Liquidity, Moral Hazard and Inter-Bank Market Collapse

TL;DR: In this article, the authors proposed a framework to analyze the functioning of the interbank liquidity market and the occurrence of liquidity crises, which relies on three key assumptions: (i) liquidity provisioning is not verifiable -it cannot be contracted upon-, (ii) banks face moral hazard when confronted with liquidity shocks-unobservable effort can help overcome the shock, and (iii) liquidity shocks are private information - they cannot be diversified away.

Abstract: This paper proposes a framework to analyze the functioning of the inter-bank liquidity market and the occurrence of liquidity crises. The model relies on three key assumptions: (i) liquidity provisioning is not verifiable -it cannot be contracted upon-, (ii) banks face moral hazard when confronted with liquidity shocks-unobservable effort can help overcome the shock-, (iii) liquidity shocks are private information - they cannot be diversified away-. Under these assumptions, the equilibrium risk-adjusted return on liquidity provisioning increases with the aggregate equilibrium volume of ex ante liquidity provision. As a consequence, banks may provision too little liquidity compared with the social optimum. Within this framework we derive two main results. First inter-bank market collapse is an equilibrium. Second such an equilibrium is more likely when (i) the individual probability of the liquidity shock is lower, (ii) ex ante competition between banks on illiquid long term assets is larger.

Summary

1. Introduction

• The financial market turmoil that has been under way since the summer of 2007 hit the core of the global financial system, the interbank market for liquidity.
• While this paper does not endeavor to account for all the features of the recent crisis, be it hard evidence or casual stories about the motivations of market players, it argues that a proper modeling of the collapse in the market for liquidity involves a close look at incentives to provision/hoard liquidity and moral hazard mechanisms in the interbank market.
• This relaxes the moral-hazard-induced liquidity constraint, raising the demand for liquidity and thereby the price of liquidity on the interbank market, which in turn raises incentives to provision liquidity ex ante.
• A common feature of this literature is that bank holdings of liquidity are not necessarily optimal.

2. Timing and Technology Assumptions

• Banks are risk neutral and maximize expected profits.
• At date 0, each bank has a unit capital endowment and two investment possibilities.
• 6The alternative arrangement under which banks would sign ex ante insurance contracts against liquidity shock is not possible here.
• Distressed banks reinvest their own liquidity plus borrowed funds in their illiquid project and deliver some effort.

3. The First-Best Allocation

• To derive the first-best allocation, the authors remove two assumptions regarding market imperfections.
• First, date 0 allocation between liquid and illiquid assets is now verifiable.
• Moreover, each distressed project in which k is reinvested yields an expected return keR(e).
• Put differently, the expected return to illiquid investments without any ex ante liquidity provision (1 − q)R is very large.
• The social planner then prefers to maximize illiquid investments.

4.1 Distressed Banks’ Optimal Demand for Liquidity

• The authors can then derive the following proposition.
• Having determined optimal borrowing and effort conditional on reinvestment, the authors can now examine whether distressed banks prefer to reinvest in their illiquid assets or to give up their illiquid project and lend their liquid holdings on the interbank market.
• Expected profits π′b from lending liquid assets on the interbank market are simply π′b = eirli because the repayment probability of distressed banks is ei.
• Given the assumption R(ei) ≥ r, d∗i is always positive and profits from reinvestment πb are always larger than profits from lending liquid assets on the interbank market.

4.2 Intact Banks’ Optimal Supply of Liquidity

• When a distressed bank delivers low effort el, the interest rate r it is charged cannot be larger than μR—otherwise, the distressed bank would not borrow—and by assumption the authors have elμR < 1.
• To make sure that the distressed bank delivers high effort eh, intact lending banks impose a liquidity constraint.
• The volume of liquidity the distressed bank can then borrow verifies the incentive constraint: eh((li + di)R − dir) ≥ el((li + di)μR − dir).
• Denoting [x]+ = max(x; 0), this condition simplifies as a borrowing constraint: di ≤ d(li) ≡ ψR [r − ψR]+ li. (9) In this case, a distressed bank’s total borrowing from the interbank market is a positive function of its ex ante liquidity provision.

