Credit Risk Transfer
Christine A. Parlour†
∗
Guillaume Plantin
‡
Tepper School of Business
Carnegie Mellon University
Pittsburgh, PA 15213
May 24, 2005
∗
We have benefitted from the comments provided by seminar participants at IHS, Vienna. The
current version of this paper is maintained at http://ozymandias.tepper.cmu.edu
†
Tel: (412) 268-5806, E-mail: [email protected]
‡
Tel: (412)-268-9823, E-mail: [email protected]
1
Abstract
We consider the effect of a Credit Risk Transfer (CRT) market on relationship banking. We extend Holmstrom Tirole (1997) by allowing banks to
receive proprietary information about a project’s success and to face a stochastic discount factor. If the project is a failure or if they have a discount factor
shock, banks want to sell their claims. We identify conditions under which the
adverse selection risk is sufficiently low so that the CRT market exists. We find
that the price of protection is the probability of default conditional on a bank’s
willingness to sell claims. Further, we identify when such a market is efficient.
A market may arise even if it is socially inefficient. We provide empirical predictions on the effect of a CRT market on promised payments in the bank and
bond market, on project size, and on the size of bond issues and bank loans.
2
Introduction
In the early 1990’s, few credit derivatives were traded. By 2004, the notional amount
of outstanding credit derivatives has been estimated at upwards of $4.8 trillion.1 Until the early 1990’s, securitization consisted mostly of the transfer of senior claims on
large loan portfolios. Subsequently, with the advent of Collateralized Debt Obligations (CDOs) and Credit Default Swaps (CDS), markets have emerged to trade more
junior exposures on less diversified risks, including first-loss positions on single names.
Therefore the secondary market for credit risks originated by lending institutions has
become dramatically more liquid over time.
How will the market’s increased liquidity affect commercial banking? An active market for credit risk transfer (CRT) eases asset-liability management for banks.
Banks can now quickly redeploy capital to the most profitable business opportunities,
and are more resilient to negative shocks. (See, for example, Greenspan, 2004, Schuermann, 2004). This additional degree of freedom is a double-edged sword, however.
In practice, banks can unbundle balance sheet management from loan or borrower
relationship management. This may reduce banks’ incentives to monitor and foster a
relationship with the borrower, and ultimately create a shift from relationship banking towards transaction-oriented banking (see, for example, Effenberger, 2003; Kiff
and Morrow, 2000 and Rule 2001).
The impact of the CRT market on banks and capital markets depends on this
tradeoff between a reduced monitoring incentive and increased liquidity. We characterize when a CRT market arises, when a CRT market is desirable, and provide
testable predictions on the effect of such a market on prices and quantities in bond
and loan markets. A burgeoning literature, including Arping (2004), Duffee and Zhou
(2001), and Morrison (2005) has investigated the consequences of credit derivatives
for relationship banking. We differ from this literature in two ways. First, in our
framework firms optimally mix equity, bonds and loans. We look at all sources of
financing because in practice, reference entities are typically large corporations with
access to diversified sources of financing. Second, and more crucially the existing literature has focussed on institutional changes that allowed trade in credit derivatives.
We complement this approach by focusing instead on the real shocks that are at the
origins of these institutional changes. More precisely, the existing literature views
credit derivatives as an exogenous expansion, ceteris paribus, of the set of feasible
contracts. The focus is on the consequences of the introduction of new tradeable
instruments. For instance, Arping (2004) formalizes credit derivatives as a relaxation of the limited-liability constraint. By receiving negative cash flows in case of
default, protection sellers reinforce the bank’s incentive for efficient liquidation. Morrison (2005) models credit derivatives as the introduction of noncontractible trades
between the bank and protection sellers before monitoring of the reference entity by
the bank takes place, which may be inefficient.
1
Estimates of the size of the market for credit derivatives appear in Bomfim (2005).
3
In our framework, by contrast, liquidity in the CRT market arises endogenously
as a response to real shocks in a fixed institutional context. This approach is consistent with the view that financial innovations arise in response to economic shocks
to financial institutions (see, e.g., Silber, 1975, 1983). We thus offer predictions regarding the relationship between real variables, and the occurrence and desirability
of liquidity in markets for credit risk. For instance, we rationalize the rise of bad
financial innovations that create excess liquidity, and also predict that some good
financial innovations may not take off even though more liquidity would be socially
desirable.
