DeanFantazzini
DepartmentofEconomics
ChairforEconomicsandEconometrics,UniversityofKonstanz(Germany)
Universitatsstrasse10,BoxD124,Zipcode78457
Email:Dean.Fantazzini@uni-konstanz.de
Abstract
Traditionalportfoliotheorybasedonmultivariatenormaldistributionassumesthatinvestorscanbenefitfromdiversificationbyinvestinginassetswithlowercorrelations.However,thisisnotwhathappensinreality,sinceitisquiteeasytoseefinancialmarketswithdifferentcorrelationsbutalmostthesamenumbersofmarketcrashes(ifwedefinemarketcrashaswhenreturnsareintheirlowestqthpercentile).Inasimilarfashion,recentempiricalstudiesshowthatinvolatileperiodsfinancialmarketstendtobecharacterizedbydifferentlevelofdependencethanoccursinquietperiods.Inordertotakeintoaccountthisreality,weresorttocopulatheoryanditsconditionaldependencemeasures,likeKendall’sTauandTaildependence.Theformersatisfiesmostofthedesiredpropertiesthatadependencemeasuremusthaveanditcandetectnon-linearassociationthatcorrelationcannotsee.Taildependencereferstothedependencethatarisesbetweenrandomvariablesfromextremeobservations.WeconsideraportfoliomadeupofthefivemostimportantfuturecontractsactuallytradedinAmericanmarketsandwetakeintoconsiderationthemostvolatileperiodofthelastdecade,thatisbetweenMarch13th2000untilJune9th2000.Weshowhowtheseconditionaldependentmeasurecanbeeasilyimplementedbothinthetraditionalmean-varianceframeworkandinmultivariateVaRestimation,withasignificantimprovementovertraditionalmultivariatecorrelationanalysis.
1Introduction
Traditionalportfoliotheorybasedonmultivariatenormaldistributionassumesthatinvestorscanbenefitfromdiversificationbyinvestinginassetswithlowercorrelations.Howeverthisisnotwhathappensinreality,sinceitisquiteeasytoseefinancialmarketswithdifferentcorrelationsbutalmostthesamenumbersofmarketcrashes(ifwedefinemarketcrashasaneventwhenreturnsareintheirlowestqthpercentile).
Correlationisagoodmeasureofdependenceinmultivariatenormaldistributionsbutithasseveralshortcomings:a)Thevariancesoftherandomvariablesmustbefiniteforthecorrelationtoexist,andforfat-taileddistributionsthiscannotbethecase(abivariatet-distributionwith2degreesoffreedom,forexample);b)Independencebetweentworandomvariablesimpliesthatlinearcorrelationiszero,buttheconverseistrueonlyforamultivariatenormaldistribution.ThisdoesnotholdwhenonlythemarginalsareGaussianwhilethejointdistributionisnotnormal,becausecorrelationreflectslinearassociationandnotnon-lineardependency;c)Correlationisnotinvarianttostrictlymonotonetransformations.Thisisbecauseitdependsnotonlyonthejointdistributionbutalsoonthemarginaldistributionsoftheconsideredvariables,sothatchangesofscalesorothertransformationsinthemarginalshaveaneffectoncorrelation.
1
Inordertoovercometheseproblemswecanresorttocopulatheory,sincecopulaecapturethosepropertiesofthejointdistributionwhichareinvariantunderstrictlyincreasingtransformation.AcommondependencemeasurethatcanbeexpressedasafunctionofcopulaparametersandisscaleinvariantisKendall’stau.Itsatisfiesmostofthedesiredpropertiesthatadependencemeasuremusthave(seeNelsen1999)anditmeasuresconcordancebetweentworandomvariables:concordancearisesiflargevaluesofonevariableareassociatedwithlargevaluesoftheother,andsmallonesoccurwithsmallvaluesoftheother;ifthisisnottruethetwovariablesaresaidtobediscordant.Itisforthisreasonthatconcordancecandetectnonlinearassociationthatcorrelationcannotsee.
Asassetlogreturndistributionsarenotnormallydistributed,theminimizationoftheportfolio’svariancedonotminimizeportfolioriskandproducethewrongcapitalallocation.Newriskmeasureshavebeenproposedtoobtainbettercapitalallocations,butatthecostofsimplicityandcomputa-tionaltractability:thisiswhymostappliedprofessionalsskipthemandprefertorelyonpreviousmethods,similartootherfinancialfields(justthinkbacktotheBlack&SholespricingformulaandGarch(1,1),whicharestillbyfarthemostusedmodelsforoptionpricingandvolatilityforecasting).
Inordertosatisfythisdemandforunderstandablemodels,weproposeheretouseKendall’staudependencemeasurewithinthetraditionalmean-varianceframework,intheplaceofthecorrelationcoefficients:thissolutionhastheadvantageofkeepingthemodeltractablebutatthesametimeconsideringthenon-lineardependencyamongtheconsideredvariables.
Inasimilarfashion,recentempiricalstudiesshowthatinvolatileperiodsfinancialmarketstendstobecharacterizedbydifferentlevelofdependencethanoccursinquietperiods.Inordertotakeintoaccountthisreality,weproposetousetheconceptofTaildependence,whichreferstothedependencethatarisesbetweenrandomvariablesfromextremeobservations.Animportantfeatureofcopulaeisthattheyallowfordifferentdegreesoftaildependence:Uppertaildependenceexistswhenthereisapositiveprobabilityofpositiveoutliersoccurringjointly,whilelowertaildependenceissymmetricallydefinedastheprobabilityofnegativeoutliersoccurringjointly.
WhatweproposeisadirectconsiderationofthisconceptinVaRmodelsbymeansofcopulatheory,astaildependencecoefficientscanbecalculatedassimplefunctionsofcopulaeparameters:ifwefollowthewell-knownRiskMetricsmultiplepositionsVaRmodel,taildependencecoefficientscanbeusedintheplaceoflinearcorrelationcoefficients.ThesameapproachcanbefollowedwithKendall’sTau,too.
WhatwedointhisworkistoanalyzethefivemostimportantfuturecontractsactuallytradedinAmericanmarkets(SP500,DowJones,Nasdaq100,EuroDollar,TBondNote)withhighfrequencydatasampledat5–minutesfrequency,takingintoconsiderationthemostvolatileperiodofthelastdecade,thatisbetweenMarch13th2000tillJune9th2000.Thisperiodoftimewitnessedthefallingofworldfinancialmarketsfollowingtheburstofthehigh-techbubble,withbigintradaydrawdownreturns.Thus,thissampleisperfectlysuitedtohighlighttheimportanceofthecopula-baseddependenceapproachcomparedtothetraditionalcorrelationanalysis.
WebuildupaportfolioandamultipleVaRpositionfollowingbothtwoapproaches,usingtheinitialpartofthesampletoestimatetheassets’weightsandthe95%(99%)VaR,andtheremainingparttocomparetheout-of-sampleperformancesofthetwoapproaches,bothintermsofrisk-returnsmeasuresandnumberofVaRexceedancesoftheeffectiveportfoliolosses.WeshowthatourapproachoutperformthecorrelationbasedonebothintermsofportfolioresultsandVaRback-testing.
Therestofthepaperisorganizedasfollows.InSection2weprovideanoutlineofcopulatheorywhileinSection3wepresentitsconditionaldependencemeasures,thatisKendall’sTauandTailDependence;inSection4weintroducethemainmethodsforcopulaestimationwhileinsection5weshowhowtousecopuladependencemeasurewithinportfoliomanagementandmultivariateVaRestimation.Section6presentstheempiricalresultsontheassetallocationproblemandVaRestimationforaportfoliooffivefuturecontractsactuallytradedinAmericanmarkets.WeconcludeinSection7.
2
2
2.1
CopulaTheory
UnconditionalCopulas
Ann-dimensionalcopulaisbasicallyamultivariatecumulativedistributionfunctionwithuniformdistributedmarginsin[0,1].LetconsiderX1,...Xntoberandomvariables,WtheconditioningvariableandHthejointdistributionfunction,andwefurtherassumethatthedistributionfunctionHhasalltherequiredderivatives,wehave:
Definition1(Unconditionalcopula):Theunconditionalcopulaof(X1,...Xn),whereX1∼F1,...Xn∼Fn,andF1...Fnarecontinuous,isthejointdistributionfunctionofU1≡F1(X1)...Un≡Fn(Xn).
ThevariablesU1...Unarethe‘probabilityintegraltransforms’ofX1...XnwhichfollowaUni-form(0,1)distribution,regardlessoftheoriginaldistribution,F.ThusacopulaisajointdistributionofUnif(0,1)randomvariables.
