Discussion Papers In Economics
And Business
The effects of asset liquidity on dynamic
bankruptcy decisions
Michi NISHIHARA,Takashi SHIBATA

Discussion Paper 19-12

September 2019

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe eﬀects of asset liquidity on dynamic bankruptcy
decisions ∗Michi NISHIHARA†,

Takashi SHIBATA‡

Abstract
We develop a dynamic bankruptcy model with asset illiquidity. In the model, a distressed firm chooses between sell-out and default, as well as its timing under the assumption that sell-out is feasible only at Poisson jump times, where the arrival rate of acquirers
stands for asset liquidity. With lower asset liquidity, the firm increases the sell-out region to mitigate the risk of not finding an acquirer until bankruptcy. Despite the larger
sell-out region, lower asset liquidity increases the default probability and decreases theequity, debt, and firm values. In the optimal capital structure, with lower asset liquidity,
the firm reduces leverage, but the cautious capital structure does not fully oﬀset the increased default risk. The stock price reaction caused by sell-out depends on the sell-out
timing. When the firm’s asset value is not suﬃciently high, the stock price jump size isan inverted U-shape with the economic state variable. Lower asset liquidity increases thejump size due to greater surprise. These results fit empirical observations.
JEL Classifications Code: G13; G32; G33.
Keywords: liquidation; illiquidity; real option; M&A.

∗

This version was written on 12 September, 2019. We thank Makoto Gotoh, Sangho Kim, and Kazutoshi

Yamazaki for helpful comments. We also thank the participants at FMA 2018 in Kyoto, 9th Global BusinessResearch Conference in Kathmandu, 27th EBES Conference in Denpasar, ORSJ 2019 spring conference in
Chiba, and 11th Global Conference in Amsterdam for their feedbacks. 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

1

1

Introduction

Asset liquidity (or asset deployability) aﬀects firms’ operating and financing decisions(e.g., Shleifer and Vishny (1992), Schlingemann, Stulz, and Walkling (2002), and OrtizMolina and Phillips (2014)). For firms in financial distress, asset illiquidity is crucial
because they can sell assets to avoid bankruptcy. As shown in Hotchkiss and Mooradian(1998) and Maksimovic and Phillips (1998), some firms are successfully acquired by otherfirms as a form of exit strategy, but others fail. For instance, the Japanese electronics
manufacturer Sharp Corporation was acquired by the Taiwanese manufacturer Foxconn
to get through financial distress, while the American retail company Circuit City filed forbankruptcy after it failed to be acquired by the American video company Blockbuster.
The objective of this paper is to explore the eﬀects of asset illiquidity on a distressed
firm’s exit decisions. We build a model of a firm with console debt as in the standard literature (e.g., Black and Cox (1976) and Leland (1994)). As in Mella-Barral and Perraudin(1997), Nishihara and Shibata (2017), and Nishihara and Shibata (2018), the firm cansell out (i.e., be acquired by another firm) to avoid bankruptcy. On sell-out, shareholdersof the firm obtain the residual value (i.e., the sales price minus the face value of debt),
while debt holders are repaid the face value of debt. On default, shareholders stop coupon
payments of debt, and the former debt holders take over the firm, where a fraction of thefirm value is lost to bankruptcy costs. Shareholders (or equivalently, managers acting
on behalf of shareholders’ interests) optimize the exit choice between sell-out and default
as well as its timing. A key diﬀerence from previous literature is an assumption thatsell-out is feasible only at Poisson jump times. This assumption captures the illiquidity
associated with searching for an acquirer.1 Throughout the paper, we regard the arrival
rate of acquirers as the firm’s asset liquidity.
We analytically derive model solutions, which are classified into the following cases.
When asset value over debt value and asset liquidity are high, the firm does not default but
sells out for a suﬃciently low state variable. When asset value over debt value is low, thefirm does not sell out but defaults at the default threshold. Notably, for the intermediateparameter values, the firm has both sell-out and default regions. If an acquirer fortunately
appears in the sell-out region until the state variable hits the default threshold, the firm
succeeds in selling out. Otherwise, the firm defaults. For relatively high asset values, thesell-out region is below a certain threshold (i.e., threshold policy of sell-out), whereas for1

Cornett, Tanyeri, and Tehranian (2011) and Wang (2018) empirically show the low predictability of M&As.

Rhodes-Kropf and Robinson (2008) also examine a merger market with search frictions.

2relatively low asset values, the sell-out region is between an interval strictly higher thanthe default threshold (i.e., interval policy of sell-out). Compared with the deterministic
choice between sell-out and default in Mella-Barral and Perraudin (1997), Nishihara andShibata (2017), and Nishihara and Shibata (2018), our result of the stochastic choice fits
empirical observations well.
In the model, we reveal the eﬀects of asset liquidity on the firm’s exit policy, default
probability, equity, debt, and firm values. With lower asset liquidity, the firm increases thesell-out region to mitigate the risk of not finding an acquirer until bankruptcy takes place.
Notably, this earnest asset sales policy cannot fully oﬀset the decreased liquidity eﬀect.
Lower asset liquidity, despite the larger sell-out region, increases the default probability
and decreases the equity, debt, and firm values. These results are consistent with empiricalevidence (e.g., Ortiz-Molina and Phillips (2014), Song and Walkling (2000), and Cornett,
Tanyeri, and Tehranian (2011)). In fact, Ortiz-Molina and Phillips (2014) show thatlower asset liquidity reduces a firm’s operating flexibility and increases the cost of capitaland the bankruptcy risk. Song and Walkling (2000) and Cornett, Tanyeri, and Tehranian(2011) show that higher acquisition probabilities (which can be interpreted as higher asset
liquidity in our model) tend to increase target stock prices.
In the optimal capital structure, with lower asset liquidity, the firm reduces its leverage
to mitigate the increased bankruptcy risk. Intriguingly, the cautious capital structure andearnest asset sales policy cannot fully oﬀset the decreased liquidity eﬀect. Lower asset
liquidity thus increases the default probability and decreases the debt and firm values. The
positive relationship between asset liquidity and leverage is consistent with the empiricalobservations in Sibilkov (2009) and Campello and Giambon (2013).
The model also yields an upward jump of the equity value at the sell-out time. The
upward jump can be interpreted as a stock price reaction to the acquisition. Our modelshows that the stock price reaction greatly depends on the sell-out timing (i.e., the economic state variable when the firm is acquired). If asset value is suﬃciently high, the jump
size decreases with the state variable due to the threshold policy of sell-out. This implies
that being acquired is more beneficial to shareholders as the firm is closer to default.
More notably, if asset value is not suﬃciently high, the jump size becomes an inverted
U-shape with the state variable due to the interval-type policy of sell-out. This suggeststhat being acquired is not very beneficial to shareholders when the distance to default iseither too short or too long. These results highlight the dependence of acquisition timing
on stock price reactions and help explain the dispersion of target stock price reactions.

3

We also show that lower asset liquidity leads to a more positive stock price reaction because of greater surprise. This result aligns with Cornett, Tanyeri, and Tehranian (2011)’s
empirical finding that stock price reactions increase with lower merger anticipation.
The remainder of this paper is organized as follows. Section 2 briefly reviews therelated literature. Section 3 introduces the model setup. After Section 4.1 shows themodel solutions for the unlevered firm, Section 4.2 shows the model solutions for thelevered firm. Section 5 demonstrates numerical examples and economic implications.
After Section 6 discusses the model extensions, Section 7 concludes the paper.

2

Literature review

This paper explores the interactions of two real options: sell-out and default. Althoughmany real options models focus on investment timing, some papers investigate exit options(for a recent review of the real options literature, see Trigeorgis and Tsekrekos (2018)).
For instance, Hagspiel, Huisman, and Nunes (2016) examine a firm’s choice between
investment and exit, whereas Guerra, Kort, Nunes, and Oliveira (2018) investigate afirm’s mothballing option before permanent exit. The above papers do not focus on alevered firm’s default decision. Regarding a default option, Black and Cox (1976) first
derive default timing, which is endogenously determined in shareholders’ interests, andLeland (1994) derives the optimal capital structure determined by the tradeoﬀ between
the tax benefits of debt and bankruptcy cost. As in most papers, we borrow these standard
frameworks of endogenous default and optimal capital structure.
By extending the standard frameworks, many papers investigate joint investment andfinancing decisions. For instance, Mauer and Sarkar (2005), Shibata and Nishihara (2012),
and Sundaresan, Wang, and Yang (2015) derive optimal investment timing and capital
structure. Morellec, Valta, and Zhdanov (2015) and Shibata and Nishihara (2015) examine optimal investment and financing choices between market debt and private debt.
Hackbarth, Mathews, and Robinson (2014) examine optimal investment, financing, andorganizational form. Charalambides and Koussis (2018) investigate optimal investment
and operation with financing by debt allowing future coupon reduction. Lukas and Thiergart (2019) explore the eﬀects of debt financing on a real option held jointly by a firm anda government. These papers examine the interactions of default and investment options.
This paper considers not an investment option but a sell-out option in addition to adefault option. In this sense, Mella-Barral and Perraudin (1997), Lambrecht and Myers

