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Federal Reserve Bank of Chicago
Optimal Debt Dynamics, Issuance Costs,
and Commitment
Luca Benzoni, Lorenzo Garlappi,
Robert S. Goldstein, Julien Hugonnier
and Chao Ying
June 12, 2020
WP 2020-20
https://doi.org/10.21033/wp-2020-20
*
Working papers are not edited, and all opinions and errors are the
responsibility of the author(s). The views expressed do not necessarily
reflect the views of the Federal Reserve Bank of Chicago or the Federal
Reserve System.
Optimal Debt Dynamics, Issuance Costs, and
Commitment∗
Luca Benzoni†
Chicago Fed
Lorenzo Garlappi‡
UBC
Julien Hugonnier¶
EPFL and Swiss Finance Institute
Robert S. Goldstein§
University of Minnesota
Chao Ying‖
University of Minnesota
June 12, 2020
∗
We thank Hengjie Ai, Kerry Back, Markus Baldauf, Marco Bassetto, Maria Chaderina, Thomas Dangl,
Peter DeMarzo, Murray Frank, Ron Giammarino, Will Gornall, Zhiguo He, Ravi Jagannathan, Charles Kahn,
Narayana Kocherlakota, Xuelin Li, Erwan Morellec, Chris Neely, Marcus Opp, Christine Parlour, Raj Singh,
Martin Szydlowski, Alberto Mokak Teguia, Fabrice Tourre, Andy Winton, Fangyuan Yu, Hongda Zhong, and
seminar participants at the 2019 NBER SI Capital Markets meeting, the 2019 CFE Conference, the 2020 SFS
Cavalcade, the University of British Columbia, and the University of Minnesota, for helpful comments. All
remaining errors are our own. The views expressed herein are those of the authors and not necessarily those of
the Federal Reserve Bank of Chicago or the Federal Reserve System.
†
Federal Reserve Bank of Chicago, 230 S. LaSalle Street, Chicago, IL 60604, USA. E-mail:
lbenzoni@frbchi.org
‡
Sauder School of Business, University of British Columbia, 2053 Main Mall, Vancouver, BC V6T 1Z2,
Canada. E-mail: lorenzo.garlappi@sauder.ubc.ca
§
University of Minnesota, Carlson School of Management, and NBER, 321 19th Ave South Minneapolis,
MN, 55455, USA. E-mail: golds144@umn.edu
¶
EPFL and Swiss Finance Institute, Route Cantonale, 1015 Lausanne, Switzerland.
E-mail:
Julien.Hugonnier@epfl.ch
‖
University of Minnesota, Carlson School of Management, 321 19th Ave South Minneapolis, MN, 55455,
USA. E-mail: yingx040@umn.edu
1
Optimal Debt Dynamics, Issuance Costs, and
Commitment
Luca Benzoni, Lorenzo Garlappi, Robert Goldstein, Julien Hugonnier and Chao Ying
Abstract
We investigate optimal capital structure and debt maturity policies in the presence of
fixed issuance costs. We identify the global-optimal policy that generates the highest
values of equity across all states of nature consistent with limited liability. The optimal
policy without commitment provides almost as much tax benefits to debt as does the
global-optimal policy and, in the limit of vanishing issuance costs, allows firms to extract
100% of EBIT. This limiting case does not converge to the equilibrium of DeMarzo and
He (2019), who report no tax benefits to debt when issuance costs are set to zero at the
outset.
1
Introduction
A large and growing literature in corporate finance investigates the firm’s optimal debt policy
in a dynamic setting. Often, the focus has been on: (i) the optimal level of debt issuance, and
(ii) the optimal debt maturity. Much of this literature has studied these questions through the
lens of the tradeoff between the cost of financial distress and the benefit of tax deductibility of
interest payments. For example, focusing on firms that are subject to issuance costs, Goldstein,
Ju and Leland (GJL, 2001) show that management can increase shareholder value by issuing
additional debt whenever leverage ratios become sufficiently low.1 In their framework, the
optimal debt issuance policy is characterized by a single state variable (i.e., inverse leverage) and
an “(s, S)” policy, with a region of inaction bounded by a lower threshold at which bankruptcy
occurs, and an upper threshold at which additional debt is issued.
More recently, DeMarzo and He (DH, 2019) investigate the optimal debt-issuance policy for
the case in which issuance costs are zero. Restricting their attention to Markov perfect equilibria
with no commitment, they demonstrate that the optimal debt issuance policy is characterized
by a locally deterministic process in which new debt is issued in all states of nature, even when
the firm is near default. Due to this aggressive issuance policy, DH show that such firms obtain
no tax benefits to debt, regardless of the choice of debt maturity.
Although the scopes of the GJL and DH papers do not overlap—being separated by whether
there are issuance costs, and whether management can commit to a particular debt policy—
there is a crucial difference in their implications: The GJL framework predicts that tax benefits
to debt are both positive and increasing as issuance costs go to zero. However, the DH framework predicts that, regardless of the debt maturity policy, there are no tax benefits to debt
when issuance costs are zero. Moreover, GJL predict that debt is issued in discrete amounts,
whereas DH predict that debt is issued continuously.
In this paper, we investigate the impact that issuance costs have on optimal debt issuance
and maturity policies for the cases with and without commitment, paying particular attention
1
Early studies of dynamic capital structure choice include Kane, Marcus, and McDonald (1984, 1985) and
Fischer, Heinkel, and Zechner (1989).
1
to the case of vanishing issuance costs. We begin by generalizing the framework of GJL to
finite maturity, non-callable debt. Debt issuance is subject to a fixed cost, the size of which
is controlled by the parameter β. We investigate optimal policies both with and without
commitment.2 For both cases, and for any given value of β, we identify the optimal (s, S)
policy in terms of a single state variable (inverse leverage (vt )) and four policy parameters: the
location of the default boundary (vb ), the location of the debt-issuance boundary (vu ), the size
of the debt issuance (γ), and the (inverse) maturity of the debt issued (ξ). For the case with
commitment, the manager can choose the values of the policy parameters (vu , γ) for all future
debt issuances. However, the choice of vB is always subject to limited liability.
In contrast, for the case without commitment, the manager has control only over current
actions, such as whether to issue additional debt today, and if so, the size of that issuance
(i.e., this period’s values of (vu , γ)), taking the decisions regarding future debt issuances as
exogenous. As in DH, when studying the no-commitment case, we restrict our attention to
Markov-perfect equilibria.
Our paper makes three contributions. First, we demonstrate that for any given issuance
cost parameter β and (inverse) maturity parameter ξ, the optimal policy for the case with
commitment generates the highest values of equity compared to all other debt issuance policies
(vb , vu , γ) consistent with limited liability. As such, we refer to this policy as the global optimal
policy. Interestingly, for sufficiently high values of β and ξ, the optimal policy is to issue no
debt, that is, vu → ∞. This occurs because debt associated with high values of ξ (i.e., shortmaturity debt) needs to be “rolled over” more often, and high values of β imply that issuance
costs are high. Hence, in this region, the present value of debt issuance costs outweighs the
present value of tax benefits of debt net of bankruptcy costs.
Second, we demonstrate that for any exogenously specified issuance cost parameter β and
2
At the risk of fighting over semantics, we follow the literature that uses the phrases “commitment” and
“no-commitment” in a somewhat relative fashion. For example, DH refer to their equilibrium as one with no
commitment in spite of the fact that they assume commitment with respect to future debt maturity decisions.
Similarly, He and Milbradt (2016) also refer to their equilibrium solution as one of no commitment in spite of
the fact that they assume firms commit to holding their future outstanding face value of debt constant. More
accurately, we can refer to the two solutions we investigate as having “more” and “less” levels of commitment.
2
(inverse) maturity parameter ξ, the optimal policy for the case without commitment is characterized by three regions: (i) As in the case with commitment, for sufficiently high values of β
and ξ, the optimal policy is to issue no debt (i.e., optimal vu = ∞); (ii) for intermediate values,
an optimal (s, S) policy exists; (iii) for sufficiently low values of β and ξ, no equilibrium (s, S)
policy exists. That no equilibrium strategy exists in region (iii) underscores the inefficiency of
myopic no-commitment policies, in contrast to the global-optimal policy that generates positive
tax benefits to debt in this region.
Third, we investigate the optimal debt issuance policy when maturity is chosen optimally. As
in DH, we restrict our attention to the case in which a firm commits to a particular amortization
rate ξ.3 Within this setting we demonstrate that, for both cases with and without commitment,
there exists an optimal maturity (1/ξ ∗ (β)) that is an increasing function of β. This result is due
to a tradeoff between tax benefits net of bankruptcy costs, and issuance costs: Issuing shortmaturity debt has the advantage that the firm can choose relatively “high leverage” today,
which creates a large tax shelter, while assuring that the firm’s future debt obligations will be
significantly lower. Hence, even if future earning before interest and taxes (EBIT) falls, the
firm is more likely to remain solvent, in turn reducing the present value of bankruptcy costs.
However, the disadvantage of shorter maturity debt is that it is associated with an increase in
expected debt issuance costs. For decreasing values of the parameter β, the first effect becomes
increasingly dominant, in turn decreasing optimal maturity. Indeed, as β goes to zero, optimal
maturity also goes to zero.
Consistent with GJL, we confirm that these policies generate significant tax benefits to debt,
and that these tax benefits increase as issuance costs drop. Moreover, we demonstrate that a
firm’s ability to choose an optimal debt maturity significantly amplifies the tax benefits to debt
compared to the GJL infinite-maturity benchmark. Indeed, for both cases with and without
commitment, we show that as issuance costs go to zero, the firm can extract 100% of the claim
3
Studying optimal maturity policy under the assumption of no commitment when debt is non-callable potentially leads to an intractable framework in which the current face value of debt for each vintage becomes a
state variable. As the focus of our paper is to understand the impact of issuance costs on debt policies in a
framework as close as possible to that of DH, we leave such an extension to future research.
3
to EBIT by combining an optimal maturity policy (ξ) with an optimal capital structure policy
(vb , vu , γ). The implication is that, in the presence of even tiny issuance costs, when firms
can choose both their capital structure and debt maturity policies optimally, the question of
whether or not a manager can commit to the global-optimal policy is of secondary importance
in terms of available tax benefits to debt.
Importantly, these results imply that in the limiting case of vanishing issuance costs, the
optimal Markov-perfect no-commitment (s, S) policy in our framework does not converge to
the zero issuance cost solution of DH, who report no tax benefits to debt, regardless of maturity
choice. The intuition for this result is the following: consider a discrete-time setting with N
periods of length ∆t such that (N × ∆t) equals some finite time interval T . Then allow for the
possibility that the optimal debt issuance policy is some combination of a locally deterministic
policy as in DH (in which the amount of debt issued in each of the N periods is of order
O(∆t)), and the (s, S) policy considered here (in which a finite amount debt is issued whenever
inverse leverage reaches some upper threshold). In the continuous-time limit ∆t → 0 (and thus
T
N = ∆t
→ ∞), any arbitrarily small (but positive) fixed issuance cost parameter generates
an infinite present value of issuance costs if debt is issued continuously as in DH. Therefore, the
locally deterministic component of the debt policy must be set to zero, leaving only the (s, S)
policy.4
The predictions of our model are in line with the empirical literature that investigates
the capital structure and maturity decisions of firms. By specifying fixed issuance costs, our
model predicts that debt will be issued in discrete (rather than continuous) amounts, consistent
with observation. Moreover, our model generates both persistence in leverage and a negative
correlation between profitability and leverage, consistent with the empirical literature (e.g.,
Titman and Wessels (1988) and Frank and Goyal (2014)). Intuitively, our model captures these
4
At β = 0, there is a discontinuity in the nature of the optimal no-commitment policy. Indeed, in unreported
results, we investigate the case in which we set β = 0 at the outset of the analysis, and allow for the possibility
that the optimal debt issuance policy is a combination of a locally deterministic policy (as in DH) and the (s, S)
policy considered here. We find that the optimal location of the upper boundary is (vu = ∞), implying that
there is no (s, S) component of optimal policy when β = 0. This result can be seen as a special case of the proof
in DH, who show that their optimal smooth policy is the unique Markov-perfect no-commitment equilibrium
when β = 0.
4
features because, when firms are in the inaction region, higher (lower) profitability increases
equity values while debt outstanding remains constant, leading to lower (higher) leverage. Also
consistent with our model’s predictions are Van Binsbergen, Graham, and Yang (2010), and
Blouin, Core, and Guay (2010), who document that firms are able to extract tax benefits to
debt. In agreement with our model’s prediction of an optimal maturity policy, Barclay and
Smith (1995) and Stohs and Mauer (1996) find that firms are not indifferent toward debt
maturity choice. We note that Fama and French (2002), Baker and Wurgler (2002), and Welch
(2004) provide evidence that shocks to capital structures are persistent, and that Leary and
Roberts (2005) attribute this persistence to the presence of adjustment costs, as opposed to
indifference toward capital structure. Graham and Harvey (2001) report survey evidence that
45% of CFOs are concerned with the tax advantage of interest deductibility (Figure 5, p. 210 and
Table 6, p. 212). That firms target specific minimum credit rating levels (e.g., Kisgen (2006,
2009)) is consistent with managers caring about the firm’s reputation in the debt markets,
given that debt issuance is a repeated game. This provides support for our claim that firms can
credibly commit to our global optimal policy. Finally, calibrating our model to be consistent
with empirical estimates of issuance costs in the range of 1-2% generates a predicted optimal
maturity in the range of 3-5 years, in line with empirical observation.
Our paper builds upon the quickly evolving literature that examines optimal dynamic capital
structure decisions of firms. There are only a few tractable frameworks in this literature, due
to the difficulty of pricing assets in an economy in which their current prices depend on the
future debt issuance policy of the firm.5 Recent papers that investigate whether firms continue
to issue debt in high leverage states, or whether they choose to repurchase outstanding debt in
this situation, include Dangl and Zechner (2016), Admati et al (2018), Xu (2018), and Malenko
and Tsoy (2020). Whereas DH note that their findings are reminiscent of the Coase conjecture
for a durable-good monopolist (Coase (1972)), our findings are analogous to the literature
5
See, for example, Titman and Tsyplakov (2007), Stebulaev (2007), Morellec, Nikolov and Schürhoff (2012),
Décamps and Villeneuve (2014), Hugonnier, Malamud and Morellec (2015), Dangl and Zechner (2016), Benzoni,
Garlappi and Goldstein (2019).
5
that identifies channels through which the Coase conjecture fails.6 Indeed, our finding that an
arbitrarily small debt issuance cost is sufficient to overturn the capital-structure and maturity
irrelevance result of DH is analogous to the findings of McAfee and Wiseman (2008), who show
that the presence of even a vanishingly small cost of capacity production is enough to counter
the Coase conjecture. Our paper also adds to the literature that investigates optimal maturity
structure.7
The rest of the paper is as follows. In Section 2 we present the model and derive expressions
for the claims to EBIT. In Section 3, for an exogenous choice of (β, ξ), we derive the optimal
debt issuance policies for both cases with and without commitment, in turn identifying regions
of the parameter space (β, ξ) for which the optimal solution is characterized by an (s, S) policy.
We also demonstrate that the optimal policy with commitment is in fact globally optimal.
In Section 4, we identify the optimal maturity structure policies ξ ∗ (β). For both cases with
and without commitment, we show that as the debt issuance parameter β approaches zero, the
optimal debt maturity also approaches zero, and moreover that the fraction of the claim to EBIT
that firms can extract approaches 100%. Section 5 solves the model with vanishing issuance
costs in the presence of commitment. Section 6 concludes. We relegate proofs to Appendix A,
and discuss our claim that the optimal policy under commitment is in fact globally optimal
in Appendix B. Appendix C identifies a supermartingale, which we argue demonstrates that
if a firm finds itself with inverse leverage ratio above the debt restructuring boundary (i.e.,
v(t− ) > vu ), then it is optimal for that firm to immediately issue a sufficient amount of debt
so that the firm’s new inverse leverage jumps into the range v(t+ ) ∈ (vb , vu ). In the Online
Appendix we (i) investigate optimal capital structure and maturity choice in the presence of
proportional costs; and (ii) investigate debt repurchase policies, and show that once realistic
issuance costs are specified, a firm cannot credibly commit to repurchase its own debt.8
6
See, for example, Bulow (1982), Kahn (1986), Ausubel and Deneckere (1989).
