Interbank Lending, Credit Risk Premia and Collateral∗
Florian HeiderMarie Hoerova†
June 8, 2009
We study the functioning of secured and unsecured interbank markets in the pres-
ence of credit risk. We allow for safe collateral, e.g. government bonds, as well as risky
collateral, e.g. mortgage-backed securities, in secured interbank transactions. Acquir-
ing liquidity in the unsecured market is costly due to credit risk premia, while the
secured segment is subject to market risk. The model illustrates how tensions in the
unsecured market and the market secured by risky collateral affect repo rates in the
market secured by safe collateral. The volatility of repo rates is exacerbated by the
scarcity of high-quality collateral. The model generates empirical predictions that are
in line with developments during the 2007-2009 financial crisis. Interest rates decouple
across secured and unsecured markets as well as across different collateral classes. We
use the model to discuss various policy responses.
JEL classification: G01, G21, D82
Keywords: Financial crisis; Interbank market; Liquidity; Credit risk
∗We thank Rafael Repullo, and seminar participants at the European Central Bank for helpful comments.
Marco Lo Duca provided excellent research assistance. The views expressed do not necessarily reflect those
of the European Central Bank or the Eurosystem.
†The authors are at the European Central Bank, Financial Research Division, Kaiserstrasse 29, D-60311
Frankfurt, email: email@example.com
Interbank markets play a key role in the financial system. They are vital for banks’ liquid-
ity management and the transmission of monetary policy. The interest rate in the unse-
cured three-month interbank market acts as a benchmark for pricing fixed-income securities
throughout the economy. Secured, or repo, markets have been a fast-growing segment of
money markets: They have doubled in size since 2002 with gross amounts outstanding of
about $10 trillion in the United States and comparable amounts in the euro area just prior
to the start of the crisis in August 2007. Since repo transactions are backed by collateral
securities similar to those used in the central bank’s refinancing operations, repo markets
are a key tool for the implementation of monetary policy.
In normal times, interbank markets are among the most liquid in the financial sector.
Rates are usually stable across secured and unsecured segments, as well as across different
collateral classes. Since August 2007, however, the functioning of interbank markets has
become severely impaired around the world. The frictions in the interbank market have
become a key feature of the 2007-09 crisis (see, for example, Allen and Carletti, 2008, and
One striking manifestation of the frictions in the interbank market has been the decou-
pling of interest rates between secured and unsecured markets. Figure 1 shows the unsecured
and secured interbank market rates for the euro area since January 2007. Prior to the out-
break of the crisis in August 2007, the rates were closely tied together. Since August 2007,
they have moved in opposite directions with the unsecured rate increasing and the secured
rate decreasing. The decoupling further deepened after the Lehman bankruptcy, and to a
lesser extent, just prior to the sale of Bear Stearns.
A second, related important feature of the tensions in the interbank market has been the
difference in the severity of the disruptions in the United States and in Europe. Figure 2
shows rates in secured and unsecured interbank markets in the United States. As in Europe,
there is a decoupling of the rates at the start of the financial crisis and a further deepening
9th Aug. 07 Bear Stearns
sold to JP Morgan
184.108.40.206.1.5. 220.127.116.11.18.104.22.168.22.214.171.124.126.96.36.199.188.8.131.52. 184.108.40.206.
2007 2008 2009
3 months unsecured
3 months secured
Figure 1: Decoupling of secured and unsecured interbank rates in the EA
after the sale of Bear Stearns and the bankruptcy of Lehman. However, the decoupling and
the volatility of the rates is much stronger than in Europe.
Why then have secured and unsecured interbank interest rates decoupled? Why has the
US repo market experienced significantly more disruptions than the euro area market? What
underlying friction can explain these developments? And what policy responses are possible
to tackle the tensions in interbank markets?
To examine these questions, this paper provides a model of interbank markets with both
secured and unsecured lending in the presence of credit risk. We model the interbank market
in the spirit of Diamond and Dybvig (1983). Banks may need to realize cash quickly due
to demands of customers who draw on committed lines of credit or on their demandable
9th Aug. 07 Bear Stearns
sold to JP Morgan
220.127.116.11. 18.104.22.168. 22.214.171.124. 126.96.36.199. 188.8.131.52.184.108.40.206.220.127.116.11. 1.5.
