Discussion Papers In Economics
And Business

Asset sale, debt restructuring, and liquidation
Michi NISHIHARA, Takashi SHIBATA

Discussion Paper 15-22

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANAsset sale, debt restructuring, and liquidation
Michi NISHIHARA, Takashi SHIBATA

Discussion Paper 15-22

August 2015

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANAsset sale, debt restructuring, and liquidation ∗Michi NISHIHARA†‡,Takashi SHIBATA§

Abstract
This paper considers a dynamic model in which shareholders of a ﬁrm in distress havea choice of whether to proceed to debt restructuring or direct liquidation at an arbitrary
time. In the model, we show the following results. Fewer asset sales, lower ﬁnancing,debt renegotiation, and running costs, a lower premium to the debt holders, a lower
cash ﬂow volatility, and a higher initial coupon increase the shareholders’ incentive tochoose debt restructuring to avoid full liquidation. In the debt renegotiation process, theshareholders arrange the coupon reduction and use equity ﬁnancing to retire a part ofthe debt value to the debt holders. The timing of debt restructuring always coincides
with that of liquidation without debt renegotiation. Most notably, the shareholders do
not prefer asset sale in debt restructuring even if they face high ﬁnancing costs. The
possibility of debt renegotiation in the future increases the initial leverage ratio in the
optimal capital structure.
JEL Classiﬁcations Code: C73; G31; G33.
Keywords: real options; asset sale; debt renegotiation; liquidation; capital structure.

∗

This version: 13 August, 2015. This work was supported by the JSPS KAKENHI (26350424, 26285071)

and the JSPS Postdoctoral Fellowships for Research Abroad. This paper was written when Michi Nishihara
was a JSPS Postdoctoral Fellow for Research Abroad (Visiting Researcher at the Swiss Finance Institute,École polytechnique fédérale de Lausanne). The authors would like to thank the Swiss Finance Institute(especially, Professor Erwan Morellec) and the JSPS Postdoctoral Fellowships for Research Abroad. This paper
was presented at 13th Viennese workshop on optimal control and dynamic games in Vienna, the Oligo workshop
in Madrid, and 27th European conference on operational research in Glasgow. The authors would like to thankthe participants for helpful comments.
†
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‡
Swiss Finance Institute, École polytechnique fédérale de Lausanne, 1015 Lausanne, Switzerland§
Graduate School of Social Sciences, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo
192-0397, Japan, E-mail: tshibata@tmu.ac.jp, Phone: 81-42-677-2310, Fax: 81-42-677-2298

1

Introduction

There has been a growing trend to investigate dynamic investment and ﬁnancing models in
corporate ﬁnance. One of the advantages of analyzing dynamic models over static models
is that we can deal with optimal timing problems more clearly. For instance, Mauerand Sarkar (2005) and Sundaresan and Wang (2007) derived the optimal investment
timing with the optimal capital structure.1 Compared to an increasing number of papers
regarding dynamic investment and ﬁnancing problems, there are not so many papers
that analyze dynamic models of divestment and deleveraging. Although a number ofthe papers, including seminal works by Anderson and Sundaresan (1996), Mella-Barral
and Perraudin (1997) and Fan and Sundaresan (2000), focus on liquidation and debt
renegotiation problems of a ﬁrm in ﬁnancial distress, they usually ignore problems related
to downsizing and continuing to operate a ﬁrm.2 In the real world, it is not unusual
to downsize and/or deleverage a ﬁrm’s operations during an economic downturn. For
example, in the Japanese electronics industry, Panasonic, Sony, and Sharp corporations
have experienced large-scale downsizing in recent years.
We reveal interactions of downsizing, debt restructuring, and liquidation in the following dynamic model. The equity holders of a levered ﬁrm in distress face a problemof choosing either debt restructuring or direct liquidation. They can also choose the optimal timing. If they choose debt restructuring, they can reduce the coupon payments
but are forced to sell a fraction of assets, where following the standard assumption (e.g.,
Mella-Barral (1999) and Gryglewicz (2009)), we assume that partial liquidation is lesseﬃcient than full liquidation in terms of asset price (economies of scale). We do not
focus on the debt holders’ strategic behaviour and suppose that the debt holders accept
the shareholders’ proposal with a suﬃcient premium compared to the value without debt
renegotiation.3 Naturally, the equity holders need to pay back more to the debt holders when they wish to reduce more coupon payments and continue the ﬁrm’s operation
longer. The equity holders also pay debt renegotiation costs, such as transaction costs,
as well as equity ﬁnancing costs that arise when the proceeds from selling assets cannot
cover all costs associated with debt renegotiation. Considering the trade-oﬀ, the equity
holders optimize the coupon reduction and its timing. The equity holders can choose
direct liquidation when it is more beneﬁcial than debt restructuring.
Our analysis of the model yields several results about when, how, and whether the ﬁrm
proceeds to debt restructuring or direct liquidation. First, we show that the timing of debt
restructuring is always equal to that of liquidation without debt renegotiation if it occurs.

1

Recently, Shibata and Nishihara (2012), Shibata and Nishihara (2015a), and Shibata and Nishihara (2015b)

extended their analysis to the cases involving debt ﬁnancing constraints.
2
A notable exception is Reindl (2013), who clariﬁed conditions under which a ﬁrm decreases its leverage
ratio along with selling assets by analyzing a dynamic game between equity and debt holders.
3This paper also explores the impact of the debt holders’ bargaining power by varying the premium.

1

This is because of the timing when the shareholders can most eﬃciently reduce the value
which they retire to the debt holders. This result is in line with Lambrecht (2001) and
Moraux and Silaghi (2014), but they do not consider asset sale and payback to the debt
holders in exchange for the coupon reduction. We show that, unlike the previous results,
the shareholders greatly decrease the coupon so that they need to pay back the partial
debt value to the debt holders. The shareholders use costly equity ﬁnancing in addition
to the proceeds from selling assets to repay the partial debt value in debt restructuring.
This is because operating the ﬁrm longer through deleveraging is more beneﬁcial to theequity holders even though they pay temporarily higher costs in debt restructuring.
A most notable result is that the shareholders do not prefer asset sale in debt restructuring. In other words, if asset sale is not forced, they adjust the coupon reduction andequity ﬁnancing to avoid asset sale. This result is diﬀerent from the following intuition:
the equity holders may wish to sell more assets to cover the costs of debt renegotiationeven if asset sale is less advantageous from the viewpoint of asset price. It is not optimalfor the shareholders to sell assets to ﬁnance the repayment value even when equity ﬁnancing is very costly. Instead of avoiding asset sale, they mitigate the coupon reduction anddecrease the repayment value to the debt holders.
Our result stems not only from the assumption of economies of scale. A more important
motivation is the timing of debt restructuring. Because the debt restructuring time isearlier than the ﬁnal liquidation time, more losses are associated with asset sale at the
debt restructuring time. Indeed, the shareholders can reduce the loss by deferring asset
sale as long as possible, i.e., until the ﬁnal liquidation time. This result is consistent withthe following empirical evidence. Maksimovic and Phillips (1998) and Maksimovic andPhillips (2001) showed that the timing of asset sale is not related to debt renegotiation
but that it is motivated by improvement of the resource allocation. Because asset saledoes not improve the resource allocation in our model, the shareholders’ unwillingness tosell assets is in line with their result. Arnold, Hackbarth, and Puhan (2013) also showedthat asset sales and investments are signiﬁcantly and positively correlated. Relatedly,
Weiss and Wruck (1998) illustrated a real-world example of ineﬃcient asset sales during
the debt restructuring process.
Another notable result is the impact of the initial coupon of debt on the shareholders’choice of whether to proceed to debt restructuring or direct liquidation. As the initial
coupon is higher, the shareholders are more likely to proceed to debt renegotiation to
avoid direct liquidation. This is because for a higher initial coupon the equity holders
can possibly reduce more coupon payments via earlier debt renegotiation. Then, they cangreatly extend the ﬁrm’s survival time, which leads to more surplus from debt renegotiation. This result predicts that larger/older ﬁrms, which tend to have more debt, aremore likely to avoid direct liquidation. This prediction is consistent with the empiricalevidence in Bris, Welch, and Zhu (2006). Although we do not consider multiple debt
renegotiations, our ﬁnding is also consistent with that of Moraux and Silaghi (2014), who2showed that due to renegotiation costs equity holders give up any further renegotiation
after the coupon is reduced to a suﬃciently low level.
We also show that, in addition to fewer asset sales and a higher initial coupon, lower
ﬁnancing, debt renegotiation, and running costs, as well as a lower premium to the debt
holders and a lower volatility, increase the shareholders’ incentive to proceed to debt
restructuring. These results are consistent with the stylized fact that larger/older/higherproductivity ﬁrms are more likely to avoid direct liquidation (e.g., Bris, Welch, and Zhu(2006), Maksimovic and Phillips (1998)).
In addition, we examine how the optimal capital structure, where the initial couponof debt is chosen to maximize the ﬁrm value, diﬀers in cases with and without debt
renegotiation in the future. In the case of taking account of debt renegotiation in the
future, the ﬁrm takes a higher coupon, leverage, and credit spread at the initial time.
This is because, by increasing the coupon, leverage, and credit spread at the initial time,
the ﬁrm can gain more beneﬁts of debt renegotiation. Similar results are also documented
in Moraux and Silaghi (2014) and Christensen, Flor, Lando, and Miltersen (2014).
The remainder of this paper is organized as follows. As a benchmark, Section 2
examines the direct and partial liquidation options of an unlevered ﬁrm and shows thatthe unlevered ﬁrm always prefers direct liquidation to downsizing under the assumption ofscales of economies. In Section 3.1, we examine the liquidation option of the levered ﬁrm.
In Section 3.2, we explore the downsizing and debt restructuring option of the levered
ﬁrm. In Section 4, we present a wide range of numerical examples and explain key results.
Section 5 concludes the paper.

