GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.223

Investment timing with fixed and proportional costs ofexternal financing

Michi Nishihara and Takashi Shibata

November 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Investment timing with fixed and proportional costs ofexternal financing∗Michi NISHIHARA†,

Takashi SHIBATA‡

Abstract
We develop a dynamic model in which a ﬁrm exercises an option to expand production
with cash balance and costly external funds. While related papers explain their results
only by numerical examples, we analytically prove the following results. In the presence of
only a proportional cost of external ﬁnancing, the ﬁrm with more cash balance invests earlier; however, the presence of both proportional and ﬁxed costs leads to a non-monotonic
relation between the investment time and cash balance. The ﬁrm with more cash balance invests later to save a ﬁxed cost, particularly when the cash balance is close to theinvestment cost. Our results can potentially account for a variety of empirical results
concerning the relation between investment volume and ﬁnancing constraints.
JEL Classifications Code: G13; G31; G32.
Keywords: Real options; investment timing; costly external ﬁnancing; growth option;
optimal stopping.

1

Introduction

Modigliani and Miller (1958) showed that ﬁnancing and investment decisions can be madeindependently in a frictionless market. Since their seminal work, a wide range of literature
has focused on investigating ﬁnancing and investment decisions in the presence of various
frictions.1 Recently, an increasing number of papers have analyzed investment timing
decisions under ﬁnancial frictions in the real options framework.2∗†

This version: 10 November, 2011.
Corresponding Author. Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka,

Osaka 560-0043, Japan, E-mail: nishihara@econ.osaka-u.ac.jp, Phone: 81-6-6850-5242, Fax: 81-6-6850-5277‡
Graduate School of Social Sciences, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo
192-0397, Japan, E-mail: tshibata@tmu.ac.jp, Phone: 81-42-677-2310, Fax: 81-42-677-2298
1
An incomplete list includes Fazzari, Hubbard, and Petersen (1988), Hoshi, Kashyap, and Scharfstein (1991),
Kaplan and Zingales (1997), Cleary (1999), Gomes (2001), and Hennessy and Whited (2005).
2
For example, the literature examined the eﬀects of liquidity constraints (Boyle and Guthrie (2003)),
shareholders-debtholders conﬂicts (Mauer and Sarkar (2005), Sundaresan and Wang (2007), Morellec and1This paper investigates an investment timing decision with costs of external ﬁnancing
in the following model. A ﬁrm owns an option to expand the scale of production by a ﬁxed
rate, where the price of the output follows a stochastic process. The investment project isﬁnanced with cash balance and costly external funds. The cash balance gradually increases
as the ﬁrm’s existent production generates cash ﬂows even before its expansion. If the
ﬁrm waits for a suﬃcient level of cash balance, the project can be ﬁnanced entirely withthe cash balance. Otherwise, the ﬁrm must rely partially on costly external ﬁnancing.
Considering the trade-oﬀ, the ﬁrm determines the optimal ﬁnancing and investment policy.
As in the standard real options literature (Dixit and Pindyck (1994)), our model
assumes the irreversibility and indivisibility of investing as frictions. In addition, ourmodel includes costs of external ﬁnancing. The ﬁnancing costs are regarded as one of
the most inﬂuential frictions (e.g., Altinkilic and Hansen (2000), Hennessy and Whited(2007)). According to the pecking order hypothesis, asymmetric information problems
associated with external funding generate higher costs; therefore, managers prefer internal
over external ﬁnance (Myers (1984), Myers and Majluf (1984)). We examine the case with
only a proportional cost and the case with both ﬁxed and proportional costs. The former
approximates investment by a large ﬁrm, whereas the latter approximates investment bya small ﬁrm (Hennessy and Whited (2007)).
Before describing the results, we emphasize that our results are analytically proved.
Most of the related papers explain their results only by numerical examples for the reason
that the complexity of the models precludes analytic results (e.g., Boyle and Guthrie(2003), Hirth and Uhrig-Homburg (2010a), Shibata and Nishihara (2012)). However, it
is more important to derive analytic results in more complicated models; the complexity
increases the possibility of computational errors and makes the parameter sensitivity
unclear. In this paper, unlike related papers, we analytically prove interesting propertiesof the ﬁrm’s optimal ﬁnancing and investment policy by employing similar techniques
to those developed in the mathematical ﬁnance literature (e.g., Broadie and Detemple(1997), Detemple (2006)). This paper contributes to the literature by demonstrating how
to derive analytic results.
The results are summarized as follows. Costs of external ﬁnancing reduce the option
value and discourage the investment compared with the case with no ﬁnancing costs.
This result is consistent with the standard view from empirical and theoretical studies
concerning costly external ﬁnancing (e.g., Hennessy, Levy, and Whited (2007)). Whencosts of external ﬁnancing are relatively low to the scale of the proﬁt expansion, the ﬁrm
may invest partially with external ﬁnancing. Otherwise, the ﬁrm always waits until the
cash balance reaches the investment cost so that the project can be ﬁnanced entirely
with internal funds. In this case, the ﬁrm receives a higher proﬁt from saving costs ofSchürhoﬀ (2011)), manager-shareholders-debtholders conﬂicts (Shibata and Nishihara (2010)), and debt capacities (Shibata and Nishihara (2012)).

