RISK-TAKING AND ASSET-SIDE CONTAGION IN AN ORIGINATE-TO-DISTRIBUTE BANKING MODEL 2010 .pdf

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RISK-TAKING AND ASSET-SIDE CONTAGION IN AN

ORIGINATE-TO-DISTRIBUTE BANKING MODEL

Andrea Pinna

WORKING PAPERS

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C O N T R I B U T I D I R I C E R C A C R E N O S

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Risk-Taking and Asset-Side Contagion in anOriginate-to-Distribute Banking Model∗

Andrea Pinna

Queen Mary, University of London

and CRENoS

Abstract

During the Subprime crisis the entire banking industry riskedcollapsing under an unprecedented lack of liquidity. This work

tries and find what channel allowed a relatively small systemicshock, the increased mortgage delinquency in the US housingmarket, to spread worldwide with such a terrific impact.I develop a model of financial contagion where banks adopt-ing the originate-to-distribute model satisfy their liquidity needsthrough repurchase agreements in the money market.I assume there are no early diers in the economy, and I look atcrises originating from inaccurate forecasting of asset returns bysome banks. The crisis may be inefficient since accurate bankswith good fundamentals are unable to roll over their debt andgo bankrupt. There is room for the regulator to make accuratebanks willing to rescue failing banks.

JEL Classification: G01, G24, G32, G33.Keywords: Repos, Haircut, Rollover, Financial Crises.

∗I thank Giovanni Cespa, Luca Deidda, Xavier Freixas, Winfried Koeniger,and seminar partecipants at the Universitat Pompeu Fabra, University of Cagliari and Queen Mary, University of London for useful comments.E-mail: [email protected]

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1 Introduction

By raising liquid assets, gathering information and spreading risks, fi-nancial markets should allow the real economy to achieve an efficientallocation of financial resources toward investment.

However, the Subprime crisis suggests that the functions performedby financial markets might be hampered because of the possibility of market-failures.

The present paper proposes a new way to model the crisis of 2007-

2009: primarily, pointing at asset mispricing as the source of the tur-moil. The aim of this approach is to capture the role “opaque” deriva-tives play in determining the possibility of a crisis.1

Secondly, I analyse the process of contagion through the market forliquidity, stressing the role strategic interdependencies play in spread-ing the financial trouble from one investment bank to the whole sys-tem.

I find that the misevaluation of risky return by some banks can leadbankrupt otherwise solvent competitors, so as to determine an ineffi-cient financial crisis.

If the realization of the return is much lower than expected by incon-

siderate banks, the latter go bankrupt and some lenders in the moneymarket suffer capital losses.

This decreases the amount of liquidity that is available to lend againstpledgeable assets for the following period.2

Pledgeable assets can therefore be undervalued on the basis of cash-in-the-market pricing, like in Allen and Gale (1994). Any decrease in theprice of collateral affects its pledgeable value, and lowers banks abilityto fund their liquidity needs.

The insolvency of some inconsiderate banks may thus make consid-erate banks illiquid.

1“The sell-side of the market (dealer banks, CDO and SIV managers) understands

the complexity of the subprime chain, while the buy-side (institutional investors) doesnot.”Gorton (2008).

2The amount of liquidity in the market for collateralized debt decreases only if potential investors have not enough time to enter the market between the discoveringof the risky return and the rollover date. However, according to Brunnermeier (2009),"Almost 25 percent of [investment banks’] total assets were financed by overnightrepos in 2007".

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This interdependence among institutions plays a role in the cost agovernment has to bear to stop the contagion.

In fact, if the government commits to inject an amount of liquiditynot exceeding the industry net shortage, the risk of contagion influ-ences the scope for vulture-like strategies and liquidity hoarding.

Since Diamond and Dybvig (1983) and Bryant (1980) seminal workson bank runs, researchers have insisted on the fragility brought aboutby the banking activity.

Allen and Gale (1988), Goldstein and Pauzner (2005), and Jacklin and

Bhattacharya (1988) enriched the analysis linking depositors’ expecta-tions to the business-cycle.

Adrian and Shin (2009), Allen and Gale (2000), Brusco and Cas-tiglionesi (2007), Freixas, Parigi and Rochet (2000), Rochet and Vives(2004), although keeping much of their framework in the same vein asprevious works on bank runs, tried and single out the channels throughwhich the run on one bank may trigger a systemic financial crisis.

These models pointed out that a domino effect is able to spread thetrouble of one bank, originated by endogenous panics or new informa-tion, to other financial institutions through cross-defaults.

Domino models account for the importance of banks providing each

other mutual insurance through the interbank market for liquidity.Nevertheless, they paint a picture of passive financial institutions,

that stand by and do nothing as the sequence of defaults unfolds.The domino approach does not consider the effect market forces and

mark-to-market accounting have on the reliability of banks.The impact of price changes on the book value of banks asset is in

fact likely to increase the negative effect of counterparties’ defaults.Allen and Gale (2004), Diamond and Rajan (2005), and Cifuentes,

Shin and Ferrucci (2005) show that the interaction among changes inasset prices and solvency requirements amplify any initial shock.

The reduction in the value of a marked-to-market balance sheet may

force a bank to respond selling its risky assets, whose value is weightedby their risk measure, with cash. That should increase prima facie thebank’s value for regulatory purposes.

However, if the asset demand is not perfectly elastic, sales induce afurther decline in its marked-to-market portfolio of assets that mightoutweigh the effect of the initial response.

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In these models contagion, far from being caused by interbank claims,results from the fall in the value of banks portfolios.

To take this approach, one needs to single out a proper source of rigidity in asset pricing in the financial market.

Shleifer and Vishny (1992) analyse the equilibrium aspect of assetsales in a generic industry and describe the coming up of fire sales,namely sales at trading price lower than the fair price of the asset, whenfinancial distress hits companies with specialized assets. When a firmmust sell assets because of financial distress, the potential buyers with

the highest valuation for the specialized asset are other firms in thesame industry, who are likely to be in a similarly dire financial situa-tion and may therefore be unable to supply liquidity.

One can imagine that the same story fits the financial market: be-cause of asymmetric information or specialised investments strategies,outsiders are willing to pay the asset less than its industry-specific value.

The asset price then decreases, the depreciation being higher for as-sets that are highly specialized or difficult to evaluate.

Adrian and Shin (2009) provide evidence in support of such a re-sult, showing that marked-to-market leverage is strongly procyclical.The authors show that institutional investors respond to changes in

the value of their portfolio moving their leverage in the same directionas the market does.

Thus market participants sell when prices slump, in a manner thatreinforces the price trend.

Because the liquidity available within the industry in the short runis given by the banks’ portfolio decisions at the initial date, substantialsales can lead to a drop in prices.

As the asset price falls, the bank has to sell an even larger proportionof its long-term assets in order to abide by solvency constraints imposedby the regulator.

The asset price can slump so dramatically that the bank cannot meet

its commitments even if it liquidates all its long-term assets. At thesame time, other banks find the value of their assets not sufficient tofulfill their commitments and start selling, so putting the trading priceunder even higher pressure.

In force of the central bank readiness, the present work focuses onthe role played by linkages among banks, namely a market to borrow

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liquidity against collateral, when depositors do not care about a dropin banks’ portfolio.

There are no early diers in the economy, and depositors do not runtheir banks because the limited convertibility keeps deposits safe.

This allows to focus on the effect the misevaluation of opaque securi-ties has on banks ability to borrow, rather than on the exogenous needfor liquidity of a portion of depositors.

The paper is organised as follows. In section 2 I give an informalaccount of the Subprime crisis. In section 3 I present the model and

characterize the banks optimal portfolio allocation. In section 4 I de-scribe the impact of interim information on the banking sector. Insection 5 I show how the misevaluation made by some banks propa-gates as a liquidity crisis. Section 6 deals with the policy implicationsof the newly found contagion channel. In section 7 I present a numericexample. Section 8 concludes.

2 Modeling the Subprime Crisis

Most extant explanations of financial crises emphasise the negative ex-ternalities on the liabilities side of the balance sheet: it is the run bydepositors that precipitates a crisis.

By virtue of the policies suggested in these works, central banks man-aged to relegate demand-driven bank crises to break out in text booksmore often than in newspapers.

Nevertheless, such a phenomenon came back in 2007 in the UK, onecentury after the most recent bank run in the British banking system,when many depositors queued to withdraw their deposits in NorthernRock.

The Subprime crisis, able to set off such secular event, questionedour understanding of financial crises.

In fact, the financial crisis of 2007-2009 does not fit either the accountgiven by models on self-fulfilling prophecies or those stemming fromshock on fundamentals.

