Discussion Papers In Economics
And Business
Optimal capital structure and bankruptcy cascades
Michi NISHIHARA,Takashi SHIBATA

Discussion Paper 20-10

July

2020

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANOptimal capital structure and bankruptcy cascades∗
Michi NISHIHARA†,

Takashi SHIBATA‡

Abstract
We examine two firms’ strategic choices of capital structure in the presence of negative
bankruptcy spillovers. The low-profitability firm (denoted by firm L) that bankrupts earlier aﬀects the high-profitability firm (denoted by firm H). Against negative bankruptcy
spillovers, firm H takes either of the two contrasting responses: decreasing leverage toprepare for operations in the worse cash flow scenario after firm L’s bankruptcy or increasing leverage to bankrupt simultaneously with firm L. Firm H takes the simultaneous bankruptcy strategy when the tax benefits of increased debt dominate the cash flows
from operations after firm L’s bankruptcy. With more negative bankruptcy spillovers, asmaller profitability diﬀerence, and lower volatility, firm H is more likely to choose thesimultaneous bankruptcy strategy. The simultaneous bankruptcy equilibrium shows anovel mechanism of bankruptcy cascades through firms’ strategic choices of capital structure with negative bankruptcy spillovers. This mechanism can potentially explain theempirical findings of bankruptcy contagion and herding behavior for corporate financial
policies.
JEL Classifications Code: G13; G32; G33.
Keywords: capital structure; bankruptcy spillovers; contagion; herding; real options.

∗This version was written in 28 July, 2020. We thank Katsumasa Nishide and Yuki Sato for helpful comments.

We also thank the participants at Nippon Finance Association 28th Annual Conference for helpful feedback.
This work was supported by the JSPS KAKENHI (Grant numbers 20K01769, JP16KK0083, JP17H02547).
†
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 1920397, Japan, E-mail: tshibata@tmu.ac.jp, Phone: 81-42-677-2310, Fax: 81-42-677-2298

1

1

Introduction

A firm’s bankruptcy can bring about negative spillovers to other firms not only throughfinancial networks (e.g., Jorion and Zhang (2007) and Jorion and Zhang (2009)) but also
through real networks such as supply chains (e.g., Hertzel, Li, Oﬃcer, and Rodgers (2008)
and Kolay, Lemmon, and Tashjian (2016)) and strategic alliances (e.g., Boone and Ivanov(2012)). In the worst cases, negative bankruptcy spillovers lead to bankruptcy cascades
over firms that are linked financially and/or operationally to each other. Although anumber of papers examine default contagion from the financial market viewpoints (e.g.,credit risks and derivatives in Duﬃe, Eckner, Horel, and Saita (2009) and Benzoni, CollinDufresne, Goldstein, and Helwege (2015), financial network stability in Allen and Gale
(2000) and Eisenberg and Noe (2001)), few papers examine this problem from a perspective of corporate financial policy.
This study points to a new perspective for bankruptcy cascades; we show that firms’
strategic capital structure choices can cause bankruptcy contagion. To do this, we extend
Leland (1994) to a setup of two firms with negative bankruptcy spillovers. The negative
bankruptcy spillovers are modeled by the assumption that cash flows of a surviving firm
decrease after the other firm’s bankruptcy. We derive the payoﬀ dominant equilibriumof the game in which the firms strategically choose initial capital structure to maximize
their firm values. The equilibrium results are summarized below.
The low-profitability firm (denoted by firm L) does not change the capital structure and bankruptcy timing from those of the nonstrategic case because the bankruptcy
spillover eﬀects do not aﬀect firm L, which bankrupts earlier than firm H does. On
the other hand, the high-profitability firm (denoted by firm H) can either decrease or
increase debt from the nonstrategic level. In the former case, firm H decreases debt exante by anticipating operations in the worse cash flow scenario after firm L’s bankruptcy(called the sequential bankruptcy equilibrium). In the latter case, firm H increases debt
ex ante to go bankrupt simultaneously with firm L (called the simultaneous bankruptcy
equilibrium). In the simultaneous bankruptcy equilibrium, firm H can maximize the tax
benefits of debt until bankruptcy through the strategic increase of debt. In other words,
firm H intentionally abandons operations after firm L’s bankruptcy and goes bankrupt
simultaneously with firm L.
Firm H’s choice depends on the tradeoﬀ between cash flows from operations after firm
L’s bankruptcy and the tax benefits of increased debt. When the tax benefits dominate
the cash flows, firm H chooses the high leverage to bankrupt with firm L. This simul-2

taneous bankruptcy equilibrium is more likely to occur for more negative bankruptcy
spillovers, a smaller profitability diﬀerence, lower market volatility, and higher market
growth rate. Compared to sequential bankruptcy, simultaneous bankruptcy reduces social welfare, which is defined as the sum of two firm values and the government value,
because firm H’s high leverage and hastened bankruptcy reduce the tax revenues of thegovernment.
The simultaneous bankruptcy equilibrium shows a novel mechanism of bankruptcy
cascades through the firms’ strategic choices of capital structure in the presence of negative bankruptcy spillovers. Indeed, this new mechanism suggests that financially and/or
operationally connected firms (e.g., firms with cross-holdings, firms in a supply chain,etc.) have ex-ante herding incentives to increase debt in order to go bankrupt together,which leads to social loss (i.e., systemic risk).
Although the empirical relevance of this mechanism has not been studied, this mechanism can potentially account for the empirical findings of default clustering. Emery andCantor (2005) show that all aﬃliates within the same corporate family often go bankrupt
together. Many papers, including Das, Duﬃe, Kapadia, and Saita (2007), Benzoni, CollinDufresne, Goldstein, and Helwege (2015), and Azizpour, Giesecke, and Schwenkler (2018),
show that default contagion can occur even for firms that do not belong to the same corporate family. Our results may also have implications regarding financial contagion andsystemic risk. Acharya and Yorulmazer (2007) show the possibility that banks get exposed
to similar risks ex ante in order to fail together. The bank herding incentive stems from a“too-many-to-fail” problem—the regulator finds it optimal ex post to bail out failed banks
when the number of failed banks is too large. We complement their results by showing
that, even in the absence of the regulator’s bailout, banks cross-holding interbank loans
have incentives to increase debt and systemic risk so that they fail together.
From a perspective of corporate financial policy, our results could potentially accountfor the empirical findings of Leary and Roberts (2014). They show that firms’ financial
policies, such as issues of new equity and debt, are partly driven by responses to theirpeers. Although the learning or competition eﬀects often explain the herding behavior
for corporate financial policies, we add an alternative contagion explanation—a firm canadjust its capital structure to that of another firm, whose bankruptcy negatively aﬀect its
operations, and aim to go bankrupt together.
Moreover, we extend the baseline model to several cases. In a setup of more than
two firms, a firm’s strategic increase of debt can amplify the negative externalities, which3

leads to another firm’s strategic increase of debt. This negative feedback loop can cause
bankruptcy cascades over multiple firms. With more negative bankruptcy spillovers, alarger cash flow scale of firm H over that of firm L, and lower merger cost and volatility,
firm H is more likely to bail out firm L. With more negative bankruptcy spillovers anda lower debt renegotiation cost and volatility, firm H is more likely to renegotiate debt.
Although bailout and debt renegotiation opportunities can prevent firm H from choosing
the simultaneous bankruptcy equilibrium with high leverage and increase social welfare,they can prevent firm H from choosing the sequential bankruptcy equilibrium with lowleverage and decrease social welfare.
Lambrecht (2001), Morellec and Zhdanov (2008), Nishihara and Shibata (2010), andMatveyev and Zhdanov (2019) are most relevant to our study. These papers extend
Leland (1994) to duopoly market models. Lambrecht (2001) examines the eﬀects of debt
on entry and exit decisions, but he does not consider the strategic capital structure choice.
Morellec and Zhdanov (2008) examine the firms’ strategic capital structure choices whenanticipating a takeover opportunity in the future. Nishihara and Shibata (2010) andMatveyev and Zhdanov (2019) examine the strategic investment timing and financing
decisions, as well as bankruptcy timing. All of these papers focus on the competitive
intra-industry eﬀects—a firm’s cash flows increase after another firm’s bankruptcy–andthey find a firm’s incentive to decrease leverage and to win industry competition. In
contrast, we focus on the negative bankruptcy spillover eﬀects of bankruptcy—a firm’s
cash flows decrease after another firm’s bankruptcy—and we find a firm’s incentive toincrease leverage and to go bankrupt together. Then, this paper complements the previous
literature by showing that negative bankruptcy spillovers can distort the capital structure
in an opposite direction from the competition eﬀects.
The remainder of this paper is organized as follows. Section 2 introduces the model
setup. After Section 3.1 derives the benchmark solutions with no bankruptcy spillover,Section 3.2 shows the equilibrium solutions with bankruptcy spillovers. In Section 3.3,
we explore the solutions in numerical examples. After Section 4 explains the extended
models, Section 5 concludes the paper.

42Model Setup

2.1

Firms until bankruptcy

We extend the standard setup (e.g., Leland (1994), Goldstein, Ju, and Leland (2001), andLambrecht and Myers (2008)) to the following model consisting of two firms. Consider
two firms (denoted by firms H and L) that face the common shock X(t). We do notnecessarily consider a duopoly market, but the two firms may have customer-supplier
relationships or belong to the same conglomerate. Firm i ∈ {H, L} receives continuous
streams of earnings before interest and taxes (EBIT) ai X(t) − bi , where ai and bi arepositive constants satisfying aH /bH ≥ aL /bL . This inequality means that firm H has
higher profitability than firm L does. As in the standard literature, the state variable
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 high to exclude a firm’s bankruptcy at the initial time. For
convergence, we assume that r > µ, where a positive constant r denotes the risk-freeinterest rate.
Following the standard literature, at time 0, firm i ∈ {H, L} issues consol debt withcoupon Ci to maximize its firm value. Shareholders of firm i continue to receive cash
flows (1 − τ )(ai X(t) − bi − Ci ) until one of the firms bankrupts, where τ ∈ (0, 1) denotes a
corporate tax rate. Throughout this paper, we assume that managers operate the firm onbehalf of shareholders’ interests, and hence we do not distinguish between shareholders
and managers.