Our framework builds upon the model of financial intermediation developed in
Holmstrom and Tirole (1997). A firm has a positive NPV long term project that can
be financed with the firm’s own equity or from two outside sources: a bank and the
bond market. The optimal financing structure maximizes the firm’s payoff subject
to the solution of a moral hazard problem: the firm may shirk which reduces the
project’s success probability.
Banks differ from the bond market in three ways. First, banks are better able
to monitor the firm than the bond market. This could be because bondholders are
typically dispersed and thus individually have a smaller incentive to monitor. Second,
while monitoring, the bank learns about the success or failure of the project. This
is an important stylized feature of relationship banking.2 Finally, the bank values
liquidity: it has a stochastic discount factor. This proxies for banks’ opportunity
cost of capital which could arise due to inherent fragility, prudential regulation or the
existence of profitable outside opportunities.
Because the bank prefers liquid assets and thus has more expensive capital, in the
firm’s optimal contract, the bank’s stake should be the minimum one that preserves
the incentive to monitor. In particular, once monitoring has taken place, bank capital
is no longer “special,” and thus the firm is indifferent as to the source of the financing.
At this point it is socially efficient that the bank transfers credit risk and recycles its
capital. We interpret resale in the secondary market as the purchase of protection in
the credit derivatives market.
Our measure of illiquidity in the credit derivatives market is the degree of adverse
selection. A bank who has monitored a firm knows if the project will succeed or fail.
Thus, investors do not know if the bank is selling the claim because of a preference
shock, or because of inside information.3 If the adverse selection problem is too severe,
trading in the credit derivatives market breaks down as in Akerlof (1960) and all loans
are illiquid. However, if there is a high probability that the bank is selling its claim for
2
In a survey on relationship banking, Boot (2000) observes,“In originating and pricing loans,
banks develop proprietary information. Subsequent monitoring of the borrower yields additional
private information. The existence of proprietary information may inhibit the marketability of these
loans.”
3
Possible inside trading in the CRT market is a major concern among practitioners as reported
in the popular press e.g., The Economist Jan 16,2003.
4
private value motives—for instance in the case of a high grade borrower, then there
is pooling and the market is liquid. We note that although a liquid CRT is socially
efficient ex post, it may not be desirable ex ante. The rise of liquidity in the CRT
market has two opposite effects on the cost of bank finance. First, interest rates on
loans are smaller because the bank no longer charges a liquidity premium. Firms need
to borrow more, however, because the ability to benefit from inside trading reduces
banks’ incentives to monitor ex ante. In sum, firms consume bank money in higher
quantities but at a lower price when there is an active CRT market. This is efficient
if and only if the outcome is a smaller cost of bank finance.
Our model delivers a number of other implications for credit derivatives and bonds
markets. We first observe that the price of protection in a secondary market for
credit risk is not the unconditional probability that the reference entity will default,
rather it is the probability of default conditional on a banks’ willingness to sell. The
unconditional and conditional probabilities differ systematically with a firm’s default
probability, thereby generating a liquidity premium in the CRT market. For high
credit quality firms (high probability of success) the difference between the conditional
and unconditional default probabilities is larger than for low credit quality firms.
The rest of our testable predictions stem from the facts that i) the CRT market
is more liquid for high grade names, ii) a more liquid CRT market implies more bank
loans, iii) and a more liquid CRT market is efficient only for not too high rated names.
Combined together, these results imply essentially that economic shocks rising the
opportunity cost of capital for banks should imply higher probabilities of defaults—
but this needs not be socially inefficient, and more cross-sectional variations for the
bank debt over arm’s length debt ratio.
The key building block of our formal model, namely that banks value liquidity
and acquire proprietary information through monitoring, follows Breton (2003). He
introduces a similar tension between monitoring and liquidity, in a different context.
In non-banking contexts, Aghion, Bolton, and Tirole (2004), and Faure-Grimaud and
Gromb (2003) develop models in which an impatient agent markets claims to her
future output. The informational efficiency of the market is an important incentive
device for the agent. By contrast in our model, liquidity is conflicting with banks’
incentives for the exact opposite reason that liquidity means the ability to trade with
uninformed agents.
1
Model
Our framework extends Holstrom and Tirole (1997) by putting more structure on the
banking sector. In a three date economy, t = 0, 1, and 2 there are three classes of
agents: a firm, a bank, and a large pool of competitive investors. All agents are risk
5
neutral and are protected by limited liability.4 Neither the risk neutral investors nor
the firm discount future cash flows.
A firm owns rights to a two period project. Every dollar invested in the project at
date 0 has a random date 2 payoff of R with probability p or 0 with probability 1 − p.
