Banks in the Venture Capital Market
Ming-Jin Jiangyz
University of Vienna
&
Marbella International University Centre
March, 2016
Abstract
This paper analyzes bank investing in the venture capital market,
which is permitted by the Gramm-Leach-Bliley Act (1999). I develop
a principal-agent model in which startup …rms have a growth opportunity and need funds to invest. The realization of non-veri…able returns
depends on the fund invested and the managers’/entrepreneurs’e¤ort,
which is unobservable to the banks. I show that convertible debt contracts can mitigate ex-post ine¢ ciency arising from …rms’strategic default due to non-veri…ability of returns. I further generalize the model
by considering moral hazard of the managers/entrepreneurs. In this
case, the optimality of convertible debt contracts for startup …rms is
shown. Finally, I develop a dynamic model in which I take into account
the relationship values of the startup …rms and their banks. I show
that in this case convertible debt contracts are optimal for startup …rms
with volatile returns, which is consistent with empirical evidence (Kaplan and Strömberg, 2003). In addition, when …rms’relationship value
is low and banks’relationship value is high, banks do not convert and
I thank the …nancial support from DIRECCIÓN GENERAL DE INVESTIGACIÓN
CIENTÍFICA Y TÉCNICA (Project ECO2010-20614).
y
I thank A. Cabrales, A. Diaz, F. Feriozzi, E. García Moran, M. Kredler, M. Mariathasan,
S. Ortigueira, H. Rachinger, C. Roessler for their advice. I am grateful to the participants
of the UC3M-Macroworkshop and the participants of seminars at IHS Vienna, University
of Lancaster, CUNEF, Universitat de Illes Balears, University of Vienna, Universität Ulm,
Hungarian Academy of Science and IWH-Halle.
z
Contact information: Department of Finance, University of Vienna, Oskar-MorgensternPlatz 1, 1090 Vienna, E-mail: [email protected]
1
thus the lending relationship between banks and …rms is maintained.
This result is also consistent with evidence that banks participate in
venture capital …nance in order to build the lending relationship early
on with potential clients (Hellman et al., 2008).
Keywords: Start-up …rms, Venture capital …nancing, Non-veri…able returns, Moral hazard, Bank-Firm relationship
JEL classi…cation: D80, D86, G14
1
Introduction
Venture capitalists (VCs) play an important role in startup …nance. Speci…cally, VCs are often considered as value-adding investors to the venture …rms
since VCs provide other supportive services to increase the value of the …rms.
Kaplan and Strömberg (2001, 2004) show that VCs expect to professionalize
the …rms. Lerner (1995), Baker and Gompers (2003), and Hochberg (2012)
…nd that VCs play an important role in determining the composition of the
board of directors. This evidence con…rms the VCs’ role in supporting the
…rms to increase the …rms’value. Besides VCs, banks also participate in venture capital …nancing1 . However, banks do not play the same role as VCs do2 .
Hellman et al. (2008) argue that banks, unlike VCs, participate in the later
rounds of venture …nancing, and thus may not be considered as value-adding
investors.
In this paper, I consider bank …nancing in the venture capital market. In
particular, in an environment that …rms’returns are not veri…able, the managers’/entrepreneurs’exerted e¤ort is unobservable and …rms are protected by
1
Since the late 80s, the 1933 Glass-Steagall Act has been partially relaxed. In particular,
the Gramm-Leach-Bliley Act (1999) allows the U.S. commercial banks to participate in
the capital market up to a certain degree. In particular, banks are allowed to hold equity
and/or participate in the venture capital market under certain restrictions. Speci…cally,
section 1843(k)(4)(H) imposes some limits on banks’activity in venture capital. Subsection
(iv) states that the banks may not "routinely manage or operate such company or entity
except as may be necessary or required to obtain a reasonable return on investment upon
resale or disposition."
2
Banks are commonly believed to play an important role in screening and monitoring.
However, due to the restrictions in Gramm-Leach-Bliley Act, they are not expected to
increase the …rms’value by actitively participating in the …rms’management.
2
limited liability when they default. Non-veri…ability of …rms’returns results
in …rms’strategic default, and hence distorts the external …nanciers’e¢ cient
lending decisions. Firms are heterogeneous in their initial assets. Their payo¤
of strategic default varies with their initial assets. Firms with small initial
assets have little to lose if they default and …le for bankruptcy. Consequently,
…rms with smaller initial assets are more likely to default strategically than
those with larger initial assets.
I …rst develop a one-period principal-agent model and consider the three
most commonly used …nancial contracts: standard debt, equity, and convertible debt contracts. The goal is to derive the optimal contracts so that in
equilibrium, the managers/entrepreneurs exert e¢ cient e¤ort and there are no
strategic defaults. I concentrate on the impact of non-veri…ability of …rms’
returns on determining the optimal contracts. I show that if …rms’ returns
are not veri…able, standard debt contracts are dominated by both equity and
convertible debt contracts for …rms with small initial assets because their managers/entrepreneurs have incentives to default strategically. Furthermore, I
generalize the model by considering additionally a moral hazard problem due
to managers’/entrepreneurs’unobservable e¤ort. In this case, convertible debt
contracts are optimal for …rms with initial assets below a threshold. A convertible debt contract gives the investors an unilateral right to convert the
debt into equity at the predetermined time and price (conversion rate). Such
a contract has properties that combine both debt and equity. Speci…cally,
it can be thought as a standard debt contract plus a call option to convert
the debt into equity when the …rm’s return has greater upside potential. I
show that this feature of convertible debt contracts can solve the problem of
managers’/entrepreneurs’strategic default arising from the non-veri…ability of
…rms’ returns. Moreover, debt dominates equity in terms of inducing e¤ort.
Consequently, convertible debt is shown to be optimal for startup …rms with
small assets.
Further, I develop a dynamic model and introduce a bank-…rm relationship
value. The managers’/entrepreneurs’decision of whether to default strategically also depends on whether they need subsequent …nancing from the same
3
banks. In other words, if the …rms depend considerably on subsequent …nancing from the same bank (i.e., …rms have a large relationship value), they would
not have incentives to default strategically since once they default, the relationship ends and they could not obtain funds in the future periods. This implies
that the incentive of strategic default is negatively related to the relationship
value. This relationship value crucially depends on the investment outcome at
the end of the …rst period. If the investment project is very successful (i.e., the
realized return is very high), the relationship value is low because …rms could
easily obtain …nance from other banks, and thus do not rely on …nance from
the same bank. On the other hand, if the project generates an intermediate return in the …rst period, …rms rely on …nance from the same bank, and thus the
relationship value is high. Finally, if the project fails and …rms go bankrupt,
the relationship between banks and …rms ends. Hence, there is no relationship
value in this case. In this dynamic setup, equity contracts are dominated by
both convertible debt and standard debt contracts because of the moral hazard problem. When comparing convertible debt contracts with standard debt
contracts, I show that when the …rm’s relationship value is low, convertible
debt contracts dominate standard debt contracts. The reason is that convertible debt contracts prevent managers’/entrepreneurs’ strategic default when
the realized return in the …rst period is very high (i.e., the relationship value
is low).