2

external ﬁnancing than a loss due to the distortion in investment timing. We derive aclear condition that forces the ﬁrm to invest entirely with internal funds. The condition
is more likely to be satisﬁed in a situation involving a smaller-scale expansion, a lower
investment cost, higher ﬁnancing costs, a smaller ﬁrm, and more cash balance.
Whether there is a ﬁxed cost of external ﬁnancing greatly aﬀects the relation between
the investment time and cash balance. First, we explain the result in the case with only
a proportional cost. In the absence of ﬁxed cost, the ﬁrm with more cash balance invests
earlier. The reasoning is as follows. An increase in the cash balance decreases the ﬁnancing
cost proportionally; hence, it decreases the threshold price above which the ﬁrm expands
production partially with external ﬁnancing. This monotonic relation is straightforwardly
consistent with the classical view of underinvestment due to ﬁnancing constraints (e.g.,
Fazzari, Hubbard, and Petersen (1988), Hubbard (1998)). In this view, a ﬁrm’s investment volume has a monotonic relation with internal funds. This paper complements theliterature by analytically proving that, in the presence of only a proportional cost, moreinternal funds accelerate investments in a dynamic model.3
Now, we consider the case with both ﬁxed and proportional costs. A ﬁxed cost, unlike
a proportional cost, plays a role in discouraging the investment by the ﬁrm with morecash balance. The intuition is as follows. An increase in the cash balance decreases thetime until the cash balance reaches the investment cost. With a shorter waiting time, theﬁrm with more cash balance can invest entirely with internal funds and save a ﬁxed cost.
This ﬁxed cost eﬀect is opposite to the proportional cost eﬀect described in the previous
paragraph. The trade-oﬀ between the two eﬀects determines the relation between theinvestment time and cash balance. We derive the result that the ﬁrm with more cash
invests later if the cash balance is close to the investment cost. This non-monotonic
relation is inconsistent with the conventional view of underinvestment due to ﬁnancing
constraints. However, our result can potentially explains empirical results against theconventional view (e.g., Kaplan and Zingales (1997), Cleary, Povel, and Raith (2007)) interms of ﬁxed and proportional costs of external ﬁnancing.
The result is also consistent with recent papers in the real options literature. Boyle andGuthrie (2003) showed that a ﬁrm with less cash balance may invest earlier to avoid therisk of a cash shortfall. In their model, a liquidity constraint, rather than ﬁnancing costs,
plays a role in leading the non-monotonic relation.4 Shibata and Nishihara (2012) concentrated on a debt capacity constraint instead of internal ﬁnancing constraints and showedthat investment thresholds have a U-shaped relation with a debt capacity constraint.
Note that these related papers demonstrate the results only in numerical examples. We
complement the literature by analytically proving the non-monotonic relation between
3

Milne and Robertson (1996) showed that investment increases with cash holdings in a dynamic dividend

and investment model. In the real options literature, Hirth and Uhrig-Homburg (2010a) and Nishihara andShibata (2010) also showed that the investment threshold decreases with internal funds.
4
Hirth and Uhrig-Homburg (2010b) extended Boyle and Guthrie (2003) to a case with ﬁnancing costs.

3

the investment time and cash balance due to ﬁxed and proportional costs of externalﬁnancing.
The remainder of this paper is organized as follows. Section 2 presents the setup andthe result in the case without ﬁnancing costs. Section 3 presents the results in the caseswith only a proportional cost as well as both ﬁxed and proportional costs. Although theprice is assumed to be a geometric Brownian motion, we discuss how the results can beextended in the case of geometric Lévy process. Section 4 presents numerical examples andexamines the comparative statics with respect to the price volatility. Section 5 concludes
the paper. All proofs appear in the appendix.

2

Preliminaries

2.1

Setup

Consider a risk-neutral ﬁrm that produces a commodity at a constant rate. The output
is sold at the market price X(t), which follows a geometric Brownian motiondX(t) = µX(t)dt + σX(t)dB(t) (t > 0),

X(0) = x,(1)

where B(t) denotes the standard Brownian motion deﬁned in a probability space (Ω, F, P)
and µ, σ(> 0) and x(> 0) are constants. For convergence, we assume that r > µ, wherer is a positive constant interest rate. Assume that the ﬁrm owns an option to expand
production to the ﬁxed scale A(> 1) at any time. If the option is exercised at time τ , theﬁrm pays a ﬁxed cost at time τ and receives an instantaneous cash ﬂow AX(t) after timeτ . Assume that the investment cost is I(> 0) if the whole amount of the cost is internallyﬁnanced. If part of the investment cost is externally ﬁnanced, the ﬁrm pays a proportional
cost C(≥ 0) and a ﬁxed cost K(≥ 0) of external ﬁnancing. The total investment cost is
expressed as I + C max(I − Y (τ ), 0) + K1{Y (τ )<I} , where Y (τ ) denotes cash balance attime τ . Until the investment time τ , cash balance Y (t) follows
dY (t) = rY (t)dt + X(t)dt, (0 < t < τ ) Y (0) = y,(2)where y(≥ 0) is a constant. Boyle and Guthrie (2003) assume dynamics of cash balance
exogenously and consider an option to initiate a new project. In contrast, we relate cash
balance Y (t) to operating cash ﬂows X(t) more directly and consider the option to expand
production. In the case of C = 0 and K = 0, the setup corresponds to a standard modelof the growth option. For a comprehensive list of typical situations ﬁtting the standard
model, refer to Dixit and Pindyck (1994). Although the standard model presumes thatthe ﬁrm needs no costs of external ﬁnancing (otherwise, it has suﬃcient internal funds),
our model considers ﬁxed and proportional costs of external ﬁnancing. Unlike Boyle and4Guthrie (2003), who concentrated on a liquidity constraint,5 we examine the eﬀects ofﬁnancing costs on optimal investment timing.
Our assumption of costly external ﬁnancing is justiﬁed as follows. In the pecking
order theory, a variety of agency and asymmetric information problems increase costs ofexternal ﬁnancing, which leads to a preference for internal over external ﬁnance (Myers(1984), Myers and Majluf (1984)). Practically, ﬁnancing costs consist of a ﬁxed cost(which is independent of the issue size) and a variable cost (which depends on the issuesize). A ﬁxed cost includes taxes, fees, and setup expenses. A variable cost increases with
the issue size primarily because more underwriting services are required for more funds
raised. In the standard view of the literature, a variable cost is convex with respect to theissue size (e.g., Altinkilic and Hansen (2000)). Hennessy and Whited (2007) estimatedthat proportional costs of equity ﬁnancing are approximately 5% (10%) for large (small)ﬁrms. They argued that a signiﬁcant level of ﬁxed costs of equity ﬁnancing may exist
only for small ﬁrms, while the convexity is statistically insigniﬁcant for any ﬁrms. In
considering their results, as well as preserving tractability of the model, we examine twocases: the case with only a proportional cost (Section 3.1) and the case with both ﬁxed
and proportional costs (Section 3.2).