Proposition1(Propertiesofaunconditionalcopula):AcopulaisafunctionCofnvari-ablesontheunitn-cube[0,1]nwiththefollowingproperties:
1.2.3.4.
TherangeofC(u1,u2,...,un)istheunitinterval[0,1];C(u1,u2,...,un)=0ifanyui=0,fori=1,2,...,n.C(1,...,1,ui,1,...,1)=ui,forallui∈[0,1]
C(u1,u2,...,un)isn-increasinginthesensethatforeverya≤bin[0,1]nthemeasure∆CbaassignedbyCtothen-box[a,b]=[a1,b1]·...·[an,bn]isnonnegative,thatis
b
∆Ca=
(εi,...,εn)∈{0,1}n
n
(−1)i=1C(ε1a1+(1−ε1)b1,...,εnan+(1−εn)bn)≥0
εi
ThisdefinitionshowsthatCisamultivariatedistributionfunctionwithuniformlydistributed
margins.Copulaehavemanyusefulproperties,suchasuniformcontinuityand(almosteverywhere)existenceofallpartialderivatives,justtomentionafew(seeNelsen(1999),Theorem2.2.4andTheorem2.2.7).Moreover,itcanbeshownthateverycopulaisboundedbytheso-calledFr´echet-Hoeffdingbounds,
max(u1+...+un–n+1,0)≤
C(u1,u2,...,un)
≤min(u1,...,un)
whicharecommonlydenotedbyWandM.Intwodimensions,bothoftheFr´echet-Hoeffdingboundsarecopulasthemselves,butassoonasthedimensionincreases,theFr´echet-HoeffdinglowerboundWisnolongern-increasing.However,theinequalityontheleft-handsidecannotbeimproved,sinceforanyufromtheunitn-cube,thereexistsacopulaCusuchthatW(u)=Cu(u)(seeNelsen(1999),Theorem2.10.12).
WenowpresenttheSklar’stheorem,whichjustifiestheroleofcopulasasdependencefunctions:Theorem1:(Sklar’stheorem):LetHdenotean-dimensionaldistributionfunctionwithmarginsF1...Fn.Thenthereexistsan-copulaCsuchthatforallreal(x1,...,xn)
H(x1,...,xn)=C(F(x1),...,F(xn))
Ifallthemarginsarecontinuous,thenthecopulaisunique.Moreover,theconverseoftheabovestatementisalsotrueanditisthemostinterestingformultivariatedensitymodeling,sinceitimpliesthatwemaylinktogetheranyn≥2univariatedistributions,ofanytype(notnecessar-ilyfromthesamefamily),withanycopulainordertogetavalidbivariateormultivariatedistribution.
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Corollary1:LetF1,...,Fndenotethegeneralizedinversesofthemarginaldistributionfunctions,thenforevery(u1,...,un)intheunitn-cube,existsauniquecopulaC:[0,1]×...×[0,1]→[0,1]suchthat
C(u1,...,un)=H(F1
(−1)
(−1)(−1)
(u1),...,Fn
(−1)
(un))
SeeNelsen(1999)foraproof,Theorem2.10.9andthereferencesgiventherein.Fromthiscorollaryweknowthatgivenanytwomarginaldistributionsandanycopulawehaveajointdistri-bution.Acopulaisthusafunctionthat,whenappliedtounivariatemarginals,resultsinapropermultivariatepdf:sincethispdfembodiesalltheinformationabouttherandomvector,itcontainsalltheinformationaboutthedependencestructureofitscomponents.Usingcopulasinthiswaysplitsthedistributionofarandomvectorintoindividualcomponents(marginals)withadependencestructure(thecopula)amongthemwithoutlosinganyinformation.ItisimportanttohighlightthatthistheoremdoesnotrequireF1andFntobeidenticaloreventobelongtothesamedistributionfamily.
ByapplyingSklar’stheoremandusingtherelationbetweenthedistributionandthedensityfunction,wecanderivethemultivariatecopuladensityc(F1(x1),,...,Fn(xn)),associatedtoacopulafunctionC(F1(x1),,...,Fn(xn)):
nn∂n[C(F1(x1),...,Fn(xn))]·fi(xi)=c(F1(x1),...,Fn(xn))·fi(xi)f(x1,...,xn)=
∂F1(x1),...,∂Fn(xn)
i=1
i=1
where
c(F1(x1),...,Fn(xn))=
f(x1,...,xn)
·,nfi(xi)
i=1
(1)
Thecopuladensitywillbelaterusedtoestimateitsparameterstorealmarketdata.
2.2ConditionalCopulas
Timeseriesanalysisisofteninterestedinrandomvariablesconditionedonsomevariables,thisiswhywenowdiscusshowtheexistingresultsincopulatheorymaybeextendedtoallowforconditioningvariables.Weconsiderherethebivariatecasesinceitwillbelaterusedforempiricalanalysis.Wewillfurthermoreassumethatthedimensionoftheconditioningvariable,W,is1.
Definition2(Conditionalcopula):Theconditionalcopulaof(X1,X2)|W,whereX1|W∼F1andX2|W∼F2,istheconditionaljointdistributionfunctionofU1≡F1(X|W)andU2≡F2(Y|W)givenW.
Atwo-dimensionalconditionalcopulaisderivedfromanydistributionfunctionsuchthattheconditionaljointdistributionofthefirsttwovariablesgiventheremainingvariablesisacopulaforallvaluesoftheconditioningvariables.Itissimpletoshowthatithasthefollowingproperties:Proposition1(Propertiesofaconditionalcopula):Atwo-dimensionalconditionalcop-ulahasthefollowingproperties:
1.2.3.4.
ItisafunctionC:[0,1]×[0,1]×W→[0,1]
C(u1,0|w)=C(0,u2|w)=0,foreveryu1,u2in[0,1]andeachw∈WC(u1,1|w)=u1andC(1,u2|w)=u2,foreveryu1,u2in[0,1]andeachw∈C(u1,u2|w)isgroundedandn-increasing.
W
4
FortheproofseePatton(2003).Itiseasytoobservethatinthiscaseatwo-dimensionalcondi-tionalcopulaistheconditionaljointdistributionoftwoconditionallyUniform(0,1)randomvariables.Theorem2(Sklar’stheoremforcontinuousconditionaldistributions):LetF1betheconditionaldistributionofX1|W,F1betheconditionaldistributionofF1|W,andHbethejointconditionaldistributionof(X1,X2)|W.AssumethatF1andF2arecontinuousinx1andx2.ThenthereexistsauniqueconditionalcopulaCsuchthat
H(x1,x2|w)=C(F1(x1|w),F2(x2|w)|w),
Conversely,ifweletF1betheconditionaldistributionofX1|W,F2betheconditionaldistri-butionofX2|W,andCbeaconditionalcopula,thenthefunctionHdefinedbyequation(1.3)isaconditionalbivariatedistributionfunctionwithconditionalmarginaldistributionsFandG(Patton,2003).
TheSklar’stheoremforconditionaldistributionsimpliesthattheconditioningvariable(s),W,mustbethesameforbothmarginaldistributionsandthecopula:ifwedonotusethesameconditioningvariableforF,GandC,thefunctionHwillnotbe,ingeneral,ajointconditional
ˆwillbethejointdistributionof(X1,X2)|(W1,W2)isdistributionfunction.TheonlycasewhenH
whenF(x1|W1)=F(x1|W1,W2)andF(x2,|W2)=F(x2,|W1,W2),thatiswhensomevariablesaffecttheconditionaldistributionofonevariablebutnottheother.
ItisstraightforwardtoseethatwecanderiveanequivalenttoCorollary1fortheconditionalcase,sothatitispossibletoextracttheimpliedconditionalcopulafromanybivariateconditionaldistribution.
2.3EllipticalCopulas
Theclassofellipticaldistributionsprovidesusefulexamplesofmultivariatedistributionsbecausetheysharemanyofthetractablepropertiesofthemultivariatenormaldistribution.Furthermore,theyallowtomodelmultivariateextremeeventsandformsofnon-normaldependencies.Ellipticalcopulasaresimplythecopulasofellipticaldistributions.IfwefollowFang,KotzandNg(1987),wehavethefollowingdefinition:
Definition3(Ellipticaldistribution):letXbean-dimensionalvectorofrandomvariablesand,forsomeµ∈nandsomenxnnonnegativedefinitesymmetricmatrixΣ,thecharacteristicfunctionϕX−µisafunctionoftheofthequadraticformtTΣt,thenXhasanellipticaldistributionwithparameters(µ,Σ,ϕ)andwewriteX∼En(µ,Σ,ϕ).