My result is consistent with empirical evidence which demonstrates the
importance of convertible debt contracts in the venture capital market. Kaplan
and Strömberg (2003) analyze 213 VC investments in 119 portfolio companies
by 14 VCs and …nd only seven of the 213 …nancing rounds do not use any
form of convertible security. In addition, my results also show that when
convertible debt contracts are o¤ered, banks do not convert if the relationship
value is high. This is consistent with the evidence that banks participate in
venture capital …nance in order to build a relationship early on with the …rms
(Hellman et al., 2008).
This paper adds to the literature on venture capital …nancing by di¤erentiating the role of banks from the one of VCs. Previous literature studies
4
the agency problem caused by double moral hazard between the …rms and
the VCs. Schmidt (2003) and Repullo and Suarez (2004) show that under a
double moral hazard problem, the optimal contracts are convertible debt contracts. This strand of literature mainly focuses on the optimal security design
of venture capital …nancing between VCs and …rms. Hence, it emphasizes the
value-adding role of VCs. However, banks play a di¤erent role in venture capital …nancing. Therefore, this framework of double moral hazard would not be
suitable for analyzing bank …nancing in the venture capital market.
The outline of this paper is the following: In Section 2, I …rst analyze the
benchmark model. Afterwards, I relax the assumption of veri…ability of …rms’
returns and characterize the optimal contracts. In Section 3, I generalize the
model by taking into account moral hazard and analyze the optimal contracts.
In Section 4, I further develop a dynamic model in which I show that convertible debt contracts are optimal for startup …rms with volatile returns. Finally,
I conclude in Section 5.
2
The Model
In this section, I present a principal-agent model. Managers/entrepreneurs
(agents) have an investment project of positive present value, but they do not
have funds to …nance the project. As a result, …rms have to obtain funds from
banks, the principals. The assumptions of the model are:
Assumption 1 Managers/entrepreneurs and banks are both risk neutral.
The …rms are heterogenous in terms of their initial assets A as collateral,
A 2 [0; 1)3 . The banks’opportunity cost of lending per unit is assumed to
be exogenous, and for simplicity, equal to one.
3
In this paper, I assume a …rm’s initial asset serves as a collateral. However, this assumption does not qualitatively change the analysis. It can be instead assumed that the …rm’s
initial asset can be liquidated without any liquidation cost, and the …rm can use (partially)
its own asset to self …nance.
5
Assumption 2 The investment project
8 H
>
if the project is very successful,
<
y=
if the project is successful,
>
:
0
if the project fails,
yields a random return,
with probability p (B; e)
with probability q
with probability (1
p (B; e)
The distribution of the realized returns is endogenous, depending on the funds
B invested in the project and managers/entrepreneurs’e¤ort e.
Assumption 3 There are two levels of managers’/entrepreneur’s e¤ort,
e 2 feL ; eH g. The e¤ort is observable. The cost is increasing in the e¤ort with
C (eL ) = 0 and C (eH ) = cH > 0.
The probability of the project being very successful p (B; e) depends on
each …rm’s capital investment B and its e¤ort e. This probability is increasing
in capital investment B and exerted e¤ort e, in particular, p0B (B; e) > 0 and
p00B (B; e) < 0 for any e, and p (B; eH ) > p (B; eL ) for any B. With probability
q, the project is successful with an intermediate return . The probability q
is exogenous, and does not depend on either B or e. However, the overall
probability of success p (B; e) + q is increasing in both B and e.
Assumption 4 The project returns of …rms are observable to both parties
(…rms and banks), however, returns are not veri…able by a third party (e.g.,
court).
Assumption 5 Firms’liability to debt, as well as the banks’liability to
the investment are limited. That is, if a …rm obtains debt from a bank, once
it defaults and …les for bankruptcy, the bank liquidates the …rm’s assets up to
the required repayment. Besides, the repayment to the bank in each state is
non-negative.
Assumption 6 banks compete à la Bertrand.
Assumption 4 is crucial in this paper. In the following analysis, I …rst
analyze the benchmark model without Assumption 4, i.e., an environment in
which …rms’returns are veri…able. Next, I analyze the e¤ect of non-veri…ability
on the optimal contracts. Further, I generalize the model by relaxing Assump6
q) :
tion 3 and consider that managers’/entrepreneurs’e¤ort is unobservable to the
banks.
2.1
The benchmark case
In this section, I derive the optimal contract as a benchmark in the environment that both managers’/entrepreneurs’ e¤ort and project returns, are observable and veri…able. Afterwards, I focus on three types of contracts which
are commonly used in practice: (1) standard debt contract, (2) equity contract
and (3) convertible debt contract.
The optimal contract outcomes are the solution to the following maximization problem:
max
B;e;s1 ;s2
p (B; e)
+ (1
H
R1 + A + q (
p (B; e)
(1)
R2 + A)
q) ( R3 + A)
c (e)
s.t.
E
l
= p (B; e) R1 + qR2 + (1
H
p (B; e)
R1 + A
R2 + A
R3 + A
q) R3
0
0
0:
B
0
(2)
(3)
(4)
(5)
where (R1 ; R2 ; R3 ) is the repayment schedule, which is contingent on the realized …rms’returns.
Lemma 1 shows that, in equilibrium, the bank’s participation constraint
always binds due to Bertrand competition.
Lemma 1 E
l
= 0:
The equilibrium borrowing amount B (ej ) depends on e¤ort e and satis…es
p0B (B (ej ) ; ej ) =
1
H
; where j = H; L:
(6)
I further assume that it is optimal to exert high e¤ort for all …rms in the
following assumption.
7
Assumption 7 Exerting eH is optimal.