This criticism constitutes the primary motivation to the present pa-per and stems from a key stylized fact: depositors do not seem to haveplayed a primary role in the crisis.

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To begin with, banks experienced difficulties before anything hap-pened to liquidity demand, and kept secret this bad news as long asthey could.3

Only after investors faced troubling information on the solvency of the banking industry in the near future, they withdrawn their endan-gered liquidity.

Furthermore, central banks and governments have succeeded in build-ing up credibility as institutions ready to do anything for the sake of retail depositors and financial stability.

Thus a consumer panic, either driven by prophecies, interim signalsor asymmetric information, does not give a satisfactory motivation forthis crisis to materialize.

Whilst all the models mentioned thus far relied on preference shocksto justify the emergence of a run, Cifuentes, Ferrucci and Shin (2005)refrain from characterising what kind of blow hits the economy.

The authors focus on the way solvency constraints create liquidityrisk in a system of interconnected financial institutions, when the lattermark their assets to market. They assume that a given shock on theasset side lets a bank violate a solvency constraint, namely a liquidityratio. The hit bank must sell part of its assets to avoid contravening the

regulation.The demand for the asset being less than perfectly elastic, sales by the

distressed institution lower their market value. Thus, assets are markedat a new lower trading price and the solvency constraints may dictatefurther disposals, leading the whole banking sector to collapse.

2.1 Literature on the crisis of 2007-2009

The paper by Cifuentes, Ferrucci and Shin (2005) belongs to a tinystrand of literature that does not rely on early diers to account for theprecipitation of a crisis.

Although the interest on modelling financial crises has gained mo-mentum since the occurrence of the Subprime crisis, the vast majority

3Shin (2008) reports on the run to Northern Rock: “The Bank of England was in-formed [of Northern Rock’s funding problems] on August 14th. From that time untilthe fateful announcement on September 14th that triggered the deposit run (i.e. for afull month), the Financial Service Authority and Bank of England sought to resolvethe crisis behind the scenes, possibly arranging a takeover by another UK bank.”

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of papers on the subject still focuses on preference shocks.This is certainly convenient, in that early diers naturally create scope

for an interbank market the crisis can spread through. However, asmentioned at the beginning of this section, preference shocks do notseems to have played a role in the materialisation of the last crisis.

Heider, Hoerova and Holthausen (2009) model an interbank marketwhere financial institutions lend money to each other in order to dealwith the traditional issue of early diers.

The authors introduce asymmetric information about counterparty

risk among banks. The private information each bank has on its riskyinvestment produces adverse selection and can lead the unsecured in-terbank market to freeze.

Diamond and Rajan (2010) show that the overhang of illiquid assetsincreases the future return of holding them in the eve of fire selling.Opaque ABSs are likely to depreciate much, in that they can be valuedonly by some specialized firms.

Thus, the prospect of buying undervalued assets in the near futureinduces endangered banks to hold them with the hope that they willappreciate before it is too late.

In a parallel work to this paper, Acharya, Gale and Yorulmazer (2010)

emphasise the role played by rollover and liquidation risk in the Sub-prime crisis. Similarly to the present paper, they allow banks to borrowliquidity through repo agreements in the money market.

In a two-state model, the authors show that in the bad scenario thepledgeable value of an asset is lower than its fundamental value. Thereason is that, when the frequency of rollover is very high, it is veryunlikely that there will be good news able to make the value of theasset jump to the good state by the rollover date.

They assume banks assets have a liquidation cost. Thus, in order toavoid the cost of bankruptcy, borrowing banks do not issue debt withface value higher than their low-state debt capacity.

Even a small decrease in the fundamental value of the asset at a rolloverdate may thus cause a large fall in the bank debt capacity. The effect canbe so important that the market for secured borrowing freezes.

To the best of my knowledge, the model by Acharya, Gale and Yorul-mazer (2010) is the only one on the Subprime crisis that does not focusneither on the infringement of regulatory provisions due to early diers

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nor on exogenous changes in haircut.On the one hand, regulators were most likely willing to loosen their

provisions to stop the crisis. On the other hand, the change in hair-cuts was undoubtedly an important channel for contagion, for lendersupdated their expectation of the counterparty risk.

The contagion channel pointed out in this paper adds to the effect of a haircut increase.

The surge of counterparty risk stemming from the discovery of as-set misevaluation from some banks, and the consequent widening of

lending margins, is neglected in the paper but would deepen the crisis.Nevertheless, haircuts are kept steady at their initial value to disen-

tangle the primary role asset misevaluation and market risk played inthe occurrence of the crisis of 2007-2009.

One novelty of the Subprime crisis was in fact the reliance of thebanking industry on short-term financing against highly opaque secu-rities that were difficult to price.

2.2 The fictitious separation between banks and SPVs

The Subprime crisis did not unfold in the traditional commercial bank-

ing sector considered in most former works on financial turmoils.The reason why the model developed in this paper alludes to depos-itors is to stress its contribution to the previous literature and makeconsistent the comparison of its results with models of contagion basedon bank runs.

To account for the crisis of 2007, the reader shall interpret deposi-tors as investors in an investment bank, whilst the bank that collectsdepositors’ liquidity can be recast as a complex economic entity madeby an investment bank and a Special Purpose Vehicles (SPV), as manyfinancial institutions set up in the years preceding the crisis of 2007.The rationale for this unified treatment is explained below.

Special vehicles are off-balance legal entities that investment bankcreated in order to attain high levels of leverage without breachingbanking regulatory provisions.

Banks are in fact allowed to invest only a limited portion of theircapital in risky assets - e.g. loans to noninstitutional borrowers.

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For regulatory purposes, the size of the risky investment is its valuein the bank trading book, weighted by a measure of its risk.

By transferring the risky investment to a SPV a bank could enjoy thereturn on the loans it issued, without making the liability appear onthe trading book and incur regulator’s intervention.

The SPV played also a second role on behalf of the parent bank. Itcould issue Asset Backed Securities (ABSs) representing claims on thecash flow paid by the pool of risky assets it received by the bank.

The SPV could either trade the securities or use them as collateral, to

borrow liquidity from investors through repurchase agreements in themoney market.

Securitization played a particularly important role in the subprimemarket, until the crisis hit. Gorton and Metrick (2010) point out that in2005 and 2006 Subprime mortgage origination was about $1.2 trillionof which 80% was securitized in different seniorities.

As a consequence of the seemingly interminable appreciation in houseprices, the cash flow to the securities was assured by the opportunityto make the borrower refinance the property before defaulting on hismortgage, just a couple of years after it was originated, or to repossessthe inflated property and recoup the whole credit.

Adopting such a strategy, the SPV gathered cash the parent bankcould use to both increase its position in the risky asset and smoothits available liquidity before the risky asset pays out its return.

Nevertheless, the SPV having a portfolio solely made of risky assets,it could not borrow liquidity under favourable terms without properguarantee.

Hence, parent banks ensured they would provide backstop liquidityif the cash flows delivered by the pool of loans appeared insufficient tofulfill the claims by investors who accepted the ABSs as collateral.4

A bank could thus originate risky loans and create a SPV, to haverisky assets bought out of the bank balance sheet. The SPV financed

the acquisition by issuing securities backed by the same loans, withthe opportunity to pay low interests on its debt in force of the twokey elements of the originate-to-distribute model: the liquidity con-duit granted by the originating bank, and the endorsement received by

4Reputational credit lines needed no capital charge under the banking regulation.

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rating agencies.In fact, ABSs issued by SPVs were not simple claims on the pool of

loans. Vehicles cooperated with rating agencies to package the loansin opaque structured securities, characterized by a credit report ratherthan by information on the asset backing them.

At the beginning of 2007, the delinquency rate of US subprime non-institutional borrowers revealed higher than expected by most investorsin securities backed by those loans. Thus, the credit ratings assigned byagencies to most ABSs proved to be wrong.5

Parent banks had to activate the liquidity backstops to their SPVsand, at the same time, the market value of ABSs held by the vehiclesfell to adjust to the new expected return.

As soon as SPVs in Europe and the US started experiencing difficul-ties, it became clear that the separation between those entities and theirparent banks was fictitious.

The originate-to-distribute model was a convenient way to overcomeregulation on leverage and increase banks’ revenues, when the riskyassets were yielding a positive return.

However, when the event regulatory provisions on leverage weremeant to protect from took place, failing SPVs and their parent banks

shown to be the same economic entity.The portfolios of ABSs having a lower market value, the single entity

bank-SPV, hit by the negative shock, faced a fall in its ability to borrow.It could then end up short of the liquidity needed to fulfill the com-

mitment with its investors, as well as with the regulator.For this reason I model the originate-to-distribute lending activity at

the root of the crisis as if the investment bank and its SPV were thesame economic entity. This rules out the fictitious separation actualbanks adopted merely for regulatory arbitrage purposes.