See, for example, Leland and Toft (1996), Leland (1998), Brunnermeier and Yogo (2009), Brunnermeier and
Oehmke (2013), He and Xiong (2012), Diamond and He (2014), He and Milbradt (2014), Abel (2016), DeMarzo,
He and Tourre (2018), Chen, Xu and Yang (2019).
8
Building on our framework, Malenko and Tsoy (2020) show that firms whose EBIT can jump downward
also cannot credibly commit to repurchase their debt, even in the absence of issuance costs.
7
6
2
The Model
In this section, we investigate the optimal debt issuance policy for a firm that is subject to a
fixed cost each time it issues debt, the magnitude of which is controlled by a parameter β. The
cost is “fixed” in that it is independent of the size of the debt issuance. For tractability, however,
we assume that issuance costs are a fraction β of the firm size. We generalize GJL to the case
of non-callable debt with exponential maturity characterized by a constant amortization rate
ξ. We investigate two cases: with and without commitment. These two cases are distinguished
by whether or not the firm can commit to a particular debt issuance policy (vu , γ) for all future
periods, or whether these policy parameters can be chosen only for the current period, with
future policy decisions made by future managers.
There are two state variables in the model. The first is EBIT (Yt ), whose risk-neutral
dynamics are exogenously specified:
dY
Y
= µ dt + σ dB Q ,
(1)
with dB Q denoting increments of a standard Brownian motion under the risk neutral measure
Q. Given the parameter restriction (r > µ), the present value (Vt ) of the claim to EBIT can
be determined from the risk-neutral expectation:
∞
Z
−r(s−t)
Q
Vt = Et
ds e
Ys =
t
Yt
.
r−µ
(2)
Since the claim to EBIT is proportional to EBIT itself, it inherits the same dynamics:
dV
V
= µ dt + σ dB Q .
(3)
As required to preclude arbitrage, these dynamics imply that the risk-neutral expected return
on the claim to EBIT equals the risk free rate:
dV + Y dt
= r dt + σ dB Q .
V
7
(4)
Due to the linear relation between Yt and Vt in equation (2), we are free to choose either as the
exogenous state variable. We find it more convenient to choose Vt .
The second state variable is the face value of outstanding debt Ft , which is characterized
by a constant amortization rate ξ and coupon rate c. During the interval (t, t + dt), ξFt dt
units of debt mature. Thus, over this interval, the sum of coupon and principal payments to
debtholders is (c + ξ)Ft dt. The effective average maturity is 1ξ .
Here we conjecture (and later verify) that the optimal debt issuance policy can be characterized in terms of a single state variable vt = Vt /Ft that can be interpreted as inverse leverage.9
There is a lower boundary vb at which it is optimal for shareholders to default. We define the
date of this event as τb . It therefore follows that vb = Vτb /Fτb . There is also an upper boundary
vu at which it is optimal to issue additional debt so that total outstanding debt scales by a
factor γ (the size of which is chosen optimally). We define the date of this event as τu . It
therefore follows that vu = Vτu /Fτu . Thus, at time τu , the level of outstanding debt jumps from
Fτu to γFτu (γ > 1), and shareholders pay a fixed cost of βVτu = βvu Fτu . Due to the presence
of fixed issuance costs, it is optimal for shareholders to follow an (s, S) policy in that no debt
is issued when inverse leverage falls in the range v ∈ (vb , vu ).
In order to price assets in this economy, it is convenient to define “period-0” as an interval
of time that includes the present moment t, and ends the first time either the lower or upper
boundary is reached. We refer to the date that period-0 ends as τ = min(τb , τu ). Each period-j,
j > 0, that follows period-0 begins at a time of debt issuance, and ends with either a subsequent
issuance or a default.
Because there are no debt issuances (or repurchases) within period-0, it follows that the
dynamics of Ft are driven only by its amortization rate:
dFt = −ξFt dt,
9
(5)
Note that V is not the sum of debt and equity values, but rather the claim to EBIT, which also includes the
claims to taxes, bankruptcy costs, and issuance costs. Still, we find it convenient to refer to v = VF as inverse
leverage.
8
with solution
Fs = Ft e−ξ(s−t)
∀s ∈ (t, τ ),
(6)
for any t in period-0. Given the dynamics of Vt and Ft , Itô’s lemma implies that the dynamics
of vt = Vt /Ft follow:
dvt
vt
= (µ + ξ) dt + σ dB Q .
(7)
Note that the (inverse) maturity parameter ξ is chosen by management, and that by choosing
ξ → ∞, the firm can virtually guarantee that future inverse leverage will reach the upper
boundary vu prior to reaching the lower boundary vb , in turn precluding default. As such, the
present value of tax benefits to debt net of bankruptcy costs is increased by choosing large
values of ξ (i.e., short maturity). However, such a policy also increases debt issuance costs.
These two mechanisms capture a key tradeoff in our model, as we discuss below.
2.1
Valuation of claims to EBIT
In our framework, the cash flows to equity (i.e., dividends), government (i.e., taxes) and outstanding debt (i.e., principal and interest payments) during the interval (t, t + dt) are:
Yt − π(Yt − cFt ) − (c + ξ)Ft dt
Equity :
Government :
Debt :
π(Yt − cFt ) dt
(c + ξ)Ft dt.
(8)
In the event of default, we assume that a proportion α of firm value is lost to bankruptcy costs.
As such, at default, the claims are:
Equity :
Government :
Debt :
Bankruptcy Costs :
0
(1 − α)πVτb
(1 − α)(1 − π)Vτb
αVτb .
(9)
9
Below, we price the claims to five different assets: equity, debt, government, bankruptcy
costs and issuance costs. Expressed in terms of risk-neutral discounted cash flows and using
Yt = (r − µ)Vt from equation (2), all of these claims take the form:
Z
Q
X(Vt , Ft ) = Et
t
τ
ds e−r(s−t) (hF Fs + hV Vs ) + e−r(τu −t) 1(τu <τ ) Hu +
b
e−r(τb −t) 1(τ
b
<τu )
i
Hb .
(10)
The first term captures the present value of cash flows prior to the firm reaching either the
upper or lower boundary, which are specified in equations (8). The second term captures the
value of the asset when the upper boundary is reached. The third term captures recovery
values conditional upon default, which are specified in equations (9). The five different assets
we investigate differ only by their factor loadings (hF , hV , Hb , Hu ), in which: (i) the loadings
(hF , hV ) are constants, and (ii) the loadings (Hb , Hu ) are linear in either (V, F ) or in the asset
values themselves. Here we determine asset values in terms of these generic factor loadings,
and then use this generic solution to price the five assets below.
Rt
−rt
−rs
Equation (10) implies that e X(Vt , Ft ) + 0 ds e
(hF Fs + hV Vs ) is a Q-martingale.
Therefore, by Itô’s lemma, the asset value satisfies the PDE:
0 = −rX + µV XV +
σ2 2
V XV V − ξF XF + hF F + hV V,
2
(11)
subject to the boundary conditions
X(vb Fτb , Fτb ) = Hb
(12)
X(vu Fτu , Fτu ) = Hu .
(13)
Because both the cash flows and state vector dynamics are linear in the state vector, it
follows that asset values are homogeneous of degree one in the state vector. Thus, we define a
single state variable that we refer to as inverse leverage vt = (Vt /Ft ), and look for a solution of
the form:
X(V, F ) = F x(v)|v=(V /F ) ,
10
(14)
where x(v) denotes asset value per unit of face value F . It follows that the partial derivatives
of X(V, F ) with respect to V and F can be expressed as:
XV
XV V
= xv
1
=
xvv
F
(15)
(16)
XF = x − vxv .
(17)
Substituting these expressions into the PDE (11), we obtain the following ODE:
0 =
σ2 2
v xvv + (µ + ξ) vxv − (r + ξ)x + hV v + hF ,
2
(18)
subject to the boundary conditions
x(vb ) =
Hb
Fτb
(19)
x(vu ) =
Hu
.
Fτu
(20)
The solution to this ODE is:
φ
ω
x(v) = Mx v + Nx v +
hV
r−µ
v+
hF
r+ξ
,
(21)
where the exponents (φ, ω) are given by
φ =
q
− (µ + ξ) +
σ2
2
2
− (µ + ξ) + 2(r + ξ)σ 2
σ2
ω =
σ2
2
σ2
2
q
− (µ + ξ) −
σ2
2
> 1
(22)
< 0.
(23)
2
− (µ + ξ) + 2(r + ξ)σ 2
σ2
The constants (Mx , Nx ) are uniquely determined by the boundary conditions:
!
Hb
x(vb ) =
Fτb
x(vu ) =
11
Hu
Fτu
(24)
!
.
(25)
With this general form of the solution, we price the five assets of interest:
Proposition 1 (Asset valuation) Given a set of debt policy parameters (vb , vu , γ, ξ):
• The value of debt d(v) = D(V, F )/F per unit of face value F can be expressed as:
d(v) = Md v φ + Nd v ω +
c+ξ
,
r+ξ
(26)
where the constants (Md , Nd ) are uniquely determined by the boundary conditions
d(vb ) = (1 − α)(1 − π)vb
(27)
d(vu ) = d(vu /γ).
(28)
• The value of equity e(v) = E(V, F )/F per unit of face value F can be expressed as:
e(v) = Me v φ + Ne v ω + (1 − π)v −
(c(1 − π) + ξ)
,
r+ξ
(29)
where the constants (Me , Ne ) are uniquely determined by the boundary conditions
e(vb ) = 0
(30)
e(vu ) = γe(vu /γ) + (γ − 1)d(vu ) − βvu .
(31)
• The value of the government claim g(v) = G(V, F )/F per unit of face value F can be
expressed as:
g(v) = Mg v φ + Ng v ω + πv −
cπ
,
r+ξ
(32)
where the constants (Mg , Ng ) are uniquely determined by the boundary conditions
g(vb ) = (1 − α)πvb
(33)
g(vu ) = γg(vu /γ).
(34)
12
• The value of the claim to bankruptcy costs b(v) = B(V, F )/F per unit of face value can
be expressed as:
b(v) = Mb v φ + Nb v ω ,
(35)
where the constants (Mb , Nb ) are uniquely determined by the boundary conditions
b(vb ) = αvb
(36)
b(vu ) = γb(vu /γ).
(37)
• The value of the claim to issuance costs i(v) = I(V, F )/F per unit of face value can be
expressed as:
i(v) = Mi v φ + Ni v ω ,
(38)
where the constants (Mi , Ni ) are uniquely determined by the boundary conditions
i(vb ) = 0
(39)
i(vu ) = βvu + γ i(vu /γ).
(40)
Proof: See Appendix A.
3
Optimal Debt Policies
In the previous section, we determined asset values for any given set of policy parameters
(vb , vu , γ, ξ). Here we investigate the optimal choice of these parameters. First, in Section 3.1
we fix the (inverse) maturity parameter ξ, and identify the optimal values of (vb , vu , γ) for
both the cases with and without commitment. We also identify regions in the parameter space
(β, ξ) for which the optimal policy is to not issue additional debt, and regions for which no
(s, S) equilibrium exists. Second, in Section 3.2 we identify the optimal debt maturity policy
parameter ξ ∗ (β) as a function of β.
13
3.1
Optimal Debt Policies for a Fixed Debt Maturity
The asset value formulas derived in the previous section assumed that the policy parameters
remain constant across all periods. This assumption is consistent with a scaling property
inherent in our framework in which, at the beginning of each period, the firm looks the same
except for its size. More generally, however, a firm’s debt policy is characterized by a set of
parameters (vb,j , vu,j , γj ), for each period j ∈ [0, ∞). Ultimately, if the equilibrium optimal
policy is characterized by a pure (s, S) strategy, then those parameters must be constant across
periods. How those parameters are chosen by the firm, however, depends upon whether a
manager can commit to a particular policy for all future periods.
In this context, it is natural to investigate two types of debt issuance policies: with and
without commitment. When investigating debt issuance with commitment, we focus on policies
in which the manager has full control over the debt issuance decisions for all future periods j.
Thus, the optimal policy with commitment is determined by first imposing that the period-0
parameters are equal to those in all other periods: (vu,j , γj ) = (vu , γ), for all j ≥ 0, and then
choosing (vu , γ) optimally. The location of the default boundary vb is also chosen optimally,
but must be consistent with limited liability. In this case, the optimal policy can be determined
by directly using the asset valuations from the previous section.
In contrast, when studying debt issuance without commitment, we focus on myopic policies in
which the manager in period-0 only gets to choose the policy parameters (vu,0 , γ0 ) for the current
period, and takes the policy parameters (vu,j , γj ) for all future periods j > 0 as exogenously
specified. Only after imposing the first order conditions for optimality on (vu,0 , γ0 ) do we set
(vu,0 , γ0 ) = (vu,j , γj ) for all j > 0 in order to identify the optimal policy parameters. Once
again, the location of the default boundary vb,0 is chosen optimally, but must be consistent with
limited liability. This analysis requires that we rewrite the pricing equations in Proposition 1 to
distinguish between claim valuations in period-0 and in subsequent periods, as we demonstrate
in Section 3.1.2.
14
3.1.1
Optimal Debt Policy with Commitment
For the case with commitment, we specify the optimal debt issuance policy as follows:
Definition 1 (Global-optimal policy with commitment) For a given set of model coefficients (µ, σ, r, π, α, β) and an inverse maturity parameter ξ, the optimal debt issuance policy
(vb∗ , vu∗ , γ ∗ ) with commitment satisfies the following system of first-order conditions (FOCs)
∂e
∂vb
∂e
∂vu
∂e
∂γ
= 0
(41)
= 0
(42)
= 0.
(43)
v=v
b
v=vu
v=vu
Appendix B verifies numerically that this policy is globally optimal in that, compared to all
other possible debt issuance policies (vb , vu , γ) consistent with limited liability, this solution
generates the highest values of equity across all values of v ∈ (vb , vu ).
At first blush, there seems to be no reason that the policy generated from the first order
conditions (FOC’s) in Definition ?? should generate a global-optimal policy in that these FOC’s
are applied only at the locations where the manager takes action (either by defaulting at vt = vb
or by issuing debt at vt = vu ). To provide intuition for our findings, it is convenient to use the
boundary condition on equity at v = vb in equation (30) to eliminate the coefficient Ne in the
equity valuation formula equation (29). This allows us to write the equity value as follows:
"
e(v) = Me v φ 1 −
v
vb
(ω−φ) #
"
+ (1 − π)v 1 −
−
v
vb
c(1 − π) + ξ
r+ξ
(ω−1) #
ω
v
1−
.
vb
(44)
Importantly, note that in equation (44), the policy parameters (vu , γ) appear only in the coefficient Me . (In contrast, vb shows up explicitly in equation (44)). Thus, the FOC’s with respect
15
to (vu , γ) for any value of v = vany are given by:
0
=
0 =
∂e
∂vu
∂e
∂γ
∝
∂Me
∂vu
(45)
∝
∂Me
.
∂γ
(46)
v=vany
v=vany
Hence, for any value v = vany , equations (45)–(46) deliver the same optimal policy parameters
(vu∗ , γ ∗ ) as those identified by equations (42)–(43). This property explains the global optimal
feature of our solution.
3.1.2
Optimal Debt Policy without Commitment
In Section 3.1.1, we assumed that a manager in period-0 had the power to commit to all present
and future debt issuance policies (vu,j , γj ) ∀j. In this section, however, we investigate optimal
policies when the manager does not have the ability to commit to future debt issuance policies.
Specifically, we set (vu,j , γj ) = (vu , γ) for j ≥ 1, which we distinguish, for now, from the period-0
controls (vu,0 , γ0 ). Under this restriction, we obtain the period-0 value of debt and equity.
For a given value of vt and policy parameters (vb , vu , γ), the period-0 value of debt d0 (v) is
determined by first calculating the value of debt d(v) for all future periods:
d(v) = Md v φ + Nd v ω +
c+ξ
,
r+ξ
(47)
where the constants (Md , Nd ) are uniquely determined by the boundary conditions
d(vb ) = (1 − α)(1 − π)vb
(48)
d(vu ) = d(vu /γ).
