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    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Explainanddemonstratetheimportance ofthetailsofthedistributions, tailcorrelationsand lowfrequency/highseverityevents ArthurCharpentier Universit´edeRennes1&´EcolePolytechnique http://blogperso.univ-rennes1.fr/arthur.charpentier/ 1
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement SCRandSolvency 2
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement SCRandSolvency 3
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement SCRandSolvency 4
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement SCRandSolvency 5
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement OnriskdependenceinQIS’s http://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 6
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement OnriskdependenceinQIS’s http://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 7
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement OnriskdependenceinQIS’s http://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 8
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement OnriskdependenceinQIS’s http://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 9
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement OnriskdependenceinQIS’s http://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 10
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Howtocapturedependenceinriskmodels? Iscorrelationrelevanttocapturedependenceinformation? Consider(seeMcNeil,Embrechts&Straumann(2003))2log-normalrisks, •X∼LN(0,1),i.e.X=exp(X� )whereX� ∼N(0,1) •Y∼LN(0,σ2 ),i.e.Y=exp(Y� )whereY� ∼N(0,σ2 ) Recallthatcorr(X� ,Y� )takesanyvaluein[−1,+1]. Sincecorr(X,Y)�=corr(X� ,Y� ),whatcanbecorr(X,Y)? 11
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Howtocapturedependenceinriskmodels? 012345 −0.50.00.51.0 Standarddeviation,sigma Correlation Fig.1–Rangeforthecorrelation,cor(X,Y),X∼LN(0,1),Y∼LN(0,σ2 ). 12
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Howtocapturedependenceinriskmodels? 012345 −0.50.00.51.0 Standarddeviation,sigma Correlation Fig.2–cor(X,Y),X∼LN(0,1),Y∼LN(0,σ2 ),Gaussiancopula,r=0.5. 13
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Whataboutofficialactuarialdocuments? 14
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Whataboutofficialactuarialdocuments? 15
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Whataboutofficialactuarialdocuments? 16
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Whataboutregulatorytechnicaldocuments? 17
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Whataboutregulatorytechnicaldocuments? 18
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Whataboutregulatorytechnicaldocuments? 19
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Whataboutregulatorytechnicaldocuments? 20
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Motivations:dependenceandcopulas Definition1.AcopulaCisajointdistributionfunctionon[0,1]d ,with uniformmarginson[0,1]. Theorem2.(Sklar)LetCbeacopula,andF1,...,Fdbedmarginal distributions,thenF(x)=C(F1(x1),...,Fd(xd))isadistributionfunction,with F∈F(F1,...,Fd). Conversely,ifF∈F(F1,...,Fd),thereexistsCsuchthat F(x)=C(F1(x1),...,Fd(xd)).Further,iftheFi’sarecontinuous,thenCis unique,andgivenby C(u)=F(F−1 1(u1),...,F−1 d(ud))forallui∈[0,1] WewillthendefinethecopulaofF,orthecopulaofX. 21
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement CopuladensityLevelcurvesofthecopula Fig.3–Graphicalrepresentationofacopula,C(u,v)=P(U≤u,V≤v). 22
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement CopuladensityLevelcurvesofthecopula Fig.4–Densityofacopula,c(u,v)= ∂2 C(u,v) ∂u∂v . 23
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Someveryclassicalcopulas •TheindependentcopulaC(u,v)=uv=C⊥ (u,v). ThecopulaisstandardlydenotedΠ,PorC⊥ ,andanindependentversionof (X,Y)willbedenoted(X⊥ ,Y⊥ ).ItisarandomvectorsuchthatX⊥L =Xand Y⊥L =Y,withcopulaC⊥ . Inhigherdimension,C⊥ (u1,...,ud)=u1×...×udistheindependentcopula. •ThecomonotoniccopulaC(u,v)=min{u,v}=C+ (u,v). ThecopulaisstandardlydenotedM,orC+ ,andancomonotoneversionof (X,Y)willbedenoted(X+ ,Y+ ).ItisarandomvectorsuchthatX+L =Xand Y+L =Y,withcopulaC+ . (X,Y)hascopulaC+ ifandonlyifthereexistsastrictlyincreasingfunctionh suchthatY=h(X),orequivalently(X,Y) L =(F−1 X(U),F−1 Y(U))whereUis U([0,1]). 24
