USING COPULAS An introduction for practitioners

1 IntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryUSING COPULASAn introduction for practitionersDANIEL BERGDnBNOR Asset ManagementNorwegian ASTIN society. Oslo - November 2008 Daniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryDnBNO R Kapitalforvaltning bredde og dybde i forvalterkompetanse, 100 analytikere og portef produktspekter - og gode dokumenterte systemer for risikostyring og - kontroll. Store volumer - kostnadseffektiv prosess med produktutvikling og 300 aarsverk. Ca NOK 500 milliarder til forvaltning. Personkunder Institusjonelle investorer Ca 500 kunder i Norge og Sverige Viktigste kundesegmenter: pensjonskasser, kommuner, bedrifter,organisasjoner/stiftelser H y raadgivningskompetanse Strategiske mandat Vital Skandia Liv Skandia FonderDaniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel copula (uranium, river flow/temp, precipitation.)

2 BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryIntro ductionMotivationFigure:Simulations from two models, both with standard normal margins and correlation BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel several applications empirical evidence has proved multinormal distributioninadequate for several reasons:.Empirical marginal distributions are skewed and of extreme co-movements, in contrast to the multinormal distributionDaniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryIntro ductionBrief historical : Hoeffding studies properties of multivariate : The word copula appears for the first time (Sklar, 1959).

3 1999: Introduced to financial applications (Embrechts et al., 1999).2008: Widely used in insurance, finance, energy, hydrology, survival analysis, BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryIntro ductionDefinition & theoremDefinition (Copula)A d-dimensional copula is a multivariate distribution functionCwith standard uniformmarginal (Sklar, 1959)Let H be a joint distribution function with margins F1,..,Fd. Then there exists a copulaC: [0,1]d [0,1]such thatH(x1,..,xd) =C(F1(x1),..,Fd(xd)).Daniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryIntro ductionUseful generald-dimensional densityhcan be expressed, for some copula densityc, ash(x1.)

4 ,xd) =c{F1(x1),..,Fd(xd)}f1(x1) fd(xd)..Non-parametric estimate forFi(xi)commonly used to transform original margins intostandard uniform:zji=bFi(xji) =Rjin+1,whereRjiis the rank ofxjiamongstx1i,.., referred to aspseudo-observationsand models based on non-parametricmargins and parametric COPULAS are referred to to use empirical copulaCnas a consistent estimator ofC(Deheuvels, 1979):Cn(u) =1n+1nXj=1I{zj1 u1,..,zjd ud}Daniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryIntro ductionUseful (x1) BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryIntro ductionAttractive copula contains all the information about the dependence between random provide an alternative and often more useful representation of multivariatedistribution functions compared to traditional approaches such as multivariate traditional representations of dependence are based on the linear correlationcoefficient - restricted to multivariate elliptical distributions.

5 Copula representations ofdependence are free of such enable us to model marginal distributions and the dependence provide greater modeling flexibility, given a copula we can obtain manymultivariate distributions by selecting different multivariate distribution can serve as a copula is invariant under strictly increasing traditional measures of dependence are measures of pairwise dependence. Copulasmeasure the dependence between alldrandom variablesDaniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryPopul ar copula copula:C (u,v) = copula:C (uv) =R 1(u) R 1(v) 12 (1 2)1/2expn x2 2 xy+y22(1 2) copula:C , (u,v) =Rt 1 (u) Rt 1 (v) 12 (1 2)1/2n1+x2 2 xy+y2 (1 2)o ( +2) COPULAS :C (u,v) = 1{ (u) + (v)}where is the copula generator.

6 Clayton copula:C (u,v) = (u +v 1) 1/ Gumbel copula:C (u,v) =exph (( lnu) + ( lnv) )1/ iDaniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryPopul ar copula familiesTake the bivariate std. normal prob. (x1,x2)Daniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryPopul ar copula by std. normal marginal 4 (x1) 4 (x2)Daniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryPopul ar copula we obtain the normal copula (u1,u2)Daniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryPopul ar copula familiesNow multiply the normal copula density with 4 (x1) (x2)Daniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryPopul ar copula we (x1,x2)

7 Daniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryPopul ar copula familiesGaussian copulauvc(u,v)Daniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryPopul ar copula familiesStudent copulauvc(u,v)Daniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryPopul ar copula familiesClayton copulauvc(u,v)Daniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryPopul ar copula familiesGumbel copulauvc(u,v)Daniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryPopul ar copula BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummarySimul ationElliptical copula: Simulatex Nd(0,R) Setu= (x).

8 Student copula: Simulatex td(0,R, ) Setu=t (x)Daniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummarySimul ationArchimedean copula: Note that the inverse of the generator is equal to the Laplace transform of aGamma variatex Ga(1/ ,1) Simulate a gamma variatex Ga(1/ ,1) SimulatediidU(0,1)variablesv1,..,vd Returnu= (1 logv1x) 1/ ,..,(1 logvdx) 1/ .Gumbel copula: Note that the inverse of the generator function is equal to the Laplace transformof a positive stable variatex St(1/ ,1, ,0), where =`cos` 2 and >1 Simulate a positive stable variatex St(1/ ,1, ,0) SimulatediidU(0,1)variablesv1,..,vd Returnu= exp logv1x 1/ ,..,exp logvdx 1/ !!Daniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummarySimul ationGeneral general we could apply the conditional marginal cdf s:Ci|1.

9 ,i 1(ui|u1,..,ui 1) = i 1C(u1,..,ui) u1 ui 1ffi i 1C(u1,..,ui 1) u1 ui a rvu1fromU(0,1),.Simulate a rvu2fromC2|1( |u1),..Simulate a rvudfromCd|1,..,d 1( |u1,..,ud 1)..Generally means simulating a rvVifromU(0,1)from whichui=C 1i|1,..,i 1(Vi|u1,..,ui 1)can be obtained, if necessary by numerical BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummarySimul ationR example> library(copulaGOF)> u=SimulateCopulae(n=10000,d=2,constructi on=list(type= opc ,copula= gumbel ),param=2)> plot(u,pch=".") BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryParam eter estimationModel:C (u1,..,ud), ,dim( ) 1 Data:xj= (xj1.)

10 ,xjd),j=1,.., densityDaniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryParam eter to by one-to-one functiongm:m=gm( ;C).Ifbmis a consistent estimator formthenb =g 1m(bm;C)is a consistent estimator for .In most cases of interest, asn : n(b ) N(0, 2(C )).Examples: Spearman s rho, Kendall s taub S=g 1 S(c S;C),b =g 1 (b ;C) S(X,Y) =12Z10Z10C(u,v)dudv 3 (X,Y) =4Z10Z10C(u,v)dC(u,v) 1 Daniel BergUsing copulasIntroductionPopular copula familiesSimulationParameter estimationModel selectionModel evaluationExamplesExtensionsSummaryParam eter estimationMaximum classical statistics, ML estimation is usually more efficient than needed since inference is based on ranks maximum pseudo-likelihood(Oakes, 1994; Genest et al.