4(2008), Nishihara and Shibata (2017), and Nishihara and Shibata (2018), who consider
the choice between sell-out (or scrap) and default, are most closely related to this paper.
Mella-Barral and Perraudin (1997) show that debt renegotiation prevents shareholders
from early default and leads to eﬃcient liquidation timing. Lambrecht and Myers (2008)
show that debt can reduce manager-shareholder conflicts and eﬃciently speed up liquidation. Nishihara and Shibata (2017) and Nishihara and Shibata (2018) show that
information asymmetries can change a firm’s exit policy from sell-out to default. However, they do not focus on the illiquidity risk of not finding an acquirer; in their models,sell-out is feasible at any time.
Although the above papers consider full liquidation with debt repayment rather thanpartial liquidation, Morellec (2001) develops the model in which a distressed firm canliquidate assets partially and sequentially until bankruptcy. He shows that for unsecured
debt, lower liquidation costs play negative roles in increasing bankruptcy risk and decreasing debt value through the channel of managerial asset disposition. Nishihara andShibata (2016) also examine partial liquidation with debt restructuring and show thatpartial asset sales decrease equity value. These models do not include the risk of not
finding an asset acquirer.
He and Xiong (2012), He and Milbradt (2014), and Chen, Cui, He, and Milbradt(2018) explore the eﬀects of debt market illiquidity on bankruptcy decisions. They showthat debt market illiquidity hastens bankruptcy timing and increases default risk via thechannel of debt rollover risk. They focus on the illiquidity of the secondary debt market
rather than real asset illiquidity; in fact, these models do not include real asset sales.
Technically, this paper treats the optimal stopping (sell-out timing) problem constrained within Poisson jump times. The unlevered firm’s sell-out timing problem in
Section 4.1 is essentially the same as the problem solved in Dupuis and Wang (2002).
By extending this mathematical framework, Hugonnier, Malamud, and Morellec (2015a),
Hugonnier, Malamud, and Morellec (2015b), and Morellec, Valta, and Zhdanov (2015)
reveal the eﬀects of the illiquidity of fund-raising on investment, cash holdings, and capital structure. They focus on capital and credit supply illiquidity rather than real asset
illiquidity.
Despite the growing literature on dynamic corporate finance issues, to our knowledge,
no paper investigates the eﬀects of asset illiquidity (i.e., the risk of not finding an acquirer)
on the dynamic bankruptcy procedure. The main contribution of this paper is to reveal
how asset illiquidity distorts a firm’s exit choice between sell-out and default, exit timing,5

bankruptcy risk, and equity, debt, and firm values, as well as optimal capital structure.
In addition, this paper also yields new implications about stock market reactions in theacquisition period.

3

Model Setup

3.1

Firm until exit

The model builds on the standard setup (e.g., Mella-Barral and Perraudin (1997), Goldstein, Ju, and Leland (2001), and Lambrecht and Myers (2008)). Consider a firm withconsole debt with coupon C, i.e., the firm continues to pay coupon C to debt holders
until bankruptcy. First, we assume an exogenous coupon C, although in Section 5 we also
examine the optimal capital structure. 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 on behalf of shareholders’ interests, and hence we
do not 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.

3.2

Asset sales opportunities

Following a Poisson process with the arrival rate λ, potential acquirers arrive and bid
P (X(t)) =

aX(t)
+θ
r−µ

(1)

for the firm, where a ∈ [0, 1) and θ > 0. We assume the linear function (1) following Lambrecht (2001), Lambrecht and Myers (2008), Nishihara and Shibata (2016), and Nishihara
and Shibata (2018). Equation (1) means that after partial assets are scrapped at a fixedprice θ, an acquirer perpetually receives cash flows aX(t) from the remaining assets, wherea ∈ [0, 1) represents that the revenues contract due to the partial liquidation. Transaction
costs and the acquirer’s gains can also be included in the parameters a and θ. Section 6.2
discusses a case in which an acquirer’s bid price is contingent on the target stock price.

6

For model tractability, we assume that the Poisson process is independent of X(t).2 If thefirm rejects an acquirer’s proposal, the firm has to wait until another acquirer arrives.
The novelty of the model is the Poisson arrival of asset sales opportunities. We regard
the arrival rate λ as the degree of asset liquidity. A lower λ means lower asset liquidity because 1/λ equals the expected duration between acquirers’ arrivals. In reality, distressedfirms do not always find acquirers that oﬀer favorable bid prices, although previous papers (e.g., Lambrecht and Myers (2008), Nishihara and Shibata (2016), and Nishihara
and Shibata (2018)) consider liquid models, which correspond to λ = ∞ in our model.
Our illiquid model can capture the eﬀects of asset illiquidity on the bankruptcy process.
Hugonnier, Malamud, and Morellec (2015a), Hugonnier, Malamud, and Morellec (2015b),
and Morellec, Valta, and Zhdanov (2015) propose dynamic investment models with illiquidity of fund-raising, which is modeled by a Poisson process, although our model focuseson the illiquidity of real asset sales rather than fund-raising.

3.3

Exit choice between sell-out and default

As in Mella-Barral and Perraudin (1997), Lambrecht and Myers (2008), and Nishihara
and Shibata (2018), we consider two types of the firm’s exit. A successful exit without
formal bankruptcy is called sell-out. On sell-out, shareholders of the distressed firm sell
all assets to an acquirer at price P (X(t)). According to the absolute priority rule (APR)of debt, debt holders are repaid the face value of debt, which equals C/r for the console
debt. Then, shareholders receive the residual value, i.e., (1 − τ )P (X(t)) − C/r. Notethat the asset redeployment also increases social welfare. Indeed, P (X(t)) > X(t)/(r − µ)holds because the residual value is nonnegative when shareholders prefers to sell out.
The other exit is called default, which leads to a formal bankruptcy. On default,
shareholders stop coupon payments to debt holders. According to the APR, debt holders
take over the firm, and shareholders receive nothing. As in the standard literature (e.g.,
Leland (1994), Mella-Barral and Perraudin (1997), and Goldstein, Ju, and Leland (2001)),
the former debt holders can operate the firm as a going concern, but a fraction α ∈ (0, 1)of the firm’s asset value is lost to the bankruptcy costs (filing fees, attorney fees, etc.).
Contrasted to sell-out, the deadweight costs of bankruptcy lead to social loss.
2

Hugonnier, Malamud, and Morellec (2015a), Hugonnier, Malamud, and Morellec (2015b), and Morellec,

Valta, and Zhdanov (2015) also assume independence between the economic state process and the Poissonprocess representing illiquidity of fund-raising. In fact, without the independence, we cannot analytically derive
any solution.

7

For simplicity, as in Lambrecht and Myers (2008), and Nishihara and Shibata (2018),
we assume no opportunity of debt renegotiation and restructuring.3 As in the standard
literature, shareholders of the distressed firm makes the exit decision in their own interests.
They do not take into account of debt in place, which leads to agency conflicts between
shareholders and debt holders, and debt is priced at the initial time under the rational
expectation of shareholders’ exit policy.

4

Model Solutions

4.1

Unlevered firm

As a benchmark, this subsection examines the all-equity firm’s exit policy. Suppose that
there is no coupon payment (i.e., C = 0). Because there is no coupon payment, there
is no possibility of default. Instead, the firm can sell out when an oﬀer is favorable.
Technically, the firm’s optimal sell-out timing problem is regarded as an optimal stopping
problem constrained to Poisson jump times, which was solved in Dupuis and Wang (2002).
We use EU (x) to denote the firm value when an asset sale opportunity is not available,
and we use E˜U (x) to denote the firm value when an acquirer arrives. We have E˜U (x) =max{(1 − τ )P (x), EU (x)} because the firm value E˜U (x) turns to EU (x) by forgoing an
asset sale opportunity. As proved in Dupuis and Wang (2002), the optimal sell-out policy
becomes the threshold policy, i.e., inf{t ≥ 0 | X(t) ≤ xsU }, and EU (x) satisfies ordinary
diﬀerential equations (ODEs):µxEU′ (x) + 0.5σ 2 x2 EU′′ (x) + λ((1 − τ )P (x) − EU (x)) + (1 − τ )x = rEU (x) (x < xsU ),(2)µxEU′ (x) + 0.5σ 2 x2 EU′′ (x) + (1 − τ )x = rEU (x) (x > xsU ),(3)

where λ((1 − τ )P (x) − EU (x)) in (2) corresponds to the fact that EU (x) changes to(1 − τ )P (x) with probability λdt in infinitesimal time interval dt in the sell-out regionx < xsU . As shown in Dupuis and Wang (2002), EUs (x) is continuously diﬀerentiable at
xsU 4 and also satisfies boundary conditions EUs (xsU ) = (1 − τ )P (xsU ), limx→0 EUs (x) < ∞,3

Several papers (e.g., Morellec, Valta, and Zhdanov (2015), and Shibata and Nishihara (2015)) distinguishbetween private debt, which can be renegotiated, and market debt, which cannot be renegotiated. In thiscontext, our model is more relevant to firms funded with market debt than firms funded with private debt.
4
This is not the smooth pasting (i.e., optimality) condition in Dixit and Pindyck (1994), but it is the same
condition as the piecewise C 2 property of the value function in Morellec, Valta, and Zhdanov (2015). Technically,
this property follows from Theorem 4.4.9 in Karatzas and Shreve (1998).