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Someveryclassicalcopulas Inhigherdimension,C+ (u1,...,ud)=min{u1,...,ud}isthecomonotonic copula. •ThecontercomotoniccopulaC(u,v)=max{u+v−1,0}=C− (u,v). ThecopulaisstandardlydenotedW,orC− ,andancontercomontoneversionof (X,Y)willbedenoted(X− ,Y− ).ItisarandomvectorsuchthatX−L =Xand Y−L =Y,withcopulaC− . (X,Y)hascopulaC− ifandonlyifthereexistsastrictlydecreasingfunctionh suchthatY=h(X),orequivalently(X,Y) L =(F−1 X(1−U),F−1 Y(U)). Inhigherdimension,C− (u1,...,ud)=max{u1+...+ud−(d−1),0}isnota copula. ButnotethatforanycopulaC, C− (u1,...,ud)≤C(u1,...,ud)≤C+ (u1,...,ud) 25
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement 0.2 0.4 0.6 0.8 u_10.2 0.4 0.6 0.8 u_2 00.20.40.60.81 Frechetlowerbound 0.2 0.4 0.6 0.8 u_10.2 0.4 0.6 0.8 u_2 00.20.40.60.81 Independencecopula 0.2 0.4 0.6 0.8 u_10.2 0.4 0.6 0.8 u_2 00.20.40.60.81 Frechetupperbound ������������������� ������������������ ������������������ ������������������ ������������������ ������������������ ������������������� ������������������ ������������������ ������������������ ������������������ Scatterplot,LowerFréchet!Hoeffdingbound ������������������ ������������������ ������������������������������������������������ ������������������ ������������������ Scatterplot,UpperFréchet!Hoeffdingbound Fig.5–Contercomontonce,independent,andcomonotonecopulas. 26
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Elliptical(Gaussianandt)copulas Theideaistoextendthemultivariateprobitmodel,X=(X1,...,Xd)with marginalB(pi)distributions,modeledasYi=1(X� i≤ui),whereX� ∼N(I,Σ). •TheGaussiancopula,withparameterα∈(−1,1), C(u,v)= 1 2π √ 1−α2 �Φ−1 (u) −∞ �Φ−1 (v) −∞ exp � −(x2 −2αxy+y2 ) 2(1−α2) � dxdy. Analogouslythet-copulaisthedistributionof(T(X),T(Y))whereTisthet-cdf, andwhere(X,Y)hasajointt-distribution. •TheStudentt-copulawithparameterα∈(−1,1)andν≥2, C(u,v)= 1 2π √ 1−α2 �t−1 ν(u) −∞ �t−1 ν(v) −∞ � 1+ x2 −2αxy+y2 2(1−α2) �−((ν+2)/2) dxdy. 27
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Archimedeancopulas •ArchimediancopulasC(u,v)=φ−1 (φ(u)+φ(v)),whereφisdecreasingconvex (0,1),withφ(0)=∞andφ(1)=0. Example3.Ifφ(t)=[−logt]α ,thenCisGumbel’scopula,andif φ(t)=t−α −1,CisClayton’s.NotethatC⊥ isobtainedwhenφ(t)=−logt. Thefrailtyapproach:assumethatXandYareconditionallyindependent,given thevalueofanheterogeneouscomponentΘ.Assumefurtherthat P(X≤x|Θ=θ)=(GX(x))θ andP(Y≤y|Θ=θ)=(GY(y))θ forsomebaselinedistributionfunctionsGXandGY.Then F(x,y)=ψ(ψ−1 (FX(x))+ψ−1 (FY(y))), whereψdenotestheLaplacetransformofΘ,i.e.ψ(t)=E(e−tΘ ). 28
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement ������������ ������������ ������������������������������������������������ !3!2!10123 !3!2!10123 ������������������������������������������������ Fig.6–Continuousclassesofrisks,(Xi,Yi)and(Φ−1 (FX(Xi)),Φ−1 (FY(Yi))). 29
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement SomemoreexamplesofArchimedeancopulas ψ(t)rangeθ (1)1 θ (t−θ−1)[−1,0)∪(0,∞)Clayton,Clayton(1978) (2)(1−t)θ[1,∞) (3)log 1−θ(1−t) t [−1,1)Ali-Mikhail-Haq (4)(−logt)θ[1,∞)Gumbel,Gumbel(1960),Hougaard(1986) (5)−loge−θt−1 e−θ−1 (−∞,0)∪(0,∞)Frank,Frank(1979),Nelsen(1987) (6)−log{1−(1−t)θ}[1,∞)Joe,Frank(1981),Joe(1993) (7)−log{θt+(1−θ)}(0,1] (8)1−t 1+(θ−1)t [1,∞) (9)log(1−θlogt)(0,1]Barnett(1980),Gumbel(1960) (10)log(2t−θ−1)(0,1] (11)log(2−tθ)(0,1/2] (12)(1 t −1)θ[1,∞) (13)(1−logt)θ−1(0,∞) (14)(t−1/θ−1)θ[1,∞) (15)(1−t1/θ)θ[1,∞)Genest&Ghoudi(1994) (16)(θ t +1)(1−t)[0,∞) 30
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Extremevaluecopulas •Extremevaluecopulas C(u,v)=exp � (logu+logv)A � logu logu+logv �� , whereAisadependencefunction,convexon[0,1]withA(0)=A(1)=1,et max{1−ω,ω}≤A(ω)≤1forallω∈[0,1]. Analternativedefinitionisthefollowing:Cisanextremevaluecopulaifforall z>0, C(u1,...,ud)=C(u 1/z 1,...,u 1/z d)z . Thosecopulaarethencalledmax-stable:definethemaximumcomponentwiseof asampleX1,...,Xn,i.e.Mi=max{Xi,1,...,Xi,n}. Remarkmoredifficulttocharacterizewhend≥3. 31