2.2

Bankruptcy and its negative spillovers

Shareholders of firm i ∈ {H, L} can stop paying coupon Ci and declare default at any
time in their own interests. They do not consider the debt value in place. At the default
time t, as in Lambrecht and Myers (2008) and Nishihara and Shibata (2019), firm i is
liquidated by the value Ui (X(t)), where the function Ui (·) is assumed to be aﬃne innumerical examples.1 We assume liquidation bankruptcy because we prefer to focus on a1

For simplicity, we assume that the functional form Ui (X(t)) is unchanged by firm j’s bankruptcy. In reality,

compared to operational cash flows, scrap value is less aﬀected by another firm’s bankruptcy. The main results

5simple case in which a bankrupt firm stops operations and causes negative spillovers to
the other firm in operations.2 We presume that debt is risky, i.e., the principal of debt
Ci /r is higher than the liquidation value Ui (X(t)).3 Then, by the absolute priority rule,debt holders receive the liquidation value Ui (X(t)), while shareholders receive nothing.
For simplicity, as in Lambrecht and Myers (2008) and Nishihara and Shibata (2019), we
assume no opportunity of debt renegotiation in the baseline model. We will also examine
a setup with debt renegotiation in Section 4.3.
After firm i’s bankruptcy, the other firm j(̸= i)’s EBIT falls to δj aj X(t) − ϵj bj if it
continues operations. The EBIT contraction parameters δj ∈ (0, 1) and ϵj (≥ 1) stand forthe negative bankruptcy spillovers, where a lower δj and higher ϵj mean stronger spillover
eﬀects. The negative externalities can arise from real and financial linkages between thetwo firms. For instance, a firm may be forced to purchase more costly parts from otherfirms when its major supply chain partner goes bankrupt. If firms cross-hold shares, afirm loses dividends after the other firm’s bankruptcy. When cross-holding debts with
infinite maturity, a firm loses coupons after the other firm’s bankruptcy. Although theloss of coupons is not counted in EBIT, it decreases cash flows. Hence, the cross-holding
model becomes almost the same model. In many cases, the negative eﬀect will not bepermanent. For example, consider firms that provide trade credit or short-term debt toeach other. Default by a firm causes bad debt loss to the other firm, but the loss is
temporary rather than permanent. The main results and implications in this paper willvary only quantitatively but not qualitatively, even if we model such temporary eﬀects.
Thus, for simplicity, we model the permanent and negative eﬀects of the other firm’s
bankruptcy on EBIT.

in this paper will remain unchanged if the bankruptcy spillover eﬀects on the liquidation value are smaller thanthat on the cash flows.
2
The assumption of liquidation bankruptcy is not essential, and we may assume reorganization bankruptcy.
The main results in this paper will remain unchanged as far as a firm’s reorganization bankruptcy reduces theother firm’s earnings. Several papers, such as Morellec, Valta, and Zhdanov (2015), Shibata and Nishihara(2015), and Antill and Grenadier (2019) examine reorganization bankruptcy with debt renegotiation.
3
Indeed, in all numerical examples, the firms issue risky debt rather than riskless debt to maximize theirfirm values.

6

2.3

Capital structure

All agents have complete information about all parameter values of both firms.4 At time
0, the equity, debt, and firm values of firm i ∈ {H, L} are priced under the rational expectation of its own bankruptcy time, the opponent firm j(̸= i)’s bankruptcy time, and thebankruptcy spillover eﬀects. Following the standard literature (e.g., Leland (1994) andGoldstein, Ju, and Leland (2001)), at time 0, firm i chooses the optimal capital structure(i.e., coupon Ci ) to maximize its firm value by the tradeoﬀ between the tax benefits andbankruptcy costs of debt. For model tractability, we do not consider the dynamic adjustment of capital structure, but Section 4.3 discusses the eﬀects of the leverage adjustment
on the model results.
The model’s key feature is that the firms make the capital structure decisions fromstrategic considerations of each other’s capital structure and bankruptcy timing, as well as
the negative bankruptcy spillovers. Morellec and Zhdanov (2008), Nishihara and Shibata(2010), and Matveyev and Zhdanov (2019) investigate the strategic decisions of capital
structure in a duopoly market, while Miao (2005) examines capital structure in a perfectlycompetitive market. These papers examine the competition eﬀects of bankruptcy—afirm’s bankruptcy increases the other firm’s cash flows. In contrast, we examine contagion
eﬀects—a firm’s bankruptcy decreases the other firm’s cash flows. By this diﬀerence, wewill show that a firm can take the capital structure which is quite diﬀerent from thatfound in previous literature.

3

Model Solutions

3.1

Nonstrategic solutions

As a benchmark, we examine two firms with no bankruptcy spillover (i.e., δi = ϵi = 1
for i = {H, L}). Hereafter, this benchmark case is called the nonstrategic case. Supposethat firm i issues debt with coupon C(≥ 0) and will bankrupt at the threshold y(> 0).
As in the standard literature (e.g., Leland (1994) and Goldstein, Ju, and Leland (2001)),

4

It might be diﬃcult for firms to precisely estimate the EBIT contraction parameters. In Section 4.4, we

discuss a case with incomplete information about bankruptcy spillovers.

7

we can calculate the following equity, debt, and firm values:∫ TN
Ei (x, y, C) = E[e−rt (1 − τ )(ai X(t) − bi − C)dt]
0
{
(
)γ (
)}
ai x
bi + Cx
ai (x ∧ y) bi + C= (1 − τ )
−
+
−
+
r−µ
r
x∧y
r−µ
r
∫ T
DiN (x, y, C) = E[e−rt Cdt + e−rT Ui (X(T ))]
0
(
)
)γ (
C
x
C
=
−
− Ui (x ∧ y) ,rx∧yr
FiN (x, y, C) = EiN (x, y, C) + DiN (x, y, C)(1 − τ )ai x (1 − τ )bi τ C
=
−
+
−
r−µ
r
r

(

x
x∧y

)γ (

(1)

(2)

)
(1 − τ )ai (x ∧ y) τ C − (1 − τ )bi+− Ui (x ∧ y) ,
r−µ
r
(3)

where T denotes the stopping time (i.e., the bankruptcy time) T = inf{t ≥ 0 | X(t) ≤ y}.
The notation x ∧ y represents the minimum of x and y, and γ denotes the negative√
characteristic root γ = 0.5 − µ/σ 2 − (0.5 − µ/σ 2 )2 + 2r/σ 2 . Throughout the paper, thesuperscript N stands for the nonstrategic case. In (1), the first two terms represent theequity value of the permanently operating firm, and the extra term represents the value ofthe option to default at the threshold y. In (2), C/r represents the riskless debt value, andthe extra term represents the expected loss of bankruptcy. Recall that on bankruptcy,
debt holders receive the liquidation value, i.e., Ui (x ∧ y), although they lose the valueof future coupon streams, i.e., C/r. The firm value (3) can be decomposed of the threeparts, i.e., the unlevered firm value, tax benefits of debt, and bankruptcy costs of debt.
The first two terms represent the unlevered firm value, while the third and forth terms
represent the tax benefits and bankruptcy costs of debt, respectively.
For debt in place, (i.e., coupon C), firm i’s shareholders optimize the bankruptcy
threshold y to maximize the equity value (1). Hence, we can derive the bankruptcy
threshold as the following function of C:
N
xN
i (C) = arg max Ei (x, y, C) =y≥0γ(r − µ)(bi + C).
(γ − 1)rai(4)

By substituting y = xN
i (C) into (1)–(3), we have the equity, debt, and firm values under
the rational expectation of the bankruptcy time. Firm i chooses its coupon C to maximize
the firm value. Then, we have the optimal coupon
CiN = arg max FiN (x, C),C≥0where we denote FiN (x, C) = FiN (x, xN
i (C), C).

(5)

We can numerically verify that in(5), FiN (x, C) is unimodal with respect to C and has a unique maximizer CiN . For

8simplicity, under the optimal capital structure, we denote the bankruptcy threshold
N
N
by xN
i = xi (Ci ).

We also denote the equity, debt, and firm values by EiN (x) =

N
N
N
N
N
N
N
N
N
EiN (x, xN
i , Ci ), Di (x) = Di (x, xi , Ci ), and Fi (x) = Fi (x, xi , Ci ), respectively.

In numerical examples, we always have


>0





∂FiN (x, C)  = 0

∂C
 <0




 =0

(C ∈ [0, CiN ))(C = CiN )(C ∈ (CiN , C̄i ))(6)(C > C̄i ),where C̄i = (γ − 1)rai x/γ(r − µ) − bi . Note that C̄i is the solution to xNi (C̄i ) = x,
which means that for C ≥ C̄i , firm i bankrupts at time 0, leading to FiN (x) = Ui (x).
In numerical examples, we can easily compute the unique maximizer CiN because of the
N > C N (cf. Figure 1) andunimodality (6). We can also numerically verify that CHL
NxN
H < xL ,(7)

which implies that compared to firm L, firm H issues more debt but defaults later becauseof its higher profitability. In theoretical analysis in Section 3.2, we assume (6) and (7).

3.2

Baseline solutions

In this subsection, we examine the baseline case with negative bankruptcy spillovers (i.e.,δi < 1, ϵi ≥ 1 for i = {H, L}). Under assumption (7), firm L, which bankrupts earlier
than firm H bankrupts, has no incentive to deviate from the nonstrategic coupon CLN ,which generates the firm value FLN (x). On the other hand, firm H, which bankrupts later
than firm L bankrupts, cannot gain the firm value FHN (x) by the nonstrategic coupon
N because firm H’s EBIT falls after firm L’s bankruptcy. Thus, firm H can potentiallyCH
N to maximize its firm value. Intuitively, firm H can decrease its coupon
deviate from CH
N to anticipate the worse cash flow scenario, where EBIT decreases after firm L’s
below CHbankruptcy. In Section 3.2.2, we will show that this straightforward result arises undercertain parameter sets. However, we will show that firm H counterintuitively increases
N under certain parameter sets.
its coupon beyond CH

3.2.1

Solutions for exogenous capital structure

In this subsection, we suppose that firms H and L have issued debt with coupons CH
and CL , respectively, at time 0. Then, we derive the firm i ∈ {H, L}’s bankruptcy

9

threshold (denoted by xi (Ci , Cj )) as well as the equity, debt, and firm values (denoted byEi (x, Ci , Cj ), Di (x, Ci , Cj ), and Fi (x, Ci , Cj )) as the functions of CH and CL . We defineδH aH CL δH aH bL+− ϵH bH ,aL
aL
aH CL aH bL
K2 (CL ) =+− bH ,aL
aL
aH CL ϵL aH bL+− bH ,
K3 (CL ) =
δL aL
δL aL
K1 (CL ) =

