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Payback criterion, hurdle rates and the gain of waiting
International Review of Financial Analysis 9 (2000) 247 ± 258
Payback criterion, hurdle rates and the gain of waiting Achim Wambach* Department of Economics, University of Munich, Ludwigstr.28 VG, 80539 Munich, Germany
Abstract In the standard literature on investment decisions, commonly observed instruments, namely the payback criterion and the use of hurdle rates, are criticised as they do not lead to a maximization of expected profits. By using the real option approach to investment under uncertainty, we show that it can be rational for the investor to use the payback criterion as a rule of thumb, because a shorter payback period indicates lower gains of waiting. In addition, we claim that the use of hurdle rates can serve a similar purpose: It allows to distinguish between projects with different values of waiting. An explicit expression for the optimal payback period and the optimal hurdle rate is given. D 2000 Elsevier Science Inc. All rights reserved. JEL classification: D21; D81; G12; L21 Keywords: Payback criterion; Hurdle rates; Gain of waiting
1. Introduction The standard literature on business studies (see e.g. Brealey & Myers, 1991) usually contains several sections on how firms decide whether to do an investment or not. Many concepts are used, out of which three we consider here: the standard net-present-value (NPV) concept, the use of hurdle rates and the payback criterion. The last concept is heavily disputed in the literature: It is not clear why an investment project with a shorter payback period should, in general, be preferred. In particular, it is very easy to construct examples where in one project the payoffs accrue at a later point in time, while for the other, the payoffs are large in the beginning but small later on. In this case, the payback criterion might allow for the latter project, but reject the first, which * Tel.: + 49-892-1802227; fax: + 49-892-1806282. E-mail address: [email protected] (A. Wambach) 1057-5219/00/$ ± see front matter D 2000 Elsevier Science Inc. All rights reserved. PII: S 1 0 5 7 - 5 2 1 9 ( 0 0 ) 0 0 0 2 8 - 4
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might not be optimal. The payback criterion systematically undervalues projects with a later payoff stream. One argument in favour of the payback rule is that if firms are constrained in their capital, it might be better to go for the investment project, which pays out earlier. However, one has to impose some form of asymmetric information to support this argument, otherwise, credit rationing will not hold. Another quite commonly heard argument is that if either the payoff or the lifetime of the project is uncertain, it is better to choose the project with the lower payback period. Although the latter is a very intuitive argument, it is not clear how a static concept, based on expected values, can be used as a means to decide between different uncertain projects. In particular, a possible finite lifetime should already be incorporated in the value of the expected payoff stream or at least in the project-specific discount rate, and it should not be the payback period, which differentiates between projects with different expected lifetimes. In a static environment, the use of a hurdle rate also seems to miss the point. Even in a deterministic dynamic scenario, there is no reason why the NPV should not be calculated by the use of the opportunity cost of capital for the firm, which, in general, is approximated by the firms weighted average cost of capital (WACC). But note the common feature of these two decision methods: Similar as the payback criterion, the use of a hurdle rate also discriminates in favour of projects with earlier payoffs. If two projects have the same NPV, calculated with the WACC, both instruments will decide against the project with the later payoff stream. However, given the criticism with these instruments, it needs to be explained why those concepts are so heavily used. Schall et al. (1978) report that more than 70% of the firms investigated work with the payback criterion. In a study by Summers (1987) for example, it was reported that firms use hurdle rates up to four times their cost of capital. The NPV concept, on the other hand, seemed to be quite a reliable guide to the decision whether to invest. However, the recent literature on investment under uncertainty (see Abel, 1983; Abel & Eberly, 1994; McDonald & Siegel, 1986; Ingersoll & Ross, 1992; Dixit & Pindyck, 1994; Ross, 1995 and references therein) has stressed that the NPV concept can be misleading; there are situations where the NPV of a project is positive, but one should still wait to do the investment. The main reason for this result is that in the standard NPV approach, there is no managerial flexibility, thus for example, the so-called option-to-wait, i.e. whether to do the investment today or tomorrow, is ignored. This option might have a positive value, it might be better to wait and see how things develop before doing the investment. The empirical evidence of the use of the NPV concept together with hurdle rates was given by Dixit and Pindyck (1995) as a strength of the new theory. This behaviour may not be `myopic' as some authors assume, but rather according to Dixit and Pindyck, it reflects the fact that a NPV much larger than zero is required to exercise the option to invest. Thus, a larger discount rate is used as a means to lower the calculated NPV. Still, even if we use this as an indication that implicitly managers follow the implications of the option-theory of investment, there are open questions which remain. First, it is not explained why the payback criterion is used at all, and second, if the hurdle rate is employed to lower the NPV in the sense of Dixit and Pindyck, why not just argue that the gross present value has to be x times as large as the costs of the project, where x is a number larger than one. Surely in this way, one would also invest only if the NPV is larger than zero.
