Journal of Macroeconomics 33 (2011) 681–689

Contents lists available at SciVerse ScienceDirect

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 a b s t r a c t

Article history:Received 18 January 2011Accepted 25 August 2011Available online 17 September 2011

JEL Classification:G3

Keywords:Real option modelInvestment timingInvestment sizeCorporate financing decisionCapacity

0164-0704/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.jmacro.2011.08.002

⇑ Tel.: +1 905 525 9140x23959; fax: +1 905 521E-mail address: sarkars@mcmaster.ca

Corporate investment is an important determinant of economic well-being. The existingliterature identifies optimal investment size and timing without the possibility of debtfinancing, as well as the effect of debt financing on investment timing without the optionto choose investment size. This paper contributes to the literature by identifying the opti-mal size, optimal timing and optimal financing for an investment when the firm controls allthree decisions (as it usually does in practice). The investment size and investment triggerare generally positively related: when investment is delayed (accelerated) it is larger(smaller) in size, thus the overall effect on investment is ambiguous. However, when taxrate or bankruptcy cost is increased, the trigger rises and size falls, hence the effect oninvestment is unambiguously negative. The effect of debt financing on investment dependson the amount of debt used; with the optimal amount of debt, investment is delayed rel-ative to the no-debt case, and this delay can be economically significant; however, theinvestment, when eventually made, will be larger in size. Overall, it is not appropriate toignore either the firm’s ability to choose investment size or its option to use debt financing,when modeling the investment decision.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

It is widely recognized in the Economics literature that corporate investment is an important determinant of economicwell-being (Bar-Ilan and Strange, 1999; DeLong and Summers, 1991,1994). A growing strand of the investment literaturetakes a real-option approach to analyzing the investment decision, as summarized nicely in Dixit and Pindyck (1994). How-ever, Hubbard (1994) points out that this ‘‘real-option’’ approach focuses on the timing of the investment but not its size orintensity. Bar-Ilan and Strange (1999) addressed this shortcoming by computing both the optimal timing and optimal inten-sity/size of the investment, and found that they often behave in unexpected ways; for instance, in response to a parameterchange, investment could be larger but delayed, or smaller but earlier. However, their model was limited to firms that use nodebt 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-maximiz-ing) ratio.

Some papers have examined the effect of debt financing on the timing of investment, e.g., Mauer and Sarkar (2005) andLyandres and Zhdanov (2006), concluding that debt financing, used optimally, lowers the investment trigger and thusaccelerates 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 optimalsize/capacity, and the optimal investment trigger. This problem has not been solved in the literature to date. Our paper

. All rights reserved.

8995.

682 S. Sarkar / Journal of Macroeconomics 33 (2011) 681–689

can be viewed as an extension of Bar-Ilan and Strange (1999) and Dangl (1999) to the case when the investment is partlydebt-financed, and an extension of Mauer and Sarkar (2005) and Lyandres and Zhdanov (2006) to include the optimal sizeof the investment.

The main results are as follows. Debt financing can have a significant impact on investment size and timing, and thedirection 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 increasein the tax rate or bankruptcy costs, investment is both delayed and smaller. Finally, there is substantial variation in optimalinvestment size. Thus, a model of investment without the option to choose investment size or the option to use debt financ-ing 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 com-putes 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 levelof 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 opportunityand must make three decisions (instead of two decisions as in Bar-Ilan and Strange, 1999): (i) when to invest, (ii) how largean investment to make, and (iii) how to finance the investment. Let the level of capital (investment) be K, and k the price perunit of capital. Thus, by paying an investment cost of kK, the firm receives in exchange the project with productive capacityQ = 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 followsthe 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 dis-counted 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 financingmix, i.e., what fractions of the investment cost will be financed by equity and debt. This decision is expressed by the choice ofthe coupon payment to debt holders, c per unit time. Thus the pre-tax profit stream is p = [Q(P � w) � c] per unit time, andthis 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 donot look good. As is standard in the corporate finance literature (e.g., Leland, 1994), shareholders declare bankruptcy whenthe output price P falls sufficiently, say to Pb. At bankruptcy, shareholders walk away with nothing (i.e., zero payoff), and thefirm’s ownership is transferred to bondholders after incurring fractional bankruptcy costs of a (0 6 a 6 1). Thus, the post-bankruptcy firm is unlevered and owned by the former bondholders. If P continues to decline, at some point the project willbe shut down permanently and the firm will exit the industry (note that both the bankruptcy option and the exit option werenot 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 beshut 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 willmake the investment, (ii) production capacity, say Q� (or equivalently, capital level K�), and (iii) debt level c�. These threechoices 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 statevariable P. Since there is no explicit time limit for the project cash flows, the project value will be independent of calendartime. Let V(P) denote the project value. Then, as in Bar-Ilan and Strange (1999), it can be shown that V(P) must satisfy theordinary differential equation (ODE):

