DCF Capital Budgeting Criteria A Broader Perspective. Robert G. Beaves and Richard W. Stolz

Transcription

1 1 DCF Capital Budgeting Criteria A Broader Perspective Robert G. Beaves and Richard W. Stolz Robert Beaves is a Professor of Finance at Robert Morris University in Pittsburgh, PA. Richard Stolz is a Professor of Finance at the University of South Carolina Upstate in Spartanburg, SC.

2 2 DCF Capital Budgeting Criteria A Broader Perspective This paper melds concepts and tools from several sources for the purpose of providing a comprehensive understanding of classic capital budgeting criteria. We begin by exploring the commonalities that exist among five popular criteria: net present value, internal rate of return, discounted payback period, modified rate of return, and profitability index. Each of these criteria is defined in terms of a single, unifying concept, cumulative present value. Algebraic relationships between NPV and each of the other criteria are explored so as to provide insight into the respective strengths and weaknesses of these criteria. By employing concepts such as project scale, return duration and net rate of return, we move the discussion of differences among criteria beyond a comparison of reinvestment assumptions. Finally, we demonstrate techniques that can be used to address the weaknesses of some DCF criteria. When used to evaluate the economic viability of a capital project, each DCF criterion provides a unique perspective. Our approach emphasizes the complementary nature of these criteria while clearly identifying their differences in a way that students can follow. Because a variety of capital budgeting criteria are likely to be encountered in practice, we feel it is especially important to equip future practitioners with a solid understanding of multiple DCF criteria.

3 3 I. INTRODUCTION The capital budgeting decision is a keystone in the teaching and practice of financial management. Over the years finance texts have demonstrated little change or variety in their treatment of capital budgeting decision criteria. In this paper we intend to impart a broader understanding of several classic decision criteria, how those criteria relate to one another, and their respective strengths and weaknesses. To facilitate this process, we rely on several concepts of that are not typically found in finance textbooks. NPV is considered to be the gold standard among capital budgeting decision criteria and for good reason. Equipped with perfect information, a decision maker will maximize firm value by rejecting projects that have negative NPV s and by ranking competing acceptable projects based on their respective NPV s; i.e., with high NPV projects preferred over those having lower NPV s. In a perfect world, NPV might be the only capital budgeting criterion a decision maker would ever need. Faced with less than perfect information, decision makers must exercise judgment and may be less inclined to rely on NPV alone. Given such an environment, valuable insight can be gained by looking at multiple decision criteria, each capable of providing a slightly different perspective of a project s investment potential. Because the time value of money must be considered in any capital investment decision, we limit our discussion to discounted cash flow (DCF) decision criteria, namely: 1) discounted payback period, 2) net present value, 3) modified rate of return, 4) profitability index, and 5) internal rate of return. We employ a unifying concept, cumulative present value (CPV), to highlight the commonalities among these criteria. We then explore the algebraic relationships between NPV and each of the other criteria

4 4 so as to clarify differences and expose weaknesses. Finally, we demonstrate techniques for addressing the shortcomings of specific criteria. We make three assumptions which allow us to focus narrowly on capital budgeting decision criteria. First, it is assumed that any capital investment project can be represented by a series of periodic, incremental cash flows that are expected to result from implementation of that project. Second, it is assumed that a hurdle rate (i.e., a minimum acceptable, risk-adjusted rate of return) can be assigned to each project. Finally, it is assumed that the decision maker's objective is to maximize owners wealth by maximizing firm value. II. CUMULATIVE PRESENT VALUE AND DCF CRITERIA We define each of our five DCF criteria in terms of a unifying concept, cumulative present value (CPV). At a specific point in time, a project s cumulative present value (CPV) is the sum of the present values of all cash flows that are expected to occur prior to or at that time. Expressed algebraically, a project s CPV at time j is: CPV j j Ct (1) t t 0 (1 k) where: C t = the project s expected cash flow at time t k = the hurdle rate assigned to the project Given a project s expected cash flows and hurdle rate, the present values and the CPV s related to those cash flows can be generated using a computer spreadsheet. In class and on exams, we generally provide students with a project s projected stream of cash flows as well as with the series of present values and CPV s derived from that

