Discussion Papers In Economics
And Business

Liquidation, fire sales, and acquirers’ private information

Michi NISHIHARA, Takashi SHIBATA

Discussion Paper 18-25

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANLiquidation, fire sales, and acquirers’ private information

Michi NISHIHARA, Takashi SHIBATA

Discussion Paper 18-25

August 2018

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANLiquidation, fire sales, and acquirers’ private information∗
Michi NISHIHARA†,

Takashi SHIBATA‡§

Abstract
We develop a dynamic model in which a distressed firm optimizes an exit choice between sell-out and default as well as its timing. We assume that the distressed firm is not
informed about the acquirer’s asset valuation. We show that the firm delays liquidation
to decrease the acquirer’s information rent. Notably, the firm can change the exit choicefrom sell-out to default when the screening cost is high. In this case, shareholders declaredefault regardless of the acquirer’s valuation, which provides the acquirer the maximum
information rent. Together with bankruptcy costs, the maximal information rent lowers
the sales price and debt recovery. This mechanism can explain many empirical findings
about fire sales and acquirers’ excess gains. Higher volatility, leverage, and asymmetric
information increase the likelihood of a fire sale, but higher bankruptcy costs could play apositive role in preventing a fire sale. With asymmetric information, the firm can reduce
debt issuance to avoid the risk of a fire sale.
JEL Classifications Code: D82; G13; G33.
Keywords: real options; screening game; fire sale; M&A; intertemporal price discrimination.

∗

Version on 7 August, 2018. The authors thank Makoto Goto and Yuan Tian for helpful comments. The

authors also thank the participants at the 23rd EBES Conference in Madrid, IRMBAM 2018 in Nice, EURO2018 in Valencia, and the seminars at Kyoto University and Osaka University for helpful feedback. This work
was supported by the JSPS KAKENHI (Grant numbers JP17K01254, 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§
Statistical Laboratory, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK

1

1Introduction

Since the seminal works by Black and Cox (1976), Leland (1994), Mella-Barral and Perraudin (1997), and Fan and Sundaresan (2000), a number of studies have investigatedcorporate bankruptcy decisions in continuous-time models. For instance, Francois andMorellec (2004), Broadie, Chernov, and Sundaresan (2007), and Antill and Grenadier(2017) investigate dynamic bankruptcy procedures of Chapter 11, while Moraux andSilaghi (2014), Christensen, Flor, Lando, and Miltersen (2014), and Silaghi (2018) investigate debt restructuring processes. He and Xiong (2012), He and Milbradt (2014),
and Chen, Cui, He, and Milbradt (2018) reveal the interactions between debt market
liquidity and bankruptcy timing via rollover risks. whereas Lambrecht and Myers (2008)
and Nishihara and Shibata (2017) focus on conflicts among managers, shareholders, anddebt holders concerning liquidation timing.
During the liquidation process, firms are frequently acquired by other parties as aform of exit (e.g., Hotchkiss and Mooradian (1998) and Maksimovic and Phillips (1998)).
However, distressed firms’ assets tend to be sold at lower prices than in normal times; thisphenomenon is called a fire sale. Since the seminal work by Shleifer and Vishny (1992),
who propose the fire sale mechanism through industry-wide financial distress, a number of papers, including Pulvino (1998), Hotchkiss and Mooradian (1998), and Acharya,
Bharath, and Srinivasan (2007), have shown empirical evidence of fire sale discounts.
Despite numerous studies on continuous-time bankruptcy models and fire sales, to ourknowledge, no paper has investigated fire sale discounts in a dynamic liquidation model.
In this paper, we study a fire sale by incorporating an acquirer’s private valuation ofa distressed firm into a dynamic liquidation model. When a distressed firm sells assets
to an outsider, as was discussed in the merger and acquisition (M&A) literature (e.g.,
Chemmanur, Paeglis, and Simonyan (2009)), asymmetric information between the seller
and acquirer matters in the transaction. We investigate the eﬀects of the acquirer’s
private information on the dynamic liquidation decision and reveal the fire sale mechanism
through the asymmetric information channel.1
We develop a dynamic model based on the standard setup in Mella-Barral and Perraudin (1997). Shareholders, who observe dynamic and stochastic changes in a distressed
firm’s cash flows, optimize an exit choice between sell-out and default as well as its timing.2 The model does not distinguish between shareholders and managers, assuming that1

Nishihara and Shibata (2018) show that a distressed firm’s private information about asset quality candelay asset sales to signal asset quality. Contrary to the previous paper, this paper assumes the counterparty’s
private information. This assumption is standard in the M&A literature. For instance, Gorbenko and Malenko(2018) and Cong (2016) develop M&A auction models involving bidders who have private asset valuation, butthey do not focus on the bankruptcy process. The assumption of buyers’ private valuation is also standard
in the literature on intertemporal price discrimination (e.g., Stokey (1979), Landsberger and Meilijson (1985),
Besbes and Lobel (2015), and Garrett (2016)).
2
Although Mella-Barral and Perraudin (1997) also examine renegotiation between equity and debt holders,2managers act in shareholders’ interests. Sell-out is a successful exit. Indeed, shareholders
sell all assets to the acquirer and obtain the residual value (i.e., the sales price minus theface 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 payments of debt, andthe former debt holders take over and liquidate the firm for the acquirer. In the case, afraction of the firm value is lost to bankruptcy costs.
In addition to the standard setup, we assume that the acquirer has private valuationof the distressed firm’s assets. To be more precise, the acquirer can take high-value
and low-value types. The distressed firm’s equity and debt holders cannot observe theacquirer’s type, although they know the ex-ante distribution of the acquirer’s types. Under
asymmetric information, a high-value type can imitate a low-value type and purchase thefirm at a low price. To prevent the imitation, shareholders can commit the oﬀer to theacquirer. By deriving the optimal oﬀer, we reveal that asymmetric information distorts
the exit choice as well as the equity, debt, firm, and acquirer’s values. The results are
summarized below.
The sales price for a high-value type is discounted from the fair price due to information rent. Shareholders of the distressed firm mitigate the acquirer’s information rent bydelaying sell-out or default timing for a low-value type. This can be regarded as a form ofintertemporal price discrimination, where Stokey (1979) and Landsberger and Meilijson(1985) show that under some conditions, a seller delays sales at low prices to discriminate between buyer types. Our result can be compared to that of Grenadier and Wang
(2005) and Shibata and Nishihara (2010), who study investment timing in the presenceof managers’ private information in real options models. They show that shareholders,
who delegate investment to managers, decrease a high-value manager’s information rent
by delaying a low-value manager’s investment. Unlike their papers, however, we focus on
the seller facing the acquirer’s private information and show that the seller decreases ahigh-value type’s information rent by delaying a low-value type’s acquisition timing.
Notably, shareholders can change the exit choice from sell-out to default if the screening
cost increases. When this happens for a high-value type, with asymmetric information,
shareholders abandon sell-out and declare default regardless of the acquirer’s type. Inthat case, debt holders cannot observe the acquirer’s type on liquidation bankruptcy;
therefore, a high-value type can purchase the firm at the maximal discount percentage.
Together with bankruptcy costs, the maximum information rent for the acquirer greatly
lowers the sales price. In other words, the acquirer’s private information leads to a wealthtransfer from debt holders to the acquirer through liquidation bankruptcy. Through theasymmetric information channel, this result can account for the mechanism of fire salediscounts.
This result is consistent with the empirical evidence of Meier and Servaes (2015).

we exclude the possibility of debt renegotiation to concentrate on the acquirer’s private information.

3

Indeed, they show that acquirers of assets in fire sale transactions earn more excess returns than in regular situations, which implies wealth transfers from sellers to acquirers
through fire sales. In addition, many empirical studies, including Hotchkiss and Mooradian (1998), Stromberg (2000), and Thorburn (2000), show that distressed firms are likely
to be acquired by industry insiders and that sales prices decrease for industry outsiders.
Compared to industry insiders’ valuation, asset valuation by industry outsiders, who havefewer business relationships with the target, is more opaque for the target. Therefore, ourasymmetric information results can explain these empirical findings.
Furthermore, our model predicts that higher volatility, leverage, and asymmetric information increase the likelihood of a fire sale, but higher bankruptcy costs could decrease
the likelihood of a fire sale. This counterintuitive impact of bankruptcy costs is explained
as follows. Higher bankruptcy costs decrease the liquidated asset value and then cut ahigh-value type’s incentive to imitate a low-value type and acquire the liquidated assets at
a low price. Although the counterintuitive result has not been tested by empirical studies,
it suggests the interesting interactions of bankruptcy costs and fire sales.
Our results are also consistent with empirical findings of premiums in M&As. Bargeron, Schlingemann, Stulz, and Zutter (2008) and Golubov and Xiong (2017) show thatprivate acquirers pay lower premiums in acquisitions than public acquirers. They also
show that higher acquirer opacity, which can stem from higher managerial ownership
and/or ownership concentration, tends to lead to lower premiums in acquisitions. Dueto fewer financial reporting requirements, private firms can conceal more private information from others than public firms. Higher opacity also increases acquirers’ private
information. Therefore, we can explain the empirical findings of acquisition premiums by
the asymmetric information channel. Although it has not been tested empirically, we also
show that discounts due to acquirers’ opacity deepen in financial distress combined withfire sales more than in regular situations.
Although we focus mainly on the liquidation process, we also explore the eﬀects ofasymmetric information on the optimal capital structure. Under asymmetric information,
the firm decreases the initial coupon, leverage, and credit spread below the corresponding
levels of the symmetric information case because the firm avoids a fire sale and alleviates
the acquirer’s information rent by taking the precautious level of debt. The relation
between asymmetric information and leverage is consistent with empirical findings of
Bharath, Pasquariello, and Wu (2009).
Because intangible asset valuation is more likely to vary by acquirer than tangible asset
valuation, a firm with more intangible assets can expect that potential acquirers have moreprivate asset valuation. Therefore, we predict that firms with less tangible assets tend
to have lower leverage ratios. Consistent with our prediction, Harris and Raviv (1991),
Rajan and Zingales (1995), and Frank and Goyal (2009) show the positive relation between
leverage and tangibility. Although the stylized argument is that lower tangibility decreases
collateral value and increases debt financing costs, we add the asymmetric information

4

channel, i.e., lower tangibility increases potential acquirers’ private information about
asset valuation and increases the risk of a fire sale.
Our contribution to the literature is four-fold. First, we complement the literature on
dynamic bankruptcy decisions (e.g., Leland (1994), Mella-Barral and Perraudin (1997),
Morellec (2001), and Gryglewicz (2011)) by revealing the eﬀects of the acquirer’s private
information. Second, we complement the literature on distressed firms’ asset fire sales(e.g., Shleifer and Vishny (1992), Maksimovic and Phillips (1998), Stromberg (2000),and Eckbo and Thorburn (2008)) by unveiling the novel interactions between fire sales
and acquirers’ private information. Third, we contribute to the literature on acquisition
premiums (e.g., Bargeron, Schlingemann, Stulz, and Zutter (2008), Massa and Xu (2013),
and Golubov and Xiong (2017)) by explaining private acquirers’ low premiums throughacquirers’ information rent. Last, we contribute to the literature on the real options
screening game (e.g., Grenadier and Wang (2005), Shibata and Nishihara (2010), andNishihara and Shibata (2017)) by analyzing target shareholders facing acquirers’ private
information rather than shareholders facing managers’ private information.
The remainder of this paper is organized as follows. We introduce the model setup in
Section 2. Sections 3.1 and 3.2 show the model solutions under symmetric and asymmetric
information, respectively. In Section 4, we provide economic implications and comparative
statics results. After Section 5 explains extensions and limitations, Section 6 concludes
the paper.

