财务风险管理Ch11PPT格式课件下载.pptx

财务风险管理Ch11PPT格式课件下载.pptx

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CorrelationsandCopulasChapter111CorrelationandCovariancelThecoefficientofcorrelationbetweentwovariablesV1andV2isdefinedaslThecovarianceisE（V1V2）E（V1）E（V2）2IndependencelV1andV2areindependentiftheknowledgeofonedoesnotaffecttheprobabilitydistributionfortheotherwheref（.）denotestheprobabilitydensityfunction3IndependenceisNottheSameasZeroCorrelationlSupposeV1=1,0,or+1（equallylikely）lIfV1=-1orV1=+1thenV2=1lIfV1=0thenV2=0V2isclearlydependentonV1（andviceversa）butthecoefficientofcorrelationiszero4TypesofDependence（Figure11.1,page235）5E（Y）XE（Y）E（Y）X（a）（b）（c）XMonitoringCorrelationBetweenTwoVariablesXandYDefinexi=（XiXi-1）/Xi-1andyi=（YiYi-1）/Yi-1Alsovarx,n:

@#@dailyvarianceofXcalculatedondayn-1vary,n:

@#@dailyvarianceofYcalculatedondayn-1covn:

@#@covariancecalculatedondayn-1Thecorrelationis6CovariancelThecovarianceondaynisE（xnyn）E（xn）E（yn）lItisusuallyapproximatedasE（xnyn）byassumingE（xn）=0,E（yn）=07MonitoringCorrelationcontinuedEWMA（exponentialweightedmovingaverage）:

@#@GARCH（1,1）:

@#@Bollerslev（1986）8PositiveSemiDefiniteConditionAvariance-covariancematrix,WW,isinternallyconsistentifthepositivesemi-definiteconditionholdsforallvectorsw9Example10Thevariancecovariancematrixisnotinternallyconsistent.Forexample,.V1andV2BivariateNormallConditionalonthevalueofV1,V2isnormalwithmeanandstandarddeviationwherem1,m2,s1,ands2aretheunconditionalmeansandSDsofV1andV2andristhecoefficientofcorrelationbetweenV1andV211MultivariateNormalDistributionlFairlyeasytohandlelAvariance-covariancematrixdefinesthevariancesofandcorrelationsbetweenvariableslTobeinternallyconsistentavariance-covariancematrixmustbepositivesemidefinite12GeneratingRandomSamplesforMonteCarloSimulation（pages207-208）l=NORMSINV（RAND（）givesarandomsamplefromanormaldistributioninExcellForamultivariatenormaldistributionamethodknownasCholeskysdecompositioncanbeusedtogeneraterandomsamples13FactorModels（page240）lWhenthereareNvariables,Vi（i=1,2,.N）,inamultivariatenormaldistributionthereareN（N1）/2correlationslWecanreducethenumberofcorrelationparametersthathavetobeestimatedwithafactormodel14One-FactorModelcontinued15lIfUihavestandardnormaldistributionswecansetwherethecommonfactorFandtheidiosyncraticcomponentZihaveindependentstandardnormaldistributionslCorrelationbetweenUiandUjisaiajGaussianCopulaModels:

@#@CreatingacorrelationstructureforvariablesthatarenotnormallydistributedlSupposewewishtodefineacorrelationstructurebetweentwovariableV1andV2thatdonothavenormaldistributionslWetransformthevariableV1toanewvariableU1thathasastandardnormaldistributionona“percentile-to-percentile”basis.lWetransformthevariableV2toanewvariableU2thathasastandardnormaldistributionona“percentile-to-percentile”basis.lU1andU2areassumedtohaveabivariatenormaldistribution16TheCorrelationStructureBetweentheVsisDefinedbythatBetweentheUs17-0.200.20.40.60.811.2-0.200.20.40.60.811.2V1V2-6-4-20246-6-4-20246U1U2One-to-onemappingsCorrelationAssumptionV1V2-6-4-20246-6-4-20246U1U2One-to-onemappingsCorrelationAssumptionExample（page241）18V1V2V1MappingtoU119V1PercentileU10.220-0.840.4550.130.6800.840.8951.64V2MappingtoU220V2PercentileU20.281.410.4320.470.6680.470.8921.41ExampleofCalculationofJointCumulativeDistributionlProbabilitythatV1andV2arebothlessthan0.2istheprobabilitythatU10.84andU21.41lWhencopulacorrelationis0.5thisisM（0.84,1.41,0.5）=0.043whereMisthecumulativedistributionfunctionforthebivariatenormaldistribution21OtherCopulaslInsteadofabivariatenormaldistributionforU1andU2wecanassumeanyotherjointdistributionlOnepossibilityisthebivariateStudenttdistribution225000RandomSamplesfromtheBivariateNormal235000RandomSamplesfromtheBivariateStudentt24MultivariateGaussianCopulalWecansimilarlydefineacorrelationstructurebetweenV1,V2,VnlWetransformeachvariableVitoanewvariableUithathasastandardnormaldistributionona“percentile-to-percentile”basis.lTheUsareassumedtohaveamultivariatenormaldistribution25FactorCopulaModelInafactorcopulamodelthecorrelationstructurebetweentheUsisgeneratedbyassumingoneormorefactors.26CreditDefaultCorrelationlThecreditdefaultcorrelationbetweentwocompaniesisameasureoftheirtendencytodefaultataboutthesametimelDefaultcorrelationisimportantinriskmanagementwhenanalyzingthebenefitsofcreditriskdiversificationlItisalsoimportantinthevaluationofsomecreditderivatives27ModelforLoanPortfoliolWemapthetimetodefaultforcompanyi,Ti,toanewvariableUiandassumewhereFandtheZihaveindependentstandardnormaldistributionslDefineQiasthecumulativeprobabilitydis