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Oncopulaparametrization •Gaussian,Studentt(andelliptical)copulas Focusesonpairwisedependencethroughthecorrelationmatrix,        X1 X2 X3 X4        ∼N        0, 1r12r13r14 r121r23r24 r13r231r34 r14r24r341        Dependencein[0,1]d ←→summarizedind(d+1)/2parameters, 32
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Oncopulaparametrization •Archimedeancopulas Initially,dependencein[0,1]d ←→summarizedinonefunctionalparameterson [0,1].Butappearslessflexiblebecauseofexchangeabilityfeatures. ItispossibletointroducehierarchicalArchimedeancopulas(seeSavu&Trede (2006)orMcNeil(2007)).LetU=(U1,U2,U3,U4), C(u1,u2,u3,u4)=φ−1 1[φ1(u1)+φ1(u2)+φ1(u3)+φ1(u4)], which,ifφiisparametrizedwithparameterαi,canbesummarizedthrough A=        1α2α4α4 α21α4α4 α4α41α3 alpha4α4α31        33
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Oncopulaparametrization •Archimedeancopulas Initially,dependencein[0,1]d ←→summarizedinonefunctionalparameterson [0,1].Butappearslessflexiblebecauseofexchangeabilityfeatures. ItispossibletointroducehierarchicalArchimedeancopulas(seeSavu&Trede (2006)orMcNeil(2007)).LetU=(U1,U2,U3,U4), C(u1,u2,u3,u4)=φ−1 4(φ4 � φ−1 2(φ2(u1)+φ2(u2)) � +φ4 � φ−1 3(φ3(u3)+φ3(u4)) � ), which,ifφiisparametrizedwithparameterαi,canbesummarizedthrough A=        1α2α4α4 α21α4α4 α4α41α3 alpha4α4α31        34
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Oncopulaparametrization •Archimedeancopulas Initially,dependencein[0,1]d ←→summarizedinonefunctionalparameterson [0,1].Butappearslessflexiblebecauseofexchangeabilityfeatures. ItispossibletointroducehierarchicalArchimedeancopulas(seeSavu&Trede (2006)orMcNeil(2007)).LetU=(U1,U2,U3,U4), C(u1,u2,u3,u4)=φ−1 4(φ4 � φ−1 2(φ2(u1)+φ2(u2)) � +φ4 � φ−1 3(φ3(u3)+φ3(u4)) � ), which,ifφiisparametrizedwithparameterαi,canbesummarizedthrough A=        1α2α4α4 α21α4α4 α4α41α3 alpha4α4α31        35
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Oncopulaparametrization •Archimedeancopulas Initially,dependencein[0,1]d ←→summarizedinonefunctionalparameterson [0,1].Butappearslessflexiblebecauseofexchangeabilityfeatures. ItispossibletointroducehierarchicalArchimedeancopulas(seeSavu&Trede (2006)orMcNeil(2007)).LetU=(U1,U2,U3,U4), C(u1,u2,u3,u4)=φ−1 4(φ4 � φ−1 2(φ2(u1)+φ2(u2)) � +φ4 � φ−1 3(φ3(u3)+φ3(u4)) � ), which,ifφiisparametrizedwithparameterαi,canbesummarizedthrough A=        1α2α4α4 α21α4α4 α4α41α3 α4α4α31        36
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Oncopulaparametrization •Archimedeancopulas Initially,dependencein[0,1]d ←→summarizedinonefunctionalparameterson [0,1].Butappearslessflexiblebecauseofexchangeabilityfeatures. ItispossibletointroducehierarchicalArchimedeancopulas(seeSavu&Trede (2006)orMcNeil(2007)).LetU=(U1,U2,U3,U4), C(u1,u2,u3,u4)=φ−1 4(φ4[φ−1 3(φ3 � φ−1 2(φ2(u1)+φ2(u2)) � +φ3(u3))]+φ4(u4)), which,ifφiisparametrizedwithparameterαi,canbesummarizedthrough A=        1α2α3α4 α21α3α4 α3α31α4 α4α4α41        37
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Oncopulaparametrization •Archimedeancopulas Initially,dependencein[0,1]d ←→summarizedinonefunctionalparameterson [0,1].Butappearslessflexiblebecauseofexchangeabilityfeatures. ItispossibletointroducehierarchicalArchimedeancopulas(seeSavu&Trede (2006)orMcNeil(2007)).LetU=(U1,U2,U3,U4), C(u1,u2,u3,u4)=φ−1 4(φ4[φ−1 3(φ3 � φ−1 2(φ2(u1)+φ2(u2)) � +φ3(u3))]+φ4(u4)), which,ifφiisparametrizedwithparameterαi,canbesummarizedthrough A=        1α2α3α4 α21α3α4 α3α31α4 α4α4α41        38
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Oncopulaparametrization •Archimedeancopulas Initially,dependencein[0,1]d ←→summarizedinonefunctionalparameterson [0,1].Butappearslessflexiblebecauseofexchangeabilityfeatures. ItispossibletointroducehierarchicalArchimedeancopulas(seeSavu&Trede (2006)orMcNeil(2007)).LetU=(U1,U2,U3,U4), C(u1,u2,u3,u4)=φ−1 4(φ4[φ−1 3(φ3 � φ−1 2(φ2(u1)+φ2(u2)) � +φ3(u3))]+φ4(u4)), which,ifφiisparametrizedwithparameterαi,canbesummarizedthrough A=        1α2α3α4 α21α3α4 α3α31α4 α4α4α41        39
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Oncopulaparametrization •Extremevaluecopulas Here,dependencein[0,1]d ←→summarizedinonefunctionalparameterson [0,1]d−1 . Further,focusesonlyonfirstordertaildependence. 40