2.2

Case with no financing costs

As a benchmark, this section brieﬂy describes the case of C = K = 0. The ﬁrm solves
the following problem:∫ τ∫ ∞sup Ex [e−rt X(t)dt +e−rt X(t)dt − e−rτ I],

τ ∈T

0

(3)

τ

where T denotes the set of all stopping times and Ex [·] denotes the expectation conditionalon X(0) = x. In (3), τ represents the time to expand the scale of production. By thestrong Markov property of X(t), (3) can be easily reduced to
(
)
A−1
x
x −rτ
+ sup E [eX(τ ) − I ],r − µ τ ∈T
r−µ
|
{z
}
=:V0,0 (x) the growth option value

where we denote the second term by V0,0 (x). The standard argument (e.g., Dixit andPindyck (1994)) proves that

)β
(
)(

(A − 1)x∗0,0

x


−I
(0 < x < x∗0,0 )

r−µ
x∗0,0
V0,0 (x) =(A − 1)x−I(x ≥ x∗0,0 ),
r−µ
5

(4)

Hirth and Uhrig-Homburg (2010b) extended Boyle and Guthrie (2003) by considering both a liquidity

constraint and ﬁnancing costs.

5

where β := 1/2 − µ/σ 2 +√(µ/σ 2 − 1/2)2 + 2r/σ 2 (> 1) is a positive characteristic root,

and x∗0,0 := β(r −µ)I/{(A−1)(β −1)} is the threshold price above which the ﬁrm expands
production.

3

Analytic Results

3.1

Case with a proportional costThis subsection examines the case of C > 0 and K = 0. This assumption applies toinvestments by large ﬁrms (Hennessy and Whited (2007)). In this case, the growth option
value, denoted by VC,0 (x, y), is expressed as)(
A−1
X(τ ) − I − C max(I − Y (τ ), 0) ],VC,0 (x, y) = sup Ex,y [e−rτ
r−µ
τ ∈T

(5)

where where Ex,y [·] denotes the expectation conditional on (X(0), Y (0)) = (x, y). The
term C max(I − Y (τ ), 0) means that a proportional cost is required when the ﬁrm is shortof cash balance. The standard argument proves that the exercise region of the option isexpressed as
SC,0 := {(x, y) ∈ R2+ | V (x, y) = (A − 1)x/(r − µ)x − I − C max(I − y, 0)}.

(6)

Now we analytically prove interesting properties of VC,0 (x, y) and SC,0 .
Consider the following problems as approximations of VC,0 (x, y):()
A−1x,y −rτVU (x, y) := sup E [eX(τ ) − I − C(I − Y (τ )) ],
r−µ
τ ∈T
and
VL (x, y) := sup E [e
x

τ ∈T

−rτ

(

)
A−1
X(τ ) − I − C(I − y) ].
r−µ

(7)

(8)

By the strong Markov property of X(t), we can easily derive the closed-form solution of(7) as follows: If A − C − 1 > 0, we have () () ( )β(A − 1 − C)x∗U
x
x


+
− (1 + C)I(0 < x1 < x∗U ) C y+

r−µ
r−µ
x∗U
VU (x, y) = (A − 1)x − I − C(I − y)(x1 ≥ x∗U ),r−µ(9)
where x∗U := β(r − µ)(1 + C)I/{(β − 1)(A − C − 1)} represents the threshold price above
which the ﬁrm expands production. Otherwise, VU (x, y) = C(x/(r − µ) + y) holds andthe growth option will be never exercised. Note that in both cases the exercise policy is
independent of Y (t). In (8), VL (x, y) is the same as (4) replaced I with I + C(I − y).
Then, the threshold price, denoted by x∗L (y), is equal tox∗L (y) =β(r − µ)(I + C(I − y)).
(β − 1)(A − 1)

6

(10)

First, we show several properties of the option value VC,0 (x, y). The following proposition shows that VL (x, y) and VU (x, y) are closed-form bounds of VC,0 (x, y).
Proposition 1 If y < I, VL (x, y) ≤ VC,0 (x, y) ≤ VU (x, y) is satisﬁed.

Otherwise,

VC,0 (x, y) = V0,0 (x, y) holds.
Next, we show several properties of the exercise region SC,0 . It immediately follows
from Proposition 1 that SC,0 includes SI which is deﬁned by
SI := {(x, y) ∈ R2+ | x ≥ x∗0,0 , y ≥ I}.

(11)

Then, we examine the properties of TC,0 := SC,0 \ SI . We can show the following lemmaand proposition.
Lemma 1
Assume that A − C − 1 > 0. It holds that
0 ≤VC,0 (x + ∆, y) − VC,0 (x, y) ≤(A − 1)∆r−µ0 ≤VC,0 (x, y + ∆) − VC,0 (x, y) ≤ C∆

(12)
(13)

for any positive constant ∆.
Proposition 2
Case (a): A − C − 1 > 0
The exercise region SC,0 is the disjoint union of SI and
TC,0 = {(x, y) ∈ R2+ | x ≥ x∗C,0 (y), y < I},(14)

where x∗C,0 (·) is a continuous and decreasing function satisfyingx∗L (y) ≤ x∗C,0 (y) ≤ x∗U
andlim x∗C,0 (y) = maxy↑I()(C + 1)rI∗
x0,0 ,
A−C −1(15)(16)

Case (b): A − C − 1 ≤ 0
The exercise region SC,0 is equal to SI , which means that TC,0 = ∅.
[Insert Figure 1 about here.]
Figure 1 illustrates the exercise region in each case.6 Note that the exercise region isconnected. The ﬁrm’s ﬁnancing and investment policy can be classiﬁed into two diﬀerent
types, depending on the relation between the scale of production expansion, A − 1, and
6

In all ﬁgures in this paper, we set the axes in the same way as Boyle and Guthrie (2003) and Hirth andUhrig-Homburg (2010b) for comparison.