(8)
(9)
(10)

and also define EiR (x, y, C) and xRi (C) by replacing ai and bi with δi ai and ϵi bi , respectively, in (1) and (4). The superscript R stands for the revised equity value and threshold
due to EBIT contraction. By (8)–(10), we can readily show that K1 (CL ) < K2 (CL ) <N
N
NR
K3 (CL ), xR
H (K1 (CL )) = xH (K2 (CL )) = xL (CL ), and xH (K3 (CL )) = xL (CL ). In otherwords, CH = K1 (CL ) is the coupon such that firm H’s revised threshold agrees with firm
L’s nonstrategic threshold. Similarly, CH = K2 (CL ) is the coupon such that firm H’s nonstrategic threshold agrees with firm L’s nonstrategic threshold, whereas CH = K3 (CL ) is
the coupon such that firm H’s nonstrategic threshold agrees with firm L’s revised threshold. The following proposition shows the firms’ policies and values for exogenously given(CH , CL ). For the proof, refer to Appendix A.
Proposition 1 Suppose that firms H and L issue debt with coupon CH and CL , respecNtively. Suppose that the initial state variable x is higher than xN
H (CH ) and xL (CH ). Firm

i’s bankruptcy threshold xi (Ci , Cj ), equity value Ei (x, Ci , Cj ), debt value Di (x, Ci , Cj ),firm value Fi (x, Ci , Cj ) are given as follows.
Sequential bankruptcy case I: CH ∈ [0, K1 (CL )).
N
xH (CH , CL ) = xR
H (CH ) < xL (CL , CH ) = xL (CL ),
)γ
(
x
R N
N
N
EH
(xL (CL ), xR
EH (x, CH , CL ) = EH (x, xL (CL ), CH ) +H (CH ), CH ),
xN
(C
)
L
L

EL (x, CL , CH ) = ELN (x, xN
L (CL ), CL ),N
DH (x, CH , CL ) = DH(x, xRH (CH ), CH ),N
DL (x, CL , CH ) = DL(x, xN
L (CL ), CL ),

(

N
N
R
FH (x, CH , CL ) = EH(x, xN
L (CL ), CH ) + DH (x, xH (CH ), CH ) +FL (x, CL , CH ) = FLN (x, xN
L (CL ), CL ).

10x
NxL (CL ))γ
R N
EH(xL (CL ), xR
H (CH ), CH ),Simultaneous bankruptcy case: CH ∈ [K1 (CL ), K3 (CL )].
N
xH (CH , CL ) = xL (CL , CH ) = xN
H (CH ) ∨ xL (CL ),NN
EH (x, CH , CL ) = EH(x, xNH (CH ) ∨ xL (CL ), CH ),N
EL (x, CL , CH ) = ELN (x, xN
H (CH ) ∨ xL (CL ), CL ),NN
DH (x, CH , CL ) = DH(x, xNH (CH ) ∨ xL (CL ), CH ),NN
DL (x, CL , CH ) = DL(x, xNH (CH ) ∨ xL (CL ), CL ),N
FH (x, CH , CL ) = FHN (x, xN
H (CH ) ∨ xL (CL ), CH ),N
FL (x, CL , CH ) = FLN (x, xN
H (CH ) ∨ xL (CL ), CL ),N
N
N
where xN
H (CH ) ∨ xL (CL ) denotes the maximum of xH (CH ) and xL (CL ).

Sequential bankruptcy case II: CH > K3 (CL ).
R
xH (CH , CL ) = xN
H (CH ) > xL (CL , CH ) = xL (CL ),N
EH (x, CH , CL ) = EH(x, xN
H (CL ), CH ),EL (x, CL , CH ) = ELN (x, xN
H (CH ), CL ) +(x
NxH (CH ))γR
ELR (xN
H (CH ), xL (CL ), CL ),N
DH (x, CH , CL ) = DH(x, xNH (CH ), CH ),N
DL (x, CL , CH ) = DL(x, xR
L (CL ), CL ),FH (x, CH , CL ) = FHN (x, xN
H (CH ), CH ),(NR
FL (x, CL , CH ) = ELN (x, xN
H (CH ), CL ) + DL (x, xL (CL ), CL ) +x
NxH (CH ))γR
ELR (xN
H (CH ), xL (CL ), CL ).

For CH < K1 (CL ), firm H’s revised threshold xR
H (CH ) is lower than firm L’s threshold xN
L (CL ). Then, firm H bankrupts later than firm L bankrupts because of low levelsof debt. For CH > K3 (CL ), firm L’s revised threshold xR
L (CL ) is lower than firm H’sthreshold xN
H (CH ). Then, firm H bankrupts earlier than firm L because of high levels of debt. We call these two cases the sequential bankruptcy cases I and II, respectively. In these cases, the equity and debt values of the firm that bankrupts first are
the same as those of the nonstrategic case (say, EL (x, CL , CH ) = ELN (x, xN
L (CL ), CL ) andN (x, xN (C ), C ) in the sequential bankruptcy cases I). The negative
DL (x, CL , CH ) = DL
L
L
Lbankruptcy spillover eﬀects aﬀect he equity and debt values of the firm that bankrupts
later (say, EH (x, CL , CH ) and DH (x, CL , CH ) in the sequential bankruptcy cases I).
The intermediate case (i.e., CH ∈ [K1 (CL ), K3 (CL )]) is most interesting. For CH ∈[K1 (CL ), K2 (CL )], firm H does not bankrupt earlier than firm L because xN
H (CH ) is not11higher than xN
L (CL ), but firm H does not postpone bankruptcy after firm L’s bankruptcy
Nbecause its revised threshold xR
H (CH ) is not lower than xL (CL ). For CH ∈ (K2 (CL ), K3 (CL )],N
firm L does not bankrupt earlier than firm H because xN
L (CL ) is lower than xH (CH ),but firm L does not postpone bankruptcy after firm H’s bankruptcy because its revisedN
threshold xR
L (CL ) is not lower than xH (CH ). Thus, firms H and L bankrupt simultaneN
ously at the bankruptcy threshold xN
H (CH ) ∨ xL (CL ). We call this case the simultaneousbankruptcy case. In this case, the equity and debt values agree with those of the nonNstrategic case with the bankruptcy threshold xN
H (CH ) ∨ xL (CL ).

With no bankruptcy spillover (i.e., δi = ϵi = 1), we have K1 (CL ) = K2 (CL ) = K3 (CL ),which means that simultaneous bankruptcy occurs only for CH = K1 (CL ) = K2 (CL ) =K3 (CL ). However, with negative bankruptcy spillovers, a firm’s bankruptcy threshold isrevised upward after the other firm’s bankruptcy. As the spillover eﬀects are stronger(i.e., a lower δi and higher ϵi ), the simultaneous bankruptcy region becomes larger.

3.2.2

Solutions for endogenous capital structure

In this subsection, we consider the two firms’ game, where firm i ∈ {H, L} optimizes Ci
to maximize its firm value Fi (x, Ci , Cj ) at time 0. We define
∗
CH
= arg

max

N )]
C∈[0,K1 (CLFH (x, C, CLN ),∗∗CH= K2 (CLN ),

(11)
(12)

∗∗ > C N follows from assumption (7). Definition (11) means that for firm L’s
where CHH
∗ is firm H’s optimal coupon in the sequential bankruptcy
nonstrategic coupon CLN , CH∗∗ is the coupon for which firm H’s nonstrategic thresholdcase I. By definition (12), CH∗∗ is firm
is equal to firm L’s nonstrategic threshold. As we will see in Appendix B, CH

H’s optimal coupon in the simultaneous bankruptcy case (cf. Figure 1). The following
proposition derives the Nash equilibria of the game. For the proof, refer to Appendix B.
Proposition 2 Suppose that the initial state variable x is higher than xN
L . The payoﬀ∗ , CN ) >dominant equilibrium is given as follows. Firm L chooses coupon CLN . If FH (x, CHL∗∗ ) holds, firm H chooses coupon C ∗ . If F (x, C ∗ , C N ) = F N (x, C ∗∗ ) holds,
FHN (x, CH
H
H
H
L
H
H
∗ or C ∗∗ . If F (x, C ∗ , C N ) < F N (x, C ∗∗ ) holds, firm
firm H chooses either coupon CH
H
H
H
L
H
H
∗∗ . If any other Nash equilibrium exists, its strategy profile (C , C )H chooses coupon CH
H
L

satisfies either
CL ∈ (CLN , C¯L ), CH = K2 (CL )12orCL ≥ C¯L , CH ≥ C¯H .

In the payoﬀ dominant equilibrium, firm L chooses the nonstrategic coupon CLN andgains the nonstrategic firm value FLN (x). Intuitively, firm L, which bankrupts earlier thanfirm H bankrupts, does not suﬀer from negative bankruptcy spillovers. Then, firm L has
no incentive to change the nonstrategic coupon because it cannot increase the firm value
beyond the nonstrategic value. Shareholders of firm L declare default at the nonstrategic
threshold xN
L , where debt holders receive all the liquidation value.
On the other hand, firm H’s optimal coupon depends on the parameter values. First,∗ , C N ) > F N (x, C ∗∗ ). In this case, firm H chooses thewe explain the case of FH (x, CH
L
HH∗ (< K (C N )), which leads to sequential bankruptcy.5 We call this outcome
low coupon CH
1L

the sequential bankruptcy equilibrium. In other words, firm H chooses a modest level ofleverage to prepare for operations in the worse cash flow scenario after firm L’s bankruptcy.
After firm L’s bankruptcy, firm H continues to operate until the state variable X(t) falls∗R∗
to the revised threshold xR
H (CH ). At xH (CH ), shareholders of firm H declare default, anddebt holders receive all the liquidation value. In this equilibrium, the expected firm value∗ , C N ) at time 0. This sequential bankruptcy equilibrium corresponds
is equal to FH (x, CHL

to the straightforward response to negative bankruptcy spillovers, as explained in the
beginning of Section 3.2.
∗ , C N ) < F N (x, C ∗∗ ). In this case, firm H
Next, we turn to the case of FH (x, CH
H
HL∗∗ (= K (C N )), which leads to simultaneous bankruptcy. We
chooses the high coupon CH
2
L∗∗
Ncall this outcome the simultaneous bankruptcy equilibrium. Note that xN
H (CH ) = xL

holds. In other words, firm H increase debt to equalize its nonstrategic bankruptcy
threshold with that of firm L. This strategic increase of debt is explained as follows.
∗∗ ], firm H’s bankruptcy threshold remains at xN , which is indeFor CH ∈ [K1 (CLN ), CHL

pendent of CH , as is the bankruptcy cost. Then, firm H maximizes the tax benefits of∗∗ (cf. F (x, C, C N ) increasing
debt by increasing its coupon to the maximum level CHH
L∗∗ ] in Figure 1). If firm H increase its coupon C∗∗
in CH ∈ [K1 (CLN ), CH
H beyond CH , thebankruptcy threshold changes to xN
H (CH ), which decreases the firm value via the increased∗∗ in Figure 1). Thus, firm H
bankruptcy cost (cf. FH (x, C, CLN ) decreasing in CH ≥ CH∗∗ to bankrupt simultaneously with firm L at xN . In
intentionally chooses the coupon CHL5∗
We can easily show that CH
is strictly lower than K1 (CLN ) when the sequential bankruptcy equilibrium ischosen. For details, see Appendix B.