Wepresentnowtwocopulaebelongingtotheellipticalfamilyandthatwillbelaterusedinempiricalapplications,theGaussianandT-copula.2.3.1
Gaussiancopula
TheGaussiancopulaisthecopulaofthemultivariatenormaldistribution.Infact,therandomvectorX=(X1,...,Xn)ismultivariatenormaliftheunivariatemarginsF1,...,FnareGaussians,andthedependencestructureamongthemarginsisdescribedbythefollowingcopulafunction
C(u1,...,un;Σ)=ΦK(Φ−1(u1),...,Φ−1(un);Σ)
(2)
whereΦkisthestandardmultivariatenormaldistributionfunctionwithlinearcorrelationmatrixΣandΦ−1istheinverseofthestandardunivariateGaussian.Whenn=2,expression(??)canbewrittenas:
−1(u)Φ−1(u)Φ
C(u1,u2;ρ)=
−∞
−∞
2π
exp
(1−ρ2)5
1
−(r2−2ρrs+s2)
2(1−ρ2)
drds,
whereρisthelinearcorrelationcoefficientbetweenthetworandomvariables.
Thecopuladensityisderivedbyapplyingequation(??):
1−1exp−xΣxn/22(x1,...,xn)f11(2π)|Σ|
c(Φ(x1),...,Φ(xn))=exp−ζ(Σ−1−I)ζ==nn1/22|Σ|√1exp−1x2fiGaussian(xi)i22πGaussian
11/2i=1i=1
whereζ=(Φ−1(u1),...,Φ−1(uK))isthevectoroftheGaussianunivariateinversedistributionfunc-tions,andui=Φ(xi).2.3.2
T-copula
Thecopulaofthemultivariatestandardizedt-Studentdistributionisthet-Studentcopulaandisdefinedasfollows:
11
C(u1,...,un;Σ,ν)=TR,ν(t−u1),...,t−un))(3)ν(ˆν(ˆwhereTΣ,νisthestandardizedmultivariateStudent’stdistributionfunction,Σisthecorrelation
1(u)denotestheinverseoftheStudent’stcumulativematrix,νarethedegreesoffreedom,t−ν
distributionfunction.Whenn=2,expression(??)canbewrittenas:
C
Studentst
(u,v,ρ)=
1−1t−ν(u)tν(v)
−∞−∞
√122π(1−ρ)exp1+
r2−2ρrs+s2)2(1−ρ2)
drds,
whereρisthelinearcorrelationcoefficientbetweenthetworandomvariables.Thecopuladensityisagainderivedbyapplyingequation(??):
−υ+n2υ+nυnζΣ−1ζ1+υΓ2Γ2υ+1υ+1,n−2ΓυΓ22ζi21+2i=1
c(t(υx1),
...,
t(υxn))
fStudent(x1,...,xn)
==|Σ|−1/2nfiStudent(xi)
i=1
(4)
1−1whereζ=(t−υ(u1),...,tυ(uK))isthevectoroftheT-studentunivariateinversedistributionfunc-tionsand.
2.4ArchimedeanCopulas
Archimedeancopulaeprovideanalyticaltractabilityandalargespectrumofdifferentdependence
measure.Thesecopulaecanbeusedinawiderangeofapplicationsforthefollowingreasons:a)Theeasewithwhichtheycanbeconstructed;b)Themanyparametricfamiliesofcopulasbelongingtothisclass;c)Thegreatvarietyofdifferentdependencestructures;d)Thenicepropertiespossessedbythemembersofthisclass.AnArchimedeancopulacanbedefinedasfollows:
Definition4(Archimedeancopula):letconsiderafunctionϕ:[0;1]→[0;1]whichiscontinuous,strictlydecreasingϕ’(u)<0,convexϕ”(u)>0,andforwhichϕ(0)=∞andϕ(1)=0.Wethendefinethepseudoinverseofϕ[−1]:[0;∞]→[0;1]suchthat:
−1
(t)for0≤t≤ϕ(0)ϕ
ϕ[−1](t)=
0forϕ(0)≤t≤∞
Asϕisconvex,thefunctionC:[0;1]2→[0;1]definedas
C(u1,u2)=ϕ−1[ϕ(u1)+ϕ(u2)]
(5)
isanArchimedeancopulaandϕiscalledthe“generator”ofthecopula.Moreover,ifϕ(0)=∞,
thepseudoinversedescribesanordinaryinversefunction(thatisϕ[−1]=ϕ(−1))andwecallϕand
6
C,astrictgeneratorandastrictArchimedeancopula,respectively.
ThemultivariateextensioncanbefoundinEmbrechtrs,LindskogandMcNeil(2001)aswellasinJoe(1997):foralln≥2,thefunctionC:[0;1]n→[0;1]definedasC(u1,...,un)=ϕ−1[ϕ(u1)+...+ϕ(un)],isann-dimensionalArchimedeancopulaifandonlyifϕ−1iscompletelymonotoneon[0,∞).
WepresentnowsomeofthemostimportantmultivariateArchimedeancopulas.1)Clayton(orCookJohnson)copula:itcorrespondstofamilyB4ofJoe(1997).Letconsiderthegeneratorϕ(t)=(t−α-1)/α,withα∈[-1,∞)\\{0}andinverseϕ−1(t)=(1+t)–1/α.Byusing(??)weget,
⎡
C(u1,...,un)=max⎣
N
−1/α
α
u−j−n+1
⎤,0⎦
j=1
However,ifα¿0thenwehaveϕ(0)=∞,andtheaboveexpressionbecome
C(u1,...,un)=
N
−1/α
αu−j−n+1
j=1
t
)α
2)Gumbelcopula:itcorrespondstofamilyB6ofJoe(1997).Thegeneratorisϕ(t)=(-ln
,withα≥1andtheinverseisϕ−1(t)=exp{-t1/α}.Wehave,
1/αααC(u1,...,un)=exp−[(−lnu1)+...+(−lnuN)]
3)Survival(orrotated)Gumbelcopula:Therotationallowsustotakeacopulaexhibit-inggreaterdependenceinthenegative(positive)quadrantandcreateonewithgreaterdependence
inthepositive(negative)quadrant.Thedistributionfunctioninthiscaseis
C(u1,...,un)=
N
j=1
1/αααuj−n+1+exp−[(−ln(1−u1)+...+(−ln(1−uN)]
3DependenceMeasures
Traditionalportfoliotheorybasedonmultivariatenormaldistributionassumesthatinvestorscan
benefitfromdiversificationbyinvestinginassetswithlowercorrelations;however,correlationisagoodmeasureofdependenceinmultivariatenormaldistributionsbutitpresentsseveralshortcomings:a)ThevariancesofX1andX2mustbefiniteforthecorrelationtoexist,andforfat-taileddistribu-tionsthiscannotbethecase(abivariatet-distributionwith2degreesoffreedom,forexample);b)Independencebetweentworandomvariablesimpliesthatlinearcorrelationiszero,buttheconverseistrueonlyforamultivariatenormaldistribution.ThisdoesnotholdwhenonlythemarginalsareGaussianwhilethejointdistributionisnotnormal,becausecorrelationreflectslinearassociationandnotnon-lineardependency;c)Correlationisnotinvarianttostrictlymonotonetransformations.Thisisbecauseitdependsnotonlyonthejointdistributionbutalsoonthemarginaldistributionsoftheconsideredvariables,sothatchangesofscalesorothertransformationsinthemarginalshaveaneffectoncorrelation.d)GiventhemarginaldistributionsF1andF2fortworandomvariablesXandY,alllinearcorrelationsbetween–1and+1cannotbeattainedthroughsuitablespecificationofthejointdistributionF(seeH¨offding,1940).
Moreover,therearealsostatisticalproblemswithcorrelation,asasingleobservationcanhaveanarbitrarilyhighinfluenceonthelinearcorrelation.Thuslinearcorrelationisnotarobustmeasure.
