Securitization and Lending Competition
David M. Frankel (Iowa State University)
Yu Jin (Shanghai University of Finance and Economics)
October 3, 2013
Abstract
We study the e¤ects of securitization on interbank lending competition.
An ap-
plicant’s observable features are seen by a remote bank, while her true credit quality
is known only to her local bank.
Without securitization, the remote bank does not
compete because of a winner’s curse.
bliss:
With securitization, in contrast, ignorance is
the less a bank knows about its loans, the less of a lemons problem it faces
in selling them. This enables the remote bank to compete successfully for applicants
with strong observables. While securitization expands lending, it also raises default
rates and leads to an ine¢ cient loan allocation.
JEL: D82, G14, G21.
Keywords: Banks, Securitization, Mortgage Backed Securities, Remote Lending, Internet Lending, Distance Lending, Lending Competition, Asymmetric Information,
Signalling, Lemons Problem, Residential Mortgages, Default Risk, Crisis of 2008.
Frankel: Department of Economics, Iowa State University, Ames, IA 50011, dfrankel@iastate.edu. Jin:
Shanghai University of Finance and Economics, Shanghai 200433, China, jinyu@shufe.edu.cn.
We thank
seminar participants at Copenhagen, Hebrew U., IDC-Herzliya, Lund, U. Minnesota, Tel Aviv, Warwick, the
2012 Workshop on Consumer Credit and Payments at FRB-Philadelphia, and the 2013 AFA annual meeting
for helpful comments.
1
1
Introduction
Securitization of conventional home mortgages began in 1970 with the founding of the Federal
Home Loan Mortgage Corporation.1
The proportion of mortgages held in market-based
instruments rose steadily from 20% in 1980 to 68% in 2008.2 Earlier evidence indicates that
securitization rose from 1975 to 1980 as well (Ja¤ee and Rosen [28, Table 2]).
Remote lending has also grown.
Petersen and Rajan [36, Figures I and II] …nd an
upwards trend in distances between small …rms and their lenders that began in about 1978
or 1979 and continued through the end of their data in 1992. The mean borrower-lender
distance in a sample of small business loans studied by De Young, Glennon, and Nigro [18,
pp. 125-6] rose from 5.9 miles in 1984 to 21.5 miles in 2001. Remote lending of residential
mortgages also rose from 1992 to 2007 (Loutskina and Strahan [31, p. 1477]).
We present a tractable theoretical model that links securitization and remote lending.
An applicant’s soft information is known only to a local bank while her hard information is
known also to a remote bank. Without securitization, the remote bank does not compete
because of a winner’s curse. With securitization, in contrast, ignorance is bliss: the less a
bank knows about its loans, the less of a lemons problem it faces in selling them.3 Under
fairly weak distributional assumptions, this enables the remote bank to compete successfully
for applicants with strong enough observables.
In our model, securitization has mixed e¤ects on social welfare. It raises the supply of
funding for worthwhile projects by connecting liquid investors with deserving loan applicants.
However, it also leads to an ine¢ cient loan allocation by encouraging lending by poorly
informed remote banks. For instance, consider two applicants in the same location. One
has a high credit score but a negative NPV project. The other has a low credit score but
1
A detailed history of securitization appears in Hill [27].
2
The source is unpublished data underlying Figure 3 in Shin [40].
3
In a prior empirical paper, Loutskina and Strahan [31] point out that banks may have an incentive to
lend remotely in order to avoid private information at the time of securitization.
2
a positive NPV project. A remote bank would favor the …rst applicant since evaluating a
project’s NPV requires soft information, which it lacks. A local bank may prefer not to fund
either applicant because it knows too much about them, which makes their loans di¢ cult
to sell. Hence, the negative-NPV project is funded while the positive-NPV project is not.
This allocation of funds is ine¢ cient.
We treat securitization as an exogenous innovation that encourages remote lending. If
instead securitization were initially possible and an exogenous barrier to remote lending were
then lifted, our model would also predict a simultaneous increase in both remote lending and
securitization.4
In practice, legal barriers to interstate banking fell gradually starting in
Maine in 1978 and ending with the federal government’s passage of the Interstate Banking
and Branching E¢ ciency Act of 1994, which abolished all remaining restrictions (Loutskina
and Strahan [31, pp. 1451-2]). Since securitization was invented earlier, these barriers may
have fallen partly in response to pressure from large banks who were eager to increase their
securitization pro…ts. Alternatively, their fall may have been due to an exogenous change
in regulatory philosophy. This is an interesting topic for future empirical research.
A number of authors have studied the e¤ects of securitization on a lender’s incentive
to screen loan applicants when screening is costly.
Bubb and Kaufman [8] show that a
monopsony loan buyer may commit not to buy all of a bank’s loans in order to accomplish
this. Hartman-Glaser, Piskorski, and Tchistyi [25] study how contingent future payments
to the bank from loan buyers can provide such an incentive. These papers focus on moral
hazard rather than adverse selection and, consequently, do not consider lending competition.
Other researchers have studied lending competition with adverse selection in the absence
of securitization. Hauswald and Marquez [26] assume that a bank’s cost of gathering soft
information is greater for more distant loan applicants.
4
Thus, banks gather more soft
Since we assume banks lack private information about their remote loans and have a lower discount
factor than investors, banks securitize all of their remote loans. Since - in our model - they securitize only
some of their local loans, removing a barrier to remote lending would raise the proportion of loans that are
securitized.
3
information about local applicants. Because of a winner’s curse, a remote bank is less likely
to compete for an applicant who lives very close to another bank.
an o¤setting e¤ect:
Our …ndings suggest
an applicant’s closeness makes it harder for her local bank to resell
her loan. This encourages remote lending. In earlier models, Rajan [37] and Sharpe [39]
assume that a bank’s soft information arises endogenously from its prior loans to borrowers.
Because of a winner’s curse, mature borrowers become captives of their home banks. Our
model suggests that securitization may upset this result.
A large literature has studied the problem of a security issuer who wishes to sell claims
on her future cash ‡ows. Myers and Majluf [34] show that an issuer may choose not to issue
equity to fund a worthwhile project if it has private information about the project’s return.
In their model, the amount of outside capital to be raised is …xed. Leland and Pyle [30]
permit a issuer to choose the amount of equity that she retains. When her information is
positive, she retains more, which is costly for her as she is risk averse.
Nachman and Noe [35] retain Myers and Majluf’s assumption of a …xed capital requirement, but permit the issuer to choose any monotonic security. Under certain assumptions,
they show that standard debt is optimal as it is the least sensitive to the issuer’s private information. In particular, the payout of debt only depends on this information when the debt
is in default.5
DeMarzo and Du¢ e [16] discard the …xed capital requirement and assume
the issuer is more impatient than security buyers. They show that standard debt is again
optimal and, as in Leland and Pyle [30], the issuer will retain more when her information
is positive. This signalling is costly for the issuer, partly since she retains more precisely
when the potential gains from trade are higher.
We show that another way for banks to issue informationally insensitive securities is to
acquire assets about which they know little. This means that a remote bank can resell a
loan more pro…tably than a local bank.
Hence, remote banks have an advantage at the
securitization stage that can o¤set their inferior loan selection at the lending stage. This
5
Biais and Mariotti [5] show that debt remains optimal if security buyers have market power and the
security is sold using an optimal trading mechanism rather than a signalling game.
4
permits remote banks to compete successfully with local banks.
Axelson [2] shows that if the security buyers are informed while the issuer is not, standard
debt is still optimal for the issuer as it minimizes the informational advantage of the buyers.
Similarly, Dang, Gorton, and Holmström [11] (DGH) show that a debt design minimizes the
incentive of future investors to acquire costly information about the security’s underlying
portfolio.
This protects the initial security buyers against cherry picking by these future
investors. In our model, an equity design is optimal for the remote bank since it permits the
bank to sell its entire portfolio. DGH’s results suggest that we may instead observe remote
banks using a debt design to securitize their loans.
The rest of the paper is as follows. Our main model is presented in section 2. Section
3 analyzes a base case without securitization, while the full model is studied in section 4.
Extensions are studied in section 5. Conclusions appear in section 6.
2
The Main Model
There is a single region that contains a unit measure of agents.
Each agent is endowed
with a project that requires one unit of capital and pays a …xed gross return of
> 1 if it
succeeds and zero otherwise. An agent has no capital of her own, so in order to implement
her project she must borrow one unit of capital from a bank.
There are two banks:
a
local bank L and a remote bank R. There is also a continuum of uninformed, deep-pocket
investors. All participants are fully rational and risk-neutral.
The success probability of the project of agent i 2 [0; 1] is the product of two independent
random variables: the agent’s idiosyncratic credit quality
and a common, region-speci…c
macroeconomic shock . Project outcomes, conditional on these success probabilities, are
independent.6
The population distribution of credit quality
is given by a function G,
which is common knowledge and represents the agents’observable features. The agents are
6
That is, the success probability of the project is
regardless of the outcomes of the other projects in
the region.
5
observably equivalent: the credit quality
and
of each agent also has the distribution G. Both
have continuous densities with support equal to [0; 1].
Initially, the local bank knows each agent’s credit quality
while the remote bank and
investors know only the distribution G. Rather than working with G and , it will usually
be more convenient to use expected credit quality
quality
type.