8and limx→∞ EUs (x)/x < ∞. The boundary condition EUs (xsU ) = (1 − τ )P (xsU ) means thatthe firm is indiﬀerent to whether to accept or reject an acquirer’s oﬀer at xsU (i.e., theoptimality of xsU ). The other boundary conditions are trivial. By solving (2) and (3) withthese boundary conditions, we have the following proposition, where we define, for y > 0,√(µ
)2 2(r + y)µβy = 0.5 − 2 +−0.5+(> 1),
(4)
2
σ
σ2
√ σ
(µ
)2 2(r + y)µγy = 0.5 − 2 −−0.5+(< 0),(5)σσ2σ2and, we simplify the notations by β = β0 and γ = γ0 . For the proof, refer to Appendix A.
Proposition 1
EU (x){
( )βλ (
)}
s

(a
−
1)x
λθ
x
rθ
(r
+
aλ
−
µ)x

U

+
(x ≤ xsU ), (1 − τ ) (r − µ)(r + λ − µ) + r + λ + xs
r
+
λ
−
µ
r
+
λ
U
=
( )γ ()}{
s
(a − 1)xU
x
x


+
+θ
(x > xsU ), (1 − τ )
s
r−µ
xU
r−µ
(6)
where the sell-out threshold xsU is defined byxsU =(r + λ − µ)((γ − βλ )r + γλ)θ(r − µ).
(r + λ)((γ − βλ )(r − µ) + (γ − 1)λ) 1 − a(7)

The firm’s optimal policy is to sell out only if an acquirer arrives at time t satisfying
X(t) ≤ xsU . For X(t) ≤ xsU , the firm is better oﬀ selling out because of a positive constantθ embedded in P (X(t)). The lower equation in (6) stands for the firm value when the firm
value is higher than the sell-out payoﬀ, i.e, EU (x) > (1 − τ )P (x). The upper equationin (6) stands for the firm value when the firm value is lower than the sell-out payoﬀ, i.e,
EU (x) ≤ (1 − τ )P (x). Then, the firm wishes to immediately sell out, but it may taketime because of asset illiquidity. To be more precise, the upper equation in (6) can be
decomposed as follows. The first and second terms of the equation stand for the expected
payoﬀ of selling out as soon as an acquirer appears. The last term of the equation stands
for the value of the option to forgo an oﬀer when it arrives at time t satisfying X(t) > xsU .
It should be noted that the sell-out threshold xsU is derived in the closed-form expression (7). By virtue of the closed-form solution, we can easily show that xsU monotonicallydecreases to the solution of the liquid model, i.e., γθ(r − µ)/(γ − 1)(1 − a) as λ → ∞.
We can also show that xsU monotonically increases to the solution of the net present value

9model, i.e., θ(r − µ)/(1 − a) as λ → 0. Similarly, we can easily show that the firm value
EU (x) converges to the corresponding solutions in the liquid and net present value models. To summarize, lower asset liquidity λ decreases the value of waiting and increases thesell-out threshold. With lower asset liquidity, the firm accepts an acquirer’s oﬀer moreeagerly because the expected waiting time for another oﬀer increases.

4.2

Levered firmThis subsection examines the levered firm’s exit policy. Suppose that the firm has issued
console debt with coupon C(> 0) at the initial time. We use E(x), D(x), and F (x) =E(x)+D(x) to denote the equity, debt, and firm values when sell-out is infeasible, while we
use Ẽ(x), D̃(x), and F̃ (x) = Ẽ(x) + D̃(x) to denote the equity, debt, and firm values whensell-out is feasible. We have Ẽ(x) = max{(1 − τ )P (x) − C/r, E(x)}5 because shareholders
optimize the choice between sell-out and forgoing an acquirer. Recall that shareholders
receive the residual value (1 − τ )P (x) − C/r on sell-out.
Clearly, E(x) is a nonnegative and nondecreasing function. E(x) is also higher than
(1 − τ )x/(r − µ) − C/r, which is the payoﬀ of perpetually operating the firm. Throughout
the paper, we suppose that E(x) is a convex function.6 Then, the two functions E(x)and (1 − τ )P (x) − C/r intersect twice at most. We can classify the functions into Cases(S), (DS), (DOS), and (D) based on the relationships between the two functions (seeFigure 1). In the upper right panel of Figure 1, we have default, sell-out, and operation
regions from left (i.e., x = 0), and we call this Case (DS). In the lower left panel, we havedefault, operation, sell-out, and operation regions from left, and we call this Case (DOS).
Intuitively, we can see from Figure 1 that Case (S) holds for the highest residual value,
followed in order by Case (DS), Case (DOS), and Case (D).
In Case (S), (i.e., when the residual value is highest) the two functions intersect once,and E(x) is strictly positive for x > 0. There is no possibility of default, and the intersection point xs is the sell-out threshold. The firm accepts an acquirer’s oﬀer for X(t) ≤ xs .
As in Section 4.1, we can derive E(x) by solving ODEs
µxE ′ (x) + 0.5σ 2 x2 E ′′ (x) + λ((1 − τ )P (x) − C/r − E(x)) + (1 − τ )(x − C) = rE(x) (x ∈ S),(8)µxE ′ (x) + 0.5σ 2 x2 E ′′ (x) + (1 − τ )(x − C) = rE(x) (x ∈ O),

5
6

There is no tax shield on repayment of the principal C/r.
We cannot mathematically prove it, but it always holds in numerical examples.

10

(9)

where S = (0, xs ) and O = (xs , ∞) are the sell-out and operation region, along with theboundary conditions E(xs ) = (1−τ )P (xs )−C/r, limx→0 E(x) < ∞, limx→∞ E(x)/x < ∞,and continuous diﬀerentiability of E(x) at xs . In ODE (8), λ((1 − τ )P (x) − C/r − E(x))
corresponds to the fact that E(x) changes to (1 − τ )P (x) − C/r with probability λdt intime interval dt in the sell-out region. Because of no default risk, the debt value D(x)equals the riskless debt value C/r.
In Case (DS), the two functions intersect once, and E(x) equals zero for x ∈ (0, xd ].
The intersection point xs is the sell-out threshold. The firm accepts an acquirer’s oﬀer
for X(t) ≤ xs . However, if no acquirer appears in the sell-out region until X(t) hits
the default threshold xd , the firm defaults. We can derive E(x) by solving ODEs (8)and (9) with the sell-out region S = (xd , xs ) and the operation region O = (xs , ∞). The
boundary conditions are E(xd ) = 0, E(xs ) = (1−τ )P (xs )−C/r, and limx→∞ E(x)/x < ∞,The condition E(xd ) = 0 follows from the assumption that shareholders gain nothing on
default, and the other conditions are the same as in Section 4.1. By the smooth pasting(optimality) condition, E(x) is continuously diﬀerentiable at xd , and as in Section 4.1,
E(x) is also continuously diﬀerentiable at xs .
In this case, there is default risk, and the debt value D(x) followsµxD′ (x) + 0.5σ 2 x2 D′′ (x) + λ(C/r − D(x)) + C = rD(x) (x ∈ S),(10)µxD′ (x) + 0.5σ 2 x2 D′′ (x) + C = rD(x) (x ∈ O),(11)

where S = (xd , xs ) and O = (xs , ∞). In ODE (10), λ(C/r − D(x)) corresponds to thefact that D(x) changes to the repayment value C/r with probability λdt in time interval
dt in the sell-out region. We can derive D(x) by solving ODEs (10) and (11). The
boundary conditions are D(xd ) = (1 − α)EU (x) and limx→∞ D(x) = C/r. The former
means that debt holders receive the unlevered firm value minus bankruptcy costs at thedefault threshold, whereas the latter means that the debt value approaches the risklessvalue for an infinitely large state variable. As in E(x), D(x) is continuously diﬀerentiable
at xs , while D(x) is continuous at xd . Note that D(x) is not continuously diﬀerentiable
at xd because the default threshold xd does not maximize D(x) but E(x).
In Case (DOS), the two functions intersect twice, and E(x) equals zero for x ∈ (0, xd ].
Notably, the sell-out region, unlike in Cases (DS) and (S), becomes the interval [xs1 , xs2 ].
The firm accepts an acquirer’s oﬀer for X(t) ∈ [xs1 , xs2 ]. However, if no acquirer appears inthe sell-out region until X(t) hits the default threshold xd , the firm defaults. We can derive
E(x) by solving ODEs (8) and (9) with S = (xs1 , xs2 ) and O = (xd , xs1 ) ∪ (xs2 , ∞). As in
Case (DS), the boundary conditions are E(xd ) = 0, E(xs1 ) = (1−τ )P (xs1 )−C/r, E(xs2 ) =11(1 − τ )P (xs2 ) − C/r, and limx→∞ E(x)/x < ∞, and E(x) are continuously diﬀerentiable
at xd , xs1 , and xs2 . We can also derive the debt value D(x) by solving ODEs (10) and(11) with S = (xs1 , xs2 ) and O = (xd , xs1 ) ∪ (xs2 , ∞). The boundary conditions areD(xd ) = (1 − α)EU (x) and limx→∞ D(x) = C/r, and we have continuous diﬀerentiabilityof D(x) at xs1 and xs2 , and continuity of D(x) at xd .
In Case (D), (i.e., when the residual value is lowest) the two functions do not intersect,and E(x) equals zero for x ∈ (0, xd ]. There is no possibility of sell-out because E(x)dominates (1 − τ )P (x) − C/r. The firm defaults when X(t) hits the default threshold xd .
We can derive E(x) by solving ODE (9) with O = (xd , ∞). Note the boundary conditions
E(xd ) = 0, limx→∞ E(x)/x < ∞, and continuous diﬀerentiability of E(x) at xd . In thisdefault only case, E(x) is essentially the same as the equity value of Leland (1994).
By choosing the maximal equity value in all cases, we have the following proposition,
where we define−γϵ = (1 − a) 1−γ ∈ (0, 1].

(12)

For the proof, refer to Appendix B.
Proposition 2
Case (S):C(1 − τ )λ≤.
rθ
τ λ + (1 − τ )(r + λ)(13)

The firm prefers to sell out.
(
)
{

C
λ
τC
(r + aλ − µ)x


−
+
θ
−
(1
−
τ
)


(r − µ)(r + λ − µ) (r
r+λ
(1 − τ )r))}( x )βλ ( (a − 1)xs


r
τC
+ s
+
θ−
(x ≤ xs ),E(x) =
x
r
+
λ
−
µ
r
+
λ
(1
−
τ
)r



{
)}
(



x
C ( x )γ (a − 1)xs
τC (1 − τ )− ++θ−(x > xs ),
r−µ
r
xs
r−µ
(1 − τ )r(14)
D(x) =

C,r

where the sell-out threshold xs is defined by(r + λ − µ)((γ − βλ )r + γλ)(r − µ)x =(r + λ)((γ − βλ )(r − µ) + (γ − 1)λ) 1 − a
s

(

τC
θ−
(1 − τ )r

)
.

(15)

Cases (DS) and (DOS) :(1 − τ )λ
C
1−τ
<
≤
.
τ λ + (1 − τ )(r + λ)rθ
τ + (1 − τ )ϵThe firm can either default or sell out.

