an introduction to credit risk modeling phần 9 pdf

... agingcreditportfolios.TheirtoolsarebasedonamodicationofMer-tonsassetvaluemodel,seeChapter3,andincludeatoolforestimatingdefaultprobabilities(CreditMonitorTM)frommarketinformationandatoolformanagingcreditportfolios(PortfolioManagerTM).ThersttoolsmainoutputistheExpectedDefaultFrequencyTM(EDF),whichcannowadaysalsobeobtainedonlinebymeansofanewlydevelopedweb-basedKMV-toolcalledCreditEdgeTM.ThemainoutputofthePortfolioManagerTMisthelossdistributionofacreditportfolio.Ofcourse,bothproductshavemanymoreinterestingfeatures,andtousitseemsthatmostlargebanksandinsuranceuseatleastoneofthemajorKMVproducts.AreferencetothebasicsoftheKMV-ModelisthesurveypaperbyCrosbie[19].CreditMetricsTMisatrademarkoftheRiskMetricsTMGroup,acom-panywhichisaspin-ooftheformerJPMorganbank,whichnowbelongstotheChaseGroup.ThemainproductarisingfromtheCreditMetricsTMframeworkisatoolcalledCreditManagerTM,whichin-corporatesasimilarfunctionalityasKMVsPortfolioManagerTM.Itiscertainlytruethatthetechnicaldocumentation[54]ofCreditMetricsTMwaskindofapioneeringworkandhasinuencedmanybank-internaldevelopmentsofcreditriskmodels.Thegreatsuccessofthemodelun-derlyingCreditMetricsTMisinpartduetothephilosophyofitsauthorsGupton,Finger,andBhatiatomakecreditriskmethodologyavailabletoabroadaudienceinafullytransparentmanner.Bothcompaniescontinuetocontributetothemarketofcreditriskmodelsandtools.Forexample,theRiskMetricsTMGrouprecentlyde-velopedatoolforthevaluationofCollateralizedDebtObligations,andKMVrecentlyintroducedanewreleaseoftheirPortfolioManagerTMPM2.0,herebypresentingsomesignicantchangesandimprovements.Returningtothesubjectofthissection,wenowdiscussthefac-tormodelsusedinKMVsPortfolioManagerTMandCreditMetricsTMCreditManagerTM.Bothmodelsincorporatetheideathateveryrmadmitsaprocessofassetvalues,suchthatdefaultorsurvivalofthermdependsonthestateoftheassetvaluesatacertainplanninghorizon.Iftheprocesshasfallenbelowacertaincriticalthreshold,calledthedefaultpointoftherminKMVterminology,thenthecompanyhasdefaulted.Iftheassetvalueprocessisabovethecriticalthreshold,thermsurvives.AssetvaluemodelshavetheirrootsinMertonsseminalpaper[86]andwillbeexplainedindetailinChapter3andalsotosomeextentinSection2.4.1. 20 03 ... agingcreditportfolios.TheirtoolsarebasedonamodicationofMer-tonsassetvaluemodel,seeChapter3,andincludeatoolforestimatingdefaultprobabilities(CreditMonitorTM)frommarketinformationandatoolformanagingcreditportfolios(PortfolioManagerTM).ThersttoolsmainoutputistheExpectedDefaultFrequencyTM(EDF),whichcannowadaysalsobeobtainedonlinebymeansofanewlydevelopedweb-basedKMV-toolcalledCreditEdgeTM.ThemainoutputofthePortfolioManagerTMisthelossdistributionofacreditportfolio.Ofcourse,bothproductshavemanymoreinterestingfeatures,andtousitseemsthatmostlargebanksandinsuranceuseatleastoneofthemajorKMVproducts.AreferencetothebasicsoftheKMV-ModelisthesurveypaperbyCrosbie[19].CreditMetricsTMisatrademarkoftheRiskMetricsTMGroup,acom-panywhichisaspin-ooftheformerJPMorganbank,whichnowbelongstotheChaseGroup.ThemainproductarisingfromtheCreditMetricsTMframeworkisatoolcalledCreditManagerTM,whichin-corporatesasimilarfunctionalityasKMVsPortfolioManagerTM.Itiscertainlytruethatthetechnicaldocumentation[54]ofCreditMetricsTMwaskindofapioneeringworkandhasinuencedmanybank-internaldevelopmentsofcreditriskmodels.Thegreatsuccessofthemodelun-derlyingCreditMetricsTMisinpartduetothephilosophyofitsauthorsGupton,Finger,andBhatiatomakecreditriskmethodologyavailabletoabroadaudienceinafullytransparentmanner.Bothcompaniescontinuetocontributetothemarketofcreditriskmodelsandtools.Forexample,theRiskMetricsTMGrouprecentlyde-velopedatoolforthevaluationofCollateralizedDebtObligations,andKMVrecentlyintroducedanewreleaseoftheirPortfolioManagerTMPM2.0,herebypresentingsomesignicantchangesandimprovements.Returningtothesubjectofthissection,wenowdiscussthefac-tormodelsusedinKMVsPortfolioManagerTMandCreditMetricsTMCreditManagerTM.Bothmodelsincorporatetheideathateveryrmadmitsaprocessofassetvalues,suchthatdefaultorsurvivalofthermdependsonthestateoftheassetvaluesatacertainplanninghorizon.Iftheprocesshasfallenbelowacertaincriticalthreshold,calledthedefaultpointoftherminKMVterminology,thenthecompanyhasdefaulted.Iftheassetvalueprocessisabovethecriticalthreshold,thermsurvives.AssetvaluemodelshavetheirrootsinMertonsseminalpaper[86]andwillbeexplainedindetailinChapter3andalsotosomeextentinSection2.4.1. 