= = . We will refer to
= E ( ) and an agent’s relative credit
as the agents’credit score and to
The support of the private type
as an agent’s private
is [0; 1= ], its mean value is one, and it has the
commonly known distribution H de…ned by H ( ) = G (
).
There are three periods. Lending competition occurs in period 1. The remote bank …rst
announces whether it is willing to lend to the agents and, if so, at what gross interest rate
r.7 We assume r is not higher than the gross project return , since o¤ering a higher rate
is equivalent to not making an o¤er.8 If the remote bank declines to lend, let r equal the
gross project return . With this convention, r is now the maximum interest rate that the
agents are willing to pay the local bank. For each agent, the local bank can then announce
a counter o¤er. If it does so, it will o¤er the interest rate r and the agent will agree. Each
bank’s loan o¤ers are publicly observed, as is the measure of projects that the bank …nances.
We assume there is an in…nitesimal chance that the loan resale market will be disrupted,
forcing a bank to hold its loans to maturity.
Since only the local bank sees the agents’
private types, a threshold strategy must be optimal: the local bank will bid r (and win)
if and only if an agent’s private type
Note that
exceeds a threshold
will in general depend on r.
In period 2, the macroeconomic shock
signal t 2 [0; 1], t
7
that the local bank chooses.
of this shock.
is realized. The local bank then sees a private
Arbitrarily low signals can occur: for any t > 0,
As the remote bank has no private information about its loans, it will sell its entire portfolio to investors.
This means that its pro…ts from di¤erent loans are additive. Hence, the remote bank would not bene…t from
o¤ering di¤erent interest rates to di¤erent agents, or from …nancing only a subset of agents who approve its
o¤er.
8
We assume an agent’s consumption cannot be negative.
6
(t) > 0. Let
( jt) denote the distribution of the shock conditional on the local bank’s
signal. We assume that
and
are continuously di¤erentiable in their arguments and have
no atoms.9 Moreover, higher signals are good news:
First Order Stochastic Dominance (FOSD). For any z 2 [0; 1], the probability
that the shock
(zjt)
does not exceed the cuto¤ z is nonincreasing in the signal t.
Finally, we assume that the expectation of the shock , conditional on the signal t taking its
minimum value of zero, is strictly positive: E [ jt = 0] > 0. Intuitively, even if it gets the
lowest macroeconomic signal, the local bank thinks there is a chance that some projects will
succeed.
After the local bank sees its signal t, each bank k = R; L selects a set of loans to securitize:
to include as collateral for its security.10 For brevity, we refer to a bank’s set of securitized
loans as its portfolio. The measure of loans in each bank’s portfolio is commonly observed.
Each bank k then chooses a proportion qk 2 [0; 1] of its portfolio to o¤er for sale.11 (We
adopt the convention that qk = 0 if bank k does not securitize any loans.) The investors
then use Bayes’s rule to assign a value pk to bank k’s portfolio.
The total sum paid by
investors for the qk units of bank k’s portfolio is thus pk qk .
In period 3, the outcomes of the funded agents’ projects are realized.
If her project
succeeds, an agent pays the interest rate r to the bank that …nanced it; her payo¤ is thus
9
10
A distribution F on [0; 1] is atomless if F is continuous and F (0) = 0.
We do not permit a bank to securitize some but not all of a given loan. This is without loss of generality.
Partial securitization is clearly suboptimal for the remote bank, which faces no lemons problem and thus
can realize all of the gains from trade with investors. As for the local bank, investors do not see the credit
quality
of a given loan.
However, as the bank securitizes entire loans in our model, investors can infer
the credit quality distribution of the bank’s security. Thus, by securitizing entire loans the local bank can
completely eliminate asymmetric information about its credit quality distribution.
Partial securitization
of particular loans thus cannot mitigate the signalling problem, which arises solely because of asymmetric
information about the subsequent macroeconomic signal (which a¤ects all loans proportionally).
11
An extension to nonequity securities is studied in section 5.3.
7
r. If her project fails or was not funded, she pays nothing and her payo¤ is zero. Each
bank k then pays the proportion qk of the repayments on loans in its portfolio to the investors
who purchased shares in this portfolio.
While periods 1 and 2 occur at the same point of real time, there is a unit of delay between
periods 2 and 3. The banks are liquidity constrained: the discount factor of investors, which
we normalize to one, exceeds the discount factor of the banks, which is denoted
2 (0; 1).
This assumption, common in the prior literature, is thought to capture the typical reason
cited for why banks sell loans: the availability of attractive alternative investments together
with the existence of regulatory capital ratios (e.g., Gorton and Haubrich [24]).
The two banks have the same cost of capital, which is normalized to one.
Suppose a
bank lends c1 units of capital in period 1 to agents who later repay the bank c3 in period 3.
Assume, moreover, that investors pay the bank c2 in period 2 in return for an equity stake
that obliges the bank to pay them c03 in period 3. Then the payo¤ of these investors is c03
and the bank’s payo¤ equals c2
c2
c03 ): its proceeds c2 from its security sales,
c1 + (c3
less its capital cost c1 , plus its discounted loan proceeds c3 , less its discounted payment to
investors c03 .
3
Base Case: Competition without Securitization
We …rst analyze a base case without securitization:
maturity.
each bank must hold every loan to
If a bank lends, at a gross interest rate r, to an agent with credit type , its
expected pro…t is r E ( )
1: the discounted interest payment r times the probability
E ( ) of project success, less the unitary cost of capital.
In the base case, only the local bank lends and it extracts the full surplus. This is due to
the winner’s curse: the banks have the same expected payo¤ from lending to a given agent,
but the local bank has superior information about this payo¤.12
Since, by assumption,
the local bank makes the second o¤er, it will slightly underbid the remote bank if a loan is
12
The winner’s curse in lending competition was …rst studied by Rajan [37] and Sharpe [39].
8
pro…table and refrain from bidding otherwise. Knowing this, the remote bank will not make
any o¤ers.
Claim 1 Without the option of securitization, only the local bank lends. The agents’payo¤s
are zero: the gross interest rate on each loan equals the gross project return . An agent
with credit quality
is …nanced if and only if her project’s discounted expected gross return,
E ( ), exceeds the unitary cost of capital.
Without securitization, an agent gets a loan if and only if her discounted expected project
return exceeds the banks’common cost of capital. Hence, the allocation of loans is e¢ cient:
one agent is funded while another is not only if the …rst agent’s project has a higher expected
return than the other agent’s.
This e¢ ciency property will not hold with securitization,
since a bank may prefer not to lend to a creditworthy agent whom it knows well. Intuitively,
the bank’s private information about the agent’s repayment probability worsens the lemons
problem the bank faces in securitizing the agent’s loan.
Our conclusion that all lending is local and the loan allocation is e¢ cient relies on our
assumption that the remote bank makes the …rst o¤er, followed by the local bank.
A
similar order of o¤ers is used by Dell’Ariccia, Friedman, and Marquez [13] and Dell’Ariccia
and Marquez [14]. Others, such as Rajan [37] and von Thadden [41], have assumed instead
simultaneous o¤ers. They generally …nd that the uninformed bank plays a mixed strategy
and sometimes wins. The extension of our model to the case of simultaneous o¤ers might
be an interesting topic for future research.
4
Full Model: Competition with Securitization
We now turn to the case of securitization. We begin in period 2. The remote bank must
securitize loans of all private types with the same probability p. But o¤ering a quantity qR
of the resulting portfolio is formally equivalent to securitizing every loan for sure and o¤ering
the quantity pqR . Hence, we can assume without loss of generality that the remote bank
9
securitizes all of its loans. Let x equal one if the remote bank o¤ers loans to the agents and
zero otherwise. The realized gross return of the remote bank’s portfolio is thus
Z
YR = xr
dH ( ) :
(1)
=0
the indicator variable x, times the interest rate r, times the integral, over all private types
in the remote bank’s portfolio, of the project success probability
.
As only the local bank sees a loan’s private type , it will securitize the loan if and only
if
is less than some securitization threshold
2 [ ; 1= ] that it chooses. The realized gross
return of the local bank’s portfolio is
Z
dH ( ) :
(2)
the interest rate r times the integral, over all private types
in the local bank’s portfolio, of
YL = r
the project success probability
=
.
As noted in section 2, both the measure 1
the measure H ( )
H ( ) of loans that the local bank makes and
H ( ) of loans that it securitizes are commonly observed. Since the
distribution function H is strictly increasing, investors can then infer the local bank’s lending
and securitization thresholds
and . Thus, when the local bank chooses its quantity qL
the distribution of private types
Let
k
in its portfolio is common knowledge.
(t) = E [Yk jt] be the expected gross return of the portfolio of bank k = L; R,
conditional on the signal t of the local bank. Let pk (qL ) be the price that investors o¤er
per unit of this portfolio. Bank L’s expected pro…t from selling the quantity qL , conditional
on its signal t, is
L
(t; qL ) = qL [pL (qL )
L
(t)] :
(3)
the quantity it sells, times the di¤erence between the price it gets and the discounted payment
it must later make. Similarly, bank R’s expected pro…t from selling the quantity qR is
h
i
b
(q
)
=
q
P
(4)
R
R
R
R
R ;
where PR =
R1
t=0
pR (qL (t)) d (t) and bR =
R1
t=0
expected payout of the bank’s loans, respectively.