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In this paper, we argue that the payback criterion and the hurdle rate can do even more. They both can be used as useful indicators of the possible ``gain of waiting'' of a project. Consider for example two projects with an uncertain profit stream for which the expected gross present values are x times the investment costs, where x is the same in both cases. However, the two projects have different expected payoff streams, the first one expects its payoff to accrue earlier than the second. Thus, the payback period of the first project will be lower than that of the second. Similarly, calculating the NPV with a hurdle rate, the NPV of the first project is larger than that of the second. But now consider the gain of waiting: as we will show in Section 2, the project with the later payoff has a larger gain of waiting, thus it is rational to do the other project first. In this case, a lower payback period and a larger NPV calculated with a hurdle rate both indicate a lower gain of waiting. To rationalise an instrument like the payback criterion, i.e., only to undertake projects that pay off in a prespecified time, it should be the case that this time is relatively stable against changes in the interest rate, or the timing of payoffs. A concept would not be very useful if for every different type of project, a different rule has to be set up. We are going to argue in this paper that the payback criterion can indeed be quite stable against changes in these parameters. In particular, for each parameter, there are two competing effects, which tend to cancel each other out. Take for example an increase in the interest rate: The optimal payback period depends positively on a parameter, which is negatively correlated to the gains of waiting. This parameter itself depends positively on the interest rate: a larger interest rate implies lower gains of waiting as the future counts less. Therefore, this effect leads to a positive correlation of the optimal payback period with the interest rate. On the other hand, cutting off a profit stream after the prespecified time (namely the payback period) leads to less profit unaccounted for in the case of a larger interest rate, as future profits matter less. This effect in turn reduces the optimal payback period. These results suggest that if an industry has developed an optimal payback rule, this rule stays relatively stable if projects have different payoff structures or if the interest rate changes. The level of uncertainty also enters the optimal payback period. Although there are no competing effects, i.e., a larger uncertainty rate reduces the optimal payback period, numerical estimates suggest that this effect is not large. Therefore, the main result of this paper is that the payback period rule can be usefully employed to account for the value of waiting of individual projects. Another explanation of the apparent irrational use of the payback criterion is given by Narayana (1985). There, a principal agent approach under asymmetric information is used. If two projects have the same NPV a manager may select that project with the quicker returns to demonstrate his ability, which can only be inferred from the vector of outputs. However, this model is not able to explain how the optimal cutoff time is determined and which factors are involved in this determination. Furthermore, it does not apply if the firm is owner-managed. In contrast to these results, in the present work, the use of the payback criterion is demonstrated to be useful for rational decision takers with symmetric information and the factors entering the cutoff time are determined. Similarly, there have been arguments brought forward to defend the use of hurdle rates as a means to weaken asymmetric information problems between the owner and the manager of a
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firm (see e.g. Antle & Eppen, 1985). If one includes, however, the possibility of postponing a project, the use of a hurdle rate can be rationalised without the need to refer to asymmetric information (Dixit & Pindyck, 1995). Ingersoll and Ross (1992) have modelled a project with a certain payoff stream in an environment where the future interest rate is uncertain. If an investment project can be postponed, the authors show that the investment should be undertaken at a prevailing interest rate lower than the internal rate of return (IRR). Or, turning the argument around, the hurdle rate used to calculate the NPV of the project is larger than the interest rate. In our context, we will consider projects where the firm's discount factor is stable, but the payoff of the project is uncertain. Then, there will be in addition to just increasing the required NPV another advantage of the use of the hurdle rate, namely to help to elicit the gain of waiting of a project. The paper is structured as follows: In section 2, we introduce the formalism of the optionto-wait approach. Here, we only consider a simple investment decision: one firm can decide to invest in an irreversible uncertain profit stream, the investment costs are fixed and have to be paid once only. The reason for this simplification (see Abel, 1983; Abel & Eberly, 1994 for more general projects) is that it captures the main ideas of the option-to-wait approach, while at the same time keeping the mathematics not too complicated. In this formalism, we derive the NPV of a given project and calculate its payback period. Section 3 is devoted to the investigation of the use of the payback criterion and its characteristic features. In section 4, the optimal hurdle rate is calculated and discussed. In section 5, we conclude. 2. The option-to-wait approach to irreversible investment under uncertainty This section gives a short introduction to the theory of investment under uncertainty. For a more detailed exposition, we refer the interested reader to the book by Dixit and Pindyck (1994). Consider a firm that has an investment opportunity. The costs of investment will be I, and doing the investment entitles the firm to receive the profit stream P(t). To model the uncertainty of the profit stream, we assume that P(t) follows a geometric Brownian motion, which implies that dP = aPdt + sPdz where dz denotes a standard Wiener process, a and s are constants. A different setup would be to model the expected present value of the project, i.e., the appropriately discounted expected cash flow, as a geometric Brownian motion (McDonald & Siegel, 1986). The project value can be seen as the market value of an asset, hence, the use of a geometric Brownian motion is less contentious here. However, to model the dependence on the occurrence of the timing of payments, we prefer to model an explicit payoff stream. The expected value of P is an increasing function of time, i.e., E[P(t)] = P0eat. a measures the drift rate of a project, therefore, a larger a implies a larger expected future payoff. If there are two projects, which have the same expected NPV, the one with the smaller a has its expected payoff accrue in the near future, while the other pays back at a later point in time. The larger s, the larger the uncertainty of the profit stream of the project is, which increases 2 over time. In particular, the variance of the payoff at time t is Var(P(t)) = P02e2at(es t ÿ 1). s describes the uncertainty rate of the individual project. As long as a > s2/2, the infimum of P(t) for all t is larger than 0, therefore, there is no fear of losing the project completely. Note that a larger uncertainty does not require a larger return of the project, because investors are
