RANKING MUTUALLY EXCLUSIVE PROJECTS: THE ROLE OF DURATION
Barney, L Dwayne Jr
This paper develops a new measure of cash-flow timing called “return duration” Numerically quite close to Macaulay duration, return duration is a straightforward function of a project’s net present value (NPV) and internal rate of return (IRR). When comparing mutually exclusive projects, differences in return duration can explain ranking conflicts between NPV and IRR. The paper also clarifies the conditions under which a manager should consider duration or generalized NPV before making investment decisions when faced with such ranking conflicts.
Managers can evaluate a project’s cash flow stream using a number of different measures: the net present value (NPV) is a dollar figure, the internal rate of return (IRR) is an annual percentage, and Macaulay duration [ 15] is a gauge of cash-flow timing. When comparing mutually exclusive alternatives, it is well known that differences in the timing of cash flows can result in a conflict between the ranking of projects using NPV and IRR. In their popular textbook, Brigham and Houston  present an example of two hypothetical projects with a ranking conflict. The two projects differ in that one of the projects has a higher IRR and has the bulk of its cash inflows occurring early in the project’s life, while the other project has a lower IRR with the bulk of its cash inflows occurring in the more distant future. For a range of discount rates the lower IRR project has the higher NPV; thus a ranking conflict exists.
The authors explain that a disagreement between NPV and IRR can arise when “the timing of cash flows from the two projects differs such that most of the cash flows from one project come in the early years while most of the cash flows from the other project come in the later years” (pp. 522 and 523). The implication is that no ranking conflict would occur if the project having the higher IRR also had the bulk of its cash inflows coming “in the later years.” As the textbook is intended to be introductory, the authors do not precisely define what it means to have “most of the cash flows” coming earlier or later. Indeed, developing a gauge of cash-flow timing that explains ranking conflicts is trickier than one might expect. A precise, numerical measure is needed and, as will be demonstrated below, the familiar Macaulay duration will sometimes not suffice.
While some of the relationships between NPV, IRR, and cash-flow timing seem intuitive-indeed, they are discussed in popular introductory textbooks-until now a precise mathematical link has not been established. Nor has there been adequate formal development of the role of duration in explaining ranking conflicts between projects. The literature on duration’s uses in capital budgeting has focused on duration as an alternative to the payback period [10-12], and on duration’s role in determining the impact of changes in the discount rate [1, 5, 7, 9]. By way of contrast, this paper examines the mathematical ties between NPV, IRR, and duration.
To clarify the links between NPV, IRR, and duration, the paper derives a new measure of duration called return duration. Using this duration measure, a single equation is obtained that relates NPV, IRR, and duration. When the discount rate and IRR of a project are close to one another, return duration will be nearly equal to Macaulay duration. However, the two measures diverge as the difference between the IRR and cost of capital becomes large.
Armed with a new measure of duration, the role duration plays in creating ranking conflicts between NPV and IRR is explored. To motivate this discussion, consider two projects requiring the same initial investment and having the same discount rate. Further, suppose one of the projects has the higher IRR and a longer duration. Will the higher IRR project necessarily have the higher NPV? Intuition seems to suggest that the answer should be yes, as a capital outlay earning a higher return over a longer time period should create more wealth than a project earning a lower return and lasting for a shorter time. But, as will be demonstrated below, the low-IRR, short-duration project may turn out to have the higher net present value-at least if duration is measured using the familiar Macaulay approach. However, when return duration is used to measure project length, a project having a higher IRR and a longer duration will always have a larger NPV than an alternative project having a lower IRR and shorter duration.
The final part of the paper identifies the conditions under which a manager should consider project duration when resolving ranking conflicts. Conventional wisdom suggests that NPV should be used when comparing mutually exclusive projects, regardless of the existence of a ranking conflict. However, if a firm faces capital constraints due to market imperfections, the firm might be required to finance future projects entirely from internally generated funds. In this case, future cash inflows that are received quickly may be used to fund positive NPV projects that would not otherwise be possible. Therefore, short-duration projects can create both direct benefits (i.e., the project’s NPV) and indirect benefits (i.e., the NPV of future projects that are possible only because the initial project returns cash quickly). Under these circumstances, project rankings using generalized net present value (GNPV)-as defined by Beaves [2, 3]-can differ from those using NPV, and GNPV is the appropriate tool to use when choosing between mutually exclusive projects.
