Essays in banking regulation [Elektronische Ressource] / vorgelegt von Maryam Kazemi Manesh

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Essays in Banking RegulationInauguraldissertationzur Erlangung des akademischen Gradeseines Doktors der Wirtschaftswissenschaftender Universit at Mannheimvorgelegt vonMaryam Kazemi ManeshJanuar 2011Dekan: Prof. Dr. Martin PeitzReferent: Prof. Dr. Ernst-Ludwig von ThaddenKorreferent: Prof. Dr. Ernst MaugTag der mundlic hen Prufung: 18. Februar 2011iiAcknowledgmentsAbove all, I would like to thank my supervisor Ernst-Ludwig von Thadden for theexcellent guidance and for the encouraging feedback he gave me on both the content ofmy research and the way of presenting it. His tight schedule notwithstanding, he wasaccessible and extremely supportive in all aspects related to the process of growing intoacademia.I would also like to thank my second supervisor Wolfgang Buehler for his insightfulcomments on the rst essay. I thank him for his kind availability and valuable supportsat the time I had just began my rst steps to writing this thesis.I very much enjoyed interacting with my colleagues at the Center for Doctoral Studiesin Economics. I would like to thank Michal Kowalik for his instructive comments on the rst and the second chapters of this thesis and the inspiring discussions about bankingtopics. I am especially grateful to Jennifer Abel-Koch and Edgar Vogel who not only ascolleagues helped for the study and writing the thesis but also as the best friends tookcare of me during hard times of settling down in my second home, Germany.

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• Publié par: universitat_mannheim
• Publié le: 01 janvier 2011
• Nombre de lectures: 30
• Langue: Deutsch
• Poids de l'ouvrage: 1 Mo

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Dr.

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Martin

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Ernst-Ludwig

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2011

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Aboveall,IwouldliketothankmysupervisorErnst-LudwigvonThaddenforthe
excellentguidanceandfortheencouragingfeedbackhegavemeonboththecontentof
myresearchandthewayofpresentingit.Histightschedulenotwithstanding,hewas
accessibleandextremelysupportiveinallaspectsrelatedtotheprocessofgrowinginto
academia.

IwouldalsoliketothankmysecondsupervisorWolfgangBuehlerforhisinsightful
commentsonthefirstessay.Ithankhimforhiskindavailabilityandvaluablesupports
atthetimeIhadjustbeganmyfirststepstowritingthisthesis.

IverymuchenjoyedinteractingwithmycolleaguesattheCenterforDoctoralStudies
inEconomics.IwouldliketothankMichalKowalikforhisinstructivecommentsonthe
firstandthesecondchaptersofthisthesisandtheinspiringdiscussionsaboutbanking
topics.IamespeciallygratefultoJenniferAbel-KochandEdgarVogelwhonotonlyas
colleagueshelpedforthestudyandwritingthethesisbutalsoasthebestfriendstook
careofmeduringhardtimesofsettlingdowninmysecondhome,Germany.Also,I
thankBj¨ornSaßforthegreatcommentsoneditingthethirdchapter.MarionLehnert
andHelgaGebauergreatlysimplifiedmylifebytakingcareofalltheadministrative
duties.

ManyothercolleaguesandfriendshaveashareinthepleasanttimesIspentinMannheim,
especiallyHeikoKarle,SebastianK¨ohne,LisandraFlach,PetraLoerkeandAlessandra
Donini,aswellasMoritzKuhn,ChristophRotheandXiaojianZhaoforourmemorable
momentsofthefirstyearscourseworks.Iamalsoindebtedtoallmycheeringfriends
outsideuniversityforwhomIdidnotalwayshavethetimetheydeserved.

Mywarmestthanksandlovegotomyfamily,whoalwaysinspiredandencouragedme
fromthefarhomelandandborewithmyabsencetheseyears,andtoHenningforbeing
thereformeandforallthehappymomentsinthepastbusymonths.

iii

tstenCon

ductiontroIn1

2ASwitchingModelinBanking
2.1Introduction..................................
2.2TheModel...................................
2.2.1NoAssetSubstitution........................
2.2.2ComparisonoftheTwoRegimes..................
2.3SwitchingStrategiesinaCrossingCase...................
2.3.1TheSwitchingModel.........................
2.3.2TheOptimalStopping-SwitchingModel..............
2.4QuantificationoftheOptimalStrategies..................
2.4.1TwoAlternativeCases........................
2.4.1.1CostlessSwitching.....................
2.4.1.2TooCostlySwitching....................
2.5NumericalExamples.............................
2.5.1GeneralCase:TwoSwitches.....................
2.5.2ACostlessSwitch...........................
2.5.3TooCostlySwitch..........................
2.6Conclusion...................................
Appendix......................................
Figures........................................

3SwitchingModelsforBanking:IsGamblingforResurrectionValid?
3.1Introduction..................................

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55810131515172123232424522526262832

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CONTENTS

3.2TheOne-PeriodModelwithDiscreteReturn................
3.3TheTwo-PeriodModelwithDiscreteReturn................
3.3.1NoRiskofInsolvency.........................
3.3.2OperatingunderRiskofBankruptcy................
3.4NumericalExamples.............................
3.4.1BankruptcyatFailure........................
3.4.2SolvencyatFailure..........................
3.5EndogenousReinvestment..........................
3.6Conclusion...................................
Appendix......................................

TheTheoriesofBankRegulationandSystemicFailures
4.1Introduction..................................
4.2TheBasicModel:FailuresinBanks.....................
4.2.1TheMoralHazard..........................
4.2.2DepositInsuranceandtheMoralHazard..............
4.2.3IntroducingtheRegulatoryActions.................
4.3TheResolutionofIndividualBankFailures.................
4.3.1TheTime-InconsistencyProblem..................
4.3.2OptimalLiquidityProvision.....................
4.3.3TakeoverasanIncentivesForRiskReduction...........
4.4RegulatingSystemicRisk..........................
4.4.1ContagionandTooMuchRelatedBanks..............
4.4.2TooManyToFail...........................
4.5Macro-PrudentialRegulationPolicies....................
4.5.1DealingwithTMTF.........................
4.5.2TooRelatedToFailandCapitalAdequacy.............
4.5.3TBTFandSystemicTaxing.....................
4.5.4Market-BasedSystemandOtherAlternatives...........
4.6RegulatoryPoliciesintheRecentCrisis...................
4.7Conclusion...................................

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1Chapter

Inductiontro

Bankingregulationhasdrownspecialattentionofpolicymakers,institutionsandthe
economistssinceyears.Theintegrationofbanksandfinancialmarketsmadefinancial
stabilityextremelydependentonthebankingsystems.Thesystemiccrisesinthepast
decadesmotivatedformsofregulationwhicharemorecloselyassociatedwithprevention
thanwithcompensation.Theneedtoincreasestabilityandtohelpprotecttheinter-
nationalfinancialsystemledtotheBankforInternationalSettlements(BIS)Accordin
1988andtheUnitedStatesofAmerica’snewFederalDepositInsuranceCorporation
ImprovementActof1991.Thesuccessoraccord,BaselII,introducedin2004,aimedto
improveupontheriskinsensitivityoftheBaselI.However,therecentcrisisraisednew
challengesforbankingregulationinadditiontowhathadbeenconsideredpreviously.
Thebankingtheoryandinparticularthetheoriesofbankingregulationhavefocused
ontheriskinessofbanks’portfolio.Therisk-takingbehaviorofbanksanddifferent
situationsofmoralhazardproblemhavebeenthemainconcentration.Thisthesisisa
collectionofthreeessayswhichapplyadifferentperspectivetoquestionsintheliterature
ofbankingregulation.Chapterstwoandthreeanalyzetherisk-takingbehaviorofabank
whichcanchoosebetweentworegimesofoperation.Chapterfouraddstheregulatorinto
thesetupandsurveysthebankingtheoriesforregulatingnotonlyanindividualbank
butalsosystemicrisk-takingamongbanks.Inthefirsttwochapters,twoportfolios
differentintheirexpectedreturnandtherisklevelareavailabletothebank.When
thebankchangestheportfolioitiscalledregimeswitching.Bothchaptersstudythe
bank’srisk-returnpreferencesintheabsenceofoutsiderintervention.Whileinchapter
twotherelationshipbetweenthecash-flowandtheregimechoiceisinvestigatedin
acontinuoustimesetup,inchapterthreetheanalysisiscarriedoutadiscretetime
setup.Theregimechoiceisexaminedwithrespecttothecapitallevelinstaticandalso
dynamicsetups.Moreover,chapterthreequestionsthestandardtheoryof“gamblingfor
resurrection”.Chapterfourcoversregulatorypoliciestocontrolabank’sriskoffailure

1

2

CHAPTER1.ODUCTIONINTRandcollectsmacro-prudentialregulatoryproposalsfordifferentrisk-takingissuesina
system.bankingTheliteraturegenerallydefinesabank-regulatorgame.Thebankoptimizesitsequity
value.Theregulatorplaysasasocialplanerandoptimizesthesocialvalueofthebank,
includingboththeequityvalueanddepositsvalue.Thekeyfeatureisthattheregulator’s
decisionneedtobeincentivecompatiblefortheequityholders.Higherriskcanincrease
theequityvalueofthebankunderdistress.However,theriskyoperatingofthebank
mayhavenegativenetpresentvalue.Astrongregulatorforcesbank-closurebeforethe
bank’snetpresentvaluebecomesnegative.Incasethattheregulatorcancommittoa
policy,sheplaysfirstandannouncestheregulatorypoliciesandtheclosurethreshold.
Giventheclosurethreshold,thebankmakesthedecisionofcapitalstructure.The
bankingregulationtheoriessolvethegamebybackwardinductiontofindtheoptimal
olicies.pregulatoryIncontrasttomostofotherstudies,thisresearchallowsforregimeswitchingduring
abank’slifetime.Anagentchangesitsinvestmentportfoliodependingonpreferences
forriskandreturn,andthecash-flowoftheinvestment.Chaptertwoanalyzesthis
switchingbehaviorforabankoptimizingequityvalueonbehalfofitsshareholders.
Havingdeposits,thebankcanchooseoneofthetworegimesofoperationineachmoment
oftime.Ifthecash-flowisbelowthedepositpayment,thebankhastoinjectmoneyor
gobankrupt.However,bankruptcyandliquidatingassetsarecostlyinthesensethatthe
depositinsurerhastorepaytodepositors.Theriskierregimereturnsahigheroutcome
butwithlowerprobability.Thisregimeraisestheequityvaluewhenthecash-flowis
low.Sincetheequityvalueisstillpositive,thebankhasthechancetooperatewith
lowcash-flow,whatcreatesrisk-incentivesforthebankunderdistress.Nevertheless,the
switchinginvolvessomecostanditisnotalwaysarationaldecisiontobearthiscost
toincreasetherisk.Forahighlevelofthecash-flow,thebankisabletopaythecost.
Thus,thebankswitchestothelessriskyprojects(whatoffersahigherexpectedreturn
withlessuncertainty).Thischaracteristicincreasestheequityvaluecomparingtothe
riskierregimeforlargelevelsofthecash-flow.Therefore,thebankwithhighcash-flow
hassufficientincentivesforrisk-reductionandalwaystakestheopportunityregardless
ofalargeswitchingcost.
Thisresultisunlikethestandardassetsubstitutionopportunity,whichinsistsonbanks’
risk-taking.Iftheswitchingcostreducesandtendstozero,themodelaboveconverges
tothestandardassetsubstitutiontheory.Theothercontributionofthesecondchapter
istobridgethegapbetweenthestochasticswitchingmodelsandthebankingtheories.
Thisproducesthresholdsintermsofthestatevariable,thecash-flow,atwhichthe
regimeofoperationchanges.
Chapterthreestudieshowbanks’willingnesstoengageinriskyinvestmentsrelatesto
theircapital.Thefactthatthecontinuoustimemodeldoesnotgiveaclosedform

3

solutionmotivatesapplyingadiscretetimesetupforfurtheranalysisofthebanks’risk-
takingbehavior.Theotherdifferencefromchaptertwoisthatthebankhastoput
efforttomonitoritscreditorsinordertoincurlessrisk.Makingasaferportfolioby
monitoringcreditorsiscostlyandthebankmayoptimallystopexertingeffort.This
meansthatthebanktakesthemoreriskyprojectwhichbringsahigheroutcomein
thelessprobablecaseofsuccess.Incaseoffailuretheriskyprojectdoesnotreturn
morethanthelessriskyregime.Thestrikingresultisthatthereisanon-monotonic
relationshipbetweenabank’srisklevelanditscapital.Thoughbankersinvestinriskier
projectsindistress,riskybutefficientprojectsarealsoattractivefor“relativelywealthy”
banks.Whenthecapitaldecreases,attheintermediatelevelthebankstopsmonitoring
creditorsandchoosesriskierinvestmentssincetheeffortcostexceedstheexpectedloss
offailure.Nevertheless,asthecapitaldecreasesfurther,thebankreversesitsstrategy
inordertobenefitfromhigherlikelihoodofsuccessbeforefallingintoextremedistress,
wherethecapitallevelissolowthatthebankcanonlysurvivebytakingmorerisk.The
effortcostofmonitoringisthemainsourceofsuchanon-monotonicriskoptimization.
Therobustnesscheckonnon-monetaryeffortcostconfirmstheresult.Inadynamic
setupthenon-monotonicityvanishesduetothebankruptcyatfailure.Thisfindingis
inlinewiththeresultofchaptertwo.Foranintertemporalinvestmentdecision,where
reinvestmentrateislow,abankhighlyindebtpreferstheriskierinvestmentinthehope
short-term.inprofithighofChapterfourreviewstheprocessinwhichbankingregulationtheoriesevolvedfromthe
individualbankregulationtowardsmacro-prudentialregulation.Thechapterbegins
withabasicsetuptoexplainhowtherisk-takingbehaviorcannotbedealtwithout
regulatoryactions.Thechapterintroducestheshareholders’risktakingintheway
discussedinchapterthree.Inaddition,thereismanagerialmoralhazardsuchthata
bank’smanagertakesmoreriskyprojectsbecauseofapecuniaryprivatebenefit.Ifthe
capitalisnotsufficientlylarge,theshareholdersdonotofferanincentivecompatible
contracttothemanager.Thedepositinsuranceguaranteesthedepositors.Taxingthe
bank’soperationpreventssubsidizingbanksbytaxpayers.However,theinsuranceortax
systemincreasetheshareholders’incentivesforrisk-taking,andtheinsurancedoesnot
reducethemanagerialmoralhazard.Theintuitionisthatthebankhastopayforthe
taxorinsuranceoutofitsprofitthatdecreasesitsequityvalue.Thesemarketfailures
requireastrongsupervisoryagenttooptimizeex-antepoliciesagainstrisk-takingand
ex-postresolutionpoliciesincaseofbankfailure.
Itisshownthatthecapitaladequacyandclosurepolicyarenotaseffectiveasexpected
duetothesocialcostofassetliquidation.Partialdepositinsurancecreatesincentivesfor
theuninsuredepositorstomonitorthebankclosely.Alikelyliquidityprovisionbythe
regulatorincreasesthebank’schartervalueandthusreducesincentivesforrisk-taking.
Despitethesepoliciesfocusedonindividualbanks,allowingahealthybanktakeovera

4

CHAPTER1.ODUCTIONINTRfailedfinancialinstitutemotivatesbankstoavoidriskyspeculativeinvestments.Infact,
thisideaconnectstheindividualbankregulationtothesystemicriskregulation.
Thesecondpartinchapterfourarguesthatifabankislarge,orifitisinterconnected
tomanyotherinstitutesand/ormanybankstakerisktogether,thenfailuretransmit
intotheentirebankingsystem.Inthatsituation,healthyinstitutesintheprivatesector
donothaveenoughendowmenttotakeoverthefailedbanks.Theseexternalitieshave
beenseeninthe2007-2009crisis,whatisstudiedinchapterfourwithstatisticsofhuge
paymentsbythegovernment.Thisinspiresbankingregulationtheoriestoconcentrate
onthesystemicriskissues.Theriskcanoriginatefromanexogenouseconomicshock
orfromanendogenousrisk-takingofbanksandtheirsystemicfailures.Thispartof
researchemphasizestherisk-takingduetomoralhazardproblemsthatarisesfromthe
balance-sheet.banks’ofsideassetRewardingschemessuchasgrantingthehealthybankstotakeoverthefailedbanks
areproventomitigatethemoralhazardproblem.Redefiningthecapitaladequacyby
takingintoaccountbanks’contributiontothecollectiverisk-shifting(amongbanks)
caneffectivelyimproveupontherisk-basedcapitalrequirementforanindividualbank
intheBaselII.Asystemictaxpolicyisconfirmedtobeoptimalforregulatingalarge
bankwhichcannotbeclosedinthecircumstancethatthesupervisoryauthorityhas
powertoexpropriatetheshareholders’ownershipandthemanagement.Indeed,for
eachregulatorypolicyrequiresanoptimalimplementationthattakesintoaccountspe-
conditions.cific

Thesethreeessaysrepresentarelevantcontributiontotheliteratureoncetheycombine
severalaspectsofbankingregulationtheories.Theassumptionsaregeneraland,thus,
theresultscanbeappliedtotherealworldsituations.Forinstance,statisticalevidences
areprovidedforthesystemicfailuresituationsinthelastchapter.Insteadofusualone-
sidedattitudeoftheliteraturetotherisk-takingbehaviorofbanks,thisresearchtakes
anunbiasedapproachtoexaminetheadvantagesanddisadvantagesofriskforbanks.
Thus,thefindingssuggestincentivemechanismsconcerningdifferentperspectivesofthe
regulationproblems.Inanutshell,thisthesisfirstemphasizesthecomplexproblems
regardingrisk-takinginthebankingsystemsanddemonstratesnonmonotonicrelation
oftheriskchoicestothebanks’decisivefactors.Next,theoptimalmethodstodealwith
theriskissuesinbothmicroandmacroscalesareanalyzed.

2Chapter

ASwitchingModelinBanking

ductiontroIn2.1

Duringitslifetime,abankmaychangeitsportfolioseveraltimes.Forinstance,an
under-capitalizedbankwhichislikelytodefaultmaychooseariskierportfoliotoincrease
itsequityvalue.Thisbehaviorisknownas“gamblingforresurrection”andmaybea
rationalstrategyifthisistheonlychancetosurvive.Apartfromthistypeofmotivation
tochangetheportfolio,abankmaygenerallychangeitsregimeofoperation.Anew
regimeofoperationmeansthatbothreturnandriskofcash-flowgeneratedbythe
bank’sassetsaredifferentfromtheinitialassetallocation.Althoughtheinitialchoice
ofstrategyhasbeenstudiedextensivelyintheliterature,regimeswitchinghasbeen
.elyrestrictivonlydiscussedInthispaperIinvestigateoptimalswitchingstrategiesofabankhavingthechoice
betweentworegimesofoperationateachmomentintimeinacontinuoustimemodel.
First,Idiscussabank’soperationundereachregimeseparately.Inmybasicsetup,
abankisinsolventwhenitcannotpaythedepositcouponoutofthecash-flow.An
insolventbankclosesdowniftheequityvalueiszero.Thisno-asset-substitutionset-up
issimilartoDecampsetal.(2004)’smodelwithouttheregulatorypart.Thischapter
studiestherelationshipbetweentheequityvalueandtheregimechoiceateachpointin
time.Themaincontributionisthatthewholecharacteristicsoftheportfoliochanges
h.switcregimeainIfindthatlessriskincreasestheequityvalueforthehighercash-flow,asaresultoflarger
expectedreturn.Theeffectisreversedforthelowcash-flowcasesuchthatlessrisk
decreasestheequityvalue.Iinterpretthelowriskregimeaswhenthebankmonitorsits
creditorstokeepthenetpresentvalue(NPV)ofinvestmentspositive.Thisfollowsfrom
the”delegatedmonitoring”ideaofDiamond(1984)wherebanksmonitorinvestments
onbehalfofthedepositors.Subsequently,thebankindistressshirksandchoosesa
riskierportfoliowithlowerexpectedreturn.Theassumptionofthehigherriskand

5

6

CHAPTER2.ASWITCHINGMODELINBANKINGlowerexpectedreturnrulesoutthefirst-orderstochasticdominanceproblem.Next,
knowingabouttheadvantageofeachregimeconditionalonthecash-flowIexamine
theswitchingstrategiesofabankintheabsenceofoutsiderintervention.Iborrowthe
assumptionofDanglandLehar(2004)regardingreversibilityofcostlyswitchingateach
time.oftoinpTohighlighttheintuitionofswitching,IfollowDecampsandDjembissi(2005)whoshow
howthetradeoffbetweenreturnandriskinfluencesassetsubstitutionbehaviorinfirms.
Banksfinancetheirinvestmentsinlargepartsbydeposits.Imperfecttransferabilityof
banks’assetsmaketheirliquidationcostly.Inaddition,profitabilityofbank’sinvest-
mentrequirescostlymonitoringbythebank.Withouttheincentiveforthebankerto
monitor,theNPVoftheinvestmentbecomesnegative.Thisillustratesthatinsuffi-
cientlycapitalizedbanksdonothavetheincentivetomonitorandtheyswitchtothe
higherriskregimeinordertoincreasetheequityvalue.
Toanalyzethebank’sstrategiesIchooseacontinuoustimeframeworkwhichisgenerally
appliedinfinancialliteraturestudyingtheswitchingbehavior.Tosolvetheswitching
model,Iapplythestochasticcontroltechniquesandthegeneralapproachofthedynamic
programmingprinciple.Sincetheexistingmathematicalmodelsdonotcombinethe
optimalswitchingstrategyandthebank’sstoppingproblem,Ihavetofillthegaps
intheanalyticalsolutionwitheconomicalintuitions.Inthesetupofthispaper,the
outflowoftheswitchingproblemisthecash-flownetofthedepositpaymentthatcanbe
negative.Therefore,thebasicassumptionsofVathandPham(2007)arenotsatisfied.
Still,theirexplicitsolutionprovidesanintuitionformycase.Thefreeboundaryproblem
relatedtothevariationalinequalitiesdividethecash-flowstatespaceintothestopping
regionandthecontinuationregion.Pham(2005b)considersthesmooth-fitprinciplefor
thevaluefunctionthroughboundariesofswitchingregions.Havingallhisassumptions,
thisprinciplegivestheboundaryconditionstofindclosedformsofvaluefunctions.Yet,
Pham(2005b)’sresultsareapplicabletomymodelonlywhentheclosurelevelofthe
cash-flowisgiven.Hence,wheretheanalyticalsolutionisnotavailableIpresentan
intuitiveconjectureaboutthemissingcharacteristicoftheobjectivefunction(NPVof
anoperatingbank)andtheboundaryconditionsintermsofthestatevariablecash-flow.
Finally,Iapplyanumericalmethodtosolvethreeexamples:thecostlyswitchingcase,
anextremecaseoftooexpensiveswitching,andtheotherextremecaseofcost-free
switching.Thesimulationsconfirmmyanalyticalresultsregardingtheuniquenessof
switchingpointsforeachregime.Whencapitaldropsbelowacash-flowthresholdthe
bankoperatingunderlessriskyregimeswitchestotheriskierregime.Ifthecapital
decreasesfurtherandfallsbelowaminimumcash-flowlevelthebankcloses.Ifthecapital
increases,aboveanotherthresholdthebankoperatingintheriskyregimeswitchestothe
lessriskyregime.Thethreethresholdsarefeasibleinthewaythattheswitchingregions
donotintersect.However,ifthereisnocostofswitchingthereisauniquethreshold
belowwhichthebankdoesnotmonitorcreditorsandoperatesinriskierregimeaslong

2.1.ODUCTIONINTR7

asthecash-flowisabovetheclosurethreshold.Iftheswitchingistoocostly,thebank
doesnotshirk(frommonitoring)whenthecapitaldecreasesbutstopsoperation.This
happenswhilerisk-reductionandoperatinginGoodregimeismoreprofitableforabank
whenthecapitalraisessufficiently.MyfindingsareinlinewithDanglandLehar(2004)
iftheexpectedreturnsdonotchangefromoneregimeofoperationtotheother.Because
inthesetupofthischaptertheexpectedreturnsplayanimportantrole,theextreme
caseoftoocostlyswitchingistotalydifferentfromtheresultofDanglandLehar(2004).
Theliteratureonthebankrisk-takingiscloselylinkedtotheliteratureonbankingreg-
ulation.Decampsetal.(2004)studythethreepillarsofBaselII(BaselCommittee
(2001))andattempttoclarifyhowmarketdisciplineandsupervisoryactioncancom-
plementcapitaladequacy.Theyassumethatabankchoosesoneoftwodifferentregimes
ofoperation,i.e.onewithahigherreturnandanotheronewithahigherrisk,atthe
verybeginningandfollowsitwithoutswitching.Theyshowhowtheregulatorysystem
canaffecttheinitialdecisionofabanktochooseasaferportfolio.
InthesetupofDanglandLehar(2004)bankersonbehalfofequity-holders1canswitch
theregimeofoperationthroughassetsubstitution.Intheirmodelswitchingiscostlyand
reversiblesuchthatonlytherisklevelcanchangeateachpointoftime.Theyassume
thattheregulatorwhoauditsthebankatrandomtimeintervalswantstopreventasset
substitutionforahigherrisk.Theregulatoryclosurethresholdsallowawell-capitalized
banktoloweritsriskandcontinuetooperateevenifthecash-flowissmallerthanit
wouldbewhenthehighriskportfoliowerechosen.Theycomparethepoweroftwo
exogenousregulations,i.e.BuildingBlock(BB)regulationandaValueatRisk(VaR)
regulatorycapitaladequacy.SinceVaRregulationisrisksensitive,itismoreefficient
thanriskinsensitiveBBcapitaladequacyinpreventinggamblingforresurrection.
Leland(1994)followstheassetsubstitutionargumentofJensenandMeckling(1976).
Hestudiestheoptimalcapitalstructureandfindsthatequity-holdersprefertomakethe
firm’sactivitiesriskierinordertoincreasethefirm’sequityvalueattheexpenseofdebt
value.Inhispaperthedebt-holdersarehurtbyhigherriskinthecaseofunprotected
debtinwhichtheequityvalueisenhancedbygreaterrisk.Buttheoppositeistrue
whenthedebtisprotectedbyapositivenetworthcovenant.Inthiscase,increasing
therisklowerstheequityvalueaswellasthedebtvalue.
LelandandToft(1996a)extendLeland(1994)’smodelandshowthatriskshiftingdis-
appearswhenthetimetomaturityofdebtisshortened,confirmingthatshort-term
debtfacilitatesthediscipliningofbankmanagers.Likewise,Leland(1998)includesa
singleswitchingtoriskyportfoliowithoutanycost.Erricson(1997)assumesaconstant
switchingcostandallowsforanirreversibleswitch.Boththesepapersfocusonthe
optimalcapitalstructurewhiletheassetsubstitutionopportunitycausesagencycosts.

1Inthisliterature,thepossiblefrictionsbetweenbankersandequity-holdersandbetweendepositors
andtheDepositInsurerFoundationareignored.

8

CHAPTER2.ASWITCHINGMODELINBANKING

Toobtainanalyticalsolutions,Iborrowthemethodsfromtheliteratureinstochastic
switchingmodels.DanglandLehar(2004)considertheelementarystoppingmodelsin
whichanagentdecidesoncontinuingorstoppingtheoperationgeneratingastochastic
outcome.Sincethesetypesofentry/exitmodelsarenotdirectlyapplicabletothe
switchingmodel,thesolutionofDanglandLehar(2004)totheswitchingproblemisby
conjecturesonthecontrollimitspolicies.2Thestoppingmodelintheleadingpaperby
BrekkeandOksendal(1994)hasbeendevelopedintoacomputationalmodelbyFackler
(2004).BayraktarandEgami(2007)applyaprobabilisticapproachtowardstheoptimal
switchingprobleminwhichthevaluefunctionischaracterizeddirectly.Theyrelyon
thesocalledcoupledoptimalstoppingproblemsinsteadofthedynamicprogramming
principle.AnalyticallytheswitchingmodelsofPham(2005b)andVathandPham(2007)are
theclosesttomymodel.VathandPham(2007)solvethegeneralswitchingmodelin
whicharegimeisbasicallyreplacedbyanotherregimeinordertomaximizeanobjective
function.Theyuseaviscositysolutionsapproachtodeterminetheoptimalinvestment
decisionforamulti-activityfirm.Theirmethodinvolvesasequenceofstoppingtimes
withregimeshifts.TheyfindtheexplicitsolutionforthetworegimescaseofBrekke
994).(1OksendalandThefollowingsectionintroducesthemodel,explainstheno-asset-substitutioncases.
Section2.3presentstheswitchingmodelandtheoptimalswitching-stoppingstrategies.
Insection2.4thestrategiesarequantifiedforseveralcases.Numericalexamplesare
explainedinsection2.5.Section2.6concludes.Theappendixincludessomeproofsand
solutions.Figuresarepresentedinthelastsection.

delMoThe2.2

Theassetvalueofthebankgeneratescash-flowxwhichisassumedtofollowageomet-
ricBrownianmotion,followingMerton(1974)andLeland(1994).Thebankermakes
decisionsoninvestmentandtheregimeofoperation.Therearetwochoicesofport-
afolios.comEacbinationhoneoftworepresenregimes.tsaTheregimeofrepresenopetativrationeandbankerthecanbankswitchcannotfromoptheeratecurrenundert
aphigherortfoliotomeanofanothercash-floateacwhandmomenalowt.erWhenrisk.theThisbankregimeismonitorscalleditsGood.creditors,itConsequenreceivtlyes,
IcalltheotherregimeBadinwhichthebankstopsmonitoring.Inthiscasetherisk

2SeePham(2005a)forasurveyontheaspectsofstochasticcontrolproblems.