2

Unlevered ﬁrm

2.1

Direct liquidation

Throughout this paper, we consider a risk-neutral ﬁrm that is receiving EBIT (Earnings
before interests and taxes) X(t) − w at time t, where X(t) is a stochastic component andw(≥ 0) is a constant running cost. Following the standard real options literature, we
assume that X(t) follows a geometric Brownian motion:dX(t) = µX(t)dt + σX(t)dB(t) (t > 0),

X(0) = x,where B(t) denotes the standard Brownian motion deﬁned in a probability space (Ω, F, P)
and µ, σ(> 0) and x(> 0) are constants. Positive constants r and τ ∈ (0, 1) denote theinterest rate and the corporate tax rate, respectively. For convergence, we assume thatr > µ. For the economic rationale behind these standard assumptions, refer to Dixit andPindyck (1994).
Consider an all-equity ﬁrm that has an option to sell whole assets (denoted by “direct
liquidation”) and gain PU (X(TU ), 1) after taxes at an arbitrary time TU . The subscript3

U stands for the unlevered case. We assume that by selling a fraction a ∈ (0, 1) of assets(denoted by “partial liquidation” or “asset sale”) at time t, the shareholders receive theproceeds PU (X(t), a) after taxes. Assume that PU (x, a) = FU (a)x + GU (a), where thefunctions FU (·) and GU (·) are non-decreasing and convex functions with FU (0) = GU (0) =0.4 The convexity means that partial liquidation destroys existing economies of scale,and hence selling assets sequentially is less proﬁtable than selling assets simultaneously.
Following most papers, including Mella-Barral (1999) and Gryglewicz (2009), we assume
the economies of scale. Also, note that in most cases, assets sold piecemeal are not as
valuable as the same assets sold as a going concern. By this assumption, we can also
diﬀerentiate our study from Reindl (2013), who do not directly relate a partial liquidation
value with a full liquidation value. We presume that the initial value x is suﬃciently highto exclude (partial) liquidation at the initial time.
Now, consider the problem of the equity holders who optimize the direct liquidation
time TU . The equity value, denoted by EU (x), which is the same as the ﬁrm value in theunlevered case, is equal toZ TUe−rt (1 − τ )(X(t) − w)dt + e−rTU PU (X(TU ), 1)],EU (x) = sup E[
TU

(1)

0

where the liquidation time TU is optimized over all stopping times. In the standard
manner (e.g., Dixit and Pindyck (1994)), we can explicitly solve problem (1) as follows.
For (1 − τ )/(r − µ) − FU (1) > 0, we have

µ µ
¶
¶
1−τ
(1 − τ )w(1 − τ )x (1 − τ )w
−rTU−−
+ sup E[e− FU (1) X(TU ) ++ GU (1) ]EU (x) =
r−µ
r
r−µ
r
TU
µ µ
¶
¶ µ ¶γ(1 − τ )x (1 − τ )w
1−τ(1 − τ )w
x
∗
=
−
+ −
− FU (1) xU ++ GU (1)
r−µ
r
r−µ
r
x∗U
(2)
for x ≥ x∗U , where γ = 1/2 − µ/σ 2 −trigger x∗U is given by
x∗U =

γ
γ−1

µ

p

(µ/σ 2 − 1/2)2 + 2r/σ 2 (< 0) and the liquidation1−τ− FU (1)
r−µ

¶−1 µ

¶
(1 − τ )w+ GU (1) .
r(3)

The optimal liquidation time is expressed as TU∗ = {t ≥ 0 | X(t) ≤ x∗U }. Throughout the
paper, following the standard literature (e.g., Leland (1994) and Mella-Barral (1999)), we
presume that (1 − τ )/(r − µ) − FU (1) > 0. Note that only when w and GU (1) are equal
to zero, the equity holders perpetually operate the ﬁrm without liquidation. In that case,
we have x∗U = 0 and EU (x) = (1 − τ )x/(r − µ), which correspond to the unlevered casein Goldstein, Ju, and Leland (2001). In the next subsection, we will derive the equity
value in the case of partial liquidation and compare it with (2) to clarify whether theshareholders prefer direct liquidation or asset sale.
4

Following the standard literature, such as Leland (1994), Mella-Barral and Perraudin (1997), Mella-Barral(1999), and Goldstein, Ju, and Leland (2001), we assume that the liquidation value is linear with respect to thestate variable.

4

2.2

Partial liquidation

We consider a problem of whether the ﬁrm should sell its assets at once or piecemeal.
In order to focus the problem, in the piecemeal case, we limit our attention to a case inwhich the ﬁrm sell assets only twice. We assume that the shareholders ﬁrst sell a fractiona ∈ (0, 1) of assets and after that liquidate the ﬁrm by selling the remaining fraction 1 − a.
We now suppose that a is a given constant.
Consider an all-equity ﬁrm that has an option to sell assets and gain PU (X(TU 1 ), a) at
an arbitrary time TU 1 . Assume that the ﬁrm’s downsizing reduces EBIT from X(t) − w
to (1 − a)(X(t) − w). The equity value, denoted by EU 1 (x), is equal toZ TU 1
EU 1 (x) = sup E[
e−rt (1−τ )(X(t)−w)dt+e−rTU 1 PU (X(TU 1 ), a))+e−rTU 1 EU 2 (X(TU 1 ))],
TU 1

0

(4)
where the downsizing time TU 1 is optimized over all stopping time, and the equity value
after downsizing, denoted by EU 2 (X(TU 1 )), is given by
Z TU 2e−r(t−TU 1 ) (1 − τ )(1 − a)(X(t) − w)dt
EU 2 (X(TU 1 )) = sup EX(TU 1 ) [
TU 2

+e

TU 1

−r(TU 2 −TU 1 )PU (X(TU 1 ), 1 − a))],(5)

where the liquidation time TU 2 is optimized over all stopping time later than TU 1 . The
notation EX(TU 1 ) [·] denotes the expectation conditional to t = TU 1 and X(t) = X(TU 1 ).
As in (2), we can explicitly solve problem (5) as follows:(1 − τ )(1 − a)X(TU 1 ) (1 − τ )(1 − a)w
EU 2 (X(TU 1 )) =
−
r−µ
r
µ µ
¶
¶µ
¶
(1 − τ )(1 − a)(1 − τ )(1 − a)w
X(TU 1 ) γ+ −− FU (1 − a) x∗U 2 ++ GU (1 − a)
r−µ
r
x∗U 2
for X(TU 1 ) ≥ x∗U 2 , where the optimal liquidation trigger x∗U 2 isx∗U 2 =

γ
γ−1

µ

1−τ
FU (1 − a)
−
r−µ
1−a

¶−1 µ

(1 − τ )w GU (1 − a)
+
r
1−a

¶
.

