may write: A B * A . In fact, this shows the link between our iso-NPV modification and the incremental IRR method and we will further demonstrate it below in Section IV. Note that the iso-NPV modification turns the comparison
11
between A and B into a comparison between A and A, which implies that the comparison is between B * and B * A . In essence, the choice depends on A . Discussion The economic intuition behind A’s iso-NPV modification in this way is that an arbitrage opportunity clearly emerges in favor of one of the projects. The question that naturally arises is as follows: Why is it that only under the dominance conditions can we ascertain no NPV-IRR ranking conflict? The answer is rooted in the basic deficiency of IRR relative to NPV. The claim that NPV(A) > NPV(B) is basically equivalent to an arbitrage opportunity claim. Namely, the preference of project A over B by NPV implies that A creates more wealth than B and therefore A must possess a higher premium over its cost10. In this context, the claim that IRR(A) > IRR(B), is entirely different from the claim that NPV(A) > NPV(B) for two reasons: 1. A rate of return per se, implies nothing about wealth creation unless it relates to defined levels of invested capital its timing and risk. 2. The alternative costs of capital (including the appropriate risk premiums inherent in them) are not part of the IRR calculations, and thus by comparing projects by their IRRs, one essentially ignores the alternative investment opportunities. In light of these two reasons, the condition IRR(A) > IRR(B), does not necessarily lead to an arbitrage opportunity unless A dominates B. The only case where we can guarantee higher wealth creation of A over B based on the rate of return ranking, is when the cost of capital is indeed irrelevant or redundant with respect to the ranking
10
If the rights to the project are marketable, A’s price will be higher than B’s in a competitive market.
12
task. We proved that it is guaranteed to be irrelevant if and only if one cash flow stream dominates the other, in which case the selection problem is trivial. The well-known Modified IRR (MIRR) generates an iso-NPV project with one outflow at t=0 and one inflow at t=n. The MIRR is simply the rate of return of an iso-NPV equivalent cash flow created by discounting all negative cash flows to time t=0, and compounding all positive cash flows to the end of the last period, i.e., t=n. Since the discounting and compounding are done at the cost of capital, the NPV is preserved, and the modified cash flows, consists of only one negative initial outlay and one positive ending cash flow. The iso-NPV modified cash flow resolves the potential problem of multiple real IRRs solution or no real IRR solution. However, as suggested by previous studies, [Shull (1992), Kierulff (2008), Balyeat, Cagle and Glasgow (2013)], to solve the potential NPV-IRR ranking conflict, one has to re-modify the equivalent cash flows and equate either the initial or the ending cash flows. As shown by Kierulff (2008) and others, for projects with equal risk levels, if the initial investment of the two competing modified cash flows are equated, then MIRR(A) > MIRR(B) if and only if NPV(A) > NPV(B). However, Kierulff’s MIRR modification, in which the initial outlays are equated, does not deal with the case of different risk levels of the two competing projects. To guarantee no NPV-IRR ranking conflict between projects with different risk levels, one has to apply our suggested cross-risk adjustment procedure prior to performing Kierulff’s modification to generate the MIRR. In doing this, one creates the dominance conditions that guarantee the elimination of NPV-IRR ranking conflict. Magni’s (2010) AIRR is computed without performing cash flows modifications. However, it may be viewed as if the original multi-period cash flow stream is 13
modified into a one period setting. Thus, under the AIRR paradigm, AIRR(A) > AIRR(B) is equivalent to NPV(A) > NPV(B) if the projects are of equal risk and if the capital streams c are PV-equivalent: PV A (c) PVB (c) . However, owing to Eq. (4), one may interpret a multi-period project as a one-period project whose beginning-of-period cash flow is PV(c) and the end-of-period cash flow is PV(c)(1+AIRR). From this point of view, when facing two projects A and B, we might consider the cash flow streams of the one-period modified projects
Am ( PV A (c), PV A (1 AIRR ( A))) and
B m ( PVB (c), PVB (1 AIRR ( B))) .