A firm also has initial equity or net assets, A. In addition to investing its own assets
in the project, the firm can solicit funds from the bank and the outside investors. To
do so, the firm offers each group a renegotiation-proof contract. The firm’s objective
is to maximize her expected return on equity.
Financial structure matters in this economy because there is a moral hazard problem in the early stage of the project. Specifically, a firm may “shirk,” in the first period
and derive a private benefit BF per dollar invested. In this case, the project fails and
pays off 0 at date 2.
Banks have three distinctive features. First, they have monitoring skills and may
induce firms to perform better. Second, through their relationship with the firm, they
may acquire information about the success of the project. Finally, bank capital is
both scarce and costly.
Active monitoring by the bank reduces the firm’s private benefit from shirking to
bF < BF . Monitoring is costly for a bank as it loses the private benefit BB associated
with not monitoring.5 One benefit of monitoring is that through it the bank acquires
proprietary information. For simplicity, we assume that private information is perfect:
if the bank monitors, it learns if the project will succeed at date 1. The assumption
that lending to and monitoring the borrower generates proprietary information is
common in the banking literature ( Rajan, 1992), and well-documented empirically
(Lummer and McConnell, 1989).
To capture the idea that bank capital is costly, we assume that the bank discounts
future cash flows. More precisely, the bank values cash flows at
π B (c0 , c1 , c2 ) = E(c0 + δ1 c1 + δ1 δe2 c2 ),
where δ1 ∈ (0, 1), and δe2 is a two point random variable whose realization at date 1 is
(
δ˜2 =
δ ∈ (0, 1) with prob.
1
with prob.
q
1 − q.
The bank’s realization of δe2 at date 1 is private information.6 This stochastic discount factor proxies for unanticipated changes in the opportunity cost of outstanding
4
Specifically, we assume that all contractible payments are bounded below by a finite amount
that we normalize to 0.
5
Alternatively, one could model the bank as incurring a private monitoring cost. The current
formulation is the simplest one that ensures all constraints bind at the optimum.
6
We have assumed that there is only one possible realization for the bank’s discount factor at
time 2. The qualitative results go through if the bank’s outside opportunities are drawn from a
known distribution F (·).
6
loans. Such changes could occur because of amendments to regulatory capital requirements or rating criteria; for example, uncertainty related to the new Basel capital accord (see Repullo and Suarez, 2004), or the new IASB accounting standards.
A specific bank’s exposure to these publically observed shocks is perforce private.
A financial institution could also receive private opportunities to invest with other
borrowers or even non lending business. Casual empiricism suggests that such opportunities have been very volatile over the last few years: for example, the booms and
busts of underwriting and lending activities for the “new economy” firms.
After the bank has received her discount factor shock and learned the realization
of the project she can sell her stake in the project in a credit risk transfer market (if
it exists). The counterparties are the risk neutral investors.7
The time line of the game is presented in figure 1.
Bank learns:
(i) opp. cost (δ)
(ii) if project succeeds
Contracts written
and
Firm Invests
Bank monitors
Firm exerts effort
t=0
CRT trading
t=1
Claims
Pay off
t=2
Figure 1: Sequence of events
Finally, we assume that all random variables are independent. To ensure interior
solutions we make the following parameter assumptions.
Assumption 1 (i) BF < 1
(ii) bF + BB < 1
(iii) 1 + BF > pR
The first two assumptions, (i) and (ii), ensure that shirking, with or without the
bank, is never socially desirable. The third assumption, ensures that the firm never
chooses to invest an infinite amount.
2
Characterization of the Optimal Contracts
Before we characterize the optimal renegotiation-proof contracts a firm offers to the
bank and outside investors, we determine conditions under which a CRT may arise:
contract specifics depend on the market’s existence. We note in passing that any
optimal contract will not be conditioned on transfers in the CRT market. This is
because monitoring activity occurs before the bank trades and is thus irrelevant for
the firm.
7
Insurance companies are the largest sellers of credit protection BIS(2004).
7
2.1
Existence of a CRT Market
A bank in the CRT market knows her private opportunity cost of future cash flows
and, if she monitored the firm, knows if the project has succeeded or failed. If a
bank always sells failed claims, then either the date 1 market breaks down (no trade)
or there is complete pooling. Thus, the bank’s stake is either illiquid (market break
down) or liquid if there is pooling.