20 03 ... Inpractice,analyticalapproximationtechniquescanbeappliedquitesuccessfullytoso-calledhomogeneousportfolios.Theseareportfolioswherealltransactionsintheportfoliohavecomparableriskcharacter-istics,forexample,noexposureconcentrations,defaultprobabilitiesinabandwithmoderatebandwidth,onlyafew(better:onesingle!)industriesandcountries,andsoon.Therearemanyportfoliossatisfy-ingsuchconstraints.Forexample,manyretailbankingportfoliosandalsomanyportfoliosofsmallerbankscanbeevaluatedbyanalyticalapproximationswithsufficientprecision.Incontrast,afullMonteCarlosimulationofalargeportfoliocanlastseveralhours,dependingonthenumberofcounterpartiesandthenumberofscenariosnecessarytoobtainsufficientlyrichtailstatisticsforthechosenlevelofconfidence.ThemainadvantageofaMonteCarlosimulationisthatitaccuratelycapturesthecorrelationsinherentintheportfolioinsteadofrelyingonawholebunchofassumptions.Moreover,aMonteCarlosimulationtakesintoaccountallthedifferentriskcharacteristicsoftheloansintheportfolio.ThereforeitisclearthatMonteCarlosimulationisthe“state-of-the-art”increditriskmodeling,andwheneveraportfoliocon-tainsquitedifferenttransactionsfromthecreditriskpointofview,oneshouldnottrusttoomuchintheresultsofananalyticalapproximation.1 .2. 3ModelingCorrelationsbyMeansofFactorModelsFactormodelsareawellestablishedtechniquefrommultivariatestatistics,appliedincreditriskmodels,foridentifyingunderlyingdriversofcorrelateddefaultsandforreducingthecomputationaleffortregard-ingthecalculationofcorrelatedlosses.Westartbydiscussingthebasicmeaningofafactor.AssumewehavetwofirmsAandBwhicharepositivelycorrelated.Forexample,letAbeDaimlerChryslerandBstandforBMW.Then,itisquitenaturaltoexplainthepositivecorrelationbetweenAandBbythecorrelationofAandBwithanunderlyingfactor;seeFig-ure1.5.Inourexamplewecouldthinkoftheautomotiveindustryas...

... (α,β)∈{(2,1/2),(5,1/5)}.Insteadofincorporatingafactormodel(aswehaveseenitinthecaseofCreditMetricsTMandKMV’sPortfolioManagerTMinSection1.2 .3) ,CreditRisk+implementsaso-calledsectormodel.However,somehowonecanthinkofasectorasa“factor-inducing”entity,or–astheCreditRisk+TechnicalDocument[18]saysit–everysectorcouldbethoughtofasgeneratedbyasingleunderlyingfactor.Inthisway,sec-tors ... Scale Risk ScaleDefault Risk onlyDefault Risk onlyMark -to- Modelof Loan ValueMark -to- Modelof Loan ValueDistance to Default (DtD)Definitionof Risk IntensityProcessMacro-economicFactorsAsset ... whereasinthePoissoncaseweobtainCorr[Li,Lj]=Cov[Λi,Λj]V[Λi]+E[Λi]V[Λj]+E[Λj].(2.24)LookingonlyatthedrivingrandomvariablesPi,PjrespectivelyΛi,Λj,weseethatinthedenominatorsof(2. 23) and(2.24)wecompareV[Pi]+E[Pi(1−Pi)]=V[Pi]+E[Pi]−E[P2i](2.25)withV[Λi]+E[Λi].Now,analogoustothedeterministiccase(2.12),wecan–evenintherandomcase–expectPiandΛitobeofthesameorderofmagnitude.Tokeepthingssimple,letusforamomentassumethatPiandΛihavethesamefirstandsecondmoments.InthiscaseEquation(2.25)combinedwith(2. 23) and(2.24)showsthattheBernoullimodelalwaysinducesahigherdefaultcorrelationthanthePoissonmodel.Buthigherdefaultcorrelationsresultinfattertailsofthecorrespondinglossdistributions.InotherwordsonecouldsaythatgivenequalfirstandsecondmomentsofPiandΛi,theexpectationsofLiandLiwillmatch,butthevarianceofLiwillalwaysexceedthevarianceofLi,therebyinducinglowerdefaultcorrelations.SothereisasystematicdifferencebetweentheBernoulliandPoissonmixturemodels.IngeneralonecanexpectthatforagivenportfoliotheBernoullimodelyieldsalossdistributionwithafattertailthanacomparably(e.g.,byafirstandsecondmomentmatching)calibratedPoissonmodel.ThisdifferenceisalsoreflectedbytheindustrymodelsfromCreditMetricsTM/KMVCorporation(PortfolioManager)andCreditSuisseFinancialProducts(CreditRisk+).InSection2.5.3wecome back to this issue.2.4 An Overview of Today’s Industry...

... thePoissonmixturemodel,herebyconfirmingourtheoreticalresultsfromSection2.3.AmoredetailedcomparisonoftheKMV-ModelandCreditRisk+canbefoundin[12].2.6LossDistributionsbyMeansofCopulaFunctionsCopulafunctionshavebeenusedasastatisticaltoolforconstruct-ingmultivariatedistributionslongbeforetheywerere-discoveredasavaluabletechniqueinriskmanagement.Currently,theliteratureontheapplicationofcopulastocreditriskisgrowingeverymonth,sothattrackingeverysinglepaperonthisissuestartsbeingdifficultifnotimpossible.Asmallandbynomeansexhaustiveselectionofpa-persprovidingthereaderwithagoodintroductionaswellaswithavaluablesourceofideashowtoapplythecopulaconcepttostandardproblemsincreditriskisLi[78,79],FreyandMcNeil [45 ],Frey,McNeil,andNyfeler [47 ],FreesandValdez [44 ],andWang[125].However,thebasicideaofcopulasissosimplethatitcanbeeasilyintroduced:2.6.1DefinitionAcopula(function)isamultivariatedistribution(function)suchthatitsmarginaldistributionsarestandarduniform.AcommonnotationforcopulaswewilladoptisC(u1, ... 1.5%0.89 64% 2.0988%Mean 5%20%Mean5% 20%0.5% 0 .49 59% 0 .49 92% 0.5% 0 .49 90% 0 .49 73%0.8% 0.8006% 0.8009% 0.8% 0.7999% 0.8051%1.5% 1.5030% 1 .49 70% 1.5%1.5023% 1.5003%Quantile 5%20%Quantile5% ... 0.5% 0.5002% 0 .49 83%0.8% 0.8000% 0.8000% 0.8% 0.8028% 0.8037%1.5% 1.5000% 1.5000% 1.5%1.50 34% 1 .49 44% Quantile 5%20%Quantile5% 20%0.5% 1. 747 0% 4. 3017% 0.5% 1.7605% 4. 3060%0.8% 2.6323%...

