Capital Markets Assignment

Read the following paper:

Optimal Debt Mix and Priority Structure: The Role of Bargaining Power* by Dirk Hackbarth, Christopher A. Hennessy, & Hayne E. Leland. April 25, 2002http://www.haas.berkeley.edu/groups/finance/Mix7C

Enter the discussion by posting your answers to the following questions (1-2 page essay):

Why is it optimal for small firms to avoid public debt markets?
Why do firms shift from bank debt into a mixture of bond market and bank debt over their life-cycle?

Conduct additional research necessary to support your discussion statements. Cite all sources of information you use. All written assignments and responses should follow APA rules for attributing sources.

Optimal Debt Mix and Priority Structure:
The Role of Bargaining Power∗

DirkHackbarth ChristopherA.Hennessy HayneE.Leland

April

ABSTRACT
This paper examines the optimal priority structure and mix of bank and bond market debt
within a continuous-time asset pricing framework. Closed-form expressions are derived for the
values of renegotiable bank debt, non-renegotiable bond market debt, equity, and levered firm
values. We show that the optimal debt structure hinges upon the ex post division of bargaining
power between the firm and bank. The optimal debt structure for firms that are weak vis-à-vi

the bank entails financing exclusively with bank debt. Strong firms find it optimal to issue

mix of bank and bond market debt, with the bank senior in the priority structure. The model
explains: (i) why it is optimal for small firms to avoid public debt markets, (ii) why large firms
employ mixed debt financing (iii) why banks are senior in the priority structure, and (iv) why
firms shift from bank debt into a mixture of bond market and bank debt over their life-cycle.
These predictions are generated within a tax shield-bankruptcy cost tradeoff model in which
the only unique feature of banks is their ability to renegotiate.

JEL Codes: G

, G

Keywords: Banking, Capital Structure, Priority Structure, and Contingent Claims Pricing.

∗PRELIMINARY AND INCOMPLETE. The authors are from the Haas School of Business, U.C. Berkeley.

I. Introduction

In the years following the publication of the Modigliani and Miller (MM, 1

) irrelevancy result,

debt tax shields and the costs of default or bankruptcy have been cited as important determinants

of a firm’s optimal capital structure. A common shortcoming of this literature is that it typically

ignores lender heterogeneity. However, debt contracts are shaped in critical ways by the type of

lender and the division of bargaining power between borrower and lender in the event ofnonpayment

of initially-promised amounts (a “renegotiation”). For instance, bank debt is typically soft in the

sense that its terms can be renegotiated without necessarily incurring formal default or bankruptcy,

while bond market debt is typically hard, in that renegotiation with dispersed creditors is costly o

impossible and may well lead to formaldefault and/or bankruptcy.1 Costs, both directand indirect,

are generally incurred in bankruptcy.2 In addition, the outcome of the debt renegotiation process

itself is influenced by the division of bargaining power between lenders and the firm whenever the

firm chooses not to make the promised debt service. Renegotiation between the firm and lenders

is also affected by the priority structure of debt, which affects the threat points of heterogeneous

lenders.

This paper examines the effect of debt heterogeneity and priority structure on the value of the

corporate tax shield, bankruptcy costs,and renegotiation costs within the context of a continuous-

time asset pricing framework. We solve for the optimal priority structure and mix of renegotiable

bank debt and non-renegotiable bond market debt. It is shown that the optimal debt struc-

ture depends critically upon the ex post division of bargaining power between the firm and bank.

Throughout the paper, the term ex ante refers to times prior to the issuance of debt contracts,

while ex post refers to times when the firm’s debt structure is in place. By optimal financial

structure, we mean the debt mix and priority structure that maximizes the ex ante value of all

marketable claims on the

firm.

We consider two polar cases regarding the distribution of bargaining power: when the firm is

strong, in which case the firm can offer reduced debt service that just avoids the bank’s demand
1Bondholders acting atomistically may hold out and preclude desirable debt restructuring; cf. Gertner and Scharf-

stein (

91), Brown, James, and Mooradian (1993), Kahan and Tuckman (1993), and Morris and Shin (2001).
Further, informational asymmetries can make Chapter

more attractive for individual bondholders than private
debt reorganizations; cf. Giammarino (1991).

2See Warner (19

7), Altman (198

), Summers and Cutler (1988), Weiss (1990), Franks and Torous (1989, 1994),
and Andrade and Kaplan (1998).

for liquidation; or when the firm is weak, in which case the bank can demand payments which just

avoid the firm’s refusing to pay and voluntarily declaring bankruptcy. In particular, we address

four related questions: (i) If the choice between bond market and bank debt is mutually exclusive,

which should be chosen? (ii) If the firm may issue both bond market and bank debt, what is the

optimal mix of the two? (iii) What is the optimal priority structure when multiple debt classes

are issued? (iv) How does the ex post division of bargaining power between the firm and lending

bank(s) affect the answer to each of these questions?

These questions are addressed using a unified continuous-time asset pricing framework that

bridges the gap between the Leland (1994) model, which treats non-renegotiable debt, and the

model of Mella-Barral and Perraudin (MBP, 1997) which features renegotiable debt. Analytical

solutions are derived for bank debt, bond market debt, equity, and levered firm values as func-

tions of the endogenously determined variables (promised coupon payments and priority structure)

and exogenous parameters (ex post bargaining power, underlying volatility, corporate tax rates,

negotiation costs, and bankruptcy costs).

The model generates several predictions that are supported by empirical observation. First,

weak firms, those ceding all bargaining power to banks in the event of renegotiation, find it optimal

to finance exclusively with bank debt. That is, for weak firms, bank debt dominates any mix

of bond market and bank debt, regardless of the priority structure under the mixed debt policy.

This result holds in the absence of any notion of monitoring or certification by banks, transaction

costs, economies of scale, and other common rationales for why small firms fail to tap public debt

markets. Second, for strong firms, those with full bargaining power in renegotiations with banks,

the optimal financial structure entails a mixture of bank and bond market debt. In particular,

the analysis indicates that there is a complementarity effect between bank debt and bond market

debt for strong firms. Thus, the model provides a rationale for the coexistence of bank debt

(indirect or intermediated finance) and bond market debt (direct finance). Third, for reasonable

parameter values, it is optimal for strong firms that optimally choose mixed debt to place bank debt

senior. Intuitively, placing the bank senior in the priority structure raises the bank’s reservation

value, which thereby commits equity to higher debt service in renegotiations. The increase in debt
3Ex post bargaining power in our model is not ex ante contractible and therefore exogenous. Note that when

renegotiations commence, both sides will want maximal bargaining power. Thus, a non-contracted “agreement” to
be weak is incentive incompatible ex post.

service raises the value of both the tax shield and bankruptcy costs. We find that the former effect

dominates for reasonable parameter values.

To the extent that one views young firms as having a weak ex post bargaining position vis-à-vis

banks, and gaining bargaining strength as they mature, the model generates a life-cycle hypothesis

for debt structure. Young firms begin by relying exclusively on bank debt. As they grow and gain

bargaining power, they shift away from bank debt, placing more reliance on bond market debt.

This prediction is consistent with observed financing patterns, and is not dependent on the notion

of bank certification. In fact, the model predicts that even if a weak firm could tap public debt

markets with fair pricing, it would be sub-optimal to do so.

Finally, when the choice between bank and bond market debt is mutually exclusive, we find

that the optimal debt structure of strong firms depends on their level of bankruptcy costs and tax

shield values. Interestingly, strong firms are ceteris paribus more likely to choose bond market

over bank debt when bankruptcy costs are high. Taken together, these results suggest the more

general conclusion that bank debt is relatively more desirable to firms ex ante when banks have a

stronger bargaining position in renegotiations. For instance, pure bank debt financing is strictly

preferred to any mixture of bond market and bank debt when banks are strong and firms are weak.

When firms are strong and optimally seek a mixture of bank and bond market financing, bank debt

seniority–which improves the threat point of the bank–is optimal ex ante for the firm. When

firms are strong and financing is mutually exclusive, bank debt will be preferred when bankruptcy

costs are low, since low bankruptcy costs raise the threat point of the bank in renegotiations.

The remainder of the paper is organized as follows. Section II contains a review of the related

empirical and theoretical literature. Section III presents the model. The contingent claims on the

levered firm are valued in Section IV. Section V analyzes the case of mutually exclusive financing

choices and Section VI contains the analysis of the mixed debt firm. Section VII concludes.

II. Literature Review

A number of papers have incorporated strategic interactions between debt and equity as well as

debt renegotiation in the context of contingent claims valuation models. Anderson and Sundaresan

(199

) examine strategic debt service in a Cox-Ross-Rubinstein binomial-tree setting, while MBP

(1997) studies a continuous-time model of debt renegotiation. More recently, Fan and Sundaresan

(2000) characterize two Nash bargaining formulations in which equity and debt bargain over the

value of the assets of the firm (debt-equity swap) or over the value of the firm (strategic debt

service).

Our model may be viewed as bridging the gap between Leland (1994) and MBP (1997). Leland

(1994) represents a unified theory of optimal leverage when the firm may only issue non-renegotiable

debt. Optimal leverage is chosen to maximize the value of tax shield benefits less bankruptcy costs.

MBP (1997) analyze a firm that may only issue renegotiable debt, with bargaining power treated as

a choice variable. In their continuous-time model, they show that strategic debt service eliminates

direct bankruptcy costs as well as inefficient liquidation. This result restores the Modigliani-Miller

irrelevance result of financial policy, since there are no tax benefits in their basic model. In the

presence of tax shields, they argue that distributing all bargaining power to the bank is optimal.

Since the Leland (1994) and MBP (1997) papers analyze situations in which the firm issues

only a single class of debt, the question of the optimal mix between bank and bond market debt

is beyond the scope of their papers. Similarly, the question of the optimal priority structure is

left unaddressed.4 The present paper stresses the notion that in many settings the distribution of

ex post bargaining power must be taken as a given by the firm, and cannot be treated as a choice

variable. Therefore, even in settings where a particular distribution of bargaining power may be

said to dominate in the sense of resulting in higher firm valuations, the firm may be unable to

commit to such a division of bargaining power, and must optimize subject to this constraint.

It has been well documented empirically that banks are generally senior in the

priority structure.