(6)

Note that EU 2 (X(TU 1 )) = FU (1 − a)X(TU 1 ) + GU (1 − a) for X(TU 1 ) < x∗U 2 .
Then, we can explicitly solve problem (4) as follows:
½ µ
¶
¾µ
¶
(1 − τ )x (1 − τ )w
(1 − τ )a(1 − τ )aw
x γ∗
EU 1 (x) =−+ −− FU (a) xU 1 ++ GU (a)
r−µ
r
r−µ
r
x∗
¶
¾µ
½ µ
¶γU 1
(1 − τ )(1 − a)wx(1 − τ )(1 − a)
− FU (1 − a) x∗U 2 ++ GU (1 − a)
+ −
r−µ
r
x∗U 2
(7)
for x ≥ x∗U 1 , where the downsizing trigger x∗U 1 isx∗U 1 =

γ
γ−1

µ

1−τ
FU (a)
−
r−µ
a

5

¶−1 µ

(1 − τ )w GU (a)
+
r
a

¶
(8)

when x∗U 1 > x∗U 2 is satisﬁed. Otherwise,½ µ¶(1 − τ )x (1 − τ )w
(1 − τ )EU 1 (x) =−+ −− FU (a) − FU (1 − a) x∗∗
U1
r−µ
r
r−µ
¾µ
¶γ
(1 − τ )w
x++ GU (a) + GU (1 − a)
r
x∗∗
U1

(9)

∗∗
for x ≥ x∗∗
U 1 , where liquidation occurs immediately after asset sale and the trigger xU 1 isdeﬁned by
x∗∗
U1 =

γ
γ−1

µ

¶−1 µ
¶
1−τ
(1 − τ )w− FU (a) − FU (1 − a)
+ GU (a) + GU (1 − a) .
r−µr

In this “piecemeal” case, we assume that the shareholders receive PU (X(TU 1 ), a)+PU (X(TU 1 ), 1−a) at the trigger x∗∗
U 1.
We can easily prove that EU 1 (x) ≤ EU (x) for an arbitrary a as follows. We consider (4) replaced PU (X(TU 1 ), a) and PU (X(TU 1 ), 1 − a) with aPU (X(TU 1 ), 1) and (1 −a)PU (X(TU 1 ), 1), respectively. We denote this value function by ẼU 1 (x). Because of theconvexity of FU (·) and GU (·), we have PU (X(TU 1 ), a) ≤ aPU (X(TU 1 ), 1) and PU (X(TU 1 ), 1−a) ≤ (1 − a)PU (X(TU 1 ), 1). Then, we have EU 1 (x) ≤ ẼU 1 (x). Because we can calculate
ẼU 1 (x) following (9) and check that ẼU 1 (x) = EU (x). Thus, we have EU 1 (x) ≤ ẼU 1 (x) =EU (x).
This result means that the all-equity ﬁrm always prefers direct liquidation to partial
liquidation. As we can see from the proof above that the reason lies in the price advantage
of selling whole assets over piecemeal assets (economies of scale). In the absence ofeconomies of scale (e.g., Reindl (2013)), a ﬁrm may prefer partial liquidation. One example
is a case in which ﬁre sales are accompanied only by ﬁnal liquidation. In the following
sections, we will explore the optimal decisions of the equity holders of the levered ﬁrm. It
is a key question to be answered whether the levered ﬁrm always chooses direct liquidation
under the assumption of economies of scale.

3

Levered ﬁrm

3.1

Direct liquidation

In this subsection, we derive the equity, debt, and ﬁrm values of the ﬁrm that chooses
direct liquidation. Consider the ﬁrm that issued console debt with coupon c and is operating with the asset size a. We denote the equity, debt, and ﬁrm values of the ﬁrm by
E(x, a, c), D(x, a, c), and V (x, a, c), respectively. As in Section 2, we consider the asset
price function PL (x, a) = FL (a)x + GL (a) for asset sale, where the functions FL (·) andGL (·) are non-decreasing and convex functions with FL (0) = GL (0) = 0. In the levered
case, the price function PL (x, a) can be diﬀerent from PU (x, a), although we do not specify
the relation.

6

Following the standard manner (e.g., Leland (1994) and Goldstein, Ju, and Leland(2001)), we have
Z T∗
L
E(x, a, c) =E[
e−rt (1 − τ )(aX(t) − aw − c)dt]
0
µ
¶µ
¶γ
(1 − τ )ax∗L (a, c) (1 − τ )(aw + c)
(1 − τ )ax (1 − τ )(aw + c)
x
=
−
−
−
r−µ
r
r−µ
r
x∗L (a, c)Z T∗L∗
D(x, a, c) =E[
e−rt cdt + e−rTL PL (X(TL∗ ), a)]0´ µ x ¶γc ³c∗
− PL (xL (a, c), a)= −rrx∗L (a, c)

(10)

(11)

V (x, a, c) =E(x, a, c) + D(x, a, c)=(1 − τ )ax (1 − τ )aw τ c
−
+
r−µ
r
r
µ
¶γ
¶µ
∗
(1 − τ )axL (a, c) (1 − τ )aw τ cx−−+− PL (x∗L (a, c), a)r−µrrx∗L (a, c)(12)

for x ≥ x∗L (a, c), where x∗L (a, c) is the liquidation trigger determined by the shareholders
who maximize (10). Actually, it is equal tox∗L (a, c) =γ(r − µ)(aw + c),
(γ − 1)ra(13)

and TL∗ = inf{t ≥ 0 | X(t) ≤ x∗L (a, c)} (we omit arguments (a, c) of TL∗ ). The equations
above presume that debt is risky, i.e., PL (x∗L (a, c), a) < c/r because we are interested only
in risky debt. This condition is satisﬁed in all numerical examples in Section 4.
When the ﬁrm with an initial coupon c0 chooses direct liquidation without debt renegotiation, its equity, debt, and ﬁrm values become E(x, 1, c0 ), D(x, 1, c0 ), and V (x, 1, c0 ),respectively. In this case, we denote by E(x) = E(x, 1, c0 ) and x∗L = x∗L (1, c) to simplify
the notations. Comparing E(x) with the equity value, denoted by EL1 (x), in the case ofdebt restructuring, the equity holders decide whether they proceed to direct liquidation or
debt restructuring. In the next subsection, we will examine the case of debt restructuring.

3.2

Debt restructuring with asset saleIn this section, we derive the equity, debt, and ﬁrm values, denoted by EL1 (x), DL1 (x),and VL1 (x), respectively, of the ﬁrm that chooses debt restructuring along with partial
liquidation. As in Section 2, we assume that the shareholders ﬁrst liquidates a fractiona ∈ (0, 1) of assets and gain PL (X(TL1 ), a) at time TL1 , and after that they liquidate theremaining fraction 1−a and gain PL (X(TL1 ), 1−a) at time TL2 . The shareholders optimize
the partial liquidation and debt restructuring time TL1 and the ﬁnal liquidation time TL2 .
We now assume a constant a, and we will also examine an optimal a in Section 4.2.1.
In the real world, the debt holders or the third party sometimes force the equity holders

7

to sell assets used as the collateral to repay the debt value in default. As documented
in Djankov, Hart, McLiesh, and Shleifer (2008), the forced asset sale is one of the major
debt enforcement frictions.
As in Lambrecht (2001), Moraux and Silaghi (2014), and Christensen, Flor, Lando, andMiltersen (2014), we assume that the shareholders demand a lump-sum and permanent
reduction in the coupon of debt. Actually, it might be diﬃcult for a ﬁrm to continuously
and marginally (e.g., Mella-Barral and Perraudin (1997) and Fan and Sundaresan (2000))
adjust the coupon because of costs associated with the negotiation, adjustment, andtransaction. For instance, Djankov, Hart, McLiesh, and Shleifer (2008) reported highcosts of debt enforcement procedures. At time TL1 , the shareholders can reduce thecoupon c0 into a new level cL1 by debt renegotiation.5 At TL1 , the equity, debt, and ﬁrm
values change to E(X(TL1 ), 1 − a, cL1 ), D(X(TL1 ), 1 − a, cL1 ), and V (X(TL1 ), 1 − a, cL1 ),∗ = inf{t ≥ 0 | X(t) ≤ x∗ = x∗ (1 −respectively. The ﬁnal liquidation time is equal to TL2
LL2∗ . The debt holders would refuse the coupona, cL1 )} under the assumption that TL1 ≤ TL2

reduction proposed by the shareholders unless it is beneﬁcial to them. For simplicity, we donot model a dynamic game between the equity and debt holders. Instead, we assume thatthe shareholders need to pay back (1 + kD ) D(X(TL1 ), 1, c0 ) − D(X(TL1 ), 1 − a, cL1 ) to
|
{z
} |
{z
}
original debt value

new debt value

satisfy the debt holders, where a parameter kD (≥ 0) measures the debt holders’ premium,
if it is positive. Although we do not model a bargaining game between equity and debt
holders, kD can be also regarded as proxy for the debt holders’ bargaining power in debt
renegotiation.6
In addition, we assume that external costs kR D(X(TL1 ), 1, c0 ), such as transaction
costs, are accompanied by debt restructuring, where kR is a non-negative parameter. Notethat kD (kR ) becomes higher (lower) as debt holdings are more concentrated. In total, attime TL1 , the shareholders pay the costs (1 + kD + kR )D(X(TL1 ), 1, c0 ) − D(X(TL1 ), 1 −a, cL1 ) in exchange for the coupon reduction from c0 to cL1 .
Now, we solve the shareholders’ problem of optimizing TL1 = inf{t ≥ 0 | X(t) ≤5This paper focuses on a ﬁnancially distressed situation, and hence, the model does not allow the shareholders

to increase the coupon when EBIT goes up. Christensen, Flor, Lando, and Miltersen (2014) assumed the callable
debt to consider the possibility of increasing debt.
6
In other words, we assume a sort of debt enforcement procedure that protects the creditors from suﬀering
loss by the shareholders’ strategic default. Although we take kD as a given parameter, it should satisfy that
PL (X(TL1 ), 1) ≤ D(X(TL1 ), 1, c0 ) ≤ (1+kD )D(X(TL1 ), 1, c0 ) ≤ c/r. (1+kD )D(X(TL1 ), 1, c0 ) is not necessarilyequal to c/r (face value) because violations of absolutely priority rule frequently occur in debt restructuring(e.g., Bris, Welch, and Zhu (2006)).