(49)
We then determine the period-0, date-t value of debt from:
d0 (vt ) = Md,0 vtφ + Nd,0 vtω +
16
c+ξ
,
r+ξ
(50)
where the constants (Md,0 , Nd,0 ) are uniquely determined from the boundary conditions
d0 (vb,0 ) = (1 − α)(1 − π)vb,0
(51)
d0 (vu,0 ) = d(vu,0 /γ0 ).
(52)
The parameters (vb , vu , γ) are outside the control of the period-0 manager, and impact the
function d(v) through the boundary conditions (48)–(49). The function d(v), in turn, affects
the value of the period-0 debt d0 (vt ) through the boundary condition (52).
Analogously, for a given value of vt and policy parameters (vb , vu , γ), we obtain the period-0
value of equity e0 (vt ) by first determining the value of equity e(v) for all future periods:
e(v) = Me v φ + Ne v ω + (1 − π)v −
c(1 − π) + ξ
,
r+ξ
(53)
where the constants (Me , Ne ) are uniquely determined from the boundary conditions
e(vb ) = 0
(54)
e(vu ) = γe(vu /γ) + γd(vu /γ) − d(vu ) − βvu .
(55)
We then determine the period-0, date-t value of equity from:
e0 (vt ) = Me,0 vtφ + Ne,0 vtω + (1 − π)vt −
c(1 − π) + ξ
,
r+ξ
(56)
where the constants (Me,0 , Ne,0 ) are uniquely determined from the boundary conditions
e0 (vb,0 ) = 0
(57)
e0 (vu,0 ) = γ0 e(vu,0 /γ0 ) + γ0 d(vu,0 /γ0 ) − d0 (vu,0 ) − βvu,0 .
(58)
The parameters (vb , vu , γ) impact the function e(v) through the boundary conditions (54)–(55).
The function e(v), in turn, affects the value of the period-0 equity e0 (vt ) through the boundary
condition (58).
Equations (50)–(52) and (56)–(58) determine the value of period-0 debt and equity for
a given set of policy parameters (vb,0 , vu,0 , γ0 ) and (vb , vu , γ). Due to the symmetry of our
17
economy, ultimately, the optimal policy parameters will be the same across all periods. The
period-0 policy is then chosen under the restriction that all future policies are identical to the
one in period-0.
Definition 2 (Optimal policy without commitment) For a given set of model coefficients
(µ, σ, r, π, α, β) and an inverse maturity parameter ξ, the optimal debt issuance policy (vb,0 , vu,0 , γ0 )
without commitment is the period-0 best response to future issuance policies (vb,j , vu,j , γj ), j ≥ 1,
that satisfies the following system of first-order conditions
0 =
0 =
∂e0
∂vu,0
∂e0
∂vb,0
(59)
v=vu,0
(60)
v=v
b,0
0 =
∂e0
∂γ0
,
(61)
v=vu,0
together with the fixed-point condition (vb,0 , vu,0 , γ0 ) = (vb,j , vu,j , γj ), for all j ≥ 1.
3.1.3
Existence of Equilibrium Debt Policies: Commitment versus no Commitment
The FOC’s (41)–(43) and (59)–(61) are necessary but not sufficient to identify an optimal
issuance policy. Two problems can arise: First, the proposed solution may not satisfy the
required second-order conditions, and thus might characterize a minimum or a saddle point.
Second, the proposed solution might be only a local maximum. In this section, for given values
of β and ξ, we identify regions for which solutions to the FOC’s are in fact the optimal policy
parameters.
Figure 1 shows that, for the case with commitment, the parameter space (β, ξ) consists of
two regions separated by an indifference curve: For sufficiently high values of β and ξ (i.e.,
sufficiently low values of (1/ξ)), it is optimal for the firm to never issue debt, and therefore
the optimal value of vu is infinity. For lower values of β and ξ, there exists an optimal (s, S)
18
debt issuance policy with parameters (vb , vu , γ). The existence of these two regions can be
understood as follows: For a given level of β, when debt maturity is low (i.e., when ξ is high),
debt needs to be “rolled over” more often in order to maintain a level of leverage that generates
significant tax benefits. Separately, for a given maturity 1/ξ, when β is high, each debt issuance
becomes more expensive. Therefore, when both β and ξ are sufficiently high, the present value
of issuance costs becomes larger than the present value of tax benefits of debt net of bankruptcy
costs. The line that separates these two regions represents the indifference curve along which
the tax benefits of debt net of bankruptcy costs equal the present value of issuance costs. For
reasons explained below, we refer to this line as the optimal indifference curve.
To provide support for this interpretation, in Figure 2 we hold ξ fixed, and plot the optimal
values of (vb , vu , vu /γ) as a function of β. As β approaches its critical value (identified by the
dashed vertical line), the optimal value of vu approaches infinity, but both vγu and vb remain
finite. Moreover, the optimal value of vb approaches the default boundary in a generalized
version of Leland (1994) model in which debt has finite maturity. Taking advantage of these
insights, in the online Appendix we show that the indifference curve can be approximated by
the formula:
(β × ξ)|
no benefit to debt
≈ rπ(1 − π).
(62)
Figure 1 shows that this approximate formula provides a good estimate for the actual
location of the curve of critical values of (β, ξ) along which tax benefits to debt net of bankruptcy
costs equal debt issuance costs. Moreover, this approximation improves for smaller values of β
and 1/ξ, for which the underlying assumptions made to derive equation (62) are more justifiable.
In contrast, for the case with no commitment, Figure 3 shows that the parameter space (β, ξ)
consists of three regions. As in the case with commitment, for sufficiently high values of β and
ξ, it is optimal for the firm to never issue debt. This is for the same reason given above: In
this region, the present value of issuance costs is larger than the present value of future tax
benefits of debt net of bankruptcy costs. For intermediate values of β and ξ, there exists an
optimal (s, S) policy characterized by the parameters (vb , vu , γ). As above, we refer to the line
19
that separates these two regions as the optimal indifference curve. Interestingly, however, for
sufficiently low values of β and ξ, we find that there is no equilibrium (s, S) strategy in that
values for (vb , vu , γ) that satisfy the relevant first order conditions either do not satisfy the
necessary second order conditions, or provide only a local maximum. The fact that there is
no (s, S) equilibrium solution for sufficiently low values of (β, ξ) underscores the inefficiency
of myopic no-commitment policies, and stands in contrast to the global-optimal policy that,
for the same values of (β, ξ), generates positive tax benefits to debt. We refer to the line that
separates this region from the region whose optimal policy is characterized by an (s, S) strategy
as the no equilibrium curve.
3.2
Optimal Debt Maturity
In the previous section, we determined the optimal policy parameters (vb∗ (ξ), vu∗ (ξ), γ ∗ (ξ)) for
a fixed inverse maturity parameter ξ. Here we choose the optimal maturity 1/ξ ∗ by maximizing the equity value at the time of issuance. The value of equity at the time of issuance is
E(vu F, γF ) + D(vu F, γF ) − D(vu F, F ) − βvu F . Using the scaling property E(V, F ) = F e(v),
D(V, F ) = F d(v), the optimal value of ξ ∗ is given by:
ξ
∗
= arg max γ e(vu /γ) + γ d(vu /γ) − d(vu ) − βvu .
(63)
ξ
As in DH, we are assuming commitment with regard to maturity in that all past debt was,
and all future debt will be, issued with inverse maturity ξ ∗ . This modeling choice allows us to
compare our results more directly to DH.
4
Results
In this section, we analyze the properties of the optimal debt issuance policies derived from our
model. Section 4.1 investigates debt and equity values. Section 4.2 analyzes the effect of the
debt issuance cost β on optimal policies. In Section 4.3 we estimate the tax benefit to debt and
in Section 4.4 we provide a decomposition of the claims to EBIT.
20
Table 1 reports the values of the coefficients that we use in our baseline model calibration.
We set the annual risk-free rate r = 4%, and the drift and volatility of the EBIT dynamics in
equation (7) to µ = 0 and σ = 22%, respectively. We assume that corporate profits are taxed
at a rate π = 20%. Following DH, we set the bankruptcy cost parameter to α = 1.10 Below,
we illustrate results for a range of values of the issuance cost parameter β. For each value of
β, we choose the inverse maturity parameter ξ according to equation (63), and then determine
the optimal parameters (vb , vu , γ) for these values of (β, ξ). Finally, we choose the coupon rate
c so that the bond is priced at par at the time of issuance.
4.1
Debt and Equity Values
In Figure 4, we report the values of (scaled) equity e(v) and debt d(v) as a function of v for
the case with commitment. We choose β = 0.0036 to reflect an issuance cost of 1% of the debt
amount issued, consistent with empirical estimates:11
βvu Fτu
βvu
set
=
= 1%,
D(vu Fτu , γFτu ) − D(vu Fτu , Fτu )
(γ − 1) d(vu )
{z
}
|
(64)
Debt amount issued
where the first equality follows from the scaling property of the debt claim. Figure 4 shows
that for values of v > (vu /γ), the value of debt is rather flat and remains close to the value of a
c+ξ
risk-free bond with the same promised cash flows drisk f ree = r+ξ
. However, the value of debt
drops quickly toward its recovery value (which, following DH, is zero in this parametrization) as
v approaches the default boundary vb . In contrast, equity exhibits positive convexity (consistent
with the smooth pasting condition) for values of v near the default boundary vb and is linear
for larger values of v.
10
DH mostly focus on the case α = 1. For the case α < 1, they find that the optimal policy includes issuing
an infinite amount of face value of debt at the default boundary, which in turn reduces the recovery rate on
debt as a fraction of face value to zero. This prediction is in stark contrast to empirical observation, in which
a typical recovery rate is approximately 40% of the face value of debt. One way to avoid this counterfactual
prediction in both our framework and theirs is to specify that covenants are in place that restrict debt issuance
beyond some level of (inverse) leverage.
11
See, for example, Altınkılıç and Hansen (2000), Hennessy and Whited (2007), Titman and Tsyplakov (2007),
and Gamba and Triantis (2008).
21
In Figure 5, we plot the difference in equity values for the cases with and without commitment. The figure confirms that, for all values of v, equity values are higher for the case with
commitment, consistent with this policy being globally optimal. More interesting, however, is
that the differences between the equity prices with and without commitment is typically twoto three orders of magnitude smaller than the typical level of the claim’s value reported in
Figure 4.12 As shown in more detail in Sections 4.3 and 4.4, this result suggests that managers who follow a no-commitment strategy can obtain most of the tax benefits afforded to the
global-optimal strategy.13
4.2
Optimal Debt Policies as a Function of β
In the next several figures, we report results associated with the optimal policy controls
vb∗ (β), vu∗ (β), γ ∗ (β), ξ ∗ (β) as a function of β. For both cases with and without commitment,
Figure 6 shows that the optimal average maturity is an increasing function of the issuance cost
parameter β. This finding is consistent with the tradeoff between net tax benefits to debt and
issuance costs. Specifically, issuing short-maturity debt increases tax benefits net of bankruptcy
costs, whereas issuing long-maturity debt reduces issuance costs. As one lowers the issuance
cost parameter β, the first channel becomes more dominant, leading to shorter optimal maturities. Interestingly, for all values of β, the optimal maturity for the case with commitment is
slightly longer than the optimal maturity for the case without commitment.
In Figures 7 and 8, we present the optimal debt issuance size parameter γ ∗ (β) and the
optimal location of the upper boundary vu∗ (β) as a function of the issuance cost parameter
β. Both with and without commitment, the optimal location of the upper boundary and the
size of the debt issuance increase monotonically with issuance costs. This is because, when
12
In both cases with and without commitment, the equity value is close to zero when inverse-leverage v
approaches the default boundary vb . Next to that point, small deviations in the equity valuations are due to
slight differences in the location of the default boundary for the policies with and without commitment.
13
Because the global-optimal policy generates higher equity values in all states of nature compared to the
no-commitment case, it can be supported in equilibrium if any deviation from the global optimal policy would
imply all future debt issuances would be priced according to the no-commitment solution. Similarly, as we
discuss in Section 5, the DH policy can serve as a punishment for the case of vanishing issuance costs. Such an
equilibrium, however, would fall outside of the Markov-perfect class that DeMarzo and He (2019) restrict their
attention to.
22
fixed issuance costs are high, it is optimal to reduce the present value of these costs by making
issuances less frequent. This can be accomplished in two ways: First, by increasing vu so that
the inverse leverage of the firm (vt ) reaches the debt issuance boundary vu less often. Second,
by increasing debt issuance size so that, after each issuance date, the firm’s inverse leverage
vu
is further away from reaching the debt issuance boundary vu again.
γ
There are two important features to note from these figures. First, as can be seen from
equation (8), the (scaled) tax flow to government can be expressed as π [(r − µ)v − c]. Hence,
c
taxes are zero when v = r−µ . In the limit of β → 0, Figure 8 shows that the optimal
c
value for vu → r−µ
, implying that the present value of the government claim goes to zero
in this limit.14 Second, in the limit β → 0, we find γ → 1, implying that the size of new debt
issuances becomes infinitesimally small. That is, in the limit of vanishing issuance costs, the
upper boundary acts like a reflecting boundary. We will use this result in the next section when
we formally investigate the limiting case β → 0.
In Figure 9, we present the optimal location of the lower boundary vb∗ (β) as a function of
the issuance cost parameter β. We see that, for a given β, the case with commitment is always
associated with a lower default boundary compared to the case without commitment. This is
because the value of equity is always higher for the case with commitment, making the option
to maintain ownership more valuable. Interestingly, however, vb∗ (β) is not monotonic in β.
This is due to the presence of two channels impacting this optimal policy function. The first
(direct) channel is driven by issuance costs: Ceteris paribus, higher issuance costs lower equity
valuations as well as the option to remain solvent. Hence, this channel alone would generate
a monotonically increasing function vb∗ (β). Indeed we find that the relation between vb∗ and β
is monotonic when we hold ξ fixed. However, there is a second (indirect) channel due to the
optimal maturity choice ξ ∗ (β). It is evident from Figure 6 that optimal maturity is increasing in
β. In those states of nature in which a firm is performing poorly, longer maturity (i.e., smaller
14
More precisely, we can show that, in the limit β → 0, the optimal values for both vb and vu go to
which leads to the present value of the government claim going to zero, as v ∈ (vb , vu ) always holds.
23
c
r−µ
,
values of ξ) implies a reduction in principal payments (ξF ) currently due, and therefore less
cash flow that shareholders need to raise to service outstanding debt. Hence, longer maturities
are associated with higher equity valuations when inverse leverage is low, which in turn leads
to a lower optimal default boundary. Figure 9 shows that this indirect channel dominates the
direct channel for values of β greater than β > 10−2.8 .
Finally, Figure 10 shows the coupon rate c as a function of β, which is chosen so that the
bond is priced at par at the time of issuance. As β is decreased, the optimal maturity 1/ξ ∗
shortens, and the bond becomes less risky. In the limit β → 0, the par coupon rate converges
to the riskfree rate r, set to 4% in our calibration.
4.3
Tax Benefits to Debt
We estimate tax benefits to debt as the ratio of the levered enterprise value to the value of an
all-equity firm at the debt issuance boundary:
TB =
E(vu F, F ) + D(vu F, F )
(1 − π)vu F
−1 =
e(vu ) + d(vu )
−1 .
(1 − π)vu
(65)
In Figure 11, we report the level of tax benefits in equation (65) as a function of the issuance
cost parameter β for three cases: (i) the case with commitment and optimal maturity, (ii)
the case without commitment and optimal maturity, and (iii) the case with commitment, but
with the restriction ξ = 0. Three points are worth noting. First, in agreement with GJL, the
tax benefits to debt increase as issuance costs decrease. Second, the tax benefits to debt are
amplified considerably when firms have the right to choose optimal debt maturity. Indeed, as β
goes to zero, for both the case with and without commitment, the optimal debt and maturity
policy allows firms to extract 100% of the claim to EBIT, that is, e(vu ) + d(vu ) → vu , and tax
vu F
π
benefits converge to TB → (1−π)v
− 1 = 1−π
= 0.25. Third, in contrast to both DH and
F
u
DeMarzo (2019), the case with commitment generates only slightly higher tax benefits to debt
than does the case without commitment. Hence, in the presence of even tiny issuance costs,
the question of whether a manager can commit to the global-optimal policy is of secondary
24
importance in terms of available tax benefits to debt when firms can choose their debt maturity
optimally.