... not. Therefore, the credit risk of the loan is neutralized andcompletely hedged. In other words, buying the put transforms therisky corporate loan3into a riskless bullet loan with face value ... Chapter3AssetValueModelsTheassetvaluemodel(AVM)isanimportantcontributiontomodernfinance.IntheliteratureonecanfindatremendousamountofbooksandpaperstreatingtheclassicalAVMoroneofitsvariousmodifica-tions.See,e.g.,Crouhy,Galai,andMark[21](Chapter9),SobehartandKeenan[1 15] ,andBohn[13],justtomentionaverysmallselectionofespeciallynicelywrittencontributions.AsalreadydiscussedinSection1.2.3andalsoinChapter2,twoofthemostwidelyusedcreditriskmodelsarebasedontheAVM,namelytheKMV-ModelandCreditMetricsTM.TherootsoftheAVMaretheseminalpapersbyMerton[86]andBlackandScholes[10],wherethecontingentclaimsapproachtoriskydebt ... e laborated.3.1 Introduction and a Small Guide to the LiteratureTheAVMinitsoriginalformgoesbacktoMerton[86]andBlackandScholes[10].Theirapproachisbasedonoptionpricingtheory,andwewill frequently...

... insurancepointofview,expectedshortfallisaveryreasonablemeasure:Definingbyc=VaRα(X)acriticallossthresholdcorrespondingtosomeconfidencelevelα,expectedshortfallcapitalprovidesacushionagainstthemeanvalueoflossesexceedingthecriticalthresholdc.Inotherwords,TCEfocussesontheexpectedlossinthetail,startingatc,oftheportfolio’slossdistribution.Thecriticalthresholdc,drivenbytheconfidencelevelα,hastobefixedbytheseniormanagementofthebankandispartofthebank’screditmanagementpolicy.Thefollowingpropositionprovidesanotherinterpretationofcoherencyandcanbefoundin[5].5.1.2 Proposition A risk ... Chapter4TheCreditRisk+ModelInSection2.4.2wealreadydescribedtheCreditRisk+modelasaPois-sonianmixturewithgamma-distributedrandomintensitiesforeachsector.InthissectionwewillexplainCreditRisk+insomegreaterdetail.ThejusticationforanotherandmoreexhaustivechapteronCreditRisk+isitsbroadacceptancebymanycreditriskmanaginginsti-tutes.EveninthenewCapitalAccord(somereferencesregardingtheBaselIIapproachareGordy[52],Wilde[1 26] ,andtheIRBconsultativedocument[103]),CreditRisk+wasoriginallyappliedforthecalibrationoftheso-calledgranularityadjustmentinthecontextoftheInternalRatings-basedApproach(IRB)ofregulatorycapitalriskweights.ThepopularityofCreditRisk+hastwomajorreasons:ãItseemseasiertocalibratedatatothemodelthanisthecaseformulti-factorassetvaluemodels.Hereweintentionallysaiditseemsbecausefromourpointofviewthecalibrationofbank-internalcreditdatatoamulti-sectormodelisingeneralneithereasiernormoredicultthanthecalibrationofamulti-factormodelonwhichanassetvaluemodelcanbebased.ãThesecondandmaybemostimportantreasonforthepopularityofCreditRisk+isitsclosed-formlossdistribution.Usingprob-abilitygeneratingfunctions,theCreditRisk+modeloers(evenincaseofmorethanonesector)afullanalyticdescriptionoftheportfoliolossofanygivencreditportfolio.ThisenablesusersofCreditRisk+tocomputelossdistributionsinaquickandstillexactmanner.Formanyapplicationsofcreditriskmodels,thisisanice -to- havefeature,e.g.,inpricingorABSstructuring.BeforegoingintothedetailsoftheCreditRisk+model,weliketopresentaquotationfromtheCreditRisk+TechnicalDocument[18]onpage ... Chapter4TheCreditRisk+ModelInSection2.4.2wealreadydescribedtheCreditRisk+modelasaPois-sonianmixturewithgamma-distributedrandomintensitiesforeachsector.InthissectionwewillexplainCreditRisk+insomegreaterdetail.ThejusticationforanotherandmoreexhaustivechapteronCreditRisk+isitsbroadacceptancebymanycreditriskmanaginginsti-tutes.EveninthenewCapitalAccord(somereferencesregardingtheBaselIIapproachareGordy[52],Wilde[1 26] ,andtheIRBconsultativedocument[103]),CreditRisk+wasoriginallyappliedforthecalibrationoftheso-calledgranularityadjustmentinthecontextoftheInternalRatings-basedApproach(IRB)ofregulatorycapitalriskweights.ThepopularityofCreditRisk+hastwomajorreasons:ãItseemseasiertocalibratedatatothemodelthanisthecaseformulti-factorassetvaluemodels.Hereweintentionallysaiditseemsbecausefromourpointofviewthecalibrationofbank-internalcreditdatatoamulti-sectormodelisingeneralneithereasiernormoredicultthanthecalibrationofamulti-factormodelonwhichanassetvaluemodelcanbebased.ãThesecondandmaybemostimportantreasonforthepopularityofCreditRisk+isitsclosed-formlossdistribution.Usingprob-abilitygeneratingfunctions,theCreditRisk+modeloers(evenincaseofmorethanonesector)afullanalyticdescriptionoftheportfoliolossofanygivencreditportfolio.ThisenablesusersofCreditRisk+tocomputelossdistributionsinaquickandstillexactmanner.Formanyapplicationsofcreditriskmodels,thisisanice -to- havefeature,e.g.,inpricingorABSstructuring.BeforegoingintothedetailsoftheCreditRisk+model,weliketopresentaquotationfromtheCreditRisk+TechnicalDocument[18]onpage 8. There we nd thatCreditRisk+focuses on modeling and managing credit...