EHT2.2.MODEL

9

increaseswhilethebanklosesthemeanvalue.3Thus,thecash-flowprocessisdenotedby

dx=µGxdt+σGxdωtheGoodregimeisoperating
µBxdt+σBxdωtheBadregimeisoperating
x(0)=x0>0.(2.1)
wherewehaveσG<σBforrisklevels,andfordriftsµG>µB.ωisawhitenoise
vThariablus,e.thedepAssumeositthrate,atalld,isagenequaltsaretorisktheriskneutralfreewithrateran.IninstanterpretingtaneousthediscounGoodtrateregimer.
asoperationunderdelegatedmonitoring,notethatthismonitoringisassumedtobe
costlessortohaveavariablecomponentwhichhasalreadybeensubtractedfromthe
originaldriftofthecash-flow.Inotherwords,µGcouldbeinterpretedasµ−mwhere
mxistheproportionofcash-flowlostinmonitoring.
Whenthebankerclosesthebankatdefault,thebank’sassetsareliquidatedforavalueof
αx,whereαisgivenexogenously.4Becauseweareinariskneutralworld,theexpected
netpresentvalueofthecash-flow(conditionalontheinformationavailableattimet)
hastocoincidewiththecurrentvalueoftheunleveredbank,
∞+Wt=Exte−rτxτdτ=xt,(2.2)
µ−rtflowherewforExallistheassetexpvaluesectationabovopezero,eratorIovneedervanariablearbitragex.Infreeordermotodel,ha5vei.e.apµositiv<re.cash-But
forarealbankholdingdeposits(leveredbank)equation(2.2)nolongerholdsbecause
ofbankruptcyrisk.Inthismodellong-termdepositsarefullyinsured,withtheface
valuenormalizedtoone.Thus,theequity-holdersareresidualclaimants.Thereis
aninstantaneousswitchingcost,k,whichispaidbytheequity-holdersatswitching
ts.momenInthefollowingsubsection,Iconcentrateonthecasewherethebankhasonlyonetype
ofregime.Thiswillintroduceincentivesofthebankforswitchingbetweentworegimes.
Throughout,Iassumeasimpleworldwherethebankoperatesintheabsenceofoutsider
tion.netervin

3Iconsiderageneralcaseratherthanonlyriskshiftingwhichissupposedinthemostofasset
literature.substitution4Notethatαisnotnecessarilylessthan1,sincethestatevariableisthecash-flow.Thisisunlike
toLeland(1994)inwhichthestatevariableistheassetvalueofthefirmandthereisafractional
bankruptcycostorcostofliquidation.
5Theintegralneverconvergesforµ>r.

10CHAPTER2.ASWITCHINGMODELINBANKING
SubstitutionAssetNo2.2.1

Iexplainthesimplestoppingproblemforthebankinthissection.Supposethebank
choosesonetypeofregimeattimet=0,andoperatesforitsentirelifetimewithout
assetsubstitutionandanychangeintheportfolio.Thus,thebankhastheonlyoption
tostopwhenoperationisnolongerbeneficial.Thatmeansassoonastheequity-holders’
wealthbecomesnegative,thebankerstopsoperatingandliquidatestheassets.
Intheabsenceofoutsiderintervention,thebankeronbehalfofequity-holdersabandons
theoperationasthecash-flowdropsbelowthresholdxC.Althoughafirmex-ante
maximizesthevalueofitsassetportfolio,aleveredbankex-post(whendepositsarein
place)maximizestheequityvalue.Inthisframeworktheearningsofthebankfromthe
cash-flow,beforedepositpaymentandextrabenefit,isdeterminedby:
τW(x)=ExCxte−rtdt+e−rτCαxC(2.3)
0wherestoppingtimeτCisarandomvariable,definedasthefirstinstantwherextfalls
belowxC,givenx0=x.ThenWisfoundtypicallybysolvingtheordinarydifferential
6(ODE)equationrW=(1/2)σ2x2Wxx+µxWx+x.(2.4)

is:utionsolgeneralTheW(x)=x+K1xγ1+K2xγ2,
µ−rwhereγ1>1,γ2<0aretherootsof:

(1/2)σ2γ(γ−1)+µγ−r=0,

to:equalusthand−(µ−(1/2)σ2)±(µ−(1/2)σ2)2+2σ2r
=γ.2σThecoefficientsK1,K2aredeterminedbytheboundaryconditions:
W(xC)=αxC,

6RefertoDixitandPindyck(1994)“EquivalentRisk-neutralValuation”,P.121-125.

(2.4)

(2.5)

(2.6)

(2.7)(2.8)

T2.2.MODELEH

and

11

x→+∞,W(x)asy−→mptoticallyr−xµ.(2.9)
Thelatterconditionattributestothecasethatthehighcash-flowpreventsbankruptcy.
Therefore,sincebankruptcyisunlikely,theearningsofthebankconvergestotheasset
valueofanunleveredbankwiththesamecash-flow.
From(2.9)wehaveK1=0,andWisdeterminedby
x1xW(x)=r−µ+(α−r−µ)xC(xC)γ2.(2.10)
Thesecondtermin(2.10)indicatestheoptionvalueassociatedwiththeirreversible
closureatxC.Ascondition(2.9)shows,thisoptionvalueconvergestozeroforahigh
w.cash-floofaluevWithasimilarapproach,Icanfindtheclosedformsforothercontingentclaims,i.e.the
marketvalueofdepositsandthemarketvalueofequity:

•Themarketvalueoftheuninsureddeposits:Incontrasttotheinsured
contractheldbythedepositors,thatisalwaysworth1,thecouponflowdprovided
bythebankisnotinsured.ThemarketvalueoftheuninsureddepositsD(x)isthe
presentvalueofcouponflowr.Wheneverthecash-flowisbelowthecouponflow,
thebankerhastoinjectmoneyintothebankinordertosurvivethesituation.
Sincethisclaimisexposedtothedefaultrisk,theinsurerbearsthedifference
betweentheinsuredvalueandthemarketvalueofthecouponflow,i.e.1−D(x).
Thisisthecurrentvalueofpossiblefutureexpendituresnecessarytoguarantee
thefullfacevaluetodepositorsincaseofbankclosure.7TheclaimD(x)satisfies
w:elobODEthe

rD=(1/2)σ2x2Dxx+µxDx+r.(2.11)
Therefore,D(x)hasapowerfunctionclosedformwithcoefficientsfoundfrom
boundaryconditions.ThefirstboundaryconditionisD(xC)=αxC=W(xC),
alsocalled“absolutepriorityrule”.Accordingtothisrule,theequity-holders
receivenothingfromtheassetvalueattheclosuretime.8Sinceahighamount
ofcash-flowrulesoutdefaultrisk,themarketvalueofdepositsconvergestothe

7ForsimplicityIassumethatthebankpaystheinsurancepremiumequal1−D(x0)attimet=0.
Yet,theinsurancepremiumcouldfollowamorecomplicatedprocess,e.g.aregularpayment.
8IfαxC>1,thenαxC−1isgiventoequity-holders.Butinthiscasethereisnouncertaintyfor
deposits.AsIseelater,alowerclosurethresholdstillincreasesthemarketvalueofequity.Thus,Ican
assumeαxC<1,andabsolutepriorityruleholds.

12

CHAPTER2.ASWITCHINGMODELINBANKING

principal.Hence,anotherboundaryconditionisx→+∞,D(x)→d/r=1.
Then,D(x)isgivenby
D(x)=1+(αxC−1)(x)γ2.(2.12)
xC

•Themarketvalueoftheequity:Beingaresidualclaim,thevalueofequityis
9ybdetermined

E(x)=W(x)−D(x)
xxCxγ2
=r−µ−1+(1−r−µ)(xC).(2.13)

AsinLeland(1994),whenthereisnoprotectionforthedebt,10bankruptcyoccursonly
ifthefirmcannotmeettherequiredinstantaneousdepositpaymentbyissuingadditional
equity,i.e.whentheequityvaluefallstozero.Ofcourse,giventheabsolutepriority
rule,theequityvalueiszeroatclosure.Maximizingthesocialvalueofthebank,W,
givesxC=0.However,thelimitedliabilityofequitypreventsxCfrombeingarbitrarily
small.11Thus,maximizingnonnegativeE(x)forallvaluesofx>xCsetstheclosure
asthreshold0<xc=γ2γ(r−−1µ)<1(γ2<0),(2.14)
2whichistheresultofthe“smooth-pasting”condition
dE/dxC|x=xc=0.(2.15)

(2.14)

(2.15)

Fromequations(2.7)and(2.14),theclosurethresholddependsonriskfreeinterestrate
randtheprocessofcash-flowsuchthat
∂γ2>0,∂γ2<0⇒γB2>γG2,(2.16)
µ∂σ∂

and

∂xc=−r−µ∂γ2<0,
∂σ(γ2−1)2∂σ

(2.16)

(2.17)

9ThevalueofequityisalsofounddirectlyfromE(x)=Ex[0τC(xt−r)e−rtdt].Thesecondorder
differentialequationrE=(1/2)σ2x2Exx+µxEx+x−randappropriateboundaryconditionsgivethe
sameclosedforms.Iwillfollowthisapproachinthenextsection.
10Thedebt(deposit)isinsuredfromthedepositor’spointofview.However,thereisnoconstraint
onthebanktomeettherequiredinstantaneousdepositpayment.
11Still,alowerclosurethresholdraisestheequityvalueasdE/dxC≤0forxC>0.

MODELEHT2.2.

13

whereγi2isparameterγ2associatedwithregimei∈{G,B}.Theclosurethresholdis
notmonotonicinthedriftasthederivativeindicates
∂xcr−µ∂γ2γ2
∂µ=−(γ2−1)2∂µ−γ2−10.(2.18)
Therefore,theclosurethresholddependsontheregimeofoperationandtheparameters.

2.2.2ComparisonoftheTwoRegimes
Havinglearnedaboutthestoppingstrategiesofabankunderasingleregime,nowthe
tworegimes(GoodandBad)canbecompared.As(2.13)shows,theequityvalueisa
convexfunctionofthecash-flow.Forasufficientlyhighvalueofthecash-flowundereach
regimei∈{G,B},Ei(x)asymptoticallyconvergestor−xµi−1whichequalstheasset
valueofanunleveredbankwiththesamecash-flownetofthedepositsprincipalofthe
leveredbank.BecauseµG>µB,theequityvalueunderregimeGoodishigherthanthe
equityvalueunderregimeBadforthehighamountofcash-flow.IfµG=µBtheclosure
ofthebankundertheBadregimeoccursbelowtheclosureundertheGoodregime.
Figure2.1showsthecaseinwhichhigherriskisalwayspreferred,sinceitincreasesthe
valueofequity,EB(x)EG(x).
AsµG>µB,andxCisnotmonotonicwithrespecttodriftµ,thedifferencebetweentwo
driftsyieldsdifferentresults.Whendµ=µG−µBishighcomparedtodσ=σG−σB,
andthedriftcoefficientoftheGoodportfolio,µG,tendstotheriskfreeinterestrater,
thebankwiththeGoodportfolioclosesatalowerthresholdthanthebankwithBad
portfolio.Further,forasufficientlyhighvalueofthecash-flowtheexpectedvalueof
equityundertheGoodregimeishigherthantheexpectedvalueofequityundertheBad
regime.Insuchacase,thebankdefinitelypreferstheGoodportfoliowiththehigher
equityvaluetotheBadportfolio,showninfigure2.2.Alternatively,µG>µBleads
toEG(x)>EB(x)forthehighcash-flowwhilenon-monotonicclosuremayresultin
xG>xB.Thentheequityvaluefunctionsofthetworegimescrossasfigure2.3shows.
Therefore,dependingonthecash-flowlevelthebankmaypreferadifferentregime.With
theintuitionsfromthefigures,thefollowingpropositionsdiscussallcases.Theproofs
endix.apptheinareProposition2.1Assumethatthebankhastwopossibleportfoliochoices:(µG,σG)and
(µB,σB),whereµG>µB,σG<σB.Thenforallxintheoperationareaofthebank,
EG(x)>EB(x)ifandonlyifxG<xB.

endix.appSeeofProProposition2.2AssumetworegimechoicesforthebanksuchthatµG=µB=µ.
ThenxG>xBandEB(x)>EG(x),forallxintheoperationareaofthebank.

14

endix.appSeeofPro

CHAPTER2.ASWITCHINGMODELINBANKING

Ifdriftsoftworegimesareequal,equityvaluefunctionsconvergeasymptoticallytothe
samelineforhighvaluesofcash-flowasfigure2.2shows.Thenextcorollaryregarding
figure2.3followsdirectlyfromthepropositionsabove.
Corollary2.1Necessaryandsufficientconditionsinordertohavecrossingequityvalue
functionsareµG>µBandxG>xB.
ProofForsufficientlyhighvalueofcash-flowwehave
x→+∞,E(x)−→r−xµ−1.
Hence∃M∈Rsuchthat∀x>M,EG(x)>EB(x)iffµG>µB.Fromproposition
2.1,∃m∈Rsuchthat∀x<m,EG(x)<EB(x)iffxG>xB,asxG=xBonly
yieldsEG(x)>EB(x)becauseofahigherdrift.Sinceequityfunctionsarecontinuous,
∃xs∈R,m≤xs≤MsuchthatEG(xs)=EB(xs).Becauseofconvexityofthetwo
equityfunctionsthecrossisunique.
Remark2.1NotethatIdonotconsideranypotentialpreferenceonstrategiesorclo-
sure.Forinstance,Decampsetal.(2004)assumethattheGoodregimedominatesthe
closuredecisionwhichisalwayspreferredtotheBadregime.Toimplementthisassump-
tion,Ineedthatr−1µG>α>r−1µB.ButinthispaperIsupposethatforallpositive
cash-flowstheexpectedvalueofthebank,operatingperpetuallyundereithertheGood
ortheBadportfolio,ispreferredtoclosure.However,thetradeoffbetweenahigher
driftandahigherriskistheimportantfeature.

TakingtheGoodregime,theclosurepointmovesbyanychangeinthedriftandrisk
levelofcash-flow.Thecrossingholdsifftheclosurepointdecreasesbyloweringthe
driftandincreasingtherisklevel.ItmeansIneedconditionsunderwhichthetotal
differentialofxCisnegative,i.e.dxC<0.Sincedµ<0anddσ>0,from(2.17),(2.18)
anddxC=∂∂xµCdµ+∂∂xσCdσ,(2.19)
weseethatdxC<0iff∂γ2
dσ<∂∂γµ2+γ2(γ2−∂γ1)2.(2.20)
dµ∂σ(r−µ)∂σ
areExcludingopportunitiestrivialforcasesadvinanwhictageshtheofeacbankhchoregime.osesForonlyaonehighvregimealueofwiththecertaincash-floty,wtherethe
Onbankerthechootherosesthand,heGoowhendtheregimecashandflowmonitorsistoolow,creditors.onlyThen,survivalisbankruptcyimportanistlessforlikelythe.
lowbank.erThecash-flohighw.riskThatguaranmeanstiestheBadnon-zeroregimeequitisyvmoaluereandattractivmakeesforopabaerationnkinpossibledistress.for

2.3.SWITCHINGSTRATEGIESINACROSSINGCASE

2.3SwitchingStrategiesinaCrossingCase

15

Thelastsectionshowedthatthetworegimesmighthaveadvantagesanddisadvantages
fordifferentvaluesofthecash-flow.Thisresultprovidestheintuitionforswitchingfrom
oneregimeofoperationtoanotherasthecash-flowvaries.Inthissection,assumethat
theparameterssatisfyinequality(2.20)whichyieldcrossingequityfunctionsasshown
2.3.figureinThebankhasthreechoicesateachmoment,i.e.theGoodportfolio,theBadportfolio
orclosure.Unlikeclosure,thetworegimescanbereversiblyabandonedatacost.This
costmaybearbitrarilyhigh,therebyrulingoutswitching.Denotethebank’sthree
possibleactionsby{Stick,Switch,Stop},forstickingtothecurrentregime,switchingto
anotherregime,andstoppingtheoperation,respectively.
andTheLehargeneralo(2004),ptimalinaswitcmohingdelmodelincludingapplieasinlump-sumthiscase.linearFollocostwingofswitcconjectureshing,of“conDangltrol
limitspolicies”indicatetheoptimaldecision.Theintuitionfromourpreviousresults
makcash-floeswconbuttrolthelimitsBadppoliciesortfolioforapplicable.lowerThecash-flobankws.prefersSincethethereGoodispaortfoliolump-sumforhighercost,
asconthetrolcash-limitsflopwisolicyintheleadsintoterioranofintervthealin[StervG,al,SB]theexertingbanksticthekstominimathelcurrenconttrol.regime.AslongIf
cash-flowxfallsbelowSG,thebankwithregimeGoodmustswitchtotheBadregime,
andifxrisesupSBthebankwithregimeBadmustswitchtotheGoodregime.
Althoughthispolicyseemsintuitivelyreasonable,suchaswitching-stoppingmodeldoes
andnotfitLehartothe(2004)formerreferentosuchtering/exitingmodelsmowhicdelshareleadingbasicallytothecondifferentroltfromlimitptheiolricies.switcDanglhing-
onestoppinghand,motodel.solveThtheus,theproblemmodeldirectlyneeds,Ianeeddirectequityvsolutionaluewithoutfunctionsantoyfindthprediction.eoptimalOn
switching-stoppingstrategies.Ontheotherhand,Icanfindtheequityvaluefunction
Theonlybyviscositpropyerbsolutionsoundaryargumenconditionstisapropresultierngapproacfromhthewhichoptimalconsidersstrategiesthetwofotheoptimiza-bank.
tionproblems(maximizingtheobjectivefunctionandfindingtheoptimalboundaries)
assimtheultaneouslyviscosit.ysolutionsTherefore,IargumenstatetstheofswitcPhamhing(200model5b).(bHisetweenargumenthettsworiskymathematicallyregimes)
fittomymodelundertheconstraintthatthebankonlyhasaswitchingproblem.

2.3.1TheSwitchingModel

IverifyswitchingstrategiesusingtheframeworkofPham(2005b)inthissection.In
ordertoexcludetheclosureproblem,supposethereisenoughsupportforthebankin
distress.Whenthereiswelfarecostofclosure,governmentsconsiderbailoutpolicies.In

16

CHAPTER2.ASWITCHINGMODELINBANKING

here,wecanassumethegovernmentrecapitalizesthebankbypublicfundsandthebank
operatesforeverwithcash-flowsaboveXSC>0.12Havingonlyaswitchingproblem,
thismodelsatisfiesassumptionsH1-H4ofPham(2005b)suchthat,

•H1.Lipschitzconditionholdsforthelineardriftandvarianceofstatevariablex.
•H2.Variancesarepositiveundereachregime.
•H3.Theout-flowfunction,x−r,isLipschitzcontinuous.
•H4.Theswitchingcostispositive,andstickingtothecurrentregimeiscostless.

MyvaluefunctionistheequityvaluedenotedbyΨi(x)forregimei.Definethediffer-
entialoperatorΔiF(x)foranyvaluefunctionF(x)underregimeias

2ΔiF(x)=µixF(x)+σ2ix2F(x).(2.21)
Theorem1.3.1ofPham(2005b)provestheexistenceoftheviscositysolution13toan
ordinarydifferentialequation.Istateitforthismodelinthenextproposition.
Proposition2.3Assumingconstantdriftandriskofthestatevariablexandthelinear
profitfunctionP(x)=x−r,foreachregimei,thevaluefunctionΨiisacontinuous
viscositysolutionon(XSC,∞)tothevariationalinequality:

min{rΨi(x)−ΔiΨi(x)−P(x),Ψi(x)−(Ψj=i(x)−k)}=0,x>XSC.(2.22)

Thismeansthatforbothregimes,wehavesupersolutionandsubsolutionproperties:

•Viscositysupersolutionproperty:foranyx¯>XSCandF∈C2(XSC,∞)s.t.x¯is
alocalminimumofΨi−F,Ψi(x¯)=F(x¯),wehave

min{rF(x¯)−ΔiF(x¯)−P(x¯),Ψi(x¯)−(Ψj=i(x¯)−k)}≥0,(2.23)

•Viscositysubsolutionproperty:foranyx¯>XSCandF∈C2(XSC,∞)s.t.x¯isa
localmaximumofΨi−F,Ψi(x¯)=F(x¯),wehave

min{rF(x¯)−ΔiF(x¯)−P(x¯),Ψi(x¯)−(Ψj=i(x¯)−k)}≤0.(2.24)

Basedon(2.22)thereexistsacontinuationareaforeachregime.Wheneverthecash-flow
isintheinterioroftheoperationarea,thebankstickstothecurrentregime.Thenthe
equityvalueisasolutiontoanODEdeterminedbythefirsttermin(2.22)beingequal
1213WSeeecanPhamlater(2005b),callXSCApptheendixclosurefortheprthreshold.oofofageneralcase.

2.3.SWITCHINGSTRATEGIESINACROSSINGCASE

17

zero.Thecontinuationareaisconnectedtotheswitchingregionwhereitisoptimalto
changetheregime.Switchingareaisaclosedsetbydefinition:assoonasthecash-flow
reachestheboundariesoftheoperationareaandfallsintheswitchingregion,thesecond
termin(2.22)willbeequaltozero.Thisequalityisthevaluematchingconditionnet
theswitchingcost.Moreover,Lemma1.4.1ofPham(2005b)addsthesmoothnessofthe
valuefunctionsintheircontinuationregion.Havingthevaluefunctionsfromthelast
proposition,Ineedtheoptimalboundaryconditionstofindtheswitching/continuation
regions.Theoptimalityconditionresultingfromtheorem1.4.1ofPham(2005b)isthe
socalledsmooth-fitproperty14overtheboundariesoftheswitchingregions.Notethat
thereisnoexplicitsolutioninPham(2005b).Idenotetheswitchingpointfromregime
GoodtoregimeBad,SGandtheswitchingpointfromBadtoGood,SB.Wedonot
knowyetiftheyexistuniquely.However,underassumptionsH1toH4proposition2.4
givesthesmooth-fitpropertyconditionsforanyswitchingpoint.

Proposition2.4Fori∈{G,B},thevaluefunctionΨiiscontinuouslydifferentiable
on(XSC,∞).Moreover,atSGandSBwehave
Ψ´G(SG)=Ψ´B(SG),(2.25)
Ψ´G(SB)=Ψ´B(SB).(2.26)

Idividethewholerangeofthecash-flowforregimei,(XSC,∞),toSWiandCi,asthe
switchingregionforregimeianditscontinuationregion,respectively.Thetwosubsets
intersectinSi.Inthisworkviscositysolutionargumentscannotdirectlyprovethatthe
continuationregionandtheswitchingregionofaregimei∈{G,B}onlyconnectas
cases(a)and(b)offigure2.4.Therefore,thecase(c)isalsopossiblesinceitsatisfiesall
resultsofPham(2005b),althoughitmightnotintuitivelybereasonableforthismodel.

2.3.2TheOptimalStopping-SwitchingModel

InordertofindtheoverallstrategiesofabankIneedtoadditsoptimalclosurede-
cisiontotheswitchingproblem,sinceintheabsenceofoutsiderinterventiondifferent
actionsactionsofandthevicebankversa.areInotcindeharacterizependent.theTheoptimalclosureswitcdecisionhingstrategiesinfluencesoftheeachswitcregimehing
includingtheclosuredecisioninthissection.Themathematicalsolutionofthecom-
15mobineddeltakingstopping-switcintoaccounhingtmothedeliseconomicafarbeylinondthistuitionswinork.thissectioConsequenn.Thetly,Ifirstdevintelopuitivthee
conjectureisthatlemma4.1ofVathandPham(2007)holdsforthestopping-switching

14Orthesmooth-pastingcondition.
15VathandPham(2007)findexplicitsolutionsforaspecialsettingofswitchingmodelwhichis
differentfromthismodel.

18

CHAPTER2.ASWITCHINGMODELINBANKING

modelsimilartothestoppingmodel.16

Conjecture1.Thevaluefunction,optimalequityΨˆi,i∈{G,B},issmoothC2
oncontinuationregionCiandsatisfies
rΨˆi(x)−ΔiΨˆi(x)−P(x)=0.
Lemma2.1Theclosurethresholdisabovezero,i.e.∃XSC>0,Ψˆi(XSC)=0for
i∈{G,B}whichistheoptimalregimeforthelowcash-flows.

ProofIfthebankneverclosesabovezerozerocash-flowandoperatesunderregimei
arbitrarilycloseto0,then0∈Ci.Fromconjecture1,Ψˆihasageneralform
xΨi(x)=r−µi−1+Ki1xγi1+Ki2xγi2.
Whenxconvergesto0,thefirstandthirdtermsconvergetozeroaswell.Thenif
Ki2=0theforthtermconvergestoinfinity.AndforKi2=0,Ψˆiconvergesto-1.By
contradiction,0∈CiandclosurethresholdXSCisabovezero.

Conjecture2.Forallx,Ψˆiismonotonicallyincreasinginx,i∈{G,B}.
Proposition2.5AsµG>µB,theswitchingregionofregimeBadisanon-emptyset.

ProofIngeneral,thebankcanswitchorclosedownwhenthecurrentregimeisno
mlongerustbhold:eneficial.whentheNomastatettervwhicariablehregimecash-floowrrisesstrategysufficienistlyoptimal,,eachthebclaimoundarycoasymptoticallyndition
convergestotheclaimonanunleveredbankwiththesamecash-flow.Itmeansthat
x→+∞,Ψˆi(x)asy−→mptoticallyx−1,i∈{G,B}.
µ−ritheeTherefore,quityvthealueofhigherthedriftbankofwiththeGhighoocdash-floregimew.triviallyAccordinglyincrea,thesesbtheettersoccialhoicvealueforhighand
cash-flowisregimeGood.OperatingundertheBadregimeabankwhichhassufficiently
highcash-flowreducestheriskandswitchestoregimeGood.Ofcoursewithahigher
costofswitchingtheswitchingpointfromBadtoGoodincreases.However,forany
0<k<∞thereexistsSB<∞suchthatΨˆB(x)=ΨˆG(x)−k(fromtheboundary
condition).ItindicatesthattheswitchingregionoftheBadregimeincludes(SB,∞),
i.e.SWB⊇(SB,∞)andSWB=.

16Inthissection,Istatesomeeconomicalconjectureswherethemathematicalproofismissing.

2.3.SWITCHINGSTRATEGIESINACROSSINGCASE

19

Lemma2.2TheoptimalclosurethresholdofthebankisxGorXSC≤xB.

ProofIfthebankfollowsoperatinginregimeGoodandneverswitchesithastoclose
optimallyatxG.WherexB<xG,havingtheswitchingopportunitythebankisable
tocontinueoperationwiththelowercash-flowsunderregimeBad.Sincetheswitching
actionisoptimalonlyifitincreasestheequityvalue,forallxintheoperationareawe
havelowerboundsEi(x)≤Ψˆi(x),i∈{G,B}.Thus,thezeroofΨˆoccursbelowthezero
ofEi,i∈{G,B}oratanequallevelofthecash-flow.Itmeansthatfortheswitching
case,ifthebankswitchestoregimeBadorstayswiththisregimeforthelowcash-flow
itstopsoperationatasmallerthresholdthantheno-substitutioncase.Forthesame
reason,havingswitchingopportunitythebankmaydecidestooperateunderregime
Goodifitcancloseatalowercash-flow.Concludingfrombothswitchingcases,forthe
closurethresholdofthestopping-switchingmodel,XSC,wehaveXSC≤xB<xG.

Lemma2.3Theswitchingregions,SWGandSWBdonotintersect.

ProofIfSWGandSWBintersect,theswitchisnotstable.Becauseifineachswitchthe
banklosesanamountofcash-flowandenterstotheswitchingareaofanotherregime,
ithastoswitchback.Thisisnotanoptimalpolicyasthebanklosesthecash-flow
continuouslyforthecostofswitchingbackandforth.Thus,onlythetwocontinuation
regions,CGandCBcanintersect.