For example, Carey (1995) finds that in the

,000 loans made between 1986 and 1993 and recorded

in theDealscan database, over 99percentof all bank loanscontaina senior priority clause, regardless

of whether or not the borrower has bond market debt outstanding. Mann (1997) and Schwart

(1997) find that in addition to taking senior positions, banks also collateralize as much of their debt

as possible and incorporate protective covenants limiting the ability of the firm to issue additional

debt.
4Hackbarth and Leland (2001) analyze the role of priority structure of a firm issuing multiple classes of non-

renegotiable bond market debt.

One of the contributions of this paper to the banking literature is that it suggests an alternative

theory regarding the optimal debt mix and priority structure that rests upon a standard tax shield-

bankruptcy cost trade-off story. The model does not invoke any notion of monitoring by the bank,

costly state verification, or asset substitution problems. Yet, it generates many of the stylized facts

that the banking literature has attempted to explain. This banking literature is discussed below.

Much of the theoretical banking literature attempts to explain why banks are senior. Diamond

(1993a, 1993b) develops a model of both adverse selection and moral hazard, in which he obtains an

optimal debt structure resulting from a trade-off between informational sensitivity and preservation

of control rent. In his model, because of the non-assignable control rent, short-term lenders have

excessive incentives to liquidate the firm at the intermediate date, which imposes a distortion on

long-term lenders. Seniority permits short-term creditors to dilute the other claims and thereby

moderate their incentives to liquidate the firm early. By invoking the assumption that renegotiation

of bond market debt is very difficult, he finds that making short-term (bank) debt senior relative

to long-term (bond market) debt improves the bank’s incentive to monitor.

Rajan (1992) arrives at the opposite conclusion regarding priority structure. The main benefit

of bank debt in his model is flexibility in times of distress. However, banks develop informational

monopolies and can distort investment incentives by demanding a share of the rents from profitable

projects as a condition for rolling over short-term loans. To weaken the bargaining power of

informed lenders, the firm grants uninformed lenders higher priority. Welch (1997) analyzes the

role of influence costs on the optimal priority structure of debt. Placing bank debt senior is optimal

in his model since influence costs are lower when the strongest creditor, ex post, is given more power

ex ante.

Repullo and Suarez (1998) abstract from Rajan’s informational monopoly problem and conclude

that the optimal priority rule entails placing informed lenders senior. Dewatripont and Tirole

(1994) demonstrate that in order to prevent borrower moral hazard, it is best for control rights in

bad states to be given to lenders rather than to equity holders since lenders have a more credible

threat to liquidate the firm. Since equity in their model is indistinguishable from junior debt, one

can draw many conclusions about the optimal debt and priority structure from their model. Rajan

and Winton (1995) and Chemmanur and Fulghieri (1994) propose models of banks as delegated

monitors, but neither analyzes the optimal debt mix or priority structure.

In a first effort to reconcile these two conflicting theories, Park (2000) develops a theory of

optimal debt structure by explicitly modeling a costly monitoring technology in a model with asset

substitution. Because thefirm’s contracting costsare reduced when lendershave stronger incentives

to monitor, firms structure debt securities to maximize the lenders’ incentives to monitor. This

is achieved only if the delegated monitor can recover all of his monitoring costs from the lending

relationship, which is feasible only if there are no lenders above him in the priority structure.

Winton (1995) analyzes the priority structure problem building on Townsend’s (1979) costly

state verification model. As in the original work, lenders are assumed to be able to commit

themselves to verification when reported income is low, so that inspections occur even when banks

know that the firm has truthfully reported low cash flow truthfully. Winton’s model implies

that junior lenders verify more often than senior lenders, which is consistent with Fama’s (1990)

intuition.

Diamond’s (1993a, 1993b) theory does not provide a role for multiple lenders holding different

debt securities of the firm. Extending the model of Hart and Moore (1998), Berglöf and von

Thadden (1994) develop a rationale for multiple creditors and seniority structure by illustrating

the role of bargaining power in a renegotiation game of symmetric information. The main insight is

that the optimal debt contract drives a wedge between short-term and long-term lenders. Bolton

and Scharfstein (1996) put forth a related model of the optimal number of creditors. In their

paper, multiple lenders provides a commitment mechanism for the borrower due to the difficulty of

renegotiating with multiple lenders of the same class.

Our model also generates a life-cycle hypothesis regarding optimal debt financing policy, which

rests upon a rather different line of causation than that invoked in the existing banking literature.

Diamond (1991) considers a setting in which banks ignore rents accruing to borrowers in late

periods, resulting in suboptimal liquidation. On the other hand, young firms are able to establish

a good reputation by borrowing from banks with monitoring skills. The Diamond model thus

predicts that firms will use bond market debt only after a favorable track record is established via

bank loans. Another closely related theory on the coexistence of financial intermediaries and bond

market finance has been developed by Bolton and Freixas (2000) within an asymmetric information

framework, absent thedebt tax shield. Similar to our approach, these authors emphasize that banks

provide value to firms since they are good at helping firms through times of financial distress.

However, the cost of intermediation creates a trade-off between bond market and bank finance.

They also obtain the result that younger firms may be constrained to bank debt, while larger and

more mature firms access both markets.

III. The Model

Theunderlying statevariableX is thefirm’sEBIT,which is assumedto followageometricBrownian

motion:

dXt = µXtdt+ σXtdWt, X0 > 0 (1)

where µ and σ are known positive constants and W is a Wiener process. All agents are risk neutral,

and there is a risk-free asset yielding a constant rate of return r > µ. Within this setting, consider

an arbitrary claim paying the state contingent flow mX+k. The value function (G) for that claim

must satisfy the following ODE:

2
σ2X2G00 (X)+µXG0 (X)−rG(X)+mX +k = 0. (2)

The general solution to this ODE is:

G(X) = K1

a +K2X

z +
mX

r−µ +
k

r
, (3)

where a < 0 and z > 1 denote the roots of the quadratic:

Q(a) ≡ 1
2
σ2a2 +(µ− 1

2
σ2)a−r = 0. (4)

All of the contingent claims priced below have solutions of this form, with suitable boundary

conditions pinning down unknown constants.

There is a linear tax at rate τ levied on corporate income, which is computed as EBIT less

instantaneous debt service.5 It is never optimal to shut-down an unlevered firm, and the implied

value of the unlevered corporation at any instant is given by V :

V (X) ≡ (1−τ)X/(r−µ). (5)

The firm may issue two classes of consol bond debt: bank debt, which carries a promised

coupon flow b, and bond market debt which carries a promised coupon flow c. Bond Market Debt
5FUTURE: Simulate effect of non-linearities in tax schedule.

(BMD) may not be renegotiated, with failure to pay the promised coupon leading to immediate

reorganization of the firm. In contrast, bank debt may be renegotiated, with the character of the

debt service depending on the bargaining power of equity vis-à-vis the bank, as well as the parties’

respective threat points. We distinguish between two types of firms. Strong Firms have full

bargaining power in renegotiation and we label their bank debt commitments Equity Bargaining

Power Debt (EBPD). Weak Firms cede all bargaining power to the bank in renegotiation and we

label their bank debt commitments Bank Bargaining Power Debt (BBPD).

A central argument in this paper is that the optimal debt mix and priority structure depends

critically on the ex post distribution of bargaining power between the firm (equity) and the bank.

We do not formally derive the relationship between bargaining power and underlying firm/bank

characteristics. Rather, the model treats the distribution of bargaining power as a “fact of life” for

the business. In our view, it is most natural to think of weak firms as being small corporations that

are possibly reliant on the bank for a portfolio of services. Strong firms are best viewed as those

that are large (perhaps too large to fail from the bank’s perspective), sophisticated, and aggressive

in negotiations due to reputational considerations.

For the remainder of the paper, the term default states refers to situations in which either BMD

is not serviced or in which the firm and the bank fail to reach agreement on debt service, bringing

about immediate reorganization, with the proceeds from sale of the firm being paid to creditors

in accordance with the Absolute Priority Rule (APR).6 States in which equity fails to make the

promised payment (b) and enters into successful renegotiations with the bank, while any BMD is

still serviced, are referred to as renegotiation states.

Following Leland (1994), there is a deadweight loss in the event of default, with the parameter

α > 0 representing net default costs as a percentage of unlevered firm value. The value of the

reorganized (liquidated) firm is denoted L, which is equal to:

L(X) ≡ (1−α)V (X) (6)

It should also be noted that in using the term “default costs,” we have in mind the broad

concept of the term which includes more than the fees paid to the attorneys as they carve up the
6The model is general enough to encompass deviations from APR, although discussion of the optimal contractual

priority structure would then necessitate some assumption regarding the behavior of the courts.

carcass. Rather, default costs include: superior ability of existing management in running the

firm; distraction of management; impaired ability to contract; and suboptimal investments in firm-

specific human capital. It should be noted that it is not assumed that the reorganized firm does not

issue debt. Rather, the creditors taking over the firm are assumed to relever optimally, implying

that α is expressed as net of the tax shield benefit associated with the reorganized firm. While the

model is general enough to allow for a negative α parameter, empirical evidence on recovery rates

on defaulted debt as a percentage of par supports the assumption that α > 0. See Ross (1997) for

a discussion.

To capture the fact that debt renegotiation is costly, there is a proportional deadweight cost

from renegotiation:

Flow Renegotiation Cost = δ[b−s(X)]+, (7)

where s represents the state contingent debt service to the bank, and δ ≥ 0.

A. Bank Debt Renegotiation

A.1. Equity Bargaining Power Debt

Consider first the dynamics of renegotiation for EBPD. In renegotiation states, equity makes take-

it-or-leave-it offers to the bank, pushing the bank down to its reservation value.7 The bank’s EBIT

contingent reservation value is denoted R, and is equal to the value the bank would receive in the

event of reorganization. We model debt priority with the parameter η ∈ [0,1], which represents
the percentage of reorganization value paid to the bank. There are a number of advantages

to this parametric approach to modeling debt priority: (i) It subsumes both Senior and Equal

Priority bank debt; (ii) It covers a broad range of sharing rules, some of which may entail court

imposed deviations from APR, which are then capitalized into asset prices in a rational expectations

equilibrium; (iii) It yields closed-form solutions for asset prices.

Assumption 1: R(X) ≡ ηL(X).
7The analysis in this section is similar to that of MBP (1997), although we employ a somewhat different optimal

control approach.