2.3 Liquidity funding in the banking industry

The effect ABSs depreciation had on the bank-SPV ability to borrowwas exacerbated by the nature of its financing contracts.

5"In the second half of 2007, Moody’s downgraded more bonds that it had over theprevious 19 years combined". The Financial Times, 18 October 2008.

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The main source of liquidity to SPVs consisted in short term - mostlyovernight - Repurchase Agreements (repos) with investors in the moneymarket.

If the borrower can not repay his debt, the lender in a repo agreementbecomes owner of the collateral and sells it in the ABS market, usingthe proceeds as repayment of its claims.

Rather than securing the funding before investing in risky loans, theentity could get the liquidity it needed to originate them - or to buythem in the secondary market - by pledging securities backed by those

loans.Banks had the opportunity to originate a loan, securitize the claims

and pledge them as collateral in a repo agreement, secure the fundsneeded for the loan to be issued and, finally, use the borrowed liquidityto issue an amount of loans far above the internal capital they investedin the risky lending activity.

The difference between interests earned on the risky loan and therepo interest paid to lenders was thus a net gain. The bank could realizeit because of the leverage secured by its SPV on the repo market.

Hence, the value of investing one unit of capital in a risky asset,namely the lending activity, was worth to the bank-SPV more than

its face value.A bank-SPV benefited from the usage value of the securities backed

by its pool of loans. This amounts to the additional liquidity, borrowedagainst ABSs issued by the SPV, the bank can invest in further remu-nerative loans.

Such a money multiplier would allow the bank to leverage indefi-nitely, if it was not for the additional guarantee money market investorsask as the liquidity cushion of the entity bank-SPV becomes less signif-icant.

In repo agreements, lenders protect themselves against possible fallin the collateral market value applying a margin between the price of

collateral at the start date of the contract and the amount of liquiditythe counterparty can borrow against the pledged asset.

In this paper the choice made by investors about this haircut is en-dogenous to the model. I assume lenders measure their exposure to theborrower in terms of Value at Risk (VaR).

The VaR of an investment is the upper bound potential losses do not

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exceed with some predetermined probability, called a confidence inter-val, according to the probability distribution of the asset liquidationvalue.

2.4 Outline of the model

The subprime crisis began with the discovery that highly structuredfinancial derivatives were overpriced.6 Starting from such stylized fact,this paper relies on wrong evaluation of opaque asset returns to propose

a new contagion channel in financial markets.Secured borrowing with high rollover frequency is shown to makethe amount of liquidity banks can raise against their illiquid collateralstrongly depend on other banks’ misevaluation.

Banks discovering highly overpriced securities in their portfolios lackpledgeable value and find it impossible to borrow the liquidity theycommitted to pay their investors.

At the first date, banks make their investment decision and com-mit to periodical payments they must deliver in the future to avoidbankruptcy.

Each institution sets the level of settlements equal to the amount of

liquidity it expects to hold at each date, according to its estimate of thereturn on risky assets and its pledgeability.If one introduces a market for liquidity, where banks can raise some

cash at the interim date against the value of their illiquid asset, thewrong assessment made by a bank can set off a contagion.

The value of pledgeable assets depends on their market value. Thus,the prevailing trading price of illiquid assets determines, at each date,the amount of cash every bank can collect against the risky assets itchose to invest in at the initial date.

If a bank hit by an interim shock goes bankrupt, its investment inthe risky asset is liquidated by lenders to recoup the face value of debt.

The new fundamental value of collateral can be too low for somelenders to avoid breaching their VaR limits. This, in a market with lim-ited participation, determines cash-in-the-market pricing and an asset

6The paternity of this mistake is not addressed here, although the moral hazardproblem with rating agencies and the lack of time series on new custom-made productsseem very good explanations.

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price below its fundamental value.In the present model, such an outcome is highly likely to take place.

In fact, the same opaqueness that contributes to the misevaluation of the risky asset makes the intervention by outsider investors costly.

The result is that the amount of cash all fundamentally solvent bankscan rise against their positions in the same illiquid investment falls.

The bankruptcy of one bank may then put in trouble some banksnon-hit from the initial shock. Other investors breach their VaR limitand the asset price falls even further, possibly endangering even banks

that were hit positively by the initial shock.A public injection of liquidity is very costly in this framework, and

it may turn out to be inefficient for the usual moral hazard concerns.7

It is then worthwhile to assess whether the regulator can set up a self enforcing coordinating mechanism in the interbank liquidity market.

The result of the model is reassuring in this respect. Differently fromwhat found by Diamond and Rajan (2009) and Acharya, Gromb andYorulmazer (2008), banks unaffected by the interim shock have no in-centive to hoard liquidity or to adopt a vulture-like strategy.

They prefer to take second best decisions on the purchase of illiquidassets previously owned by inconsiderate banks to bail out themselves.8

The threat of contagion is part of this mechanism. It can save thegovernment the cost of injecting liquidity in the attempt to preventthe financial sector from freezing the functions it performs for the realeconomy.

In fact, when a crisis spreads through the channel pointed out in thispaper, the only way a government has to stop the contagion is by buy-ing pledgeable assets to sustain their price. Public intervention has tokeep the price inflated, in order to maintain the pledgeable value of banks portfolio to a sustainable level until the illiquid investment paysits return to solvent but otherwise illiquid banks.

7"In a survey of 120 banks in 24 developed countries in the 1980’s and 1990’s,Goodhart (1995) found that two out of three failed banks were bailed out (...) PatrickHonohan and Klingebiel (2000) found that, on average, countries spend 12.8 percentof their GDP cleaning up their banking systems." Gorton and Huang (2004).

8This analysis is performed in a framework with n = 2 banks in the present paper,whilst the issue of coordination when n > 2 is being assessed in a parallel work inprogress " Liquidity Shortages and the Viability of a Superfund ".

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The purpose of the present work is to formalise the above mentionedchannel for contagion and banks incentive to bailout the industry.

3 The model

Consider a multi-region economy lasting for three dates t = 0, 1, 2.There is no discounting and the risk free rate is normalised to zero.

The economy is populated by depositors, banks, funds, and nonin-stitutional borrowers. Projects undertaken by the latter type of players

are the only source of return in the model.Each region i = 1,..., n is endowed at the initial date with one unit of

liquidity, proportionally owned by a continuum of depositors.

3.1 Players

a) Depositors

Among depositors there are no early diers, to use the jargon of previousliterature. Their preferences are homogeneous and described by theutility function

U d (c t ) = min(c 1, c 2)

where c t is the amount of liquidity a depositor can use to afford con-sumption at date t .

Thus, there are no shocks in the demand for liquidity, and depositorsprefer to smooth their consumption over periods.

Depositors from each region have the opportunity to put their en-dowment in the regional banking industry, in order to access its in-vesting technology and afford at each future date t = 1,2 the highestpossible amount of consumption good.

Since there are no early diers, depositors do not run their banks onthe basis of coordination problems. Furthermore, since they strictly

prefer smooth consumption over time, a contract that specifies c 1

= c 2

is optimal. Such a contract is akin to limited convertibility and there-fore depositors never run their bank.

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b) Noninstitutional borrowers

Noninstitutional borrowers face a long-term entrepreneurial projectand get from banks the external capital needed to undertake it.9 Dif-ferently from depositors, these agents can deal with banks from anyregion.

They borrow the money needed at the initial date, and their abilityto repay the loan at the final date depends on the project outcome.

Borrowers have limited liability but are homogeneous and need ex-

ert no effort to affect a project probability of success, nor can theymisrepresent the project return.Thus there are no agency problems between banks and borrowers.

The latter are just a black box banks can access remunerative projectsthrough.

c) Banks

Regional banking industries are perfectly competitive. Banks receivefrom each depositor a fee ∆ for managing her liquidity.

In order to outperform its competitors, every bank has to offer de-posit contracts maximising depositors’ utility in its region. Thus, banks

maximize the amount of liquidity they commit to give depositors ateach date.

As mentioned above, banks can lend part of depositors’ liquidityto noninstitutional borrowers. Each bank takes its lending decisionupon the loans repayment it expects at the final date, having observeda region-specific signal on the realization of exogenous factors that af-fect the project return.

Banks optimization problem amounts to select what share of de-posits to invest in lending to noninstitutional borrowers and what tobe held as cash, together with a commitment on the periodic paymentc t paid to depositors at dates t = 1,2.