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assumed to be risk-neutral. However, as we will show later on, a larger s implies a larger gain of waiting. If the uncertainty is larger, it is probably better to wait for some time, even if the decision taker is risk-neutral. The question the investor now faces is: What is the optimal P* when she should undertake the investment? To answer this, it is helpful to consider the investment decision as a real option in analogy to an American call on a dividend paying stock, i.e., the investor has the option to pay I and receive the payoff stream P(t), but she does not need to exercise the option now, thus, the `option-to-wait' approach to investment. In order to calculate the optimal point of investment, we have to borrow some mathematics already developed in the theory of dynamic optimization under uncertainty. This works as follows: consider a firm that has already done the investment. Its value is the gross-present-value of the investment, which is Z 1 P ÿrt P te dt 1 GPV P E r ÿ a 0 where r is the discount rate of the firm, P is the known profit flow at time t = 0. As we do not assume that firms are risk-averse, r does not include a term proportional to the uncertainty of the project. Instead, it can be interpreted as the cost of capital a firm or industry faces, thus, it depends positively on the real interest rate, and in addition, there is also a dependency on the industry specific risk rate. A discussion on the specific value of r can be found in McDonald and Siegel (1986) and Dixit and Pindyck (1994). Now, consider a firm that has not undertaken the investment so far. Its value can be calculated by using Bellman's principle: V P EV P dPeÿrdt
2
that is the value of the investment option now is equal to the value of the discounted investment option tomorrow, where the payoff level has changed from P to P + dP. Eq. (2) holds as long as the investment option is not exercised, that is as long as P and P + dP are lower than P*, the strike price. The expectation on the right hand side can be expanded by using Ito's Lemma: 1 00 2 ÿrdt ÿrdt 0 EV P dPe E V P V PdP V P dP e 3 2 By using E[dP] = aPdt and E[(dP)2] = s2P2dt, expanding Eq. (3) in dt, we get the differential equation [Eq. (4)] for the value function: 1 ÿrV P aPV 0 P s2 P2 V 00 P 0 2 This can be solved to yield Eq. (5): V P A1 Pb1 A2 Pb2
4
5
where b1(2) are the roots of the quadratic equation 1 ÿr ab s2 b b ÿ 1 0 2
6
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We choose the numbering in such a way that b1 > 0, b2 < 0. As long as r > a, which we assume in the following, it can be shown that b1 > 1. A1(2) has to be determined by outside constraints. One such constraint is that if P = 0, the value of the project should be zero (V(0) = 0). This can be seen by noting that if P = 0, the stochastic process will never return to positive values of P again. Given that b2 is negative, this constraint implies that A2 has to be zero. To calculate A1, one has to consider the value of V at the optimal point of investment, which we denote by P*. Here, two equations have to be fulfilled [Eq. (7)]: First, the value of the option should be equal to the GPV minus investment costs, as one exchanges the option with the project. Therefore, if the option value were larger, it would not be rational to pay I now and receive the payoff stream. If the option value were lower, one should have invested already earlier. A second equation, which has to be satisfied at the optimal investment point is the so-called smooth pasting condition (see Dixit, 1993). This implies that the derivative of V with respect to P at the optimal point (P*) is the same as the derivative of GPV with respect to P at the same point. Thus, the following two equations define A1 and P*: V P GPV P ÿ I
7
V 0 P GPV0 P Using the expression for GPV from Eq. (1), this yields: b1 P r ÿ aI b1 ÿ 1 P ÿ I P ÿb1 A1 rÿa
8
P* is the critical point where it is optimal to invest, i.e., there are no further gains of waiting, the option should be exercised now. Note that this point cannot be derived by a usual NPV argument. The NPV of the project for a given P is given by Eq. (9): NPV P
P ÿI rÿa
9
As (b1)/(b1 ÿ 1) is larger than 1, the optimal investment point surely gives a positive NPV, but it can be considerably larger than the lowest P at which the NPV is positive.1 Thus, here, we have established the point that the NPV is not a sufficient means to decide on an investment opportunity. Before proceeding to the next section, let us finally calculate
1
Dixit and Pindyck (1994) have estimated that for sensible values of a, r and s, the factor (b1)/(b1ÿ1) can take on values of 2 and even larger.
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the expected payback period2 of the investment if it would be undertaken at a given level P. This time PP(P) is defined by "Z # PP P ÿrt P te dt ÿ I 0 10 E 0
From Eq. (10), it follows that ÿ1 I ln 1 ÿ r ÿ a PP P rÿa P
11
3. Payback criterion as a decision instrument From the discussion in Section 2, it is clear that the optimal payback period (PP*) of a project is finite. In the traditional view of investment, a payback period of infinity, which implies a NPV of 0, should have been enough to undertake the investment. However, as argued above, the NPV has to be larger than 0, therefore, it follows that PP* has to be finite. However, to say more, we have to consider explicitly the gain of waiting per unit of investment costs (GoW). For P > P*, GoW is 0, the project should directly be exercised. As long as P < P*, the gain of waiting is given by the option value minus the NPV of the exercised project [Eq. (12)]: P ÿ I =I 12 GoW A1 Pb1 ÿ rÿa where A1 is given in Eq. (8). Inserting A1 and P* from the first equation in Eq. (8), this reduces to b1 P P b1 ÿ1 ÿb1 13 b1 ÿ GoW b1 ÿ 1 I r ÿ a I r ÿ a The gain of waiting depends on two parameters, the factor b1 and the relative gross present value (RGPV) of the project, P/(I(r ÿ a)) [Eq. (13)]. Although b1 does also depend on the drift rate a and the discount rate r, it is useful to disentangle those two effects. Taking the derivative of GoW with respect to b1 we arrive at Eq (14): d b ÿ1 P 14 GoW GoW ln 1 db1 b1 I r ÿ a Using Eq. (8), it follows that as long as P is smaller than P*, the derivative is negative. This has the following implication: Taking the derivative of Eq. (6) with respect to a, s and r, it can be shown that b1 depends negatively on a and s, but positively on r. Thus, a lower drift
2
There is another version of the payback period that takes the equality of the undiscounted sum of profits with investment costs. We do not consider this rule here.
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or uncertainty rate reduces the gain of waiting, given that the GPV stays constant. The reverse is true with r. Taking the derivative of GoW with respect to the relative GPV, it can be shown that a larger RGPV reduces the GoW. This also makes sense intuitively, as a larger RGPV makes a project more profitable. The possible gain due to investing later cannot offset the loss due to forfeiting the profit in this period. But now consider the payback period, given in Eq. (11). PP depends negatively on the relative GPV and on the factor (r ÿ a), which is the dividend rate. Thus, the dependency of the payback period on the discount rate and the drift rate mimics that of the gain of waiting. This result is summarised in Fig. 1. The results obtained so far are true for a specific structure of the investment project. However, we think that the insight provided is more general. A larger interest rate for example increases the firm-specific discount rate. Thus, the NPV of a given project falls. With that, the gain of waiting increases as does the payback period. However, there is a second effect. A larger interest rate increases b1 and thus reduces the gain of waiting. Similarly, a larger interest rate increases the convenience yield r ÿ a and thus decreases the payback period, as the negative effect of cutting off the project at time PP is reduced. Hence, the project-specific payback period can be used as a guide to the individual gain of waiting. Having discussed the project-specific payback period, let us now turn to the payback criterion. Firms specify a time such that only projects whose payback period is lower than this time are undertaken, and this holds for a variety of projects. With the model given above, this optimal time for each project can be calculated by setting P = P* in Eq. (11). The optimal payback period (still as a function of a, r and s) is then: PP
ln b1 rÿa
15
For the payback criterion to make sense, one would like to have PP* be relatively stable against changes in the parameters. Otherwise, a different PP* would be needed for each individual project, thus offsetting the initial motivation of having one rule for many projects. Note first, that I does not enter PP*. The optimal payback rule is independent of the actual cost of the project. This term has cancelled out as a larger I has two effects on an investment project: If I is larger the absolute gain of waiting is increased, as waiting
Fig. 1. The effect of changes in the discount rate of the firm and the drift rate of the process on the gain of waiting and the payback period.