Return duration is the effective number of years the initial investment in a project will earn a compounded annual return equal to the project’s IRR. This period will generally not be equal to the number of years until the last cash flow. For example, consider a project that costs $ 1,000 today, and pays $ 1,200 in year one and $ 1 in year ten. The project’s IRR is 20.0194 percent. While the project generates cash flows over a ten-year period, the initial $1,000 investment will earn a return of 20.0194 percent for only one year. To see this, note that the year 1 cash flow of $ 1,200 can be separated into an economic return of $200.19 and a return of principal of $999.81 (much as a mortgage payment can be divided into interest expense and principal). The remaining $0.19 investment will earn a compounded annual return of 20.0194 percent through the end of year ten, when the $ 1 cash flow is paid. However, the effective return in years two through ten on the $999.81 principal payment in year one will depend on the IRR the firm can earn when reinvesting these funds at the end of year one.
The variable [tau] in Eq. (3) and Eq. (4) is the project’s return duration. When a project has only one cash inflow, its return duration is equal to the number of periods until the single cash flow is received, as is its Macaulay duration. For a project with more than one cash inflow, return duration will be less than the number of periods until the last cash inflow is received and also will not generally be equal to Macaulay duration.
One shortcoming of return duration is that it requires a project to have a unique IRR. This requirement may not be overly restrictive, though, as Bey  notes “the frequency of multiple IRRs occurring may be much less than textbooks often imply” (p. 86). Even if a project has more than one IRR, Cannaday, Colwell, and Paley  show how to determine which rate is the relevant IRR for capital budgeting purposes. Thus, for projects with more than one internal rate of return, the appropriate IRR to use in Eq. (3) can be determined.
Equation (9) shows that return duration and Macaulay duration will not always be equal, which is unsurprising as they measure different aspects of a cash flow stream’s life. As defined before, return duration measures the effective number of years the initial investment in a project earns a compounded annual return equal to the project’s IRR. Macaulay duration is a measure of project length that is typically used to estimate the PV change in response to a discount rate change. Equation (9) shows that the Macaulay duration of a cash flow stream-at the discount rate k-is equal to the return duration at that discount rate less a term that reflects the sensitivity of return duration to a change in the discount rate.
In the literature on fixed-income securities, it is well known that Macaulay duration decreases as the discount rate increases (dD/dk
Because the term ln[(l + r)/(l + k)] can be either positive or negative, Eq. (9) implies that return duration will sometimes be larger than Macaulay duration, and sometimes smaller. If r is less than k-meaning the project’s NPV is negative-this term is negative, and return duration will be larger than Macaulay duration. If r is greater than k-meaning the project’s NPV is positive-the term is positive, and return duration will be less than Macaulay duration.
Figure 1 illustrates these relations between return and Macaulay duration-over a range of discount rates from 5 percent to 12 percent-for a project requiring an initial $1,000 investment, and offering an annual cash inflow of $162.7454 for ten years (IRR = 10 percent). This figure shows that return duration is less than Macaulay duration when the discount rate is less than 10% (the positive NPV range), converges toward Macaulay duration as the discount rate approaches 10 percent, and is greater than Macaulay duration if the discount rate exceeds 10 percent. Across the entire range, the decrease in return duration in response to a discount rate increase is approximately one-half the decrease in Macaulay duration.
RANKING CONFLICTS BETWEEN IRR AND NPV
Equation (4) is a single equation that relates duration, NPV, and IRR. Using this equation, one can readily determine the return duration of a capital budgeting project from its NPV, IRR, and the cost of capital. Most importantly, having a single equation relating these measures facilitates an improved understanding of the role duration plays in creating ranking conflicts between mutually exclusive projects.