The priority structure for many debt contracts when APR is followed entails non-linearity in

the reservation value function:

Bank Senior ⇒ R(X) = min{b/r, L(X)} (8)
Equal Priority ⇒ R(X) = min{b/r, (b/(b+c))L(X)}
Bank Junior ⇒ R(X) = min{b/r, max{0, L(X)−c/r}}.

However, for EBPD, the assumed linear sharing rule in default subsumes both the Bank Senior

and Equal Priority cases shown above.8 The reasoning is as follows:

• For EBPD, if there is no renegotiation, then the analysis of the mixed debt firm issuing the
pair (b,c) is subsumed in the case of the firm financing exclusively with BMD carrying the

coupon b+c.

• For Senior EBPD, if there is renegotiation, it must be the case that L(X) < b/r on the renegotiation region, otherwise the bank rejects any offer less than b. Therefore, setting

η = 1 produces the correct debt service and correct valuations of all contingent claims on the

firm.

• For Equal Priority EBPD, if there is renegotiation, it must be the case that (b/(b+c))L(X) < b/r on the renegotiation region, otherwise the bank rejects any offer less than b. Therefore,

setting η = b/(b+c) produces the correct debt service and correct valuations of all contingent

claims on the firm.

• Junior bank debt (To Be Added). Results thus far suggest this is suboptimal.

The incentive compatible (IC) strategic debt service function for EBPD is denoted bs. It is
inappropriate to think of bs as being optimal from an ex ante perspective. Rather, bs is optimal for
equity, ex post, given that in order for the bank to accept an offer, the value of its claim cannot be

below R(X). The essence of the problem is that even if bs is not optimal ex ante, equity cannot
commit to an alternative stream of promised payments. This points out the key difference between

EBPD and BMD in the model. EBPD offers financial flexibility, which preserves value in terms

of reducing expected bankruptcy costs and maintaining the ability of the firm to exploit existing
8The priority structure for BBPD is shown to be moot, as mixed debt finance is dominated.

tax shields. BMD has superior commitment value, keeping the flow tax shield benefit high in all

non-default states.

Below, the term Xd(b,c) refers to the endogenous default-triggering EBIT level that is chosen

by equity for an arbitrary mix of EBPD and BMD, with the shorthand Xd often being used for the

sake of brevity. Letting E denote the value of equity, the Bellman equation in non-default states

(X > Xd) is:
9

rE(X) = max
s(X)∈S(X)

(1−τ)[X −c−s(X)]−δ[b−s(X)]+ (9)

+µXE0 (X)+
1

2
σ2X2E00 (X) .

where:

S(X) ≡ {s(X) : s(X) < b ⇒ B (X) ≥ R(X)}. (10)

Note that the state-contingent feasible set S simply represents the set of all strategic debt

payments such that if the promised coupon is not paid, the value of bank debt is weakly above

the reservation value. Dropping terms on the right-side of the Bellman that do not involve the

strategic debt service term, equity’s problem reduces to:

bs(X) ≡ arg min
s(X)∈S(X)

s(X)(1−τ)+ δ[b−s(X)]+. (11)

This expression (11) indicates that the IC debt service under EBPD is that which minimizes the

sum of the after-tax cost of debt service and renegotiation costs, subject to the constraint that debt

service offers below b are accepted by the bank.

For the remainder of the paper, it is assumed that δ is sufficiently low so that the total cost of

debt service, inclusive of renegotiation costs, is increasing in s.1

Assumption 2: δ < (1−τ)

We now characterize the IC strategic debt service that obtains when the parameters of the problem

are such that renegotiation occurs. The function B denotes the value of bank debt. Reference

to (11) indicates that when equity has maximum bargaining power, the IC debt service entails
9As in MBP (1997), attention is confined to debt service functions that are piecewise right-continuous functions

of the state variable. This avoids problems associated with deviations on sets of measure zero.
10If δ ≥ (1 − τ), then bs(X) = b in all non-default states, which is subsumed in the analysis of the Bond Market

Debt financed firm.

B(X) = R(X) in the renegotiation region. Based on (2), the function B must satisfy the following

ODE on the region (Xd,∞) :
1

2
σ2X2B00 (X)+µXB0 (X)−rB (X)+s(X) = 0. (

)

Substituting in R and its derivatives for B implies that in the renegotiation region, strategic debt

service for EBPD is:

bs(X) = η (1−α)(1−τ)X. (

)
The equation (13) is informative about the role that priority plays in the context of EBPD. When

the bank is placed higher in the priority structure (η is increased) the flow debt service under EBPD

is higher. Bargaining power is not changed, but the threat point of the bank in instantaneous

renegotiations is higher, implying that while equity retains full bargaining power, it cannot push

the bank as hard. Similarly, if bankruptcy costs (α) are lower, the IC debt service is higher due

to the higher threat point of the bank.

The IC switch point for entering into renegotiation is denoted Xs, with arguments (b,c). Letting

the subscripts L and H denote claim values for X < Xs and X ≥ Xs, respectively, it follows that for EBPD:11

BL (X) = η (1−α)V (X) ∀X ∈ [Xd,Xs) (

)
BH (X) = A1X

a +A2X
z + b/r ∀X ∈ [Xs,∞). (

)

The unknowns for this problem are (A1,A2,Xs), which are derived using the following bound-

ary conditions, which represent value matching, smooth pasting and asymptotic conditions, respec-

tively:

BH (Xs) = η (1−α)V (Xs) , (

)

∂BH (Xs)

∂X

η (1−α)(1−τ)
r−µ , (

)

lim
X→∞

BH (X) = b/r. (18)

The solution yields the following bank debt values and renegotiation threshold for EBPD:

BL(X) = η (1−α)V (X) (19)
11The valuation formulas assume the existence of an endogenous default threshold. In the case of no default, the

term Xd is simply replaced with a zero.

BH(X) =

·
1− 1

1−a

µ
X

¶a¸

(20)

Xs =

µ
r−

µ
r

µ
a

a−1
¶·

η (1−α)(1−τ)
¸
. (

)

Finally, it is worth noting that the IC debt service function must exhibit a jump at the point Xs

since:

lim
X↑Xs

bs(Xs) = µr−µ
r

¶ µ
a

a−1
¶
b < b. (

)

The existence of a default trigger for equity (Xd > 0) under EBPD depends on the existence

of negotiation costs (δ > 0) and/or a BMD coupon commitment (c > 0). Intuitively, for EBPD

default is triggered when equity is unwilling to pay the flow costs associated with servicing BMD and

renegotiation costs that are necessary to maintain its claim on positive cash-flows in good states.

Absent fixed costs associated with BMD and negotiation costs, default never occurs because equity

incurs no flow cost in keeping alive its option. Of course, if renegotiation costs are sufficiently high,

renegotiation does not occur. Additionally, for strong firms financing with a mixture of EBPD

and BMD, renegotiation does not occur if c is sufficiently high relative to b. This is because for

EBPD, the optimal renegotiation threshold Xs as given in (21) does not depend on c, whereas

increases monotonically in c.

The following cases characterize the range of possible equilibria under EBPD:

• For the firm financed exclusively with EBPD, if δ = 0, default never occurs. There exists an
EBIT level Xs such that bs(X) = η (1−α)(1−τ)X on the interval (0, Xs) and bs(X) = b for
all X ≥ Xs. This equilibrium is depicted in Figure 1.

• If δ or c is positive, ∃ Xd such that equity declares default.

• There is no renegotiation of EBPD for (δ,c) sufficiently high relative to b.

• For positive δ or c, and (δ,c) sufficiently low relative to b, ∃ Xd such that default occurs
endogenously; ∃ Xs > Xd such that bs(X) = η (1−α)(1−τ)X on (Xd, Xs) and bs(X) = b
for all X ≥ Xs. This equilibrium is depicted in Figure 2.

A.2. Bank Bargaining Power Debt

With BBPD, in the event that equity fails to pay b and enters into renegotiation, it is the bank that

makes take-it-or-leave it offers. In renegotiation, equity is pushed down to its reservation value,

which is zero under limited liability. Since equity value is everywhere zero on the renegotiation

region, satisfaction of (2) implies the cash-flow to equity is zero in renegotiation states. That is,

equity pays all free cash-flow, X−c, to the bank. So long as the bank is willing to service the BMD
coupon commitment (c) as well as pay any negotiation costs, reorganization is avoided. Therefore,

under BBPD, it is the bank that chooses when to enter into reorganization, whereas under EBPD

it is equity that chooses when to enter into reorganization. The bank’s endogenous reorganization

threshold under BBPD is denoted Xb(b,c).

Lemma 1 proves that equity’s endogenous renegotiation threshold under BBPD for the mixed

debt policy (b,c) is identical to the default threshold chosen by equity facing the BMD coupon

commitment b + c. Intuitively, from the perspective of equity, there is no difference between

default and entering into renegotiations with a bank that has full bargaining power, since both

outcomes generate a claim worth zero.

Lemma 1 For Bank Bargaining Power Debt, equity’s optimal switch point for entering into rene-

gotiation satisfies Xs(b,c) = Xd(0,b+c).

Proof. The cash-flow to equity in non-renegotiation states under BBPD for the pair (b,c)

is identical to that under exclusive BMD finance with coupon ec = b + c. Default on BMD and
renegotiation of the BBPD entail E[Xs(b,c)] = E[Xd(0,ec)] = 0. Result follows from the value
matching and smooth pasting conditions being identical.

To characterize the optimal financial structure of the firm in light of the distribution of bargain-

ing power, the constituent pieces of the firm must be priced. The value of the marketable claims

of the levered firm (v) is equal to the sum of the values of equity (E), bond market debt (C), and

bank debt (B):

v(X) = E(X)+C(X)+B(X). (

)

Lemma 2 proves that the optimal financial policy for a weak firm that is able to issue BBPD entails

financing with bank debt exclusively. This result holds regardless of the assumed values of the

parameters (τ,α,δ,σ) and regardless of the assumed priority structure on any proposed mixed debt

financing arrangement.

Lemma 2 For the Weak Firm, financing exclusively with Bank Bargaining Power Debt dominates

any policy involving nonzero Bond Market Debt, regardless of priority structure.