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implies not paying I now. However, waiting longer implies waiting for larger values of P, i.e., P* has increased, so that at the optimal point the payback period of the project has not changed. Similar to the discussion above, the factors r and a have two competing effects on PP*: they change the discount factor (rÿa), but they have the opposite effect on b1. This offsetting behaviour makes the payback period relatively stable against changes in the underlying parameters a and r. s enters the optimal payback period only via b1, so there is no competing effect. However, as we will show below, numerical estimates suggest that PP* does not vary much if s changes. In Figs. 2 and 3, it is shown how PP* changes if a and r (respectively s) change. We have done the calculation for values of s around 0.2, which is approximately equal to the annual volatility of the New York Stock Exchange Index. a varies around 0.09, approximately equal to the annual expected rate of growth of the NYSE Index (see Dixit & Pindyck, 1994). r is varied around 0.16, the only requirement being that it has to be larger than a. In Fig. 2, it can be seen that for a variety of values for r(s), PP* stays approximately constant around a value of 6 years. Numerically, the elasticity of the payback period with respect to a lies between 0.05 and 0.4. The value is larger if a approaches r and/or s is low. The elasticity of PP* with respect to variations in s also lies between 0.05 and 0.4, well below 1. It increases if a is low or s is large. Concerning changes in r, the elasticity varies between 0.5 and 0.65, where the latter values are taken for larger values of r, and are relatively independent of a and s. Although the elasticity with respect to r is slightly larger (but still well below 1), this does not imply that changes in the interest rate have necessarily a strong effect on PP*. Using the CAPM for example, the discount rate, which is the firm (or industry) specific cost of capital, includes the specific risk-factor of the industry. Thus, changes in the interest rate, which is only a part of the discount rate, change PP* by a lesser amount.
Fig. 2. PP*(a), s is held fixed at 0.2. r changes from 0.12 (upper curve) to 0.14, 0.16, . . ., 0.24 (lower curve).
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Fig. 3. PP*(a), r is held fixed at 0.16. s changes from 0.10 (upper curve) to 0.15, 0.20, . . ., 0.35 (lower curve).
4. Hurdle rate and the gain of waiting The IRR of the project given above is defined in Eq. (16): IRR a P=I
16
The optimal hurdle rate (HR*) is chosen in such a way that at P*, it coincides with the IRR, that is: 1 HR a P =I r a s2 b1 17 2 where the last equality follows from Eqs. (6) and (8). One benchmark case is a = 0, where the use of a hurdle rate is solely driven by the uncertainty of the project. Here, we get that the IRR has to exceed the firm-specific discount rate by 1/2s2b1. With s at 0.2 and r varying between 8% and 16%, this difference lies between 4% and 5%. This can be compared to the result pby Ross (1995), where the interest rate is uncertain and follows the Ito equation: dr sr rdz with z being a Wiener process. In that case, the difference between the IRR and the p optimal yield, i.e., the interest rate below which it is optimal to exercise the project, is s 2. For an interest rate of 6% and an annual proportional deviation of 20%, this difference is 3.5%. Although the uncertainty is quite different in the two models, it seems reassuring that for realistic parameter values the results are similar. If r increases, the difference (HR* ÿ r) increases as b1 depends positively on r. A change in s has two competing effects, it raises s2 and lowers b1, but the net effect is always positive. Also, a influences the difference in two ways, if there were no uncertainty, an increase in a would increase the difference, a result already known since Jorgenson's work on optimal investment behaviour (Jorgenson, 1963). But an increase in a lowers the effect of uncertainty
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(db1/da < 0). If the project is expected to increase stronger in value the negative effect of uncertainty is weakened. If there are two investment projects with the same NPV but different drift rates, we know from Section 3 that the gain of waiting is larger for that one with the larger drift rate. If one now calculates the NPV with the hurdle rate instead of the lower discount rate, the project with the larger drift rate has a lower value. Thus, the NPV calculated with a hurdle rate can be used as an indicator of the gain of waiting of a project. 5. Concluding remarks We have combined the recent literature on investment under uncertainty with the classic concepts of the payback criterion and the hurdle rate. We have shown that it can be rational to refer to one of those instruments as a rule of thumb to decide whether an investment project should be undertaken. In contrast to earlier arguments concerning the usefulness of the payback period as a decision instrument, where uncertainty or capital constraints were given as reasons, we have shown that the payback period can indicate the possible gain of waiting, i.e., a larger payback period indicates that it might be better to wait and see how things develop. Similar, as already argued by Ingersoll and Ross (1992), Dixit and Pindyck (1994), and Ross (1995), the use of a hurdle rate can be rationalised. Apart from being used merely as a means to increase the required NPV of a project, we have shown that the hurdle rate can be usefully employed to discriminate between projects with different payoff structures and different gains of waiting. Although the model used is specialÐwe solely concentrated on a geometric Brownian motion process for the payoff structureÐwe think that the insight gained by this model is more general. A lower payback period is an indicator of lower gains of waiting; if a project is such that it will pay off quickly, it is about time to undertake it. On the other hand, if the payback period is large, it is probably better to wait before doing the investment. Theoretical reasoning and numerical estimates suggest that the optimal payback period stays relatively constant if fundamental parameters change. These parameters may either be project specific like the expected growth rate or the uncertainty rate or firm or industry specific like the discount rate, which itself depends on the interest rate. In Eqs. (15) and (17), an explicit expression for the optimal payback period and the optimal hurdle rate as functions of the discount rate, the growth rate and the uncertainty is given. For realistic parameter values, the optimal payback period is around 5 to 7 years, while the hurdle rate lies in between 4% and 5%. The functional forms for these two expressions might be usefully employed to test the predictions of the theory. Acknowledgments I thank Franz Benstetter, Ray Rees and Bernd Rudolph for helpful comments and suggestions.