8∗ ) and c (≤ c ) as follows:
xL1 }(≤ TL2
0
L1

Z TL1
EL1 (x) = sup E[
e−rt (1 − τ )(X(t) − w − c0 )dt +e−rTL1 {PL (X(TL1 ), a){z}|
TL1 ,cL1|0{z}asset pricevalue before asset sale− {(1 + kD + kR )D(X(TL1 ), 1, c0 ) − D(X(TL1 ), 1 − a, cL1 )} + E(X(TL1 ), 1 − a, cL1 )}]
|
{z
} |
{z
}
total costs
value after asset sale½
(1 − τ )x (1 − τ )(w + c0 )
=
−
+ sup
V (xL1 , 1 − a, cL1 ) + PL (xL1 , a)r−µrxL1 ,cL1¶¾µ(1 − τ )xL1 (1 − τ )(w + c0 )
x γ
,
(14)
− (1 + kD + kR )D(xL1 , 1, c0 ) −
+
r−µ
r
xL1
where we reduce the original problem to the problem (14) of optimizing the debt restructuring trigger, denoted by xL1 ∈ [x∗L2 , x], and the coupon cL1 ∈ [0, c0 ].
Next, we consider the shareholders’ behavior at the debt renegotiation trigger xL1
more carefully. At xL1 , they need the funds
EF = (1 + kD + kR )D(xL1 , 1, c0 ) − D(xL1 , 1 − a, cL1 ) − PL (xL1 , a)|{z
} | {z }total costs(15)

asset price

if EF is positive. Formally, they use equity ﬁnancing to raise the funds. In fact, however,
it is diﬃcult and costly for especially smaller/younger ﬁrms during an economic downturnto use external equity ﬁnancing (e.g., Greenwald, Stiglitz, and Weiss (1984), Bernanke andGertler (1989), and Bernanke, Gertler, and Gilchrist (1996)). Taking the stylized fact into
account, we assume the following equity ﬁnancing cost kF . The shareholders need to pay
the proportional ﬁnancing cost
kF max{EF, 0}(16)when using equity ﬁnancing, where kF (≥ 0) is a constant. The parameter kF is higher forsmaller/younger ﬁrms in a worse economy. If the proceeds from selling assets cover the
total costs of the debt renegotiations, the shareholders do not raise the funds. Unlike inthe unlevered case, the ﬁnancing costs may be a ﬁrm’s great motive for selling assets eventhough partial liquidation is less advantageous in terms of asset price. This point will beclosely explored in Section 4.2.1. In the presence of the equity ﬁnancing cost, we modify
the objective function of the problem (14) with the subtractive term (16) ×(x/xL1 )γ .
Note that if the ﬁnancing cost kF goes to inﬁnity, the problem is subject to the constraint(1 + kD + kR )D(xL1 , 1, c0 ) − D(xL1 , 1 − a, cL1 ) ≤ PL (xL1 , a).

(17)

When a = 0, kD = 0, and the ﬁnancing constraint (17) are assumed, our model corresponds to the debt renegotiation model of Lambrecht (2001) and the single debt renegotiation model of Moraux and Silaghi (2014) (the case where the shareholders have fullbargaining power).
The equity holders obtain the higher value of E(x) and EL1 (x). Note that the inequality between E(x) and EL1 (x) does not depend on x, and hence, the shareholders

9have no incentive to change their decision on the way to liquidation. As will be shown in
Section 4, unlike in the unlevered case in Section 2, the decision whether to proceed to
debt restructuring or direct liquidation depends on the parameter values.
We derive also the debt and ﬁrm values, denoted by DL1 (x) and VL1 (x) respectively,in the debt restructuring case. We denote the optimal debt restructuring time, debt
∗ , x∗ , x∗ , and c∗ . Using
restructuring trigger, ﬁnal liquidation trigger and coupon by TL1
L1 L2
L1∗ , x∗ , and c∗ , we have
TL1
L1
L1
Z T∗
L1
∗
∗
∗
DL1 (x) = E[e−rt c0 dt + e−rTL1 {(1 + kD )D(X(TL1), 1, c0 ) − D(X(TL1), 1 − a, c∗L1 )0∗+ D(X(TL1), 1 − a, c∗L1 )}]´ µ x ¶γc ³c∗
= −− (1 + kD )D(xL1 , 1, c0 )
r
r
x∗L1

VL1 (x) = EL1 (x) + DL1 (x)½(1 − τ )x∗L1 (1 − τ )w τ c0
−
+
r−µ
r
r
¾µ
¶γ
x
− V (x∗L1 , 1 − a, c∗L1 ) + PL (x∗L1 , a) − kR D(x∗L1 , 1, c0 )x∗L1(1 − τ )x (1 − τ )w τ c0
=
−
+
−
r−µ
r
r

(18)

(19)

for x ≥ x∗L1 . We presume that c/r ≥ (1 + kD )D(x∗L1 , 1, c0 ), which is satisﬁed in all
numerical examples in Section 4. Also note that in the debt restructuring case, the ﬁnal
∗ = inf{t ≥ 0 | X(t) ≤ x∗ = x∗ (1 − a, c∗ )}.
liquidation occurs at time TL2
L2
L
L1

In the remainder of this section, we add an explanation why this paper does not assume
the shareholders who maximize the ﬁrm value and share the surplus with the debt holders
through a bargaining game. In the case of ﬁrm value maximization, the ﬁrm does not needto proceed to debt renegotiation but just continues to operate permanently or liquidate
the ﬁrm at a very low liquidation trigger x∗V . In other words, the ﬁrm can choose the“renegotiation” trigger xL1 at 0 or x∗V with setting the “new” coupon cL1 at c0 . Indeed,
when we consider the ﬁrm value maximization problem, the ﬁrm value becomes½¾ µ ¶γ(1 − τ )xV(1 − τ )w τ c0x
(1 − τ )x (1 − τ )w τ c0−+
+ sup −+−+ PL (xV , 1)
r−µ
r
r
r−µ
r
r
xV
xV
½
¾ µ ¶γ∗(1 − τ )xV(1 − τ )x (1 − τ )w τ c0
(1 − τ )w τ c0
x


−
+
+ −
+
−
+ PL (x∗V , 1)


r
r
r−µ
r
r
x∗V
 r−µ
=
(if (1 − τ )w/r − τ c0 /r + GL (1) > 0)
 (1 − τ )x − (1 − τ )w + τ c0 (otherwise),
r−µ
r
r
(20)
where the liquidation trigger x∗V is deﬁned by
µ
¶−1 µ
¶
γ
1−τ
(1 − τ )w τ c0∗
xV =− FL (1)−+ GL (1) ,γ−1 r−µrr(21)

if (1 − τ )w/r − τ c0 /r + GL (1) > 0. Note that (21) is of the same form as (3) except for
the tax advantage τ c0 /r. When (1 − τ )w/r − τ c0 /r + GL (1) ≤ 0, the ﬁrm continues tooperate perpetually.

10

In Mella-Barral and Perraudin (1997) and Mella-Barral (1999), due to τ = 0, the ﬁrm
liquidates at the liquidation trigger like (3) through debt renegotiation, which maximizes
the ﬁrm value. On the other hand, in the presence of tax advantages of debt, most of theliterature restricts beneﬁts of tax savings when the equity and debt holders cooperate tomaximize the ﬁrm value. For example, Fan and Sundaresan (2000) considered the debt
renegotiation process assuming no tax advantage of debt while the equity and debt holders
cooperate to adjust the coupon. Without this sort of restriction, in the basic models,such as those provided by Leland (1994) and Goldstein, Ju, and Leland (2001) (note thatw = 0), the equity and debt holders could continue to operate the ﬁrm perpetually andenjoy tax savings without potential default costs. Thus, in our model, it is not meaningful
to consider the ﬁrm value maximization problem (20), allowing tax advantage of debt and abargaining problem between the equity and debt holders. As in Lambrecht (2001), Reindl(2013), Christensen, Flor, Lando, and Miltersen (2014), Moraux and Silaghi (2014), we
consider the equity holders’ value maximization problem (14).

