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1、CorrelationsandCopulasChapter10RiskManagementandFinancialInstitutions2e,Chapter10,Copyright©JohnC.Hull20091CorrelationandCovarianceThecoefficientofcorrelationbetweentwovariablesV1andV2isdefinedasThecovarianceisE(V1V2)−E(V1)E(V2)RiskManagementandFinancialInstitutions2e,Chapter10,Copyright©John
2、C.Hull20092IndependenceV1andV2areindependentiftheknowledgeofonedoesnotaffecttheprobabilitydistributionfortheotherwheref(.)denotestheprobabilitydensityfunctionRiskManagementandFinancialInstitutions2e,Chapter10,Copyright©JohnC.Hull20093IndependenceisNottheSameasZeroCorrelationSupposeV1=–1,0,or+
3、1(equallylikely)IfV1=-1orV1=+1thenV2=1IfV1=0thenV2=0V2isclearlydependentonV1(andviceversa)butthecoefficientofcorrelationiszeroRiskManagementandFinancialInstitutions2e,Chapter10,Copyright©JohnC.Hull20094TypesofDependence(Figure10.1,page204)RiskManagementandFinancialInstitutions2e,Chapter10,Cop
4、yright©JohnC.Hull20095E(Y)XE(Y)E(Y)X(a)(b)(c)XMonitoringCorrelationBetweenTwoVariablesXandYDefinexi=(Xi−Xi-1)/Xi-1andyi=(Yi−Yi-1)/Yi-1Alsovarx,n:dailyvarianceofXcalculatedondayn-1vary,n:dailyvarianceofYcalculatedondayn-1covn:covariancecalculatedondayn-1ThecorrelationisRiskManagementandFinanci
5、alInstitutions2e,Chapter10,Copyright©JohnC.Hull20096CovarianceThecovarianceondaynisE(xnyn)−E(xn)E(yn)ItisusuallyapproximatedasE(xnyn)RiskManagementandFinancialInstitutions2e,Chapter10,Copyright©JohnC.Hull20097MonitoringCorrelationcontinuedEWMA:GARCH(1,1)RiskManagementandFinancialInstitutions2
6、e,Chapter10,Copyright©JohnC.Hull20098PositiveFiniteDefiniteConditionAvariance-covariancematrix,W,isinternallyconsistentifthepositivesemi-definiteconditionholdsforallvectorswRiskManagementandFinancialInstitutions2e,Chapter10,Copyright©JohnC.Hull20099ExampleThevariancecovariancematrixisnotinter
7、nallyconsistentRiskManagementandFinancialInstitutions2e,Chapter10,Copyright©JohnC.Hull200910V1andV2BivariateNormalConditionalonthevalueofV1,V2isnormalwithmeanandstandarddeviationwherem1,,m2,s1,ands2aretheunconditionalmeansandSDsofV1andV2andri