12

(16)

Case (DS):0(x ≤ xd ),


(
)
}
{



C
λ
τC
(r + aλ − µ)x (1 − τ )− +θ−+ A1 xβλ + A2 xγλ(x ∈ (xd , xs ]),(r − µ)(r + λ − µ)rr+λ(1 − τ )r
E(x) =


{
(
)}



x
C ( x )γ (a − 1)xs
τC (1 − τ )− ++θ−(x > xs ),
r−µ
r
xs
r−µ
(1 − τ )r
(1 − α)EU (x) C− B1 xβλ − B2 xγλD(x) =r( x )γ C− B3 srx(17)(x ≤ xd ),(x ∈ (xd , xs ]),(18)(x > xs ),where we denotexsγλ F1 − xdγλ F2,xdβλ xsγλ − xsβλ xdγλ
xdβλ F2 − xsβλ F1,A2 = dβ sγx λ x λ − xsβλ xdγλ
(
)
C
λ
τC
(r + aλ − µ)xd
+ −
θ−,F1 = −(r − µ)(r + λ − µ)rr+λ(1 − τ )r()(a − 1)xs
r
τC
F2 =
+
θ−
,
r+λ−µ r+λ(1 − τ )r(γ − γλ )xsγλ (C/r − (1 − α)EU (xd ))B1 =,(βλ − γ)xsβλ xdγλ − (γλ − γ)xdβλ xsγλ(βλ − γ)xsβλ (C/r − (1 − α)EU (xd )),B2 =(βλ − γ)xsβλ xdγλ − (γλ − γ)xdβλ xsγλ
A1 =B3 = B1 xsβλ + B2 xsγλand the default and sell-out thresholds xd , xs are the solutions tor + aλ − µ+ βλ A1 xdβλ −1 + γλ A2 xdγλ −1 = 0,
(19)
(r − µ)(r + λ − µ)()
r + aλ − µ
1 + γ(a − 1)γτCsβλ −1sγλ −1+ β λ A1 x
+ γ λ A2 x=
+ s θ−.
(r − µ)(r + λ − µ)r−µx
(1 − τ )r(20)

13

Case (DOS):


 0


{
}


x
C

β
γ

(1 − τ )− + A1 x + A2 x



r−µ
r


{
(
)
}
E(x) =(r + aλ − µ)x
C
λ
τC
βλ
γλ

(1 − τ )− +θ−+ A3 x + A4 x(r − µ)(r + λ − µ)rr+λ(1 − τ )r




{
(
)}


x
C ( x )γ (a − 1)xs2τC (1 − τ )
− +
+θ−
r−µ
r
xs2
r−µ
(1 − τ )r
(1 − α)EU (x)




C


− B1 xβ − B2 xγr
D(x) =C− B3 xβλ − B4 xγλr( x )γ C− B5 s2rx(21)(x ≤ xd ),(x ∈ (xd , xs1 )),(x ∈ [xs1 , xs2 ]),(22)(x > xs2 ),where we denote
xsγ F1 − xdγ F2,
xdβ xsγ − xsβ xdγ
xdβ F2 − xsβ F1A2 = dβ sγ,x x − xsβ xdγ
xs2γλ F3 − xs1γλ F4A3 = s1β s2γ,x λ x λ − xs2βλ xs1γλ
xs1βλ F4 − xs2βλ F3,A4 = s1β s2γx λ x λ − xs2βλ xs1γλ
xdCF1 = −+ ,r−µrs1(a − 1)x
τC
F2 =
+θ−
,
r−µ
(1 − τ )r()(a − 1)xs1
λ
τC
F3 =
+
θ−
,
r+λ−µ
r+λ
(1 − τ )r()(a − 1)xs2
r
τC
F4 =
+
θ−
,
r+λ−µ
r+λ
(1 − τ )rA1 =and the default and sell-out thresholds xd , xs1 , xs2 are the solutions toxd+ βA1 xdβ + γA2 xdγ = 0,
(23)
r−µ
xs1
(r + aλ − µ)xs1(24)+ βA1 xs1β + γA2 xs1γ =+ βλ A3 xs1βλ + γλ A4 xs1γλ ,r−µ(r − µ)(r + λ − µ))((r + aλ − µ)xs2(1 + γ(a − 1))xs2
τC
s2βλ
s2γλ
.
+ βλ A3 x+ γλ A4 x=+γ θ−(r − µ)(r + λ − µ)r−µ(1 − τ )r(25)

14(x ≤ xd ),(x ∈ (xd , xs1 )),(x ∈ [xs1 , xs2 ]),(x > xs2 ),The coeﬃcients Bi (i = 1, 2, . . . , 5) are the solution of the linear system:
xdβ B1 + xdγ B2 =C
− (1 − α)EU (xd ),r(26)

xs1β B1 + xs1γ B2 − xs1βλ B3 − xs1γλ B4 = 0,(27)

βxs1β B1 + γxs1γ B2 − βλ xs1βλ B3 − γλ xs1γλ B4 = 0,(28)

xs2βλ B3 + xs2γλ B4 − B5 = 0,(29)

βλ xs2βλ B3 + γλ xs2γλ B4 = 0.

(30)

Case (D):
C
1−τ
>
.
rθ
τ + (1 − τ )ϵ(31)

The firm prefers to default.


 0

{

) (

)}

x
C (x γxdC (1 − τ )
− +
−
+
d
r−µ
r
r−µ
r
x


 (1 − α)EU (x)(x ≤ xd ),()D(x) =C ( x )γ C−− (1 − α)EU (xd )(x > xd ),rr
xd
E(x) =(x ≤ xd ),(x > xd ),(32)

where the default threshold xd is defined byxd =γ(r − µ)C.
(γ − 1)r(33)

In each case, Ẽ(x) and D̃(x) equal (1 − τ )P (x) − C/r and C/r, respectively, in thesell-out region, whereas Ẽ(x) and D̃(x) equal E(x) and D(x), respectively, in the operation
region. The firm values F (x) and F̃ (x) equal E(x) + D(x) and Ẽ(x) + D̃(x), respectively.

We can see from (13) that Case (S) (i.e., the sell-out only case), tends to hold forlower coupon, higher scrap value, and higher asset liquidity. In Case (S), the firm’s
optimal policy is to sell out only if an acquirer arrives at time t satisfying X(t) ≤ xs .
Debt is riskless, and E(x) is similar to EU (x) in Proposition 1. The sell-out threshold xs
is lower than xsU because of the term τ C/(1 − τ )r (cf. (15) and (7)). This is because thetax benefits of coupon payments are lost after sell-out.
On the other hand, we can see from (31) that Case (D), i.e., the default only case,tends to hold for higher coupon and lower asset value. Because of ∂ϵ/∂σ ≥ 0, a higher σalso increases the region of Case (D). The impact of σ is the same as shown by Nishihara

15

and Shibata (2018).7 In Case (D), the firm’s optimal policy is to default once X(t) hits
the default threshold xd . On bankruptcy, debt holders take over the firm, but a fraction
α of the firm value is lost as bankruptcy costs. There is no possibility of sell-out, andhence, E(x) and D(x) are essentially the same as shown by Leland (1994). In this case,the default threshold xd is derived in the closed form (33).
Cases (DS) and (DOS) are novel. These intermediate cases do not appear in any
previous literature. Indeed, in the previous models allowing asset sales at any time (e.g.,
Mella-Barral and Perraudin (1997), Nishihara and Shibata (2017), and Nishihara andShibata (2018)), the firm sells out if (31) does not hold. We can see from (16) that thenovel cases tend to appear for intermediate coupon and asset value, as well as for lower
market volatility. In these cases, the firm’s exit choice between sell-out and default is notdetermined at the initial time but depends on the realized paths of X(t) and the Poissonprocess. The firm succeeds in sell-out only if an acquirer appears in the sell-out region
until X(t) hits the default threshold xd . Otherwise, the firm defaults.
The stochastic exit choice is contrasted with the deterministic choice in Mella-Barral
and Perraudin (1997), Nishihara and Shibata (2017), and Nishihara and Shibata (2018).
This result fits empirical observations well. In fact, before bankruptcy, many firms search
for acquirers that will save them. Some firms (e.g., Sharp Corporation) succeed in being
acquired, while other firms (e.g., Circuit City) fail and become bankrupt. More generally, M&A literature, such as Cornett, Tanyeri, and Tehranian (2011) and Wang (2018),
empirically shows the low predictability of targets of acquisitions.
We cannot analytically prove but can numerically show (also intuitively from Figure 1)
that Case (DS) holds for relatively lower coupon and higher asset value, while Case (DOS)holds for relatively higher coupon and lower asset value. According to our computations
for a wide range of parameter values, we always have a unique solution either in Case
(DS) or in Case (DOS).
Now, we see E(x) and D(x) in Case (DS) more closely. The lower equation in (17)stands for the equity value in the operation region, i.e, E(x) > (1 − τ )P (x) − C/r. The
center equation in (17) stands for the equity value in the sell-out region, i.e, E(x) ≤(1 − τ )P (x) − C/r. The first three terms of the center equation represent the expected
payoﬀ of selling out as soon as an acquirer appears. The last two terms, i.e., A1 xβλ +A2 xγλ

7Intuitively, a higher σ increases the default option value than the sell-out option value because the convexityof the default option is stronger than that of the sell-out option. By the similar logic, Kort, Murto, and Pawlina(2010) show that higher volatility is more likely to lead to lumpy rather than stepwise investment.