Remark2.2SupposethereexisttwoswitchingpointsforregimeBad,mBandMB
wheremB<MBandcontinuationareaCB=(mB,MB).Definetheswitchingregion
oftheGoodregimeasanintervalSWG=(mG,MG).Fromlemma2.3wemusthave
thatSWG⊆CB,showninfigure2.5.Then,ifthecash-flowdecreasesfromahighvalue
M>MBtoalowerlevelm<mBweseethattwotimesswitchingismorecostlythan
operatingalongMtomonlyunderregimeGood.Withmultipleswitchesthebank
facesmultiplecostsofswitching
ΨˆG(m)<ΨˆG(mG)=ΨˆB(mG)−k
<ΨˆB(MB)−k=ΨˆG(MB)−2k
<ΨˆG(M)−2k.

WithoutswitchalongMtom,theequityvaluedecreasessmoothly.Anyothercase
oftwoormoreswitchingpointsforeachregimedecreasestheequityvaluewiththe
sameintuitionasthecaseinfigure2.5,unlessalowerclosurethresholdcanbeachieved.
Albeit,ifthebankcouldcloseatalowerlevelofcash-flowunderregimeGood,there
wouldbenoincentivetoswitchtotheBadregimeinthemeantime.Astheincentive
forchoosingregimeBadisalowerclosure,wecanexcludeallmultipleswitchingswhich

20

CHAPTER2.ASWITCHINGMODELINBANKING

aretoocostly.Doingso,thereisatmostoneswitchingpointforeachregimeandthe
switchingregionsareconvexsets.Thatmeanstheoptimalcontinuationareaofeach
regimeisaconvexsettooasshowninparts(a)and(b)offigure2.4.
Proposition2.6AssumeoptimalSGandSBexistuniquely.Thenswitchingpointsare
differentandSG<SB.

ProofThehigherdriftoftheGoodregimeincreasesthesocialvalueandtheequityvalue
ofthebankwhenthecash-flowrisessufficientlysuchthattheseclaimsasymptotically
convergetotheclaimsonanunleveredbankwiththesamecash-flow.Hence,above
somecertainlevelofthecash-flow,abankoperatingunderregimeGoodstickstoit.
IfthebankwasoperatingunderregimeBadwouldswitchtoregimeGoodabovethis
level.Thus,thethresholdlevelisanupperboundaryoftheoperationareaoftheBad
regime.Undertheassumptionofuniquenessofswitchingpoints,CBisaconvexsetwith
lowerboundaryXSC.ItfollowsthattheupperboundaryisSBandCB=(XSC,SB).
Further,thecontinuationareaoftheGoodregimeisconvexsinceSGisunique.Having
monotonicallyincreasingequityvalues,SG>XSCsincebydefinitionΨˆB(XSC)=0and
ΨˆB(XSC)=0while
ΨˆB(SG)=ΨˆG(SG)+k>0(k>0).
Fromlemma2.3,theoperationareaofregimeGoodisasupersetofSWB=[SB,∞).
Therefore,thebankshouldsticktooperationunderregimeGoodaboveSGandCG=
(SG,∞),whereSG≤SB.
IfswitchingfromGoodtoBadandfromBadtoGoodoccuratthesamevalueof
cash-flow,i.e.SG=SB=S,thenwehave

From(2.27),wehave

ΨG(S)=ΨB(S)−k,ΨB(S)=ΨG(S)−k,
Ψ´G(S)=Ψ´B(S).

ΨG(S)=ΨB(S)−k
=ΨG(S)−k−k
=ΨG(S)−2k.

(2.27)(2.28)

whichholdsonlyifk=0.Bycontradictiontok>0,thefirstassumptionisnotsatisfied,
i.e.SG=SB.

Accordingly,abankwithregimeGoodgamblesforresurrectionwhenindistressand
increasestheriskloosingtheexpectedreturn.Thenhigherriskwillhelptoclosethe
bankWheninthealowercash-flolevweloffallsbcash-floeloww.SG,Figurethe2.7bankshoopwseratingtheinoptimaltheGoostrategiesdregimeoftheswitcbank.hes

2.4.QUANTIFICATIONOFTHEOPTIMALSTRATEGIES

21

totheBadregime.HavingtheBadregime,whenthecash-flowdropsbellowXSC,the
bankwillclose.Butifthecash-flowincreasesandraisesaboveSB,thebankswitches
fromBadtoGoodinordertobenefitfromlargerreturn.

3.Conjecturea)Anincreaseink>0decreasesvaluefunctionsΨˆi,i∈{G,B}.
b)Whenswitchingiscostlessk=0,theequityΨioreachesmaximumvalue,i.e.∀x,Ψio(x)>
Ψˆi(x),i∈{G,B}.

Theintuitionisthattheequityvaluesareamericanoptionlikeclaims.Thehigher
thestrikeprice,thelowertheoptionvalue.Thatmeanstheequityvaluedecreaseswhen
switchingcostincreases.Therefore,whenk=0theequityvaluesaremaximal.

Remark2.3Whenswitchingcostishighthebankdoesnotswitchunlessinexcessive
need.Therefore,Iexpectthehigherk,thebankswitchesatalowerleveltoBadregime,
i.eatalowerSG.Similarlyfortheotherwayofswitching,abankwithregimeBad
postponesswitchingtoalargervalueofcash-flow.Itmeansthatthehigherk,thelarger
SB.Interval(SG,SB)expandsbyincreasingkwhileequityvaluesdecrease.However,
SGispreventedfrombeingarbitrarilysmallsinceSG>XSC>0.Hence,thereexistsk∗
suchthatanyk>k∗rulesoutswitchingfromGoodtoBad.Thebankerstopsoperation
underregimeGoodinsteadofexpensiveswitch.ThenwehaveSWB=(SB,+∞)and
SWG=.ThisgivesthatCB=(XSC,SB)andCG=(xG,+∞)asfigure2.6shows.

Remark2.4Inthelimitwhenk=0,itfollowsthatSG=SB=S.Havingthecostless
switchingopportunity,theargumentofremark2isnolongervalid.Atanylevelof
cash-flow,thebankcaninstantlyswitchwithoutanycosttoincreasetheequityvalue
distress.esurvivor

2.4QuantificationoftheOptimalStrategies

Applyingasystemofoptimalityconditions,ImodeltheequityvalueoftheBadregime
forthecash-flowboundedinaninterval[XSC,SB]andtheequityvaluefortheGood
regimeintheinterval[SG,+∞).AsthesolutionofthesecondorderODEfrom(2.22),
wehavethegeneralclosedformsoftheequityvaluefunctions:
ΨG(x)=x−1+K1xγG1+K2xγG2,(2.29)
µ−rGΨB(x)=r−xµB−1+K3xγB1+K4xγB2.(2.30)

22

CHAPTER2.ASWITCHINGMODELINBANKING

Addtheboundaryconditionsforallbarriers:

ΨG(SG)=ΨB(SG)−k,(2.31)
ΨB(SB)=ΨG(SB)−k,(2.32)
x→+∞⇒ΨG(x)asy−mptotical→ly.x−1,(2.33)
µ−rGΨB(XSC)=0.(2.34)
ThenK1=0from(2.33),andIfindtherestofcoefficientsversusSG,SBandXSCfrom
thesystemofequationsbelow:
rX−SµCB−1+K3XSγCB1+K4XSγCB2=0
r−SµGG−1+K2SGγG2=r−SµGB−1+K3SGγB1+K4SGγB2−k(2.35)
rS−BµB−1+K3SBγB1+K4SBγB2=r−SµBG−1+K2SBγG2−k
Sincethecoefficientsarecomplicatedfunctionsofbarriers,Ipresentthemindetailin
theappendix.Giventhecoefficientsintheclosedformsoftheequityvaluefunctions,I
usetheoptimalityconditions(2.25),(2.26)andthesmooth-fitpropertybellowinorder
todetermineSG,SBandXSC,

Ψ´B(XSC)=0.(2.36)
Thefollowingsystemofnon-linearequationsdeterminesallbarrierssimultaneously:
11+K2γG2SGγG2−1=+K3γB1SGγB1−1+K4γB2SGγB2−1(2.37)
r−µGr−µB
1+K3γB1SBγB1−1+K4γB2SBγB2−1=1+K2γG2SBγG2−1(2.38)
r−µBr−µG
1+K3γB1XγSBC1−1+K4γB2XSγCB2−1=0(2.39)
µ−rB

SubstitutingforK2,K3andK4in(2.37)-(2.39),Ihaveasetofthreenonlinearequations
tosolveforthethreeunknownvariables.Sincetheequationsalsocontaincrossmultipli-
cationsoftheunknownvariables,ananalyticalsolutionisnotpossible.17Suchmodels
canonlybesolvednumerically.However,thenextpropositionconfirmstheexpected
result.

17forxThe<r,closedmakeitformimpoftheossibleequittoyvapplyaluethepointsapproacouththatbyVXSathC>and0.PhamThisprop(2007)ertytoandgetneexplicitgativeout-flosolution.w

2.4.QUANTIFICATIONOFTHEOPTIMALSTRATEGIES23
Proposition2.7Insystemofequations(2.37)-(2.39)thereexistXSC,SGandSBsuch
that:(1)XSC>0,
(2)SG=0andSG=XSC,
(3)SB<∞.
endix.appSeeofPro2.4.1TwoAlternativeCases
Beforefindingnumericalevidencesforthegeneralstopping-switchingmodel,Iexplain
twocasesinwhichthegeneralmodelnolongerfits.Remarks2.3and2.4discusstwo
specialcasesfortheswitchingcostk.Equations(2.37)-(2.39)cannotgivesolutionsfor
k>k∗orwherek=0.Therefore,inthissubsectionIadjusttheframeworkofthe
modelforeachofthetwoboundarycases.18
hingSwitcCostless2.4.1.1Fork=0thatSG=SB=S,theclosedformsoftheequityvaluefunctionsarethe
sameas(2.29)-(2.30).Buttheboundaryconditionsneedtochange,aswehaveonly
threeboundaryconditionsforfourunknownKj’s,
ΨˆG(S)=ΨˆB(S),(2.40)
x→+∞⇒ΨˆG(x)asy−mptotical→ly.x−1,(2.41)
µ−rGΨˆB(xSC)=0.(2.42)
From(2.41),againK1=0in(2.29).Sinceonly(2.40)and(2.42)arenotenoughfor
determiningK2,K3andK4,Ihavetoaddsmooth-pastingcondition(2.28).
Givencoefficients,theoptimalitycondition(2.35)determinestheclosurethreshold.
However,anotheroptimalityconditionisnecessaryinordertofindtheswitchingpoint.
Toachieveanoptimalclosureandswitchingstrategytogether,Iusethefollowingopti-
malitycondition:19ˆˆ
∂∂ΨSG−∂∂XΨG|x=XSC∂∂XSSC=0,(2.43)
CSwhere∂X∂(ΨˆB(XSC))
∂SSC=∂(ΨˆB∂(SXSC)).(2.44)
X∂CS18Iexcludetherigorousdetailsofcalculationasnonlinearsystemsofequationsdonotgiveanyexplicit
solution.19(1998)LelandSee

24CHAPTER2.ASWITCHINGMODELINBANKING
Sincethesmoothfitpropertyissatisfiedattheswitchingpoint,optimizingequityvalue
fortheGoodregimeyieldsoptimalityoftheequityvalueundertheBadregime.Con-
dition(2.43)takesintoaccountthetotaldifferentialofthevaluefunctionandcondition
(2.44)considerstheoptimalityofclosurewithrespecttotheswitchingstrategy.The
systemofnon-linearequations(2.28)and(2.43),substitutingfor(2.44),indicatesthe
ts.oinpcritical2.4.1.2TooCostlySwitching
Supposek>k∗andthebankintheBadregimeneverswitches.Theassociatedequity
valuefunctionisgivenby(2.13)and(2.14),
xxxGΨˆG(x)=EG(x)=r−µG−1+(1−r−µG)(xG)γG2(2.45)
xG=γG2γ(r−−µ1G).(2.46)
2GTheclosedformofΨˆBis(2.30)withcoefficientsindicatedby(2.32)and(2.34).Then
wehaveoptimalityconditions(2.26)and(2.35).
ExamplesNumerical2.5Thissectionpresentsthreeexamplesoftheswitchingmodelcombinedwithclosurefor
thegeneralandspecialcases.First,weshouldlookatthecrossingbehavioroftwo
equityvaluefunctions,assumingnoswitch.TheparametersinrealvaluesareinTable
1.arametersP1.ableTParameterrσGσBµGµB
Value0.10.080.20.030.02
Table2showstheclosurepointsunderthetwodifferentregimeswithincentivesfor
switchingsinceclosuressatisfyxG>xB.Figure2.8sketchestheequityfunctions.Then
Ibuildthepropersystemofequationsforthegeneralcaseandalternativessubstituting
forparameters.NotethatnonlinearequationsbasicallyhavemultiplesolutionsandI
havetochoosethefeasiblesolution.
Table2.ACrossingCase
γG1γG2γB1γB2xGxBEB(xG)
2.79714-11.17212.23607-2.236070.06424920.05527860.023886

EXAMPLESNUMERICAL2.5.

25

2.5.1GeneralCase:TwoSwitches
SBSince.HothewevBader,fromregime(2.22)alwaaysswitcnecessaryhestoandGoosufficiend,SBtc<+ondition∞andfortheswitchigherhingkfromtheGolargerod
toBadis0<k≤ˆΨB(SG)−ΨˆG(SG)suchthatSGitselfisdependentonk.Thesolution
intable3showsthatSG>xGdoesnotnecessarilyholdsinceswitchingopportunity
increasestheequityvalues,ΨˆG>EGandΨˆB>EB.Evenk>EB(xG)givestwo
ts.oinphingswitc

Table3.CombinedSwitchingandClosure

k10−150.010.020.050.10.15

XSC0.0523540.0527550.0527460.0531880.0537150.054141

SG0.1034700.0837860.0780510.0685650.0601240.054774

SB0.1034710.1161890.1204770.1330160.1582410.190163

Theinterval(SG,SB)expandsbyincreasingk.Whentheswitchingcostislarge,the
bankwaitstillitisnecessarytochangetheregimeofoperation.Hence,itswitchesto
theBadregimeatamorestressfullevel.Withk=0.15theswitchingpointSGisvery
bcloseeingtolessthethanclosureXSCwhicthreshold.hisMyinfeasible.tryfor20kThelargerbankthanoporeratingequaltounder0.156BadendedregimeupalsoSG
theswitccasehesoftokGo=o0.d02.whenitApplyingcanlocowvveraluestheofcostkIandfindthistwoswitcincreaseshingSBp.ointsFigureconv2.9ergeshosucwsh
thatfork=10−15theyareextremelyclose.

hSwitcCostlessA2.5.2Whenk=0,costlessswitcheshappenatthesamelevelofcash-flow,SG=SB=S.The
bankinregimeBadswitchestoGoodifthecash-flowrisesS,whilethebankoperating
underregimeGoodswitchestoBadassoonasthecash-flowfallsbellowS.Then,ifthe
cash-flowdecreasesfurtherandreachesXSCthebankwithBadregimestopsoperating.
Underaboveparameters,asfigure2.10showsthetwocriticalpointsareS=0.1034702
andXSC=0.0523536.Onecouldexpectthesevaluesfromthegeneralcaseabove.The
closurethresholdobtainsthesmallestvalue,comparingtotable3.Theswitchingpoint
Sisinsideintervals[SG,SB]kforallk>0.
20Unfortunately,theanalysiscannotexplainthisboundaryfork.

26

CHAPTER2.ASWITCHINGMODELINBANKING

2.5.3TooCostlySwitch

Table4givesriskreductionresultsofourexampleunderhighswitchingcosts.

Table4.TooCostlySwitchingandClosure

k0.160.20.51100

XSC0.05420280.05440570.05496850.05514540.0552782

SB0.1973260.2282210.5108231.01398101.305

Figure2.11sketchescasek=0.16.AskgrowstheBadregimeisstillbeneficialsince
XSC<xG.Weseeintable4thatthehighertheswitchingcostthelargertheclosure
thresholdandswitchingpoint.Thecriticalpointsarealsolargerthanthegeneral
casewithtwoswitches.ByincreasingthecostswitchingfromGoodtoBadbecomes
unprofitable.However,riskreductionisstillvaluablebutatalargerlevelofcash-flow.

Conclusion2.6

Thispaperdevelopsacontinuoustimemodeltoverifybanks’risk-takingbehavior.Two
regimesofoperationareavailabletoarepresentativebankateachmomentoftime.The
differenceisinbothreturnandrisklevelsoftheportfoliochosenundereachregime.It
isassumedthatthebankisalreadyoperatinginthemarket.Thequestionishowthe
operationshouldcontinuefurtherintime.Investigatingthegamblingforresurrection
rationaleshowsthatwhenthecash-flowdecreasesbelowacertainlevelthebanktakes
morerisk(regimeBad).Ifthecash-flowraisesabovealargerthresholdthebankswitches
tothelessriskyregimewhichgeneratesahigherexpectedreturn(regimeGood).The
cash-flowthresholdsarenamedswitchingpoints.Optimalswitchingstrategiesforregime
GoodandBadarepresented.Insolvencyisdefinedasifthecash-flowfallsbelowthe
depositpaymentateachmoment.Thedepositsarefullyinsured.Intheseverecase
ofinsolvencythatthebank’sequityvalueszero,thebankgoesbankruptandstops
eration.opThisresearchextendstheliteratureonrisk-takingbehaviorinthesensethatitincludes
thechangeinthereturninadditiontotheriskchanges.Hence,advantagesofaregime
createsincentivesforthebanktostickonitorswitchtoanotherregime.Inthisregard,
thepaperpromotestheswitchingmodelofDanglandLehar(2004)whoonlystudythe
riskyassetsubstitutionproblem.Inmymodel,theswitchingsbehaviorisaresultofthe
tradeoffbetweenthereturnandrisk.Optimallythereexistsatmostoneswitchingpoint
foreachregime.However,theswitchingcostaffectsoptimalstrategiesofswitchingand

2.6.ONCLUSIONC27

closureincaseofbankruptcy.Ahighswitchingcostrulesoutrisktakingbylowering
switchingcriteriaandincreasingclosurethreshold.Itinfluencesriskreductionincentives
bypushingtheswitchingpointupward.Costlessswitchingendsupinoneswitching
point,abovewhichthebankoperatesunderGoodregime.Belowthiscriteria,thebank
operatesunderregimeBad,unlessthecash-flowdropsattheclosurethreshold.
Thestopping-switchingproblemwhichoptimizesthebank’sswitchingandclosurestrate-
giesdoesnothaveaclosedformsolution.Theextremecaseswheretheswitchingcost
istoohighortheswitchingiscostlesscannotbesolvedastheboundarycasesofthe
generalsetup.Thus,IcombinetheswitchingmodelofPham(2005b),thebasicstop-
pingmodel,andeconomicalintuitionsinordertooptimizestrategiesofabankforeach
case.Foundingaverificationtheoremforeachcaseisassevereasinventingastochastic
controlmodelwhichcouldsolvethegeneralswitching-stoppingproblem.Thisshould
bedoneasfurtherstochasticcontrolstudies.
Theentranceproblem,i.e.underwhichregimeabankstartsoperationatt=0is
leftforfurtherresearch.Theinitialregimedependsontheinitialcapital.Havingthe
relationshipbetweenthecash-flowandthecapital,theresultsofswitching-stopping
modelcanexplainthisproblemonlypartially:weneedatleastpositivecapitalwhich
requiresx0>XSC.Ifx0∈SWi,i∈{G,B}thebankoptimallystartsunderregimej.
Incaseofbankruptcy,the“lenderoflastresort”(LOLR)hastobearalldeposits.Thus,
thehighriskofinsolvencyisnotinfavorofthesupervisoryagencywhohastoplayasthe
LOLR.Thisresearchexcludesoutsiders’intervention.Yet,thesetupcanbeextended
toincludetheregulator’sroleasthesocialplannerwhomaximizesthetotalsurplusof
eration.opbankthe

28endixApp

CHAPTER2.ASWITCHINGMODELINBANKING

ProofofProposition2.1.WheretheequityvalueinregimeGoodisalwayshigher
thaninregimeBad,itistrivialthattheclosurethresholdislower.Fortheotherway,
ifxG<xB,whileµG>µB,thenforthefirstderivativesE´(.)wehave
∀x>xB>xG,
xx11GxE´G(x)−xE´B(x)=x(−)−γG2(−1)()γG2
r−µGr−µBr−µGxG
xBxγB2
+γB2(r−µB−1)(xB)
>xB(1−1)−γG2(xG−1)(x)γG2
r−µGr−µBr−µGxB
+γB2(xB−1)(x)γB2
r−µBxB
xxBG>((r−µG−r−µB)
xGxBxγB2
−γG2(r−µG−1)+γB2(r−µB−1))(xB)
xG=((r−µG)(1−γG2)+γG2−
0(xB)(1−γB2)+γB2)(x)γB2
r−µBxB
0.0=Therefore,E´G(x)>E´B(x),∀x>xB;andsinceEG(xB)>EB(xB)=0,forallpossible
cashflowsxwehaveEG(x)>EB(x).
ProofofProposition2.2.IfµG=µB=µ,thendµ=0in(2.19).Itfollows
from(2.17)thatxG>xB.Therefore,∀x>xG,
EG(x)−EB(x)=(1−π−xG)(x)γG2−(1−π−xB)(x)γB2
rr−µxGrr−µxB
<(1−π−xB)((x)γG2−(x)γB2)
rr−µxGxB
<(1−π−xB)((x)γG2−(x)γG2)
rr−µxGxB
<(1−π−xB)((x)γG2−(x)γG2)
rr−µxGxG
.0=

APPENDIX2.6.

Solutionstosystemofequations(2.35):
K2=(r2(SBγB2SGγB1−SBγB1SGγB2−k(SBγB2+SGγB2)XSγCB1+kSγBB1XSγCB2+
kSGγB1XSγCB2)+SB(SGγB2XSγCB1−SGγB1XSγCB2)(µG−µB)+k(−SGγB2XSγCB1
+SGγB1XSγCB2)µGµB+SBγB2(SGγB1µG(XSC+µB)−XSγCB1(SG(µG−µB)
+kµGµB))+SBγB1(−SGγB2µG(XSC+µB)+XSγCB2(SG(µG−µB)+
kµGµB))+r(k(SγGB2XSγCB1−SGγB1XSγCB2)(µG+µB)+SBγB2(kXSγCB1
γγγ(µG+µB)−SGB1(XSC+µG+µB))+SBB1(−kXSCB2(µG+µB)
+SGγB2(XSC+µG+µB))))/((SBγB2SGγG2XSγCB1−SBγB1SGγG2XSγCB2+SBγG2
(−SGγB2XSγCB1+SGγB1XSγCB2))(r−µG)(r−µB))

K3=(XSγCB2(k(SBγG2+SGγG2)(r−µG)(r−µB)−(−SBγG2SG+SBSGγG2)
(µG−µB))+(SγB2SγG2−SγG2SγB2)(r−µG)(r−XSC−µB))
GBGB/((SγB2SγG2XγB1−SγB1SγG2XγB2+SγG2(−SγB2XγB1+
BGSCBGSCBGSC
SGγB1XSγCB2))(r−µG)(r−µB))

K4=(XSγCB1(−k(SBγG2+SγGG2)(r−µG)(r−µB)+(−SBγG2SG+SBSGγG2)
(µG−µB))−(SγB1SγG2−SγG2SγB1)(r−µG)(r−XSC−µB))/
GBGB((SγB2SγG2XγB1−SγB1SγG2XγB2+SγG2(−SγB2
BGSCBGSCBG
XSγCB1+SGγB1XSγCB2))(r−µG)(r−µB))

29

30

CHAPTER2.ASWITCHINGMODELINBANKING

ProofofProposition2.7.Proofbycontradictionforeachcase:
(1)ByreplacingcoefficientsK2,K3andK4in(2.39)andorderingpowersofXSCinthe
evhaIequation

(XSγCB1+1(r−µG)(1−γB1)(SBγG2SGγB2−SBγB2SGγG2)+

XSγCB1γB1(r−µG)(r−µB)(SBγG2SGγB2−SBγB2SGγG2)+

XSγCB1+γB2(γB1−γB2)(k(r−µG)(r−µB)(SGγG2+SBγG2)+(µG−µB)

(SBSGγG2−SGSBγG2))+XSγCB2+1(r−µG)(−1+γB2)(SBγG2SGγB1−SBγB1SGγG2)+

XSγCB2γB2(r−µG)(r−µB)(SBγB1SGγG2−SBγG2SGγB1))/

((XSγCB1+1(SBγB2SGγG2−SBγG2SGγB2)+

XγB2+1(SγG2SγB1−SγB1SγG2))(r−µG)(r−µB))=0(2.47)
SCBGBG
Simplifyingthisequationtermbytermwithrespecttothedenominator,weseethat
thelefthandsideoftheequationgivesthefollowinglimit
.−lim1−γB2γB2
XSC→0r−µXγB1−γB2+1
GBGBBXSC+(SγG2SγSBC1−SγB1SγG2)
(SBγB2SGγG2−SBγG2SGγB2)
WhenXSC→0,thefactthatthelimitmustgoto0givesγB2=0andγB2−1=0
tradict.conhwhic

APPENDIX2.6.

31

(2)Similartopart(1),Ireplacethecoefficientsin(2.38)andorderitversusthepowers
ofSG.21ThelimitofresultingequationwhenSG→0isequalzeroif
SBγB1(r−µG)(r−µB)
γB2=γB1or(XSC)=−k(r−µG)(r−µB)−XSC(r−µG)
and:r=µGorr=µBork=0orγB2=0orSB=∞orXSC=0
and:µG=µB.
Thesenecessaryconditionsareinconsistentandnotsatisfied.Therefore,SG=0.
AssumeSG=XSC=S,rewrite(2.38)and(2.39)replacingforthecoefficientsandS.
Wemusthaveequation(2.38)plusequation(2.47)equalszero.Thisgives:
S=(1−k)γG2(r−µG)
1−γ2GFirstofallwefindS<0fork>1.Next,k=1yieldsS=0whichisimpossible.
Then,fork<1thetwosmooth-pastingconditions(2.38)and(2.39)nolongerhold.
Thecontradictionrejectsthehypothesis.

(3)BySBconvergingto∞,theequation(2.37),substitutingforcoefficientsK2,K3and
K4,holdsifµG=µBorγB1=1.Sincethelattercannotbetrue,equalityofthedriftsis
thenecessarycondition.Underthiscondition,(2.47)indicatesthatXSCisexactlythe
closurethresholdinno-switchcasefortheBadregime.Followingthisresult,SGfrom
(2.38)isγB2
SG=xBkγG2(1−γB2)
γG2−γB2
SuchSGmaycauseSG<xB.Despitetheresultisconsistentwithproposition2.2,the
assumptionofunequaldriftsrejectsthehypothesisofSG→∞.

21Sincetheresultingequationismorerigorousthanhelpful,Idonotmentionithere.

32

Figures

CHAPTER2.ASWITCHINGMODELINFigure2.1:Theequityvaluevs.thecash-flow,whereµG=µB.

BANKIFigure2.2:Theequityvaluevsthecash-flowwhereµG>µBandxG<xB.

NG

FIGURES2.6.

Figure2.3:Valueofequityvsthecash-flowwhereµG>µBandxG>xB.

33

opFigureerates2.4:underExamplesregimeiofasconlongtinasuationx∈Ciandandswitcitwillhingswitcareashtoforaregimeregimej=ii∈as{soG,onB}as.xThe∈SWbanki.

Figure2.5:Anexampleofmorethanoneswitchingpointforeachregime.

34

2.6:Figure

2.7:Figure

OnlyoneCHAPTERhingswitcp2.t:oinASWITCHINGfromregimeBadMODELtoregimeINGoBANKIodOptimalswitchingdecisionsandtheoptimaloperationregions.