Assumption 7 implies that, in equilibrium, a …rm’s expected pro…t if exerting eH is strictly higher than the expected pro…t if exerting eL for any A:
p (B (eH ) ; eH )
> p (B (eL ) ; eL )
H
H
+q
+q
B (eH ) + A
cH
(7)
B (eL ) + A;
equivalently,
H
>
(B (eH ) B (eL )) + cH
:
p (B (eH ) ; eH ) p (B (eL ) ; eL )
(8)
Proposition 2 The optimal contract (B; R1 ; R2 ; R3 jA) is a state-contingent
contract. In equilibrium,
(1) B = B (eH ) = B , where p0B (B ; eH ) =
1
H
;
(2) Due to risk neutrality, repayment schemes
(R1 ; R2 ; R3 jR2
+ A; R3
A; R1 = R1 (R2 ; R3 ))
are not unique.
Proposition 2 shows that the optimal borrowing amount B positively depends on H .
Due to the risk-neutrality of the managers/entrepreneurs and the banks,
the optimal repayment scheme is indeterminate. According to the Modigliani
and Miller (1958) Theorem on the irrelevance of …rms’ …nancial structure,
any equilibrium repayment scheme that satis…es Proposition 2, is optimal.
Therefore, standard debt, equity and convertible debt contracts are all optimal
in the benchmark case. Note that in this paper, I assume that the costs of
signing di¤erent types of contracts are the same. Without loss of generality4 ,
I assume that the cost is equal to zero. If the costs were di¤erent, the contract
with the lowest cost would be optimal.
Assumption 8 The cost of signing standard debt, equity and convertible
debt contracts equals to zero.
4
As long as the costs of signing di¤erent types of contracts are equal, there is no di¤erence
between assuming cost = 0 or cost = c (> 0), where c is constant.
8
In the following sections, I analyze these three types of contracts which are
commonly used in practice –standard debt, equity and convertible debt contracts –and show that if …rms’returns and e¤ort are observable and veri…able,
all three types of contracts are optimal.
2.1.1
Standard debt contract
A standard debt contracts (B; rjA) speci…es the …rm’s borrowing amount B
and the corresponding interest rate r for a …rm with initial asset A. In equilibrium, the contract (B; rjA) satis…es the bank’s participation constraint,
p (B; e) Br + qBr + (1
p (B; e)
q) min (Br; A)
B
0;
(9)
the …rms’limited liability constraints,
H
Br + A
Br + A
min (Br; A) + A
(10)
0
(11)
0
(12)
0;
and the …rm’s expected pro…t is maximized.
Proposition 3 The equilibrium standard debt contract (B; rjA) is optimal for
all risk-neutral …rms, where in equilibrium
(1) B = B
(2) r = 1 for A
B ; and r =
B
(1 p q)A
(p +q)B
> 1 for A < B , where
5
p = p (B ; eH ) :
For unconstrained …rms (…rms with initial assets A
B ), in equilibrium,
both limited liability constraints do not bind. Besides, they have enough initial
assets to repay fully even if the project fails. Their banks always obtain the
full repayment, thus, their debt is secured, and the equilibrium interest rate
for the unconstrained …rms is equal to one.
For constrained …rms (…rms with initial assets A < B ), Equation (12)
binds in equilibrium. Therefore, if the project fails, the …rm goes bankrupt,
5
In the following analysis in this paper, I denote p = p (B ; eH ) :
9
and its asset is liquidated by the bank. The bank cannot receive the full
repayment once the project fails, thus, the equilibrium interest rate is higher
than one in order to satisfy the bank’s participation constraint. Besides, the
equilibrium interest rate for a constrained …rm decreases when the …rm’s initial
asset increases.
Given that the …rms’returns are observable and veri…able by a third party,
under standard debt contracts, …rms do not default strategically (i.e., they
do not default when they are able to repay). If the project succeeds, since
the realized return can be veri…ed, …rms’ assets will be liquidated up to the
required repayment. Banks will obtain the same full repayment as if the
managers/entrepreneurs did not default.
2.1.2
Equity contract
An equity contract (B; sjA) speci…es the investment amount B that the bank
invests and the share s (0 < s < 1) of the …rm(’s value) that the bank obtains
(at the end of the period). Speci…cally, the pro…t the bank obtains is sy.
Besides cash ‡ow right, banks, as equity holders, also have control right. In
other words, they can dismiss the old manager/entrepreneur6 and hire a new
manager (Fluck (2010)). If banks dismiss the manager/entrepreneur, they
bear a dismissal cost. As for the manager/entrepreneur, once she is dismissed,
her return is zero.
In equilibrium, the contract (B; sjA) for a given A satis…es the banks’
participation constraint,
p (B; e) s
H
+ A + qs ( + A) + (1
p (B; e)
q) sA
B
0
(13)
and the …rms’expected pro…ts are maximized.
Proposition 4 The equilibrium equity contract (B; sjA) given an initial asset
A is optimal, where in equilibrium
(1) B = B
6
Gramm-Leach-Bliley Act (1999) restricts banks (or bank a¢ liates), as an equity holder,
to rountinely manage or operate the …rms in which they invest. However, replacing the old
manager of the …rm does not constitute routine management (12 CFR 225.171 (d)).
10
(2) s =
p
B
:
+q +A
H
Under equity contracts, the manager/entrepreneur does not default. The
bank, as an equity holder, automatically obtains its share of the …rm’s pro…t.
2.1.3
Convertible debt contract
De…nition 5 Convertible debt contracts
A convertible debt contract is a standard debt contract plus a call option
which gives the …nancier a unilateral right to convert the debt to equity at a
predetermined time and at a predetermined conversion rate.
A convertible debt contract (B; i; jA) for a …rm with initial asset A spec-
i…es the borrowing amount B and interest rate i and a predetermined conversion rate (0 <
< 1). Speci…cally, the bank receives y if the bank
converts debt to equity. Once a bank converts, it becomes the equity holder
of the …rm and the properties of an equity contract applies. The equilibrium
convertible debt contract maximizes the …rm’s expected pro…t subject to the
bank’s participation constraint,
p (B; e) max Bi;
+ (1
H
+A
+q max (Bi; ( + A))
p (B; e) q) max (min (A; Bi) ; A) B
(14)
0;
and the three limited liability constraints of the …rm,
H
max Bi;
H
+A
+A
max (Bi; ( + A)) +A
max (min (A; Bi) ; A) +A
0
(15)
0
(16)
0:
(17)
In equilibrium, if the project is either successful or very successful, managers/entrepreneurs have no incentive to default strategically because if they
did so, the banks could either convert the debt to equity and dismiss the manager, or even if they did not convert, they could go to the court and verify
the …rms’ return. If the project fails, in equilibrium, the constrained …rms
11
always default, and the banks will not convert because
A < A. As for the
unconstrained …rms, they have no incentives to default, and the banks obtain
the full repayment.
