An-analytic-derivation-of-the-cost-of-deposit-insu

JournalofBankingandFinance1(1977)3-11.©North-HollandPublishingCompanyANANALYTICDERIVATIONOFTHECOSTOFDEPOSITINSURANCEANDLOANGUARANTEESAnapplicationofmodernoptionpricingtheoryRobertC.MERTON*ProJessorofFinance,SloanSchoolofManagement,MIT,Cambridge,MA02139,U.S.A.Itisnotuncommoninthearrangementofaloantoincludeaspartofthefinancialpackageaguaranteeoftheloanbyathirdparty.Examplesareguaranteesbyaparentcompanyofloansmadetoitssubsidiariesorgovernmentguaranteesofloansmadetoprivatecorporations.AlsoincludedwouldbeguaranteesofbankdepositsbytheFederalDepositInsuranceCorporation.Aswithotherformsofinsurance,theissuingofaguaranteeimposesaliabilityorcostontheguarantor.Inthispaper,aformulaisderivedtoevaluatethiscost.Themethodusedistodemonstrateanisomorphiccorrespondencebetweenloanguaranteesandcommonstockputoptions,andthentousethewelldevelopedtheoryofoptionpricingtoderivetheformula.1.IntroductionTheessentialfunctionsofabankaretolendmoneytofirmsandindividualsandtoserveasarisklessrepositoryfortheshort-termfundsoffirmsandindividuals.Thebankchargesinterestontheloansandpaysinterestorprovidesnoncashservicestodepositorsfortheuseoftheirfunds.Thetraditionaladvantagestodepositorsofusingabankratherthanmakingdirectmarketpurchasesoffixed-incomesecuritiesareeconomiesofscale,smallertransactionscosts,liquidity,andconvenience.However,theseareimportantadvantagesonlyifdepositscanbetreatedasriskless.Otherwise,todeterminewhichbanktouse,thedepositormustassumetheroleofasecurityanalystandanalyzethebalancesheetsofthebank,itsmanagement,andoverallmarketconditionstodeterminetherisks.Evenifsuchanalysesareperformed,itwouldbeprudentforthedepositortodiversifyhisholdingsacrossmanybanks.Moreover,theseanalysesandholdingswouldhavetoberevisedasconditionschanged.Hence,forthesmalldepositorparticularly,therearelargeinformationandsurveillancecoststobesavediftheinstitutionalstructureofthebankweresuchthatthesafetyofthedepositswasassuredwithoutrequiringtheseanalyses.Whilethiscouldbeaccomplishedbyrequiringthatbanksinvestonlyinshort-term*AidfromtheNationalScienceFoundationisgratefullyacknowledged.4R.C.Merton,Thecostofdepositinsuranceandloanguaranteesgovernmentsecurities,sucharequirementwouldnotallowbankstoperformtheirotlaerfunction,whichistolendmoneytofirmsandindividuals.Moreover,sucharestrictionwouldrequiresurveillancetoensurecompliance.Whilesuchsurveillancecouldbecarriedoutinacentralizedfashion(e.g.byagovernmentregulatoryagency),thefinancialburdenfromnoncompliancewouldstillbebornebythedepositor.Asensiblealternativechoicewouldbetohavethird-partyguaranteeswherethecapabilityandwillingnessofthatpartytomeetitsobligationsarebeyondquestion.Forthescaleofthebankingsystem,thisalmostcertainlymeansthatthethird-partywouldbethegovernmentoroneofitsagencies.Whiletheseguaranteescouldtaketheformofguaranteesontheloansmadebybanks,alessexpensiveandmoreefficientalternativeformistoguaranteethedeposits.Indeed,oneobserveswidespreaduseofsuchdepositguaranteesalthoughtheirinstitutionalformsdiffer.IntheUnitedStates,thereistheseparately-fundedFederalDepositInsuranceCorporation(FDIC)forcommercialbanksandtheFederalSavingsandLoanInsuranceCorporationforsavingsandloanassociations.Inothercountriesthegovernmentmayownthebanksortheremaysimplybeawidespreadbeliefthatthegovernmentwillactto,ineffect,guaranteedeposits.Indeed,itisprobablyfairtosaythatalthoughtheFDICisseparatelyfunded,thereisafurtherbeliefthattheU.S.governmentwouldtakethenecessaryactionstoprotectdepositorsintheeventofamajordefaultbybanksthatbankruptedtheFDIC.Totheextentsuchimplicitguaranteesarepoliticallybinding,theyimposeacostontheguarantorwhichisessentiallythesameasforexplicitguarantees.Inthispaper,asystematictheoryfordeterminingthesecostsispresented.Aswillbeshown,techniquesforsolvingthisproblemcanbefoundintheseeminglyunrelatedareaofFinancecalledOptionPricingtheory.Indeed,thepropertiesofdepositinsuranceviewedasasecurityareisomorphictothoseofaputoption.1Thisisfortunatebecauserecentresearchhasledtoamajor'breakthrough'inthedevelopmentofanoptionpricingtheory.Inaseminalpaper,BlackandScholes(1973)developedanexplicitformulaforpricingcalloptions2oncommonstocks.Whiletherehadbeenanumberofearliereffortsdatingbackatleastto1900,theBlack-Scholesanalysisisamajoradvancementfortworeasons.First,theimportantassumptionsrequiredtoderivetheformulaaresubstantiallyweakerthanintheearlierefforts.Second,theinputsrequiredbytheformulaareeitherdirectlyobservablevariablesorones1Theessentialtermsofthecontractaredefinedlaterinthepaper.Forfurtherdiscussionofputoptions,seeBlackandScholes(1973),Merton(1973a,b),BrennanandSchwatrz(1977),andParkinson(1977).2Acalloptiongivesitsownertherighttobuyaspecifiednumberofsharesofagivenstockataspecifiedpricepershareonorbeforeaspecifieddate.ForanalternativederivationoftheBlack-Scholesmodel,seeMerton(1973a).Smith(1976)providesanexcellentsurveyarticleoftheresearchonoptionpricingtheory.R.C.Merton,Thecostofdepositinsuranceandloanguarantees5thatcanbereadilyestimated.Hence,theformulacanbeempiricallytested,andifprovedsatisfactory,itcanthenbeusedasapractic