16

stand for the value of the options to default and to delay sell-out. The upper equation in(17) means that the equity value is zero in the default region.
Debt holders are repaid the face value on sell-out, while they cannot fully recover theface value on bankruptcy. Hence, debt is risky. The lower equation in (18) represents
the debt value in the operation region. The term C/r stands for the value of perpetually
receiving coupons, whereas the second term measures the possibility of X(t) hitting thesell-out threshold xs . The center equation in (18) stands for the debt value in the sell-out
region. The term C/r stands for the value of perpetually receiving coupons, whereas thelast two terms, i.e., B1 xβλ + B2 xγλ measure the possibility of X(t) hitting either xd or
xs . The upper equation in (18) means that debt holders receive the unlevered firm value
subtracted by bankruptcy costs at the default time.
Next, we turn to E(x) and D(x) in Case (DOS). The main diﬀerence from Case (DS)is that the sell-out region is the interval [xs1 , xs2 ]. Because of the lower residual value
from sell-out in Case (DOS), unlike in Case (DS), shareholders are better oﬀ forgoingan acquirer’s oﬀer for X(t) < xs1 . Then, we have two disconnected operation regions(xd , xs1 ) and (xs2 , ∞) (cf. Figure 1). This is why E(x) and D(x) are expressed as thefour equations in (21) and (22). All the equations can be explained as in Case (DS);we omit the detailed explanations. The existence of the intermediate operation region(xd , xs1 ) is similar to the result of Guerra, Kort, Nunes, and Oliveira (2018) who examine
a mothballing option before permanent exit. Although their model considers neither debt
financing nor asset illiquidity, they show that the operation region can exist between themothballing region and the exit threshold.
In Cases (DS) and (DOS), the firm’s exit choice between sell-out and default is stochastic. The following proposition analytically derives the default probability, denoted by
P d (x).
Proposition 3 Suppose that µ − 0.5σ 2 < 0.
Case (S):P d (x) = 0.
Case (DS):




1


d
P (x) =
G1 xβλ−r + G2 xγλ−r



 G
3

(x ≤ xd ),(x ∈ (xd , xs ]),(34)(x > xs ),where xd and xs are the default and sell-out thresholds in Case (DS) of Proposition 2,

17and the coeﬃcients G1 , G2 , G3 are defined by
γλ−r xsγλ−r,
βλ−r xsβλ−r xdγλ−r − γλ−r xdβλ−r xsγλ−r
βλ−r xsβλ−r
G2 =,βλ−r xsβλ−r xdγλ−r − γλ−r xdβλ−r xsγλ−r

G1 = −

G3 = G1 xsβλ−r + G2 xsγλ−r .

Case (DOS):



1





 G1 x1−2µ/σ2 + G2
d
P (x) =

G3 xβλ−r + G4 xγλ−r




 G
5

(x ≤ xd ),(x ∈ (xd , xs1 )),(x ∈ [xs1 , xs2 ]),(35)(x > xs2 ),where xd , xs1 , and xs2 are the default and sell-out thresholds in Case (DOS) of Proposition
2, and the coeﬃcients Gi (i = 1, 2, . . . , 5) are the solution to the linear system:2

xd1−2µ/σ G1 + G2 = 1,2

xs11−2µ/σ G1 + G2 − xs1βλ−r G3 − xs1γλ−r G4 = 0,2

(1 − 2µ/σ 2 )xs11−2µ/σ G1 + G2 − βλ−r xs1βλ−r G3 − γλ−r xs1γλ−r G4 = 0,xs2βλ−r G3 + xs2γλ−r G4 − G5 = 0,
βλ−r xs2βλ−r G3 + γλ−r xs2γλ−r G4 = 0.
Case (D):P d (x) = 1.
Similarly, we can derive P d (x) for µ − 0.5σ 2 ≥ 0, but we omit the expressions to avoid
redundancy.
Note that P d (x) is constant for x ≥ xs in Case (DS) and for x ≥ xs2 in Case (DOS)because X(t) hits xs and xs2 almost surely under the assumption of µ − 0.5σ 2 < 0. X(t)also hits the default threshold xd almost surely, but a Poisson jump may occur in thesell-out region before X(t) hits the default threshold xd . Hence, P d (x) decreases with x
for x ∈ (xd , xs ] in Case (DS) and for x ∈ (xd , xs2 ] in Case (DOS). Under the assumptionof µ − 0.5σ 2 < 0, the sell-out probability is 1 − P d (x).
As explained after Proposition 2, contrasted with the deterministic choice (i.e., P d (X(t)) =0 or 1) in the previous models (e.g., Mella-Barral and Perraudin (1997), Nishihara andShibata (2017), and Nishihara and Shibata (2018)), the stochastic dynamics of P d (X(t))

18is more realistic. In the next section, we will examine the eﬀects of asset liquidity on thefirm’s default and sell-out policies, equity, debt, and firm values, and default probability
in full detail.

5

Numerical analysis and implications

In this section, we conduct numerical analysis, including comparative statics with respect
to asset liquidity λ and volatility σ. We set the baseline parameter values in Table 1.
Asset liquidity λ is set at 2, which means that an asset sales opportunity is expected toarrive every half year. The other parameter values are standard and similar to those of
Nishihara and Shibata (2018).
For the baseline parameter values, Case (DOS) holds true, and we have the default
and sell-out thresholds xd = 0.432, xs1 = 0.472, and xs2 = 0.796, as well as the equity anddebt values E(x) = 5.308 and D(x) = 16.647. Debt is risky, i.e., D(x) = 16.647 < C/r =16.667, and the default probability is P d (x) = 0.0065. Figure 2 shows a sample path ofthe equity value E(X(t)) with no oﬀer in the sell-out region. We depict the dynamics ofE(X(t)) rather than X(t) because it can be interpreted as the stock price dynamics until
bankruptcy. Note that the stock price equals E(X(t))/N , where N denotes the number ofshares. In the figure, the sell-out region is between E(xs1 ) = 2.846 and E(xs2 ) = 0.0953.
The firm attempts to sell out in the region, but in this example, no acquirer appears in
the region. Then, the firm defaults when the equity value falls zero, i.e., E(xd ) = 0. If
an acquirer arrives in the sell-out region, the firm sells out, and then the equity value
E(X(t)) jumps upward to (1 − τ )P (X(t)) − C/r.

5.1

Impacts of asset liquidity

Figure 3 depicts the default and sell-out thresholds, the equity, debt, and firm values, andthe default probability for 1/λ ∈ (0, 2], where 1/λ equals the expected duration (year)
between acquirers’ arrivals. The other parameter values are set in Table 1. In Figure 3,
we also depict the sell-out threshold of the liquid model (i.e., λ = ∞), denoted by xsb ,for comparison. For all 1/λ ∈ (0, 2], Case (DOS) holds true. The right-hand side of (13)monotonically increases up to 1 − τ as λ increases. For the baseline parameter values,
we have 1 − τ < C/rθ ≤ (1 − τ )/(τ + (1 − τ )ϵ), and hence, Case (DOS) always holds. If
C/rθ < 1 − τ holds, a higher λ leads to Case (S).
In the top-left panel, the sell-out threshold xs2 is higher than xsb of the liquid model,

19and the sell-out region [xs1 , xs2 ] expands with higher 1/λ. In other words, with lower
liquidity, the firm accepts an acquirer’s oﬀer more eagerly to mitigate the risk of not
finding an acquirer until bankruptcy. In the top-right and bottom-right panels, the default
threshold xd and the default probability P d (x) increase in 1/λ. This implies that thefirm’s earnest asset sales policy does not fully oﬀset the decreased liquidity, and lower
asset liquidity increases the default risk.
These results are similar to those of Hugonnier, Malamud, and Morellec (2015b) who
consider the model in which a firm can increase debt to receive more tax benefits only
at Poisson jump times. They show that with a lower arrival rate of creditors, the firm
decreases the debt restructuring threshold to mitigate the risk of not finding creditors;
despite this eﬀort, the restructuring probability decreases. This means that the firm’s
earnest debt restructuring policy does not fully oﬀset the decreased credit supply. Although they focus on debt expansion in economic expansion, we focus on asset sales in
economic downturn. He and Xiong (2012), He and Milbradt (2014), and Chen, Cui,He, and Milbradt (2018) show that the illiquidity of the secondary debt market hastensbankruptcy timing and increases the default probability via the channel of rollover risk.
Unlike in their papers, we study the eﬀects of real asset illiquidity. Morellec (2001) showsthat for unsecured debt, lower asset liquidation cost, which leads to managerial asset
disposition, increases the bankruptcy probability and decreases the debt value. Unlike
Morellec (2001), we focus on sell-out, where debt holders are repaid the face value of
debt; hence, our model does not have the possibility for asset disposition to decrease thedebt value.
Finally, the center panels of Figure 3 show that both the debt value D(x) and theequity value E(x) decrease in 1/λ. The decrease in D(x) is consistent with the stylized
fact (e.g., Shleifer and Vishny (1992) and Sibilkov (2009)). Shleifer and Vishny (1992)
argue that lower asset liquidity decreases the post-default value debt holders receive, andour model captures the eﬀect because the post-default value (1−α)EU (x) decreases in 1/λ.
In addition, our model suggests that lower asset liquidity also decreases D(x) through theincreased default probability channel. Furthermore, E(x) decreases in 1/λ because lower
asset liquidity decreases shareholders’ sell-out option value. This result is consistent with
empirical evidence in Ortiz-Molina and Phillips (2014), Song and Walkling (2000), andCornett, Tanyeri, and Tehranian (2011). Ortiz-Molina and Phillips (2014) show that real
asset illiquidity reduces a firm’s operating flexibility and through this channel increases
the cost of capital. Song and Walkling (2000) and Cornett, Tanyeri, and Tehranian (2011)20