7

a proportional cost of external ﬁnancing, C. Consider Case (a) in which a proportional
cost is relatively low. The ﬁrm invests partially with external ﬁnancing when the output
price, X(t), exceeds the threshold x∗C,0 (Y (t)). Proposition 2 shows a monotonicity in thethreshold x∗C,0 (Y (t)) with respect to cash balance Y (t). The reason is as follows. An
increase in Y (t) decreases a ﬁnancing cost, C(I − Y (t)), which enables the ﬁrm to invest
earlier. Accordingly, the presence of the proportional cost of external ﬁnancing leads to astraightforward result that the ﬁrm with more cash balance invests earlier.
In Case (b), the ﬁrm always waits until the cash balance reaches the whole amount of
the investment cost. The reasoning is as follows. As long as Y (t) < I, the ﬁrm receives
cost savings of CX(t)dt + r(1 + C)Idt and loses (A − 1)X(t)dt by deferring investment
by an inﬁnitesimally short period dt. If A − C − 1 ≤ 0, we have CX(t)dt + r(1 + C)Idt −(A − 1)X(t)dt > 0 for any X(t), which means that the cost saving eﬀect always dominates
the loss. Then, the ﬁrm has no incentive to invest with external ﬁnancing. Below, we
describe major determinants of entirely internal ﬁnancing. Clearly, low ﬁnancing costs andmuch cash balance are the determinants. Since a small ﬁrm suﬀers from relatively higher
ﬁnancing costs (Hennessy, Levy, and Whited (2007)), a small ﬁrm is more likely to invest
entirely with internal ﬁnancing compared with a large ﬁrm. In addition, a small-scale
expansion with low investment cost increases the possibility of entirely internal ﬁnancing.
For example, suppose that C = 0.1. While a large-scale investment with A = 1.5 andI = 100 leads to Case (a), a small-scale investment with A = 1.1 and I = 20 leads to Case(b).
It should be noted that we, unlike most of the related papers, analytically prove theexistence of continuous and decreasing thresholds x∗C,0 (Y (t)). It is worth deriving theanalytic results in a complicated problem involving multiple state variables because thecomplexity increases the possibility of computational errors. Although some of the techniques used in the proof are enlightened by the mathematical ﬁnance literature (e.g.,
Broadie and Detemple (1997), Detemple (2006)), the proofs are newly developed in thispaper. While the mathematical ﬁnance studies analyzed the exercise regions of American
options that involve a multi-dimensional geometric Brownian motion7 , the stochastic process Y (t) in our model is not a geometric Brownian motion; instead, it is but deﬁned by(2). Furthermore, the payoﬀ function of problem (5) is not convex, which makes the proofs
more diﬃcult. This paper contributes to the literature from this technical viewpoint.
Proposition 2 shows that ﬁnancing costs of external ﬁnancing discourage the investment compared with the case with no ﬁnancing costs. This is consistent with empiricalresults (e.g., Hennessy, Levy, and Whited (2007)). The monotonic relation between theinvestment time and cash balance is consistent with the conventional view of underinvestment due to ﬁnancing constraints. Indeed, many empirical and theoretical papers haveshown that a ﬁrm with less internal funds invests less than a ﬁrm with suﬃcient internal
7

Nishihara (2011) showed the properties of multiple real options using similar techniques.

8funds (e.g., Fazzari, Hubbard, and Petersen (1988), Hubbard (1998)). Milne and Robertson (1996) examined a ﬁrm’s dynamic policy of dividend and investment; they showedthat the investment level increases with cash holdings. Similar results are seen in the real
options literature. Nishihara and Shibata (2010) showed that a ﬁrm delays the investment
when it must rely more heavily on debt ﬁnancing than the optimal level of the leverage.
Hirth and Uhrig-Homburg (2010a) showed that the investment threshold is decreasing inthe ﬁrm’s liquid funds.
On the other hand, empirical ﬁndings against the monotonic relation between theinvestment volume and internal funds have been identiﬁed (e.g., Kaplan and Zingales(1997), Moyen (2004), Cleary, Povel, and Raith (2007)). These studies have shown thathaving more internal funds does not necessarily increase the investment level. In the next
section, we will show that the inclusion of a ﬁxed cost of external ﬁnancing can lead tothe non-monotonic relation between the investment time and cash balance.

3.2

Case with fixed and proportional costs

This section examines the case of C > 0 and K > 0. In this case, the option value,denoted by VC,K (x, y), is expressed as()
A−1x,y −rτ
VC,K (x, y) = sup E [eX(τ ) − I − C max(I − Y (τ ), 0) − K1{Y (τ )<I} ],
r−µ
τ ∈T
(17)
where 1{Y (τ )<I} denotes the deﬁning function. Note that the exercise region of the option
isSC,K := {(x, y) ∈ R2+ | VC,K (x, y) = (A−1)x/(r−µ)−I−C max(I−y, 0)−K1{y<I} }. (18)
We can readily show the following proposition regarding the properties of VC,K (x, y).
Proposition 3 VC,0 (x, y) − K ≤ VC,K (x, y) ≤ VC,0 (x, y) is satisﬁed. If A − C − 1 ≤ 0,
VC,K (x, y) = VC,0 (x, y) is satisﬁed. If y ≥ I, VC,K (x, y) = V0,0 (x, y) is satisﬁed.
Next, we analytically prove the properties of the exercise region SC,0 . Proposition 3
shows that SC,0 includes SI . Then, we concentrate on the properties of TC,K := SC,K \ SI .
Proposition 4
Case (a-K): A − C − K/I − 1 > 0
The exercise region SC,K is the disjoint union of SI and TC,K satisfying
TC,K ⊂ TC,0 ∩ R+ × [0, I − K/(A − C − 1)),(19){(x, y) ∈ R2+ | x ≥ x∗C+K/(I−y),0 (y), y < I − K/(A − C − 1)} ⊂ TC,K ,