0:5r2P2V 00ðPÞ þ lPV 0ðPÞ � 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 projectvalue

VðPÞ ¼ ð1� sÞQ Pðq� lÞ �

wq

� �þ A0Pc1 þ A1Pc2 ð3Þ

S. Sarkar / Journal of Macroeconomics 33 (2011) 681–689 683

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:

0:5r2cðc� 1Þ þ lc� q ¼ 0 ð4Þ

and are given by

c1 ¼ 0:5� l=r2 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2q=r2 þ ð0:5� l=r2Þ2

qð5Þ

c2 ¼ 0:5� l=r2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2q=r2 þ ð0:5� l=r2Þ2

qð6Þ

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 Pðq� lÞ �

wq

� �þ A1Pc2 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=c2Þ ð8Þ

A1 ¼ �ð1� sÞQðPeÞ1�c2

c2ðq� lÞ ð9Þ

This completes the project valuation. Note that, in Bar-Ilan and Strange (1999), the project value had a simpler expressionbecause their model did not include the exit option. However, in order to be consistent with ‘‘limited liability,’’ we felt itnecessary 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 andequity, being contingent claims on the firm, will also be functions of the state variable P, and their values must also satisfythe 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 thedebt value D(P):

DðPÞ ¼ c=qþ D0Pc1 þ D1Pc2 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 (forP ?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=qþ D1Pc2 ð11Þ

As discussed above, when P falls to Pb, shareholders declare bankruptcy, leaving the bondholders with the assets of thefirm after incurring fractional bankruptcy costs of a; thus, bondholders are left with (1 � a) times the project value at bank-ruptcy, giving the boundary condition:

DðPbÞ ¼ ð1� aÞVðPbÞ

which gives the constant

D1 ¼ ½ð1� aÞVðPbÞ � c=q�ðPbÞ�c2 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 Pðq� lÞ �

wþ c=Qq

� �þ E0Pc1 þ E1Pc2 ð13Þ

684 S. Sarkar / Journal of Macroeconomics 33 (2011) 681–689

The first term gives the equity value without the bankruptcy option, and the other two terms capture the effects of the bank-ruptcy 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 Pðq� lÞ �

wþ c=Qq

� �þ E1Pc2 ð14Þ

To complete the valuation of debt and equity, we must identify the constant E1 and the bankruptcy trigger Pb. This can bedone form the two boundary conditions associated with the bankruptcy boundary P = Pb. The two boundary conditions arethe same as above, i.e., value-matching and smooth-pasting:

Value-matching : EðPbÞ ¼ 0 ð15Þ

Smooth-pasting : E0ðPbÞ ¼ 0 ð16Þ

Solving Eqs. (15) and (16), we get:

Pb ¼ðwþ c=QÞð1� l=qÞ

ð1� 1=c2Þð17Þ

E1 ¼ �ð1� sÞQðPbÞ1�c2

c2ðq� lÞ ð18Þ

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 Prises to the trigger level P�. The optimal trigger is the one that maximizes the total value of the levered firm (debt plus equi-ty). 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 beF(P); then, it can be shown that F(P) must satisfy the ODE (2) with p = 0. The solution is:

FðPÞ ¼ F0 Pc1 þ F1 Pc2 ð19Þ

where F0 and F1 are constants to be determined from the boundary conditions. For very small values of P(P ? 0), the optionbecomes 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 F0Pc1 ð20Þ

This option exercise trigger or boundary is P = P�. The associated boundary conditions are value-matching and smooth-pasting, as in the earlier sections. However, the firm will now maximize its total value, hence the value-matching conditionwill 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 F0ðP�Þc1 ¼ DðP�Þ þ EðP�Þ � kQb ð21Þ

and the smooth-pasting condition will be:

F’ðP�Þ ¼ F0c1ðP�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

F0 ¼ ½DðP�Þ þ EðP�Þ � kQb�ðP�Þ�c1 ð23Þ

and an equation that has to be solved (numerically, since there is no analytical solution) for P�:

scq� ð1� sÞQw

qþ ð1� sÞQP�ð1� 1=c1Þ

ðq� lÞ þ ðD1 þ E1Þð1� c2=c1ÞðP�Þc2 ¼ kQb 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 scaleor capacity Q. An optimal choice of Q would maximize the value of the option to invest, F0Pc1 , for any P; this is equivalent tomaximizing the value of F0. We therefore set dF0

dQ ¼ 0 in order to identify the optimal capacity Q�. This gives, after somesimplification:

Fig. 1.l = 0.01

S. Sarkar / Journal of Macroeconomics 33 (2011) 681–689 685

ð1� sÞ P�

ðq� lÞ �wq

� �� kbQb�1 þ ðP�Þc2

dE1

dQþ dD1

dQ

� �¼ 0 ð25Þ

where

dE1

dQ¼ E1

w=c þ c2=Q1þ Qw=c

� �

dD1

dQ¼ ð1� aÞA1

Qþ cc2ð1� aÞð1� sÞ

QqðPbÞc2

11� 1=c2

� 11þ c=ðQwÞ �

Qwcc2ð1� c2Þ

� 1ð1� aÞð1� sÞð1þ Qw=cÞ

� �

Eq. (25) can be solved (numerically) for the optimal capacity Q�. However, in order to identify both optimal investmenttrigger 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 in-crease value by reducing its tax obligations, since interest payments are tax deductible; hence debt has a valuable tax shieldassociated with it, which makes the investment more attractive. Two, there are bankruptcy costs associated with debt, whichreduces project value and thus makes the project less attractive. The overall effect of debt financing on the investment deci-sion (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, weuse 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 (ora = 0.5). In addition, we have two parameters that did not have a role in the Bar-Ilan and Strange (1999) model, the corporatetax 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, aresomewhat 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. Forinstance, 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 fromthose of an unlevered firm, and can be lower or higher depending on how much debt financing is used. The level of debtfinancing 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 projectvalue is given by the value of the option to invest, which is F0Pc1 . 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 isseen that F0 first rises and then falls as c is increased, giving an inverted-U shaped relationship; this can be expected from thetwo valuation effects associated with debt financing, discussed at the beginning of this section; for small levels of debt thetax shield effect dominates hence value rises, while for large levels of debt the bankruptcy cost effect dominates hence valuefalls. Hence there is a unique debt level (or leverage ratio) which maximizes value. Eye estimation indicates that this optimal

0

1

2

3

4

5

6

0 1 2 3 4 5 6Coupon c

Investment Trigger P* Capacity Q

Leverage Ratio Value F0

The Effect of debt financing on the investment decision (investment trigger P�, capacity Q, and value F0). The following parameter values are used:, q = 0.05, r = 0.05, k = 4, w = 1, b = 2, s = 0.15 and a = 0.5.

686 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 sum-marized in:

7.1. Result 1

Debt financing can potentially have a significant effect on the timing and intensity of investment, with the direction andmagnitude 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 max-imizes the total firm value (equity plus debt), i.e., c� = Argmax {D(P,c) + E(P,c)}. The first order condition for this optimizationis dDðPÞ

dc þdEðPÞ

dc ¼ 0, which simplifies to:

srþ Pc2

dD1

dcþ dE1

dc

� �¼ 0 ð26Þ

Differentiating with respect to c and simplifying, we get:

dD1

dc¼ c2½1� ð1� aÞð1� sÞ�=ð1þ Qw=cÞ � 1

rðPbÞc2and

dE1

dc¼ ð1� sÞ

rðPbÞc2

Substituting into Eq. (26) and simplifying, we get an equation that must be satisfied by the optimal coupon level:

Pb

P

� �c2

þ c2½1� ð1� aÞð1� sÞ�sð1þ Qw=cÞ ¼ 1 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 financ-ing decision is made when the investment is made, i.e., at P = P�. Thus, the optimal c will satisfy the equation:

Pb

P�

� �c2

þ c2½1� ð1� aÞð1� sÞ�sð1þ Qw=cÞ ¼ 1 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�), wehave to simultaneously solve (numerically) Eqs. (24), (25), and (28). With the above base case parameter values, the solu-tions to Eqs. (24), (25), and (28) are as follows: optimal debt level c� = 2.3113, optimal investment trigger P� = 1.7604 andoptimal 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 alarger capacity), which is the opposite of the traditional (exogenous-capacity) result that optimal debt financing results inearlier investment (Mauer and Sarkar, 2005; Lyandres and Zhdanov, 2006). This is because, when Q is endogenous as inour model, debt financing allows the company to choose a larger capacity, which results in delayed investment (since theinvestment cost is an increasing function of capacity). The above effects of leverage (confirmed by the comparative staticresults 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-debtcase).