Banks choose their portfolio allocation and commitment with de-positors under a solvency constraint. A bank goes bankrupt if, at any

9Borrowers can be consumers who want to buy a home, to account for thecategoryof borrowers the 2007-2009 crisis generated from, as well as other agents involved inentrepreneurial ventures.

15



date, its available liquidity is lower than the amount it had committedto pay.

The capital banks in each region can use to finance entrepreneurialprojects goes beyond the unit of liquidity they collect from depositors.Banks can in fact turn to the money market in order to find additionalliquidity they want to invest in further loans to noninstitutional bor-rowers.

d) Money market funds

Money market funds are awash with liquidity and can only invest inshort-term debt securities issued by banks.

They are subject to a "VaR equal zero" constraint. Whenever a fundbreaches the constraint it exits the market.

These investors lack specialized skills needed to assess the return onentrepreneurial projects. Their lending decision is thus taken on thebasis of common knowledge.

3.2 Investment technologies

a) Storage technology

The storage technology is simply cash, that is a 1-period investmentyielding one unit of liquidity per unit of liquidity invested.

b) Deposit Technology

The deposit technology is a 2-period fixed commitment contract thatmay differ among regions.

It entitles depositors to get from banks in their region i a constantamount of liquidity c t

iat dates t = 1,2 per unit of liquidity deposited

at the initial date.

c) Entrepreneurial projects

Entrepreneurial projects are homogeneous 2-period investments.Their return depends solely on the realization of exogenous factors at

the final date. The statistical distribution of these random variables can

16



be inferred, at the initial date, on the basis of historical data available toall players.

At the interim date uncertainty is fully resolved and the return onentrepreneurial projects at the final date is common knowledge.

d) Lending technology

The lending technology is available only to banks. It involves grantingliquidity to noninstitutional borrowers at the initial date, in order to

receive the face value S of the loan two periods later.The return on entrepreneurial projects being random, noninstitu-tional borrowers’ ability to repay the loan is stochastic. I assume loansrepayment distributes as ̃s ∼N ( ̄s , σ 2

s) at the final date.

All players share a common prior belief about the realization of thefuture uncertain return ̃s on noninstitutional borrowing.

Banks from region i are able to form a posterior belief ̄si about loansrepayment in force of region-specific skills.

Noninstitutional borrowers’ repayment can be securitized by banksto issue ABSs. Without any loss of generality and to keep the modelas simple as possible, I assume each unit of liquidity lent by banks is

securitized into one unit of ABS.

e) Repo technology

Banks use their ABSs as collateral to leverage in the money marketusing the repo technology. This is a 1-period contract giving banksthe opportunity to borrow liquidity from money market funds againsttheir pool of loans.

In a repo agreements banks sell their ABSs to a money market fundin exchange for cash, promising to repurchase them one period later.

When the bank buys back its securities it has to pay a prearrangedprice.

This amounts to the ABS market value at the sell date, plus an inter-est rate r over the liquidity received in the first place.

Funds set the interest rate as remuneration for the risk they under-take. I assume there exist an ABS price p and an interest rate r on repotransactions such that funds liquidity is sufficient to accommodate ABSsupply.

17



Repo financing being short term, banks may roll over the repo agree-ment at the interim date to obtain further liquidity. However, the newinformation available on the realization of noninstitutional loans re-payment affects the market value of ABS.10 Thus, banks’ ability to raiseliquidity through the repo technology varies accordingly.

Depending on the bank cash stock and the new amount of liquidityavailable through the repo technology at the interim date, a bank canbe unable to fulfill the obligation it has with its counterparty in themoney market.

In case their counterparty is unable to repurchase the collateral, fundsrefuse to roll over the repo and sell the securities at current marketprice, to try and recoup the part of repo repayment that banks liquid-ity does not cover.

Nevertheless, the ABS market price at the repurchase date can beinsufficient for funds to satisfy their VaR constraint.

In order to protect against this market risk, lenders in the moneymarket apply a margin h between the market value of pledged assetsand the amount of liquidity they pay for them.

This “haircut”determines the mapping between a bank’s investmentin the lending technology and its pledgeability.

3.3 Information

Banks screen entrepreneurial projects in order to forecast the expectedrepayment from noninstitutional borrowers. Region-specific skills per-mit banks to update the common prior belief E ( ̃s ) on the basis of infor-

mative signals ˜ f i = s + ε̃i of the actual final repayment s of the lendingtechnology, where ε̃i ∼ N (0, σ 2

ε), s and ε̃i are independent, and errors

are independent across banks.The uninformed expectation ̄s0 = E ( ̃s ) made by money market funds,

who can not observe any regional signal to update the prior belief, may

therefore differ from the informed expectations ̄si = E i ̃s | f i

made bybanks in regions i = 1,..., n.

10Full revelation of the final return at the interim date determines an abrupt adjust-ment in the price of securities backed by the lending technology to its fundamentalvalue. In future versions of the paper I will focus on the adjustment of this marketvalue when information is only progressively revealed to market participants.

18



Banks and depositors in each region share the same signal.11 Regionsare indeed a fiction I make to match investment banks with depositorsthat believe in their assessment of the risky return. Banks with a spe-cific signal on the risky return compete only with other banks sharingthat same signal - i.e. dealing in the same region - for depositors whotrust that piece of information.

The assumption on matching between banks and depositors in a re-gion permits to model the banking sector as competitive and yet toallow banks receiving a relatively bad signal to face positive demand,

although they offer lower future consumption.Since banks in a region are perfectly competitive and share the same

information and technologies, they are all alike and take the same in-vestment decision.

Thus, to simplify notation, I can look at one representative bank forevery region without any loss of generality.

Relying on the heterogeneity in posterior beliefs, I label regionalbanks from the most optimistic bank 1 to the most cautious bank n: ̄s1 > ̄s2 > ... > ̄sn .

3.4 Time Structure

Initial date: Bank i observes the signal f i and forms its posteriorbelief ̄si . It chooses what part λi of its capital to invest in the storagetechnology in order to remunerate depositors at t = 1, and what part(1− λi ) to invest in the lending technology to get the risky return att = 2.

Depositors sharing bank i signal put their unit of liquidity in thebank to receive an amount of liquidity c t

iat dates t = 1,2.

Banks securitize their loans to noninstitutional borrowers and sellthem to money market funds through repo agreements that specify aninterest rate r and a haircut h.

Interim date: Uncertainty over the enterpreneurial project is resolvedand the repayment by noninstitutional borrowers is known to be s .

11In this model a signal can be thought of as a proprietary forecast model ordatabase.

19



Bank i must pay depositors the sum c 1i

it specified in the contract atthe initial date.

Banks must repurchase their securities and may roll over the repotechnology, raising an amount of liquidity that equals the pledgeablevalue of their securitized loans at current market price.

Final date: The entrepreneurial project yields its return and banksreceive from noninstitutional borrowers the repayment s on loans.

Banks must repay their debt and give depositors the sum c 2i

they

committed to at the initial date.For the sake of clarity, the following timeline summarises the time-structure of the model:

t = 0Players form uninformed expectation ̄s

Bank receives private signal f i

Bank offers deposit contract c 1,2i

Customers deposit liquidity

Bank invests into the risky asset S

Bank pledges ABSs in repos

t = 1Risky return s is public

Bank pays c 1i

to customers

Bank repurchases ABSs

Bank rolls over the repo

t = 2

Risky return realizes

Bank pays c i to customers

Bank repurchases the ABSs

3.5 ABS pledgeability in the repo market

The pledgeability of bank’s assets is endogenous to the model and isdetermined by the need for money market funds to limit the VaR of their positions.

Looking at the time series of the entrepreneurial project, funds can

make up the likely scenarios for the ABS price at the repurchase date.This practice is relevant for risk management purposes, in case the bor-rowing bank is unable to repay its debt at the rollover date.

A fund in the money market wants to make sure that a borrowingbank will repay the whole face value of its secured loan, that is therepurchase price, at the exogenous confidence level α.

20



For instance, a fund writes the repo contract to lend liquidity againstAAA ABSs, so as not to suffer any loss with 1−α% probability.

To do so, funds look at the time series of AAA ABS market value andset a haircut h to offset their statistical loss in the α% worst scenario.

Since banks can invest their capital in either risky loans or liquidity,there is a trade-off between a bank leverage and its credit worthiness.

In fact, the ability of a money market fund to recoup the full value of its credit in bad scenarios decreases with the share of capital the bankinvested in the risky technology.