If there is pooling, then investors believe that the bank is selling either because
the project failed (which occurs with probability 1 − p) or that the project succeeded
but the bank received an attractive outside opportunity. This occurs with probability
pq. Thus, if there is pooling, outside investors value one promised date-2 dollar at a
price r where
pq
.
r=
1 − p + pq
For there to be trade in this market, the price r must be high enough so that
banks with discount factor shocks are willing to sell at this price. Thus,
Lemma 1 A CRT market exists if and only if δ ≤ r =
pq
.
1−p+pq
The opportunity cost of outstanding loans has to be sufficiently high (δ sufficiently
low) so that a bank would be willing to accept the pooling discount to liquidate
its claim: Financial innovation arises when the bank’s private cost of bearing the
risk until maturity is greater than the illiquidity premium. Thus, we should expect
to see risk transfer vehicles arise when banks are faced with unanticipated growth
opportunities.
Further, observe that for a fixed value of δ, the higher the ex ante probability
of success (p) the higher the price at which outside investors are willing to buy the
future claims. Thus, the easier it is to sustain a pooling equilibrium and the higher
the probability that a CRT market can exist. Alternatively, CRT markets are easier
to sustain for high credit rated firms (high p). This corresponds to casual empiricism:
active CDS markets exist for highly rated names but not for those below investment
grade. For example, a 2003 study by Fitch cited in Bomfim (2005) estimate that 90%
of CDS were written on investment grade bonds, with 28% BBB, 28% A, 15% AA
and 21% AAA. Only 8 % were below investment grade.
The price at which outside investors are willing to buy future claims can be used
to determine the price of protection. In the CDS market, conditional on a default
event, the protection buyer receives either the par value of the claim and delivers
the bonds or the difference between the notional amount and the post default market
value of the reference asset. In this model, if the firm defaults the claims are worthless.
Thus, for both cash or physical settlement the default payment is zero. Consider an
obligation to pay $1 at date 2 if the firm does not default and zero otherwise. If
a CRT market exists, this obligation is worth r. Thus, a claim that pays off $1 if
8
the firm does default and 0 otherwise, must be worth 1 − r. Thus, we can interpret
(1−p)
1 − r = 1−p+pq
as the price of protection.
It is immediate that in this framework, the price of protection for a fixed investment size is not the default probability (1 − p). Rather, it is the probability that
the firm will default conditional on the bank selling the claim. This is always weakly
higher than the default probability. Frequently, empirical studies use the price of
protection to infer the market’s assessment of default probability and do not account
for the fact that a liquid CRT market only exists if, in addition to default, claims are
traded for private reasons.
Figure 2 depicts the default probability and the price of protection.
1 HH
HH
H
HH
H
HH
H
priceof protection
1−p
1−p+pq
HH
H
HH
H
HH
H
HH
H
HH
H
default probability
(1 − p)
HH
H
HH
H
HH
H
HH
H
HH
0
p : the probability the project succeeds
1
Figure 2: The price of protection and default probabilities
Clearly, for p = 1 or p = 0, the price of protection is equal to the default probability. However, for p ∈ (0, 1) the price of protection is strictly higher than the default
probability. To explore this difference, let ∆(p) denote the difference between the
price of protection and the default probability.
∆(p) = (1 − r) − (1 − p)
(1 − p) p(1 − q)
=
1 − p + pq
≥ 0
9
Thus, ∆(p) is the liquidity premium, or that part of the price of protection that
is not explained by default probabilities.
Lemma 2 If a CRT exists, then
(i) The liquidity premium, ∆(p) is concave in p, the probability of no-default.
(ii) The ratio of the liquidity premium to the default probability, ∆(p)
is increasing,
1−p
convex in the probability of no–default.
Thus, the liquidity premium for junk bonds (low p) is increasing in the credit
rating. While, for investment grade bonds (high p), the liquidity premium is decreasing in the credit rating. However, the liquidity premium as a fraction of the default
probability is increasing in the credit rating.
2.2
Optimal Contracts
We now characterize how the existence of a CRT market affects the renegotiation–
proof contract offered by the firm to the different lenders. As we have observed,
such contracts do not contain restrictions on the CRT market. The bank is hit by a
private shock and receives private information after monitoring has taken place. How
the bank and investors trade in the secondary market is therefore ex post irrelevant
for the firm. So any clause that restricts CRT activity would not be renegotiation
proof. Typically, such clauses are not included in lending contracts. The fact that
they do not appear in our paper distinguishes it from Morrison (2005) in which the
non-contractibility of CRT is assumed and Aghion, Bolton, and Tirole (2004) who
assume full commitment.