... 5.2.2CapitalAllocationw.r.t.Value-at -Risk CalculatingriskcontributionsassociatedwiththeVaRriskmeasureisanaturalbutdifficultattempt,sinceingeneralthequantilefunc-tionwillnotbedifferentiablewithrespecttotheassetweights.Undercertaincontinuityassumptionsonthejointdensityfunctionoftheran-domvariablesXi,differentiationofVaRα(X),whereX=iwiXi,isguaranteed.Onehas(see[122])∂VaRα∂wi(X)=E[Xi|X=VaRα(X)].(5.1)Unfortunately,thedistributionoftheportfoliolossL=wiˆLi,asspecifiedatthebeginningofthischapter,ispurelydiscontinuous.ThereforethederivativesofVaRαintheabovesensewilleithernotexistorvanishtozero.Inthiscasewecouldstilldefineriskcontribu-tionsviatheright-hand-sideofEquation(5.1)bywritingγi=E[ˆLi|L=VaRα(L)]−E[ˆLi].(5.2)Foraclearerunderstanding,notethat∂E[L]∂wi=E[ˆLi]andmi=1wiγi=ECVaRα.Additionallyobserve,thatforalargeportfolioandonanappropriatescale,thedistributionofLwillappeartobe“closetocontinuous”.Unfortunately,eveninsuch“approximatelygood”cases,thelossdis-tributionoftenisnotgiveninananalyticalforminordertoallowfordifferentiations.RemarkFortheCreditRisk+model,ananalyticalformofthelossdistributioncanbefound;seeSection2.4.2andChapter4foradis-cussionofCreditRisk+.Tasche[121]showedthatintheCreditRisk+framework ... to a rating class, i.e., firms with defaultrates less than or equal to 0.002% are mapped to AAA, 0.002% to 0.04% corresponds to AA, etc. The historical frequencies of changesfrom one range to ... thatisbasedonextremelosssituationstoasingletransaction,sincetheriskinasingletransactionmightbedrivenbyshort-termvolatilityandnotbythelong-termviewofextremerisks.Thesecondreasonismoredrivenbythecomputationalfeasibilityofexpectedshortfall.Inthe“binaryworld”ofdefaultsimulations,toomanysimulationsarenecessaryinordertoobtainapositivecontributionconditionalonextremedefaulteventsforallcounterparties.Thebasicresultofthesimulationstudyisthatanalyticcontributionsproduceasteepergradientbetweenriskyandlessriskyloansthantailriskcontributions.Inparticular,loanswithahighdefaultprobabil-itybutmoderateexposureconcentrationrequiremorecapitalintheanalyticcontributionmethod,whereasloanswithhighconcentrationrequirerelativelymorecapitalintheshortfallcontributionmethod.TransactionViewThefirstsimulationstudyisbasedonacreditportfolioconsideredindetailin[105].Theparticularportfolioconsistsof40counterparties.Ascapitaldefinition,the99%quantileofthelossdistributionisused.WithintheMonte-Carlosimulationitisstraightforwardtoevaluateriskcontributionsbasedonexpectedshortfall.Theresultingriskcontribu-tionsanditscomparisontotheanalyticallycalculatedriskcontribu-tionsbasedonthevolatilitydecompositionareshowsinFigure5.2.In...

... pletemarkets,butitisnotclearwhethertheseconditionsholdforthecreditmarketornot.Ifacrediteventisbasedonafreelyobservablepropertyofmarketprices,suchascreditspreads,thenwebelievethatconventionalderivativepricingmethodologymaybeapplicable.Creditderivativesarebilateralfinancialcontractsthatisolatespecificaspectsofcreditriskfromanunderlyinginstrumentandtransferthatriskbetweentwocounterparties.Byallowingcreditrisktobefreelytraded,riskmanagementbecomesfarmoreflexible.Therearelotsofdifferenttypesofcreditderivatives,butweshallonlytreatthemostcommonlyusedones.Theycouldbeclassifiedintotwomaincategoriesaccordingtovaluation,namelythereplicationproducts,andthedefaultproducts.Theformerarepricedoffthecapacitytoreplicatethetrans-actioninthemoneymarket,suchascreditspreadoptions.Thelatterarepricedasafunctionoftheexposureunderlyingthesecurity,thede-faultprobabilityofthereferenceasset,andtheexpectedrecoveryrate,suchascreditdefaultswaps.Anotherclassificationcouldbealongtheirperformanceasprotection-likeproducts,suchascreditdefaultoptionsandexchange-likeproducts,suchastotalreturnswaps.Inthenextsectionswedescribethemostcommonlyusedcreditderivativesandillustratesimpleexamples.Foramoreelaborateintroductiontothedifferenttypesofcreditderivativesandtheiruseforriskmanagementsee[ 68, 107];fordocumentationandguidelineswereferto[61].7.1 ... pletemarkets,butitisnotclearwhethertheseconditionsholdforthecreditmarketornot.Ifacrediteventisbasedonafreelyobservablepropertyofmarketprices,suchascreditspreads,thenwebelievethatconventionalderivativepricingmethodologymaybeapplicable.Creditderivativesarebilateralfinancialcontractsthatisolatespecificaspectsofcreditriskfromanunderlyinginstrumentandtransferthatriskbetweentwocounterparties.Byallowingcreditrisktobefreelytraded,riskmanagementbecomesfarmoreflexible.Therearelotsofdifferenttypesofcreditderivatives,butweshallonlytreatthemostcommonlyusedones.Theycouldbeclassifiedintotwomaincategoriesaccordingtovaluation,namelythereplicationproducts,andthedefaultproducts.Theformerarepricedoffthecapacitytoreplicatethetrans-actioninthemoneymarket,suchascreditspreadoptions.Thelatterarepricedasafunctionoftheexposureunderlyingthesecurity,thede-faultprobabilityofthereferenceasset,andtheexpectedrecoveryrate,suchascreditdefaultswaps.Anotherclassificationcouldbealongtheirperformanceasprotection-likeproducts,suchascreditdefaultoptionsandexchange-likeproducts,suchastotalreturnswaps.Inthenextsectionswedescribethemostcommonlyusedcreditderivativesandillustratesimpleexamples.Foramoreelaborateintroductiontothedifferenttypesofcreditderivativesandtheiruseforriskmanagementsee[ 