5. The Decentralized Equilibrium

• In the previous section, the authors derived the optimal date 1 decision rules for intact and distressed banks in terms of lending, borrowing, and effort.
• Based on these results, the authors now turn to the optimal date 0 liquidity provision policy in order to characterize the different equilibria of the economy.
• Recall that ex ante liquidity provisions are observable, so that the size of illiquid projects as well as reinvestment needs, assuming a shock has occurred, are also observable.
• The implementation of a borrowing constraint by intact banks on distressed banks requires the additional (implicit yet standard) assumption that total interbank borrowing is observable by lenders.
• The following two subsections are devoted to laying down the conditions under which each of these two situations can be an equilibrium.

5.1 The Full-Reinvestment Equilibrium

• 1.1 Optimal Ex Ante Liquidity Provision with Full Reinvestment.
• The equilibrium with distressed banks achieving full reinvestment exists if and only if two conditions are met:.
• The authors can then derive the following proposition.
• This drives up the interbank market interest rate, which provides incentives for banks to provision liquidity ex ante.
• The full-reinvestment equilibrium is therefore more likely when the liquidity shock is more likely, a property the authors refer to as the virtue of bad times.

5.2 The Credit-Rationing Equilibrium

• In the equilibrium described in the previous subsection, distressed banks are able to carry out full reinvestment thanks to their relatively large ex ante liquidity provision.
• The function ∂Eπi∂li is potentially nonmonotonic in the interest rate on the interbank market.
• The authors can then derive the following proposition.
• When the liquidity shock is less likely, banks provision less liquidity ex ante and invest more in illiquid assets.
• Credit rationing reduces the demand for liquidity and thereby depresses the return on ex ante liquidity provision for intact banks.

5.3 Multiple Equilibria and the General Equilibrium Externality

• When ex ante liquidity provisioning is low, then both liquidity supply and liquidity demand are relatively low.
• Hence, with little provisioning, the demand for liquidity is also low.
• The expected return on ex ante liquidity provisioning is then high.
• Outside this region the equilibrium is unique.
• The multiple equilibria property can be examined in a diagram representing aggregate liquidity supply Ls and aggregate 10The interbank-market-collapse equilibrium could be eliminated if banks could sign contracts contingent on the volume of date 0 liquidity provisioning.

6.1 Aggregate Shocks

• While this model shows that the fragility of the market for liquidity does not necessarily stem from the presence of aggregate shocks, it can easily be extended to allow for such shocks.
• Suppose, for instance, that the individual probability q to face a liquidity shock can take different values, F denoting the cumulative distribution function for q.
• Then when q < q, which happens with probability F (q), the interbank market collapse is the unique equilibrium and therefore happens with probability one.
• On the one hand, it raises the return to illiquid investments and hence raises banks’ incentives to invest in illiquid assets.
• As a consequence, the probability q below which the market-collapse equilibrium exists tends to decrease.

6.2 Interim Liquidation of Illiquid Assets

• Let us assume instead that distressed banks can liquidate (part of) their illiquid projects with a strictly positive liquidation value.
• Hence, its profit conditional on reinvestment being successful is (li +ρvi +di)R−rdi, while eh is both the effort the distressed bank undertakes and the probability that reinvestment is successful.
• Second, the distressed bank cannot liquidate more than a fraction α of its illiquid project.
• Moreover, interim liquidation could well happen at the same time or even after distressed banks borrow on the interbank market.
• When the share α of illiquid assets that distressed banks can liquidate is sufficiently low, there is still a credit-rationing equilibrium in which some liquidity is traded on the interbank market as opposed to the previous credit-rationing equilibrium where a total collapse of the interbank market takes place.

7. Policy Implications

• In this section the authors investigate whether and how policy can avoid a collapse of the interbank market.
• First the authors look at ex post interventions, i.e., policies that take place after the interbank market has collapsed.
• Then the authors focus on ex ante interventions, i.e., interventions aiming at preventing the collapse of the interbank market.

7.1 Ex Post Interventions

• There are basically two types of interventions that can take place after the interbank market has collapsed: liquidity injections and changes in interest rates which modify the return on the liquid technology.
• It can also influence the cost of liquidity by modifying short-term interest rates.
• In other words, unless the central bank has access to a monitoring technology that market participants do not have access to, liquidity injections are doomed to fail.
• Given that banks make no ex ante liquidity provision in the equilibrium with a market collapse, the reduction in interest rates does not modify distressed banks’ borrowing capacity, which remains at zero.
• Hence, interest rate cuts are most effective when banks have made relatively large ex ante liquidity provisions.