Proposition 6 Given an initial asset A, the equilibrium convertible debt contract (B; i; jA) is optimal, where
(1) B = B for all …rms,
(2) i = 1 for A
(3)
Moreover,
B
+A
H
A B
<
B ; and i 2 [ (B
(p (
for A
B ; and
A<B
(1 p q)A)( +A)
+A)+q( +A))B
= B (1p p(
H
; r ] for A < B ;
q)A qB i
+A)
H
for A < B .
:
In Proposition 7, I further show that for constrained …rms, the equilibrium
interest rate is weakly lower under convertible debt contract compared with
the equilibrium interest rate under standard debt contract.
Proposition 7 For A < B , the equilibrium interest rate r under a standard
debt contract is weakly higher than the interest rate i under a convertible debt
contract.
Proposition 7 shows that, for a constrained …rm, the interest rate under
convertible debt is lower than the one under standard debt contract. The
intuition is that under convertible debt contracts, banks have the right to
convert, hence, banks could use this right to share the …rm’s pro…ts when the
…rm has a very high return. This, ex ante compensates the banks’ received
repayment when the …rm’s return takes an intermediate value. As a result, the
convertible debt resembles the properties of a state-contingent debt contract.
This leads to a lower interest rate under convertible debt.
Moreover, comparing equity contracts with convertible debt contracts, Proposition 8 shows that in equilibrium, the conversion rate under a convertible debt
contract is lower than the equilibrium share in the equity contract.
12
Proposition 8 For A < B , the equilibrium share s under an equity contract
is strictly higher than the equilibrium conversion rate
under a convertible
debt contract.
Proposition 8 shows that the equilibrium share under equity contracts is
higher than the equilibrium conversion rate under convertible debt contracts.
The reason is that, under equity contracts, banks would simply, and necessarily, share the …rm’s pro…t, even in the case of the …rm having a zero return.
However, under convertible debt, banks are more ‡exible and could choose not
to be the equity holders when the …rm’s return is zero. In this case, banks
remain debt holders, who liquidate the …rm and obtain the …rm’s initial assets.
Consequently, when the …rm’s return is zero, the banks could obtain a larger
payo¤ under convertible debt than under equity, and thus, the equilibrium
conversion rate is lower than the equilibrium share under equity contracts.
In this section, I have shown that standard debt, equity and convertible
debt contracts are all optimal when …rms’ returns are veri…able. Since under veri…ability for all three types of contracts, managers/entrepreneurs have
no incentives to default strategically. Besides, under risk-neutrality of both
managers/entrepreneurs and their banks, …rms are indi¤erent among all three
contracts since, in equilibrium, …rms’expected pro…ts are equal and are maximized under all three types of contracts. Further, I have compared these three
types of contracts and demonstrated that convertible debt contracts are hybrid
contracts of standard debt and equity contracts. As a result, the equilibrium
interest rate under convertible debt is lower than the equilibrium interest rate
under standard debt, and the equilibrium conversion rate under convertible
debt is lower than the equilibrium share under equity contracts.
2.2
Non-veri…able …rms’returns
In this section, I consider that …rms’project returns are not veri…able by a third
party (Assumption 3). For unconstrained …rms (A B ), all three types of
contracts are still optimal. First, under standard debt contracts, unconstrained
…rms do not have incentives to default strategically. Unconstrained …rms have
13
enough assets such that even if they default, the banks still obtain the full
repayment B . Moreover, since convertible debt and equity contracts eliminate
the incentive of ex-post strategic default, they are both optimal under nonveri…ability of …rms’returns.
However, for constrained …rms, standard debt contracts are not optimal
when the …rms’ returns are not veri…able. The reason is that constrained
…rms have incentives to default strategically if standard debt contracts are
o¤ered. Specially, if the realized return is H or , the managers/entrepreneurs
have incentives to lie to the banks. Suppose that the realized return is
H
or , under standard debt contracts, if the manager/entrepreneur lies and
pretends that the realized return is zero, the …rm’s asset is liquidated, and its
pro…t is equal to the realized return, which is still higher than the pro…t the
managers/entrepreneurs would get if they repay truthfully:
H
|{z} > |
default
|{z} > |
default
H
B
{z r + A};
(18)
do not default
B{zr + A}:
(19)
do not default
As a result, equilibrium standard debt contracts must mitigate the managers’/entrepreneurs’incentives to default strategically. That is, if the project
succeeds, the equilibrium repayment must not exceed the …rms’ initial asset
A. Therefore, the equilibrium borrowing amount B equals the …rms’ initial
assets A. Hence, if …rms’returns are not veri…able, standard debt contracts
are not optimal for constrained …rms.
Equity contracts and convertible debt contracts are both immune to ex-post
strategic default. An equity holder could dismiss the manager/entrepreneur
if the manager/entrepreneur lies.
This threat of dismissal eliminates the
incentive of strategic default under an equity contract, since if the manager/entrepreneur is dismissed, her return is zero. Thus,
0
|{z}
<
default & be dismissed
(1 s ) y :
| {z }
(20)
do not default
Convertible debt contracts grant the banks a unilateral right to convert debt to
equity. As analyzed in Section 2.1.3, the equilibrium convertible debt contract
14
resembles a state-contingent contract if the conversion is exercised. Under
convertible debt contracts, the manager/entrepreneur does not have incentives
to default strategically, since banks can convert and become equity holders.
In the following Lemma, I show that banks would convert for certain if the
manager/entrepreneur defaults.
Lemma 9 Under convertible debt contracts, banks convert and dismiss the
manager/entrepreneur if she defaults strategically as long as the dismissal cost
is not too large.
In summary, in the case of non-veri…ability of …rms’returns, standard debt
contracts are weakly dominated by the other two types of contracts.
Proposition 10 Under non-veri…ability of …rms’returns,
(1) for unconstrained …rms, standard debt, equity and convertible debt are
optimal,
(2) for constrained …rms, standard debt contracts are dominated. Equity
and convertible debt contracts are optimal.
So far, I have shown that under non-veri…ability of …rms’returns, standard
debt contracts are dominated by convertible debt and equity contracts, in
which case both are optimal. In next section, I further relax Assumption 3.
3
Generalized Model: Moral hazard
In this section, I generalize the model and relax the assumption of observable …rm e¤ort (Assumption 3). The moral hazard problem is generated by
the dependence of the distribution of the project returns on the …rms’e¤ort
choice, which is unobservable to the banks. In order to provide incentives
to the managers/entrepreneurs to exert high e¤ort, it is necessary to let the
managers/entrepreneurs bear some risk of the project. In the following analysis, I again focus on the three types of contracts and derive the equilibrium
contracts under moral hazard.