68, 107];fordocumentationandguidelineswereferto[61].7.1 ... Chapter7CreditDerivativesCreditderivativesareinstrumentsthathelpbanks,nancialinstitu-tions,anddebtsecurityinvestorstomanagetheircredit-sensitivein-vestments.Creditderivativesinsureandprotectagainstadversemove-mentsinthecreditqualityofthecounterpartyorborrower.Forex-ample,ifaborrowerdefaults,theinvestorwillsuerlossesontheinvestment,butthelossescanbeosetbygainsfromthecreditderiva-tivetransaction.Onemightaskwhybothbanksandinvestorsdonotutilizethewell-establishedinsurancemarketfortheirprotection.Themajorreasonsarethatcreditderivativesoerlowertransactioncost,quickerpayment,andmoreliquidity.Creditdefaultswaps,forinstance,oftenpayoutverysoonaftertheeventofdefault1;incon-trast,insurancestakemuchlongertopayout,andthevalueoftheprotectionboughtmaybehardtodetermine.Finally,aswithmost-nancialderivativesinitiallyinventedforhedging,creditderivativescannowbetradedspeculatively.Likeotherover-the-counterderivativese-curities,creditderivativesareprivatelynegotiatednancialcontracts.Thesecontractsexposetheusertooperational,counterparty,liquidity,andlegalrisk.Fromtheviewpointofquantitativemodelingwehereareonlyconcernedwithcounterpartyrisk.Onecanthinkofcreditderivativesbeingplacedsomewherebetweentraditionalcreditinsur-anceproductsandnancialderivatives.Eachoftheseareashasitsownvaluationmethodology,butneitheriswhollysatisfactoryforpric-ingcreditderivatives.Theinsurancetechniquesmakeuseofhistoricaldata,as,e.g.,providedbyratingagencies,asabasisforvaluation(seeChapter6).Thisapproachassumesthatthefuturewillbelikethepast,anddoesnottakeintoaccountmarketinformationaboutcreditquality.Incontrast,derivativetechnologyemploysmarketinformationasabasisforvaluation.Derivativesecuritiespricingisbasedontheassumptionofrisk-neutralitywhichassumesarbitrage-freeandcom-1EspeciallyundertheISDAmasteragreement,cf.[61].â2003...

... eventoccurredduringthelifetimeofthereferencenote,theissuerpaysthefullprincipalbacktotheinvestor.SointhisexampleonecouldsummarizeaCLNasasyntheticbondwithanembeddeddefaultswap.Inoursecondexample,aninvestor,whohasnoaccesstothecreditderivativesmarketorisnotallowedtodooff-balancesheettransactions,wantstoinvestinacreditdefaultswap,sellingprotectiontotheownerofsomereferenceasset.ThiscanbeachievedbyinvestinginaCLNinthesamewayasdescribedinourfirstexample.Notethatfromtheinvestor’spointofviewtheCLNdealdiffersfromadefaultswapagreementbythecashpaymentmadeupfront.Inadefaultswap,noprincipalpaymentsareexchangedatthebeginning.Anothercommonwaytoset-upaCLNisprotectionbuying.Assumethatabankisexposedtothedefaultriskofsomereferenceasset.Thiscouldbethecasebymeansofanassetonthebalancesheetofthebankorbymeansofasituationwherethebankistheprotectionsellerinacreditdefaultswap.Inbothcasesthebankhastocarrythereferenceasset’sdefaultrisk;seeFigure7.8.ThebankcannowissueaCLN to ... eventoccurredduringthelifetimeofthereferencenote,theissuerpaysthefullprincipalbacktotheinvestor.SointhisexampleonecouldsummarizeaCLNasasyntheticbondwithanembeddeddefaultswap.Inoursecondexample,aninvestor,whohasnoaccesstothecreditderivativesmarketorisnotallowedtodooff-balancesheettransactions,wantstoinvestinacreditdefaultswap,sellingprotectiontotheownerofsomereferenceasset.ThiscanbeachievedbyinvestinginaCLNinthesamewayasdescribedinourfirstexample.Notethatfromtheinvestor’spointofviewtheCLNdealdiffersfromadefaultswapagreementbythecashpaymentmadeupfront.Inadefaultswap,noprincipalpaymentsareexchangedatthebeginning.Anothercommonwaytoset-upaCLNisprotectionbuying.Assumethatabankisexposedtothedefaultriskofsomereferenceasset.Thiscouldbethecasebymeansofanassetonthebalancesheetofthebankorbymeansofasituationwherethebankistheprotectionsellerinacreditdefaultswap.Inbothcasesthebankhastocarrythereferenceasset’sdefaultrisk;seeFigure7.8.ThebankcannowissueaCLN to ... will bereduced to an extent reflecting the amount of risk transferred to thecapital market. Both effects, and additional tax and other benefits canhelp a bank to refinance a loan portfolio at...