(20)

and

9

where x∗C+K/(I−y),0 (y) denotes the threshold of problem (5) replaced C with C +K/(I −y).
In particular, there exist (x, y1 ) ∈ SC,K , (x, y2 ) ∈/ SC,K , and (x, y3 ) ∈ SC,K satisfying
y1 < y 2 < y 3 .
Case (b-K): A − C − K/I − 1 ≤ 0
The exercise region is equal to SI .
[Insert Figure 2 about here.]
Figure 2 illustrates the exercise region in each case. The ﬁrm defers the investment sothat the project can be ﬁnanced entirely with internal funds if costs of external ﬁnancing
are suﬃciently high (Case (b-K)). Otherwise, the ﬁrm may invest partially with costly
external ﬁnancing (Case (a-K)). A ﬁxed cost K > 0 increases the possibility of Case (b-K)
by the additional term K/I and discourages the investment, compared with the case with
only a proportional cost. Similar to the case with only a proportional cost, a smaller-scale
expansion, a lower investment cost, higher ﬁnancing costs, a smaller ﬁrm, and more cash
balance increase the possibility that the ﬁrm invests entirely with internal ﬁnancing.
Below we explain Case (a-K) in which a ﬁxed cost of external ﬁnancing generates
quite a diﬀerent result from that of Case (a) in Proposition 2. The key result is thatR+ × [I − K/(A − C − 1), I) is not included in the exercise region SC,K . Before cash
balance, Y (t), reaches the critical level, I − K/(A − C − 1), the ﬁrm may expand the scaleof production for a suﬃciently high price of the output. Once the cash balance reaches
the critical level prior to the investment, the ﬁrm always defers the investment until the
cash balance is equal to the whole amount of the investment cost.
A ﬁxed cost provides a greater incentive for the ﬁrm with more cash balance to wait
and invest entirely with internal funds. Indeed, the ﬁrm with more cash balance can save
a ﬁxed cost with a shorter deferment of the investment. On the other hand, there always
exists the proportional cost eﬀect; an increase in cash balance reduces a proportional
cost of ﬁnancing and increases the incentive to invest earlier. The two conﬂicting eﬀects
determine the relation between the investment time and cash balance. Proposition 4 showsthat the ﬁxed cost eﬀect dominates the proportional cost eﬀect when the cash balance is
suﬃciently close to the investment cost. Consequently, contrary to the conventional view(or the result in the case with only a proportional cost), the ﬁrm with more cash balance
invests later in the situation. As easily seen, the ﬁxed cost eﬀect is weak when the cash
balance is far from the investment cost. Thus, the presence of both ﬁxed and proportionalcosts leads to the non-monotonic relation between the investment time and cash balance.
Proposition 4 has the potential to explain the non-monotonic relation observed inempirical studies (e.g., Kaplan and Zingales (1997), Cleary, Povel, and Raith (2007)), in
terms of ﬁxed and proportional costs of external ﬁnancing. This result is also consistent
with the following ﬁndings in the real options literature.8 , Boyle and Guthrie (2003)
8

Apart from dynamic investment models, Cleary, Povel, and Raith (2007) showed both theoretically andempirically a U-shaped relation between the investment level and internal funds.

10

showed that a ﬁrm with less cash balance may invest earlier to avoid the risk of a cash
shortfall. In their model, investment is possible only when a liquidity constraint is satisﬁed.
The liquidity constraint leads to a V-shaped relation between the investment thresholdand cash balance. Hirth and Uhrig-Homburg (2010b) extended Boyle and Guthrie (2003)
to a case involving ﬁnancing costs and showed the possibility of a variety of relations.
Shibata and Nishihara (2012) showed that, in a dynamic investment and capital structure
model, investment thresholds have a U-shaped relation with a debt issuance constraint.
Although these papers explain their results only by numerical examples, we analytically
prove the non-monotonic relation between the investment time and cash balance.

3.3

The output price following a geometric Lévy process

The results obtained in the previous subsections hold true when the output price follows ageometric Brownian motion, as well as when it follows a geometric Lévy process. The classof geometric Lévy processes, unlike a geometric Brownian motion, includes processes withjumps and can account for fat tails and skewness of probability distributions of the output
price. Assume that the output price X(t) := xeZ(t) , where Z(t) is a Lévy process (i.e.,a process with stationary independent increments), satisﬁes the convergence condition:
E1 [X(t)] = eµt with µ < r. In the case with no ﬁnancing costs, the results are seen in
Boyarchenko (2004). Although the problem precludes a closed-form solution of V0,0 (x) andx∗0,0 , we still obtain an investment threshold as proved in Mordecki (2002). A diﬃculty inextending the results in the previous subsections to the case of a geometric Lévy process
is that the generating operator of (X(t), Y (t)) includes the integrals corresponding to
the jumps (refer to (Øksendal and Sulem 2007)). However, we obtain the same relation,
Lf (x, y) − rf (x, y) ≤ 0 ⇔ x ≥ (C + 1)rI/(A − C − 1), as in the case with no jumps using
the linearity of f (x, y) deﬁned by (23). As a result, we can trace all the proofs of thelemma and propositions in Sections 3.2 and 3.3. All technical details are omitted.

4

Numerical examples

As mentioned in Section 1, the main contribution of this paper is to analytically prove theproperties of the ﬁrm’s optimal ﬁnancing and investment policy (Section 3). This section
supplements the analytic results by presenting the comparative statics results with respect
to the volatility σ in numerical examples. The base parameter values except for σ are set
as follows:r = 0.07, µ = 0.03, I = 100, (X(0), Y (0)) = (10, 50), A = 1.5, C = 0.1, K = 1.

(21)

Cases (a) and (a-K) are satisﬁed because A − C − K/I − 1 = 0.39 > 0 hold. In thecomputation, we make a tri-nomial lattice model that approximates to a a geometric
Brownian motion (1), and we use a value function iteration algorithm.