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 shouldincrease value (the tax shield will dominate bankruptcy costs). The capacity Q increases by about 14% when debt financing isused. The investment trigger P� is only about 2% higher when leverage is used, which does not seem that significant. How-ever, we can gauge its effect on investment timing by computing how much the investment is delayed because of the higherinvestment trigger. Suppose the current value of the state variable is P0 = 1.7256. Then the unlevered firm will invest rightaway (since its investment trigger is 1.7256), but the levered firm will wait until P rises to its trigger level of 1.7604. As dis-cussed in Mauer and Ott (2000), the expected time for the variable P to go from P0 to P1 is given by the expression:

EðTÞ ¼ lnðP1=P0Þl�0:5r2 . With the appropriate substitutions in the above expression, we get the expected time to investment of the

optimally levered firm, EðTÞ ¼ lnð1:7604=1:7256Þ0:01�0:5ð0:05Þ2

¼ 2:28 years; that is, using debt financing will delay the investment by two

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.

S. Sarkar / Journal of Macroeconomics 33 (2011) 681–689 687

8. Comparative static results

Table 1 summarizes the comparative static results with a wide range of parameter values. It can be noted that the invest-ment trigger P� is closely related to the investment size Q�. In most cases, the two are positively related; that is, both rise orfall 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 1Comparative static results.

P Q� F0 Leverage ratio of levered firm (%)

No Lev Lev No Lev Lev No Lev Lev

Base case 1.7256 1.7604 2.4587 2.8077 3.1626 3.8285 70.84

r0.025 1.4384 1.4549 1.6958 1.9744 2.2353 2.8782 83.890.05 1.7256 1.7604 2.4587 2.8077 3.1626 3.8285 70.840.075 2.2166 2.2700 3.7633 4.2284 4.6977 5.5052 63.510.10 3.0737 3.1546 6.0408 6.7062 7.0275 8.0741 58.600.125 4.8493 4.9840 10.7579 11.8313 10.5734 11.9765 54.750.15 10.5057 10.8084 25.7823 28.1335 16.3628 18.3213 51.40

l0.0 1.3789 1.3874 0.8240 0.8912 0.3013 0.3378 53.550.005 1.4953 1.5152 1.4072 1.5671 1.0595 1.2302 62.040.01 1.7256 1.7604 2.4587 2.8077 3.1626 3.8285 70.840.015 2.2882 2.3407 4.8212 5.5789 8.5605 10.7372 78.110.02 4.4621 4.5654 13.6784 15.9297 23.6171 30.3816 83.42

q0.03 3.2268 3.3203 13.6010 15.6735 38.6075 47.7095 74.770.04 2.0520 2.1008 4.6112 5.2877 8.8256 10.7751 72.480.05 1.7256 1.7604 2.4587 2.8077 3.1626 3.8285 70.840.06 1.5699 1.5974 1.5652 1.7812 1.4335 1.7242 69.610.07 1.4775 1.5004 1.0987 1.2467 0.7526 0.9006 68.63

w0.50 0.8628 0.8802 1.2293 1.4039 10.4796 12.6864 70.840.75 1.2942 1.3203 1.8440 2.1058 5.1998 6.2948 70.841.00 1.7256 1.7604 2.4587 2.8077 3.1626 3.8285 70.841.25 2.1570 2.2005 3.0733 3.5097 2.1505 2.6033 70.841.50 2.5884 2.6406 3.6880 4.2116 1.5692 1.8997 70.84

k2 1.7256 1.7604 4.9173 5.6154 6.3251 7.6571 70.843 1.7256 1.7604 3.2782 3.7436 4.2168 5.1047 70.844 1.7256 1.7604 2.4587 2.8077 3.1626 3.8285 70.845 1.7256 1.7604 1.9669 2.2462 2.5301 3.0628 70.846 1.7256 1.7604 1.6391 1.8718 2.1084 2.5524 70.84

a0.33 1.3376 1.3571 0.9758 1.0366 2.5129 2.8544 66.600.40 1.4462 1.4702 1.2355 1.3434 2.5725 2.9827 68.150.50 1.7256 1.7604 2.4587 2.8077 3.1626 3.8285 70.840.60 2.4281 2.4877 1.8223 14.5835 5.9891 7.7629 74.090.67 4.0948 4.2095 136.17 181.81 16.5761 23.0709 76.65