2007 2008 2009
3 months unsecured
3 months secured
Figure 2: Decoupling of secured and unsecured interbank rates in the US
deposits. Banks in need of liquidity can borrow from banks with a surplus of liquidity as
in Bhattacharya and Gale (1987) and Bhattacharya and Fulghieri (1994). Banks’ profitable
but illiquid assets are risky. Hence, banks may not be able to repay their interbank loan
giving rise to credit risk. To compensate lenders, borrowers have to pay a premium for funds
obtained in the unsecured interbank market.
In addition to the choice between the liquid (cash) and the illiquid asset (loans), banks
can invest in bonds. Bonds provide a long-run return but unlike the illiquid asset, they can
also be traded for liquidity in the short-term. We first consider the case of safe bonds, e.g.
government bonds. Since unsecured borrowing is costly due to credit risk, banks in need of
liquidity will sell bonds to reduce their borrowing needs. We assume that government bonds
are in fixed supply and that they are scarce enough not to crowd out the unsecured market.
Credit risk will affect the price of safe government bonds since banks with a liquidity surplus
must be willing to both buy the bonds offered and lend in the unsecured interbank market.
In equilibrium there must not be an arbitrage opportunity between secured and unsecured
We then introduce risky bonds, e.g. mortgage-backed securities. The realization of the
risky bond return becomes known when banks trade liquidity and safe bonds. Risky bond
returns lead to aggregate risk that spills over to the market for safe bonds. In consequence,
even the price of safe bonds will be volatile.
Our modeling assumptions are designed to reflect the insights from broad analyses of
the 2007-09 financial crisis. First, risk, and the accompanying fear of credit default, which
was created by the complexity of securitization, is at the heart of the financial crisis (see
Gorton, 2008, 2009). Second, illiquidity is a key factor contributing to the fragility of modern
financial systems (see, for example, Diamond and Rajan, 2008a, and Brunnermeier, 2009).
Hence, we employ the model of banking introduced by Diamond and Dybvig (1983) that
allows us to consider the tradeoff between liquidity and return in bank’s portfolio decisions.
A further advantage of this model is that it naturally creates a scope for interbank markets
(see Bhattacharya and Gale, 1987, and Bhattacharya and Fulghieri, 1994).1
This paper is part of the growing literature analyzing the ability of interbank market
to smooth out liquidity shocks. Our model builds on Freixas and Holthausen (2004) who
examine the scope for the integration of unsecured interbank markets when cross-country
information in noisy. They show that introducing secured interbank markets reduces interest
rates and improves conditions when unsecured markets are not integrated, however their
introduction may hinder the integration process.
The role of asymmetric information about credit risk as a factor behind tensions in
the unsecured interbank markets is emphasized in Heider, Hoerova and Holthausen (2009).
1An important complement to liquidity within the financial sector is the demand and supply of liquidity
within the real sector (see Holmstr¨ om and Tirole, 1998).
They derive various regimes in the interbank markets akin to the developments prior to and
during the 2007-2009 financial crisis. Bolton, Santos, and Scheinkman (2009) also examine
the role of asymmetric information and distinguish between outside and inside liquidity
(asset sales versus cash), which connects to our analysis where banks hold liquid and illiquid
securities, and where safe and risky claims on illiquid assets can be traded in exchange for
liquidity. Brunnermeier and Pedersen (2009) similarly distinguish between market liquidity
and funding liquidity. In our model, banks can obtain funding liquidity in the secured and
unsecured interbank markets by issuing claims on illiquid assets, i.e. assets with limited
A recent paper by Allen, Carletti, and Gale (2009) presents a model of a market freeze
without credit risk or unsecured interbank markets. Banks can stop trading due to aggregate
liquidity risk, i.e. banks hold similar rather than offsetting positions. Aggregate shortages
are also examined in Diamond and Rajan (2005) where bank failures can be contagious due
to a shrinking of the pool of available liquidity. Freixas, Parigi, and Rochet (2000) analyze
systemic risk and contagion in a financial network and its ability to withstand the insolvency
of one bank. In Allen and Gale (2000), the financial connections leading to contagion arise
endogenously as a means of insurance against liquidity shocks. Illiquidity can depress lending
and low prices for illiquid assets go hand in hand with high returns on holding liquidity in
Diamond and Rajan (2009). Potential buyers may want to wait for asset prices to decline
further. At the same time, the managers of selling banks may want to gamble for resurrection.