Then, AIRR ( A) AIRR ( B) is equivalent to NPV(A) > NPV(B) if either the initial capital of A m and B m are equal (i.e., PV A (c) PVB (c) ) or their ending cash flows are equal (i.e., PV A (1 AIRR ( A)) PVB (1 AIRR ( B)) ). If these conditions are met, one of the cash flow streams clearly dominates the other as per our general claim. If the projects risks are different, Magni suggests to consider the one period discounted risk premiums (see Eq. 5). Following an iso-NPV re-modification of the cash flow stream in order to equate the risk and the initial or ending cash flow values, the AIRR and MIRR provide projects ranking that are consistent with NPV ranking. Then both adhere to our wealth maximization dominance rule. Resolving the NPV- IRR ranking conflict by MIRR may lead to modifying at least one of the cash flow streams in a way that could be remote from the original cash flow, and as such may seem unintuitive, mainly to practitioners. Thus, the goal of minimizing the iso-NPV modifications seems plausible and appealing. Accordingly, our proposed method is consistent with wealth maximization as well as with AIRR and MIRR, but is potentially much more intuitive than MIRR . As for AIRR, it requires modifying the invested capital 14
stream, and it is not clear how to do it. In particular, if one makes use of the oneperiod projects A m and B m , as explained above, it is not clear how one should change the initial invested capital or the ending cash flows of A m and B m .
IV. A Numerical Example The example below shows that following a risk adjustment and following the equating modification of the initial, (or ending) cash flow, the well-known MIRR resolve to the NPV-IRR ranking conflict as it is based on a modified cash flow stream that is a special case of our dominance criterion. As noted above, the AIRR measure, does not require any cash flow modification, but may nonetheless be viewed as an IRR measure of a modified one period cash flow stream. Viewed through this lens one may interpret it as a special case of our dominance condition. INSERT TABLE 1 HERE The cash flows of projects A and B and their NPV and IRR values are presented in Table 1 lines 2 and 3, respectively. We assume that project B is riskier than project A, and accordingly, the cost of capital of project A is 5% and that of B is 8%. We denote project B after its cash flows were cross-risk adjusted to A’s level of risk, as B. Line 4 of Table 1 shows B ’s cash flows11. For example, B ’s cross-risk adjusted cash flow in year 3 was calculated as follows: 32.6456 1.05 1.083 30 .
11
For convenience, B’s cross-risk adjusted cash flows were chosen as round numbers. The original cash flows were therefore “grossed up” by reversing the cross-risk adjustment procedure. For example, B’s original cash flow in year 3 was calculated as follows: 30 1.08 1.053 32.6456 . The result is that B’s cross-risk adjusted cash flow in year 3 is 30.
15
The ranking conflict of the original cash flow streams is evident: IRR(A)=17.72% < IRR(B)=42.30%, while NPV(A)=38.44 > NPV(B)=31.00. Line 5 shows project A’s invested capital at the end of each year of the project’s life, calculated using Equation (1).12 The present value of A’s invested capital is given in line 6 and is calculated as follows: 4
(10)
PV A (c) t 0
ct ( A) 317.43 (1 r ) t
Note that the cash flow stream of A in line 1 is iso-NPV modified in line 7 to become the cash flow of a one-period investment project with an initial investment of -317.4 at t=0 and a single return of 373.67 at t=1. The modification retains the original NPV of 38.44 and the one period yield of 17.72%. Line 8 presents the invested capital of the original project B and line 9 shows that the present value (using r=8%) of the invested capital is equal to 97.61. Line 10 presents the invested capital of B. While Bs invested capital is lower than that of B’s at all points of time (except at t=0 and t=4), the present value of the invested capital stream (line 11) is identical to that of line 9. This is due to the lower discount rate used in calculating the present values (8% was used for line 8 and 5% was used for line 10). Line 12 presents the one period modified cash flows of the cross-risk adjusted B project that retain the original net present value and rate of return measures:
12
The invested capital as defined in Equation (1), employs a simplifying assumption that the IRR is the rate of return in each of the projects periods, hence we obtain AIRR=IRR. This is tantamount to assuming certain "deprecation" values across periods. When the rates of returns are not equal across periods, the AIRR is generally not equal to the project’s IRR [see Magni (2010)]. Different periodic depreciation as well as different periodic costs of capital, yield different present values of invested capital. By Magni's approach, one may modify the cash flows to equate the present value of the invested capital of the two competing projects and if the cost of capital is the same for the two competing projects, then NPV(A) > NPV(B) if and only if AIRR(A) > AIRR(B). As demonstrated in this study, for equal risk alternatives, Magni’s implied modifications of capitals can be reframed in terms of modifications of cash flows so as to be consistent with our dominance conditions.