In this section, return duration is utilized to explain ranking conflicts between the IRR and NPV of two projects requiring the same initial investment. As was suggested in the paper’s introduction, comparing projects using IRR and Macaulay duration will not guarantee an NPV-consistent ranking. This can be seen with a simple example involving the comparison of two mutually exclusive projects: Project 4-1 involves an initial cash outlay and then has increasing cash inflows occurring over a period of four years, while Project 10-D has decreasing cash flows occurring over a ten-year period. Both projects require a $1,000 investment (at time zero), and have the same discount rate. Project 4-1 has cash inflows of $100, $320, $425, and $925 in years 1 to 4, and no cash inflows thereafter. Project 10-D has cash inflows of $385, $350, $300, and $215 in years 1 to 4, $100 per year in years 5 to 10, and no cash inflows thereafter. Table 1 lists the Macaulay duration (D), the return duration ([tau]), and the NPV for each cash flow stream over a range of possible discount rates from 5 percent to 23 percent.
Table 1 identifies three differences between Macaulay duration and return duration. First, return duration is less than Macaulay duration when r is greater than k, is larger than Macaulay duration when r is less than k, and converges to Macaulay duration as r approaches k. To illustrate, consider Project 4-1, which has an IRR of 19.91 percent. The return duration is not defined at a discount rate of 19.91 percent. However, at a discount rate that is very near 4-I’s IRR, such as 20.07 percent, the two duration measures are nearly equal at 3.057.
Second, Table 1 shows that return duration estimates are less sensitive to changes in interest rates than Macaulay duration estimates. The return durations range from 3.045 (when k is 23 percent) to 3.123 (when k is 5 percent) for Project 4-1, and from 2.801 to 3.209 for Project 10-D. The Macaulay durations range from 3.032 to 3.185 for Project 4-1, and from 2.749 to 3.615 for Project 10-D. As demonstrated in Appendix C, Macaulay duration will vary approximately twice the amount of return duration as the discount rate changes. The Macaulay duration changes in Table 1 are not exactly twice the return duration changes because the relation in Appendix C is derived for discount rate changes in the neighborhood of the IRR.
Finally, Table 1 shows that Macaulay and return duration can rank projects in conflicting orders. At discount rates above 14.13 percent, Project 4-1 has a longer duration than Project 10-D using either measure, even though it only produces cash flows over a four-year period. At rates below 8.79 percent, Project 4-1 has the shorter duration. At discount rates between 8.79 percent and 14.13 percent Project 4-1 has the longer return duration, but the shorter Macaulay duration.
Because Macaulay and return duration can rank projects in the opposite order, the two measures are not always equally effective at explaining the relation between the NPV and IRR of two projects. Intuition suggests that a project will add more to shareholder wealth if it has a larger IRR and a longer duration, meaning it produces a higher return lasting for a longer time period. As shown in Table 1, for discount rates between 11.51 percent and 14.13 percent, the relations between NPV, IRR, and Macaulay duration for Projects 4-1 and 10-D do not conform to this intuition. For example, at a discount rate of 12 percent the Macaulay duration of Project 10-D is 3.211, compared to a shorter Macaulay duration of 3.125 for Project 4-1. And, Project 10-D’s IRR is greater than that of 4-1. But, surprisingly, Project 10-D’s higher IRR and longer Macaulay duration do not combine to create a higher NPV. Instead, Project 10-D’s NPV is $234.2, slightly less than 4-I’s NPV of $234.7.
In contrast, the relations between NPV, IRR, and duration are more understandable when return duration is used. Still referring to a discount rate of 12 percent, Table 1 shows that Project 10-D has the shorter return duration. Although Project 10-D has a higher IRR than does Project 4-1, these returns will last for a shorter period (as measured by return duration), making it possible for Project 10-D to have the lower NPV. Thus, the relations between the projects’ NPV, IRR, and return duration do not conflict with intuition.
In summary, it is well know that NPV, IRR, and duration can rank projects in different orders, and this fact is often used as a justification to disregard a project’s IRR. However, our analysis suggests that these ranking conflicts do not indicate a flaw in any of the metrics. Instead, each metric measures a different attribute of a cash flow stream, and when used together, return duration and IRR can provide an intuitive understanding of how a project’s NPV is created. For example, when evaluating a project with a very high positive NPV estimate, the tools described in this paper can be used by managers to gain insights about whether the NPV is created by an IRR far in excess of the cost of capital, a long duration, or both. The plausibility of such an NPV estimate will then depend on the strength of the firm’s competitive advantages and the intensity of competition in its industry, as these factors will constrain both the size and duration of the IRR a firm’s investments can earn.