11[Insert Figure 3 about here.]
Figure 3 plots the exercise regions with varying levels of σ in the cases with only aproportional cost and with both ﬁxed and proportional costs. We can see from Figure
3 that a higher σ decreases the exercise regions in both cases. This implies that theinvestment threshold and the option value increase with σ. Thus, the eﬀect of the volatility
σ on the investment policy remains unchanged from that of the standard model with noﬁnancing costs. In the upper panel of Figure 3, there is a gap between limy↑I x∗C,0 (y) =(C + 1)rI/(A − C − 1) and x∗0,0 for σ = 0.1, 0.15, whereas limy↑I x∗C,0 (y) is equal to x∗0,0for σ = 0.2, 0.25, 0.3. Since x∗0,0 is increasing with σ, a higher σ reduces a gap between
limy↑I x∗C,0 (y) and x∗0,0 (cf. Propostion 2). In the lower panel of Figure 3, the boundaryof TC,K has an asymptote y = I − K/(A − C − 1). As proved in Propostion 4, the option
is not exercised for Y (t) ∈ [0, I − K/(A − C − 1)).

5

Conclusion

This paper investigated a ﬁrm’s option to expand the scale of production by a ﬁxed rate.
We assumed that the project is ﬁnanced with cash balance, which is increasing with time,and external funds that may require proportional and ﬁxed costs. We, unlike most of
the related papers, analytically proved the properties of the ﬁrm’s optimal ﬁnancing andinvestment policy. The results are summarized as follows.
Costs of external ﬁnancing reduce the option value and discourage the investmentcompared with the case with no ﬁnancing costs. When costs of external ﬁnancing are
relatively low to the scale of the proﬁt expansion, the ﬁrm may invest partially withexternal ﬁnancing. Otherwise, the ﬁrm always waits until the cash balance reaches theinvestment cost so that the project can be ﬁnanced entirely with internal funds. The
entirely internal ﬁnancing is likely to be adopted in a small-scale expansion by a small
ﬁrm with more cash balance. In the presence of only a proportional cost, the ﬁrm withmore cash balance invests earlier; however, the presence of both proportional and ﬁxed
costs leads to a non-monotonic relation between the investment time and cash balance.
Our results can potentially account for ambiguous results in empirical studies regarding
the relation between investment volume and ﬁnancing constraints.

Acknowledgment
This work was supported by KAKENHI (22710146, 22710142, 23310103, 21241040).

12

A

Proof of Proposition 1

Since Y (t) monotonically increases from the initial value y, we have for y < I
I − Y (t) ≤ max(I − Y (t), 0) ≤ I − yat any time t. This implies that VL (x, y) ≤ VC,0 (x, y) ≤ VU (x, y) holds for y < I. Clearly
we have VC,0 (x, y) = V0,0 (x, y) for y > I.

B

Proof of Lemma 1

First, we show (12). Note that E1 [·] represents the expectation with X(t) starting from
X(0) = 1. For any positive constant ∆, we have
VC,0 (x + ∆, y)()∫ τ(A − 1)(x + ∆)1 −rτrτr(τ −s)= sup E [eX(τ ) − I − C max(I − e y −e(x + ∆)X(s)ds, 0) ]
r−µ
τ ∈T
0
(
∫ τ
(A − 1)x
1 −rτrτ
er(τ −s) xX(s)ds, 0)≤ sup E [eX(τ ) − I − C max(I − e y −
r−µ
τ ∈T
0
)
∫ τ
(A − 1)∆+
X(τ ) + C∆er(τ −s) X(s)ds ]
r−µ
0
)
(
∫ τ
(A
−
1)x
r(τ −s)1 −rτrτexX(s)ds, 0) ]≤ sup E [eX(τ ) − I − C max(I − e y −
r−µ
τ ∈T
0
|
{z
}
=VC,0 (x,y)∫ τr(τ −s))(A − 1)∆e
X(s)ds ]+ sup E [eX(τ ) + C∆
r−µ
τ ∈T
0
(
)
(∫ ∞
)
∫ ∞
(A − 1)∆
1 −rτ
−rs
−rs
=VC,0 (x, y) + sup E [eX(τ ) + C∆
e X(s)ds −e X(s)ds ]
r−µ
τ ∈T
0
τ
C∆
(A − 1 − C)∆=VC,0 (x, y) ++sup E1 [e−rτ X(τ )]
(22)
r−µ
r−µ
τ ∈T
|
{z
}
1

−rτ

(

=1

(A − 1)∆=VC,0 (x, y) +,r−µwhere in (22) supτ ∈T E1 [e−rτ X(τ )] = 1 follows from µ < r.
Next, we show (13). For any positive constant ∆, we have
VC,0 (x, y + ∆))(∫ τ
(A − 1)rτ
r(τ −s)x −rτ
X(τ ) − I − C max(I − e (y + ∆ −eX(s)ds, 0) ]= sup E [e
r−µ
τ ∈T
0
(
)
∫ τ
(A − 1)x −rτ
rτ
r(τ −s)rτ
≤ sup E [eX(τ ) − I − C max(I − e y −eX(s)ds, 0) + e C∆ ]
r−µ
τ ∈T
0
(
)
∫ τ
(A − 1)x −rτ
rτ
r(τ −s)= sup E [eX(τ ) − I − C max(I − e y −eX(s)ds, 0) ] +C∆
r−µ
τ ∈T
0
|
{z
}
=VC,0 (x,y)=VC,0 (x, y) + C∆.