10
R
(t) d (t) are the expected value and
There is symmetric information between the remote bank and the investors. Moreover,
there are gains from trade since the investors are more patient than the bank. Hence, if only
the remote bank lends (
1), it will sell all of its loans: qR will equal one. The remote
bank’s pro…ts from loan sales in this case are simply the di¤erence 1
in discount rates
times the expected payout E (YR ) = r E ( ) of the bank’s loan portfolio. (Recall that the
mean E ( ) of the private types is one.)
Assume now that the local bank makes some loans (
< 1).
We use the following
de…nition of equilibrium in this subgame.
De…nition 2 A Bayes-Nash equilibrium of the subgame that begins in period 2, if the local
bank made some loans in period 1, is a measurable quantity function qL (t) chosen by the local
bank, a quantity qR chosen by the remote bank, and a pair pL (qL ), pR (qL ) of measurable price
functions chosen by investors such that:
1. Best Response: qL (t) 2 arg maxq2[0;1]
L
(t; q) and qR 2 arg maxq2[0;1]
R
(q), almost
surely;
2. Bayesian Updating: pL (qL (t)) = E [
L
(t) jqL (t)] and pR (qL (t)) = E [
R
(t) jqL (t)],
almost surely.
The equilibrium is separating if the following condition also holds.
3. Separation: pL (qL (t)) =
L
(t) almost surely.
We restrict to separating equilibria, which satisfy all three conditions. This restriction
uniquely determines the banks’ behavior and pro…ts.
Using the formula for PR above,
condition 2 of De…nition 2, and the law of iterative expectations,
Z 1
Z 1
b
PR =
E [ R (t) jqL (t)] d (t) =
R (t) d (t) = R > 0:
t=0
t=0
Hence, by (4), qR = 1: the remote bank sells all its loans.
from Proposition 2 in DeMarzo and Du¢ e [16].
11
The next claim then follows
Claim 3 The subgame that begins in period 2, if the local bank made some loans in period
1, has a unique separating Bayes-Nash equilibrium, with the following properties.
1. If the remote bank lent in period 1, it sell all its loans in period 2.
R
securitization pro…ts are (1
) r E ( ) =0 dH ( ).
Its expected
2. The equilibrium quantity and price functions for the local bank’s loans are qL (t) =
1
E[ jt=0]
E[ jt]
1
and pL (qL ) =
L
(0) = (qL )1 .
3. Bank L’s expected pro…ts from selling its loans, conditional on its signal t, are
(1
)r
Z
dH ( )
E [ jt = 0]
=
E [ jt]
!11
:
By part 2 of Claim 3, when the local bank is most pessimistic, it sells all of its loans:
qL (0) = 1. As its signal t rises, the local bank sells progressively less in order to signal higher
quality. This indeed leads to a higher price. However, the quantity falls so fast that the
bank’s securitization pro…ts (which equal the realized gains from trade as investor payo¤s are
zero) decline in its signal t. But the maximum potential gains from trade (1
) E (YL jt)
are increasing in the signal t.13 In this sense, the signalling outcome is quite ine¢ cient, as
DeMarzo and Du¢ e [16] note.
We now step backwards to the lending stage (period 1). By part 3 of Claim 3, the local
bank’s securitization pro…ts are increasing in its securitization threshold . Thus, it will set
equal to the upper bound on an agent’s private type:
= 1= :
(5)
A bank’s expected pro…t (excluding its securitization pro…ts) from lending to an agent with
private type
13
is the discounted expected gross loan return r
E ( ) minus the unitary cost
The maximum gains from trade are just the investors’ valuation E (YL ju) less bank L’s valuation
E (YL ju).
12
of capital. When it lends, the local bank does not know its signal t of the shock , so by
Claim 3 and equation (5) its expected total pro…ts are14
Z 1=
(r
1) dH ( )
L ( ; r) =
(6)
=
where
= E ( ) + (1
and
2
E [ jt = 0]
=E4
(7)
)
E [ jt]
!11 3
5
(8)
is the expected proportion E [qL (t)] of its loan portfolio that the local bank sells.
An
increase in the informativeness of the local bank’s signal t lowers this expected proportion
and hence the local bank’s expected total pro…ts
The expected proportion
L.
lies between zero and the lowest possible conditional expec-
tation of the shock :
Claim 4 0 <
< E [ jt = 0].
Proof. By assumption, E [ jt = 0] > 0, so
> 0. Since E ( jt) is minimized when t takes
1
its lowest value of zero,
<
E[ jt=0]
E[ jt=0]
1
= E [ jt = 0].
Moreover, by Claim 4 and equation (7),
0<
<
(9)
< E ( ) < 1:
The local bank chooses its lending threshold
to maximize its total pro…ts
L
( ; r). By
(6), its optimal lending threshold is
= (r
14
)
1
(10)
:
This is a simpli…cation of
L
( ; r) =
Z
1=
[ r E( )
1] dH ( ) + (1
)r
=
13
"Z
1=
=
#
2
E [ jt = 0]
dH ( ) E 4
E [ jt]
!11 3
5:
If the remote bank lends, it sells all of its loans. Its expected pro…t
R
(r) from o¤ering
an interest rate r not greater than the gross project return is just its direct lending pro…t
R
( r E ( ) 1) dH ( ) plus its securitization pro…ts:
=0
Z
(r E ( ) 1) dH ( )
(11)
R (r) =
=0
by part 1 of Claim 3, where the local bank’s lending threshold
depends on the remote
bank’s o¤er r via equation (10).
Let r be the interest rate r at which the local bank is just willing not to lend to the agent
with the highest private type: at which its lending threshold
equals the upper bound 1=
on . We refer to r as the deterring rate. By (10), it is de…ned by
(12)
r = 1= :
By equations (7) and (8), the deterring rate r does not depend on the credit score .
An explicit solution can be obtained if the distribution of private types is not too concave,
and its concavity is nondecreasing in . More formally, we assume:
No Cream Skimming (NCS). For all private types
H 00 ( ) =H 0 ( ) is greater than
in the interior of the support of H,
1 and is increasing in .
Under NCS, if the remote bank competes, it prefers to charge an interest rate that is low
enough to deter the local bank from cream-skimming.
Equation (12) then provides an
explicit solution for the interest rate. The formal result is as follows.
Claim 5 Assume No Cream Skimming.
1. If the remote bank competes, it o¤ers the interest rate r = min f ; r g and lends to all
private types .
2. If r
type
, then the local bank is just willing not to bid for the best agent, whose private
is
= 1= . If r > , the local bank strictly prefers not to bid for any agent.
14
By Claim 5, if the remote bank competes, then it o¤ers an interest rate that is low enough
to deter the local bank from lending even to an agent with the highest possible private type
= 1= . Hence, in equilibrium the local bank does not “cream skim”: lend to agents with
high private types but not to all agents. This fact, which relies on NCS, allows us to solve
analytically for the interest rates that the banks o¤er. It is consistent with the observation
of Agarwal and Hauswald [1] that internet lenders charge low rates partly in order to prevent
cream skimming:15
Arm’s-length debt is less readily available but carries lower rates because
competition among symmetrically informed banks, which rel[ies] on public information, not only drive[s] down its price but also restrict[s] access to credit to
minimize adverse selection. [Agarwal and Hauswald [1]]
Under NCS, the remote bank’s pro…ts from competing are
R
(r) = min f ; r g E ( )
1;
by (11) and Claim 5. It will compete if this is positive: if the credit score
(13)
exceeds the
threshold
=
1
:
min f ; r g E ( )
(14)
Intuitively, if the remote bank competes it lends to all agents at an interest rate min f ; r g
that does not depend on the mean credit quality .16 However, the agents’mean repayment
15
Broecker [7] shows that a similar e¤ect occurs when each bank has a noisy private signal of an applicant’s
credit quality.
In his model, the banks simultaneously bid interest rates to an applicant, where a bank’s
bid is decreasing in its signal. In a symmetric equilibrium, a bank attracts an applicant if and only if all
other banks got worse signals than the bank. Thus, if a bank raises its interest rate slightly, it loses the best
applicants among those who would have accepted its initial, lower rate. This gives each bank an incentive
to lower its rate in order to prevent cream-skimming by other banks.
16
The deterring rate r is the rate at which the local bank breaks even from lending to the agent with the
highest credit quality
= 1. Hence, r does not depend on the mean credit quality .
15
probability - and thus the remote bank’s pro…t from competing - is increasing in . Thus,
the remote bank follows a threshold strategy in .
If the credit score
is less than the threshold , the remote bank does not compete. In
this case, an agent’s willingness to pay r is just the gross project return
local bank will lend to an agent if her private type
equivalently, if her credit quality
exceeds the threshold
=[
]
1
or,
] 1 . If it lends, then by
exceeds the threshold [
=
so by (10) the
1
E[ jt=0]
E[ jt]
Claim 3, it will sell a proportion
1
of its loans to investors. We have proved the
following result.
Theorem 6 Assume No Cream Skimming.
1. If the agents’credit score
an agent’s credit quality
is less than , then the remote bank does not compete. If
is not less than the lending threshold [
=
] 1 , the local
bank o¤ers her a loan at an interest rate equal to the gross project return
and the
agent accepts. On seeing its private signal t of the macroeconomic shock , the local
1
bank then sells a proportion
2. If
E[ jt=0]
E[ jt]
1
of these loans.