THE RELEVANCE OF DURATION TO THE CAPITAL BUDGETING DECISION
If a firm must choose between two mutually exclusive projects-one with a higher NPV and the second with a higher IRR and a shorter duration-the firm is essentially selecting between projects offering different economic benefits. The first project will create the highest economic value, as measured by NPV, but the second project will return cash quicker. The relative importance of these economic benefits to the firm’s capital budgeting decision will depend on whether the firm faces capital constraints.
In perfect capital markets, a firm will be able to raise sufficient external funds to invest in all future positive NPV projects. Therefore, if the firm must choose between two projects requiring the same initial investment today, the firm should choose the project offering the higher NPV-regardless of the relative IRR and duration of the two projects.
In contrast, market imperfections can either limit the amount of new funds the firm can obtain, or increase the cost of these new funds. In the most extreme cases, these imperfections can make it impossible for a firm to raise additional funds at any cost. For example, Peterson and Rajan [18, 19] argue that credit rationing can make it difficult for some small firms to obtain new bank loans, even at a high cost. For other firms, market imperfections-such as information or transaction costs-can make external funds more expensive than retained earnings (see Myers and Majluf  and Myers ). If capital constraints are present, internal cash flows may be a firm’s primary source of funds for future investments, and projects that return cash quickly may provide the financing for future positive NPV projects that would not otherwise be possible.
If a firm faces multi-period capital constraints, the firm should select the sequence of projects that will create the most value, which is a decision requiring the firm to consider the future trajectory the selection of the initial project will start. The total value of a sequence of projects depends on: 1) the initial project’s direct NPV; 2) the timing of the initial project’s cash flows; and 3) the expected return from future projects. These factors are captured by the generalized net present value (GNPV) of a project, as developed by Beaves [2, 3].
A project’s GNPV is closely related to its modified internal rate of return (MIRR). The MIRR can be defined as the discount rate that sets a project’s GNPV equal to zero if all reinvested cash flows will earn a return equal to the cost of capital, k. Thus, to find a project’s MIRR using Eq. (12), Z would be equal to k in the numerator, the GNPV would be set equal to zero, and the discount rate in the denominator would be the interest rate that satisfies the equality (i.e., it is the MIRR).
If the IRR on future investments is equal to the cost of capital, Eq. ( 12) can be simplified to show that the GNPV will exactly equal the project’s NPV, meaning the two measures will rank projects in the same order. However, if the return on future investments, Z, is large enough, it is possible for a shorter-duration, lower-NPV project to have a larger GNPV than a second project. The larger GNPV of a short-duration project can be the result of receiving cash inflows earlier, providing the financial capital necessary to fund future high-return projects.
Intuitively, high reinvestment rates (greater than Z) tilt the scales in favor of projects returning funds more quickly, thereby facilitating the reinvestment of internally generated dollars in attractive future ventures. If the return on reinvested funds is greater than the breakeven rate, the project with the shorter duration will have the higher GNPV, while the longer duration project will have the higher GNPV if the return on reinvested funds is less than the breakeven rate.
To illustrate how knowledge of the breakeven reinvestment rate might inform a capital budgeting decision, consider Project A and Project B as summarized in Table 2. Project A requires an initial investment of $ 100, and then produces cash inflows of $35, $35, 35, and $57.50 in years 1 through 4, respectively. Project B requires an initial investment of $100, and produces cash flows of $50 in year 1 and $35 in each of years 2, 3, and 4. Project A has an IRR of 20.7089 percent, while Project B’s IRR is 21.7514 percent. Table 2 lists the NPV, return duration, and Macaulay duration of Projects A and B at discount rates ranging from 10 percent to 25 percent. The two projects have the same positive NPV of $14.1 at a discount rate of 14.4714 percent.