2.8:Figure

rivialTcases.NG

2.6.FIGURESFigure

2.9:

A

general

case

in

optimal

ing,hing-stoppswitc

k

=

02..0

35

36

CHAPTERFigure

Figure

2.11:

2.10:

oT

o

2.ACostless

costly

SWITCHINGhing.switc

hing,switc

k

=

MODEL16..0

INBANKING

3Chapter

SwitchingModelsforBanking:Is
GamblingforResurrectionValid?

ductiontroIn3.1

“Gamblingforresurrection”hasbeencoveredinthefinancialaswellaspoliticaleco-
nomicsliterature.Theconventionalwisdomisthatanagentunderdistressgambles
andtakeshigherriskinordertosurvive.Understandardassumptionthatabankcan
chooseoneoftwodifferentregimesofoperation,i.e.oneportfoliowithahigherreturn
andanotherportfoliowithahigherrisk,therelationshipbetweenabank’scapitaland
risklevelismonotonic.Thecontributionofthispaperistoshowthatthemonotonicity
doesnotalwaysholdtrue.
Inthefirstpart,thispaperexaminestheendogenouschoiceofrisk-returnregimeforrisk-
neutralbankerswhomaximizetheequityvalue.Thebank’swealthisnormalizedwith
respecttothedepositvalue,andthereishenceaonetoonerelationshipbetweenthe
netwealthandthecapital(equity).Withlimitedliabilityandfullyinsureddepositsthe
bankincreasesitsrisk,loosingpartoftheexpectedreturnoftheinvestment.However,
itcanthenoperateunderasaferregimewhenhavingenoughwealth.Inadiscrete
timestaticmodelthe“cutoffvalues”belowwhichabanktakesmoreriskarefoundin
termsofthenetwealth.Therisk-takingstrategyismainlyinfluencedbythecostof
efforttoreducetherisk.Ifthebankcouldoperatewithoutanextracosttomonitor
itscreditors,itwouldincreasetheriskonlybelowauniqueleveloftheinitialwealth.
However,ifthebankhastopayforhavingabetterchancetosucceedinitsoperation,
theoptimalrisk-takingstrategyisobtainedbymultiplecutoffvalues.Infact,abank
maygobankruptatfailurewithahigherleveloftheinitialwealthunderthelessrisky
portfoliobecauseoftheeffortcost.Thiscanleadtothemultiplecutoffvaluepolicy.

37

38

CHAPTER3.ISGAMBLINGFORRESURRECTIONVALID?

Thesecondpartofthispaperextendsthesetuptoadynamicmodeltoinvestigate
theintertemporalrisk-takingbehavior.Thetwoperiodsofthemodelareconnected
toeachotherthroughthedividendpolicy.Atthebeginningofeachperiodthebank
choosesarisk-returnregimeofoperationbymaximizingthenetpresentvalue(NPV).
Themainfindinginthispartisthatbesidegamblingforresurrectionabankmayreduce
itsriskbyswitchingfromtheriskyregimetothesaferregimeifthelowriskportfolio
issufficientlyadvantageous.Theoptimalrisk-returnchoicevariesdependingonthe
capitallevelandthedividendpayment.Theriskreductionorgamblingforresurrection
strategyisimpactedalsobytheinterestratewhichdeterminesthedepositpayment.
Duetotheassumptionthatfailurebringsbankruptcy,theuniquenessofacutoffvalue
isrobustwithrespecttotheeffortcost.
Risk-takingbehaviorhasbeenwidelydiscussedintheliterature,mostlyasabasefor
regulationstudies.Decampsetal.(2004)verifyBasel(II)regulatorypoliciesina
continuous-timemodelbutwithoutdynamicregimeswitching.Ifollowthemtospecify
effortcosttothesaferregime.Thediscrete-timemodelofmypaperwhichincludes
endogenousregimeswitchingcontributestothisavenueofstudies.Agoodreferenceof
switchingmodelsisthecontinuous-timemodelofDanglandLehar(2004).Theyassume
standardgamblingforresurrectionwithtwoswitchingpointswhichareidenticalinthe
absenceofswitchingcost.Nevertheless,acontinuoustimemodeldoesnotnecessarily
haveclosedformsolutions.Thisobstaclemakesitimpossibletoverifythebank’sen-
dogenousportfoliochoiceanalyticallyinageneralcontinuoustimemodel.Therefore,in
thischapterIstructurethebank’sprobleminadiscrete-timemodel.
ThemodelpresentedinthischapterisgeneralizedcomparingtoDanglandLehar(2004)
inthesensethatbothreturnandrisklevelofportfoliochangefromoneregimeto
another.Further,Idefinenodeficiencyassumptiononanyregimetotheextentthat
eventheriskierregimemayhavepositivenetpresentvalue(NPV).Thisisdifferent
fromtheassumptioninDecampsetal.(2004)whichgivesprioritytoliquidationrather
thanoperatingundertheriskierregime.Alsoincreditrationing,StiglitzandWeiss
(1981)assigntheriskierprojectsforbeinginefficient.Incontrary,inthispaperIdiscuss
advantagesanddisadvantagesofeachregime,freefromdeficiencyassumptions.
Thefocusofstudiesinsomeotherliteratureisontheriskofcreditorsorfirms.In
thatview,thebankhastotakeamonitoringpositiontoavoidriskycreditors.Stiglitz
andWeiss(1981)relatetherisk-takingbehaviorofcreditorstotheinterestrateand
analyzecreditrationing.Intheirpaper,asdemandandsupplyofloansarefunctionsof
theinterestrate,itplaystheroleofscreeningdeviceforthebank.Highinterestrates
attractriskierborrowersanddecreasethebank’sexpectedprofit.Thebankisreluctant
totakeriskandmonitorsitscreditorsthroughtheinterestrateatwhichtheyarewilling
toborrow.Inthelender-borrowerrelationship,BerlinandMester(1992)defineabank’s
roleinpreventingafirm’sgamblingforresurrection.Thebankmayreceiveanoisysignal

3.2.THEONE-PERIODMODELWITHDISCRETERETURN

39

indicatingthesuccess/failureofthefirminordertothenallow/restrictrenegotiationof
ts.enanvcoloanMyworkisintheclassofstudiesconcernedwithregulatingthebanks’risk-takingand
dealdirectlywiththeriskincentives.DiamondandDybvig(1983)studybankdeposit
contractsandrisk-takingincentivesofbank-managerswhichleadtospeculativebank-
runs.MailathandMester(1994)solvethebank-regulatorygameinatwoperiodmodel
inwhichthebankaccessestoonerisk-freeandoneriskyasset.Theyexplainregulatory
forbearanceandhowtheregulatoryagencycannotcommitexantetoatoughclosure
policy.Thebanktakeshigherriskandtheregulatorwantstoimposeclosurebeforethe
netpresentvalueofthebank’sassetsbecomenegative.However,fromasocialwelfare
perspective,itisalmostalwaysoptimaltoletanunder-capitalizedbankcontinueto
operate.Thisgeneratesbadincentivesforthebankersfromanexantepointofview
totakerisk.AcharyaandYorulmazer(2007)observeaherdingbehavioramongmany
bankstoincreasetheriskasaresultofthemanagers’moralhazard,andtheregulator’s
problemregardingclosurepolicy.CordellaandYeyati(2003)analyzethemoralhazard
problemwithinamulti-periodmodelbutassumeindependentrisk-takingstrategyin
eachperiod.Theexistingrisk-takingstudiescanbesummerizedaseitheranagent
choosesbetweenasafeassetandariskyinvestmentinaoneperiodmodelorifa
dynamicmodelisprsented,eachperiodisindependentandunaffectedbyotherperiods.
Thismotivatesmyworktochallengetheclassicalideaofgamblingforresurrection,
allowingfullyendogenousrisk-takingbehavior.
Thefollowingsectionsetsupaoneperiodmodeltodeterminetheswitchingcutoff
valuesandtheassociatedpolicies.Section3.3developsthesetuptoatwoperiodmodel.
Insection3.4theoptimaldividendpolicyisinvestigated.Section3.5presentssome
numericalexamplestoillustratetheresult.Section3.6concludes.Theappendixincludes
theproofofaremark.

3.2TheOne-PeriodModelwithDiscreteReturn

Assumeariskneutralworld.TheinitialstatusofthebankisW0whichconsistsofinitial
equity,A0≥0,anddepositprincipalnormalizedtoone,i.e.W0=A0+1.Deposits
arefullyinsuredandshareholdershavelimitedliability.Operationofthebankhasa
constantreturnstoscaletechnologywithrateofreturn(RR)ziunderregimei∈{0,1}.
Attheendofperiod,stochasticvariablezireturnsRiincaseofsuccesswhichoccurs
withprobabilityPiunderregimei.Incaseoffailure,therateorreturnofregimeiisri,
Ri>0withprobabilityPi,
zi=ri>−1withprobability1−Pi.(3.1)

40

CHAPTER3.ISGAMBLINGFORRESURRECTIONVALID?

FollowingStiglitzandWeiss(1981),assumethatregime1hashigherexpectedreturn
andmoreconcentrateddistributionthanregime0.Thismeansthatµ1>µ0where
µi=PiRi+(1−Pi)ributR1<R0andr0≤r1.ItmustbethenP1>P0.Further,
Iassumethatregime1bearsmonetaryeffortcoste≥0.Thecostcanbeinterpreted
asexpensesofmonitoringcreditors(delegatedmonitoringasinDiamond(1984)).By
definition,noneofthetworegimesisessentiallyinefficientorlesspreferable.Theequity
valueattimet=1is,
W1=max(0,(W0−ie)(1+zi)−C).(3.2)
Inordertocreateincentivesforthebanktomonitoritscreditors,theaddedvalueof
higheffortregimeshouldexceeditscost,i.e.forW0≥1,
W0(1+µ1)−W0>e(1+µ1)⇔e<1+µ1µ.(3.3)
1Theriskfreeinterestrateinthemarketis0<rf<1,whererf<Rifori∈{0,1}.
Generallythebankfacestwooptimizationproblems.Firstitmustdecideonhowmuch
capitaltoinvestinariskyregime,andseconditmustdecidewhichregimetotake.Due
totheriskneutralityassumption,thebankinvestseitherallthewealthornothing.We
canthustranslatethebank’sfirstproblemtoanentrancedecision.Thebankdecides
toenterthemarketandstartsoperationiftheexpectedprofitofinvestinginarisky
regime1ismorethantheexpectedvalueofsavingtheinitialcapital.Definediscountrate
β=1+rf,thebanktakeanyriskyregimeiif,
A0≤βE(W1,i).(3.4)

(3.4)

Forthebanktostartoperation,anecessaryconditionisthatthebankmustbesol-
ventiftheoperationsucceeds.Otherwise,thebankwouldneverchoosethatregimeof
operation.Thisrequiresforanyregimei,

CW0≥1+R+ie,i∈{0,1}.(3.5)
iSince1+CR0<1+CR1+e,thehigheffortregimemaycauseinsolvencywheretheloweffort
operationissolvent.
Yet,iftheNPVisnegativethebankhasincentivetooperateunderariskyregime
becauseofahigherreturn(comparingtotheriskfreeRR)incaseofsuccess.Therefore,
solvencyandprofitabilityofariskyregimeiisthenecessaryandsufficientconditionfor
thebanktotakedepositandstartoperationunderthatregime,
A0≤βPi((A0+1−ie)(1+Ri)−C).(3.6)

3.2.THEONE-PERIODMODELWITHDISCRETERETURN

41

TosolvetheinequalityforA0,thesignofβPi(1+Ri)−1isimportant.Thistermcanbe
interpretedastheNPVofsuccessreturnforinvestingoneunitinriskyregimei,although
theeffortcostofregime1mustbeconsideredtoo.Manypapers,e.g.MailathandMester
(1994),assumethatariskyregimehasnegativeNPV.Withtheirassumptiontheresult
isgamblingforresurrection:ifβPi(1+Ri)<1,thebankchoosesriskyinvestmenti
wherethecapitalisbelowathreshold,
A0≤βPi(C−(1−ie)(1+Ri)).(3.7)
βPi(1+Ri)−1
whichismeaningful(positive)iffitsnumeratorisnegative.Alternatively,supposesuc-
cessofanyriskyregimeisprofitable,

βPi(1+Ri)>1,i∈{0,1}.(3.8)

Contrarytotheclassicalidea,thebankoperatesunderariskyregimeiff,
A0≥βPi(C−(1−ie)(1+Ri)).(3.9)
βPi(1+Ri)−1
Theuniquecut-offvaluepolicyobtainsfortheentrancedecisionifthenumeratoris
positive.Otherwise,thebankstartsoperatingunderariskyregimeforanypositive
levelofinitialcapital.1When(3.9)holds,thebankentersthemarketandtakesdeposits.
Hence,fortheinitialwealthwehave,
W0≥Gi=βPi(C+ie(1+Ri)−1).(3.10)
βPi(1+Ri)−1

Nowfocusonthebank’ssecondproblem:optimizationofrisk-returnregime.Regime
iisriskyforsomelevelofinitialwealthifitsfailurebringsoutinsolvency,i.e.W0<
1+Cri+ie.Withoutlossofgeneralitysuppose1+Cr0<1+Cr1+e.Againthehigheffort
regimethoughsucceedswithahigherprobabilitycausesinsolvencyincaseoffailure
forinitialwealth1+Cr0<W0<1+Cr1+ewherethebankundertheloweffortregimeis
solvent.2Correspondingly,theregimechoiceisdifferentineachpossiblecase.Thebank
optimallychoosesregimei0atthebeginningoftheperiodtomaximizetheexpected
valueofitsequity,
E(W1,i0)≥E(W1,i),i0=i∈{0,1}.(3.11)
Thefollowingpropositiondescribesthebank’soptimalstrategy.

1CordellaandYeyati(2003)havesimilarresult,albeitwithrespecttorateofreturnsandinterest
rate.21+µ1>WithoutP0(1+theR0),butassumptiontheresultregardingdoesthenotchange.thresholds,proposition3.1hastoberepeatedforacase

42

CHAPTER3.ISGAMBLINGFORRESURRECTIONVALID?

Proposition3.1(I)SupposeforexpectedreturnsP1(1+R1)>P0(1+R0).
(a)IfP1(1+R1)>1+µ0,theregimechoiceischaracterizedbyauniquecutoff,i.e.for
eachsetofparametersthereexistsonlyonecutoffvalue,intermsofnetwealth,above
whichthebanktakeslessrisk.Figure3.1showspossiblecases.
(b)IfP1(1+R1)<1+µ0,multiplecutoffvaluescharacterizetheoptimalregimechoices
intwooutoffivefeasibleordersofthresholds,S1,S2,S3,1+Cr0and1+Cr1+e,asshown
byfigure3.2.Ineachmultiple-cutoffstrategythreecutoffvaluesintermsofnetwealth
existsuchthatthebank’sregimechoicediffersfromlowertohigherthaneach.Inother
cases,uniquecutoffvaluegivestheoptimalregimechoices.
(II)IfP0(1+R0)>P1(1+R1),thenuniquecutoffvaluepolicyisoptimal.

Proof(I.a)Asfailureofaregimemaycausebankruptcy,findinitialwealthlevels
anassoyciatedreturntoofeaceachhlikregime,elyitfailureisinsolvcase.entThreeonlyatcasesfailurearepofossible:regimethe1,itbankisisinsolvsolvenenttatfora
failure.Compareexpectedprofitsundertworegimechoices.

1.Supposethebankissolventforallreturns,i.e.W0≥1+Cr1+e.From(11)regime
1makesthebankbetteroffiff

(W0−e)(1+µ1)−C≥W0(1+µ0)−C.
Consequently,thebankchoosesregimei0=1iff
W0≥S1=e(1+µ1)/(µ1−µ0),
andi0=0otherwise.

(3.12)

2.When1+Cr0≤W0<1+Cr1+e,failureofregime1makesthebankinsolventbecause
ofeffortcostbutatreturnr0thebankisstillsolvent.Itchoosesregimei0=1iff
P1((W0−e)(1+R1)−C)≥W0(1+µ0)−C⇔
W0≥S2=(P1−1)C+eP1(1+R1)(3.13)
P1(1+R1)−1−µ0

3.Neitherr0,norr1yieldsolvency,i.e.W0<1+Cr0.Thebankprefersthehigheffort
iffregime

P1((W0−e)(1+R1)−C)≥P0(W0(1+R0)−C)⇔
C(P1−P0)+eP1(1+R1)
W0≥S3=P1−P0+P1R1−P0R0.(3.14)

3.2.THEONE-PERIODMODELWITHDISCRETERETURN43
Inspiteofhavingthreethresholds,noticethatS1>1+Cr1+econtradictsS2<1+Cr1+e
ersa,vviceandCS2<1+r1+e⇔
((P1−1)(1+r1)−[P1(1+R1)−1−µ0])C<
(P1(1+R1)−1−µ0)(1+r1)
e[(P1(1+R1)−1−µ0)−P1(1+R1)]
P1(1+R1)−1−µ0
Ce(1+µ0)
⇔1+r1>µ1−µ0.
Therefore,whereS1>1+Cr1+e,S2isnotfeasibleasS2>1+Cr1+e.Thenoptimalregime
is0forall1+Cr0<W0<1+Cr1+e.Yet,S2>1+Cr0contradictsS3<1+Cr0andviceversa.

C1+r0<S2⇔
C(P1(1+R1)−1−µ0−(P1−1)(1+r0))
<(P1(1+R1)−1−µ0)(1+r0)
eP1(1+R1)
P1(1+R1)−1−µ0

C(P1(1+R1)−P0(1+R0)−(P1−P0)(1+r0))
⇔<(1+r0)(P1(1+R1)−1−µ0)
eP1(1+R1)
P1(1+R1)−1−µ0

(3.15)

(3.16)

(3.17)

⇔C(P1(1+R1)−P0(1+R0)−(P1−P0)(1+r0))<
(1+r0)(P1(1+R1)−P0(1+R0))
eP1(1+R1)
P1(1+R1)−P0(1+R0)(3.17)
C⇔S3>1+r0,
Thus,theoptimalregimeis0forallW0≤1+Cr0.ItmeansthatS1istheuniquecutoff
value.Inasimilarway,weenduphavingS2orS3asauniquecutoffvalueiffoneof
themisfeasible,rulingoutfeasibilityoftwoothers.

44

CHAPTER3.ISGAMBLINGFORRESURRECTIONVALID?

grossFigure3.1:success-returnSwitchingofregimestrategy1einaxceedsoneexpperioecteddmoreturndelofwithregimediscrete0.return,expected

(I.b)UnderassumptionP1(1+R1)<1+µ0,theinequality(3.13)changes.Inother
words,nowregime1istheoptimalchoiceforW0≤S2,andregime0isoptimaloth-
erwise.Incontrasttopart(I.a),inequalities(3.16)and(3.17)reverse.HereS2>1+Cr0
requiresS3<1+Cr0,andS2<1+Cr1+edirectsto1+Cr1<S1.IfS1isfeasible,then
feasibilityofS2makesS3feasible,case(a)infigure3.2.However,S2mightbelower
than1+Cr0.Thenfor1+Cr0≤W0<1+Cr1+ewehavei0=0.ButitmeansthatS3isabove
Candtheoptimalregimeis0whereW<C.Case(b)showstheuniquecutoff
v1+ralu0eS1.WithfeasibleS3,feasibilityofS20rather1+r0thancase(a)canalsoleadtocase
(c)whereS1isabove1+Cr1andstillinfeasible.Hence,threecutoffvaluesareS3,S2and
1+Cr1+e.Incase(d),S1isCinfeasiblebut1+Cr1<SC1.ThisdemandsS2<1+Cr1+e,butwe
haveinfeasibilityofS2<1+r0<S3.Therefore,1+r1istheuniquecutoffvCalue.Thelast
possiblecaseis(e)infigure3.2.There,feasibleS3appearswithS2>1+r+ewhich
bringsoutS1infeasible.Hence,S3istheuniquecutoffvalue,asbelowS2b1etweentwo
boundaries1+Cr0and1+Cr1+etheoptimalstrategyisregime1.

(II)Inasimilarapproachas(I.a),weobservethebankisreluctanttoexerteffort.Since
S3<0,below1+Cr0theoptimalchoiceisregime0.Still,ifP1(1+R1)>1+µ0,then
S2>1+Cr0becauseS3<0<1+Cr0.Inthiscase,eitherS1orS2istheuniquecutoffvalue
(cases(a)and(b)infigure3.1).ButP1(1+R1)<1+µ0causesS2<1+Cr0.Therefore,
eitherS1isfeasibleandtheuniquecutoffvalueoritisonlyabove1+Cr1,making1+Cr1+e
theuniquecutoffvalue(cases(b)and(d)infigure3.2).

Remark3.1Inpart(I.b)ofproposition3.1,theexpectedreturnofthehigheffort
regimeequalsitsreturninthelikelycaseofsuccessandthatislowerthantheexpected
notreturntooofsmalltheriskyandthisregime.motivItoatesccursforwhentakingthehigherreturnriskofofregimeregime0in0.caseThus,ofiffailurefailureis

3.2.HTEONE-PERIODMODELWITHDISCRETERETURN45

Figure3.2:Switchingstrategyinaoneperiodmodelwithdiscretereturn,expected
returnofregime0surpassexpectedgrosssuccess-returnofregime1.

makesthebankinsolventunderregime1butsolventforregime0,asaboveS2,the
bankprefersregime0.However,effortcostissmallenoughsuchthatthebankexerts
efforttogainmorethroughhigherprobabilityofsuccess.Thiscanbeseeninthe
intervalbetweenfeasibleS2andS3.Insolvencyatfailureofeitherregime0or1makes
regime1moreinterestingsinceitsfailurehasalowerprobability.Underassumption
P1(1+R1)<P0(1+R0)inpart(II),inexpectationthebankismoreprofitableif
succeedsinregime0thaninregime1.Hence,itisreluctanttochoosethesaferregime.
Thisbringsoutauniquecutoffpolicy,withalargecutoffvaluecomparingtoallother
cases.

Remark3.2Inthebenchmarkcasewithouteffortcost,i.e.e=0,theoptimalregime
isi0=1becauseofitshigherexpectedreturn,aslongasW0≥1+Cr0.Notethatnowthe
orderchangesforboundaries1+Cr1<1+Cr0.Withinthesamemethodasproposition3.1,
cutoffvaluesarefoundbaseduponthesimilarassumptions.Nevertheless,theoptimal
regimechoiceisgivenbyauniquecutoffvalueforeachcase.Theproofisincludedin
endix.appthe

Comparingtheresultsofremark3.2andproposition3.1,weseethateffortcostplaysan
importantroleforthebank’schoiceoftheregimeofoperation.Theclassicalgambling

46

CHAPTER3.ISGAMBLINGFORRESURRECTIONVALID?

forresurrectionobtainsaslongasthereisnoeffortcost.Thisresultisforinstanceseen
inDanglandLehar(2004)ifchangingfromoneregimetoanother,i.e.regime-switch,
costsnothing.Whiletheswitchingcostyieldsseparatethresholdsfortheriskreduction
andgamblingforresurrection,theeffortcostforaportfoliowithhigherexpectedreturn
inducesnon-monotoneregimechoices.Decampsetal.(2004)consideraneffortcost
intheircontinuoustimemodel,buttheyassumenegativeNPVoftheriskierregime
whichmakesitworsethannooperation.Therefore,theyobtainastandardgambling
forresurrectioncutoffpolicy.Themodelinthispapershowsthateffortcostontheone
handandnodeficiency(nonegativeNPV)ofriskyregimesontheotherhandarethe
sourceofamultiplicityofcutoffvalues.
Remark3.3RobustnessCheckofProposition3.1forNon-monetaryEffortCost:Al-
ternatively,theremightbenon-monetaryeffortcostforregime1,whichdoesnotaffect
thereturnoftheportfoliobutinflictsanadditionalmonitoringcostonthebank.Con-
sequently,thebankhastopayeattheendoftheperiodandtheequityvalueis

(3.18)

W1=(1+zi0)W0−i0e−C.(3.18)
Thenthesolvencyvalueintermsofnetwealthis
W0≥i0e+C.(3.19)
z+1i0Assumethat1+Cr0<e1++rC1.Withthesamevalueofe,regimechoicecutoffvaluesturnout
tobesmallercomparedtotheoriginalmodelwithmonetarycost,sincetheeffortcostis
paidoutoncereturnsarerealized.Forhighlevelofnetwealthabovee1++rC1,betweentwo
boundariesandbelow1+Cr0,thebankbringseffortiffW0is,respectivelyineachinterval,
abovethefollowingcutoffvalues,

Sˆ1=µ1−eµ0,(3.20)
(P1−1)C+eP1
Sˆ2=P1(1+R1)−(1+µ0)(3.21)
C(P1−P0)+eP1
Sˆ3=P1(1+R1)−P0(1+R0).(3.22)
Nonetheless,proposition3.1includingsomemulti-cutoffstrategiesissatisfied.

Wecaninterpretregimestrategiesintheoneperiodmodelasashort-rundecisionin
adynamicmodel.However,theshort-rundecisioncanbedifferentfromthelong-run
thedecision.endofIfpateriodthebbutalsoeginningaofstreameachofperiofuturedtheprofitsbanktheoptimizesdecisionnotforonlyriskthetakingprofitmaofy

3.3.THETWO-PERIODMODELWITHDISCRETERETURN

Figure3.3:TheTime-lineoftwo-periodmodel

47

change.Theintuitionisthatthebankfacesanintertemporaldecisiononitsprofit.For
instance,theprofitofoneperiodoperationaddsonthebank’swealthwhichdetermines
theregimeofoperationforthenextperiod.Sincetheresultofoneperiodinfluences
nextperiods,thebankhastotakeintoaccounttheconsequencesofitstoday’sdecision
onthefuture.Tocapturetheintertemporaleffects,inthefollowingsectionIanalyze
theoptimalbehaviorofbanksinadynamicsetupoftwoperiods.

3.3TheTwo-PeriodModelwithDiscreteReturn

Supposetherearethreedates,t=0,1and2.Atthebeginningofperiodone,t=o,
thebankreceivesdepositsnormalizedto1thatithastopaybackinequalpayments
Catt=1,2.Havinginitialequity,theinitialwealthW0exceedsprincipal.3Atthe
endofthefirstperiodthebankhastopaydividendoutofpositiveprofit.Thedividend
isassumetobeanexogenouslygivenfractionofthefirstperiodoutcomelessdeposit
payment,(1−δ)(Y1−C)>0.Theremainingwealth,δ(Y1−C)>0,coverseffort
costandgeneratesoutcomeinthenextperiod.4Att=2residualprofits(afterdeposit
payment)arepaidtoshareholders.Therefore,thebankonbehalfoftheshareholders
aimstooptimizethedividendofthefirstperiodaddedtothefinalprofit.Thediscount
rateis0<β<1.Therateofreturn(RR)iszigivenbyequation(3.1).Figure3.3
sketchesthetimingofthemodel.

3Alternatively,depositscouldbedefinedasbeingrolledovereachperiod.Thiswouldhowevernot
results.theaffect4Notethat0<δ<1representthereinvestmentratio.Thisnotationmakesthefurthercalculations
simpler.

48

CHAPTER3.ISGAMBLINGFORRESURRECTIONVALID?

AgainIassumethatthereismonetaryeffortcosteforregime1.Thebankmayswitchat
t=1toadifferentregimeofoperationinthesecondperiod,i.e.i0=i1.Differentform
DanglandLehar(2004),Iabstractfromswitchingcostsinthepresentmodelinwhich
thelowriskinduceseffortcosts.Thesecondperiodoperationandregimechoiceare
knownfromtheoneperiodmodel.Thefirstperioddecisionsareaffectedbythesecond
period,asthebankhastoconsideritsnetpresentvalueoftwoperiodsoperation.The
beststrategyattimet=0isthesolutiontothefollowingoptimizationproblem:

maxi0E((1−δ)max(Y1−C,0)+βmax(Y2−C,0)),(3.23)

where

Y1=(1+zi0)(W0−i0e),
W1=max(0,δ(Y1−C)),
Y=(1+zi1)(W1−i1e)forW1>0,
2otherwise.0

(3.24)(3.25)(3.26)

Thewealthofthebankmustcoverthedepositpayment,dividendandthecostof
operationunderanappropriateregime.Aftertherealizationofthereturnattheendof
eachperiodthebankissolventifitsassociatedlevelofwealthislargerthanthedeposit
payment.However,att=1afterpayingfordepositanddividendthebankmaynot
haveenoughmoneyforcontinuationundereitherregime.Thenthebankclosesandpays
theremainingwealthtotheshareholders.Hence,whenmakingitsdecisionregarding
i0andi1thebankmustconsidercostsandsolvency.ThatmeansforsomerangeofW0
evensuccessinthefirstperiodisnotenoughtocontinueoperation.Forsuchvalues,the
modelreducestotheoneperiodmodelalreadydiscussedinthelastsection.Therefore,
asecondperiodwillonlyberelevantifY2−C>0.Indeed,firstitisrequiredY1−C>0
whichisverifiedinthelastsection.Thesufficientcondition

Y2−C>0⇔
(δ((1+zi0)(W0−i0e)−C)−i1e)(1+zi1)−C>0(3.27)

extendstoseveralcasesundereachsomeoutcomecannotbesolvent.Thereversecases,
whereY2<C,canbedescribedby
Czi1<δ((1+zi0)(W0−i0e)−C)−i1e−1.(3.28)
Thisdeterminestherelationbetweenthereturnsofthefirstperiodandthesecond
periodwhichdoesnotbringsolvencyforthebankattheendoftwoperiodsoperation.

