Discussion Papers In Economics
And BusinessCorporate investment, financing, and exit modelwith an earnings-based borrowing constraintMichi NISHIHARA
Takashi SHIBATA
Chuanqian ZHANG

Discussion Paper 21-13

September 2021

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANCorporate investment, ﬁnancing, and exit model with anearnings-based borrowing constraint∗
Michi NISHIHARA†,

Takashi SHIBATA‡,

Chuanqian ZHANG§

Abstract
We study the eﬀects of an earnings-based borrowing constraint (EBC) on a ﬁrm’s investment,
ﬁnancing, and exit decisions by developing a real options model with EBC. We highlight how EBC
aﬀects the decisions and values diﬀerently from a liquidation value-based borrowing constraint(LBC). Unlike under LBC, the ﬁrm delays investment to increase the cap of debt under EBC.
Investment reversibility (or equivalently, liquidation value) does not largely aﬀect the ﬁrm with
EBC, although it greatly aﬀects the ﬁrm with LBC. Unlike LBC, EBC loosens with higher volatility
because higher volatility delays investment, which increases the cap of debt. With low investment
reversibility and high volatility, EBC is preferable to LBC from a ﬁrm value perspective, andin case of ﬁnancial distress, the ﬁrm goes into reorganization bankruptcy rather than liquidation
bankruptcy. This also implies a positive relation between the prevalence of EBC and reorganization
bankruptcy. Our results are largely consistent with empirical observations.
JEL Classiﬁcations Code: G13; G32; G33.
Keywords: real options; borrowing constraints; capital structure; bankruptcy.

∗

This version was written on 21 September, 2021. This work was supported by the JSPS KAKENHI (Grant numbers

JP20K01769, JP21H00730). We thank Makoto Goto, Haejun Jeon, Sudipto Sarkar, and Yuan Tian for helpful comments.
We also thank the participants at FMA 2020 in Kyoto (online), Real Options Seminar 2021 in Osaka (online), and ORSJ
2021 fall conference in Fukuoka (online).
†
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 Management, 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§
Cotsakos College of Business, William Paterson University, 1600 Valley Road, Wayne, New Jersey 07470, USA,
Email: zhangc4@wpunj.edu

11Introduction

Economics literature, including Kiyotaki and Moore (1997) and Bernanke, Gertler, and Gilchrist(1999), has traditionally investigated the eﬀects of liquidation (or asset) value-based borrowing (orlending) constraint (hereafter, denoted by LBC) on corporate investment and ﬁnancing decisions,as well as on business cycles. Under LBC, a ﬁrm faces a debt issuance capacity based on theliquidation value of its speciﬁc assets (e.g., real estate and equipment). On the other hand, recent
empirical studies, such as Lian and Ma (2021), Kermani and Ma (2020b), and Drechsel (2020),
have shown the prevalence of earnings-based borrowing constraint (hereafter, denoted by EBC)
over LBC. According to Lian and Ma (2021), 80% of debt in the United States is issued basedon EBC, while 20% is issued based on LBC. Under EBC, debt capacity is based on operating
earnings—earnings before interest, taxes, depreciation, and amortization (EBITDA) are typicallyused—rather than asset values.
This paper aims to theoretically show the eﬀects of EBC on corporate investment, ﬁnancing,and exit decisions. In particular, we focus on how EBC aﬀects the corporate policies and values
diﬀerently from LBC. We build a standard real options model (e.g., Shibata and Nishihara (2012)
and Sundaresan, Wang, and Yang (2015)), which combines McDonald and Siegel (1986)’s optimal
investment timing model and Leland (1994)’s optimal leverage model. The model considers aﬁrm that has an option to invest by incurring investment costs along with issuing consol debt.
Based on Abel and Eberly (1996), Abel, Dixit, Eberly, and Pindyck (1996), and Shibata andNishihara (2018), we assume partial investment reversibility, where a part of investment costs willremain as liquidation value. On bankruptcy, a fraction of ﬁrm value is lost, and debt holders, whotake over the ﬁrm, choose between liquidating or operating the ﬁrm by comparing the liquidationand going-concern values. The former is called liquidation bankruptcy, while the latter is called
reorganization bankruptcy. In addition to this standard setup, we impose EBC at the investment
time. For comparison, we also investigate a model with LBC rather than EBC.1 The model results
are summarized below.
With very tight EBC, the ﬁrm relinquishes risky debt issuance but resorts to riskless debt.
The possibility of riskless debt ﬁnancing is consistent with empirical observations by Corbae andD’Erasmo (2021). Reasonable levels of tightness of EBC (implied by Lian and Ma (2021) andDrechsel (2020)) lead to realistic levels of leverage, which are lower than that of the unconstrainedmodel. With less tightness of EBC, the ﬁrm increases risky debt to increase ﬁrm value. This highleverage makes the ﬁrm choose reorganization bankruptcy rather than liquidation bankruptcy incase of ﬁnancial distress. Reorganization bankruptcy by the ﬁrm with high leverage is consistent
with empirical evidence of Bris, Welch, and Zhu (2006) and Corbae and D’Erasmo (2021). EBC
can increase the investment threshold beyond that of the riskless debt ﬁnancing case, and the1

The model with LBC is essentially the same as in Shibata and Nishihara (2018). Shibata and Nishihara (2018)

examine not only investment timing, but also investment size. However, LBC does not aﬀect the optimal investment
size.