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Naturalpropertiesfordependencemeasures Definition4.κismeasureofconcordanceifandonlyifκsatisfies •κisdefinedforeverypair(X,Y)ofcontinuousrandomvariables, •−1≤κ(X,Y)≤+1,κ(X,X)=+1andκ(X,−X)=−1, •κ(X,Y)=κ(Y,X), •ifXandYareindependent,thenκ(X,Y)=0, •κ(−X,Y)=κ(X,−Y)=−κ(X,Y), •if(X1,Y1)�PQD(X2,Y2),thenκ(X1,Y1)≤κ(X2,Y2), •if(X1,Y1),(X2,Y2),...isasequenceofcontinuousrandomvectorsthat convergetoapair(X,Y)thenκ(Xn,Yn)→κ(X,Y)asn→∞. 41
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Naturalpropertiesfordependencemeasures Ifκismeasureofconcordance,then,iffandgarebothstrictlyincreasing,then κ(f(X),g(Y))=κ(X,Y).Further,κ(X,Y)=1ifY=f(X)withfalmostsurely strictlyincreasing,andanalogouslyκ(X,Y)=−1ifY=f(X)withfalmost surelystrictlydecreasing(seeScarsini(1984)). Rankcorrelationscanbeconsidered,i.e.Spearman’sρdefinedas ρ(X,Y)=corr(FX(X),FY(Y))=12 �1 0 �1 0 C(u,v)dudv−3 andKendall’sτdefinedas τ(X,Y)=4 �1 0 �1 0 C(u,v)dC(u,v)−1. 42
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Historicalversionofthosecoefficients SimilarlyKendall’stauwasnotdefinedusingcopulae,butastheprobabilityof concordance,minustheprobabilityofdiscordance,i.e. τ(X,Y)=3[P((X1−X2)(Y1−Y2)>0)−P((X1−X2)(Y1−Y2)<0)], where(X1,Y1)and(X2,Y2)denotetwoindependentversionsof(X,Y)(see Nelsen(1999)). Equivalently,τ(X,Y)=1− 4Q n(n2−1) whereQisthenumberofinversions betweentherankingsofXandY(numberofdiscordance). 43
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement !2.0!1.5!1.0!0.50.00.51.0 !0.50.00.51.01.5 ���������������� X Y !2.0!1.5!1.0!0.50.00.51.0 !0.50.00.51.01.5 ���������������� X Y Fig.7–Concordanceversusdiscordance. 44
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Alternativeexpressionsofthosecoefficients Notethatthosecoefficientscanalsobeexpressedasfollows ρ(X,Y)= � [0,1]×[0,1] C(u,v)−C⊥ (u,v)dudv � [0,1]×[0,1] C+(u,v)−C⊥(u,v)dudv (thenormalizedaveragedistancebetweenCandC⊥ ),forinstance. ThecaseoftheGaussianrandomvector If(X,Y)isaGaussianrandomvectorwithcorrelationr,then(Kruskal(1958)) ρ(X,Y)= 6 π arcsin �r 2 � andτ(X,Y)= 2 π arcsin(r). 45
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement FromKendall’tautocopulaparameters Kendall’sτ0.00.10.20.30.40.50.60.70.80.91.0 Gaussianθ0.000.160.310.450.590.710.810.890.950.991.00 Gumbelθ1.001.111.251.431.672.002.503.335.0010.0+∞ Plackettθ1.001.572.484.006.6011.421.144.1115530+∞ Claytonθ0.000.220.500.861.332.003.004.678.0018.0+∞ Frankθ0.000.911.862.924.165.747.9311.418.220.9+∞ Joeθ1.001.191.441.772.212.863.834.568.7714.4+∞ Galambosθ0.000.340.510.700.951.281.792.624.299.30+∞ Morgensteinθ0.000.450.90-------- 46
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement FromSpearman’srhotocopulaparameters Spearman’sρ0.00.10.20.30.40.50.60.70.80.91.0 Gaussianθ0.000.100.210.310.420.520.620.720.810.911.00 Gumbelθ1.001.071.161.261.381.541.752.072.583.73+∞ A.M.H.θ1.001.111.251.431.672.002.503.335.0010.0+∞ Plackettθ1.001.351.842.523.545.127.7612.724.266.1+∞ Claytonθ0.000.140.310.510.761.061.512.143.195.56+∞ Frankθ0.000.601.221.882.613.454.475.827.9012.2+∞ Joeθ1.001.121.271.461.691.992.393.004.036.37+∞ Galambosθ0.000.280.400.510.650.811.031.341.863.01+∞ Morgensteinθ0.000.300.600.90------- 47
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement ������������������ ������������������ ���������������� ���������������� !2024 !2024 Margesgaussiennes Fig.8–SimulationsofGumbel’scopulaθ=1.2. 48
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement ������������������ ������������������ ���������������� ����������������� !2024 !2024 Margesgaussiennes Fig.9–SimulationsoftheGaussiancopula(θ=0.95). 49
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement TailcorrelationandSolvencyII 50
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement TailcorrelationandSolvencyII 51
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Strongtaildependence Joe(1993)defined,inthebivariatecaseataildependencemeasure. Definition5.Let(X,Y)denotearandompair,theupperandlowertail dependenceparametersaredefined,ifthelimitexist,as λL=lim u→0 P � X≤F−1 X(u)|Y≤F−1 Y(u) � , =lim u→0 P(U≤u|V≤u)=lim u→0 C(u,u) u , and λU=lim u→1 P � X>F−1 X(u)|Y>F−1 Y(u) � =lim u→0 P(U>1−u|V≤1−u)=lim u→0 C� (u,u) u . 52