4

Numerical analysis and implications

4.1

Basic results

In this section, we compute problem (14) with the ﬁnancing costs (16) for the base parameter values as follows:7
r = 0.06, µ = 0.01, σ = 0.2, τ = 0.15, x = 2, w = 0.5, a = 0.1, c0 = 2, kD = kR = 0.05, kF = 0.1.
(22)
We also deﬁne the asset price by PL(x, a) = 0.6a1.01 x/(r − µ) + 2a1.01 in the base case.

Note that for x = 1 the full liquidation value PL (x, 1) = 0.6x/(r − µ) + 2 is the same as
0.7x/(r − µ), which is often used as the liquidation value of the basic literature.8
For the base parameter (22), we have Table 1, where LV = D(x∗L , 1−a, c∗L1 )/V (x∗L , 1−a, c∗L1 ) (leverage ratio), CS = c∗L1 /D(x∗L , 1 − a, c∗L1 ) − r (credit spread), and EF = (1 +kD +kR )D(x∗L1 , 1, c0 )−D(x∗L1 , 1−a, c∗L1 )−PL (x∗L1 , a) (newly issued equity value). Because
EL1 (x) = 5.91 > E(x) = 5.58, the equity holders choose debt renegotiation. At the debt
renegotiation trigger x∗L1 = 1.25, they reduce the coupon from c0 = 2 to c∗L1 = 1.07;
instead, they need to pay the total costs (1 + kD + kR )D(x∗L1 , 1, c0 ) − D(x∗L1 , 1 − a, cL1 ) =3.02. The proceeds from asset sale, PL (x∗L1 , a) = 1.44, are not suﬃcient to cover the costs,
and hence they raise new equity ﬁnancing EF = 1.58. Then, the new leverage ratio and
7

The growth rate µ is set at a small value because we focus on the ﬁrm approaching liquidation in an

economic downturn. Following the standard literature, we set r and τ . The initial state x is not substantial
because it can be normalized. In Section 4.2, we will show the comparative statics with respect to the otherparameter values.
8
In the parameter values, the liquidation trigger is around 1. We also computed many results, changing theparameter values of 0.6, 1.01, and 2. The results are straightforward and are omitted from the paper.

11credit spread become LV = 0.846 and CS = 0.0207, respectively. The ﬁrm continues
to operate until the state variable X(t) hits the ﬁnal liquidation trigger x∗L2 = 0.842.
Although this is a typical outcome, several values depend on the parameter values. In thenext subsection, we will explain the comparative statics results. Now, we explain several
robust ﬁndings which hold true regardless of parameter values.
In the base case, because of EF = 1.58 > 0, the shareholders need equity ﬁnancing to
pay the costs associated with debt renegotiation. According to our computation for a wide
range of parameter values, this result remains true even when the equity ﬁnancing cost
kF is very high. Of course, the shareholders suﬃciently mitigate the coupon reduction to
decrease EF as kF increases. However, EF is always positive unless kF is inﬁnite. To
our knowledge, there has been no paper that shows the eﬃciency of equity ﬁnancing in
debt restructuring. Most of the literature about debt renegotiation, including Lambrecht(2001) and Moraux and Silaghi (2014), considered neither the partial retirement of debt
nor equity ﬁnancing. Reindl (2013), who considered the problem of buying back debt anddeleveraging along with selling asset, denied the possibility of equity ﬁnancing.
(Insert Figures 1 and 2 around here.)
Next, in Table 1 we ﬁnd that the debt restructuring trigger x∗L1 is exactly equal
to the original liquidation trigger x∗L without debt renegotiation. This means that theleverage ratio without debt renegotiation increases up to 1 right before debt renegotiation.
We made numerous computations in addition to presented examples and veriﬁed thatx∗L1 = x∗L in all cases. For instance, Figure 1 shows that x∗L1 = x∗L = 1.25 without regard
to the fraction of asset sale a, although, as will be checked in Figure 3 of Section 4.2.1, the
new coupon c∗L1 varies with a. While this result does not hold in debt renegotiation models
with a temporary reduction in the coupon (e.g., Mella-Barral and Perraudin (1997) and
Fan and Sundaresan (2000)), it is in line with the results in previous models with a lumpsum and permanent reduction in the coupon. Actually, Lambrecht (2001) and Morauxand Silaghi (2014) showed the same result, although they did not consider either asset
sale or debt retirement. In Reindl (2013), the debt restructuring timing can be earlier
depending on the asset price. This is because his assumption, unlike our assumption ofPL (x, a), does not directly relate partial liquidation value and the bankruptcy value, andhence partial liquidation can be more proﬁtable than full liquidation. In our view, debt
restructuring always takes place at the original liquidation trigger as long as we assume
economies of scale in production relating asset sale with the full liquidation value.
Using Figure 2, we explain the rationale behind the result that x∗L1 = x∗L . Figure 2
depicts the debt value D(xL1 , 1, c0 ) as a function of xL1 . For xL1 ≥ x∗L , D(xL1 , 1, c0 ) is
concave and increasing. Because of the concavity, the shareholders can eﬃciently decrease
the repayment value by waiting until X(t) reaches x∗L . They have no incentive to strategically default before the original liquidation time because of the high repayment value.
On the other hand, the shareholders do not delay the debt renegotiation timing after theoriginal liquidation time because they do not wish to continue to pay the initial coupon12c0 any longer. The debt value D(xL1 , 1, c0 ) coincides with a linear function PL (xL1 , 1)
for xL1 ≤ x∗L = 1.25, and hence there is no incentive for the shareholders to delay debt
renegotiation any longer. Accordingly, the equity holders optimally choose the same debt
renegotiation timing as the liquidation timing, i.e., x∗L1 = x∗L .
Based on the result above, from now on we suppose that x∗L1 = x∗L and examine theequity holders’ decision in more details. Under the condition, we can reduce (14) to thefollowing:µ ¶γ ½(1 − τ )x (1 − τ )(w + c0 )x
+ PL (x∗L , a)−
+ sup V (x∗L , 1 − a, cL1 )
∗
r−µ
r
x
cL1
L
¾ µ ¶γ
∗
(1
−
τ
)x
(1
−
τ
)(w
+
c
)
x
0
∗
L
− (1 + kD + kR )PL (xL , 1) −
+
,
(23)
r−µ
r
x∗L

EL1 (x) =where we used D(x∗L , 1, c0 ) = PL (x∗L , 1). In the absence of ﬁnancing costs, the condition
under which debt renegotiation is preferred is as follows:
EL1 (x) > E(x)⇔V (x∗L , 1 − a, c∗L1 ) + PL (x∗L , a) − (1 + kD + kR )PL (x∗L , 1) > 0.

(24)

For a = w = kD = kR = 0, we can ﬁnd c < c0 such that D(x∗L , 1, c) = PL (x∗L , 1) because
D(x∗L , 1, c) is hump-shaped with respect to c ≤ c0 (e.g., Leland (1994), Lambrecht (2001),
and Figure 1 of Moraux and Silaghi (2014)). Using this c, we have
V (x∗L , 1, c∗L1 ) − PL (x∗L , 1) ≥V (x∗L , 1, c) − PL (x∗L , 1)=E(x∗L , 1, c)>0.
That is, without asset sale, operating costs, and costs associated with debt restructuring,
the shareholders always prefer debt renegotiation to liquidation. This result remains
unchanged even in the presence of a ﬁnancing cost because we have EF = 0 for thesame c. The similar results were found in many papers (e.g., Lambrecht (2001), Reindl(2013), and Moraux and Silaghi (2014)). On the other hand, if any of a, w, kD , and kR
are positive, the shareholders may proceed to direct liquidation. In the next subsection,
we will examine how the parameter values aﬀect the equity holders’ decision of whether
to proceed to debt restructuring or direct liquidation.

4.2

Comparative statics

This section analyzes comparative statics with respect to parameters a, c0 , σ, kF , kD , andw, and it reveals how these parameters aﬀect the shareholders’ optimal decision. Above
all, the impacts of the fraction of asset sale a and the initial coupon c0 are novel anddiscussed in detail. In each ﬁgure, the other parameter values are ﬁxed at the base case(22). Each ﬁgure contains six panels which show equity values (EL1 (x) and EL (x)),

13new coupons (c∗L1 ), new and original liquidation (renegotiation) triggers (x∗L2 and x∗L (=x∗L1 )), new leverage ratios (LV = D(x∗L , 1 − a, c∗L1 )/V (x∗L , 1 − a, c∗L1 )), new credit spread(CS = c∗L1 /D(x∗L , 1 − a, c∗L1 ) − r), and the newly issued equity value (EF = (1 + kD +kR )D(x∗L1 , 1, c0 ) − D(x∗L1 , 1 − a, c∗L1 ) − PL (x∗L1 , a)). In some ﬁgures, we show other values
to explore the results more closely.