2

investment threshold decreases with less tightness of EBC. The investment delay EBC causes is
consistent with Adler (2020) and Kariya (2020)’s empirical evidence, which shows that EBC reduces
corporate investment. This eﬀect of EBC on investment timing is contrasted with that of LBC. In
fact, Shibata and Nishihara (2018) show that LBC can accelerate investment earlier than that of
the unconstrained model.
The eﬀects of investment reversibility on the investment and ﬁnancing decisions under EBC arequite diﬀerent from those under LBC. The cap of debt is not related to liquidation value under
EBC, although the cap of debt is based on liquidation value under LBC. Then, higher investment
reversibility does not largely aﬀect EBC, leverage, investment timing, and ﬁrm value under EBC,
although it greatly relaxes LBC, increases leverage and ﬁrm value, and accelerates investment
under LBC. These diﬀerent eﬀects under EBC and LBC are consistent with empirical observationsof Kermani and Ma (2020b). The comparison results also imply that for the ﬁrm with lower
investment reversibility, EBC tends to be preferable to LBC. This result is also consistent with
Kermani and Ma (2020b) and Lian and Ma (2021)’s empirical ﬁndings that EBC is more prevalent
for ﬁrms with lower liquidation value. Under EBC, the ﬁrm with lower investment reversibility
tends to choose reorganization bankruptcy in case of ﬁnancial distress. Then, the model predicts
a positive relation between the prevalence of EBC and reorganization bankruptcy through low
liquidation value. This prediction is consistent with empirical evidence of Lian and Ma (2021) andKermani and Ma (2020b).
The eﬀects of cash ﬂow volatility on the ﬁrm with EBC is also diﬀerent from those with LBC.
Higher volatility does not change the cap of debt under LBC, but it increases the cap of debt
under EBC because higher volatility increases the investment threshold. Through this investment
timing channel, EBC is less binding with higher volatility, but LBC is more binding with higher
volatility. Then, contrary to the results in unconstrained and LBC models, higher volatility induces
the ﬁrm with EBC to take higher leverage, which tends to lead to reorganization bankruptcy incase of ﬁnancial distress. The comparison results also imply that for the ﬁrm with higher volatility,
EBC tends to be preferable to LBC. These volatility eﬀects are novel and have not been tested inempirical literature.
At the end of this section, we brieﬂy explain contributions to the related literature. This paper
is closely related to the real options literature on investment and ﬁnancing problems. Mauer andSarkar (2005) develop an investment timing model of a levered ﬁrm and show how agency conﬂicts
between shareholders and debt holders aﬀect investment timing. Sundaresan and Wang (2007) andSundaresan, Wang, and Yang (2015) analytically derive optimal investment timing with optimal
capital structure and show the eﬀects of optimal leverage on investment timing. Hackbarth andMauer (2012) develop an investment timing model with multiple debt issues and explore optimal
debt priority structure. Shibata and Nishihara (2012) and Shibata and Nishihara (2018) examine
investment and ﬁnancing models with exogenous debt capacity and LBC, respectively, and showthat the debt borrowing constraints can counter-intuitively accelerate investment. However, to ourknowledge, no papers investigate an investment and ﬁnancing problem with EBC. This paper is the3

ﬁrst to analyze a real options model with EBC and show how EBC aﬀects the corporate investment,
ﬁnancing, and exit decisions.
This paper also contributes to the recent literature on EBC. Lian and Ma (2021), Kermani
and Ma (2020b), and Kermani and Ma (2020a) fully investigate how EBC binds ﬁrms in practice.
These papers argue that ﬁrms that are expected to choose reorganization bankruptcy rather thanliquidation bankruptcy in case of ﬁnancial distress tend to be bound by EBC rather than LBC.
Adler (2020) and Kariya (2020) show empirical evidence that EBC reduces corporate investment.
However, these papers focus mainly on empirics, and to our knowledge, no papers develop dynamic
corporate ﬁnance theory under EBC. Thus, we complement the EBC literature by theoreticallyshowing the eﬀects of EBC on the investment timing, capital structure, exit timing, and exit typein the dynamic model.
The remainder of this paper is organized as follows. Section 2 introduces the model setup.
Section 3 formulates and solves the ﬁrm’s investment and ﬁnancing problem under EBC. Section 4
explores the model solutions in full details in numerical examples and illustrates empirical implications. We also highlight how EBC aﬀects the corporate investment, ﬁnancing, and exit decisions
diﬀerently from LBC. Section 5 concludes the paper.

2

Model Setup

2.1

Entry to the market

The model builds on the standard setup (e.g., Shibata and Nishihara (2012) and Sundaresan, Wang,
and Yang (2015)) of investment with optimal capital structure based on tradeoﬀ theory. Consider
a ﬁrm that has an option to invest in a new project by incurring capital expenditure I(> 0) (cf.
McDonald and Siegel (1986) and Dixit and Pindyck (1994)). At the investment time, the ﬁrm canuse debt ﬁnancing. Following the standard literature (e.g., Black and Cox (1976), Leland (1994),
and Goldstein, Ju, and Leland (2001)), we consider consol debt with coupon C. After investment,
the ﬁrm receives continuous streams of earnings before interest and taxes (EBIT) X(t), where X(t)follows a geometric Brownian motiondX(t) = µX(t)dt + σX(t)dB(t) (t > 0),

X(0) = x,where B(t) denotes the standard Brownian motion deﬁned in a ﬁltered probability space (Ω, F, P, {Ft })and µ, σ(> 0) and x(> 0) are constants. Assume that the initial value, X(0) = x, is suﬃciently
low to exclude the ﬁrm’s entry at the initial time. For convergence, we assume that r > µ, where apositive constant r denotes the risk-free interest rate. The corporate tax rate τ ∈ (0, 1) is applied
for X(t) − C.
The ﬁrm optimizes both investment time T i and coupon C (which determines leverage) to maximize the investment option value. At investment time T i , an earnings-based borrowing constraint(EBC)
Dd (X(T i ), C) ≤ ϕE X(T i )4(1)

is enforced, where Dd (X(T i ), C) and ϕE denote the risky debt value at time T i and the tightness
parameter of EBC, respectively. Throughout the paper, we use superscript d in the risky debt
ﬁnancing case.2 In EBC (1), we use EBIT just after investment because in this model, the ﬁrm
receives no EBIT until investment. If we consider a growth option model, we can use EBIT justbefore investment in EBC. Technically, it does not matter whether to use EBIT before or afterinvestment by adjusting the level of ϕE . If the ﬁrm can optimally choose debt issuance timing
apart from investment timing, the ﬁrm would issue debt just after investment rather than before
investment to increase the cap of debt.3 Hence, the main results of this paper will remain unchanged
in a growth option model.
EBC (1) means that the cap of debt is based on EBIT. This type of EBC is the most prevalent
among various types of EBC (see Lian and Ma (2021) and Drechsel (2020)). Another popular
EBC type is an interest coverage constraint, which sets a cap on the ratio of interest payments toearnings (see Greenwald (2019)), and this type can be modeled as4
C ≤ ϕE X(T i ).

(2)

However, this paper’s main results will remain unchanged, whether we assume EBC (1) or (2).
According to Lian and Ma (2021), EBC (especially for bank debt) can be periodically checked after
the debt issuance time. For model tractability, we assume debt with inﬁnite maturity and EBC
only at the debt issuance time, although debt rebalancing with short-term debt under periodic
EBC will be an interesting issue for future research.

