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Optimal size, optimal timing and optimal financing of an investment
Journal of Macroeconomics 33 (2011) 681–689
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Optimal size, optimal timing and optimal financing of an investment Sudipto Sarkar ⇑ DSB 302, McMaster University, Hamilton, ON, Canada L8S 4K4
a r t i c l e
i n f o
Article history: Received 18 January 2011 Accepted 25 August 2011 Available online 17 September 2011 JEL Classification: G3 Keywords: Real option model Investment timing Investment size Corporate financing decision Capacity
a b s t r a c t Corporate investment is an important determinant of economic well-being. The existing literature identifies optimal investment size and timing without the possibility of debt financing, as well as the effect of debt financing on investment timing without the option to choose investment size. This paper contributes to the literature by identifying the optimal size, optimal timing and optimal financing for an investment when the firm controls all three decisions (as it usually does in practice). The investment size and investment trigger are generally positively related: when investment is delayed (accelerated) it is larger (smaller) in size, thus the overall effect on investment is ambiguous. However, when tax rate or bankruptcy cost is increased, the trigger rises and size falls, hence the effect on investment is unambiguously negative. The effect of debt financing on investment depends on the amount of debt used; with the optimal amount of debt, investment is delayed relative to the no-debt case, and this delay can be economically significant; however, the investment, when eventually made, will be larger in size. Overall, it is not appropriate to ignore either the firm’s ability to choose investment size or its option to use debt financing, when modeling the investment decision. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction It is widely recognized in the Economics literature that corporate investment is an important determinant of economic well-being (Bar-Ilan and Strange, 1999; DeLong and Summers, 1991,1994). A growing strand of the investment literature takes a real-option approach to analyzing the investment decision, as summarized nicely in Dixit and Pindyck (1994). However, Hubbard (1994) points out that this ‘‘real-option’’ approach focuses on the timing of the investment but not its size or intensity. Bar-Ilan and Strange (1999) addressed this shortcoming by computing both the optimal timing and optimal intensity/size of the investment, and found that they often behave in unexpected ways; for instance, in response to a parameter change, investment could be larger but delayed, or smaller but earlier. However, their model was limited to firms that use no debt financing, although virtually all firms in practice use some debt financing. This paper extends the Bar-Ilan and Strange (1999) model to the case when the investment is financed with both debt and equity, used in the optimal (value-maximizing) ratio. Some papers have examined the effect of debt financing on the timing of investment, e.g., Mauer and Sarkar (2005) and Lyandres and Zhdanov (2006), concluding that debt financing, used optimally, lowers the investment trigger and thus accelerates investment. However, these models assume fixed investment size, which is typically not the case in real life, as pointed out by Bar-Ilan and Strange (1999) and Dangl (1999). This paper contributes to the investment literature by identifying the optimal investment decision (size and timing) when financed (possibly partly) by debt; that is, we compute the optimal mix of debt and equity financing, the optimal size/capacity, and the optimal investment trigger. This problem has not been solved in the literature to date. Our paper ⇑ Tel.: +1 905 525 9140x23959; fax: +1 905 521 8995. E-mail address: [email protected] 0164-0704/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2011.08.002
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can be viewed as an extension of Bar-Ilan and Strange (1999) and Dangl (1999) to the case when the investment is partly debt-financed, and an extension of Mauer and Sarkar (2005) and Lyandres and Zhdanov (2006) to include the optimal size of the investment. The main results are as follows. Debt financing can have a significant impact on investment size and timing, and the direction and magnitude of the effect depends on the amount of debt used. When the optimal amount of debt is used, the investment trigger rises or investment is delayed (relative to the no-debt case), and the delay can be substantial, e.g., a ‘‘base case’’ delay of two and a quarter years. However, when the investment is eventually made, it is larger in size. In general, investment size and investment trigger are positively related, i.e., delayed (accelerated) investment is larger (smaller) in size, as in Bar-Ilan and Strange (1999). However, there are two exceptions – when responding to an increase in the tax rate or bankruptcy costs, investment is both delayed and smaller. Finally, there is substantial variation in optimal investment size. Thus, a model of investment without the option to choose investment size or the option to use debt financing is not appropriate. The rest of the paper is organized as follows. Section 2 describes the model, which is similar to Bar-Ilan and Strange (1999) but with the addition of limited liability and debt financing. Section 3 computes the project value, and Section 4 computes the value of the levered firm’s debt and equity. Section 5 identifies the optimal investment trigger for a given capacity, and Section 6 identifies the optimal capacity. Section 7 examines the effect of debt financing and identifies the optimal level of debt, Section 8 presents the comparative static results, and Section 9 concludes. 2. The model The starting point of our model is similar to Bar-Ilan and Strange (1999). A firm is considering an investment opportunity and must make three decisions (instead of two decisions as in Bar-Ilan and Strange, 1999): (i) when to invest, (ii) how large an investment to make, and (iii) how to finance the investment. Let the level of capital (investment) be K, and k the price per unit of capital. Thus, by paying an investment cost of kK, the firm receives in exchange the project with productive capacity Q = Ka, where 0 6 a 6 1 and Q is the number of units produced per unit time. The investment cost can then be written as kQb, where b = 1/a. The marginal cost of production (or the variable cost) is w per unit. The output price P is stochastic and follows the lognormal process:
dP=P ¼ ldt þ rdz
ð1Þ
where l and r are the drift and volatility, and dz the increment of a standard Wiener process. Future cash flows are discounted at a constant discount rate of q (we assume q > l to preclude infinite values). Thus far our model is the same as Bar-Ilan and Strange (1999). In addition, the firm also has to decide on the financing mix, i.e., what fractions of the investment cost will be financed by equity and debt. This decision is expressed by the choice of the coupon payment to debt holders, c per unit time. Thus the pre-tax profit stream is p = [Q(P w) c] per unit time, and this profit stream is taxed at the corporate tax rate of s. After the project is accepted, shareholders can declare bankruptcy if the firm keeps making losses and future prospects do not look good. As is standard in the corporate finance literature (e.g., Leland, 1994), shareholders declare bankruptcy when the output price P falls sufficiently, say to Pb. At bankruptcy, shareholders walk away with nothing (i.e., zero payoff), and the firm’s ownership is transferred to bondholders after incurring fractional bankruptcy costs of a (0 6 a 6 1). Thus, the postbankruptcy firm is unlevered and owned by the former bondholders. If P continues to decline, at some point the project will be shut down permanently and the firm will exit the industry (note that both the bankruptcy option and the exit option were not available in Bar-Ilan and Strange, 1999). Let the exit trigger be Pe, i.e., when the price falls to this level the project will be shut down permanently; we make the standard assumption that the salvage value at exit is zero (Mauer and Ott, 2000; Mauer and Sarkar, 2005). Both the triggers Pb and Pe will be determined optimally in our model. Thus the solution to the problem consists of (i) the investment trigger price, say P, on reaching which the company will make the investment, (ii) production capacity, say Q (or equivalently, capital level K), and (iii) debt level c. These three choices are made optimally and simultaneously in our model. 3. Project value We first value the project by itself, i.e., without the effects of leverage. The project value will be a function of the state variable P. Since there is no explicit time limit for the project cash flows, the project value will be independent of calendar time. Let V(P) denote the project value. Then, as in Bar-Ilan and Strange (1999), it can be shown that V(P) must satisfy the ordinary differential equation (ODE):
0:5r2 P 2 V 00 ðPÞ þ lPV 0 ðPÞ qVðPÞ þ p ¼ 0
ð2Þ
where p is the profit or cash flow per unit time from the project. Here, p = (1 s)Q(P w). Solving Eq. (2), we get the project value
VðPÞ ¼ ð1 sÞQ
P w þ A0 P c1 þ A1 Pc2 ð q lÞ q
ð3Þ
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where A0 and A1 are constants to be determined by boundary conditions, and c1 and c2 are the positive and negative roots, respectively, of the quadratic equation:
0:5r2 cðc 1Þ þ lc q ¼ 0
ð4Þ
and are given by