15
3.1
Standard debt contract
For unconstrained …rms, the equilibrium standard debt contracts are the solution to the following problem:
max
B;e;r
p (B; e)
+ (1
H
Br + A +q (
p (B; e)
(21)
Br + A)
q) ( Br + A) C (e)
s.t.
E l = p (B; e) Br + qBr+ (1
p (B; e)
Proposition 11 For A
B
0
(22)
Br + A
0
(23)
Br + A
0
(24)
min(Br; A) + A
(p (B; eH )
q) Br
p (B; eL ))
(25)
0
H
cH :
(26)
B , under non-veri…ability of …rms’ return and
moral hazard, the equilibrium standard debt contract (B; rjA) is optimal, where
(1) the equilibrium borrowing is optimal, B = B ;
(2) the equilibrium interest rate r = 1:
The equilibrium standard debt contracts for unconstrained …rms are the
same as the ones in Proposition 3. The unconstrained …rms pay a …xed repayment B and keep the rest of the return. Under Assumption 7, exerting eH
is optimal. Unconstrained …rms have incentives to exert high e¤ort, and the
incentive compatibility constraint (Equation (26)) does not bind.
For constrained …rms, …rst-best standard debt contracts shown in Proposition 3 will not be o¤ered in equilibrium since the constrained …rms would
default strategically due to non-veri…ability of …rms’returns. As a result, in
equilibrium, standard debt contracts are dominated by equity and convertible
debt contracts.
16
3.2
Equity contract
If equity contracts are o¤ered in equilibrium, the equilibrium equity contracts
must solve the following problem:
max
p (B; e)
B;e;s
+ (1
H
H
s
s ( + A) +A)
(27)
q) sA
(28)
+A + A +q (
p (B; e) q) ( sA + A) C (e)
s.t.
p (B; e) s
H
+ A + qs ( + A) + (1
H
H
s
p (B; e)
+A +A
s ( + A) + A
sA + A
(p (B; eH )
0
(29)
0
(30)
0
(31)
0
H
p (B; eL ))
B
s
H
cH :
(32)
Proposition 12 Under non-veri…ability of …rms’ return and moral hazard,
there exists a threshold A1 such that the equilibrium equity contracts (B; sjA)
is optimal for A
A1 , and
(1) the equilibrium borrowing B = B ,
(2) the equilibrium share s =
where A1 satis…es
(p
For A
pL )
B
;
+q +A
p
H
H
(1
s (A1 )) = cH :
A1 , the equilibrium equity contracts are the same as the ones
stated in Proposition 4. However, for …rms with smaller A (< A1 ), there will
be no equity contracts o¤ered in equilibrium due to the incentive compatibility
constraint (Equation (32)).
17
3.3
Convertible debt contract
The equilibrium convertible debt contracts are the solution to the following
problem:
max
B;e;
H
p (B; e)
+q (
H
max Bi;
+A
(33)
+A
max (Bi; ( + A)) + A)
+ (1
p (B; e)
q) (0
max (min (A; Bi) ; A))
C (e)
s.t.
H
p (B; e) max Bi;
+ (1
p (B; e)
H
+A
+ q max (Bi; ( + A))
q) max (min (A; Bi) ; A)
H
max Bi;
+A
+A
(34)
0
(35)
0
max (Bi; ( + A)) + A
0
(36)
max (min (A; Bi) ; A) + A
0
(37)
(p (B; eH )
p (B; eL )) [
H
H
max Bi;
+ max (min (A; Bi) ; A)]
+A
+A
(38)
cH
Proposition 13 Under non-veri…ability of …rms’ return and moral hazard,
there exists a threshold A1 such that the equilibrium convertible debt contract
(B; i; jA) is optimal for A
(1) B = B
(2) i = 1 for A
(3)
B
H
+A
A2 , and
B and i = r for A2
for A
B and
=
A<B ;
B i
( H +A)
=
B (1 p q)A
(p +q)( H +A)
for A2
A<
B , where A2 satis…es
(p
pL ) (1
(A2 ))
H
+ A2 = c H :
Proposition 13 demonstrates that convertible debt contracts are optimal
for any A
A2 if managers’/entrepreneurs’e¤ort is unobservable.
18
I show that, for unconstrained …rms, all three types of contracts are still
optimal even if taking into account the moral hazard problem since the incentive compatibility constraint (Equation (38)) does not bind for larger …rms.
For constrained …rms, standard debt contracts are not optimal because the
managers/entrepreneurs have incentives to default strategically. Equity contracts and convertible debt contracts both solve this ex-post strategic default.
However, here due to the moral hazard problem, equity contracts are not optimal for …rms with initial assets A smaller than a threshold A1 . For a …rm with
small initial asset A < A1 , in order to induce the manager/entrepreneur to exert high e¤ort, the share s assigned to the bank cannot be too high. However,
if the share is not high enough, the bank’s participation constraint (Equation
(33)) is violated (i.e., the bank has negative pro…ts) because the share s is too
low. As a result, in equilibrium, there is no equity contract o¤ered to …rms
with A < A1 . This means that there exists a trade-o¤ between ex-post and
ex-ante e¢ ciency. In order to prevent ex-post ine¢ ciency caused by unobservable e¤ort, the ex-ante e¢ ciency would be lost, that is, …rms with small initial
assets will not be able to obtain the funds to invest in projects with positive
present value. Convertible debt contracts, due to moral hazard, result in the
same trade-o¤ as equity contracts face. However, the ex-ante ine¢ ciency is
less severe in convertible debt contracts than in equity contracts. Because in
equilibrium,
A2
< s, there exists another threshold A2 such that for a …rm with
A < A1 , convertible debt contracts are optimal.
Lemma 14 A2 < A1 :
In summary, unconstrained …rms have no incentives to default strategically ex-post even though the …rms’returns are not veri…able. Besides, unconstrained …rms are not constrained by the incentive compatibility constraints
even though there is a moral hazard problem due to unobservable …rm e¤ort.
Therefore, for unconstrained …rms, standard debt, equity and convertible debt
contracts are all optimal. However, this result does not hold for constrained
…rms. First of all, constrained …rms always have incentives to default strategically if the …rms’ returns are not veri…able. The solutions to this ex-post
19
strategic default are either to let the bank be an equity holder or a convertible debt holder. In other words, both equity and convertible debt contracts
can e¤ectively solve the ex-post strategic default if the …rms’returns are not
veri…able. However, equity contracts are dominated by convertible debt contracts for …rms with initial assets A2
contract for a …rm with A2
A < A1 because there is no equity
A < A1 such that the incentive compatibility
constraint of the …rm and the bank’s participation constraint are both satis…ed
at the same time. Therefore, while for constrained …rms with A A1 , equity
and convertible debt contracts are both optimal, for constrained …rms with
A2
A < A1 , only convertible debt contracts are optimal.