2.2

Exit from the market

Following the standard literature (e.g., Abel and Eberly (1996), Abel, Dixit, Eberly, and Pindyck(1996), and Shibata and Nishihara (2018)), we assume partial investment reversibility, where afraction k ∈ (0, 1) of capital expenditure I will remain as liquidation value. Higher k stands forhigher investment reversibility. In the presence of liquidation value kI > 0, the ﬁrm chooses one of
the following three exit forms when EBIT X(t) deteriorates.
The ﬁrst type is exit without bankruptcy. We call this sellout. In sellout, the distressed ﬁrm’s
shareholders liquidate all assets by kI. According to the absolute priority rule (APR) of debt, debt
holders are repaid the face value of debt, which equals C/r for the consol debt. This corresponds
to a standard debt covenant (e.g., Morellec (2001) and Lambrecht and Myers (2008)) that restricts
the disposition of assets unless debt holders are fully compensated. Shareholders receive positive
residual value, i.e., (1 − τ )kI − C/r ≥ 0. In this case, there is no deadweight costs of bankruptcy,
although the level of k reﬂects ineﬃciency in asset liquidation.
The second type is liquidation bankruptcy. In liquidation bankruptcy, the ﬁrm’s shareholders
do not liquidate the ﬁrm due to negative residual value, i.e., (1 − τ )kI − C/r < 0. Instead,
We will derive Dd (X(T i ), C) in (9) in Section 3.2.
The behavior of debt ﬁnancing after investment is consistent with empirical evidence of Drechsel (2020).
4
Plausible levels of ϕE in (2) are diﬀerent from those in (1).

2

3

5

shareholders stop coupon payments to debt holders and receive nothing. Debt holders take overthe ﬁrm based on the APR and liquidate all assets immediately to gain liquidation value (1 − α)kI,
where a fraction α ∈ (0, 1) of the ﬁrm value is lost to deadweight costs of bankruptcy (e.g., ﬁling
and attorney fees). Debt holders choose liquidation bankruptcy if this liquidation value is higher
than the going-concern value in the third type.
The third type is bankruptcy without immediate liquidation.

We call this reorganization

bankruptcy. In reorganization bankruptcy, the ﬁrm’s shareholders do not liquidate the ﬁrm due tonegative residual value, i.e., (1 − τ )kI − C/r < 0. As in liquidation bankruptcy, shareholders stop
coupon payments to debt holders and receive nothing. Debt holders take over the ﬁrm based on the
APR and operate the ﬁrm as a going concern. As in liquidation bankruptcy, a fraction α ∈ (0, 1)of the ﬁrm value is lost to deadweight costs of bankruptcy, i.e., EBIT shrinks to (1 − α)X(t) after
reorganization. Debt holders choose reorganization bankruptcy if this going-concern value is higher
than the liquidation value.
Note that if we set k = 0, the exit model is essentially the same as in Leland (1994) andGoldstein, Ju, and Leland (2001), where only the third type of exit arises. 5 In the presence of
positive liquidation value kI, as in Mella-Barral and Perraudin (1997), Lambrecht and Myers(2008), and Shibata and Nishihara (2018), the two other possibilities arise. The three potential exit
forms are consistent with Corbae and D’Erasmo (2021)’s empirical observations. In practice, thereorganization bankruptcy procedure is not as simple as the above. Indeed, especially in Chapter
11 in the United States, reorganization bankruptcy frequently accompanies debt renegotiation(e.g., coupon reductions) between shareholders and debt holders. For more detailed modeling ofChapter 11, refer to Broadie, Chernov, and Sundaresan (2007) and Antill and Grenadier (2019).
Although our model simpliﬁes the reorganization bankruptcy procedure, the model can capture
the interactions between EBC and exit forms. Empirical papers motivate our analysis on theinteractions, including Lian and Ma (2021) and Kermani and Ma (2020b), which argue a positive
correlation between EBC and reorganization bankruptcy.

3Model Solution

3.1

Debt holders’ choice between liquidation and reorganization bankruptcy

As in the standard literature (e.g., Sundaresan and Wang (2007), Shibata and Nishihara (2012),
and Sundaresan, Wang, and Yang (2015), and Shibata and Nishihara (2018)), we solve the model
backward. First, we derive the ﬁrm’s going-concern for debt holders. Suppose that the ﬁrm goes
into reorganization bankruptcy at time T d satisfying X(T d ) = xd (C), where xd (C) denotes thebankruptcy threshold, which will be later speciﬁed as a function of coupon C in (8) in Section 3.2.

5

Leland (1994) and Goldstein, Ju, and Leland (2001) do not explicitly distinguish between liquidation and reorgani-zation bankruptcy, but the ﬁrm value on bankruptcy agrees with that of the third type, i.e., the unlevered ﬁrm value
multiplied by (1 − α).

6At bankruptcy time T d , the going-concern value, denoted by G(xd (C)), becomes
Z Tl
G(x (C)) = sup E[d

T l ≥T de−r(t−T ) (1 − τ )(1 − α)X(t)dt + e−rT (1 − τ )(1 − α)kI]
d

l

Td

 d
γ 

 d
x (C)
x (C)
xl
+
kI −
= (1 − τ )(1 − α)
r−µ
xl
r−µ

(3)

for xd (C) ≥ xl , where T l = inf{t ≥ T d | X(t) ≤ xl } and xl denote the liquidation time andthreshold (optimized by former debt holders), respectively, and we can easily derivexl =γ(r − µ)kI
,
γ−1

(4)

following the standard real options literature (cf. the value matching and smooth pasting conditionsq2
2
in Dixit and Pindyck (1994)). Notation γ = 0.5 − µ/σ − (µ/σ 2 − 0.5) + 2r/σ 2 denotes thenegative characteristic root. Note that after reorganization bankruptcy, the ﬁrm operates as an
all-equity ﬁrm and pays the corporate tax until liquidation. In (3), (1 − τ )(1 − α)xd (C)/(r −µ) denotes the value of operating the ﬁrm perpetually, and the extra term is the value of theliquidation option.6 For xd (C) ≤ xl , we have G(xd (C)) = (1 − τ )(1 − α)kI, which means liquidation
immediately after reorganization bankruptcy. However, due to corporate tax τ , this value is lower
than that of liquidation bankruptcy, i.e., (1 − α)kI. Note that debt holders can obtain (1 − α)kI
by choosing liquidation bankruptcy directly. Thus, debt holders choose reorganization bankruptcy
for G(xd (C)) > (1 − α)kI, where xd (C) > xl is satisﬁed. They choose liquidation bankruptcy forG(xd (C)) ≤ (1 − α)kI. The debt holders’ choice max{(1 − α)kI, G(xd (C))} will be relevant in debt
valuation (9) in Section 3.2.
Higher investment reversibility k increases (1 − α)kI more than G(xd (C)). Higher coupon C
increases xd (C) (cf. (8) in Section 3.2), and higher xd (C) increases G(xd (C)) in (3). Then, withlower k and with higher C and xd (C), the ﬁrm is more likely to go into reorganization bankruptcy.7
These results are consistent with empirical evidence. For instance, Kermani and Ma (2020a) showthat ﬁrms in the industries with high liquidation value (e.g., transportation industries) tend to go
into liquidation bankruptcy. Bris, Welch, and Zhu (2006) and Corbae and D’Erasmo (2021) showthat reorganization bankruptcy is more prevalent for ﬁrms with higher levels of EBITDA and debt.