c1 ¼ 0:5 l=r2 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2q=r2 þ ð0:5 l=r2 Þ2
ð5Þ
c2 ¼ 0:5 l=r2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2q=r2 þ ð0:5 l=r2 Þ2
ð6Þ
The first term in Eq. (3) represents the project value without the exit option, and the other terms capture the exit option. When P reaches very high levels (P ? 1), the exit option is worthless, hence the project value must equal the first term, i.e., V(P) ? (1 s)Q[P/(q l) w/q]. This implies A0 = 0 in Eq. (3), hence the project value can be written as
VðPÞ ¼ ð1 sÞQ
P w þ A1 Pc2 ðq lÞ q
ð7Þ
There is one boundary (P = Pe), and the two unknowns A1 and Pe can be computed from the following boundary conditions (see Dixit and Pindyck, 1994, for details of these boundary conditions): (i) Value-matching or continuity condition, which requires that, at the boundary, the project value must equal the payoff, i.e., V(Pe) = 0; (ii) Smooth-pasting or optimality condition, which requires that the derivatives be equal at the boundary, i.e., V0 (Pe) = 0. Solving these two boundary conditions, we get the two unknowns:
Pe ¼ wð1 l=qÞ=ð1 1=c2 Þ A1 ¼
ð1 sÞQ ðPe Þ1c2 c2 ðq lÞ
ð8Þ ð9Þ
This completes the project valuation. Note that, in Bar-Ilan and Strange (1999), the project value had a simpler expression because their model did not include the exit option. However, in order to be consistent with ‘‘limited liability,’’ we felt it necessary to include the exit option in the model. 4. Valuation of levered firm’s debt and equity Here we value the securities (debt and equity) of the operating levered firm after the investment has been made. Debt and equity, being contingent claims on the firm, will also be functions of the state variable P, and their values must also satisfy the ODE (2) with the appropriate values of p. Debt: The cash flow to debt holders is the coupon payment, hence p = c. Solving ODE (2) with this value of p, we get the debt value D(P):
DðPÞ ¼ c=q þ D0 Pc1 þ D1 Pc2
ð10Þ
The term c/q is the value of the bond if risk-free, and the other terms capture the default risk. For very high values of P (for P ? 1), the bond is essentially risk-free, hence D(P) ? c/q, which implies D0 = 0. Thus, debt value can be written as
DðPÞ ¼ c=q þ D1 Pc2
ð11Þ
As discussed above, when P falls to Pb, shareholders declare bankruptcy, leaving the bondholders with the assets of the firm after incurring fractional bankruptcy costs of a; thus, bondholders are left with (1 a) times the project value at bankruptcy, giving the boundary condition:
DðPb Þ ¼ ð1 aÞVðP b Þ which gives the constant
D1 ¼ ½ð1 aÞVðPb Þ c=qðPb Þc2
ð12Þ
Equity: The cash flow or after-tax net profit (i.e., after coupon payment) to equity holders is given by p = (1 s) [Q(P w)c]. Solving ODE (2) with this p, we get the equity value E(P):
EðPÞ ¼ ð1 sÞQ
P w þ c=Q þ E0 Pc1 þ E1 Pc2 ðq lÞ q
ð13Þ
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The first term gives the equity value without the bankruptcy option, and the other two terms capture the effects of the bankruptcy option. When P is very large (P ? 1), bankruptcy becomes very unlikely, hence E(P) ? (1 s)Q[P/(q l) (w + c/Q)/ q], which implies E0 = 0. Hence, equity value can be written as
EðPÞ ¼ ð1 sÞQ
P w þ c=Q þ E1 Pc2 ð q lÞ q
ð14Þ
To complete the valuation of debt and equity, we must identify the constant E1 and the bankruptcy trigger Pb. This can be done form the two boundary conditions associated with the bankruptcy boundary P = Pb. The two boundary conditions are the same as above, i.e., value-matching and smooth-pasting:
Value-matching : EðPb Þ ¼ 0
ð15Þ
Smooth-pasting : E0 ðPb Þ ¼ 0
ð16Þ
Solving Eqs. (15) and (16), we get:
Pb ¼
ðw þ c=Q Þð1 l=qÞ ð1 1=c2 Þ
ð17Þ
ð1 sÞQ ðP b Þ1c2 c2 ðq lÞ
ð18Þ
E1 ¼
This completes the valuation of the levered company’s equity and debt. 5. The investment decision As in Bar-Ilan and Strange (1999) and Dangl (1999), the firm has the option to invest, and exercises the option when P rises to the trigger level P. The optimal trigger is the one that maximizes the total value of the levered firm (debt plus equity). The value of the option to invest will be a function of the state variable P, as in the above papers. Let the option value be F(P); then, it can be shown that F(P) must satisfy the ODE (2) with p = 0. The solution is:
FðPÞ ¼ F 0 P c1 þ F 1 Pc2
ð19Þ
where F0 and F1 are constants to be determined from the boundary conditions. For very small values of P(P ? 0), the option becomes essentially worthless (since P follows a lognormal process, it will remain at zero once it reaches that level). That is, as P ? 0, we have F(P) ? 0, which implies F1 = 0 in Eq. (19). Thus, we can write the option value as:
FðPÞ ¼ F 0 Pc1
ð20Þ
This option exercise trigger or boundary is P = P . The associated boundary conditions are value-matching and smoothpasting, as in the earlier sections. However, the firm will now maximize its total value, hence the value-matching condition will equate the option value at exercise to the sum of equity and debt values at exercise less the cost of option exercise (investment cost):
FðP Þ ¼ F 0 ðP Þc1 ¼ DðP Þ þ EðP Þ kQ
b
ð21Þ
and the smooth-pasting condition will be:
F’ðP Þ ¼ F 0 c1 ðP Þc1 1 ¼ D0ðP Þ þ E0ðP Þ
ð22Þ
The two boundary conditions (Eqs. (21) and (22)) are solved for the two unknowns F0 and P, giving
F 0 ¼ ½DðP Þ þ EðP Þ kQ ðP Þc1 b
ð23Þ
and an equation that has to be solved (numerically, since there is no analytical solution) for P :
sc ð1 sÞQw ð1 sÞQP ð1 1=c1 Þ b þ þ ðD1 þ E1 Þð1 c2 =c1 ÞðP Þc2 ¼ kQ ðq lÞ q q
ð24Þ
We now have, for a given capacity Q, the optimal investment trigger P. 6. Optimal capacity So far, Q has been treated as a given exogenous parameter. In real life, however, the firm has to choose the operating scale or capacity Q. An optimal choice of Q would maximize the value of the option to invest, F 0 P c1 , for any P; this is equivalent to 0 maximizing the value of F0. We therefore set dF ¼ 0 in order to identify the optimal capacity Q. This gives, after some dQ simplification:
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ð1 sÞ
P w dE1 dD1 kbQ b1 þ ðP Þc2 þ ¼0 ðq lÞ q dQ dQ
ð25Þ
where
dE1 w=c þ c2 =Q ¼ E1 1 þ Qw=c dQ dD1 ð1 aÞA1 cc2 ð1 aÞð1 sÞ 1 1 Qw 1 ¼ þ 1 1=c2 1 þ c=ðQwÞ cc2 ð1 c2 Þ ð1 aÞð1 sÞð1 þ Qw=cÞ dQ Q Q qðPb Þc2 Eq. (25) can be solved (numerically) for the optimal capacity Q. However, in order to identify both optimal investment trigger and optimal investment size (i.e., P and Q), Eqs. (24) and (25) must be solved simultaneously. 7. Effect of debt financing and the optimal debt level When the investment is financed (partly) with debt, there are two main effects. One, using debt allows the firm to increase value by reducing its tax obligations, since interest payments are tax deductible; hence debt has a valuable tax shield associated with it, which makes the investment more attractive. Two, there are bankruptcy costs associated with debt, which reduces project value and thus makes the project less attractive. The overall effect of debt financing on the investment decision (timing and size) will depend on the amount of debt financing used. To illustrate this effect, we solve Eqs. (24) and (25) numerically for different debt levels c. For the numerical solutions, we use the same parameter values used by Bar-Ilan and Strange (1999), i.e., l = 0.01, q = 0.05, r = 0.05, k = 4, w = 1, and b = 2 (or a = 0.5). In addition, we have two parameters that did not have a role in the Bar-Ilan and Strange (1999) model, the corporate tax rate s and bankruptcy cost a; both of these are taken from Leland (1994), so that s = 0.15 and a = 0.5. The numerical results are illustrated in Fig. 1. First, if the firm uses no debt financing (i.e., c = 0), the output is as follows: P = 1.7256, Q = 2.4587, and F0 = 3.1626. (Note that these results, while close to the Bar-Ilan and Strange (1999) results, are somewhat different because of corporate tax and the exit option). As c (leverage) is increased, both the investment trigger P and the capacity Q initially decline and subsequently rise. For instance, with c = 0.22 (leverage ratio of about 10%), we get P = 1.6676 and Q = 2.3047, and with c = 3.7 (leverage ratio 75%), we get P = 1.9528 and Q = 3.5097. Thus, the investment trigger and capacity of a levered firm can be quite different from those of an unlevered firm, and can be lower or higher depending on how much debt financing is used. The level of debt financing is up to the company, since it controls the financing decision.How much debt financing will the company use? To answer this question, we need to look at the behavior of the project value. At any point prior to investment, the project value is given by the value of the option to invest, which is F 0 Pc1 . The company wants to maximize this value for any given P, hence it should maximize the value of the parameter F0. Fig. 1 therefore also shows how the value of F0 varies with c. It is seen that F0 first rises and then falls as c is increased, giving an inverted-U shaped relationship; this can be expected from the two valuation effects associated with debt financing, discussed at the beginning of this section; for small levels of debt the tax shield effect dominates hence value rises, while for large levels of debt the bankruptcy cost effect dominates hence value falls. Hence there is a unique debt level (or leverage ratio) which maximizes value. Eye estimation indicates that this optimal
6 5 4 3 2 1 0
0
1
2
3
4
5
6
Coupon c Investment Trigger P*
Capacity Q
Leverage Ratio
Value F0
Fig. 1. The Effect of debt financing on the investment decision (investment trigger P, capacity Q, and value F0). The following parameter values are used: l = 0.01, q = 0.05, r = 0.05, k = 4, w = 1, b = 2, s = 0.15 and a = 0.5.