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Gaussiancopula 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 LandRconcentrationfunctions Lfunction(lowertails)Rfunction(uppertails) GAUSSIAN ● ● Fig.10–LandRcumulativecurves. 53
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Gumbelcopula 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 LandRconcentrationfunctions Lfunction(lowertails)Rfunction(uppertails) GUMBEL ● ● Fig.11–LandRcumulativecurves. 54
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Claytoncopula 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 LandRconcentrationfunctions Lfunction(lowertails)Rfunction(uppertails) CLAYTON ● ● Fig.12–LandRcumulativecurves. 55
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Studenttcopula 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 LandRconcentrationfunctions Lfunction(lowertails)Rfunction(uppertails) STUDENT(df=5) ● ● Fig.13–LandRcumulativecurves. 56
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Studenttcopula 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 LandRconcentrationfunctions Lfunction(lowertails)Rfunction(uppertails) STUDENT(df=3) ● ● Fig.14–LandRcumulativecurves. 57
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Estimationoftaildependence 58
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Estimating(strong)taildependence From P≈ P � X>F−1 X(u),Y>F−1 Y(u) � P � Y>F−1 Y(u) �foruclosedto1, asforHill’sestimator,anaturalestimatorforλisobtainedwithu=1−k/n, �λ (k) U= 1 n �n i=11(Xi>Xn−k:n,Yi>Yn−k:n) 1 n �n i=11(Yi>Yn−k:n) , hence �λ (k) U= 1 k n� i=1 1(Xi>Xn−k:n,Yi>Yn−k:n). �λ (k) L= 1 k n� i=1 1(Xi≤Xk:n,Yi≤Yk:n). 59
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Asymptoticconvergence,howfast? 0.00.20.40.60.81.0 0.00.20.40.60.81.0 (Upper)taildependence,Gaussiancopula,n=200 Exceedanceprobability 0.0010.0050.0500.500 0.00.20.40.60.81.0 Logscale,(lower)taildependence Exceedanceprobability(logscale) Fig.15–ConvergenceofLandRfunctions,Gaussiancopula,n=200. 60
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Asymptoticconvergence,howfast? 0.00.20.40.60.81.0 0.00.20.40.60.81.0 (Upper)taildependence,Gaussiancopula,n=200 Exceedanceprobability 0.0010.0050.0500.500 0.00.20.40.60.81.0 Logscale,(lower)taildependence Exceedanceprobability(logscale) Fig.16–ConvergenceofLandRfunctions,Gaussiancopula,n=2,000. 61
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Asymptoticconvergence,howfast? 0.00.20.40.60.81.0 0.00.20.40.60.81.0 (Upper)taildependence,Gaussiancopula,n=200 Exceedanceprobability 0.0010.0050.0500.500 0.00.20.40.60.81.0 Logscale,(lower)taildependence Exceedanceprobability(logscale) Fig.17–ConvergenceofLandRfunctions,Gaussiancopula,n=20,000. 62
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Weaktaildependence IfXandYareindependent(intails),forulargeenough P(X>F−1 X(u),Y>F−1 Y(u))=P(X>F−1 X(u))·P(Y>F−1 Y(u))=(1−u)2 , orequivalently,logP(X>F−1 X(u),Y>F−1 Y(u))=2·log(1−u).Further,ifX andYarecomonotonic(intails),forulargeenough P(X>F−1 X(u),Y>F−1 Y(u))=P(X>F−1 X(u))=(1−u)1 , orequivalently,logP(X>F−1 X(u),Y>F−1 Y(u))=1·log(1−u). =⇒limitoftheratio log(1−u) logP(Z1>F−1 1(u),Z2>F−1 2(u)) . 63
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Weaktaildependence Coles,Heffernan&Tawn(1999)defined Definition6.Let(X,Y)denotearandompair,theupperandlowertail dependenceparametersaredefined,ifthelimitexist,as ηL=lim u→0 log(u) logP(Z1≤F−1 1(u),Z2≤F−1 2(u)) =lim u→0 log(u) logC(u,u) , and ηU=lim u→1 log(1−u) logP(Z1>F−1 1(u),Z2>F−1 2(u)) =lim u→0 log(u) logC�(u,u) . 64
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Gaussiancopula 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 Chidependencefunctions lowertailsuppertails GAUSSIAN ● ● Fig.18–χfunctions. 65
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Gumbelcopula 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 Chidependencefunctions lowertailsuppertails GUMBEL ● ● Fig.19–χfunctions. 66
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Claytoncopula 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 Chidependencefunctions lowertailsuppertails CLAYTON ● ● Fig.20–χfunctions. 67