3.2

Shareholders’ choice between sellout and default

In this subsection, we examine shareholders’ exit choice between sellout and default. Suppose thatthe ﬁrm issues consol debt with coupon C at investment time T i satisfying X(T i ) = xi , wherexi denotes the investment threshold. Shareholders of the ﬁrm choose sellout for positive residual
value, i.e., (1 − τ )kI − C/r ≥ 0. Let C s be C s = r(1 − τ )kI, which means the maximum coupon

6

For k = 0, we have G(xd (C)) = (1 − τ )(1 − α)xd (C)/(r − µ), which agrees with the ﬁrm value on bankruptcy in

Leland (1994) and Goldstein, Ju, and Leland (2001).
7
Antill and Grenadier (2019) also show the same results. Although our model simpliﬁes reorganization bankruptcy
process, the model can capture the stylized results on the choice between liquidation and reorganization bankruptcy.

7

within the sellout region. For C ∈ [0, C s ], the equity value, denoted by E s (xi , C), becomes


Z Ts
C
s i
−r(t−T i )−rT s]
E (x , C) = sup E[e(1 − τ )(X(t) − C)dt + e(1 − τ )kI −r
T s ≥T i
Ti
 i

γ 

C
τC
xs (C)xxi= (1 − τ )
− +
kI
−
−
r−µ
r
xs (C)(1 − τ )rr−µ(5)

for xi ≥ xs (C), where T s = inf{t ≥ T i | X(t) ≤ xs } and xs (C) denote the sellout time andthreshold (optimized by shareholders), respectively, and we can easily deriveτCγ(r − µ)sx (C) =kI −.
(γ − 1)(1 − τ )r(6)

in the standard manner. Throughout the paper, the superscript s denotes the sellout case. Note that
xs (C) > 0 follows from C ≤ C s . In (5), (1 − τ )(xi /(r − µ) − C/r) denotes the value of operating theﬁrm perpetually, and the remaining term denotes the value of the sellout option, where shareholders
lose the tax beneﬁts of debt and future cash ﬂows instead of obtaining the constant liquidation
value. In (6), xs (C) decreases in C because the tax beneﬁts of debt increase in C. There isno possibility of bankruptcy, and hence, the debt value is the riskless value Ds (xi , C) = C/r.
Although there is no possibility of sellout in models without the constant liquidation value (e.g.,
Leland (1994) and Goldstein, Ju, and Leland (2001)), sellout can potentially arise in models withﬁxed liquidation value (e.g., Mella-Barral and Perraudin (1997), Lambrecht and Myers (2008), andShibata and Nishihara (2018)). Sellout by the ﬁrm with low debt level (i.e., C ≤ C s ) is consistent
with empirical ﬁndings in Corbae and D’Erasmo (2021).
On the other hand, shareholders choose default for negative residual value, i.e, C > C s .
Bankruptcy by the ﬁrm with high debt level is consistent with empirical ﬁndings in Corbae andD’Erasmo (2021). In this case, the equity value, denoted by E d (x, C), becomes
Z Tdi
E d (xi , C) = sup E[
e−r(t−T ) (1 − τ )(X(t) − C)dt]
T d ≥T i= (1 − τ )



Ti

xi
C
− +
r−µ
r



xi
xd (C)γ C
xd (C)
−
r
r−µ


(7)

for xi ≥ xd (C), where T d = inf{t ≥ T i | X(t) ≤ xd } and xd (C) denote the default time andthreshold (optimized by shareholders), respectively, and we can easily derivexd (C) =γ(r − µ)C.
(γ − 1)r(8)

in the standard manner. In (7), (1 − τ )(xi /(r − µ) − C/r) denotes the value of operating the ﬁrm
perpetually, and the remaining term reﬂects the value of the default option. Equations (7) and(8) are essentially the same as in Black and Cox (1976), Leland (1994) and Goldstein, Ju, andLeland (2001) because investment reversibility k does not matter to the equity value in the risky
debt ﬁnancing case. Following the standard literature (e.g., Leland (1994) and Goldstein, Ju, andLeland (2001)), shareholders do not take into account debt in place but choose xd (C) in their own
interests. This leads to agency conﬂicts between shareholders and debt holders, and debt is priced
under the rational expectation of shareholders’ default timing. The debt value becomes

γ 

C
xi
C
d
Dd (xi , C) =+max{(1
−
α)kI,
G(x
,
(C))}
−
r
xd (C)r8(9)

where max{(1 − α)kI, G(xd (C))} reﬂects debt holders’ choice between liquidation and reorganization bankruptcy, as shown in Section 3.1. In (9), the ﬁrst term stands for the riskless debt value,
whereas the remaining term reﬂects loss from bankruptcy.