> , the remote bank o¤ers the interest rate min f ; r g. The local bank does not
compete, and all agents accept the remote bank’s o¤er.
Moreover, the remote bank
sells its entire portfolio of loans.
Theorem 6 is illustrated in Figure 1. An agent’s credit quality , which only the local
bank sees, appears on the vertical axis. Her credit score
is depicted on the horizontal axis.
Without securitization, the local bank lends to agents whose her credit quality
the threshold [ E ( )]
1
exceeds
by Claim 1: only agents in areas A0 and A3 are funded. With
securitization, the remote bank lends to agents with credit scores
area A4 now get funding as well. For agents with credit scores
(by Claim 4) its lending threshold to [
] 1:
> . Hence, agents in
< , the local bank lowers
funding is extended also to agents in area
A1 . Thus, securitization stimulates lending both by making remote lending possible and by
leading local banks to lower their standards.
16
Figure 1:
Comparative statics are as follows. The remote bank lends if and only if agents’observables are strong enough: when
> . This occurs more often, and the remote bank o¤ers a
higher interest rate, if the local bank is expected to be better informed about the macroeconomic shock:
(respectively, r ) is increasing (decreasing) in
by (7), (8), (12), and (14).
Intuitively, if the local bank knows more about local conditions, it sells a lower proportion
of its loans on average (part 3 of Claim 3), which lowers its expected securitization pro…ts
(equations (6) and (7)). This permits the remote bank to raise its interest rate o¤er while
still preventing the local bank from cream-skimming: the deterring rate r rises. Able to
charge a higher interest rate, the remote bank competes more often: it lowers its credit score
17
threshold . This also implies that if the local bank’s expected private information about
local economic conditions grows (if
falls), agents with strong observables su¤er by paying
higher interest rates to remote banks (r rises), while agents with intermediate observables
bene…t by switching from local to remote banks ( falls).
Our model assumes that a bank cannot bid on the other bank’s security. Clearly, only
the local bank might pro…t from doing so since only it has an information advantage (the
signal t) that o¤sets its greater impatience vis-a-vis investors. When the remote bank lends,
it sells its entire portfolio to investors for a price equal to the portfolio’s expected payout
r E [ ]. The local bank’s valuation of this portfolio is at most r E [ jt = 1]. Hence, in
the above equilibrium, the local bank can never pro…t from bidding on the remote bank’s
portfolio if
E [ jt = 1] < E [ ] :
that is, if its degree of patience
(15)
is less than the maximum informational advantage that it
gets from its private signal t of the macroeconomic shock . If (15) holds, then the above
outcome remains an equilibrium if banks can bid on each others’securities.
4.1
Welfare
We now show that securitization unambiguously raises welfare in the case of agents who
have observables that are too weak to get remote loans. However, it has ambiguous welfare
e¤ects when agents have strong observables: while securitization connects them with liquid
investors, it also transfers their loan decisions to the remote bank, which does not screen as
well.
Without securitization, only the local bank lends. An agent’s payo¤ is zero: the gross
interest rate on the loan equals the gross project return . An agent with private type
gets a loan if and only if her discounted expected gross project return,
E ( ), exceeds
the bank’s unitary cost of capital. Moreover, lending to this agent creates a social surplus
equal to this discounted expected return less the unitary cost of capital. Social welfare is
18
just the expectation of this surplus over all private types :
Z 1=
N
max f0;
E ( ) 1g dH ( ) :
SW =
=0
With securitization, there are two cases. If
< , the remote bank does not lend, so
r = : the agents get zero. Hence, social welfare equals the local bank’s pro…t L ( ; )
R 1=
which by (6) and (10) equals =0 max f0;
1g dH ( ). This exceeds SW N since, by
(7),
> E ( ):
securitization raises welfare.
Intuitively, the local bank still makes all
the loans, so the loan allocation is still e¢ cient, but securitization allows more agents to get
loans, raising welfare.
If
>
, the remote bank o¤ers the interest rate r = min f ; r g.
The local bank
does not compete, and every agent accepts the remote bank’s o¤er. Moreover, the remote
bank securitizes all of its loans. Since all projects are …nanced, the mean project success
R 1=
probability is P = =0 E ( ) dH ( ). Investors pay the bank rP for its loans, so the bank’s
pro…t is rP
1. The expected payo¤ of an agent with private type
the project return
is the product of (a)
less the interest rate r and (b) the agent’s project success probability
. Thus, the agents’mean payo¤ is (
r) P . Social welfare is the sum of these payo¤s,
R 1=
which is P 1 = =0 [
E ( ) 1] dH ( ).
This di¤ers from social welfare in the absence of securitization (SW N ) in two ways. First,
the gross project return is no longer discounted by , which tends to raise welfare. However,
as the local no longer lends, a project is funded even if its expected gross return
E [ ] is
less than the unitary cost of capital. This is an e¢ ciency loss. Whether welfare rises or
falls depends on which e¤ect is stronger.
4.2
Empirical Implications
Our model yields several predictions that are consistent with prior empirical …ndings.
1. Securitization Stimulates Lending.
As in Shin [40], securitization leads to ex-
panded lending by connecting liquid investors with loan applicants. In Figure 1, areas
A1 and A4 are added.
There is considerable evidence that the securitization boom
19
in the 2000s led to expanded lending (Demyanyk and Van Hemert [17]; Krainer and
Laderman [29]; Mian and Su… [32]).
2. Securitization Favors Remote Lenders, who Securitize More. The introduction of securitization enables the remote bank to compete for some loans. Moreover,
this bank sells all of its loans while the local bank retains a portion of its loan portfolio.
Loutskina and Strahan [31] …nd that as securitization rose, the market share of concentrated lenders - those which originate at least 75% of their mortgages in one MSA - fell
from 20% to 4% from 1992 to 2007. Moreover, concentrated lenders retain a higher
proportion of their loans. Finally, when they expand to new MSA’s, these lenders are
more likely to sell their remote loans than those made in their core MSA’s.17
3. Remote Borrowers have Strong Observables but High Conditional Default
Rates. In Figure 1, agents who borrow from the remote bank have stronger observables ( > ) than those who borrow locally ( < ).
But the remote bank makes
worse lending decisions since it does not know an agent’s credit quality . So conditional on the credit score , remote borrowers are more likely to default.18 Loutskina
and Strahan [31, p. 1456] …nd that concentrated lenders (de…ned above) have lower
loan losses despite lending to applicants who are riskier in terms of loan to value ratios. Agarwal and Hauswald [1] …nd that applicants with strong observables tend to
apply online for loans, while in-person applicants tend to be those with weaker observables but positive estimates of the bank’s soft information about them.
Moreover,
online loans default more than observationally equivalent in-person loans. De Young,
Glennon, and Nigro [18] …nd that banks that lend remotely have higher default rates.
17
Our model suggests that similar …ndings should hold for internet vs. in-person loans. To our knowledge,
this question has not yet been studied empirically.
18
More precisely, an agent whose credit score
slightly exceeds the remote bank’s lending threshold
is
much more likely to default than an agent whose credit score is slightly below this threshold, as only the
latter agent is screened by the local bank.
20
4. Securitization Lets Borrowers with Strong Observables Get Cheap Remote
Loans. Securitization enables the remote bank to lend to agents with strong observables. Under No Cream Skimming, it o¤ers a low interest rate to these agents in order
to prevent cream skimming by the local bank. In contrast, the local bank can demand
a high interest rate from a high quality agent whose observables are weak since this
agent cannot get a remote loan. This has two implications. First, the securitization
boom in the 2000s should have strengthened the (negative) relation between borrower
observables and interest rates.
Rajan, Seru, and Vig [38] …nd that borrower credit
scores and LTV ratios explain just 9% of interest rate variation among loans originated
in 1997-2000 but 46% of this variation among loans originated in 2006. Second, remote
loans should carry lower interest rates. Agarwal and Hauswald [1] …nd that internet
loans carry lower interest rates than in-person loans. Degryse and Ongena [12] …nd
that interest rates decrease with the distance between small …rms and their lenders in
Belgium. Mistrulli and Casolaro [33] …nd the same relation among business lines of
credit in Italy.
5. Securitization Raises Conditional and Unconditional Default Rates.
curitization leads banks to lend to agents in areas A1 and A4 in Figure 1.
agents have uniformly lower credit qualities
SeThese
than agents in areas A0 and A3 , who
borrow without securitization. This raises both conditional (on the credit score ) and
unconditional default rates.
These predictions are con…rmed by empirical research.
Rajan, Seru, and Vig [38] …nd that conditional default rates rose between 1997-2000
and 2001-6. Demyanyk and Van Hemert [17] …nd that conditional and unconditional
default rates rose from 2001 to 2007.
6. Securitized Loans Have Higher Conditional Default Rates than Retained
Loans. A decrease in the local bank’s private information about the shock
the expected proportion
its lending threshold [
raises
of loans that it sells and, by (7), leads the bank to lower
] 1 : it lends to less creditworthy agents. Hence, securitized
21
loans should have higher default rates than retained loans, conditional on observables.
Krainer and Laderman [29] …nd that controlling for observables, privately securitized
loans default at a higher rate than retained loans. Elul [19] …nds that securitized loans
perform worse than observationally similar unsecuritized loans.