4.2.1

Fraction of asset sale a(Insert Figure 3 around here.)
Figure 3 shows the panels with varying levels of the fraction of asset sale a. We cansee from the top left panel that EL1 (x) monotonically decreases with a. For a ≥ 0.34,
the shareholders prefer to liquidate the ﬁrm because of the ineﬃciency in the partial asset
sale in debt renegotiation. According to a number of computations, we ﬁnd that partial
liquidation always destroys the equity value. Partial liquidation is less eﬃcient not only
because PL (x, ·) is convex but also because for x = x∗L , which is larger than x∗L2 , the equity
holders pay the fundamental costs of partial liquidation, i.e., (a − 0.6a1.01 )x/(r − µ). In
other words, they wish to defer all liquidation costs until X(t) hits the ﬁnal liquidation
trigger x∗L2 . If the shareholders can optimize a in debt renegotiation, they always choose
a = 0 (debt reorganization with no asset sale). Instead, they optimally adjust the coupon
reduction and use equity ﬁnancing. Actually, as checked in the bottom right panel of
Figure 3, selling assets adversely increases the newly issued equity value because it reduces∗ ) more than the increase in P (x∗ , a).
the new debt value D(x∗L , 1 − a, CL1
L L(Insert Figure 4 around here.)
Even if a ﬁnancing cost kF is higher (in an extreme case inﬁnite) or PL (x, ·) is linear,
they do not prefer a positive a. Figure 4 shows EL1 (x) and E(x) with varying levels
of a, where kF is set at 0.5, 1, and 1.5. The left and right panels show the results for
PL (x, a) = 0.6a1.01 x/(r − µ) + 2a1.01 (base case) and PL (x, a) = 0.6ax/(r − µ) + 2a (linearcase), respectively. Although the regions of direct liquidation increase with higher kF , theequity holders do not voluntarily sell assets in order to repay the partial debt value.
This result is diﬀerent from the ﬁnding of Reindl (2013), who argued that the possibility of selling assets causes deleveraging. Again, in our view, this is because partial
liquidation can be more proﬁtable than full liquidation in the setup of Reindl (2013).
Our result can potentially account for several empirical ﬁndings. Actually, Maksimovic
and Phillips (1998) and Maksimovic and Phillips (2001) showed that the frequency ofasset sales does not increase during the debt reorganization process. They argued thatlower-productivity ﬁrms are more likely to sell assets to higher-productivity ﬁrms whenindustry output increases, which leads to improvement in the resource allocation. Notethat our model focuses only on a ﬁrm’s behaviour in declining economies and assumes thatsome costs are associated with liquidation. Arnold, Hackbarth, and Puhan (2013) also
showed that asset sales are related to an increase in investment and resource allocation.

14

Relatedly, Weiss and Wruck (1998) documented the case of Eastern Airlines, in which
ineﬃcient asset sales caused Chapter 11’s failure.
The values x∗L2 , x∗L , LV, and CS are almost constants over a (note the scale of the vertical axis in the ﬁgure). This is because the debt service ratio (1 − a)/c∗L1 is almost constant
while c∗L1 decreases with a (see the top right panel of Figure 3). That is, the fraction ofasset sale a does not greatly inﬂuence the capital structure after debt restructuring.

4.2.2

Initial coupon c0(Insert Figure 5 around here.)
Figure 5 shows the panels with varying levels of initial coupons c0 . Naturally, EL1 (x)and E(x) decrease with a higher c0 , while x∗L2 and x∗L increase. Interestingly, in the
top left panel, for c0 ≤ 1.65 the shareholders prefer liquidation to debt renegotiation.
This is because, for a higher c0 , the equity holders can reduce the coupon more (see thediﬀerence between the two lines in the top right panel), which leads to more surplus fromdebt renegotiation. Accordingly, a higher c0 increases the incentive to proceed to debt
restructuring rather than direct liquidation.
This result may align with the “too big to fail” theory. Indeed, our model suggeststhat larger/older ﬁrms with more debt gain more surplus from succeeding in debt renegotiation and avoiding full liquidation. Note that due to information asymmetry problems,
smaller/younger ﬁrms tend to have limited access to debt (e.g., Bernanke, Gertler, andGilchrist (1996)). Our prediction is also consistent with the stylized fact that larger/older
ﬁrms are more likely to proceed to debt renegotiation rather than direct liquidation (e.g.,
Bris, Welch, and Zhu (2006)). Although we do not consider multiple debt renegotiations,
our result is also similar to the result of Moraux and Silaghi (2014). They showed that
due to renegotiation costs equity holders give up any more renegotiation round after the
coupon is reduced to a suﬃciently low level. We will also return to this point while
examining another aspect in Section 4.3.
The sensitivities of c0 on LV and CS are counter-intuitive (see the middle right andbottom left panels of Figure 5). The ﬁrm with a higher c0 will be less risky after debt
restructuring. The reason can be explained by the top right and middle left panels. Indeed,
we ﬁnd from the panels that, for a higher c0 , the coupon reduction c0 − c∗L1 is larger andthe ﬁrm’s survival time after debt restructuring is longer. Because of this eﬃcient capital
restructuring on the early timing, the ﬁrm with a higher c0 can be less risky after debt
renegotiation. On the other hand, due to the large coupon reduction, the ﬁrm needs more
equity ﬁnancing to pay back to the debt holders (see the bottom right panel).

4.2.3

Volatility σ(Insert Figure 6 around here.)
Figure 6 shows the panels with varying levels of cash ﬂow uncertainty σ. In the top

15and middle left panels, EL1 (x) and E(x) (x∗L and x∗L2 ) increase (decrease) with a higher
σ. This corresponds to the standard volatility eﬀect (e.g., Dixit and Pindyck (1994)) thata higher σ increases the option value of waiting and delays the liquidation timing. Morenotably, we ﬁnd that the increase in E(x) (decrease in x∗L ) dominates that of EL1 (x)(x∗L2 ). As a result, E(x) exceeds EL1 (x) for σ ≥ 0.35. The reason is that the convexityof shareholders’ option to liquidate is stronger in the direct liquidation case than in thepiecemeal liquidation case. Kort, Murto, and Pawlina (2010) also documented a similar
logic in the context of stepwise investment. Our result is consistent with the empiricalevidence in Favara, Schroth, and Valta (2012) and Favara, Morellec, Schroth, and Valta(2014). They showed that debt renegotiation decreases the convexity of shareholders’claim and their incentives for risk-taking.
The ﬁnding that a ﬁrm with a high σ is more likely to liquidate without debt renegotiation is also consistent with the following empirical evidence. Smaller/younger ﬁrms
tend to have a higher σ, and such ﬁrms are more likely to proceed to direct liquidation(e.g., Bris, Welch, and Zhu (2006)). We can see from LV and CS that the ﬁrm with ahigher σ is more risky, despite that the newly arranged coupon c∗L1 is lower. The graphsof LV and CS are similar to those presented by Shibata and Nishihara (2015b). In thebottom right panel, the result on EF is not monotonic but unimodal.

4.2.4

Financing cost kF(Insert Figure 7 around here.)
Figure 7 shows the panels with varying levels of equity ﬁnancing costs kF . EL1 (x)monotonically decreases with kF . In this example, once kF increases beyond 0.8, theshareholders prefer direct liquidation to debt restructuring. In the ﬁgure, we ﬁnd that ahigher kF increases c∗L1 and decreases EF ; nevertheless, the ﬁnancing cost (16) increases(see Cost in the bottom right panel). That is, with a higher equity ﬁnancing cost, theshareholders give up a larger coupon reduction. Then, the time interval between debt
renegotiation and ﬁnal liquidation becomes shorter with a higher kF . Indeed, the middle
left panel shows that a higher kF increases the ﬁnal liquidation trigger x∗L2 while it does
not change the renegotiation trigger x∗L . As well as the liquidation probability, LV and CS
increase with a higher kF . Especially smaller/younger ﬁrms tend to face more diﬃculty inaccessing external ﬁnancing during a recession (e.g., Greenwald, Stiglitz, and Weiss (1984),
Bernanke and Gertler (1989), and Bernanke, Gertler, and Gilchrist (1996)). Taking this
into account, our result is consistent with the empirical evidence that, during a recession,
weaker ﬁrms are likely to proceed to direct liquidation rather than debt renegotiation(e.g., Bris, Welch, and Zhu (2006)).