Proof. The optimal debt policy maximizes v. Consider an arbitrary mixed debt policy (b,c)

with an arbitrary priority rule f that maps L to payments to the bank. Now consider the pure

BBPD policy setting eb = b+c. From Lemma 2 it follows that the value of equity in the two firms
is identical in all states. Let (es,eB) denote the bank debt service and bank debt value under pure
BBPD finance. The mixed debt firm is represented by (s,B,C). Note that in non-reorganization

states, es(X) = s(X) + c. On this same region, letting Φ be an indicator for renegotiation, flow
renegotiation costs are identical for both firms and equal to Φ(X)δ[eb−es(X)]. The stopping time
problem for the bank under pure BBPD is:

B(X0) ≡ max
T

·Z T
0
e−rt[es(Xt)−Φ(Xt)δ(eb−es(Xt))]dt+e−rTL(XT)¸ .

The stopping time problem for the bank under mixed debt is:

B(X0) ≡ max
T
E0

·Z T
0
e−rt[s(Xt)−Φ(Xt)δ(b−s(Xt))]dt+e−rTf(L(Xt))

¸
.

Let T∗m ≡ inf{t ≥ 0 : Xt = Xb (b,c)} represent the optimal stopping time for the bank under mixed
debt. Using the adding up constraint for recoveries in default, the equality of renegotiation costs,

and the fact that es(X) = s(X)+ c in non-reorganization states we have:
B(X0)+C(X0) = E0

“Z T∗m
0

e−rt[es(Xt)−Φ(Xt)δ(eb−es(Xt))]dt+e−rT∗mL(XT∗m)
#
≤ eB(X0).

Lemma 2 suggests the following intuition for the dominance of pure BBPD finance over any

mixed debt policy for weak firms. For any mixed debt policy (b,c), the firm can achieve the same

equity valuation by issuing BBPD with couponeb = b+c. However, since it is the bank that decides
when to terminate renegotiations, through its choice of Xb, it is optimal to avoid ex post inefficient

bank decision rules that emerge when the choice of Xb generates an externality to other classes

of debt. Similarly, BBPD dominates pure BMD finance since the option to remain open under

BBPD has weakly positive value ex post.

There is an alternative graphical interpretation regarding why pure BBPD dominates any mixed

debt or pure BMD policy for weak firms. Assume that negotiation costs are zero. Consider

first the choice between issuing pure BMD versus pure BBPD when there are no negotiation costs.

Assuming that the BMD policy entails coupon c, consider issuing BBPD with promised debt service

b = c. Figure 3 depicts the debt service under BBPD. Note that the debt service under the BMD

contract is represented by the horizontal segment. Clearly, the BBPD contract dominates since it

entails higher tax shield benefits as well as lower bankruptcy costs. Figure 4 illustrates why BBPD

dominates an arbitrary mixed debt policy entailing (b,c). The schedule beginning at Xb(b,c) > 0

depicts total debt service under the mixed policy. If the firm had financed with pure BBPD foreb = b+c no default would occur, implying bankruptcy costs are reduced. Further, additional tax
shield benefits are realized, as the firm is able to shield income on the interval (0,Xb).

Pricing of Bank Bargaining Power Debt Due to Lemma 2, in the remainder of the paper

discussion of weak firms is confined to those financing exclusively with BBPD. We now solve for

the price of BBPD and the bank’s optimal reorganization threshold Xb. The cash-flow to the bank

in renegotiation states is equal to debt service, denoted by s(X), less renegotiation costs, if any:

Cash-Flow to Bank = s(X)−δ[b−s(X)]+ (

)
=

½
b for non-renegotiation states
X −δ[b−X] for renegotiation states

Below, we prove that an endogenous reorganization threshold for the bank does not exist if δ = 0.

For δ > 0, the general solution for the value of BBPD is as follows:

BL (X) = A3X
a +A4X

z +
X(1+ δ)

r−µ −

δb

r
∀X ∈ [Xb,Xs) (

)

BH (X) = A5X
a +A6X

z + b/r ∀X ∈ [Xs,∞). (

)

Based on Lemma 1, the optimal renegotiation point for equity (Xs) is derived below in (

where the BMD model is solved. Taking Xs as given, the unknowns for this problem are

(A3,A4,A5,A6,Xb) which are derived using the following boundary conditions:

BL (Xb) = (1−α)V (Xb) , (

)
∂BL (Xb)

∂X
= (1−α)V 0(Xb), (

)

BL(Xs) = BH(Xs) (

)

∂BL (Xs)

∂X
=

∂BH (Xs)

∂X
(

)

lim
X→∞

BH (X) = b/r. (

)

It is interesting to note that since it is the bank that chooses when to enter into reorganization,

we have imposed value matching and smooth pasting conditions for the bank value function, with

B pasting up smoothly to the reservation value function (R) at the point Xb. The solution is as

follows:12

BL(X) =
X(1+ δ)

r−µ −
δb
r
·
1− 1

(1−a)
µ
X

¶a¸
∀X ∈ [Xb,Xs) (32)

BH (X) =

·
δb

r(1−a)
¸·
X

¸a
−
·
(1+ δ)b

r(1−a)
¸·
X
Xs

¸a
+ b/r ∀X ∈ [Xs,∞). (33)

Xb =

·
δ

δ + τ + α(1−τ)]
¸
Xs (

)

=
µ
a

1−a
¶µ

δb
r

¶·
δ + τ + α(1−τ)

r−µ
¸−1

(

)

Lemma 3 follows from inspection of the equation for Xb:

Lemma 3 For Bank Bargaining Power Debt, ∃Xb > 0 iff δ > 0.

In terms of the optimal reorganization point, one can think of the bank as solving an optimal

stopping problem which involves an irreversible switch from the stream of cash-flows associated

with the BBPD to that associated with the reorganized firm. The reorganized firm generates the

perpetual cash-flow (1−α)(1−τ)X. The difference in instantaneous cash-flows is:

X[δ + τ + α(1−τ)]−δb. (36)

The second expression for Xb embeds this cash-flow differential.

In summary, the following cases characterize the range of possible equilibria under BBPD:

• For the firm financed with BBPD, if δ = 0, default never occurs and∃Xs such that s(X) = X
on the interval (0, Xs) and s(X) = b for all X ≥ Xs.

• If δ > 0, ∃ Xb such that the bank chooses to enter reorganization; ∃ Xs > Xb such that
s(X) = X on (Xb, Xs) and s(X) = b for all X ≥ Xs.

12The expression for Xs is derived below in (51).

IV. Valuation of the Levered Firm

Having characterized debt service functions for both EBPD and BBPD we can now solve for the

constituent pieces of levered firm value. It is easily verified that the sum of instantaneous cash

flows to equity, bond market debt, and bank debt is:

Firm Cash Flow = (1−τ)X + τ[c+s(X)]−δ[b−s(X)]+. (

)

This implies that the value of the levered firm can be expressed as the sum of the unlevered firm

value, plus the value of the tax shield (TB), less negotiation costs (N), less bankruptcy costs (BC):

v(X) = V (X) + TB(X) − N(X) − BC(X) (

)

Before proceeding, the price of a particular primary claim will be of use for valuing a number of

claims on the firm. Consider a primary claim paying $1 at the first-passage time from above of

the EBIT process X to some threshold X∗. Based on (2), it may be verified that for X ≥ X∗, this
claim’s value is:

Hitting Claim =
µ
X

X∗

¶a
(

)

A. Tax Shield

The value of the tax shield generated by bank and bond market debt is readily computed once one

recognizes that the tax shield value is equal to the tax rate τ multiplied by the expected present

value of future debt service. For the case of EBPD this latter value is equal to EBPD plus BMD

less the creditors’ claims on recoveries in the event of default. To determine the value of the

recovery claim held by the firm’s creditors, one multiplies the value of a hitting claim that pays off

at Xd times the value of payoffs in default, L(Xd). Therefore, the tax shield value is:

TB(X) =

·
C(X)+B(X)−L(Xd)

µ
X
Xd

¶a¸
for EBPD. (

)

The value of the tax shield in the case of BBPD is computed similarly, with negotiations costs,

N(X), being added back to the value of the BBPD in order to compute the value of debt service:

TB(X) = τ

·
B(X)+N(X)−L(Xb)

µ
X
Xb

¶a¸
for BBPD.

B. Bankruptcy Costs

For EBPD the deadweight loss in the event of default is given by αV (Xd). Therefore, we have the

following expression for bankruptcy costs:

BC(X) = αV (Xd)

µ
X
Xd

¶a
for EBPD.

Similarly:

BC(X) = αV (Xb)

µ
X
Xb

¶a
for BBPD.

C. Negotiation Costs

The negotiation cost function satisfies (2), with δ[b − s(X)] representing the flow term in the
renegotiation region, while there is zero flow cost for X ≥ Xs. The following boundary conditions
pin down the solution for EBPD. For BBPD, one simply replaces Xd with Xb in the first boundary

condition.

NL (Xd) = 0, (

)

NL (Xs) = NH (Xs) , (

)
∂NL (Xs)

∂X
=

∂NH (Xs)

∂X
, (

)

lim
X→∞

N (X) = 0. (

)

For EBPD the solution for the negotiation cost function is:

NL(X) = δ
b

·
1−

µ
1−

µ
a

a−z
¶µ

Xd
Xs

¶z¶µ
X

¶a
−
µ

a
a−z
¶µ
X
Xs

¶z¸
(

)

−δ

µ
η (1−α)(1−τ)

r−µ
¶·
X −

µ
Xd −

µ
a−1
a−z

¶
Xs

µ
Xd
Xs

¶z¶µ
X
Xd

¶a
−
µ
a−1
a−z

¶
Xs
µ
X
Xs

¶z¸
NH(X) = δ

b
r

·µµ
a

a−z
¶µ
Xd
Xs

¶z
−1
¶µ

X
Xd
¶a
−
µ
z
a−z
¶µ
X
Xs

¶a¸
(

)

+δ

µ
η (1−α)(1−τ)

r−µ
¶·µ

Xd −
µ
a−1
a−z

¶
Xs
µ
Xd
Xs
¶z¶µ
X
Xd

¶a
−
µ
1−z
a−z

¶
Xs
µ
X
Xs

¶a¸
For BBPD the solution for the negotiation cost function is:

NL(X) = δ
b
r
·
1−
µ
1−
µ
a

a−z
¶

µ
Xb
Xs

¶z¶µ
X
Xb
¶a
−
µ
a
a−z
¶µ
X
Xs

¶z¸
(

)