5
Extensions
We now consider several variations of the basic model. These are a model with no macroeconomic signal; a model in which No Cream Skimming is replaced by a monotone hazard
rate property; a model with security design; and a model with a continuum of local banks.
5.1
No Macroeconomic Signal
A model with no macroeconomic signal is equivalent to the special case of our model in
which the conditional expected value of the shock, E [ jt], is independent of the signal t.
This implies
=
= E ( ) by (7) and (8), so the remote bank’s lending threshold
is one
by (14): the remote bank never lends. As the local bank does not face a lemons problem at
the securitization stage, the remote bank cannot compete at the lending stage.
One way to reintroduce a lemons problem is to modify the model so that only the local
bank knows the distribution of private types
in its loan portfolio. A simple model with
this property is one with a single agent whose private type
is known only to the local bank.
If the local bank lends, investors know only that the agent’s private type
does not lie below
the local bank’s lending threshold. They do not know the precise value of . Hence, the
local bank may still retain part of the loan in order to signal that
is high. As the remote
bank does not need to signal, it may still be able to compete with the local bank.
We analyze such a model in our online appendix (Frankel and Jin [21]). Remote lending
can still occur. However, unlike our main model, there are multiple separating equilibria.
Intuitively, with a single agent the market does not observe the local bank’s lending threshold
. Hence, if the local bank deviates (e.g., by selling a higher than expected proportion of
22
its loan), the market may conclude that this threshold is zero: the loan has no chance of
being repaid. Such punishing beliefs can prevent the local bank from securitizing more than
an arbitrary proportion (including zero) of its loans. This permits multiple equilibria with
di¤erent such arbitrary proportions.19
5.2
Replacing No Cream Skimming
We now consider the e¤ect of replacing No Cream Skimming by a monotone hazard rate
property.
Our main …nding is robust to this change: the remote bank will lend only if
the agents have good enough observables. However, in this case the local bank may now
(successfully) compete for some of these agents. This means that some agents with strong
observables - those whose actual credit qualities
are also high - may instead borrow locally.
However, social welfare does not depend on which bank lends to a given agent: the welfare
implications of our main model continue to hold.
Consider a family (G )
is common knowledge.
2[0;1]
of distributions of credit quality , where the parameter
Assume that an increase in
raises the distribution of
in the
following sense.20
Monotone Hazard Rate Property (MHRP). For threshold
h ( 0) =
G0
(
G (
0)
0)
Like the credit score
0
2 (0; 1), the hazard rate
is increasing in .
in the main model, the parameter
represents the agents’observables.
The following claim implies that, as in the main model, the remote bank lends only to agents
with strong enough observables.
19
We also show that there is a unique equilibrium that survives the D1 re…nement of Banks and Sobel
[4]. There is no remote lending in this equilibrium. However, it is hard to justify the strong restrictions
on beliefs that D1 imposes. We discuss this issue further in the online appendix; see also Fudenberg and
Tirole [23, p. 460].
20
The monotone hazard rate property is stronger than …rst order stochastic dominance, but weaker than
the monotone likelihood ratio property (Nachman and Noe [35, n. 13, p. 19]).
23
Claim 7 Assume MHRP. If the remote bank’s pro…ts from competing are nonnegative for
a given parameter , then they are strictly positive for any higher parameter.
An intuition is as follows.
The analysis of our main model through and including
equation (11) does not rely on No Cream Skimming (NCS). Hence, this analysis also holds
if we replace NCS with MHRP. In particular, if the remote bank o¤ers r, the local bank
competes (and wins) if and only if the credit quality
=
1
exceeds the threshold (r )
by (10). Importantly, this threshold does not depend on observables : as the local bank
knows an agent’s actual credit quality , it does not care additionally about her observable
features.
The remote bank is left with those agents whose credit qualities
Under MHRP, the conditional expectation E [ j
old
0
and in particular for
in observables
0
0]
is increasing in
are at most (r ) 1 .
for any given thresh-
= (r ) 1 .21 Thus, for any given interest rate r, an increase
raises the mean credit quality of those agents who accept the o¤er r of the
remote bank. This makes o¤ering any given interest rate r more pro…table for the remote
bank, which therefore competes if and only if
is high enough.
The following variant of Theorem 6 is immediate.
Theorem 8 Assume MHRP. There is a threshold
2 < with the following properties.
< , then the remote bank does not compete. If an agent’s credit quality
1. If
not less than the lending threshold [
rate equal to the gross project return
is
=
] 1 , the local bank o¤ers her a loan at an interest
and the agent accepts. On seeing its private
1
signal t of the macroeconomic shock , the local bank sells a proportion
E[ jt=0]
E[ jt]
1
of these loans.
2. If
>
R (r ) 1
=0
, the remote bank o¤ers an interest rate r
(r E ( )
1) dG ( ).
exceed the threshold
21
that maximizes its pro…ts
The local bank lends to agents whose private types
that is de…ned in (10). The remote bank lends to private types
A proof of this well known property appears in the appendix (p. 38).
24
1
< .
On seeing its signal t, the local bank sells a proportion
E[ jt=0]
E[ jt]
1
of its
loans, while the remote bank sells all its loans.
While the remote bank still follows a threshold strategy, it may not charge an interest
rate low enough to deter the local bank from lending. Thus, while all agents get loans if
the remote bank competes, agents with high enough credit quality may instead borrow from
the local bank.
From a welfare point of view it is not important which bank …nances a
given agent. Hence, the welfare implications of our main model continue to hold if NCS is
replaced by MHRP.
Both NCS and MHRP imply that a threshold strategy is optimal for the remote bank.
Under NCS, if the remote bank competes for agents with given observables, it o¤ers the
deterring rate: the interest rate that just deters the local bank from lending to the agent
in this group who has the highest possible credit quality. This highest credit quality does
not depend on an agent’s observables, which can always be trumped by the bank’s soft
information. However, the remote bank does not know an applicant’s actual credit quality.
All it sees is her observables. Stronger observables imply that a loan is more likely to be
repaid. Accordingly, the remote bank will lend if and only if an applicant’s observables are
strong enough.
Under MHRP, if the remote bank o¤ers an agent the interest rate r, the local bank
competes (and wins) if and only if the agent’s credit quality exceeds a threshold that depends
on r but not on the agent’s observables. The reason is that the agent’s credit quality, which
the local bank knows, is a su¢ cient statistic for the local bank’s pro…ts from lending to the
agent. The remote bank is left with those agents whose credit qualities lie below the local
bank’s threshold. Under MHRP, the mean credit quality of these agents is increasing in their
observables. Thus, for any given interest rate r, an improvement in an agent’s observables
raises her mean credit quality condition on her accepting the remote bank’s o¤er.
This
makes o¤ering r more pro…table for the remote bank: the remote bank competes if and only
if an agent’s observables are strong enough.
25
5.3
Ex Post Security Design
We now modify the main model of section 2 to permit ex post security design, in which the
local bank designs a security after it sees its signal (DeMarzo [15]).22 Our qualitative results
remain intact.
However, security design permits the local bank to signal its information
more e¢ ciently. This strengthens the local bank’s position at the lending stage and thus
makes remote lending less likely. Moreover, as in DeMarzo [15], the local bank issues simple
debt whose face value is decreasing in its signal. This contrasts with the main model, in
which only equity can be issued.
The modi…ed model is as follows.
Period 1 (lending competition) is unchanged.
period 2, the local bank …rst sees its signal t of the shock .
In
Each bank k = R; L then
announces a function Fk that speci…es the payment Fk (Yk ) that investors will receive given
the gross return Yk of bank k’s loan portfolio. We assume that a bank must sell its entire
security, as selling a portion q < 1 of a security that pays Fk (Yk ) is equivalent to selling
all of a security that pays qFk (Yk ). We restrict to monotone securities, for which both the
security payout Fk (Yk ) and the portion Yk
Fk (Yk ) of its portfolio return that bank k keeps
are nondecreasing in the portfolio return Yk .23
We make the following technical assumptions. First, the signal density
0
(t) = d (t) =dt
is bounded and Lipschitz continuous:
Lipschitz- . There are constants k0 ; k1 2 (0; 1) such that for all signals t; t0 2 [0; 1],
0
22
(t)
k0 and j
0
(t)
0
(t0 )j
k1 jt
t0 j.
The working paper version of this paper (Frankel and Jin [20]) instead considers ex ante security design:
the local bank designs its security, sees its signal, and decides how many shares of its security to sell. The
results are essentially the same in the two cases.
23
The former can be justi…ed by assuming that the bank can hide debts.
Thus, if Fk were decreasing,
bank k could borrow money to in‡ate Yk , pay investors the lower payout, and then return the loan. The
second assumption follows from free disposal. See DeMarzo [15, p. 18]. The e¤ect of relaxing monotonicity
is not known for the case of ex post security design. With ex ante design, standard debt is no longer optimal
if monotonicity is dropped for a class of conditional distribution functions
26
(Nachman and Noe [35, n. 3]).
Moreover, the conditional distribution
, of the shock
given the signal t, is Lipschitz
continuous and has some minimum sensitivity to its arguments:
Lipschitz- . There are constants k2 ; k3 2 (0; 1) such that for all ; t 2 [0; 1],
@ ( jt)
2 (k2 ; k3 ) and
@
@ ( jt)
2 [k2 (1
@t
(16)
(17)
) ; k3 ) :
Equation (16) implies that even if the signal is zero, the shock has a positive expectation:
E [ jt = 0] > 0. As for (17), a higher signal t is good news about the shock and thus lowers
( jt).
quantity
Accordingly, the absolute sensitivity of
@ ( jt)
.