A firm with assets A, facing a project which succeeds with probability p and
generates return R, maximizes expected profits π F = pRF I − A, where RF is her per
unit investment return. She chooses RB , the per unit investment return to the bank,
and L the loan that she solicits. In addition, she determines the size of the project
I, and M , the amount to borrow from outside investors. These are promised RM per
dollar invested. Thus, her choice variables are {I, L, M, RF , RB }. In what follows we
compare the firm’s contracting problems under different parameter assumptions.
First, suppose that bank monitoring is inefficient. Then, the firm maximizes her
surplus subject to her incentive-compatibility constraint (Equation 1) and investors’
participation constraint (Equation 2). These are
pRF I ≥ BF I,
p(R − RF )I ≥ I − A.
(1)
(2)
As in Holmstrom and Tirole (1997), these constraints are binding at the optimum.
10
Lemma 3 If the firm borrows only from outside investors, then the expected profit of
the firm, π F , the optimal investment I and the optimal outside investment stake, M
are
!
pR − 1
F
π =
A.
1 − pR + BF
A
I =
1 + BF − pR
!
pR − BF
M = A
.
1 + BF − pR
Now suppose that bank monitoring is efficient. There are two cases; the fundamentals are such that there is an active CRT market and second, that there is not.
If there is no active CRT market, the firm chooses a contract:
pRF
δ1 Eδ2 pRB
L
p (R − RF − RB ) I
{z
|
≥
≥
≤
≥
M axI,L,M,RF ,RB {pRF I − A}
s.t.
bF
BB
δ1 Eδ2 pRB I
M
(3)
(4)
(5)
(6)
}
RM
I ≤ M +A+L
I, L, M, RF , RB ≥ 0
(7)
(8)
Equations (3) and (4) are the incentive-compatibility constraints of the firm and the
bank, respectively. Equation (3) ensures that the firm, if monitored, will not shirk.
Equation (4) means that the bank’s payoff is higher if it monitors the firm. Equations
(5) and (6) are the participation constraints of the bank and the outside investors.
Equation (7) is a resource constraint.
Lemma 4 If the firm borrows Bank Capital in addition to outside investors, and if
1 Eδ2
1 Eδ2
there is no CRT market, then if 1 + BB 1−δ
< pR < 1 + BB 1−δ
+ bF , the profits
δ1 Eδ2
δ1 Eδ2
F
to the firm, π , the project size I, the amount borrowed from the bank, L, and the
amount borrowed from the market, M , are


πF
bF
− 1 A
= 
1 Eδ2
)
1 − pR + bF + BB ( 1−δ
δ1 Eδ2
I =
A
1 Eδ2 )
)
1 − pR + bF + BB ( 1−δ
δ1 (Eδ2 )
M = I(1 − BB ) − A
L = BB I.
11
Now, suppose that a CRT market exists, and a bank can sell claims to period 2
cash flows at a price r. The firm then solves:
pRF
δ1 pRB
L
p(R − RF − RB )I
I
I, L, M, RF , RB
≥
≥
≤
≥
≤
≥
M axI,L,M,RF ,RB {pRF I − A}
s.t.
bF
BB + δ1 rRB
δ1 pRB I
M
M +A+L
0
(9)
(10)
The existence of a CRT market changes the bank’s incentive-compatibility (9)
and participation constraints (10) through changes in its discount factor. A bank
faced with a liquidity shock (or good outside opportunity), in the face of an active
CRT market, may want to sell her existing claim. In addition, a bank that knows
the project failed will also sell its stake at time 1. Thus, the existence of the market
changes a bank’s expected period 2 discount factor. With probability (1 − p + pq), the
pq
bank believes it will sell its claim at price r = 1−p+pq
. With probability p(1−q), it will
not sell its claim but value it at a discount rate of 1. Thus, the bank now values cash
flows that payoff at time 2 at a rate of δ1 . The change in expected discount rate affects
a bank’s behavior through both the participation and the incentive compatibility
constraint.
It is instructive to compare these new constraints to the IC and PC constraints
that obtain without a CRT market, namely (4), and (5). The bank’s participation
constraint is easier to meet with this higher discount factor because the cost of bank
capital is lower: The price of the loan no longer incorporates a liquidity premium.