3.3

Investment and ﬁnancing decisions

Finally, we examine the ﬁrm’s investment and ﬁnancing decisions. Following the standard literature(e.g., Leland (1994), Sundaresan and Wang (2007), and Shibata and Nishihara (2018)), investment
time T i and coupon C are chosen to maximize the ﬁrm value. This is because the initial ﬁrm value
agrees with the ex-ante shareholders’ value when debt is fairly priced.
We ﬁrst consider the problem subject to riskless debt ﬁnancing, i.e., C ∈ [0, C s ]. For riskless
debt, debt holders have no reason to impose EBC, and hence, we consider no ﬁnancial constraints. 8
The riskless ﬁrm value at time 0, denoted by V s (x), becomesV s (x) =supT i ≥0,C∈[0,C s ]

= sup
xi ≥x

E[e−rT (E s (X(T i ), C) + C/r) − I)]i x β  (1 − τ )x

|

xi

r−µ

τ Cs
+
−I +
r



xi
xs (C s )γ {z(1 − τ )xs (C s ) τ C s(1 − τ )kI −
−
r−µ
r


,
}

=v s (x,xi )(10)where T i = inf{t ≥ 0 | X(t) ≥ xi } and xi denote the investment time and threshold (optimizedq
2
by shareholders), respectively. Notation β = 0.5 − µ/σ 2 + (µ/σ 2 − 0.5) + 2r/σ 2 denotes the
positive characteristic root. We have (10) because E s (X(T i ), C) + C/r monotonically increases
in C. Indeed, E s (xi , C) + C/r (see (5)) can be decomposed as the unlevered ﬁrm value and tax
beneﬁts of debt, and C s maximizes the tax beneﬁts of debt. Unfortunately, we cannot analyticallysolve xi in problem (10). When we deﬁne v s (x, xi ) as the objective function of problem (10), theﬁrst-order condition becomes ∂v s (x, xi )/∂xi = 0.
We next consider the problem subject to risky debt ﬁnancing, i.e., C > C s . In this case, EBC(1) is imposed. The risky ﬁrm value at time 0, denoted by V d (x), becomesV d (x) = x βsupxi ≥x,C>C si|x(E d (xi , C) + Dd (xi , C) − I)
{z
}

(11)

=v d (x,xi ,C)subject to
Dd (xi , C) ≤ ϕE xi .

(12)

As in Leland (1994), E d (xi , C) + Dd (xi , C) can be decomposed as the unlevered ﬁrm value, tax
beneﬁts of debt, and bankruptcy costs; hence, the tradeoﬀ between the tax beneﬁts of debt andbankruptcy costs determines C under EBC (12). Unfortunately, we cannot analytically solve xi
and C in problem (11) subject to EBC (12). Let v d (x, xi , C) as the objective function of problem(11). We will numerically solve problem (11) with no constraints, where the ﬁrst-order condition
8

Although we do not endogenously explore the rationale of EBC, debt holders usually impose EBC to mitigate default

risk. Therefore, EBC is relevant only in the risky debt ﬁnancing case.

9

Table 1: Borrowing constraints and bankruptcy types.
(a) Cap of debt.

(b) Value on bankruptcy.

EBC

ϕE x

Reorganization)(1−α)x
G(x) = (1−τr−µ+ (Option value)

LBC

ϕL (1 − α)kI

Liquidation(1 − α)kI

becomes ∂v d (x, xi , C)/∂xi = 0 and ∂v d (x, xi , C)/∂C = 0. If the solution satisﬁes EBC (12) andC > C s , EBC is not binding. Otherwise, EBC is binding, where the ﬁrst-order condition becomes∂D d (xi , C)∂v d (x, xi , C)−λ− ϕE = 0,∂xi∂xi∂v d (x, xi , C)
∂D d (xi , C)
−λ
= 0,
∂C
∂C
Dd (xi , C) − ϕE xi = 0,
where λ is a positive Lagrange multiplier.9 The ﬁrm chooses between riskless and risky debt ﬁnancing by comparing V s (x) and V d (x), and hence, the initial ﬁrm (option) value is equal toV (x) = max{V s (x), V d (x)}.
In the next section, we also examine a model with LBC to highlight the diﬀerences between theeﬀects of EBC and LBC on the corporate investment, ﬁnancing, and exit decisions. In the LBC
model, we assume a borrowing constraint for risky debt as
Dd (xi , C) ≤ ϕL (1 − α)kI,(13)

where ϕL denotes the tightness parameter of LBC. LBC (13) means that the cap of debt is based on
liquidation value (1−α)kI, which is consistent with Kiyotaki and Moore (1997)’s model. The modelwith LBC (13) is essentially the same as in Shibata and Nishihara (2018). Table 1 summarizes thecaps of debt under EBC and LBC as well as the reorganization and liquidation values. Although
the cap of debt under LBC is based on the liquidation value, the cap of debt under EBC isclosely related to the reorganization value (see Table 1). Indeed, G(x) is approximately equal to(1 − τ )(1 − α)x/(r − µ) (i.e., a linear function of x) for low k because the option value (i.e., thesecond term in (3)) is nearly zero. As we will show in the next section, this relation leads to a
positive relation between the prevalence of EBC and reorganization bankruptcy.

4

Numerical analysis and implications

In this section, we conduct numerical analyses, including comparative statics with respect to tightness of EBC ϕE , investment reversibility k, and volatility σ. We set the baseline parameter values
in Table 2, where the values of r, µ, σ, τ , and α are standard in dynamic corporate ﬁnance literatureand reﬂect a typical S&P ﬁrm (e.g., Morellec (2001) and Arnold (2014)). We set the investment
reversibility k = 0.23 based on empirical evidence that average liquidation value of plant, property,
9

We remove the possibility C → C s because v d (x, xi , C s ) is lower than v s (x, xi ).

10

Table 2: Baseline parameter values.
r

µ

σ

τ

α

k

ϕE

I

x

0.05

0.01

0.2

0.15

0.4

0.23

4

10

0.5

and equipment (PPE) and working capital is 23% of the book value in Kermani and Ma (2020a).
We set the tightness parameter ϕE = 4 based on empirical observations (e.g., Lian and Ma (2021)
and Drechsel (2020)) that the cap of debt under EBC is usually set between 3 and 5 times EBITDA.
For the baseline parameter values, EBC is binding. The ﬁrm invests at xi = 0.996 and issues
debt with coupon C = 0.2047(> C s = 0.0978). At the investment time, the leverage LV =Dd (xi , C)/(E d (xi , C) + Dd (xi , C)) is 0.1833, and the credit spread CS = C/D d (xi , C)) − r is
0.0014. The ﬁrm goes into reorganization bankruptcy when X(t) falls to xd = 0.0941. Afterreorganization bankruptcy, former debt holders operate the ﬁrm until X(t) falls to xl = 0.0529. At
time 0, the ﬁrm value V (x) becomes 3.276.
As a benchmark case, we also compute the ﬁrst-best solution in the absence of EBC. In thiscase, the ﬁrm invests at xi = 0.9728 and issue debt with coupon C = 0.6062. At investment time,
LV is 0.4856, and CS is 0.0075. The ﬁrm goes into reorganization bankruptcy when X(t) falls to
xd = 0.2787. After reorganization bankruptcy, former debt holders operate the ﬁrm until X(t) falls
to xl = 0.0529. At time 0, the ﬁrm value V (x) becomes 3.421.
When we compare the results in the two cases, we ﬁnd the following eﬀects of EBC on thecorporate investment, ﬁnancing, and exit decisions. EBC greatly decreases C, LV , and CS. These
deleverage eﬀects of EBC are consistent with empirical observations that debt holders mitigate
default risk by imposing EBC. Although Leland-type models (e.g., Leland (1994) and Goldstein,Ju, and Leland (2001)) tend to generate much higher leverage than those observed in the real world(e.g., Graham (2000) and Frank and Goyal (2009)), reasonable tightness of EBC generates observed
leverage levels. EBC also delays investment and decreases the ﬁrm value through the deleverage
eﬀect.