4.4
A Decomposition of Claims to EBIT
Figures 12 and 13 show the values of all claims to EBIT as a function of the issuance cost
parameter β for the cases with and without commitment, respectively. As noted previously,
tax benefits net of bankruptcy costs are maximized for debt with zero maturity (ξ → ∞).
In contrast, the present value of issuance costs are minimized for debt with infinite maturity
(ξ → 0). As β → 0, the former channels dominates. As Figures 12 and 13 show, for both cases
with and without commitment, the claims to issuance costs, bankruptcy costs, and government
all go to zero as β vanishes, in turn allowing the firm to extract 100% of EBIT by following an
optimal capital structure and maturity policy.
The fact that the present value of the claim to issuance costs vanishes with β is mostly
intuitive, as the cash flows to this claim are linear in β.15 As noted above, the government
claim vanishes as β → 0 because both vb and vu (and hence, all v ∈ (vb , vu )) approach the value
c
in this limit, which in turn implies zero taxes paid to government. Finally, the claim to
r−µ
bankruptcy costs vanish because, as β → 0, the optimal inverse maturity parameter ξ → ∞,
implying, via equation (7), that the drift of the stochastic process (1/dt)E dv
= (µ + ξ) goes
v
to infinity and, hence, that the default probability vanishes.
5
The Case of Vanishing Issuance Costs
In Section 3.1.3, we focused on the larger values of β in Figure 2 to motivate our claim for the
existence of a critical value β ∗ at which shareholders become indifferent to issuing additional
debt. Here, we focus on the smaller values of β in Figure 2 to motivate our claim that, with
commitment, in the limit of vanishing issuance costs (i.e., β → 0), there exists a well-defined
15
Admittedly, the frequency in which inverse leverage reaches the upper boundary becomes arbitrarily large
in the β → 0 limit. However, we find numerically that the product (β ξ ∗ (β)) converges to zero as β goes to zero.
25
limit for the policy parameters (vb , vu , γ) for any given value of ξ.16 Indeed, Figures 14, 15,
and 16 show optimal values of (vb , vu , γ) for extremely small values of β. Clearly, these policy
parameters appear to converge to limiting values. In this section, we identify this limit more
formally.
To begin, note that the general pricing ODE (equation (18)) and the lower boundary conditions for debt and equity (equations (27) and (30)) are independent of the parameters (vu , γ).
That is, these parameters appear only in the upper boundary conditions (equations (28) and
(31)). Here we identify these upper boundary conditions for the case β → 0.
Importantly, Figure 16 demonstrates that, as the issuance cost parameter β → 0, the
optimal debt issuance size parameter γ → 1. Therefore, in order to identify the limiting case,
we consider a Taylor series expansion around ≡ (γ − 1). In terms of , we can express the
upper boundary condition for debt in equation (28) as:
(1 + )D(vu F, F ) = D(vu F, F + F ) = D(vu F, F ) + F DF (V, F )|V =v
uF
.
(66)
Taking the limit → 0, this relation simplifies to
D(vu F, F )
→0
=
F DF (V, F )|V =v
uF
.
(67)
Applying equations (14) and (17) to equation (67), we obtain a boundary condition that has a
form typical of problems with a reflecting boundary:
dv (vu ) = 0.
(68)
This equation can also be derived by taking a Taylor series expansion of equation (28). Equation (68) can be interpreted as precluding an arbitrage opportunity in that the price of debt
per unit face value must be the same just prior to (i.e., at date τu− ) and just after (i.e., at date
τu+ ) the debt issuance.
A similar argument holds for the upper boundary condition on equity. Specifically, setting
β = 0 in equation (A.10), and Taylor expanding about ≡ (γ − 1), this boundary condition for
16
We restrict our attention to the case with commitment because, for the case with no commitment, as shown
in Figure 3, for a fixed value of ξ, as we lower β, we enter a region in which no (s, S) equilibrium exists.
26
debt reduces to:
0 = (EF (V, F ) + DF (V, F ))|V =v
uF
.
(69)
Applying equations (17) and (68) to equation (69), we obtain
vu ev (vu ) = e(vu ) + d(vu ).
(70)
This condition can also be derived from equation (31) by setting β = 0, and then performing
a Taylor series expansion on = (γ − 1). Moreover, note that, using equations (17) and (68),
we can rewrite equation (69) as the first order condition that characterizes the no-commitment
equilibrium of DH:
−EF (V, F )|V =v
uF
= d(vu ) = − e(vu ) − vu ev (vu ) ,
(71)
where the first equality follows from equation (67) and the last equality follows from equation (70). This expression captures the no-arbitrage condition that the firm keeps issuing debt
until, at the margin, the decline in equity value (−EF ) equals the dollar amount raised per unit
of face value, d(vu ). Note, however, that this condition holds only at the point v = vu for the
case with commitment, whereas it holds for all values of v in the no-commitment policy studied
by DH.
In Figure 17, we present the value of debt d(v) as a function of inverse leverage v for the
case β → 0 and, as in DH, ξ = 0.2. For comparison, we also plot the value of debt for the
no-commitment case of DH. This figure shows that, for all values of v, the value of debt is higher
in our model with commitment than in the DH no-commitment equilibrium even in the case of
vanishing issuance costs, β → 0. This is because our debt policy is more conservative, which
leads to lower future default probabilities, and thus to higher debt valuations for all values of v.
On this figure, we also plot −EF (V, F ) = −(e(v) − vev (v)). Under the global-optimal solution,
the value of debt is higher than the marginal value of equity for all values of v less than vu .
In contrast, as already mentioned, the condition (−EF (V, F ) = d(v)) holds for all values of v
in the DH no-commitment equilibrium. A myopic manager would interpret d(v) > −EF (v)
27
as an arbitrage opportunity to issue additional debt. However, such a manager would not be
accounting for the fact that any deviation from the optimal policy would have significant impact
on the pricing of debt, and on the optimal debt issuance policies for other values of the state
variable v. Thus, the apparent arbitrage opportunity is in fact a mirage, and shareholder value
is maximized by following the optimal policy that we have identified.
In Figure 18, we show the value of equity e(v) as a function of inverse leverage v and we
contrast it to that of the DH no-commitment solution. As previously noted, the value of equity
in the case with commitment is higher across all states of nature: e(v) > eDH (v), for all v. Hence,
there is no incentive for shareholders to ever deviate from the policy with commitment if the
punishment is that, going forward, debt would be priced according to the DH no-commitment
policy. Note that, because debtholders observe the size of the debt issuance and pay fair value
for their claim, at the date of issuance, they are indifferent to the debt issuance policy chosen
by the firm. Therefore, debtholders can credibly threaten to punish any deviation from the
optimal policy by pricing debt according to the DH no-commitment equilibrium. As a result,
shareholders would gain zero cash benefit at the date of the deviation,17 and would be left
with an equity claim that has lower valuation than under the optimal policy. In this sense, the
global-optimal policy is the subgame perfect equilibrium, and the policy that value-maximizing
managers will follow.
6
Conclusion
Within a standard tradeoff setting, we investigate the optimal dynamic capital structure and
debt maturity policies of firms. We focus on two elements: (i) issuance costs, and (ii) optimal
policies with and without commitment. For the case with commitment, we identify the globaloptimal policy that maximizes shareholders’ value across all policies consistent with limited
liability. For the case without commitment, we show that the optimal policy generates almost
as much tax benefits to debt as those obtained under the global-optimal policy. For both cases,
17
Shareholders would obtain zero cash benefit because, as shown by DH, the optimal level of debt issuance
at the deviation date is characterized by the locally deterministic process.
28
a reduction in issuance costs is associated with a decrease in the optimal maturity and an
increase in the tax benefits to debt. Indeed, as issuance costs go to zero, the firm can extract
100% of the claim to EBIT. Hence, in the limiting case of vanishing issuance cost, our optimal
policy does not converge to the optimal policy of either DeMarzo and He (2019) or DeMarzo
(2019), whose models of no commitment predict that, regardless of debt maturity choice, there
are no tax benefits to debt available to shareholders.
The predictions of our model are qualitatively consistent with the findings of the empirical
capital structure literature. The fact that debt is issued in discrete amounts is a signature
characteristic of fixed issuance costs. Our prediction that firms are able to extract tax benefits
to debt is consistent with the empirical findings of van Binsbergen, Graham, and Yang (2010),
and Blouin, Core, and Guay (2010). That firms are not indifferent toward debt maturity
choice is consistent with the findings of Barclay and Smith (1995) and Stohs and Mauer (1996).
Consistent with the findings of Titman and Wessels (1988) and Frank and Goyal (2014), our
model generates a negative correlation between profitability and leverage, and predicts that
leverage ratios are persistent. Intuitively, this is because when firms are in the inaction region,
higher (lower) profitability increases equity values while debt outstanding remains constant,
leading to lower (higher) leverage. That firms target specific minimum credit rating levels
(e.g., Kisgen (2006, 2009)) is consistent with managers caring about the firm’s reputation in
the debt markets, given that debt issuance is a repeated game. This provides support for our
claim that firms can credibly commit to our global optimal policy. Finally, when calibrated to
estimated issuance costs, our model generates quantitatively accurate predictions for optimal
debt maturity.
We acknowledge that our analysis abstracts from other important mechanisms that would
allow our framework to better match observed debt issuance dynamics. Such mechanisms include investment opportunities, jumps in state vector dynamics, time-variation in debt market
liquidity, asymmetric information between managers and debtholders, and other market imperfections. We leave the study of these important issues to future research.
29
A
Proof of Proposition 1
Debt valuation. Expressed in terms of risk-neutral expected cash flows, the debt value is:
Z
Q
τ
−r(s−t)
ds e
D(Vt , Ft ) = Et
(c + ξ)Fs
t
1
+Et e
1(τu <τ )
D(vu Fτu , γFτu )
b
γ
i
h
−r(τb −t)
.
+EQ
e
1
(1
−
α)(1
−
π)v
F
b τb
(τ <τu )
t
Q
−r(τu −t)
(A.1)
b
The first term captures the present value of cash flows prior to the firm reaching either the
upper or lower boundary. The second term captures the fact that when the upper boundary is
reached and the face value of debt jumps from Fτu to γFτu , old bondholders have a claim to
1
of the total debt claim, as it is assumed that all debt is issued pari-passu. The third term
γ
captures recovery conditional upon default.
Comparing equation (10) and equation (A.1), the debt claim is described by the factor
loadings:
hF = (c + ξ)
(A.2)
hV
(A.3)
= 0
Hb = (1 − α)(1 − π)vb Fτb
1
D(vu Fτu , γFτu ).
Hu =
γ
(A.4)
(A.5)
It therefore follows from equation (21) that the bond price can be expressed as:
d(v) = Md v φ + Nd v ω +
c+ξ
,
r+ξ
(A.6)
where the constants (Md , Nd ) are uniquely determined by the boundary conditions
d(vb ) = (1 − α)(1 − π)vb
(A.7)
d(vu ) = d(vu /γ),
(A.8)
30
Equity valuation. The value of equity can be determined via the risk-neutral expectation:
Z
τ
−r(s−t)
Q
E(Vt , Ft ) = Et
ds e
(1 − π)Ys − (c(1 − π) + ξ) Fs
(A.9)
t
Q
+Et
e
−r(τu −t)
1(τu <τ
b
)
.
E(vu Fτu , γFτu ) + D(vu Fτu , γFτu ) − D(vu Fτu , Fτu ) − βvu Fτu
The first term captures the present value of the claim to cash flows (i.e., dividends) throughout
period-0, which ends at date τ = min(τb , τu ). The second term captures the fact that when
the upper boundary is reached, shareholders still hold the entire equity claim, but now with
the face value of debt scaled by a factor of γ. In addition, shareholders receive as dividend the
value of the new debt issuance (net of issuance costs).
Comparing equation (10) and equation (A.9), the equity claim is described by the factor
loadings:
hF = −(c(1 − π) + ξ)
hV
= (1 − π)(r − µ)
Hb = 0
Hu = E(vu Fτu , γFτu ) + D(vu Fτu , γFτu ) − D(vu Fτu , Fτu ) − βvu Fτu .
(A.10)
It therefore follows from equation (21) that the value of equity per unit face value can be
expressed as:
e(v) = Me v φ + Ne v ω + (1 − π)v −
c(1 − π) + ξ
,
r+ξ
(A.11)
where the constants (Me , Ne ) are uniquely determined by the boundary conditions
e(vb ) = 0
(A.12)
e(vu ) = γe(vu /γ) + (γ − 1)d(vu ) − βvu ,
(A.13)
Government Claim. The value of the government claim can be determined via the risk31
neutral expectation:
τ
Z
−r(s−t)
Q
G(Vt , Ft ) = Et
ds e
π (Ys − cFs )
(A.14)
t
h
i
h
−r(τu −t)
Q
+EQ
e
1
G(v
F
,
γF
)
+
E
e−r(τb −t) 1(τ
u τu
τu
(τu <τ )
t
t
b
b
<τu )
i
(1 − α)πvb Fτb .
The first term captures the present value of the claim to cash flows (i.e., taxes) throughout
period-0, which ends at date τ = min(τb , τu ). The second term captures the fact that when the
upper boundary is reached, the government still holds the entire tax claim, but now with the
face value of debt scaled by a factor of γ. The third term captures recovery conditional upon
default.
Comparing equation (10) and equation (A.14), the government claim is described by the
factor loadings:
hF = −cπ
hV
= π(r − µ)
Hb = (1 − α)πvb Fτb
Hu = G(vu Fτu , γFτu ).
(A.15)
It therefore follows from equation (21) that the value of the government claim per unit face
value can be expressed as:
g(v) = Mg v φ + Ng v ω + πv −
cπ
,
r+ξ
(A.16)
where the constants (Mg , Ng ) are uniquely determined by the boundary conditions
g(vb ) = (1 − α)πvb
(A.17)
g(vu ) = γg(vu /γ).
(A.18)
Claim to Bankruptcy Costs. The value of the claim to bankruptcy costs can be determined
via the risk-neutral expectation:
Q
h
−r(τb −t)
B(Vt , Ft ) = Et e
1(τ
b
<τu )
αvb Fτb
i
h
i
−r(τu −t)
+ Et e
1(τu <τ ) B(vu Fτb , γFτb ) ,
Q
b
32
(A.19)
which can be interpreted as the risk-neutral expected cash flows going to the bankruptcy cost
claim conditional upon default.
Comparing equation (10) and equation (A.19), the claim to bankruptcy costs is described
by the factor loadings:
hF = 0
hV
= 0
Hb = αvb Fτb
Hu = B(vu Fτu , γFτu ).
(A.20)
It therefore follows from equation (21) that the value of the claim to bankruptcy costs per unit
face value can be expressed as:
b(v) = Mb v φ + Nb v ω ,
(A.21)
where the constants (Mb , Nb ) are uniquely determined by the boundary conditions
b(vb ) = αvb
(A.22)
b(vu ) = γb(vu /γ) .
(A.23)
Claim to Issuance Costs. The value of the claim to issuance costs can be determined via
the risk-neutral expectation:
h
I(Vt , Ft ) = EQ
e−r(τu −t) 1(τu <τ
t
b
)
i
βvu Fτu + I(vu Fτu , γFτu ) .
(A.24)
This equation states that when the upper boundary is reached, the claim to issuance cost
receives the cash flow βvu Fτu in addition to a claim to future issuance costs after the debt
issuance, which are captured recursively via the term I(vu Fτu , γFτu ).
Comparing equation (10) and equation (A.24), the claim to issuance costs is described by
33
the factor loadings:
hF = 0
hV
= 0
Hb = 0
Hu = βvu Fτu + I(vu Fτu , γFτu ).
(A.25)
It therefore follows from equation (21) that the value of the claim to issuance costs per unit
face value can be expressed as:
i(v) = Mi v φ + Ni v ω ,
(A.26)
where the constants (Mi , Ni ) are uniquely determined by the boundary conditions
B
i(vb ) = 0
(A.27)
i(vu ) = βvu + γ i(vu /γ) .