@t
Now, as
cannot require that
@ ( jt)
@t
(0; t) and
to t is represented by the nonnegative
(1; t) equal zero and one, independent of t, we
be sensitive to the signal t for all shocks .
However, we can
require that as
moves away from zero or one, this sensitivity rises at least linearly in .
The factor (1
) ensures this property as it is approximately equal to
of
= 0 and to 1
in a neighborhood of
in a neighborhood
= 1.
We also assume that the partial derivatives of the conditional distribution function
are
Lipschitz continuous in the signal t:
Lipschitz Partial Derivatives. There is a k4 2 (0; 1) such that for all ; t0 ; t00 2 [0; 1],
o
n
0
00
( jt)
@ ( jt)
< k4 jt0 t00 j.
max @ @( jt ) @ (@ jt ) ; @ @t
@t
t=t0
Finally, we assume that
t=t00
satis…es the following strengthening of First Order Stochastic
Dominance:
Hazard Rate Ordering. For all t0 > t00 ,
1
1
( jt0 )
( jt00 )
is increasing in
2 [0; 1].
This condition is weaker than the monotone likelihood ratio property, which is commonly
assumed in signalling games (see DeMarzo [15, p. 14]).
27
By (1) and (2), the realized gross return YR and YL of the remote and local banks’
portfolios (of securitized loans) equal yR and yL respectively, where the parameters yR =
R
xr =0 dH ( ) and
Z
yL = r
dH ( )
(18)
=
equal the maximum gross returns of the banks’respective portfolios and are common knowledge at the security design stage.24
We …rst consider the remote bank’s security design problem. Let FLt equal the security
designed by the local bank when its signal is t. Let E [x ( ) jFLt ] denote the expectation of a
function x ( ) given what the local bank’s security design FLt reveals about the signal t.25 Let
pR (FR ; FLt ) denote the price E [FR (yR ) jFLt ] that investors o¤er for the remote bank’s security FR when the local bank announces the security FLt . The unconditional expected price of
R1
the remote bank’s security is just t=0 pR (FR ; FLt ) d (t) which, by the law of iterated expectations, equals the unconditional expected payout E [FR (yR )] to investors. From this we sub-
tract the discounted unconditional expected payout to investors E [FR (yR )] to obtain the
remote bank’s unconditional expected securitization pro…ts
R
(FR ) = (1
) E [FR (yR )].
To maximize this, the remote bank simply sets FR (yR ) equal to yR : it sells a 100% equity
stake in all the loans that it made.
Turning now to the local bank, assume that its portfolio is nonempty, whence the maximum return yL of its portfolio is positive.
bank’s expected securitization pro…ts
L
Given its security FLt and signal t, the local
(FLt ; t) equal the price pL (FLt ) of its security less
the discounted conditional expected payout to investors E [FLt (yL ) jt].26 De…ne the func24
The only endogenous variables they depend on are the local bank’s lending threshold
and securitization
threshold , which are common knowledge once the local bank announces its portfolio (p. 10).
25
In particular, in a separating equilibrium FLt reveals t so E [x (t) jFLt ] equals x (t).
26
As the remote bank’s security FR does not convey any information to investors, it is not an argument
of the price function pL .
28
tion
1
y
f (m; t) =
1
@E[minfm;y gjt]
@t
Pr
>
m
jt
y
=
y
1
R my
=0
1
@ ( jt)
d
@t
m
jt
y
:
(19)
Consider the following initial value problem:
Initial Value Problem (IVP). The di¤erential equation
dmy
dt
= f (my ; t) with my : [0; 1] !
<, together with the initial value my (0) = y.
Claim 9 Assume Lipschitz- , Lipschitz- , Lipschitz Partial Derivatives, and Hazard Rate
Ordering.
1. For any y > 0, there exists a unique solution my to IVP, which is strictly positive,
decreasing in t, and homogeneous of degree one in y: my = ym1 .
2. It is an equilibrium for the local bank to issue simple debt with face value given by the
function myL . This security has the payout function FLt (YL ) = min fYL ; myL (t)g. In
this equilibrium, the local bank’s unconditional expected securitization pro…t, which we
denote E [
yL
L ],
equals the expected gains from trade (1
) E [min fyL ; myL (t)g] from
the issuer’s security, where the latter expectation is taken with respect to both t and .
Proof. Follows from Claim 2 and Theorem 3 in Frankel and Jin [22].
While there may be other signalling equilibria, there are good reasons to focus on this one.
First, if the signal t and shock
come from discrete distributions, there is a unique equilib-
rium that satis…es the Intuitive Criterion of Cho and Kreps [10] (DeMarzo [15]). Moreover,
this unique equilibrium converges to the equilibrium of Claim 9 as the gaps between signals
and shocks shrink to zero (Frankel and Jin [20]). Finally, the Intuitive Criterion has found
experimental support in the work of Brandts and Holt [6] and Camerer and Weigelt [9].
The local bank’s total expected pro…ts b L ( ; r) from choosing the lending threshold
consist of the discounted expected portfolio return E [yL ], plus the bank’s expected securitization pro…ts E [
yL
L ],
less its cost of loaned funds 1
H ( ). However, by the homogeneity
of the face value function my , expected securitization pro…ts are also homogeneous of degree
29
one in y: E [
yL
L ]
= yL E [
1
L ].
Thus, by part 2 of Claim 9 and (18), the local bank’s total
expected pro…ts can be written simply as
Z
b L ( ; r) =
(r
=
where
b
b = E [ ] + (1
1) dH ( )
(20)
)b
(21)
and b = E [min f ; m1 (t)g]. It follows that, as in our main model, the local bank securitizes
all of its loans:
= 1= .
By (19), the constants b and b depend only on the discount factor , the distribution
of the signal t, and the conditional distribution / of the shock
given t. Moreover, the
quantity b lies between zero and the mean shock :
Claim 10 0 < b < E ( ).
Proof. Follows from Lipschitz-
and part 1 of Claim 9.
Finally, by Claim 10 and equation (21),
0 < b < b < E ( ) < 1:
(22)
Hence, our analysis of lending competition in the main model applies unchanged to this
version. In particular, Claim 10 and equations (20), (21), and (22) are formally identical
to the corresponding results in our main model: Claim 4 and equations (6), (7), and (9).
The only changes are that
and
are replaced by the analogous quantities b and b . This
implies parts 1 and 2 of the following result, where
0
=
1
minf ;b
r gE( )
and rb = 1=b. Part
3 (proved in the appendix) states that security design makes remote lending less likely by
raising the local bank’s securitization pro…ts. However, remote lending can still occur.
Theorem 11 Assume Lipschitz- , Lipschitz Partial Derivatives, Hazard Rate Ordering,
and No Cream Skimming. Also assume that, in the security design subgame, the local bank
always plays the equilibrium described in Claim 9.
outcome.
30
Then the full game has the following
1. If
0
<
, then the remote bank does not compete. If an agent’s credit quality
is
=
not less than the lending threshold [ b] 1 , the local bank o¤ers her a loan at an interest
rate equal to the gross project return
and the agent accepts. On seeing its private
1
signal t of the macroeconomic shock , the local bank sells a proportion
1
E[ jt=0]
E[ jt]
of these loans.
2. If
0
>
, the remote bank o¤ers the interest rate min f ; rb g. The local bank does not
compete, and all agents accept the remote bank’s o¤er.
Moreover, the remote bank
sells its entire portfolio of loans.
3. The constant b exceeds the analogous constant
0
5.4
= :
b<
1
in the main model. If b >
security design has no e¤ect on the extent of remote lending.
, then
0
1
, then
If instead
> : security design raises the remote bank’s lending threshold.
Multiple Local and Remote Banks
We now modify our main model (section 2) to incorporate multiple local and remote banks.
While greater competition does lead to lower interest rates, it does not alter the set of projects
that are …nanced. Hence, the welfare results in section 4.1 are robust to this change.
There is now also a continuum of remote banks i 2 [0; 1] and local banks j 2 [0; 1]. The
remote banks …rst make simultaneous o¤ers. Let ri be the o¤er of remote bank i; if this
bank makes no o¤er, let ri = 1. We assume that if any remote bank makes an o¤er, the
minimum such o¤er r exists.27 If no remote bank makes an o¤er, let r = .
The local banks then make simultaneous o¤ers. Let r
from bank j; if no such o¤er is made, let r
that the minimum r = min fr
j
j
j
be the o¤er that agent
= 1. For each private type , we assume
: j 2 [0; 1]g of the local banks’ bids exists.
(respectively, r > r), we assume that agent
gets
If r
r
borrows from the local (remote) bank with
the lowest o¤er; if more than one local (remote) bank made this o¤er, she ‡ips a coin to
choose among them.
27
For instance, this rules out the following pro…le of o¤ers: ri = 1 + i for i > 0 and ri = 1 for i = 0.
31
We …rst analyze a base case without securitization: each bank must hold every loan to
maturity.