Depending on the exposure q the bank has to noninstitutional bor-rowers, a lender in the money market sets the haircut h on repo suchthat:

λ + q p t (1− v)≥ q p t (1− h)(1 + r ). (1)

Where λ is the amount of a bank’s initial capital invested in the stor-age technology, q is the number of ABS issued by the bank, p t is themarket value of the security, r is the repo rate and v is the percentagedecrease of the ABS price in the α% worst scenario, according to theunconditional probability distribution of ̃s .

On the left hand side of the inequality is the sum of the liquidity λ

the bank decided to hold at the interim date and the market value, inthe worst α% scenario, of the portfolio of assets it pledges.On the right hand side is the face value of repo debt at the interim

date.The solution to inequality (1) gives a value for the minimum hair-

cut h that depends on r , v, p and on the bank’s portfolio allocationbetween storage and lending technologies:

h ≥r + v − λ

q p

1 + r . (2)

Funds face a trade off between protecting themselves from losses byapplying a higher haircut and earning interests on a larger amount of liquidity.

Thus, the optimal haircut lenders can apply is the lowest value al-lowed by inequality (2) - i.e. the same formula with equals sign.

When claims on the 2-period investment are offered as collateral in

21



the repo market each bank can ultimately borrow, at any date t , anamount of liquidity specified by the pledgeability function

B t (λ, r , v, p t ) = (1− h)q p t

=

1−r + v − λ

q p t

1 + r

q p t . (3)

Lemma 1 As the bank risky investment q increases, a lower percentage

of the collateral market value is pledgeable in repo agreements with moneymarket funds. The gap between the two values, the haircut, increases withthe statistical asset depreciation v.

Proof. The derivative of the pledgeable value per unit of market value with respect to the size of the risky investment is

∂ (1− h)

∂ q =−

λ

p t q 2(1 + r )< 0.

The haircut moves with the statistical percentage change of the collateral

market value according to the partial derivative

∂ h

∂ v=

1

1 + r > 0.

Figure 1 shows the shape of the pledgeability function for some val-ues of v, when all other parameters are held constant.

3.6 Investment and contract design

Each competitive bank wants to maximize its future payments to de-positors choosing the optimal portion of capital (1− λ) to be investedin the lending technology.

As the exposure to the risky technology is chosen, the current priceof ABSs and its past volatility, together with the repo rate, determinethrough equality (3) the pledgeable value of the bank portfolio of loans.

22



Thus, a bank leverage is endogenously determined by its expectationon the risky return.

The bank’s budget constraint at the initial date is

q ≤ (1−λ) + q p0(1− h)

where q are the units of liquidity it lends to noninstitutional borrow-ers, (1−λ) is the capital allocated to lending activity, and q p0(1− h) isthe amount of liquidity the bank can borrow by pledging its ABSs at

current market price.The maximum amount of money the bank lends to its noninstitu-tional borrowers - i.e. the number of ABSs it issues - is then

q =1−λ

1− p0(1− h)(4)

The amount of loans a bank can issue on the basis of its portfolioallocation is given, together with the haircut on its repo, by the simul-taneous solution to inequality (2) with equals sign and of equality (4):

h =

r +v− λ

q p0

1+r

q = 1−λ

1− p0(1−h)

Thus, depending on the optimal portfolio allocation (λ, 1− λ), a bankwill issue an amount of financing to noninstitutional investors

q =1 + r (1−λ)

(1 + r )− p0(1− v)(5)

and it is applied by investors a margin on repo transactions

h = p0(1−λ)(r + v) + λ( p0 − 1)

p0(1 + r (1−λ)). (6)

In order for the portfolio allocation to be non trivial for banks, it isnecessary that the money market be concerned with the fulfilment of the VaR constraint.

23



If that was not the case banks would be free to leverage without alimit. They could raise an infinite amount of liquidity to invest in thelending technology, regardless of the share of deposits invested in loans.

Thus, the marginal return the liquidity invested in the risky asset isexpected to yield in the α% worst scenario has to be lower than therepayment to be made for the same unit of liquidity in the money mar-ket.

For this reason, throughout the model it is assumed that

(1 + r ) > p0

(1− v). (7)

This assumption ensures that the market value of pledged collateralmay be insufficient in order for lenders to limit their potential lossesunder the VaR level at the rollover date. Thus, the following resultholds:

Lemma 2 When lenders in repo transactions are concerned with the riskof the borrowing bank being unable to repay its debt, the haircut they applyincreases with the bank investment in the lending technology.Proof. From equation 6,

∂ h

∂ λ=

p0(1− v)− (1 + r )

p0(λr − (1 + r ))2< 0.

Since the value of the haircut is decreasing in the portion of capital de-voted to the riskless technology, it increases with the risky position of the borrower.

Banks optimal decision problem can be solved by backward induc-tion.

Bank i observes a signal f i and chooses to invest a portion (1−λi ) of its deposits in the lending technology at the initial date.

Equality (5) determines the investment q i in the lending technologybank i affords using the leverage available through the repo technology.The bank borrows q i p0(1− hi ) from money market funds and lendsthis external capital, plus its investment (1− λi ), to noninstitutionalborrowers.

24



The bank is left with liquidity λ it carries on to period 1, when therepayment to money market investors is q i p0(1− hi )(1 + r ).

Bank i has a posterior belief on the repayment ̄si = E i ̃s | f i

fromnoninstitutional borrowers. Thus, at the initial date, it chose the opti-mal allocation (λi , 1− λi ) expecting the value of its ABSs to change atthe interim date by

δi = ̄si

p0− 1. (8)

Considering the expected price change, bank i expects to be able toraise from the money market at time 1, when the repo agreement hasto be rolled over, an amount of liquidity

q i p0(1 + δi )(1− hi ).

The investment in the storage technology at initial date, together withthe difference between the liquidity borrowed and repaid to moneymarket funds at date 1, are the liquidity bank i has available to pay itsdepositors the first periodic installment c i .

Thus, bank i enters the final period with an amount of liquidity

λi + q i p0(δi − r )(1− hi )− c i . (9)

At the final date the entrepreneurial project yields its return and thelending technology pays the return that bank i expected at the initialdate to be ̄si .

The bank has to repay an amount q i p0(1+δi )(1−hi )(1+r ) to moneymarket funds, and depositors must be paid their second installment c i .

At the final date bank i has then an amount of liquidity

λi + q i p0(δi − r )(1− hi )−q i p0(1+ δi )(1− hi )(1+ r ) + q i ̄si −2c i (10)

Since bank i wants to maximize the payment to its depositors, no liq-uidity is to be left in the bank at the end of the contract. Furthermore,money market funds must take their optimal decision and all playerssatisfy their solvency constraints in expected terms.

25



Thus, bank i decision problem at the initial date is:

maxλi

: c i =λi + q i p0(δi − r )(1− hi ) + q i ( ̄si − p0(1 + δi )(1− hi )(1 + r ))

2

(11)

s .t . c i≤ λi + q i p0(δi − r )(1− hi ) (12)

hi = p0(1−λi )(r + v) + λi ( p0 − 1)

p0(1 + r (1−λi ))

(13)

q i =1 + r (1−λi )

(1 + r )− p0(1− v)(14)

Equation (11) stems from the condition that at the end of the finalperiod the bank must not hold any spare liquidity. Inequality (12) is thebank budget constraint derived from (9) and ensures that, at the end of period t = 1, the bank has sufficient liquidity to pay its depositors.

Constraints (13) and (14) come from the optimization problem of money market investors (6) and from bank’s leveraging at the initialdate (5) respectively.

The optimal portfolio allocation is trivial when solvency constraint(12) is not binding.

If this is the case, the optimization problem has a bang-bang solution.The optimal portfolio allocation sees bank i investing all its deposits

in the risky lending technology - i.e. λi = 0 - when its a posteriorbelief ̄si is high enough to offset the costs of collateralized borrowingstemming from r and v.

If instead the bank expects the loans repayment to be relatively lowit invests everything in cash, and λi = 1.

If the solvency constraint is binding, the optimal portfolio allocationis given by

λ∗i

= p0((1 + r )( p0(1− v) + ̄si v)− ̄si (1− v))

p0( p0 r − ̄si )(1− v) + ̄si ((1 + r )(1 + p0v)− ( p0 − 1)).(15)

Lemma 3 Banks whose expectation on ̄si is more biased towards opti-mistic results choose higher exposure to the risky investment. They commit

26



to higher periodical payments to their depositors.

Proof. Bank i optimal investment in the storage technology is lower, the higher is the expected return on the risky investment:

∂ λ∗i

∂ ̄si

=( p0)2(2 + r )(1− v)((1 + r )− p0(1− v))

(( p0)2 r + 2 s1 + 2 p0 si + r si − ( p0)2 r v + 2 p0 si v + p0 si r v)2< 0

where

(1 + r )− p0(1− v) > 0

under the assumption, made in (7), that lenders be concerned with the repayment of their financing.