(A.28)
Global Optimality of the Commitment Policy in Definition 1
In this section we verify the global optimality of the policy (vb∗ , vu∗ , γ ∗ ) for any given inverse
maturity parameter ξ. We proceed in two steps. First, we show that, for any given vb , the
FOC’s (42)–(43) determine the same optimal values for (vu , γ) regardless of whether they are
evaluated at vu or at any value vt ∈ (vb , vu ). Second, we show numerically that vb∗ is globally
optimal under limited liability.
Step 1: For any given vb , the lower boundary condition (30) implies:
c (1 − π) + ξ
φ
Ne =
− (1 − π) vb − Me vb vb−ω .
r+ξ
34
(B.1)
Therefore, we can express the equity value in equation (29) as
"
ω−φ #
ω
v
v
φ
e (v, Me , vb ) = Me v 1 −
+ (1 − π) v − vb
vb
vb
−
c (1 − π) + ξ
r+ξ
ω
v
1−
.
vb
(B.2)
Note that the coefficient Me in equation (B.2) is uniquely determined by the policy parameters
(vb , vu , γ). Further note that, because (vu , γ) appear in equation (B.2) only through Me , it
follows that if one could choose these parameters at any value of v = vany ∈ (vb , vu ), one would
solve the FOC:
0 =
0 =
∂e
∂vu
∂e
∂γ
"
φ
= vany
1−
v=vany
"
φ
= vany
1−
v=vany
vany
vb
vany
vb
ω−φ #
ω−φ #
∂Me
.
∂vu
∂Me
.
∂γ
(B.3)
(B.4)
vany ω−φ
Because vany 1 − v
6= 0, ∀vany ∈ (vb , vu ), it follows that these FOC’s reduce to
φ
b
∂Me
∂Me
= 0 and
= 0.
∂vu
∂γ
(B.5)
This proves that the optimal policy parameters are independent of the current value of v = vany .
Step 2: In the previous step, we demonstrated that the optimal values of (vu∗ (vb ), γ ∗ (vb )) are
independent of v, and functions only of vb . As such, it is convenient to define
ce (v ) ≡ Me (v , v ∗ (v ), γ ∗ (v ))
M
b
b
b
b
u
ce (v ), v .
eb(v, vb ) ≡ e v, M
b
b
(B.6)
Using the chain rule of calculus, we can write
c
c
∂e v, Me (vb ), vb ∂ M
ce ∂e v, Me , vb
∂b
e(v, vb )
≡
+
ce
∂vb
∂vb
∂vb
∂M
v
(B.7)
35
Now, because:18
ce (v ), vb
∂e v, M
b
ce
∂M
ce , vb
∂e v, M
∂vb
h
i
= v ω v φ−ω − vbφ−ω > 0,
= −
v
vb
(B.8)
ω h
i
(1 − π̄) (1 − ω) + Me (φ − ω) vbφ−1 < 0,
(B.9)
it follows that, if
ce
∂M
∂vb
< 0, then
∂b
e(v,vb )
∂vb
< 0. This in turn would imply that the equity claim
v
is maximized by choosing the smallest value of vb consistent with limited liability, which is
the value of vb consistent with the FOC in equation (41). Hence, our problem is reduced to
showing that
ce
∂M
∂vb
< 0 holds in our economy. Although the term Me has a closed-form analytical
expression in terms of the debt policy parameters (vb , vu , γ), it does not lend itself to an easy
interpretation. Therefore, we resort to a numerical argument for the rest of this analysis, using
the model coefficients in Table 1.
In Figure B.1 we plot Me vb , vu∗ (vb ) , γ ∗ (vb ) as a function of vb , where vu∗ (vb ) and γ ∗ (vb ) are
given by the FOC’s (42)–(43). The plot shows that Me vb , vu∗ (vb ) , γ ∗ (vb ) is indeed decreasing
in vb . The black dot on the line marks the highest value of Me for which the equity value is
consistent with limited liability for all v > vb . Furthermore, Figure B.2 reports the derivative
of the equity value,
∂e
∂v
, as a function of vb . As this figure shows, the value of vb that gives
v=v
b
the highest value of Me consistent with limited liability satisfies the smooth pasting condition
∂e
∂v
= 0.
(B.10)
v=v
b
Moreover the value vb∗ that satisfies condition (B.10) is the same as that obtained from the
FOC’s (41)–(43). Therefore, it follows that our policy generates the highest values of equity
for vt > vb∗ , since all other solutions for equity that satisfy limited liability will have an equity
18
c (v ) ≥ 0 is available upon request.
The proof of M
e
b
36
valuation of zero at vb∗ . We conclude that the debt issuance policy (vb∗ , vu∗ , γ ∗ ) that satisfies the
FOC’s (41)–(43) generates an equity valuation that is globally optimal.
C
Optimal Policy for vt > vu
In the main text we identify optimal capital structure strategies characterized by an (s, S)
policy with a single inaction region (vb , vu ). In this Appendix, we provide numerical support
for the claim that if a firm finds itself with an inverse leverage ratio v(t− ) > vu , then it is optimal
for that firm to immediately issue new debt so that its inverse leverage jumps into the region
v(t+ ) ∈ (vb , vu ).19 Note that the condition v(t− ) > vu implies that the firm’s inverse leverage is
even higher than that value vu for which the firm finds it optimal to pay the debt issuance cost
in order to increase their tax benefits. Thus, it seems intuitive that if inverse leverage is higher
than vu , then the firm has even more incentive to issue debt immediately.
First, we identify the value of equity based on our proposed optimal policy. Consider a
firm in a state (Vt , Ft ) such that (Vt /Ft ) > vu . Under the proposed policy, the face value of
debt will jump from Ft to F̂t = (Vt /v̂), so that inverse leverage jumps from v(t− ) = (V /F ) to
v(t+ ) = v̂ v(t− ) , which is a function of v(t− ) . Under this policy, we have the relations:
D(V, F )
= d(v̂)
F
"
E(V, F ) = max E V, F̂ + F̂ − F
F̂
D(V, F̂ )
F̂
#
− βV .
(C.1)
The first equation captures a no-arbitrage condition in that the price per unit face value of debt
just prior to the issuance must equal the price just after the issuance. The second equation
also captures a no-arbitrage condition in that the value of the equity in state (V, F ) should
equal the sum of: (a) the value of equity in state (V, F̂ ), plus (b) proceeds raised from the debt
19
We thank Peter DeMarzo, Zhiguo He, and Fabrice Tourre for suggesting that we pursue this analysis.
37
issuance, minus (c) issuance costs. Proceeds raised from debt issuance equal the product of (i)
the quantity (F̂ − F ) of debt issued, multiplied by (ii) the price per unit of debt—the latter is
equal to D(V, F̂ )/F̂ .
Looking for solutions of the form:
D(V, F ) = F d(v = (V /F ))
(C.2)
E(V, F ) = F e(v = (V /F )),
(C.3)
we can express these relations as:
d(v) = d(v̂)
(C.4)
e(v) = max {(v/v̂) e(v̂) + (v/v̂) d(v̂) − d(v̂) − βv} .
(C.5)
v̂
The optimal v̂ is determined by the following first order condition of equation (C.5):
d0 (v̂)
v̂e0 (v̂) − e(v̂) + v̂d0 (v̂) − d(v̂)
=
,
v̂ 2
v
(C.6)
which implies the optimal v̂ depends on v. We find that v̂(v) ∈ (vb , vu ) for any v > vu under all
β such that an (s, S) policy exists. Equation (C.5) gives the value of scaled equity for values
of v > vu under our proposed optimal policy.
Next, we provide numerical support for our claim that this proposed strategy is indeed
optimal. We do this two ways. First, we examine multiple strategies in which we posit additional
inaction regions (v, v) for which, if the current value of firm’s inverse leverage falls into the
range vt ∈ (v, v), no debt is issued. When reaching either boundary (v, v), we consider both
possibilities that when these boundaries are reached, the firm issues sufficient debt so that
inverse leverage immediately jumps into either: i) the region (v, v), or ii) into the region (vb , vu ).
Given any explicit strategy, we then determine the value of equity following such a strategy
using the same types of calculations used above. In spite of investigating millions of different
parameter choices for (v, v), we never identified a single example in which our proposed optimal
policy did not generate a higher equity valuation.
38
In addition to investigating millions of specific cases, here we investigate a more general
approach that provides additional evidence that, when the firm’s inverse leverage is above the
restructuring boundary (v(t− ) ≥ vu ), the optimal policy is to issue debt immediately so that
the new inverse leverage jumps into the region v(t+ ) ∈ (vb , vu ). Specifically, we first identify the
proposed optimal policy as described in equation (C.1). We then compare this equity value to
that obtained if the manager waits a period dt, in turn receiving the dividend accrued over that
period, and then restructures a period dt from now. The value of this alternative strategy is:
Y (1 − π) − F (c(1 − π) + ξ) dt + (1 − r dt)EQ [E(V + dV, F + dF )]
= (Y (1 − π) − F (c(1 − π) + ξ)) dt
σ2 2
+(1 − r dt) E(V, F ) + µV EV dt + V EV V dt − ξF EF dt .
2
(C.7)
Note that E(V + dV, F + dF ) is the value of equity provided that the firm restructures immediately at time t + dt.
Subtracting E(V, F ) from both sides, we study whether:
?
0 > (Y (1 − π) − F (c(1 − π) + ξ)) − rE(V, F ) + µV EV +
σ2 2
V EV V − ξF EF . (C.8)
2
Basically, we are investigating whether or not we have identified a supermartingale, which would
imply that issuing debt immediately dominates any strategy that involves waiting.
Using our scaling argument, this simplifies to
?
0 > v(r − µ)(1 − π) − c(1 − π) − ξ − re + µvev +
σ2 2
v evv − ξ(e − vev ).
2
(C.9)
Numerically, after we set the grid on {vi }Ii=1 ∈ (vu , ∞), we approximate the first and second
order derivatives via the central differences:
ev (vi ) ≈
e (vi+1 ) − e (vi−1 )
e (vi+1 ) − 2e (vi ) + e (vi−1 )
; evv (vi ) ≈
24v
(4v)2
where 4v = vi − vi−1 .
39
In addition to equation (C.9), to satisfy the verification theorem of the HJB equation, we
check whether the following equation holds for all V :
0 = max
−E (V, F ) ; ME (V, F ) − E (V, F ) ;
−rE (V, F ) + Y (1 − π) − F (c (1 − π) + ξ) + LE (V, F )
,
(C.10)
where ME (V, F ) is defined in equation (C.1) and
LE (V, F ) = µV EV (V, F ) +
σ2 2
V EV V (V, F ) − ξF EF (V, F ) .
2
Dividing both sides of equation (C.10) by F we find,
0 = max −e (v); Me (v) − e (v); −re (v) + v (r − µ) (1 − π) − (c (1 − π) + ξ) + Le (v)
{z
} |
{z
}
| {z } |
term A
term B
term C
(C.11)
where
Le (v) = µvev (v) +
σ2 2
v evv (v) − ξ (e (v) − vu ev (v)) .
2
Under both commitment and no-commitment, we verify numerically that equation (C.11) holds
for all v and β for which an (s, S) policy exists.
Figure B.3 shows the value of all terms in equation (C.11) as a function of v for the economy
without commitment. We interpret this result to imply that issuing debt immediately dominates
any strategy that involves waiting, regardless of whether that waiting strategy involves waiting
a particular amount of time, or whether it involves waiting to reach some pre-specified barrier
before issuing debt.
40
Parameter
Symbol Value
Annual risk-free rate
r
0.04
Annual asset drift
µ
0
Annual asset volatility
σ
0.22
Corporate tax rate
π
0.2
Loss given default
α
1
Table 1: Baseline Model Coefficients. The table shows the values of the model coefficients
in the baseline calibration. We choose the coupon rate c so that the bond is priced at par at
the time of issuance.
2.5
2
1.5
1
0.5
0
10 -4
10 -3
10 -2
Figure 1: Debt issuance with commitment. For the case with commitment, this figure
identifies the region in which it is optimal to issue additional debt according to the (s, S) policy
with parameters (vb , vu , γ). The dashed line shows the approximate theoretical location of the
debt issuance indifference curve, βξ = rπ(1 − π), that separates the (s, S) from the no-debt
issuance regions.
41
12
10
8
6
4
2
0
10 -6
10 -5
10 -4
10 -3
10 -2
10 -1
Figure 2: Optimal policy parameters with commitment: Fixed maturity. For ξ = 0.2,
this figure shows the parameter values (vu , vu /γ, vb ) as a function of the issuance cost parameter
β. It also shows the default boundary vb,no issuance for a firm that is restricted from issuing debt
in the future (equation (C.6) in the online Appendix). The vertical dashed line at β ∗ identifies
the value of β for which optimal vu goes to infinity. For this β ∗ value, there are no tax benefits
to debt, and therefore no additional debt is issued.
18
16
14
12
10
8
6
4
2
0
10-6
10-5
10-4
10-3
10-2
Figure 3: Debt issuance without commitment. For the case without commitment, this
figure identifies three regions. The region that falls below the red dotted line refers to the case
in which vu = ∞, and thus it is never optimal to issue additional debt, as in Leland (1994).
The region between the red dotted line and the blue solid line is characterized by an optimal
(s, S) policy with parameters (vb,0 , vu,0 , γ0 ). Points in the region above the solid blue line do not
possess a pure (s, S) strategy equilibrium. The dashed line shows the approximate theoretical
location of the debt issuance indifference curve, βξ = rπ(1 − π), that separates the (s, S) region
from the no-debt issuance region.
42
4
3.5
3
2.5
2
1.5
1
0.5
0
0
1.5
3
4.5
6
Figure 4: Debt and equity values with commitment. This figure shows debt and equity
values for the case with commitment, equations (A.6) and (A.11), respectively, when β = 0.0036.
This parameter value reflects a fractional issuance cost of 1%. The other parameter values are
reported in Table 1.
0.004
0.0035
0.003
0.0025
0.002
0.0015
0.001
0.0005
0
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Figure 5: Difference between equity values under commitment and no commitment.
This figure shows the difference in equity values for the cases with and without commitment,
(e(v) − eN C (v)), when β = 0.0036. This parameter value reflects a fractional issuance cost of
1% for the case with commitment. When calculating the difference e(v) − eN C (v), we use the
coupon rate c and inverse maturity ξ from the optimal policy with commitment. The other
parameter values are reported in Table 1.
43
7
6
5
4
3
2
1
0
10 -6
10 -5
10 -4
10 -3
10 -2
Figure 6: Optimal maturity. This figure shows the optimal debt maturity for the cases with
and without commitment as a function of β. The other parameter values are in Table 1.
8
7
6
5
4
3
2
1
0
10 -6
10 -5
10 -4
10 -3
10 -2
Figure 7: Optimal debt issuance size. This figure shows the optimal debt issuance parameter
γ for the cases with and without commitment as a function of β. The other parameter values
are in Table 1.
44
18
16
14
12
10
8
6
4
2
0
10 -6
10 -5
10 -4
10 -3
10 -2
Figure 8: Optimal debt issuance boundary. This figure shows the location of the optimal
debt issuance boundary vu for the cases with and without commitment as a function of β. We
also plot the function c/(r − µ) for the case with commitment. The coupon c is chosen so that
debt is issued at par. As can be seen in Figure 10, the function c/(r −µ) for the no commitment
case is very similar to that for the case with commitment, and therefore we omit it for clarity.
The other parameter values are reported in Table 1.
1.14
1.12
1.1
1.08
1.06
1.04
1.02
1
0.98
10 -6
10 -5
10 -4
10 -3
10 -2
Figure 9: Optimal default boundary. This figure shows the location of the optimal default
boundary vb for the cases with and without commitment as a function of β. It also reports the
function c/(r − µ) for the case with commitment. The coupon c is chosen so that debt is issued
at par. As can be seen in Figure 10, the function c/(r − µ) for the no commitment case is very
similar to that for the case with commitment, and therefore we omit it for clarity. The other
parameter values are reported in Table 1.
45
0.0414
0.0412
0.041
0.0408
0.0406
0.0404
0.0402
0.04
10 -6
10 -5
10 -4
10 -3
10 -2
Figure 10: Coupon rate. This figure shows the coupon rate c such that the bond is priced at
par at the time of issuance. The other parameter values are in Table 1.