−δ
µ

1
r−µ
¶·
X −

µ
Xb −

µ
a−1
a−z
¶
Xs
µ
Xb
Xs
¶z¶µ
X
Xb
¶a
−
µ
a−1
a−z
¶
Xs
µ
X
Xs
¶z¸
NH(X) = δ
b
r
·µµ
a

a−z
¶µ
Xb
Xs

¶z
−1
¶µ
X
Xb
¶a
−
µ
z
a−z
¶µ
X
Xs

¶a¸
(

)

+δ

µ
1

r−µ
¶·µ

Xb −
µ
a−1
a−z

¶
Xs
µ
Xb
Xs
¶z¶µ
X
Xb
¶a
−
µ
1−z
a−z
¶
Xs
µ
X
Xs
¶a¸

D. Bond Market Debt Value

The value function for bond market debt satisfies (2) with flow payment equal to c on the non-

default region. The boundary conditions are value matching at Xd, and an asymptotic condition:

C (Xd) = (1−η)(1−α)V (Xd), (

)
lim
X→∞

C (X) = c/r. (

)

The solution for the value of BMD is given by:

C(X) =

r
·
1−
µ
X
Xd

¶a¸
+(1−η)(1−α)V (Xd)

µ
X
Xd

¶a
(51)

E. Equity Value

Finally, having derived each of the terms in the levered firm value expression (38), the value of

equity can be expressed as a residual based on (23). In the case of EBPD, the following valuation

equation holds on the interval [Xd,∞) :

E(X) = V (X)−V (Xd)
µ
X

¶a
−(1−τ)

·
B(X)+C(X)−L(Xd)

µ
X
Xd

¶a¸
−N(X) for EBPD. (

)

In the case of BBPD, equity is worth zero for X ≤ Xs, since it is pushed down to its reservation
value in renegotiations. The following pricing equation holds on the interval (Xs,∞) :

E(X) = V (X)−V (Xb)
µ
X

¶a
− (1−τ)

·
B(X)+N(X)−L(Xb)
µ
X
Xb
¶a¸
for BBPD.

The two expressions for the value of equity under EBPD and BBPD have intuitive interpretations.

The first two terms represent the value of holding an unlevered firm that will be worthless at

the first-passage of the EBIT process to the relevant reorganization threshold. The third term

represents the expected present value of the after-tax cost of total debt service. The fourth term

is renegotiation costs, which is paid by equity in the case of EBPD but is paid by the bank in the

case of BBPD.

The remaining unknown variable for the case of EBPD is the optimal default threshold, Xd(b,c).

The brute force method of solving for equity value and the endogenous default threshold is to solve

for the flow payments to equity and then utilize the following boundary conditions:

EL(Xd) = 0 (

)

EL(Xs) = EH(Xs) (

)

∂EL (Xs)

∂X
=

∂EH (Xs)

∂X
(

)

lim
X→∞

E (X) =
(1−τ)X
r−µ −

(1−τ)(b+c)
r

(

)

∂EL (Xd)

∂X
= 0 (

)

However, it can be verified that the solution (52) necessarily satisfies the first four boundary con-

ditions. The smooth pasting condition at default must therefore be exploited in solving for Xd.

Using (21), the endogenous default-triggering EBIT level under an arbitrary mixture of EBPD and

BMD is:

Xd(b,c) =
a(r−µ)[δb+(1−τ)c]

(a−1)r(1−τ)[1−η (1−α)(1−τ −δ)] (

)

V. Optimal Debt Structure under Mutually Exclusive Financing

A. Analysis of Strong Firm

A.1. Bond Market Debt Finance

We begin with an analysis of the bond market debt financed firm, confirming that the qualitative

features of the Leland (1994) framework carry over to our model. The discussion will be brief, since

the main objective of this section is to contrast the bond market and bank debt firms. Dropping

terms in (38) that involve bank debt, the expression for the value of the marketable claims of the

corporation is given by:

v(X) = V (X)+
τc

r
µ
1−
µ
X
Xd

¶a¶
−αV (Xd)

µ
X
Xd

¶a
. (

)

Value matching and smooth pasting conditions for equity yield the following solution for the en-

dogenous default-triggering EBIT level as a function of the promised coupon payment:

Xd(0,c) =

µ
r−µ
r
¶ µ
a

a−1
¶
c. (

)

Differentiating the previously derived expression for the value of bond market debt given in (51),

with respect to c and setting it equal to zero yields the bond market debt coupon capacity, denoted

cmax:

cmax (X0) =
r(a−1)X0
a(r−µ) [1−a(τ + α(1−τ))]

1/a
. (

)

Substituting this expression back into the pricing equation yields the bond market debt value ca-

pacity:

Cmax (X0) =
X0 [1−a(α +(1−α)τ)]1/a

(r−µ) . (

)

Figure 5 parallels Figure 1 in Leland (1994) presenting comparative statics results for the value

of bond market debt as a function of leverage for varying degrees of underlying cash flow risk. It

is easy to see that a debt capacity cmax obtains. Perhaps surprisingly, it can be seen that the

firm’s bond market debt coupon capacity is increasing in EBIT risk, but the bond market debt value

capacity decreases in the riskiness of the underlying cash flows.

We can compute the optimal coupon by solving the first-order condition ∂v(X0;c∗)/∂c = 0:

c∗(X0) =
r(a−1)X0
a(r−µ)

·
τ −a(τ + α(1−τ))

¸1/a
(

)

Substituting this expression back into the total firm value expression yields the ex ante value of an

optimally levered bond market debt firm:

v∗ (X0;0,c∗) =
X0
r−µ

(
1−τ + τ

·
τ −a(τ + α(1−τ))
τ

¸1/a)
(

)

Figure 6 depicts the optimal coupon level c∗. Although high volatility firms have a higher

debt coupon capacity than low volatility firms, the optimal choice of leverage goes in the opposite

direction in that it is greater for low volatility firms than it is for high volatility firms. Altogether,

less risky firms attain a higher total levered firm value by issuing a higher promised coupon level

relative to riskier firms.

A.2. Equity Bargaining Power Debt Finance

We begin this subsection by verifying that some of the key results in MBP (1997) relating to the

case in which equity has full bargaining power are also valid in our modeling framework.

Lemma 4 If c = δ = 0, the Strong Firm financing with EBPD never defaults.

Proof. Follows from the fact that default is not optimal when the cash-flow to equity is

everywhere positive.

Lemma 5 For the Strong Firm, if τ = 0, δ = 0, and α > 0, then the optimal financial structure

entails no Bond Market Debt, and the firm is indifferent regarding the level of b.

Proof. Issuing BMD entails default costs, whereas EBPD entails no such costs.

Lemma 6 For the Strong Firm, if δ = 0 and τ > 0, it is optimal to increase b up to the bank debt

capacity, bmax, which satisfies B[X0;bmax] = L(X0). The ex ante firm value for any b ≥ bmax is
V (X0)[1+ τ(1−α)].

Proof. Since there are no negotiation costs and default never occurs, the firm solves the problem

max
b

τB[X0;b].

The intuition for bank debt capacity in this setting is as follows. From the equation for the

endogenous renegotiation threshold given in (21), it can be seen that Xs is increasing in b. For a

fixed initial EBIT level, low levels of b are consistent with X0 > Xs(b). However, for b sufficiently

high, the bank recognizes that equity will push it down to the reservation value L(X0) immediately,

assuming equity has maximum bargaining power. This implies that increasing the promised coupon

will not raise the value of debt, since the reservation value is invariant to b. More formally, the

bank debt capacity, bmax, satisfies:

Xs (b
max) = X0 ⇔ bmax =

µ
r

r−µ
¶ µ

a−1
a

¶
(1−α) (1−τ) X0. (

)

We now proceed to a discussion of some numerical results. Figure 7 depicts the bank’s reservation

value R and the value of bank debt B as functions of EBIT for varying levels of bankruptcy costs,

assuming that equity has full bargaining power. In the model, the strategic debt service to the

bank in the renegotiation region depends crucially on the level of bankruptcy costs (and corporate

taxes, for which the effect is essentially the same). The dashed rays from the origin represent the

bank’s threat point or, more formally, reservation value function R(α, ·). The construction of the
incentive compatible switch point Xs(α, ·) implies that the bank debt value B is tangent to R(α, ·)
up to Xs(α, ·). Note that ceteris paribus, firms with higher bankruptcy costs enter negotiations
with their bank sooner. The figure clearly depicts the fact that the bank is pushed down to

its reservation value in the renegotiation region, where B(X) = R(X), with value matching and

smooth pasting being exhibited at Xs.

Figure 8 depicts the value of EBPD (B) and BMD (C) as functions of EBIT (X) evaluated at

the same promised coupon, b = c = 20 in the presence of negotiation costs. Note that default

occurs earlier with BMD than with EBPD. However, for points to the right of Xd, the value of

BMD exceeds that of EBPD. Figure 9 reports comparative static results for the price of bank debt

when the firm’s EBIT volatility varies. EBPD is worth weakly more when the firm’s operations

are less risky. This is not surprising given that in Figure 7 it was shown that EBPD is concave in

EBIT. It is also interesting to note that the higher the underlying volatility, the longer the firm

waits before entering into renegotiation.

Figures 10-12 explore some comparative statics for the effects of key parameters on the total

leveraged firm value v. Lemma 6 demonstrated that with δ = 0, it is optimal for the firm to lever

up to its debt capacity. Figure 10 illustrates the effect variations in negotiation costs. It can be

seen that the introduction of negotiation costs generates an interior optimal coupon, with higher

negotiation costs implying a lower firm value and a lower optimal coupon b∗.

Figure 11 depicts the effect of variations in bankruptcy costs. It can be seen that the firm’s debt

capacity and optimal coupon are inversely related to bankruptcy costs. In other words, the model

predicts that firms with lower bankruptcy cost have more access to bank debt and will optimally

choose to borrow more. Figure 12 indicates that firm value is decreasing in underlying cash flows’

volatility.

Finally, the effect of an increase in perpetual coupon payments from b = 10 to b̃ = 15 on strategic

debt service is illustrated in Figure 13. Provided negotiation costs are strictly positive, an increase

in the promised coupon: (i) shifts the default-triggering EBIT level up ∂Xd (b,0)/∂b > 0; (ii)

results in a loss in tax shield benefits in the region between the previous switch point Xs and the

new switch point X̃s > Xs; and (iii) brings about an increase in tax shield benefits in the non-

renegotiation region, i.e., for all X ∈ [X̃s,∞). This provides the underlying basis for the interior
optimum b∗.