3.3.THETWO-PERIODMODELWITHDISCRETERETURN

3.3.1NoRiskofInsolvency

49

Fornowfocusontheveryspecialcasewhereinitialwealthissufficientlyhighsuchthat
allreturnsofthefirstandthesecondperiodaresolvent.From(3.25)itmeansthatfor
allvaluesofzi0andzi1wherei0,i1∈{0,1},initialwealthshouldexceed
W0>T(zi0,zi1)=C(1+δ(1+zi1))+e(i1+i0δ(1+zi0)).(3.29)
δ(1+zi0)(1+zi1)δ(1+zi0)
Proposition3.2Ifinitialwealthofthebanksatisfies(3.29),theuniquecutoffvalue
policyholdsifS1>T(zi0,zi1).Otherwisethebankneverchoosesregime0.Therefore,
risk-returnchoiceofeachperiodisindependentofanotherperiod.

ProofTofindswitchingstrategies,themodelissolvedbybackwardinduction.The
wsolutionealth.totheConsequensecondtlyfropmeriod(3.12),isthethesamebankasoptheeratesonepundereriodmoregimedeli1with=1Wiff1Was1≥initialS1,
i.e.

W1≥e(1+µ1),(3.30)
µ−µ01andregime0otherwise.Tosolvethefirstperiodoptimizationproblem,assumethe
bankoperatesunderagivenregimei1inthesecondperiod.Plug(3.24)-(3.26)in(3.23).
SinceY2>C,theobjectivefunctionfollows

maxi0E[(1−δ)((1+zi0)(W0−i0e)−C)+
β((1+zi1)(δ((1+zi0)(W0−i0e)−C)−i1e)−C)]
whichis,usingequation(3.25),equivalentto

maxi0(1−δ)[(1+µi0)(W0−i0e)−C]
+β[(1+µi1)(δ((1+µi0)(W0−i0e)−C)−i1e)−C]
yieldsRearranging

maxi0(1+µi0)[1−δ+βδ(1+µi1)]W0
−((1+µi0)[1−δ+βδ(1+µi1)]i0+β(1+µi1)i1)e
−(1−δ+β+βδ(1+µi1))C.

.31)(3

(3.32)

Theoptimalregimechoiceoft=0maximizesthenetpresentvalueoftwoperiodsfor
anygivenregimeinthesecondperiod.Therefore,foreachi1thebankisbetteroffby

50

CHAPTER3.ISGAMBLINGFORRESURRECTIONVALID?

choosingi0=1iftheexpectedvalueoftwoperiodsunderi0=1isbetterthanorequal
totheexpectedvalueunderi0=0.Thisconditioncanbesimplifiedto

(1+µ0)[1−δ+βδ(1+µi1)]W0≤
(1+µ1)[1−δ+βδ(1+µi1)]W0−(1+µ1)[1−δ+βδ(1+µi1)]e,

esgivhwhic

W0≥e(1+µ1).(3.33)
µ−µ01ThethresholdisidenticaltoS1whichimpliesthatthefirstperioddecisionisindependent
ofthesecondperiod.
Hence,Icanconsidereachperiodinisolation.Thisisaresultoftheassumptionthat
thedividendratioisgivenexogenously.Sincethebankissolventforallreturns,the
futureoutcomedoesnotaffectthecurrentsituation.However,thefeasibilitycondition
requiresthethresholdtobegreaterthanT(zi0,zi1)fori0,i1∈{0,1}.Ifnot,thebank
choosesonlytheregimewithahigherexpectedreturn.
Inadditiontotheindependenceoftheregimechoicesinthetwoperiodsforthespecial
caseabove,thecutoffvalueisonlyaffectedbyaveragereturns.Noreturnmakesthe
bankinsolventandtheriskisirrelevant.Thevariancesandreturnintervalsthusdonot
appearintheregimechoicedecisionsforthissituation.Thebankchoosestheloweffort
regimeonlyifitcannotaffordtheeffortcostassociatedwiththehighmeanreturn.The
effectofvariancesinregimestrategiesareexaminedinthenextsubsectionthatinvolves
someriskofinsolvency.

3.3.2OperatingunderRiskofBankruptcy

Whentheoutcomeofthefirstperiodislowsuchthatthebankneedshigheroutcome
inthesecondperiod,condition(3.29)iscrucial.Forsomecases,failuremaycause
insolvency,butevensuccessreturnmaynotbesufficientforonemoreperiodoperation.
Thegeneralsettingisexploredinthenextsectionaspartofnumericalexample,since
equation(3.29)extendstotoomanyconditionswhichcannotbesolvedinageneral
case.Toobtainanalyticalsolutionsandgainintuition,Ihavetolimitthesettingtoa
simplebenchmark.Now,assumetheextremecaseinwhichthebanklosestotalwealth
andgoesbankruptifitfails.ItmeansthatRRzifrom(3.1)returnsri=−1,i=0,1
incaseoffailure.Thensimilartoproposition3.2,wecomparetheexpectedreturnsof
operationundertwoalternativeregimes.Thebankassertseffortiffthesuccessofregime
1ismoreprofitablethansuccessofregime0,

P1[(1+R1)(W1−e)−C]≥P0[(1+R0)W1−C].

3.3.THETWO-PERIODMODELWITHDISCRETERETURN

51

Thatgivestheuniquecutoffpointofthesecondperiod(henceintermsofW1),below
whichthebankchoosesriskierregime,
C(P1−P0)+eP1(1+R1)
W1≥S3=P1(1+R1)−P0(1+R0).(3.34)
Inthenextstep,Iapplybackwardinductiontosolveforthebank’sregimechoicesin
thefirstperiod.Proposition3.3describesswitchingandcutoffstrategiesundertherisk
ofbankruptcy,i.e.(3.29)doesnotholdor(3.27)isviolated.

Proposition3.3Whenri=−1,i=0,1,foranexogenousδ∈(0,1),theunique-cutoff
policyoptimizesrisk-returnregimeofthefirstperiod.Thereexistanonemptyswitching
area,intermsofnetwealth.

ProofThebankchoosesi0bymaximizingobjectivefunction(3.23)whichgives

maxi0E[(1−δ)max((1+zi0)(W0−i0e)−C,0)+
βmax((1+zi1)(δ((1+zi0)(W0−i0e)−C)−i1e)−C,0)](3.35)

Theoptimalchoiceisaffectedbyi1sincenotallreturnshavepositivevalueforthe
bank.Although,i1isknownbythethresholdin(3.34)att=1,thebankneedsto
realizeitatt=0.Theoperationcontinuesforthesecondperiodonlyaftersuccessat
thefirstperiod.Hence,substituteW1from(3.25)and(3.26)in(3.34),wehavei1=1
iffW0≥Qi0suchthat
Q=C(P1−P0+δ(P1(1+R1)−P0(1+R0)))+eP1(1+R1)+ie.(3.36)
i0δ(P1(1+R1)−P0(1+R0))(1+Ri0)0
SinceQ0<Q1,thebankatt=0findsitsoptimalchoiceofthefirstperiod,
(I)i1=1iffW0≥Q1where
Q=C(P1−P0+δ(P1(1+R1)−P0(1+R0)))+eP1(1+R1)+e,(3.37)
1δ(P1(1+R1)−P0(1+R0))(1+R1)
(II)i1=0iffW0<Q0where
Q0=C(P1−P0+δ(P1(1+R1)−P0(1+R0)))+eP1(1+R1),(3.38)
δ(P1(1+R1)−P0(1+R0))(1+R0)
(III)i1=i0iffQ0≤W0<Q1,i.e.[Q0,Q1]isanonemptysubsetofswitchingarea.

52

CHAPTER3.ISGAMBLINGFORRESURRECTIONVALID?

Now,Ianalyzethechoiceofthefirstperiodrisk-returnregime,i0,ineachofthethree
intervals.FirstassumeW0≥Q1,thenhigheffortregimeisoptimaliff
P1[(1−δ)((1+R1)(W0−e)−C)+βP1((1+R1)(δ((1+R1)
(W0−e)−C)−e)−C)]≥P0[(1−δ)((1+R0)W0−C)+
βP1((1+R1)(δ((1+R0)W0−C)−e)−C)].
ThresholdU1obtainssuchthati0=1iffW0≥U1,
U1=(C(P1−P0)(1−δ+δβP1(1+R1)+βP1)+
eP1(1+R1)(1−δ+δβP1(1+R1)+β(P1−P0)))/
((P1(1+R1)−P0(1+R0))(1−δ+δβP1(1+R1))).(3.39)
Next,ifQ0≤W0<Q1thebankswitchesfromtheregimeithasatt=0toanother
regimeatt=1.Thus,therearetwooptionsofregimecombination:(i0=1,i1=0)
and(i0=0,i1=1).Thebankisbetteroffbytheformerregimecombinationiff
P1[(1−δ)((1+R1)(W0−e)−C)+βP0((1+R0)δ
((1+R1)(W0−e)−C)−C)]≥P0[(1−δ)((1+R0)W0−C)+
βP1((1+R1)(δ((1+R0)W0−C)−e)−C)].
ThisgivesthresholdU2forW0,belowwhichthebankassertsnoeffortandtakeshigher
risk,U2=(C((P1−P0)(1−δ)+δβP1P0(R0−R1))+eP1(1+R1)(1−δ
−βP0+δβP0(1+R0)))/((P1(1+R1)−P0(1+R0))(1−δ)).(3.40)
ForlowinitialwealthW0<Q0,thebankchoosesi0=1att=0,thoughitundertakes
higherriskinthesecondperiod,iff
P1[(1−δ)((1+R1)(W0−e)−C)+βP0((1+R0)(δ((1+R1)
(W0−e)−C))−C)]≥P0[(1−δ)((1+R0)W0−C)+βP0
((1+R0)(δ((1+R0)W0−C))−C)].
ThisdemandsW0≥U3withthreshold
U3=(C(P1−P0)(1−δ+δβP0(1+R0)+βP0)+
eP1(1+R1)(1−δ+δβP0(1+R0)))/
((P1(1+R1)−P0(1+R0))(1−δ+δβP0(1+R0))).(3.41)

3.3.THETWO-PERIODMODELWITHDISCRETERETURN53
Nevertheless,onlyfeasiblethresholdsarecutoffvaluesthatrequiresthemtosatisfy
U3≤Q0,Q0≤U2<Q1andQ1≤U1.NotethatQ0=Q1,andallthethresholds
cannotbeequal.Q0andQ1arebothcontinuous,decreasingandconvexinδ,
dQ1−(C(P1−P0)+eP1(1+R1))
dδ=δ2(P1(1+R1)−P0(1+R0))(1+R1)<0,(3.42)
dQ0−(C(P1−P0)+eP1(1+R1))
dδ=δ2(P1(1+R1)−P0(1+R0))(1+R0)<0,(3.43)
22ddδQ21>0,ddδQ20>0.(3.44)
HavingequaldenominatorsinU1andU3,highprobabilityofsuccessandexpectedreturn
inregime1bringsU1>U3.Moreover,thesetwomonotonethresholdshavemonotone
firstderivativeswithrespecttoδ.AlsoU2isincreasingandconvexinδ,
U1−U3=(Cβ[(P1−P0)(1−δ)+δβP0P1(R1−R0)]+
eP1(1+R1)(P1−P0))/((P1(1+R1)−P0(1+R0))
(1−δ+δβP0(1+R0))(1−δ+δβP1(1+R1)))>0,(3.45)
dU1=βP1(P1−P0)(C+e(1+R1))(1−βP1(1+R1)),(3.46)
dδ(P1(1+R1)−P0(1+R0))(1−δ+βP1(1+R1))2
dU2=βCP0P1(R0−R1)+βeP0P1R0(1+R1)(3.47)
dδ(1−δ)2(P1(1+R1)−P0(1+R0))
dU3βP0(P1−P0)C(1−βP0(1+R0))
dδ=(P1(1+R1)−P0(1+R0))(1−δ+βP0(1+R0))2,(3.48)
d2U2>0(3.49)
2dδHence,asfunctionsofδ,eachofU1,U2andU3intersectQ0andQ1onlyonce.
IverifythattheintersectionofU1andQ1,denotedbyδ1,isidenticaltotheintersection
ofU2andQ1.Itisarootoftheequation
C[δP0(R0−R1)(1−δ+δβP1(1+R1))−(1−δ)(P1−P0)]=e(1+R1)
[(1−δ)(P1−δP0+P0(1+R0))+δβP0P1(1+R1)(1−δ(1+R0))].(3.50)
ThisisequivalenttoU2=Q1,aswell.WhenU1>Q1,theLHSinequation(3.50)
islargerthanitsRHS.ThisimpliesU2>Q1,andviceversa.Thus,feasibilityofU1
demandsinfeasibilityofU2andtheotherwayaround.

54

CHAPTER3.ISGAMBLINGFORRESURRECTIONVALID?

Similarly,IfindthatU2=Q0occursatδ0whichisthesolutiontothefollowingequation,
C[δP0(R0−R1)(1−δ+δβP0(1+R0))−(1−δ)(P1−P0)]=
eP1(1+R1)(δ(1+R0)−1)(1−δ+δβP0(1+R0)).(3.51)
ThisequationimposesU3=Q0aswell.IftheLHSislargerthantheRHSinthis
equationthenU3<Q0whichgivesQ0>U2.Conversely,feasibilityofU2makesU3
infeasible.Comparethetwoequations,coefficientsofδin(3.50)arelargerthanthose
in(3.51).Therefore,δ1>δ0,whichcompletesthesufficientconditionsforfeasibilityof
onlyoneofthethresholdspergivenδ.Itfollowsthatgivenδ,onlyoneofthresholds
Uj,j=1,2,3istheuniquecutoffvalue.

Remark3.4Themodelwithnon-monetarycostisanalogoustothiscase.

ExamplesNumerical3.4

Inordertoillustratetheresults,Ipresentthreebenchmarksetsofparametersforthe
t3.3.wo-pNeerioxt,dImoinvdel.estigateThetfirstwomoreexampleisgeneralinlinecaseswithinthewhichsettingthebankpresentedcaninbepropsolventositionat
regime.aoffailure

ailureFatBankruptcy3.4.1Figure3.4showsregimechoicesandswitchingstrategiesfordifferentinitialwealthlevels
andinvestmentpolicywithouteffortcoste,asitdoesnotinfluencethegeneralityofthe
result.Inthiscasewehaver0=r1=−1.
Whentheeffortcostisequalto0thebanktakesregime1foralowerlevelofinitialwealth
sincetheexpectedreturnishigherateachlevelofwealth.Inaddition,lowerdividend
ratio,i.e.higherδ,createsmoreincentivetoundertakerisk.Tosummarize,thereare
twotrade-offs.Theintertemporaloneisassociatedtothedividendandreinvestment
decision.Thoughthereinvestmentratioisexogenous,thebankneedstodecideabout
postponingeitherhigherriskorhigherreturn.Anothertrade-offisbetweenhigher
probabilityofsuccessorhighersuccessreturn.Theoptimalcombinationofregime
choicesfortwoperiodsisaresultoftwotrade-offs.
Whenthedividendratioislow,failuredoesnotbringalargelosstothebankwhose
wealthisalsosmall.Sincetheprofitofsuccessislow,thebankbehavesindifferent
betweenfailureandthelowdividend.Thus,itgamblesforresurrectionfirst.Ifthe
banksucceedsithassufficientwealthandplayssafeinthesecondperiod.Inthisarea
weobserveriskreductionfromperiodonetotwo.

EXAMPLESNUMERICAL3.4.

55

Figure3.4:Inter-temporalSwitchingstrategiesinatwo-periodmodelwithouteffort
cost,i.e.e=0,showingfourpossibleregime-combinationsintheoptimalregionw.r.t
initialwealthW0andreinvestmentratioδ

ForthewealthlevelbetweenQ0andQ1,thebankcanaffordtheeffortcostandtakes
thesaferregimeforhighdividend.Inthatcase,ithastoplayriskyinthesecondperiod
becausereinvestmentislowsuchthatthebankdoesnothavesufficientwealthtostart
thesecondperiodunderthesaferregime.Thisareaisincludedforthegamblingfor
resurrectionstrategyfromthefirstperiodtothesecond.
Inaddition,notethattheswitchingareaisactuallyasuper-setof[Q0,Q1],depending
onδ.Giventhedividendratio,ifU2isfeasible(δ0<δ<δ1),bothtwo-wayswitching
strategiesaretakeninthisrangeofwealth.

3.4.2SolvencyatFailure

Supposeri>−1,i∈{0,1}.Thethresholdsofproposition3.1givepossiblecutoff
pointsforthesecondperiod.Considerthefollowingexamples:
Example1.Takeparameterset:

R1r1P1R0r0P0βδe
0.75−0.10.650.9−0.30.450.90.990.3

(3.52)

56

CHAPTER3.ISGAMBLINGFORRESURRECTIONVALID?

Sincethedepositisfullyinsured,thepaymentequalsC=β+1β2.Notethat(3.8)isnot
trueforregime0butissatisfiedunderregime1.Therefore,thebankmaytakeregime0
forW0≤G0=3.31.Butregime1isworthierthannooperationforallpositivewealth
becauseG1=−14.7693.Checkingtheassumptionsofproposition3.1andSjs,wefind
case(b)offigure3.2withuniquecutoffvalueS1=2.05forthesecondperiod.Inorder
tohaveW1>S1thebankneedsforeachregimeanditsoutcomeofthefirstperiod,zi0,
/δS+C1W0>Q(zi0)=1+zi0+i0e.(3.53)
WeseethatQ(R0)<Q(R1)<Q(r1)<Q(r0).Next,Iverifywhethereachcombination
ofoutcomesoftwoperiodsissolvent.ItmeansthatIcompute16thresholdsfrom(3.29)
foralloutcomesoftwoperiods.LocatethemonintervalsmadebyQ(zi0)sfortheinitial
wealth.Findoptimalregimeineachintervalboundedtothedescribedthresholdsby
comparingexpectedprofitsofsolventoutcomes.Attheendtheuniquecutoffpointis
2.24,whereQ(R1)<2.24<Q(r1).Forall1<W0<2.24thebankoperatesunder
riskierregimewhileifitwasoperatingforonlyoneperioditwouldtakesaferregime
alreadyaboveS1<2.24.Inotherwords,havingtheopportunitytooperateforone
moreperiodthebanktakesthesaferregimeatalargercapitalcomparingtothecaseof
oneperiodoperation.
parametersAssume2.ExampleR1r1P1R0r0P0βδe
0.650.10.650.7500.30.570.990.2(3.54)
Now,foroneperiodoperation(e.g.secondperiod)wehavecase(a)offigure3.2with
threeconsistentcutoffpointsS3=1.106,S2=1.158andS1=1.253.Consequently,
therearefourthresholdsQ(zi0),i0∈{r0,r1,R0,R1}foreachcase.Alsotakeintoaccount
16thresholdsfromequation(3.29).Thenumericalsolutiondeterminestheuniquecutoff
policyasmanythresholdsareinfeasible.Thecutoffvalueis1.76belowwhichthebank
choosesregimei0=0.
Ariskyinvestmentisworthyinregime0(basedon(3.10))forW0≤G0=1.154,and
inregime1ifW0≤G1=1.193where(3.8)doesnotholdtrue.Onlytwothresholds
T(R0,R0)=1.007andT(R0,R1)=1.145arebelowGis.Still,GisarebellowallQ(zi0)s.
Hence,thebankoperatesundertheriskyregimeinthesecondperiod.Thebankcan
survivetwoperiodsonlyifitchoosestheriskyregimeatt=0andsucceeds.Thus,
operatingforonlyoneperiodunderregime1producesahigherexpectedprofitthan
operatingunderregime0inthefirstperiodandhopingtosucceedandcontinuefor
thesecondperiod.SinceinthisexampleS2<G1<G0<S1,thebankdecidesabout
regimesbasedonSjsandoperatesforonlyoneperiodbelowGis.Toconclude,thebank
operatesforoneperiodunderregime0whereS2≤W0<G0.Butittakesthesafer

REINVESTMENTGENOUSENDO3.5.

regimeforS3≤W0<S2andtheriskierregimebelowS3.

57

Comparisonofthetwoexamplesshowsthatwhentheriskfreeinterestrateishigh(dis-
countfactorβissmall)thedepositpaymentislarge.Therefore,inoperation,thebank
losescapital.Asfaraspossible,itoperatesoneperiodandtakesrisknon-monotonically
intermsofthecapitallevel.Otherwise,ifthedepositpaymentislowitcanbesolvent
atfailureandhaslesstendencytowardsrisk.Then,itcanoperatefortwoperiodsand
followstheuniquecutoffpolicyatthefirstperiod.

3.5EndogenousReinvestment

Tocompletetheoptimizationproblemofthebank,Iincludeitsdividendpolicyandfind
optimalδ.Thisdecisionismadeatt=1simultaneouswiththeregimechoicedecision.
Thebankmustbesolventbythenandthefirstperiodoutcomeisrealizedsuchthat
Y1>C.Theobjectivefunctionisstill(3.31).Forsimplicity,weassumer0=r1=−1.
Foreveryi1,theoptimalδobtainsfrom
maxδ(1−δ)(Y1−C)+βP(zi1)((1+zi1)(δ(Y1−C)−i1e)−C).(3.55)
Rearrangeitforδ∗,
δ∗=argmaxδ(βP(zi1)(1+zi1)−1)(Y1−C)+(Y1−C)
−Iδ>0[βP(zi1)(i1e(1+zi1)+C)].(3.56)
Sincethisequationisalinearfunctionofδ,thereinvestmentratiodependsonthesign
ofitscoefficientin(3.50).Noticethatthisisthesameproblemasthebankhasatt=0
whenitdecidestoenterthegame.Thebankreinvestsallofitscapitalinariskyregime
iff(3.8)holdstrue,βPi(1+Ri)−1>0.Therefore,ifabank∗operatedforoneperiod,
fromsection3.2,thesolventbankwouldpaynodividendandδ=1.
Ifthebankreinvest,itscapitalatt=1isW1=Y1−C.Inordertohaveaprofitable
investmentinregimei,theNPVshouldbepositive,
W1≥βPi(ie(1+Ri)+C).(3.57)
βPi(1+Ri)−1
Therefore,foroperatingunderregime1thebankneeds
CPβW1≥G0=βP0(1+0R0)−1,(3.58)

(3.58)

58

CHAPTER3.ISGAMBLINGFORRESURRECTIONVALID?

andforoperatingunderregime0itshouldbethat,
W1≥G1=βP1(e(1+R1)+C).(3.59)
βP1(1+R1)−1
Stillthebankchoosesbetweenregime0and1att=1basedoncutoffpolicy.Feasibility
ofcutoffvalueS1requiresS1>G0andS1>G1.Bothconditionsaresatisfiedwhere
C(P1−P0−βP0P1(R0−R1))>eP1(1+R1).(3.60)
βP0(1+R0)−1
Knowingallaboutthesecondperiod,thebankfindsitsfirstperiodinvestmentstrategies
bybackwardinduction.
Endogenouslyoptimizationofthereinvestmentreducestoabang-bangpolicyofreinvest-
ingallornothing,asagentsareriskneutral.Whenthebankreinvestsalltheoutcome
ofthefirstperiod,theproblemissimilartotheregimechoiceoptimizationinthetwo
periodmodel.Theoptimalstrategycanagainbecharacterizedbyuniquecutoffpolicy
asassumptionsofproposition3.3hold.Thismakestheendogenousreinvestmentmodel
anotherrobustnesschecktothefindingsintheprevioussections.

Conclusion3.6

Thisworkquestionsgamblingforresurrectionandverifiesexistenceofanon-monotonic
relationshipbetweenthecapitallevelofabankanditsportfoliorisk.Thestandard
rationaleofbankstakingriskunderdistressisviolatedinastaticmodelwhichcompares
twodifferentregimesofoperation.Thisisbeyondplentyofstudieswhichfocusonly
onselectingriskyorrisk-freeasset,e.gMailathandMester(1994).Indeed,thecost
ofefforttoreducetheriskfromoneriskyregimetothelessriskyoneplaysthemain
roletoruleoutstandardcutoffpolicy.Theriskislessinoneportfoliosincethebank
exertsefforttomonitorcreditors.Theriskierprojectshoweverproducealargerreturn
ifsucceedwhiletheprobabilityoffailureandassociatedlossesarelarger.Iobserve
twotypesofrisk-returnstrategies.Thefirsttypeisinlinewiththestandardrationale:
theriskneutralbankerchoosestheriskierassetwhenthecapitaldecreases,inorder
tobenefitfromahigherreturnincaseofsuccessandhopetosurvivethedistress.
However,inthesecondtype,therisk-takingdecisiondependsontheinitiallevelat
whichthecapitalbeginstodecay.First,whenthecapitaldecreasesfromahighlevel,
abankwithlessriskfacesbankruptcyincaseoffailure,becausethemonitoringeffort
ispaidsoutofthecapital.Therefore,thebankstopsmonitoringandtheriskincreases.
Riskymortgagesareexamplesofsuchbehaviorinbanks.Furtherthecapitaldecreases,
thebankoperatingunderariskyregimealsogoesbankruptincaseoffailure.Thus,the
bankchangesitsportfoliototheonewithlessriskoffailure.Nevertheless,withvery

3.6.ONCLUSIONC59

lowcapitalatwhichneithermonitoringnorriskyprojectscansurviveincaseoffailure,
thebankgoesonoperatingunderriskyregime.
Inthedynamicmodel,contrarytoDanglandLehar(2004),therisk-takingandtheinter-
temporalswitchingstrategiesareendogenous(notforcedbyaregulator)andinfluence
eachother.Undertheirhypothesisofgamblingforresurrection,thecutoffvalueofthe
risk-takingpolicyisidenticaltotheswitchingpoint,whentheswitchingcostisomitted.
Therefore,noswitchmeansthebankchoosesacertainfixedregime.Inthispaper,if
aswitchingareaexistitincludescutoffvalues.Theswitchingareaofeachregimeisa
continuoussetwithatmostonepointofintersectionwiththeswitchingareaoftheother
regime.Dependingonthedividendratio,eachswitchingareamaynarroworwidenand
onemaydisappear.Whenbothexisttheyintersectonauniquecutoffvalue.Low
dividendraisesgamblingforresurrectionbuthighdividendcausesthebanktoreduce
riskinthefirstperiod.Yettheimpactisreversedafterpayingdividendoutsincethereis
nooutsideinvestorandthebankispoor.Monitoringcostincreasesrisk-takingincentive
gentlybutdoesnothaveastructuraleffect.
Thefindingsregardingnon-monotonicrisk-takingpoliciescontributetothebanking
regulationliterature.Banksfinancetheirinvestmentsinlargepartsbydeposits.Imper-
fecttransferabilityofbanks’assetsmakebanks’liquidationcostly.Tomakeprofitthe
bankneedstospendonmonitoringthecreditors.Yet,withlimitedliabilityandinsuf-
ficientcapitalthebankshirksinordertoincreasetheequityvalue.Fromaregulatory
pointofview,theclosurepolicywithasufficientlyhighcapitalratiorequirementwould
eliminatetherisk-takingincentives.However,astrictregulatoryclosurepolicyisnot
sociallyoptimal.Theregulatorshouldnotonlyprotectthedepositorsbutalsooptimize
thesocialvalueofthebank.Thisway,thesupervisoryagencyendsupinlargescale
forbearanceincaseofacrisis.Instead,myresultsproposemoreaccuratescreeningof
risksinthebanks,inthefirstplace.Thepossiblemethodscouldbethemoremarket
basedapproaches,forinstancepartialprivateinsuranceandrisk-basedtaxing.

60CHAPTER3.ISGAMBLINGFORRESURRECTIONVALID?
endixApp

Proofofremark3.2:Suppose1+µ1>P0(1+R0).If1+Cr1≤W0<1+Cr0,thebank
takesregime1abovethresholdS10definedbelow,
P1((1+R1)W0−C)+(1+P1)((1+r1)W0−C)≥
P0((1+R0)W0−C)(3.61)
0C(1−P0)
W0≥S1=1+µ1−P0(1+R0).(3.62)
TheLHSandRHSof(3.61)aretheexpectedreturnsunderregime1and0,respectively.
IfW0<1+Cr1theLHSofinequality(3.61)reducestoonlyP1((1+R1)W0−C).Assuming
P1(1+R1)−P0(1+R0)>0,theoptimalregimeis0belowathreshold,
C(P1−P0)
S20=P1(1+R1)−P0(1+R0).(3.63)
Tocheckthefeasibilityofthethreshold,comparethemtoboundariesWeseethat
S10<1+Cr0followsfromassumptionµ0<µ1.Yetforanotherboundarywehave
CS10>1+r1⇔µ1−µ0<(r1−r0)(1−P0),(3.64)
equalshwhic

CS20≥1+r1,ifP1(1+R1)>P0(1+R0)(3.65)
CS20<1+r1,ifP1(1+R1)<P0(1+R0).(3.66)
In(3.66)however,S20<0.Itmeansthati0=0forW0<1+Cr1andS10isfeasibleand
theuniquecutoffvalue.ButwhenP1(1+R1)>P0(1+R0),eitherS20orS10isfeasible.
Accordingly,weendupinunique-cutoffpolicy.
Ifconditions1+µ1<P0(1+R0)isviolated,thenS01<0andinfeasible.Thesame
holdsforS20ifP1(1+R1)<P0(1+R0).Itfollowsthattheonlycutoffvalueis1+Cr1.
Yet,forP1(1+R1)>P0(1C+R0),seethatS201+Cr1.Hencethebanktakesregime0
belowuniquecutoffvalue1+r1.