2

Model Setup

2.1

Firm until exit

The model builds on the standard setup of Mella-Barral and Perraudin (1997), Goldstein,Ju, and Leland (2001), 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. Although in Section 3.3, we discuss the optimal capital structure, in thebaseline model we exogenously assume coupon C to focus on a firm close to bankruptcy.
The firm receives continuous streams of earnings before interest and taxes (EBIT), X(t),where X(t) follows a geometric Brownian motiondX(t) = µX(t)dt + σX(t)dB(t) (t > 0),

X(0) = x,where B(t) denotes the standard Brownian motion defined in a filtered probability space(Ω, F, P, {Ft }) and µ, σ(> 0) and x(> 0) are constants. Assume that the initial value,
X(0) = x, is suﬃciently high to exclude the firm’s exit at the initial time. For convergence,
we assume that r > µ, where a positive constant r denotes the risk-free interest rate. We
assume that managers operate the firm in shareholders’ interests, and hence we do not5distinguish between shareholders and managers. Shareholders continue to receive cash
flows (1 − τ )(X(t) − C) until exit, where τ ∈ (0, 1) denotes a corporate tax rate.

2.2

Asset valuation by the acquirer

Consider a potential acquirer that is interested in purchasing the firm. As in Lambrecht(2001), Lambrecht and Myers (2008), and Nishihara and Shibata (2016), we assume thatthe acquirer evaluates the firm’s assets by
Pi (X(t)) =ai X(t)+θr−µ(i = L, H),(1)

where ai ∈ (0, 1) and θ > 0 are constants. In (1), the acquirer can take two types: i = H(high-value type) and i = L (low-value type), where ∆a = aH − aL > 0. The ex-ante
probability of a high-value type is q ∈ (0, 1).3 The asset value (1) can be interpreted asfollows. After partial assets are scrapped at a fixed price θ, the acquirer can perpetually
receive cash flows ai X(t) from the remaining assets, where ai ∈ (0, 1) means that the revenues contract due to the partial liquidation.4 The parameter ai reflects not only revenues
from the acquired business but also synergies in acquisition. A high-value type can more
eﬃciently utilize the assets than a low-value type; therefore, aH is higher than aL .
All information except the acquirer’s type is common to all agents. Section 3.1 examines the symmetric information model, in which all agents can observe the acquirer’s typei at the initial time, whereas Section 3.2 examines the asymmetric information model, in
which neither shareholders nor debt holders can verify the acquirer’s type i. In the latter,as in Gorbenko and Malenko (2018) and Cong (2016), the acquiring firm has private information on how to utilize assets and gain synergies in acquisition. For instance, private
information is great for an unlisted acquirer because of the low corporate transparency.
It is also greater when an acquirer (especially industry outsider) has fewer prior business
relationships with the target.5

2.3

Exit choice between sell-out and default

Following Mella-Barral and Perraudin (1997), Lambrecht and Myers (2008), Nishihara
and Shibata (2017), and Nishihara and Shibata (2018), we consider the following two
types of the firm’s exit. A successful exit without formal bankruptcy is called sell-out. In
3
4

The type is drawn and learned by the acquirer at the initial time.
If we take account of transaction costs and the acquirer’s gain in the transaction, the asset price decreases

to (1 − kT − kA )(ai X(t)/(r − µ) + θ), where kT (> 0) and kA (> 0) stand for the transaction costs and theacquirer’s return, respectively. In such a case, we have only to replace ai and θ with a′i = (1 − kT − kA )ai andθ′ = (1 − kT − kA )θ, respectively. The acquirer gains (1 − τ )kA (ai X(t)/(r − µ) + θ) by the transaction.
5
In this case, the seller is also likely to have private information about its own assets. This paper concentrates
on an acquirer’s private information, while Nishihara and Shibata (2018) reveal the eﬀects of a target’s private
information on bankruptcy procedure.

6the sell-out case, the distressed firm’s shareholders sell all assets to the acquirer. To be
more precise, the seller makes a take-it-or-leave-it oﬀer, and the acquirer accepts it if thetransaction value is non-negative for the acquirer.6 In Sections 3.1 and 3.2, we will showthat the oﬀer price is equal to (1) under symmetric information, while the sales pricecan be lower than (1) under asymmetric information. Following the absolute priority
rule (APR) of debt, debt holders are repaid the face value of debt, which equals C/rfor the console debt. Shareholders receive the residual value (i.e., the sales price minus
C/r). For simplicity, as in Lambrecht and Myers (2008), Nishihara and Shibata (2017),
and Nishihara and Shibata (2018), we assume no opportunity for debt renegotiation andrestructuring.7
The other exit is default causing a formal bankruptcy. In this 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 thefirm, while shareholders receive nothing. As in Lambrecht and Myers (2008), we focus
only on liquidation bankruptcy, in which former debt holders are forced to liquidate thefirm immediately after default. This assumption may approximate bankruptcy law insuch countries as Sweden (see Eckbo and Thorburn (2008)). In general, debt holders may
choose operating concern bankruptcy rather than liquidation bankruptcy by comparing
their costs. For example, Bris, Welch, and Zhu (2006) document that in the United States,
small and/or young firms tend to choose liquidation bankruptcy (chapter 7) rather thanreorganization (chapter 11). Therefore, our assumption is likely to hold for such firms. In
Section 5.3, we also discuss the results without this restriction.
On bankruptcy, debt holders instantly scrap or sell the firm to the acquirer. Following the standard literature (e.g., Leland (1994), Goldstein, Ju, and Leland (2001), andLambrecht and Myers (2008)), a fraction α ∈ (0, 1) of the firm’s asset value is lost to
bankruptcy costs (filing fees, attorney fees, etc.), which decreases the acquirer’s valuation
to (1 − α)Pi (X(t)). Note that as will be shown in Section 3.2, the sales price can be lower
than (1 − α)Pi (X(t)) under asymmetric information.
As in Mella-Barral and Perraudin (1997), Lambrecht and Myers (2008), Nishihara
and Shibata (2017), and Nishihara and Shibata (2018), shareholders of the distressed firmchoose between sell-out and default, its timing, and the oﬀer price in the sell-out case in
6

The model is more relevant to target-initiated deals than acquirer-initiated deals. Masulis and Simsir (2015)

show that targets in financial distress are likely to initiate transactions than in normal times.
7
Several papers, including Mella-Barral and Perraudin (1997), Sundaresan and Wang (2007), Christensen,
Flor, Lando, and Miltersen (2014), Nishihara and Shibata (2016), and Silaghi (2018), examine the debt renegotiation and restructuring process in dynamic bankruptcy models. It could be interesting for a future study toinvestigate the eﬀects of asymmetric information on the debt renegotiation and restructuring procedure. Several
papers, including Hackbarth, Hennessy, and Leland (2007), Morellec, Valta, and Zhdanov (2015), and Shibata
and Nishihara (2015), distinguish between private debt, which can be renegotiated, and market debt, whichcannot be renegotiated, investigating optimal financing and investment problems. Our model is more relevant
to firms with market debt than firms with private debt.

7

their own interests. They do not take into account of debt in place, which leads to agency
conflicts between equity and debt holders. Debt is priced at the initial time under therational expectation of shareholders’ liquidation policy.

3

Model Solutions

3.1

Symmetric information

As a benchmark, we solve the distress firm’s shareholders’ exit choice problem undersymmetric information. At the initial time, shareholders commit to a price schedule(sL , sH , xL , xH , pL , pH ) for the acquirer, where for the acquirer’s type i, si ∈ {0, 1} stands
for the indicator function of sell-out (i.e., si equals 1 in the sell-out case and 0 in thedefault case), while xi ∈ (0, x) and pi ∈ [0, ∞) stand for the exit threshold and the oﬀer
price, respectively. For si = 1 (sell-out), the acquirer with type i can purchase the firm at
the price pi at time Ti = inf{t ≥ 0 | X(t) ≤ xi }. If the acquirer accepts the oﬀer price pi ,the transaction occurs, i.e., debt holders are retired the face value C/r, and shareholders
receive the residual value (1 − τ )pi − C/r.8 Note that repayment of the face value C/rgenerates no tax shield. In the sell-out case, as in Mella-Barral and Perraudin (1997), theequity value at the initial time becomes()
Ce (1 − τ )(X(t) − C)dt + e(1 − τ )pi −
]
E[
r
0
(
( )γ (
))
x
C
x
xi
τC
=(1 − τ )− +−+ pi −
,
(2)
r−µ
r
xi
r−µ
(1 − τ )r√
where γ = 1/2 − µ/σ 2 − (µ/σ 2 − 1/2)2 + 2r/σ 2 (< 0). In (2), the first, second, and
∫ Ti

−rt

−rTi

last terms stand for the expected values of infinite streams of EBIT, infinite streams of
coupon payments, and the sell-out option, respectively. For si = 0 (default), shareholders
make no oﬀer and declare default at Ti = inf{t ≥ 0 | X(t) ≤ xi }. In the default case,as in Goldstein, Ju, and Leland (2001), the distress firm’s equity value at the initial time
becomes∫ Ti

e−rt (X(t) − C)dt](1 − τ )E[0))(( )γ (
xi
x
C
x
C
−
.
=(1 − τ )
− +
+
r−µ
r
xi
r−µ
r

(3)

In (3), the first, second, and last terms stand for the expected values of infinite streamsof EBIT, infinite streams of coupon payments, and the default option, respectively.

8

We always have (1 − τ )pi − C/r ≥ 0 when shareholders prefer to sell out rather than default. Accordingly,

we do not need to impose a limited liability condition.

8By (2) and (3), we have the ex-ante equity value maximization problem as follows:{( )γ (
(
))}
x
C
x
xL
C
C
∗
E (x) = s max(1 − q)(1 − τ )−− ++ + sL pL −L ,sH ,xL ,
r−µ
r
xL
r−µ
r
(1 − τ )rxH ,pL ,pH
{
(
(
)γ (
))}
x
C
x
xH
C
C
+ q(1 − τ )− +−+ + sH pH −
(4)
r−µ
r
xH
r−µ
r
(1 − τ )r
where (sL , sH , xL , xH , pL , pH ) are optimized subject to the acquirer’s participation conditions (PCs)sL (PL (xL ) − pL ) ≥ 0(5)sH (PH (xH ) − pH ) ≥ 0.

