Discussion Papers In Economics
And Business

Dynamic bankruptcy procedure with asymmetric
information between insiders and outsiders
Michi NISHIHARA, Takashi SHIBATA

Discussion Paper 17-18

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDynamic bankruptcy procedure with asymmetric
information between insiders and outsiders
Michi NISHIHARA, Takashi SHIBATA

Discussion Paper 17-18

July 2017

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDynamic bankruptcy procedure with asymmetric
information between insiders and outsiders∗Michi NISHIHARA†,

Takashi SHIBATA‡

Abstract
We develop a dynamic model in which a distressed ﬁrm optimizes the bankruptcy
choice and its timing. When the distressed ﬁrm’s shareholders sell the assets, they are
better informed about the asset value than outsiders are. We show that this asymmetric
information can delay the asset sales to signal asset quality to outsiders. More debt andlower asset value can reduce the signaling cost and mitigate the asset sales delay. Notably,
we show that the ﬁrm changes the bankruptcy choice from selling out to liquidation
bankruptcy when the signaling cost associated with selling out is high. This distortion in
the bankruptcy choice greatly lowers the debt value, whereas it has a weak impact on theequity value.
JEL Classifications Code: D82; G13; G33.
Keywords: bankruptcy; adverse selection; asymmetric information; signaling game; real
options; M&A.

∗

This version: 10 July, 2017. We thank Junichi Imai, Haejun Jeon, Mami Kobayashi, Yue Kuen Kwok,

Yuki Sato, Ryuta Takashima, and Neng Wang for their helpful suggestions. We also thank the participantsat the 18th Asia Paciﬁc Management Conference, the Winter Workshop on Operations Research, Financeand Mathematics in 2017, the Spring National Conference of Operations Research Society of Japan in 2017,
the 5th Asian Quantitative Finance Conference, and Nippon Finance Association 25th Annual Conference for
helpful feedback. This work was supported by the JSPS KAKENHI (Grant number JP26350424, JP26242028,
JP17K01254, JP16KK0083, JP17H02547) and the JSPS Postdoctoral Fellowships for Research Abroad. This
paper was partially written when Michi Nishihara was a JSPS Postdoctoral Fellow for Research Abroad (Visiting
Researcher at the Swiss Finance Institute, École polytechnique fédérale de Lausanne). The authors thankthe Swiss Finance Institute (especially, Erwan Morellec) and the JSPS Postdoctoral Fellowships for Research
Abroad.
†Corresponding Author. Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka,
Osaka 560-0043, Japan, E-mail: nishihara@econ.osaka-u.ac.jp, Phone: 81-6-6850-5242, Fax: 81-6-6850-5277‡
Graduate School of Social Sciences, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo
192-0397, Japan, E-mail: tshibata@tmu.ac.jp, Phone: 81-42-677-2310, Fax: 81-42-677-2298

1

Introduction

Since the seminal works by Black and Cox (1976), Leland (1994), Mella-Barral and Perraudin (1997), and Fan and Sundaresan (2000), an increasing number of studies investigate corporate bankruptcy decisions in continuous-time models. In dynamic bankruptcy
models, prior works examine bankruptcy timing, debt renegotiation, liquidation, agency
conﬂicts between equity- and debt holders, and so on.1 However, no study incorporates
the stylized fact that a distressed ﬁrm has diﬃculty selling assets due to asymmetric information about asset quality (e.g., Hotchkiss and Mooradian (1998), Gilson, Hotchkiss,
and Ruback (2000), and Povel and Singh (2006)) into dynamic bankruptcy models.
To our knowledge, this is the ﬁrst study that incorporates asymmetric information
between insiders and outsiders in a dynamic bankruptcy model. In this novel framework,
we address several questions. How can a distressed ﬁrm use asset sales timing as a signaling
tool to resolve informational issues? How does asymmetric information aﬀect bankruptcy
timing and the procedure, as well as the debt and equity values?
Our model builds on the standard setup in Mella-Barral and Perraudin (1997). Shareholders of a distressed ﬁrm make a bankruptcy choice between selling out and default, as
well as its timing.2 The model does not distinguish between shareholders and managers,
assuming that managers act in shareholders’ interests. Selling out is a rather successful
exit. Indeed, shareholders sell all assets and obtain the residual value, that is, the sales
price minus the face value of debt, while debt holders are repaid the face value of debt.
On the other hand, default is an unsuccessful exit. Shareholders stop coupon paymentsof debt, and the former debt holders take over the ﬁrm and can either instantly sell assets(called liquidation bankruptcy) or operate the ﬁrm (called operating concern bankruptcy).
A fraction of the ﬁrm value is lost to bankruptcy costs associated with ownership change.
We add asymmetric information about asset quality to this standard setup. To be
precise, the ﬁrm’s shareholders are better informed than outsiders about whether theﬁrm’s running cost is high or low.3 The market value of assets will be higher as therunning cost is lower. Shareholders cannot directly transmit information about whether
the ﬁrm is a high- or low-cost type to outsiders. Although outsiders cannot directly
observe the ﬁrm’s type, they can guess the ﬁrm’s type through the sales timing.
In the model, we derive a separating equilibrium where the low-cost ﬁrm can separate
itself from the high-cost ﬁrm through its bankruptcy choice and timing, and the ﬁrm’s
1

An incomplete list includes Mella-Barral (1999), Francois and Morellec (2004), Broadie, Chernov, andSundaresan (2007), Lambrecht and Myers (2008), Gryglewicz (2011), Moraux and Silaghi (2014), Christensen,
Flor, Lando, and Miltersen (2014), Correia and Poblacion (2015), and Nishihara and Shibata (2016).
2
Although Mella-Barral and Perraudin (1997) also examine renegotiation between equity- and debt holders,
we exclude the possibility of debt renegotiation to focus on asymmetric information between insiders andoutsiders.
3
For simplicity, we consider the case with two types. Our key ﬁndings remain unchanged, even if we considera case with a continuum of types, following Grenadier and Malenko (2011).

2

type is perfectly revealed to outsiders. In equilibrium, the low-cost ﬁrm’s bankruptcy
choice and timing can change with asymmetric information, while the high-cost ﬁrm’s
bankruptcy choice and timing remain unchanged. In other words, the low-cost ﬁrms pay
all costs due to asymmetric information.
First, we show that asymmetric information can delay the low-cost ﬁrm’s sales timing
because the ﬁrm signals its asset quality to outsiders by delaying sales until the point at
which the high-cost ﬁrm cannot imitate the low-cost ﬁrm’s sales. In the delayed sales case,only shareholders pay signaling costs, and debt holders suﬀer no loss due to asymmetric
information because they are retired the face value of debt. Although a number of studies
investigated a distressed ﬁrm’s asset sales with depressed prices (cf. ﬁre sales in Shleiferand Vishny (1992), Pulvino (1998), and Eckbo and Thorburn (2008)), this is the ﬁrst to
reveal that a distressed ﬁrm can potentially avoid ﬁre sales due to information issues bydelaying asset sales.
Although our result is novel in the context of liquidation timing, the mechanism isconsistent with that of the prior literature on dynamic trading between informed sellers
and uninformed buyers (e.g., Janssen and Roy (2002), Fuchs and Skrzypacz (2013), andFuchs, Green, and Papanikolaou (2016)). Actually, in dynamic models, unlike the static
adverse selection models (cf. Akerlof (1970)), the sales timing becomes a signal of asset
quality to outsiders. The empirical evidence in Adelino, Gerardi, and Hartman-Glaser(2017) supports the result that informed sellers signal quality by delaying sales in themortgage market.
With respect to the delayed sales timing, we ﬁnd two results that diﬀer from thosein the standard literature on dynamic liquidation timing models. One is the impact
of existing debt on the sales timing. Under symmetric information, the sales timing is
independent of debt because the face value of debt is retired to creditors (cf. Mella-Barral
and Perraudin (1997)). In contrast to this standard result, we show that higher debt canaccelerate sales. Indeed, higher debt decreases the residual value from selling out andhence decreases the incentive for the high-cost ﬁrm to imitate the low-cost ﬁrm’s sales.
Thus, higher debt can play a positive role in alleviating the delay in asset sales timing.
We also ﬁnd a counter-intuitive impact of asset value on the sales timing. Undersymmetric information, higher asset value straightforwardly accelerates asset sales. However, we show that higher asset value can delay asset sales under asymmetric information.
This is because higher asset value increases the residual value from selling out and hence
increases the incentive for the high-cost ﬁrm to imitate the low-cost ﬁrm’s sales. Thus,
higher asset value can play a negative role in intensifying the delay in asset sales timing.
Next, we show that the low-cost ﬁrm changes its bankruptcy choice from selling out toliquidation bankruptcy when the signaling cost by delaying sales is higher than the direct
cost, that is, the asset value minus the face value of debt. This result strongly contrasts
prior ﬁndings that shareholders prefer to default if and only if the asset value is lower than
the face value of debt. The failure to sell out lowers the ﬁrm value and sales price due