show that a stock price increases as the probability of being acquired increases. Note thatthe arrival rate λ can be interpreted as the probability of being acquired.
So far, we have examined the eﬀects of asset liquidity with a fixed initial coupon C.
How do the results change if the firm can precisely anticipate the degree of asset liquidity
and optimize the capital structure at the initial time? As in the standard literature (e.g.,
Leland (1994)), the firm chooses the optimal capital structure to maximize the firm value
F (x), taking account of the trade-oﬀ between the tax benefits of debt and bankruptcy
costs.
Figure 4 shows the optimal coupon C, leverage LV (= D(x)/F (x)), and credit spread
CS(= C/D(x) − r) for 1/λ ∈ (0, 2]. In the figure, the optimal C and LV decreases in
1/λ, although CS increases in 1/λ. These results are interpreted as follows. Lower asset
liquidity increases the bankruptcy risk, and as a result, the firm reduces debt financing.
Despite the firm’s more cautious capital structure, the default risk increases (see CS in
Figure 4 and P d (x) in Figure 5) with decreased asset liquidity because the direct impact of
decreased asset liquidity dominates the impact from the reduced leverage. These results
are consistent with the empirical findings of Sibilkov (2009), Campello and Giambon(2013), and Ortiz-Molina and Phillips (2014), who show that leverage is positively related
to asset liquidity.
Figure 5 depicts the default and sell-out thresholds, the equity, debt, and firm values,
and the default probability for 1/λ ∈ (0, 2] under the optimal capital structure. Case
(DOS) holds for 1/λ ∈ (0, 1.25], while Case (DS) holds for 1/λ ∈ (1.25, 2]. This is because
a higher 1/λ decreases the optimal coupon, which changes from Case (DOS) to Case (DS).
Aside from the default threshold xd and the equity value E(x), Figure 5 is similar to
Figure 3. In the top-left panel of Figure 5, the sell-out region expands with 1/λ. With
lower liquidity, in addition to using a more cautious capital structure, as in a case with afixed coupon, the firm accepts an acquirer’s oﬀer more eagerly to mitigate the risk of not
finding an acquirer. In the bottom-right panel, the default probability P d (x) increases in
1/λ, although the increase is mitigated from that of a case with a fixed coupon (cf. P d (x) in
Figure 3). In other words, the cautious capital structure and earnest asset sales policy do
not fully oﬀset the decreased liquidity, and then lower asset liquidity increases the default
probability. Due to the increased bankruptcy risk, in the center-right and bottom-leftpanels, the debt and firm values monotonically decrease in 1/λ. These results align withthe empirical evidence (e.g., Ortiz-Molina and Phillips (2014), Song and Walkling (2000),
and Cornett, Tanyeri, and Tehranian (2011)).

21

In the top-right panel, the default threshold xd decreases in 1/λ, and in the center-leftpanel, the equity value E(x) decreases in 1/λ. These results are contrary to those of acase with a fixed coupon (cf. xd and E(x) in Figure 3). The explanation follows. A higher
1/λ decreases the optimal C, which plays roles in decreasing xd and increasing E(x).

5.2

Impacts of cash flow volatility

Figure 6 depicts the default and sell-out thresholds, the equity, debt, and firm values, andthe default probability for σ ∈ [0.15, 0.35]. The other parameter values are set in Table 1.
Case (DOS) holds for σ ∈ [0.15, 0.289), while Case (D) holds for σ ∈ [0.289, 0.35]. Recall
that by ∂ϵ/∂σ ≥ 0 and (31), a higher σ leads to Case (D).
In the top-left panel, a higher σ narrows the sell-out region, whereas in the top-rightpanel, a higher σ decreases the default threshold xd . These results are consistent with thestandard real options theory (e.g., Dixit and Pindyck (1994)) that a higher σ increases
the option value of waiting.
In the center panels, a higher σ increases the equity value E(x) and decreases thedebt value D(x). The wealth transfer from debt holders to shareholders is consistent
with the standard result (e.g., Jensen and Meckling (1976) and Eisdorfer (2008)). In thebottom-left panel, the sensitivity of the firm value F (x) to σ depends on this trade-oﬀ.
Most notably, the change from Case (DOS) to Case (D) at σ = 0.289 greatly decreases
F (x) because the change greatly reduces D(x). The sharp decrease in D(x) stems fromthe sharp increase in the default probability P d (x). In fact, in the bottom-right panel,
P d (x) sharply increases in σ ≈ 0.289. These results lead to empirical implications thata slight increase in volatility can greatly increase default risk and decrease debt and firm
values.
Now, we examine the impacts of σ under the optimal capital structure. Figure 7
shows the optimal coupon C, leverage LV , and credit spread CS for σ ∈ [0.15, 0.35]. In
the figure, C and LV decrease in σ, whereas CS forms an inverted U-shape. The negative
correlation between σ and LV is consistent with the standard theory (e.g., Harris andRaviv (1991) and Leland (1994)). A higher σ can reduce CS due to the lower LV , but it
can increase CS due to more volatile cash flows. The ambiguous sensitivity of CS stems
from this tradeoﬀ.
Figure 8 depicts the default and sell-out thresholds, the equity, debt, and firm values,
and the default probability for σ ∈ [0.15, 0.35] under the optimal capital structure. Unlike
in a case with a fixed coupon, Case (DOS) holds for all σ ∈ [0.15, 0.35]. This is because

22(16) is satisfied, even for a higher σ, due to the reduced coupon. In the top panels, thesell-out and default thresholds xs1 , xs2 , and xd decrease in σ. As in the case with a fixedcoupon, this result aligns with the standard real options result.
In the center panels, as in the case with a fixed coupon, a higher σ leads to a wealthtransfer from debt holders to shareholders. There are no sharp decreases in D(x) and F (x)because the firm can adjust the optimal capital structure in accordance with volatilitychanges. In the bottom-left panel, unlike in the case with a fixed coupon, F (x) monotonically increases in σ. This is because the firm chooses the capital structure to maximize
F (x) and a higher σ increases the sell-out and default option values embedded in F (x). In
the bottom-right panel, as in CS in Figure 7, the default probability P d (x) is an inverted
U-shape based on the trade-oﬀ between the reduced leverage and more volatile cash flows.

5.3

Stock price reactions at the sell-out time

In this subsection, we explore the market reaction when the firm sells out. In the model,
an acquirer pays P (X(t)) at acquisition, and then, the equity value jumps from E(X(t))
to (1 − τ )P (X(t)) − C/r, which can be interpreted as the stock price reaction or theacquisition premium. Recall that E(X(t)) ≤ (1 − τ )P (X(t)) − C/r always hold for X(t)in the sell-out region. The positive stock price reaction is consistent with well-known facts.
In fact, targets gain abnormal returns (around 20%) in M&As (e.g., Huang and Walkling(1987), Hotchkiss and Mooradian (1998), and Boone and Mulherin (2007)). Below, we
explore the key drivers of the positive stock price reaction.
Figure 9 depicts the jump size, defined by ((1 − τ )P (X(t)) − C/r − E(X(t)))/E(X(t)),
with respect to X(t) for varying levels of 1/λ and σ. In the left panels, the other parametervalues are set in Table 1, and Case (DOS) holds true. In the right panels, we replace C = 1
with C = 0.9 to see the results in Case (DS). There is no jump for x ∈/ [xs1 , xs2 ] in Case
(DOS) and for x ∈/ [xd , xs2 ] in Case (DS) because the firm forgoes an acquirer’s oﬀer.
In the left panels, i.e., Case (DOS), the jump size is an inverted U-shape with respect
to X(t). By Proposition 2, we have (1 − τ )P (X(t)) − C/r − E(X(t)) = 0 for the sellout thresholds xs1 and xs2 , whereas we have (1 − τ )P (X(t)) − C/r − E(X(t)) > 0 for
X(t) ∈ (xs1 , xs2 ). In other words, the interval policy of sell-out in Case (DOS) leads tothe inverted U-shaped jump size with respect to X(t). On the other hand, in the rightpanel, i.e., Case (DS),8 the jump size increases infinitely as X(t) approaches xd because
the denominator E(xd ) equals zero. According to our computations for a wide range of
8

Only for σ = 0.25, Case (DS) holds, and the jump size is an inverted U-shape with X(t).

23parameter values, in Cases (DS) and (S), the jump size monotonically decreases in X(t).
We see these results more closely below.
The jump size is rewritten as (1 − τ )P (X(t))/E(X(t)) − C/rE(X(t)) − 1. The firstterm (1 − τ )P (X(t))/E(X(t)) decreases in X(t). Indeed, the firm has a stronger incentive
to sell out to the acquirer for lower X(t) because sell-out involves partial liquidation withthe fixed scrap price θ (called the sell-out eﬀect). The second term −C/rE(X(t)) increasesin X(t) because the denominator E(X(t)) increases in X(t) (called the debt repaymenteﬀect). The impact of X(t) on the jump size is determined by the tradeoﬀ between thetwo eﬀects. Recall that increased asset value (or decreased coupon) changes the region
from Case (DOS) to Cases (DS) and (S) (cf. Figure 1). When the target asset value
is suﬃciently high compared to debt value (i.e., Cases (DS) and (S)), the sell-out eﬀect
dominates the debt repayment eﬀect. Hence, the jump size decreases in X(t). When thetarget asset value is not suﬃciently high compared to debt value (i.e., Case (DOS)), thesell-out eﬀect is stronger for relatively high X(t), but the debt repayment eﬀect is stronger
for relatively low X(t). Hence, the jump size is an inverted U-shape with respect to X(t).
Our results highlight the significance of acquisition timing on stock price reaction. The
jump size greatly depends on the acquisition timing, even if we consider a transaction
between the same two firms. To put it roughly, we have the following predictions. If
the target asset value is relatively low, the stock price jump size is an inverted U-shapewith the distance to default. If the target asset value is relatively high, the stock pricejump size decreases in the distance to default. Although these have not been empiricallytested to date, our model can potentially account for the dispersion of target stock pricereactions.
The debt value also jumps upward from D(X(t)) to the riskless debt value C/r at thesell-out time in Cases (DS) and (DOS), whereas the debt value remains C/r in Case (S).
With higher X(t), D(X(t)) approaches C/r (see (18) and (22)), and then the jump size
(C/r − D(X(t)))/D(X(t)) decreases. In other words, the debt value jump size increases
as the economic state variable deteriorates. Because acquisition saves debt holders frombankruptcy risk in our model, this result is straightforward. Hence, we omit depicting afigure of the debt value jump.
Figure 9 also shows that a higher 1/λ and a lower σ increase the equity value jumpsize. Because E(X(t)) decreases in 1/λ (cf. E(x) in Figure 3), a higher 1/λ increases thediﬀerence between E(X(t)) and the sell-out payoﬀ (1 − τ )P (X(t)) − C/r. As a result,the jump size increases in 1/λ. Intuitively, with lower asset liquidity, the acquisition24