0.25
0.2
0.15
0.1
0.05
0
10 -6
10 -5
10 -4
10 -3
10 -2
Figure 11: Tax benefits to debt. This figure reports the tax benefits to debt under the
optimal policy parameters for the cases with and without commitment as a function of β. It
also reports the tax benefits to debt for the case with commitment when the inverse maturity
parameter is set to ξ = 0. The other parameter values are in Table 1.
46
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10 -6
10 -5
10 -4
10 -3
10 -2
Figure 12: Claim values with commitment. For the model with commitment, this figure
reports the value of the sum of debt and equity (d(vu ) + e(vu )), bankruptcy costs claim b(vu ),
government claim g(vu ), and issuance cost claim i(vu ) at the debt issuance boundary vu as a
function of the debt issuance cost parameter β. The other parameter values are in Table 1.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10 -6
10 -5
10 -4
10 -3
10 -2
Figure 13: Claim values without commitment. For the model without commitment,
this figure reports the value of the sum of debt and equity (d(vu ) + e(vu )), bankruptcy costs
claim b(vu ), government claim g(vu ), and issuance cost claim i(vu ) at the debt issuance boundary vu as a function of the debt issuance cost parameter β. The other parameter values are in
Table 1.
47
1.0073
1.00725
1.0072
1.00715
1.0071
1.00705
10 -12
10 -11
10 -10
10 -9
10 -8
10 -7
10 -6
Figure 14: Optimal lower boundary for fixed maturity. This figure reports the optimal
policy parameter vb for β < 10−6 . The horizontal line is the optimal value of vb calculated for
the limiting case β → 0 determined from equations in Section 5.
2.5
2.49
2.48
2.47
2.46
2.45
2.44
2.43
10 -12
10 -11
10 -10
10 -9
10 -8
10 -7
10 -6
Figure 15: Optimal upper boundary for fixed maturity. This figure reports the optimal
policy parameter vu for β < 10−6 . The horizontal line is the optimal value of vu calculated for
the limiting case β → 0 determined from equations in Section 5.
48
1.045
1.04
1.035
1.03
1.025
1.02
1.015
1.01
1.005
1
0.995
10 -12
10 -11
10 -10
10 -9
10 -8
10 -7
10 -6
Figure 16: Optimal debt issuance size for fixed maturity. This figure reports the optimal
policy parameter γ for β < 10−6 .
1.2
1
0.8
0.6
0.4
0.2
0
0
0.7
1.4
2.1
2.8
Figure 17: Debt valuation in absence of issuance cost: commitment vs. no commitment. For ξ = 0.2, this figure shows the value of the debt claim d(v) for the case with
commitment in the inaction region v ∈ (vb , vu ). We set the coupon rate to c = 0.0407 to
guarantee that debt is priced at par at the time of issuance. The figure also shows the value of
the debt claim for the no-commitment case studied by DH. Because the DH solution does not
have an upper boundary vu , the figure reports a truncated version of their function. The other
parameter values are in Table 1.
49
1.2
1
0.8
0.6
0.4
0.2
0
0
0.7
1.4
2.1
2.8
Figure 18: Equity valuation in absence of issuance cost: commitment vs. no commitment. For ξ = 0.2, this figure shows the value of the equity claim e(v) for the case with
commitment in the inaction region v ∈ (vb , vu ). We set the coupon rate to c = 0.0407 to
guarantee that debt is priced at par at the time of issuance. The figure also shows the value
of the equity claim for the no-commitment case studied by DH. Because the DH solution does
not have an upper boundary vu , the figure reports a truncated version of their function. The
other parameter values are in Table 1.
50
0.055
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.8
0.9
1
1.1
1.2
1.3
Figure B.1: Optimality of the default boundary. This figure shows Me vb , vu∗ (vb ) , γ ∗ (vb )
is decreasing in vb , where vu∗ (vb ) , γ ∗ (vb ) are solved from the FOC approach for any given vb .
The issuance cost parameter is set to β = 0.0036, which is chosen to match a fractional debt
issuance cost of 1% under the optimal maturity ξ ∗ = 0.2976. The other parameter values are
reported in Table 1.
2
1
0
-1
-2
-3
-4
0.8
0.9
1
1.1
1.2
1.3
Figure B.2: Slope of equity value e(v) at vb . This figure shows
of equity value
the derivative
∂e
∗
∗
∗
∗
as a function of vb , with Me ≡ Me vb , vu (vb ) , γ (vb ) and vu (vb ) , γ (vb ) are solved
∂v v=v
b
from the FOC approach for any given vb . The issuance cost parameter is set to β = 0.0036,
which is chosen to match a fractional debt issuance cost of 1% under the optimal maturity
ξ ∗ = 0.2976. The other parameter values are reported in Table 1.
51
0
10 -3
0
-2
-5
-4
-10
-6
0
5
10
0
1
-0.1
0.5
-0.2
0
-0.3
0
5
10
0
5
10
-0.5
-0.4
-1
0
5
10
Figure B.3: The verification of HJB equation. This figure shows the value of each term of
equation (C.11) when β = 0.0036 for the economy without commitment. The other parameter
values are in Table 1 of the main text.
52
References
Abel, A. B., 2016, “Investment with Leverage,” Working Paper, Wharton School of the University of Pennsylvania.
Admati, A. R., P. M. DeMarzo, M. F. Hellwig, and P. Pfleiderer, 2018, “The leverage ratchet
effect,” The Journal of Finance, 73(1), 145–198.
Altınkılıç, O., and R. S. Hansen, 2000, “Are there economies of scale in underwriting fees?
Evidence of rising external financing costs,” Review of Financial Studies, 13(1), 191–218.
Ausubel, L., and R. Deneckere, 1989, “Reputation in bargaining and durable goods monopoly,”
Econometrica, pp. 511–531.
Baker, M., and J. Wurgler, 2002, “Market timing and capital structure,” The Journal of Finance, 57(1), 1–32.
Barclay, M. J., and C. W. Smith, 1995, “The maturity structure of corporate debt,” Journal
of Finance, 50.2, 609–631.
Benzoni, L., L. Garlappi, and R. S. Goldstein, 2018, “Asymmetric Information, Dynamic Debt
Issuance, and the Term Structure of Credit Spreads,” working paper, Working paper. The
Federal Reserve Bank of Chicago.
Blouin, J., J. E. Core, and W. Guay, 2010, “Have the tax benefits of debt been overestimated?,”
Journal of Financial Economics, 98(2), 195–213.
Brunnermeier, M. K., and M. Oehmke, 2013, “The maturity rat race,” The Journal of Finance,
68(2), 483–521.
Brunnermeier, M. K., and M. Yogo, 2009, “A note on liquidity risk management,” American
Economic Review, 99(2), 578–83.
Bulow, J. I., 1982, “Durable-goods monopolists,” Journal of Political Economy, 90(2), 314–332.
Chen, H., Y. Xu, and J. Yang, 2019, “Systematic risk, debt maturity, and the term structure
of credit spreads,” working paper, National Bureau of Economic Research.
Coase, R. H., 1972, “Durability and monopoly,” The Journal of Law and Economics, 15(1),
143–149.
Dangl, T., and J. Zechner, 2016, “Debt maturity and the dynamics of leverage,” CFS Working
Paper.
Dècamps, J.-P., and S. Villeneuve, 2014, “Rethinking Dynamic Capital Structure Models with
Roll-Over Debt,” Mathematical Finance, 24, 66–96.
53
DeMarzo, P., and Z. He, 2019, “Leverage Dynamics without Commitment,” forthcoming, Journal of Finance.
DeMarzo, P., Z. He, and F. Tourre, 2018, “Sovereign Debt Ratchets and Welfare Destruction,” Working paper, Stanford University, University of Chicago, and Copenhagen Business School.
DeMarzo, P. M., 2019, “Presidential Address: Collateral and Commitment,” The Journal of
Finance, 74(4), 1587–1619.
Diamond, D. W., and Z. He, 2014, “A theory of debt maturity: the long and short of debt
overhang,” The Journal of Finance, 69(2), 719–762.
Duffie, D., and D. Lando, 2001, “Term structures of credit spreads with incomplete accounting
information,” Econometrica, 69(3), 633–664.
Fama, E. F., and K. R. French, 2002, “Testing trade-off and pecking order predictions about
dividends and debt,” The Review of Financial Studies, 15(1), 1–33.
Fischer, E. O., R. Heinkel, and J. Zechner, 1989, “Dynamic capital structure choice: Theory
and tests,” The Journal of Finance, 44(1), 19–40.
Frank, M. Z., and V. K. Goyal, 2014, “The profits–leverage puzzle revisited,” Review of Finance,
19(4), 1415–1453.
Gamba, A., and A. Triantis, 2008, “The Value of Financial Flexibility,” The Journal of Finance,
63(5), 2263–2296.
Goldstein, R., N. Ju, and H. Leland, 2001, “An EBIT-Based Model of Dynamic Capital Structure,” The Journal of Business, 74(4), 483–512.
Graham, J. R., and C. R. Harvey, 2001, “The theory and practice of corporate finance: Evidence
from the field,” Journal of Financial Economics, 60(2-3), 187–243.
He, Z., and K. Milbradt, 2014, “Endogenous liquidity and defaultable bonds,” Econometrica,
82(4), 1443–1508.
He, Z., and W. Xiong, 2012a, “Debt financing in asset markets,” American Economic Review,
102(3), 88–94.
, 2012b, “Dynamic Debt Runs,” Review of Financial Studies, 25, 1799–1843.
Hennessy, C. A., and T. M. Whited, 2007, “How Costly Is External Financing? Evidence from
a Structural Estimation,” The Journal of Finance, 62(4), 1705–1745.
Hugonnier, J., S. Malamud, and E. Morellec, 2015, “Credit market frictions and capital structure dynamics,” Journal of Economic Theory, 157, 1130–1158.
54
Kahn, C., 1986, “The durable goods monopolist and consistency with increasing costs,” Econometrica, pp. 275–294.
Kane, A., A. J. Marcus, and R. L. McDonald, 1984, “How big is the tax advantage to debt?,”
The Journal of Finance, 39(3), 841–853.
, 1985, “Debt policy and the rate of return premium to leverage,” Journal of Financial
and Quantitative Analysis, 20(4), 479–499.
Leary, M. T., and M. R. Roberts, 2005, “Do firms rebalance their capital structures?,” The
Journal of Finance, 60(6), 2575–2619.
Leland, H. E., 1994, “Corporate debt value, bond covenants, and optimal capital structure,”
The Journal of Finance, 49(4), 1213–1252.
, 1998, “Agency costs, risk management, and capital structure,” The Journal of Finance,
53(4), 1213–1243.
Leland, H. E., and K. B. Toft, 1996, “Optimal capital structure, endogenous bankruptcy, and
the term structure of credit spreads,” The Journal of Finance, 51(3), 987–1019.
Malenko, A., and A. Tsoy, 2020, “Optimal Time-Consistent Debt Policies,” Working paper,
Boston College.
McAfee, R. P., and T. Wiseman, 2008, “Capacity choice counters the Coase conjecture,” The
Review of Economic Studies, 75(1), 317–332.
Merton, R. C., 1974, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,”
Journal of Finance, 29, 449–470.
Modigliani, F., and M. H. Miller, 1958, “The cost of capital, corporation finance and the theory
of investment,” The American, 1, 1321–1346.
Morellec, E., B. Nikolov, and N. Schürhoff, 2012, “Corporate governance and capital structure
dynamics,” The Journal of Finance, 67.3, 803–848.
Stohs, M. H., and D. C. Mauer, 1996, “The determinants of corporate debt maturity structure,”
Journal of Business, pp. 279–312.
Strebulaev, I. A., 2007, “Do tests of capital structure theory mean what they say?,” The Journal
of Finance, 62(4), 1747–1787.
Titman, S., and S. Tsyplakov, 2007, “A dynamic model of optimal capital structure,” Review
of Finance, 11(3), 401–451.
Titman, S., and R. Wessels, 1988, “The determinants of capital structure choice,” The Journal
of Finance, 43(1), 1–19.
55
Van Binsbergen, J. H., J. R. Graham, and J. Yang, 2010, “The cost of debt,” The Journal of
Finance, 65(6), 2089–2136.
Welch, I., 2004, “Capital structure and stock returns,” Journal of Political Economy, 112(1),
106–131.
Xu, H., 2018, “Scooping Up Own Debt On the Cheap: The Effect Of Corporate Bonds Buyback
on Firm’s Credit Condition,” Working Paper, University of Illinois at Urbana Champaign.
56
Optimal Debt Dynamics, Issuance Costs, and
Commitment
Internet Appendix
Luca Benzoni, Lorenzo Garlappi, Robert S. Goldstein, Julien Hugonnier and
Chao Ying
This Online Appendix contains additional analysis to accompany the manuscript. Section A
investigates the firm’s optimal debt issuance policy in the presence of proportional, rather than
fixed, debt issuance costs. Section B considers (s, S) policies in which the lower boundary is a
debt-repurchase/equity-issuance boundary and compares them to the global-optimal policy of
Section 3.1.1 of the main text. Section C derives an approximate formula for the location of
the indifference boundary shown in Figure 1 that separates the regions where additional debt
issuance is optimal, and where it is not.
A
Optimal Debt Dynamics with Proportional Issuance
Costs
In this section, we investigate the optimal debt issuance policy when the firm is subject to
proportional debt issuance costs. The case with no commitment has been studied in earlier
versions of DeMarzo and He (DH, 2019), but we add it here to the case with commitment in
order to facilitate comparison between the two cases.
A.1
Economy without Commitment
Following DH, we conjecture that the optimal debt issuance policy in a setting with no commitment is locally deterministic. We therefore express the state variable dynamics in the form:
dV
= µV dt + σV dB
(A.1)
dF = (g(v) − ξ) F dt.
(A.2)
Using the definition of inverse leverage vt = (Vt /Ft ), Itô’s lemma yields
dv = (µ + ξ − g(v)) v dt + σv dB.
(A.3)
In the presence of proportional costs, under the assumption that the debt issuance policy
g(v) is always (weakly) positive, the value of equity can be expressed as:
Z
Q
E(Vt , Ft ) = Et
τb
−r(s−t)
ds e
Ys (1 − π) − Fs (c(1 − π) + ξ) + (1 − β)g(vs )D(Vs , Fs ) .
t
(A.4)
1
In equation (A.4), the term proportional to β represents the funds raised from the debt issuance
that is lost to issuance costs. Below, we identify a constraint for the parameter β so that g(v)
is (weakly) positive for all values of v. In the special case β = 0, this framework reduces to the
DH model. Equation (A.4) implies that
−rt
e
t
Z
−rs
ds e
E(Vt , Ft ) +
Ys (1 − π) − Fs (c(1 − π) + ξ) + (1 − β)g(vs )D(Vs , Fs )
0
is a Q-martingale, hence the equity claim satisfies the PDE:
0 = max
g(v)
2
−rE + µV EV + σ2 V 2 EV V + (g(v) − ξ)F EF +
Yt (1 − π) − Ft (c(1 − π) + ξ) + (1 − β)g(vt )D(Vt , Ft ).
.
(A.5)
The first-order condition with respect to g(v) yields:
0 = F EF + (1 − β)D(Vt , Ft ).
(A.6)
Substituting this expression into equation (A.5), we find that the equity claim satisfies:
σ2 2
0 = −rE + µV EV + V EV V − ξF EF + Yt (1 − π) − Ft (c(1 − π) + ξ) .
2
(A.7)
As in the main text, the value of equity is homogeneous of degree one in its arguments.
Thus, we look for a solution of the form:
E(Vt , Ft ) = Ft e(vt = Vt /Ft ) .
(A.8)
Using the relations:
EV
EV V
= ev
1
evv
=
F
EF = e − vev ,
(A.9)
(A.10)
(A.11)
and Yt = (r − µ)Vt , we can rewrite the PDE in equation (A.7) as the ODE
0 =
σ2 2
v evv + (µ + ξ)vev − (r + ξ)e + (1 − π)(r − µ)v − (c(1 − π) + ξ) .