4.2.5

Premium to the debt holders kD(Insert Figure 8 around here.)16

Figure 8 shows the panels with varying levels of premiums kD , which the debt holders
receive. Although this paper does not model any bargaining game, kD can be proxy forthe debt holders’ bargaining power in debt renegotiation. As is shown clearly, EL1 (x)monotonically decreases with kD . In this example, for kD ≥ 0.091, the shareholders
prefer liquidation to debt renegotiation. This negative impact of kD on debt renegotiationis straightforward and consistent with the previous ﬁndings regarding the debt holders’bargaining power (e.g., Moraux and Silaghi (2014) and Favara, Morellec, Schroth, andValta (2014)). If the debt holders have full bargaining power, they optimally choose thecritical premium kD = 0.091, which is the cross point of EL1 (x) and E(x). Note that the
debt holders gain more in debt renegotiation than in direct liquidation as long as kD islarger than 0.
In Figure 8, c∗L1 , x∗L2 , x∗L , LV , and CS do not depend on kD . These are obvious fromthe equations (23) and (16). Indeed, the shareholders choose c∗L1 = arg maxcL1 V (x∗L , 1 −a, cL1 )−kF D(x∗L , 1−a, cL1 ) regardless of the values of kD . This means that the premium tothe debt holders does not inﬂuence the capital structure after debt renegotiation. Lastly,
as is easily checked by (15), EF increases with kD (see the bottom right panel) because
the shareholders need to pay back more to the debt holders. We omit the comparative
statics with respect to the renegotiation cost kR because its eﬀects are quite similar to
those of kD .

4.2.6

Running cost w(Insert Figure 9 around here.)
Figure 9 shows the panels with varying levels of running costs w. In the top left panel,a higher w decreases both EL1 (x) and E(x). The eﬀect on EL1 (x) is larger because theﬁrm that chooses debt renegotiation will operate and suﬀer from the running cost after theoriginal liquidation time. In this example, for w ≥ 0.62 the shareholders prefer liquidation
to debt renegotiation. The impact of w is opposite from that of c0 (cf. Figure 5), althoughboth w and c0 are costs to the equity holders. This diﬀerence stems from the fact that w,
unlike c0 , cannot be reduced in debt renegotiation.
In the middle left panel, we ﬁnd that both x∗L2 and x∗L increase with w. Furthermore,
the distance between the two triggers monotonically decreases with w. This means thatthe ﬁrm with a higher w is more risky even when it succeeds in debt renegotiation. LV
and CS in the middle right and the bottom left panels also support this result. Our
result is consistent with the stylized fact that lower-productivity ﬁrms tend to fail in debt
renegotiation and proceed to liquidation (e.g., Maksimovic and Phillips (1998)).

4.3

Optimal capital structure

So far, we have changed parameter values with the initial coupon c0 ﬁxed at 2 in thebase case except for Figure 5. We also check the comparative statics with c0 maximizing

17

the ﬁrm value at the initial time. The ﬁrm chooses an initial coupon c0 for which theshareholders can choose whether to proceed to debt restructuring or direct liquidation.
Taking account of the shareholders’ future behavior contingent on c0 , we need to compute
an optimal c0 .
For the base parameter (22) where c0 replaced the optimal coupon, we have Table 2,
where the ﬁrst and second rows present the optimal capital structure in the case allowing
debt restructuring (DR) and the case with direct liquidation only (L), respectively. LV0
and CS0 denote the leverage ratio and credit spread at the initial time. In the DR case,
we have the optimal coupon c0 = 1.95, which will lead to debt renegotiation rather thandirect liquidation in the future.9 Table 2 shows that the ﬁrm that will proceed to debt
renegotiation has a higher c0 , LV0 , and CS0 than in the direct liquidation case. This isbecause, as shown in Figure 5, the ﬁrm can gain more surplus from debt renegotiation
when the initial coupon is high. This may be related to a moral hazard problem caused byprospective debt renegotiation. Our result is similar to the previous ﬁndings. For instance,
Moraux and Silaghi (2014) showed that a ﬁrm increases the initial coupon, taking accountof debt renegotiation in the future. Christensen, Flor, Lando, and Miltersen (2014) also
showed that the possibility of debt renegotiation increases the initial leverage ratio because
it reduces the default costs, which is the negative impact of debt ﬁnancing.
(Insert Figure 10 around here.)
The comparative statics results, except for σ in the previous subsection, remain similareven when we take the optimal c0 at the initial time, and thus they are omitted here.
Figure 10 shows the panels with varying levels of σ. Diﬀerently from Figure 6 in Section
4.2.5, the ﬁrm proceeds to debt renegotiation regardless of σ. We see from the top rightpanel that the optimal c0 increases with σ. Because of the increased c0 , the equity holders
can make enough of a coupon reduction (see the gap between c0 and cL1 in the top rightpanel) to gain more proﬁts from debt renegotiation than from direct liquidation. Notethat the new coupon cL1 , unlike in Figure 6, increases with σ, but the gap between c0
and cL1 increases more than in Figure 6. The eﬃciency of debt restructuring is also seenin the gap between x∗L and x∗L2 in the middle left panel of Figure 10, compared to thatof Figure 6. In the middle right panel, both LV and LV0 decrease with σ in the optimal
capital structure case, diﬀerently from Figure 6. CS, CS0 and EF have similar shapes tothose of Figure 6.
(Insert Figure 11 around here.)
When we optimally adjust the capital structure, the timing of adjustment also inﬂuences the results. To see the impact, we present the comparative statics with respect tothe initial state variable x. It is worth noting that in the previous subsection with c0
ﬁxed, the results do not depend on x. Figure 11 shows the panels with varying levels of x.

9

As will be seen in Figure 11 later, an optimal coupon, which maximizes the ﬁrm value, may lead to direct

liquidation depending on the parameter values.

18

For x ≤ 1.52, the ﬁrm is better oﬀ leaving the equity holders to choose liquidation rather
than debt renegotiation.10 Because of this, EL1 (x), c∗0 , x∗L , LV0 and CS0 jump at x = 1.52.
All graphs, especially LV0 and CS0 , have quite diﬀerent shapes for x ∈ (1.52, 1.75) fromthose for larger x. This is because in the region the ﬁrm dares to choose a higher c0 tomake the equity holders proceed to debt renegotiation rather than liquidation. Recall
that with a higher c0 the equity holders are more likely to proceed to debt restructuring(see Figure 5). The ﬁrm wishes to choose a lower c0 , but then the equity holders would
liquidate the ﬁrm. Because of this, the ﬁrm chooses high c0 (hence, LV0 and CS0 ) whenx is close to 1.52. This motivation greatly distorts the result for x ∈ (1.52, 1.75).
Once x decreases below 1.52, the ﬁrm is better oﬀ setting a low c0 and leaving theshareholders to choose direct liquidation, rather than a suﬃciently high c0 to lead to debt
renegotiation. Then, the optimal c∗0 jumps down when x decreases below the threshold. At
the same threshold, the (ex-post) equity value adversely jumps up due to the downward
jump in c∗0 . On the other hand, the ﬁrm value VL1 (x), which can be regarded as theex-ante equity value, continuously increases with x. As in Moraux and Silaghi (2014), ourresult predicts that the ﬁrm is more likely to proceed to direct liquidation when the latest
capital adjustment occurs for a lower state variable.

5

Conclusion

In this paper, we investigated whether and how the shareholders of a ﬁrm in distressproceed to direct liquidation or debt restructuring along with partial liquidation. We
showed the following results.
In debt restructuring, the shareholders arrange the coupon reduction and use equity
ﬁnancing to retire the partial debt value. The optimal debt restructuring time is always
equal to the original liquidation time without debt renegotiation because the shareholders
can most eﬃciently decrease the debt repayment value on this timing. Most notably, even
if they face high ﬁnancing costs, the shareholders do not prefer partial liquidation in debt
restructuring because the costs arise earlier. Instead, they prefer to adjust the coupon
reduction at a small level so that they do not need to pay back a signiﬁcant amount to thedebt holders. Our result about the ineﬃciency of asset sale is consistent with empiricalﬁndings.
Fewer forced asset sales, lower ﬁnancing, renegotiation, and running costs, a lower
premium to the debt holders, a lower volatility, and more debt increase the shareholders’
incentive to choose debt renegotiation to avoid full liquidation. These results suggestthat larger/older/higher-productivity ﬁrms are more likely to proceed to debt renegotiation, which is supported by the empirical evidence. When the ﬁrm optimizes its capital
structure, taking account of debt renegotiation in the future, it chooses a higher coupon,10

For x ≤ 1.52 (liquidation region), the ﬁgure does not depict c∗L1 , x∗L2 , LV, CS, and EF .

19

leverage, and credit spread than in the case without debt renegotiation. The ﬁrm tends
to proceed to direct liquidation rather than another debt restructuring shortly after thecapital adjustment.

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21

1.5
x∗L1
1.4

1.3

1.2

1.1

1

0

0.05

0.1

0.15
a

0.2

0.25

0.3

Figure 1: The debt renegotiation trigger x∗L1 with varying levels of the fraction of asset sale a.
The other parameter values are set at the base case (22). Note that debt renegotiation, rather
than direct liquidation, is chosen for these parameter values.