announcement brings a greater surprise to the market, leading to a more positive stock
price reaction. This result is consistent with empirical evidence of Cornett, Tanyeri, andTehranian (2011), who show that lower merger anticipation increases market reactions at
the announcement. Because E(X(t)) increases in σ (cf. E(x) in Figure 6), a higher σdecreases the diﬀerence between E(X(t)) and (1 − τ )P (X(t)) − C/r. Hence, the jump
size decreases in σ.
Lastly, Figure 10 depicts the jump size ((1 − τ )P (X(t)) − C/r − E(X(t)))/E(X(t)) for
varying levels of 1/λ and σ under the optimal capital structure, where the initial coupon
C is chosen to maximize the initial firm value. The other parameter values are set in Table
1. Case (DS) holds for 1/λ = 1.5, whereas Case (DOS) holds for the other parametervalues. Note that the optimal C decreases in 1/λ and σ (cf. C in Figures 4 and 7); hence,the sell-out payoﬀ (1 − τ )P (X(t)) − C/r increases in 1/λ and σ. The comparative statics
result with respect to 1/λ under the optimal capital structure is similar to that of thecase with a fixed coupon. Indeed, the jump size increases in 1/λ. Unlike in the case witha fixed coupon, the jump size slightly increases in σ. This is because a higher σ, which
decreases the optimal C, increases (1 − τ )P (X(t)) − C/r more than E(X(t)).

6Extensions

6.1

Search cost

In the baseline model, we assume there is no cost for searching for an acquirer. In thissubsection, we suppose that the firm has to pay flows of cost K while searching for an
acquirer. ODE (8) is replaced with
µxE ′ (x)+0.5σ 2 x2 E ′′ (x)+λ((1−τ )P (x)−C/r−E(x))+(1−τ )(x−C−K) = rE(x) (x ∈ S),
where S denotes the sell-out region. In the operation region, the firm does not have tosearch for an acquirer, meaning that ODE (9) remains unchanged. The problem can besolved as in Section 4. Clearly, with a higher K, the firm is more likely to default thanin the baseline model. The main results in the previous sections remain unchanged in thepresence of search cost.

6.2

Oﬀer price contingent on the stock priceIn the baseline model, an acquirer’s oﬀer price P (X(t)) (see (1)) depends on the statevariable X(t), but it is not contingent on the current equity value (or stock price) E(X(t)).

25

In this subsection, we suppose that an acquirer pays ξP (x) + (1 − ξ)E(x), where ξ ∈ (0, 1]
can be interpreted as the bargaining power of the distressed firm. Note that the oﬀer
price agrees with P (x) for ξ = 1. ODE (8) is replaced with
µxE ′ (x) + 0.5σ 2 x2 E ′′ (x) + λ((1 − τ )(ξP (x) + (1 − ξ)E(x)) − C/r − E(x))+ (1 − τ )(x − C) = rE(x) (x ∈ S)

(36)

because shareholders gain (1 − τ )(ξP (x) + (1 − ξ)E(x)) − C/r at the sell-out time, whereas
ODE (9) remains unchanged. We can rewrite (36) asµxE ′ (x) + 0.5σ 2 x2 E ′′ (x) + λ̃((1 − τ )P̃ (x) − C/r − E(x)) + (1 − τ )(x − C̃) = rE(x) (x ∈ S),(37)
where we define λ̃ = λ(τ +(1−τ )ξ), P̃ (x) = ξP (x)/(τ +(1−τ )ξ), and C̃ = C/(τ +(1−τ )ξ).
Note that λ̃ = λ, P̃ (x) = P (x), and C̃ = C hold for ξ = 1.
We can solve the problem as in Section 4 because ODE (37) is of the same form as
ODE (8). We have ∂ λ̃/∂ξ > 0, ∂ P̃ (x)/∂ξ > 0, and ∂ C̃/∂ξ < 0. Then, lower bargaining
power ξ plays the same role as lower asset liquidity and asset value, as well as higher
coupon. This implies that the firm is more likely to default than in the baseline model.
The main results in the previous sections remain unchanged in this case.

7

Conclusion

In this paper, we developed the dynamic bankruptcy model with illiquidity of sell-out
opportunities. In the model, we analytically derived the firm’s exit policy, its timing,the default probability, and the equity, debt, and firm values. The main implications are
summarized below.
With lower asset liquidity, the firm accepts an acquirer’s oﬀer more eagerly to mitigate
the risk of not finding an acquirer until bankruptcy takes place, but this earnest asset
sales policy does not fully oﬀset the decreased liquidity eﬀect. Lower asset liquidity thenincreases the default probability and decreases the equity, debt, and firm values. In the
optimal capital structure, the firm reduces leverage with lower asset liquidity, but thiscautious capital structure does not fully oﬀset the decreased illiquidity eﬀect either. The
stock price reaction caused by sell-out greatly depends on the sell-out timing. Intriguingly,
when the target asset value is not suﬃciently high, the stock price jump size can be aninverted U-shape with the state variable. Lower asset liquidity leads to a more positive
stock price reaction due to greater shock. Our results fit empirical findings about asset
illiquidity well.

26

A

Proof of Proposition 1

A nonnegative solution to ODE (3) with limx→∞ EUs (x)/x < ∞ is expressed as (1 −τ )x/(r −µ)+U1 xγ , where U1 is an unknown coeﬃcient. Then, by EUs (xsU ) = (1−τ )P (xsU ),we have(
U1 = (1 − τ )

1
xsU

)γ (

)
(a − 1)xsU
+θ ,
r−µ

which leads to equation (6) for x > xsU .
A nonnegative solution to ODE (2) with limx→0 EUs (x) < ∞ is expressed as(1 − τ )(r + aλ − µ)x (1 − τ )λθ++ U2 xβλ ,(r − µ)(r + λ − µ)r+λwhere U2 is an unknown coeﬃcient. Then, by EUs (xsU ) = (1 − τ )P (xsU ), we have( )βλ ()(a − 1)xsU
1rθ
U2 = (1 − τ )
+
,
xsU
r+λ−µ
r+λ
which leads to equation (6) for x ≤ xsU .
Finally, by continuous diﬀerentiability of EUs (x) at xsU , we have
(
)
(
)
xsU
(a − 1)xsU(1 − τ )(r + aλ − µ)xsU(a − 1)xsU
rθ
+γ
+θ =
+ βλ
+
,
r−µ
r−µ
(r − µ)(r + λ − µ)
r+λ−µ
r+λ
by which we can derive xsU as (7).

B

Proof of Proposition 2First, we derive the solution in each case, and we will choose the appropriate solution later.
In Case (S), E(x) can be derived as in Appendix A, and hence, we omit the derivationof E(x). Clearly, the debt value equals D/r because of no default risk. In Case (D),E(x) and D(x) are essentially the same as the well-known solutions in Goldstein, Ju, andLeland (2001), and hence, we omit the derivation of E(x) and D(x).
Consider Case (DS). The solution to ODE (8) with boundary conditions E(xd ) = 0
and E(xs ) = (1−τ )P (xs )−C/r is expressed as (17) for x ∈ (xd , xs ]. The solution to ODE(9) with boundary conditions E(xs ) = (1 − τ )P (xs ) − C/r and limx→∞ E(x)/x < ∞ isexpressed as (17) for x > xs . Continuous diﬀerentiability of E(x) at xd and xs leads to (19)and (20), respectively. A general solution to ODE (10) is expressed as (18) for x ∈ (xd , xs ].
A general solution to ODE (11) with boundary condition limx→∞ D(x) = C/r is expressedas (18) for x > xs . Continuity and continuous diﬀerentiability at xs lead to
xsβλ B1 + xsγλ B2 = B3 ,(38)

βλ xsβλ B1 + γλ xsγλ B2 = γB3 ,

(39)

27

respectively. By substituting (38) into (39), we have(βλ − γ)xsβλ B1 + (γλ − γ)xsγλ B2 = 0.

(40)

By (40) and boundary condition D(xd ) = (1 − α)EU (xd ), we have the expressions of B1
and B2 in Proposition 2. The expression of B3 equals (38).
Next, consider Case (DOS). The solution to ODE (9) with boundary conditions
E(xd ) = 0 and E(xs1 ) = (1 − τ )P (xs1 ) − C/r is expressed as (21) for x ∈ (xd , xs1 ).
The solution to ODE (8) with boundary conditions E(xs1 ) = (1 − τ )P (xs1 ) − C/r andE(xs2 ) = (1 − τ )P (xs2 ) − C/r is expressed as (21) for x ∈ [xs1 , xs2 ]. The solution to ODE(9) with boundary conditions E(xs2 ) = (1 − τ )P (xs2 ) − C/r and limx→∞ E(x)/x < ∞ isexpressed as (21) for x > xs2 . Continuous diﬀerentiability of E(x) at xd , xs1 , and xs2 leadsto (23), (24), and (25), respectively. A general solution to ODE (11) is expressed as (22)for x ∈ (xd , xs1 ). A general solution to ODE (10) is expressed as (22) for x ∈ [xs1 , xs2 ]. A
general solution to ODE (11) with boundary condition limx→∞ D(x) = C/r is expressedas (22) for x > xs2 . Boundary condition D(xd ) = (1 − α)EU (xd ), continuity and continuous diﬀerentiability at xs1 , and continuity and continuous diﬀerentiability at xs2 lead to(26)–(30), respectively.
Now, we consider which case holds. Note that (14) is nonnegative if and only if (13)holds. Indeed, as x → 0, (14) decreases to
(
)}
{
λ
τC
C
θ−
,
(1 − τ ) − +rr+λ(1 − τ )rwhich is nonnegative if and only if (13) holds. In this case, default is not optimal for any
X(t) because (14) dominates the default payoﬀ 0. This removes the possibility of Cases(DS), (DOS), and (D), and hence, we have Case (S) under condition (13).
Suppose that (13) does not hold. By the above argument, (14) is negative for asuﬃciently low x, which means that Case (S) does not hold. The remaining candidates are
Case (DS), (DOS), and (D). Note that (32) dominates the sell-out payoﬀ (1−τ )P (x)−C/r
for any x(> 0) if and only if (31) holds. In fact, we define)}{(x
C ( x )γxdC
− ((1 − τ )P (x) − C/r)f (x) = (1 − τ )
− +
−
+
r−µ
r
r−µ
r
xd
(1 − τ )(1 − a)x ( x )γ (1 − τ )CτC=++
− (1 − τ )θdr−µ(1 − γ)rr
x