2
2
(A.12)
Moreover, we can write the first order condition in equation (A.6) as
D(V, F )
= −
F
1
1−β
EF =
1
1−β
(vev − e) .
(A.13)
In terms of the free coefficients (Me , Ne ), the solution of the ODE (A.12) is
φ
ω
e(v) = Me v + Ne v + (1 − π)v −
c(1 − π) + ξ
r+ξ
,
(A.14)
where the exponents (φ, ω) are given in equations (22)–(23) of the main text. Imposing the
boundary conditions:
de
= (1 − π)
v→∞ dv
lim
e(vb ) = 0,
(A.15)
and using the restrictions φ > 1 and ω < 0, we find that the scaled equity function is:
e(v) = (1 − π) v − vb
v
vb
ω
−
c(1 − π) + ξ
r+ξ
ω
v
1−
.
vb
(A.16)
Note that the value of equity in this economy is independent of the proportional issuance cost
parameter β, and it is identical to the value of equity in the DH economy.
Smooth pasting or, equivalently, the first-order condition
∂e
∂vb
= 0, identifies the location
v=v
b
of the optimally-chosen default boundary:
vb =
c(1 − π) + ξ
(1 − π)(r + ξ)
ω
ω−1
.
(A.17)
This allows us to rewrite scaled equity value as:
ω
1
vb v
e(v) = (1 − π) v − vb 1 −
−
.
ω
ω vb
(A.18)
To determine the scaled debt value, we look for a solution of the form:
D(Vt , Ft ) = Ft d(vt = Vt /Ft ) .
3
(A.19)
Using the equity value in equation (A.16), we simplify the first order condition (A.13) to
d(v) =
Except for the term
1
1−β
1
1−β
[vev − e] = vb
1−π
1−β
ω
1
v
1−
1−
.
ω
vb
(A.20)
, this formula is identical to the one derived by DH. Thus, in contrast
to the equity claim, which is independent of β, the debt claim is increasing in β. The intuition
for this result is that, as we demonstrate below, higher issuance costs reduces the aggressiveness
at which the firm issues new debt in the future. That is, g(v) is a decreasing function of the
issuance cost parameter β. As emphasized in equation (A.3), smaller values of g(v) increase the
drift in the dv dynamics, in turn reducing the probability that v reaches the default boundary
vb in the future. Lower default probabilities generate higher debt prices.
To determine the optimal policy g(v), we follow DH in deriving the ODE for the debt claim.
The debt price is given by the risk-neutral expectation:
τb
Z
Q
−(r+ξ)(s−t)
ds e
d(vt ) = (c + ξ) Et
.
(A.21)
t
Because e−(r+ξ)t d(vt ) + (c + ξ)
Rt
0
ds e−(r+ξ)s is a Q-martingale, it follows that the scaled debt
function satisfies the ODE:
σ2 2
v dvv + (µ + ξ − g(v))vdv − (r + ξ)d + (c + ξ).
0 =
2
(A.22)
Differentiating equation (A.7) with respect to F , and using the first-order condition −EF =
(1 − β)d, we find:
σ2 2
0 = (1 − β) (r + ξ)d − (µ + ξ)vdv − v dvv − (c(1 − π) + ξ) .
2
(A.23)
Equations (A.22) and (A.23) yield the optimal debt issuance policy:
g(v) =
πc − β(c + ξ)
(1 − β)(vdv )
=
πc − β(c + ξ)
vb (1 − π)(1 − ω)
v
vb
−ω
.
(A.24)
As predicted above, g(v) is decreasing in β, which explains why debt prices are increasing in
β. In the limit of β going to zero, g(v) in equation (A.24) converges to the optimal DH policy.
4
So far we have assumed that the debt issuance policy is (weakly) positive for all values of
v. From the numerator of equation (A.24), this assumption holds when
πc
.
c+ξ
β ≤
(A.25)
This restriction is equivalent to
1
1−β
≤
c+ξ
c(1 − π) + ξ
,
(A.26)
or, using the expression for vb in equation (A.17):
vb
1−π
1−β
1
c+ξ
1−
≤
.
ω
r+ξ
(A.27)
Using inequality (A.27) in the expression for debt value d(v) in equation (A.20), we conclude
that the bond price remains below the risk-free bond price, as required to preclude arbitrage
opportunities:
d(v) ≤
A.2
c+ξ
r+ξ
ω
v
c+ξ
1−
<
.
vb
r+ξ
(A.28)
Economy with Commitment
In the case of proportional issuance costs, the debt issuance size is infinitesimal. To capture
this property, we conjecture that the optimal policy with commitment is characterized by an
inaction region described by the four parameters (vb , vu , γ, ξ) and study its limit γ → 1+ .
As in the main text, we define τb and τu to be the first times vt reaches vb and vu , respectively,
and τ = min(τb , τu ). For dates t ∈ (0, τ ), there is no debt issuance. For a given debt issuance
policy, the PDE and the boundary conditions for the debt claim are identical to those in the
main text. From equation (26), the debt value is
d(v) = Md v φ + Nd v ω +
5
c+ξ
,
r+ξ
(A.29)
with boundary conditions given by equations (27)–(28),
d(vb ) = (1 − α)(1 − π)vb
(A.30)
d(vu ) = d(vu /γ).
(A.31)
The value of equity can be determined via the risk-neutral expectation:
Z τ
Q
−r(s−t)
(1 − π)Ys − (c(1 − π) + ξ) Fs
E(Vt , Ft ) = Et
ds e
(A.32)
t
+Et e−r(τu −t) 1(τu <τ
E(vu Fτu , γFτu ) + (1 − β) D(vu Fτu , γFτu ) − D(vu Fτu , Fτu )
.
Q
b
)
The first term captures the present value of the claim to cash flows (i.e., dividends) throughout
period-0, which ends at date τ = min(τb , τu ). The second term captures the fact that when the
upper boundary is reached, shareholders still hold the entire equity claim, but now with the
face value of debt scaled by a factor of γ. In addition, shareholders receive the value of the new
debt issuance (net of issuance costs) as dividend.
Rt
−rs
−rt
(1 − π)Ys − (c(1 − π) + ξ) Fs is a Q-martingale, the
Because e E(Vt , Ft ) + 0 ds e
equity claim satisfies the PDE:
0 = −rE + µV EV +
σ2 2
V EV V − ξF EF + V (r − µ)(1 − π) − F (c(1 − π) + ξ) ,
2
(A.33)
with boundary conditions:
E(vb F, F ) = 0
E(vu F, F ) = E(vu F, γF ) + (1 − β) (D(vu F, γF ) − D(vu F, F )) .
(A.34)
We look for a solution of the form E(V, F ) = F e v = VF Using the relations in equations (A.9)–
(A.11) we can rewrite the PDE as an ODE:
0 =
σ2 2
v evv + vev (µ + ξ) − e(r + ξ) + v(r − µ)(1 − π) − (c(1 − π) + ξ) ,
2
(A.35)
subject to the boundary conditions
e(vb ) = 0
(A.36)
e(vu ) = γe(vu /γ) + (1 − β)(γ − 1)d(vu ),
(A.37)
6
where we have used the relation d(vu ) = d(vu /γ) in the last equation.
The solution is:
e(v) = Me v φ + Ne v ω + (1 − π)v −
(c(1 − π) + ξ)
,
r+ξ
(A.38)
where the constants (Me , Ne ) are uniquely determined by the boundary conditions in equations (A.36)–(A.37).
Under the conjecture that the size of the optimal debt issuance is infinitesimal, we set
γ = (1 + ) and investigate the case → 0+ . Therefore, the boundary conditions for debt and
equity at the upper boundary simplify to:
dv (vu ) = 0
(A.39)
vu ev (vu ) = e(vu ) + (1 − β)d(vu ).
(A.40)
The optimal debt issuance policy is determined as follows: for a given ξ, optimal boundaries
are determined by the first-order conditions1
∂e
∂vb
∂e
∂vu
= 0
v=v
b
= 0.
(A.41)
v=vu
We then identify the value of ξ that maximizes equity value e(v) at time of issuance, where
e(v) satisfies the ODE (A.35) with boundary conditions (A.36) and (A.40). For any value of
∗
∗
∗
the proportional cost parameter β, we identify the optimal debt policy vb (β), vu (β), ξ (β) .
A.3
Results for the Cases with and without Commitment
In this section, we investigate model predictions for the cases with and without commitment
under proportional costs. In the analysis that follows we set the coupon rate c so that debt is
1
As in the fixed cost case, we can show that the solution with commitment corresponds to a global-optimal
∂e
policy in that the solution to 0 = ∂v
generates the same value for vu∗ independent of choice of vany .
u
v=vany ;v
b
7
priced at par at the time of issuance. The other parameter values are in Table 1 of the main
text.
A.3.1
Debt and Equity Values
Figures A.1 and A.2 show the debt and equity values with commitment for β = 0.01, a value
consistent with empirical estimates of debt issuance costs. In both figures we fix ξ ∗ = 0.2505
to maximize the equity value at the time of issuance. Finally, we set c = 0.0406 to guarantee
that debt is issued at par. For comparison, we also plot the value of debt and equity for the
no-commitment case. Since, without commitment, ξ is undetermined, in this case we use the
optimal value ξ ∗ of the commitment policy and its associated coupon c.
With commitment, the value of debt in Figure A.1 is higher than that without commitment
for any inverse leverage v. The difference quantifies the value of committing to a less aggressive
debt issuance policy that lowers future default probability and increases debt value. In the
E
F
,
same figure, we also plot the marginal value of equity (adjusted for issuance cost), − 1−β
which, by equation (A.13) is given by
vev (v)−e(v)
.
1−β
We compute the functions e(v) and ev (v)
using the optimal policy with commitment. For all v values the debt value with commitment
(the continuous blue line) is higher than the marginal value of equity (the dashed-blue line),
except for v = vu where −EF (V, F ) = (1−β)d(vu ). In contrast, consistent with DH, the value of
debt without commitment (continuous red line) satisfies the condition −EF (V, F ) = (1−β)d(v)
for all values of v.
In Figure A.2, the value of equity with commitment exceeds the no commitment equity
value for all v. Similar to the case of fixed issuance costs, shareholders would have no incentive
to deviate from the commitment policy. Hence the no-commitment issuance policy can serve as
a credible punishment to sustain the commitment equilibrium. Indeed, any deviation from the
optimal policy would lead to future debt issues being priced according to the no-commitment
equilibrium. Shareholders would gain zero cash benefit at the date of the deviation, and be left
with an equity claim that has lower valuation than under the optimal policy.
8
A.3.2
Optimal Debt Policies as a Function of β
In Figures A.3–A.6, we report the optimal policy vb∗ (β), vu∗ (β), ξ ∗ (β) and the coupon rate c
as a function of β.
For the case with commitment, the optimal average maturity in Figure A.3 is an increasing
function of the proportional cost parameter β. This results from the trade off between the
tax benefits net of bankruptcy costs (which favor short-maturity debt) and debt issuance costs
(which favor long-maturity debt). The effect of net tax benefits dominates for lower values of β,
leading to shorter optimal maturities. For the case without commitment, regardless of maturity,
the gain from additional tax shields is offset by increased default costs. Hence, consistent with
the findings of DH, shareholders are indifferent toward the debt maturity structure. This
implies that there is no optimal maturity in the economy with proportional cost under the
no-commitment policy.
For the case with commitment, Figure A.4 shows that the upper boundary vu∗ (β) increases
monotonically with the issuance cost β. This is because, when issuance costs are high, it is
optimal to reduce the present value of these costs by issuing less frequently. To this end, the
firm increases the location of vu so that the inverse leverage vt reaches the upper boundary vu
less often. For the case with no commitment, the upper boundary is infinite for all values of β
and therefore it does not appear in the figure.
In Figure A.5, we report the lower boundary vb∗ (β) for both cases with and without commitment. As in Section A.3.1, in both cases, we use the optimal maturity 1/ξ ∗ and coupon
c obtained under commitment. The default boundary with commitment is always lower than
that without commitment. This is because commitment increases cash flows to shareholders,
making the option to retain ownership more valuable. Interestingly, the functional dependence
of the default boundary on β is rather different for the two cases. With commitment, vb∗ (β)
is a hump-shaped function of β, similar to what we find in the main text for the case of fixed
issuance costs. Two channels affect the shape of the default boundary vb∗ (β). The first (direct)
channel is due to issuance costs. A higher β implies lower equity valuations ceteris paribus,
making the option to remain solvent less valuable. Hence, this channel alone would generate a
monotonically increasing relation for vb∗ (β). However, there is a second (indirect) channel due to
9
the optimal maturity choice 1/ξ ∗ (β), which is increasing in β. In those states of nature in which
a firm is performing poorly, a longer maturity implies less cash flow that shareholders need to
raise to service debt in place. Hence, the maturity channel leads to higher equity valuations
in bad states of nature, leading to a lower optimal default boundary. Figure A.5 shows that,
with commitment, this indirect channel dominates the direct channel for values of β greater
than β > 10−2.4 . In contrast, without commitment vb∗ (β) is a decreasing function of β. This is
because for the case with no commitment, the equity value is independent of β. Hence the first
(direct) channel has no impact and the second (indirect) channel dominates.2
Finally, Figure A.6 shows the coupon rate c as a function of β, chosen so that the bond is
priced at par at the time of issuance. Similar to the case of fixed costs, a lower β is associated
with a shorter optimal maturity 1/ξ ∗ , which makes the bond less risky. In the limit β → 0, the
par coupon rate converges to the riskfree rate r = 4%.
A.3.3
Tax Benefit of Debt
We define the tax benefits to debt as the ratio of the levered enterprise value at the debt
issuance boundary and the value of an all-equity firm, that is:
TB =
e(vu ) + d(vu )
− 1.
(1 − π)vu
(A.42)
In Figure A.7, we show the tax benefits from equation (A.42) as a function of the proportional
cost coefficient β. The case with commitment always generates strictly positive tax benefits to
debt, while, as previously mentioned, there are no tax benefits to debt under no commitment.
Consistent with intuition, the tax benefit of debt under commitment increases as the proportional costs decrease. We also plot the level of tax benefits under the restriction ξ = 0 for the
case with commitment. We get the similar conclusion as the fixed costs that the tax benefits
to debt are amplified considerably when firms have the right to choose optimal debt maturity.
Indeed, for a fixed ξ, the default boundary vb∗ (β) in equation (A.17) is independent of β under the optimal
no-commitment policy.
2
10
B
Equity Valuation with Debt Repurchase
In Section 2 of the main text, we derived a global-optimal (s, S) policy in which the lower boundary is a default boundary, and compared its properties to those of optimal no-commitment policies that also involve a default boundary. In this section, instead, we investigate (s, S) policies
in which the lower boundary is a debt-repurchase/equity-issuance boundary and compare their
properties to the global-optimal policy of Section 2. Specifically, we consider the possibility
that, in each period, there is a lower boundary v` at which the firm will scale down the level of
outstanding debt by a factor ψ ≤ 1. The implication is that debt becomes risk-free, as inverse
leverage always stays within the region vt ∈ (v` , vu ) for which equity values remain positive.
The value of outstanding debt with coupon c and amortization rate ξ is therefore
Z ∞
c+ξ
−r(s−t)
Q
ds e
(c + ξ)Fs =
D(Vt , Ft ) = Et
Ft .
r+ξ
t
(B.1)
In analogy with Section 2, we define τu as the time when the upper boundary is reached, τ` as
the time when the lower boundary is reached, and τ = min(τ` , τu ).
Up to this point, we have assumed full tax loss offset in that, when EBIT falls below
interest payments (Yt < cFt ), our specification for the dividend in equation (A.9) of the main
text implies that government pays π(cFt − Yt ) to the firm.3 In reality, firms do not receive cash
infusions from government, but rather only a tax loss carryforward, whose present value may be
significantly less than what this specification implies. Therefore, in this section, we investigate
the situation in which the cash flow (i.e., dividends) to equity is specified as:
Yt − π(Yt − cFt ) − (c + ξ)Ft Yt > cFt
Div(Yt , Ft ) =
Yt − (c + ξ)Ft
Yt < cFt .