7.2 Ex Ante Interventions

• A regulator can affect the banks’ date 0 allocation of capital by imposing a liquidity ratio, requiring that banks invest at least some fraction of their portfolio in liquid assets.
• Yet imposing a liquidity ratio is equivalent to writing such a contingent contract between the regulator and banks, stating that the bank would be shut down if the share of liquid assets was lower than a given threshold.
• Imposing such a regulation in this type of model therefore ends up giving discretion to the regulator, which can be costly for reasons outside the scope of this paper (e.g., in terms of capture of the regulator by the regulated agents).
• In particular, the central bank can raise the return to liquid assets between date 0 and date 1 to raise banks’ incentives to invest in liquid assets and thereby prevent the collapse of the interbank market at date 1.
• Given that the right-hand side of the above inequality is decreasing in the interest rate r, the above inequality always holds if it holds for the lowest possible interest rate, i.e., when ehr = 1.

8. Conclusion

• The model the authors analyzed in this paper provides a framework for analyzing the occurrence of liquidity crises and discussing policy responses to situations of interbank market collapse.
• When banks achieve full reinvestment, they always provision the first-best volume of liquidity, and the equilibrium interest rate on the interbank market is r∗ = min{r1; r2}.
• There may be two types of such equilibria.
• The equilibrium with zero ex ante liquidity provision and interbank market collapse is therefore the only equilibrium, when it exists, in the credit-rationing regime.

Figures

Figure 1. Timing of the Model

Figure 2. Aggregate Supply of and Aggregate Demand for Liquidity

Full text

Liquidity, Moral Hazard, and
Interbank Market Collapse
∗
Enisse Kharroubi
a
and Edouard Vidon
a,b
a
Banque de France
b
International Monetary Fund
This paper proposes a framework to analyze the functioning
of the interbank liquidity market and the occurrence of liquid-
ity crises. The model relies on three key assumptions: (i) ex
ante investment in liquid assets is not verifiable—it cannot be
contracted upon, (ii) banks face moral hazard when confronted
with liquidity shocks—unobservable effort can help overcome
the shock, and (iii) liquidity shocks are private information—
they cannot be diversified away. Under these assumptions,
the aggregate volume of capital invested in liquid assets is
shown to exert a positive externality on individual decisions
to hoard liquid assets. Due to this property, the collapse of
the interbank market for liquidity is an equilibrium. More-
over, such an equilibrium is more likely when the individual
probability of the liquidity shock is lower. Banks may there-
fore provision too few liquid assets compared with the social
optimum.
JEL Codes: D53, D82, D86.
∗
We thank Nuno Cassola, Edouard Challe, Denis Gromb, Arvind Krish-
namurthy, Henri Pag`es, Jean-Charles Rochet, Rafael Repullo, Tano Santos,
Jean Tirole, and two anonymous referees for helpful comments and suggestions.
All remaining errors are ours. A previous version of the paper was circulated
under the title “Increasing Returns in Inter-Bank Liquidity Market.” The views
expressed herein are those of the authors and should not be attributed to the
Banque de France, the Eurosystem, or the International Monetary Fund, its
executive board, or its management. Corresponding author: Enisse Kharroubi.
Address: Banque de France. 49-1374. 1, rue de la Vrilli`ere. 75049 Paris cedex 01.
E-mail: enisse.kharroubi@banque-france.fr. Tel: + 33 1 42 92 47 39. Fax: + 33 1
42 92 49 50.
51