3to bankruptcy costs. Notably, we ﬁnd that, in this liquidation bankruptcy case, unlikein the delayed sales case, debt holders suﬀer severe losses, although shareholders’ loss issmall. To our knowledge, this is the ﬁrst study that reveals how equity and debt holders
pay information costs due to asymmetric information in the bankruptcy procedure.
Several empirical ﬁndings are consistent with our results. For instance, Hotchkiss andMooradian (1998), Stromberg (2000), and Thorburn (2000) show that distressed ﬁrms
are more likely to be acquired by better informed ﬁrms, including former owners. Thisﬁnding aligns with our result that the ﬁrm is more likely to sell out under symmetric
information than under asymmetric information. Stromberg (2000) shows that asset sales
to less informed ﬁrms lowers sales prices, while Thorburn (2000) shows that debt holders
recover more when former owners buy back ﬁrms. These ﬁndings support our resultthat asymmetric information can lead to liquidation bankruptcy, where ﬁrm sell assets at
depressed prices due to bankruptcy costs, and debt holders suﬀer severe losses.
Our contribution to the literature is three-fold. First, we complement the literature on
dynamic bankruptcy decisions (e.g., Leland (1994), Mella-Barral and Perraudin (1997),
Lambrecht and Myers (2008), and Gryglewicz (2011)) by showing several new resultsfrom asymmetric information between insiders and outsiders. Second, we complement
the literature on the asset sales of distressed ﬁrms (e.g., Shleifer and Vishny (1992),
Maksimovic and Phillips (1998), Stromberg (2000), and Eckbo and Thorburn (2008)) by
showing the possibility that a distressed ﬁrm can avoid selling assets at depressed prices
by delaying the asset sales timing. Lastly, we complement the literature on dynamic
signaling models (e.g., Janssen and Roy (2002), Grenadier and Malenko (2011), Daleyand Green (2012), and Fuchs, Green, and Papanikolaou (2016)) by showing that a similar
mechanism holds in the context of dynamic liquidation timing problems.
The remainder of this paper is organized as follows. We introduce the model setup
in Section 2. After Section 3.1 shows the model solution under symmetric information,
Section 3.2 shows the equilibrium under asymmetric information. In Section 4, we demonstrate the economic implications along with numerical examples. Section 5 concludes thepaper.

2

Model Setup

2.1

Firm until bankruptcy

The model builds on the standard setup of Mella-Barral and Perraudin (1997) and Lambrecht and Myers (2008). Consider a ﬁrm with console debt with coupon C, that is,
the ﬁrm continues to pay coupon C to debt holders until bankruptcy. The ﬁrm receives
continuous streams of earnings before interest and taxes (EBIT) X(t) − wi , where X(t)4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. We assume that the initial value,
X(0) = x, is suﬃciently high to exclude the ﬁrm’s bankruptcy at time 0. For convergence,
we assume that r > µ, where a positive constant r denotes the risk-free interest rate.
The running cost wi (≥ 0) can take two types: wi = wL (low-cost type) and wi = wH
(high-cost type), where wL < wH . All agents know the prior probability of the lowcost type, q ∈ (0, 1) as well as all information (e.g., X(t) and C) except for the ﬁrm’s
type i. Under symmetric information, all agents know the ﬁrm’s type i (or equivalently,
shareholders can directly prove the ﬁrm’s type to other agents at no cost), whereas underasymmetric information, only shareholders know the ﬁrm’s type i and cannot directly
prove the ﬁrm’s type to other agents. We assume that managers act in the shareholders’interests, and hence we do not distinguish between shareholders and managers.

2.2

Bankruptcy choice between selling out and default

As in Mella-Barral and Perraudin (1997) and Lambrecht and Myers (2008), we examine
the ﬁrm’s bankruptcy choice between selling out (liquidation without default) and default.
In the sales case (closing the business by scrapping and/or selling assets to outsiders), theﬁrm sells its total assets and receives the sales price from bidder(s). As in Lambrecht(2001), Lambrecht and Myers (2008), and Nishihara and Shibata (2016), we assume thatthe market value of assets is a linear function
aX(t) bwi
−
+θ
r−µ
r

(i = L, H),(1)

where a ∈ [0, 1), b ∈ (0, 1), and θ ≥ 0 are constants.4 We assume that all agents know theparameter values of a, b, and θ. We can interpret (1) as follows. Bidders are competitive.5
After partial assets are scrapped and liquidated at a ﬁxed price θ, a bidder can perpetually
receive cash ﬂows aX(t) − bwi from the remaining assets, where a < 1 and b < 1 meanthat both the revenues and costs contract due to the decrease in assets. The parametera may include synergies in acquisition, that is, an increase in the acquirer’s cash ﬂows
in the related business. Following the absolute priority rule (APR) of debt, debt holders
4

Although Mella-Barral and Perraudin (1997) assume a rather simple form, that is, a = 0, we assume thatthe asset value can depend on the state variable X(t). However, we do not directly model ﬁre sales, as Shleiferand Vishny (1992) and Pulvino (1998), among others, discuss. They argue that a distressed ﬁrm sells assets at
depressed prices because potential bidders in the same industry tend to be ﬁnancially distressed as well.
5
Our model presumes asset sales by auction rather than by negotiation. In the real world, a ﬁrm in distressis more likely to be acquired by competitive bidding than in a non-distressed situation. For example, refer toHotchkiss and Mooradian (1998) and Eckbo and Thorburn (2008).

5

are repaid the face value of debt, which equals C/r in the case of the console debt.
Shareholders receive the residual value, that is, (1) minus C/r.
On the other hand, in the default case, shareholders declare default and stop paying
coupon C on the debt; thereafter, debt holders lose coupon payments. Following the APR,
at the time of default, debt holders take over the ﬁrm, while shareholders receive nothing.
Following the standard literature, including Leland (1994), Goldstein, Ju, and Leland(2001), and Lambrecht and Myers (2008), a fraction α ∈ (0, 1) of the ﬁrm’s asset value
is lost to the bankruptcy costs (ﬁling fees, attorney fees, etc.) associated with ownershipchanges at the time of default. Then, the ﬁrm value, which former debt holders take over,becomes (1 − α)Ui (X(t)), where Ui (X(t)) denotes the unlevered ﬁrm value with runningcost wi . Former debt holders either operate the ﬁrm as a going concern or sell all assets
instantly by (1 − α)× (1).

3Model Solutions

3.1

Symmetric information

As a benchmark, we solve the problem under symmetric information. Outsiders observe
the ﬁrm’s cost wi and evaluate the assets using (1) depending on the type i and statevariable X(t). When the ﬁrm chooses to sell out (we denote the sales case by the subscript1), the equity value, as in Mella-Barral and Perraudin (1997), becomes the following valuefunction:()∫ T
aX(T ) bwi
C
−rt
−rT
e (X(t) − wi − C)dt + e−+θ−]Ei,1 (x) = sup E[
r−µ
r
r
T
0
(
)
x
wi + C(a − 1)X(T ) (1 − b)wi=−
+ sup E[e−rT
+
+θ ]
r−µ
r
r−µ
r
T
(
) (
)
x
wi + Cx γ (a − 1)xi,1 (1 − b)wi
=
−
+ sup
+
+θ
(i = L, H),
r−µ
r
xi,1
r−µ
r
xi,1
(2)
where γ = 1/2 − µ/σ 2 −√(µ/σ 2 − 1/2)2 + 2r/σ 2 (< 0), and sales time T (we optimize

trigger xi,1 ) runs over stopping times.6
When the ﬁrm chooses to default (we denote the default case by the subscript 2),
the equity value, as in Leland (1994) and Goldstein, Ju, and Leland (2001), becomes the6

We always have aX(T )/(r − µ) − bwi /r + θ − C/r ≥ 0 when the ﬁrm chooses to sell out rather than default.

Accordingly, we do not need to impose a limited liability condition in problem (2).

6following value function:∫ T
Ei,2 (x) = sup E[
e−rt (X(t) − wi − C)dt]
T
0
(
)
x
wi + C
X(T ) wi + C−rT=−+ sup E[e
−
+
]
r−µ
r
r−µ
r
T
(
)γ (
)
xi,2
x
wi + Cx
wi + C
=
−
+ sup
−
+
r−µ
r
xi,2
r−µ
r
xi,2

(i = L, H),(3)

where default time T (we optimize trigger xi,2 ) runs over stopping times.
As Décamps, Mariotti, and Villeneuve (2006) show, max{Ei,1 (x), Ei,2 (x)} is equal to
the result by the dynamic choice between selling out and default if X(0) = x is higher than
the sales trigger.7 Then, by solving (2) and (3), and comparing Ei,1 (x) and Ei,2 (x), we
obtain the optimal choice between selling out and default. For the proof of the proposition,
see Appendix A.
Proposition 1 Symmetric information. For each type i ∈ {L, H}, we have the following results. If

(

1
1−a

) −γ

1−γ

≥

wi + C(1 − b)wi + rθ(4)holds, shareholders choose to sell out. The optimal sales trigger isxi,1 =

γ(r − µ) (1 − b)wi + rθ.
(γ − 1)r
1−a(5)

The equity value is
wi + Cx−+
Ei,1 (x) =
r−µ
r

(

x
xi,1

)γ (

)
(a − 1)xi,1 (1 − b)wi
+
+θ .
r−µ
r

(6)

The debt value is Di,1 (x) = C/r (risk-less debt).
If (4) does not hold, shareholders choose to default. The optimal default trigger isxi,2 =

γ(r − µ)(wi + C).
(γ − 1)r(7)

The equity value is
wi + Cx−+
Ei,2 (x) =
r−µ
r
If

(

1
1−a

) −γ

1−γ

<

(

x
xi,2

)γ (
)
xi,2
wi + C−+.
r−µr

wi + C1≤(1 − b)wi + rθ
1−a

(8)

(9)

holds, the debt value isCDi,2 (x) =
−
r
7

(

x
xi,2

)γ (

C
− (1 − α)
r

(

axi,2
bwi
−
+θ
r−µ
r

))
.