If a bank lends, at a gross interest rate r, to an agent with credit type , its
expected pro…t is r E ( )
1: the discounted interest payment r times the probability
E ( ) of project success, less the unitary cost of capital. The proof of the following claim,
which follows that of Claim 1, is omitted.
Claim 12 Without the option of securitization, only the local banks lend.
credit quality
An agent with
is …nanced if and only if her project’s discounted expected gross return,
E ( ), exceeds the unitary cost of capital. Such an agent pays the interest rate [ E ( )] 1 .
She receives positive pro…ts while her lender’s pro…ts are zero.
A comparison with Claim 1 shows that introducing more banks does not a¤ect the set of
projects that are funded. It merely transfers rents from the banks to the agents.
We now turn to the e¤ect of securitization.
We assume the local banks belong to a
cooperative that pools and securitizes their loans.28 After the lending stage, the cooperative
sees the macroeconomic signal t and sells a portion Q (t) of its portfolio with the goal of
maximizing its securitization pro…ts. The cooperative’s pro…ts are divided among the local
banks in a manner to be described below.
To facilitate comparison with our main model, we restrict attention to equilibria in which
each local bank j bids on an agent if and only if her private type
threshold
j
2 <+ . As the local banks win all ties with the remote banks, the cooperative’s
portfolio must then consist of all private types
and may depend on r).29
28
is not less than some
= minj
j
(which we assume exists
If any remote bank competes, those who bid r each receive a
By assigning the issuance decision to a cooperative, we sidestep the technical issues that arise in signalling
games when multiple senders see the same signal (see Bagwell and Ramey [3]). As the local banks are small,
…xed costs of issuance would create an incentive to issue their loans through a cooperative. As information
among them is symmetric, bargaining costs would be minimal. However, an explicit model of the bargaining
process that would lead to such an agreement is outside the scope of this paper.
29
If
exceeds the highest private type 1= , the cooperative’s portfolio is empty.
32
representative sample of the agents whose private types
are less than , while those who
bid higher do not lend. Investors observe the measure of loans in each portfolio, and thus
can infer the threshold .30
The securitization stage is equivalent to that of our main model, with the cooperative
playing the role of the local bank. Hence, if any remote banks compete, the total expected
R
pro…ts of the remote banks that o¤er the minimum bid r sum to =0 (r E ( ) 1) dH ( )
as in (11). Each remote bank that bids r gets an equal share of these pro…ts since they all
make an equal proportion of remote loans.
1
As for the cooperative, the proportion of its portfolio that it sells is qL (t) =
by part 2 of Claim 3.
1
The unconditional expected payout to investors that derives from
the securitization of a loan to private type
by equation (8).
E[ jt=0]
E[ jt]
is E [r
E [ jt] qL (t)] which equals r
Since the investors are competitive and risk-neutral, the securitization
of the marginal borrower
=
raises the cooperative’s securitization revenue by r
.
By gradually lowering the marginal type from 1= (its upper bound) to zero, one …nds
that securitizing each private type
for each private type
lent to agent .
this loan is r
adds securitization revenue r
. We assume that
, the cooperative pays this marginal revenue to the bank that
Later, when project returns are realized, the expected gross return of
E ( ), of which the expected amount r
remainder (whose expectation is r
is paid to investors.
The
]) is paid to bank j, which discounts this
[E ( )
payment at the rate . In this way, the cooperative passes all revenues on to its member
banks.
Bank j’s total expected discounted pro…t from lending to private type
is thus
[E ( )
], which equals r
1 by (7). The pro…ts of all of the
R 1=
local banks thus sum to = (r
1) dH ( ), which is identical to the local bank’s pro…t
r
1+ r
function in our main model (equation (6)) except that the interest rate r in that equation is
30
As investors also observe the interest rate of every loan, they may also be able to infer the private type
of individual loans in the cooperative’s portfolio. However, the cooperative would not bene…t from splitting
these loans into separate securities since their expected gross returns r
shock .
33
are all proportional to a common
replaced by the interest rate r that here is o¤ered to agent .
In equilibrium, the local banks bid the interest rate r of any private type
the point where their expected pro…t r
banks o¤er the interest rate r = (
1 from lending to
1
)
to private type
down to
is zero. Hence, all local
as long as this rate does not
exceed an agent’s willingness to pay r. Otherwise, they do not compete for agent . The
lending threshold
must therefore satisfy (
)
1
= r or, equivalently,
= (r
) 1 . Thus,
the local banks have the same lending threshold as in our main model (equation (10)).
As for the remote banks, their collective pro…ts from o¤ering the minimum interest rate
are given by (11) as in the main model. In the special case in which r also does not
r
exceed the deterring rate r = 1= (equation (12)),
R
(r) =
Z
1= by (10), so (11) simpli…es to
1=
(r
E( )
1) dH ( ) = r E ( )
(23)
1
=0
as E ( ) = 1. In particular, remote pro…ts from the interest rate min fr ; g are
R
where
=
1
minf ;
1 gE(
)
(min fr ; g) = min fr ; g E ( )
is de…ned in equation (14).
Hence, if
(24)
>
then it is pro…table
for remote banks to o¤er the interest rate min fr ; g, thus deterring the local banks from
competing. However, they do not actually o¤er this rate: competition among them drives
their o¤er down to the rate r = [ E ( )]
1
at which their pro…ts r E ( )
1 are zero.
We have shown the following modi…cation of Theorem 6.
Theorem 13 Assume No Cream Skimming. With securitization:
1. If
< , the remote banks do not compete. If an agent’s credit quality
less than the lending threshold [
interest rate r = (
)
1
=
is not
] 1 , the agent gets a loan from a local bank at the
. Agents with lower credit qualities do not get o¤ers.
On seeing its private signal t of the macroeconomic shock , the local cooperative sells
1
a proportion
E[ j0]
E[ jt]
1
of its loans.
34
2. If
> , the remote banks do compete.
Their lowest bid r is [ E ( )] 1 , which lies
in (1; min fr ; g) and gives them zero pro…ts:
R
(r) = 0.
The local banks do not
compete, and all agents accept an o¤er from a remote bank.
Moreover, the remote
banks sell their entire portfolio of loans.
A comparison of Claim 12 and Theorem 13 with Claim 1 and Theorem 6, respectively,
shows that whether or not banks can securitize, the introduction of multiple remote and
local banks merely lowers the interest rates paid by borrowers. We conclude that greater
interbank competition does not alter the e¤ects of securitization on the set of projects that
are …nanced and thus on social welfare.
6
Conclusions
In Leland and Pyle [30], an issuer sells a security to a continuum of risk-neutral, uninformed
investors. Before choosing how much to sell, the issuer sees private information about her
security’s value. In equilibrium, she varies the amount that she sells in order to signal her
information to investors. This is costly for her since she sells less of her security when the
gains from trade are higher. DeMarzo and Du¢ e [16] show how the issuer can reduce these
costs by designing a security whose payout is not very sensitive to her private information.
The central insight of our paper is that an issuer can accomplish the same goal by
acquiring assets about which she has little private information. In particular, a bank may
lend to a remote loan applicant about whom it knows only hard information such as the
credit score and loan-to-value ratio. Since the bank lacks soft information, it can sell this
loan without costly signalling. This gives remote banks an advantage over local banks that
can more than o¤set the remote banks’poorer screening ability at the lending stage. We also
show, under two alternative assumptions, that the remote bank follows a threshold strategy:
it lends if and only if an agent’s observables are su¢ ciently strong.
Our model has mixed welfare implications.31
31
Without securitization, only local banks
We limit ourselves here to properties that hold under both NCS and MHRP.
35
lend. Agents get loans if their credit quality is su¢ ciently high. Hence, the loan allocation
is e¢ cient. However, investors have no way to get their capital to worthy loan applicants.
With securitization, agents with weak observables can borrow from local banks if their
credit quality exceeds a new, lower threshold.
This extension of credit improves welfare.
On the other hand, agents with strong observables are funded by remote banks regardless
of their actual credit quality. This has ambiguous welfare e¤ects: some new projects are
worthwhile but others are not. Remote lending also leads to an ine¢ cient loan allocation.
A medium quality agent with weak observables may be turned down by both banks while a
low quality agent with strong observables is funded by a remote bank.
Our model yields several empirical predictions that are con…rmed by recent research.
Securitization stimulates lending in general and remote lending in particular. Remote lenders
securitize a higher proportion of their loans. Remote borrowers have strong observables and
pay low interest rates but have high conditional default rates.
Securitization raises both
conditional and unconditional default rates. Finally, in cross section, securitized loans have
higher conditional default rates than unsecuritized loans.
Our model follows the literature in assuming that banks securitize because they are less
patient than investors. However, if banks are risk-averse, they might instead securitize in
order to reduce their exposure to local macroeconomic shocks.32 This bene…t would likely
be smaller for the local bank for two reasons:
asymmetric information prevents it from
selling its entire portfolio, and the price of its security varies with its signal which creates
risk. Hence, securitization should still favor remote lending. It should also favor applicants
whose project outcomes are more correlated with local macroeconomic shocks, such as real
estate developers: a risk averse bank would be especially hesitant to lend to such applicants
if it could not securitize. This is an interesting question for future research.
32
This assumes that investors are either less risk averse or better able to construct diversi…ed portfolios.
36
A
Proofs
Proof of Claim 1: If the remote bank competes, it must make the …rst o¤er.