A.3. Equity Bargaining Power and Bond Market Debt Comparison

The purpose of this section is to compare EBPD and BMD, when the choice between the two is

mutually exclusive.13 We begin by comparing EBPD and BMD when the firm may choose the

optimal coupon commitment for each, respectively. Under the base case parameter values there

exist threshold values of bankruptcy costs α∗ and corporate taxes τ∗ at which the firm is indifferent

between EBPD and BMD. Figure 14 reports valuations applicable to optimally levered EBPD and

BMD firms as α is varied. For the optimal promised coupon payments b∗ (α, ·) and c∗ (α, ·) , firms
with bankruptcy costs of 31% or less prefer EBPD. At first the result may seem counter-intuitive.

After all, casual intuition suggests that issuing a hard claim like BMD is less attractive when

bankruptcy costs increase. There is nothing wrong with this intuition, as far as it goes. That

is, an increase in α reduces the optimal coupon commitment under BMD and the total firm value

attained. However, increases in bankruptcy costs also make EBPD less attractive and lower the

value attained. As Lemma 6 demonstrates, this effect is present even when the issuance of EBPD

entails no default (when δ = 0). This is because increases in α reduce the value of the tax shield

created by EBPD. Recall that under EBPD, the bank is pushed down to its reservation value in

renegotiations. Increases in α rotate the function R clockwise downwards, implying lower debt

service under EBPD.

Figure 15 reports valuations applicable to optimally levered EBPD and BMD firms as τ is

varied. At the promised coupon levels b∗ (τ, ·) and c∗ (τ, ·) firms with corporate taxes of less than
26% achieve a higher value with bank debt. Finally, Figure 16 traces out the locus of points

at which the firm is indifferent between the two sources of capital so long as they are mutually
13This case would be of particular interest when there are substantial fixed costs associated with tapping either

source of debt capital.

exclusive. If δ = 0, the following condition identifies parameters such that EBPD dominates

BMD, when both are optimally levered:

(1−α)(1−τ) > {1−a [τ + α(1−τ)]/τ}1/a . (

)

Finally, by parametrically varying the firm’s EBIT volatility another empirical prediction emerges.

In line with previous theoretical and empirical studies, we obtain the result that safer firms find it

optimal to tap public debt markets, while riskier firms are more likely to be financed by banks.

Turning now to the analysis of equal promised coupon payments to the bank and the bond

market, Figures 17—19 confirm in principle all of the previous findings for optimally levered firms.

Assuming that δ = 0, we can derive an analytic expression for the cut-off contour under equal

promised payments to EBPD and BMD. The following condition defines parameter values such

that the EBPD financed firm with the same perpetual coupon promise as the BMD firm achieves

higher value (when both are evaluated at X > Xs):

[(1−α)(1−τ)]−a > (1−a)1/a. (

)

Figure 19 confirms the intuition that ceteris paribus bank debt is more desirable for lower nego-

tiation costs δ; that is, a decrease in the negotiation costs δ implies that more (α,τ) pairs lead to

the optimality of EBPD for mutually exclusive financing decisions.

B. Analysis of Weak Firm

This section presents numerical results for the weak firm. We confine analysis to issuance of

BBPD, since for a weak firm the issuance of BMD is dominated, as was shown in Lemma 2. To

begin, we parallel the analysis of EBPD by plotting BBPD as a function of EBIT when the cost

of negotiation with the bank changes (Figure 20) and when the firm’s cash flow riskiness varies

(Figure 21). It is important to note that under BBPD, the bank becomes the residual claimant

on the firm’s cash flows in the renegotiation region. The value function for BBPD therefore pastes

smoothly to the reservation function R at the endogenous reorganization threshold Xb. Figure 21

illustrates that for any positive negotiation costs (δ > 0) both the default-triggering EBIT level Xb

and the incentive compatible switch point Xs are decreasing in the firm’s riskiness σ. Also, the

value function for BBPD exhibits a convex-concave shape, illustrating that the bank also becomes

risk-loving in some states.

As was already noted, under BBPD the bank becomes the residual claimant in the renegotiation

region. The has an important implication for the equity value function, which is depicted in Figure

22. In particular, the value of equity is equal to zero in the renegotiation region since the bank

pushes equity down to its reservation value. At equity’s incentive compatible switch point for

entering into renegotiations (Xs), the equity value function exhibits value-matching and smooth-

pasting.

Figure 23 illustrates that lower negotiation costs generate a higher optimal coupon under BBPD.

Figure 24 indicates that the value attained by the optimally levered firm is higher when the volatility

ofunderlyingEBITis reduced. TheeffectofvolatilityonfirmvalueunderBBPDseemsqualitatively

similar to the results reported in Figure 5 for firm value under BMD. However, it is interesting to

note that the increase in firm value stemming from lower volatility is achieved via different means,

depending on the type of debt issued. For BMD, the optimal coupon commitment increases when

volatility is reduced. For BBPD, on the other hand, the optimal coupon commitment decreases

when volatility is reduced.

VI. Optimal Debt Structure of the Mixed Debt Firm

This section confines discussion to the case of the strong firm since Lemma 2 demonstrates that

the optimal debt structure for the weak firm entails exclusive BBPD financing. We begin with

an analysis of the optimal mix of EBPD and BMD when the bank is placed senior in the priority

structure. Analysis of pari passu treatment follows. Comparison of ex ante firm valuations

achieved under the optimal debt mix for each priority rule, respectively, pins down the optimal

priority structure.

A. Senior Bank Debt

The following result is useful before proceeding to the numerical results for senior bank debt.

Lemma 7 For a Strong Firm issuing the mixed debt pair (b,c) such that there is no renegotiation

region, the ex ante firm value is equivalent to that under BMD finance with ec = c+ b.

Proof. If there is no renegotiation, then bank debt represents a hard claim from the perspective

of equity, implying that the optimal default threshold is based upon facing a fixed flow cost c + b

at each instant. This is the same situation facing a firm that has promised ec = c+ b to the bond
markets.

Intuitively, no renegotiation region exists when the bank is senior and the debt mix is weighted

heavily towards BMD. In such cases:

R[Xd(b,c)] =
(1−α)(1−τ)Xd(b,c)

r−µ >
b

r
, (

)

implying that no renegotiation occurs, since the bank rejects any debt service payment less than

b. Therefore, full analysis of levered firm valuations for the strong firm is conducted as follows.

Senior bank debt is examined by setting η = 1 and solving the model for parameters (b,c) are such

that there exists a renegotiation region. Other (b,c) pairs are subsumed under the exclusive BMD

financing case.

The analysis of the model in this subsection has been accompanied by several consistency

checks of the numerical scheme. First, the theoretical results for the pricing of claims for the

mixed-debt firm are predicated on the assumption that there exists a non-trivial renegotiation

region. Therefore, we need to impose the following regularity condition in our numerical analysis

of the

mixed debt firm:

Condition 1: Xd (b,c, ·) < Xs (b, ·) ∀ (b,c) ∈ IR2+ (

)

Secondly, it was stated that for certain (b,c) pairs the firm may be in the position to issue safe, senior

bank debt. These cases are subsumed in the analysis of the BMD firm. To avoid any pathological

cases, the numerical scheme consequently needs to contain a second regularity condition for the

mixed debt firm:

Condition 2: L [Xd (b,c, ·)] < b/r ∀ (b,c) ∈ IR2+ (70)

Figures 25 indicates that for the strong firm there exists an interior optimum debt mix (b∗,c∗) =

(24.09,23.63), when the bank is senior in the priority structure. Although the promised coupon to

the bank is higher, the initial value of BMD (334.94) exceeds that on EBPD (315.31). Regardless of

the measure, the optimal debt mix when the bank is senior is relatively even. Substantial increases

in firm value are achieved when the firm can exploit a mix of EBPD and BMD, rather than relying

on one source exclusively. The first line below reports the ex ante firm value attained when the

firm may mix EBPD and BMD, assuming the bank is senior. The second and third lines report

valuations under exclusive BMD and EBPD finance, respectively:

v (b∗,c∗) = 844.09 (71)

v(c∗) = 809.

v(b∗) = 752.01.

An interesting feature of the model is that it may be used to generate estimates regarding the

magnitude of the benefit the firm enjoys when it gains access to a previously untapped source of

funds. The results above indicate that firms without access to the bond markets reap large gains

when this constraint is removed. Note that this result holds in the absence of any notion that bond

markets are “cheaper.” In this model all claims are priced fairly, with all claimants discounting

flows at the rate r. All of the benefit from mixed debt finance comes from the complementarity

effect. Firm value increases by 4.3% when mixed debt is employed, rather thanfinancing exclusively

with BMD. The increase in firm value is 12.2% when mixed debt is employed, rather than financing

exclusively with EBPD.

The complementarity effect is due the relationship between bank and bond market debt coupon

payments and the default-triggering EBIT level Xd (b,c, ·). Therefore, we fix the default-triggering
EBIT level at Xd (b,c, ·) = 5 in Figure 33 and solve for the (b,c) pairs such that our root equation
is satisfied. This analysis reveals that bank and bond market debt have an uneven contribution to

the endogenous bankruptcy boundary and hence a complementarity effect of the nature discussed

above obtains. Secondly, in comparing these numerical results, it can be seen that placing bank

debt senior in the priority structure leads to higher total firm value. Lastly, at a broader level

these findings show that our model yields a product differentiation result for financial markets in

the following sense. In order to maximize firm value, the role of banks is to issue genuinely soft

debt such that they offer a financial product, which is as “distant” as possible from hard bond

market debt. Thus we provide a rationale for the coexistence of bank debt (indirect/intermediated

finance) and bond market debt (direct finance).14

14See for instance Bolton and Freixas (2000) for a similar result. These authors obtain coexistence of bank and
bond market debt from an asymmetric information model of financial intermediation.

The individual value components of the total leveraged firm are reported in Figures 27—32.

Valuations for bank B and bond market debt C are depicted as functions of coupon payments to

the bank b and to the bond market c in Figures 31 and 32, respectively. Our earlier discussion on

comparative statics results concerning debt capacity of the all soft (hard) debt firm are confirmed

by these graphs.

B. Equal Priority Bank Debt

The following result will be useful before proceeding to the numerical results for the case of equal

priority bank debt.

Lemma 8 Under equal priority with endogenous default, it is impossible for the firm to issue safe

EBPD or BMD.