(10)

When X(0) = x is smaller than the sales trigger, the optimal policy can be the dynamic choice as follows.

Shareholders choose to default when X(t) drops to a lower threshold, while they choose to sell out when X(t)rises to a higher threshold. For details, see Décamps, Mariotti, and Villeneuve (2006).

7

If (9) does not hold, the debt value is
(
) (
)
C
x γ C
Di,2 (x) =−− (1 − α)Ui (xi,2 ) ,
r
xi,2
r

(11)

where Ui (X(t)) is defined by (6) with C = 0, that is, the unlevered firm value with runningcost wi .
The results for a = 0 are essentially the same as those of Mella-Barral and Perraudin(1997); the results in the default case are essentially the same as those of Goldstein, Ju,
and Leland (2001). Note that in (6) and (8), x/(r − µ) − (wi + C)/r correspond to the
expected value of inﬁnite streams of cash ﬂows X(t) − (wi + C). The third terms in (6)and (8) are the values of the option to sell out and default, respectively. In (10) and(11), the ﬁrst term C/r is the face value of debt, that is, the risk-less debt value, while
the second term indicates the discount due to default risk. As we show in Appendix B,
Di,2 (x) is below the face value due to default risk.
For low C (compared to the asset value), (4) holds. In this case, shareholders directlysell the total assets to outsiders, and debt holders are retired the face value of debt.
Although we refer to this case selling out, Mella-Barral and Perraudin (1997) call thiscase no bankruptcy because the debt is riskless. The existing debt imposes no eﬃciency
loss, that is, the ﬁrm value Ei,1 (x) + C/r is equal to the maximum value, which is theunlevered ﬁrm value8 , because the ﬁrm avoids incurring bankruptcy costs. In short, theﬁrm is optimally liquidated with no costs.
For intermediate C, (9) holds, and xi,2 ≤ xi,1 holds in this region. In this case,shareholders declare default at the default trigger xi,2 , and debt holders are not retired
the face value of debt but instead take over the ﬁrm. Former debt holders sell the total
assets as soon as X(t) decreases below the sales trigger xi,1 .9 Due to xi,2 ≤ xi,1 , theysell the total assets immediately after taking over the ﬁrm, although the asset value isless than the face value. Following Mella-Barral and Perraudin (1997), we call this caseliquidation bankruptcy which can be related to Chapter 7 (liquidation) bankruptcy in the
United States. The existing debt imposes an eﬃciency loss. Indeed, due to bankruptcy
cost α and delayed liquidation, the ﬁrm value Ei,2 (x) + Di,2 (x) is discounted from theunlevered ﬁrm value. In short, the ﬁrm ineﬃciently liquidates by incurring additionalcosts.
For high C, (9) does not hold, and xi,2 > xi,1 holds. In this case, shareholders declaredefault at the default trigger xi,2 , and debt holders are not retired at the face value ofdebt, but instead take over the ﬁrm, after which the former debt holders operate theﬁrm as a going concern until X(t) decreases below the sales trigger xi,1 . Following MellaBarral and Perraudin (1997), we call this case operating concern bankruptcy. Operating
8

Following Mella-Barral and Perraudin (1997) and Lambrecht and Myers (2008), we omit the tax beneﬁts

of debt from the model. Then, the unlevered ﬁrm value agrees with the maximum ﬁrm value.
9
Even after default, the sales trigger remains unchanged from xi,1 deﬁned by (5) because both cash ﬂow andasset value are multiplied by (1 − α).

8concern bankruptcy can be related to Chapter 11 (reorganization) bankruptcy in the
United States, although we do not model debt renegotiation and restructuring process.10
The existing debt imposes an eﬃciency loss. In fact, due to bankruptcy cost α, the ﬁrm
value Ei,2 (x) + Di,2 (x) is discounted from the unlevered ﬁrm value.
Proposition 1 shows that a ﬁrm with less debt and higher asset value tends to sell out
without default. In addition, if the ﬁrm goes into bankruptcy, less debt and higher asset
value tend to lead to liquidation bankruptcy rather than operating concern bankruptcy.
These results are consistent with the stylized fact that smaller/younger ﬁrms with lower
leverage ratios are more likely to go into Chapter 7 bankruptcy rather than Chapter 11
bankruptcy (e.g., Bris, Welch, and Zhu (2006)). Another interesting result from Proposition 1 is the impact of cash ﬂow volatility σ. Because ∂γ/∂σ > 0, condition (4) is more
likely to hold for lower σ. Intuitively, a higher σ increases the option value of default
more than the option value of selling out because the convexity of the option to default
is stronger than that for selling out.11 This implies that a ﬁrm with higher cash ﬂow
volatility tends to fail to sell out. Although this is not seen in Mella-Barral and Perraudin(1997), who assume a = 0, this result is consistent with the stylized fact that higher cash
ﬂow volatility is more likely to lead to an unsuccessful bankruptcy.

3.2

Asymmetric information

Although Section 3.1 assumed that outsiders have perfect knowledge of the distressed
ﬁrm’s asset quality, this is not the case in the real world. Many ﬁrms have diﬃculty
selling assets at a fair price, especially during ﬁnancial distress. In particular, small
and/or private ﬁrms, which have less transparency and disclosure, pay costs due to strong
asymmetric information. For example, refer to Gilson, Hotchkiss, and Ruback (2000),
Povel and Singh (2006), and Hotchkiss, John, Mooradian, and Thorburn (2008) on thismatter.
Now, suppose that asymmetric information exists between the ﬁrm’s insiders and outsiders. Only shareholders know the ﬁrm’s type i, while outsiders do not observe the ﬁrm’s
type. Intuitively, shareholders of the high-cost ﬁrm may have an incentive to imitate thelow-cost ﬁrm’s sales timing and receive a higher asset value (1) with i = L, whereas shareholders of the low-cost ﬁrm have no incentive to imitate the high-cost ﬁrm. Shareholders
receive nothing when they declare default rather than selling out. Then, shareholdersof the high-cost ﬁrm have no incentive to imitate the low-cost ﬁrm if the low-cost ﬁrm
10

A number of papers, including Sundaresan and Wang (2007), Moraux and Silaghi (2014), Christensen, Flor,

Lando, and Miltersen (2014), Shibata and Nishihara (2015), and Nishihara and Shibata (2016), incorporate thedebt renegotiation and restructuring process into dynamic bankruptcy timing models. It could be interesting fora future study to investigate the eﬀects of asymmetric information on debt renegotiation and the restructuring
process.
11
By the same logic, Kort, Murto, and Pawlina (2010) show that higher volatility is more likely to lead to
lumpy rather than stepwise investment.

9chooses to default. The default time (the default trigger xi,2 deﬁned by (7)), which is
not aﬀected by asymmetric information, reveals the ﬁrm’s type to outsiders. This meansthat former debt holders, who take over the ﬁrm after default, have no concerns about
asymmetric information. This reasoning leads to the following proposition.
Proposition 2 Low-cost firm default case.
Case (I): Suppose that

(

1
1−a

) −γ

1−γ

<

wL + C,(1 − b)wL + rθ

(12)that is, shareholders of the low-cost firm choose to default under symmetric information.
The bankruptcy choice and timing, as well as the equity and debt values, of both types offirms are unchanged from those in the case of symmetric information.
When the face value of debt is high compared to asset value, (12) is likely to hold, andasymmetric information has no eﬀects on the bankruptcy procedure. This is straightforward because asymmetric information matters only in the sales case. More interestingly,
high volatility σ removes losses due to asymmetric information because (12) is more likely
to hold for higher σ. Lambrecht and Myers (2008) and Nishihara and Shibata (2017)
argue that risky debt mitigates manager-shareholder conﬂicts. Although they focus on
manager-shareholder conﬂicts, Proposition 2 complements their results by showing that
more debt and higher risks can mitigate insider-outsider conﬂicts.
Next, suppose that (12) does not hold. Although there can be two types of equilibria,
namely, separating (informative) equilibrium and pooling (uninformative) equilibrium,
following Janssen and Roy (2002), Grenadier and Malenko (2011), and Adelino, Gerardi,
and Hartman-Glaser (2017), we focus only on the least-cost separating equilibrium for the
low-cost ﬁrm.12 In equilibrium, shareholders of the low-cost ﬁrm optimize the bankruptcy
choice and timing within the policies that can separate the low-cost ﬁrm from the high-costﬁrm.
When the low-cost ﬁrm prefers to sell out in the separating equilibrium, as in (2), theequity value of the low-cost ﬁrm is()∫ T
aX(T ) bwL
C
s
−rt
−rT
EL (x) = sup E[
e (X(t) − wL − C)dt + e
−
+θ−
]
r−µ
r
r
T
0
(
)
x
wL + C(a − 1)X(T ) (1 − b)wL−rT=−
+ sup E[e
+
+θ ]
r−µ
r
r−µ
r
T
( )γ ()(a − 1)xsL (1 − b)wLx
wL + C
x
=
−
+ sup
+
+
θ
,
(13)
r−µ
r
xsL
r−µ
r
xsL
where the sales trigger xsL is optimized subject to the following incentive compatibility

12

Most of the literature concentrates on the least-cost separating equilibrium because the standard reﬁnement

criteria in Cho and Kreps (1987) eliminate the pooling equilibria. For details, see Grenadier and Malenko (2011).