s0.05 1.7256 1.7404 2.7479 2.8597 3.9505 4.1441 57.310.10 1.7256 1.7512 2.6033 2.8327 3.5456 3.9741 65.710.15 1.7256 1.7604 2.4587 2.8077 3.1626 3.8285 70.840.20 1.7256 1.7688 2.3140 2.7839 2.8014 3.6979 74.580.25 1.7256 1.7766 2.1694 2.7610 2.4622 3.5777 77.55

a0.00 – 1.7437 – 2.8246 – 4.0144 84.840.25 – 1.7545 – 2.8131 – 3.8913 75.830.50 – 1.7604 – 2.8077 – 3.8285 70.840.75 – 1.7643 – 2.8041 – 3.7879 67.421.00 – 1.7673 – 2.8014 – 3.7562 64.83

Note: F0: denotes the effective value of the investment opportunity (the actual value of the option to invest is F0Pc1 ). No Lev: denotes a firm that is unable touse 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.

688 S. Sarkar / Journal of Macroeconomics 33 (2011) 681–689

Delayed entry is unfavorable to investment while larger size or capacity is favorable to investment; similarly, acceleratedentry is favorable to investment while smaller capacity is unfavorable to investment. In these cases, therefore, the overalleffect on investment is ambiguous. This is what Bar-Ilan and Strange (1999, p. 58) refer to as the ‘‘perverse comparative staticresult’’ 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 firmresponds 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 compar-ative static result’’ of Bar-Ilan and Strange (1999) is not observed for these two parameters. Another exception is the case ofthe unit investment cost parameter k: as k is increased, P� remains unchanged while Q� falls. In this case, the firm responds toa 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 (invest-ment 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 investmentsize Q, and lower leverage ratio; higher growth rate l results in higher P� and Q and higher leverage ratio; higher discountrate q results in lower P� and Q and higher leverage ratio; and higher operating cost w results in higher P� and Q but leavesthe leverage ratio unchanged.

A higher k makes investing more costly, hence Q is smaller, but P� (and leverage ratio) remain unchanged. A higher a (orlower b) reduces the cost of investing, as a result of which Q (hence P�) increases; it also increases the leverage ratio. A highertax rate s makes the investment less attractive, reducing Q and increasing P�; but it makes debt more attractive (higher taxshield), hence leverage ratio rises. Finally, a higher bankruptcy cost a also makes the investment less attractive, reducing Qand 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 financ-ing. 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 lev-ered and the no-debt cases. Also, there is substantial variation in the optimal investment size/capacity, particularlywith respect to the parameters r, l, q and a. Thus, when modeling the corporate investment decision, it would beinappropriate to ignore either the debt financing option or the option to choose the optimal investment size, sinceboth 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 pol-icies are well worth studying. There is a strand of the Economics literature that identifies the optimal investment size andtiming for firms that are unable to use debt financing, e.g., Bar-Ilan and Strange (1999) and Dangl (1999). Another strand ofthe literature (Mauer and Sarkar, 2005; Lyandres and Zhdanov, 2006) examines the effect of debt on investment timing whenthe company cannot choose the size of the investment (i.e., the size is pre-specified). This paper combines the two strands toidentify the optimal investment policy when the firm has both the option to use debt financing and choose the investmentsize. That is, we compute the optimal size, optimal timing and optimal financing arrangement for an investment, using areal-option model similar to the papers mentioned above. This is the most realistic scenario, since the firm generally getsto choose the size or capacity of the investment, as well as the financing mix (the combination of debt and equity to financethe investment).

Because the three optimizing decisions are taken simultaneously, the model is too complicated to allow closed-form solu-tions, and we had to resort to numerical solutions. Nevertheless, the numerical results are repeated for a wide range ofparameter 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 magni-tude of this effect depends on the amount of debt financing used.

S. Sarkar / Journal of Macroeconomics 33 (2011) 681–689 689

� 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., Mauerand 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 valueresults 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 invest-ment size, we conclude that it would be inappropriate to ignore either the debt financing option or the option to choose theoptimal investment size when modeling the investment decision of a company, as both seem to have a significant effect onthe investment decision.

Acknowledgments

The author thanks the editor Theodore Palivos and an anonymous referee for helpful suggestions. Financial support fromthe 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.