16
NPV(B)=NPV(B)=31 and IRR(B)=38.34%. Lines 7 and 12 demonstrate that the ranking conflict is yet not resolved even though project B’s cash flows were crossrisk adjusted. The one period modified cash flows of the competing projects in lines 7 and 12 differ at two time points (-317.43 versus -97.61 at t=0 and 373.67 versus 135.03 at t=1). Therefore, the single period cash flows in lines 7 and 12 do not satisfy the dominance condition, and indeed the ranking problem persists. To assure the absence of ranking conflict, we must equate the single period cash flows at either t=0 or at t=1 thereby creating dominance. In line 13 and 14 we use isoNPV re-modifications of the one period modified cash flow of B in order to equate its initial investment to that of A’s (as shown in line 13) or to equate the ending one period modified cash flow of B to that of A’s (as shown in line 14). The result is that A’s one period cash flows dominate B ’s one period cross-risk adjusted cash flows by both NPV and IRR: They both signal the preference of A over B and thus over B:
NPV ( A) 38.44 31.00 NPV ( B ) NPV ( B) and (by using the cash flows in line 13):
IRR( A) AIRR ( A) 17.72% 15.25% MAIRR ( B ) Where MAIRR ( B ) is the AIRR ( B ) following the iso-NPV rescaling of the initial invested capital in B* to become equal to that of A’s. One can also solve the ranking conflict by using the cash flows in line 14 rather than those of line 13 to obtain: AIRR ( A) 17.72% 15.02% MAIRR ( B )
Note that the choice to equate the beginning cash flows of the projects or the ending cash flows is arbitrary. Indeed, line 14 of Table 1 presents B*’s cash flow after it was iso-NPV modified so that its ending (rather than the beginning) cash flow is 17
equal to that of A. The IRR ranking once again does not conflict the NPV ranking. The resulting IRR is now 15.02% and it generates equal NPV to that of our previous calculation as it now applies to a higher invested capital. Namely,
0.1502 0.05 0.1525 0.05 324.88 317.67 31 1.05 1.05 As Magni (2010) (2013) showed, the AIRR approach does not require the modification of the cash flow stream. Indeed, the AIRR of the cash flow stream of B may be calculated as follows:
AIRR ( B ) 0.05
30
20 5 30 25 2 3 1.05 1.05 1.05 1.05 4 1.05 0.1525 317.43
In this equation, the cost of capital (5%) is that of project A and of the cross-risk adjusted project B, and the denominator (317.43) is equal to the present value of A’s invested capital. Although the AIRR does not require the modification of a project’s cash flow after it has been risk-adjusted, it may be viewed as if the cash flow stream has been modified by compressing the project's multi-period cash flow stream into an equivalent single-period iso-NPV investment. Ranking projects by AIRR necessarily leads to correct value creation ranking if the original cash flows are of equal risk or the cash flows were cross-risk adjusted prior to calculating the AIRR and, in addition, the present value of the invested capital of the competing projects are equated prior to calculating the AIRR.
The MIRR solution is presented in Table 2. INSERT TABLE 2 HERE
18
Lines 2 to 4 of Table 2 simply replicate the corresponding lines of Table 1. Lines 5 to 7 present the modified cash flows (MCFs) and the respective MIRRs. Once again, the NPV-IRR ranking conflict is not resolved, as MCF(A) differs from MCF(B*) at two points in time (t=0 and t=4), and we obtain MIRR(A) = 13.23% < MIRR(B*) = 23.24% where NPV(A) = 38.44 > NPV(B*) = 31.00. To resolve the ranking conflict we perform a re-modification of B*’s cash flow (see line 8), and equate B* 's initial investment with that of A’s to be 109.07 while preserving B* 's NPV. The result is that the ending inflow of the re-modified MCF(B*) is only 170.25 compared with MCF(A), which equals 179.31. Only at this stage when the competing modified cash flows adhere to our dominance rule (compare lines 5 and 8) the MIRR rank the projects consistently with their value creation (i.e., MIRR(B*) is only 11.78% while MIRR(A) is 13.23%).