The relative GNPVs of the two projects will depend on the projected size of the IRR on reinvested funds. If this IRR is higher than the crossover rate of 14.4714 percent, Project B-with the higher IRR but the shorter duration (measured using either Macaulay or return duration)-will have the higher GNPV. For example, if the return on reinvested funds is 15 percent, Project A will have an accumulated (via reinvestment) wealth of $197.27 after its last cash flow is received in four years, while Project B will have an accumulated wealth of $ 197.58 at this time. Both numbers are calculated using Eq. (11), the project cash flows as defined in Table 2, a value of Z equal to 15 percent, and a total life T of 4 years. Because Project B has the larger accumulated wealth, it will have the higher GNPV as long as each project has the same discount rate. If the discount rate is, say, 10 percent, Project B will have the higher GNPV ($34.95 vs. $34.74 for Project A-both numbers calculated using Eq. (12), Z = 15 percent, and T =4 years), despite having a lower NPV at this discount rate (see Table 2). This example shows that the future reinvestment rate required to choose a shorter-duration, lower-NPV, but higher IRR project does not have to be all that high. In this example, the reinvestment rate that equates the GNPV of the two projects was much less than the IRR of both projects.
To summarize, when capital budgeting is taking place in perfect capital markets, one needs to consider only the NPV to choose between mutually exclusive alternatives. However, when capital constraints are present and the firm has the ability to earn investment IRRs in excess of its cost of capital, the GNPV of a project may also be relevant to the capital budgeting decision. In such instances, NPV and GNPV may rank projects in different orders if NPV and IRR rank the projects differently. By calculating a breakeven reinvestment rate, the firm obtains a benchmark for comparison purposes. If the firm expects it can earn a rate of return on reinvested funds that exceeds the breakeven rate, then the project having the shorter duration will have the higher GNPV and is the proper choice. If the breakeven reinvestment rate is so high that it is unlikely to be obtainable on reinvested dollars, then the project with the longer duration will have the higher GNPV.
This paper introduces a new measure of cash flow duration, called return duration, and illustrates its mathematical links to Macaulay duration. Return duration provides the conceptual link between a project’s internal rate of return and its net present value. Having a single equation relating duration, IRR, and NPV aides in understanding how cash flow timing differences can create ranking conflicts. Using return duration, a project having a higher IRR and a longer duration will necessarily have a higher NPV when compared to a lower-IRR, shorter-duration project. This intuitively appealing result surprisingly does not always hold with Macaulay duration.
Besides introducing the new measure of duration, the paper clarifies duration’s role in capital budgeting decisions when projects are being evaluated in less than perfect markets. If capital budgeting always occurred in the idealized textbook world of perfect capital markets, then the presence of ranking conflicts would be of no particular concern to the analyst. When comparing mutually exclusive projects, firms would maximize shareholders’ wealth by choosing the project having the highest NPV, while rates of return and duration would be a secondary concern. However, if a firm faces capital constraints that will extend to future years, the firm must consider duration, IRR, and the potential IRR on future projects-in addition to NPV-when making investment decisions. Suggesting that capital markets might be less than perfect, numerous surveys [13, 14, 20] have shown that managers do in fact consider alternative measures-such as IRR, duration, and payback period-along with NPV when making capital budgeting decisions. By describing when and how duration is relevant to the capital budgeting decision, this paper helps reconcile capital budgeting theory to capital budgeting practice.
We thank John Daley, Lynda Livingston, Alex Wilson, and Harry White for comments on previous drafts of this paper.
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L. Dwayne Barney Jr.
Professor of Finance, Boise State University, Boise, Idaho
Morris G. Danielson
Assistant Professor of Finance, St. Joseph’s University, Philadelphia, Pennsylvania
DR. L. DWAYNB BARNEY is a professor of finance at Boise State University, Boise, Idaho (email@example.com) where he is formerly Chairman of the Department of Marketing and Finance. he received his M.S. and Ph.D. in Economics from Texas A&M University. Dr. Barney’s research interests are in the areas of asset valuation and risk and uncertainty, and included among his publications are articles in the Journal of Risk and Insurance, Journal of Economic Dynamics and Control, International Review of Economics and Finance, and The Journal of Financial Research.
DR. MORRIS G. DANIELSON is an assistant professor of finance at St. Joseph’s University, Philadelphia, Pennsylvania. he received his Ph.D. in Finance from the University of Washington. Dr. Danielson’s research explores the relations between fundamental measures of firm performance and firm valuation. Included among his publications are articles in the Financial Analysts Journal; Journal of Accounting, Auditing, and Finance; Journal of Corporate Finance; Review of Accounting Studies; and Financial Management.
Copyright Institute of Industrial Engineers 2004
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