10

condition (ICC)wH + C
x
−
+
r−µ
r
|

(

x
xsL

)γ (

)
(a − 1)xsL wH − bwL++ θ ≤ max{EH,1 (x), EH,2 (x)} .
r−µ
r
|
{z
}
{z
}
Value by truthful action

Value by imitation(14)
Throughout the paper, the superscript s represents the separating equilibrium. The lefthand side of (14) is the expected value of the high-cost ﬁrm’s shareholders who imitate
the low-cost ﬁrm’s sales timing and receive the higher asset value axsL /(r − µ) − bwL /r + θ,
whereas the right-hand side of (14) is the expected value of the high-cost ﬁrm’s shareholders who truthfully take the ﬁrst-best policy derived in Proposition 1. Under ICC,
shareholders of the high-cost ﬁrm are better oﬀ following the ﬁrst-best policy as thehigh-cost type rather than imitating the low-cost ﬁrm’s sales timing.13 Thus, throughthe sales trigger xsL , outsiders verify the low-cost type and pay the higher asset valueaxsL /(r − µ) − bwL /r + θ to the low-cost ﬁrm.
Recall that shareholders of the low-cost ﬁrm can gain EL,2 (x) by choosing default
because the high-cost ﬁrm has no incentive to imitate the low-cost ﬁrm’s default timing.
Accordingly, shareholders of the low-cost ﬁrm gain max{ELs (x), EL,2 (x)}. By solving
problem (13) subject to (14) and comparing ELs (x) with EL,2 (x), we have the least-cost
separating equilibrium as follows. For the proof, see Appendix C.
Proposition 3 Separating equilibrium. Suppose that (12) does not hold.
Case (II): Suppose that xL,1 defined by (5) with i = L satisfies ICC (14). The bankruptcy
choices and timing, as well as the equity and debt values, of both types of firms do notchange from those of the symmetric information case.
Case (III): Suppose that xL,1 dose not satisfy ICC (14). We define xsL ∈ (0, xL,1 ) byequating ICC (14), that is, the solution to( )γ ()(a − 1)xsL wH − bwL
1
+
+θ
xsL
r−µ
r
(
)γ (
) (
)γ (
)
(a − 1)xH,1 (1 − b)wH
xH,2
1
1
wH + C= max{
+
+θ ,
−
+
},
xH,1
r−µ
r
xH,2
r−µ
r
(15)
where xH,1 and xH,2 are defined by (5) and (7) with i = H, respectively. Suppose that( )γ () ()) ((a − 1)xsL (1 − b)wL
xL,2
1
1 γ
wL + C
+
+θ ≥
+
,
(16)
−
xsL
r−µ
r
xL,2
r−µ
r
where xL,2 is defined by (7) with i = L. Shareholders of the low-cost firm choose to sellout. The sales trigger is xsL , and the equity value is( )γ ()(a − 1)xsL (1 − b)wL
wL + C
x
x
s
−
+
+
+θ .
EL (x) =
r−µ
r
xsL
r−µ
r
13

(17)

We consider only ICC for the high-cost ﬁrm because we can easily check that the solution satisﬁes ICC forthe low-cost ﬁrm, that is, the low-cost ﬁrm has no incentive to imitate the high-cost type.

11

The debt value remains unchanged from DL,1 (x) = C/r (risk-less debt). The bankruptcy
choice and timing, as well as the equity and debt values, of the high-cost firm do notchange from those of the symmetric information case.14
Case (IV): Suppose that xL,1 does not satisfy ICC (14) and that (16) does not hold.
Shareholders of the low-cost firm choose to default. The default trigger is xL,2 , and theequity and debt values of the low-cost firm are EL,2 (x) and DL,2 (x), defined by (8) and(10) with i = L, respectively. The bankruptcy choice and timing, as well as the equity anddebt values, of the high-cost firm are unchanged from those of the symmetric informationcase.
In Case (II), the low-cost ﬁrm has no incentive to imitate the high-cost ﬁrm’s ﬁrst-bestsales trigger xL,1 . This case occurs mainly because the ﬁrm’s types are discrete. We do
not have this case when we consider a continuum of types. We omit the explanation here
because the results are trivial.
In Case (III), the low-cost ﬁrm signals the ﬁrm’s type to outsiders by decreasing thesales trigger. Figure 1 illustrates the mechanism that determines the sales trigger xsL
and equity value ELs (x) in this case. The parameter values are set in Table 1. In theﬁgure, the upper curve shows (13) as a function of xsL , while the lower curve shows theleft-hand side of ICC (14) as a function of xsL . Without ICC (14), the equity value isthe ﬁrst-best value EL,1 (x) = 17.538 with the sales trigger xL,1 = 1.03. However, thelower curve is above max{EH,1 (x), EH,2 (x)} = EH,2 (x) = 11.381 for xL,1 = 1.03, andhence outsiders cannot distinguish between low- and high-cost ﬁrms. The low-cost ﬁrm
decreases the sales trigger from xL,1 = 1.03 to xsL = 0.851, at which the lower curve crosses
max{EH,1 (x), EH,2 (x)} = EH,2 (x) = 11.381, to signal the ﬁrm’s type to outsiders. Dueto the second-best trigger xsL = 0.851, the low-cost ﬁrm’s equity value ELs (x) = 17.402 is
below EL,1 (x) = 17.538, but it is still above EL,2 (x) = 17.296.
This logic is in line with that of the literature on dynamic trading between informed
sellers and uninformed buyers (e.g., Janssen and Roy (2002), Fuchs and Skrzypacz (2013),
and Fuchs, Green, and Papanikolaou (2016)). The delay in sales makes it more costly
for the high-cost ﬁrm to feign the low-cost type. The low-cost ﬁrm can reveal its type tooutsiders by delaying the sales timing to the point at which the high-cost ﬁrm does notimitate the low-cost ﬁrm’s policy.15 Although in Case (III), the signaling cost decreases
the equity value from EL,1 (x) to ELs (x), shareholders of the low-cost ﬁrm are better oﬀ
selling out because ELs (x) > EL,2 (x). Note that lower C, higher θ, higher a, and lower b
14

In Case (III), outsiders’ belief is given as follows: The ﬁrm is a low-cost type at probability one for any

sales at a trigger in (x, xsL ], while the ﬁrm is a high-cost type at probability one for the other policies. It is easy
to check the single crossing condition. In Cases (I), (II), and (IV), outsiders’ belief is trivial.
15
Recently, several papers, including Grenadier and Malenko (2011), Morellec and Schürhoﬀ (2011), andBustamante (2012), develop real options models under asymmetric information between insiders and outsiders.
They focus on the investment and ﬁnancing timing as a signaling tool, whereas we examine the asset sales
timing as a signaling tool.