This result is consistent with the stylized fact that the probability of using
convertible debt is negatively related with the …rm size (Lewis, Rogalski and
Seward (1999)).
4
A dynamic model
So far, I have shown that in a static one-period model, under non-veri…able
…rms’ returns and unobservable e¤ort of …rms, standard debt contracts are
strictly dominated by equity and convertible debt contracts due to the incentive of strategic default. However, in practice, startup …rms may not default
strategically since they face a trade-o¤ between short term gain (if the managers/entrepreneurs default strategically) and long term relationship gain (if
they repay truthfully and maintain the relationship with their banks). In the
one-period model above, for constrained …rms, convertible debt contracts are
optimal. Banks are indi¤erent between converting or not when the return is
H
and do not convert when the return is either
or 0. However, in practice,
a bank may prefer converting if the …rm’s future expected value is higher than
the bank’s long-term relationship gain from remaining as a debt holder, and
vice versa.
In this section, I consider a dynamic model in which the trade-o¤s between short-term and long-term gain of both …rms and banks are taken into
account. I incorporate the relationship values (RVB , RVF ) for a bank and a
20
…rm respectively, as well as a future expected value (F EV ) of a …rm.
A …rm’s relationship value arises only if the …rm’s future funding comes
from the same bank. After the realization of the investment outcome at the end
of the …rst period, the …rm needs funds to invest again in the future periods.
Whether the …rm can obtain the fund from the same bank depends on the
realized outcome in the …rst period. In particular, RVF depends negatively on
the …rms’realized return in the …rst period y,
RVF (y) > 0 and RVF0 (y) < 0;
for y > 0;
(39)
and
(40)
RVF (0) = 0:
In order to capture how RVB , RVF and F EV determine the optimal …nancial contract, I assume H is very high. Hence, the …rm does not rely on
funding from the same bank since a very successful …rm has a larger set of
outside options.
Assumption 9
H
is su¢ ciently high and RVF
H
<B i
A for all A.
As for banks, for simplicity, I assume that RVB is independent of the …rm’s
return in the …rst period as long as y > 0,
RVB (y) > 0 and RVB0 (y) = 0;
for y > 0;
(41)
and RVB is zero if the …rm goes bankrupt,
(42)
RVB (0) = 0:
On the other hand, F EV positively depends on the realized return in the
…rst period and equals zero if the …rm goes bankrupt. It could be expected
that a …rm which succeeds initially would also be more successful in the future.
It is also clear that only equity holders are eligible to share the …rm’s future
expected value. Speci…cally, I assume,
F EV (y) > 0 and F EV 0 (y) > 0;
21
for y > 0;
(43)
and
(44)
F EV (0) = 0:
In the one-period model, since there is no relationship value, the …rm always defaults if a standard debt contract is o¤ered. However, in the dynamic
model, under standard debt contracts, the …rm only defaults strategically if
the following inequalities hold:
H
Br + A + RVF
H
H
<
(45)
;
(46)
Br + A + RVF ( ) < :
Under Assumption 9, Equation (45) holds since i
defaults strategically when the realized return is
H
r. Hence, a …rm always
. When the realized return
is , whether the …rm defaults or not depends on the value of . In particular,
(A) managers/entrepreneurs default if Br A > RVF ( ), and
(B) managers/entrepreneurs do not default if Br
A < RVF ( ) :
For satisfying case (A), standard debt contracts are dominated regardless
of banks’relationship value. In case (B), managers/entrepreneurs only default
strategically when y = H and do not when y = . Therefore, standard debt
contracts are optimal regardless of banks’relationship value.
When convertible debt contracts are o¤ered, managers/entrepreneurs do
not default strategically, and banks make conversion decisions. In the oneperiod model, banks are indi¤erent between converting or not when y = H ,
and never convert in the other two cases. In the dynamic setup, banks’conversion decisions additionally depend on RVB (y) and F EV (y). The following
two conditions describe the banks’conversion decisions.
The bank converts if and only if
Bi + RVB (
H
)< (
H
+ A) + F EV (
H
Bi + RVB ( ) < ( + A) + F EV ( );
);
(47)
(48)
and banks never convert if y = 0, since
A > A:
22
(49)
From Equation (47) and Equation (48), banks’conversion decisions depend
on the following conditions:
(a) Banks always convert when y > 0, if
(50)
Bi + RVB < ( + A) + F EV ( );
(b) banks only convert when y =
H
( + A) + F EV ( ) < Bi + RVB (
, if
H
)< (
H
+ A) + F EV (
H
);
(51)
and
(c) banks never convert, if
Bi + RVB (> (
H
+ A) + F EV (
First, suppose case (B) for the …rm holds (i.e., for
H
):
(52)
such that the …rm’s
relationship value is high), and consider case (a) and (b) for the bank. For
the …rms with a small initial asset, A < A2 , convertible debt contracts are
dominated by standard debt contracts in both cases. In case (c), since banks
do not convert under convertible debt contracts, both convertible and standard
debt contracts are optimal.
Secondly, suppose case (A) for the …rm holds (i.e., for such that the …rm’s
relationship value is low), in all cases (a), (b) and (c), standard debt contracts
are dominated due to the ex-post strategic default. Hence, only convertible
debt contracts are optimal.
Table 1 summarizes the optimal contracts in the dynamic setup with a
su¢ ciently high
H
.
Ta b le 1 : O p tim a l c o ntra c ts d e p e n d in g o n b a n k s’ a n d …rm s’ re la tio n sh ip va lu e s
high RVF ( )
low RVF ( )
(lower )
(higher )
standard debt
high RVB
&
convertible debt
convertible debt
low RVB
standard debt
convertible debt
23
This result leads to the following conjecture.
Conjecture 15 In the dynamic model, for
H
high enough and a given func-
tion RVF (y) where RVF0 (y) < 0, for y > 0,
(1) if
is close to
H
or if
is close to zero (i.e., the returns are of high
volatility) regardless of the value of RVB (y) ; convertible debt contracts are
optimal,
(2) if
is su¢ ciently smaller than
H
and su¢ ciently larger than zero (i.e.,
the returns are of low volatility), and
(a) RVB (y) is high, standard debt contracts and convertible debt contracts
are both optimal,
(b) RVB (y) is low, only standard debt contracts are optimal.