Chapter4

andTheSystemicTheoriesFofailuresBankRegulation

4.1ductiontroIn

Thispapersurveystherecentliteratureonbankregulation,inparticularforregulating
systemicrisk.Traditionally,therehasbeenmicro-prudentialbankingregulationfocusing
onindividualbanksandtherisktheyhold.Asanexample,thesurveyofBhattacharya
etal.(1998)coverstheliteratureintheeconomicsofbankregulationpriortoBaselII.
Subsequently,untilaround,thefocusoftheliteraturewasontheoptimalcombination
andimplementationoftheBaselIIaccords.The2007-2009financialcrisis,however,has
highlightedtheinterdependenciesinthebankingsectorandinthefinancialindustryas
awhole.Asaconsequence,systemicriskissueshavebeeninthefocusoftherecent
theoriesonbankingregulationandstudiesconcentratedonmacro-prudentialregulation
strategies.Theselatestexperiencesprovidethemotivationtoreviewhowthebankingregulation
theorieshavebeenprogressing.Hence,thecontributionofthissurveyistoconnectthe
previousbankregulationliteraturethathasfocusedonasingleentitywiththemost
recentideasontakingsystemicriskintoaccount.Thefocusistoshowhowthelatter
iscomplementarytotheformerandinwhichdirectionsbothstrainsoftheacademic
progress.shoulddiscussionsFirstinabasicsetup,Idiscussbankfailures.Thefundamentalproblemsthatpoten-
tiallyleadtobankdefaultintheexpenseofdepositorsareaddressed.Thesearethe
shareholders’risk-takingandmanagerialmoralhazard.Toprotectdepositorsthesuper-
visoryauthorityhastoregulatebanksonbehalfofdepositors.However,theauthority
facestime-inconsistencyprobleminsolvingthis.Theregulatorwhowantsex-anteto
reducetherisk-takingincentivesbythreateningtoliquidateassetsincaseofdefault,

61

62

CHAPTER4.THETHEORIESFOBANKTIONREGULAmighthavetoforegotheliquidationex-postbecauseofthehighsocialcost.Solutions
tothistime-inconsistencyproblemarepresentedinthefirstpartofthisarticle.Hereby,
Imostlyconcentrateonthelateststudiesinwhichrisk-basedapproaches,e.g.BaselII,
considered.areThesecondpartofthearticleexploresthetopicsofregulatingsystemicrisk.Widespread
bank-failures,namedsystemicfailure,bringexternalitiesintothefinancialsystem.The
externalitiesconsistofbank-runscontagionandmassivebank-failuressuchthatnopri-
vateinstituteisabletocompensateforthelosses.Inordertopreventthis,governments
havetotakethesystemicrisksintoaccountforregulation.Inbanks,systemicrisk
mayoriginatefromeithertheliabilityortheassetsideofthebalance-sheet.Bank-runs
areexamplesofexogenousshockswhichappearintheliabilityside.Moralhazardand
riskyinvestmentsgenerateriskintheassetside.Inthispaper,thecomponentsofboth
idiosyncratic(exogenous)shocksonbanksandthecontributionofbanks(endogenous
shocks)tosystemiccrisesarestudied.Toovercomethetime-inconsistencyproblemin
eachofthesecases,ex-anteoptimalmacro-prudentialregulationpoliciesarerequired.
Myarticlecoversthistopicfollowingtheliteraturewhichspecificallyreferstotheexpe-
rienceoftherecentcrisis,itsoriginsandconsequences.Forthisreason,afterpresenting
thesystemicriskregulatoryproposals;e.g.grantinghealthybanks,systemicrisksen-
sitivecapitaladequacyandtaxing,Ireviewthestatisticsabouttheresolutionpolicies
appliedinthepastcrisisevents.
Inordertoshowhowthissurveyrelatestotheexistingliterature,Ifirstgiveanoverview
ofprevioussurveystudies.Inanintegratedmodel,Bhattacharyaetal.(1998)analyze
differentdepositinsurancerelatedmoralhazardsandregulatorypolicies.Reasonsfor
theexistenceofbanksarediscussedontheassetsideaswellastheliabilitysideofthe
balancesheet.Ononehand,theyexplainthedelegatedmonitoringideaofDiamond
(1984)thatbanksmonitorthecreditorsonbehalfofdepositorsatalowercostthannon-
intermediatedbilateralcontractingbetweeninvestorsandentrepreneurs.Ontheother
hand,theargumentofDiamondandDybvig(1983)ispresentedthattheintermediaries
contributetoimproverisksharingandprovideliquiditybetterthannon-intermediated
casewhereinvestorswouldhavetowaitforthepayoffsfromthelongterminvestments.
BasedonDiamondandDybvig(1983),thegovernments’depositinsurancecanprevent
panicbank-runs.However,theinsuredbankshaveincentivesformoralhazardinthe
sensetokeeplowerliquidreserveandtoseekriskierportfolios.Theregulatorypolicies
areneededtoattacktheserisk-takingactivities.
In1988,theBaselIaccordintroducedthecapitalrequirementstoruleoutincentives
forrisktaking.However,studiessuchasGennotteandPyle(1991)andBootand
Greenbaum(1993)showparticularsituationsinwhichstringentcapitalconstraintsdo
notreducetheriskinbankingsector.BesankoandKanatas(1996)emphasizethat
whentheinsideandoutsideequitiesareextremelydistinguishablethehighercapital

4.1.ODUCTIONINTR63

requirementcanreducethebank’sincentivetomonitortheborrowersandincreaserisk.
FurtherstudiesonfairpricedandrisksensitivedepositinsurancebyforinstanceChan
etal.(1992)illustratethatthemoralhazardcannotbepreventediftheregulatorcan
notobservethebank’srisk.
Toconfrontprivateinformationproblems,partialdepositinsuranceissuggestedasa
regulatoryinstrumentwhichbringsforthmarketdisciplineastheactuarieshaveto
measurethebank’sriskandalsouninsureddepositorsmonitorthebank.Peters(1994)
pointsoutthattheinformeduninsureddepositors,withtheirownendowmentsatrisk,
willmonitoranddisciplinebanksbetterthangovernmentalregulatorsdo.
Risk-basedcapitaladequacyisanothercuretothemoralhazardissue.Thisideapro-
videdsupportsfortheFederalDepositInsuranceCorporation(FDIC)ImprovementAct
of1991intheUSandalsoevolvedintheBaselIguidelines.FDICpromptcorrective
actionmandatesprogressivepenaltiesagainstbanksthatexhibitprogressivelydeterio-
ratingcapitalratios.Bankclosureisconsideredasathreattoreduceincentivesforrisk
inthislaw.DahlandSpivey(1995)investigatebanks’effortsforrecoveryunderthe
closurethreatforcedbytheFDIC.Theyfindthatthedeterminationoffailureforan
undercapitalizedbankisbetterdefinedintermsofthebanks’capacityforrecoverythan
thelikelihoodforfurtherdecaysincethebankcanrecapitalizequicklybyequityinfusion.
EmpiricalstudiesassessingthecostandbenefitoftheFDICpromptcorrection,suchas
JonesandKing(1995),suggestthattherisk-basedcapitalstandardsshouldimproveto
betterrecognizethecreditriskoftroubledbanks.
Assessingregulatoryclosurepolicyshowslessefficiencyasexpected,thoughitisnot
sociallyoptimaleither.Forinstance,BootandThakor(1993)arguethattheregulator
caresaboutitsreputationanddoesnotexertclosurewhenitisneededandthisagain
raisestherisktakingbybanks.Insteadofatoughclosurepolicy,Friesetal.(1997)
proposeoptimalreorganizationofthebankandclosurerulebesidefairpricingofdeposit
guarantees.Theotherbranchofstudiesfocusingonmoralhazardissues,e.g.Leland
(1994)andLelandandToft(1996b),worksonthecapitalstructuretopreventasset
substitution.The1988BaselIcapitalframeworkevolvedovertime.TheBaselcommitteeissuedan
amendmenttorefinetheframeworktoaddressrisks,e.g.marketrisks,otherthancredit
risk.Accordingly,bankswereallowed,subjecttostrictquantitativeandqualitative
standards,touseinternalvalue-at-riskmodelsasabasisformeasuringtheirmarketrisk
capitalrequirements.InJune1999thecommitteeissuedaproposalforanewcapital
adequacyframework.Afternearlysixyearsofchallengingworks,theBaselIIcapital
frameworkwasreleasedinJune2004.Itconsistsofthreepillars:minimumcapital
adequacyexpandedstandardsofthe1988Accord;supervisoryreview;andregulatory
closuretostrengthenmarketdiscipline.

64

CHAPTER4.THETHEORIESFOBANKTIONREGULAFollowingtheworksonhowtorefineregulatorystrategieswhichresultedintheBasel
IIstandards,mostofliteraturepriortotherecentcrisishasfocusedonhowtomixthe
threepillarsofBaselIIandimprovetheirimplementation.Prescott(2004)verifiesthat
thebankshaveincentivesnottorevealthetruelevelofrisk.Stochasticauditisfound
tobemoreeffectivecomparingtotheperiodicalbanks’reportsoftheirrisks.Inamodel
ofoptimalbankclosurewithstochasticauditBhattacharyaetal.(2006)findanoptimal
combinationofcapitalrequirement,closureruleandfrequentauditwhichcaneliminate
risk-takingincentiveforbanks.Decampsetal.(2004)andDanglandLehar(2004)take
asimilarapproachtowardsthegamblingforresurrectionproblemforbanksindistress.
FRepullourther,andthereSuarezareresearc(2004)hfoescusonondetailsloanofpricingtheBaandselIIdemimplemenonstratethattation.theForbanksinstance,which
adopttheinternalratingbasedontheBaselIIattractlowriskfirmsbyreductionin
.srateloantheirConsideringbanksasliquidity-creators,thebankfragilityissuerelatesalsotothecapital
marketrisksandthemarket-drivenfragility.Inthisregard,BootandThakor(2008)
reviewtheexistingliteratureontheinterbankrelationshipaswellastheintegrationof
banksandmarkets.Intherecentyearsofcrisissomeempiricalresearchesexamined
theeffectivenessofBaselII.Alsotherehavebeenstudiesabouttheregulatorypolicies
inemergingmarkets.Theviewonbanksasinstitutionsthatarecloselyrelatedto
eachotherandtotheentireeconomydirectedrecentstudiestofocusontheanalysisof
regulation.bankingsystemicThisnewstrainintheliteraturemotivatesanewsurveytocollecttheirfindingsandex-
plainstrategiesdifferendeptendonregulatorythebacproposalskgroundinanproblems,analyticalwhetherframewitisork.anTheexogenousoptimalshockorregulatoryone
ofthemoralhazardissues.Idescribethepossibleregulatoryconfrontationsrelatedto
eachcategoryofproblems.
Thepaperfollowsinthenextfivesections.Section4.2outlinesthebasicsetupandbank
canfailureapplyproblemtos.Inindividualsectionbanks.4.3,ISectionreviewdiff4.4erentdiscussespoliciesthethateffectstheofregulatorysystemicauthoritfailures.y
SectionSubsequen4.6tly,includessectionthe4.5statisticspresentsofthetheregrecentulatorycrises.propAtosalslast,addressingsection4.7systemicsummarizesrisk.
concludes.and

4.2TheBasicModel:FailuresinBanks

Marketfailurescanprovidetheintuitionfortheexistenceofasupervisoryauthority.
Thissectionconsistsofthebasicsetupthatallowsforastringentanalysisofthepar-
ticularproblemscausingmarketfailuresinthebankingsector.Thedetailsconcerning
possibleregulatoryactionsareinvestigatedinthenextsection.

4.2.THEBASICMODEL:FAILURESINBANKS

65

Inariskneutralworld,assumearepresentativebankreceives1unitofdepositatdate
0.HavingequityEthebank’stotalwealthA=1+Ecanbeinvestedinaliquidrisk
freeassetwhichreturnsr>0,perunitofinvestmentatdate1.Themanagerworking
forshareholders,canalternativelygiveloanstoriskycredits.Tomonitorcreditorsand
haveless1risk,thebankhastobearcoste≥0drawnoutofthewealthatt=1.This
regimeofoperation,denotedbyi=1,generatesR>rperunitoftheinvestment
withprobabilityP1orzerootherwise,attimet=1.Ho2wever,becauseoftheeffort
cost,theshareholdersorthemanagermaydecidetoshirk(i=0)whichincreasesthe
risk,reducingtheprobabilityofsuccesstoP0<P1.The3shareholderscannotobserve
themanager’sdecisionuntilthereturnisrealizedatdate1.Iftheyaskthemanager
tostopmonitoringshewilldo,buttheycannotforcethemanagertomonitor.This
happenssincethemanagerreceivessomenon-pecuniarybenefitQ>0iftheshirking
regimesucceedsandgeneratesR−q,q≥0,perunitoftheinvestmentattheendof
d.rioeptheThedepositorsarepaidafixedamountDatdate1.Riskyregimeibringsthetotal4
expectedprofitattheendofperiod,

Πi=Pimax(0,A(R−(1−i)q)−ie−D).(4.1)

Thereturnishigherundermoreriskyregimewhenitsucceeds,i.e.R−q≥R−ewhich
requirese≥q.Butshirkingisinefficientinthesensethatitstotalreturnislessthan
monitoringregime:R−q+Q<R.
Fortheexpectedprofitofriskyoperationtobepositive,thecapitalshouldexceedsome
thresholds,

E≥D−R+q+i(e−q).
R−(1−i)q

(4.2)

Ifthebankcoulddefinethedepositpaymentendogenously,themanagerwouldpropose
Dsuchthatthebankwouldbesolventincaseofsuccess.Thismeansfrom(4.2)forthe
depositpayment

D≤(1+E)(R−(1−i)q)−ie.(4.3)

Sincethebankshouldmotivatedepositors,theyhavetopaythematleastthesameas
theriskfreereturn,i.e.D≥r.Thenitfollows

1Thedecisionorplanofoperatingwithaspecificportfolioisknownastheregimeofoperation.
2Tostopmonitoringthecreditors.
3Thedepositorshavenoinformationatall.
4Inthisriskneutralsetup,theinvestmentdecisionisoptimallytoinvestallinariskyassetor
nothing.Thishasbeenjustifiedinmysecondpaper.

66

CHAPTER4.THETHEORIESOFBANKREGULATION

(1+E)(R−(1−i)q)−ie≥r.(4.4)

Onemainproblemofhighrisk(lowprobabilityofsuccess)isanegativenetpresentvalue
(NPV).Thisoccurswhenthereisnomonitoringonthecreditors,whereforeachunit
ofinassumevDestmen+et>P0(DR.5−q)Then,−r<the0.GivshareholdersendepositratherpaymenpreferstD,lesswithoutriskiffΠloss≥ofΠ,generalitwhichy
01requiresRfortheR−qtotalcapitallevel,
E≥(P1−P0)(D−R)+P1e−P0q=Eˆ.(4.5)
(P1−P0)R+P0q

HazardMoralThe4.2.1Thechoiceofregimedependsonhowtheshareholderscompensatethemanagertowork
fortheirinterest.Thispartofworkoutlinesthepossiblemanagerialcontractthatthe
shareholderscanofferandmotivatethemanagertooperateintheirfavoriteregime,
thoughitmayberisky.
Whenthemanagerispaidacertainsalarysindependentofherperformance,theex-
pectedprofitoftheshareholdersin(4.1)changessubstitutingA−sforA.Thenthe
shareholderstakethelessriskyregimeforlargecapitallevels,i.e.E≥Eˆ+s.Ifthecap-
italwasnotsufficientlyhightheshareholderswouldprefermorerisk.WhereE<Eˆ+s,
theywouldonlymakeafixedpaymentstothemanageratthebeginningoftheperiod
aslessasheroutsideoptionutilityinordertohaveherinthefirm.Thenthemanager
wouldworkforherprivatebenefitandshirkwhichwouldbealsointheinterestofthe
shareholders.However,iftheshareholderswanttohavelessriskthemanager’ssalaryshoulddepend
ontheperformancewhichinfluencesthesuccessandfailureofthebank.Tomotivatefor
thelessriskyregimetheshareholdersofferanincentivecompatible(IC)contracttothe
managerpayingasalarysonlywhenthebank’soperationsucceedsandthemanager’s
expectedprofitunderregimei=1ishigherthanunderregimei=0:
P1s≥P0(s+Q)⇐⇒s≥P1QP−0P0.(4.6)
NotethattheshareholdersmakesuchacontractifffortheirexpectedprofitΠ1(s)≥
Π0(s)where
Πi(s)=Pimax(0,A(R−(1−i)q)−ie−D−s).(4.7)
5Otherwise,thecalculationchangesbutnottheresult.

(4.7)

4.2.THEBASICMODEL:FAILURESINBANKS

Forthiswehaveanewcapitalrequirement,
E≥(P1−P0)(D−R+s)+P1e−P0q=E∗.
(P1−P0)R+P0q

67

(4.8)

Anotheroptionfortheshareholdersistodefinethesalaryasashareoftheprofit6,
si=αΠiwherePiiisdefinedin(4.1).Iftheshareholders’expectedreturn(1−α)Πiis
higherunderregimei=1,theICconstraintchangestos1>s0+P0Q,or

α(P1[AR−e−D])≥α(P0[A(R−q)−D])+P0Q.(4.9)

Sinceαisindependentfromtheregimechoice,theshareholdersdecisionisredundant
totheonepresentedin(4.5).Therefore,ifE>Eˆ,fractionαmustsatisfy
QP0α≥[A(P1−P0)R+P0q]−eP1−D(P1−P0)=αˆ.(4.10)

4.2.2DepositInsuranceandtheMoralHazard

Incaseofinsolvency,evenifalltheoutcomeofitsoperationgoestothedepositors,
itislessthanthepromiseddepositpayment.Thus,theexcessiveriskofdefaultisat
theexpenseofdepositorsintheabsenceofanyguarantee.Thissubsectionexamines
whetherprotectingdepositorsisaneffectivestrategytopreventfailureinthebanking
sector.Adefaultedbankhastogobankruptandtheshareholdersandthemanagerreceive
nothing.Afairpriceddepositinsurancecanprotectthedepositors.Theinsurercanbe
aprivatecompanyorthestate.However,wheneverthehighamountofdepositpayment
isnotmanageablebytheprivateinsurerthestatehastointervene.Sincealargescale
defaultcaninfluencetheentireeconomy,thestatehastobearthisresponsibility.
Forhavingafairedpriceinsurance,thepremiumequalstheexpectedvalueoftheworst
i.e.case,default

m=D(1−P0).(4.11)

whichshouldbealsosubtractedfromtheprofitfunctionin(4.1).Thestatecould
thenlevyataxonthebankequaltom.Thispreventssubsidizationofthebankby
taxpayers,incaseofadefault.However,theinsurancepremiumortaxwouldshiftthe
capitalrequirementforlessrisk-takingtoEˆ+m.Regardingmanagerialcontract,the
fixedICsalarysdoesnotchangesincetheICconstraint(4.6)remainsthesame.

6SeeforinstanceAcharyaandYorulmazer(2007)thatIexplaininsection4.4.

68

CHAPTER4.THETHEORIESOFBANKREGULATION

Ifthemanagerispaidashareoftheprofit,theshareholders’decisionisasabovebut
theICconstraint(4.9)changes.Inotherwords,A−mappearsinsteadofAinboth
sidesof(4.9).Thisraisestheminimumrequiredmanagerialsharebecausenowweneed
QP0∗α≥[(A−m)(P1−P0)R+P0q]−eP1−D(P1−P0)=α.(4.12)
whereα∗>αˆ.Thisshowsthattheinsuranceortaxsystemincreasesthesharehold-
ers’risk-takingincentivesbyshiftingtheminimumcapitalthresholdupward.Withan
analogousargument,theinsurancedoesnotreducethemanagerialmoralhazard.

4.2.3IntroducingtheRegulatoryActions

Accordingtotheresultofthelastsubsection,thereisaneedforaregulatoryagency
whichnotonlyprovidesresolutionsincaseoffailurebutalsohaspowertoforceex-ante
olicies.preductionriskForlowcapitallevelstheshareholdersprefertheriskierregimeofoperationandletthe
managerfulfiltheirinterest.Asastraightforwardresultoftheshareholders’risk-taking
beingrelatedtothecapitallevel,theregulatormayofferacapitaladequacyruleto
preventtherisk-taking.Itfollowsthattheregulatorclosesthebankwherethecapital
isbelowtherequiredlevelA∗orAˆdependingonthemanagerialcontract.
Mysimplesetupdoesnotincludeanybankruptcycost.Intherealworld,anybank
failureinfluencesitscreditorsanddepositors.Thecreditorswillnotreceivefurther
investment.Andforthedepositors,theycannotfollowtheirplantousethepayment.
Forinstance,acompanymaystopitsdevelopmentsinceithasnotreceivedthedeposit
payment,orreceiveditlateronlythroughtheinsurancepayment.Thesesocialcosts
maketheclosurepolicynottobeex-postoptimal.MailathandMester(1994)describe
howtheclosurepolicycannotbeimposed.Yet,giventheincentivecompatiblecontract
thebankwithlessriskyloansmaydefaulttoo.Thus,thebankingsystemdemands
foroptimizingtheresolutionpoliciesratherthanonlyclosure.Freixas(1999)considers
partiallyinsureddepositsandexaminestheliquidityprovisionpolicywhereclosureisnot
ex-postoptimal.FreixasandRochet(2010)concentrateonintroductionofasystemic
taxthatrequiresaregulatoryauthoritywiththepowertoreplacethemanagerand
shareholders.Thefollowingsectionsdescribehowanoptimalresolutionpolicydependsonwhetheritis
anindividualbankfailureorasystemicfailure.Thenextsectionsurveystheresolution
policiesforindividualbankdefaults.Explainingregulatorystrategiesfocusedonasingle
bankmakesaproperbackgroundforextendingthemodelfurthertoexaminesystemic
crises.Moresystemicriskregulatorypoliciesarepresentedinsection4.5.

4.3.THERESOLUTIONOFINDIVIDUALBANKFAILURES

69

4.3TheResolutionofIndividualBankFailures

Thissectionaddressesthepossibleinterventionpoliciesofastrongregulatory/supervisory
authoritytoresolvecostlybankruptcies.Themainproblemthattheregulatorfaceswhen
takingactionagainstariskyorfailedbankisknownastime-inconsistency.Thisprob-
lemandthealternativestodealwithitareexplainedinthenextsubsections.Ineach
case,thebasicset-updescribedabovemaychangeslightlytofittherequirements.For
instance,thetimehorizonandriskaversion/neutralitymaydiffer.Weseehowpolicy
implicationsmaychangefromonesituationandsetofassumptionstotheother.

Time-InconsistencyThe4.3.1Problem

Astrongregulatorshouldhavethepowertoshutdowntheoperationofabankwhichis
takingexcessiverisk,asthisex-postreactioncaninfluenceex-antetheinvestmentofthe
bank.Inorderfortheregulatortohavetheopportunityofsupervisoryvisittothebank,
Imustconsideratimehorizonmorethanoneperiod.MailathandMester(1994)assume
thattheregulatorhastwooptionsatdate1andthebankhastwoperiodsofoperation
iftheregulator,visitingatdate1,letsitoperateforonemoreperiod.Thissubsection
analyzesthemodelofMailathandMester(1994)whichlooksintotheeffectivenessof
theregulatoryclosurepolicy.
Iexcludethefrictionbetweenthemanagerofthebankandtheshareholders.Suppose,
thebankdecidesonlybetweentheliquidassetwithcertainreturnr(safe)andregime
ofoperationi=1(risky).Thebankreceives1unitofdepositatthebeginningofeach
periodandinvestsall.Theriskyassetsmatureattheendofthesecondperiod.Forless
complicationassumetheriskyloansarefreeofthecostofmonitoringeffort,i.e.e=0.
TheinefficiencyoftheriskyassetisdefinedashavingnegativeNPV,P1R<r.Thereis
afixedcostofclosureC,bornebytheregulatorwhorepaysfullythedepositorsofthe
bank.failedNotethattheregulatorisredundantifthebanktakesnoriskyinvestmentforthe
twoperiods(i1,i2)=(safe,safe).Ifthebanklosesinthesecondperiodithastopay
everythingevenoutoftheprofitofthefirstperiod.Therefore,itprefers7(risky,safe)
strictlyto(risky,risky)ifftheNPVislargerfor(risky,safe)thanfor(risky,risky),

P1[(R−1)+(r−1)]>2p12(R−1).(4.13)

First,supposethebanktakesthestrategyofswitchingfromoneregimetotheother
atthebeginningofthesecondperiod.Theregulatorypolicyistoclosethebankifit
choosessafeforthefirstperiodbecausethebankwouldotherwisechooseriskyforthe

7(risky,safe)and(safe,risky)areequivalent.

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CHAPTER4.THETHEORIESOFBANKREGULATION

secondperiod.Hence,theoptimalsolutionisforthebanktoplay(risky,safe)andfor
theregulatortoleaveitopen.
Next,considerthebankalwaysplaysriskyinthesecondperiod,ifitisopen.Ifthe
regulatorclosesthebanktheexpectedcostfortheregulatorwillbe(1−P1)+Cthat
istheclosurecostplusthedepositpayment.Ifthebankremainsopentheregulator’s
expectedcostis2(1−P1)2+2(1−P1)P1(2−R)+C(1−P12).Thefirsttermisthe
expecteddepositpaymentwhenthebanklosesinbothperiods.Thesecondtermisthe
expectedcostifthebanksucceedsinoneofthetwoperiods.Andthelasttermisthe
costofclosureincasethebanklosesinatleasoneofthetwoperiods.Comparingthe
regulator’sexpectedcostsofpolicies,weseethatthebankwillbeclosedifftheexpected
costofclosureislessthanleavingthebankopen,
C<(1−P1)(1−2P1(R−1)).(4.14)
2P1When(14)doesnohold,thebankplays(risky,risky)becausetheregulatorwillplayopen.
Otherwiseif(4.14)issatisfied,thebankchoosesbetweentakingriskfacingliquidation
orstayingundercertainty.Butthentheregulatorknowsthatthebankwouldplayrisky
inthesecondperiod.Inasimilarmethodwecanfindthatthebankplaysriskyatthe
firstperiodandtheregulatorclosesitiff
C<(1−P1)(2−r).(4.15)
P1

(4.15)

Therefore,thecostofclosureistheimportantvariabletotheregulator.Sinceclosure
maybelesscostlyinthefuture,theregulatorcannotcommitex-antetobesevere.When
thedepositvalueofabankisveryhighthesocialcostofclosurebecomeslargethat
mayleadtonon-liquidationandbailout.Thispolicygeneratesmoral-hazardincentives.
This”toobigtofail(TBTF)”problemismorediscussedinthenextsectionasitisnot
onlyanissueofasinglebank,butalsomayaffectthebankingsystem.