However, the impact of liquidity on the incentive-compatibility constraint is ambiguous. First, the bank does not discount date-2 cash flows with a liquidity premium,
which reduces the cost of incentives. However a bank who does not monitor and
consumes private perquisites can now sell the loan in the CRT market. This makes
shirking more attractive and thus the IC is more difficult to satisfy. In sum, with
an active CRT market, the firm needs to borrow higher quantities from the bank,
but each dollar is borrowed at a lower cost. Whether borrowing higher quantities at
a lower price reduces the total cost of informed capital, and is therefore efficient is
ambiguous. Just because a CRT market can be sustained, does not necessarily mean
that it will be efficient.
p
p 1−δ1
; 1 + BB p−r
<
Lemma 5 If there is a liquid CRT and if max bF + BB δ11 p−r
δ1
p 1−δ1
F
pR < 1 + BB p−r δ1 + bF , then the surplus to the firm, π , the size of the project, I,
12
M , L are

πF = 

bF
p
1 + bF − pR + BB ( p−r
)

1−δ1
δ1
− 1 A

1
I = A
p
1 + bF − pR + BB ( p−r
)
p
L = BB (
)I
p−r
!
(1 − BB )p − r
M = I
− A.
p−r
1−δ1
δ1

We will analyze the model under parameter restrictions that ensure the firm
chooses both bank and market financing for its project. These restrictions satisfy
the non–negativity constraint in the specification of the contracting problems.
If there is no CRT we require
1 + BB
1 − δ1 Eδ2
1 − δ1 Eδ2
< pR < 1 + BB
+ bF ,
δ1 Eδ2
δ1 Eδ2
(11)
and if there is a CRT,
max bF + BB
1
δ1
p
p 1 − δ1
; 1 + BB
p−r
p − r δ1
!
< pR < 1 + BB
p 1 − δ1
+ bF .(12)
p − r δ1
In both cases, the left-hand side amounts to a restriction that the NPV is sufficiently high for bank monitoring to be feasible and the right-hand side ensures that
the NPV is sufficiently low so that the optimal investment size is finite.
Equivalently, these conditions can be viewed as restrictions on the probability
of success, p. Clearly, all else equal, for BB sufficiently small and bF sufficiently
b δb .
large, these parameter restrictions are met for all (p, δ) in a neighborhood of p,
Empirically, the variation in default probabilities for investment grade bonds is extremely low. Hamilton (2001) estimates average five year cumulative default rates for
investment grade bonds as 0.82% and 18.56% for speculative grade bonds.8
3
Analysis and Testable Implications
In equilibrium, the firm offers contracts to both the outside investors and the bank
that maximizes her surplus. The outside investors are always held to their participation constraint. By contrast, for the bank both incentive compatibility and
8
The Bank for International Settlements (2004) reports that the CDS market for 5 year maturities
is the most liquid.
13
participation constraints are binding. In all cases, as these differ depending on the
CRT regime so too does the bank’s informational rent. This may reduce the amount
that firms can pledge to the bond market and may result in smaller aggregate investment. Social surplus is maximized when the firm makes the highest profit. Given
the constant returns to scale technology we can alternatively consider the size of the
project. Depending on parameters, the investment size may be greater or smaller
with the presence of the CRT market. Therefore, there are parameters for which a
CRT market can arise, but it results in lower aggregate investment and is thus socially
inefficient.
(1−q)(δ1 −p)
pq
a liquid CRT market can exist. If δ ≤ (1−p)−q(δ
the
Proposition 1 If δ ≤ 1−p+pq
1 −p)
CRT market is efficient. Further, there exists a p̂ < 1, so that for p > p̂ any liquid
CRT market is inefficient.
In figure, 3, we illustrate the two conditions of proposition 1. The condition for
efficiency is the threshold below which aggregate investment is larger with a CRT.
There are four possibilities in this economy. The intersection of the two curves defines
p̂.
δ
(1−q)δ1
1−qδ1
illiquid
inefficient
illiquid
efficient
liquid
inefficient
liquid
efficient
p probability of success
Figure 3: Efficiency and Liquidity
δ1
The reason that a CRT can be inefficient is due to its effect on the cost of bank
capital. A CRT reduces the required return on bank loans but increases the incentivecompatible stake that banks have to lend.
14
Proposition 2 The cost of bank capital per unit of total investment is lower if there
is an active CRT market than if there is not. Specifically, RLB I |CRT < RLB I |N o .
The required return on bank loans is lower if there is a CRT because the bank no
longer requires a liquidity premium. Thus, this should be reflected in the amount of
money firms have to pay to banks. Specifically,
Corollary 1 All else equal, names that are traded in a CRT market pay a lower rate
on bank loans.
Thus, there are two effects. First, firms which have a higher credit rating are more
to have a lower cost of bank capital and they are more likely to have their names
traded which ceteris paribus reinforces the decrease in the cost of bank capital.