Conjecture 15 implies that given the function of RVF (y) ; if the …rm’s return is less volatile, the types of optimal contracts depend on the banks’relationship value RVB (y). If RVB (y) is high, both convertible debt and standard
debt contracts are optimal. If RVB (y) is low, only standard debt contracts are
optimal due to the Bertrand competition among banks. On the other hand,
if the …rm’s return is more volatile, only convertible debt contracts are optimal independent of the banks’ relationship value. This results is consistent
with empirical evidence that convertible debt contracts are commonly used
for high-tech startup …rms with more volatile returns (Kaplan and Strömberg,
2000).
In addition, the result also shows that, if banks have a relatively high
relationship value RVB , but the …rm has a low relationship value RVF (y)
when y = , banks …nance these startup …rms only under convertible debt
contracts without actual conversion. This is also consistent with the evidence
that banks participate in venture capital …nance in order to build the lending
relationship early on with potential clients (Hellman et al., 2008) in order to
enjoy the high relationship value in the future.
However, it is worth noting that if both the banks’ and the …rms’ relationship value are low (i.e., in the case of (A)+(a)), convertible debt contracts
with actual conversion are optimal, and banks would be equity holders of the
24
…rms and share the …rms’future expected value. This implies that in this case,
banks are competing with VCs in …nancing startup …rms. However, under the
restrictions imposed by the Gramm-Leach-Bliley Act, banks are not allowed
to engage in routinely managing the …rms. Hence, banks become inferior in
…nancing these startup …rms, and consequently lack incentives to develop expertise in …nancing them.
5
Conclusion
This paper analyzes optimal …nancial contracts of …rms with heterogeneous
initial assets. I start with developing a one-period principal-agent model with
both risk-neutral …rms and banks and show that under non-veri…ability of
…rms’returns, …rms with small initial assets have incentives to default strategically under standard debt contracts. Equity and convertible debt contracts
can prevent the ex-post strategic default.
Further, I generalize the model and consider the case with moral hazard
caused by managers’/entrepreneurs’unobservable e¤ort. For …rms with small
initial assets, only convertible debt contracts are optimal. Standard debt contracts and equity contracts are dominated because of ex-post strategic default
and moral hazard, respectively.
Finally, I consider a dynamic setup in which the relationship value between
startup …rms and their banks, as well as the future expected value for equity
holders, are taken into account. I show that equity contracts are dominated
by convertible debt and standard debt contracts due to moral hazard. I conclude that if the investment project generates high-volatility returns, convertible debt contracts dominate standard debt contracts. My result is consistent
with empirical evidence showing that convertible debt contracts are pervasively adopted in venture capital …nance (Kaplan and Strömberg, 2000). In
addition, my result shows that banks …nance startup …rms even though the
…rms have low relationship value, and they do not convert in order to continue
the lending relationship as long as the banks’relationship value is high. This
is consistent with the fact that banks participate in venture capital …nance
25
with the objective to build relationships with their clients at an early stage
(Hellman et al., 2008). On the other hand, if the banks’ relationship value
is low, banks would convert and become equity holders. However, due to the
restrictions imposed by the Gramm-Leach-Bliley Act, banks are disadvantageous in the latter relative to VCs. It could cause banks to limit themselves
in their …nancing and not to develop the expertise for funding certain startup
…rms. Therefore, it would be important to analyze costs and bene…ts of the
Gramm-Leach-Bliley Act more thoroughly. Moreover, it could be important
to endogenize the startup …rms’ choice of the …nancing source. These two
extensions would enrich the studies in venture capital …nance and are left for
future research.
26
6
Appendix
Proof. Lemma 1
Suppose that B; R1 ; R2 ; R3 jA is an equilibrium contract and it yields a
positive expected pro…t to the lender,
E
l
B; R1 ; R2 ; R3 jA > 0:
Consider another contract (B 0 ; R10 ; R20 ; R30 jA) ; where B 0 = B, R10 = R1
R20
= R2
pro…t:
" and
E
R30
l
= R3
",
" so that the lender still has non-negative expected
B; R1 ; R2 ; R3 jA > E
l
(B 0 ; R10 ; R20 ; R30 jA) > 0:
However, this contract (B 0 ; R10 ; R20 ; R30 jA) gives the …rm higher expected return.
Hence, B; R1 ; R2 ; R3 jA would not be an equilibrium contract. Thus, the
equilibrium contract satis…es E
= 0.
l
Proof. Proposition 2
The optimal contract solves the following problem
H
max p (B; e)
R1 + A + q (
B;e
R2 + A) + (1
p (B; e)
q) ( R3 + A)
s.t.
E
l
= p (B; e) R1 + qR2 + (1
H
p (B; e)
R1 + A
R2 + A
R3 + A
In equilibrium, E
l
q) R3
B
0
(PC)
(LL1)
0
(LL2)
0
(LL3)
0:
= 0. Hence, from (PC),
p (B; e) R1 + qR2 + (1
p (B; e)
q) R3 = B:
Plugging this into the …rm’s objective function, and for (R1 ; R2 ; R3 ) satisfying
(LL) and (IC), B and e solve
max p (B; e)
H
+q +A
max p (B; e)
H
+q
B;e
B;e
(p (B; e) R1 + qR2 + (1
B + A:
27
p (B; e)
q) R3 )
C (e)
Under Assumption 7, in equilibrium, e = eH . Therefore, the optimal borrowing
amount
B (eH ) = B ;
satis…es
p0B (B ; eH ) =
1
H
:
Since …rms and their banks are risk-neutral, the optimal repayment scheme
(s1 ; s2 ; s3 ) is indeterminate, as long as it satis…es LL1, LL2, LL3 and the
following relation:
R1 =
B
(1
p (B ; eH ) q) R3
p (B ; eH )
qR2
:
Proof. Proposition 3 (Standard debt contracts)
The equilibrium standard debt contract is the solution to the following
problem
max
B;e
H
p (B; e)
+ (1
Br + A + q (
p (B; e)
q) ( Br + A)
Br + A)
C (e)
s.t.
p (B; e) Br + qBr + (1
H
p (B; e)
Br + A
Br + A
Br + A
0
0
0:
q) Br
B
0
(PC)
(LL1)
(LL2)
(LL3)
As before, under Assumption 7, e = eH , and thus the equilibrium borrowing
amount satis…es B (eH ) = B .
For unconstrained …rms (A
B ), both LL1 and LL2 do not bind and the
equilibrium r = 1.
For constrained …rms (A < B ), only LL2 binds. Hence, in equilibrium,
we obtain
B
(1 p (B ; eH ) q) A
r =
> 1:
(p (B ; eH ) + q) B
28
All …rms’expected pro…ts are maximized.