Proof. Suppose to the contrary that either class is safe. Then it must be the case that

L[Xd(b,c)] ≥ b+cr and there is no renegotiation. With no renegotiation the endogenous default
threshold is the same as that pertaining to a firm issuing bond market debt exclusively, with

promised payment ec = c+b. But L[Xd(0,ec)] < ecr. This is a contradiction. Corollary 9 A renegotiation region always exists for the Strong Firm issuing equal priority EBPD.

Figure 26 indicates that for the strong firm there exists an interior optimum debt mix (b∗,c∗) =

(5.89,32.53), when the bank is pari passu in the priority structure. However, the debt mix is

slanted towards BMD under pari passu, whereas it was roughly equal for senior bank debt. More

modest increases in firm value are achieved when the firm can exploit a mix of EBPD and BMD,

rather than relying on one source exclusively. The first line below reports the ex ante firm value

attained when the firm may mixes EBPD and BMD, when the bank is pari passu. The second and

third lines report valuations under exclusive BMD and EBPD finance, respectively:

v (b∗,c∗) = 816.15 (72)

v(c∗) = 809.30

v(b∗) = 752.01

Under equal priority, firm value increases by 0.8% when mixed debt is employed, rather than

financing exclusively with BMD. The increase in firm value is 8.5% when mixed debt is employed,

rather than financing exclusively with EBPD. For our baseline parameter values, placing the bank

senior in the priority structure dominates pari passu, with the ex ante firm values equal to 844.09

and 816.15, respectively. The increase in firm value associated with moving away from pari passu

is 3.4%.

Why does placing the bank senior generate superior ex ante firm valuations? Table 1 suggests

an answer. Placing the bank senior in the priority structure offers costs and benefits relative to

pari passu treatment. On the one hand, placing the bank senior leads to an increase in bankruptcy

costs equal to 4.78. The increase in bankruptcy costs is generated by the increase in the optimal

default threshold from 8.11 to 8.76. The optimal default threshold increases because placing the

bank senior raises the bank’s threat point in instantaneous negotiations, leading to higher requisite

debt service. Outweighing this cost is the benefit of high ex post debt service in terms of combined

tax shield increases and renegotiation cost reductions. Most of the benefit is generated by the

increase in the value of the tax shield, which increases in value by 32.27 when the bank is placed

senior in the priority structure.

VII. Conclusions/Extensions

• Non-linear tax schedule.

• Agency problems, e.g. endogenous volatility as in Leland (1998).

• Costly monitoring by banks.

• Monitoring of bank debt.

• Maturity structure, as in Leland and Toft (1996).

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Figure 2: Strategic debt service s(X) as a function of EBIT X in case of Equity
Bargaining Power Debt (EBPD). It is assumed that either there are negotiation
costs (δ > 0) or the firm issued Bond Market Debt (c > 0). Note that for positive
δ or c, and (δ,c) sufficiently low relative to b, ∃ Xd(b,c;δ) such that default occurs
endogenously and a nontrivial renegotiation region exists: Xd(b,c;δ) < Xs.

– X
6
s(X)

c
:
:
:
:

Xd(0,c)

¡
¡
¡
¡
¡
¡
¡

¡¡

b = c

Figure 3: Strategic debt service s(X) as a function of EBIT X in case of

Bank

Bargaining Power Debt (BBPD). It is assumed that there are no negotiation costs
(δ = 0) and that the promised coupon payments under Bank Bargaining Power Debt
and Bond Market Debt are identical: b = c. Note that the defaul-triggering EBIT
level under BMD is equal to the switch point under BBPD, i.e., Xd(0,c) = Xs(b).

– X
6
s(X)

b̃ = b+c

c
:
:
:
:
s

Xs(b,c)

¡
¡
¡
¡
¡
¡
¡

¡
¡
¡
¡
¡
¡
¡s:

:
:
:
:
:.

Xb(b,c)

Figure 4: Strategic debt service s(X) as a function of EBIT X in case of Bank
Bargaining Power Debt b̃ = b+c versus strategic debt service of a mixed debt firm
financed by Bank Bargaining Power Debt b and Bond Market Debt c. Note that the
incentive compatible switch point Xs is the same under both debt structures, but
the latter is associated with a strictly positive default point Xb(b,c). It is assumed
that there are no negotiation costs (δ = 0).

0 20 40 60 80

100

200

400

600

800

cmax

Figure 5: Bond market debt value C as a function of promised coupon payments c
when the firm’s EBIT volatility varies: σ = 10% (long-dashed line), σ = 20% (solid
line), and σ = 30% (short-dashed line). It is assumed X0 = 20, r = 6%, µ = 4%,
α = 50%, τ = 35%, η = 0. Note that the hard debt capacity cmax depends on firm’s
riskiness σ. Cf. Figure 1 in Leland (1994).

0 20 40 60 80
c

700

750

800

850

900

Figure 6: Total firm value v of a Bond Market Debt firm as a function of promised
coupon payments c when EBIT volatility varies: σ = 10% (long-dashed line), σ =
20% (solid line), and σ = 30% (short-dashed line). It is assumed X0 = 20, r = 6%,
µ = 4%, α = 50%, τ = 35%, η = 0. Note that the optimal leverage c∗ depends on
firm’s riskiness σ. Cf. Figure 6 in Leland (1994).

0 10 20 30 40 50
X

50
100

150

200

250

300

350

400

B
dna
R

Figure 7: Equity Bargaining Power Debt B and bank’s reservation value R as a
function of EBIT X for a given coupon level b as the firm’s bankruptcy costs vary:
α = 25% (long dashed line), α = 50% (solid line), and α = 75% (short-dashed line).
It is assumed that X0 = 20, b = 20, r = 6%, µ = 4%, σ = 20%, τ = 35%, δ = 0,
η = 1. Note that both the bank’s reservation value R (thin short-dashed lines) and
the incentive compatible switch point Xs depend on the firm’s bankruptcy costs α.

0 10 20 30 40 50
X
50
100
150
200
250
300
350
400

B
dna
C

Xd Xs

Figure 8: Equity Bargaining Power Debt B as a function of EBIT X for a given
coupon level b. It is assumed that X0 = 20, b = 20, r = 6%, µ = 4%, σ = 20%,
α = 50%, τ = 35%, δ = 10%, η = 1. The Bond Market Debt C as a function
of EBIT X for the same coupon level b = c is depicted by the dashed line. Note
that ceteris paribus default occurs earlier with Bond Market Debt than with Equity
Bargaining Power Debt.

0 10 20 30 40 50
X
50
100
150
200
250
300
350
400
B

=Xd Xs

Figure 9: Equity Bargaining Power Debt B as a function of EBIT X for a given
coupon level b as EBIT volatility varies: σ = 10% (long-dashed line), σ = 20% (solid
line), and σ = 30% (short-dashed line). It is assumed X0 = 20, b = 20, r = 6%,
µ = 4%, α = 50%, τ = 35%, δ = 0%, η = 1. Note that the default-triggering EBIT
level Xd is equal to zero in all cases because the negotiation cost δ is equal to zero,
but the incentive compatible switch point Xs still varies with the firm’s riskiness σ.

0 5 10 15 20 25
b

680

700

720

740

760

bmax

Figure 10: Total firm value v of an Equity Bargaining Power Debt firm as a function
of promised coupon b when the firm’s negotiation cost varies: δ = 10% (long-dashed
line), δ = 20% (solid line), and δ = 30% (short-dashed line). It is assumed X0 = 20,
r = 6%, µ = 4%, σ = 20%, α = 50%, τ = 35%, η = 1. Note that optimal leverage
b∗ depends on the size of negotiation costs δ, but not the firm’s debt capacity bmax.

0 10 20 30 40
b

680
700
720
740
760

780

800
v
bmaxb*

Figure 11: Total firm value v of an Equity Bargaining Power Debt firm as a function
of promised coupon b when the firm’s bankruptcy costs vary: α = 25% (long dashed
line), α = 50% (solid line), and α = 75% (short-dashed line). It is assumed that
X0 = 20, r = 6%, µ = 4%, σ = 20%, τ = 35%, δ = 10%, and η = 1. Note that both
optimal leverage b∗ and debt coupon capacity bmax depend on the firm’s bankruptcy
costs α.

0 5 10 15 20 25 30 35
b

660

680
700
720
740
760
v

b bmaxb* bmaxb* bmaxb* bmax

Figure 12: Total firm value v of an Equity Bargaining Power Debt firm as a function
of promised coupon b when the firm’s EBIT volatility varies: σ = 10% (long-dashed
line), σ = 20% (solid line), and σ = 30% (short-dashed line). It is assumed X0 = 20,
r = 6%, µ = 4%, α = 50%, τ = 35%, δ = 10%, η = 1. Note that both optimal
leverage b∗ and debt capacity bmax depend on the firm’s riskiness σ.

0 5 15 20
X

2
4
6
8

12
14
s

Xd X
~
d

Loss

Gain

Xs X
~
s

Figure 13: “Loss” versus “Gain” of bank debt’s strategic service flow s with equity
making take-it-or-leave-it offers as a function of EBIT X when leverage increases
from b = 10 to b̃ = 15. It is assumed X0 = 20, r = 6%, µ = 4%, σ = 20%, α = 50%,
τ = 35%, δ = 10%, η = 1. Note that both the default-triggering EBIT level Xd
and the incentive compatible switch point Xs are (strictly) increasing in the firm’s
leverage choice: Xd < X̃d and Xs < X̃s.

0 0.1 0.2 0.3 0.4

0.5

alpha

780
800

820

840

860

880

Bank

Bond

Figure 14: Trade-off in terms of total (optimally levered) firm value when choosing —
as an exclusive source of financing — between Equity Bargaining Power Debt (dashed
line) and Bond Market Debt (solid line) as a function of bankruptcy costs α. An
optimal degree of leverage, i.e., promised coupon payments b∗ (α, ·) and c∗ (α, ·),
X0 = 20, r = 6%, µ = 4%, σ = 30%, τ = 35%, and δ = 10% are assumed.

0.1 0.2 0.3 0.4 0.5
tau

650

700
750
800
850
900

950

v*
Bank
Bond

Figure 15: Trade-off in terms of total (optimally levered) firm value when choosing
— as an exclusive source of financing — between Equity Bargaining Power Debt
(dashed l ine ) and Bond Market Debt ( sol id line ) as a function of corp orate taxes τ .
An optimal degree of leverage, i.e., promised coupon payments b∗ (τ, ·) and c∗ (τ, ·),
X0 = 20, r = 6%, µ = 4%, σ = 30%, α = 50%, and δ = 10% are assumed.