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S. Sarkar / Journal of Macroeconomics 33 (2011) 681–689
debt level is about c = 2.3. In the next section, we identify the optimal debt level c more precisely. The above effects are summarized in: 7.1. Result 1 Debt financing can potentially have a significant effect on the timing and intensity of investment, with the direction and magnitude of the effect depending on the amount of debt used. 7.2. Optimal debt level The optimal debt level c can be identified as in Leland (1994): for any given value of P, the optimal coupon level maximizes the total firm value (equity plus debt), i.e., c = Argmax {D(P,c) + E(P,c)}. The first order condition for this optimization is dDðPÞ þ dEðPÞ ¼ 0, which simplifies to: dc dc
s r
þ Pc2
dD1 dE1 ¼0 þ dc dc
ð26Þ
Differentiating with respect to c and simplifying, we get:
dD1 c2 ½1 ð1 aÞð1 sÞ=ð1 þ Qw=cÞ 1 dE1 ð1 sÞ and ¼ ¼ dc dc rðPb Þc2 rðP b Þc2 Substituting into Eq. (26) and simplifying, we get an equation that must be satisfied by the optimal coupon level:
c2 Pb c ½1 ð1 aÞð1 sÞ þ 2 ¼1 P sð1 þ Qw=cÞ
ð27Þ
where Pb is given by Eq. (17). Eq. (27) gives the optimal c for any given P, and is solved numerically. In our model, the financing decision is made when the investment is made, i.e., at P = P. Thus, the optimal c will satisfy the equation:
c2 Pb c ½1 ð1 aÞð1 sÞ þ 2 ¼1 P sð1 þ Qw=cÞ
ð28Þ
(Incidentally, this is the same as maximizing the parameter F0). Therefore, in order to identify the optimal leverage (or c), optimal investment timing (P) and optimal capacity (Q), we have to simultaneously solve (numerically) Eqs. (24), (25), and (28). With the above base case parameter values, the solutions to Eqs. (24), (25), and (28) are as follows: optimal debt level c = 2.3113, optimal investment trigger P = 1.7604 and optimal capacity Q = 2.8077. The corresponding value is F0 = 3.8285 and leverage ratio 70.84%. Comparing these figures with the unlevered case, we find that the optimally levered firm will invest later (but will have a larger capacity), which is the opposite of the traditional (exogenous-capacity) result that optimal debt financing results in earlier investment (Mauer and Sarkar, 2005; Lyandres and Zhdanov, 2006). This is because, when Q is endogenous as in our model, debt financing allows the company to choose a larger capacity, which results in delayed investment (since the investment cost is an increasing function of capacity). The above effects of leverage (confirmed by the comparative static results in Section 8 below) can be summarized in: 7.3. Result 2 Optimal debt usage in financing the investment generally results in larger but delayed investment (relative to the no-debt case). How large are the effects of debt financing? Comparing the above outputs with the unlevered case, we note that the value (F0) increases by about 21% when the firm uses leverage optimally; this is not surprising since optimal debt financing should increase value (the tax shield will dominate bankruptcy costs). The capacity Q increases by about 14% when debt financing is used. The investment trigger P is only about 2% higher when leverage is used, which does not seem that significant. However, we can gauge its effect on investment timing by computing how much the investment is delayed because of the higher investment trigger. Suppose the current value of the state variable is P0 = 1.7256. Then the unlevered firm will invest right away (since its investment trigger is 1.7256), but the levered firm will wait until P rises to its trigger level of 1.7604. As discussed in Mauer and Ott (2000), the expected time for the variable P to go from P0 to P1 is given by the expression: 1 =P 0 Þ EðTÞ ¼ lnðP l0:5r2 . With the appropriate substitutions in the above expression, we get the expected time to investment of the
¼ 2:28 years; that is, using debt financing will delay the investment by two optimally levered firm, EðTÞ ¼ lnð1:7604=1:7256Þ 0:010:5ð0:05Þ2 and a quarter years; this is a fairly substantial delay. Thus, although the effect on the investment trigger might seem small, the effect on investment timing can be economically significant.
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8. Comparative static results Table 1 summarizes the comparative static results with a wide range of parameter values. It can be noted that the investment trigger P is closely related to the investment size Q. In most cases, the two are positively related; that is, both rise or fall together. This is not surprising, since a larger size results in higher investment cost (recall that investment cost = kQb, where b P 1). Thus, when investment is delayed (P rises) it is larger (Q also rises); and when investment is accelerated (P falls) it is smaller (Q also falls). Although the investment is delayed, when it is eventually made, it will be larger in size. Table 1 Comparative static results. Q
P No Lev Base case
Lev
F0
No Lev
Lev
No Lev
Leverage ratio of levered firm (%) Lev
1.7256
1.7604
2.4587
2.8077
3.1626
3.8285
70.84
1.4384 1.7256 2.2166 3.0737 4.8493 10.5057
1.4549 1.7604 2.2700 3.1546 4.9840 10.8084
1.6958 2.4587 3.7633 6.0408 10.7579 25.7823
1.9744 2.8077 4.2284 6.7062 11.8313 28.1335
2.2353 3.1626 4.6977 7.0275 10.5734 16.3628
2.8782 3.8285 5.5052 8.0741 11.9765 18.3213
83.89 70.84 63.51 58.60 54.75 51.40
1.3789 1.4953 1.7256 2.2882 4.4621
1.3874 1.5152 1.7604 2.3407 4.5654
0.8240 1.4072 2.4587 4.8212 13.6784
0.8912 1.5671 2.8077 5.5789 15.9297
0.3013 1.0595 3.1626 8.5605 23.6171
0.3378 1.2302 3.8285 10.7372 30.3816
53.55 62.04 70.84 78.11 83.42
0.03 0.04 0.05 0.06 0.07
3.2268 2.0520 1.7256 1.5699 1.4775
3.3203 2.1008 1.7604 1.5974 1.5004
13.6010 4.6112 2.4587 1.5652 1.0987
15.6735 5.2877 2.8077 1.7812 1.2467
38.6075 8.8256 3.1626 1.4335 0.7526
47.7095 10.7751 3.8285 1.7242 0.9006
74.77 72.48 70.84 69.61 68.63
w 0.50 0.75 1.00 1.25 1.50
0.8628 1.2942 1.7256 2.1570 2.5884
0.8802 1.3203 1.7604 2.2005 2.6406
1.2293 1.8440 2.4587 3.0733 3.6880
1.4039 2.1058 2.8077 3.5097 4.2116
10.4796 5.1998 3.1626 2.1505 1.5692
12.6864 6.2948 3.8285 2.6033 1.8997
70.84 70.84 70.84 70.84 70.84
k 2 3 4 5 6
1.7256 1.7256 1.7256 1.7256 1.7256
1.7604 1.7604 1.7604 1.7604 1.7604
4.9173 3.2782 2.4587 1.9669 1.6391
5.6154 3.7436 2.8077 2.2462 1.8718
6.3251 4.2168 3.1626 2.5301 2.1084
7.6571 5.1047 3.8285 3.0628 2.5524
70.84 70.84 70.84 70.84 70.84
a 0.33 0.40 0.50 0.60 0.67
1.3376 1.4462 1.7256 2.4281 4.0948
1.3571 1.4702 1.7604 2.4877 4.2095
0.9758 1.2355 2.4587 1.8223 136.17
1.0366 1.3434 2.8077 14.5835 181.81
2.5129 2.5725 3.1626 5.9891 16.5761
2.8544 2.9827 3.8285 7.7629 23.0709
66.60 68.15 70.84 74.09 76.65
1.7256 1.7256 1.7256 1.7256 1.7256
1.7404 1.7512 1.7604 1.7688 1.7766
2.7479 2.6033 2.4587 2.3140 2.1694
2.8597 2.8327 2.8077 2.7839 2.7610
3.9505 3.5456 3.1626 2.8014 2.4622
4.1441 3.9741 3.8285 3.6979 3.5777
57.31 65.71 70.84 74.58 77.55
– – – – –
1.7437 1.7545 1.7604 1.7643 1.7673
– – – – –
2.8246 2.8131 2.8077 2.8041 2.8014
– – – – –
4.0144 3.8913 3.8285 3.7879 3.7562
84.84 75.83 70.84 67.42 64.83
r 0.025 0.05 0.075 0.10 0.125 0.15
l 0.0 0.005 0.01 0.015 0.02
q
s 0.05 0.10 0.15 0.20 0.25
a 0.00 0.25 0.50 0.75 1.00
Note: F0: denotes the effective value of the investment opportunity (the actual value of the option to invest is F 0 P c1 ). No Lev: denotes a firm that is unable to use debt financing. Lev: denotes a firm that uses the optimal mix of debt and equity financing. Base case parameter values: l = 0.01, q = 0.05, r = 0.05, k = 4, w = 1, b = 2 (or a = 0.5), s = 0.15 and a = 0.5.