... (α,β)∈{(2,1/2),(5,1/5)}.Insteadofincorporatingafactormodel(aswehaveseenitinthecaseofCreditMetricsTMandKMV’sPortfolioManagerTMinSection1.2 .3) ,CreditRisk+implementsaso-calledsectormodel.However,somehowonecanthinkofasectorasa“factor-inducing”entity,or–astheCreditRisk+TechnicalDocument[18]saysit–everysectorcouldbethoughtofasgeneratedbyasingleunderlyingfactor.Inthisway,sec-tors ... Scale Risk ScaleDefault Risk onlyDefault Risk onlyMark -to- Modelof Loan ValueMark -to- Modelof Loan ValueDistance to Default (DtD)Definitionof Risk IntensityProcessMacro-economicFactorsAsset ... whereasinthePoissoncaseweobtainCorr[Li,Lj]=Cov[Λi,Λj]V[Λi]+E[Λi]V[Λj]+E[Λj].(2.24)LookingonlyatthedrivingrandomvariablesPi,PjrespectivelyΛi,Λj,weseethatinthedenominatorsof(2. 23) and(2.24)wecompareV[Pi]+E[Pi(1−Pi)]=V[Pi]+E[Pi]−E[P2i](2.25)withV[Λi]+E[Λi].Now,analogoustothedeterministiccase(2.12),wecan–evenintherandomcase–expectPiandΛitobeofthesameorderofmagnitude.Tokeepthingssimple,letusforamomentassumethatPiandΛihavethesamefirstandsecondmoments.InthiscaseEquation(2.25)combinedwith(2. 23) and(2.24)showsthattheBernoullimodelalwaysinducesahigherdefaultcorrelationthanthePoissonmodel.Buthigherdefaultcorrelationsresultinfattertailsofthecorrespondinglossdistributions.InotherwordsonecouldsaythatgivenequalfirstandsecondmomentsofPiandΛi,theexpectationsofLiandLiwillmatch,butthevarianceofLiwillalwaysexceedthevarianceofLi,therebyinducinglowerdefaultcorrelations.SothereisasystematicdifferencebetweentheBernoulliandPoissonmixturemodels.IngeneralonecanexpectthatforagivenportfoliotheBernoullimodelyieldsalossdistributionwithafattertailthanacomparably(e.g.,byafirstandsecondmomentmatching)calibratedPoissonmodel.ThisdifferenceisalsoreflectedbytheindustrymodelsfromCreditMetricsTM/KMVCorporation(PortfolioManager)andCreditSuisseFinancialProducts(CreditRisk+).InSection2.5.3wecome back to this issue.2.4 An Overview of Today’s Industry...

... not. Therefore, the credit risk of the loan is neutralized andcompletely hedged. In other words, buying the put transforms therisky corporate loan3into a riskless bullet loan with face value ... F0implicitlybymeansofdiscountingthefacevalueFataratehigherthantherisk-freerate.Thepayoutofdebttotheobligorattimet=0willthenbesmallerthemoreriskytheobligor’sbusinessis.Atypicalstrategyofdebtholders(e.g.,alendingbank)istheat-tempttoneutralizethecreditriskbypurchasingsomekindofcreditprotection.Inourcaseasuccessfulstrategyistobuyasuitablederiva-tive.Forthispurpose,debtholderstakealongpositioninaputoptiononAwithstrikeFandmaturityT;seealsoFigure3.1.Table3.2showsthat ... to an important conclusion: Taking the hedge into account,the portfolio of debt holders consists of a put option and a loan. Itsvalue at time t = 0 is D0+ P0(A0, σA, F, T, r). The risk- free...

... less than or equal to 0.002% are mapped to AAA, 0.002% to 0.04% corresponds to AA, etc. The historical frequencies of changesfrom one range to another are estimated from the history of changesin ... 5.2.2CapitalAllocationw.r.t.Value-at -Risk CalculatingriskcontributionsassociatedwiththeVaRriskmeasureisanaturalbutdifficultattempt,sinceingeneralthequantilefunc-tionwillnotbedifferentiablewithrespecttotheassetweights.Undercertaincontinuityassumptionsonthejointdensityfunctionoftheran-domvariablesXi,differentiationofVaRα(X),whereX=iwiXi,isguaranteed.Onehas(see[122])∂VaRα∂wi(X)=E[Xi|X=VaRα(X)].(5.1)Unfortunately,thedistributionoftheportfoliolossL=wiˆLi,asspecifiedatthebeginningofthischapter,ispurelydiscontinuous.ThereforethederivativesofVaRαintheabovesensewilleithernotexistorvanishtozero.Inthiscasewecouldstilldefineriskcontribu-tionsviatheright-hand-sideofEquation(5.1)bywritingγi=E[ˆLi|L=VaRα(L)]−E[ˆLi].(5.2)Foraclearerunderstanding,notethat∂E[L]∂wi=E[ˆLi]andmi=1wiγi=ECVaRα.Additionallyobserve,thatforalargeportfolioandonanappropriatescale,thedistributionofLwillappeartobe“closetocontinuous”.Unfortunately,eveninsuch“approximatelygood”cases,thelossdis-tributionoftenisnotgiveninananalyticalforminordertoallowfordifferentiations.RemarkFortheCreditRisk+model,ananalyticalformofthelossdistributioncanbefound;seeSection2.4.2andChapter4foradis-cussionofCreditRisk+.Tasche[121]showedthatintheCreditRisk+framework ... Insofarasalowerratingpresentsahighercreditrisk,Jarrowetal.[64]formulatedthecondition:(viii)j≥kmijisanondecreasingfunctionofiforeveryfixedk,whichisequivalenttorequiringthattheunderlyingMarkovchainbestochasticallymonotonic.Notethatrowandcolumnmonotonytowardsthediagonal(properties(vi)and(vii))impliesstochasticmonotonybutnotviceversa.Theproblemwiththiswishlististhatonecannotexpecttheseprop-erties to be satisfied by transition...