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References Abel, A. B. (1983). Optimal investment under uncertainty. American Economic Review, 73, 228 ± 233. Abel, A. B., & Eberly, J. C. (1994). A unified model of investment under uncertainty. American Economic Review, 84, 1369 ± 1384. Antle, R., & Eppen, G. (1985). Capital rationing and organisational slack in capital budgeting. Management Science, 3, 163 ± 174. Brealey, R. A., & Myers, S. C. (1991). Principles of Corporate Finance. New York: McGraw-Hill. Dixit, A. K. (1993). The art of smooth pasting. Fundamentals of Pure and Applied Economics (Vol. 55). Chur, Switzerland: Harwood Academic Publishers. Dixit, A. K., & Pindyck, R. S. (1994). Investment Under Uncertainty. Princeton, NJ: Princeton University Press. Dixit, A. K., & Pindyck, R. S. (1995). The New Option View of Investment. Working Paper, Alfred P. Sloan School of Management. Ingersoll, J. E., & Ross, S. A. (1992). Waiting to invest: investment and uncertainty. Journal of Business, 65, 1 ± 30. Jorgenson, D. (1963). Capital theory and investment behavior. American Economic Review, 53, 247 ± 259. McDonald, R., & Siegel, D. R. (1986). The value of waiting to invest. Quarterly Journal of Economics, 101, 707 ± 727. Narayana, M. I. (1985). Observability and the payback criterion. Journal of Business, 58, 305 ± 322. Ross, S. A. (1995). Uses, abuses, and alternatives to the net-present-value rule. Financial Management, 24(3), 96 ± 102. Schall, L. D., Sundem, G. L., Geigsbeek, W. R. Jr. (1978). Survey and analysis of capital budgeting methods. Journal of Finance, 33(1), 281 ± 287. Summers, L. H. (1987). Investment incentives and the discounting of depreciation allowances. Pp. 295 ± 304 in M. Feldstein (Ed.), The Effects of Taxation on Capital Accumulation. Chicago, IL: University of Chicago Press.
Payback criterion, hurdle rates and the gain of waiting Achim Wambach* Department of Economics, University of Munich, Ludwigstr.28 VG, 80539 Munich, Germany
Abstract In the standard literature on investment decisions, commonly observed instruments, namely the payback criterion and the use of hurdle rates, are criticised as they do not lead to a maximization of expected profits. By using the real option approach to investment under uncertainty, we show that it can be rational for the investor to use the payback criterion as a rule of thumb, because a shorter payback period indicates lower gains of waiting. In addition, we claim that the use of hurdle rates can serve a similar purpose: It allows to distinguish between projects with different values of waiting. An explicit expression for the optimal payback period and the optimal hurdle rate is given. D 2000 Elsevier Science Inc. All rights reserved. JEL classification: D21; D81; G12; L21 Keywords: Payback criterion; Hurdle rates; Gain of waiting
1. Introduction The standard literature on business studies (see e.g. Brealey & Myers, 1991) usually contains several sections on how firms decide whether to do an investment or not. Many concepts are used, out of which three we consider here: the standard net-present-value (NPV) concept, the use of hurdle rates and the payback criterion. The last concept is heavily disputed in the literature: It is not clear why an investment project with a shorter payback period should, in general, be preferred. In particular, it is very easy to construct examples where in one project the payoffs accrue at a later point in time, while for the other, the payoffs are large in the beginning but small later on. In this case, the payback criterion might allow for the latter project, but reject the first, which * Tel.: + 49-892-1802227; fax: + 49-892-1806282. E-mail address: [email protected] (A. Wambach) 1057-5219/00/$ ± see front matter D 2000 Elsevier Science Inc. All rights reserved. PII: S 1 0 5 7 - 5 2 1 9 ( 0 0 ) 0 0 0 2 8 - 4
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might not be optimal. The payback criterion systematically undervalues projects with a later payoff stream. One argument in favour of the payback rule is that if firms are constrained in their capital, it might be better to go for the investment project, which pays out earlier. However, one has to impose some form of asymmetric information to support this argument, otherwise, credit rationing will not hold. Another quite commonly heard argument is that if either the payoff or the lifetime of the project is uncertain, it is better to choose the project with the lower payback period. Although the latter is a very intuitive argument, it is not clear how a static concept, based on expected values, can be used as a means to decide between different uncertain projects. In particular, a possible finite lifetime should already be incorporated in the value of the expected payoff stream or at least in the project-specific discount rate, and it should not be the payback period, which differentiates between projects with different expected lifetimes. In a static environment, the use of a hurdle rate also seems to miss the point. Even in a deterministic dynamic scenario, there is no reason why the NPV should not be calculated by the use of the opportunity cost of capital for the firm, which, in general, is approximated by the firms weighted average cost of capital (WACC). But note the common feature of these two decision methods: Similar as the payback criterion, the use of a hurdle rate also discriminates in favour of projects with earlier payoffs. If two projects have the same NPV, calculated with the WACC, both instruments will decide against the project with the later payoff stream. However, given the criticism with these instruments, it needs to be explained why those concepts are so heavily used. Schall et al. (1978) report that more than 70% of the firms investigated work with the payback criterion. In a study by Summers (1987) for example, it was reported that firms use hurdle rates up to four times their cost of capital. The NPV concept, on the other hand, seemed to be quite a reliable guide to the decision whether to invest. However, the recent literature on investment under uncertainty (see Abel, 1983; Abel & Eberly, 1994; McDonald & Siegel, 1986; Ingersoll & Ross, 1992; Dixit & Pindyck, 1994; Ross, 1995 and references therein) has stressed that the NPV concept can be misleading; there are situations where the NPV of a project is positive, but one should still wait to do the investment. The main reason for this result is that in the standard NPV approach, there is no managerial flexibility, thus for example, the so-called option-to-wait, i.e. whether to do the investment today or tomorrow, is ignored. This option might have a positive value, it might be better to wait and see how things develop before doing the investment. The empirical evidence of the use of the NPV concept together with hurdle rates was given by Dixit and Pindyck (1995) as a strength of the new theory. This behaviour may not be `myopic' as some authors assume, but rather according to Dixit and Pindyck, it reflects the fact that a NPV much larger than zero is required to exercise the option to invest. Thus, a larger discount rate is used as a means to lower the calculated NPV. Still, even if we use this as an indication that implicitly managers follow the implications of the option-theory of investment, there are open questions which remain. First, it is not explained why the payback criterion is used at all, and second, if the hurdle rate is employed to lower the NPV in the sense of Dixit and Pindyck, why not just argue that the gross present value has to be x times as large as the costs of the project, where x is a number larger than one. Surely in this way, one would also invest only if the NPV is larger than zero.