Thus, for high expected values of the risky return, a bigger portion of the bank’s capital is devoted to the risky asset.

The amount of liquidity c i bank i commits to pay its depositors at eachdate is then given by (11), once the optimal value λ∗

iis determined. The

periodic payment bank i commits to pay its depositors increases with the optimal risky investment:

∂ c ∗i

∂ λi=

r (( p0)2(1− v)− si (1 + p0v))

2 p0((1 + r )− p0(1− v)) < 0.

Since the optimal risky investment increases with its expected return, bankswith higher expectation ̄si commit to higher periodic payments to their de-

positors.

4 Effect of systemic shock

At date t = 1 the systemic shock is realised and the risky return s is

common knowledge.If s < ̄s1, the return of the risky asset is lower than its most optimistic

expected value at t = 0, and one or more optimistic banks are forced toviolate their contracts.

27



Banks j is unable to abide by the contract whenever ̄s j > s . If this is

the case, in fact:

c j =λ j + q j p

0(δ j − r )(1− h j ) + q j ( ̄s j − p0(1 + δ j )(1− h j )(1 + r ))

2

>λ j + q j p

0(δ s − r )(1− h j ) + q j ( s − p0(1 + δ s )(1− h j )(1 + r ))

2= c s

j

whereδ s =

s

p0− 1 < δ j

is the change in ABS price due to the interim information on its finalreturn.

The value for c s j

is therefore higher than the liquidity bank j has

available after the shock hit.The unexpected gain G j bank j realizes is the liquidity it has in excess

once depositors’ claims are fulfilled:

G j =c s j − c j =(2 s j − p0 r (1− v))( s − ̄s j )

p0( p0 r − s j )(1− v) + s j (1− p0 + (1 + r )(1 + p0v))(16)

Although the issue is not addressed in the present paper, the bias of banks evaluation can well depend on their governance: financial in-stitution investing on the basis of higher expectations about the riskyinvestment enjoy greater competitiveness in the banking industry butbear more risk.

Therefore, capital structure, managerial incentives and banks riskmanagement play a role in determining what scenario theeconomy-wide banking industry faces.

Assuming for the sake of simplicity that the economy is divided inonly two regions, three different scenarios can arise.The relevant scenario depends on the bias each bank had towards

more optimistic or pessimistic values.

28



4.1 Scenario 1: considerate banking sector

In the first scenario, all banks made a correct or pessimistic estimate of s . Thus, at the initial date they were able to keep a sufficient amountof liquidity to escape bankruptcy afterwards:

c s1≥ c 1, c s

2≥ c 2

4.2 Scenario 2: inconsiderate banking sector

The second scenario arises if at date t = 0 both banks had incentive tooverexpose their portfolio to the risky investment.

c s1

< c 1, c s2

< c 2

The whole system goes bankrupt, and there is no way to collect amongbanks the sum needed to face the systemic impact of the shock. In thisscenario, public intervention is necessary to avoid the whole bankingsector from shutting down.

However, every single bank goes bankrupt in this scenario as a con-sequence of its wrong investment in the risky technology. There is nocontagion channel involved in the crisis.

4.3 Scenario 3: idiosyncratic effect of systemic shock

In the third scenario, the difference among banks ex-post beliefs allowsthe shock to produce an idiosyncratic impact. The optimistic bank 1is insolvent, whereas bank’s 2 bias towards a more cautious evaluationallows it to face the shock without going bankrupt.

This is the scenario the model is focused on: the idiosyncratic impact

of a systemic shock whereby

c s1

< c 1, c s2≥ c 2.

The reason why this paper focuses on scenario 3 rather than scenario 2is twofold.

29



Primarily, the considerate - or pessimistic - bank correctly evaluatedthe return on risky technology. Yet, as I shall show, it can be forced tobankruptcy by the overvaluation made by the other bank.

Secondly, the banking industry may have enough liquidity to copewith the crisis, in scenario 3, without need for intervention by the gov-ernment. This possibility is briefly investigated in Section 6 for its pol-icy implications.

Under scenario 3 only the optimistic bank is insolvent. Bank 2 hav-ing set up lower-profile deposit contracts, the effect of the shock on its

ability to pay is an excess of liquidity available at the final date, whenthe return s - for bank 2 unexpectedly high - is paid by noninstitutionallenders.

Yet, the pledgeable value of bank 2 assets depends on the price theABS is traded for.

If the interim information on the risky return is good enough forlenders of the insolvent banks to recoup the face value of their cred-its at the new ABS fundamental value, the amount of liquidity that isavailable for lending does not vary in the model. The collateral in repoagreement is priced efficiently.

Solvent banks can roll over their debt and fulfill their commitment

until loans are repaid at the final date.However, when the interim information is negative - i.e. s < p0 -

lenders can breach their VAR limit. If this is the case, a lender "breaksthe buck" and can not participate to the money market anymore.

Thus, if the new information on the liquidation value of the riskyasset is such that

δ s <−v,

the liquidity that is available in the money market decreases becauselenders of insolvent banks face losses above their VaR limit. This re-sults in "cash-in-the-market" pricing for the securities pledged in repo

agreement.

12

The issuing of ABSs being chosen by banks at the initial date, thenumber of traded securities at the interim date does not vary. The total

12For a comprehensive treatment of cash-in-the-market pricing see Allen and Gale(1994).

30



supply of the pledgeable asset is

Q =n

i=1

q i

Nevertheless, the liquidity that is available in the money market totrade the risky asset decreases because some investors left the moneymarket.

The available liquidity L amounts to the repayment funds received

from solvent banks at the end of the first repo:

L = p0n

i=k+1

q i

where k is the number of insolvent banks.Only lenders of the n − k less optimistic banks in the industry sur-

vived in the money market.Thus, the maximum price ˆ p money market investors are able to pay

for the pledgeable asset at the interim date is

̂p = LQ

= p0n

i=k+1 q ini=1

q i(17)

that is lower than the asset fundamental value if

ki=1

q i >

p0 − s n

i=k+1

q i .

The higher the risky investment of the k overoptimistic banks is, themore likely the price of the pledgeable asset is to be below its funda-mental value at the interim date.

Lemma 4 The difference between fundamental value of the risky asset and its price at the interim date increases with the position held by overop-timistic banks, in proportion to the total supply.

31



Proof. From (17), the underpricing at the interim date is

̂p − s

s=

p0

s

ni=k+1

q ini=1

q i− 1

=

p0

s− 1

ni=k+1

q i

Q−

ki=1

q i

Q(18)

Hence, the higher is the holding of the k inconsiderate banks relatively to

the total quantity of risky assets issued at the initial date, the higher is the difference between the fundamental value of the asset and its cash-in-the-market price.

Corollary 5 The pessimistic misevaluation made by solvent banks at the initial date limits the depreciation of the asset at the interim date.

This result is in line with the too-big-too-fail doctrine: the impact of fire sales depending on the amount of assets held by insolvent banksrelatively to the whole supply, the size of the insolvent institution iscritical in determining the impact of the misevaluation on the bankingsystem.

More optimistic banks are precisely those holding bigger quantitiesof the pledgeable asset. Thus, the event of a sudden liquidation of theirwhole portfolio of risky assets is likely to put their market value underpressure.

This calls a solvent bank for strategic concerns.On the one hand, if assets of the optimistic banks are liquidated at

fire-sale price, the pessimistic banks has the opportunity to buy theassets for a lower price p1.

On the other hand, by paying a very low price, the solvent banksuffers a decrease in the value it is able to pledge to get liquidity in themoney market.

Thus, as I shall show in the following section, the considerate bankcan go bankrupt even though the fundamental value of the investmentis greater than it expected at the initial date.

32



5 Contagion

I start by showing a benchmark case, labeled L, where the amount of liquidity banks can borrow by pledging their assets does not changeover time. Instead of relying on short-term funding, banks are forcedto sign at the initial date long-term repo agreements.

In this case, banks only “bet”on the final return of the risky invest-ment.

The repo market for liquidity here is not a way for banks to increase

the return to their investors, relying on a future increase of collateralmarket value. Banks only look at the repayment from the lending tech-nology at the final date.

Hence, bank i ’s objective function doesn’t take into account the ef-fect of asset appreciation (depreciation) on its ability to borrow. Theportfolio allocation problem is the same described by equations 11-14,now with δi = 0.