5 5 cash flow stream. The information provided in Table 1 for Projects A, B, and C is representative of the information we would provide students for the purpose of project analysis. Discounted Payback Period The discounted payback period (DPP) represents the length of time required to recoup the firm s investment in a project from cash flows discounted at the project s hurdle rate. The point in time at which a project experiences its final negative CPV represents the beginning of the period during which the DPP ends provided that the project s subsequent CPV s are nonnegative. A project whose final CPV is negative has no DPP. Consider Projects A, B, and C in Table 1. Each has a projected life of 4 years and a hurdle rate of 15%. The DPP s of Projects A and B end during year four (between times 3 and 4) because the final negative CPV of each occurs at time 3. Likewise, the DPP of Project C occurs during year 3 since its final negative CPV occurs at time 2. If we assume that a project generates discounted cash flow at a uniform rate throughout a particular period, we can interpolate a point at which the DPP would end. For example, given that Project A s CPV is -24,256 at time 3 and its period 4 cash flow has a present value of 31,488, the DPP of Project A could be estimated as follows: DPPA 24, years 31,488 (2)

6 6 Net Present Value The net present value (NPV) of a project represents the change in the firm s value that, in theory, would result if that project were implemented. A project's NPV is calculated by summing the present values of all of its cash flows as discounted using the project's hurdle rate. The NPV of an n-period project is: NPV n Ct (3) t t 0 (1 k) Note that a project's NPV and its final CPV are one and the same. Projects A, B, and C in Table 1 all have the same NPV of $7,232, indicating that implementation of any one of these projects would be expected to increase the firm's value by $7,232. Modified Rate of Return The modified rate of return (MRR) represents the average periodic rate of return earned over a project s life on the funds required to finance that project. The funds required by a project are assumed to earn the project's IRR while in the project and to earn the project's hurdle rate when not required by the project. As defined here, MRR differs from the modified internal rate of return (MIRR) found in many textbooks. The essential difference is that MRR recognizes that a project s positive cash flows can be used to offset any subsequent negative cash flows the project might have (Lin, 1976; Beaves, 1988). To calculate a project s MRR requires knowledge of that project s scale and its terminal value. For a given hurdle rate, a project s scale (S) represents the time 0 value of all funds required to fund that project and is equal to the absolute value of the project s minimum (most negative) CPV (Beaves and Stolz, 2005). Because an

7 7 investment project has at least one negative CPV by definition, the minimum CPV of an investment project must be negative. In contrast, the MIRR formula found in most textbook effectively scales a project based on the sum of the present values of all of a project s negative cash flows. In practice, a detailed project analysis might produce quarterly or even monthly cash flow projections. Projecting cash flows for shorter periods increase the likelihood of a mixed cash flow stream wherein at least one negative flow occurs subsequent to a positive flow. The CPV-based definition of project scale used herein does not overstate the size of the investment required for projects having mixed cash flow streams. Using that definition, Projects A, B, and C in Table 1 each have a scale of $150,000. Project A's minimum CPV occurs at time 0 whereas the minimum CPV s of Projects B and C occur at times 1 and 2 respectively. A project s terminal value is defined as the time n value (i.e., the sum of the future values compounded at the project s hurdle rate) of all cash flows that occur after the project experiences its minimum CPV (Beaves, 1988). For example, Project C has two cash flows that occur after time 2, the time at which its minimum CPV occurs. Project C s terminal value is the sum of the future values of those cash flows: TV (300, ) ( 70,000) 275,000 (4) Calculation of a project s terminal value can simplified by relying on the fact that terminal value is equal to the future value, at time n, of the sum of the project s scale (S) and its NPV (Beaves and Stolz, 2005): TV ( S NPV ) (1 k) n (5)

8 8 Projects A, B, and C in Table 1 all have the same scale, NPV, project life, and hurdle rate and thus have the same terminal value which is calculated as follows: TV 150,000 7, ,000 (6) Given a project s scale and terminal value, that project s MRR is determined as follows (Lin, 1976; Beaves, 1988; Beaves and Stolz, 2005): MRR TV S 1 n 1.0 (7) Because Projects A, B, and C all have the same $150,000 investment scale, the same $275,000 terminal value, and the same four-year life they have the same MRR: ,000 MRR or 16.36% (8) 150,000 A project s MRR can be found with a financial calculator using the following steps: 1) Enter the sum of the project's scale and its NPV as [PV] (note, most calculators require that PV and FV have opposite signs). 2) Enter project's hurdle rate as [I/Y]. 3) Enter the number of periods in the project's life as [N]. 4) Solve for terminal value by computing [FV]. 5) Without clearing the calculator, substitute the project s scale as [PV]. 6) Then solve for MRR by computing [I/Y]. Our students have little problem using financial calculators to solve for MRR and find that solving for MRR in this manner provides additional insight into the relationships that exist among a project s scale, terminal value, NPV, hurdle rate, and MRR.