These two effects feed on each other and may lead to a market freeze.
Rationales for central bank intervention in the interbank market are examined in Acharya,
Gromb, and Yorulmazer (2008) and Freixas, Martin, and Skeie (2008). In Acharya et al.,
market power makes it possible for liquidity-rich banks to extract surplus from banks that
need liquidity. A central bank provides an outside option for the banks suffering from
such liquidity squeezes. In Freixas et al., multiple equilibria exist in interbank markets,
some of which are more efficient than others. By steering interest rates, a central bank
can act as a coordination device for market participants and ensure that a more efficient
equilibrium is reached. Freixas and Jorge (2008) examine how financial imperfections in the
interbank market affect the monetary policy transmission mechanism beyond the classical
The remainder of the paper is organized as follows. In Section 2, we describe the set-up
of the model. In section 3, we solve the benchmark case when banks can only trade in the
unsecured market. In Section 4, we allow banks to invest in safe bonds. In Section 5, we
introduce risky bonds and market risk. In Section 6, we present empirical implications and
relate them to the developments during the 2007-09 financial crisis. In Section 7, we discuss
policy responses to mitigate the tensions in interbank markets and in Section 8 we offer
concluding remarks. All proofs are in the Appendix.
2 The model
There are three dates, t = 0,1, and 2, and a single homogeneous good that can be used for
consumption and investment. There is no discounting between dates.
Consumers and banks. There is a [0,1] continuum of consumers. Every consumer has
an endowment of 1 unit of the good at t = 0. Consumers deposit their endowment with a
bank at t = 0 in exchange for a demand deposit contract which promises them consumption
c1 if they withdraw at t = 1 (“impatient” consumers) or c2 if they withdraw at t = 2
(“patient” consumers), as in Diamond and Dybvig (1983).
There is a [0,1] continuum of risk neutral, profit maximizing banks. We assume that
the banking industry is perfectly competitive. Thus, banks make zero profits in equilibrium
and maximize pay-out to depositors. Deposits are fully insured by deposit insurance and no
bank runs occur.2
Liquidity shocks. Banks are uncertain about the liquidity demand they will face at
2We abstract from any risk sharing issues and take the institutions of banking (and interbank markets)
model of secured and unsecured interbank lending in the presence of credit risk. Credit risk
premia in the unsecured market also affect the price of riskless bonds when they are used to
manage banks’ liquidity shocks. Moreover, if the collateral in the secured market is subject
to aggregate risk, the resulting market risk also impacts the price of riskless bonds.
Going forward, our analysis points to a number of issues for further research. First, the
size of the banking sector relative to the amount of collateral matters. We saw that the
presence of bonds reduces the amount banks have to borrow in unsecured markets which is
vulnerable to credit risk concerns. The positive effect of bonds is stronger when the ratio
of banks’ balance sheet to the value of bonds is larger. Hence, the interplay between the
relative size of banking, financial markets and the economy deserves further attention.