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3.1Kendall’sTau
Inordertoavoidtheaboveproblems,wehavetoturntorankcorrelation;however,wefirstneedtodefinetheconceptsofconcordanceanddiscordance:
Definition5(Concordance):Observations(xi;yi)and(xj;yj)areconcordant,ifxi Concordancearisesiflargevaluesofonevariableareassociatedwithlargevaluesoftheother,andsmallonesoccurwithsmallvaluesoftheother;ifthisisnottruethetwovariablesaresaidtobediscordant.Itisforthisreasonthatconcordancecandetectnonlinearassociationthatcorrelationcannotsee. Ifwehave(X1;Y1)and(X2;Y2)independentandidenticallydistributedrandomvectors,thepopulationversionofKendall’stauτ(X;Y)rankcorrelationisdefinedby τ(X,Y)=P[(X1−X2)(Y1−Y2)>0]−P[(X1−X2)(Y1−Y2)<0] thatis,τ(X,Y)isanestimateoftheprobabilityofconcordanceminustheprobabilityofdiscor-dance.ThesampleversionofthemeasureofdependenceknownasKendall’stauisdefinedintermsofconcordanceasfollows: Definition6(Kendall’sTau-sampleversion):Letusdenotearandomsampleofnobservations{(X1,Y1);(X2,Y2),....,(Xn;Yn)}fromavector(X,Y)ofcontinuousrandomvariables.Therearedistinctpairs(Xi;Yi)and(Xj;Yj)ofobservationsinthesample,andeachpairiseitherconcordantordiscordant;ifwecallcthenumberofconcordantpairs,anddthenumberofdiscordantpairs,thenanestimateofKendall’srankcorrelationforthesampleisgivenby n−d=(c−d)/,whichcanbealternativelyexpressedasfollowsτ=cc+d2 τˆ= n 2−1 i τ(X,Y)canbeconsideredameasureofthedegreeofmonotonicdependencebetweenXandY,whereaslinearcorrelationmeasuresthedegreeoflineardependenceonly. ThegeneralisationofKendall’sTauton¿2dimensionsisanalogoustotheprocedureforlinearcorrelation,wherewehaveanxnmatrixofpairwisecorrelation.Themainpropertiesofthisdependencemeasurearereportedhere: Theorem3(Kendall’sTauproperties):LetXandYberandomvariableswithcontinuousdistributionsF1andF2,jointdistributionHandcopulaC.Thefollowingaretrue: (1)τ(X,Y)=τ(Y,X). (2)IfXandYareindependentthenτ(X,Y)=0.(3)–1≤τ(X,Y)≤+1 (4)ForT:R→RstrictlymonotonicontherangeofX,τ(X,Y)satisfies τ(X,Y)ifTincreasing τ(T(X),Y)= −τ(X,Y)ifTdecreasingProof:(1)to(3)followsfromthefactthatτ(X,Y)isameasureofconcordance,whilefor(4)seeEmbrechts,McNeilandStraumann(1999)andreferencestherein. Kendall’sTaucanbeexpressedintermsofcopulafunctions,thussimplifyingcalculus. 8 Proposition2(Kendall’sTauforcopulae):τ(X,Y)=4Proof:SeeNelsen(1999). 11 0 0 C(u,v)dC(u,v)−1 EvaluatingKendall’sTaurequirestheevaluationofadoubleintegralandforellipticaldistri-butionsthisisnotaneasytask:thisproblemwassolvedbyLindskog,McNeil,andSchmock(2002)whoprovedthistheorem: Proposition3(Kendall’sTauforellipticaldistributions):LetX∼En(µ,Σ,ϕ).Ifi,j∈{1,...,p}satisfyP{Xi=µi}¡1andP{Xj=µj}¡1,then 2 arcsinρij(P{Xi=x})2(6)τ(Xi,Xj)=1− π x∈R wherethesumextendsoverallatomsofthedistributionofXiandρijisthecorrelationcoefficient. Ifinadditionrank(Σ)≥2,then(??)simplifiesto2 τ(Xi,Xj)=1−(P{Xi=µi})2arcsinρij π whichfurthersimplifiesifP{Xi=µi}=0, 2 arcsinρxyπ Proof:SeeLindskog,McNeil,andSchmock(2002) τ(X,Y)= (7) ForanArchimedeancopulathesituationissimpler,inthatτ(X,Y)canbeevaluateddirectlyfromthegeneratorofthecopula.Archimedeancopulasareeasytoworkwithbecauseexpressionswithafunctionofonlyoneargument(thegenerator)canoftenbeemployedratherthanexpressionswithafunctionoftwoarguments(thecopula). Proposition4(Kendall’sTauforArchimedeanCopulae):LetXandYberandomvariableswithanArchimedeancopulaChavinggeneratorϕ.Thepopulationversionofτ(X,Y)forXandYisgivenby τ(X,Y)=1+4 01 ϕ(t) dtϕ(t) Proof:SeeGenestandMacKay(1986) StraightforwardcalculationsshowthatforClaytonandGumbelcopulaewehavethefollowingresults: Clayton:τ(X,Y)=α/(α+2) Gumbel=Rotatedgumbel=τ(X,Y)=1-α−1 3.2TailDependence Aswepreviouslysaw,linearcorrelationpresentmanyproblems,mainlywhenweworkwithheavy-taileddistributions.CopulasfunctionspresentthenicepropertytobeabletomodelTaildependence:thismeasurereferstothedependencethatarisesbetweenrandomvariablesfromextremeobserva-tions;itiseasytoseethatinvolatileperiodsfinancialmarketstendstobecharacterizedbydifferentlevelofdependencethanoccursinquiteperiods.Animportantfeatureofcopulaeisthattheyallow 9 fordifferentdegreesoftaildependence:Uppertaildependenceexistswhenthereisapositiveprob-abilityofpositiveoutliersoccurringjointly,whilelowertaildependenceissymmetricallydefinedastheprobabilityofnegativeoutliersoccurringjointly. Definition7(Uppertaildependence):Let(X1,X2)beabivariatevectorofcontinu-ousrandomvariableswithmarginaldistributionfunctionsF1andF2,thecoefficientofuppertaildependenceofXandYis: −1−1−1−1(u)|X1>F1(u)]=limPr[X1>F1(u)|X2>F2(u)]λU=limPr[X2>F2 u→1 u→1 =lim −1−1 (u),X1>F1(u)]Pr[X2>F2−1Pr[X1>F1(u)]u→1 =lim −1−1−1−1 (u)]−Pr[X2≤F2(u)]+Pr[X2≤F2(u),X1≤F1(u)]1−Pr[X1≤F1−11−Pr[X1≤F1(u)]u→1 =lim (1−2u+C(u,u)) (1−u)u→1 =λU providedalimitλU∈[0,1]exists.Here,F−1(u)=inf{x|F(x)≥u},u∈(0,1). X1andX2aresaidtobeasymptoticallydependentintheuppertailifλU∈(0,1]andasymp-toticallyindependentifλU=0.Uppertaildependenceexistswhenthereisapositiveprobabilityofpositiveoutliersoccurringjointly. Definition8(Lowertaildependence):Let(X1,X2)beabivariatevectorofcontinu-ousrandomvariableswithmarginaldistributionfunctionsF1andF2,thecoefficientoflowertaildependenceofX1andX2is: −1−1−1−1(u)|X1≤F1(u)]=limPr[X1≤F1(u)|X2≤F2(u)]=limλL=limPr[X2≤F2 u→0 u→0 C(u,u) =λL u→0u providedalimitλL∈[0,1]exist.X1andX2aresaidtobeasymptoticallydependentinthelowertail ifλL∈(0,1]andasymptoticallyindependentifλL=0.Lowertaildependenceexistswhenthereisapositiveprobabilityofnegativeoutliersoccurringjointly.3.2.1 TaildependenceforEllipticalcopulae. Whendealingwithcopulaewithoutclosedformexpressions,suchastheGaussianandtheStudent’stcopula,analternativeversionisprovidedbyEmbrechts,LindskogandMcNeil(2003).FirstbyapplyingDel‘Hopitaltheorem,weobtain: ∂∂ λU=lim(1-2u+C(u,u))/(1-u)=−lim−2+C(x1,x2)|x1=x2=u+C(x1,x2)|x1=x2=v u→1∂x1∂x2u→1 Then,bythedefinitionofcopulafunctionandbytheexpressionofthecopuladensity,wehave: ∂∂C(u,v),P(U≤u|V=v)=∂vC(u,v)P(V≤v|U=u)=∂u∂∂P(V>v|U=u)=1−∂uC(u,v),P(U>u|V=v)=1−∂vC(u,v) Rearrangingtheexpressionaboveweobtain: λU=lim[P(V>u|U=u)]+lim[P(U>u|V=u)] u→1u→1 Inthecaseofexchangeablecopulas(i.e.C(u,v)=C(v,u))wecansimplifyasfollows: λU=2lim[P(V>u|U=u)] u→1 10 Finally,byusingtheprobabilityintegralproblem,inthecaseofadistributionfunctionFwithinfiniterightendpoint(suchastheGaussianoftheStudent’stdistribution)weget: λU=2lim[P(F−1(V)>u|F−1(U)=u)]=2lim[P(Y>u|X=u)] u→∞u→∞ whereweusedSklar’sTheorem(X1,X2)∼C(F(x1),F(x2)). LetapplytheseresultstoGaussianandT-Copula: 1)GaussianCopula.Usingstandardresultinthestatistictheory,wehave λU=2lim[P(X2>x|X1=x)]= x→∞ √ −ρx−ρx 1−Φ√2=2lim1−Φx√1=2lim=01+ρ1−ρx→∞x→∞ GiventheradialsymmetrypropertyoftheGaussiandistribution,wecanconcludethatalso thelowertaildependencecoefficientisnull,soconfirmingtheasymptoticindependenceintailoftheGaussiancopula. 