It does
not know the borrower’s private type . Since the local bank knows , the local bank can
tell which of the remote bank’s o¤ers are pro…table for the local bank and which are not.
However, in the absence of securitization the two banks have common values: the value of
lending to a borrower is simply her discounted expected repayment less the common cost of
capital. Thus, the local bank will slightly underbid the remote bank’s pro…table o¤ers and
refrain from bidding on the unpro…table ones.
in lending only to unpro…table borrowers.
an o¤er to the agent.
interest rate of
As a result, the remote bank will succeed
Knowing this, the remote bank will not make
But given this, the local bank can charge the maximum possible
and any of the agent will agree. The local bank will do so if the resulting
discounted expected repayment,
Proof of Claim 5: Part 1.
an interest rate r
E ( ), exceeds its unitary cost of capital. Q.E.D.
Suppose the remote bank does compete.
to maximize
r < min f ; r g, then by (10)
R
(r) and, moreover,
exceeds
= 1= > 0 so
Then it chooses
0 at this optimal r. If
R
1=
0
E ( ) dH ( ) > 0:
R (r) =
=0
R
(r)
the remote bank’s optimal r is at least min f ; r g. It remains to show that its optimal r is
at most min f ; r g. If r
, we are done since min f ; r g =
remote bank competes (chooses r
). Finally, if r <
and, by assumption, the
then no r 2 (r ; ] can maximize
the remote bank’s pro…ts by the following result.
Lemma 14 Assume No Cream Skimming.
Let
(10). For any r 2 [r ; ] satisfying
0, the slope
R
(r)
R
be given by (11) where
0
R
is given by
(r) (which denotes the right
derivative if r = r and the left derivative if r = ) exists and is negative.
Proof. By (11),
R
(r) = c 1 I (r) where c = [ E ( )]
1
and I (r) =
R
=0
(r
c) dH ( ).
By (10) and (12),
= (r ) 1 = r =r
= 1= for all r 2 [r ; ], so I (r) =
R r =r
R r
1
(r
c)
dH
(
).
With
the
change
of
variables
s
=
r
,
I
(r)
=
(s c) H 0 rs ds.
r s=0
=0
37
Thus, the slope exists:
1
r2
I 0 (r) =
Z
r
(s
c)
s=0
"
s
r
H0
+ H 00
H 0 rs
s
r
s
r
#
H0
!
s
ds ;
r
where I 0 (r) denotes the right derivative if r = r and the left derivative if r = . Changing
i
h 0
R r =r
H ( )+H 00 ( )
1
0
dH ( ) . For any functions '0 ( )
variables back, I (r) = r
(r
c)
H0( )
=0
and '1 ( ), let E ('0 ) and Cov ('0 ; '1 ) denote the expectation of '0 and covariance of
'0 and '1 , both conditioned on
where x ( ) = r
1
E
r
lying in [0; r =r]. Then I 0 (r) =
c and y ( ) =
H 0 ( )+H 00 ( )
H0( )
.
(xy) H ( r =r),
By de…nition of covariance, E (xy) =
Cov (x; y) + E (x) E (y). By assumption E (x) =
I(r)
H( r =r)
0. By NCS, E (y) > 0 and
Cov (x; y) > 0, so E (xy) > 0, whence I 0 (r) < 0 as claimed.
Part 2. By (10) and (12),
=
1
r =r. Substituting for r,
=
1
1
r = min f ; r g
:
the local bank does not lend to any private types. Moreover, the local bank strictly prefers
(not) to lend to agents whose types
1
exceed (respectively, are less than)
r =r. If r
,
then r = min f ; r g = r : the local bank is just willing not to bid for the best agent, whose
private type
is 1= . If r > , then r =
< r : the local bank strictly prefers not to bid
for any agent. Q.E.D.Claim 5
Proof of Claim 7: We …rst show that MHRP implies a monotone conditional expectation
property. For any
<
0
when
0,
let E [ j <
0]
given that
has the distribution G .
Lemma 15 Assume MHRP. For any
Proof. Let
denote the conditional expectation of
0
2 (0; 1), E [ j <
0]
is increasing in .
2 (0; 1) and 0 < 1 . De…ne f ( 0 ; 0 ; 1 ) = E 1 [ j
R 0
g ( 0 ; 0 ; 1 ) = exp
h 1 ( ) d f ( 0 ; 0 ; 1 ). Clearly, g (0; 0 ;
=0
0
E 0[ j
0]
1)
0]
= f (0;
0;
1)
h 1 ( 0) f ( 0;
0;
1) ;
Moreover,
@
f ( 0;
@ 0
0;
1)
=(
0
E 0[ j
0 ]) [h
1
( 0)
h 0 ( 0 )]
so
@
g ( 0;
@ 0
0;
1) = (
0
E 0[ j
0 ]) [h 1 ( 0 )
38
h 0 ( 0 )] exp
Z
0
h 1 ( )d
=0
;
and
= 0.
which is strictly positive for all
so f ( 0 ;
Fix
0;
1)
0
> 0 by MHRP. Hence, for all
0
> 0, g ( 0 ;
0;
1)
> 0,
> 0.
and thus
= E ( ). De…ne z = E ( ). The remote bank will de…nitely lose money
if it o¤ers an interest rate r below its cost of capital.
to interest rates r
if and only if
Hence, we can restrict attention
If the remote bank o¤ers r, the local bank competes (and wins)
1.
= (r
)
1
by (10) or, equivalently, if
> (r ) 1 .
=
Thus,
the remote bank’s maximum pro…ts from competing are ( ) = maxr2[1; ] ( (r; )) where
R (r ) 1
(r; ) = =0 (r z 1) dG ( ). Now consider a such that ( )
0. Then there
(r; )
= rzE
j < (r ) 1
1 is
((r ) 1 )
nonnegative. But for …xed r 2 [1; ], ! (r; ) is increasing in by Lemma 15. Thus for all
must exist an r 2 [1; ] for which the ratio ! (r; ) =
0
2 ( ; 1), ! (r; 0 ) > 0, so
(r; 0 ) > 0, whence
G
( 0 ) > 0. Q.E.D.Claim 7
Proof of Theorem 11: Parts 1 and 2 are proved in the text. For part 3, it su¢ ces to show
that b > . De…ne v 1 (m1 (t) ; t) = E [min f ; m1 (t)g jt], so b = E [v 1 (m1 (t) ; t)] by the law
of iterative expectations. By (8), it su¢ ces to show that when t = 0, v 1 (m1 (t) ; t)
1
E [ jt] =
E [ jt = 0].33 The case t = 0 holds since m1 (0) = 1 > . As for t > 0, E [ jt = 0] is independent of t and m1 (t) is decreasing in t. Hence, it su¢ ces to show that when m1 (t) < 1,
0<
1
d
dt
v 1 (m1 (t) ; t)
1
v 1 (m1 (t) ; t)
By (19) and IVP,
dm1 (t)
dt
=
1
E [ jt]
E [ jt]
1
1
1 v2 (m (t);t)
,
v11 (m1 (t);t)
= (1
so
d
dt
)
d 1
v (m1 (t) ; t)
dt
v 1 (m1 (t) ; t)
[v 1 (m1 (t) ; t)] =
1
+
d
E
dt
[ jt]
:
E [ jt]
(25)
v21 (m1 (t) ; t), whence
(25) can be rewritten as
d
E
dt
v21 (m1 (t) ; t)
:
(26)
v 1 (m1 (t) ; t)
Rc
Let c 2 [0; 1]. Integrating by parts, v 1 (c; t) = c
( jt) d , so (26) must hold if, for
=0
[ jt]
0<
E [ jt]
c < 1,
d2
0<
ln
dtdc
33
Z
c
[1
=0
( jt)] d
"
d
1
Rc
=
dt
[1
=0
(cjt)
( jt)] d
#
.
(27)
The notation "f0 (t) = f1 (t) when t = 0" means that f0 (t) exceeds (equals) f1 (t) when t exceeds
(equals) zero.
39
0
By HRO, for all
>
00
,
( 00 jt)
( 0 jt)
1
1
00
is decreasing in t. Integrating over
<
0
,
R
0
=0 [1
1
( jt)]d
( 0 jt)
is decreasing in t, which implies (27). Q.E.D.Theorem 11
Proof of Theorem 13: Let re denote min fr ; g.
1. If
< , then remote pro…ts
re
r
R
are negative at r = re by (24). As (23) is increasing in
r , remote pro…ts are negative for lower interest rates as well. Hence, remote
banks will not bid r
let r0 = inf fr 2 (r ; ] :
fact that
R
(r ) =
R
re.
R
If re =
(r)
we are done.
In this case,
0g. If r0 exists, then by continuity of
(e
r) < 0, r0 must exceed r and satisfy
by Lemma 14, there must be an " > 0 such that
This contradicts the de…nition of r0 .
R
Else re = r < .
R
R
R
(r0 ) = 0. But then
(r) > 0 for all r 2 (r0
Hence, r0 does not exist:
and the
"; r0 ).
for all r 2 (r ; ],
(r) < 0. Thus, the remote banks will not compete.
2. If
in r
> , remote pro…ts from bidding re are positive by (24). Thus, as (23) is increasing
re
r , the remote banks must bid the interest rate r = [ E ( )]
which (23) equals zero.
1
2 (1; re) at
Q.E.D.Theorem 13
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