9 9 Profitability Index A project s PI is the ratio of the time 0 value (i.e., present value) of its terminal value to its scale (Beaves and Stolz, 2005). Of course the time 0 value of a project s terminal value is equal to the sum of that project s investment scale and its NPV. PI TV (1 k) n S NPV S S (9) Because Project s A, B, and C all have the same investment scale and NPV, they also have the same PI: 157,232 PI (10) 150,000 Internal Rate of Return The internal rate of return (IRR) of a project is that discount rate for which the project s NPV (i.e., its final CPV) equals zero. The IRR is internal in the sense that it represents the average rate of return that funds earn while invested in the project itself. A project s IRR does not reflect returns earned on funds subsequent to their release from the project. For example, Project A releases $55,073 at times 1, 2 and 3. Project A s IRR is unaffected by whatever "external" returns those funds might earn subsequent to their release from Project A. Unlike our other criteria, a project s IRR can be calculated without knowledge of that project s hurdle rate. Nonetheless, capital budgeting decisions based on a project s IRR must be made with reference to the hurdle rate assigned to that project. The IRR's of Project's A, B, and C are respectively 17.36%, 17.55%, and 17.99%. As Table 2 illustrates, the final CPV of each project is zero when the present values of its cash

10 10 flows are calculated using that project s IRR as the discount rate. The use of the project s IRR as the discount rate in Table 2 naturally results in a different set of CPV s than those reported in Table 1 where the project s hurdle rate was used. Further, the use of a particular rate to calculate time values generally suggests an ability to invest or borrow at that rate. This implication is at the root of the argument that the IRR criteria assumes an ability to reinvest at the IRR whereas the other four criteria assume an ability to reinvest at the project s hurdle rate. The IRR of a project is difficult to conceptualize and to interpret. It is critical to understand that the IRR is simply a dollar-weighted, average rate of return. As such, a project s IRR does not represent a constant rate of return that is generated by that project period-after-period (Bailey, 1959). Thus the IRR does not represent the actual rate earned during any period shorter than the project s life. Further, a project s IRR provides no information as to the amounts invested in the project at any point in time between time 0 and the project's termination. For example, the fact that Project A s IRR is 17.36% provides little useful information regarding either the actual rates of return earned by funds invested in that project during years 1, 2, or 3 or the amount of funds invested or remaining in that project at times 1, 2, or 3. Finally, it is well known that some projects have multiple IRR s whereas others have none.

11 11 III. USING DCF CRITERIA TO MAKE DECISIONS Accept/Reject Decisions The accept/reject decision addresses the economic viability of a project. Simply put, would implementation of the project be consistent with the objective of maximizing firm value? Each DCF criterion has its own rules for accepting or rejecting a project. 1. Net present value: Accept a project if its NPV is greater than or equal to zero; otherwise, reject. 2. Internal rate of return: Accept a project if its IRR is greater than or equal to the project s hurdle rate; otherwise, reject. 3. Modified rate of return: Accept the project if its MRR is greater than or equal to the project s hurdle rate; otherwise, reject. 4. Profitability index: Accept the project if its PI is greater than or equal to one; otherwise, reject. 5. Discounted payback period: Accept the project if its DPP occurs during its life (i.e., is less than or equal to n periods); otherwise, reject. We refer to this as the modified payback rule because DPP is generally not used in this manner. For projects that have a unique IRR, all five of these decision rules will provide the same accept/reject decisions. In any case, the NPV, DPP, MRR, and PI criteria will always give the same accept/reject decision.