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Studenttcopula 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 0.00.20.40.60.81.0 Chidependencefunctions lowertailsuppertails STUDENT(df=3) ● ● Fig.21–χfunctions. 68
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Applicationinriskmanagement:Loss-ALAE 0.00.20.40.60.81.0 0.00.20.40.60.81.0 Loss AllocatedExpenses Fig.22–Lossesandallocatedexpenses. 69
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Applicationinriskmanagement:Loss-ALAE 0.00.20.40.60.81.0 0.00.20.40.60.81.0 LandRconcentrationfunctions Lfunction(lowertails)Rfunction(uppertails) Gumbelcopula ● ● 0.00.20.40.60.81.0 0.00.20.40.60.81.0 Chidependencefunctions lowertailsuppertails Gumbelcopula ● ● Fig.23–LandRcumulativecurves,andχfunctions. 70
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Applicationinriskmanagement:car-household 0.00.20.40.60.81.0 0.00.20.40.60.81.0 Carclaims Householdclaims Fig.24–MotorandHouseholdclaims. 71
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Applicationinriskmanagement:car-household 0.00.20.40.60.81.0 0.00.20.40.60.81.0 LandRconcentrationfunctions Lfunction(lowertails)Rfunction(uppertails) Gumbelcopula ● ● 0.00.20.40.60.81.0 0.00.20.40.60.81.0 Chidependencefunctions lowertailsuppertails Gumbelcopula ● ● Fig.25–LandRcumulativecurves,andχfunctions. 72
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement CaseofArchimedeancopulas ForanexhaustivestudyoftailbehaviorforArchimedeancopulas,see Charpentier&Segers(2008). •uppertail:functionofφ� (1)andθ1=−lim s→0 sφ� (1−s) φ(1−s) , ◦φ� (1)<0:tailindependence ◦φ� (1)=0andθ1=1:dependenceinindependence ◦φ� (1)=0andθ1>1:taildependence •lowertail:functionofφ(0)andθ0=−lim s→0 sφ� (s) φ(s) , ◦φ(0)<∞:tailindependence ◦φ(0)=∞andθ0=0:dependenceinindependence ◦φ(0)=∞andθ0>0:taildependence 0.00.20.40.60.81.0 05101520 73
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Measuringrisks? thepurepremiumasatechnicalbenchmark Pascal,Fermat,Condorcet,Huygens,d’AlembertintheXVIIIthcentury proposedtoevaluatethe“produitscalairedesprobabilit´esetdesgains”, <p,x>= n� i=1 pixi= n� i=1 P(X=xi)·xi=EP(X), basedonthe“r`egledesparties”. ForQu´etelet,theexpectedvaluewas,inthecontextofinsurance,thepricethat guaranteesafinancialequilibrium. Fromthisidea,weconsiderininsurancethepurepremiumasEP(X).Asin Cournot(1843),“l’esp´erancemath´ematiqueestdonclejusteprixdeschances” (orthe“fairprice”mentionedinFeller(1953)). Problem:SaintPeterburg’sparadox,i.e.infinitemeanrisks(cf.natural catastrophes) 74
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement thepurepremiumasatechnicalbenchmark ForapositiverandomvariableX,recallthatEP(X)= �∞ 0 P(X>x)dx. 0246810 0.00.20.40.60.81.0 Expectedvalue Lossvalue,X Probabilitylevel,P Fig.26–ExpectedvalueEP(X)= � xdFX(x)= � P(X>x)dx. 75
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement frompurepremiumtoexpectedutilityprinciple Ru(X)= � u(x)dP= � P(u(X)>x))dx whereu:[0,∞)→[0,∞)isautilityfunction. Examplewithanexponentialutility,u(x)=[1−e−αx ]/α, Ru(X)= 1 α log � EP(eαX ) � , i.e.theentropicriskmeasure. SeeCramer(1728),Bernoulli(1738),vonNeumann&Morgenstern (1944),Rochet(1994)...etc. 76
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Distortionofvaluesversusdistortionofprobabilities 0246810 0.00.20.40.60.81.0 Expectedutility(powerutilityfunction) Lossvalue,X Probabilitylevel,P Fig.27–Expectedutility � u(x)dFX(x). 77
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Distortionofvaluesversusdistortionofprobabilities 0246810 0.00.20.40.60.81.0 Expectedutility(powerutilityfunction) Lossvalue,X Probabilitylevel,P Fig.28–Expectedutility � u(x)dFX(x). 78
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement frompurepremiumtodistortedpremiums(Wang) Rg(X)= � xdg◦P= � g(P(X>x))dx whereg:[0,1]→[0,1]isadistortedfunction. Example •ifg(x)=I(X≥1−α)Rg(X)=VaR(X,α), •ifg(x)=min{x/(1−α),1}Rg(X)=TVaR(X,α)(alsocalledexpected shortfall),Rg(X)=EP(X|X>VaR(X,α)). SeeD’Alembert(1754),Schmeidler(1986,1989),Yaari(1987),Denneberg (1994)...etc. Remark:Rg(X)mightbedenotedEg◦P.Butitisnotanexpectedvaluesince Q=g◦Pisnotaprobabilitymeasure. 79