This yields a result for the optimal liquidity

λ Li

= p0(1− v)− ̄si

2 p0(1− v)− (2 + r + r ̄ si ),

together with an investment in the lending technology and issuing of ABSs

q L =2 + r

2 p0(1− v)− (2 + r + r ̄ si ).

The gain to the bank at the end of the second period is then

G Li

=(2 + r )( ̄si − s)

2 p0(1− v)− (2 + r + r ̄ si )(19)

= q H i

( ̄si − s ).

Noninstitutional lending being nonnegative, bank i ’s gain is positivewhenever its initial expectation on the risky return is lower than itsrealization.

If banks are given the opportunity to trade bank 1 assets at cash-in-

33



the-market price p1 < p0, bank gain becomes

G L,T i

=(2 + r )( ̄si − s )

2 p0(1− v)− (2 + r + r ̄ si )+

(2 + r )( s − p1)

2[(1 + r )− p0(1− v)] + r ( ̄s−i − 1)(20)

where G L,T i

labels the ex post gain in the benchmark case with trade,

and ̄s−i is the optimistic bank expectation on the risky return.

Proposition 1 Without interim rollover, the repo market does not act as

channel for contagion from inconsiderate banks to other borrowers in the repo market.

Moreover, the fact that misevaluating banks go bankrupt and their portfolios are liquidated is always beneficial to solvent banks.Proof. The additional gain is given by the difference between equalities(20) and (19):

∆G L =(2 + r )( s − p1)

2[(1 + r )− p0(1− v)] + r ( ̄s−i − 1).

The denominator is positive under the assumption made in equation (7)

noticing that for ̄s−i < 1 a bank would not invest in the risky technology.Thus, solvent banks gain from insolvencies resulting in the market underpricing ABSs at the interim date.

If the pledgeability of an asset is not affected by underpricing at theinterim date, the fact that a distressed institution suddenly offers itswhole portfolio of assets is beneficial to banks that took their invest-ment decision over a more considerate evaluation.

The result is in line with those by Acharya, Gale and Yorulmazer(2010) and Carlin, Lobo and Viswanathan (2007). Banks have incentiveto act as vultures and hoard liquidity with the expectation of using it inthe near future to buy assets with positive return at fire sale price.

When the market prices the pledgeable asset on the basis of its ex-pected return, there is no contagion in the banking sector even thoughbanks have to roll over their debt at the interim date. On the contrary,considerate banks can raise more liquidity than they expected at thetime of their portfolio allocation.

34



Banks with a pessimistic expectation on the return end up at theinterim date with some spare liquidity given by equation (16).

On top of that, they have the funding needed to buy some assetspreviously owned by the insolvent optimistic banks.

As long as the asset is priced at its fundamental value - that is publicinformation at the interim date - solvent banks pay an ABS exactlywhat they are going to receive from it at the final period. Thus, theydo not gain nor lose anything from trading.

However, if the market prices efficiently the asset, there is no room

for predatory liquidity hoarding in the first place.Under the specification of this model a contagion channel is at work

when banks roll over their debt at the interim date. When the marketprices pledgeable assets inefficiently, banks with a correct belief on therisky return go bankrupt.

The effect of asset depreciation on the liquidity of solvent banks chal-lenges previous results on liquidity hoarding and changes the incentivesof financial institutions in the midst of a crisis.

If at the interim date, following bankruptcy of the optimistic bank,the market price falls to p1 < ¯ s2, the final amount of liquidity bank 2has available is higher than needed to honour its commitments.

Under cash-in-the-market pricing, labeled AG, this yields consideratebank 2 a final unexpected amount of spare liquidity

G AG2

=c AG2−c 2 =

r ( p0(1− v) + ¯ s2)( ¯ s2 − p1) + ¯ s2(2 + r )( s − ¯ s2)

p0( ¯ s2 − p0 r )(1− v) + ¯ s2( p0 − 1)− (1 + r ) ¯ s2(1 + p0v))(21)

that is always positive.Pessimistic banks have in fact the opportunity to buy a remunerative

asset at low price and, having less pledgeable value at the interim date,need pay a lower amount of interests in the following period.

Nevertheless bank 2 may fail at the interim date because it is unable

to raise the liquidity needed to remunerate its investors and roll overthe first repo.

Proposition 2 If fire sales lower the price of pledgeable assets below the liquidation value expected by solvent banks, the latter go bankrupt.

35



Proof. From equation (12), at the interim date

c 2 = λ2 + q 2 p0(δ2 − r )(1− h2)

whilst the liquidity available to the bank at that date is

L12

= λ2 + q 2 p0(δ AG − r )(1− h2)

where

δ AG =

p1

p0 − 1

is the price change of the pledgeable asset in case of cash-in-the-market pric-ing. The liquidity available to the bank is not sufficient to cover its needs -that is, L1

2< c 2 - whenever

δ F S < δ2. (22)

Thus, from equations 8 and 22, the amount of liquidity c 12

the considerate bank has available at t = 1 is lower than needed to fulfill depositors’ and lenders’ claims whenever p1 < ¯ s2

The condition that considerate banks had an expectation on the totalreturn of the risky asset higher than its cash-in-the-market price at theinterim date is sufficient for the liquidity crisis to unleash.

This happens notwithstanding the fundamental value of consideratebanks portfolio is greater than necessary to ensure their solvency.

The insolvency of misevaluating banks has thus the potential to spreadas liquidity crisis endangering otherwise solvent institutions.

6 Policy implications

Section 5 showed that an institution can be driven to bankruptcy re-

gardless of the fundamental value of its portfolio, if the pledgeable assetdepreciates sufficiently to be lower than the expectation of consideratebanks on its final return.

The most optimistic bank fails as a consequence of the misevaluationit made the optimal risky investment at the initial date. The trouble of the first failing bank may spread to other institutions in the banking

36



sector because of indirect linkages through the repo market for liquid-ity.

The fall in the market value of banks portfolio is the only conta-gion channel in the model. A government can prevent the mistakeof some banks from becoming a systemic liquidity crisis by inflatingbanks pledgeable value. It can do so either by accepting the asset in liq-uidation as collateral for public lending, or by keeping its market priceto a value that is compatible with the liquidity needs of solvent banks.

However, the government issuing financing to private financial insti-

tution is a source of moral hazard for the industry and can come at acost.

If the government does not intervene, banks are left to fight aloneagainst the contagion. In the paper by Acharya, Gromb and Yorul-mazer (2009) the outcome is deadly for the banking industry. Banksworsen the crisis by strategically providing insufficient liquidity to in-stitutions in needs, in order to induce fire sales and buy cheap assets.

In the model of originate-to-distribute banking developed here, theoutcome is far more desirable, because banks are hurt by the default of a competitor.

In what follows, I perform a preliminary analysis of the policy im-

plications resulting from asset-side contagion through the repo market.This exercise is done neglecting the issue of coordination within n > 2banks and the comparison with different public intervention strategies.Both topics are left to future research.

Proposition 3 Solvent banks have the opportunity to bail out themselvesby paying the asset of insolvent banks an inflated price b > p1 when

q 2(1− h) ¯ s2

(1− h)(q 2 + q 1)− q 1≤

(q 2 + q 1) s − h(1 + r )q 2 ¯ s2

(1− h)(1 + r )(q 2 + q 1), (23)

Proof. In order to avoid contagion, solvent banks must clear the market for the pledgeable asset at a bailout price b that allows themselves to bor-row, against their collateral, the amount of liquidity needed to satisfy the

solvency constraints.

37



The solvency constraint with the bailout price at the interim date is

λ −q 2(1− h) p0(1 + r ) + q 2(1− h) ¯ s2

≤ λ −q 2(1− h) p0(1 + r ) + (q 2 + q 1)(1− h)b − q 1b (24)

On the left hand side of the inequality is the amount of liquidity considerate banks need to hold after the bailout, in order to fulfill the commitmentsthey signed at the initial date.

On the right hand side is the liquidity they will be holding after the

bailout, given a new market price b for the pledgeable asset.This gives a condition on the bailout price

b ≥q 2(1− h) ¯ s2

(1− h)(q 2 + q 1)− q 1.

Thus, the bailout price must be high enough for the increase in pledgeable value of the initial portfolio to offset the cost of the new purchase.

However, a high pledgeable value at the interim date translates into ahigh interest to be paid at the final date. The solvency constraint at the

final date is

q 2 ¯ s2 − (1 + r )(1− h)q 2 ¯ s2 ≤ (q 2 + q 1) s − (1 + r )(1− h)(q 2 + q 1)b .

Once more, on the left hand side of the inequality is the amount of liquid-ity considerate banks need to hold after the bailout in order to fulfill the commitments they signed at the initial date.

