财务风险管理Ch11.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