... pletemarkets,butitisnotclearwhethertheseconditionsholdforthecreditmarketornot.Ifacrediteventisbasedonafreelyobservablepropertyofmarketprices,suchascreditspreads,thenwebelievethatconventionalderivativepricingmethodologymaybeapplicable.Creditderivativesarebilateralfinancialcontractsthatisolatespecificaspectsofcreditriskfromanunderlyinginstrumentandtransferthatriskbetweentwocounterparties.Byallowingcreditrisktobefreelytraded,riskmanagementbecomesfarmoreflexible.Therearelotsofdifferenttypesofcreditderivatives,butweshallonlytreatthemostcommonlyusedones.Theycouldbeclassifiedintotwomaincategoriesaccordingtovaluation,namelythereplicationproducts,andthedefaultproducts.Theformerarepricedoffthecapacitytoreplicatethetrans-actioninthemoneymarket,suchascreditspreadoptions.Thelatterarepricedasafunctionoftheexposureunderlyingthesecurity,thede-faultprobabilityofthereferenceasset,andtheexpectedrecoveryrate,suchascreditdefaultswaps.Anotherclassificationcouldbealongtheirperformanceasprotection-likeproducts,suchascreditdefaultoptionsandexchange-likeproducts,suchastotalreturnswaps.Inthenextsectionswedescribethemostcommonlyusedcreditderivativesandillustratesimpleexamples.Foramoreelaborateintroductiontothedifferenttypesofcreditderivativesandtheiruseforriskmanagementsee[ 68, 107];fordocumentationandguidelineswereferto[61].7.1 ... pletemarkets,butitisnotclearwhethertheseconditionsholdforthecreditmarketornot.Ifacrediteventisbasedonafreelyobservablepropertyofmarketprices,suchascreditspreads,thenwebelievethatconventionalderivativepricingmethodologymaybeapplicable.Creditderivativesarebilateralfinancialcontractsthatisolatespecificaspectsofcreditriskfromanunderlyinginstrumentandtransferthatriskbetweentwocounterparties.Byallowingcreditrisktobefreelytraded,riskmanagementbecomesfarmoreflexible.Therearelotsofdifferenttypesofcreditderivatives,butweshallonlytreatthemostcommonlyusedones.Theycouldbeclassifiedintotwomaincategoriesaccordingtovaluation,namelythereplicationproducts,andthedefaultproducts.Theformerarepricedoffthecapacitytoreplicatethetrans-actioninthemoneymarket,suchascreditspreadoptions.Thelatterarepricedasafunctionoftheexposureunderlyingthesecurity,thede-faultprobabilityofthereferenceasset,andtheexpectedrecoveryrate,suchascreditdefaultswaps.Anotherclassificationcouldbealongtheirperformanceasprotection-likeproducts,suchascreditdefaultoptionsandexchange-likeproducts,suchastotalreturnswaps.Inthenextsectionswedescribethemostcommonlyusedcreditderivativesandillustratesimpleexamples.Foramoreelaborateintroductiontothedifferenttypesofcreditderivativesandtheiruseforriskmanagementsee[ 68, 107];fordocumentationandguidelineswereferto[61].7.1 ... Chapter7CreditDerivativesCreditderivativesareinstrumentsthathelpbanks,nancialinstitu-tions,anddebtsecurityinvestorstomanagetheircredit-sensitivein-vestments.Creditderivativesinsureandprotectagainstadversemove-mentsinthecreditqualityofthecounterpartyorborrower.Forex-ample,ifaborrowerdefaults,theinvestorwillsuerlossesontheinvestment,butthelossescanbeosetbygainsfromthecreditderiva-tivetransaction.Onemightaskwhybothbanksandinvestorsdonotutilizethewell-establishedinsurancemarketfortheirprotection.Themajorreasonsarethatcreditderivativesoerlowertransactioncost,quickerpayment,andmoreliquidity.Creditdefaultswaps,forinstance,oftenpayoutverysoonaftertheeventofdefault1;incon-trast,insurancestakemuchlongertopayout,andthevalueoftheprotectionboughtmaybehardtodetermine.Finally,aswithmost-nancialderivativesinitiallyinventedforhedging,creditderivativescannowbetradedspeculatively.Likeotherover-the-counterderivativese-curities,creditderivativesareprivatelynegotiatednancialcontracts.Thesecontractsexposetheusertooperational,counterparty,liquidity,andlegalrisk.Fromtheviewpointofquantitativemodelingwehereareonlyconcernedwithcounterpartyrisk.Onecanthinkofcreditderivativesbeingplacedsomewherebetweentraditionalcreditinsur-anceproductsandnancialderivatives.Eachoftheseareashasitsownvaluationmethodology,butneitheriswhollysatisfactoryforpric-ingcreditderivatives.Theinsurancetechniquesmakeuseofhistoricaldata,as,e.g.,providedbyratingagencies,asabasisforvaluation(seeChapter6).Thisapproachassumesthatthefuturewillbelikethepast,anddoesnottakeintoaccountmarketinformationaboutcreditquality.Incontrast,derivativetechnologyemploysmarketinformationasabasisforvaluation.Derivativesecuritiespricingisbasedontheassumptionofrisk-neutralitywhichassumesarbitrage-freeandcom-1EspeciallyundertheISDAmasteragreement,cf.[61].â2003...

... eventoccurredduringthelifetimeofthereferencenote,theissuerpaysthefullprincipalbacktotheinvestor.SointhisexampleonecouldsummarizeaCLNasasyntheticbondwithanembeddeddefaultswap.Inoursecondexample,aninvestor,whohasnoaccesstothecreditderivativesmarketorisnotallowedtodooff-balancesheettransactions,wantstoinvestinacreditdefaultswap,sellingprotectiontotheownerofsomereferenceasset.ThiscanbeachievedbyinvestinginaCLNinthesamewayasdescribedinourfirstexample.Notethatfromtheinvestor’spointofviewtheCLNdealdiffersfromadefaultswapagreementbythecashpaymentmadeupfront.Inadefaultswap,noprincipalpaymentsareexchangedatthebeginning.Anothercommonwaytoset-upaCLNisprotectionbuying.Assumethatabankisexposedtothedefaultriskofsomereferenceasset.Thiscouldbethecasebymeansofanassetonthebalancesheetofthebankorbymeansofasituationwherethebankistheprotectionsellerinacreditdefaultswap.Inbothcasesthebankhastocarrythereferenceasset’sdefaultrisk;seeFigure7.8.ThebankcannowissueaCLN to ... eventoccurredduringthelifetimeofthereferencenote,theissuerpaysthefullprincipalbacktotheinvestor.SointhisexampleonecouldsummarizeaCLNasasyntheticbondwithanembeddeddefaultswap.Inoursecondexample,aninvestor,whohasnoaccesstothecreditderivativesmarketorisnotallowedtodooff-balancesheettransactions,wantstoinvestinacreditdefaultswap,sellingprotectiontotheownerofsomereferenceasset.ThiscanbeachievedbyinvestinginaCLNinthesamewayasdescribedinourfirstexample.Notethatfromtheinvestor’spointofviewtheCLNdealdiffersfromadefaultswapagreementbythecashpaymentmadeupfront.Inadefaultswap,noprincipalpaymentsareexchangedatthebeginning.Anothercommonwaytoset-upaCLNisprotectionbuying.Assumethatabankisexposedtothedefaultriskofsomereferenceasset.Thiscouldbethecasebymeansofanassetonthebalancesheetofthebankorbymeansofasituationwherethebankistheprotectionsellerinacreditdefaultswap.Inbothcasesthebankhastocarrythereferenceasset’sdefaultrisk;seeFigure7.8.ThebankcannowissueaCLN to ... will bereduced to an extent reflecting the amount of risk transferred to thecapital market. Both effects, and additional tax and other benefits canhelp a bank to refinance a loan portfolio at...