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Delayed entry is unfavorable to investment while larger size or capacity is favorable to investment; similarly, accelerated entry is favorable to investment while smaller capacity is unfavorable to investment. In these cases, therefore, the overall effect on investment is ambiguous. This is what Bar-Ilan and Strange (1999, p. 58) refer to as the ‘‘perverse comparative static result’’ that the investment size (intensity) often behaves differently than the timing of investment. Interestingly, for the two leverage-related parameters, tax rate (s) and bankruptcy cost (a), we get the opposite result; that is, when s or a is raised, we see that P rises (investment delayed) and Q falls (investment size reduced). Thus, the firm responds to the unfavorable situation of higher tax rate or bankruptcy cost by both delaying entry and reducing capacity, both actions unfavorable to investment. That is, the effect on investment is unambiguously negative. The ‘‘perverse comparative static result’’ of Bar-Ilan and Strange (1999) is not observed for these two parameters. Another exception is the case of the unit investment cost parameter k: as k is increased, P remains unchanged while Q falls. In this case, the firm responds to a higher k by reducing the investment size but not changing the entry trigger. The above observations lead to: 8.1. Result 3 The optimal investment size and optimal investment trigger are generally positively related, i.e., when P rises (investment delayed) Q also rises (larger investment), and when P falls (investment accelerated) Q also falls (smaller investment). The exceptions occur when the parameters tax rate, bankruptcy cost and unit investment cost vary. Most of the other results are as expected, e.g., higher volatility r results in higher investment trigger P, larger investment size Q, and lower leverage ratio; higher growth rate l results in higher P and Q and higher leverage ratio; higher discount rate q results in lower P and Q and higher leverage ratio; and higher operating cost w results in higher P and Q but leaves the leverage ratio unchanged. A higher k makes investing more costly, hence Q is smaller, but P (and leverage ratio) remain unchanged. A higher a (or lower b) reduces the cost of investing, as a result of which Q (hence P) increases; it also increases the leverage ratio. A higher tax rate s makes the investment less attractive, reducing Q and increasing P; but it makes debt more attractive (higher tax shield), hence leverage ratio rises. Finally, a higher bankruptcy cost a also makes the investment less attractive, reducing Q and increasing P; it also makes debt less attractive, hence leverage ratio falls. Finally, the following interesting relationships can be noted from the comparative static results in Table 1: (1) In every single case, the investment trigger with debt financing exceeds the investment trigger without debt financing, thus confirming Result 2 for all parameter combinations examined. This result is just the opposite of the traditional (exogenous-capacity) literature, e.g., Mauer and Sarkar (2005) and Lyandres and Zhdanov (2006). (2) Also in every single case, the capacity or investment size with debt financing exceeds the capacity without debt financing. This explains point 1 above: debt financing makes the investment more attractive (because of the tax shield), hence the investment size is larger; however, since a larger investment is more costly, it results in delayed investment. (3) Thus, the optimal use of debt financing affects investment positively (via the larger investment) as well as negatively (via the delayed investment); the overall effect of debt financing on investment is ambiguous. (4) Finally, we note from Table 1 that there can be significant differences in investment size and timing between the levered and the no-debt cases. Also, there is substantial variation in the optimal investment size/capacity, particularly with respect to the parameters r, l, q and a. Thus, when modeling the corporate investment decision, it would be inappropriate to ignore either the debt financing option or the option to choose the optimal investment size, since both can have a significant impact on the investment decision. 9. Conclusions Corporate investment is an important determinant of economic growth and well-being, hence corporate investment policies are well worth studying. There is a strand of the Economics literature that identifies the optimal investment size and timing for firms that are unable to use debt financing, e.g., Bar-Ilan and Strange (1999) and Dangl (1999). Another strand of the literature (Mauer and Sarkar, 2005; Lyandres and Zhdanov, 2006) examines the effect of debt on investment timing when the company cannot choose the size of the investment (i.e., the size is pre-specified). This paper combines the two strands to identify the optimal investment policy when the firm has both the option to use debt financing and choose the investment size. That is, we compute the optimal size, optimal timing and optimal financing arrangement for an investment, using a real-option model similar to the papers mentioned above. This is the most realistic scenario, since the firm generally gets to choose the size or capacity of the investment, as well as the financing mix (the combination of debt and equity to finance the investment). Because the three optimizing decisions are taken simultaneously, the model is too complicated to allow closed-form solutions, and we had to resort to numerical solutions. Nevertheless, the numerical results are repeated for a wide range of parameter values, to ensure that the results are robust. The main results are as follows: Debt financing can have a significant effect on the investment decision (both timing and size); the direction and magnitude of this effect depends on the amount of debt financing used.
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When the firm can choose the scale of operations and uses the optimal amount of debt financing, investment is delayed (relative to the no-debt case); this is the opposite of the result from the existing (exogenous-size) literature, e.g., Mauer and Sarkar (2005) and Lyandres and Zhdanov (2006). However, with optimal debt financing, when the (delayed) investment is made, it is on a larger scale (again, relative to the no-debt case). The ‘‘perverse comparative static results’’ discussed in Bar-Ilan and Strange (1999) is also found to exist in our model in most cases; however, for two parameters (corporate tax rate and bankruptcy cost), an increase in the parameter value results in delayed investment as well as smaller investment. There is substantial variation in the optimal investment size/capacity, particularly with respect to the parameters r, l, q and a. Given the significant effect of debt financing on the investment decision, and the substantial variation in optima investment size, we conclude that it would be inappropriate to ignore either the debt financing option or the option to choose the optimal investment size when modeling the investment decision of a company, as both seem to have a significant effect on the investment decision. Acknowledgments The author thanks the editor Theodore Palivos and an anonymous referee for helpful suggestions. Financial support from the Social Science and Humanities Research Council (SSHRC) of Canada is also acknowledged. References Bar-Ilan, A., Strange, W.C., 1999. The timing and intensity of investment. Journal of Macroeconomics 21, 57–77. Dangl, T., 1999. Investment and capacity choice under uncertain demand. European Journal of Operational Research 117, 415–428. DeLong, J.B., Summers, L.H., 1991. Equipment investment and economic growth. Quarterly Journal of Economics 106, 445–502. DeLong, J.B., Summers, L.H., 1994. How strongly do developing economies benefit from equipment investment? Journal of Monetary Economics 32, 395– 415. Dixit, A., Pindyck, R.S., 1994. Investment Under Uncertainty. Princeton University Press, Princeton, NJ. Hubbard, R.G., 1994. Investment under uncertainty: keeping one’s options open. Journal of Economic Literature 32, 1816–1831. Leland, H., 1994. Risky debt, bond covenants and optimal capital structure. Journal of Finance 49, 1213–1252. Lyandres, E., Zhdanov, A., 2006. Accelerated Investment in the Presence of Risky Debt, Working Paper. Mauer, D.C., Ott, S., 2000. Agency costs, under-investment, and optimal capital structure: the effect of growth options to expand. In: Brennan, M., Trigeorgis, L. (Eds.), Project Flexibility, Agency, and Competition. Oxford University Press, New York, pp. 151–180. Mauer, D.C., Sarkar, S., 2005. Real options, agency conflicts, and optimal capital structure. Journal of Banking and Finance 29, 1405–1428.