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In this paper, we argue that the payback criterion and the hurdle rate can do even more. They both can be used as useful indicators of the possible ``gain of waiting'' of a project. Consider for example two projects with an uncertain profit stream for which the expected gross present values are x times the investment costs, where x is the same in both cases. However, the two projects have different expected payoff streams, the first one expects its payoff to accrue earlier than the second. Thus, the payback period of the first project will be lower than that of the second. Similarly, calculating the NPV with a hurdle rate, the NPV of the first project is larger than that of the second. But now consider the gain of waiting: as we will show in Section 2, the project with the later payoff has a larger gain of waiting, thus it is rational to do the other project first. In this case, a lower payback period and a larger NPV calculated with a hurdle rate both indicate a lower gain of waiting. To rationalise an instrument like the payback criterion, i.e., only to undertake projects that pay off in a prespecified time, it should be the case that this time is relatively stable against changes in the interest rate, or the timing of payoffs. A concept would not be very useful if for every different type of project, a different rule has to be set up. We are going to argue in this paper that the payback criterion can indeed be quite stable against changes in these parameters. In particular, for each parameter, there are two competing effects, which tend to cancel each other out. Take for example an increase in the interest rate: The optimal payback period depends positively on a parameter, which is negatively correlated to the gains of waiting. This parameter itself depends positively on the interest rate: a larger interest rate implies lower gains of waiting as the future counts less. Therefore, this effect leads to a positive correlation of the optimal payback period with the interest rate. On the other hand, cutting off a profit stream after the prespecified time (namely the payback period) leads to less profit unaccounted for in the case of a larger interest rate, as future profits matter less. This effect in turn reduces the optimal payback period. These results suggest that if an industry has developed an optimal payback rule, this rule stays relatively stable if projects have different payoff structures or if the interest rate changes. The level of uncertainty also enters the optimal payback period. Although there are no competing effects, i.e., a larger uncertainty rate reduces the optimal payback period, numerical estimates suggest that this effect is not large. Therefore, the main result of this paper is that the payback period rule can be usefully employed to account for the value of waiting of individual projects. Another explanation of the apparent irrational use of the payback criterion is given by Narayana (1985). There, a principal agent approach under asymmetric information is used. If two projects have the same NPV a manager may select that project with the quicker returns to demonstrate his ability, which can only be inferred from the vector of outputs. However, this model is not able to explain how the optimal cutoff time is determined and which factors are involved in this determination. Furthermore, it does not apply if the firm is owner-managed. In contrast to these results, in the present work, the use of the payback criterion is demonstrated to be useful for rational decision takers with symmetric information and the factors entering the cutoff time are determined. Similarly, there have been arguments brought forward to defend the use of hurdle rates as a means to weaken asymmetric information problems between the owner and the manager of a
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firm (see e.g. Antle & Eppen, 1985). If one includes, however, the possibility of postponing a project, the use of a hurdle rate can be rationalised without the need to refer to asymmetric information (Dixit & Pindyck, 1995). Ingersoll and Ross (1992) have modelled a project with a certain payoff stream in an environment where the future interest rate is uncertain. If an investment project can be postponed, the authors show that the investment should be undertaken at a prevailing interest rate lower than the internal rate of return (IRR). Or, turning the argument around, the hurdle rate used to calculate the NPV of the project is larger than the interest rate. In our context, we will consider projects where the firm's discount factor is stable, but the payoff of the project is uncertain. Then, there will be in addition to just increasing the required NPV another advantage of the use of the hurdle rate, namely to help to elicit the gain of waiting of a project. The paper is structured as follows: In section 2, we introduce the formalism of the optionto-wait approach. Here, we only consider a simple investment decision: one firm can decide to invest in an irreversible uncertain profit stream, the investment costs are fixed and have to be paid once only. The reason for this simplification (see Abel, 1983; Abel & Eberly, 1994 for more general projects) is that it captures the main ideas of the option-to-wait approach, while at the same time keeping the mathematics not too complicated. In this formalism, we derive the NPV of a given project and calculate its payback period. Section 3 is devoted to the investigation of the use of the payback criterion and its characteristic features. In section 4, the optimal hurdle rate is calculated and discussed. In section 5, we conclude. 2. The option-to-wait approach to irreversible investment under uncertainty This section gives a short introduction to the theory of investment under uncertainty. For a more detailed exposition, we refer the interested reader to the book by Dixit and Pindyck (1994). Consider a firm that has an investment opportunity. The costs of investment will be I, and doing the investment entitles the firm to receive the profit stream P(t). To model the uncertainty of the profit stream, we assume that P(t) follows a geometric Brownian motion, which implies that dP = aPdt + sPdz where dz denotes a standard Wiener process, a and s are constants. A different setup would be to model the expected present value of the project, i.e., the appropriately discounted expected cash flow, as a geometric Brownian motion (McDonald & Siegel, 1986). The project value can be seen as the market value of an asset, hence, the use of a geometric Brownian motion is less contentious here. However, to model the dependence on the occurrence of the timing of payments, we prefer to model an explicit payoff stream. The expected value of P is an increasing function of time, i.e., E[P(t)] = P0eat. a measures the drift rate of a project, therefore, a larger a implies a larger expected future payoff. If there are two projects, which have the same expected NPV, the one with the smaller a has its expected payoff accrue in the near future, while the other pays back at a later point in time. The larger s, the larger the uncertainty of the profit stream of the project is, which increases 2 over time. In particular, the variance of the payoff at time t is Var(P(t)) = P02e2at(es t ÿ 1). s describes the uncertainty rate of the individual project. As long as a > s2/2, the infimum of P(t) for all t is larger than 0, therefore, there is no fear of losing the project completely. Note that a larger uncertainty does not require a larger return of the project, because investors are