12 12 The DPP as a Filter The modified DPP rule described above is likely to be new to readers, but does allow the DPP criterion to be consistent with the other DCF criteria in terms of accept/reject decisions. In practice, the DPP is often used as a filter to prevent consideration of projects that have a DPP extending beyond a specified maximum acceptable period. If at least one project under consideration has a project life that extends beyond that maximum acceptable period, this practice may filter positive NPV projects from consideration. Firms often employ maximum allowable DPP s to exclude projects that generate positive NPV s by relying on large, positive cash flows late in the project s life. Reliance upon more distant cash flows is viewed to be too speculative or risky in a rapidly changing, competitive environment. This issue could be addressed by requiring that all proposals be presented with a life equal to or less than some maximum number of years. Assuming a reasonable maximum life is set, any inconsistencies with the goal of value maximization would be limited. There are methods for dealing with the speculative nature of distant future cash flows while remaining consistent with the objective of maximizing firm value (Beaves, 1993). Ranking Acceptable Projects The need to rank acceptable project arises when a decision maker must choose from among several acceptable projects or sets of acceptable projects. This might occur when acceptable projects are mutually exclusive or if capital is being rationed. The ranking rules generally associated with our five criteria are as follows:

13 13 1. Net present value: choose the project with the highest NPV. 2. Internal rate of return: choose the project with the highest IRR. 3. Modified rate of return: choose the project with the highest MRR. 4. Discounted payback period: choose the project with the shortest DPP. 5. Profitability Index: choose the project with the highest PI. Comparing Criteria: The NPV criterion has long been considered to be the "gold standard" of capital budgeting decision criteria. Because a project s NPV directly measures the incremental effect on firm value of implementing that project, choosing the alternative that has the highest NPV is tantamount to choosing the alternative that maximizes firm value. To understand why other DCF criteria provide project rankings that are not NPV-consistent, it is instructive to explore how each relates algebraically to NPV. Note that there is no algebraic relationship between NPV and DPP. Although a project whose DPP is shorter than its life will have a positive NPV, the size of that NPV is independent of the length of the project s DPP. Under no circumstances can the DPP criterion be relied upon to provide NPV-consistent project rankings. Equation 9 above can be manipulated algebraically to show that PI and NPV are related as follows: NPV S( PI 1.0) (11) As equation 11 suggests, a project s scale (S) is reflected in its NPV but not in its PI. As a result, PI cannot be relied upon to provide NPV-consistent rankings of projects that differ in scale. The PI criterion reflects only the relative change in value provided by a

14 14 project in contrast to the NPV criterion which represents the absolute amount of the change. To gain essential insight into the relationship between NPV and IRR, we rely on a concept called return duration (Barney and Danielson, 2004). For a given hurdle rate (k), a project s return duration is the number of periods that funds would have to remain invested at that project s IRR to produce its PI. The return duration of a project can be calculated as follows (Barney and Danielson, 2004; Beaves and Stolz, 2005): 1 IRR D lnpi ln 1 k (12) Recall Project B which has a PI of , an IRR of %, and a 15% hurdle rate. Thus Project B s return duration is calculated as follows: D ln ln years (13) 1.15 In other words, given hurdle rate of 15%, investing funds at a rate of % for years will generate a PI of Employing the concept of return duration, NPV and IRR are related as follows: NPV 1 IRR S 1 k D 1.0 (14) Equation 14 suggests that the NPV criterion reflects the project s scale, return duration and hurdle rate whereas the IRR criterion does not. Because IRR fails to capture these factors, it cannot be relied upon to provide NPV consistent ranking of projects that differ in scale, return duration or hurdle rate. To demonstrate the relationship presented in equation 14, recall that Project B s NPV is $7,232, its scale is $150,000, its IRR is %, its return duration is years, and its hurdle rate is 15%:

15 15 NPV , $7, (15) Note that when each dollar of Project B s scale earns an average rate of return equal to its IRR ( %) for a period equal to its return duration ( years), Project B s NPV of $7,232 is produced. As noted earlier, the MRR criterion is based on the assumption that funds earn the project's IRR while they are in the project and the project s hurdle rate when unneeded by the project. In other words, a project s MRR is simply a time-weighted average of its IRR and its hurdle rate that can be expressed as follows (Beaves and Stolz, 2005): 1 D n D n MRR (1 IRR) (1 k) 1.0 (16) Recall that project B s hurdle rate is 15%, its IRR is %, and its return duration is years. Thus Project B s MRR could be calculated as: MRR ( ) (1.15) % (17) Note that equation 16 implies that if a project s IRR were equal to its hurdle rate, its MRR would also equal its IRR. MRR and IRR are also equal for any project whose return duration and projected life are the same (i.e., any project having only two cash flows).