(6)

Throughout the paper, we denote the symmetric information case by the superscript ∗.
Recall that in (4), q ∈ (0, 1) denotes the prior probability of the high type. PCs (5)and (6) mean that in the sell-out case, the oﬀer price pi is not lower than the acquirer’svaluation Pi (xi ). Clearly, in the sell-out case, the optimal oﬀer prices are set atpi = Pi (xi ) =

ai xi
+θ
r−µ

(i = L, H),(7)so that they equate PCs (5) and (6). By substituting (7) into (4), we have E ∗ (x) =∗ (x), where the ex-post equity values E ∗ (x) (i = L, H) are defined by(1 − q)EL∗ (x) + qEHi{( )γ (
))}
(
x
C
x
xi
C
C
ai xi
Ei∗ (x) = max(1−τ )− +−+ + si+θ−.
xi ,si
r−µ
r
xi
r−µ
r
r−µ
(1 − τ )r(8)

Equity value (8) can be derived in the same manner as Proposition 1 in Nishihara andShibata (2018). First, we solve problem (8) with si = 1. By the first order condition, we
can easily derive the equity value(( )γ ())
x
Cx
τC(1 − ai )xsisEi (x) = (1 − τ )
− +
+θ−
−
r−µ
r
xsi
r−µ
(1 − τ )r(i = L, H),(9)and the sell-out trigger
xsi =

γ r−µ
γ − 1 1 − ai

(
θ−

τC
(1 − τ )r)(i = L, H)(10)for θ − τ C/(1 − τ )r > 0. If θ − τ C/(1 − τ )r ≤ 0 holds, we have Eis (x) = (1 − τ )(x/(r − µ) −C/r), i.e., there is no value of the sell-out option. Throughout the paper, we denote thesell-out case by the superscript s. Note that both (9) and (10) depend on the acquirer’s
type i. In the sell-out case, debt holders are repaid the face value of debt. Then, the debt
value is equal to the face value C/r, regardless of type i.
Next, we solve problem (8) with si = 0. The problem does not depend on the acquirer’s
type i. By the first order condition, we can easily derive the equity value
(
))
(
xd
C ( x )γ
C
x
−
,
− +
+
E d (x) = (1 − τ )
r−µ
r
r−µ
r
xd

9

(11)

and the default trigger
xd =γ(r − µ) C
.
γ−1 r

(12)

Throughout the paper, we denote the default case by the superscript d. Neither (11) nor(12) depends on the acquirer’s type i because shareholders do not sell the firm to theacquirer but declare default at the threshold xd . In this case, the debt value at the initialtime becomes∫ Tdde−rt Cdt + e−rT (1 − α)Pi (X(T d ))]Did (x) = E[
0())(
C ( x )γ C
ai xd
=
−
− (1 − α)+θ(i = L, H),
r
r
r−µ
xd

(13)

where T d = inf{t ≥ 0 | X(t) ≤ xd }. In (13), the first term C/r is the face value of debt,
while the second term means that on default, debt holders lose streams of coupon payments
in exchange for liquidation value (1 − α)Pi (xd ). The liquidation value is equal to theacquirer’s evaluation, where proportion α of the firm value is lost on bankruptcy. Then, thedebt value Did (x), unlike E d (x), depends on acquirer’s type i. When shareholders preferto default, the asset value is lower than the face value under the plausible assumption ofτ ≤ α. This implies that debt holders cannot fully recover the face value, i.e., Did (x) is
lower than C/r.9
By (9), (11), and Ei∗ (x) = max{Eis (x), E d (x)}10 , we can reveal shareholders’ optimal
choice between sell-out and default. Actually, we show that E d (x) ≤ Eis (x) holds if andonly if C/rθ ≤ Bi∗ holds, where
Bi∗ =

1−τ
,
τ + (1 − τ )b∗i−γb∗i = (1 − ai ) 1−γ(i = L, H).

(14)
(15)

For the proof, see Appendix A. The corresponding debt values are denoted by Di∗ (x) (i =L, H).
Proposition 1 Symmetric information. The optimal oﬀer (s∗L , s∗H , x∗L , x∗H , p∗L , p∗H ) is
given below.

9For α < τ , the debt value can be higher than the face value because debt holders, unlike shareholders, donot suﬀer from the corporate tax rate τ .
10
As in Décamps, Mariotti, and Villeneuve (2006), the initial choice max{Eis (x), E d (x)} is equal to the result
by the dynamic choice if X(0) = x is higher than the sell-out trigger. When X(0) = x is lower than the sell-out
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, seeDécamps, Mariotti, and Villeneuve (2006).

10

Case (I-S): C/rθ ≤ BL∗ . Sell-out to any type.
s∗L = s∗H = 1,x∗i = xsi (i = L, H),p∗i = Pi (xsi ) (i = L, H),
Ei∗ (x) = Eis (x) (i = L, H),∗∗DL(x) = DH(x) = C/r.
∗ . Sell-out to only a high-value type.
Case (II-S): BL∗ < C/rθ ≤ BH

s∗L = 0, s∗H = 1,x∗L = xd , x∗H = xsH ,p∗H = PH (xsH ),∗sEL∗ (x) = E d (x), EH(x) = EH
(x),
∗
d
∗
DL
(x) = DL(x), DH(x) = C/r.
∗ < C/rθ. Default to any type.
Case (III-S): BHs∗L = s∗H = 0,x∗L = x∗H = xd ,∗EL∗ (x) = EH(x) = E d (x),Di∗ (x) = Did (x) (i = L, H).
Note that in each case, the ex-ante equity and debt values are E ∗ (x) = (1 − q)EL∗ (x) +∗ (x) and D ∗ (x) = (1 − q)D ∗ (x) + qD ∗ (x), respectively. The ex-ante firm value F ∗ (x)qEH
L
H

is the sum of equity and debt values, i.e., F ∗ (x) = E ∗ (x) + D∗ (x).
Proposition 1 agrees with Proposition 1 in Nishihara and Shibata (2018) for τ = 0,
while it corresponds to the standard solution in Mella-Barral and Perraudin (1997) forτ = ai = 0. The default case agrees with the standard solution in Goldstein, Ju, andLeland (2001). As in Mella-Barral and Perraudin (1997), Lambrecht and Myers (2008),
and Nishihara and Shibata (2018), Proposition 1 shows that shareholders prefer to sellout for low debt levels compared to asset value, while they prefer to default for high debt
levels compared to asset value. Naturally, the sell-out region is wider for a high-value typethan for a low-value type (cf. Case (II-S)).
As in Nishihara and Shibata (2018), Proposition 1 reveals the impacts of volatility σon the exit choice, although Mella-Barral and Perraudin (1997) and Nishihara and Shibata(2017), who assume a = 0, have no impact of σ. Actually, by (14) and ∂γ/∂σ > 0, we
have ∂Bi∗ /∂σ < 0 (i = L, H). The negative sensitivities imply that a higher σ increases
the incentive for shareholders to choose default rather than sell-out. The reason is thatthe convexity of the default option is stronger than that of the sell-out option for ai > 0.

11

Due to the stronger convexity, a higher σ increases the default option value more than
the sell-out option value.11 Our result is consistent with the stylized fact that greater cash
flow uncertainty is more likely to lead to an unsuccessful and costly bankruptcy.

3.2

Asymmetric information

This section examines the asymmetric information model, in which the acquirer has private valuation of the distressed firm. Under asymmetric information, shareholders cannot
always gain the first-best payoﬀ in Proposition 1. For example, in Case (I-S), for the
first-best oﬀer (s∗L , s∗H , x∗L , x∗H , p∗L , p∗H ), a high-value type imitates a low-value type andgains ∆ax∗L at time TL∗ = inf{t ≥ 0 | X(t) ≤ x∗L }.
As in Section 3.1, we consider shareholders who commit to a price schedule (sL , sH , xL , xH , pL , pH )at the initial time. For the price schedule, the acquirer responds in its own interests. We
have only to compare three types of oﬀers, i.e., oﬀers with (sL , sH ) = (1, 1), (0, 1), and(0, 0) because any oﬀer with (sL , sH ) = (1, 0) is dominated by an oﬀer with (sL , sH ) =(1, 1) or (0, 1). We can also easily show that for oﬀers with (sL , sH ) = (1, 1) and (0, 1),
an optimal menu exists within menus that lead to the acquirer’s truthful action (cf. therevelation principle in Bolton and Dewatripont (2005)). In the baseline model, we assume
that shareholders can credibly commit to sell-out and default thresholds at the initialtime. In the asymmetric information model, unlike in the symmetric information model,
the assumption of credible commitment is critical for the results. Section 5.1 discusses
a case in which shareholders cannot credibly commit to the default threshold. For simplicity, we also assume that aL ≥ qaH , although Section 5.2 presents the solutions foraL < qaH . As will be explained later, under the assumption of aL ≥ qaH , debt holders,
who take over the bankrupt firm, prefer to sell the firm to any type of acquirer rather
than scrapping.
First, we derive the optimal oﬀer with (sL , sH ) = (1, 1) (i.e., sell-out to any type). We
have the ex-ante equity value maximization problem as follows:( )γ (
)}
{
C
x
τC
xL
x
(1,1)
− +
−
+ pL −
E(x) = xmax
(1 − q)(1 − τ )L ,xH ,
r−µ
r
xL
r−µ
(1 − τ )rpL ,pH
{
(
)γ (
)}
x
C
x
xH
τC
+ q(1 − τ )− +−+ pH −
,
(16)
r−µ
r
xH
r−µ
(1 − τ )r
where (xL , xH , pL , pH ) are optimized subject to the incentive compatibility conditions(ICCs) for low- and high-value types
( )γ
)
(
x
x γ
(PL (xL ) − pL ) ≥(PL (xL ) − pH )
xL
xH
)
(
( )γ
x γ
x
(PH (xH ) − pH ) ≥(PH (xL ) − pL )
xH
xL
11

(17)
(18)

Similar mechanisms can be seen in Décamps, Mariotti, and Villeneuve (2006) and Kort, Murto, and Pawlina(2010). In the former, a higher volatility leads to large-scale rather than small-scale investment, and in the latter,a higher volatility leads to lumpy rather than stepwise investment.

12and PCs
PL (xL ) − pL ≥ 0(19)

PH (xH ) − pH ≥ 0.

(20)

The objective function of problem (16) is equal to that of problem (4) with (sL , sH ) =(1, 1). The key diﬀerence from the symmetric information problem lies in ICCs (17) and(18). The left-hand side of (17) stands for a low-value type’s expected gain from truthfulaction, while the right-hand side of (17) stands for a low-value type’s expected gain byimitating a high-value type. Then, (17) ensures that a low-value type truthfully accepts
the oﬀer price pL at time TL = inf{t ≥ 0 | X(t) ≤ xL }. Similarly, (18) ensures that a highvalue type truthfully accepts the oﬀer price pH at time TH = inf{t ≥ 0 | X(t) ≤ xH }. For
the acquirer’s type i, equity and debt holders receive (1−τ )pi −C/r and C/r, respectively,
at time Ti .
Next, we derive the optimal oﬀer with (sL , sH ) = (0, 1) (i.e., sell-out only to a highvalue type). We have the ex-ante equity value maximization problem as follows:( )γ (
)}
{
C
x
xL
C
x
(0,1)
− +
−
+
E
(x) = max (1 − q)(1 − τ )xL ,xH ,pH
r−µ
r
xL
r−µ
r
{
(
)γ (
)}
x
x
xH
C
τC
+ q(1 − τ )−,− ++ pH −r−µrxH
r−µ(1 − τ )r
where (xL , xH , pH ) are optimized subject to ICCs
(
)
x γ
0≥
(PL (xL ) − pH )
xH
(
)
( )γ
x γ
x
(PH (xH ) − pH ) ≥(1 − α)(PH (xL ) − PL (xL ))
xH
xL

(21)

(22)
(23)

and PCPH (xH ) − pH ≥ 0.