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assumed to be risk-neutral. However, as we will show later on, a larger s implies a larger gain of waiting. If the uncertainty is larger, it is probably better to wait for some time, even if the decision taker is risk-neutral. The question the investor now faces is: What is the optimal P* when she should undertake the investment? To answer this, it is helpful to consider the investment decision as a real option in analogy to an American call on a dividend paying stock, i.e., the investor has the option to pay I and receive the payoff stream P(t), but she does not need to exercise the option now, thus, the `option-to-wait' approach to investment. In order to calculate the optimal point of investment, we have to borrow some mathematics already developed in the theory of dynamic optimization under uncertainty. This works as follows: consider a firm that has already done the investment. Its value is the gross-present-value of the investment, which is Z 1 P ÿrt P te dt 1 GPV P E r ÿ a 0 where r is the discount rate of the firm, P is the known profit flow at time t = 0. As we do not assume that firms are risk-averse, r does not include a term proportional to the uncertainty of the project. Instead, it can be interpreted as the cost of capital a firm or industry faces, thus, it depends positively on the real interest rate, and in addition, there is also a dependency on the industry specific risk rate. A discussion on the specific value of r can be found in McDonald and Siegel (1986) and Dixit and Pindyck (1994). Now, consider a firm that has not undertaken the investment so far. Its value can be calculated by using Bellman's principle: V P EV P dPeÿrdt
2
that is the value of the investment option now is equal to the value of the discounted investment option tomorrow, where the payoff level has changed from P to P + dP. Eq. (2) holds as long as the investment option is not exercised, that is as long as P and P + dP are lower than P*, the strike price. The expectation on the right hand side can be expanded by using Ito's Lemma: 1 00 2 ÿrdt ÿrdt 0 EV P dPe E V P V PdP V P dP e 3 2 By using E[dP] = aPdt and E[(dP)2] = s2P2dt, expanding Eq. (3) in dt, we get the differential equation [Eq. (4)] for the value function: 1 ÿrV P aPV 0 P s2 P2 V 00 P 0 2 This can be solved to yield Eq. (5): V P A1 Pb1 A2 Pb2
4
5
where b1(2) are the roots of the quadratic equation 1 ÿr ab s2 b b ÿ 1 0 2
6
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We choose the numbering in such a way that b1 > 0, b2 < 0. As long as r > a, which we assume in the following, it can be shown that b1 > 1. A1(2) has to be determined by outside constraints. One such constraint is that if P = 0, the value of the project should be zero (V(0) = 0). This can be seen by noting that if P = 0, the stochastic process will never return to positive values of P again. Given that b2 is negative, this constraint implies that A2 has to be zero. To calculate A1, one has to consider the value of V at the optimal point of investment, which we denote by P*. Here, two equations have to be fulfilled [Eq. (7)]: First, the value of the option should be equal to the GPV minus investment costs, as one exchanges the option with the project. Therefore, if the option value were larger, it would not be rational to pay I now and receive the payoff stream. If the option value were lower, one should have invested already earlier. A second equation, which has to be satisfied at the optimal investment point is the so-called smooth pasting condition (see Dixit, 1993). This implies that the derivative of V with respect to P at the optimal point (P*) is the same as the derivative of GPV with respect to P at the same point. Thus, the following two equations define A1 and P*: V P GPV P ÿ I
7
V 0 P GPV0 P Using the expression for GPV from Eq. (1), this yields: b1 P r ÿ aI b1 ÿ 1 P ÿ I P ÿb1 A1 rÿa
8
P* is the critical point where it is optimal to invest, i.e., there are no further gains of waiting, the option should be exercised now. Note that this point cannot be derived by a usual NPV argument. The NPV of the project for a given P is given by Eq. (9): NPV P
P ÿI rÿa
9
As (b1)/(b1 ÿ 1) is larger than 1, the optimal investment point surely gives a positive NPV, but it can be considerably larger than the lowest P at which the NPV is positive.1 Thus, here, we have established the point that the NPV is not a sufficient means to decide on an investment opportunity. Before proceeding to the next section, let us finally calculate
1
Dixit and Pindyck (1994) have estimated that for sensible values of a, r and s, the factor (b1)/(b1ÿ1) can take on values of 2 and even larger.
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the expected payback period2 of the investment if it would be undertaken at a given level P. This time PP(P) is defined by "Z # PP P ÿrt P te dt ÿ I 0 10 E 0
From Eq. (10), it follows that ÿ1 I ln 1 ÿ r ÿ a PP P rÿa P
11
3. Payback criterion as a decision instrument From the discussion in Section 2, it is clear that the optimal payback period (PP*) of a project is finite. In the traditional view of investment, a payback period of infinity, which implies a NPV of 0, should have been enough to undertake the investment. However, as argued above, the NPV has to be larger than 0, therefore, it follows that PP* has to be finite. However, to say more, we have to consider explicitly the gain of waiting per unit of investment costs (GoW). For P > P*, GoW is 0, the project should directly be exercised. As long as P < P*, the gain of waiting is given by the option value minus the NPV of the exercised project [Eq. (12)]: P ÿ I =I 12 GoW A1 Pb1 ÿ rÿa where A1 is given in Eq. (8). Inserting A1 and P* from the first equation in Eq. (8), this reduces to b1 P P b1 ÿ1 ÿb1 13 b1 ÿ GoW b1 ÿ 1 I r ÿ a I r ÿ a The gain of waiting depends on two parameters, the factor b1 and the relative gross present value (RGPV) of the project, P/(I(r ÿ a)) [Eq. (13)]. Although b1 does also depend on the drift rate a and the discount rate r, it is useful to disentangle those two effects. Taking the derivative of GoW with respect to b1 we arrive at Eq (14): d b ÿ1 P 14 GoW GoW ln 1 db1 b1 I r ÿ a Using Eq. (8), it follows that as long as P is smaller than P*, the derivative is negative. This has the following implication: Taking the derivative of Eq. (6) with respect to a, s and r, it can be shown that b1 depends negatively on a and s, but positively on r. Thus, a lower drift
2
There is another version of the payback period that takes the equality of the undiscounted sum of profits with investment costs. We do not consider this rule here.
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or uncertainty rate reduces the gain of waiting, given that the GPV stays constant. The reverse is true with r. Taking the derivative of GoW with respect to the relative GPV, it can be shown that a larger RGPV reduces the GoW. This also makes sense intuitively, as a larger RGPV makes a project more profitable. The possible gain due to investing later cannot offset the loss due to forfeiting the profit in this period. But now consider the payback period, given in Eq. (11). PP depends negatively on the relative GPV and on the factor (r ÿ a), which is the dividend rate. Thus, the dependency of the payback period on the discount rate and the drift rate mimics that of the gain of waiting. This result is summarised in Fig. 1. The results obtained so far are true for a specific structure of the investment project. However, we think that the insight provided is more general. A larger interest rate for example increases the firm-specific discount rate. Thus, the NPV of a given project falls. With that, the gain of waiting increases as does the payback period. However, there is a second effect. A larger interest rate increases b1 and thus reduces the gain of waiting. Similarly, a larger interest rate increases the convenience yield r ÿ a and thus decreases the payback period, as the negative effect of cutting off the project at time PP is reduced. Hence, the project-specific payback period can be used as a guide to the individual gain of waiting. Having discussed the project-specific payback period, let us now turn to the payback criterion. Firms specify a time such that only projects whose payback period is lower than this time are undertaken, and this holds for a variety of projects. With the model given above, this optimal time for each project can be calculated by setting P = P* in Eq. (11). The optimal payback period (still as a function of a, r and s) is then: PP
ln b1 rÿa
15
For the payback criterion to make sense, one would like to have PP* be relatively stable against changes in the parameters. Otherwise, a different PP* would be needed for each individual project, thus offsetting the initial motivation of having one rule for many projects. Note first, that I does not enter PP*. The optimal payback rule is independent of the actual cost of the project. This term has cancelled out as a larger I has two effects on an investment project: If I is larger the absolute gain of waiting is increased, as waiting
Fig. 1. The effect of changes in the discount rate of the firm and the drift rate of the process on the gain of waiting and the payback period.