13∗∗ , C N ) = F N (x, C ∗∗ ) at
this equilibrium, the expected firm value is equal to FH (x, CHL
HHtime 0. This simultaneous bankruptcy equilibrium corresponds to the counterintuitive
response to negative bankruptcy spillovers, as explained in the beginning of Section 3.2.
Indeed, firm H increases debt beyond the nonstrategic level rather than decreasing debt
in the presence of negative bankruptcy spillovers.
In another equilibrium, if it exists, firms H and L issue debt with coupon CH =∗∗ ) and C (> C N ), respectively, or issue debt to bankrupt immediately. In
K2 (CL )(> CHLL

any case, they bankrupt simultaneously. In other words, both firms issue more debt andsimultaneously bankrupt earlier than in the payoﬀ dominant equilibrium. This equilibrium
can be regarded as a worse equilibrium than the payoﬀ dominant simultaneous bankruptcy
equilibrium.

3.2.3

Empirical implications

In this subsection, we explain the empirical implications from Proposition 2. It determines
firm H’s capital structure choice of whether firm H’s cash flows after firm L’s bankruptcy
are suﬃciently high compared to the tax benefits of increased debt. Notably, when the
cash flows are insuﬃcient, firm H aims to cause simultaneous bankruptcy. More negative
bankruptcy spillovers (i.e., a lower δi and higher ϵi ) and more tax benefits (i.e., a higher
τ ) increase firm H’s incentive to choose the simultaneous bankruptcy equilibrium. To
our knowledge, no paper shows that firms’ strategic capital structure choices lead tobankruptcy cascades. Our results provide a novel mechanism of bankruptcy contagion—afirm has an ex-ante herding incentive to increase debt and to go bankrupt simultaneouslywith another firm on which it financially and/or operationally depends.
Obviously, the simultaneous bankruptcy equilibrium is likely to arise for aﬃliates inthe same corporate family. Emery and Cantor (2005) show the empirical evidence ofbankruptcy cascades over aﬃliates belonging to the same corporate family or conglomerate. Similarly, we predict that the simultaneous bankruptcy equilibrium applies for firms
belonging to the same supply chain network (cf. Hertzel, Li, Oﬃcer, and Rodgers (2008)
and Kolay, Lemmon, and Tashjian (2016)). For instance, in the Japanese automotive
industry, a large automaker such as Toyota has developed a strong supply chain networkwith specific auto parts suppliers called “keiretsu” companies. Keiretsu companies depend
greatly on the large automaker in terms of sales and profits, and hence, these companies
are unlikely to generate suﬃcient earnings if the large automaker goes into financial distress. Then, we predict that keiretsu companies can adjust capital structure to align their

14

bankruptcy timing with that of their leader.
Trade-credit linkages between firms in the same supply chain network are akin tointerbank loan linkages between banks. Indeed, a bank’s bankruptcy can lead to negative
spillovers to other banks through interbank loan linkages (cf. Allen and Gale (2000)
and Eisenberg and Noe (2001)).6 Schepens (2016) shows the empirical evidence of bankcapital structure driven by the tax benefits of debt, although the tax benefits are often
overlooked in the banking literature. In recent years, Hugonnier and Morellec (2017) andSundaresan and Wang (2015) have examined the bank capital structure by extending the
setup of Leland (1994). The main diﬀerence from the nonfinancial corporate model of thispaper is to include deposits in addition to equity and debt, but we have similar results
even in the banking models where the tax benefits of debt are present. Thus, we argue
that banks can adjust capital structure ex ante in order to go bankrupt together, whichcauses systemic banking crises. Acharya and Yorulmazer (2007) also show the possibility
that banks take the same risk ex ante to fail together. In their model, the driver of theherding behavior is a too-many-to-fail problem—the regulator finds it ex post optimal
to bail out failed banks when the number of failed banks is too large. Banks try to failsimultaneously in order to be bailed out by the regulator. We complement their arguments
by showing that only in the presence of negative bankruptcy spillovers (e.g., interbank
loan linkages) do banks have incentives to increase debt and to go bankrupt together.
Many papers, including Das, Duﬃe, Kapadia, and Saita (2007), Benzoni, CollinDufresne, Goldstein, and Helwege (2015), and Azizpour, Giesecke, and Schwenkler (2018),
show the empirical evidence that default contagion is not limited to firms with counterparty risk. In some cases, the mechanism of the simultaneous bankruptcy equilibrium maybe the story behind these findings, although the empirical relevance has yet to be studied.
In addition, the mechanism can potentially explain the peer eﬀects in corporate financial
policies. For instance, Leary and Roberts (2014) show that firms’ financial policies (e.g.,
leverage, equity and debt issuance, etc.) are aﬀected by their peer firms’ financial policies.
The learning or competition eﬀects often explain the herding behavior, our results add analternative explanation—a firm that anticipates negative externalities from bankruptcy
of its peer firm in the same industry (cf. Lang and Stulz (1992)) can adjust its capital
structure in response to its peer firm to align the bankruptcy timing.
Lastly, empirical evidence shows that there is substantial inter- and intra-variation inleverage (e.g., MacKay and Phillips (2005)). Our model can also explain a wide variation
6

As explained in Section 2.2, we have the same result even in a setup with temporary spillover eﬀects.

15

in leverage through the diﬀerence between the sequential and simultaneous bankruptcy∗ , C N ) = F N (x, C ∗∗ ) in Proposition 2).
equilibria (cf. the multiple equilibria for FH (x, CHL
HH

Miao (2005) and Morellec and Zhdanov (2008) explain the dispersion in leverage throughthe competitive intra-industry eﬀects. They show that in oligopoly markets, a firm candecrease leverage to survive longer or to win competition. The sequential bankruptcy
equilibrium, where firm H decreases leverage in preparation of operations in the future
worse cash flow scenario, is similar to the results of Miao (2005) and Morellec and Zhdanov(2008). However, the simultaneous bankruptcy equilibrium is opposite from the previous
results.7 In fact, unlike in the competitive intra-industry model, firm H increases leverage
to go bankrupt simultaneously with firm L. Thus, this paper complements the previous
literature by showing the opposite capital structure stemming from the contagion eﬀect.

3.3

Numerical analysis

3.3.1

The baseline results

We numerically examine the payoﬀ dominant equilibrium in Proposition 2 in full details.
We omit depicting the other type of equilibria (i.e., worse versions of the simultaneousbankruptcy equilibrium) because their properties are similar to those of the payoﬀ dominant simultaneous bankruptcy equilibrium. The baseline parameter values are set in Table
1, where the values of r, µ, σ, and τ are standard in dynamic corporate finance literatureand reflect a typical S&P firm (e.g., Morellec (2001) and Arnold (2014)). The liquidation value function is set by Ui (x) = 0.5(ai x/(r − µ) − bi /r). The bankruptcy spillover
eﬀects are represented by δi = 0.94(< 1) (i = H, L). We set the baseline values of ai , bi ,∗ , C N ) andand δi (i = H, L) to make firm H’s values in the two equilibria (i.e., FH (x, CHL∗∗ )) almost equal because we want to examine how the two equilibria switch due
FHN (x, CH

to the parameter values.
Figure 1 shows the firm value functions FH (x, C, CLN ), FHN (x, C), and FLN (x, C) withrespect to coupon C. In the figure, we see that FiN (x, C) (i = H, L) are unimodal and
N = 0.958 and C N = 0.748,
satisfy assumption (6). Their maximums are attained at CH
LNand the bankruptcy thresholds are xN
H = 0.679 and xL = 0.874, which satisfy assumption(7). This implies that with no spillover eﬀect, firm H issues more debt but defaults laterbecause of its higher profitability than firm L. As shown in Proposition 2, in the payoﬀ
7

Static models, such as Brander and Lewis (1986) and Fulghieri and Nagarajan (1996), argue that higher

leverage can lead to competitive advantages in oligopoly markets, although they do not examine the contagion
eﬀects.

16

Table 1: Baseline parameter values.
r

µ

σ

τ

aH /bHaL /bL

δi

ϵi

x

1/0.4

1/1

0.94

1

2

0.06 0.01 0.2 0.15

xH

Table 2: Baseline results.
CH (x) LVH (x) CSH (x) EH (x) DH (x)FH (x)SW (x)

0.874

1.348

0.589

0.017912.09617.30629.40254.734xL

CL (x)LVL (x)CSL (x)EL (x)DL (x)FL (x)

0.874

0.748

0.426

0.023412.096

8.967

21.063dominant equilibrium, firm L’s financing and bankruptcy policies as well as the firm value
are unchanged with bankruptcy spillovers.
On the other hand, firm H’s value FH (x, C, CLN ) is diﬀerent from the nonstrategic∗∗ =firm value FHN (x, C). See the top panel of Figure 1. We have K1 (CLN ) = 1.243, CH

K2 (CLN ) = 1.349, and K3 (CLN ) = 1.46. For C < 1.243, We have the sequential bankruptcy
case I in Proposition 1 for C < 1.243, the simultaneous bankruptcy case for C ∈ [1.243, 1.46],
N∗ = arg max
and the sequential bankruptcy case II for C > 1.46. We have CHC∈[0,1.243] FH (x, C, CL ) =N = 0.958) as the optimal coupon in the sequential bankruptcy case. As we ex0.931(< CHplained after Proposition 2, for CH ∈ [1.243, 1.349], firm H’s bankruptcy threshold staysN
at xN
L = 0.874, and hence, FH (x, C, CL ) monotonically increases in this region because

the tax benefits of debt increase. For CH ≥ 1.349, FH (x, C, CLN ) agrees with FHN (x, C)∗∗ = 1.349 as the optimal coupon in theand monotonically decreases. Then, we have CH

simultaneous bankruptcy case.
∗∗ ) =The simultaneous bankruptcy equilibrium arises because the firm value FHN (x, CH∗∗ , C N ) = 29.402 is higher than F (x, C ∗ , C N ) = 29.394. In equilibrium, firm
FH (x, CH
H
L
H
L∗∗ = 1.349(> C N = 0.958) and goes bankrupt
H issues debt with the high coupon CHH

at xN
L = 0.874 simultaneously with firm L. Table 2 summarizes the baseline values inequilibrium. If firm H did not take into account strategic considerations, firm H would
N = 0.958 and operate in the worse cash flow
issue debt with the nonstrategic coupon CH
N
scenario after firm L’s bankruptcy until the bankruptcy threshold xR
H (CH ) = 0.722 isreached. However, in equilibrium, firm H, that anticipates the negative spillover eﬀects,
strategically increases debt to bankrupt simultaneously with firm L. In other words, firm
H’s strategic increase of debt causes bankruptcy cascades in this numerical example.