The simple dominance resolution of the NPV-IRR Ranking Conflict
In Table 3 line 4 we present the cash flow stream of A”. At points in time t = 1,2,3,4, the cash flows are equal to the difference CFt A CFt B* . At t=0, the initial investment of A” is equal to the present value of its own cash flow stream at times t = 1,2,3,4: 5 30 40 20 CF0A" 77.45 2 3 4 1.05 1.05 1.05 1.05
The NPV and IRR of the incremental cash flow stream A” are 0 and 5%, respectively. INSERT TABLE 3 HERE
19
Line 5 presents A s cash flow stream, which is identical to A’s cash flows at points of time t=1,2,3,4. A’s cash flow stream is equal to the sum of the cash flow streams of B* (line 3) and A (line 4) at all points of time: A [107.45,40,10,60,70] [30,20,5,30,25] [77.45,20,5,30,45]
The result is an iso-NPV modified cash flow of B*, with future cash flow stream identical to that of A’s but the initial investment (107.45) is higher than A’s (100). It follows that more initial capital must be invested in A to generate the same future cash flow stream as A’s. Thus, the NPV of A is lower than that of A by 7.45 and the IRR of A (14.71%) is lower than that of A (17.72%). Below we show that the one period discounted premium of A ’s IRR ( 14.71%) over the cost of capital (5%) is a weighted average of the one period discounted premiums of the IRRs of B* (38.34%) and A'' (5%) over the cost of capital. The weights are based on the present value of the invested capital at t=0 of A' (335.15), and the presents values of the invested capital of streams of A'' and B* (237.54 and 97.61 respectively) which sum up to 335.15. Namely,
( IRRA r ) ( IRRA" r ) ( IRRB* r ) PV (cA | r ) PV (cA" | r ) PV (cB* | r ) 1 r 1 r 1 r ( IRRA 0.05) (0.05 0.05) (0.3834 0.05) 335.15 237.54 97.61 1.05 1.05 1.05 Thus:
IRR A 0.05 (0.3834 0.05)
97.61 0.1471 14.71% 335.15
V. Conclusions NPV ranks investment projects by their amount of value creation. Comparing projects by their IRRs may be inconsistent with value creation, hence, NPV-IRR
21
ranking conflict is generally possible. However, if one of the competing alternatives has the same or higher expected cash flows at all points of time and the same or lower risk than the other alternative, then such ranking conflict is not possible. We define such a relationship between projects’ cash flow streams as “dominance” of the preferred cash flow over the other. In general, dominance relationships between original cash flow streams are very rare and conflicting NPVIRR ranking are frequent. However, dominance conditions may be easily generated by using a two-step iso-NPV modification that eliminates the conflict. First, one of the alternative streams is cross-risk adjusted using a procedure that equates its risk to the risk of the alternative stream. Second, an iso-NPV modification is performed on one of the cash flow streams, so that all the future cash flows from time t=1 to time t=n, are equated to the cash flows of the alternative project. When the competing projects have identical risk level (either originally or after performing the cross-risk adjustment), the AIRR as well as the MIRR resolve the ranking conflict, but as we demonstrated, they are, in fact only special cases that adhere to the proposed dominance rule. The MIRR, is often based on cash flows far remote from the original cash flows and thus may be quite different from the original cash flow's IRR. In addition, it is not independent of the cost of capital and thus cannot serve as hurdle rates for the cost of capital. Even if we disregard the fact that alternative rates of return, AIRR and MIRR, may be quite different from the original IRR, we are still left with a major deficiency that they could mislead investors to regard them as cutoff rates for the cost of capital. While our procedure also uses the cost of capital to modify one of the cash flow streams, the economic intuition behind the comparison of the rate of return of the
21
modified stream with the rate of return of the unmodified stream is straightforward. Considering the alternative rates of return in the market (i.e., the cost of capital), one of the cash flow streams emerges as generating the same future cash flows at the same or lower risk, but with lower initial investment than the competing alternative. Such dominance is intuitive and appealing and is well reflected in the higher rate of return of the dominating project.
Acknowledgments We wish to acknowledge the very helpful comments and suggestions of an anonymous referee. We also wish to thank the Ono Research Institute in Finance (ORIF) for its generous support, the seminar participants at Ono Academic College for their comments and suggestions on an earlier draft of the paper. Remaining errors are our responsibility.
References Balyeat, R.B., J. Cagle and P. Glasgo (2013), "Teaching MIRR to Improve Comprehension of Investment Performance Evaluation Techniques". Journal of Economics and Finance education 12, 39-50. Bierman, H. and S. Smidt, 2007, "The Capital Budgeting Decision", 9th Ed. New York, N.Y. Routledge. Ben-Horin, M. and Y. Kroll (2010) "IRR, NPV and PI Ranking: Reconciliation". Advances in Financial Education 8, 88-105. Ben-Horin, M., Kroll, Y. (2012). "The Limited Relevance of the Multiple IRRs". The Engineering Economist 57, 101 –118.