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Journal of Macroeconomics journal homepage: www.elsevier.com/locate/jmacro
Optimal size, optimal timing and optimal financing of an investment Sudipto Sarkar ⇑ DSB 302, McMaster University, Hamilton, ON, Canada L8S 4K4
a r t i c l e
i n f o
Article history: Received 18 January 2011 Accepted 25 August 2011 Available online 17 September 2011 JEL Classification: G3 Keywords: Real option model Investment timing Investment size Corporate financing decision Capacity
a b s t r a c t Corporate investment is an important determinant of economic well-being. The existing literature identifies optimal investment size and timing without the possibility of debt financing, as well as the effect of debt financing on investment timing without the option to choose investment size. This paper contributes to the literature by identifying the optimal size, optimal timing and optimal financing for an investment when the firm controls all three decisions (as it usually does in practice). The investment size and investment trigger are generally positively related: when investment is delayed (accelerated) it is larger (smaller) in size, thus the overall effect on investment is ambiguous. However, when tax rate or bankruptcy cost is increased, the trigger rises and size falls, hence the effect on investment is unambiguously negative. The effect of debt financing on investment depends on the amount of debt used; with the optimal amount of debt, investment is delayed relative to the no-debt case, and this delay can be economically significant; however, the investment, when eventually made, will be larger in size. Overall, it is not appropriate to ignore either the firm’s ability to choose investment size or its option to use debt financing, when modeling the investment decision. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction It is widely recognized in the Economics literature that corporate investment is an important determinant of economic well-being (Bar-Ilan and Strange, 1999; DeLong and Summers, 1991,1994). A growing strand of the investment literature takes a real-option approach to analyzing the investment decision, as summarized nicely in Dixit and Pindyck (1994). However, Hubbard (1994) points out that this ‘‘real-option’’ approach focuses on the timing of the investment but not its size or intensity. Bar-Ilan and Strange (1999) addressed this shortcoming by computing both the optimal timing and optimal intensity/size of the investment, and found that they often behave in unexpected ways; for instance, in response to a parameter change, investment could be larger but delayed, or smaller but earlier. However, their model was limited to firms that use no debt financing, although virtually all firms in practice use some debt financing. This paper extends the Bar-Ilan and Strange (1999) model to the case when the investment is financed with both debt and equity, used in the optimal (value-maximizing) ratio. Some papers have examined the effect of debt financing on the timing of investment, e.g., Mauer and Sarkar (2005) and Lyandres and Zhdanov (2006), concluding that debt financing, used optimally, lowers the investment trigger and thus accelerates investment. However, these models assume fixed investment size, which is typically not the case in real life, as pointed out by Bar-Ilan and Strange (1999) and Dangl (1999). This paper contributes to the investment literature by identifying the optimal investment decision (size and timing) when financed (possibly partly) by debt; that is, we compute the optimal mix of debt and equity financing, the optimal size/capacity, and the optimal investment trigger. This problem has not been solved in the literature to date. Our paper ⇑ Tel.: +1 905 525 9140x23959; fax: +1 905 521 8995. E-mail address: [email protected] 0164-0704/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2011.08.002
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can be viewed as an extension of Bar-Ilan and Strange (1999) and Dangl (1999) to the case when the investment is partly debt-financed, and an extension of Mauer and Sarkar (2005) and Lyandres and Zhdanov (2006) to include the optimal size of the investment. The main results are as follows. Debt financing can have a significant impact on investment size and timing, and the direction and magnitude of the effect depends on the amount of debt used. When the optimal amount of debt is used, the investment trigger rises or investment is delayed (relative to the no-debt case), and the delay can be substantial, e.g., a ‘‘base case’’ delay of two and a quarter years. However, when the investment is eventually made, it is larger in size. In general, investment size and investment trigger are positively related, i.e., delayed (accelerated) investment is larger (smaller) in size, as in Bar-Ilan and Strange (1999). However, there are two exceptions – when responding to an increase in the tax rate or bankruptcy costs, investment is both delayed and smaller. Finally, there is substantial variation in optimal investment size. Thus, a model of investment without the option to choose investment size or the option to use debt financing is not appropriate. The rest of the paper is organized as follows. Section 2 describes the model, which is similar to Bar-Ilan and Strange (1999) but with the addition of limited liability and debt financing. Section 3 computes the project value, and Section 4 computes the value of the levered firm’s debt and equity. Section 5 identifies the optimal investment trigger for a given capacity, and Section 6 identifies the optimal capacity. Section 7 examines the effect of debt financing and identifies the optimal level of debt, Section 8 presents the comparative static results, and Section 9 concludes. 2. The model The starting point of our model is similar to Bar-Ilan and Strange (1999). A firm is considering an investment opportunity and must make three decisions (instead of two decisions as in Bar-Ilan and Strange, 1999): (i) when to invest, (ii) how large an investment to make, and (iii) how to finance the investment. Let the level of capital (investment) be K, and k the price per unit of capital. Thus, by paying an investment cost of kK, the firm receives in exchange the project with productive capacity Q = Ka, where 0 6 a 6 1 and Q is the number of units produced per unit time. The investment cost can then be written as kQb, where b = 1/a. The marginal cost of production (or the variable cost) is w per unit. The output price P is stochastic and follows the lognormal process:
dP=P ¼ ldt þ rdz
ð1Þ
where l and r are the drift and volatility, and dz the increment of a standard Wiener process. Future cash flows are discounted at a constant discount rate of q (we assume q > l to preclude infinite values). Thus far our model is the same as Bar-Ilan and Strange (1999). In addition, the firm also has to decide on the financing mix, i.e., what fractions of the investment cost will be financed by equity and debt. This decision is expressed by the choice of the coupon payment to debt holders, c per unit time. Thus the pre-tax profit stream is p = [Q(P w) c] per unit time, and this profit stream is taxed at the corporate tax rate of s. After the project is accepted, shareholders can declare bankruptcy if the firm keeps making losses and future prospects do not look good. As is standard in the corporate finance literature (e.g., Leland, 1994), shareholders declare bankruptcy when the output price P falls sufficiently, say to Pb. At bankruptcy, shareholders walk away with nothing (i.e., zero payoff), and the firm’s ownership is transferred to bondholders after incurring fractional bankruptcy costs of a (0 6 a 6 1). Thus, the postbankruptcy firm is unlevered and owned by the former bondholders. If P continues to decline, at some point the project will be shut down permanently and the firm will exit the industry (note that both the bankruptcy option and the exit option were not available in Bar-Ilan and Strange, 1999). Let the exit trigger be Pe, i.e., when the price falls to this level the project will be shut down permanently; we make the standard assumption that the salvage value at exit is zero (Mauer and Ott, 2000; Mauer and Sarkar, 2005). Both the triggers Pb and Pe will be determined optimally in our model. Thus the solution to the problem consists of (i) the investment trigger price, say P, on reaching which the company will make the investment, (ii) production capacity, say Q (or equivalently, capital level K), and (iii) debt level c. These three choices are made optimally and simultaneously in our model. 3. Project value We first value the project by itself, i.e., without the effects of leverage. The project value will be a function of the state variable P. Since there is no explicit time limit for the project cash flows, the project value will be independent of calendar time. Let V(P) denote the project value. Then, as in Bar-Ilan and Strange (1999), it can be shown that V(P) must satisfy the ordinary differential equation (ODE):
0:5r2 P 2 V 00 ðPÞ þ lPV 0 ðPÞ qVðPÞ þ p ¼ 0
ð2Þ
where p is the profit or cash flow per unit time from the project. Here, p = (1 s)Q(P w). Solving Eq. (2), we get the project value
VðPÞ ¼ ð1 sÞQ
P w þ A0 P c1 þ A1 Pc2 ð q lÞ q
ð3Þ
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where A0 and A1 are constants to be determined by boundary conditions, and c1 and c2 are the positive and negative roots, respectively, of the quadratic equation:
0:5r2 cðc 1Þ þ lc q ¼ 0
ð4Þ
and are given by
c1 ¼ 0:5 l=r2 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2q=r2 þ ð0:5 l=r2 Þ2