52 International Journal of Central Banking December 2009
1. Introduction
The financial market turmoil that has been under way since the sum-
mer of 2007 hit the core of the global financial system, the interbank
market for liquidity. While this paper does not endeavor to account
for all the features of the recent crisis, be it hard evidence or casual
stories about the motivations of market players, it argues that a
proper modeling of the collapse in the market for liquidity involves
a close look at incentives to provision/hoard liquidity and moral haz-
ard mechanisms in the interbank market. In addition, it makes sense
to do so in a framework where banks can actually fail and default
on their borrowing. Both of these assumptions are strongly vindi-
cated by salient features of the recent crisis. Many observers have
argued that securitization may have provided the wrong incentives
regarding the monitoring of underlying asset quality, in a clear-cut
case of moral hazard. In addition, recent developments have shown
that bank failure scenarios are only too realistic.
We investigate the possible role of insufficient ex ante liquidity
provision in paving the way to an interbank market collapse. We
thus highlight the benefits of situations where banks set aside large
amounts of liquid assets in order to better deal with shocks affecting
their illiquid investments. By liquidity provisions, we mean specifi-
cally holdings of assets that can be used to safely transfer wealth over
a short period of time. This may be seen as a form of “balance-sheet
liquidity.” In practice, such liquid holdings could be remunerated
reserves held at the central bank, or short-term Treasury securities.
1
Indeed, the secular decline in the share of liquid assets on banks’ bal-
ance sheets is a striking stylized fact that has been underscored by
Goodhart (2008) as a troubling feature of risk management. A situ-
ation where market and funding liquidity appeared to be high may
thus have hidden vulnerabilities stemming from limited holdings of
liquid assets.
Against such a background, this paper shows that across equilib-
ria, the risk-adjusted return on liquid assets can be increasing with
1
We do not model a risk-free asset market as such; however, we will simply
assume that a technology providing a risk-free rate of return is available as an
alternative to illiquid investments on the one hand, and to interbank lending on
the other hand.

Vol. 5 No. 4 Liquidity, Moral Hazard 53
the aggregate volume of such assets in the economy. When a bank
faces a liquidity shock, it needs to reinvest in its impaired assets.
Moreover, success in reinvestment depends on the effort the bank
undertakes. When it has provisioned a large volume of liquidity ex
ante, reinvestment is mostly financed through internal funds. Hence,
the distressed bank pays particular attention to improving the prob-
ability that reinvestment succeeds. Consequently, the moral hazard
problem is mitigated and the distressed bank benefits from a large
capacity to borrow liquidity on the interbank market. This tends
to raise the demand for liquidity and hence the price of liquidity,
which in turn raises incentives to provision liquid assets ex ante. As
a result, both the risk-adjusted return on liquidity provisioning and
the total volume of liquidity in the economy are large.
By contrast, with low ex ante liquidity provision, the argument is
reversed: the moral hazard problem is amplified through the afore-
mentioned channel—reinvestment is mostly financed through exter-
nal funds. Intact lending banks then impose a tight constraint on
the volume of liquidity distressed banks can borrow on the inter-
bank market so as to restore their incentives to deliver effort. This,
however, reduces the demand for liquidity and drives down the price
of liquidity, which in turn depresses banks’ incentives to provision
liquidity ex ante. Consequently, the risk-adjusted return on liquid-
ity provisioning and the total volume of liquidity in the economy are
low. The two polar cases of high and low liquidity provisions can
therefore both be equilibrium outcomes.
Turning to comparative statics, the credit-rationing equilibrium
happens to be more likely when the liquidity shock is less likely. We
call this property the curse of good times. When the probability of
facing the liquidity shock is low, banks reduce their liquidity holdings
because they are less likely to need these liquid assets for reinvest-
ment. This tightens the moral-hazard-induced liquidity constraint,
reducing the demand for liquid assets and thereby the return on
liquid assets on the interbank market, which in turn reduces incen-
tives to provision liquidity ex ante. Conversely, the equilibrium with
large liquidity provision and high risk-adjusted return is more likely
when the liquidity shock is more likely, a property we call the virtue
of bad times. When the probability of facing the liquidity shock
is high, banks raise their liquidity holdings because they are more
likely to need these provisions for reinvestment. This relaxes the