(24)

The objective function of problem (21) is equal to that of problem (4) with (sL , sH ) =(0, 1). As in problem (16), ICCs separate low- and high-value types. If the acquirer is ahigh-value type, it accepts the oﬀer price pH at TH . At the sales time TH , shareholders
receive (1 − τ )pH − C/r, whereas debt holders receive the face value C/r. Otherwise, theacquirer forgoes the oﬀer and waits until the default time TL . Because of this separation,
the acquirer’s type is revealed to debt holders at TL . On bankruptcy, debt holders set thesales price (1 − α)PL (xL ), which equates PC, to maximize their recovery. Recall that afraction α ∈ (0, 1) of the firm’s asset value is lost as bankruptcy costs.
Last, we derive the optimal oﬀer with (sL , sH ) = (0, 0) (i.e., default to any type).
Because shareholders do not sell assets to the acquirer, they are indiﬀerent to the acquirer’s
type. Therefore, the default trigger xi (i = L, H) is unchanged from the first-best default
trigger xd (see (12)). The equity value is also unchanged from E d (x) (see (11)). Because13

of the pooling property, debt holders, who liquidate assets on bankruptcy, do not know theacquirer’s type. At the threshold xd , debt holders must set the oﬀer price for the acquirer
without knowing the acquirer’s type. Debt holders can sell assets at price (1 − α)PL (xd )to any type of acquirer. On the other hand, debt holders can set price (1 − α)PH (xd ),although a low-value type rejects the oﬀer. Then, they will scrap assets by (1 − α)θ at
probability 1 − q, i.e., the probability of a low-value type. In this case, the expected gain
becomes (1 − q)(1 − α)θ + q(1 − α)PH (xd ). Under the baseline parameter assumption ofaL ≥ qaH , we have (1−α)PL (xd ) ≥ (1−q)(1−α)θ +q(1−α)PH (xd ). That is, debt holders
prefer to set the sales price (1 − α)PL (xd ) so that any type can purchase the assets. Thus,d (x) regardless of the acquirer’s type. Section 5.2 examines
the debt value is equal to DL

the case of aL < qaH , where debt holders prefer to set the sales price (1 − α)PH (xd ).
Comparing the ex-ante equity values E (1,1) (x), E (0,1) (x), and E d (x), we have the op∗∗ ∗∗ ∗∗ ∗∗ ∗∗timal oﬀer (s∗∗L , sH , xL , xH , pL , pH ) in the following proposition. Throughout the paper,

we denote the asymmetric information case by the superscript ∗∗. Actually, we can showthat(1,1) (x) (C/rθ ≤ B ∗∗ )


L
 E
(1,1)
(0,1)
d
∗∗ )
(0,1)
∗∗
max{E
(x), E(x), E (x)} =E(x) (BL < C/rθ ≤ BH
 E d (x)∗∗ ),(C/rθ > BH

(25)

where
1−τ
(< BL∗ ),τ + (1 − τ )b∗∗
L
1
−
τ
∗∗
∗
BH
=
∈ (BL∗ , BH),
τ + (1 − τ )b∗∗H
) −γ(
1 − aL + ∆aq/(1 − q) 1−γ
∗∗
,
bL =
1 + (1 − α)∆aq/(1 − q)() 1−γ
1 − (1 − q)(1 + (1 − α)∆aq/(1 − q))γ 1−γ∗∗1−γbH = (1 − aH ).
qBL∗∗ =

(26)
(27)
(28)
(29)

For the proof, see Appendix B. In the proposition, we denote the high-type acquirer’s∗∗
value by A∗∗
H (x). The low-type acquirer’s value AL (x) is always equal to 0.
∗∗ ∗∗ ∗∗ ∗∗ ∗∗Proposition 2 Asymmetric information. The optimal oﬀer (s∗∗L , sH , xL , xH , pL , pH )is given below.

14

Case (I-A): C/rθ ≤ BL∗∗ . Sell-out to any type.
∗∗s∗∗
L = sH = 1,

(
)
γ
r−µ
τC
s
θ−
, x∗∗
H = xH ,γ − 1 1 − aL + ∆aq/(1 − q)
(1 − τ )r( s )γ
xH
∆ax∗∗
∗∗
∗∗
s
L
,
p∗∗
=
P
(x
),
p
=
P
(x
)
−
L L
H H
L
H
∗∗
xL
r−µ
{
(
) (
)}
x
C
x γ (aL − 1)x∗∗
τC∗∗L
EL (x) = (1 − τ )
− +
+θ−
,
r−µ
r
x∗∗
r−µ
(1 − τ )rL()
x γ (1 − τ )∆ax∗∗∗∗s
L
EH (x) = EH (x) −
,
x∗∗
r
−
µ
L
x∗∗
L =

∗∗
∗∗
DL
(x) = DH(x) = C/r,()
x γ (1 − τ )∆ax∗∗∗∗
L
AH (x) =
.
x∗∗
r
−
µ
L
∗∗ . Sell-out to only a high-value type.
Case (II-A): BL∗∗ < C/rθ ≤ BH∗∗s∗∗
L = 0, sH = 1,γ(r − µ)C
, x∗∗ = xsH ,γ − 1 r(1 + (1 − α)∆aq/(1 − q)) H( s )γxH(1 − α)∆ax∗∗∗∗s
L,
pH = PH (xH ) −
x∗∗
r−µ
L
{
(
)}
) (
x∗∗
x
C
x γ
C
∗∗
L
EL (x) = (1 − τ )
− +
+
,
−
r−µ
r
x∗∗
r−µ
r
L
(
)γ
(1 − τ )(1 − α)∆ax∗∗
x
∗∗
s
L
EH
(x) = EH
(x) −
,
x∗∗
r
−
µ
L
(
) (
(
))
aL x∗∗C
x γ C∗∗
L
DL (x) =−− (1 − α)
+θ
,
r
x∗∗
r
r−µ
L
x∗∗
L =

∗∗
DH
(x) = C/r,()
x γ (1 − τ )(1 − α)∆ax∗∗∗∗
L
AH (x) =
.
x∗∗
r
−
µ
L
∗∗ < C/rθ. Default to any type.
Case (III-A): BH∗∗s∗∗
L = sH = 0,∗∗dx∗∗
L = xH = x ,∗∗EL∗∗ (x) = EH(x) = E d (x),
∗∗
∗∗
d
DL
(x) = DH(x) = DL(x),( x )γ (1 − τ )(1 − α)∆axd
A∗∗.
H (x) =r−µxd

Note that in each case, the ex-ante equity and debt values are E ∗∗ (x) = (1 − q)EL∗∗ (x) +∗∗ (x) and D ∗∗ (x) = (1 − q)D ∗∗ (x) + qD ∗∗ (x), respectively. The ex-ante firm value
qEH
L
H

15

F ∗∗ (x) is the sum of equity and debt values, i.e., F ∗∗ (x) = E ∗∗ (x) + D∗∗ (x), whereas theex-ante acquirer’s value is A∗∗ (x) = qA∗∗
H (x).
For low levels of debt compared to asset value, Case (I-A) holds. At the initial time,∗∗shareholders commit to the price schedule (i.e., high price p∗∗
H at high threshold xH and low∗∗
price p∗∗
L at low threshold xL ). This schedule can be regarded as a kind of intertemporal

price discrimination (cf. Stokey (1979) and Landsberger and Meilijson (1985)). In the
optimal oﬀer, shareholders decrease a high-value type’s incentive to imitate a low-valuestype by delaying the asset sales at the low price, i.e., x∗∗
L < xL . Mathematically, adecrease in xL decreases a high-value acquirer’s information rent (x/xL )γ ∆axL more than
the seller’s payoﬀ (x/xL )γ (−(1 − ai )xL /(r − µ) + θ − τ C/(1 − τ )r). Then, the sellers can
optimally decrease the acquirer’s information rent by decreasing xL . This mechanism issimilar to that of Landsberger and Meilijson (1985), in which a monopolist seller delays
sales at low prices to discriminate between buyer types if the seller’s time discount is lower
than that of the buyers.
For intermediate levels of debt, Case (II-A) holds, where shareholders commit as fol∗∗∗∗lows: They oﬀer the firm at price p∗∗
H at threshold xH and default at xL . Note thatthe bankruptcy cost α decreases a high-value type’s information rent. As in Case (I-A),
shareholders can optimally decrease the acquirer’s information rent by decreasing xL . Interestingly, both Cases (I-S) and (II-A) hold for BL∗∗ < C/rθ ≤ BL∗ . See Table 1. In thisregion, asymmetric information changes the exit choice from sell-out to default. This isbecause the screening cost is higher than the residual value which shareholders receivefrom the transaction.
Although their mechanisms are diﬀerent, Nishihara and Shibata (2017) and Nishihara
and Shibata (2018) report similar changes in the exit choice. In the former, shareholders
can cut managers’ private information rent about sell-out costs by changing the exit choicefrom sell-out to default. In the latter, shareholders, who have private information about
asset quality, can change the exit choice from sell-out to default due to the signaling costassociated with sell-out.
For high levels of debt, Case (III-A) holds, where shareholders abandon selling assets
via the truth-revealing mechanism but declare default at threshold xd . Interestingly,∗∗ < C/rθ ≤ B ∗ (see Table 1). In this region,
both Cases (II-S) and (III-A) hold for BH
H

shareholders’ screening cost is higher than the residual value in the asset sales for a highvalue type, and hence, they change the exit choice from sell-out to default. In other∗ < C/rθ,words, shareholders refuse to pay the screening cost and choose default. For BH

shareholders’ exit decision and timing as well as the equity value are not aﬀected byasymmetric information. In this region, the equity value is unchanged from the first-bestvalue.
∗∗ monotonically decrease in ∆a. A
From (26) and (27), we can show that BL∗∗ and BH

higher ∆a increases shareholders’ screening cost, and hence, they tend to choose default∗∗ /∂α > 0. As
rather than sell-out. We can also easily show that ∂BL∗∗ /∂α < 0 and ∂BH

16was discussed above, in Case (II-A) a higher α decreases the acquirer’s information rent.
Then, for a higher α, shareholders’ screening cost decreases in Case (II-A), which increases
the region of Case (II-A).
Corollary 1 Exit timing.
Case (I-A): C/rθ ≤ BL∗∗ . Sell-out to any type.
∗
s
∗∗
∗
s
x∗∗
L < x L = xL < x H = xH = xH

Case (II-A): Sell-out to only a high-value type.
Case (I-S): BL∗∗ < C/rθ ≤ BL∗ .
d
∗
s
∗∗
∗
s
x∗∗
L < x < x L = xL < x H = xH = xH∗∗ .
Case(II-S): BL∗ < C/rθ ≤ BH
∗
d
∗∗
∗
s
x∗∗
L < x L = x < x H = xH = xH

Case (III-A): Default to any type.
∗∗ < C/rθ ≤ B ∗ .
Case (II-S): BH
H
∗∗
∗
d
∗
s
x∗∗
L = xH = xL = x < xH = xH
∗ < C/rθ.
Case (III-S): BH
∗∗
∗
∗
d
x∗∗
L = xH = xL = xH = x

Corollary 1 shows that the firm’s liquidation timing is delayed due to the acquirer’s private
information. Below, we examine each case in full detail. In Case (I-A), the sell-out triggersx∗∗
L is lower than the first-best sell-out trigger xL because shareholders delay the asset

sales at the low price to decrease a high-value type’s information rent. In Case (II-A),d
the default trigger x∗∗
L is lower than the first-best default trigger x . As in Case (I-A),

shareholders delay default for a low-value type to decrease a high-value type’s information
rent. Note that for BL∗∗ < C/rθ ≤ BL∗ , the default trigger x∗∗
L is lower than the first-bestsell-out trigger x∗L = xsL . In this region, asymmetric information does not only change
the exit choice from sell-out to default but also delays the liquidation timing. In Case∗∗d(III-A), the default trigger x∗∗
L = xH is the same as the first-best default trigger x .
∗∗ < C/rθ ≤ B ∗ , the default trigger x∗∗ is lower than the first-best sell-out trigger
For BHH
Hx∗H = xsH , which implies that asymmetric information does not only change the exit choice∗ < C/rθ,from sell-out to default but also delays the liquidation timing. Only for BH

asymmetric information has no impact on the liquidation timing.
As was explained after Proposition 2, the delayed liquidation timing is related to themechanism of intertemporal price discrimination (e.g., Landsberger and Meilijson (1985)).