4.1

How EBC and LBC bind the ﬁrm

In this subsection, we examine how EBC binds the ﬁrm. The top left panel of Figure 1 shows thebinding and nonbinding regions for varying levels of tightness ϕE and investment reversibility k.
For comparison, the top right panel shows those of LBC. In the left panel, with higher ϕE , EBC
is less likely to bind the ﬁrm. For realistic values ϕE ∈ [3, 5] (implied by Lian and Ma (2021) andDrechsel (2020)), EBC is binding.
Notably, k does not largely aﬀect whether EBC is binding in the top left panel of Figure 1,
although k greatly aﬀects whether LBC is binding in the right panel.10 This diﬀerence arises fromthe diﬀerence of the right-hand sides of EBC (12) and LBC (13) (cf. Table 1). Indeed, the cap
10

The eﬀect of k on LBC is the same as in Shibata and Nishihara (2018).

11

of debt is not related to k under EBC, although the cap of debt stems from liquidation value kI
under LBC. Then, unlike under LBC, k does not largely aﬀect whether EBC binds the ﬁrm. Thisobservation also leads to the following implications about ﬁrm value. For the ﬁrm with lower k,
EBC tends to be preferable to LBC because EBC tends to be looser than LBC.11 This result is
consistent with empirical evidence of Kermani and Ma (2020b) and Lian and Ma (2021). In fact,
Kermani and Ma (2020b) show that EBC is more prevalent for ﬁrms with lower liquidation value,and Lian and Ma (2021) show that EBC is more frequent for countries with bankruptcy laws thatfacilitate reorganization rather than liquidation (which means countries with relatively high costsof liquidation).
The bottom left panel of Figure 1 shows the binding and nonbinding regions for tightness ϕE
and volatility σ. For comparison, the bottom right panel shows those of LBC. EBC is less likely to
be binding with higher σ, although LBC is more likely to be binding with higher σ.12 This diﬀerenceis explained as follows. As it is well known in the real options literature (e.g., Dixit and Pindyck(1994)), higher σ increases the option value of waiting and investment threshold xi . An increasein xi increases both sides of EBC (12), and the right-hand side eﬀect dominates. In other words,
higher σ relaxes EBC through the investment delay channel. On the other hand, an increase in xi
increases the left-hand side of LBC (13), but does not change the right-hand side of LBC (13). Thatis, the cap of debt in LBC (13) does not depend on xi ; hence, higher σ tightens LBC through theinvestment delay channel. From the ﬁrm value perspective, with higher σ, the ﬁrm tends to prefer
EBC to LBC. This also means that when the economic environment is more volatile, ﬁrms with
EBC can be more advantageous than those with LBC. These results show the notable diﬀerence
between the impacts of σ on EBC and LBC through the investment timing channel, which leads
to new empirical predictions.

4.2

Tightness of EBC

Figure 2 plots the investment threshold xi , coupon C, exit thresholds xd , xl , xs , ﬁrm value V (x),leverage LV , and credit spread CS with varying levels of tightness of EBC ϕE . In all ﬁgures,xl denotes the liquidation threshold both in reorganization and liquidation bankruptcy. That is,
xl < xd indicates reorganization bankruptcy, while xl = xd indicates liquidation bankruptcy. For
ϕE < 2.1, the ﬁrm invests at xi = 0.9918 and issues riskless debt with coupon C s = 0.0978. In
this region, CS is 0 because of riskless debt. That is, with very tight EBC, the ﬁrm does not rely
on risky debt but resorts to riskless debt. This result is consistent with that of LBC in Shibata
and Nishihara (2018). The result is also consistent with that of Corbae and D’Erasmo (2021), whoshow that ﬁrms with low levels of debt can exit without declaring bankruptcy (i.e., sellout in thispaper).
For ϕE ∈ [2.1, 10.8], the ﬁrm issues risky debt although EBC binds it. In this region, C, xd , V (x), LV,
11This preference comes only from the ﬁrm value (ex-ante shareholder value) perspective. However, in reality, negoti-

ations between shareholders and debt holders determine such debt constraints and terms.
12
The eﬀect of σ on LBC is the same as in Shibata and Nishihara (2018).