However, while the per unit cost of bank capital drops if a name is traded on a
CRT market, to ensure monitoring is incentive compatible, the fraction of the project
funded by bank debt must be higher. In particular, if there is an active CRT market,
then, ceteris paribus, the proportion of the project financed by bank debt is higher,
and the proportion of public debt is lower. Since bank capital is expensive, this
reduces the surplus pledgable to bondholders. Thus,
Proposition 3 (i) All else equal, the ratio of bank debt over total project size, LI is
higher if there is an active CRT market. (ii) All else equal, the ratio of public debt
over total project size, MI is smaller if there is an active CRT market.
Thus, while the per unit cost bank capital is lower, a greater quantity is required
to ensure the bank has proper incentives. If the net effect of these two effects is a
reduction in the total interest paid to the bank, then more surplus is available to
pledge to raise outside (bond financing). In this case CRT is efficient. However, if
there is an increase in the total interest paid to the bank, then the result is inefficient.
Corollary 2 The existence of trade in a CRT market is efficient if and only if it
reduces the cost of bank finance—namely the total interest charges on loans—for that
name.
In the economy firms differ by their default probabilities. Thus our results can be
interpreted in the cross section. We have observed that for a fixed banking sector,
there is a minimum probability of success (or credit rating) for which CRT exists. Let
pc denote this probability. The following proposition studies variations in the debt
ratios in the cross-section of credit ratings.
Proposition 4 (i) The ratio of bank debt over total project size is invariant to the
probability of success (credit rating) up to a threshold, pc , after which it is increasing
convex.
(ii) The ratio of market debt over total project size is increasing in the probability of
success (credit rating) up to a threshold, pc , after which it is increasing, concave.
15
Two empirical observations follow directly from this comparative static.
Corollary 3 All else equal, names that are the most actively traded in CRT markets
should finance a higher fraction of their investments with bank loans.
Corollary 4 All else equal, names that are the most actively traded in CRT markets
should finance a lower fraction of their investments with bonds.
The total project size also differs across firms that have access to a CRT. In
particular, if there is no CRT market, then the optimal investment is increasing in p.
Thus, firms with a higher credit rating, would have proportionally higher project size.
However, once a CRT market opens, firms whose default probability is sufficiently
low (high credit rating) will restrict the amount that they invest because the cost of
providing incentives to the bank becomes prohibitively high.
e so that for p < p,
e the optimal
Proposition 5 If a CRT exists, then there is a p,
e
investment size is increasing, convex in p, the probability of success, and for p > p,
the optimal investment size is decreasing in p.
Consider an economy with a fixed distribution of default probabilities. Then,
consider an exogenous change in δ, the opportunity cost of bank capital. A decrease
in δ increases the range of firms for whom CDS can be traded, further this is efficient
(leads to larger Investment sizes) as it affects the lower credit rated firms. Thus,
paradoxically, an increase in the cost of bank capital can lead to larger investment
being undertaken by lower quality firms. Thus, the average default risk per dollar
lent must have increased. Notice, that this is not necessarily inefficient: It may still
lead to an increase in aggregate expected profits.
Corollary 5 All else equal, an increase in banks’ cost of capital should imply an
increase in default rates. This needs not be ex ante inefficient or undesirable.
4
Conclusion
We have presented a simple model that captures the tradeoffs in the CRT market.
Banks can sell loans if they are faced with an attractive outside investment opportunity. However, the existence of a CRT may not be socially efficient, in that firms
may restrict the amount that they borrow.
In a complete market, with no frictions agents are indifferent between different
financial instruments and markets and prices. If markets have arisen, or the traded
financial instruments have changed, this suggests that they arise as the solution to a
friction. In the case of CRT, we posit that because informed bank capital is scarce,
markets have evolved which allow banks to separate balance sheet management and
borrower management.
16
Of course, all economic agents react optimally to this new market. In equilibrium,
this market changes the optimal contracts offered by firms to banks and the bond
market. This may result in outcomes that are socially inefficient. In particular,
firms may choose smaller project sizes. Further, the relationship between prices and
quantities in the market has changed. In our model, the promised return on debt
instruments change not because of changes in a particular firm’s default probability,
but because the cost of bank capital has changed and thus the optimal contract. This
suggests that the fact that a CRT market exists for some firms and not for others is
an important conditioning events for those seeking to infer default probabilities from
the data.