17

3.3.2The impacts of bankruptcy spillovers

Now, we explore the comparative statics with respect to the key parameters. We focus
on firm H’s results in the payoﬀ dominant equilibrium because firm L’s results are thesame as those of the nonstrategic case. Figure 2 shows firm H’s bankruptcy threshold
xH , coupon CH , leverage LVH (= DH (x)/FH (x)), credit spread CSH (= CH /DH (x) − r),equity value EH (x), debt value DH (x), firm value FH (x), and social welfare SW (x) withvarying levels of the spillover eﬀect parameter δH . Recall that a lower δH means morenegative bankruptcy spillover eﬀects. The other parameter values are set in Table 1. The
social welfare SW (x) is defined by the sum of the two firm values and government value
through the corporate tax revenue.8
In Figure 2, the values other than FH (x) jump at δH = 0.9417. This is because firm H∗∗ (i.e., the simultaneous bankruptcy equilibrium) for δ ≤ 0.9417 and C ∗ (i.e.,chooses CHHH

the sequential bankruptcy equilibrium) for δH > 0.9417. A lower δH decreases cash flows∗∗ and toafter firm L’s bankruptcy, and hence, it increases firm H’s incentive to choose CH

bankrupt simultaneously with firm L. It is straightforward that xH , CH , LVH , and CSH
are higher in the simultaneous bankruptcy case than in the sequential bankruptcy case.
In the panels of CH and LVH , a higher δH , which means weaker externalities, can decrease
CH and LVH by shifting from the simultaneous bankruptcy equilibrium to the sequential
bankruptcy equilibrium, although within the sequential bankruptcy region, a higher δHstraightforwardly increases CH and LVH . In other words, firm H’s capital structure is
nonmonotonic with respect to the bankruptcy spillover eﬀects.
In the panels of EH (x) and DH (x) in Figure 2, we see that EH (x) (DH (x)) in the simultaneous bankruptcy equilibrium is lower (higher) than those in the sequential bankruptcy
equilibrium. This is explained by the higher coupon in the simultaneous bankruptcy
equilibrium. In fact, the higher coupon decreases residual payoﬀs to shareholders and accelerates bankruptcy, and hence, it decreases EH (x). The higher coupon increases coupon
payments although it speeds up bankruptcy. The former eﬀect dominates the latter eﬀect,
and hence, the higher coupon increases DH (x). Then, a higher δH can increase EH (x)and decrease DH (x) by shifting from the simultaneous bankruptcy equilibrium to thesequential bankruptcy equilibrium, although within the sequential bankruptcy region, ahigher δH decreases EH (x) and increases DH (x). As in the capital structure, the equity
and debt values are nonmonotonic with respect to the bankruptcy spillover eﬀects.
8

The main results are unchanged even if we include the consumer surplus which decreases by a firm’s

bankruptcy.

18

Lastly, the panel of SW (x) shows that SW (x) jumps upward at δH = 0.9417. In
the simultaneous bankruptcy equilibrium, SW (x) is lower than that of the sequential∗∗ and bankruptcy threshold xN
bankruptcy equilibrium because firm H’s high coupon CHL

reduce the tax revenues of the government. This suggests that firm H’s strategic choiceof simultaneous bankruptcy reduces the social welfare.

3.3.3

The impacts of profitability

Figure 3 shows the equilibrium results with varying levels of firm H’s operating cost bH .
The other parameter values are set in Table 1, where firm L’s operating cost bL is set at∗
1. The values other than FH (x) jump at bH = 0.391. This is because firm H chooses CH∗∗ (i.e., the simultaneous(i.e., the sequential bankruptcy equilibrium) for bH < 0.391 and CH

bankruptcy equilibrium) for bH ≥ 0.391. A higher bH decreases firm H’s advantage overfirm L in terms of profitability, and hence, it decreases cash flows by operations after firm∗∗ to bankrupt
L’s bankruptcy. Then, for a higher bH , firm H is more likely to choose CH

simultaneously with firm L.
Notably, Figure 3 shows that CH , LVH , and DH (x) are nonmonotonic with respect
to bH . This is explained by the shift from the simultaneous bankruptcy equilibrium tothe sequential bankruptcy equilibrium. In fact, CH , LVH , and DH (x) jump upward at
bH = 0.391 by shifting from the simultaneous bankruptcy equilibrium to the sequential
bankruptcy equilibrium, although they monotonically decrease in each equilibrium region.
Most of the structural models (e.g., Leland (1994)) based on the tradeoﬀ theory predict a
positive relation between leverage and profitability (i.e., CH , LVH , and DH (x) decreasing
in bH ), although empirical studies (e.g., Titman and Wessels (1988) and Frank and Goyal(2015)) show a negative relation. Our results for CH , LVH , and DH (x) can potentiallyexplain the negative relation between leverage and profitability by shifting from the simultaneous bankruptcy equilibrium to the sequential bankruptcy equilibrium. Indeed,
our model suggests that higher profitability leads a firm to take lower leverage to prepare
for operating in a worse cash flow scenario in the future.
In the panel of EH (x), EH (x) monotonically decreases in bH , and EH (x) jumps downward at bH = 0.391 by shifting from the the simultaneous bankruptcy equilibrium to thesequential bankruptcy equilibrium. In the panel of SW (x), we can also see a downward∗∗ and bankruptcy
jump of SW (x) at bH = 0.391. This is because firm H’s high coupon CH

threshold xN
L reduce the tax revenues of the government.

19

3.3.4The impacts of cash flow volatility

Figure 4 shows the equilibrium results with varying levels with varying levels of the market
volatility σ. The other parameter values are set in Table 1. The values other than FH (x)∗∗ (i.e., the simultaneous bankruptcy
jump at σ = 0.213. Indeed, firm H chooses CH∗ (i.e., the sequential bankruptcy equilibrium) for σ >equilibrium) for σ ≤ 0.213 and CH

0.213. A lower σ increases firm L’s default distance, and hence, it increases the tax benefit
eﬀects until firm L’s bankruptcy and decreases the future cash flow eﬀects after firm L’s∗∗ and to enjoy
bankruptcy. Then, a lower σ increases firm H’s incentive to choose CH∗ to prepare operations
the tax benefits until firm L’s bankruptcy rather than choosing CH

after firm L’s bankruptcy. Although we omit depicting a figure, a higher market growthrate µ, that increases firm L’s default distance, also increases firm H’s incentive to choose
the simultaneous bankruptcy equilibrium. These results are counterintuitive and lead tonew empirical predictions that all else being equal, bankruptcy cascades are more likely
to hit a firm with a lower volatility and higher growth rate of cash flows.
The panel of LVH shows leverage monotonically decreasing in σ, which is consistent
with the standard result (e.g., Titman and Wessels (1988) and Leland (1994)). Leverage
jumps downward at σ = 0.213 by shifting from the simultaneous bankruptcy equilibrium to the sequential bankruptcy equilibrium. More notably, the panel of CSH presents
the nonmonotonic shape because of the capital structure change at σ = 0.213. In fact,
CSH jumps downward at σ = 0.213 by shifting from the simultaneous bankruptcy equilibrium to the sequential bankruptcy equilibrium, although it monotonically increase in eachequilibrium region. Most of the structural models (e.g., Leland (1994)) shows a positive
relation between credit spread and volatility because the leverage decreasing with higher
volatility does not fully oﬀset the bankruptcy risk increasing with higher volatility. The
same logic accounts for CSH increasing with higher σ for each equilibrium region. However, in our model, at σ = 0.213, the firm changes the capital structure policies from thehigh-risk policy that leads to simultaneous bankruptcy to the low-risk policy that leads
to sequential bankruptcy. Then, unlike in the previous results, CSH can decrease in σ.
The panels of EH (x) and DH (x) show that a higher σ leads to a wealth transfer fromdebt holders to shareholders. This result is consistent with the standard result (e.g.,
Leland (1994)). Note that the shift from the simultaneous bankruptcy equilibrium to thesequential bankruptcy equilibrium at σ = 0.213 amplifies the wealth transfer. In the panelof SW (x), we can also find an upward jump of SW (x) at σ = 0.213. This is because∗∗ and bankruptcy threshold xN reduce the tax revenues of thefirm H’s high coupon CHL20

government.
To summarize, the numerical examples show that a small diﬀerence in parametervalues can greatly change firm H’s capital structure by switching between the simultaneousbankruptcy equilibrium and the sequential bankruptcy equilibrium. A switch between thetwo equilibria also causes several comparative static results diﬀerent from the previous
results of the standard structural model, leading to new empirical predictions. The novel
result of the simultaneous bankruptcy equilibrium is more likely to occur for more negative
bankruptcy spillovers, a smaller profitability diﬀerence, lower market volatility, and higher
market growth rate.

4

Extensions

4.1

More than two firms

We can easily extend the baseline model consisting of two firms to the following setup withn firms. Firm i ∈ {1, 2, , . . . , n} receives EBIT ai X(t) − bi , where a1 /b1 < a2 /b2 < · · · <an /bn , while no firm is bankrupt, whereas firm i’s EBIT decreases to δ(k)ai X(t) − ϵ(k)bi ,where the EBIT contraction parameter δ(j) ∈ (0, 1) (ϵ(j) ≥ 1) decreases (increases) with
the number of bankrupt firms, k. For simplicity, we assume that the negative bankruptcy
spillover eﬀects are symmetric to all firms.
As in Section 3.2, we can derive the Nash equilibria for the game of n firms. Most notably, the simultaneous bankruptcy equilibrium can arise as follows. The least-profitability
firm 1 chooses the nonstrategic coupon and bankruptcy threshold. Firm 2 increases debt
beyond the nonstrategic level to bankrupt simultaneously with firm 1 because it anticipates the bankruptcy spillover eﬀects δ(1) and ϵ(1). Firm 3 increases debt beyond thenonstrategic level to bankrupt simultaneously with firms 1 and 2 because it anticipates
the eﬀects δ(2) and ϵ(2) amplified by firm 2’s bankruptcy. This procedure repeats until
firm n chooses the high coupon to bankrupt simultaneously with firms 1, 2, . . . , n − 1, due
to the amplified eﬀects δ(n − 1) and ϵ(n − 1). A firm’s strategy to increase debt andto go bankrupt simultaneously with lower-profitability firms amplifies negative externalities, leading to another higher-profitability firm’s strategic debt increase and simultaneousbankruptcy. Through this negative feedback loop channel, bankruptcy cascades spreadover multiple firms.