4.3.2OptimalLiquidityProvision

Whenbankclosureandliquidationofassetsarenotex-postoptimal,acentralbankmay
finditessentialtoprovideliquiditytothebank.Thisideahasbeenaddressedasthe
LenderofLastResort(LOLR).Thequestioninthispartoftheworkisthathowthe
liquidityprovisioncanoptimallysolvethetime-inconsistencyproblemwhenthecostof
rge.laisbankruptcyDespiteallargumentsagainstLOLRthatitwillcausethecentralbanktofacethecon-
depsequenendstonmoralthehazarddegreeptoroblemwhichaandbank’sincreasingdepositsrisk,areFreixasinsured.F(1999)reixasclaims(1999)thatinvtheestigatesresult

4.3.THERESOLUTIONOFINDIVIDUALBANKFAILURES

71

twopossiblesourcesofrisk,i.eexogenousandendogenous,andwhethertheregulatory
policyshouldchangefromonecasetotheother.Hesortsoutanefficientimplemen-
tationofliquidityprovisionbasedonacostbenefitanalysis.Themaindifferencesto
thesetupfromsection4.2arethatβpercentofdepositsareuninsuredandthebankis
investingonlyintheriskyasset(i=1).
Anegativeexogenousliquidityshockwhichcausesfailureofriskyloansbringsfinancial
distress.Thetimehorizonisoneperiod,attheendofwhichthecentralbankreacts
incaseofadefault.Thepromisedpaymenttoinsuredanduninsureddepositsare
(1−β)(1+rD)andβ(1+rL),respectively.Theexpectedvalueofthebankunder
regulatorybailoutorliquidation(incaseofinsolvency)isdenotedbyVLandVC.Since
theliquidationvalueisnon-zero,thefairpricedinsurancepremiumchangesto8

m=(1−P1)max[(1−β)(1+rD)−VL,0]

andthesubsidybybailoutsumsupto

S=β(1+rL)−max[VL−(1−β)(1+rD),0],

(4.16)

(4.17)

assumedtobepositive.LetΔbethedifferencebetweencostsofcontinuationand
liquidation.TheregulatorydecisiondependsonΔwhichisdecreasinginclosurecostC.
Whenthecentralbankhasnocommitmentforclosure,sinceCisincreasinginbank’s
wealth,A=E+1,forsomerangeofparameterstheTBTFproblemholdsinthesense
thatifabankwithassetAisbailedout,alllargerbankswouldbeoptimallyrescued.
Assumethatthecentralbankmakescommitmenttoaspecificregulatoryresolution
policy.Letθ>0betheprobabilitythatthecentralbankrescuesthebank.The
optimalregulatorypolicyisdeterminedbymaximizingtotalsurplusofthebank’sactions
subjecttotheincentivecompatibilityconditionwhichrequiresahigherbankprofitunder
bailoutforanygivenβ.Freixas(1999)assumesthatCisincreasinginβ.Thus,hefinds
thateithertobailoutortouseamixedstrategy(betweenliquidationandbailoutwith
θ>0)isoptimaldependingontheamountofuninsureddebt,β.Themixedstrategy
isinterpretedas”constructiveambiguity”,whichhadpreviouslyonlybeenjustifiedin
el.leveconomicsmacroInthesecondpart,Freixas(1999)takesintoaccountthemoralhazardproblemwhere
therisklevelischosenendogenously.Theresultabouttheoptimalregulatoryissimilarly
dependentonβaslongasthemonitoringeffortcostisnotconsidered.FirstIdescribehis
generalsetupwithendogenousrisktaking,thenexplainhowthemonitoringassumption
influencestheoptimalregulatoryactions.
Thebankhasacontinuumofrisklevelsandchoosestheprobabilityofsuccessatacost
ϕ(P),assumedtobestrictlyincreasing,convexandtwicedifferentiable.Thedifference

8(4.11).toitCompare

72

CHAPTER4.THETHEORIESOFBANKREGULATION

tomysetupis9thatinhismodelthedifferentiabilityofϕ(P)isnecessaryforoptimizing
theandriskthelevel.manager.Indeed,Theinhisprobabilitpapeyroftheregrossisnreoturnagencyx=x(problemβ)isbPet=wPeen(x).theThesharehprofitoldersof
thebankfortotalreturnequals,

Π=P[x(β)−β(1+rL)−(1−β)(1+rD)]−m−ϕ(P).(4.18)
ThefirstbestPˆcanbeobtainedfromx(β)=dϕdP(P).However,sincerateofreturnto
uninsureddepositsrLandmshouldbealreadyadjustedrationallyforP,thefirstorder
condition(maximizingΠ)yieldshigherrisk,i.e.smallerprobabilityofsuccessthanPˆ.
RewriteprofitfunctiontoobservetherelationshipbetweenparametersθandP,

Π=Π0+(1−P)θS,(4.19)

whereΠ0istheexpectedprofitthebankwouldgainintheabsenceofanysubsidy(if
nobailout).ConcavityofΠ(resultingfromconvexityofϕ(P))andthederivativeofthe
firstorderconditionshowthatPisdecreasinginθ.Itmeansthatliquidationismore
frequentasthebailoutpolicywouldincreasethebank’sriskiness10.Inaddition,welfare
analysisshowsthattakingmoreriskdecreasessocialsurplusesofbailoutpolicy.Yet,
theoptimalpolicyforthecentralbankiseitherasystemicbailoutoramixedstrategy.
Inthecasethattheeffortlevelofthebankdeterminestherisk,theprobabilityofsuc-
pcessolicyPisandsimilaritscosttoϕotherarethcases;ushofunctionswever,ofFeffortreixaslev(1999)elev(aberifiesoundedthatvmoralalue).Thehazardoptimaleffect
apprescueearsadbankifferenwithtly.smallThough,leveltheofsocialuninsuredcostofdebt;abankruptcylargerimpliesamountthatofitisuninsuredoptimaldebtto
generatesaclosermonitoringofthebankbyitscreditors.Hence,themoralhazardef-
fectdebtwandorkstighcounterter-balamonitoring.nced.11AThismixedisthestrategycasestimalsoulateswherebanksthetoLOLRkeepismoableretouninsuredcommit
tobank’sbailoutriskiness,withsomedecreasespositivemonitoringprobabilityeffort.Nevanderttheheless,marginalthebbailoutenefitspofolicyrescuingincreasesbanks.the

TheworkofCordellaandYeyati(2003)onthemoralhazardproblemfocusesonthe
valueeffectofbail-outpolicywherethecentralbankannouncesandcommitsex-anteto
rescuebanksintimesofexogenousmacroeconomicshocks.Theprobabilityofsuccess
notonlydependsontheriskchoiceofthebank,butalsoisaffectedbyastatedependent
termη,whichisunobservablebythecentralbank.Inadynamicmulti-periodsetup,

9Insection2ofthispaperIconsideramonitoringeffortcosttobringahigherprobabilityofsuccess.
Thus,inmymodelϕ(P)takesonlytwovalues:eforP1but0forP0.
10Sufficientconditionsareconsideredtoavoidcornersolutions.
11Theresultisinlinewiththerationalethatsubordinateddebthelpstohaveabetterbanking
discipline.

4.3.THERESOLUTIONOFINDIVIDUALBANKFAILURES

73

theriskyinvestmentreturnsxwithprobabilityP(x,η)=ηP(x)where0≤η≤1isi.i.d
andP(x)isdecreasing.12
Withfulldepositinsurance,intheabsenceofbail-outpolicy,itisshownthatquite
intuitivelythebankneverchooseslowerriskthansociallyoptimal.Thecentralbank
followsaconstructiveambiguityapproach.Theshareholdersmayrecapitalizethebank
incaseoffailurebyraisingcapitalinthecapitalmarket,evenifthecentralbankdoes
notbailout.Inthenon-recapitalizationscenario,theprobabilityofbailoutθbecomes
anegativefunctionofη.Thenastate-independentbail-outpolicy,θ(η)=θ,increases
risk-takingofthebankaswecouldexpect.However,regardlessofthebank’sdecisionon
capitalization,thecentralbankminimizestherisk.Theoptimalrisk-minimizingbailout
policyisobtainedbyathresholdηˆbelowwhichthecentralbankrescuesthebankwith
certaintyandletsitfailotherwise.Underthisstrategy,thebankalwaystakesriskmore
thanoptimallevel.Analternativeoptimizingapproachistomaximizingthecentral
bank’sobjective,whichconsidersthepossibleefficiencycostofbailout.Thisapproach
bringsaboutsimilarregulatorypolicywithathresholdatleastaslargeastherisk-
minimizingthreshold.Inotherwords,thecentralbankisneverlessgenerousthanthe
.olicyprisk-minimizingAccordingly,constructiveambiguityisbeneficialtoruleoutthemoralhazardproblem
arisingfromthebank’sendogenousrisktaking.However,onoccasionofmacroeco-
nomicshocks,systemicalinterventionofthecentralbankcontingentontheexogenous
conditionsisdesirableasitcreatesrisk-reducingvalueeffects.

4.3.3TakeoverasanIncentivesForRiskReduction

Besideclosureandbailoutpolicies,thesupervisoryagencymayallowfortakeoverofthe
failedbankahealthyfinancialinstitute.Thispolicyhasbeenpromotedasanincentive
program.Inadynamicmodel,PerottiandSuarez(2002)arguesthatasolventbank
canbuyafailedinstitutionandbenefitfromtheincreaseinitschartervalue.
InthesetuppresentedbyPerottiandSuarez(2002),anewbranchofthebankentersthe
marketonarandombasisdeterminedbytheregulator.Theregulatordecidesalsohow
toresolvethefailures.Ifbothbranchesfail,shewillemploytwonewbankerandlets
themtocompeteinaduopoly.Butifonlyonebankfails,sheshouldoptimizewhether
toallowfortakeoverbytheotherbranch.
Foreachbank,thereturntoaprudentlendingiscertain.Thereisanopportunity
forspeculativelendingwhichgeneratesextrareturnbutleavesthebankexposedto
exogenoussolvencyshocks.Monopolyismoreprofitableforabankduetotheabsence

12TosimplifythemodelthebankonlychoosesMarkovstrategiesinrisk-taking.Thissimplification
makesaclosedformsolutionpossiblebutreducestheproblemtoaspecificcaseinwhichrisk-takingin
eachperiodisindependentofandhasnoimpactonotherperiods.

74

CHAPTER4.THETHEORIESOFBANKREGULATION

ofcompetitionbuttherentcomesatacost.Thestochasticentryofthenewbranch
turnsmonopolytoduopoly.Therefore,thelendingstructureofeachbranchimpactsthe
otherbank.Abankmayspeculateinamonopolybutinduopolyitcanbeallowedto
buythefailedbranchifitissolvent.Thus,inaduopolythebankhaslessincentivefor
speculativelendingbecauseoftherewardforbeingsolvent.Bytakeoverthesurvived
bankistemporarilyamonopolist.Thehigherrentinthiscasemakesanewbranch
willingtoenterthemarket.
Thesupervisoryagencyasasocialplaneroptimizestheentranceandtakeoverpolicy,
minimizingthesociallossesincaseoffailures.Itleadstoallowingtakeoverandimple-
mentinganoptimummixtureofprudenceandcompetitionthroughanadequatelevelof
newentryrate.Thiswaybanksconvertfromspeculativelendingintostrategicdecisions
inordertoremainsolvent.

RiskSystemicRegulating4.4

Failureofasubstantialpartoftheeconomy,meaningalargeinstituteormanysmallones,
areconsideredassystemicfailures.Mostlytheregulationpolicieshavesofarfocusedon
individualbank’srisk.Therefore,insolvencyofabankisdealtwithaccuratelyinnormal
times.However,inadditionthereisariskofsystemicfailuresthatleadtoseverecrisis.
Therecentcrisisraisedattentiontotheneedforrestructuringregulatorystrategiesin
ordertotakeaccountofsystemicrisks.Thissectionstateswhyitisnecessarytoregulate
systemicfinancialcrisesandinvestigatestheexternalitiesinvolvedinasystemicfailure.
In2008thestatesletLehmanBrothersfailinordertolimitmoralhazardrisk-taking.
Onthecontrarytothegovernment’sinterest,itledtoaseriouscollapseofthefinancial
system.Eventually,failureofthislargefinancialinstitutespreadtoasignificantpart
oftheeconomythroughdirectandindirectinterconnectionstootherinstitutes.Then,
thesecondexternalityappeared.Noprivatesector,includingbanksandinsurancecom-
panies,couldtakeoverandcompensateforthelargescalefailuresofmanybanksand
institutes.Hence,therecentcrisisshowstracesofexternalitiesintwomaindirections.Thefirst
externalityisthespilloverriskofonebankonotherbanks.Thesecondisthecollective
failuresofbankswherehealthybankscannottakethemover.Theformerisdiscussed
withinamodelofcontagionandthelatterasthetoo-many-to-failproblem,inthe
subsections.wingfolloIfthebankingsystemisindangertocollapsealltogether,naturallythesupervisory
hastotakeprecautionaryreactions.AfterthefailureofLehman,thegovernmentcould
notletanyotherlargefinancialinstitutefail,despitethefactthatthebail-outpolicy
strengthenedmoral-hazard.Thecostsandinconveniencesongovernmentsandsuper-

4.4.REGULATINGSYSTEMICRISK

75

visoryauthoritiesdemonstrateneedsformacro-prudentialregulatorystrategiesthatis
thetopicofthenextsection.

4.4.1ContagionandTooMuchRelatedBanks

Toillustratethefirstexternalityeffectabankfailurehasonthebankingsystem,I
refertothecaseoftransmittingbank-run,namedcontagion.AllenandGale(2000)
studythefragilityofabankingsystem,wherebankrunsspreadinthesystem.Their
modelisnotableformypurposesinceitseparatestheinter-bankstructurefromthe
risk-takingbehavior.Thisapproachhelpstoemphasizethespilloverexternalityand
avoidscomplexitycausedbytheriskoptimizationchallenges.AllenandGale(2000)
considertheliquidityprovider13roleofbankswhichmaximizetheirdepositors’utility.
Assumetherearefourbankseachoperatinginadifferentregion,denotedbyA,B,C
andD.Forsimplicity,supposeeachbankhasnoequity,i.e.E=0,atdate0.The1
unitofdeposit(providedbythedepositorsofthesameregion)attimet=0istheonly
availablesourceofwealthtoeachbank.Depositorsdemandd1andd2atdates1and2,
respectively.14However,eachbankreceivesearlydemandswithprobabilitywHorwL
ineachregionatdate1,where0<wH<wL<1.Themanagerworksforthebank
withoutmoralhazardproblem.Theliquidityproblemraisesfromthebanksinvestment
inanilliquidassetwhichtakestwoperiodstomature.Itmeansthateachbankinvests
amountLinanassetwhichreturnsR>1per1unitatt=2.Therefore,abank
mayhavetoliquidateassetsprematurelytopaytodepositors.Liquidatingoneunitof
investmentproduces0<λ<1unitatt=1.
Eachbankdecidesabouttheinter-bankmarket,aninvestmentportfolioandadeposit
contract.Supposeacompletemarketinwhicheverybankhasdepositsineachofother
regions.Sinceallregionsandconsumersareequivalent,withoutlossofgenerality
assumeinregionsAandCthereareearlyconsumerswithlowprobability,butinB
andDwiththehighprobability.Inthecompletemarketbankscaneasilytransfertheir
excesssupplyoftheliquidassettotheregionswithexcessdemandsatdate1.Suppose
everybankhasdepositz=(wH−γ)/2ineachbankofthreeotherregions,where
γ=wH2+wL.Now,thebankshavetochooseonlythedepositcontracts,d1,d2,andthe
riskyinvestmentL.Eachbankmaximizestheexpectedutilityofconsumersattime
t=0inthefollowingway,

γu(d1)+(1−γ)u(d2).

(4.20)

13TheframeworkofDiamondandDybvig(1983).
14Thedepositorscouldnotexpectthesamelevelofutilityabankintheirregionbringsthemin
.autarky

76

CHAPTER4.THETHEORIESOFBANKREGULATION

Asthetotalofdepositors’consumptionineachperiodisaconstant,itisoptimalforthe
banktoholdtheliquidasset(byitselforasdepositinotherregions)fortheearlydeposit
demands,i.e.γd1≤r(1−L).Itgivesthefeasibilityconstraintofthesecondperiod,
(1−γ)d2≤RL.Theobjectivefunction(4.20)increasesaslongastheconsumptioncan
beshiftedfromearlydepositdemandtothelatedemandsusingtheliquidasset.The
firstorderconditionisobtained,u´(d1)≥u´(d2),whereu´(.)isthefirstderivative.This
conditionstopsshiftingdepositdemanduntild1≤d2whichisanincentiveconstraintfor
thedepositorswhowaitlonger.Otherwise,thedepositorswithlatedemandwouldbe
betteroffwithdrawingatdate1.Thisoptimizationproblemisthesameasifacentral
planneroptimizesrisksharing.AllenandGale(2000)callthisoptimalallocationafirst
bestallocation,whichisalsoincentiveefficientasseenabove.Thereisnobank-runand
liquidation.prematureforneednoConsideraperturbstatewhichoccurswithprobabilityzero,suchthateachbankinB,
C,andDreceivesearlydepositdemandswithprobabilityγ,buttheycometothebank
inregionAwithprobabilityγ+,>0.Ifabankisinsolventitmayliquidatesomeof
theilliquidassettomeetitscommitmenttoearlydepositdemand.Butitpreferstopay
outofliquidassetsatfirst,andnextliquidatesthedepositsinotherbanks.Ifneither
liquidasset,nordepositliquidationhelps,the15bankwillliquidatetheilliquidassetsat
date1.Thisiscalledliquidation”pekingorder”.
Inordertopreventarun16indate2,abankwithafractionwofearlydepositsdemands
hastokeepatleast(1−w)d1/Runitsoftheilliquidasset.Sincetheamountofilliquid
assetis1−L,thehighestamountthatcanbeliquidatedatt=1is(1−L−(1−w)d1/R)
whichproducesthebuffer

b(w)≡λ(1−L−(1−w)d1/R)(4.21)

Aslongastheamountofilliquidassetabankneedstoliquidateislessthanthisbuffer,
thebankisinsolventbutnotbankrupt.Intheperturbcase,theassetsofthebankin
regionAarevaluedatr(1−L)+λL+3zd1atdate1.Thelasttermcomesfromits
depositsinthreeotherregions,asineachonedepositsarevaluedatd1.Theliabilities
ofbankAarevaluedat(1+3z)qA,whereqjisthevalueofabank’sdepositsinregion
j.Balancingassetsandliabilities,qAisfound:
qA=r(1−L)+λL+3zd1.(4.22)
z3+1ThebankinregionAisbankruptwhenever,

d1≤b(γ+).

15Itholdsforsmallλ.
16Thebankcannotpaytothelatedepositdemandsfully.

(4.22)

(4.23)

4.4.REGULATINGSYSTEMICRISK

77

ThelosstoeveryotherbankbecauseofbankruptcyinAisz(d1−qA).Thosebanks
willnotbebankruptiffthelossislessthantheirbuffer,
z(d1−qA)≤b(γ).(4.24)

ImaginethatregionAwasonlyconnectedtoB,andConlytoD.Then,bankrunin
onesectorofthemarketwouldnevertransmittoanothersector.AllenandGale(2000)
investigatethefragilityofsystemwhereregionsareincompletelyconnectedwhereeach
bankhasdepositonlyinoneneighborbank.RegionAhasdeposits2z=wH−γinB.
SimilarlyBinC,CinDandregionDhasdeposits2zinregionA.Undertheassumption
thattwobanksreceiveearlydepositdemandswithhighprobabilityandtwootherwith
lowones,thefirstbestallocationisstillachieved.Thereasonisthattheobjective
functionisthesameas(4.20)andthebudgetconstraintsforhighliquidityshocksat
dates1and2are,respectively,

wHd1=r(1−L)+(wH−γ)d1
[(1−wH)+(1−wH)d2]=RL,

whichequalthepreviousconstraints,

γd1=r(1−L)
(1−γ)d2=RL.

(4.25)(4.26)

Similarlyfortheregionswithlowliquidityshocksthesamebudgetconstraintshold.
Thus,inthiscasetheliquiditytransfermakesthefirstbestallocationpossible.Never-
theless,intheperturbsituationwhere(4.23)issatisfiedforregionA,thespilloverto
regionDcanbelargeenoughthat,
2z(d1−qA)>b(γ).(4.27)

Notethatbothinequalities(4.24)and(4.27)arepossible.Itmeansthat,forasetof
parametersthecompletemarketcanbesafefrombank-runcontagionbutnottheincom-
pletemarket.AsbankinAisbankrupt,itsassetsarevaluedlessthand1.Therefore,
thedepositofregionDinAisnotsufficientanditmustliquidatemorethanthesafe
buffer.ThisinturncausesbankruptcyforthebankinD.Inasimilarwaythelosses
transfertothebankinregionCandthentooneinB.Accordingly,allbanksconnected
byachainofoverlappingbankliabilitiesmustgobankrupt.
Asseenfragilityofasystemisdifferentundercompleteandincompletemarkets,but
thereisnotamonotonerelation.Actually,thelevelofinter-connectionamongbanks
determineshowacontagioncanspread.

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CHAPTER4.THETHEORIESOFBANKREGULATION

IftheoutcomeinAllenandGale(2000)wererisky,similartomysetupinsection4.2,
thereturnsindifferentregionswouldnotbeperfectlycorrelated.Inthatcasebanks
wouldgainfromrisksharingandtheywouldholdclaimoneachother.Ex-ante,the
firstbestcouldbeachieved.But,ex-postrisksharingwouldnotbepossiblewhenthe
returnswereknown,asarrangementsandbankruptcyruleswouldnotworkproperly,in
additiontothecomplicatedanalysis.Thisexplainsinsomeextendthemorecomplexity
ofcontagionintherealworld,thatitleadstodifficultyofdealingwithcrisis.
Thisdiscussionofcontagionandfinancialfragilityconcentratesontheliabilitystructure
ofbanks.TheinterbankrelationshasbeenaddressedtogetherwiththeTBTFproblem.
Thesourcetothesystemicriskinbothissuescanbetheassetsideofthebanks’balance-
sheets.RochetandTirole(1996)investigatetheTBTFandsuggestpeermonitoring
amongcommercialbanks.Intheirwork,TBTFoccursifthepeermonitoringstarts
aftertheliquidityshockanditmoredependsonthesizeofinterbankloansthansizeof
theindividualfailedbank.FurtherworkoninterbankmarketisdoneinFreixasetal.
(2000).Theyshowthatononehand,interbankcreditlinesreducethecostofholding
reservestocopewithliquidityshocks.Ontheotherhand,acontagionisinevitable
intheseconnectionlines.Insolvencyofonebankaffectsthestabilityofthebanking
systembecauseofacoordinationfailure,evenifotherbanksaresolvent.Moreover,
thesubsidygeneratedinthenetworkofcross-liabilitiesallowstheinsolventbankto
continueitsweakperformance.Ifthecentralbankdecidestoliquidatethisbank,it
hastocompensateforpaymentsofthedefaultingbanktothedependingbanks.Here
twocoursesareavailable,inefficientliquidationofcounterpartsofaninsolventbankor
bailoutthedefaultingbank.Therefore,Freixasetal.(2000)resultinamoralhazard
.TBTFasproblem

4.4.2TooManyToFail

Asoftregulatorwhoex-antelowersmonitoringcapacityorex-postrescuesinsolvent
banks,not-beingsufficientlygenerous,triggersbankstocolludeondisclosingtheirlosses.
Thus,manybanksrollovertheirbadloanspassivelyratherthantoannouncebankruptcy
againstdefaults.Consequently,theregulatormayneedtorepeatrescueorrecapital-
izationinthefuture.Mitchell(1997)explainsthisissueandcallit“toomanytofail”
(TMTF).AcharyaandYorulmazer(2007)analyzestheherdingbehaviorofbanksleadingto
TMTF.Theirworkfocusesonbanks’inter-correlationofrisk-takingandcoversthe
threemainregulatoryactions:closure,bailoutandtake-over.Thissubsectionstudies
theeffectivenessofthesepoliciesindealingwiththeTMTF.
Thecollectivefailureofmanybankshavebeenanalyzedinthepresenceofregulatory
actionswhichfocusonindividualbanks.Theseregulatorysystemshavebeenlimitedly

4.4.REGULATINGSYSTEMICRISK

79

effectiveastheycouldnotpreventsystemicfailures.Thissubsectionincludesasimilar
setupthattheregulatorypoliciesexistbutdonottargetsystemicrisks.Weseehow
bankstakeadvantageofitandinitiatewidespreadfailures.Tocomplementtheproblem
discussedinthelastsubsection,thefocusisontherisktakingbehaviorofbanksin
thispart.Further,thebanks’herdingonrisktakingcontributestothesecondtypeof
externalityinalargecrisis.
Considertwoequal-sizedbanksAandBoperatingeachfortwoperiods.Eachperiod
hasbasicallythesetupofsection4.2.Butitisadaptedforlettingtheregulatorintothe
modelandafewsimplifications.Theregulatormayinterveneattimet=1.Theonly
sourceoffundforeachbankis1unitofdepositperperiod.Depositsaredebtcontracts
withmaturityofoneperiod.Thebanksbenefitfromfulldepositinsuranceonlyinthe
firstperiodwhichcostsad1,a>0,whered1isthedepositreturnatt=1.Thend2
denotesthedepositreturnatt=2.
Moralhazardofabankmanagerisdefinedasbefore.However,assumetheprobabilities
ofsuccessdonotchangebymoralhazard,i.e.P0=P1=P¯,buttheprobabilitydepends
ontheperiod.P¯1andP¯2standfortheprobabilityofsuccessindependentofwhomowns
thebank’sassetsatt=1,2,respectively.Further,ignoretheeffortcost,i.e.e=0.
ICconstraint(4.10)indicatesthatthebankerneedsaminimumshareofα=qQnotto
zard.hamoralcommitDefineliquidationassellingthebanktooutsiderswhogeneratesonlyR−δinthesuccess
state.AcharyaandYorulmazer(2007)assumeδ<qwhichmeansoutsiderscanmanage
thebankbetterthanthemoralhazardcasebutarenotasproductiveasthebankers.
Thisisinlinewiththeliteraturethatthemoralhazardrisk-takingisthemostsevere
caseintermsofsocialwelfareasitsoutcomeistheleast.
Thebankschoosetheirinterbankcorrelation,ρ∈{0,1}whichreferstothecorrelation
oftheirrespectivereturns.Whereasρ=0,thetwobanksbelongtotwodifferent
industries,andρ=1meansthattheychoosethesameindustry.HavingtwobanksA
andBintheeconomy,4possiblestatesattimet=1aregiven:SS,SF,FS,FF,
whileSandFrecallingsuccessandfailureofbankAandB,respectively.Beinginthe
sameindustrythejointprobabilitiesofthe4casesare,P¯1,0,0,1−P¯1.However,ifthe
banksareindependent(twodifferentindustries)thenthejointprobabilityofeachstate
isgivenbymultiplyingtheprobabilitiesofthetwooutcomes.
Toshowthatsurvivingbankwillalwaysbuythefailedbank,takeintoaccountfollowing
assumptions:(i)withoutlossofgenerality,bankAhasthebargainingpowertooffer
tobuybankB,(ii)bankAwillaccesstodepositorsofbankBafterpurchase,and(iii)
depositinsuranceiscostlytotheregulatorwhenthereisabankfailure.Thesurviving
bank,Awillalwaysbuythefailedbankinaprice,ψ=P¯2(R−δ)−1equaltowhat
outsidersatmostwouldpayinstatesSF.Thesurvivingbank’sexpectedprofitfrom
theinvestmentinassetsofthefailedbankwillbeP¯2R−1.Therefore,itpurchasesthe

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CHAPTER4.THETHEORIESOFBANKREGULATION

failedbankandreceivesthediscount(P¯2δ).Thisresolutionpolicyisoptimalalsoforthe
regulator.Themisallocationcostiszero,comparingtothemisallocationcostofselling
totheoutsiders,P¯2δ.Thefiscalcostfortheregulatorisa(d1−ψ)inbothcases.The
bank’sbailoutpolicyincludesnomisallocationcostbutfiscalcostad1.
InthisuniquesubgameperfectequilibriumtheregulatorneverintervenesinstateSS.
InstateFF,ifbothbanksaresoldtooutsiders,theregulator’sobjectivefunctionis
E(V2L)=2[P¯2(R−δ)−1]−a(2d1−2ψ),(4.28)
andifbotharebailedout,ittakesthevalue
E(V2B)=2(P¯2R−1)−a(2d1),(4.29)
ofcourseitisalreadyassumedthatthebank’smanagerhasaminimumshareofαin
eachbailedoutbank.Bailingoutonebankandliquidatingtheotherone,theobjective
functiontakesavaluebetweenE(V2L)andE(V2B).Astwobanksaretakensymmetri-
cally,theregulatortakesthesameactiontowardseitherofthem.Hence,bothbanks
areliquidatedifE(V2L)≥E(V2B)whichgivesδ≤δ∗,whereδ∗=aP(¯2P¯2(1+R−a)1).Otherwise,
theregulatorbailsthemoutandtakesashareυineachbank’sequityυ<(1−α).
Knowingtheregulator’sstrategy,wefindthebanks’decisionontheinterbankcorrela-
tion,whichistheirinvestmentproblemofdate0.Notethatabank’sfirstperiodprofit
P¯1R−d1isindependentofρ.Therefore,itsexpectedprofitoftwoperiods,

E(π1)+E(π2(ρ))(4.30)

isoptimizedwithrespecttoρasE(π2(ρ))maximizes.Whentwobanksinvestinthe
17,industrysameE(π2(1))=P¯1E(π2ss)+(1−P¯1)E(π2ff).(4.31)
tiate,differentheyifButE(π2(0))=P¯12E(π2ss)+P¯1(1−P¯1)E(π2sf(0))+(1−P¯1)2E(π2ff),(4.32)
whereE(π2sf(0))=E(π2ss)+P¯2δ,aswediscussedbeforethatthesurvivingbankreceives
adiscountP¯2δbybuyingthefailedbankinstateSF.Thechoiceofinterbankcorrelation
isdeterminedbythetradeoffbetweenthisdiscountandthesubsidyatbeingbailedout
,FFstateinE(π2(1))−E(π2(0))=P¯1(1−P¯1)[E(π2ff(0))−P¯1δ].(4.33)

17Theoutcomesareonlysuccessforbothorfailureofbothbanks.