Second, our analysis abstracted from risk sharing concerns. Banks were simply maximiz-
ing the total amount of demandable liabilities. Even without risk sharing, we obtain a credit
risk premium in unsecured interbank markets. Introducing risk aversion of banks’ customers
is beyond the scope of this paper and constitutes a fruitful avenue for further research. With
respect to the spillover of credit and market risk across interbank markets, we anticipate
that risk aversion adds an additional risk premium that would exacerbate the tensions that
Third, we assumed that the various shocks in our model are uncorrelated. The financial
crisis has made painfully clear that in reality, the risk embedded in banks’ illiquid assets, their
liquidity needs and shocks to collateral values are interlinked. The challenge will therefore be
to model and analyze the joint distribution of the risks in banks’ balance sheets, especially
“at the tail”. Banks’ risk management practices have to take into account the forces affecting
different collateral classes and the market’s response in times of stress when liquidity and
high quality collateral is scarce.
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Proof of Lemma 1
The interbank market is active so that the Lagrange multipliers on the feasibility
constraints on Lkare zero. Let µkbe the Lagrange multiplier on the budget constraint.
The first-order condition for a lender w.r.t. Llis:
p2(1 + r) − µl= 0(30)
while the first-order condition for a borrower w.r.t. to Lhis:
−p(1 + r) + µh= 0.(31)
Proof of Lemma 2
since otherwise all available cash at t = 1 is reinvested and nothing can be paid or lent out.
The first-order condition for a type-k bank w.r.t. to γ1
1be the Lagrange multipliers on 0 ≤ γ1
k. The constraint γ1
k≤ 1 cannot be binding
(1 − α)(p − µk) + µk
1= 0 (32)
Substituting µh= p(1 + r) (Lemma 1) yields:
(1 − α)(−pr) = −µh
since left hand side is negative. It cannot be zero since α = 1 cannot be optimal. A type-h
bank would have to finance its entire need for liquidity by borrowing in the interbank
market at a rate r > 0 whereas it could just store some liquidity without cost using the
short-term asset. Since −µh
Consider now the case of a lender. Substituting µl= p2(1 + r) (Lemma 1) into (32) yields:
1< 0 we have γ1
(1 − α)p(1 − p(1 + r)) = −µl
Again, α = 1 cannot be optimal. A type-l bank cannot invest everything into the illiquid
asset and still lend in the interbank market. Hence, γ1
assume that a type-l bank does not reinvest into the liquid asset when the condition holds
as an equality).
l= 0 if and only if p(1 + r) ≥ 1 (we
Proof of Lemma 3
Using the binding budget constraints from the optimization programs (1) and (2) (Lemma
1) to substitute for Lland Lhin the market clearing condition πlLl= πhLhand using
k= 0 (Lemma 2) gives the result.
Proof of Proposition 1
Competition requires that
Rα + p(1 + r)[(1 − α) − λlc1] − (1 − λl)c2= 0
Rα − (1 + r)[λhc1− (1 − α)] − (1 − λh)c2= 0
where we have used the results from Lemma 1 and 2. The pay-out by solvent lenders and
borrowers must be such that they make zero profits. If they did not, then another bank
could offer a slightly higher combination of early and late pay-out (ˆ c1,ˆ c2) and attract all
consumers. Using the result on c1from Lemma 3, using one condition to solve for c2and
substituting back into the other condition gives the desired result.
Proof of Lemma 4
The first-order condition of a borrower with respect to reinvesting into the liquid asset at
t = 1, γ1
?(1 − α − β) + βS
where µhis the marginal value of liquidity for a borrower (given by Lemma 1) and µh
the multiplier on the feasibility constraint γ1
since we are considering interior portfolio allocations, 1 − α − β > 0. Since
µh= p(1 + r) > p, we have that µh
borrower does not reinvest into the liquid asset.
From the first-order condition of a lender with respect to bond purchases at t = 1, γ2
?(1 − α − β) + βS
where µlis the marginal value of liquidity for a lender (given by Lemma X) and µl
multiplier on the feasibility constraint γ2
reinvested and nothing can be paid or lent out. Since µl= p2(1 + r), the first-order
condition holds iff
The yield of the bond at t = 1 must be less or equal to the expected return of unsecured
interbank lending (given that the unsecured interbank market is open).