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Graham and Harvey, 2002, "How Do CFOs Make Capital Budgeting and Capital Structure Decisions?", Journal of Applied Corporate Finance, 15, 8-22. Hazen, G.B. (2003) "A New Perspective on Multiple Internal Rates of Return". The Engineering Economist 48, 31–51. Lin, Steven A. 1976. “The Modified Internal Rate of Return and Investment Criterion.” Engineering Economist 21, 237-247 Kierulff, H. (2008), "MIRR: A Better Measure". Business horizon 51, 321-329. Magni, C.A. (2010), "Average Internal Rate of Return and Investment Decisions: A New Perspective". The Engineering Economist 55, 150–180. Magni, C.A. (2013), "The Internal Rate-of-Return approach and the AIRR paradigm: A refutation and a corroboration”, The Engineering Economist 58, 73–111. Petry, G. H., J. Sprow,(1993)"The theory and practice of finance in the 1990s", The Quarterly Review of Economics and Finance 33,359-381. Payne, J. D., W. C. Heath and I.R. Gale, (1999), “Comparative Financial practice in the USA and Canada: Capital Budgeting and Risk Assessment techniques”, Financial Practice and Education, 9, 16-24. Osborne, M. (2010) A resolution to the NPV-IRR debate? Quarterly Review of Economics and Finance, 50, 234-239. Shull, David, (1992), “Efficient Capital Project Selection Through a Yield-Based Capital Budgeting Technique", Engineering Economist 38 (1) (fall):1-18.
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Table 1: Ranking projects A and B using Magni's AIRR. Project B’s cash flows are presented before (B) and after (B) they were cross-risk adjusted*. 1
Time (Years) t
0
1
2
3
4
IRR/AIRR
NPV
2
CF(A)
-100
40
-10
60
70
17.72%
38.44 (r=5%)
3
CF(B)
-30
20.57
-5.28
32.64
27.98
42.30%
31.00 (r=8%)
4
CF(B)
-30
20
-5
30
25
38.34%
31
5
Invested Capital in A
100
77.72
101.49
59.46
0
6
Present value of A’s invested capital (r=5%)
317.43
7
A's equivalent iso-NPV one-period cash flow
-317.43
373.67
17.72%
38.44
8
Invested Capital in B
30
22.12
36.76
19.66
0
9
Present value of B’s invested capital (r=8%)
97.61
10
Invested capital in B
30
21.5
34.75
18.07
0
11
Present value of B invested capital (r=5%)
97.61
12
B ‘s iso-NPV one-period cash flow
-97.61
135.03
38.34%
31
13
Iso NPV modification of the one-period equivalent cash flow of B by equating its initial invested capital to that of the one-period cash flow of A
-317.43
365.85
15.25%
31
14
Iso NPV modification of the one-period equivalent cash flow of B by equating its ending cash flow to that of the one-period equivalent cash flow of A
-324.88
373.67
15.02%
31
In this Table, we compute the standardized AIRR using the AIRR-consistent modification of cash flows which generates and
24
B m . The same results would be obtained using Magni’s (2013) Eqs. (29) and (30).
Am
Table 2: Ranking Projects A and B by Cross-Risk Adjusting and Re-modifying MIRR 1
Time (Years) t
0
1
2
3
4
IRR
2
CF(A)
-100
40
-10
60
70
17.72%
38.44 (r=5%)
3
CF(B)
-30
20.57
-5.28
32.64
27.98
42.30%
31.00 (r=8%)
4
CF(B)
-30
20
-5
30
25
38.34%
31.00 (r=5%)
5
MCF(A)
-109.07
0
0
0
179.31
13.23%
38.44 (r=5%)
6
MCF(B)
-34.54
0
0
0
89.15
26.76%
31.00 (r=8%)
7
MCF(B)
-34.54
0
0
0
79.65
23.24%
31.00 (r=5%)
8
Re-modification of MCF(B) that equates its initial outlay to that of MCF(A)
-109.07
0
0
0
170.25
11.78%
31 (r=5%)
25
MIRR
NPV
Table 3: Equating the Future Cash Flow Stream of B* with that of A 1
Time (Years) t
0
1
2
3
4
IRR
NPV
2
CF(A)
-100
40
-10
60
70
17.72%
38.44 (r=5%)
3
CF(B)
-30
20
-5
30
25
38.34%
31.00 (r=5%)
4
CF(A’’)
-77.45
20
-5
30
45
5.00%
0.00 (r=5%)
5
CF(A’) = CF(B*) + CF(A”)
-107.45
40
-10
60
70
14.71%
31.00 (r=5%)
26