ð5Þ
c2 ¼ 0:5 l=r2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2q=r2 þ ð0:5 l=r2 Þ2
ð6Þ
The first term in Eq. (3) represents the project value without the exit option, and the other terms capture the exit option. When P reaches very high levels (P ? 1), the exit option is worthless, hence the project value must equal the first term, i.e., V(P) ? (1 s)Q[P/(q l) w/q]. This implies A0 = 0 in Eq. (3), hence the project value can be written as
VðPÞ ¼ ð1 sÞQ
P w þ A1 Pc2 ðq lÞ q
ð7Þ
There is one boundary (P = Pe), and the two unknowns A1 and Pe can be computed from the following boundary conditions (see Dixit and Pindyck, 1994, for details of these boundary conditions): (i) Value-matching or continuity condition, which requires that, at the boundary, the project value must equal the payoff, i.e., V(Pe) = 0; (ii) Smooth-pasting or optimality condition, which requires that the derivatives be equal at the boundary, i.e., V0 (Pe) = 0. Solving these two boundary conditions, we get the two unknowns:
Pe ¼ wð1 l=qÞ=ð1 1=c2 Þ A1 ¼
ð1 sÞQ ðPe Þ1c2 c2 ðq lÞ
ð8Þ ð9Þ
This completes the project valuation. Note that, in Bar-Ilan and Strange (1999), the project value had a simpler expression because their model did not include the exit option. However, in order to be consistent with ‘‘limited liability,’’ we felt it necessary to include the exit option in the model. 4. Valuation of levered firm’s debt and equity Here we value the securities (debt and equity) of the operating levered firm after the investment has been made. Debt and equity, being contingent claims on the firm, will also be functions of the state variable P, and their values must also satisfy the ODE (2) with the appropriate values of p. Debt: The cash flow to debt holders is the coupon payment, hence p = c. Solving ODE (2) with this value of p, we get the debt value D(P):
DðPÞ ¼ c=q þ D0 Pc1 þ D1 Pc2
ð10Þ
The term c/q is the value of the bond if risk-free, and the other terms capture the default risk. For very high values of P (for P ? 1), the bond is essentially risk-free, hence D(P) ? c/q, which implies D0 = 0. Thus, debt value can be written as
DðPÞ ¼ c=q þ D1 Pc2
ð11Þ
As discussed above, when P falls to Pb, shareholders declare bankruptcy, leaving the bondholders with the assets of the firm after incurring fractional bankruptcy costs of a; thus, bondholders are left with (1 a) times the project value at bankruptcy, giving the boundary condition:
DðPb Þ ¼ ð1 aÞVðP b Þ which gives the constant
D1 ¼ ½ð1 aÞVðPb Þ c=qðPb Þc2
ð12Þ
Equity: The cash flow or after-tax net profit (i.e., after coupon payment) to equity holders is given by p = (1 s) [Q(P w)c]. Solving ODE (2) with this p, we get the equity value E(P):
EðPÞ ¼ ð1 sÞQ
P w þ c=Q þ E0 Pc1 þ E1 Pc2 ðq lÞ q
ð13Þ
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The first term gives the equity value without the bankruptcy option, and the other two terms capture the effects of the bankruptcy option. When P is very large (P ? 1), bankruptcy becomes very unlikely, hence E(P) ? (1 s)Q[P/(q l) (w + c/Q)/ q], which implies E0 = 0. Hence, equity value can be written as
EðPÞ ¼ ð1 sÞQ
P w þ c=Q þ E1 Pc2 ð q lÞ q
ð14Þ
To complete the valuation of debt and equity, we must identify the constant E1 and the bankruptcy trigger Pb. This can be done form the two boundary conditions associated with the bankruptcy boundary P = Pb. The two boundary conditions are the same as above, i.e., value-matching and smooth-pasting:
Value-matching : EðPb Þ ¼ 0
ð15Þ
Smooth-pasting : E0 ðPb Þ ¼ 0
ð16Þ
Solving Eqs. (15) and (16), we get:
Pb ¼
ðw þ c=Q Þð1 l=qÞ ð1 1=c2 Þ
ð17Þ
ð1 sÞQ ðP b Þ1c2 c2 ðq lÞ
ð18Þ
E1 ¼
This completes the valuation of the levered company’s equity and debt. 5. The investment decision As in Bar-Ilan and Strange (1999) and Dangl (1999), the firm has the option to invest, and exercises the option when P rises to the trigger level P. The optimal trigger is the one that maximizes the total value of the levered firm (debt plus equity). The value of the option to invest will be a function of the state variable P, as in the above papers. Let the option value be F(P); then, it can be shown that F(P) must satisfy the ODE (2) with p = 0. The solution is:
FðPÞ ¼ F 0 P c1 þ F 1 Pc2
ð19Þ
where F0 and F1 are constants to be determined from the boundary conditions. For very small values of P(P ? 0), the option becomes essentially worthless (since P follows a lognormal process, it will remain at zero once it reaches that level). That is, as P ? 0, we have F(P) ? 0, which implies F1 = 0 in Eq. (19). Thus, we can write the option value as:
FðPÞ ¼ F 0 Pc1
ð20Þ
This option exercise trigger or boundary is P = P . The associated boundary conditions are value-matching and smoothpasting, as in the earlier sections. However, the firm will now maximize its total value, hence the value-matching condition will equate the option value at exercise to the sum of equity and debt values at exercise less the cost of option exercise (investment cost):
FðP Þ ¼ F 0 ðP Þc1 ¼ DðP Þ þ EðP Þ kQ
b
ð21Þ
and the smooth-pasting condition will be:
F’ðP Þ ¼ F 0 c1 ðP Þc1 1 ¼ D0ðP Þ þ E0ðP Þ
ð22Þ
The two boundary conditions (Eqs. (21) and (22)) are solved for the two unknowns F0 and P, giving
F 0 ¼ ½DðP Þ þ EðP Þ kQ ðP Þc1 b
ð23Þ
and an equation that has to be solved (numerically, since there is no analytical solution) for P :
sc ð1 sÞQw ð1 sÞQP ð1 1=c1 Þ b þ þ ðD1 þ E1 Þð1 c2 =c1 ÞðP Þc2 ¼ kQ ðq lÞ q q
ð24Þ
We now have, for a given capacity Q, the optimal investment trigger P. 6. Optimal capacity So far, Q has been treated as a given exogenous parameter. In real life, however, the firm has to choose the operating scale or capacity Q. An optimal choice of Q would maximize the value of the option to invest, F 0 P c1 , for any P; this is equivalent to 0 maximizing the value of F0. We therefore set dF ¼ 0 in order to identify the optimal capacity Q. This gives, after some dQ simplification:
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ð1 sÞ
P w dE1 dD1 kbQ b1 þ ðP Þc2 þ ¼0 ðq lÞ q dQ dQ
ð25Þ
where
dE1 w=c þ c2 =Q ¼ E1 1 þ Qw=c dQ dD1 ð1 aÞA1 cc2 ð1 aÞð1 sÞ 1 1 Qw 1 ¼ þ 1 1=c2 1 þ c=ðQwÞ cc2 ð1 c2 Þ ð1 aÞð1 sÞð1 þ Qw=cÞ dQ Q Q qðPb Þc2 Eq. (25) can be solved (numerically) for the optimal capacity Q. However, in order to identify both optimal investment trigger and optimal investment size (i.e., P and Q), Eqs. (24) and (25) must be solved simultaneously. 7. Effect of debt financing and the optimal debt level When the investment is financed (partly) with debt, there are two main effects. One, using debt allows the firm to increase value by reducing its tax obligations, since interest payments are tax deductible; hence debt has a valuable tax shield associated with it, which makes the investment more attractive. Two, there are bankruptcy costs associated with debt, which reduces project value and thus makes the project less attractive. The overall effect of debt financing on the investment decision (timing and size) will depend on the amount of debt financing used. To illustrate this effect, we solve Eqs. (24) and (25) numerically for different debt levels c. For the numerical solutions, we use the same parameter values used by Bar-Ilan and Strange (1999), i.e., l = 0.01, q = 0.05, r = 0.05, k = 4, w = 1, and b = 2 (or a = 0.5). In addition, we have two parameters that did not have a role in the Bar-Ilan and Strange (1999) model, the corporate tax rate s and bankruptcy cost a; both of these are taken from Leland (1994), so that s = 0.15 and a = 0.5. The numerical results are illustrated in Fig. 1. First, if the firm uses no debt financing (i.e., c = 0), the output is as follows: P = 1.7256, Q = 2.4587, and F0 = 3.1626. (Note that these results, while close to the Bar-Ilan and Strange (1999) results, are somewhat different because of corporate tax and the exit option). As c (leverage) is increased, both the investment trigger P and the capacity Q initially decline and subsequently rise. For instance, with c = 0.22 (leverage ratio of about 10%), we get P = 1.6676 and Q = 2.3047, and with c = 3.7 (leverage ratio 75%), we get P = 1.9528 and Q = 3.5097. Thus, the investment trigger and capacity of a levered firm can be quite different from those of an unlevered firm, and can be lower or higher depending on how much debt financing is used. The level of debt financing is up to the company, since it controls the financing decision.How much debt financing will the company use? To answer this question, we need to look at the behavior of the project value. At any point prior to investment, the project value is given by the value of the option to invest, which is F 0 Pc1 . The company wants to maximize this value for any given P, hence it should maximize the value of the parameter F0. Fig. 1 therefore also shows how the value of F0 varies with c. It is seen that F0 first rises and then falls as c is increased, giving an inverted-U shaped relationship; this can be expected from the two valuation effects associated with debt financing, discussed at the beginning of this section; for small levels of debt the tax shield effect dominates hence value rises, while for large levels of debt the bankruptcy cost effect dominates hence value falls. Hence there is a unique debt level (or leverage ratio) which maximizes value. Eye estimation indicates that this optimal
6 5 4 3 2 1 0
0
1
2
3
4
5
6
Coupon c Investment Trigger P*
Capacity Q
Leverage Ratio
Value F0
Fig. 1. The Effect of debt financing on the investment decision (investment trigger P, capacity Q, and value F0). The following parameter values are used: l = 0.01, q = 0.05, r = 0.05, k = 4, w = 1, b = 2, s = 0.15 and a = 0.5.