12

tend to lead to Case (III). In other words, when the face value of debt is lower than theasset value, the ﬁrm tends to adopt the policy of delayed sales.
A number of papers, including those by Shleifer and Vishny (1992), Hotchkiss andMooradian (1998), Stromberg (2000), and Eckbo and Thorburn (2008), both theoreticallyand empirically investigate asset sales at discounted prices by ﬁrms in ﬁnancial distress.
Nevertheless, to our knowledge, no paper shows that a distressed ﬁrm can potentiallydelay asset sales to keep asset prices higher. Thus, our result complements the literature
on distressed ﬁrms’ ﬁre sales by showing that a distressed ﬁrm can avoid ﬁre sales bydelaying the asset sales timing as a signaling tool under asymmetric information.
In Case (IV), the signaling cost in the sales decreases the equity value below the equity
value in the default case, that is, ELs (x) is lower than EL,2 (x). In other words, the signaling
cost is higher than the direct cost, that is, the asset value minus the face value of debt.
Then, shareholders of the low-cost ﬁrm give up selling out and resort to a default. They
do not change the default trigger from xL,2 because the high-cost ﬁrm has no incentive toimitate the low-cost ﬁrm’s default timing. Although the default trigger does not change
from xL,2 , the equity value decreases from the ﬁrst-best value EL,1 (x) to EL,2 (x) with thechange in bankruptcy choice.
In Case (IV), liquidation bankruptcy occurs, in other words, debt holders sell the ﬁrm
immediately after taking over the ﬁrm. Note that outsiders observe that the ﬁrm is the
low-cost type at the default trigger xL,2 , but asymmetric information aﬀects the debt
value. In fact, the debt value changes from the ﬁrst-best value DL,1 (x) = C/r to DL,2 (x),deﬁned by (10) with i = L, with asymmetric information. Case (IV), where neither (16)nor (12) holds, tends to hold when C is balanced with levels of θ, a, and b. In other words,
when the face value of debt is close to the asset value, under asymmetric information, theﬁrm can change the bankruptcy choice from selling out to liquidation bankruptcy. This is
in sharp contrast with the standard result that a ﬁrm goes into default if and only if theasset value is lower than the face value of debt (cf. Mella-Barral and Perraudin (1997)).
Next, we examine costs due to asymmetric information between insiders and outsiders.
Note that the high-cost ﬁrm’s equity and debt values do not change with asymmetric information. We denote by EL∗ (x) and DL∗ (x) the equity and debt values under symmetric∗∗ (x) the equity and debt values uninformation, respectively, and denote by EL∗∗ (x) and DL

der asymmetric information, respectively. Proposition 3 immediately yields the following
corollary.
Corollary 1 Informational cost.
Cases (I) and (II):EL∗ (x) − EL∗∗ (x) = 0
DL∗ (x) − DL∗∗ (x) = 0

13

Case (III):)(a − 1)xL,1 (1 − b)wL
+
+θ
xL,1
r−µ
r
( )γ ()(a − 1)xsL (1 − b)wLx−+
+ θ (> 0)
xsL
r−µ
r

EL∗ (x) − EL∗∗ (x) =

(

x

)γ (

∗
∗∗
DL
(x) − DL(x) = 0Case (IV):
(

)γ (

)
(a − 1)xL,1 (1 − b)wL
+
+θ
xL,1
r−µ
r
)γ (
(
)
xL,2
x
wL + C
−
−
+
(> 0)
xL,2
r−µ
r
) (
(
(
))
axL,2
x γ C
bwL
∗
∗∗
DL (x) − DL (x) =− (1 − α)
−
+θ
xL,2
r
r−µ
r
EL∗ (x) − EL∗∗ (x) =x

In Cases (I) and (II), there is no informational cost because the low-cost ﬁrm conducts
the ﬁrst-best policy. In Case (III), the signaling cost equals the informational cost, andhence, shareholders of the low-cost ﬁrm pay the cost. Actually, the sales trigger decreases
from xL,1 to xsL , which lowers the equity value from EL,1 (x) to ELs (x). On the other hand,debt holders pay no cost because, as in the symmetric information case, they are retired
the face value of debt. The delay in the sales timing does not aﬀect the debt value.
Case (IV) is the most interesting case, where both equity and debt holders pay the
informational cost. As we explained above, shareholders give up selling out and lose theresidual value, that is, the asset value minus the face value of debt. Although theoretically,
C/r − (1 − α)(axL,2 /r − µ − bwL /r + θ) can be negative16 , this term is always positive
for plausible bankruptcy cost α ∈ [0.1, 0.5]17 according to our numerical analysis. Thisimplies that debt holders cannot recover the face value of debt by selling assets. Thus,
both equity and debt holders suﬀer losses due to asymmetric information. In Section 4.1,
we will see how equity and debt holders share the informational cost in Case (IV).

4

Economic implications

4.1

Comparative statics

We set the baseline parameter values in Table 1. We report the baseline results in Table
2 for the parameter values. Recall that the superscript ∗ denotes symmetric information,
while the superscript ∗∗ denotes asymmetric information. Under symmetric information,
16∗∗
This leads to the abnormal result that DL(x) > C/r, which means that the debt value becomes the facevalue plus the expected gain due to default. Nishihara and Shibata (2017) show that the abnormal result canhold for plausible parameter values with asymmetric information between managers and shareholders.
17
Recall that a fraction α ∈ (0, 1) of the ﬁrm’s asset value is lost at the time of default.

14the low-cost ﬁrm prefers to sell out at xL,1 = 1.03, whereas the high-cost ﬁrm prefers todefault at xH,2 = 1.28. Naturally, the low-cost ﬁrm’s equity and debt values are higher∗ (x) is equal to the facethan those of the high-cost ﬁrm. The low-cost ﬁrm’s debt value DL

value C/r = 16.667, whereas the high-cost ﬁrm’s debt value is discounted due to default
risk.
Under asymmetric information, Case (III) holds. The low-cost ﬁrm lowers the sales
trigger from xL,1 = 1.03 to xsL = 0.851 to signal asset quality to outsiders. Shareholders
pay the signaling cost EL∗ (x) − EL∗∗ (x) = 0.136, whereas debt holders pay no cost due toasymmetric information. Notably, the informational cost is quite low, that is, (EL∗ (x) −EL∗∗ (x))/EL∗ (x) = 0.00775, although the impact on the sales trigger is large, that is,(xL,1 − xsL )/xL,1 = 0.174.

4.1.1

Eﬀects of coupon C

Figure 2 shows the equity and debt values, as well as the informational costs and bankruptcy
triggers with varying levels of coupon C. Case (III) holds for C ≤ 1.04. In this regionsof the top right panel, we have x∗L = xL,1 = 1.03 and x∗∗
L = xL . Under asymmetric

information, the low-cost ﬁrm sells out at the sales trigger xsL (< xL,1 = 1.03) to signal
the ﬁrm’s type to outsiders. The high-cost ﬁrm chooses to sell out for C ≤ 0.94, while
it chooses to default for C > 0.94. In the top right panel, the high-cost ﬁrm’s change inbankruptcy choice generates a kink of x∗∗
L at C = 0.94 through ICC.
Case (IV) holds for C ∈ (1.04, 1.06]. In this region of the top right panel, we have∗∗sx∗L = xL,1 = 1.03 and x∗∗
L = xL,2 , and hence xL jumps from xL to xL,2 at C = 1.04.
The debt value DL∗∗ (x) also jumps from C/r to DL,2 (x) at C = 1.04. The low-cost ﬁrmchanges its bankruptcy choice from selling out to liquidation bankruptcy with asymmetric
information. From another viewpoint, the maximum risk-free debt level decreases fromC = 1.06 to C = 1.04 due to asymmetric information. When we relate the maximum
risk-free debt level to debt capacity, the result is consistent with the stylized fact thatstronger asymmetric information (e.g., a higher ratio of intangible assets to tangible assets)
decreases the debt capacity.
Case (I) holds for C > 1.06. In this region of the top right panel, we have x∗L = x∗∗
L =∗ (x) also jumps
xL,2 , and hence x∗L jumps from xL,1 to xL,2 at C = 1.06. The debt value DL

from C/r to DL,2 (x) at C = 1.06. In this case, the low-cost ﬁrm’s ﬁrst-best bankruptcy
choice is default, and asymmetric information does not aﬀect the bankruptcy procedure.
For C ∈ (1.06, 1.2] in the top right panel, x∗L = x∗∗
L = xL,2 is less than xL,1 = 1.03,
which means liquidation bankruptcy. Note that we have a region of operating concern
bankruptcy for much higher C.
We can see a novel result in the top right panel. Indeed, the sales trigger xsL increases
in C ∈ (0.94, 1.04] in Case (III). This result is not found in the existing literature. The
standard literature (e.g., Mella-Barral and Perraudin (1997)) shows that as in (5), the15sales timing is independent of C because the ﬁrm repays the face value of debt at the timeof sales. With asymmetric information, however, C inﬂuences the sales timing through
ICC. The mechanism is explained below. For C ∈ (0.94, 1.04], the high-cost ﬁrm prefers
to default in the ﬁrst-best case. While a higher C increases the right-hand side of (15),it does not change the left-hand side of (15). Then, a higher C increases xsL by (15). In
other words, the low-cost ﬁrm can increase xsL because a higher C decreases the incentive
for the high-cost ﬁrm to imitate the low-cost ﬁrm’s sales timing.
Although Lambrecht and Myers (2008) also show that a higher C speeds up closure
without default, the mechanism is quite diﬀerent from ours. They focus on managershareholder conﬂicts rather than asymmetric information between insiders and outsiders.
In their model, managers receive a fraction of the cash ﬂows until liquidation, and due
to managerial rents, managers have an incentive to delay liquidation. A higher C speeds
up the closure because a higher C decreases managerial rents and the incentive to delay
closure.
We also ﬁnd some interesting results in the bottom right panel. Informational costEL∗ (x) − EL∗∗ (x) decreases in C in Cases (III) and (IV) because a higher C decreases
the signaling cost. Recall that under symmetric information, the ﬁrm value in the sales
case agrees with the maximum value, which is the unlevered ﬁrm value. On the other
hand, under asymmetric information, the ﬁrm value with C = 0.94 is the highest because
informational cost EL∗ (x) − EL∗∗ (x) decreases in C in Cases (III). As we show in Corollary
1, in Case (IV), debt holders also pay an informational cost. Notably, we can see from∗ (x) − D ∗∗ (x) is much higher than E ∗ (x) − E ∗∗ (x) inthe bottom right panel that DL
L
L
L

Case (IV). This suggests that debt holders rather than shareholders suﬀer severe losses
when asymmetric information triggers liquidation bankruptcy that does not occur undersymmetric information.