12and CS increases in ϕE . This means the straightforward result that with less tightness of EBC,
the ﬁrm issues more and riskier debt to increase ﬁrm value. This result is also consistent with thatof LBC in Shibata and Nishihara (2018). Note that realistic levels of ϕE (i.e., 3 to 5 by Lian and
Ma (2021) and Drechsel (2020)) lead to realistic levels of LV (i.e., about 0.2 by Graham (2000)
and Frank and Goyal (2009)) in the bottom panels. As explained in Section 3.1, with higher C,
the ﬁrm is more likely to go into reorganization bankruptcy rather than liquidation bankruptcy
in case of ﬁnancial distress. This can be seen in the center-left panel of Figure 2, where xd = xl
(liquidation bankruptcy) holds for ϕE ∈ [2.1, 3.7], and xd > xl (reorganization bankruptcy) holds
for ϕE > 3.7. That is, with less tightness of EBC, the ﬁrm issues more debt; hence, it will go
into reorganization bankruptcy at the higher threshold. These results are consistent with empiricalevidence of Bris, Welch, and Zhu (2006) and Corbae and D’Erasmo (2021). In fact, they show thatﬁrms with higher EBITDA and leverage tend to go into reorganization bankruptcy rather thanliquidation bankruptcy.
More notably, in the top-left panel of Figure 2, xi is not monotonic with respect to ϕE . Indeed,xi is higher for ϕE ∈ [2.1, 4.7] than xi = 0.9918 for ϕE < 2.1 (riskless debt). Note that the ﬁrm
suﬀers from no constraints in riskless debt ﬁnancing. When the ﬁrm replaces riskless debt ﬁnancing
with risky debt ﬁnancing, EBC binds the ﬁrm. It is clear from EBC (12) that higher xi increases
the cap of debt. Then, under tight EBC (i.e., ϕE ∈ [2.1, 4.7]), the ﬁrm increases xi beyond thatof the riskless debt case to relax the cap of debt. For ϕE ∈ [4.7, 10.8], xi lies between xi = 0.9728(the unconstrained case) and xi = 0.9918 (the riskless debt case). Although xi has a kink at
ϕE = 3.7 (on which the bankruptcy types change), xi decreases in ϕE . That is, tighter EBC delays
corporate investment. This result is consistent with Adler (2020) and Kariya (2020)’s empiricalevidence, which shows that tighter EBC reduces corporate investment. More generally, the result
is consistent with the stylized facts that limited access to debt ﬁnancing tend to prevent ﬁrms
from investing (e.g., Hoshi, Kashyap, and Scharfstein (1991) and Whited (1992)). EBC’s eﬀect on
investment is contrasted with that of LBC in Shibata and Nishihara (2018). Shibata and Nishihara(2018) show that the investment threshold is U-shaped with respect to the tightness of LBC. In
particular, the investment threshold can be lower than that of the unconstrained case. Contrasted
with LBC’s ambiguous eﬀects on investment timing, EBC clearly delays investment.

4.3

Impacts of investment reversibility

Figure 3 plots the investment threshold xi , coupon C, exit thresholds xd , xl , xs , ﬁrm value V (x),leverage LV , and credit spread CS with varying levels of investment reversibility k. In the depicted
region, EBC is binding. For k < 0.43, the ﬁrm invests and issues risky debt. As explained in
Section 3.1, with higher k, the ﬁrm is more likely to go into liquidation bankruptcy rather thanreorganization bankruptcy in case of ﬁnancial distress. This can be seen in the center-left panel of
Figure 3, where xd > xl (reorganization bankruptcy) holds for k ≤ 0.24, and xd = xl (liquidation
bankruptcy) holds for k ∈ (0.24, 0.43). Combining this result with the result that the ﬁrm with low
k prefers EBC to LBC as shown in Section 4.1, our model predicts a positive relation between the13

prevalence of EBC and reorganization bankruptcy through low liquidation value. This prediction
is consistent with empirical evidence of Lian and Ma (2021) and Kermani and Ma (2020b).
Notably, xi , C, and LV are almost constant with varying levels of k(< 0.43). Unlike in LBC(13), the cap of debt in EBC (12) does not depend on k. Although higher k increases debt value
after default (through this channel, V (x) slightly increases in k), the eﬀects of k on xi , C, and LV
through this channel are weak. In the bottom-right panel of Figure 3, CS sharply decreases ink(< 0.43). This is because higher k does not aﬀect debt issuance but increases debt value after
default. That is, higher k mitigates the risk of debt by increasing debt recovery after bankruptcy.
These eﬀects of k of xi , C, LV , and CS are contrasted with those under LBC in Shibata andNishihara (2018). Under LBC, higher k greatly increases C and LV because it directly increases
the cap of debt in (13). Then, under LBC, xi decreases in k, while CS increases in k, because of theleverage eﬀect. Our results are consistent with empirical observations by Kermani and Ma (2020b).
They show that total borrowing does not depend on liquidation value for large ﬁrms and ﬁrms withpositive earnings (these ﬁrms are likely to be unconstrained or constrained by EBC), while totalborrowing increases with liquidation values for small ﬁrms and ﬁrms with negative earnings (theseﬁrms are likely to be constrained by LBC).
On the other hand, for k ≥ 0.43, the ﬁrm uses riskless debt ﬁnancing. As explained in Section
3.3, the optimal coupon is equal to C s = r(1 − τ )kI in the riskless debt case. Then, higher k
increases C s and LV , and this leverage eﬀect decreases xi and increases xs and LV . These eﬀects
are stronger than that of k < 0.43 because higher k directly increases the cap of riskless debt.
Hence, for k ≥ 0.43, V s (x) increases beyond V d (x), and the ﬁrm prefers investment with risklessdebt ﬁnancing. In the bottom-right panel of Figure 3, CS remains 0 for k ≥ 0.43 because of riskless
debt.

4.4

Impacts of cash ﬂow volatility

Figure 4 plots the investment threshold xi , coupon C, exit thresholds xd , xl , xs , ﬁrm value V (x),leverage LV , and credit spread CS with varying levels of cash ﬂow volatility σ. In the depicted
region, EBC is binding, and the ﬁrm uses risky debt ﬁnancing. In the center-left panel, xd = xl
(liquidation bankruptcy) holds for σ ≤ 0.172, while xd > xl (reorganization bankruptcy) holds for
σ > 0.172. This means that higher σ is more likely to induce the ﬁrm to go into reorganization
bankruptcy rather than liquidation bankruptcy. This result is explained by the investment timing
and EBC as follows. In the top-left panel, xi increases in σ. This is consistent with the standard
volatility eﬀect that higher σ increases the option value of waiting and the hurdle rate for investment(e.g., Dixit and Pindyck (1994)). An increase in xi relaxes the cap of debt in EBC (12), and then,the ﬁrm issues more debt to beneﬁt from the leverage eﬀect. In fact, in the top-right and bottomleft panels, C and LV (xi ) increases in σ. As explained in Section 3.1, higher C induces the ﬁrm
to go into reorganization bankruptcy rather than liquidation bankruptcy. Combining this resultwith the result that the ﬁrm with high σ prefers EBC to LBC as shown in Section 4.1, our model
predicts a positive relation between the prevalence of EBC and reorganization bankruptcy through14high σ. The positive relation is consistent with that of Section 4.3, as well as empirical evidence ofLian and Ma (2021) and Kermani and Ma (2020b).
These eﬀects of σ on C, LV , and the choice between liquidation and reorganization are contrary
to those of Antill and Grenadier (2019). They study the optimal capital structure and bankruptcy
choice by developing a more detailed model of the reorganization bankruptcy procedure, but theirmodel includes neither optimal investment timing nor debt issuance constraint. Then, as in Leland(1994), in their model, higher volatility decreases optimal coupon and leverage; hence, it induces theﬁrm to go into liquidation bankruptcy rather than reorganization bankruptcy. Even if we consider
optimal investment timing in the unconstrained and LBC models, higher σ decreases leverage,
leading the ﬁrm to go into liquidation bankruptcy rather than reorganization bankruptcy. Unlike
this paper, Shibata and Nishihara (2015) investigate the optimal choice between bank debt (whichis renegotiable) and market debt (which is nonrenegotiable). They show that with higher σ, theﬁrm tends to prefer bank debt to market debt. If we relate renegotiable and nonrenegotiable debt
to reorganization and liquidation bankruptcy, Shibata and Nishihara (2015)’s results are consistent
with those of the current paper.