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Distortionofvaluesversusdistortionofprobabilities 0246810 0.00.20.40.60.81.0 Distortedpremiumbetadistortionfunction) Lossvalue,X Probabilitylevel,P Fig.29–Distortedprobabilities � g(P(X>x))dx. 80
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Distortionofvaluesversusdistortionofprobabilities 0246810 0.00.20.40.60.81.0 Distortedpremiumbetadistortionfunction) Lossvalue,X Probabilitylevel,P Fig.30–Distortedprobabilities � g(P(X>x))dx. 81
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement someparticularcasesaclassicalpremiums Theexponentialpremiumorentropymeasure:obtainedwhentheagent asanexponentialutilityfunction,i.e. πsuchthatU(ω−π)=EP(U(ω−S)),U(x)=−exp(−αx), i.e.π= 1 α logEP(eαX ). Esscher’stransform(seeEsscher(1936),B¨uhlmann(1980)), π=EQ(X)= EP(X·eαX ) EP(eαX) , forsomeα>0,i.e. dQ dP = eαX EP(eαX) . Wang’spremium(seeWang(2000)),extendingtheSharpratioconcept E(X)= �∞ 0 F(x)dxandπ= �∞ 0 Φ(Φ−1 (F(x))+λ)dx 82
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Riskmeasures ThetwomostcommonlyusedriskmeasuresforarandomvariableX(assuming thatalossispositive)are,q∈(0,1), •Value-at-Risk(VaR), VaRq(X)=inf{x∈R,P(X>x)≤α}, •ExpectedShortfall(ES),TailConditionalExpectation(TCE)orTail Value-at-Risk(TVaR) TVaRq(X)=E(X|X>VaRq(X)), Artzner,Delbaen,Eber&Heath(1999):agoodriskmeasureis subadditive, TVaRissubadditive,VaRisnotsubadditive(ingeneral). 83
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Riskmeasures:apratitionner(mis)understanding 84
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Riskmeasures:apratitionner(mis)understanding 85
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Riskmeasures:apratitionner(mis)understanding 86
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Riskmeasures:apratitionner(mis)understanding 87
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Riskmeasures:apratitionner(mis)understanding 88
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Riskmeasuresanddiversification AnycopulaCisboundedbyFrchet-Hoeffdingbounds, max �d� i=1 ui−(d−1),0 � ≤C(u1,...,ud)≤min{u1,...,ud}, andthus,anydistributionFonF(F1,...,Fd)isbounded max �d� i=1 Fi(xi)−(d−1),0 � ≤F(x1,...,xd)≤min{F1(x1),...,Ff(xd)}. Doesthismeansthecomonotonicityisalwaystheworst-casescenario? Givenarandompair(X,Y),let(X− ,Y− )and(X+ ,Y+ )denote contercomonotonicandcomonotonicversionsof(X,Y),dowehave R(φ(X− ,Y− )) ? ≤R(φ(X, Y) ) ? ≤R(φ(X+ ,Y+ )). 89
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Tchen’stheoremandboundingsomepurepremiums Ifφ:R2 →Rissupermodular,i.e. φ(x2,y2)−φ(x1,y2)−φ(x2,y1)+φ(x1,y1)≥0, foranyx1≤x2andy1≤y2,thenif(X,Y)∈F(FX,FY), E � φ(X− ,Y− ) � ≤E(φ(X,Y))≤E � φ(X+ ,Y+ ) � , asprovedinTchen(1981). Example7.thestoplosspremiumforthesumoftworisksE((X+Y−d)+)is supermodular. 90
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Example8.Forthen-yearjoint-lifeannuity, axy:n�= n� k=1 vk P(Tx>kandTy>k)= n� k=1 vk kpxy. Then a− xy:n�≤axy:n�≤a+ xy:n�, where a− xy:n�= n� k=1 vk max{kpx+kpy−1,0}(lowerFrchetbound), a+ xy:n�= n� k=1 vk min{kpx,kpy}(upperFrchetbound). 91
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Makarov’stheoremandboundingValue-at-Risk InthecasewhereRdenotestheValue-at-Risk(i.e.quantilefunctionoftheP&L distribution), R− ≤R(X− +Y− )�≤R(X+Y)�≤R(X+ +Y+ )≤R+ , wheree.g.R+ canexceedthecomonotoniccase.Recallthat R(X+Y)=VaRq[X+Y]=F−1 X+Y(q)=inf{x∈R|FX+Y(x)≥q}. Proposition9.Let(X,Y)∈F(FX,FY)thenforalls∈R, τC−(FX,FY)(s)≤P(X+Y≤s)≤ρC−(FX,FY)(s), where τC(FX,FY)(s)=sup x,y∈R {C(FX(x),FY(y)),x+y=s} and,if˜C(u,v)=u+v−C(u,v), ρC(FX,FY)(s)=inf x,y∈R {˜C(FX(x),FY(y)),x+y=s}. 92
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement 0.00.20.40.60.81.0 !4!2024 ���������������������������������� Sommede2risquesGaussiens Fig.31–Value-at-Riskfor2GaussianrisksN(0,1). 93
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement ������������������������ ������� ���������������������������������� ���������������������������� Fig.32–Value-at-Riskfor2GaussianrisksN(0,1). 94
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement ������������������ �������� ���������������������������������� ������������������������ Fig.33–Value-at-Riskfor2GammarisksG(3,1). 95