4.1.2

Eﬀects of scrap value θFigure 3 shows the equity and debt values, as well as the informational costs and bankruptcy
triggers with varying levels of scrap value θ.

Case (I) holds for θ < 12.32, where∗x∗L = x∗∗
L = xL,2 = 0.75 holds. Case (IV) holds for θ ∈ (12.32, 12.52], where xL = xL,1∗∗∗s
and x∗∗
L = xL,2 = 0.75 hold. Case (III) holds for θ > 12.52, where xL = xL,1 and xL = xL

hold. As in Figure 2, we have a kink at θ = 13.66 because the high-cost ﬁrm chooses todefault for θ ≤ 13.66 and to sell out for θ > 13.66. For θ ∈ [12.52, 15], xL,2 is less thanxL,1 , which indicates liquidation bankruptcy.
We ﬁnd a novel result in the top right panel. The sales trigger xsL decreases in θ ∈(12, 52, 13.66], meaning that a higher scrap value θ delays the sales timing. This result is
novel and is not reported in the existing literature. Indeed, this is opposite to the standard
result (cf. Mella-Barral and Perraudin (1997) and Lambrecht and Myers (2008)) that ahigher scrap value θ accelerates the sales timing. We can explain the mechanism though

16

ICC as follows. For θ ∈ (12, 52, 13.66], the high-cost ﬁrm prefers to default in the ﬁrst-best
case; hence, its ﬁrst-best equity value is independent of θ. While the right-hand side of(15) is constant, the left-hand side of (15), that is, the imitation value, increases in θ.
Then, the sales trigger xsL decreases in θ. In other words, the low-cost ﬁrm lowers xsL
because a higher θ increases the incentive for the high-cost ﬁrm to imitate the low-cost
ﬁrm’s sales timing. Note that for θ > 13.66, both the left- and right-hand sides of (15)increase in θ, and xsL straightforwardly increases in θ.
The bottom right panel indicates several results. Informational cost EL∗ (x) − EL∗∗ (x)increases in θ in Cases (III) and (IV). Only in Case (IV), debt holders also pay an infor∗ (x) − D ∗∗ (x) is much higher
mational cost. As in the bottom right panel of Figure 2, DL
Lthan EL∗ (x) − EL∗∗ (x) in Case (IV). The impacts of parameters a and b are similar to thoseof θ, and we omit their depiction.

4.1.3

Eﬀects of cash ﬂow volatility σ

Figure 4 shows the equity and debt values, as well as the informational costs and bankruptcy
triggers with varying levels of volatility σ. Case (III) holds for σ ≤ 0.223, where x∗L = xL,1s
and x∗∗
L = xL hold. For σ ∈ [0.1, 0.223], xL,2 is below xL,1 , which indicates liquidation

bankruptcy. As in Figures 2 and 3, we have a kink at σ = 0.175 because the high-cost
ﬁrm chooses to sell out for σ ≤ 0.175 and to default for σ > 0.175. Case (IV) holds forσ ∈ (0.223, 0.236], where x∗L = xL,1 and x∗∗
L = xL,2 hold. Case (I) holds for σ > 0.236,
where x∗L = x∗∗
L = xL,2 holds.
In the top panels, regardless of whether the information is symmetric or asymmetric, we can see that a higher σ increases the option value of bankruptcy and decreases
the bankruptcy trigger. This aligns with the standard volatility eﬀects (e.g., Dixit andPindyck (1994)) that a higher σ increases the option value of waiting and delays the exercise of the option. In the left panels, we ﬁnd that a higher σ causes asset substitution
from debt holders to shareholders. This result is consistent with the standard result (e.g.,
Jensen and Meckling (1976) and Shibata and Nishihara (2010)).
In the bottom right panel, we can see more interesting results. Indeed, EL∗ (x) − EL∗∗ (x)increases in σ ∈ [0.1, 0.175] and decreases in σ ∈ [0.175, 0.223] in Case (III). We can explain
the non-monotonic result by the option convexity. As we explain in the last of Section
3.1, due to the option convexity, a higher σ increases the option value of default morethan the option value of selling out. For σ ∈ [0.175, 0.223], the high-cost ﬁrm chooses todefault under symmetric information, and hence, a higher σ increases the ﬁrst-best value
more than the imitation value. Thus, a higher σ mitigates ICC for σ ∈ [0.175, 0.223]. On
the other hand, for σ ∈ [0.1, 0.175], the high-cost ﬁrm chooses to sell out under symmetric
information. Due to the size eﬀect, a higher σ increases the imitation value more thanthe ﬁrst-best value. Thus, a higher σ tightens ICC for σ ∈ [0.1, 0.175]. We can also seethat as in the bottom right panels of Figures 2 and 3, DL∗ (x) − DL∗∗ (x) is much higher

17than EL∗ (x) − EL∗∗ (x) in Case (IV), that is, debt holders suﬀer much greater losses due tothe distortion in the bankruptcy choice than shareholders do.

4.1.4

Eﬀects of bankruptcy cost α

As seen in the bottom right panels of Figures 2–4, we ﬁnd that in Case (IV), debt holders
pay rather high informational costs compared to equity holders. We wonder if bankruptcy
cost α = 0.3 causes this result because the post-default asset value falls to (1 − α) times
the pre-default value. In this subsection, we investigate the eﬀects of α on the debt value
and informational costs. Figure 5 shows the debt values and informational costs withvarying levels of bankruptcy cost α. To focus on Case (IV), we set θ = 12.5. We set theother parameter values besides α and θ in Table 1. We omit equity values and bankruptcy
triggers, which are independent of α.
∗ (x)−D ∗∗ (x) > E ∗ (x)−E ∗∗ (x) holds, even for α = 0,
In the right panel, we ﬁnd that DL
L
L
L∗ (x) − D ∗∗ (x) increases linearly with α. For a realistic α ∈ [0.1, 0.5], D ∗ (x) −and that DL
L
LDL∗∗ (x) is much higher than EL∗ (x)−EL∗∗ (x). In conclusion, we argue that debt holders pay
much higher informational costs than shareholders when under asymmetric information,
shareholders change the bankruptcy choice from selling out to default. Although weassume that shareholders cannot directly transmit asset quality to outsiders, we now
suppose that they can inform outsiders of asset quality with a transmission cost. In sucha case, shareholders transmit asset quality and sell the ﬁrm to outsiders if and only if thetransmission cost is lower than EL∗ (x) − EL∗∗ (x). However, shareholders do not care about∗ (x) − D ∗∗ (x). Then, shareholders can greatly
the debt holders’ informational cost DL
Ldamage debt holders by choosing default in their self-interest.
In this study, we do not consider renegotiation between equity and debt holders. In
reality, however, debt holders may negotiate with shareholders to sell the ﬁrm at trigger
xL,2 without formal bankruptcy to save the bankruptcy costs associated with ownershipchanges. In such a case, debt holders may be able to decrease the informational cost to∗∗ (x) with α = 0, but they are likely to pay a renegotiation cost.
DL∗ (x) − DL

4.2

Testable implications and related empirical ﬁndings

Our analysis in the model yields several new predictions. We summarize them here.
1. Firms sell out later to signal asset quality to outsiders.
2. Firms with more debt can incur lower signaling costs and sell out earlier.
3. Firms with higher asset values can incur higher signaling costs and sell out later.
4. Asymmetric information can change the bankruptcy choice from selling out to liquidation bankruptcy.
5. In liquidation bankruptcy triggered by asymmetric information, debt holders suﬀer
severe losses, while shareholders suﬀer limited losses.

18

Predictions 1–3 discuss asset sales timing, while predictions 4 and 5 discuss liquidation bankruptcy stemming from asymmetric information. Predictions 2 and 3 have notbeen tested in prior studies to date. Adelino, Gerardi, and Hartman-Glaser (2017) ﬁnd
a positive relation between time-to-sale and mortgage performance. Although they examine privately-securitized mortgages instead of distressed ﬁrms, their empirical evidence
supports prediction 1, that is, informed sellers can signal quality and obtain higher prices
by delaying sales.
We can also ﬁnd empirical evidence consistent with predictions 4 and 5. For example, Hotchkiss and Mooradian (1998), Stromberg (2000), and Thorburn (2000) showthat distressed ﬁrms are likely to be acquired by well-informed ﬁrms, including former
owners. This is in line with prediction 4 that the ﬁrm is more likely to sell out under symmetric information than under asymmetric information. Furthermore, Stromberg (2000)
shows that asset sales to industry outsiders decrease sales prices, while Thorburn (2000)
demonstrates that debt holders recover more when former owners buy back ﬁrms. Theseﬁndings support predictions 3 and 4 that asymmetric information can lead to liquidation
bankruptcy, where the ﬁrm sells the assets at depressed prices, and debt holders suﬀer
severe losses. On the other hand, the empirical literature has not tested how equity- anddebt holders share informational costs due to asymmetric information.