21

4.2

Bailout

In the presence of negative externalities, firm H can potentially bail out firm L. For
instance, Mistrulli (2011) observes that a bank bails out another bank aﬃliated with thesame conglomerate. Yang, Birge, and Parker (2015) document bailout examples throughsupply chain networks. Now, we examine a model in which firm H can merge firm L when9firm L goes bankrupt at xN
L . Suppose that the merger requires the sunk cost M C(> 0).

After the merger, firm H receives EBIT (aH +aL )X(t)−bH −bL and pays coupon CH +CLN
until bankruptcy. As in (4), we can derive the merged firm H’s bankruptcy threshold asN
xM
H (CH , CL ) =γ(r − µ)(bH + bL + CH + CLN ),(γ − 1)r(aH + aL )(13)

where the superscript M stands for the merged firm.
At time 0 prior to merger, firm H’s equity and debt values can be derived as)(
x γ N N M
M
N
N
N
EH (x, CH , CL ) = EH (x, xL , CH ) +(EH (xL , xH (CH , CLN ), CH )
xN
L
M
N
N
+ ELN (xN
L , xH (CH , CL ), CL ) − M C),
M
N
NDH(x, CH , CLN ) = DH(x, xM
H (CH , CL ), CH ).

(14)
(15)

In (14), the first term represents the value before merger and the second term represents
the value after merger. Debt value (15) implies that original debt holders of firm H
continues to receive coupon CH after merger until bankruptcy. Firm H chooses CH
to maximize the firm value (i.e., the sum of (14) and (15)) in the bailout case. Firm
H decides whether to bail out firm L by comparing the firm values with and without
bailout. Although it is costly for firm H to bail out firm L, firm H can bail out to avoid
the negative bankruptcy spillover eﬀects.
Clearly, more negative spillover eﬀects and a lower merger cost increases firm H’s
incentive to bail out firm L. When firm H’s EBIT scale is larger than that of firm L(i.e., aH /aL >> 1 and bH /bL >> 1), firm H is more likely to bail out firm L. This isbecause firm H can avoid relatively large spillover eﬀects with relatively small distortion
NN(i.e., xM
H (CH , CL ) is close to xH (CH )). For instance, our model predicts that a large

automaker, such as Toyota, is likely to bail out a smaller keiretsu supplier if its bankruptcy
greatly damages the automaker’s production line via the supply chain channel.
9

For robustness check, we also examined an alternative model in which firm H can provide subsidies for firm

L to delay firm L’s bankruptcy. In fact, some firms provide more favorable pricing for financially distressedcustomers (e.g., see Yang, Birge, and Parker (2015)). The results are the same as those of the merger model
presented in this subsection.

22

We now explore the impacts of cash flow volatility σ.

Figure 5 shows firm H’s

MMM
bankruptcy threshold xM
H , coupon CH , leverage LVH , credit spread CSH , equity value
M (x), debt value D M (x), firm value F M (x), and social welfare SW M (x) with varying
EH
H
H

levels of σ. For comparison, we also depict the results in the baseline model with nobailout by dashed lines. For the baseline parameter values in Table 1, no bailout occurs
even if we set M C = 0. To see the possibility of bailout changing with σ, we quadruple
the firm H’s EBIT scale (i.e., aH = 4 and bH = 1.6) in Table 1.
In Figure 5, firm H bails out firm L for σ ≤ 0.238, whereas firm H does not bailout firm L (i.e., the solid and dashed lines align) and sequential bankruptcy occurs for
σ > 0.238. For a lower σ, firm H is more likely to bail out firm L partially because bailout
is a more eﬀective measure to resolve the simultaneous bankruptcy equilibrium rather than
the sequential bankruptcy equilibrium. In fact, the simultaneous bankruptcy equilibrium,
which occurs in the case with no bailout for σ ≤ 0.213, is fully replaced by bailout. We canMMM
see from Figure 5 that for σ ≤ 0.213, bailout leads to lower xM
H , CH , LVH , and CSH thanthose for the case with no bailout. In other words, firm H reduces leverage to anticipate
its operations after bailout of firm L. The panel of SW M (x) shows that for σ ≤ 0.213,
bailout, which delays bankruptcy, increases social welfare.
On the other hand, for 0.213 < σ ≤ 0.238, the sequential bankruptcy equilibriumin the baseline case is also replaced by bailout. In this region, bailout leads to higherM
MM
xM
H , CH , LVH , and CSH than those for the case with no bailout. This is because firm H

does not prepare for operations in the worse cash flow scenario after firm L’s bankruptcy
but rather anticipates bailout of firm L. The panel of SW M (x) shows that, for 0.213 < σ ≤0.238, bailout, which accelerates bankruptcy, decreases social welfare through the channelof tax revenues of the government. These results are counterintuitive and contrary to theresults for σ ≤ 0.213.

4.3

Debt renegotiation

Although the baseline model focuses on non-renegotiable debt, firm L’s bankruptcy may
give a debt restructuring opportunity to firm H. In reality, debt holders sometimes accept
debt renegotiation apart from the absolute priority rule to avoid bankruptcy costs when
the business environment deteriorates through an inevitable incident.10 In this subsection,
we consider a model in which firm H can renegotiate debt at firm L’s bankruptcy time.
10

This type of debt renegotiation is more likely to occur in a case with temporary losses rather than permanent

losses from another firm’s bankruptcy.

23

Assume that debt renegotiation requires sunk cost DC(> 0). At threshold xN
L , the firm
can renew the coupon to maximize the firm value. In debt renegotiation, shareholders
gain the portion η of the surplus, while debt holders gain the portion 1 − η of the surplus,
where η ∈ [0, 1] stands for the bargaining power of shareholders.
R
D = arg max
R N
The renewed coupon is CH
C≥0 FH (xL , xH (C), C), and the renewed bankruptcy
R
R
D
threshold is xR
H (CH ). Recall that FH (·, ·, ·) and xH (·) denote the revised firm value andbankruptcy threshold, i.e., (3) and (4) with δi ai and ϵi bi rather than ai and bi . The
superscript D stands for the debt renegotiation case. If the debt renegotiation surplus is
R
D
D
N
R
positive, i.e., FHR (xN
L , xH (CH ), CH ) − FH (xL , xH (CH ), CH ) − DC > 0, firm H prefers to

restructure debt. In this case, firm H’s equity and debt values at time 0 can be derived
as

(
D
N
EH
(x, CH , CLN ) = EH(x, xRH (CH ), CH ) +

x
xN
L

)γ
R
D
D
η(FHR (xN
L , xH (CH ), CH )R
− FHN (xN
(16)
L , xH (CH ), CH ) − DC),
(
)γ
x
D
N
R
D
D
DH
(x, CH , CLN ) = DH(x, xR(1 − η)(FHR (xN
H (CH ), CH ) +L , xH (CH ), CH )xN
L
R
− FHN (xN
L , xH (CH ), CH ) − DC).

(17)

In (16), the first and second terms represent the equity value with no debt renegotiationand the shareholders’ gain from debt renegotiation, respectively. Similarly, in (17), thefirst and second terms represent the debt value with no debt renegotiation and the debt
holders’ gain from debt renegotiation, respectively. Firm H chooses CH to maximize thefirm value (i.e., the sum of (16) and (17)) in the debt renegotiation case.
Clearly, more negative spillover eﬀects and a lower debt renegotiation cost, which
increase the debt renegotiation surplus, increases firm H’s incentive to choose debt renegotiation. We now explore the impacts of cash flow volatility σ. Figure 6 shows firm H’s
DDDbankruptcy threshold xD
H , coupon CH , leverage LVH , credit spread CSH , equity value
D (x), debt value D D (x), firm value F D (x), and social welfare SW D (x) in the case allowEH
HH

ing debt renegotiation with varying levels of σ. For comparison, we also depict the resultsin the baseline model with no debt renegotiation by dashed lines. To see the possibilityof debt renegotiation changing with σ, we set DC = 3, and the other parameter values
are set in Table 1.
In Figure 6, firm H renegotiate debt for σ ≤ 0.298, whereas firm H does not renegotiate
debt (i.e., the solid and dashed lines align) for σ > 0.298. In both cases, firm H operates
after firm L’s bankruptcy. For a lower σ, firm H is more likely to renegotiate debt
partially because debt renegotiation is a more eﬀective measure to resolve the simultaneous24

bankruptcy equilibrium rather than the sequential bankruptcy equilibrium. In fact, thesimultaneous bankruptcy equilibrium, which occurs in the case with no debt renegotiationfor σ ≤ 0.213, is fully replaced by debt renegotiation. The panel of xD
H shows thatD
debt renegotiation greatly decreases xD
H below xH . In the panel of CH , we find that for
D is equal to C ∗∗ . This is because firm H, that anticipates future debt
σ ≤ 0.298, CHH∗∗renegotiation at firm L’s bankruptcy threshold xN
L , chooses the initial coupon CH to

maximize the tax benefits until debt renegotiation.
D show that LV D and CS D are lower than LV and CS
The panels of LVHD and CSH
HH
H
H
D are higher than LV and CS for 0.213 < σ ≤ 0.298.
for σ ≤ 0.213, while LVHD and CSH
H
H

Contrary to the region σ ≤ 0.213, in the region 0.213 < σ ≤ 0.298, firm H, whichanticipates debt renegotiation in the future, issues more debt at time 0 than it doesin the sequential bankruptcy equilibrium in the baseline model. The panel of SW D (x)shows that debt renegotiation increases social welfare for σ ≤ 0.213. In this region, theshift from the simultaneous bankruptcy equilibrium to the debt renegotiation equilibrium
delays firm H’s bankruptcy and increases both values of firm H and the government.
On the other hand, the panel of SW D (x) shows that debt renegotiation decreases social
welfare for 0.213 < σ ≤ 0.298, yet it delays firm H’s bankruptcy and increases firm H’svalue. This result is counterintuitive but is explained as follows. In this region, with debt
D = C ∗∗
renegotiation, firm H enjoy the tax benefits of debt by setting the high coupon CHH

at time 0. However the high coupon decreases tax revenues of the government until firm
L’s bankruptcy. In other words, debt renegotiation, which increases firm H’s tax benefitsof debt, causes a wealth transfer from the government to firm H, decreasing total social
welfare via the decrease in the government value.
The results in the debt renegotiation case also yield implications about capital structure adjustment and debt maturity. For simplicity, this paper assumes that firms issue
debt with infinite maturity at time 0. In the real world, firms can use short-term debt
rather than long-term debt and adjust capital structure. The leverage adjustment withshort-term debt plays the same role as debt renegotiation. As in the debt renegotiationcase, firm H is more likely to issue short-term debt and adjust leverage for more negative spillover eﬀects, lower debt issuance and adjustment costs, and a lower cash flow
volatility. The dynamic leverage adjustment can prevent firm H from choosing the simultaneous bankruptcy equilibrium with high leverage and increase social welfare, but it can
prevent firm H from choosing the sequential bankruptcy equilibrium with low leverageand decrease social welfare.