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement ������������������������ �������� ���������������������������������� ������������������������ Fig.34–Value-at-Riskfor2GammarisksG(3,1). 96
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Themorecorrelated,themorerisky? Willtheriskoftheportfolioincreasewithcorrelation? Recallthefollowingtheoreticalresult: Proposition10.AssumethatXandX� areinthesameFr´echetspace(i.e. Xi L =X� i),anddefine S=X1+···+XnandS� =X� 1+···+X� n. IfX�X� fortheconcordanceorder,thenS�TVaRS� forthestop-lossor TVaRorder. AconsequenceisthatifXandX� areexchangeable, corr(Xi,Xj)≤corr(X� i,X� j)=⇒TVaR(S,p)≤TVaR(S� ,p),forallp∈(0,1). SeeM¨uller&Stoyen(2002)forsomepossibleextensions. 97
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Themorecorrelated,themorerisky? Consider •dlinesofbusiness, •simplyabinomialdistributiononeachlineofbusiness,withsmallloss probability(e.g.π=1/1000). Let    1ifthereisaclaimonlinei 0ifnot ,andS=X1+···+Xd. WillthecorrelationamongtheXi’sincreasetheValue-at-RiskofS? Consideraprobitmodel,i.e.Xi=1(X� i≤ui),whereX� ∼N(0,Σ),i.e.a Gaussiancopula. AssumethatΣ=[σi,j]whereσi,j=ρ∈[−1,1]wheni�=j. 98
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Themorecorrelated,themorerisky? Fig.35–99.75%TVaR(orexpectedshortfall)forGaussiancopulas. 99
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Themorecorrelated,themorerisky? Fig.36–99%TVaR(orexpectedshortfall)forGaussiancopulas. 100
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Themorecorrelated,themorerisky? Whataboutotherriskmeasures,e.g.Value-at-Risk? corr(Xi,Xj)≤corr(X� i,X� j)�VaR(S,p)≤VaR(S� ,p),forallp∈(0,1). (seee.g.Mittnik&Yener(2008)). 101
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Themorecorrelated,themorerisky? Fig.37–99.75%VaRforGaussiancopulas. 102
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Themorecorrelated,themorerisky? Fig.38–99%VaRforGaussiancopulas. 103
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Themorecorrelated,themorerisky? Whatcouldbetheimpactoftaildependence? Previously,weconsideredaGaussiancopula,i.e.tailindependence.Whatifthere wastaildependence? ConsiderthecaseofaStudentt-copula,withνdegreesoffreedom. 104
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Themorecorrelated,themorerisky? Fig.39–99.75%TVaR(orexpectedshortfall)forStudentt-copulas. 105
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Themorecorrelated,themorerisky? Fig.40–99%TVaR(orexpectedshortfall)forStudentt-copulas. 106
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Themorecorrelated,themorerisky? Fig.41–99.75%VaRforStudentt-copulas. 107
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Themorecorrelated,themorerisky? Fig.42–99%VaRforStudentt-copulas. 108
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Themorecorrelated,themorerisky? 109
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement OntheCEIPSrecommendations 110
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement OntheCEIPSrecommendations 111
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement OntheCEIPSrecommendations 112
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement OntheCEIPSrecommendations 113
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement OntheCEIPSrecommendations 114
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement OntheCEIPSrecommendations 115
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Afirstconclusion 116
    • ArthurCHARPENTIER-Extremesandcorrelationinriskmanagement Anotherpossibleconclusion •(standard)correlationisdefinitivelynotanappropriatetooltodescribe dependencefeatures, ◦inordertofullydescribedependence,usecopulas, ◦sincemajorfocusinriskmanagementisrelatedtoextremalevent,focuson taildependencemeausres, •whichcopulacanbeappropriate? ◦Ellipticalcopulasofferaniceandsimpleparametrization,basedonpairwise comparison, ◦Archimedeancopulasmightbetoorestrictive,butpossibletointroduce HierarchicalArchimedeancopulas, •Value-at-Riskmightyieldtonon-intuitiveresults, ◦needtogetabetterunderstandingaboutValue-at-Riskpitfalls, ◦needtoconsideralternativedownsideriskmeasures(namelyTVaR). 117