28

and havemin f (x)x≥xd=f (xd (1 − a)1/(γ−1) )

(41)

=

(1 − τ )(1 − a)xd (1 − a)1/(γ−1)(1 − τ )CτC+ (1 − a)γ/(γ−1)+− (1 − τ )θr−µ
(1 − γ)rr=((1 − τ )(1 − a)γ/(γ−1) + τ )C− (1 − τ )θ,rwhich is positive if and only if (31) holds. We obtained (41) by solving the first ordercondition (i.e., f ′ (x) = 0) because of convexity of f (x). Under condition (31), sell-out is
not optimal for any X(t) because (32) dominates the sell-out payoﬀ. This removes thepossibility of Cases (DS) and (DOS), and hence, we have Case (D) under (31). Note that
(1 − τ )λ1−τ<.
τ λ + (1 − τ )(r + λ)
τ + (1 − τ )ϵfollows from ϵ ∈ (0, 1].
We now assume (16). By the above argument, (32) does not dominate the sell-out
payoﬀ, which removes the possibility of Case (D). Thus, the remaining candidates are
Case (DS) and (DOS) under condition (16).

C

Proof of Proposition 3

Clearly, we have P d (x) = 0 in Case (S), and P d (x) = 1 follows from µ − 0.5σ 2 < 0 inCase (D). Suppose that (16) holds. The default probability P d (x) satisfies the following
ODEs:µxP d′ (x) + 0.5σ 2 x2 P d′′ (x) + λ(−P d (x)) = 0 (x ∈ S),(42)µxP d′ (x) + 0.5σ 2 x2 P d′′ (x) = 0 (x ∈ O),(43)

where S and O stand for the sell-out and operation regions, respectively. In ODE (42),λ(−P d (x)) corresponds to the fact that P d (x) changes to 0 with probability λdt in timeinterval dt in the sell-out region.
Now, consider Case (DS). A general solution to ODE (42) is expressed as (34) forx ∈ (xd , xs ]. A general solution to ODE (43) with boundary condition limx→∞ P d (x) < ∞is expressed as (34) for x > xs . As in derivation of D(x) in Case (DS) in Appendix B, bycontinuity at xd and xs and continuous diﬀerentiability at xs , we have G1 , G2 , and G3 in
Proposition 3.

29

Next, consider Case (DOS). A general solution to ODE (43) is expressed as (35) forx ∈ (xd , xs1 ). A general solution to ODE (42) is expressed as (35) for x ∈ [xs1 , xs2 ]. A
general solution to ODE (43) with boundary condition limx→∞ P d (x) < ∞ is expressed as(35) for x > xs2 . As in derivation of D(x) in Case (DOS) in Appendix B, by continuity at
xd , xs1 , and xs2 , as well as continuous diﬀerentiability at xs1 and xs2 , we have the linear
equations of Gi (i = 1, 2, . . . , 5) in Proposition 3.

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33

Table 1: Baseline parameter values.
r

µ

σ

τ

α

a

θ

C

λ

x

0.06 0.01

0.2

0.15 0.3 0.5 15

1

2

1

Figure 1: Four cases depending on E(x) and (1 − τ )P (x) − C/r. The notations O, S, and D
stand for the operation, sell-out, and default regions, respectively.

34

Figure 2: A sample path of the equity value E(X(t)). The notations O, S, and D stand for theoperation, sell-out, and default periods, respectively.

35

0.5
0.8

xd

0.48
0.7
x s2
x s1

0.6

0.46

x sb

0.44

0.5
0.4

0.42
0

0.5

1

1.5

2

0

0.5

1/

1

1.5

2

1/
16.7

5.35
E(x)

D(x)

16.65
16.6

5.3

16.55
16.5

5.25

16.45
5.2

16.4
0

0.5

1

1.5

0

2

0.5

1

1.5

2

1/

1/
0.1

22

P d(x)

F(x)

0.08
21.9
0.06
21.8
0.04
21.7

0.02

21.6

0
0

0.5

1

1.5

2

0

1/

0.5

1

1.5

2

1/

Figure 3: Comparative statics with respect to 1/λ. The other parameter values are set in Table
1. Case (DOS) holds for all 1/λ ∈ (0, 2].

36

0.8

1.05
C

LV

0.78
1

0.76
0.74

0.95
0.72
0.9

0.7
0

0.5

1

1.5

2

0

1/

0.5

1

1.5

2

1/

10-4

2

CS

1.5

1

0.5

0
0

0.5

1

1.5

2

1/

Figure 4: Optimal capital structure with respect to 1/λ. Case (DOS) holds for 1/λ ∈ (0, 1.25],
while Case (DS) holds for 1/λ ∈ (1.25, 2]. The other parameter values are set in Table 1.

37

1
xd

0.5

0.9
0.8

0.45

0.7
0.4

x s2
x s1

0.6
0.5

0.35

x sb

0.4

0.3
0

0.5

1

1.5

2

0

0.5

1/

1

1.5

2

1/
18

E(x)

D(x)

17.5

6

17
5.5

16.5
16

5

15.5
4.5

15
0

0.5

1

1.5

2

0

0.5

1/

1

1.5

2

1.5

2

1/
0.02
P d(x)

F(x)

22
0.015
21.9

0.01

21.8

0.005

21.7

0
0

0.5

1

1.5

0

2

1/

0.5

1

1/

Figure 5: Comparative statics with respect to 1/λ under the optimal capital structure. Case
(DOS) holds for 1/λ ∈ (0, 1.25], while Case (DS) holds for 1/λ ∈ (1.25, 2]. The other parametervalues are set in Table 1.

38

0.46

0.9
0.8

x

s2

x

s1

xd

0.44
0.42

xsb
0.7

0.4

0.6

0.38
0.36

0.5
0.34
0.4
0.15

0.2

0.25

0.3

0.32
0.15

0.35

0.2

0.25

0.3

0.35

17

7
E(x)

D(x)

16.5

6.5
16
6

15.5

5.5

15
14.5

5
14
4.5
0.15

0.2

0.25

0.3

13.5
0.15

0.35

22.5

0.25

0.3

0.35

0.25

0.3

0.35

1
F(x)

P d (x)

22

0.8

21.5

0.6

21

0.4

20.5

0.2

20
0.15

0.2

0.2

0.25

0.3

0
0.15

0.35

0.2

Figure 6: Comparative statics with respect to σ. The other parameter values are set in Table
1. Case (DOS) holds for σ ∈ [0.15, 0.289), while Case (D) holds for σ ∈ [0.289, 0.35].

39

1.1

0.85
C

LV

1.05

0.8

1

0.75

0.95

0.7

0.9

0.65

0.85
0.15

10

0.2

0.25

0.3

0.35

0.25

0.3

0.35

0.6
0.15

0.2

0.25

0.3

0.35

10-5
CS

9.5
9
8.5
8
7.5
7
0.15

0.2

Figure 7: Optimal capital structure with respect to σ. The other parameter values are set in
Table 1. Case (DOS) holds for all σ ∈ [0.15, 0.35].

40

1
0.8

x s2
x s1

0.6

x sb

0.5

0.6

xd

0.4

0.4

0.3

0.2
0.15

0.2

8

E(x)

0.25

0.3

0.2
0.15

0.35

0.2

0.25

0.3

0.35

18
D(x)

17
7
16

6

15

5
4
0.15

23

0.2

0.25

0.3

14
0.15

0.35

9
F(x)

0.2

0.25

0.3

0.35

0.25

0.3

0.35

10-3

8.5

P d(x)

8

22.5

7.5
7

22

6.5
21.5
0.15

0.2

0.25

0.3

6
0.15

0.35

0.2

Figure 8: Comparative statics with respect to σ under the optimal capital structure. The otherparameter values are set in Table 1. Case (DOS) holds for all σ ∈ [0.15, 0.35].

41

2

1/ =0.5
1/ =1
1/ =1.5

1.2

1/ =0.5
1/ =1
1/ =1.5

1.5

1
0.8

1
0.6
0.4

0.5

0.2
0

0
0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

X(t)

0.6

0.8

1

X(t)
2

1
=0.15
=0.2
=0.25

0.8

=0.15
=0.2
=0.25

1.5

0.6
1
0.4
0.5

0.2
0

0
0

0.2

0.4

0.6

0.8

1

0

X(t)

0.2

0.4

0.6

0.8

1

X(t)

Figure 9: Stock price reaction. This figure depicts the jump size (E(x) − (1 − τ )P (x) −C/r)/E(x). In the left panels, the other parameter values are set in Table 1, and Case (DOS)holds. In the right panels, we set C = 0.9, where Case (DOS) holds only for σ = 0.25, andCase (DS) holds for the other parameter values.

422
1/ =0.5
1/ =1
1/ =1.5

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

0.2

0.4

0.6

0.8

1

X(t)
2
=0.15
=0.2
=0.25

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

0.2

0.4

0.6

0.8

1

X(t)

Figure 10: Stock price reaction under the optimal capital structure. This figure depicts thejump size ((1 − τ )P (X(t)) − C/r − E(X(t)))/E(X(t)) under the optimal capital structure. The
other parameter values are set in Table 1. Case (DS) holds for 1/λ = 1.5, while Case (DOS)holds for the other parameter values.

43