17

The mechanism is also similar to that of Grenadier and Wang (2005) and Shibata andNishihara (2010). They study investment timing in the presence of managers’ private information in real options models. They show that shareholders, who delegate investment
to a manager, decrease a good-type manager’s information rent by delaying a bad-typemanager’s investment. Unlike their papers, we focus on the exit timing with the acquirer’s
private information, and we show that shareholders decrease the high-value type’s information rent by delaying the low-value type’s acquisition of assets.
Nishihara and Shibata (2017) reveal that asymmetric information between managers
and shareholders aﬀect the default and sell-out timing. Unlike in this paper, they showthat managers’ private information accelerates the default timing. The key diﬀerenceis that managers’ information rent about the firm’s running cost lasts until the default
time; therefore, shareholders can cut managers’ information rent by accelerating default.
Because Nishihara and Shibata (2017) consider two types of private information (i.e., running and sell-out costs), the sell-out timing can be either accelerated or delayed dependingon the trade-oﬀ of two types of information rent.
Nishihara and Shibata (2018) also show that the seller’s private information can delay
sell-out in the signaling model. Unlike in this paper, they focus on the seller’s private
information about the firm’s asset quality and show that the seller can delay sell-out
to signal asset quality and sell assets at a high price to the acquirer. Lambrecht andMyers (2008) show that manager-shareholder conflicts can delay the firm’s exit timing,
but they do not consider asymmetric information. In their model, managers, who decide
the exit timing unless shareholders take collective action, can delay liquidation to increase
managerial rents. In a setup with an uninformed seller versus informed bidders, Cong
(2016) and Cong (2017) examine auction timing of assets with embedded real options.
They show that the seller prefers to delay auction timing to reduce the winning bidder’sinformation rent, although they do not focus on the corporate bankruptcy process.
Next, we examine discounts in the oﬀer price for a high-value type to relate our model
to studies on fire sale discounts. Ignoring the scrap value θ, which does not depend on
the acquirer’s type, we define the discount percentage of the usable assets by
DC =PH (xsH ) − p∗∗
H
PH (xsH ) − θin Cases (I-A) and (II-A) and byDC =(1 − α)PH (xd ) − p∗∗
PH (xd ) − PL (xd )H=(1 − α)(PH (xd ) − θ)PH (xd ) − θin Case (III-A), where p∗∗
H stands for a high-value type’s purchase price. Recall that indCase (III-A), the purchase price p∗∗
H is not the high price (1 − α)PH (x ) but the low price(1 − α)PL (xd ).
Corollary 2 Discounts for a high-value type.

18

Case (I-A): C/rθ ≤ BL∗∗ . Sell-out to any type.
( ∗∗ )1−γ
xL
∆a
DC =
.
xsH
aH
∗∗ . Sell-out to only a high-value type.
Case (II-A): BL∗∗ < C/rθ ≤ BH

( ∗∗ )1−γxL(1 − α)∆a
DC =
.
xsH
aH
∗∗ < C/rθ. Default to any type.
Case (III-A): BH

DC =

∆a
.
aH

Corollary 2 shows how shareholders mitigate the discount for a high-value type throughthe optimal price schedule. Indeed, without commitment, a high-value type could imitatea low-value type, and DC would be at the maximum level (i.e., the valuation wedge∆a/aH = (PH (xsL ) − PL (xsL ))/(PH (xsL ) − θ)). By the optimal commitment, DC in Cases(I-A) and (II-A) are lower than ∆a/aH . In Corollary 2, we can also see that the decreased∗
threshold x∗∗
L (< xL ) plays a role in reducing DC. On the other hand, in Case (III-A), shareholders do not commit to a price schedule but declare default at threshold xd ,where the acquirer’s type is not revealed to equity and debt holders. Upon liquidation
bankruptcy, debt holders sell the firm at the low price; threfore, DC agrees with the
maximum level ∆a/aH . In Section 4.1, we discuss implications by relating DC in Case
(III-A) to fire sale transactions.
These results contrast with those of Nishihara and Shibata (2018), who examine theeﬀects of the seller’s private information on liquidation procedure. They show that theseller can signal asset quality and sell the firm at a fair price by delaying the sell-out
timing. Even when the signaling cost is too high for shareholders to choose sell-out, thedefault time, which depends on the seller’s private information, reveals the seller’s type.
In other words, in Nishihara and Shibata (2018), DC always equals zero regardless ofdefault or sell-out. In this paper, however, the default time, which is independent of
the acquirer’s private information, does not reveal the acquirer’s type, which leads tothe maximal discount on liquidation bankruptcy. Even in the sell-out case, shareholders
discount the oﬀer price and give information rent to a high-value type.
Corollary 3 Equity and debt values.
Case (I-A): C/rθ ≤ BL∗∗ . Sell-out to any type.
∗∗∗s
EL∗∗ (x) < EL∗ (x) = ELs (x) < EH(x) < EH(x) = EH(x), E ∗∗ (x) < E ∗ (x)
∗∗
∗∗
∗
∗
DL
(x) = DH(x) = D∗∗ (x) = DL(x) = DH(x) = D∗ (x) = C/rCase (II-A): Sell-out to only a high-value type.

19

Case (I-S): BL∗∗ < C/rθ ≤ BL∗ .
∗∗∗sEL∗∗ (x) < E d (x) ≤ EL∗ (x) = ELs (x) < EH(x) < EH(x) = EH(x), E ∗∗ (x) < E ∗ (x)
∗∗
∗
∗
DH
(x) = DL(x) = DH(x) = D∗ (x) = C/r∗∗ (x) < D ∗ (x) and D ∗∗ (x) < D ∗ (x) hold if and only if
DL
L
L
(
)
aL x∗∗
CL(1 − α)
+θ <
r−µ
r

(30)

holds.
∗∗ .
Case(II-S): BL∗ < C/rθ ≤ BH∗∗∗
sEL∗∗ (x) < EL∗ (x) = E d (x) < EH(x) < EH(x) = EH(x), E ∗∗ (x) < E ∗ (x)
∗
d
∗∗
∗
DL
(x) = DL(x), DH(x) = DH(x) = C/r∗∗ (x) > D ∗ (x) and D ∗∗ (x) > D ∗ (x) hold if and only ifDLL( )γ (
(
)) (
) (
(
))
aL x∗∗
1
C
aL xd
1 γ C
L
− (1 − α)+θ>− (1 − α)
+θ
r
r−µ
x∗∗
r
r−µ
xd
L

(31)

holds.
Case (III-A): Default to any type.
∗∗ < C/rθ ≤ B ∗ .
Case (II-S): BH
H∗∗∗sEL∗∗ (x) = EH(x) = EL∗ (x) = E d (x) < EH(x) = EH(x), E ∗∗ (x) < E ∗ (x)
∗∗
∗∗
∗
d
∗
DL
(x) = DH(x) = D∗∗ (x) = DL(x) = DL(x), DH(x) = C/r∗∗ (x) < D ∗ (x) and D ∗∗ (x) < D ∗∗ (x) hold if and only if
DH
H
(
)
aL xd
C
(1 − α)
+θ <
r−µ
r

(32)

holds.
∗ < C/rθ.
Case (III-S): BH∗∗∗
EL∗∗ (x) = EH(x) = E ∗∗ (x) = EL∗ (x) = EH(x) = E ∗ (x) = E d (x)
∗∗
∗∗
∗
d
∗
d
DL
(x) = DH(x) = D∗∗ (x) = DL(x) = DL(x) < DH(x) = DH(x), D∗∗ (x) < D∗ (x)Corollary 3 shows that shareholders suﬀer loss from asymmetric information except∗ < C/rθ. Shareholders suﬀer loss from the discount for a high-value
in Case (III-S): BH

type and the delayed exit time for a low-value type. In notable regions BL∗∗ < C/rθ ≤ BL∗∗∗ < C/rθ ≤ B ∗ (see Table 1), shareholders also suﬀer from the change in the exit
and BHH∗ < C/rθ, the equity value is unchanged
choice from sell-out to default. For Case (III-S): BH

from the first-best value because the first-best policy is default, which is irrelevant to theacquirer’s private information.

20

On the other hand, the impacts of asymmetric information on debt holders depend on
parameter values. In Case (I-A), debt holders suﬀer no loss because the debt is riskless.
In Case (II-A), debt holders can either decrease or increase their expected payoﬀ byasymmetric information. For Case (I-S): C/rθ ∈ [BL∗∗ , BL∗ ), (30) tend to hold for plausibledsparameter values12 because bankruptcy cost α and threshold x∗∗
L (< x < xL ) decrease

the right-hand side of (30). Then, we argue that in this region, asymmetric information∗∗ ), (31)makes debt riskier and lowers the debt value. For Case (II-S): C/rθ ∈ [BL∗ , BHdusually holds in numerical examples because the decreased threshold x∗∗
L (< x ) lowers thedefault probability. Then, in this region, asymmetric information has a positive eﬀect on
the debt value. Last, (32) holds for τ ≤ α. Then, in Case (III-A), debt holders tend to
suﬀer loss due to asymmetric information because they liquidate the firm at a low priceeven to a high-value type.
Nishihara and Shibata (2017) and Nishihara and Shibata (2018) show similar results
regarding information costs for low and intermediate levels of debt. For high levels ofdebt, however, Corollary 3 diverges from findings of Nishihara and Shibata (2017) andNishihara and Shibata (2018). Indeed, in Nishihara and Shibata (2018), neither equity
nor debt holders pay signaling costs when a good-type firm’s first-best decision is default.
In Nishihara and Shibata (2017), shareholders always pay information rent to managers
until bankruptcy, whereas its eﬀects on debt value can be either positive or negative.

4

Economic implications

4.1

Fire sales

The results in the previous section are closely related to empirical findings about fire salesof distressed firms. Shleifer and Vishny (1992) explain fire sales through the following
mechanism. When a firm enters liquidation bankruptcy, potential bidders, who suﬀer fromthe same industry shocks, are likely to also be financially distressed. Therefore, ineﬃcientbidders (say, industry outsiders), rather than eﬃcient bidders (say, industry insiders), canpurchase the distressed firm’s assets. Because of this ineﬃciency, the distressed firm isforced to sell assets at lower prices than in normal times. The empirical evidence mostly
supports the fire sale theory. For instance, Pulvino (1998) finds that the fire sale discountsof aircrafts are deepened to a non-airline during industry distress, and Acharya, Bharath,and Srinivasan (2007) show that creditor recoveries are depressed during industry distress.
In terms of acquirers’ private information, we can explain why asset sales to industry
outsiders are so ineﬃcient. When an industry outsider who has no prior business relation12

For α < τ , (30) may not hold, which means that the debt value can be higher than its face value. Even

without asymmetric information, this abnormal result can arise for α < τ (see footnote 9). However, we cannot
find the abnormal result in any numerical example satisfying τ ≤ α. Therefore, we conclude that the abnormal
result does not arise for plausible parameter values.

21

ship with the target acquires the target’s assets, the target is unlikely to know how theacquirer utilizes the assets or how much business synergy is gained in the acquisition.13
Therefore, the asymmetric information model is more relevant to asset sales to industry
outsiders than asset sales to industry insiders.
Even when the firm sells out without formal bankruptcy, asymmetric information
decreases the sales price by the acquirer’s information rent (see Cases (I-A) and (II-A)
in Corollary 2). Worse, a combination of bankruptcy and asymmetric information can∗∗ < C/rθ ≤ B ∗ (see Table 1). In
further decrease the sales price. Consider a region BH
H

the absence of asymmetric information (e.g., sale to an industry insider), the sales pricefor a high-value acquirer equals the fair value PH (xsH ). With asymmetric information(e.g., sale to an industry outsider), the firm changes the exit choice from sell-out toliquidation bankruptcy, where a faction α of the asset value is lost as bankruptcy costs.
Furthermore, debt holders’ inability to operate the firm as a going concern increases
the acquirer’s information rent to the maximum level (see Case (III-A) in Corollary 2).
Because of the two eﬀects, asymmetric information decreases the sales price from PH (xsH )to (1 − α)PL (xd ).
Nishihara and Shibata (2018), who examine the seller’s private information, also showthat asymmetric information causes bankruptcy, which lowers the asset value due tobankruptcy costs. However, in addition to their view of bankruptcy costs, this paper
unveils how greatly the acquirer’s private information decreases the sales price. Our
result also predicts that fire sale transactions are good opportunities for industry outsiders,
who can purchase assets at favorable prices. This prediction is consistent with empiricalfindings of Meier and Servaes (2015). They show that acquirers of assets in fire saletransactions earn more returns than in regular situations, which implies wealth transfersfrom sellers to acquirers through such transactions.
Hotchkiss and Mooradian (1998), Stromberg (2000), and Thorburn (2000) show that
distressed firms are likely to be acquired by industry insiders, including former owners.
Stromberg (2000) shows that asset sales to industry outsiders decrease sales prices. Thorburn (2000) shows that debt holders recover more when former owners buy back firms.
These empirical findings are largely consistent with our results in the previous section.
Indeed, Corollary 3 proves that shareholders always prefer the symmetric information caseto the asymmetric information case and that in Case (III-A), debt holders suﬀer greatloss due to asymmetric information.
Lastly, Corollary 1 also generates a new empirical prediction that has not been tested
in the prior literature on fire sales. In our view, distressed firms can mitigate loss due toacquirers’ private information by delaying asset sales. Therefore, distressed firms tend todelay asset sales to industry outsiders compared to asset sales to industry insiders.
13

As Nishihara and Shibata (2018) discussed, asset sales to an industry outsider also increase the seller’s

private information. Although Nishihara and Shibata (2018) focus on the seller’s private information, thispaper focuses on the acquirer’s private information.