Proof. Proposition 4 (Equity contracts)
The equilibrium equity contract is the solution to the following problem
max
H
p (B; e)
B;e
+ (1
H
s
p (B; e)
+A +A +q(
q) ( sA + A)
s ( + A) + A)
C (e)
s.t
p (B; e) s
H
+ A + qs ( + A) + (1
H
H
s
p (B; e)
+A +A
s ( + A) + A
sA + A
q) sA
B
(PC)
0
(LL1)
0
(LL2)
0
(LL3)
0:
From Lemma 1, (PC) binds in equilibrium. Besides, due to Assumption 7,
e = eH , and thus, equilibrium B = B which satis…es p0B (B ; eH ) =
1
H
. The
equilibrium share satis…es
B
s =
H
p (B ; eH )
+q +A
:
Proof. Proposition 6 (Convertible debt contracts)
For unconstrained …rms, the equilibrium convertible contract solves the
following problem
max
B;e; ;i
p (B; e)
+q (
+ (1
H
max
H
+ A ; Bi + A
max ( ( + A) ; Bi) + A)
p (B; e)
q) ( max ( A; Bi) + A)
s.t.
p (B; e) max
H
+ (1
q) max ( A; Bi)
p (B; e)
+ A ; Bi + q max ( ( + A) ; Bi)
29
B
0
(PC)
H
H
max
+ A ; Bi + A
max ( ( + A) ; Bi) + A
max ( A; Bi) + A
(LL1)
0
(LL2)
0
(LL3)
0:
Since unconstrained …rms are able to fully repay in any case, in equilibrium,
i = 17 and B = B . As for equilibrium conversion rate , we consider the
following four cases:
H
(I) (
(II) (
(III) (
+ A) > ( + A) > A
H
B
+ A) > ( + A) B > A
+ A) B > ( + A) > A
H
(IV) B
(
H
+ A) > ( + A) > A
Case (I), (II) and (III) will not satisfy the equilibrium
since in these cases,
banks cannot commit not to convert ex-post, and thus banks’expected pro…t
is larger than zero (PC does not bind). Therefore, the equilibrium should
B
:
+A
satisfy case (IV)’s condition, and
H
For constrained …rms, we know that when y = 0, banks never convert.
Hence in order to derive the equilibrium
three cases:
(i) (
(ii) (
H
H
+ A) > ( + A)
and i, we consider the following
B i> A
+ A) > B i > ( + A) > A
(iii) B i
( H + A) > ( + A) > A
From (ii), we can derive the relationship between
only convert when y =
H
and i. In this case, banks
, and do not convert in the other two states. Hence,
=
B
(1
p
H
p
q) A
+A
qB i
:
From (i) and (iii), we derive the upper and lower bound of the equilibrium
interest rate i . In particular, from (i)
i
7
( + A) (B
(1 p
q) A)
;
H
B p
+ A + q ( + A)
i is the nominal interest factor, it must be as least equal to one.
30
and from (iii)
i
B
r =
(1 p
q) A
:
(p + q) B
Proof. Proposition 10 (Non-veri…able …rms’returns)
For unconstrained …rms, …rms always can fully repay, hence, even under
non-veri…able returns, they do not have incentives to default strategically.
Therefore, standard debt, equity, and convertible debt contracts all can implement the optimal contracts in Proposition 2.
Equity and convertible debt contracts are immune to non-veri…ability.
Hence, in equilibrium, both contracts implement the optimal contracts derived in Proposition 2. However, standard debt contracts are not optimal for
constrained …rms because constrained …rms have incentives to default strategically. Therefore, under standard debt contracts, in equilibrium, B = A, and
the constrained …rm’s expected pro…ts is p (A; eH ) + q .
We can show that
p (B ; eH )
H
+q
H
>
B + A > p (A; eH )
H
+q ;
if and only if
B
p (B ; eH )
For any constrained …rm (A < B ),
(1) p(B ;eBH ) A
p(A;eH )
H
>
A
:
p (A; eH )
B A
p(B ;eH ) p(A;eH )
if
is decreasing in A and
(2) at A = 0, p (B ; eH )
H
B > 0 holds
Condition (2) is always true since we assume that the project is of a positive
net present value. Next, we show (1) also holds.
@
B A
p(B ;eH ) p(A;eH )
@A
p0 (A; eH ) (B ; eH A)
=
(p (B ; eH )
< 0
(p (B ; eH )
p (A; eH ))2
because for any A < B ;
p0 (A; eH ) >
p (B ; eH )
B
31
p (A; eH )
A
p (A; eH ))
is always true, since the function p (:) is strictly concave. Therefore, (1) & (2)
both hold and thus, the following condition holds:
H
p (B ; eH )
+q
B + A > p (A; eH )
H
+q :
In conclusion, for A < B , standard debt contracts are dominated by equity
and convertible debt contracts.
Proof. Proposition 11
The proof corresponds to the one of Proposition 3 since the Incentive Compatibility (IC) constraint does not bind for any unconstrained …rms.
Proof. Proposition 12
The proof follows from Proposition 4 and (IC) bind for …rms with A < A1 .
Proof. Proposition 13
For unconstrained …rms, the proof follows from Proposition 6.
For constrained …rms, due to the additional (IC) constraint, convertible
debt contracts share the features of standard debt contracts. In equilibrium,
i = r , and thus the equilibrium
=
satis…es
B r
B
(1 p
q) A
=
;
H
+A
(p + q)
+A
H
and (IC) binds for …rms with A < A2 ,
(p
pL ) (1
(A2 ))
H
+ A2 = c H :
Proof. Lemma 14 (Two initial asset thresholds)
Let p denote p (B ; eH ). A2 is the threshold under convertible debt contracts such that
(p
pL ) (1
where
(A) =
(A2 ))
H
+ A2 = c H ;
B
(1 p
q) A
:
H
(p + q)
+A
32
A1 is the threshold under equity …nancing such that
(p
pL )
H
(1
s (A1 )) = cH ;
where
s (A) =
H
p
B
:
+q +A
In order to show that A2 < A1 , we need to show that for any A 2 [0; 1),
(p
pL )
H
(1
s (A)) > (p
pL ) (1
The above condition holds as long as s (A) >
Since both s (A) >
we can show that:
(A))
+A :
(A) for any A:
(A) are strictly and monotonically decreasing in A,
(1) when A ! 0;
s (0) =
H
B
p
H
>
+q
(2) when A ! 1;
B
(p + q)
H
=
(0) ;
s ! 0;
and consequently,
!
1
p
q
< 0:
p +q
Hence,
s (A) >
(A)
for all A 2 [0; 1):
33
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