54 International Journal of Central Banking December 2009
moral-hazard-induced liquidity constraint, raising the demand for
liquidity and thereby the price of liquidity on the interbank market,
which in turn raises incentives to provision liquidity ex ante. Hence,
when the probability of the liquidity shock is intermediate, multiple
equilibria emerge: large (resp. low) aggregate investment in liquid
assets tends to raise (resp. reduce) the return on liquid assets and
thereby raise (resp. reduce) individual incentives to invest in liquid
assets.
Finally, the paper investigates how policy can prevent or dampen
a collapse of the market for liquidity. The main result is that policies
aimed at tackling the collapse of the interbank market ex post—i.e.,
after the collapse has happened—are unlikely to reach their goal. In
particular, liquidity injections as well as interest rate cuts cannot
help distressed banks overcome their liquidity shocks. By contrast,
ex ante policies, especially those that modify the relative return of
liquid assets compared with illiquid assets, can be successful in pre-
venting a collapse of the interbank market. In other words, monetary
policy, by setting short-term interest rates which provide incentives
to invest in liquid assets, can be helpful in reducing the occurrence
of liquidity crises. Regulatory policies requiring liquidity provision
can also be useful.
The model in this paper builds on the standard literature on
moral hazard and liquidity crisis. The demand for liquidity is mod-
eled in a basic, standard fashion, similar to that of Holmstr¨om and
Tirole (1998). Agents (in our case, banks) with long-term assets
face stochastic liquidity shocks which trigger a reinvestment need
and a moral hazard problem: success in reinvestment depends on
unobservable effort by banks.
2
We, however, depart from this sem-
inal paper in an important way, by assuming that idiosyncratic
liquidity shocks cannot be diversified away: this opens the door
to an interbank market where liquidity can be reallocated interim.
Because of this feature, our framework is closely related to the
2
The main alternative modeling of liquidity is based on the Diamond and
Dybvig (1983) approach—enriched by Diamond and Rajan (2001)—in which
banks with illiquid assets supply liquidity to consumers through liquid deposits
(funding liquidity). While this approach can account for bank runs that have
taken place during the current financial crisis, the Holmstr¨om and Tirole (1998)
approach, focused on market liquidity, seems more relevant given the particular
initial circumstances of the crisis.

Vol. 5 No. 4 Liquidity, Moral Hazard 55
model of liquidity demand developed by Caballero and Krishna-
murthy in a series of papers (in particular, Caballero and Krish-
namurthy 2004) dealing with access to international financing. Our
model shares their features that (i) idiosyncratic shocks cannot be
written into insurance contracts, generating the need for domestic
financial transactions, and (ii) borrowers cannot transfer the full
surplus generated by reinvestment resources. Likewise, we therefore
have situations where private decisions are biased against hoarding
liquidity.
Our paper is connected to the literature on interbank markets, as
a mechanism for managing, and potentially eliminating, risks stem-
ming from idiosyncratic liquidity shocks. Bhattacharya and Gale
(1987) in particular studied the case where neither banks’ invest-
ments in the illiquid technology nor liquidity shocks are observ-
able. In their framework, banks have an incentive to underprovision
liquidity ex ante and free-ride the common pool of liquidity. Rochet
and Tirole (1996) adapted the Holmstr¨om-Tirole framework to the
interbank market in order to study systemic risk and “too-big-to-
fail” policy. The existence of interbank market imperfections has
been established empirically by Kashyap and Stein (2000), which
showed the role of liquidity positions, the so-called “liquidity effect.”
Building on such evidence, Freixas and Jorge (2008) analyzed the
functioning of the interbank market in order to show the conse-
quences of its imperfections for monetary policy. In particular, they
established the relevance of heterogeneity in banks’ liquid asset hold-
ings for policy transmission.
Our work is also related to recent work on liquidity crises. A
recent strand of literature has explored the propagation of crises
through banks’ balance sheets, while treating the level of liquidity
held by banks as endogenous. This approach builds on Allen and
Gale’s (1998) analysis of distressed liquidation of risky assets, to
explore the mechanism whereby anticipation of fire-sale pricing of
such assets determines banks’ ex ante portfolio allocation. Allen and
Gale (2004) as well as Acharya, Shin, and Yorulmazer (2007, 2009)
have concentrated on this interaction between equilibrium liquid-
ity and endogenously determined fire sales. In particular, Acharya,
Shin, and Yorulmazer (2007) showed that banks’ holdings of liquid-
ity may be too low or too high compared with the social optimum,
depending on the pledgeability of their assets and the possibility to

Frequently asked questions

Q1. What have the authors contributed in "Liquidity, moral hazard, and interbank market collapse" ?

This paper proposes a framework to analyze the functioning of the interbank liquidity market and the occurrence of liquidity crises. 

Q2. What future works have the authors mentioned in the paper "Liquidity, moral hazard, and interbank market collapse" ?

These are possible research avenues for future work. However, since by assumption the authors have eh ≥ 1 − q, this condition can not be satisfied. There may be two types of such equilibria. Having determined banks ’ date 1 decisions, the authors can turn to banks ’ date 0 problem, which writes as max l ( 1 − q ) [ ( 1 − l ) R + [ ( 1 − l ) β + l ] max { ehr ; 1 } ] + qehr R [ 1 − ψ ] r − ψR ( l + ρα ( 1 − l ) ).