5

Conclusion

In this study, we examined corporate bankruptcy decisions in a contingent claim model.
We reveal how asymmetric information about asset quality between a ﬁrm’s insiders andoutsiders aﬀects the bankruptcy choice between selling out and default, their timing,and the debt and equity values. This paper contributes to the literature on dynamic
bankruptcy models, the asset sales of distressed ﬁrms, and dynamic signaling games byshowing the following novel results.
The low-cost ﬁrm can delay the sales timing to signal asset quality to outsiders. This
result suggests that distressed ﬁrms can potentially avoid ﬁre sales by delaying asset sales.
In contrast to the standard results, more debt and lower asset value can accelerate thelow-cost ﬁrm’s asset sales timing because they reduce the high-cost ﬁrm’s incentive toimitate the low-cost ﬁrm. When the signaling cost in asset sales is higher than the direct
cost, that is, the asset value minus the face value of debt, the low-cost ﬁrm changes thebankruptcy choice from selling out to liquidation bankruptcy, which greatly lowers theﬁrm value. In this case, debt holders suﬀer severe losses, although shareholders suﬀer
limited losses. The existing literature does not report such results, and they can accountfor several empirical ﬁndings.

19

A

Proof of Proposition 1

By the ﬁrst order condition in (2), we have (5) and (6). Similarly, by the ﬁrst ordercondition in (3), we have (7) and (8). The inequality (6) ≥ (8) is equivalent to
(
) (
) (
) (
)
xi,2
1 γ (a − 1)xi,1 (1 − b)wi
1 γ
wi + C
+
+θ ≥
−
+
xi,1
r−µ
r
xi,2
r−µ
r
(
)γ (
) (
)γ
1−a
(1 − b)wi1
wi + C⇔+θ ≥(1 − b)wi + rθr
wi + Cr⇔(4).
Suppose that (4) does not hold. By the discussion above, the ﬁrm chooses to default
at the default trigger xi,2 deﬁned by (7). Now, we can derive the debt value at the default
trigger xi,2 as follows:)(∫ T
aX(T ) bwi−rt−rT
sup Exi,2 [−+θ ]e (1 − α)(X(t) − wi )dt + e(1 − α)
r−µ
r
T
0
(
(
))
xi,2
wi
(a − 1)X(T ) (1 − b)wi−rT=(1 − α)−+ sup E[e
+
+θ ]
r−µ
r
r−µ
r
T
(
))
( x )γ ( (a − 1)x̃ (1 − b)w
xi,2
wi
i,2
i
−
+ sup
+
+θ
=(1 − α)
r−µ
r
x̃
r
−
µ
r
x̃(≤xi,2 )
(
)

axi,2
bwi


(1
−
α)
−
+
θ
(xi,2 ≤ xi,1 )

r−µ
r
(
(
) (
))
=

xi,2
xi,2 γ (a − 1)xi,1 (1 − b)wiwi (1 − α)−+++θ(xi,2 > xi,1 ),
r−µ
r
xi,1
r−µ
r
(18)
where in (18), xi,1 and xi,2 are deﬁned by (5) and (7), respectively. Equation (18) meansthat the former debt holders sell the total assets instantly after taking over the ﬁrm if
and only if xi,2 ≤ xi,1 holds. By (5) and (7), we have
xi,2
wi + C= (1 − a).
xi,1(1 − b)wi + rθ(19)

By (19) and the upper equation in (18), we can derive the debt values (10) if (9) holds.
By (19) and the lower equation in (18), we can derive the debt values (11) if (9) does nothold. The proof is complete.

B

The proof of Di,2(x) < C/r under symmetric in-formation.
Because α > 0, we have
Ei,1 (x) + Di,1 (x) ≥ Ei,2 (x) + Di,2 (x).

20

(20)

By substituting Di,1 (x) = C/r into (20), we have
Di,2 (x) ≤ Ei,1 (x) − Ei,2 (x) + C/r.

(21)

Suppose that (4) does not hold, that is, the ﬁrm chooses to default. In this case, we have
Ei,1 (x) < Ei,2 (x), and hence, by (21), we have Di,2 (x) < C/r.

C

Proof of Proposition 3.

Case (II): xL,1 is the solution to the unconstrained problem (2). When xL,1 satisﬁes(14), xL,1 is the solution to problem (13) subject to (14).
Case (III): In the binding case, the optimal solution to problem (13) subject to (14) isequal to (14). By equating (14), we obtain two candidates xsL ∈ (0, xL,1 ) and x̃sL (> xL,1 )for the optimal solution, where( )γ ()(a − 1)xsL wH − bwL
1
+
+θ
(22)
xsL
r−µ
r
( )γ ()(a − 1)x̃sL wH − bwL
1
=
+
+
θ
(23)
x̃sL
r−µ
r
(
)γ (
) (
)γ (
)
(a − 1)xH,1 (1 − b)wH
xH,2
1
1
wH + C= max{
+
+θ ,
−
+
}
xH,1
r−µ
r
xH,2
r−µ
r
holds. Using (22) = (23), we have( )γ ()(a − 1)xsL (1 − b)wL
1
+
+θ
xs
r−µ
r
( L )γ () ( )γ(a − 1)xsL wH − bwL
1
1
wH − wL
=
+
+θ −
s
s
x
r−µ
r
x
r
( L )γ () ( L )γs(a − 1)x̃L wH − bwL
1
1
wH − wL
=
+
+θ −
s
s
x̃
r−µ
r
x
r
( L )γ () ( L )γs(a − 1)x̃L wH − bwL
1
1
wH − wL
>
+
+θ −
s
s
x̃
r−µ
r
x̃L
r
( L )γ ()
s(a − 1)x̃L (1 − b)wL
1
=
+
+θ ,
x̃sL
r−µ
r

(24)

(25)
(26)

where we use xsL < x̃sL and γ < 0 in (25). Because (24) > (26), the objective value for xsL
in the constrained problem (13) is higher than the objective value for x̃sL . Then, xsL is the
optimal solution.
By (16), we have( )γ ()(a − 1)xsL (1 − b)wLx
wL + C
x
−
+
+
+θ
r−µ
r
xsL
r−µ
r
(
) (
)
xL,2
x
wL + Cx γ
wL + C
≥
−
+
−
+
r−µ
r
xL,2
r−µ
r

ELs (x) == EL,2 (x).

21

Then, shareholders prefer to sell out at the trigger xsL . In the sales case, the debt isriskless.
Case (IV): When (16) does not hold, we have ELs (x) < EL,2 (x). Then, shareholders
prefer to default at the trigger xL,2 . Because (12) does not hold in Case (IV), we havewL + C≤(1 − b)wL + rθ

(

1
1−a

) −γ

1−γ

≤

1
,
1−a

(27)

which leads to xL,2 ≤ xL,1 (liquidation bankruptcy). Then, we have the equity and debt
values, EL,2 (x) and DL,2 (x) deﬁned by (8) and (10), respectively. The proof is complete.

References
Adelino, M., K. Gerardi, and B. Hartman-Glaser, 2017, Are lemons sold ﬁrst? Dynamic
signaling in the mortgage market, Discussion Paper, American Finance Association
Annual Meeting.
Akerlof, G., 1970, The market for lemons: Quality uncertainty and the market mechanism,
Quarterly Journal of Economics 84, 488–500.
Black, F., and J. Cox, 1976, Valuing corporate securities: Some eﬀects of bond indenture
provisions, Journal of Finance 31, 351–367.
Bris, A., I. Welch, and N. Zhu, 2006, The costs of bankruptcy: Chapter 7 liquidation
versus chapter 11 reorganization, Journal of Finance 61, 1253–1303.
Broadie, M., M. Chernov, and S. Sundaresan, 2007, Optimal debt and equity values in
the presence of chapter 7 and chapter 11, Journal of Finance 62, 1341–1377.
Bustamante, M., 2012, The dynamics of going public, Review of Finance 16, 577–618.
Cho, I., and D. Kreps, 1987, Signaling games and stable equilibria, Quarterly Journal of
Economics 102, 179–221.
Christensen, P., C. Flor, D. Lando, and K. Miltersen, 2014, Dynamic capital structurewith callable debt and debt renegotiations, Journal of Corporate Finance 29, 644–661.
Correia, R., and J. Poblacion, 2015, A structural model with explicit distress, Journal of
Banking and Finance 58, 112–130.
Daley, B., and B. Green, 2012, Waiting for news in the market for lemons, Econometrica
80, 1433–1504.
Décamps, J., T. Mariotti, and S. Villeneuve, 2006, Irreversible investment in alternative
projects, Economic Theory 28, 425–448.
Dixit, A., and R. Pindyck, 1994, Investment Under Uncertainty (Princeton University
Press: Princeton).