25

4.4

Incomplete information

So far, we have assumed that firms H and L have complete information about all parameter values. In reality, it may be diﬃcult to know the bankruptcy spillover eﬀects
prior to bankruptcy. Then, we examined a model in which the firms know the spillover
parameters δi ∈ (0, 1) and ϵi ≥ 1 as some distributions. Unlike in the baseline model, firm
H chooses C to maximize the expectation of FH (x, C, CLN ) with respect to δH and ϵH in∗∗ in the simultaneous bankruptcy case remains unchanged. We computed the(11), yet CH

results for uniformly distributed δH and ϵH . We verified that the eﬀects of incomplete
information are very small and that the main results in the baseline models robustly holdtrue. Hence, we omitted depicting a figure.

4.5

Bidimensional state process

Although the baseline model assumes that firms H and L face the common shock X(t), it
is more realistic to assume the bidimensional geometric Brownian motion with a positive
correlation for two firms’ cash flows. For the bidimensional state process, we cannot
obtain analytical forms for the equity, debt, and firm values. Hence, it is harder tocompute the equilibrium results. In particular, unlike in the baseline model, the order ofbankruptcy for a given strategy profile (CH , CL ) is not known at time 0 but depends on
the sample path of the state process X(t). Then, not only firm H but also firm L can
anticipate the negative bankruptcy spillover eﬀects. This leads to the possibility that,through strategic considerations, both firms can either decrease debt for preparation of
the worse cash flow scenario or increase debt to bankrupt simultaneously. We believe
that, with a suﬃciently high correlation, the results remain quite similar to the baseline
results; two types of equilibria (i.e., high- and low-leverage equilibria) arise depending on
the parameter values.
In the bidimensional setup, one may consider a model in which a firm can optimize thecorrelation between the two firms’ EBIT in addition to capital structure. In fact, a firm
could change the correlation by investing in diﬀerent projects. The choice of correlation is
examined in the banking model in Acharya and Yorulmazer (2007). They show that banks
can increase the interbank correlation in cash flows by choosing similar sets of loans. In
their model, banks herd in order to fail and to be bailed out together in the presence of a
too-many-to-fail problem. The same herding behavior can arise in our setup, even though
we do not consider a bailout opportunity. Indeed, a firm has an incentive to increase thecorrelation coeﬃcient to one in order to go bankrupt at exactly the same time with the26other firm and to avoid the negative bankruptcy spillover eﬀects.

5Conclusion

This paper develops the optimal capital structure model of two firms with negative externalities of bankruptcy. We examine the firms’ capital structure and bankruptcy timing
choices in equilibrium. The equilibrium results are summarized below.
The low-profitability firm L’s capital structure and bankruptcy timing remain unchanged from those of the nonstrategic case. On the other hand, the high-profitability
firm H takes either of the two contrasting capital structures: lower leverage than the nonstrategic leverage to prepare for operations after firm L’s bankruptcy (i.e., the sequential
bankruptcy equilibrium) or higher leverage than the nonstrategic leverage to bankrupt
simultaneously with firm L (i.e., the simultaneous bankruptcy equilibrium). The capital
structure choice is determined by the tradeoﬀ between the cash flows from operations
after firm L’s bankruptcy and the tax benefits of increased debt. With more negative
bankruptcy spillovers, a smaller profitability diﬀerence, and lower volatility, firm H is
more likely to choose the simultaneous bankruptcy equilibrium. The model can yield
nonstandard comparative static results by switching between the sequential and simultaneous equilibria.
Most notably, the simultaneous bankruptcy equilibrium shows a novel mechanism ofbankruptcy cascades—firms with negative bankruptcy spillovers, such as firms with crossholdings and firms in the same supplier chain, can intentionally increase leverage ex ante
to go bankrupt together. Although the empirical relevance of this mechanism has yet to
be studied, this mechanism can potentially account for the empirical findings of default
clustering, financial contagion, and herding behavior for corporate financial policies.

A

Proof of Proposition 1

Recall that we define xRi (C) by replacing ai and bi with δi ai and ϵi bi , respectively, in (4).
R
Then, xN
i (C) < xi (C) follows from δi < 1 and ϵi ≥ 1.

Sequential bankruptcy case I: CH ∈ [0, K1 (CL )).

27

By (4) and (8), we haveγ(r − µ)(ϵH bH + CH )(γ − 1)rδH aH
γ(r − µ)(ϵH bH + K1 (CL ))<(γ − 1)rδH aH

xR
H (CH ) == xN
L (CL ).

(18)

By (18), we have the bankruptcy thresholds xH (CH , CL ) = xR
H (CH ) and xL (CL , CH ) =xN
L (CL ), as well as the inequality xH (CH , CL ) < xL (CL , CH ). Then, the equity and debt
values of firm L are the same as those in the nonstrategic case. We can calculate theequity and debt values of firm H as follows:
∫ T1

∫ T2

(1 − τ )(aH X(t) − bH − CH )dt +e−rt (1 − τ )(δH aH X(t) − bH − CH )dt]
T1
(
)γ
x
N
R N
= EH
(x, xN
(C
),
C
)
+
EH
(xL (CL ), xR
L
H
L
H (CH ), CH ),
xN
(C
)
L
L
∫ T2
DH (x, CH , CL ) = E[e−rt CH dt + e−rT2 UH (X(T2 ))]EH (x, CH , CL ) = E[

e

−rt

0

0
N
= DH
(x, xRH (CH ), CH ),where we define firm L’s bankruptcy time T1 = inf{t ≥ 0 | X(t) ≤ xN
L (CL )} and firm H’sbankruptcy time T2 = inf{t ≥ 0 | X(t) ≤ xR
H (CH )}.
Simultaneous bankruptcy case: CH ∈ [K1 (CL ), K3 (CL )].
N
In the same manner as derivation of (18), we can show that xR
H (CH ) ≥ xL (CL ). By(4) and (8), we can show thatγ(r − µ)(bH + CH )(γ − 1)raHγ(r − µ)(bH + K3 (CL ))≤(γ − 1)raH

xH (CH ) == xRL (CL ).

(19)

The two inequalities imply that firms H and L have the same bankruptcy thresholdN
NN
xN
H (CH ) ∨ xL (CL ). Then, we have the equity value Ei (x, Ci , Cj ) = Ei (x, xH (CH ) ∨N
NN
xN
L (CL ), Ci ) and debt value Di (x, Ci , Cj ) = Di (x, xH (CH ) ∨ xL (CL ), Ci ) for i = {H, L}.

Sequential bankruptcy case II: CH > K3 (CL ).
R
As in derivation of (19), we can show that xN
H (CH ) > xL (CL ). This leads to xH (CH , CL ) =R
xN
H (CH ) and xL (CL , CH ) = xL (CL ). By exchanging the roles of firms H and L in thesequential bankruptcy case I, we can derive the expressions of the equity and debt values
in this case.

28

B

Proof of Proposition 2First, we prove that the strategy profile∗(CH, CLN )∗∗∗(FH (x, CH, CLN ) > FHN (x, CH
))

(20)

∗
∗∗
, CLN ) or (CH, CLN )(CH∗∗∗
(FH (x, CH, CLN ) = FHN (x, CH
))

(21)

∗∗
(CH
, CLN )∗∗∗(FH (x, CH, CLN ) < FHN (x, CH
))

(22)

is a Nash equilibrium of the game, and we will later show that it is the payoﬀ dominantequilibrium. Consider firm H’s response CH to firm L’s coupon CLN . By Proposition 1,
we have firm H’s value

N
N
R

EH
(x, xNL (CL ), CH ) + DH (x, xH (CH ), CH )


(
)

γ

R (xN (C ), xR (C ), C ),
 + NxEH(CH ∈ [0, K1 (CLN ))),
L
H
H
L
H
x
(C
)
N
L
L
FH (x, CH , CL ) =∗∗ ]),
FHN (x, xN(CH ∈ [K1 (CLN ), CH

L , CH )



 N
∗∗ ),
FH (x, CH )(CH > CH
(23)
Note that this function is continuous with respect to CH . By (3), FHN (x, xN
L , CH ) monoton∗∗ ]. By the unimodality (6), F N (x, C ) monotonically
ically increases in CH ∈ [K1 (CLN ), CH
H
H
N . By (4) and assumption (7), we havedecreases in CH > CH

(γ − 1)raH xNH− bHγ(r − µ)(γ − 1)raH xNL− bH<γ(r − µ)

N
CH
=

∗∗
= K2 (CLN ) = CH.

Hence, we have
∗∗
CH
= arg

max

N)
CH ≥K1 (CLFH (x, CH , CLN )

(24)

∗ for
(cf. Figure 1). By (11) and (24), firm H’s best response to firm L’s coupon CLN is CH∗ , C N ) > F N (x, C ∗∗ ), C ∗ or C ∗∗ for F (x, C ∗ , C N ) = F N (x, C ∗∗ ), and C ∗∗ forFH (x, CH
H
L
H
H
H
H
H
L
H
H
H
∗ , C N ) < F N (x, C ∗∗ ).
FH (x, CHL
H
H∗ or C ∗∗ . We have
Next, we consider firm L’s best response to firm H’s coupon CH
H
N
∗∗
N
∗
N
N
N
xN
L = xH (CH ) > xH (CH ). Hence, firm L gains FL (x, CL ) by choosing coupon CL . By

δL < 1 and ϵL ≥ 1, we have FLN (x, CL ) ≥ FL (x, CL , CH ) for any CL and CH . Then, by
the unimodality (6), we have∗∗∗
FLN (x, CLN ) > FLN (x, CL ) ≥ max{FL (x, CL , CH), FL (x, CL , CH
)}

29

(25)

for any CL ̸= CLN . Inequality (25) implies that firm L’s best response to firm H’s coupon∗ or C ∗∗ is C N . We have completed the proof that the strategy profile (20)–(22) is aCH
HLNash equilibrium of the game.
Now, we examine other equiilibria. Suppose that another Nash equilibrium (CH , CL )exists. If CL = CLN holds, the above argument shows that (CH , CL ) agrees with thestrategy profile (20)–(22). Suppose that CL < CLN . By (4), we immediately haveNxN
L (CL ) < xL .