13

C

Proof of Proposition 2First, consider the case of A − C − 1 > 0. Fix (x, y) ∈ TC,0 and (x′ , y ′ ) satisfying x ≤ x′
and y ≤ y ′ < I. Using Lemma 1, we have
VC,0 (x′ , y ′ ) = VC,0 (x′ , y ′ ) − VC,0 (x, y ′ ) + VC,0 (x, y ′ ) − VC,0 (x, y) + VC,0 (x, y)(A − 1)(x′ − x)
+ C(y ′ − y) + VC,0 (x, y)r−µ(A − 1)(x′ − x)
(A − 1)x=+ C(y ′ − y) +− I − C(I − y)r−µr−µ(A − 1)x′=− I − C(I − y ′ ),r−µ≤where the last inequality implies (x′ , y ′ ) ∈ TC,0 . This proves that TC,0 is expressed as (14)with the decreasing function x∗C,0 (·). By Proposition 1, we immediately obtain inequality(15).
Next, we will show (16). Clearly we have limy↑I x∗C,0 (y) ≥ x∗0,0 . Denote the payoﬀ
function of problem (7) asf (x, y) :=(A − 1)x− I − C(I − y).
r−µ

(23)

We haveLf (x, y) − rf (x, y) ≤ 0(A − 1)µx⇔+ C(x + ry) − r
r−µ(C + 1)rI,⇔x ≥A−C −1()(A − 1)x− I − C(I − y) ≤ 0
r−µ
(24)

where L denotes the generating operator of (X(t), Y (t)). Since the general theory of
optimal stopping ensures LVC,0 (x, y) − rVC,0 (x, y) ≤ 0 (refer to Peskir and Shiryaev(2006)), by (24) f (x, y) is not equal to VC,0 (x, y) for x < (C + 1)rI/(A − C − 1) andy < I. In other words, the option is not exercised in the region {(x, y) ∈ R2+ | x <(C + 1)rI/(A − C − 1), y < I}. This proves that limy↑I x∗C,0 (y) ≥ (C + 1)rI/(A − C − 1).
[Insert Figure 4 about here.]Now, suppose that (C + 1)rI/(A − C − 1) ≤ x∗0,0 < limy↑I x∗C,0 (y) (see Figure 4).
We can lead to contradiction as follows. Consider problem (7) with a ﬁnite maturity T .
Generally, the exercise region of an American option converges to the region Lf − rf ≤ 0,
where L is the generating operator and f is the payoﬀ function, when the remaining
life of the option goes to zero (refer to Detemple (2006)). Then, because of (24), theexercise region of problem (7) with a ﬁnite maturity T converges to {(x, y) ∈ R2+ | x ≥(C + 1)rI/(A − C − 1)} when T ↓ 0. Consider the exercise region of problem (5) fora ﬁxed x satisfying x∗0,0 < x < limy↑I x∗C,0 (y) and y ↑ I. Note that τI := inf{t ≥ 0 |14

X(t) ≥ x∗0,0 , Y (t) ≥ I} converges to 0 as y ↑ I. Accordingly, the exercise region of the
problem (5) for the ﬁxed x and y ↑ I converges to that of problem (7) with T ↓ 0. Thisimplies that limy↑I x∗C,0 (y) = (C + 1)rI/(A − C − 1), which contradicts the assumptionof (C + 1)rI/(A − C − 1) < limy↑I x∗C,0 (y). Similarly we can lead to contradiction if
x∗0,0 < (C + 1)rI/(A − C − 1) < limy↑I x∗C,0 (y) is supposed. Thus, we complete the proofof (16).
We can show the continuity of x∗C,0 (·) as follows. By Lemma 1 we have the continuity of
VC,0 (x, y). Since VC,0 (x, y) and (A − 1)x/r − µ − I − C max(I − y, 0) are both continuous,
SC,0 is a closed set. Then, we have limϵ↓0 (x∗C,0 (y + ϵ), y + ϵ) ∈ SC,0 , which leads tolimϵ↓0 x∗C,0 (y + ϵ) ≥ x∗C,0 (y). We have limϵ↓0 x∗C,0 (y + ϵ) ≤ x∗C,0 (y) because x∗C,0 (·) is
decreasing. Thus, we obtain the right-continuity of x∗C,0 (·). Now, suppose that there
exists y(< I) satisfying x∗C,0 (y) < limϵ↓0 x∗C,0 (y − ϵ). We can lead to contradiction as
the same method as the proof of (16). Consider the exercise region of problem (5) fora ﬁxed x satisfying x∗C,0 (y) < x < limϵ↓0 x∗C,0 (y − ϵ) and y − ϵ. Note that inf{t ≥ 0 |X(t) ≥ x∗C,0 (Y (t))} converges to 0 as ϵ ↓ 0. Then, the exercise region converges to thatof problem (7) with T ↓ 0. This implies that limϵ↓0 x∗C,0 (y − ϵ) = (C + 1)rI/(A − C − 1),which contradicts (C + 1)rI/(A − C − 1) ≤ x∗C,0 (y) < limϵ↓0 x∗C,0 (y − ϵ). Thus, we obtainthe left-continuity of x∗C,0 (·).
Lastly, consider the case of A − C − 1 ≤ 0. In this case, we have for any (x, y) ∈ R2+
Lf (x, y) − rf (x, y) = −(A − C − 1)x + (C + 1)rI > 0,
where L is the generating operator of (X(t), Y (t)) and f (x, y) is deﬁned by (23). Since
LVC,0 (x, y) − rVC,0 (x, y) ≤ 0 holds by the general theory of optimal stopping, f (x, y) does
not agree with VC,0 (x, y). This implies that TC,0 = ∅ and SC,0 = SI .

D

Proof of Proposition 3

By 0 ≤ K1{Y (t)<I} ≤ K we have VC,0 (x, y) − K ≤ VC,K (x, y) ≤ VC,0 (x, y). Suppose thatA − C − 1 ≤ 0. It follows from Proposition 2 and the optimality of VC,K (x, y) that()
A−1
x,y −rτI
VC,0 (x, y) = E [eX(τI ) − I ](25)
r−µ≤ VC,K (x, y),(26)

where τI denotes the ﬁrst hitting time to SI = {(x, y) ∈ R2+ | x ≥ x∗0,0 , y ≥ I}. Then, we
obtain VC,0 (x, y) = VC,K (x, y). Clearly we have VC,K (x, y) = V0,0 (x, y) for y ≥ I.