17
5
Appendix
Proof of Lemma 1
A bank will sell a failed project at any price r ≥ 0. A bank with a successful
project values it at δ2 R, thus will sell it if rR ≥ δ2 R, or r ≥ δ2 . If r < δ, then only
the banks who know that the project is unsuccessful will sell, thus claims are worth
0. If r ≥ δ2 , then with probability 1 − p, the bank knows the project is a failure, and
with probability pq, the project was a success, but the bank got a shock. Thus, if
. The result follows.
r ≥ δ2 , the expected value of $ 1 promised at date 2 is pq+(1−p)0
pq+(1−p)
Proof of Lemma 2
. Thus,
Recall, ∆(p) = (1−p)p(1−q)
1−p(1−q)
(i)
d∆(p)
(1 − 2 p + p2 (1 − q)) (1 − q)
=
dp
(1 + p (−1 + q))2
Here,
∆(p)
1−p
=
d
p(1−q)
.
1−p+pq
∆(p)
1−p
∆(p)
1−p
(dp)2
!
=
dp
d2
Clearly
=
p (1 − q) (1 − q)
1−q
+
≥0
2
1 − p + pq
(1 − p + p q)
2 p (1 − q) (1 − q)2 2 (1 − q) (1 − q)
+
≥ 0.
(1 − p + p q)3
(1 − p + p q)2
Proof of Lemma 3
If the firm does not use Bank capital, then the firm’s incentive compatibility
constraint and outside investors participation constraints are:
pRF I ≥ BF I,
p(R − RF )I ≥ I − A.
(13)
(14)
As in Holmstrom and Tirole (1997), these constraints are clearly optimally binding. The optimal contract follows.
Proof of Lemma 4
18
All the constraints are binding. The result follows.
Proof of Lemma 5
All constraints bind. The result follows.
Proof of Proposition 1
pq
Define r = 1−p+pq
. r is increasing and convex in p through the origin. Define
δ∗ =
(1−q)(δ1 −p)
.
(1−p)−q(δ1 −p)
(1 − q)(1 − δ1 )
dδ ∗
= −
dp
(1 − p + q(p − δ1 ))2
(15)
1
. Further, when p = δ1 , δ ∗ = 0, and
Further, δ ∗ has a positive intercept of (1−q)δ
1−qδ1
∗
when δ = 0, p = δ1 . Thus, there is a unique intersection point less than one.
p̂ =
1 + δ1 (1 − 2q) +
q
(1 − δ1 )(1 − δ1 (1 − 4q + 4q 2 ))
2(1 − q)
Proof of Proposition 2
The promised return per dollar invested is RLB I . Thus,
CRT
No
RB
= pδ11 if there is a CRT market and RB
= pEδ12 δ1 if there is not.
CRT
No
.
> RB
As Eδ2 < 1, RB
Further,
CRT
dRB
dp
2 CRT
d RB
(dp)2
= −
=
1
p2 δ1
2
δ1 p3
For no CRT, then
No
dRB
1
= − 2
dp
p Eδ2 δ1
No
dRB
1
= 2 3
dp
p Eδ2 δ1
19
(16)
Proof of Proposition 3
(i) The ratio of bank debt over total project size if there is an active CRT Market
is
LCRT
I CRT
=
pBB
,
p−r
(17)
and if there is not,
Lno
= BB .
I no
CRT
(18)
no
Thus, LI CRT ≥ LI no if p ≥ p − r, which always holds.
(ii) The ratio of market debt over total project size, if there is no CRT market is
M no
1
,
=
pR
−
b
−
B
F
B
I no
δ1 Eδ2
(19)
and if there is one,
M no
I no
≥
M CRT
I CRT
M CRT
I CRT
= pR − bF − BB
p−r
,
p
which always holds.
if Eδ2 >
Proof of Proposition 4
CRT
(i) LI CRT is proportional to
q
> 0.
(1−q)(1−p)2
M no
p
.
p−r
p
.
δ1 (p − r)
The derivative of
(20)
p
p−r
with respect to p is
CRT
no
= R. Whereas,
(ii) d Idp
2q
−BB (1−p)3 (1−q) < 0.
d MCRT
I
dp
Proof of Proposition 5
If there is no CRT, then I no =
q
= R − BB (1−q)(1−p)
2 , with a rate of increase of
A
1−δ1 Eδ2 ) .
)
1−pR+bF +BB ( δ (Eδ
)
1
p.
This is increasing, convex in
2

If there is a CRT, then I CRT = A 

1
p
1+bF −pR+BB ( p−r
)
1−δ1
δ1
. This is increasing,
convex in p for
R > BB
(1 − δ1 )
q
δ1 (1 − q)(1 − p)2
e so that for p > p,
e this will not hold.
Thus, there is a p,
20
(21)
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22