25

D(xL1,1,c0)

20

15

10

0.5

0.75

1

1.25
xL1

1.5

1.75

2

Figure 2: The debt value D(xL , 1, c0 ) with varying levels of xL . The parameter values are set
at the base case (22).

22Table 1: Base case.
x∗L
x∗L1
x∗L2
LV

EL1 (x)

E(x)

c∗L1

5.91

5.58

1.07 1.25 1.25

CS

0.842 0.846 0.0207

EF
1.58

Table 2: Optimal capital structure.
VE

c0 , c∗L1

LV0 , LV

CS0 , CSx∗L , x∗L2

EF

DR

30.4

6.21

1.95, 1.05

0.796, 0.852

0.0205, 0.0209

1.23, 0.833

1.55

L

30.1

9.58

1.51, N/A

0.682, N/A

0.0134, N/A

1.01, N/A

N/A

23

6.1

1.2
∗

EL1

6

E

5.9

1

5.8

0.9

5.7

0.8

5.6

0.7

5.5

0

0.1

0.2
a

0.3

cL1

1.1

0.4

∗

(1−a)/cL1

0

1.3

0.1

0.2
a

0.3

0.4

0.8475
LV

0.847

1.2
1.1

x∗L2

0.8465∗xL

0.846
0.8455

1

0.845
0.9
0.8

0.8445
0

0.1

0.2
a

0.3

0.844

0.4

0.0208

0

0.1

0.2
a

0.3

0.4

1.62
CS
1.6
EF

1.58
0.02071.56
1.54
0.0206

0

0.1

0.2
a

0.3

1.52

0.4

0

0.1

0.2
a

0.3

0.4

Figure 3: Comparative statics with respect to the fraction of asset sale, a. The other parametervalues are set at the base case (22).

24

6

6

5.5

5.5

5

5
EL1 (kF=0.5)

EL1 (kF=0.5)

E

4.5

0

0.1

E

0.2
a

0.3

4.5

0.4

6

6

5.5

5.5

5

4.5

0

0.2
a

0.3

0.4

0.3

0.4

0.3

0.4

5

0

EL1 (kF=1)

EL1 (kF=1)

E

E

0.1

0.2
a

0.3

4.5

0.4

0

6

6

5.5

5.5

5

0.1

0.2
a

5
EL1 (kF=1.5)

EL1 (kF=1.5)

E

4.5

0.1

0

0.1

E

0.2
a

0.3

4.5

0.4

0

0.1

0.2
a

Figure 4: EL1 (x) and E(x) with respect to the fraction a and the ﬁnancing cost kF . The
left and right panels show the values for PL (x, a) = 0.6a1.01 x/(r − µ) + 2a1.01 (base case) andPL (x, a) = 0.6ax/(r − µ) + 2a (linear case). The other parameter values are set at the basecase (22).

25

15

3
EL1
2.5

E
10

∗
L1

c

2

c0
1.5
5
1
0

1

1.5

2
c0

2.5

0.5

3

2

1

1.5

2
c0

2.5

3

1
x∗
L2
x∗L

1.8
1.6

LV
0.95

1.4

0.9

1.2

0.85

1
0.8

0.8
1

1.5

2
c0

2.5

0.75

3

0.03

1

1.5

2
c0

2.5

3

2.4
CS

EF

2.2

0.028

2

0.026

1.8
0.024
1.6
0.022

1.4

0.02
0.018

1.2
1

1.5

2
c0

2.5

1

3

1

1.5

2
c0

2.5

3

Figure 5: Comparative statics with respect to the initial coupon c0 . The other parameter values
are set at the base case (22).

26

14

1.2
c∗

E

L1

12

L1

E

1.15

10
8

1.1
6
4

1.05

2
0
0.1

0.15

0.2

0.25
σ

0.3

0.35

1
0.1

0.4

2

0.15

0.2

0.25
σ

0.3

0.35

0.4

0.9
∗
x
L2
x∗L

1.5

LV
0.88

0.86
1
0.84
0.5
0.1

0.15

0.2

0.25
σ

0.3

0.35

0.82
0.1

0.4

0.07

0.15

0.2

0.25
σ

0.3

0.35

0.4

1.8
EF

CS

0.06

1.6

0.05
0.04

1.4
0.03
0.02

1.2

0.01
0
0.1

0.15

0.2

0.25
σ

0.3

0.35

1
0.1

0.4

0.15

0.2

0.25
σ

0.3

0.35

0.4

Figure 6: Comparative statics with respect to the cash ﬂow volatility σ. The other parametervalues are set at the base case (22).

27

6.1
EL1

6

1.3

E
∗

5.9

cL1

1.2

5.8
1.1

5.7
5.6
5.5

1
0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

k

0.6

0.8

1

k

F

F

1.4

0.94

1.3

0.92

1.2

∗

0.9

x∗

0.88

xL2

1.1

LV

L

1

0.86

0.9

0.84

0.8

0.82

0.7

0

0.2

0.4

0.6

0.8

0.8

1

0

0.2

0.4

kF

0.6

0.8

0.04

2.5
EF
Cost

CS
0.035

2

0.03

1.5

0.025

1

0.02

0.5

0.015

1

kF

0

0.2

0.4

0.6

0.8

0

1

kF

0

0.2

0.4

0.6

0.8

1

kF

Figure 7: Comparative statics with respect to the ﬁnancing cost kF . The other parametervalues are set at the base case (22).

28

1.07

6.4

∗

EL1
E

6.2

1.068

6

1.066

5.8

1.064

5.6

1.062

5.4

0

0.02

0.04

0.06

0.08

cL1

1.06

0.1

0

0.02

0.04

0.091

k

D

0.06

0.08

0.1

k

D

1.3
LV

1.2

0.8468

x∗L2
∗
xL

1.1

0.846610.84640.9

0.8462

0.8

0

0.02

0.04

0.06

0.08

0.846

0.1

0

0.02

0.04

kD

0.06

0.08

0.1

kD

0.02082.5
CS

EF

0.020820.02081.5
0.020710.02070.0207

0

0.02

0.04

0.06

0.08

0.5

0.1

kD

0

0.02

0.04

0.06

0.08

0.1

kD

Figure 8: Comparative statics with respect to the premium to the debt holders kD . The otherparameter values are set at the base case (22).

29

1.5

12
EL1

10

∗

cL1

1.4

E

1.3

8

1.2
6
1.1
4

1

2
0

0.9
0

0.2

0.4

0.6

0.8

0.8

1

0

0.2

0.4

w

1.8

0.8

1

1

1.6

x∗L2

0.95

1.4

xL

∗

0.9

1.2

0.85

1

0.8

0.8

0.75

0.6

0.7

0.4

0.6
w

0

0.2

0.4

0.6

0.8

0.65

1

LV

0

0.2

0.4

w

0.6

0.8

1

w

2
EF

CS

0.026

1.8

0.024
0.022

1.6

0.02
1.4

0.018
0.016

1.2

0.014
0.012

0

0.2

0.4

0.6

0.8

1

1

w

0

0.2

0.4

0.6

0.8

1

w

Figure 9: Comparative statics with respect to the running cost w. The other parameter values
are set at the base case (22).

30

40

3

30

2.5
EL1

20

∗

cL1

2

VL1

c∗
0

10

0
0.1

1.5

0.15

0.2

0.25
σ

0.3

0.35

1
0.1

0.4

1.5

0.15

0.2

0.25
σ

0.3

0.35

0.4

1
∗
x
L2
x∗L

LV
LV0

0.95
0.9

1

0.85
0.8
0.75

0.5
0.1

0.15

0.2

0.25
σ

0.3

0.35

0.7
0.1

0.4

0.06

0.15

0.2

0.25
σ

0.3

0.35

0.4

1.6
EF

1.55

0.05

1.5
0.04

1.45

0.03

1.4

CS
CS0

0.02

1.35
1.3

0.01
0
0.1

1.25
0.15

0.2

0.25
σ

0.3

0.35

0.4

0.1

0.15

0.2

0.25
σ

0.3

0.35

0.4

Figure 10: Comparative statics with respect to volatility σ in the optimal capital structurecase. The other parameter values are set at the base case (22).

31

50

3
∗

E

L1

40

VL1

2

20

1.5

10

1

1

1.5

∗
0

c

30

0

cL1

2.5

2
x

2.5

0.5

3

2

1

1.5

2
x

2.5

3

0.95
x∗L2
∗
xL

1.5

LV
LV0

0.9
0.85
0.8

1
0.75
0.5

1

1.5

2
x

2.5

0.7

3

0.024

1

1.5

2
x

2.5

3

2.2
EF

CS
CS0

0.022

2

0.02

1.8

0.018
1.6
0.016
1.4
0.014

1

1.5

2
x

2.5

3

1

1.5

2
x

2.5

3

Figure 11: Comparative statics with respect to the initial state variable x in the optimal capital
structure case. The other parameter values are set at the base case (22).

32