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implies not paying I now. However, waiting longer implies waiting for larger values of P, i.e., P* has increased, so that at the optimal point the payback period of the project has not changed. Similar to the discussion above, the factors r and a have two competing effects on PP*: they change the discount factor (rÿa), but they have the opposite effect on b1. This offsetting behaviour makes the payback period relatively stable against changes in the underlying parameters a and r. s enters the optimal payback period only via b1, so there is no competing effect. However, as we will show below, numerical estimates suggest that PP* does not vary much if s changes. In Figs. 2 and 3, it is shown how PP* changes if a and r (respectively s) change. We have done the calculation for values of s around 0.2, which is approximately equal to the annual volatility of the New York Stock Exchange Index. a varies around 0.09, approximately equal to the annual expected rate of growth of the NYSE Index (see Dixit & Pindyck, 1994). r is varied around 0.16, the only requirement being that it has to be larger than a. In Fig. 2, it can be seen that for a variety of values for r(s), PP* stays approximately constant around a value of 6 years. Numerically, the elasticity of the payback period with respect to a lies between 0.05 and 0.4. The value is larger if a approaches r and/or s is low. The elasticity of PP* with respect to variations in s also lies between 0.05 and 0.4, well below 1. It increases if a is low or s is large. Concerning changes in r, the elasticity varies between 0.5 and 0.65, where the latter values are taken for larger values of r, and are relatively independent of a and s. Although the elasticity with respect to r is slightly larger (but still well below 1), this does not imply that changes in the interest rate have necessarily a strong effect on PP*. Using the CAPM for example, the discount rate, which is the firm (or industry) specific cost of capital, includes the specific risk-factor of the industry. Thus, changes in the interest rate, which is only a part of the discount rate, change PP* by a lesser amount.
Fig. 2. PP*(a), s is held fixed at 0.2. r changes from 0.12 (upper curve) to 0.14, 0.16, . . ., 0.24 (lower curve).
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Fig. 3. PP*(a), r is held fixed at 0.16. s changes from 0.10 (upper curve) to 0.15, 0.20, . . ., 0.35 (lower curve).
4. Hurdle rate and the gain of waiting The IRR of the project given above is defined in Eq. (16): IRR a P=I
16
The optimal hurdle rate (HR*) is chosen in such a way that at P*, it coincides with the IRR, that is: 1 HR a P =I r a s2 b1 17 2 where the last equality follows from Eqs. (6) and (8). One benchmark case is a = 0, where the use of a hurdle rate is solely driven by the uncertainty of the project. Here, we get that the IRR has to exceed the firm-specific discount rate by 1/2s2b1. With s at 0.2 and r varying between 8% and 16%, this difference lies between 4% and 5%. This can be compared to the result pby Ross (1995), where the interest rate is uncertain and follows the Ito equation: dr sr rdz with z being a Wiener process. In that case, the difference between the IRR and the p optimal yield, i.e., the interest rate below which it is optimal to exercise the project, is s 2. For an interest rate of 6% and an annual proportional deviation of 20%, this difference is 3.5%. Although the uncertainty is quite different in the two models, it seems reassuring that for realistic parameter values the results are similar. If r increases, the difference (HR* ÿ r) increases as b1 depends positively on r. A change in s has two competing effects, it raises s2 and lowers b1, but the net effect is always positive. Also, a influences the difference in two ways, if there were no uncertainty, an increase in a would increase the difference, a result already known since Jorgenson's work on optimal investment behaviour (Jorgenson, 1963). But an increase in a lowers the effect of uncertainty
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(db1/da < 0). If the project is expected to increase stronger in value the negative effect of uncertainty is weakened. If there are two investment projects with the same NPV but different drift rates, we know from Section 3 that the gain of waiting is larger for that one with the larger drift rate. If one now calculates the NPV with the hurdle rate instead of the lower discount rate, the project with the larger drift rate has a lower value. Thus, the NPV calculated with a hurdle rate can be used as an indicator of the gain of waiting of a project. 5. Concluding remarks We have combined the recent literature on investment under uncertainty with the classic concepts of the payback criterion and the hurdle rate. We have shown that it can be rational to refer to one of those instruments as a rule of thumb to decide whether an investment project should be undertaken. In contrast to earlier arguments concerning the usefulness of the payback period as a decision instrument, where uncertainty or capital constraints were given as reasons, we have shown that the payback period can indicate the possible gain of waiting, i.e., a larger payback period indicates that it might be better to wait and see how things develop. Similar, as already argued by Ingersoll and Ross (1992), Dixit and Pindyck (1994), and Ross (1995), the use of a hurdle rate can be rationalised. Apart from being used merely as a means to increase the required NPV of a project, we have shown that the hurdle rate can be usefully employed to discriminate between projects with different payoff structures and different gains of waiting. Although the model used is specialÐwe solely concentrated on a geometric Brownian motion process for the payoff structureÐwe think that the insight gained by this model is more general. A lower payback period is an indicator of lower gains of waiting; if a project is such that it will pay off quickly, it is about time to undertake it. On the other hand, if the payback period is large, it is probably better to wait before doing the investment. Theoretical reasoning and numerical estimates suggest that the optimal payback period stays relatively constant if fundamental parameters change. These parameters may either be project specific like the expected growth rate or the uncertainty rate or firm or industry specific like the discount rate, which itself depends on the interest rate. In Eqs. (15) and (17), an explicit expression for the optimal payback period and the optimal hurdle rate as functions of the discount rate, the growth rate and the uncertainty is given. For realistic parameter values, the optimal payback period is around 5 to 7 years, while the hurdle rate lies in between 4% and 5%. The functional forms for these two expressions might be usefully employed to test the predictions of the theory. Acknowledgments I thank Franz Benstetter, Ray Rees and Bernd Rudolph for helpful comments and suggestions.
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