2)Student’stCopula.Itcanbeshownthatinthiscasewehave(seeEmbrechts,McNeil,Straumann,1999): λU=2lim[P(X2>x|X1=x)]= x→∞ √√1−ρ=2-2tν+1ν+1·√1+ρ=λL whichisincreasinginρanddecreasinginν.Asthenumberofdegreesoffreedomgoestoinfinity,λU tendsto0forρ<1.ThiscanbeexplainedbyconsideringtheasymptoticbehaviorofaStudent’stdistributedrandomvariablewhichconvergesindistributiontoastandardnormalvariate.3.2.2 TaildependenceforArchimedeancopulae. 1 ForcopulasoftheArchimedeanclass,theformulasforthecoefficientoftaildependencearethefollowing: ∂C(u,u)∂C(u,u) ,λL=lim u→1u→0∂u∂u WereportherethetaildependencecoefficientsfortheGumbelandrotatedGumbelcopulathatwewilluselaterforempiricalapplications: λU=2−lim 1)GumbelCopula:Thiscopulahasonlypositiveuppertaildependence,whichmeanstheprobabilitythatbothvariablesareintheirrighttailsispositive.RememberingthatCGumbel(u1,u2)=exp{-[(-lnu1)α+(-lnu2)α]1/α},thetaildependencecoefficientsare: λU=2–2 1/α λL=0,whereαisthecopulaparameter 2)Survival(orrotatedGumbel):AGumbelSurvivalcopulaisitsmirrorimageandhaspositivelowertaildependence:theprobabilitythatbothvariablesareintheirlefttailsispositive.Needlesstosay,theGumbelSurvivalcopulaisfarmoreinterestingforfinancialmodeling.Inthiscasewehave, λU=0λL=2–2 1 1/β ,whereβisthecopulaparameter ForaformalproofseeCasellaandBerger(2002) 11 4EstimationProcedures WewillconsiderarandomsamplerepresentedbythetimeseriesX=(x1t,x2t,...xNt),t=1...T,whereNstandsforthenumberofunderlyingassetsincludedandTrepresentsthenumberofobser-vationsavailable. 4.1Exactmaximumlikelihoodmethod(oronestagemethod) LetfbethedensityofthejointdistributionF: f(x;α1,...,·αn;θ)=c(F1(x1;α1),...,F1(xn;αn))· ni=1 fi(xi;αi) (8) wherefiistheunivariatedensityofthemarginaldistributionFiandcisthedensityofthecopula givenbythefollowingexpression: ∂nC(u1,u2,...,un;θ) c(u1,...,u2;θ)= ∂u1∂u2...∂un WesupposetohaveasetofTempiricaldataofnfinancialassetlog-returns,andletΘ=(α1,...,αn;θ)betheparametervectortoestimate,whereαi,i=1,...,nisthevectorofparametersofthemarginaldistributionFiandθisthevectorofthecopulaparameters.Thelog-likelihoodfunctionisthefollowing: l(Θ)= Tt=1 log(c(F1(x1,t;α1),...,Fn(xn,t;αn);θ))+ Tnt=1i=1 logfi(xi,t;αi,t) (9) ˆtheparametervectoristheonewhichmaximize(??):Θˆ=argmaxl(Θ)ˆTheMLestimatorΘof ˆMLbethemaximumlikelihoodestimator.ThenitverifiesthepropertyofasymptoticLetθ normalityandwehave(Durrelmanetal.2000): √ ˆML−θ0)→N(0,I−1(θ0))T(θ withI(θ0)theinformationmatrixofFischer. 4.2TheInferenceFunctionsforMarginsmethod(IFM–ortwostagemethod) AccordingtotheIFMmethod,theparametersofthemarginaldistributionsareestimatedseparately fromtheparametersofthecopula.Inotherwords,theestimationprocessisdividedintothefollowingtwosteps: estimatingtheparametersαi,i=1,...,nofthemarginaldistributionsFiusingtheMLmethod: αˆi=argmaxl(αi)=argmax i Tt=1 logfi(xi,t;αi) whereliisthelog-likelihoodfunctionofthemarginaldistributionFi; estimatingthecopulaparametersθ,giventheestimationsperformedinstep1): ˆ=argmaxlc(θ)=argmaxθ Tt=1 log(c(F1(x1,t;αˆ1),...,Fn(xn,t;αˆn);θ)) (10) wherelcisthelog-likelihoodfunctionofthecopula.LiketheMLestimatoritverifiestheproperties ofasymptoticnormality(JoeandXu,1996): √ ˆIFM−θ0)→N(0,V−1(θ0))T(θ 12 whereV(θ0)idtheInformationmatrixofGodambe.Ifwedefinethescorefunctioninthefollowingwayg(θ)=(∂α1l1,...,∂αNlN,∂θlc),theGodambeinformationmatrixtakestheform(Joe,1997): V(θ0)=D−1M(D−1)T whereD=E[∂g(θ)T/∂θ]andM=E[g(θ)Tg(θ)].NotethattheIFMmethodcouldbeviewedasaspecialcaseoftheGMMwithanidentityweightmatrix. 4.3TheCanonicalMaximumLikelihood(CML)method TheCMLmethoddiffersfromtheIFLmethodbecausenoassumptionsaremadeaboutthepara-metricformofthemarginaldistributions.Theestimationprocessisperformedintwosteps:u1t,uˆ2t,...,uˆTt)1.Transformingthedataset(x1t,x2t...xNt),t=1...Tintouniformvariates(ˆ ˆn(·)definedasfollows:usingtheempiricaldistributionsF T1ˆn(·)=F1{Xnt≤ T t=1 •) (11) where1 {X≤•}represent theindicatorfunction. 2.Estimatingthecopulaparametersasfollows: Tt=1 ˆ=argmaxθ log(c(ˆu1,t,...,uˆn,t);θ) (12) ThepreviousGodambeinformationmatrixorthesemi-parametriccovariancematrixbyGen-est,GhoudiandRivest(1995)canbeusedforinferencepurposes:ifthemarginaldistributionsare correctlyspecified,thetwoapproachesgivethesameresults. Remark1:CMListhebestestimator,becausetherearenoassumptionsonthemargins.Ifweusewrongmargins,MLEandIFMwill‘modify’thedependencefunction.LookatthisniceexampletakenbyRoncalli(2000):BivariatedistributionFwithNormalcopula(ρ=0,5)andtwot-studentmargins(F1=t1andF2=t2).IfwefitthedistributionwithaNormalcopulaandtwoGaussianmargins,weget: Figure1:CMLvsIFMandML 13 5 5.1 PortfoliomanagementandVaRapplications Portfoliomanagement Introduction 5.1.1 Themean-varianceapproachforportfolioselectionfirstproposedbyMarkowitz(1952)isveryintu-itiveand,duetoitssimplicity,isbyfarthemostusedinthefinancialsector.Theadvantageofusingthevariancefordescribingportfolioriskisprincipallyduetothesimplicityofthecomputation,butfromthepointofviewofriskmeasurementthevarianceisnotasatisfactorymeasure.First,thevarianceisasymmetricmeasureandconsidergainsandlossesinthesameway.Second,thevarianceisinappropriatetodescribetheriskoflowprobabilityevents,asforexamplethedefaultrisk.Finally,mean-variancedecisionsareusuallynotconsistentwiththeexpectedutilityapproach,unlessreturnsarenormallydistributedoraquadraticutilityindexischosen. AsalreadysuggestedbyMarkowitz(1959),otherriskmeasurescanbeusedinthemean-riskapproach,suchasvalue-at-risk(VaR)andexpected-shortfall(ES).However,itispossibletoshowthatundertheassumptionthatreturnsarenormallydistributed,theefficientfrontiersresultingfromthemean-VaRandfromthemean-ESoptimizationaresubsetsofthemean-varianceefficientfrontier.TheequivalenceoftheseoptimizationproblemsundermultivariatenormaldistributionwasfirststatedbyRockafellarandUryasev(1999,Proposition4.1).Leippold(2001)and,foramoregeneralframework,Leippold,Vanini,andTroiani(2002)consideredtheimpactofvalue-at-riskandexpected-shortfalllimitsonthemean-varianceportfolioallocationandhasshownformultivariateGaussianreturnsthatVaRandESconstraintsreducesthemean-variancesetofefficientportfolioallocations.However,whenarisk-freeassetisavailable,thesetofefficientportfoliosresultingfrommean-VaRorfrommean-ESportfolioselectionareidenticaltothemean-varianceefficientfrontier.Finally,H¨urlimann(2002)provedtheequivalenceofmean-ESandmean-varianceanalysisforamoregeneralclassofdistributionfunctionforthereturns,thatistheclassofellipticaldistributions. Forthesereasons,wedecidetorelyonthetraditionalmean-varianceapproach.However,wepresenthereapossiblevariationbyusingKendall’sTauinsteadoflinearcorrelation,inordertoimproveitsperformancewhendealingwithassetsnotellipticallydistributed,thatistheusualcasewhenweareworkwithhighfrequencydatasets. Moreover,aswe’veseeninsection3,linearcorrelationisagoodmeasureofdependenceinmultivariatenormaldistributionsbutitpresentsfourimportantshortcomings:a)Thevariancesoftherandomvariablesmustbefiniteforthecorrelationtoexist;b)Independencebetweentworandomvariablesimpliesthatlinearcorrelationiszero,buttheconverseistrueonlyforamultivariatenormaldistribution;c)Correlationisnotinvarianttostrictlymonotonetransformations.d)GiventhemarginaldistributionsF1andF2fortworandomvariablesXandY,alllinearcorrelationsbetween-1and+1cannotbeattainedthroughsuitablespecificationofthejointdistributionF. Kendall’sTauprovetobeagoodalternative,wherethemainadvantagesofrankcorrelationoverordinarycorrelationaretheinvarianceundermonotonictransformationsandthefactthatperfectdependencecorrespondstocorrelationsof+1and–1.Anotheradvantageisthatrankcorrelationisquiterobustagainstoutliers.Themaindisadvantageisthatrankcorrelationsdonotlendthemselvestothesameelegantvariance-covariancemanipulationsasdolinearcorrelations,sincetheyarenotmoment-basedmeasures.However,aswe’veseeninparagraph3.1,Kendall’sTaucanbeeasilyobtainedasafunctionofcopulaparameters,andinsomecaseslikemultivariatedistributionwhichpossessesasimpleclosed-formcopula,liketheGumbelcopula,momentsmaybedifficulttodeterminewhilecalculationofrankcorrelationisamuchmoreeasierwork.5.1.2 Copula’sconditionalKendall’sTauforportfoliomanagement Theclassicaloptimizationproblemofaportfoliomanageristhefollowingone: 14 2σp,t+1=minwt+1Σt+1wt+1 wt+1 (a)(b) (c) s.t.: wt+11=1wt+1µt+1=g whereg=portfoliotargetreturn; wt+1=optimalvectorofweightsattimet+1; Σt+1=forecastedvariance-covariancematrixofassetreturnsattimet+1;µt+1=vectorofexpectedreturnsattimet+1.andtheusualvariancecovariancematrixΣt+1ismadeupasfollows(2·2example): 2σi,t+1·ρij,t+1σi,t+1·σj,t+1 ρ·ij,t+1σj,t+1·σi,t+1 2σj,t+1 whereρij,t+1·σi,t+1·σj,t+1isthecovariancebetweenassetiandassetjattimet+1. Asadirectconsequenceofwhatwesaidinparagraphs3.1and5.1.1,wewanttoproposeheretomodifythecovariancematrixofassetreturnsΣt+1asfollows: 2σi,t+1·τij,t+1σi,t+1·σj,t+1 ·τij,t+1σj,t+1·σi,t+1 2σj,t+1 ij,t+1 wherewesubstitutedthelinearcorrelationρwithKendall’sTauτ ij,t+1. ThegeneralprocedurewefollowedtogettheforecastsofbothvariancesandKendall’sTau coefficients(orlinearcorrelation’s)isathree-stepapproachsimilartoEngleandSheppard(2001)andTseandTsui(2002)fordynamicconditionalcorrelationmodels,andEmbrechtsetal.(2003a,b)andLingHu(2003)forcopulamodeling.It’sbasicallythesemi-parametricCMLmethoddescribedinparagraph4.3: Definition9(Globalminimumvarianceportfolio):Thestepstoestimatetheoptimalvectorwt+1are: 1.EstimateanARMA–GARCHmodelfortheconditionalmeansandvariances,byusingt-studentornormaldistributederrors,whereinthelattercasewehaveQMLproperties;→Gettheforecastedvariancesσi,t+1,σj,t+1;2.EstimatethestandardizedresidualsandcomputetheempiricalCDF,assumingindependence: T1ˆn(·)=1{Xnt≤F T t=1 •) Asfinancialreturnsareusuallynoti.i.d.,wehavetousethestandardizedresidualstoestimatecopulasandusetheasymptoticresultsinGenestetal.(1995)fori.i.d.observations. 3.Computethelog-likelihoodfunctionofthecopula(??),withu1andu2replacedbytheempirical ˆ1(·)andFˆ2(·)respectively,whereweassumethatthecopulas’parametersθfollowady-CDFsF namicsimilartothatusedforARMA-GARCHmodels.→Gettheforecastedlinearcorrelationsρij,t+1orKendall’sTauτij,t+1,buildΣt+1andestimatetheglobalminimumvarianceportfoliominimizing(a)s.t.(b)ItisimportanttorememberthatSklar’stheoremforconditionaldistributions(Theorem2)impliesthattheconditioningvariable(s)mustbethesameforbothmarginaldistributionsandthecopula:ifwedonotusethesameconditioningvariableforF1,F2andC,thefunctionHwillnotbe,ingeneral,ajointconditionaldistributionfunction(Patton,2003). 15 5.2 5.2.1 Valueatrisk Introduction:theunivariatecase. ValueatRiskorVaRisaconceptdevelopedinthefieldofriskmanagementthatisdefinedastheminimumamountofmoneythatonecouldexpecttolosewithagivenprobabilityoveraspecificperiodoftime.WhileVaRiswidelyused,itis,nonetheless,acontroversialconcept,primarilyduetothediversemethodsusedinobtainingVaR,thewidelydivergentvaluessoobtainedandthefearthatmanagementwillrelytooheavilyonVaRwithlittleregardforotherkindsofrisks.TheVaRconceptembodiesthreefactors: 1.Agiventimehorizon.Ariskmanagermightbeconcernedaboutpossiblelossesoveroneday,oneweek,etc.2.VaRisassociatedwithaprobability.ThestatedVaRrepresentstheminimumpossiblelossoveragivenperiodoftimewithagivenprobability.3.Theactualamountofmoneyinvested. Ifwecall∆V(l)thechangeinthevalueoftheassetsinthefinancialpositionfromttot+landFl(x)thecumulativedistributionfunctionof∆V(l),VaRisformallydefinedasfollows: Definition10(VaR):WedefinetheVaRofalongpositionovertimehorizonlwithprobabilitypas p=Pr[∆V(l)≤VaRt(p,l)]=Fl(VaRt(p,l)). whereVaRisdefinedasanegativevalue(loss).ManyfinancetextbooksdefinetheVaRasapositivevalue. Theholderofshortpositionsuffersaloss,whenthevalueoftheassetincreases[i.e.∆V(l)¿0].HencetheVaRforashortpositionisdefinedas, p=Pr[∆V(l)≥VaRt(p,l)]=1-Pr[∆V(l)≤VaRt(p,l)]=1-Fl(VaRt(p,l)). ThepreviousdefinitionshowthatVaRisconcernedwithtailbehaviourofthecdfFl(x).ForanyunivariatecdfFl(x)andprobabilityp,suchthat0¡p¡1,thequantityxp=inf{x|Fl(x)≥p}iscalledthep-thquantileofFl(x),whereinfdenotesthesmallestrealnumbersatisfyingFl(x)≥p.Ifthecdfisknown,thenVaRissimplyitsp-thquantiletimesthevalueofthefinancialposition:howeverthisisnotknowninpracticeandmustbeestimated. IfweconsideranARMA-GARCHmodeltomodelthemeanrtandvolatilityσtandweassumethattheresidualsareGaussian,thentheconditionaldistributionofrt+1giventheinformationavailableattimetisN[ˆrt+1,ˆσt+1]whererˆt+1andσˆt+1are1-stepaheadforecast.Itdirectlyfollows σt+1,andingeneralthatthe5%quantileisthenrˆt+1–1.65ˆ ˆi,t+1−zασˆi,t+1VaRt+1=r (13) IfweinsteadassumethattheresidualsfollowastandardizedStudent-tdistributionwithυ degreesoffreedom,thenthequantileis ˆt+1−t∗ˆt+1VaRt+1=rυ,aσ (14) wheret∗υ,aistheα-thquantileofastandardizedStudent-tdistributionwithυdegreesoffreedom. 