15

E

Proof of Proposition 4First, suppose that A − C − K/I − 1 > 0. By Proposition 3 we have for any (x, y) ∈ TC,K
VC,0 (x, y) ≤ VC,K (x, y) + K(A − 1)x− I − C(I − y) − K + K
r−µ(A − 1)x=− I − C(I − y),r−µ=where the last inequality implies (x, y) ∈ TC,0 . This proves that TC,K ⊂ TC,0 .
Fix any (x, y) ∈ R2+ satisfying I − K/(A − C − 1) ≤ y < I. Consider the ﬁrst hitting
time to SI , denoted by τI . Since Y (t) monotonically increases from the initial point
Y (0) = y, we have
VC,K (x, y) ≤ sup E
τ ∈T
|
=E

x,y

x,y

[e

−rτ

(

[e

−rτI

(

(
)
)
A−1
K
X(τ ) − I − C +max(I − Y (τ ), 0) ]
r−µ
I −y
{z
}
=VC+K/(I−y),0 (x,y))
A−1
X(τI ) − I ]
r−µ

(27)

≤ VC,K (x, y),where by Proposition 2 we have (27) because of A − C − K/(I − y) − 1 ≤ 0. The lastinequality implies that (x, y) ∈/ TC,K , which leads to TC,K ⊂ R+ × [0, I − K/(A − C − 1)).
This completes the proof of (19).
Next, ﬁx any (x, y) ∈ R2+ satisfying x ≥ x∗C+K/(I−y),0 (y) and y < I − K/(A − 1 − C).
Since Y (t) monotonically increases from the initial point Y (0) = y, we have
(
)
)
(
K
A−1
x,y −rτX(τ ) − I − C +max(I − Y (τ ), 0) ],VC,K (x, y) ≤ sup E [e
r−µ
I −y
τ ∈T
|
{z
}
=VC+K/(I−y),0 (x,y)()
K(A − 1)x=−I − C +(I − y)r−µI −y(A − 1)x− I − C(I − y) − K,=r−µwhere the last inequality implies (x, y) ∈ TC,K , and, therefore, we obtain (20).
Lastly, consider the case of A − C − K/I − 1 ≤ 0. We use the same technique as theproof of (19). Fix any (x, y) ∈ R2+ satisfying y < I and denote by τI the ﬁrst hitting timeto SI . Because Y (t) ≥ 0 holds for any time t, we have
(
(
)
)
A−1
K
x,y −rτ
VC,K (x, y) ≤ sup E [eX(τ ) − I − C +max(I − Y (τ ), 0) ]
r−µ
I
τ ∈T
|
{z
}
x,y

=E

[e

−rτI

(

=VC+K/I,0 (x,y))
A−1
X(τI ) − I ]r−µ≤VC,K (x, y),

16

(28)

where by Proposition 2 we have (28) because of A − C − K/I − 1 ≤ 0. The last inequality
proves that (x, y) ∈/ TC,K , and, then, we have TC,K = ∅ and SC,K = SI .

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18

y;ƚͿ

‫ݔ‬௎‫כ‬‫כ‬‫ݔ‬଴଴‫ݔ‬஼‫ כ‬ǡ଴ ሺܻሺ‫ݐ‬ሻሻ‫ݔ‬௅‫ כ‬ሺܻሺ‫ݐ‬ሻሻ

Ϭ

/

z;ƚͿ

/

z;ƚͿ

y;ƚͿ

‫ݔ‬௎‫כ‬‫ ܥ‬൅ ͳ ‫ܫݎ‬‫ܣ‬െ‫ܥ‬െͳ
‫כ‬
‫ݔ‬଴଴

‫ݔ‬஼‫ כ‬ǡ଴ ሺܻሺ‫ݐ‬ሻሻ‫ݔ‬௅‫ כ‬ሺܻሺ‫ݐ‬ሻሻ

Ϭ

y;ƚͿ

ܵூ

‫כ‬
‫ݔ‬଴଴

/

Ϭ

z;ƚͿ

Figure 1: The exercise region in the case with only a proportional cost. The upper panel
illustrates Case (a) satisfying x∗0,0 ≥ (C + 1)rI/(A − C − 1). The middle panel illustrates Case(a) satisfying x∗0,0 < (C + 1)rI/(A − C − 1). In this case, there is a gap between limy↑I x∗L (y)and x∗0,0 . The lower panel illustrates Case (b).

19

y;ƚͿ

‫כ‬
‫ݔ‬஼ା௄
Ȁሺூି௒ሺ௧ሻሻǡ଴ ሺܻሺ‫ ݐ‬ሻሻ

ܶ஼ǡ௄
‫כ‬

‫ݔ‬଴଴

Ϭ

‫ݔ‬஼‫ כ‬ǡ଴ ሺܻሺ‫ݐ‬ሻሻܵூ/Ͳ<ͬ;ͲͲϭͿ

z;ƚͿ

/

y;ƚͿ

ܵூ

‫כ‬

‫ݔ‬଴଴

/

Ϭ

z;ƚͿ

Figure 2: The exercise region in the case with both fixed and proportional costs. The upper
panel illustrates Case (a-K). The lower panel illustrates Case (b-K).

20

50

40

TC,0

σ=0.1
σ=0.15
σ=0.2
σ=0.25
σ=0.3

SI

X(t)

30

20

10

0

40

60

80
Y(t)

100

120

50

40

TC,K

σ=0.1
σ=0.15
σ=0.2
σ=0.25
σ=0.3

SI

X(t)

30

20

10

0

40

60

80
Y(t)

100

120

Figure 3: The comparative statics with respect to the volatility σ. The upper and lower
panels illustrate the cases with only a proportional cost and both fixed and proportional costs,
respectively. The parameter values other than σ are set at the base case (21).

21

y;ƚͿ

‫ݔ‬஼‫ כ‬ǡ଴ ሺܻሺ‫ݐ‬ሻሻ

ǆ
‫כ‬
‫ݔ‬଴଴

Ǉ/

Ϭ

z;ƚͿ

Figure 4: The assumption of x∗0,0 < limy↑I x∗C,0 (y). The dot represents the initial point (x, y)satisfying x∗0,0 < x < limy↑I x∗C,0 (y) and y ≈ I.

22