风险管理与金融机构课件Ch10.ppt
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风险管理与金融机构课件Ch10.ppt
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CorrelationsandCopulasChapter10RiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull20091CorrelationandCovariancelThecoefficientofcorrelationbetweentwovariablesV1andV2isdefinedaslThecovarianceisE(V1V2)E(V1)E(V2)RiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull20092IndependencelV1andV2areindependentiftheknowledgeofonedoesnotaffecttheprobabilitydistributionfortheotherwheref(.)denotestheprobabilitydensityfunctionRiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull20093IndependenceisNottheSameasZeroCorrelationlSupposeV1=1,0,or+1(equallylikely)lIfV1=-1orV1=+1thenV2=1lIfV1=0thenV2=0V2isclearlydependentonV1(andviceversa)butthecoefficientofcorrelationiszeroRiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull20094TypesofDependence(Figure10.1,page204)RiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull20095E(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-1ThecorrelationisRiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull20096CovariancelThecovarianceondaynisE(xnyn)E(xn)E(yn)lItisusuallyapproximatedasE(xnyn)RiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull20097MonitoringCorrelationcontinuedEWMA:
GARCH(1,1)RiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull20098PositiveFiniteDefiniteConditionAvariance-covariancematrix,WW,isinternallyconsistentifthepositivesemi-definiteconditionholdsforallvectorswRiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull20099ExampleThevariancecovariancematrixisnotinternallyconsistentRiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull200910V1andV2BivariateNormallConditionalonthevalueofV1,V2isnormalwithmeanandstandarddeviationwherem1,m2,s1,ands2aretheunconditionalmeansandSDsofV1andV2andristhecoefficientofcorrelationbetweenV1andV2RiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull200911MultivariateNormalDistributionlFairlyeasytohandlelAvariance-covariancematrixdefinesthevariancesofandcorrelationsbetweenvariableslTobeinternallyconsistentavariance-covariancematrixmustbepositivesemidefiniteRiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull200912GeneratingRandomSamplesforMonteCarloSimulation(pages207-208)l=NORMSINV(RAND()givesarandomsamplefromanormaldistributioninExcellForamultivariatenormaldistributionamethodknownasCholeskysdecompositioncanbeusedtogeneraterandomsamplesRiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull200913FactorModels(page209)lWhenthereareNvariables,Vi(i=1,2,.N),inamultivariatenormaldistributionthereareN(N1)/2correlationslWecanreducethenumberofcorrelationparametersthathavetobeestimatedwithafactormodelRiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull200914One-FactorModelcontinuedRiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull200915lIfUihavestandardnormaldistributionswecansetwherethecommonfactorFandtheidiosyncraticcomponentZihaveindependentstandardnormaldistributionslCorrelationbetweenUiandUjisaiajGaussianCopulaModels:
CreatingacorrelationstructureforvariablesthatarenotnormallydistributedlSupposewewishtodefineacorrelationstructurebetweentwovariableV1andV2thatdonothavenormaldistributionslWetransformthevariableV1toanewvariableU1thathasastandardnormaldistributionona“percentile-to-percentile”basis.lWetransformthevariableV2toanewvariableU2thathasastandardnormaldistributionona“percentile-to-percentile”basis.lU1andU2areassumedtohaveabivariatenormaldistributionRiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull200916TheCorrelationStructureBetweentheVsisDefinedbythatBetweentheUsRiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull200917-0.200.20.40.60.811.2-0.200.20.40.60.811.2V1V2-6-4-20246-6-4-20246U1U2One-to-onemappingsCorrelationAssumptionV1V2-6-4-20246-6-4-20246U1U2One-to-onemappingsCorrelationAssumptionExample(page211)RiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull200918V1V2V1MappingtoU1RiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull200919V1PercentileU10.220-0.840.4550.130.6800.840.8951.64V2MappingtoU2RiskManagementandFinancialInstitutions2e,Chapter10,CopyrightJohnC.Hull20
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