16 16 The following relationship between a project s NPV and its MRR is derived from equations 5 and 7 above: NPV 1 MRR S 1 k n 1.0 (18) Equation 18 suggests that NPV reflects the project s scale (S), and its hurdle rate in ways not captured by the MRR criterion. Thus MRR alone does not provide sufficient information about the incremental value provided by competing projects to allow a ranking of projects that have different scales or different hurdle rates. Addressing Ranking Issues: The IRR rule for ranking projects fails to provide NPV-consistent rankings because dollar-weighted rates such as IRR are not generally comparable across investment alternatives or projects. In contrast, time-weighted rates of return such as MRR are comparable for projects that have the same project scale and hurdle rate. A project s IRR could be transformed into a time-weighted rate of return by assuming a reinvestment rate equal to its IRR. In other words, one could assume that funds devoted to the project would earn the project s IRR not only while in the project, but also when outside (or not required by) the project. Such an assumption would, however, directly contradict the assumption that a project s hurdle rate represents the relevant reinvestment opportunities. As noted above, the IRR, MRR, and PI criteria fail to reflect a project s scale. For example, if one were to double each of a project s cash flows, the NPV of resulting project would be twice that of the original project, yet its MRR, IRR, and PI would be

17 17 unchanged from the original project. As a consequence, choosing the project or set of projects that have the highest IRR, MRR, or PI does not necessarily maximize firm value if that project s scale is less than those of competing projects. Methods do exist to adjust the MRR and PI criteria to reflect differences in project scale (Athanasopoulos, 1978; Shull, 1992). A scale-adjusted PI can be used to provide NPV-consistent project rankings. Nonetheless, a scale-adjusted MRR will necessarily produce NPV-consistent rankings where all competing projects share the same hurdle rate. A recent publication employs a concept called the net rate of return to address the problem of using a rate criterion to rank projects that do not share the same hurdle rate (Bosch et al, 2007). A quick example demonstrates 1) how the MRR criteria can be adjusted for scale and 2) how a net MRR can be employed to provide NPV-consistent rankings of projects that have different hurdle rates and scales. Consider Projects D and E presented in Table 3. Project D has a hurdle rate of 12% and a project scale of $84,464 whereas Project E has a hurdle rate of 17% and a scale of $120,000. Note that both have the same NPV, $6,872 and would thus be considered to be equally attractive. Project E s MRR is 18.64% whereas Project D s MRR is 14.21%. Project D s MRR can be scale-adjusted to Project E s scale ($120,000) by substituting $120,000 into the MRR calculation in place of project D s actual scale. Thus project D s scale-adjusted MRR would be: MRR ,872 (1.12) ,000 (19)

18 18 We now have two MRR s scaled to $120,000 for which net MRR s can be calculated by applying the following formula: net MRR 1 MRR 1 k 1.0 (20) Project scale-adjusted, net MRR would be: net MRR (21) Project E s net MRR is: net MRR (22) After adjusting for scale, the net MRR s of project s E and D are identical. In other words, net MRR s for MRR s which have been adjusted to the same scale will provide NPV-consistent rankings of competing projects. Although a net IRR can be calculated in a similar fashion, net IRR s do not necessarily provide NPV-consistent rankings. Net IRR s, like the IRR s from which they are derived, represent dollarweighted rather than a time-weighted rates and as such are not directly comparable from project to project without adjustment for differences in return duration as well as differences in scale. Indeed, Equation 16 above suggests that adjusting a project s IRR for return duration simply produces that project s MRR. Thus we conclude that a project s net MRR is a much more useful metric than its net IRR. CONCLUSION It is hard to overstate the importance of capital budgeting to the understanding and practice of managerial finance. We believe it is important to give students a firm

19 19 understanding of various criteria that can be used to make capital budgeting decisions. To that end, we have found that the concept of CPV provides a useful, unifying foundation for teaching and demonstrating the complementary nature of DCF capital budgeting criteria. Once that complementary nature is grasped, students develop a better understanding of the particular strengths and weaknesses of each criterion. This is especially important because studies have shown that many practitioners rely on decision criteria other than NPV. (Cohen and Yagil, 2007; Broun et al, 2004; Graham and Harvey, 2001). We feel confident that our students can make a case for the correctness of NPV in ranking projects as compared to PI, IRR, DPP, or MRR. We also believe that they will be well-equipped to answer questions such as when will we get our money out of this project or what rate of return will our money earn if we implement this project. Although some may view these questions as irrelevant from a theoretical point of view, such a response is unlikely to persuade one s supervisor. Focus on NPV to the exclusion of other DCF criteria ignores the information enrichment that multiple DCF criteria, properly applied, can bring to the capital budgeting decision process. The added insight that can be gained about a project s potential is particularly relevant where decision makers lack perfect knowledge and must rely to some extent on judgment.