(4.33)

4.4.REGULATINGSYSTEMICRISK

81

Iftheregulatorliquidatestwobanksatt=1,i.e.δ≤δ∗,bankschoosethehighestlevel
ofcorrelationatt=0.Otherwise,ifbanksarebailedout,E(π2ff)=(1−υ)(P¯2R−1)
exceedssubsidy18P¯2δifandonlyif,
¯δP2∗υ<υ=1−(P¯2R−1).(4.34)
Thus,ifυ∗>1−α,forabailoutstrategyofυ<1−αbankstakesρ=1.Butif
υ∗≤1−α,theyherdwheretheregulatortakesverylowshareυ<υ∗.Tomakebanks
differentiateunderabailoutpolicytheregulatorhastotakeυ∈[υ∗,1−α].
However,theex-anteoptimalpolicymaydifferfromtheregulator’sex-postpolicies.
ThelossesinstateFFinspiretheregulatortoimplementclosurepoliciesthatmini-
mizesex-antetheprobabilityofthisstate.Itmeansthattheexpectedtotal19outputof
thebankingsectorismaximizedwhenbanksinvestindifferentindustries.Incase
δ≤δ∗,obviouslytheex-anteandex-postpoliciesarethesame.Inthemorecrucial
caseofδ>δ∗,theregulatorneedstotakeadilutionυ>υ∗topreventherding.Where
υ∗<1−α,theregulatorcantakeυ=υ∗toprovideincentiveforbankstodeviate
andstillcontinuewithoutmoralhazard.Nevertheless,themostconsiderablecaseis
whenυ∗>1−α.AcharyaandYorulmazer(2007)findasetofparametersunderwhich
ex-anteitisoptimaltoliquidatebothbanks,asδ<qandtheliquidationcostsare
smallerthantheagencycost.Butdiscussedabovethatex-postitisoptimaltobailout
bothbanks,sincetheregulatorisex-postonlymaximizingtheprofitsatstateFF.The
regulatorex-anteobjectstoreducethelikelihoodofjoint-failure.Shemaygiveupsome
ofitsprofitandimposesatougherliquiditypolicyinordertoincentivelesscorrelation
betweenbanks.Hence,stateFFincludestimeinconsistencyproblemforlargeδ.

AcharyaandYorulmazer(2007)comparetoo-big-to-failandtoo-many-to-fail,assuming
twobanksasymmetricintheirsizes.WithoutlossofgeneralityletbankAbethelarger
bankwhichtakesdepositmorethan1unit.Theresultchangesduetotheassumption
thatthelargebankhasenoughcapitaltobuythesmallbankbutthesmallbankdoes
nothaveenoughfundtoacquirethelargebank’sassets.Therefore,onlyinstateSFthe
survivinglargebankbuysthesmallbank.InstateFS,ifδ>δ∗theregulatorbailsout
thefailedlargebank,sinceliquidatingtooutsidersisamisallocation.Otherwise,where
δ≤δ∗theregulatorliquidatesanyfailedbanktooutsiders.Intheirpaper,Acharyaand
Yorulmazer(2007)showthatstateFFissimilartothesymmetriccase,unlessforlarge
δ.Forδ≤δ∗,thesmallbankisactuallyindifferentbetweenhighandlowcorrelation,as
itcannottakeovertheassetsoftheotherbank.Butthebigbankdifferentiatesitself,
asitgetsalwaysanextrabenefitinstateSFthanstateSS.Inthecontrary,forδ>δ∗,

18Thesubsidyofdifferentiating,likelysurvivingandbuyingtheotherbankinstateSF.
19Itcanbeverifiedbycomputingthetotalexpectedoutputgeneratedbybanks,netofliquidation
and/orbailoutpolicy,inasimilarapproachasabove.

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CHAPTER4.THETHEORIESOFBANKREGULATION

sincethesmallbankhasnoopportunitytoaccessthefailedlargebank’sassets,only
itsbailoutsubsidyatstateFFmatters.Thebailoutsubsidyforthelargebankdoes
notbankfails.increaseThiswhengivestheincensmalltivesbanktofailsthetosmallo,bankwhereastoitherddoeswithforthethebigsmallbank.bankifThus,thethebig
inter-bankcorrelationobtainsbymixedstrategiesandthereisnoequilibriuminpure
andstrategyGupta.(1987)Accordinglyand,theBarronTMTandFValevmostly(2000)affectonsmallUSbanks.banks’lendingEmpiricalbehwaviororksofpriorJainto
thedebtcrisisof1982-1984supporttheresults.

4.5Macro-PrudentialRegulationPolicies

Thissectionaddressesprudentialregulationpoliciesdealingwiththesystemicrisk.The
focusisonthethreesevercasesofmoralhazard:1)TMTF2)Toomuchrelatedtofail
and3)TBTF.Thefirstandthethirdcaseshavebeenbrieflymentionedintheprevious
thesesectionsissues.andthisThecasesectionoftoconcenomuctrateshonrelatedrathertoofailrefersmacro-prudentoatialhighlyapproacinhestoterconnectedwards
pbankingoliciesagainstsystemtheliabletodistributedcontagionrisk.andEffectivitdistributingyofeacthehrisk.regulatoryNextstrategysubsectionsisintroanalyzedduce
withrespecttothesourceofmoralhazard.

TMTFwithDealing4.5.1

AcharyaandYorulmazer(2008)concentratedirectlyonthetime-inconsistencyproblem
ascreateofincenTMTF.tivesTheyfortakshowingthatlessrisk.grantingWiththesurvivingsetupofbanksActoharytakaeandoverYtheorulmazerfailed(2007)banks
fornbanksinaneconomy,thissubsectionanalyzestherewardingpolicy.
Whentoomanybanksareindefault,thesurvivingbanksmaynothaveenoughliquidity
toacquirelargeamountofassetsofallthefailedones.Therefore,thepriceofassets
fallsinthemarketsuchthatoutsidersofthebankingsectorcanpurchasesomeofthe
orderfailedtobbanks’eableassets.topurcEvenhaseifallthethesurvivingfailedbanks’bankwanassets,ttotheyissuewillequitneedytotoraisecompfundensatein
theturnoutsidersreducestheastheirpriceforcompassetsetitorsofinthethesurvivingmarketbanforksbfailedecausebanks’theyhaassets.vetoThissellwillequitiny
atlowaerthediscounmarkt.Hence,et-clearingtheypricewillstillandnottheaccesshighertheenoughtotalfund.misalloThecationmorecost.failed20banks,the
where20Theδistotalthelossmisalloincationreturncostgeneratedequalsbythenumoutsiders.beroffailurestimesaconstantmisallocationcostP¯2δ,

4.5.MACRO-PRUDENTIALREGULATIONPOLICIES

83

Liquidatingtooutsidersisnotex-postoptimalinawelfareanalysisperspective.How-
ever,bailingoutfailedbanksincursafiscalcostintheworkofAcharyaandYorulmazer
(2008).Thus,theregulatorex-postoptimallybailsoutsomeofthefailedbanksaslong
asthemarginalcostofbailoutislessthanthemisallocationcost.Alternatively,sup-
posethattheregulatorprovidessufficientliquiditytosurvivingbankstobuythesame
optimalnumberoffailedbanks.Fromthepointofviewofsocialwelfare,theregulator
hastopaythesameamountofinsurancecostandthetotalmisallocationcostisnot
morethanbefore,assurvivingbanksaretheefficientusers.Therefore,theex-postsocial
welfarecostwiththealternativepolicyisasequalasthedirectbailoutpolicy.
Ex-antetheregulatorwishestoavoidtoomanyfailures.Thetimeinconsistencyproblem
arisesasshewantsex-antetoavoidherdingamongbanksbythreatofliquidatingto
outsidersbuthastoex-postbailoutthefailedbanks.Tomitigateherding,theregulator
takhazard.es21dilutionTheinsametheresultequityoffollothewsbawheniledtheoutrequbanksireddepliquidendenittyisonprothevidedseverittoyofmsurvivingoral
banks.Ytheet,theoutsidersendohawmenve,tothefpriceoutsidersandtheinfluencesnumbtheerofherdingfailedincenbankstives.theyThetogetherlessendowithwmenthet
survivingbankscanacquiredecreases.Theregulatorhastoprovideliquidityforeven
surplussmallernofumtakbereingofoverfailures.failedThisbanksincreasesraises.Inbanks’turn,incenthetivesregulatortodifferencantaktiateeaassmtheiraller
dilutiontocontrolherdingoveralargerrangeofα.
Asurvivingbankusesitsfirstperiodprofit,R−d1,topurchasefailedbanks.Whenthis
resource,availabletoeachsurvivingbankatdate1,exceedsthemaximumpriceoutsiders
wouldpayforpurchasingafailedbank,ψ,thebankcanpurchaselargeramountoffailed
banks’assets.Asurvivingbankbenefitsmorefromtheliquidityprovisionpolicy,asits
purchasesurplusoutweighsthesubsidyofbailoutpolicy.
Thisway,theregulatorencouragesbankstodifferentiatethroughratherrelaxliquida-
intionterbankstrategycorr.elationComparingbyatosmallerex-poststakeoptimalinthebailoutbailedpoutolicy,banks.shecanToimplemesummarize,ntlowtheer
alsoliquiditydominatesprovisionthenotbailoutonlypolicydiminishesfromantheex-anlikteelihoostandpdofoint.aggregatebankingcrisisbut

4.5.2TooRelatedToFailandCapitalAdequacy

Theexternalityfromonebank’sinvestmenttootherbanks’,broadensprudentialbanking
regulationstudiestowardsamultiple-bankdesign.Oneextensionapproachistotake
intoaccountbanks’correlationintheexistingregulationstrategies.Thissubsection
21Asdiscussedinsection4.2,thechoiceofwithrespectto∗andαdependsontherelationbetween
.qandδ

84

CHAPTER4.THETHEORIESOFBANKREGULATION

describestwoproposalswhichconcerncapitaladequacy.Asdiscussedintheprevious
sections,theex-postoptimalclosurepoliciessufferfromtime-inconsistencyproblem.
Thisdirectsustoex-anteoptimaldesigns.Capitalrequirementisthenthecoreofsuch
olicies.pregulatoryAcharya(2009)studytheexternalitiesofabankfailureonsurvivingbanks.Inatwo
bankssetupsimilartoAcharyaandYorulmazer(2007),ifbankBfails,afractionς<1of
itsdepositorsmigratetobankA.Sincetheoverallinvestmentintheeconomyreduces,
thereturnonthesafeassetraises.Thisincreasesthecostofdepositsonsurviving
banks,becausethereturntodepositorsequalsthereturntothesafeassetinequilibrium
(otherwise,therewouldbenoinvestmentorshort-sellonit.).
Besidethe”recessionaryspill-over”,thereisapositiveexternality.Havingmoredeposi-
tors,bankAcanexpandandacquirealsothehumancapitalofbankB.Infact,itscostof
investmentdecreasestoσpercent,andsodoestheinvestmentintheriskyasset.Thus,
thetotaleffectoftwo(negativeandpositive)externalitiesmakesthedifferencebetween
theprofitinstateSFandtheprofitinstateSS,i.e.E(πsf)−E(πss).Thisvaluewhich
determinesthebank’schoiceoninterbankcorrelation,isbydefinitiondecreasinginς
butincreasinginσ.Foranyσ,athresholdς∗(σ)canbefoundbelowwhichthetotal
externalityisnegativeandbankshavenoincentivetodifferentiate.Thissituationcan
alsoholdforsufficientlyhighinvestmentcostσ∗(ς),givenς.Weendupincollective
riskshifting,i.e.highρ,forlargeσand/orsmallς,andlowcorrelationotherwise.
Asdiscussedinsection4.2,individualbankswithlowchartervalue(wealth)takeshigher
risk.Nowthesystemicriskshiftingduetotheircorrelationisextratotheindividual
failurerisk.Bydefinition,thelossofjointfailureislargerthananindividualbankfailure.
Thisprovokestheneedforregulatoryactionsagainstbothsystemicandindividual
risk-takinginAcharya(2009).Consequently,theregulator’sclosurepolicies(including
liquidityprovision)shouldexhibitlessforbearanceinthejointfailure.
Regardingprudentialtreatmentstopenalizecollectiverisk-takingandTMTFphenom-
ena,ex-antemechanismssuchascapitalrequirementcanbeeffectivelyimproved.Since,
thecollectiverisk-shiftingisbasedonexternalities,amyopiccapitaladequacyregula-
tion,independentofρ,canatbestmitigateindividualrisk-shifting.
Acharya(2009)showsthatacapitaladequacyregulation,increasinginthecorrelation
ofbanks’portfolioandindividualportfoliorisk,moderatesbanks’systemicrisk-shifting.
ThenegativeexternalityinstateSFincentivesbankstoincreasetheprobabilityofstate
SSbytakinghighcorrelation.However,thecapitaladequacywhichdependsonthe
endogenousnegativeexternalityinducesthecostofcapitalinthatcase.Hence,thehigh
costofcapitalcounteractsthenegativeexternality.Accordingly,theproposalamends
themyopiccapitalrequirementstrategy.Itsuggeststhatbanksshouldholdmorecapital
andtakeintoaccountthegeneralriskineconomyinadditiontotheirspecificrisk.

4.5.MACRO-PRUDENTIALREGULATIONPOLICIES

85

Thenextproposalcontainsratherpracticalviewtothecapitaladequacystrategies.
Themainintuitionisagainaboutconsideringeachbank’scontributiontoasystemic
crisis.Thesystemicriskregulatorcanbecomparedtoaseniormanagerwhowantsto
preventfinancialdistressinafirm.Sheappliesriskmanagementtechnicstomeasure
eachdivision’scontributiontothetotalriskofthefirm.Theequityisassumedapublic
goodtotheentirefirm.Therefore,eachunitmustbechargedaccordingtotheequity
valueusedtosupportit.Acharyaetal.(2009)implysimilarapproachforregulating
crisisinthebankingsystem.22Assystemicriskisdefinedtooccurendogenously,each
measured.istributionconbank’sCurrentregulationpoliciesshouldbeadjustedtoconsidersystemicriskinthebanking
system.Capitaladequacyisthusasanintuitiveregulatoryinstrumentimposedto
dependoneachbank’smeasureofthesystemicriskcontribution.Forinstance,theBasel
IIcapitalrequirementmultipliedbythismeasuredsystemicfactorisanimprovement,
consistenttothediscussionabove.TheproposalisinfactanintroductiontotheBasel
IIIregulatoryaccords.However,itcanbeenforcedefficientlyundercircumstancesthat
limitthecyclicalityprobleminthesystemicriskmeasurementandtheissueoffake
erage.levindecrease

4.5.3TBTFandSystemicTaxing

Sinceabigcomplexbankcannotbeliquidated,anaturalprudentialstrategyistotaxits
activitiesthatbringnegativeexternalitieswiththeintuitiontodiscouragethebehavior
leadingtosystemicrisk.Further,theaccumulatedtaxthencouldbeusedtofundthe
lossesofthesystemiccrisis.However,fromsection4.2weknowthatfortaxingbeing
effectiveagainstrisk-takingaproperdesignisnecessary.FreixasandRochet(2010)plan
asystemictaxtodealwiththeextremeandrareeventoflargelossesinaSystemically
ImportantFinancialInstitution(SIFI).
IncaseofaSIFIfailure,apublicsupervisoryinterventionisneededsincenoprivate
insurancecancoverthelossesC,neithertheshareholderswanttorecapitalizeastheir
expectedNPVisnegative.Inamulti-periodsetup,takeintoaccountthemanager’s
moralhazarddiscussedinsection4.2.TheSIFIgeneratesafixedpositivecashflowµ
ineachperiod.Butitmayfailwithaverysmallprobabilityτwhichincreasesbydτ
becauseofthemanager’smoralhazard.Anothermainfrictionbetweenthemanager
andshareholdersisthatsheismoreimpatientashisdiscountfactorξMissmallerthan
.ξshareholders’Existenceofastrongandindependentsystemicriskauthoritywhichhasthepowerto
restructurethebankandtheownershipisnecessaryinFreixasandRochet(2010),to
(2009)22Adetaileddescribefollodiscussionwingonmethotheds:appvliedriskalue-at-riskmanagemenandexptectedmethodsishortfall,sbeyondstressthistestssurvandey.Acaggregateharyaetriskal.
risk.systemicpricingandscenarios,

86

CHAPTER4.THETHEORIESOFBANKREGULATION

establishex-anteoptimalregulation.Theauthorityhastobearacostforrestructuring
thebankΓ.Asdiscussedinsection4.2,toavoidmoralhazard,themanager’ssalary
sheshouldreceivdepesendabononushisspattheerformanceendofandeachsatisfysuccessfulincentivpeeriodcompatibilitthatyconstraint.Hence,

(1−τ)(s+ξMω)≤(1−τ−dτ)(s+ξMω)+Q.(4.35)

Still,sheshouldbeofferedaonetimepaymentGwhenshesignsthecontract.Thisfills
thegapbetweenhisreservedutilityU(histrainingcost)andtheexpectedcontinuation
payoffω,i.e.G=U−ω,whereω=(1−τ)(s+ξM).
Thesystemicriskauthorityexpropriatestheshareholdersafterthecrisis.Theregulator
sellsthebanktonewshareholders,naturallyinapriceequaltotheirexpectedbenefit.
Themanager,pricei.e.istheΠ−expGectedwherevΠalue=ofµ−theT+bank(1π−τnet)(−ofs+theΠ).one-timeTheofferregulatorytothecostnewof
expectedrestructuringcostforreducesthebregulatorecauseofinsellingcaseofthecrisis,bank.T=τHence,[C+theξ(Γ−systemic(Π−Gtax))].equaltothe
ThecleverproposalofFreixasandRochet(2010)istoconsidertheregulatoroffering
agraceperiodtothenewmanagerafterthecrisis.Itmeansthatifimmediatelyaf-
terarestructuringthebankfailsthemanagerwillnotbefiredandtheshareholders
arenotexpropriated,butthebankisbailedout.Forthemanagernottotakemoral
hazardinthisperiodtheminimumbonusoftheperiodisQ/(dτ)whichislargerthan
ωfrom(4.31).Inreturn,heronetimepaymentreducesbecauseofbiggerbonusin
thegraceperiod.Afterthegraceperiod,everythingisbacktothecontractmentioned
ofabove.restructurinNevgertheless,(immediatelyguaranteeingafteragracepreviousperiodisrestructuring)sociallyξ(bUe+neficialΓ)isiffhigherthetotalthancostthe
∗∗costminimofumloadingcontintheuationcompsalaryensationfromofthe(4.35).Inmanagerother(ξw−ξords,M)ωun,derwheresuchωiscondittheionthemanager’sone
periodgracecontractissociallymorebeneficial.
Tofindtheoptimalcontract,FreixasandRochet(2010)controlfortheoptimalprobabil-
ityofthebankbeingrestructured.Furthermore,thequestioniswhetherthemanager’s
paymentcontractisoptimalwithrespecttoherperformance.Theregulatoroptimizes
thetotalsocialsurplusofthebank.Thoughnomanagerialpaymentincaseofcrisis
minimizesthemanagerialrisk-takingincentives,acrisisimpliesrestructuringthebank
whichiscostlytothesystemicriskauthority.Thetrade-offbringsthesolutiontothe
Freixasproblem,andasRothechetsuffic(2010)ientsolverequirementhetsrecursivforehavingdynamicagracepprogrammingeriodisproblemexplainedtoabjustifyove.
wtheasnooptimalitsupyervisoryoftheofconthetractregulator,withonethegracenewperiod.shareholdersTheinwterestingouldprefuseointtoisthatcompifensatethere
thenewmanager.Therobustnessoftheresultisalsoverifiedforlargerτ.

4.6.REGULATORYPOLICIESINTHERECENTCRISIS

4.5.4Market-BasedSystemandOtherAlternatives

87

Abovementionedmethodsmaketheregulatorresponsibleformeasuringrisksandimple-
mentationofresolutionpolicies.Alternatively,aninsuranceagainstonlysystemicpart
oftheriskwouldbeamarket-basedcomplementarysystem.Theinsurermustcompute
theriskandincaseofcrisiscanpaypartoflossestothefinancialstabilityregulator,
institute.thetodirectlynotAsAcharyaetal.(2009)discuss,tohandlethecrisisamonginsurancesystem,the
theinsurerslenderonlyoflastprovideresort.covHoeragewevforer,athesmallpinsuranceercentofcompanieslosses.wTheouldregulatorinspecthasthetostillsystemicbe
riskthanofuneaderchfixedbankrecarefullygulatoryandfeesorregularlycapitalsuchrequirementhatts.banksThishavweaylessthebincenanktivweouldtogamelimit
itssystemicriskandprovidemoretransparencytodecreasetheinsurancepremia.The
insurer’sregulator.pricingNoteprothatvidesthealsoinsurancemoresysteminformationcanbeforcomthebinedpublicandandimptheosedfinancialtogetherstabilitwithy
thesystemic-risk-basedcapitaladequacyortaxingpolicies.Therefore,apublic-private
systemwouldworkmoreeffectivelybothinexaminingthesystemicriskandtheninthe
t.enevcrisisrare

4.6RegulatoryPoliciesintheRecentCrisis

Aftersurveyingtheregulatorypolicies,itistimetoinvestigatewhathavebeensofar
doneinthepastcrisis.ThissectionpresentstheUSregulatorydataonbankand
financialinstitutionsfailures.Thesamplestartsfrom1934butthemainfocusison
therecentcrisesof2007-2009anditscomparisontothepast.Thesourceofdatais
FDIC’sFailuresandAssistanceTransactionsdatabase.23Unfortunately,detaileddata
onbailoutarenotavailablebuttherearedataaboutotherresolutions.
Theresolutiontransactionsareinthreemaincategories:1)assistanceinwhichinsti-
tution’schartervaluesurvives,2)failurewithterminationofthechartervalue,and3)
payout,wheretheinsurerpaysthedepositorsdirectlyandplaceassetsintheliquidat-
ingreceivership.Assistancetransactionsincludetransactionswhereahealthyinstitu-
tionacquirestheentirebridgebank-typeentitybutcertainotherassetsweremoved
toliquidatingreceivership,oropenbankassistancetransactionsunderasystemicrisk
determination.InabridgebanktransactiontheFDICitselfactstemporarilyastheac-
quirer.Itprovidesuninterruptedservicetobankcustomerswhilehavingsufficienttime
tomarkettheinstitution.Reprivatizationasmanagementtakeoverwithorwithoutas-
sistanceattakeover,followedbyasale,isveryrareinthedata.Thesecondcategory
containsalltypesof”PurchaseandAssumption”(P&A)agreements.Intheseresolution
23Thedataforyear2010isuptoAugust,20.

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CHAPTER4.THETHEORIESOFBANKREGULATION

transactionsthehealthyinstitutionpurchasessomeorallofassetsofafailedinstitution
andassumessomeoralloftheliabilities,includingallinsureddeposits.24
Figures4.1to4.4shownumberofalltransactionsofthethreecategoriesinfourtime
intervals.Figure4.5putsthemalltogetherinordertomakecomparisonpossible.The
secondcategorytransactionsareknownas”failure,merger”inthefigures.Thetrend
ofaveragetotaldepositsinfailure(categorytwoandthree)andassistancetransactions
provideinformationonvolumes.Inallyearsfrom1934to1979,thetotaldeposits
underassistancesumsuptoabout6$billion.Compareitwithyearsafter.Inthe
80stheaveragetotaldepositunderassistanceismuchhigherthanunderfailures.Itis
increasingandthepeakis1.5$billionin1989.Thisissowhilethenumberofassistance
isalwaysverysmall.Itmeansthatmostlylargebankshavebeenunderassistance.Huge
numberoffailuresisseeninthe80sthatisreversedinthe90s.Thetrendofsystemic
failureisdecreasinginthe90sandsodothetrendofaveragetotaldeposits.However,a
relativelylargervolumeofdepositswereunderfailuretransactionthanassistance.Since
2000therewasnotmuchproblemsinthebankingsystemuntil2008and2009.Though
thenumberoffailuresandassistanceisnotaslargeasthe80s,theaveragetotaldeposits
isenormous.Withlownumberofassistancetransactions,uptoabout6$and14$billion
arespenttoassisttotaldepositsperbankin2008and2009,respectively.Notethat
the7$billionbailouttothefinancialsystemoftheUSisextratothesetransactions.
Theimportantroleofbanksaleisobservable.However,thesystemicshocksweresuch
extremethattheyaremostlycoveredbyhugecostforthegovernment,i.e.theregulatory
.yauthorit

Conclusion4.7

Thispapersurveysthedevelopmentofbankingregulationtowardssystemicriskregu-
lationintherecentyears.Regulatingasinglebankinnormaltimeshavebeenwidely
studied.Regulationstrategiesagainstabank’srisk-takingandresolutionpoliciesincase
ofafailurearewelloptimized.However,theyhavebeenlimitedtoindividualbanks’
problems.Preventingorresolvingasystemiccrisisrequiresdifferentpolicies.Ex-antepoliciessuch
ascapitaladequacy,taxingand/ordepositinsuranceshouldadjustforthispurpose.De-
pendenceoftheadequatecapitalrationotonlyoneachbank’sriskbutalsoonbanks’
correlationwoulddecreasebanks’herdinginrisk-taking.Computingeachbank’scontri-
butiontothesystemicriskinaproperrisk-managementmethod,thecapitaladequacy
orinsurancepremiumshoulddependonthismeasurementtoo.Systemictaxingfora
substantiallyimportantinstituteinaneconomywoulddiminishtheriskiftheregulator
isstrongenoughtoexpropriatetheownership.Taxingandpartialinsurancecanalso
24AbridgebanktransactionisalsoatypeofP&A.

4.7.ONCLUSIONC89

providefundingforthelosses.Privateinsurancecompanieswouldalsomonitorbanks’
fromactivitiestramnsmittedorecloselyloss,anthedpricepartialpremiainsurancemoregivcarefentoully.banksToprotectshouldtheonlycovinsuranceerafrasystemction
risk.systemictheirof

Ex-postcrisisresolutionsshouldalsobeex-anteoptimal.Sinceatacrisis,assetliquida-
tionisnotex-postoptimalinmajorityoffailures,forbearancepoliciesshouldencourage
risk-reduction.Inotherwords,directbailoutwouldhighlyincreasemoralhazardand
mustbeprohibited.Researchesproposetakeoverofafailedbankbyahealthyinsti-
tuteshouldbeallowedandalsogranted.Itmeansthat,theregulatorshouldprovide
powliquiditersytohealthaysurvivbanksalinofvolthevescrisistheforsamepurcsocialhasingcostasfailedadirectinstitutes.bailout,Thebutpolicyhasthethatgreatem-
advantagethatreducescollectiverisk-takingamongbanks.

Still,thereismuchspaceforfurtherdevelopmentofmacro-prudentialregulation.Fur-
therresearchcouldforinstanceconsidertheinterbankrelation.Atfailureofsome
avbanks,oidhowtransmission?couldtheirHowconnectionshouldthistoinotherterconnectionpartsofbefinancialex-antesystemoptbimallyecontrolledregulated?to
Besideopenquestionsregardinginterbankrelations,implementationofexistingpropos-
palsolicyisorequallycombiimpnationortant.ofpTheoliciesdepracticalpendswayonthethesupeconomervisoryyandauthoritalsoylegalshouldsystemimsp.oseThisa
providesbroadareaofresearchinbothappliedandtheoreticaltopics.

90

Figures

CHAPTER4.THETHEORIESFOBANKFigure4.1.USBankResolutions1934-1979.

Figure4.2.USBankResolutionsinthe80s.

TIONREGULAAssistanceTransactionsinclude:A/Atransactionswhereassistancewasprovidedtothe
acquirerwhopurchasedtheentireinstitution,orwhereassistancewasprovidedundera
systemicriskdetermination;andtheinstitution’schartersurvived.

4.7.FIGURESFigure

4.3.

4.4.Figure

US

Bank

Resolutions

in

the

90s.

2000.sinceResolutionsBankUS

91

AssistanceTransactionsinclude:A/Atransactionswhereassistancewasprovidedtothe
acquirerwhopurchasedtheentireinstitution,orwhereassistancewasprovidedundera
systemicriskdetermination;andtheinstitution’schartersurvived.

92

Figure

CHAPTER4.5.

US

4.Bank

THETHEORIESResolutions

FOBANK1980-August

0201

TIONREGULA

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enslaufLeb

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