(26)

By the unimodality (6), we have FLN (x, CLN ) > FLN (x, CL ). By the optimality of CL , wehave FL (x, CL , CH ) ≥ FL (x, CLN , CH ). Then, we have
FLN (x, CLN ) > FLN (x, CL ) ≥ FL (x, CL , CH ) ≥ FL (x, CLN , CH ),which leads toN
xN
H (CH ) > xL .

(27)

Note that if (27) does not hold, firm L deviates from CL to CLN and increases the firm valueNfrom FL (x, CLN , CH ) to FLN (x, CLN ). By (26) and (27), we have xN
H (CH ) > xL (CL ), whichNmeans that firm H’s value becomes FHN (x, CH ). By (6), (7), and (27), we have CH > CH
′ (< C ) satisfying xN (C ′ ) > xN (C )and ∂FHN (x, CH )/∂C < 0. Then, we can take CH
L
H
L
H
H
′ ) > F N (x, C ). Thus, Cand FHN (x, CH
H
H is not firm H’s best response, which leads toHcontradiction.
Next, suppose that CL ∈ (CLN , C¯L ). By assumption (6), we have ∂FLN (x, CL )/∂C < 0.
N
If xN
H (CH ) < xL (CL ) holds, firm L can increase the firm value by decreasing the coupon

below CL , which contradicts with the optimality of CL . Hence, we have xN
H (CH ) ≥N
NxN
L (CL ), which implies that firm H’s value becomes FH (x, CH ). We have CH > CH by
N
CL > CLN , xN
H (CH ) ≥ xL (CL ), and assumption (7). Then, by the unimodality (6), weN
have ∂FHN (x, CH )/∂C < 0. If xN
H (CH ) > xL (CL ) holds, firm H can increase the firm value

by decreasing the coupon below CH , which contradicts with the optimality of CH . Hence,N
we have xN
H (CH ) = xL (CL ), i.e., CH = K2 (CL ). In fact, this strategy profile (CH , CL )becomes a Nash equilibrium, if any firm cannot increase the firm value by decreasing the
coupon and choosing the sequential bankruptcy.
N
Suppose that CL ≥ C¯L , i.e., x ≤ xN
L (CL ). If xH (CH ) < x holds, firm L can increase

the firm value by decreasing the coupon below C¯L , which contradicts with the optimality¯
of CL . Hence, we have xN
H (CH ) ≥ x, i.e., CH ≥ CH . This strategy profile (CH , CL )becomes a Nash equilibrium, if any firm cannot increase the firm value by decreasing the
coupon and choosing sequential bankruptcy.

30

Lastly, we show that the strategy profile (20)–(22) is the payoﬀ dominant equilibrium.
Consider another Nash equilibrium (CH , CL ). As we proved above, CL > CLN and CH >∗∗ > C N hold, and firm i’s value is F N (x, C ) for i ∈ {H, L}. By the unimodality (6),CHiiH∗∗ ) > F N (x, C ). The proof is complete.
we have FLN (x, CLN ) > FLN (x, CL ) and FHN (x, CH
H
H

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29.8
F H (x,C,CN
)
L

29.7

FN
(x,C)
H

29.6
29.5
29.4
29.3
29.2
29.1
29
0.5

CH
0.6

0.7

0.8

0.9

CN
H
1

1.1

1.2

1.3 C 1.4
H

1.5

K1(CN
)
L

C

21

FN
(x,C)
L

20.9
20.8
20.7
20.6
20.5
20.4
20.3
20.2
20.1
20
0.5

0.6

0.7

CN
L

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

C

Figure 1: Firms H and L’s values. This figure shows FH (x, C, CLN ), FHN (x, C), and FLN (x, C) as
functions of coupon C. The parameter values are set in Table 1.

34

0.85

1.3

xH

0.8

1.2

0.75

1.1
1

0.7
0.9

C H (x)

0.92

0.94

0.96

0.98

1

0.9

0.92

0.94

H

0.96

0.98

1

H

0.6
LVH (x)CSH (x)

0.016

0.55

0.014
0.5
0.012
0.45
0.9

0.92

0.94

0.96

0.98

0.01
0.9

1

0.92

0.94

H

17

EH (x)

15

16

14

15

13

14
0.92

0.94

1

0.96

0.98

13
0.9

1

DH (x)

0.92

0.94

H

0.96

0.98

1

0.96

0.98

1

H

SW(x)

55.6

FH (x)

29.6

0.98

H

16

0.9

0.96

55.4
55.2

29.5

55
29.4
0.9

54.8
0.92

0.94

0.96

0.98

0.9

1

H

0.92

0.94
H

Figure 2: Comparative statics with respect to the spillover parameter δH . This figure shows
firm H’s bankruptcy threshold xH , coupon CH , leverage LVH , credit spread CSH , equity value
EH (x), debt value DH (x), firm value FH (x), and social welfare SW (x). The other parametervalues are set in Table 1. Simultaneous bankruptcy occurs for δH ≤ 0.9417, while sequential
bankruptcy occurs for δH > 0.9417.
35

1.4

xH

0.8

C H (x)

1.2
0.7
0.6
0.2

1
0.8
0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

0.8

1

bH

bH
0.6
LVH (x)

0.55

0.02

CSH (x)

0.5
0.015
0.45
0.4
0.2

0.4

0.6

0.8

0.01
0.2

1

0.4

bH

0.6

bH
EH (x)17

DH (x)

16

16

14

15
14

12

13

10

0.2

0.4

0.6

0.8

1

0.2

0.4

bH

0.6

0.8

1

bH
60
FH (x)

30

SW(x)

55
25
50
20
0.2

0.4

0.6

0.8

1

0.2

bH

0.4

0.6

0.8

1

bH

Figure 3: Comparative statics with respect to firm H’s operating cost bH . This figure shows
firm H’s bankruptcy threshold xH , coupon CH , leverage LVH , credit spread CSH , equity value
EH (x), debt value DH (x), firm value FH (x), and social welfare SW (x). The other parametervalues are set in Table 1. Sequential bankruptcy occurs for bH < 0.391, while simultaneousbankruptcy occurs for bH ≥ 0.391.
36

1
1.3

xH

C H (x)

1.2

0.8

1.1
0.6

0.15

1
0.2

0.25

0.3

0.6

0.9
0.15

0.35

0.03

LVH (x)

0.2

0.25

0.3

0.35

0.25

0.3

0.35

CSH (x)

0.025

0.5

0.02
0.4
0.15

0.015
0.2

0.25

0.3

0.35

0.15

0.2

20

20
EH (x)DH (x)

15

15

10
0.15

0.2

0.25

0.3

10
0.15

0.35

57

29.7

FH (x)

29.6

0.2

0.25

0.3

0.35

0.25

0.3

0.35

SW(x)

56

29.5
29.4

55

29.3
0.15

0.2

0.25

0.3

54
0.15

0.35

0.2

Figure 4: Comparative statics with respect to market volatility σ. This figure shows firm H’sbankruptcy threshold xH , coupon CH , leverage LVH , credit spread CSH , equity value EH (x),debt value DH (x), firm value FH (x), and social welfare SW (x). The other parameter values
are set in Table 1. Simultaneous bankruptcy occurs for σ ≤ 0.213, while sequential bankruptcy
occurs for σ > 0.213.
37

1
xM
H

CM
(x)
H

5

xH

0.8

C H (x)

4.5
0.6

0.15

4
0.2

0.25

0.6
0.5

0.3

0.35

0.15

0.2

LV M
(x)
H

0.03

CS M
(x)
H

LV H (x)0.025

CS H (x)

0.25

0.3

0.35

0.25

0.3

0.35

0.02
0.015

0.4

0.01
0.15

0.2

0.25

0.3

0.35

0.15

0.2

80
DM
(x)
H

70
70

EM
(x)
H

60

E H (x)

50

50
40
0.15

DH (x)

60

0.2

0.25

0.3

119

0.35

FM
(x)
H

118.5

40
0.15

154

F H (x)

118

0.2

0.25

0.3

0.35

0.25

0.3

0.35

SW M (x)
SW(x)

152

117.5
150

117
0.15

0.2

0.25

0.3

0.35

0.15

0.2

Figure 5: Bailout. Comparative statics with respect to cash flow volatility σ. This figure showsM
MM
firm H’s bankruptcy threshold xM
H , coupon CH , leverage LVH , credit spread CSH , equity
MM
value EH(x), debt value DH(x), firm value FHM (x), and social welfare SW M (x) in the casewith bailout. The parameter values other than aH /bH = 4/1.6 and M C = 1 are set in Table 1.
Firm H bails out firm L for σ ≤ 0.238, while firm H does not bail out firm L for σ > 0.238.
38

1
xD
H

0.8

1.4

xH

1.2

0.6
0.4
0.15

CD
(x)
H
C H (x)

1
0.2

0.25

0.3

0.35

0.15

0.2

0.25

0.3

0.35

LV D
(x)
H

0.6

0.03

LV H (x)

0.025

0.5

0.02
0.4
0.15

CS D
(x)
H

0.015
0.2

0.25

0.3

0.35

0.15

CS H (x)

0.2

0.25

0.3

0.35

20

20

DD
(x)
H
DH (x)

15

15
ED
(x)
H
E H (x)

10
0.15

0.2

0.25

0.3

0.35

10
0.15

0.2

0.25

0.3

0.35

57
FD
(x)
H

30

F H (x)

56

29.5

29
0.15

55

0.2

0.25

0.3

0.35

54
0.15

SW D (x)
SW(x)

0.2

0.25

0.3

0.35

Figure 6: Debt renegotiation. Comparative statics with respect to cash flow volatility σ. This
D
D
D
figure shows firm H’s bankruptcy threshold xD
H , coupon CH , leverage LVH , credit spread CSH ,D
Dequity value EH(x), debt value DH(x), firm value FHD (x), and social welfare SW D (x) in thecase with debt renegotiation. The parameter values other than DC = 3 are set in Table 1.
Firm H renegotiates debt for σ ≤ 0.229, while firm H does not renegotiate debt for σ > 0.229.
39