22

4.2

Premium in acquisition

Corollary 2 shows that the acquirer with private information can purchase the firm at
the discounted price. Due to fewer financial reporting requirements, unlisted firms cankeep more private information about synergies in acquisition than public firms. Higher
managerial ownership and/or ownership concentration of acquirers also increase corporate
opacity, generating more private information. Therefore, our result provides a testable
implication that acquisition premiums are lower when acquirers are unlisted and lesstransparent. This implication is consistent with empirical observations below.
Bargeron, Schlingemann, Stulz, and Zutter (2008) and Golubov and Xiong (2017)
show that private acquirers pay lower premiums in acquisitions than public acquirers.
Bargeron, Schlingemann, Stulz, and Zutter (2008) state that most acquisitions by private
firms are cash deals rather than stock deals, whereas approximately half of transactions
by public firms are stock deals according to Chemmanur, Paeglis, and Simonyan (2009).
Although Bargeron, Schlingemann, Stulz, and Zutter (2008) argue that the main reason
is that private firms have no publicly traded equity to oﬀer in acquisitions, this paper
complements their view. In our view, private acquirers prefer cash deals with favorable
prices to conceal their private information concerning synergies in acquisition (for details,
refer to Appendix 5.4).
Regarding ownership structures, Bargeron, Schlingemann, Stulz, and Zutter (2008)
show that among public firms, those with higher managerial ownership pay lower premiums in acquisitions. Golubov and Xiong (2017) also show that private acquirers, especiallythose with higher ownership concentration, pay lower premiums despite higher productivity after acquisition. Battigalli, Chiarella, Gatti, and Orlando (2017) show the negative
correlation between acquirers’ opacity and acquisition premium. These findings can be
explained by Corollary 2, i.e., discounted prices due to the acquirer’s private information.
Finally, Corollary 2 leads to another testable prediction as follows: The diﬀerencein premiums for public and private acquirers (or equivalently, transparent and opaqueacquirers) becomes larger in liquidation bankruptcy than in regular situations. This implication sheds light on the interactions between fire sales and acquirers’ characteristics,
although it has not been empirically tested.

4.3

Comparative statics

This section examines the eﬀects of parameter values on the results, especially in firesales (i.e., Case (III-A)). Clearly, a higher level of debt compared to asset value and more
asymmetric information change the region from Case (I-A) to Case (III-A) via Case (IIA), increasing the likelihood of a fire sale. Now we numerically show comparative statics
with respect to volatility σ and bankruptcy cost α because the impacts of σ and α are
non-trivial and interesting. We set the baseline parameter values in Table 2. Most of theparameter values are standard and similar to those of Nishihara and Shibata (2018). We

23

set C = 1, which is close to the threshold between Cases (II-A) and (III-A), to identify
how σ and α aﬀect the region of Case (III-A), i.e., the likelihood of a fire sale.
For these parameter values, we have the baseline results in Table 3. We have BL∗∗ =∗∗ = 1.124 < B ∗ = 1.196. Then, Cases (II-A)
0.922 < BL∗ = 0.982 < C/rθ = 1.111 < BH
H

and (II-S) hold (cf. Table 1), which means that shareholders sell assets only to a high-value∗∗ ∗∗∗∗type. In Table 3, we can see that asymmetric information decreases x∗∗
L , pL , pH , E (x),and F ∗∗ (x) below the corresponding values in the symmetric information case. On theother hand, D∗∗ (x) is higher than D∗ (x). The reason is that asymmetric information
decreases x∗∗
L and then decreases the default probability. The decrease in the equity value
dominates the increase in the debt value; therefore, F ∗∗ (x) < F ∗ (x) holds in the baseline
example.

4.3.1

Eﬀects of cash flow volatility σ

Figure 1 shows how the sell-out and default regions change with levels of volatility σ.
All thresholds monotonically decrease in σ, which means that a higher σ increases thepossibility of default rather than sell-out. As was explained after Proposition 1, this result
is consistent with the stylized fact that a higher σ increases the likelihood of unsuccessful
and costly bankruptcy. In addition, by relating Case (III-A) to a fire sale, we argue thata higher σ increases the likelihood of a fire sale.
Now we set the baseline parameter values in Table 2, where C/rθ = 1.111, and moreclosely examine comparative statics with respect to σ. Figure 2 depicts exit thresholds
and sales prices, as well as equity, debt, firm, and acquirer’s values for σ ∈ [0.1, 0.4].
Under asymmetric information, we have Case (II-A) for σ < 0.214 and Case (III-A) forσ ≥ 0.214, whereas under symmetric information, we have Case (II-S) for σ < 0.29 andCase (III-S) for σ ≥ 0.29. As was shown in Corollary 1, in Case (II-A), x∗∗
L is lower thanx∗L = xd because shareholders decrease x∗∗
L to mitigate the acquirer’s information rent. Atd∗d∗
σ = 0.214, x∗∗
L jumps upward to xL = x and equals xL = x in Case (III-A). The graph
∗
s
∗∗
∗∗
∗∗
of p∗∗
L = (1−α)PL (xL ) is the same as that of xL . On the other hand, xH equals xH = xH

in Case (II-A) and falls to xd at σ = 0.214. As was shown in Corollary 2, in Case (II-A),∗
the sales price p∗∗
H is slightly lower than pH due to the acquirer’s information rent. Atd
σ = 0.214, p∗∗
H greatly jumps downward because of the low threshold x , bankruptcy costα, and the maximal discount DC (cf. Corollary 2). We can interpret the large diﬀerence∗
between p∗∗
H and pH for σ ∈ [0.214, 0.29) as a fire sale caused by asymmetric information.

For σ ≥ 0.29, the firm goes into bankruptcy even under symmetric information, and∗hence, the diﬀerence between p∗∗
H and pH mitigates but is still larger than that of Case

(II-A).
Even in the presence of asymmetric information, a higher σ increases E ∗∗ (x) because
it increases the value of shareholders’ exit option. On the other hand, a higher σ decreasesD∗∗ (x) because it increases the default probability. Interestingly, in Case (II-A), D∗∗ (x)

24

∗
d
is slightly higher than D∗ (x). This is because x∗∗
L , which is lower than xL = x , decreases

the default risk. At σ = 0.214, D∗∗ (x) and F ∗∗ (x) fall sharply, respectively. At this point,A∗∗ (x) jumps upward, and A∗∗ (x) monotonically increases in σ. From these observations,
we argue that a higher σ causes a wealth transfer from debt holders to shareholders andthe acquirer. In particular, combined with asymmetric information, a higher σ tends toincrease the likelihood of a fire sale, which reduces not only debt values but also firm
values.

4.3.2

Eﬀects of bankruptcy cost α

Figure 3 shows how the sell-out and default regions change with levels of bankruptcy
cost α. Thresholds Bi∗ (i = L, H) (see (14)) are constant under symmetric information
because shareholders, who have limited liability, do not pay bankruptcy costs. However,asymmetric information changes shareholders’ indiﬀerence to α. As was explained after
Proposition 2, a higher α decreases the acquirer’s information rent in Case (II-A). There∗∗ ] monotonically increases in α. Notably, thefore, the region of Case (II-A): (BL∗∗ , BH∗∗ generates a counterintuitive implication that a higher α canmonotonic increase in BH

play a positive role in decreasing the likelihood of a fire sale.
Now we set the baseline parameter values in Table 2 and more closely examine comparative statics with respect to α. Figure 4 depicts exit thresholds and sales prices, as wellas equity, debt, firm, and acquirer’s values for α ∈ [0, 0.5]. Under asymmetric information,
we have Case (III-A) for α ≤ 0.112 and Case (II-A) for α > 0.112, whereas Case (II-S)d∗holds regardless of α under symmetric information. In Case (III-A), x∗∗
L equals xL = x

and jumps downward at α = 0.112. As was shown in Corollary 1, in Case (II-A), shared∗holders decrease x∗∗
L below xL = x to mitigate the acquirer’s information rent. In Case

(II-A), a higher α increases x∗∗
L because a higher α decreases the acquirer’s information∗∗
rent. However, a higher α decreases p∗∗
L = (1 − α)PL (xL ) because the eﬀect of multiplierds∗∗(1 − α) dominates the increase of x∗∗
L . In Case (III-A), xH equals x (< xH ) and jumps

upward to x∗H = xsH at α = 0.112. Notably, at this point, p∗∗
H , which is much lower thanp∗H in Case (III-A), jumps upward nearly to p∗H because of the high threshold x∗∗
H , nobankruptcy cost, and the mitigated DC (cf. Corollary 2). In Case (II-A), a higher αslightly increases p∗∗
H . From these observations, we have a counterintuitive implicationthat a higher α can prevent a fire sale and keep the sales price higher.
In the four lower panels, we can also see the counterintuitive eﬀects of bankruptcy
cost α. Indeed, E ∗∗ (x) monotonically increases in α ≥ 0.112. D∗∗ (x) and F ∗∗ (x) jumpupward at α = 0.112, but except for the jumps, they monotonically decrease with a higher
α. Interestingly, in Case (II-A), D∗∗ (x) is slightly higher than D∗ (x) because of x∗∗
L , whichis lower than x∗L = xd . On the other hand, A∗∗ (x) monotonically decreases with α andjumps downward at α = 0.112. These results suggest that a higher α causes a wealthtransfer from the acquirer to equity (and debt) holders. Then, we conclude that combined25with asymmetric information, a higher α could play a positive role in hindering fire saletransactions and improving equity, debt, and firm values.