The first-order condition of a borrower with respect to bond purchases at t = 1, γ2
?(p − µh) + µh
h≥ 0. Note that (1 − α − β) + βS
1> 0 and thus γ1
h= 0. As in the case without bonds, a
− µl) + µl
l≥ 0. Note that the feasibility constraint
l≤ 1 must be automatically satisfied since otherwise all available cash at t = 1 is
≤ p(1 + r) (35)
?(1 − α − β) + βS
− µh) + µh
have that µh
2is the multiplier on the feasibility constraint γ2
2> 0 and hence γ2
h≥ 0. Due to condition (35), we
h= 0. A borrower also does not reinvest using bonds.
The first-order condition of a borrower with respect to bond sales at t = 1, βS
4are the Lagrange multipliers on 0 ≤ βS
µh= p(1 + r), we have
h+ Y (γ2
h− 1)?− µhP1
h− 1??+ µh
h≤ 1. Using γ1
h= 0, γ2
h= 0 and
?− Y + (1 + r)P1] + µh
Due to condition (35), the term is squared brackets is positive. For the condition to hold, it
must be that µh
4> 0, and hence βS
h= 1. The borrower sells all his bonds at t = 1.
Proof of Lemma 5
Market clearing for bonds at t = 1 requires that:
?(1 − α − β) + βS
where we have used βS
Since borrowers sell bonds, lenders must buy them.
Given that γ2
bond purchases (34) requires that
h= 1 and γ2
h= 0. Market clearing therefore requires that γ2
l> 0, and hence µl
2= 0, the lender’s first-order condition with respect to
= p(1 + r)(36)
The yield on safe bonds must be equal to the expected return on risky interbank loans as
both markets are open.
The first-order condition of a lender with respect to bond sales at t = 1, βS
4are the Lagrange multipliers on 0 ≤ βS
It cannot be that P1> Y since lenders would not want to buy any bonds at t = 1. When
P1< Y then µl
generality. To see, this set P1= Y and p(1 + r) = 1 (see condition (36)) into the lender’s
problem at t = 1 (equations (12) and (13)):
l+ Y (γ2
l− 1)?− µlP1
l− 1??+ µl
l≤ 1. Using (36), the condition
?(P1− Y )γ1
3> 0 and hence βS
l= 0. If P1= Y then we can let βS
l= 0 without loss of
p[Rα + (1 − α − β) +β
P0Y − λlc1− (1 − λl)c2](38)
where we substituted the budget constraint into the objective function using Ll. The
objective function is independent of βS
The first-order condition of a lender with respect to reinvesting into the liquid asset at
t = 1, γ1
(1 − α − β)p(1 −Y
where we used βS
have ruled out that P1> Y . When P1= Y , the lender’s problem is independent of γ1
(38)) and we can set γ1
P1) + µl
l= 0, the lender’s marginal value of liquidity µl= p2(1 + r) and (36). We
l= 0 without loss of generality. When P1< Y , then µl
1> 0 and
Proof of Lemma 6
As in the proof of Lemma 3. The extra element is the presence of γ2
bought by banks with a liquidity surplus. But we can use the condition on market clearing
in the bond market (18) to solve for γ2
l, the amount of bonds
Proof of Proposition 3
Analogous to the proof of Proposition 1. Competition requires that
Rα + γ2
P1(1 − α − β) + p(1 + r)[(1 − γ2
l)(1 − α − β) − λlc1] +β
P0Y − (1 − λl)c2= 0
Rα − (1 + r)[λhc1− (1 − α − β) −β
where we have used the results from Lemma 1, 4 and 5. The amount of bonds purchased γ2
is given by market clearing in the bond market (equation (16), or after simplification, (18)).
Using the result on c1from Lemma 6, using one condition to solve for c2and substituting
back into the other condition gives the desired result.
P0P1] − (1 − λh)c2= 0
Proof of Lemma 7
The first-order condition with respect to the allocation into the government bond, β,
Solving the first-order condition for P0yields
+ p(1 + r)(−πl− πhP1
P0) − (1 + r)πh(1 −P1
P0)] = 0
1 + r+ πh(1 − p)P1
Using (17) to substitute for
since 1 − πh= πl.
1+rresults in P0= P1
which gives the desired result