22

Eckbo, B., and K. Thorburn, 2008, Automatic bankruptcy auctions and ﬁre-sales, Journalof Financial Economics 89, 404–422.
Fan, H., and S. Sundaresan, 2000, Debt valuation, renegotiation, and optimal dividend
policy, Review of Financial Studies 13, 1057–1099.
Francois, P., and E. Morellec, 2004, Capital structure and asset prices: Some eﬀects ofbankruptcy procedures, Journal of Business 77, 387–411.
Fuchs, W., B. Green, and D. Papanikolaou, 2016, Adverse selection, slow-moving capital,
and misallocation, Journal of Financial Economics 120, 286–308.
Fuchs, W., and A. Skrzypacz, 2013, Costs and beneﬁts of dynamic trading in a lemonsmarket, Discussion Paper, Stanford University.
Gilson, S., E. Hotchkiss, and R. Ruback, 2000, Valuation of bankrupt ﬁrms, Review of
Financial Studies 13, 43–74.
Goldstein, R., N. Ju, and H. Leland, 2001, An EBIT-based model of dynamic capital
structure, Journal of Business 74, 483–512.
Grenadier, S., and A. Malenko, 2011, Real options signaling games with applications tocorporate ﬁnance, Review of Financial Studies 24, 3993–4036.
Gryglewicz, S., 2011, A theory of corporate ﬁnancial decisions with liquidity and solvency
concerns, Journal of Financial Economics 99, 365–384.
Hotchkiss, E., K. John, R. Mooradian, and K. Thorburn, 2008, Bankruptcy and theresolution of ﬁnancial distress, Available at SSRN: 1086942.
Hotchkiss, E., and R. Mooradian, 1998, Acquisitions as a means of restructuring ﬁrms inchapter 11, Journal of Financial Intermediation 7, 240–262.
Janssen, M., and S. Roy, 2002, Dynamic trading in a durable good market with asymmetric
information, International Economic Review 43, 257–282.
Jensen, M., and W. Meckling, 1976, Theory of the ﬁrm: Managerial behavior, agency
costs and ownership structure, Journal of Financial Economics 3, 305–360.
Kort, P., P. Murto, and G. Pawlina, 2010, Uncertainty and stepwise investment, European
Journal of Operational Research 202, 196–203.
Lambrecht, B., 2001, The impact of debt ﬁnancing on entry and exit in a duopoly, Review
of Financial Studies 14, 765–804.
Lambrecht, B., and S. Myers, 2008, Debt and managerial rents in a real-options model ofthe ﬁrm, Journal of Financial Economics 89, 209–231.
Leland, H., 1994, Corporate debt value, bond covenants, and optimal capital structure,
Journal of Finance 49, 1213–1252.
Maksimovic, V., and G. Phillips, 1998, Asset eﬃciency and reallocation decisions ofbankrupt ﬁrms, Journal of Finance 53, 1495–1532.

23

Mella-Barral, P., 1999, The dynamics of default and debt reorganization, Review of Financial Studies 12, 535–578.
Mella-Barral, P., and W. Perraudin, 1997, Strategic debt service, Journal of Finance 52,
531–556.
Moraux, F., and F. Silaghi, 2014, Inside debt renegotiation: Optimal debt reduction,
timing and the number of rounds, Journal of Corporate Finance 27, 269–295.
Morellec, E., and N. Schürhoﬀ, 2011, Corporate investment and ﬁnancing under asymmetric information, Journal of Financial Economics 99, 262–288.
Nishihara, M., and T. Shibata, 2016, Asset sale, debt restructuring, and liquidation,
Journal of Economic Dynamics and Control 67, 73–92.
Nishihara, M., and T. Shibata, 2017, Default and liquidation timing under asymmetric
information, European Journal of Operational Research 263, 321–336.
Povel, P., and R. Singh, 2006, Sale-backs in bankruptcy, Journal of Law, Economics, andOrganization 23, 710–730.
Pulvino, T., 1998, Do asset ﬁre sales exist? An empirical investigation of commercial
aircraft transactions, Journal of Finance 53, 939–978.
Shibata, T., and M. Nishihara, 2010, Dynamic investment and capital structure undermanager-shareholder conﬂict, Journal of Economic Dynamics and Control 34, 158–
178.
Shibata, T., and M. Nishihara, 2015, Investment-based ﬁnancing constraints and debt
renegotiation, Journal of Banking and Finance 51, 79–92.
Shleifer, A., and R. Vishny, 1992, Liquidation values and debt capacity: A market equilibrium approach, Journal of Finance 47, 1343–1366.
Stromberg, P., 2000, Conﬂicts of interest and market illiquidity in bankruptcy auctions:
Theory and tests, Journal of Finance 55, 2461–2492.
Sundaresan, S., and N. Wang, 2007, Investment under uncertainty with strategic debt
service, American Economic Review, Papers & Proceedings 97, 256–261.
Thorburn, K., 2000, Bankruptcy auctions: Costs, debt recovery, and ﬁrm survival, Journalof Financial Economics 58, 337–368.

24

Table 1: Baseline parameter values.
r

µ

σ

x

wH

wL

q

a

b

θ

α

C

0.06 0.01 0.2

2

1

0.5 0.5

0.5

0.5 130.3

1

Table 2: Baseline results.
∗EL∗∗ (x) EH(x) DL∗ (x)

xsL

xH,2

EL∗ (x)1.03 0.851

1.28

17.538xL,1EL,1(x)17.402

11.381 16.667

DL∗∗ (x)∗DH(x)

16.66714.404

18

s
EL(x)
17

16
Low−cost equity value
High−cost equity value
by immitation

15
14
13
12
EH,2(x)
11
10
9
0.5

0.6

0.7

0.8 xs 0.9
L

1 xL,1 1.1
Sales trigger

1.2

1.3

1.4

1.5

Figure 1: The low-cost firm’s equity value and high-cost firm’s imitation value as functions of
the sales trigger in Case (III) of Proposition 3. The figure illustrates how to determine the sales
trigger xsL and the equity value ELs (x) in the separating equilibrium. The parameter values are
set in Table 1, where max{EH,1 (x), EH,2 (x)} = EH,2 (x) holds.

25

22

1.1
E∗

L

20
18

0.9

16

0.8

14
0.8

0.9

1
C

20

D∗L

18

D∗∗
L

∗

1

E∗∗
L

1.1

0.7
0.8

1.2

xL
x∗∗
L

0.9

1.5

1
C

1.1

1.2

1.1

1.2

E∗L−E∗∗L

C/r∗

DL−D∗∗

1

L

16
0.5
14
12
0.8

0.9

1
C

1.1

0
0.8

1.2

0.9

1
C

Figure 2: Comparative statics with respect to C. The other parameter values are set in Table
1. Cases (III), (IV), and (I) hold for C ≤ 1.04, C ∈ (1.04, 1.06], and C > 1.06, respectively.
1.2
E∗L
E∗∗
L

18.2
18

xL

1

x∗∗
L

17.8

0.9

17.6

0.8

17.4

0.7

17.2
10

11

12

13

14

15

∗

1.1

10

11

12

θ

13

14

17

1.5
E∗L−E∗∗

16.5

L∗
DL−D∗∗
L

1

16

D∗

L

15.5

D∗∗

0.5

L

15
14.5
10

15

θ

C/r
11

12

13

14

0
10

15

θ

11

12

13

14

15

θ

Figure 3: Comparative statics with respect to θ. The other parameter values are set in Table
1. Cases (I), (IV), and (III) hold for θ < 12.32, θ ∈ (12.32, 12.52], and θ > 12.52, respectively.

26

20

1.4

∗

EL

19

1.2

E∗∗
L

18

1

17

0.8

16

0.6

15
0.1

0.4
0.1

∗

xL
x∗∗
L

0.15

0.2
σ

0.25

0.3

17

0.15

2

0.2
σ

0.25

0.3

0.25

0.3

E∗L−E∗∗
L

1.5

16

D∗L

L

L

1

D∗∗

15

D∗ −D∗∗

L

0.5

C/r
14
0.1

0.15

0.2
σ

0.25

0
0.1

0.3

0.15

0.2
σ

Figure 4: Comparative statics with respect to σ. The other parameter values are set in Table
1. Cases (III), (IV), and (I) hold for σ ≤ 0.223, σ ∈ (0.223, 0.236], and σ > 0.236, respectively.
17

2.5
∗

EL−E∗∗
L

16.5

2

16

1.5

15.5

1

D∗L

15

D∗L−D∗∗
L

0.5

D∗∗
L

C/r
14.5

0

0.1

0.2

0.3

0

0.4

α

0

0.1

0.2

0.3

0.4

α

Figure 5: Comparative statics with respect to α in Case (IV) of Proposition 3. To focus on
Case (IV), we set θ = 12.5. The other parameter values are set in Table 1.

27