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debt level is about c = 2.3. In the next section, we identify the optimal debt level c more precisely. The above effects are summarized in: 7.1. Result 1 Debt financing can potentially have a significant effect on the timing and intensity of investment, with the direction and magnitude of the effect depending on the amount of debt used. 7.2. Optimal debt level The optimal debt level c can be identified as in Leland (1994): for any given value of P, the optimal coupon level maximizes the total firm value (equity plus debt), i.e., c = Argmax {D(P,c) + E(P,c)}. The first order condition for this optimization is dDðPÞ þ dEðPÞ ¼ 0, which simplifies to: dc dc
s r
þ Pc2
dD1 dE1 ¼0 þ dc dc
ð26Þ
Differentiating with respect to c and simplifying, we get:
dD1 c2 ½1 ð1 aÞð1 sÞ=ð1 þ Qw=cÞ 1 dE1 ð1 sÞ and ¼ ¼ dc dc rðPb Þc2 rðP b Þc2 Substituting into Eq. (26) and simplifying, we get an equation that must be satisfied by the optimal coupon level:
c2 Pb c ½1 ð1 aÞð1 sÞ þ 2 ¼1 P sð1 þ Qw=cÞ
ð27Þ
where Pb is given by Eq. (17). Eq. (27) gives the optimal c for any given P, and is solved numerically. In our model, the financing decision is made when the investment is made, i.e., at P = P. Thus, the optimal c will satisfy the equation:
c2 Pb c ½1 ð1 aÞð1 sÞ þ 2 ¼1 P sð1 þ Qw=cÞ
ð28Þ
(Incidentally, this is the same as maximizing the parameter F0). Therefore, in order to identify the optimal leverage (or c), optimal investment timing (P) and optimal capacity (Q), we have to simultaneously solve (numerically) Eqs. (24), (25), and (28). With the above base case parameter values, the solutions to Eqs. (24), (25), and (28) are as follows: optimal debt level c = 2.3113, optimal investment trigger P = 1.7604 and optimal capacity Q = 2.8077. The corresponding value is F0 = 3.8285 and leverage ratio 70.84%. Comparing these figures with the unlevered case, we find that the optimally levered firm will invest later (but will have a larger capacity), which is the opposite of the traditional (exogenous-capacity) result that optimal debt financing results in earlier investment (Mauer and Sarkar, 2005; Lyandres and Zhdanov, 2006). This is because, when Q is endogenous as in our model, debt financing allows the company to choose a larger capacity, which results in delayed investment (since the investment cost is an increasing function of capacity). The above effects of leverage (confirmed by the comparative static results in Section 8 below) can be summarized in: 7.3. Result 2 Optimal debt usage in financing the investment generally results in larger but delayed investment (relative to the no-debt case). How large are the effects of debt financing? Comparing the above outputs with the unlevered case, we note that the value (F0) increases by about 21% when the firm uses leverage optimally; this is not surprising since optimal debt financing should increase value (the tax shield will dominate bankruptcy costs). The capacity Q increases by about 14% when debt financing is used. The investment trigger P is only about 2% higher when leverage is used, which does not seem that significant. However, we can gauge its effect on investment timing by computing how much the investment is delayed because of the higher investment trigger. Suppose the current value of the state variable is P0 = 1.7256. Then the unlevered firm will invest right away (since its investment trigger is 1.7256), but the levered firm will wait until P rises to its trigger level of 1.7604. As discussed in Mauer and Ott (2000), the expected time for the variable P to go from P0 to P1 is given by the expression: 1 =P 0 Þ EðTÞ ¼ lnðP l0:5r2 . With the appropriate substitutions in the above expression, we get the expected time to investment of the
¼ 2:28 years; that is, using debt financing will delay the investment by two optimally levered firm, EðTÞ ¼ lnð1:7604=1:7256Þ 0:010:5ð0:05Þ2 and a quarter years; this is a fairly substantial delay. Thus, although the effect on the investment trigger might seem small, the effect on investment timing can be economically significant.
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8. Comparative static results Table 1 summarizes the comparative static results with a wide range of parameter values. It can be noted that the investment trigger P is closely related to the investment size Q. In most cases, the two are positively related; that is, both rise or fall together. This is not surprising, since a larger size results in higher investment cost (recall that investment cost = kQb, where b P 1). Thus, when investment is delayed (P rises) it is larger (Q also rises); and when investment is accelerated (P falls) it is smaller (Q also falls). Although the investment is delayed, when it is eventually made, it will be larger in size. Table 1 Comparative static results. Q
P No Lev Base case
Lev
F0
No Lev
Lev
No Lev
Leverage ratio of levered firm (%) Lev
1.7256
1.7604
2.4587
2.8077
3.1626
3.8285
70.84
1.4384 1.7256 2.2166 3.0737 4.8493 10.5057
1.4549 1.7604 2.2700 3.1546 4.9840 10.8084
1.6958 2.4587 3.7633 6.0408 10.7579 25.7823
1.9744 2.8077 4.2284 6.7062 11.8313 28.1335
2.2353 3.1626 4.6977 7.0275 10.5734 16.3628
2.8782 3.8285 5.5052 8.0741 11.9765 18.3213
83.89 70.84 63.51 58.60 54.75 51.40
1.3789 1.4953 1.7256 2.2882 4.4621
1.3874 1.5152 1.7604 2.3407 4.5654
0.8240 1.4072 2.4587 4.8212 13.6784
0.8912 1.5671 2.8077 5.5789 15.9297
0.3013 1.0595 3.1626 8.5605 23.6171
0.3378 1.2302 3.8285 10.7372 30.3816
53.55 62.04 70.84 78.11 83.42
0.03 0.04 0.05 0.06 0.07
3.2268 2.0520 1.7256 1.5699 1.4775
3.3203 2.1008 1.7604 1.5974 1.5004
13.6010 4.6112 2.4587 1.5652 1.0987
15.6735 5.2877 2.8077 1.7812 1.2467
38.6075 8.8256 3.1626 1.4335 0.7526
47.7095 10.7751 3.8285 1.7242 0.9006
74.77 72.48 70.84 69.61 68.63
w 0.50 0.75 1.00 1.25 1.50
0.8628 1.2942 1.7256 2.1570 2.5884
0.8802 1.3203 1.7604 2.2005 2.6406
1.2293 1.8440 2.4587 3.0733 3.6880
1.4039 2.1058 2.8077 3.5097 4.2116
10.4796 5.1998 3.1626 2.1505 1.5692
12.6864 6.2948 3.8285 2.6033 1.8997
70.84 70.84 70.84 70.84 70.84
k 2 3 4 5 6
1.7256 1.7256 1.7256 1.7256 1.7256
1.7604 1.7604 1.7604 1.7604 1.7604
4.9173 3.2782 2.4587 1.9669 1.6391
5.6154 3.7436 2.8077 2.2462 1.8718
6.3251 4.2168 3.1626 2.5301 2.1084
7.6571 5.1047 3.8285 3.0628 2.5524
70.84 70.84 70.84 70.84 70.84
a 0.33 0.40 0.50 0.60 0.67
1.3376 1.4462 1.7256 2.4281 4.0948
1.3571 1.4702 1.7604 2.4877 4.2095
0.9758 1.2355 2.4587 1.8223 136.17
1.0366 1.3434 2.8077 14.5835 181.81
2.5129 2.5725 3.1626 5.9891 16.5761
2.8544 2.9827 3.8285 7.7629 23.0709
66.60 68.15 70.84 74.09 76.65
1.7256 1.7256 1.7256 1.7256 1.7256
1.7404 1.7512 1.7604 1.7688 1.7766
2.7479 2.6033 2.4587 2.3140 2.1694
2.8597 2.8327 2.8077 2.7839 2.7610
3.9505 3.5456 3.1626 2.8014 2.4622
4.1441 3.9741 3.8285 3.6979 3.5777
57.31 65.71 70.84 74.58 77.55
– – – – –
1.7437 1.7545 1.7604 1.7643 1.7673
– – – – –
2.8246 2.8131 2.8077 2.8041 2.8014
– – – – –
4.0144 3.8913 3.8285 3.7879 3.7562
84.84 75.83 70.84 67.42 64.83
r 0.025 0.05 0.075 0.10 0.125 0.15
l 0.0 0.005 0.01 0.015 0.02
q
s 0.05 0.10 0.15 0.20 0.25
a 0.00 0.25 0.50 0.75 1.00
Note: F0: denotes the effective value of the investment opportunity (the actual value of the option to invest is F 0 P c1 ). No Lev: denotes a firm that is unable to use debt financing. Lev: denotes a firm that uses the optimal mix of debt and equity financing. Base case parameter values: l = 0.01, q = 0.05, r = 0.05, k = 4, w = 1, b = 2 (or a = 0.5), s = 0.15 and a = 0.5.