16 5.2.2Themultivariatecase. TheconceptofVaRisaveryappealingonebecauseitcanbedevelopedforanykindofportfolioandcanbeaggregatedacrossportfoliosofdifferentkindsofinstruments.ThisdoesnotimplythatestimatingVaRforaportfolioisasimpleprocess,becausethedependencesacrossassetclassesmustbeaccountedfor:sofar,correlationrepresentedthemostusedtoolforevaluatingdependence,but,aswehaveseen,itcannotbethebestwaytodoitduetononlineardependenceamongfinancialassets. Whenwedealwithaportfolioofassets,VaRestimationcanbecomeverydifficultduetothecomplexityofjointmultivariatemodeling.Someapproacheshavebeenproposed,seeGiotandLaurent(2003)forareview,butmostofthemarerathercomplicatedtoimplementandcangivesimilarresultstosimplermethods.Moreover,recentempiricalstudiesshowthatinvolatileperiods,financialmarketstendtobecharacterizedbydifferentlevelsofdependencethanoccurinqueitperiods.Inordertotakeintoaccountthisreality,weproposetousetheconceptofTaildependence,whichreferstothedependencethatarisesbetweenrandomvariablesfromextremeobservations.Animportantfeatureofcopulasisthattheyallowfordifferentdegreesoftaildependence:Uppertaildependenceexistswhenthereisapositiveprobabilityofpositiveoutliersoccurringjointly,whilelowertaildependenceissymmetricallydefinedastheprobabilityofnegativeoutliersoccurringjointly. WhatweproposeisadirectconsiderationofthisconceptinVaRmodelsbymeansofcopulatheory,astaildependencecoefficientscanbecalculatedassimplefunctionsofcopulaeparameters:ifwefollowthewell-knownRiskMetricsmultiplepositionsVaRmodel,lowertaildependencecoeffi-cientscanbeusedattheplaceoflinearcorrelationcoefficients.Forthesereasons,wewanttopresentherethefollowingtwomethodologies: Definition11(AmodifiedRiskMetricsmultiplepositionsmodel):TheRiskMetricsmodelisbyfarthemostusedbyfinancialprofessionalsandweconsiderithereforthisreason,howeverwithaslightmodificationtotakeintoaccountcopula’sconditionaldependencemeasures,suchasKendall’sTauandTaildependence. Letconsidertheglobalminimumvarianceportfoliomadeupofmassetsandestimatedasdescribedinparagraph5.1.TheeuroamountinvestedinassetiisWi,t+1=wi,t+1W,whereWisthetotaleuroamount,andwi,t+1istheoptimalshareofassetiintheportfolioattimet+1:inthiscasetheVaRi,t+1forthesingleassetiis ri,t+1–zασˆi,t+1)orVaRi,t+1=Wi,t+1·(ˆri,t+1–t∗VaRi,t+1=Wi,t+1·(ˆυ,a σˆi,t+1) asinparagraph5.2.1.Then,thetraditionalRiskMetricsgeneralizationofVaRwhendealingwitha portfolioofmassetsis(seeTsay2002,andLongerstaeyandMore1995), mm 2VaRp,t+1=VaRi,t+1+2ρijVaRi,t+1VaRj,t+1(15) i=1 i VaRoftheithinstrument. Inordertopreservethissimpleframework,weproposetomodify(10)bysubstitutingthecorrelationρijwiththefollowingconditionaldependencemeasures,estimatedviaadynamiccopulaspecification: a)theforecastedconditionalcorrelationρˆij,t+1: mm 2VaRi,tρˆij,t+1VaRi,t+1VaRj,t+1VaRp,t+1=+1+2 i=1 i 17 b)theforecastedconditionalKendall’sTauτˆijt+1: mm 2VaRp,t+1=VaRi,t+1+2τˆij,t+1VaRi,t+1VaRj,t+1 i=1 i ˆij,t+1:c)theforecastedconditionallowerTaildependenceλ m=VaR2 i=1 mi i,t+1 +2 ˆij,t+1VaRi,t+1VaRj,t+1λ (18) Definition12(Multivariatecopulamodeling):Consideragaintheglobalminimumvari-anceportfoliomadeupofmassetsestimatedasdescribedinparagraph5.1.Theeuroamount investedinassetiisWi,t+1=wi,t+1W,whereWisthetotaleuroamount,andwi,t+1istheoptimalshareofassetiintheportfolioattimet+1:Letwt+1betheoptimalvectorofsharesfortimet+1,Rt+1thevectorofrealreturns,µt+1thevectorofexpectedreturnscomputedbyanARMAmodel,andΣt+1theforecastedvariance-covariancematrixestimatedviaadynamicparametriccopulaandGARCHmodels,aspreviouslyexplainedinparagraph5.1.Then, Rp,t+1=wt+1’Rt+1,E[Rp]=wt+1’µt+1,V[Rp]=σ2p=wt+1’Σt+1wt+1 TheVaRoftheportfolioatlevelαistheminimalamountthatcanbelostwithprobability lessthanα,VaRp,t+1(α)=W·qp,t+1(α),whereqp,t+1(α)isdeterminedbyPr[Rp,t+1 ]·VaRp,t+1=[wt+1µt+1+zα(wt+1Σt+1wt+1) W(19) withzαtheleftα%-quantileoftheN(0,1)distribution.Undertheassumptionofmultivariate Student-tinnovations,theone-step-aheadVaRisobtainedbyreplacingzαbyt∗υ,a,instead,where∗tυ,aisthep-thquantileofastandardizedStudent-tdistributionwithυdegreesoffreedom, ∗1/2 ]·WVaRp,t+1=[wt+1µt+1+tυ,a(wt+1Σt+1wt+1) (20) Itisagainclearthatthisapproachemphasizesthesemi-parametricapproachwe’vefollowedso far,whereweconsideraparametriccopula,aparametricjointdistribution(normalorT-Student),butwedonotmodelthemarginals,eventhoughweemphasisthattheyareconnectedbyequation(??).ThischoiceismostlyjustifiedbythefactthatasfarasweareconcernedwithportfolioallocationandVaR,dependencesacrossassetsplaythemainroleintoday’sfinancialindustry:moreover,misspecificationofmarginalscanleadtodangerousbiasesindependencemeasuresestimation,aswe’veseeninparagraph4.3.Thisiswhythesemi-parametricapproachisquicklybecomingthemajorstandardinjointmultivariatemodeling. Wewouldalsoliketohighlightafewcommentsregardingskewedmodels,bothunivariateandmultivariate,thatwehaveexcludedsofar: Remark2:someportfoliomanagerspointedoutthatskewedmodelspresentsomehiddendangersduetothefactthatapositiveornegativeskewnessishighlydependentontheparticularsampleoftimeconsideredintheanalysis.Forexample,assetsusuallypresentedpositiveskewnesstilltheyear2000,whilelaterweobservedashifttonegativeskewnessduetofallingmarkets:ifyouhadestimatedtheVaRquantilesusingaskewedmodelandmarketdatatillthebeginningof2000,youwouldhavefoundconservativeestimatesforthelefttailandaggressiveonesfortherighttail,wheninrealityfinancialprofessionalsneededjusttheopposite!Moreover,asallpractitionersknow,thisshiftinskewnessispresentnotonlyinmacrodatabutalsoatthehighfrequencylevel,whereassetpricescanfollowmicro-trendswhicharejusttheoppositethegeneralmacrotrend.Asimple 18 Figure2:Nasdaq100distributions 140120100806040200-0.015-0.010-0.0050.0000.0050.010Series: NDSample 1 1000Observations 1000Mean -0.000115Median 0.000000Maximum 0.010605Minimum -0.018268Std. Dev. 0.003494Skewness -0.077014Kurtosis 3.890615Jarque-Bera 34.03834Probability 0.000000300250200150100500-0.01250.00000.01250.0250Series: NDSample 1001 2000Observations 1000Mean -9.42E-05Median 0.000000Maximum 0.032888Minimum -0.021710Std. Dev. 0.004863Skewness 0.517710Kurtosis 6.918418Jarque-Bera 684.4207Probability 0.000000exampleisgivenbythefirst2000observationofourNasdaq100futuredataset(Figure2):thefirst 1000presentednegativeskewness(duetomarketfalling),whileobservationsspanningfrom1001to2000presentpositiveskewness(duetothemarketrebound)! Forthesereasonsfinancialprofessionalprefertousesymmetricdistributionswhicharemorestableagainstthesechangesinskewness,ratherthanskewedmodels,whichcanprovidebetterdatafittingbutcompletelywrongestimatesifmarketsconditionsarechanging. 