On the right hand side is the liquidity they will be holding after the bailout. In force of the realization of the risky return this would be higher than expected.

However, the bank has to pay additional interests for the liquidity bor-rowed at the interim date to bail out inconsiderate banks.

To satisfy the solvency constraint at the final date, the bailout price res-

cuing banks can afford at the interim date is

b ≤(q 2 + q 1) s − h(1 + r )q 2 ¯ s2

(1− h)(1 + r )(q 2 + q 1)

38



As mentioned before, this result contrasts with the findings of Acharyaet al. (2008) and Carlin et al. (2007). The threat of contagion throughthe repo market prevents banks from adopting vulture-like strategies.

Beside the potential for self-bailout found under the feasibility con-dition (23) in Scenario 3, this result makes public intervention cheaper.

Out of the feasibility condition specified by Proposition 3 in fact, thepolicy implication of the model is more general: when the governmentcommits not to inject in the banking industry an amount of liquidityhigher than what institutions need to stop the contagion, the direct

threat a systemic crisis constitutes to solvent institutions induces thelatter to carry a partial public intervention.

Corollary 6 When condition (23) is not satisfied, banks have still incen-tive to join the government in an effort to prevent the contagion of insol-vent banks bankruptcy to the banking sector.

Contagion is thus avoided with the government committing to exert the lowest effort compatible with the industry succeeding to prevent a liquiditycrisis.

Corollary 6 ensures that even in scenario 2, where there is not enough

spare liquidity in the banking sector to remedy the wrong portfolioallocation chosen by misevaluating institutions, solvent banks do nothoard the liquidity injected by the government to sustain ABS price.

7 Numeric example

Consider an economy where agents need to borrow money in order tobuy a home.

Each borrower needs one unit of liquidity and are able to repay itonly two periods later.

Banks can issue the loan and simultaneously borrow liquidity in re-

purchase agreements, credits with the home buyers being the securi-tized collateral for the transaction.

One period later, public data on mortgage delinquency at the finalperiod are being released.

The time series of the market value for the relevant class of HomeEquity ABS shows that in the 1% worst scenario the security had lost

39



4.8631% of its value during one period.13

Financial markets currently price the ABS 1.021. Thus, under theassumption of no discounting, the market expected return from themortgage is 2.1% after the two periods. The repo rate for one period is0.08%.14

Assume there are two regions in the economy, each with one bankand a continuum of home buyers.

Bank in region 1 is optimistic about the return on mortgages. Ac-cording to its forecast, the vast majority of borrowers will be able to

repay their debt at the final date.Bank 1 expects the data on delinquency being released the following

period to confirm its belief that the actual return on loans will be 2.5%.Bank 2 is less optimistic. According to its forecast model on mortgagedelinquency, the return of the loans will be only 1.5%.

The optimal portfolio allocation for Bank 1 is to invest the 21% of its capital in loans to home buyers. It faces a haircut of 2.65% and isable to promise its depositors two periodic payments of 0.89 each - i.e.a 79% return.

Bank 2 being less optimistic on the realization of the risky return,it only invests the 5% of its portfolio in the risky asset and promises

its depositors a periodic payment of 0.72 that corresponds to a 44%return.

One period later, the public report shows that the return on loanswill be 2%.

The contract bank 1 proposed to its investors is unfeasible. In pe-riod 2 the bank has only 0.69 units of liquidity to pay them. Thus,bank 1 will be insolvent at time 2 and lenders in the money market areunwilling to roll over its debt at the interim period.

The outcome is much different for bank 2. New information showsthat it has a surplus of liquidity at the final period. The risky returnbeing now known to be 2%, the present value of its assets is more than

sufficient to roll over its debt.Moreover, the high final return will leave the bank with some spare

liquidity available after all commitments are fulfilled (Proposition 1).

13Weekly data on Barclays ABS Home Equity for the period Jan 2000-Oct 200714According to Gorton and Metrick (2009), in the first half of 2007 AAA asset-

backed securities traded below LIBOR.

40



However, the pledgeable value of collateral depending on its marketvalue, bank 2 ability to roll over its debt is endangered by the lossesbank 1 determined to money market investors.

Since in this example there are only two banks, the default by one in-stitution has the effect of decreasing the market price of the pledgeableasset by a great extent.

According to the model, the cash-in-the-market pricing of the ABSdetermines a new market value of 0.4992.

The pledgeable value of its collateral being below its initial expecta-

tion, bank 2 lacks the liquidity necessary to repay the debt taken out atthe initial date.

This happens notwithstanding the f undamental value of its invest-ment is higher than needed for its solvency (Proposition 2).

The only way bank 2 has to avoid defaulting is to buy bank 1 assetsfor an artificially high price.

With the values posited in this numerical example a self-bailout is notfeasible. However, it is sufficient that the information available at theinterim date on mortgage delinquency determined a return on loans of 2.2% to change this prediction.

Bank 2 can then pay bank 1 ABSs a price b ≥ 1.0382, so as to have

the liquidity necessary at time 1 to repay the initial debt. At the sametime, this bailout price has to be b ≤ 1.0438 in order for the bank to beable to repay its commitment at the final date (Proposition 3).

With the values assumed in the new example, the outcome of themodel is overly reassuring. Although the insolvency of some banks canlead bankrupt otherwise solvent banks, the latter have the opportunityto bail out themselves by buying the assets being liquidated.

With a the original set of values the feasibility condition in Proposi-tion 3 is not satisfied, but the result is still hopeful for the stability of the economy.

Bank 2 has not enough liquidity available to clear the market for the

ABS at a sufficiently high price at the interim date to avoid bankruptcy,thus public intervention is necessary to prevent a liquidity crisis to un-fold.

However, since solvent banks have a direct interest in clearing themarket at a sufficiently high price, under the results of the model publicintervention comes to a lower cost than expected otherwise.

41



In fact, the government has the opportunity to prevent the conta-gion by buying only part of the assets being liquidated by the insolventbank.

Bank 2 has then incentive to participate to the bailout with all itsspare liquidity, buying at the same bailout price the remaining assets toavoid bankruptcy.

The numeric example was deliberately designed to show the resultsof the model when those are less likely to have effect. In fact, the cir-cumstance that more than half of the whole capital of the banking in-

dustry was invested in mispriced assets is very unlikely to arise.When the overoptimistic investment is confined to a smaller portion

of the capital of the banking sector, the contagion channel is still inplace.

However, the amount of private spare liquidity held by banks willingto help the government is greater than in the example shown above.

This translates into the government having to inject less liquidity tosustain the price of pledgeable assets and stop the contagion.

8 Concluding remarks

Far from solving the debate on what allowed the shock in the US sub-prime mortgage delinquency rate in 2007 to spread worldwide withsuch a terrific effect, this paper shades a new light on the possibilityfor the mistake of some institutions to precipitate a systemic liquiditycrisis.

Neglecting the issue of early diers, I focused on the misevaluationof opaque financial derivatives as a source of the crisis. Since limitedconvertibility at the interim date is optimal to depositors, there are nobank runs in the model.

Moreover, because the model rules out crossed claims among banks,there is no room for domino-contagion.15

The main result of the paper is that a market providing liquidityagainst collateral may act as a channel for contagion.

15Banks are only indirectly linked one to each other through the repo market. Thus,differently from what happens in domino models, a bank is unable to protect fromcontagion by mean of an appropriate choice of its counterparty.

42



The composite effect of opaque assets and short term collateralizedborrowing is able to propagate the negative effect of asset mispricingby any overoptimistic institution to the whole banking sector.

This new channel adds to the domino effect and to the change inhaircut that are well known in the previous literature.

Differently from what found in previous papers on predatory liquid-ity hoarding, banks have no incentive to adopt vulture-like strategies.

The existence of a market for collateralized borrowing introduces astrategic interdependence that limits the advantage solvent banks can

gain from the bankruptcy of their competitors.When banks rely on the repo market for much of their liquidity

needs, the additional return they get from cheap collateral is offset bythe amount of liquidity they must give up on the money market.

The decrease in asset value can thus make solvent banks illiquid, andlead them bankrupt because of their inability to meet the solvency con-straint.

Three stylised facts about the Subprime crisis are consistent with thepresent model: banks-SPVs relied on collateralized borrowing againstopaque assets to increase their leverage, a negative shock on mortgage-backed securities return reduced the value of those securities, and bank-

ing institutions had difficulties to fund their periodical payments.Rather than assuming a surge in haircut to give account of the after-

math of the 2007-2009 crisis, the value of that margin is derived endoge-nously in the model. This paves the way to a more general analysis of liquidity funding in the banking sector.

43



Figure 1: Pledgeability function.

44



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