20 20 REFERENCES Athananasopoulos, PJ, 1978, A Note on the Modified Internal Rate of Return and Investment Criterion, The Engineering Economist (Winter), Bailey, M., 1959, Formal Criteria for Investment Decisions, Journal of Political Economy (October), Barney, L.D. and M.G. Danielson, 2004, Ranking Mutually Exclusive Projects: The Role of Duration, The Engineering Economist (Summer), Beaves, R.G., 1988, "Net Present Value and Rate of Return: Implicit and Explicit Reinvestment Assumptions," The Engineering Economist (Spring), Beaves, R.G., 1993, The Case for a Generalized Net Present Value Formula, The Engineering Economist (Winter), Beaves, R.G. and R.W. Stolz, 2005, Defining Project Scale, The Engineering Economist (Spring), pp Bosch, M., J. Montllor-Serrats and M. Tarrozon, 2007, NPV as a Function of the IRR: The Drivers of Investment Projects, Journal of Applied Finance (Fall/Winter), pp Brounen, D., A. de Jong, and K. Koedijk, 2004, Corporate Finance in Europe: Confronting Theory with Practice, Financial Management (Winter), Cohen, G. and J. Yagil, 2007, A Multinational Survey of Corporate Financial Policies, Journal of Applied Finance (Spring/Summer), Fisher, I., 1930, The Theory of Interest, New York, NY, Macmillan Co. Graham, J.R. and C.R. Harvey, 2001, The Theory and Practice of Corporate Finance: Evidence from the Field, Journal of Financial Economics (May/June), Hirshleifer, J., 1958, On the Theory of Optimal Investment Decision, Journal of Political Economy (August), Lin, S., 1976, The Modified Rate of Return and Investment Criterion, The Engineering Economist (Summer), Renshaw, E., 1957 A Note on the Arithmetic of Capital Budgeting Decisions, The Journal of Business, Vol 30, No. 12 (April, 1956), pp

21 Shull, D.N., 1992, Efficient Capital Project Selection Through a Yield-Based Capital Budgeting Technique, The Engineering Economist (Fall),

22 22 TABLE 1 CPV s for Projects A, B, and C discounting at 15% hurdle rate Project A Time Cash flow PV(Cash flow) Cumulative PV 0-150, , , ,073 47, , ,073 41,643-60, ,073 36,211-24, ,073 31,488 7,232 Project B Time Cash flow PV(Cash flow) Cumulative PV 0-80,000-80,000-80, ,500-70, , ,000 90,737-59, ,000 36,163-23, ,050 30,332 7,232 Project C Time Cash flow PV(Cash flow) Cumulative PV 0-100, , , ,300 39,391-60, ,220-89, , , ,255 47, ,000-40,023 7,232

23 23 TABLE 2 CPV s for Project s A, B, and C discounting at respective IRR s Project A (IRR = %) Time Cash flow PV(Cash flow) Cumulative PV 0-150, , , ,073 46, , ,073 39,982-63, ,073 34,067-29, ,073 29,026 0 Project B (IRR = %) Time Cash flow PV(Cash flow) Cumulative PV 0-80,000-80,000-80, ,500-68, , ,000 86,840-61, ,000 33,859-27, ,050 27,782 0 Project C (IRR = %) Time Cash flow PV(Cash flow) Cumulative PV 0-100, , , ,300 38,394-61, ,220-84, , , ,650 36, ,000-36,121 0

24 24 TABLE 3 CPV s for Project s D and E, discounting at respective hurdle rates Project D (hurdle rate = 12%) Time Cash flow PV(Cash flow) Cumulative PV 0-80,000-80,000-80, ,000 4,464-84, ,000 39,860-44, ,000 35,589-9, ,000 15,888 6,872 Project E (hurdle rate = 17%) Time Cash flow PV(Cash flow) Cumulative PV 0-120, , , ,000 34,188-85, ,000 36,526-49, ,000 31,219-18, ,735 24,940 6,872