4.3.3

Optimal capital structure

To focus on the behavior of a nearly bankrupt firm, which could be far from the optimal
capital structure, we have analyzed the bankruptcy problem for a fixed level of coupon
C. In this section, we explore how asymmetric information aﬀects the firm’s optimal
capital structure. Following the standard literature (e.g., Leland (1994), Goldstein, Ju,
and Leland (2001), and Shibata and Nishihara (2018)), we set C to maximize the initial
firm value. The parameter values other than C are set at the baseline parameter values
in Table 2. Table 4 shows the results under symmetric and asymmetric information. In
Table 4, the leverage and credit spread at the initial time are denoted by LV = D(x)/V (x)and CS = C/D(x) − r, respectively.
With asymmetric information, the firm chooses C = 1.011, which leads to Case (IIA), while under symmetric information, the firm chooses C = 1.526, which leads to Case
(III-S). We can also see in Table 4 that asymmetric information decreases LV and CS.
Then, we conclude that the firm reduces debt financing under asymmetric information.
The mechanism is explained as follows. As was explained in Corollaries 2 and 3, asset
sales in Case (III-A) greatly decrease the debt value due to the most depressed sales price.
The firm lowers the coupon level C to avoid the future cost of a fire sale in Case (III-A).
By decreasing C, the firm chooses Case (II-A) and alleviates the acquirer’s information
rent.
The eﬀects of asymmetric information on the capital structure contrast with those of
Lambrecht and Myers (2008) and Nishihara and Shibata (2017). In their papers, managershareholder conflicts increase debt issuance because a higher level of debt accelerates
bankruptcy and decreases managerial rents. On the other hand, Shibata and Nishihara(2010) focus on managers’ private information on the investment timing rather than thebankruptcy timing and show that manager-shareholder conflicts increase coupon payments
as well as the investment and default triggers but do not aﬀect the leverage and credit
spread.
Our result of decreased leverage with asymmetric information can potentially accountfor the capital structure puzzle that observed real-world leverage ratios are significantly
lower than those predicted by the trade-oﬀ theory (e.g., see Graham (2000)). Morellec(2001) shows that the possibility of partial asset sales before bankruptcy can decreasea leverage ratio in the trade-oﬀ model to a practical level. This paper complements his
result by showing that the possibility of a future fire sale can lead to a practical leverage
ratio even if debt covenants restrict the disposition of assets until bankruptcy.
More notably, the result can also explain the relation between leverage and tangibility
through the asymmetric information channel. Valuations of intangible assets, such as

26

intellectual property, are more likely to vary by acquirer than tangible assets, such as
land, equipment, and inventory. Therefore, firms with more intangible assets can expectthat potential acquirers have more private information about asset valuation, even though
they do not know whether potential acquirers are industry outsiders and/or private firms.
Our capital structure result predicts that firms with more tangible assets tend to havehigher leverage ratios. This prediction is consistent with empirical findings about thepositive relation between leverage and tangibility in Harris and Raviv (1991), Rajan andZingales (1995), and Frank and Goyal (2009). The previous literature explains that lower
tangibility decreases collateral value and increases debt issuance costs. We complement
this stylized view by adding another channel that lower tangibility increases potential
acquirers’ private information about asset valuation and increases the future cost of a firesale.
Our result is also consistent with that of Bharath, Pasquariello, and Wu (2009). Theyshow that asymmetric information tends to decrease leverage ratios using several market
microstructure measures. Note that their market microstructure measures, which capture asymmetric information between informed and uniformed traders, include potential
acquirers’ private information.

5

Extensions and limitations

5.1

Shareholders who cannot commit to the default time

The standard literature (e.g., Leland (1994) and Mella-Barral and Perraudin (1997))
assumes that shareholders cannot commit the default time to debt holders, which leads
shareholders to choose the default time to maximize the ex-post equity value rather than
the firm value. This paper is also based on the same assumption. On the other hand, ourasymmetric information model assumes that shareholders are able to commit not only thesales time but also the default time to the acquirer. Indeed, in Case (II-A) of Proposition
2, shareholders decrease a high-value type’s information rent by committing to the default∗
threshold x∗∗
L (< xL ) for a low-value type. This section considers an alternative case inwhich shareholders cannot credibly commit to the default time for the acquirer.
As in Proposition 2, we solve the problem in three cases, namely, (sL , sH ) = (1, 1), (0, 1),
and (0, 0). For (sL , sH ) = (1, 1) and (0, 0), the solutions are unchanged. For (sL , sH ) =(0, 1), without commitment of the default trigger, shareholders cannot change the default
trigger from xd , which maximizes the ex-post equity value. Then, we have the ex-ante27

equity value as follows:
E (0,1) (x)

{
(
)}
x
C ( x )γxd
C= max (1 − q)(1 − τ )− +−+xH ,pH
r−µ
r
r−µ
r
xd
(
)γ (
{
)}
x
C
x
xH
τC
+ q(1 − τ )− +−+ pH −
,
r−µ
r
xH
r−µ
(1 − τ )r(33)

where (xH , pH ) are optimized subject to ICCs
(
)
x γ
0≥
(PL (xL ) − pH )xH)(( x )γx γ(PH (xH ) − pH ) ≥(1 − α)(PH (xd ) − PL (xd ))xH
xd
and PCPH (xH ) − pH ≥ 0.
By solving problem (33), we have the following solutions:
d
∗∗
s
x∗∗
L = x , xH = xH ,( s )γxH(1 − α)∆axd∗∗s
pH = PH (xH ) −r−µxdEL∗∗ (x) = E d (x),∗∗sEH(x) = EH(x) −( x )γ (1 − τ )(1 − α)∆axd
xd

r−µ

,

∗∗
d
DL
(x) = DL(x),∗∗DH(x) = C/r,
( x )γ (1 − τ )(1 − α)∆axd
.
A∗∗
(x)
=
H
r−µ
xd

Because shareholders cannot decrease the acquirer’s information rent by changing
xL , E (0,1) (x) in the non-commitment model is lower than that of the baseline model.
Therefore, the region of Case (II-A) contracts, and it may vanish for some parametervalues. However, the following key results remain unchanged from the commitment model.
Asymmetric information delays the sales time for a low-value type and lowers the sales
price for a high-value type. When the screening cost is high, shareholders can change
the exit choice from sell-out to default. If shareholders choose default regardless of theacquirer’s type, the acquirer’s information rent reaches the maximum level, and debt
holders suﬀer from the depressed sales price. In the optimal capital structure, the firm
decreases debt issuance to avoid the risk of a fire sale.

5.2

Debt holders who prefer to scrap assets

To focus on the acquirer’s gain, the baseline model assumes that aL ≥ qaH . This assumption is critical to our results regarding the acquirer’s gain. Now we consider the case of28

aL < qaH . Shareholders’ exit decisions are essentially unchanged from the baseline model,
whereas debt holders can change their liquidation policy. In Case (III-A), former debt
holders set the oﬀer price (1 − α)PH (xd ) instead of (1 − α)PL (xd ). They scrap the firm
and gain the scrap value (1 − α)θ unless the acquirer pays (1 − α)PH (xd ). Indeed, theexpected gain from this policy, i.e., q(1 − α)PH (xd ) + (1 − q)(1 − α)θ, is higher than the
expected gain from the baseline policy, i.e., (1−α)PL (xd ). Then, in Case (III-A), unlike inthe baseline model, the acquirer’s information rent is not maximal but zero. Although nowealth transfer is transferred from debt holders to the acquirer, debt holders suﬀer greatloss from asymmetric information in the scrap case. The key results except the acquirer’s
gain in Case (III-A) remain unchanged from the baseline model.

5.3

Debt holders who can operate the firm after bankruptcy

The baseline model presumes liquidation bankruptcy where former debt holders are forced
to either sell out or scrap the bankrupt firm immediately after default. This assumption
is critical to our implications for fire sales. In general, debt holders may choose operating
concern bankruptcy rather than liquidation bankruptcy. This section supposes that former
debt holders can receive EBIT flows (1 − α)X(t) by operating the post-bankruptcy firm
themselves. They can also optimize the sales timing to the acquirer.
In Case (III-A), debt holders’ behavior changes from the baseline model. Actually,
after bankruptcy without separation, debt holders do not have to liquidate assets at a low
price. As with shareholders with C = 0 in Case (I-A), they can optimize the screening
oﬀer to the acquirer with private information. The former debt holders’ optimal oﬀer isessentially the same as the oﬀer in Case (I-A) with C = 0 of Proposition 2. They cut
the information rent to a high-value type by decreasing the sell-out trigger to a low-value
type. Then, even in Case (III-A), debt holders can mitigate the acquirer’s information
rent and avoid great loss.
The results in Case (II-A) also diﬀer from the baseline model. Mostly, shareholders do
not change the default trigger from the first-best trigger xd for a low-value type because
the former debt holders’ sales trigger after bankruptcy determines a high-value type’s
imitation value. The acquirer’s information rent tends to decrease because a low-value
type’s acquisition timing is later than that of the baseline model. The implications otherthan fire sale discounts remain unchanged from the baseline model.

5.4

Acquirer’s method of payment

We have assumed that the acquisition is paid with cash. In reality, some acquisitionsare (partially) paid through stocks and other mediums (e.g., Chemmanur, Paeglis, andSimonyan (2009)). This section discusses the acquirer’s medium of payment. From theacquirer’s viewpoint, the use of cash maximizes the information rent. Without financial

29

constraints, the acquirer prefers to use cash. In the presence of financial constraints,
however, the acquirer may be forced to use not only cash but also stock. When a lowvalue type uses stock in the acquisition of assets, a high-value type cannot achieve theinformation rent derived in Proposition 2. Indeed, the imitation value (cf. the righthand sides of (18) and (23)) decreases because stock of a high-value type who imitates a
low-value type is undervalued as a low-value type by the target firm. The optimal mix ofcash and stock can be determined by the trade-oﬀ between information rent and financing
costs.
On the other hand, shareholders of the target prefer a stock oﬀer to a cash oﬀer because
a stock oﬀer can reduce the acquirer’s information rent. Similarly, they can reduce theacquirer’s information rent by payment contingent on the performance (e.g., earnout).
Although this paper concentrates on the acquirer’s private information, in reality, theseller is also likely to have private information (see Chemmanur, Paeglis, and Simonyan(2009) and Nishihara and Shibata (2018)). In such a case, the seller’s information rent
decreases with use of stock or contingent payment. Thus, the medium of payment depends
on which party has the right to determine the medium, which party’s private information
is greater, and the severity of the acquirer’s financing constraints.

6

Conclusions

In this paper, we examined a dynamic model in which a distressed firm optimizes anexit choice between sell-out and default as well as its timing. In the model, we revealed
how the acquirer’s private information about asset valuation distorts shareholders’ exit
choice, exit timing, as well as equity, debt, firm, and acquirer’s values. The results underasymmetric information are summarized as follows.
Shareholders of the distressed firm decrease the liquidation threshold for a low-value
type and decrease the sales price for a high-value type to mitigate the acquirer’s information rent. Notably, shareholders can change the exit choice from sell-out to default if thescreening cost is high. In such a situation, in addition to bankruptcy costs, the acquirer’s
information rent, which reaches the maximum level, greatly decreases the asset price anddebt value. Through this asymmetric information channel, we explained the mechanismof a fire sale on liquidation bankruptcy. Furthermore, we showed that higher volatility,
leverage, and asymmetric information increase the likelihood of a fire sale, but higher
bankruptcy costs could counterintuitively decrease the likelihood of a fire sale. Regarding
the optimal capital structure, with asymmetric information, the firm takes a precautious
leverage ratio to avoid the risk of a fire sale. Our results fit well empirical findings aboutfire sales, acquirers’ gain, and capital structure.

30

A

Proof of Proposition 1

Below, we will prove that E d (x) ≤ Eis (x) holds if and only if C/rθ ≤ Bi∗ holds. Defineθ̃ = θ −τC.
(1 − τ )rFor θ̃ ≤ 0, i.e., C/rθ ≥ (1 − τ )/τ , E d (x) > Eis (x) holds. Suppose that θ̃ > 0. By (9) and(11), the inequality E d (x) ≤ Eis (x) is equivalent to
(

⇔
⇔
⇔
⇔
⇔
⇔

1
xd

)γ (
) ( )γ ()
1(ai − 1)xsi
xd
C
−
+
≤
+
θ̃
r−µ
r
xsi
r−µ
( )γ
( )γ
C
θ̃
1
1
≤
(1 − γ)rxsi
1−γxd( d )γ
x
C
≤
xs
rθ̃
( i
)
C(1 − ai ) γ
C
≤
rθ̃
rθ̃
C
≤ b∗−1
i
rθ̃
(
)
C
τC
∗−1
≤ bi
θ−
r
(1 − τ )r
C
1−τ
≤
= Bi∗
rθ
τ + (1 − τ )b∗i

The proof is complete.

B

Proof of Proposition 2.