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Delayed entry is unfavorable to investment while larger size or capacity is favorable to investment; similarly, accelerated entry is favorable to investment while smaller capacity is unfavorable to investment. In these cases, therefore, the overall effect on investment is ambiguous. This is what Bar-Ilan and Strange (1999, p. 58) refer to as the ‘‘perverse comparative static result’’ that the investment size (intensity) often behaves differently than the timing of investment. Interestingly, for the two leverage-related parameters, tax rate (s) and bankruptcy cost (a), we get the opposite result; that is, when s or a is raised, we see that P rises (investment delayed) and Q falls (investment size reduced). Thus, the firm responds to the unfavorable situation of higher tax rate or bankruptcy cost by both delaying entry and reducing capacity, both actions unfavorable to investment. That is, the effect on investment is unambiguously negative. The ‘‘perverse comparative static result’’ of Bar-Ilan and Strange (1999) is not observed for these two parameters. Another exception is the case of the unit investment cost parameter k: as k is increased, P remains unchanged while Q falls. In this case, the firm responds to a higher k by reducing the investment size but not changing the entry trigger. The above observations lead to: 8.1. Result 3 The optimal investment size and optimal investment trigger are generally positively related, i.e., when P rises (investment delayed) Q also rises (larger investment), and when P falls (investment accelerated) Q also falls (smaller investment). The exceptions occur when the parameters tax rate, bankruptcy cost and unit investment cost vary. Most of the other results are as expected, e.g., higher volatility r results in higher investment trigger P, larger investment size Q, and lower leverage ratio; higher growth rate l results in higher P and Q and higher leverage ratio; higher discount rate q results in lower P and Q and higher leverage ratio; and higher operating cost w results in higher P and Q but leaves the leverage ratio unchanged. A higher k makes investing more costly, hence Q is smaller, but P (and leverage ratio) remain unchanged. A higher a (or lower b) reduces the cost of investing, as a result of which Q (hence P) increases; it also increases the leverage ratio. A higher tax rate s makes the investment less attractive, reducing Q and increasing P; but it makes debt more attractive (higher tax shield), hence leverage ratio rises. Finally, a higher bankruptcy cost a also makes the investment less attractive, reducing Q and increasing P; it also makes debt less attractive, hence leverage ratio falls. Finally, the following interesting relationships can be noted from the comparative static results in Table 1: (1) In every single case, the investment trigger with debt financing exceeds the investment trigger without debt financing, thus confirming Result 2 for all parameter combinations examined. This result is just the opposite of the traditional (exogenous-capacity) literature, e.g., Mauer and Sarkar (2005) and Lyandres and Zhdanov (2006). (2) Also in every single case, the capacity or investment size with debt financing exceeds the capacity without debt financing. This explains point 1 above: debt financing makes the investment more attractive (because of the tax shield), hence the investment size is larger; however, since a larger investment is more costly, it results in delayed investment. (3) Thus, the optimal use of debt financing affects investment positively (via the larger investment) as well as negatively (via the delayed investment); the overall effect of debt financing on investment is ambiguous. (4) Finally, we note from Table 1 that there can be significant differences in investment size and timing between the levered and the no-debt cases. Also, there is substantial variation in the optimal investment size/capacity, particularly with respect to the parameters r, l, q and a. Thus, when modeling the corporate investment decision, it would be inappropriate to ignore either the debt financing option or the option to choose the optimal investment size, since both can have a significant impact on the investment decision. 9. Conclusions Corporate investment is an important determinant of economic growth and well-being, hence corporate investment policies are well worth studying. There is a strand of the Economics literature that identifies the optimal investment size and timing for firms that are unable to use debt financing, e.g., Bar-Ilan and Strange (1999) and Dangl (1999). Another strand of the literature (Mauer and Sarkar, 2005; Lyandres and Zhdanov, 2006) examines the effect of debt on investment timing when the company cannot choose the size of the investment (i.e., the size is pre-specified). This paper combines the two strands to identify the optimal investment policy when the firm has both the option to use debt financing and choose the investment size. That is, we compute the optimal size, optimal timing and optimal financing arrangement for an investment, using a real-option model similar to the papers mentioned above. This is the most realistic scenario, since the firm generally gets to choose the size or capacity of the investment, as well as the financing mix (the combination of debt and equity to finance the investment). Because the three optimizing decisions are taken simultaneously, the model is too complicated to allow closed-form solutions, and we had to resort to numerical solutions. Nevertheless, the numerical results are repeated for a wide range of parameter values, to ensure that the results are robust. The main results are as follows: Debt financing can have a significant effect on the investment decision (both timing and size); the direction and magnitude of this effect depends on the amount of debt financing used.
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When the firm can choose the scale of operations and uses the optimal amount of debt financing, investment is delayed (relative to the no-debt case); this is the opposite of the result from the existing (exogenous-size) literature, e.g., Mauer and Sarkar (2005) and Lyandres and Zhdanov (2006). However, with optimal debt financing, when the (delayed) investment is made, it is on a larger scale (again, relative to the no-debt case). The ‘‘perverse comparative static results’’ discussed in Bar-Ilan and Strange (1999) is also found to exist in our model in most cases; however, for two parameters (corporate tax rate and bankruptcy cost), an increase in the parameter value results in delayed investment as well as smaller investment. There is substantial variation in the optimal investment size/capacity, particularly with respect to the parameters r, l, q and a. Given the significant effect of debt financing on the investment decision, and the substantial variation in optima investment size, we conclude that it would be inappropriate to ignore either the debt financing option or the option to choose the optimal investment size when modeling the investment decision of a company, as both seem to have a significant effect on the investment decision. Acknowledgments The author thanks the editor Theodore Palivos and an anonymous referee for helpful suggestions. Financial support from the Social Science and Humanities Research Council (SSHRC) of Canada is also acknowledged. References Bar-Ilan, A., Strange, W.C., 1999. The timing and intensity of investment. Journal of Macroeconomics 21, 57–77. Dangl, T., 1999. Investment and capacity choice under uncertain demand. European Journal of Operational Research 117, 415–428. DeLong, J.B., Summers, L.H., 1991. Equipment investment and economic growth. Quarterly Journal of Economics 106, 445–502. DeLong, J.B., Summers, L.H., 1994. How strongly do developing economies benefit from equipment investment? Journal of Monetary Economics 32, 395– 415. Dixit, A., Pindyck, R.S., 1994. Investment Under Uncertainty. Princeton University Press, Princeton, NJ. Hubbard, R.G., 1994. Investment under uncertainty: keeping one’s options open. Journal of Economic Literature 32, 1816–1831. Leland, H., 1994. Risky debt, bond covenants and optimal capital structure. Journal of Finance 49, 1213–1252. Lyandres, E., Zhdanov, A., 2006. Accelerated Investment in the Presence of Risky Debt, Working Paper. Mauer, D.C., Ott, S., 2000. Agency costs, under-investment, and optimal capital structure: the effect of growth options to expand. In: Brennan, M., Trigeorgis, L. (Eds.), Project Flexibility, Agency, and Competition. Oxford University Press, New York, pp. 151–180. Mauer, D.C., Sarkar, S., 2005. Real options, agency conflicts, and optimal capital structure. Journal of Banking and Finance 29, 1405–1428.