(B.2)
With a slight abuse of notation, we rewrite this formula in terms of our state vector (Vt , Ft ),
where Vt = (r − µ)Yt by equation (2) of the main text:
(1 − π)(r − µ)Vt − (c(1 − π) + ξ) Ft
Div(Vt , Ft ) =
(r − µ)V − (c + ξ) F
t
t
3
Vt >
cFt
r−µ
Vt <
cFt
.
r−µ
(B.3)
Note that, for the model investigated in the main text, EBIT never falls below interest payments, so the
tax-loss offset constraint imposed here would not bind anyway.
11
As this specification implies no tax-loss carry-forward, the actual situation falls between this
specification and the one that allows full tax loss offset. However, for the case we study below,
we argue that this specification is the more relevant case. This is because, in what follows, the
optimal policy will be to choose the location of the upper boundary such that the firm’s EBIT
always falls below coupon payments made to debtholders. As such, the company never pays
taxes. Under this scenario, it is more realistic to assume no tax-loss carry-forward rather than
a tax rebate from the government.
The value of equity can be determined as the risk-neutral expectation:
Z τ
Q
−r(s−t)
E(Vt , Ft ) = Et
ds e
Div(Ys , Fs )
t
Q
+Et
e
Q
+Et
−r(τu −t)
1(τu <τ
e−r(τ` −t) 1(τ
`
`
)
<τu )
c+ξ
E(vu Fτu , γFτu ) + (γ − 1)
Fτu − βvu Fτu
r+ξ
c+ξ
Fτ` − βE v` Fτ`
.
E(v` Fτ` , ψFτ` ) − (1 − ψ)
r+ξ
(B.4)
The first term captures the present value of the claim to dividends from the current date t
until date τ = min(τ` , τu ). The second term captures the fact that when the upper boundary
is reached, shareholders still hold the entire equity claim, but now with the face value of debt
scaled by a factor γ ≥ 1. In addition, shareholders receive as dividend the value of the new
debt issuance (net of issuance costs), which, because debt is risk-free, has price per unit face
c+ξ
. The third term captures the fact that when the lower boundary is reached,
value of r+ξ
shareholders still hold the entire equity claim, but now with the face value of debt scaled by a
factor ψ ≤ 1. In addition, shareholders pay for the repurchase of the (risk-free) debt, and an
equity issuance cost controlled by the parameter βE .
Rt
−rt
−rs
Equation (B.4) implies that e E(Vt , Ft ) + 0 ds e Div(Vs , Fs ) is a Q-martingale, therefore, by Itô’s Lemma, we have:
0 = −rE + µV EV +
σ2 2
V EV V − ξF EF + Div(V, F ).
2
(B.5)
As in Section 2, because both the cash flows to equity and state vector dynamics are linear in
the state vector, the value of equity is homogeneous of degree one in the state vector. Thus,
12
we look for a solution of the form:
V
E(V, F ) = F e v =
.
F
(B.6)
This allows us to reexpress the PDE in equation (B.5) as the coupled ODEs:
0 =
σ2 2
v evv
2
+ vev (µ + ξ) − e(r + ξ) + v(r − µ) − (c + ξ) ,
0 =
σ2 2
v evv
2
+ vev (µ + ξ) − e(r + ξ) + v(r − µ)(1 − π) − (c(1 − π) + ξ) ,
c
,
r−µ
c
v>
,
r−µ
v<
(B.7)
subject to the boundary conditions
c+ξ
e(v` ) = ψe(v` /ψ) − (1 − ψ)
− βE v`
r+ξ
c+ξ
e(vu ) = γe(vu /γ) + (γ − 1)
− βvu .
r+ξ
(B.8)
(B.9)
Under this specification, the value of equity is given by:
(
e(v) =
Me+ v φ + Ne+ v ω + (1 − π)v −
c+ξ
Me− v φ + Ne− v ω + v − r+ξ
(c(1−π)+ξ)
r+ξ
v>
v<
c
r−µ
c
,
r−µ
(B.10)
where the constants (Me+ , Me− , Ne+ , Ne− ) are uniquely determined from the boundary conditions
in equations (B.8)–(B.9), and the value- and slope-matching conditions4
limc e (v) =
v↑
r−µ
limc ev (v) =
v↑
r−µ
limc e (v)
v↓
limc ev (v) .
v↓
(B.11)
r−µ
(B.12)
r−µ
We investigate this model under two different parameterizations: with and without issuance
costs.
c
c
These value- and slope-matching conditions assume that v` < r−µ
and vu > r−µ
. If instead v` and vu are
c
both either above or both below r−µ , then these additional conditions are not required.
4
13
B.1
Debt Repurchase in the Absence of Issuance Costs
For the case β = 0, βE = 0, the value of equity possesses a simple analytic solution. Specifically,
because (i) the claim to bankruptcy costs is zero when debt is risk-free, and (ii) the government’s
claim to taxes can be reduced to zero if the debt issuance threshold is chosen such that vu ≤
c
,
r−µ
that is, coupon payments always exceed EBIT, then the value of equity is equal to the value of
the EBIT claim minus the value of debt. Hence, under the assumption vu ≤
c
,
r−µ
which implies
that the firm never pays any taxes because coupon payments always exceed EBIT, we have
erepurchase (v) = v −
c+ξ
r+ξ
.
(B.13)
Now, for this equation to be compatible with limited liability, we must restrict the lower boundary v` such that:
0 ≤ erepurchase (v` ) = v` −
c+ξ
r+ξ
,
(B.14)
which implies:5
v` ≥
c+ξ
r+ξ
.
(B.15)
We then compare the value of the equity claim under the debt repurchase policy to the equity value under the global optimal policy for all v ∈ (v` , vu ). As the firm is free to choose
v` arbitrarily close to vu , ultimately, this is equivalent to comparing value of equity in the
c
bond-repurchase case with its value under the global-optimal policy at vu = r−µ
. That is,
shareholders are better off by following a debt repurchase policy if
erepurchase
c
r−µ
=
c
r−µ
−
c+ξ
r+ξ
> eglobal
c
r−µ
.
(B.16)
We find that this condition holds for a range of parameters. As one example, Figure A.8 shows
c+ξ
c
that this condition holds for the parametrization (ψ = 1, γ = 1, ξ = 0.2, v` = 12 r+ξ
+ r−µ
, vu =
5
More generally, we impose the following conditions: (i)
v
v` ≤ γu ≤ vu
14
c+ξ
r+ξ
≤ v` ≤ vu ≤
c
r−µ ;
(ii) v` ≤
v`
ψ
≤ vu ; and (iii)
c
).
r−µ
Any value of v` that satisfies erepurchase (v` ) > eglobal (v` ) guarantees that shareholders will
be better off under the repurchase policy v` . The next section, however, shows that this result
is not robust to the presence of realistic security issuance costs.
B.2
Debt Repurchase in the Presence of Issuance Costs
In the previous section, EBIT falls below the promised coupon payments. Hence, over each
interval dt the firm must issue equity of order O(dt) in order to avoid default. Here we relax
the counterfactual assumption that the firm can issue equity at no cost. For tractability, we
impose that the firm incurs equity issuance costs only when issuing a finite amount of equity
at v` , but not on issuances of infinitesimal size. Accounting for these additional equity issuance
costs would serve only to reduce equity valuation even further, thus making equity value even
lower under the debt-repurchase policy in the presence of realistic issuance costs.
Consistent with the empirical literature (e.g., Altınkılıç and Hansen (2000), Hennessy and
Whited (2007), Titman and Tsyplakov (2007), and Gamba and Triantis (2008)), we calibrate
the model so that equity issuance costs are approximately 7% and debt issuance costs are
1% of the amount raised. Calibrated to these issuance costs, we find that the model with debt
repurchase generates values of equity at the lower boundary v` that are lower than equity values
under the global optimal solution for all possible (s, S) debt issuance and repurchase policies
(v` , vurep , γ, ψ). It follows that the value of equity under the debt-repurchase policy will always
be lower than under the global-optimal policy without debt repurchase. Figure A.9 provides
an example. This result implies that, in the presence of realistic equity issuance costs, it is not
in shareholders’ best interest to repurchase debt at the v` boundary. Hence, debtholders would
not be willing to purchase the firm’s debt at risk-free prices.
This analysis shows that the presence of equity issuance costs provides one explanation for
why firms are unable to issue risk-free debt. Many other mechanisms exist. For example, if the
manager is more informed about the firm’s asset value than creditors (e.g., Duffie and Lando
(2001)), and the true asset value is sufficiently low, then a manager acting in the best interest
of shareholders could not credibly commit to repurchasing debt according to a policy based on
15
public information. As a second example, the possibility that firm value could jump well below
the promised debt repurchase boundary may lead shareholders to renege on their promise.
C
Derivation of the Approximate Debt Issuance Indifference Curve
We identify an approximate formula for the location of the indifference boundary shown in
Figure 1 that separates the regions where additional debt issuance is optimal, and where it is
not. Our approach is motivated by Figure 2, which shows how (vb , vu , γ) change as β → β ∗ ,
where β ∗ is a specific point on the indifference boundary for a given value of ξ. Specifically, we
find that vu → ∞, whereas vb approaches the case with no debt issuance, and (vu /γ) is only
slightly larger than vb (and hence, γ is only slightly less than (vu /vb )). The economic intuition
for these findings is as follows: as we increase β, the optimal policy reduces issuance costs by
making both vu and γ larger, as increasing each reduces the number of times the debt issuance
boundary is reached. The threshold vb converges to the default boundary for the case of no
debt issuance, vb, no issuance , because, as β → β ∗ , vu → ∞ and therefore the firm does not issue
any further debt.
With these insights, we identify an approximate formula for the location of the indifference
curve. Assume the firm is at a debt-issuance boundary (V = vu F, F ), and is about to issue
debt with face value ∆F . Thus, the state vector moves from (vu F, F ) to (vu F, F + ∆F ),
vu
implying that inverse leverage jumps from vu to 1+(∆F/F
. Thus, we have identified the
)
relation γ = (1 + (∆F/F )), or equivalently, ∆F = F (γ − 1)
Now, if the debt issued is approximately risk-free, then the present value of bankruptcy
costs is small, and therefore the net tax benefit of the new debt approximately equals its tax
savings, which is a product of the tax rate π and the coupon payment paid at a date-t after
the debt issuance date c ∆F e−ξt :
Z
tax benefit =
∞
−rt
dt e
−ξt
πc ∆F e
= ∆F
0
cπ
r+ξ
≈ F (γ − 1)
rπ
r+ξ
,
(C.1)
where this last line holds because c ≈ r when debt is approximately risk-free, and coupon is
16
chosen so that debt is issued at par. Moreover, for all values of β that we examine, we have
the condition ξ r. Hence, we approximate further that
rπ
tax benefit ≈ F (γ − 1)
.
ξ
(C.2)
As noted above, γ becomes large as we approach the critical value β ∗ (ξ). Therefore, we approximate (γ − 1) ≈ γ. Further, as we see in Figure 2,
γ ≈ vvu . Hence, we approximate tax benefits as:
vu
γ
≈ vb . Therefore, we approximate
b
tax benefit ≈ F
vu
vb
rπ
ξ
,
(C.3)
Now, the cost of this debt issuance is:
issuance cost = βvu F.
(C.4)
Comparing equations (C.3) and (C.4), after dividing both sides by vu F , we see that net tax
benefits approximately equal issuance costs when:
rπ
= β.
vb ξ
(C.5)
To get an estimate of vb , we determine the location of the optimal default boundary under
the assumption of no future debt issuance. Such a model is effectively that of Leland (1994)
generalized to finite maturity (1/ξ). As in Benzoni, Garlappi, and Goldstein (2018), we find:
ω
1
c (1 − π) + ξ
vb,no issuance =
.
(C.6)
ω−1
1−π
r+ξ
Again, using the approximation ξ r and noting that ω is large in magnitude for large ξ, the
optimal location of the lower boundary simplifies to
1
vb ≈
.
1−π
(C.7)
Substituting the last expression into equation (C.5), we find an approximate solution for the
location of the boundary at which the firm is indifferent to issuing additional debt or not:
βξ ≈ rπ(1 − π).
17
(C.8)
1.2
1
0.8
0.6
0.4
0.2
0
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Figure A.1: Debt valuation with proportional costs: commitment vs. no commitment. This figure shows the value of the debt claim d(v) for the cases with and without
commitment and v ∈ (vb∗ , vu∗ ), with the proportional cost parameter β = 0.01 under the optimal
inverse maturity ξ ∗ = 0.2505 and coupon rate c = 0.0406 to guarantee that debt is issued at
v (v)
par. The curve ve(v)−e
is derived using the value of equity under commitment. The figure
1−β
shows only a truncated version of the no-commitment solution because, without commitment,
the optimal policy has no upper boundary vu . The other parameter values are in Table 1 of
the main text.
1.2
1
0.8
0.6
0.4
0.2
0
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Figure A.2: Equity valuation with proportional costs: commitment vs. no commitment. This figure shows the value of the equity claim e(v) for the cases with and without
commitment and v ∈ (vb∗ , vu∗ ), with the proportional cost parameter β = 0.01 under the optimal
inverse maturity ξ ∗ = 0.2505 and coupon rate c = 0.0406 to guarantee that debt is issued at
par. The figure shows only a truncated version of the no-commitment solution because, without
commitment, the optimal policy has no upper boundary vu . The other parameter values are in
Table 1 of the main text.
18
4
3.5
3
2.5
2
1.5
1
0.5
0
10 -6
10 -5
10 -4
10 -3
10 -2
Figure A.3: Optimal maturity. This figure shows the optimal debt maturity 1/ξ with commitment as a function of the proportional cost parameter β. The other parameter values are
in Table 1 of the main text.
2.4
2.2
2
1.8
1.6
1.4
1.2
1
10 -6
10 -5
10 -4
10 -3
10 -2
Figure A.4: Optimal debt issuance boundary. This figure shows the location of the optimal
debt issuance boundary vu with commitment as a function of the proportional cost parameter
β. The other parameter values are in Table 1 of the main text.
19
1.25
1.2
1.15
1.1
1.05
1
10 -6
10 -5
10 -4
10 -3
10 -2
Figure A.5: Optimal default boundary. This figure shows the location of the optimal default
boundary vb for the cases with and without commitment as a function of the proportional cost
parameter β. The other parameter values are in Table 1 of the main text.
0.0406
0.0405
0.0404
0.0403
0.0402
0.0401
0.04
10 -6
10 -5
10 -4
10 -3
10 -2
Figure A.6: Coupon rate. This figure shows the coupon rate c such that the bond is priced
at par at the time of issuance. The other parameter values are in Table 1 of the main text.
20
0.25
0.2
0.15
0.1
0.05
0
10 -6
10 -5
10 -4
10 -3
10 -2
Figure A.7: Tax benefits to debt. This figure shows the tax benefits to debt under the optimal
policy with commitment as a function of the proportional cost parameter β. It also reports the
tax benefits to debt for the case with commitment when the inverse maturity parameter is set
to ξ = 0. The other parameter values are in Table 1 of the main text.
21
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
1.015
1.025
1.035
Figure A.8: Debt repurchase and zero issuance cost. This figure shows the value of equity
under both the risk-free and defaultable debt cases, where both equity and debt issuance costs
are zero.
For the risk-free debt, we use the following parameters: ψ = 1, γ = 1, ξ = 0.2,
c
c
1 c+ξ
, vurep = r−µ
. We choose the coupon rate c = 0.0407 so that the bond is
v` = 2 r+ξ + r−µ
priced at par with defaultable debt. We only plot part of the equity value for defaultable debt,
which has the upper boundary vu = 2.4358. The other parameter values are in Table 1 of the
main text.
22
4
3.5
3
2.5
2
1.5
1
0.5
0
2
4
6
Figure A.9: Debt repurchase and positive issuance cost. This figure shows the value of
equity under both the risk-free and defaultable debt cases, where both equity and debt issuance
costs are positive. The issuance cost parameter is set to β = 0.0036, which is chosen to match a
fractional debt issuance cost of 1% under the optimal maturity ξ = 0.2976. The equity issuance
cost is set to βE = 0.001, which is chosen to match a fractional equity issuance cost of 7%. For
the risk-free debt, we choose the following parameters: ψ = 0.9797, γ = 2.2437, v` = 1.5292,
vurep = 3.4325. The other parameter values are in Table 1 of the main text.
23
@FedFRASER