We derive the equity values E (1,1) (x) (Step 1) and E (0,1) (x) (Step 2) and compare themwith E d (x) (Step 3).
Step 1. Solution for (sL , sH ) = (1, 1): Sell-out to any type.
∗∗ ∗∗ ∗∗We derive the optimal solution (x∗∗L , xH , pL , pH ) to problem (16) without constraints(17) and (20), and we will prove that the solution also satisfies (17) and (20) later. In the
optimum, clearly constraints (18) and (19) are binding, and hence, we have
)
( )γ
(
x
x γ
(PH (xH ) − pH ) =(PH (xL ) − pL )xHxL
PL (xL ) − pL = 0

(34)
(35)

We denote θ̃ = θ − τ C/(1 − τ )r. By substituting (34) and (35) into the objective function31

(16), the problem can be reduce to{( )γ ()}
xCx(1 − aL )xL
max (1 − q)(1 − τ )
− +
−
+ θ̃
xL ,xH
r−µ
r
xL
r−µ
{
(
) ( )γ
) (
}
x
C
x γ
(1 − aH )xHx∆axL+ q(1 − τ )− +−+ θ̃ −
r−µ
r
xH
r−µ
xL
r−µ
{
( )γ ()
xCx(1 − aL + ∆aq/(1 − q))xL=(1 − τ )− + (1 − q) max
−
+ θ̃
xL
r−µ
r
xL
r−µ
(
) (
)}
x γ
(1 − aH )xH+q max
−
+ θ̃
.
xH
xH
r−µ
Because E d (x) clearly dominates E (1,1) (x) and E (0,1) (x) for θ̃ ≤ 0, from now on we
presume that θ̃ > 0. By the first order condition, we can easily derive the optimal
solutionx∗∗L =(r − µ)θ̃γ, x∗∗ = xsH ,γ − 1 1 − aL + ∆aq/(1 − q) H

where xsH is defined by (10) with i = H, and by (34) and (35), we have( s )γ
xH
∆ax∗∗
∗∗
∗∗
∗∗
s
L
pL = PL (xL ), pH = PH (xH ) −
.
x∗∗
r
−
µ
L

(36)

(37)

By (36) and (37), the solution satisfies (17) and (20). Then, this is the optimal solution
to problem (16) with all constraints, and we have the ex-ante equity value
{
}
(
)γ
(
)γ
x
C
x
θ̃
x
θ̃
E (1,1) (x) = (1 − τ )− + (1 − q)
+q
,
r−µ
r
x∗∗
1−γ
xsH
1−γ
L

(38)

We can easily derive the equity and debt values in Case (I-A) in Proposition 2.
Step 2. Solution for (sL , sH ) = (0, 1): Sell-out to only a high-value type.
∗∗ ∗∗As in the previous case, we derive the optimal solution (x∗∗L , xH , pH ) to problem (21)without constraints (22) and (24), and we will prove that the solution also satisfies (22)and (24) later. Clearly, in the optimum, constraint (23) is binding, and hence, we have
)
( )γ
(
x γ
x
(PH (xH ) − pH ) =(1 − α)(PH (xL ) − PL (xL ))
(39)
xH
xL
By substituting (39) into the objective function (21), the problem can be reduce to{( )γ (
)}
xL
x
C
x
C
max (1 − q)(1 − τ )−− ++xL ,xH
r−µ
r
xL
r−µ
r
)γ (
}
{
(
) ( )γ(1 − aH )xHx
Cxx(1 − α)∆axL−+ q(1 − τ )− ++ θ̃ −
r−µ
r
xH
r−µ
xL
r−µ
( )γ (){
x(1 + (1 − α)∆aq/(1 − q))xL C
xC−=(1 − τ )− + (1 − q) max
+
xL
r−µ
r
xL
r−µ
r
(
)γ (
)}
x
(1 − aH )xH+q max
−
+ θ̃
xH
xH
r−µ
By the first order condition, we can easily derive the optimal solution
x∗∗
L =

C
γ(r − µ), x∗∗ = xsH ,γ − 1 r(1 + (1 − α)∆aq/(1 − q)) H

32

(40)

and by (39) and (40), we havesp∗∗
H = PH (xH ) −

(

xsH
x∗∗
L

)γ

(1 − α)∆ax∗∗
L
.
r−µ

(41)

∗ , the solution also satisfies
By (40) and (41), the solution satisfies (24). For C/rθ < BH∗∗∗(22) because xsH > xd > x∗∗
L holds. We will later check that BH < BH . Then, thissolution agrees with the optimal solution to problem (21) with all constraints, and wehave the ex-ante equity value
}
{
(
)γ
(
)γ
x
x
x
C
C
θ̃
E (0,1) (x) = (1 − τ )− + (1 − q)+q
. (42)r−µrx∗∗(1 − γ)rxsH
1−γ
L
We can easily derive the equity and debt values in Case (II-A) in Proposition 2.
Step 3. Calculating max{E (1,1) (x), E (0,1) (x), E d (x)}.
Below, we distinguish between x∗∗
L in (36) and (40) by superscripts (1, 1) and (0, 1).
By (11) and (42), E (0,1) (x) ≤ E d (x) holds if and only if
(
)γ
(
)
( )γ
C
1
C
1 γ θ̃
1(1 − q)+q≤sd∗∗(0,1)
(1
−
γ)r
x
1
−
γ
(1
−
γ)r
x
xL
H
(
(
)γ )
( d )γ
1 1−q
xd
x
C
≤
−
⇔
s
∗∗(0,1)
xH
q
q
rθ̃
xL
(
)γ
(1 − aH )C
1 − (1 − q)(1 + (1 − α)∆aq/(1 − q))γ C
⇔
≤
q
rθ̃
rθ̃
C
≤
⇔
b∗∗−1
H
rθ̃
1−τ
C
∗∗
⇔
BH =
≤ .
τ + (1 − τ )b∗∗rθH∗∗ < B ∗ follows from b∗∗ > b∗ .
Note that BH
H
H
H

By (38) and (42), E (1,1) (x) ≤ E (0,1) (x) holds if and only if
(
(
)γ
)γ
1
1
C
θ̃
≤
∗∗(1,1)∗∗(0,1)
1
−
γ
(1
−
γ)r
xL
xL
( ∗∗(0,1) )γxLC⇔≤∗∗(1,1)rθ̃xL()γ
C(1 − aL + ∆aq/(1 − q))C⇔≤rθ̃(1 + (1 − α)∆aq/(1 − q))
rθ̃
C
⇔
b∗∗−1
≤
L
rθ̃
C
1
−
τ
≤
⇔
BL∗∗ =∗∗
τ + (1 − τ )bLrθ
∗Note that BL∗∗ < BL∗ follows from b∗∗
L > bL .
∗∗ as follows. Consider a case where C/rθ = B ∗ . We
We can also show that BL∗ ≤ BH
L

have ELs (x) = E d (x). On the other hand, by definition, E (1,1) (x) is not lower than ELs (x).
Then, we have E (0,1) (x) > E (1,1) (x) ≥ ELs (x) = E d (x) for C/rθ = BL∗ (> BL∗∗ ). This∗∗ . Thus, we have (s∗∗ , s∗∗ ) = (1, 1) for C/rθ ≤ B ∗∗ , (s∗∗ , s∗∗ ) = (0, 1) for
implies BL∗ < BH
L
H
L
L
H
∗∗ ], and (s∗∗ , s∗∗ ) = (0, 1) for C/rθ > B ∗∗ , The proof is complete.
C/rθ ∈ (BL∗∗ , BH
L
H
H

33

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37

Table 1: Cases under symmetric and asymmetric information.
∗∗
∗∗
∗
∗
] (BH
, BH
] (BH
, +∞)
C/rθ [0, BL∗∗ ] (BL∗∗ , BL∗ ] (BL∗ , BH
Sym.

(I-S)

(I-S)

(II-S)(II-S)(III-S)

Asym.

(I-A)

(II-A)(II-A)(III-A)(III-A)r

Table 2: Baseline parameter values.
µστ
x a H aL
θ
α

0.06

0.01

0.2 0.15

2

0.5 0.25 15

qC

0.3 0.5

1xL

xH

Table 3: Baseline results.
pL
pH
E(x)
D(x)

0.5

0.724

12.25

22.235 20.634

16.39137.0240

Asym. 0.426

0.724

11.989 21.564 20.558

16.43736.9950.0621

E(x)

D(x)

F (x)

A(x)
0

Sym.

F (x)

A(x)

Table 4: Optimal capital structure.
C

CS

xL

xH

0.0068

0.763

0.763

13.171 15.841 14.419

22.85837.278

Asym. 1.011 0.449 0.00087

0.43

0.722

12.006 21.523 20.406

16.60837.015 0.0638

Sym.

LV

1.526 0.613

38

pL

pH

1.35
B∗∗
L

1.3

B∗∗
H

1.25

B∗L
B∗H

1.2
1.15
1.1
1.05
1
0.95
0.9
0.85
0.1

0.15

0.2

0.25
σ

0.3

0.35

0.4

Figure 1: Thresholds Bi∗∗ and Bi∗ (i = L, H) with respect to σ. The other parameter values
are set in Table 2.

39

1

1

0.8

x∗∗
L

0.8

x∗∗
H

0.6

x∗L

0.6

x∗H

0.4

0.4

0.2

0.2

0
0.1

0.2

σ

0.3

0
0.1

0.4

13

p∗L

20

11.5

15
0.2

σ

0.3

0.3

0.4

p∗∗
H

25

12

11
0.1

σ

30

p∗∗
L

12.5

0.2

10
0.1

0.4

24

p∗H

0.2

σ

0.3

18

D∗∗(x)

17

22

0.4

D∗(x)

16
∗∗

20

E (x)

15

E∗(x)
18
0.1

0.2

σ

0.3

14
0.1

0.4

39

σ

0.3

0.4

0.3

0.4

0.2
∗∗

F (x)
38

0.2

0.15

F∗(x)

A∗∗(x)

0.1
37
36
0.1

0.05
0.2

σ

0.3

0
0.1

0.4

0.2

σ

Figure 2: Comparative statics with respect to σ. The other parameter values are set in Table 2.
Cases (II-A) and (II-S) hold for σ < 0.214. Cases (III-A) and (II-S) hold for σ ∈ [0.214, 0.29).
Cases (III-A) and (III-S) hold for σ ≥ 0.29.

40

1.25

1.2

1.15
B∗∗
L

1.1

B∗∗
H
B∗L

1.05

B∗H

1

0.95

0.9

0

0.05

0.1

0.15

0.2

0.25
α

0.3

0.35

0.4

0.45

0.5

Figure 3: Thresholds Bi∗∗ and Bi∗ (i = L, H) with respect to α. The other parameter values
are set in Table 2.

41

0.55

x∗∗
L

0.8

x∗L

0.7

0.5

0.45

0.4

0

0.1

0.2

α

0.3

0.4

0.6

x∗∗
H

0.5

x∗H

0.4

0.5

20

0

0.1

0.2

α

0.3

0.4

0.5

25

15
10

p∗∗
L

5

p∗L

0

0

0.1

0.2

20

p∗∗
H
p∗H

α

0.3

0.4

15

0.5

E∗∗(x)

20.7

0.1

0.2

α

0.3

0.4

0.5

17

∗

E (x)

20.65

0

D∗∗(x)

16.8

D∗(x)

16.6
20.6
16.4
20.55
20.5

16.2
0

0.1

0.2

α

0.3

0.4

16

0.5

37.5

0

0.1

0.2

α

0.3

0.5

0.2

A∗∗(x)

0.15
37

0.1

F∗∗(x)

0.05

F∗(x)
36.5

0.4

0

0.1

0.2

α

0.3

0.4

0

0.5

0

0.1

0.2

α

0.3

0.4

0.5

Figure 4: Comparative statics with respect to α. The other parameter values are set in Table
2. Cases (III-A) and (II-S) hold for α ≤ 0.112, while Cases (II-A) and (II-S) hold for α > 0.112.

42