... [105 ]L.Overbeck.Allocationofeconomiccapitalinloanportfolios.InU.Franke,W.Hăardle,andG.Stahl,editors,MeasuringRiskinComplexStochasticSystems.Springer,NewYork,2000. [106 ]W.R.Pestman.MathematicalStatistics.deGruyter,1998. [107 ]PriceWaterhouseCoopers.ThePriceWaterhouseCoopersCreditDerivativesPrimer,1999. [108 ]D.RevuzandM.Yor.ContinuousMartingalesandBrownianMotion.Springer-Verlag,1991.ChapterIV(3.13). [109 ]J.A.Rice.MathematicalStatisticsandDataAnalysis.DuxburyPress,2ndedition,1995.[ 110] W.SchmidtandI.Ward.Pricingdefaultbaskets .Risk, 15(1):111114,January2002.[111]Ph.J.Schoenbucher.Factormodelsforportfoliocreditrisk.Preprint,UniversityofBonn,Germany,2001.[112]J.Skarabot.Assetsecuritizationandoptimalassetstructureoftherm.WorkingPaper,July2001.[113]A.Sklar.Fonctionderepartition`andimensionetleurmarges.PublicationsdelInsituteStatistiquedelUniversitedeParis,8:229231,1959.[114]A.Sklar.Randomvariables,jointdistributionfunctionsandcop-ulas.Kybernetika,9:449460,1973.[115]J.R.SobehartandS.C.Keenan.Anintroductiontomarket-basedcreditanalysis.MoodysRiskManagementServices,November1999.[116]Standard&Poors.GlobalCBO/CLOCriteria.[117]Standard&Poors.GlobalSyntheticSecuritiesCriteria.[118]Standard&Poors.Standard&PoorsCorporateRatingsCrite-ria1998.[119]W.Stromquist.Rootsoftransitionmatrices.DanielH.WagnerAssociates,1996.PracticalPaper.[120]D.Tasche.Riskcontributionsandperformancemeasurement.http://www.ma.tum.de/stat/,1999.â2003 ... [90]Moody’sInvestorsService.ThedoublebinomialmethodanditsapplicationtoaspecialcaseofCBOstructure,March1998.[91]Moody’sInvestorsService.StabilityofRatingsofCBO/CLOTranches:WhatDoesItTaketoDowngradeaCBOTranche?,November1998.[92]Moody’sInvestorsService.AnotherPerspectiveonRiskTrans-ferenceandSecuritization,July1999.[93]Moody’sInvestorsService.Moody’sApproachtoratingmultisec-torCDOs,September2000.[94]Moody’sInvestorsService.Moody’sApproachtoRatingMulti-sectorCDOs,September2000.[95]Moody’sInvestorsService.DefaultandRecoveryRatesofCor-porateBondIssuers:2000,February2001.[96]Moody’sInvestorsService.CollateralizedDebtObligationsPer-formanceOverviewCompilation,March2002.[97]D.Murphy.Keepingcreditundercontrol .Risk, 9,September1996.[98]S.R.Neal.Creditderivatives:Newfinancialinstrumentsforcon-trollingcreditrisk.EconomicReview,1996.[99]R.Nelsen.AnIntroductiontoCopulas.Springer,NewYork,1999. [100 ]C.NelsonandA.Siegel.Parsimoniousmodelingofyieldcurves.JournalofBusiness,60:473–489,1987. [101 ]P.Nickell,W.Perraudin,andS.Varotto.Ratings-versusequity-basedcreditriskmodeling:anempiricalanalysis.http://www.bankofengland.co.uk/workingpapers/,1999.Workingpaper. [102 ] J. Norris. Markov ... [16]D.R.CoxandD.Oakes.AnalysisofSurvivalData.ChapmanandHall,1984.[17]J.C.Cox,J.E.IngersollandS.A.Ross.Atheoryoftermstruc-tureofinterestrates.Econometrica,53:385–407,1985.[18]CreditSuisseFinancialProducts.CreditRisk+–ACreditRiskManagementFramework,1997.[19]P.Crosbie.Modelingdefaultrisk.KMVCorporation,http://www.kmv.com,1999.[20]M.Crouhy,D.Galai,andR.Mark.Acomparativeanalysisofcurrentcreditriskmodels.JournalofBanking&Finance,24:59–117,2000.[21]M.Crouhy,D.Galai,andR.Mark.RiskManagement.McGraw-Hill,2000.[22]M.Crouhy,D.Galai,andR.Mark.Prototyperiskratingsystem.JournalofBanking&Finance,25:41–95,2001.[23]S.R.Das.Structurednotesandderivativesembeddedsecurities.EuromoneyPublicationsPLC,1996.[24]M.Denault.Coherentallocationofriskcapital.http://www.risklab.ch/Papers.html,1999.[25]...