5Conclusion

This paper develops a dynamic investment and ﬁnancing model with EBC. The model captures aﬁrm’s decisions on investment timing, debt issuance, leverage, exit timing, and exit choice among
sellout, liquidation bankruptcy, and reorganization bankruptcy. Through the model analyses, weshow the eﬀects of EBC on the ﬁrm’s decisions and values are quite diﬀerent from those of LBC.
The results are summarized below.
Although very tight EBC induces the ﬁrm to resort to riskless debt, plausible levels of EBC
lead the ﬁrm to use risky debt ﬁnancing with realistic levels of leverage. The ﬁrm can increase
the cap of debt under EBC by delaying investment, although investment timing is not related
to the cap of debt under LBC. Then, unlike with LBC, the ﬁrm with EBC delays investment toutilize more debt ﬁnancing. This investment delay that EBC causes is consistent with empiricalﬁndings. Investment reversibility does not largely aﬀect the ﬁrm with EBC, although it greatly
aﬀects the ﬁrm with LBC. The diﬀerence implies that the ﬁrm with low investment reversibility,
which leads to reorganization bankruptcy rather than liquidation bankruptcy in case of ﬁnancial
distress, prefers EBC to LBC. This also leads to a positive relation between the prevalence of EBC
and reorganization bankruptcy through low liquidation value. These results also are consistent with
empirical evidence. Higher volatility increases the cap of debt under EBC by delaying investment,
although the cap of debt under LBC does not depend on investment timing. Then, contrary to
the results in unconstrained and LBC models, under EBC, the ﬁrm with higher volatility increases
leverage and chooses reorganization bankruptcy in case of ﬁnancial distress. These volatility eﬀects
through the investment timing channel entail new empirical predictions.

15

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17

20

11.4
11.2

Nonbinding

15

Nonbinding

E

L

11
10

10.8

Binding

10.6
10.4
0.1

5

Binding
0.2

0.3

0.4

0.5

0
0.1

0.6

0.2

0.3

k

0.4

0.5

0.6

k

16

8.4
8.2

14

Nonbinding
Nonbinding

L

E

8
12

7.8
10

8
0.1

Binding

0.15

7.6

0.2

0.25

7.4
0.1

0.3

Binding

0.15

0.2

0.25

0.3

Figure 1: Binding and nonbinding regions under EBC and LBC. The left panels show the binding
and nonbinding regions under EBC for varying levels of tightness ϕE , investment reversibility k, andvolatility σ, whereas the right panels show the binding and nonbinding regions under LBC for varying
levels of tightness ϕL , investment reversibility k, and volatility σ.

18

1.01

0.6
x

C

i

0.5

1
0.4
0.99

0.3
0.2

0.98
0.1
0.97

0
2

4

6

8

10

12

2

4

6

E

8

10

12

8

10

12

8

10

12

E

0.3
xd
xl

3.4

0.25
0.2

xs

3.35

0.15

V(x)

3.3

0.1
3.25

0.05
0

3.2
2

4

6

8

10

12

2

4

6

E

E

8

10-3
CS

LV

0.4
6
0.3
4
0.2
2

0.1
0

0
2

4

6

8

10

12

2

E

4

6
E

Figure 2: Comparative statics with respect to tightness ϕE . The ﬁgure plots the investment threshold
xi , coupon C, exit thresholds xd , xl , xs , ﬁrm value V (x), leverage LV , and credit spread CS. The
ﬁrm chooses riskless debt for ϕE < 2.1, risky debt with liquidation bankruptcy for ϕE ∈ [2.1, 3.7], andrisky debt with reorganization bankruptcy for ϕE > 3.7. EBC is not binding for ϕE > 10.8.

19

1

0.26
C
x

0.98

i

0.24

0.96

0.22

0.94

0.2

0.92

0.18
0

0.2

0.4

0.6

0

0.2

k

0.4

0.6

k

3.36

0.1

V(x)

3.34

0.08
0.06

xd
xl

0.04

xs

3.32
3.3
3.28

0.02
3.26
0
0

0.2

0.4

0.6

0

0.2

k

0.4

0.6

k
10-3

1.5
0.24

CS

LV

0.23
1

0.22
0.21
0.2

0.5

0.19
0.18
0
0

0.2

0.4

0.6

0

k

0.2

0.4

0.6

k

Figure 3: Comparative statics with respect to investment reversibility k. The ﬁgure plots the investment threshold xi , coupon C, exit thresholds xd , xl , xs , ﬁrm value V (x), leverage LV , and credit
spread CS. The ﬁrm chooses risky debt with reorganization bankruptcy for k ≤ 0.24, risky debt withliquidation bankruptcy for k ∈ (0.24, 0.43), and riskless debt for k ≥ 0.43. In all the regions, EBC isbinding.
20

1.4

0.3
C

xi

1.3
1.2

0.25

1.1
1
0.2
0.9
0.8
0.7
0.1

0.15

0.2

0.25

0.15
0.1

0.3

0.15

0.2

0.25

0.3

4.5

0.1

V(x)
0.09

x

4

d

xl

0.08

3.5

xs

0.07
3
0.06
2.5

0.05
0.04
0.1

0.15

0.2

0.25

2
0.1

0.3

7

0.1838

0.15

0.2

0.25

10 -3

LV

CS

6
0.18360.3

5
4

0.18343
2

0.1832

1
0.183
0.1

0.15

0.2

0.25

0
0.1

0.3

0.15

0.2

0.25

0.3

Figure 4: Comparative statics with respect to volatility σ. The ﬁgure plots the investment threshold
xi , coupon C, exit thresholds xd , xl , xs , ﬁrm value V (x), leverage LV , and credit spread CS. The
ﬁrm chooses risky debt with liquidation bankruptcy for σ ≤ 0.172 and risky debt with reorganization
bankruptcy for σ > 0.172. In all the regions, EBC is binding.

21