0 0.2 0.4 0.6 0.8
alpha

0.1

0.2

0.3

0.4

0.5

ta
u

Bank
Bond

Figure 16: Implicit trade-off based on total (optimally levered) firm value when
choosing between Equity Bargaining Power Debt and Bond Market Debt as an
exclusive s ource of financing i n terms of bankruptcy costs α and corp orate taxes
τ when EBIT riskiness varies: σ = 15% (long-dashed line), σ = 20% (solid line),
σ = 25% (short-dashed line). An optimal degree of leverage, i.e., promised coupon
payments b∗ (α,τ, ·) and c∗ (α,τ, ·), X0 = 20, r = 6%, µ = 4%, and δ = 10% are
assumed. Note that ceteris paribus it is more likely to be optimal for riskier firms
to choose Equity Bargaining Power Debt over Bond Market Debt and vice versa.

0 0.1 0.2 0.3 0.4 0.5
alpha

760

770

780

790

v
Bank
Bond

Figure 17: Trade-off in terms of total (equally levered) firm value when choosing —
as an exclusive source of financing — between Equity Bargaining Power Debt (dashed
line) and Bond Market Debt (solid line) as a function of bankruptcy costs α. An
equal degree of leverage, i.e., promised coupon payments b = c = 30, and X0 = 20,
r = 6%, µ = 4%, σ = 30%, τ = 35%, δ = 10% are assumed.

0.1 0.2 0.3 0.4 0.5
tau
700
750
800
850
900
v
Bank
Bond

Figure 18: Trade-off in terms of total (equally levered) firm value when choosing —
as an exclusive source of financing — between Equity Bargaining Power Debt (dashed
line) and Bond Market Debt (solid line) as a function of corporate taxes τ. An
equal degree of leverage, i.e., promised coupon payments b = c = 30, and X0 = 20,
r = 6%, µ = 4%, σ = 30%, α = 50%, δ = 10% are assumed.

0 0.2 0.4 0.6 0.8
alpha
0.1
0.2
0.3
0.4
0.5
ta
u
Bond
Bank

Figure 19: Implicit trade-off based on total (equally levered) firm value when choos-
ing between equity bargaining power bank and Bond Market Debt as an exclusive
source of financing in terms of bankruptcy costs α and corporate taxes τ when bank
debt’s negotiation costs vary: δ = 5% (long-dashed line), δ = 10% (solid line),
δ = 15% (short-dashed line). An equal degree of leverage, i.e., promised coupon
payments b = c = 30, and X0 = 20, r = 6%, µ = 4%, σ = 30% are assumed. Note
that ceteris paribus it is more likely to be optimal for firms with lower negotiation
costs to choose Equity Bargaining Power Debt over Bond Market Debt and vice
versa.

0 2 4 6 8 10
X

50
100
150
200
250
300
B
Xs

Figure 20: Bank Bargaining Power Debt B as a function of EBIT X for a given
coupon level b when the firm’s negotiation cost varies: δ = 5% (long-dashed line),
δ = 10% (solid line), and δ = 20% (short-dashed line). It is assumed X0 = 20,
b = 20, r = 6%, µ = 4%, σ = 20%, α = 50%, and τ = 35%. Note that in
the renegotiation region the bank is the residual claimant on the firm’s cash flows
and hence at the default-triggering EBIT level Xb the value and the slope of bank
bargaining power debt are equal to the corresponding value and slope of the firm’s
restructuring function (1−α)(1−τ)X/(r−µ) (thin short-dashed line).

0 2 4 6 8 10 12 14
X

50
100
150
200
250
300
350
400
B
Xs

Figure 21: Bank Bargaining Power Debt B as a function of EBIT X for a given
coupon level b as EBIT volatility varies: σ = 10% (long-dashed line), σ = 20% (solid
line), and σ = 30% (short-dashed line). It is assumed X0 = 20, b = 20, r = 6%,
µ = 4%, α = 50%, τ = 35%, and δ = 10%. Note that for any nontrivial negotiation
costs (δ > 0) both the default-triggering EBIT level Xb and the incentive compatible
switch point Xs are decreasing in the firm’s riskiness σ.

4 4.2 4.4 4.6 4.8 5 5.2 5.4
X

1
2
3
4
5
6
E
Xs

Figure 22: Equity E under Bank Bargaining Power Debt as a function of EBIT X.
It is assumed that X0 = 20, r = 6%, µ = 4%, σ = 20%, α = 50%, τ = 35%, and
δ = 10%. Note value-matching and smooth-pasting obtain at the switch point Xs.

0 20 40 60 80 100 1

20
b

700
750
800
850
900
950

1000

v
b*

Figure 23: Total firm value v of a Bank Bargaining Power Debt firm as a function
of promised coupon b when the firm’s negotiation cost varies: δ = 5% (long-dashed
line), δ = 10% (solid line), and δ = 20% (short-dashed line). It is assumed X0 = 20,
r = 6%, µ = 4%, σ = 20%, α = 50%, and τ = 35%. Note that optimal leverage b∗

depends on the size of negotiation costs δ.

0 20 40 60 80 100 120
b
700
750
800
850
900
950
1000
v
b*

Figure 24: Total firm value v of a Bank Bargaining Power Debt firm as a function
of promised coupon b when the firm’s EBIT volatility varies: σ = 10% (long dashed
line), σ = 20% (solid line), and σ = 30% (short-dashed line). It is assumed that
X0 = 20, r = 6%, µ = 4%, α = 50%, τ = 35%, and δ = 10%. Note that optimal
leverage b∗ depends on the firm’s riskiness σ.

0
10
20
30
b

0
10

20
30

40
50

c
650
700
750
800
850
v
0
10
20
30
b

Figure 25: Total firm value v as a function promised coupon payments to the bank
b and to the bond market c when equity holders have all the bargaining power vis-
à-vis the bank and the bank debt is senior (η = 1). It is assumed that X0 = 20,
r = 6%, µ = 4%, σ = 20%, α = 50%, τ = 35%, and δ = 10%, which implies that
the unlevered firm is worth V (X0) = (1−τ)X0/(r−µ) = 650. Note that there
exists an interior firm value optimum for both classes of debt v (b∗,c∗).

0
10
20
30

b
0

10
20

30
40

50
c
650
700
750
800
850
v
0
10
20
30
b

Figure 26: Total firm value v as a function promised coupon payments to the bank
b and to the bond market c when equity holders have all the bargaining power and
soft and hard debt are pari passu, i.e., equal priority: η = b/(b+ c). It is assumed
that X0 = 20, r = 6%, µ = 4%, σ = 20%, α = 50%, τ = 35%, δ = 0.1, which
implies that the unlevered firm is worth V (X0) = (1−τ)X0/(r−µ) = 650. Note
that there exists an interior firm value optimum for both classes of debt v (b∗,c∗).

0
10
20
30
b
0
10
20
30
40
50
c
0
50
100
150
200
250

0
10
20
30
b

Figure 27: Tax benefits TB as a function promised coupon payments to the bank
b and to the bond market c when equity holders have all the bargaining power vis-
à-vis the bank and the bank debt is senior (η = 1). It is assumed that X0 = 20,
r = 6%, µ = 4%, σ = 20%, α = 50%, τ = 35%, and δ = 10%.

0
10
20
30
b
0
10
20
30
40
50
c
0
50
100
150
200

0
10
20
30
b

Figure 28: Bankruptcy costs BC as a function promised coupon payments to the
bank b and to the bond market c when equity holders have all the bargaining power
vis-à-vis the bank and the bank debt is senior (η = 1). It is assumed that X0 = 20,
r = 6%, µ = 4%, σ = 20%, α = 50%, τ = 35%, and δ = 10%.

0
10
20
30
b
0
10
20
30
40
50
c
0
5
10
15
N
0
10
20
30
b
0
5
10
15
N

Figure 29: Negotiation costs N aas a function promised coupon payments to the
bank b and to the bond market c when equity holders have all the bargaining power
vis-à-vis the bank and the bank debt is senior (η = 1). It is assumed that X0 = 20,
r = 6%, µ = 4%, σ = 20%, α = 50%, τ = 35%, and δ = 10%.

0
10
20
b
0
10
20
30
40
50
c
0
200
400
600
E
0
10
20
b
0
200
400
600
E

Figure 30: Equity value E as a function promised coupon payments to the bank b
and to the bond market c when equity holders have all the bargaining power vis-
à-vis the bank and the bank debt is senior (η = 1). It is assumed that X0 = 20,
r = 6%, µ = 4%, σ = 20%, α = 50%, τ = 35%, and δ = 10%.

0
10
20
b
0
10
20
30
40
50
c
0
100
200
300
B
0
10
20
b
0
100
200
300
B

Figure 31: Bank debt value B as a function promised coupon payments to the bank
b and to the bond market c when equity holders have all the bargaining power vis-
à-vis the bank and the bank debt is senior (η = 1). It is assumed that X0 = 20,
r = 6%, µ = 4%, σ = 20%, α = 50%, τ = 35%, and δ = 10%.

0
10
20
b
0
10
20
30
40
50
c
0
100
200
300
400
C
0
10
20
b
0
100
200
300
400
C

Figure 32: Hard debt value C as a function promised coupon payments to the bank
b and to the bond market c when equity holders have all the bargaining power vis-
à-vis the bank and the bank debt is senior (η = 1). It is assumed that X0 = 20,
r = 6%, µ = 4%, σ = 20%, α = 50%, τ = 35%, and δ = 10%.

0 5 10 15 20 25
b
10
11
12
13
14
15
c

Xd(b,c )= 5

Figure 33: Complementarity effect between Equity Bargaining Power Debt and
Bond Market Debt. Feasible pairs of Equity Bargaining Power Debt and Bond
Market Debt (b,c) when the equity’s endogenous default-triggering EBIT level is
fixed at Xd (b,c;δ) = 5 and the firm’s negotiation costs vary: δ = 5% (long-dashed
line), δ = 10% (solid line), δ = 15% (short-dashed line). It is assumed that X0 = 20,
r = 6%, µ = 4%, α = 50%, τ = 35%, and σ = 20%.

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