6Empiricalanalysis WeconsiderthefivemostimportantfuturecontractsactuallytradedinAmericanmarkets(Sp500,DowJones,Nasdaq100,EuroDollar,TBondNote)withhighfrequencydatasampledat5–min-utesfrequency,takingintoconsiderationthemostvolatileperiodofthelastdecade,thatisbetweenMarchthe13th2000tillJunethe09th2000.Thisperiodoftimesawthefallingofworldfinancialmarketsfollowingtheburstingofthehigh-techbubble,withbigintradaydrawdownreturns.Thus,thissampleisperfectlysuitedtohighlighttheimportanceofthecopula-baseddependenceapproachcomparedtothetraditionalcorrelationanalysis. WebuildupaportfolioandamultipleVaRpositionfollowingboththetwoapproaches,usingtheinitialpartofthesampletoestimatetheassets’weightsandthe95%(99%)VaR,andtheremainingparttocomparetheout-of-sampleperformancesofthetwoapproaches,bothintermsofriskmeasuresandnumberofVaRexceedancesoftheeffectiveportfoliolosses.Weshowthatourap-proachoutperformthecorrelationbasedonebothintermsofportfolioresultsandVaRback-testing.Weanalysethesefuturesbyconsideringtheintradaydataincludedbetween9.30and15.00(NewYorktime):wechosethistimesamplebecausetheEuro$futurecontractistradeduntil15.00intheafternoon.Forthesamereason,somedaysthatwerenotpresentinallfivefutureswherecancelledoutinordertohaveacommonsampleamongallfutures.Moreover,thefirsttradewhichtookplaceeverydayafter9.30inthemorninghasbeendeletedinordertoanalyseonlytheintradaybehaviouroftheconsideredseries.Thedescriptivestatisticsofthe(raw)log-returnsarepresentedinTable1. 6.1Univariatemodeling Aspointedoutinpreviousparagraphs,youneedi.i.dobservationsinordertoestimatecopulae.Toreachthisgoal,anAR(6)-GARCH(1,1)filterwitht-distributederrorswasappliedtotheraw5-minutesfuturesreturns.Thesixlagswerefoundtobesignificantonalmostallcases2andthey 2 Forsakeofbrevity,wedonotreportheretheestimatedcoefficients.However,theseresultsareavailablefromtheauthors. 19 Table1:Descriptivestatisticsoffutureslog-returns DowJones(DJ) MeanMaximumMinimumStd.Dev.SkewnessKurtosisJarque-BeraP-valueJBObservations2.98e−050.0071-0.00960.00140.02825.4337839.60.00003400S&P500(SP)1.89e−050.0072-0.01020.0015-0.00745.73151057.00.00003400Nasdaq100(ND)-6.74e−050.0329-0.02170.00400.26126.79762081.70.00003400Euro-Dollar(EC)2.44e−060.0048-0.00540.0007-0.17947.07952375.90.00003400T-BondNote(US)-3.13e−060.0039-0.00360.00060.10625.66091009.40.00003400canbeexplainedbymeansoftwowellknownreasons:1.Bid-askbounce; 2.Delaysinnewsdeliverytothepublic.Asitisknown,quotedataareinsomecasesdelayedby10/15minutes,accordingtotheusedtradingplatformandsimilarlysomefinancialnewsismadepublicwithadelayrangingfromafewminutestohalfanhour.ItisawellknownfactthatasingleBloombergorReutersplatform,withthemostimportantnewsfeatures,costsabout$30.000-$100.000peryearandnotalltraderscanaffordthat.TheLjung-BoxQ-statisticsrelativetothefutureslog-returns,standardizedresidualsandstan-dardizedsquaredresiduals,plustheBDStestforindependenceappliedtothestandardizedresiduals,arereportedinTable2.WeremindthattheBDStestisaportmanteautestfortimebaseddepen-denceinaseriesandcanbeusedtocheckwhethertheresidualsareindependentandidenticallydistributed.Underthenullhypothesisofindependence,weexpectthisstatistictobeclosetozero(seeBrock,Dechert,ScheinkmanandLeBaron-1996). Table2:Testsstatistics:futureslog-returnsandstandardizedresiduals LB(50)-LB(50)-StdRawreturnsresidualssquared (∗)DowJones(DJ)90.41540.1(∗)32.64 (∗)(∗)S&P500(SP)85.571174.538.54 (∗)(∗)Nasdaq100(ND)73.781391.427.71 (∗∗)(∗)Euro-Dollar(EC)65.96147.245.22 (∗)T-BondN.(US)40.54250.736.94 (*)H0rejectedat1%level;(**)H0rejectedat10%levelLB(50)-Rawreturns LB(50)-Stdresidu-alssquared 5.3311.1019.4526.4927.29 BDS(50)(P-Value)0.320.310.280.390.34 Allstandardizedresidualsdisplaythedesiredi.i.d.propertiesnecessaryforcopulaestimation,sowecannowestimatetheempiricalCDF(Step2inProposition5), T1ˆn(·)=1{Xnt≤F T t=1 •) andproceedtocomputethelog-likelihoodfunctionsofthebivariatedynamiccopulae(Step3inProposition5). 20 6.2 6.2.1 Multivariatemodeling Copulamodeling. TheellipticalcopulaeweuseherearetheNormalcopulaandtheT-copula,whosedensityinthebivariatecasearereportedhere: 1)Normalcopula: 11[(Φ−1(u1))2+(Φ−1(u2))2−2ρ·Φ−1(u1)·Φ−1(u2)]−12−12 ·exp[(Φ(u1))+(Φ(u2))]·exp−·c(Φ(x1),Φ(x2);ρ)=2(1−ρ2)2(1−ρ2)1 whereρisthecorrelationcoefficientandΦ−1istheinverseofthestandardunivariateGaussian distributionfunction. 2)T-copula: −υ+nυ−122+(t−1(u))2−2ρ·t−1(u)·t−1(u)ΓΓυ+2(t(u))1212υυυυ221+c(tυ(x1),tυ(x2);ρ,υ)=2(1−ρ2)·υΓυ+11−ρ22· 12t−υ(u1)) 1+ υ 12(t−υ(u2)) 1+ υ υ+121whereρisthecorrelationcoefficient,t−υistheinverseofthestandardunivariateT-studentdistribu-tionfunction,whileυarethedegreesoffreedom. TheArchimedeancopulaweuseistheMixed-Gumbel:mixedcopulaeareacommonwayto modelbothupperandtaildependencewithcopulaethatwouldotherwisemodelonlyoneofthetwodependencemeasure(seeEmbrechtsandDias2003,LingHui2003,Patton2003).ThesimpleGumbelcopulamodelonlyuppertaildependence,whiletheRotatedCopulaonlythelowertaildependence:ifwemixthemwecanmodelbothtaildependences,nestingsymmetryasaspecialcaseandnotbyconstructionasforellipticalcopulae.Themixed-gumbelCDFandPdfarereportedhere:3)Mixed-Gumbelcopula(weexpresscopulaparametersasafunctionoftaildependencecoefficientsλU,λL): CDF:C(u1;u2;θ)=0.5*CGumbel(u1,u2;λU)+0.5*CRotatedGumbel(u1,u2;λL)where CGumbel(u1,u2;λU)=exp{-((-logu)θ1+(-logu)θ1)1/θ1},CRotatedGumbel(u1,u2;λL)=u1+u2−1+CGumbel(1-u1,1-u2;λL) and θ1=f(λU),θ2=f(λL) PDF:c(u1;u2;θ)=0.5*cGumbel(u1,u2;λU)+0.5*cRotatedGumbel(u1,u2;λL)where c Gumbel CGumbel(u1,u2)(logu1·logu2)θ1−1θ1/θ1 (u1,u2;λ)=·((−logu1)θ+θ1−1)1+(−logu2)1)θθ2−1/θ1 uv((−logu1)1+(−logu2)1) U cRotatedGumbel(u1,u2;λL)=cGumbel(1-u1,1−u2;λL) and θ1=f(λU),θ2=f(λL) 21 Aswe’llseeinthenextparagraph,themixed-Gumbelprovetobethebestcopulainsomecases,confirmingpreviousresultsinEmbrechtsandDias(2003)andLingHui(2003):however,itpresentstheseriouslimitationtomodelonlypositivedependencebyconstruction.Toovercomethisproblem,weconsideralsothemixed-Normalcopula,whichisamixtureoftherotated-GumbelandtheNormalcopula.Thischoiceisjustifiedbythefactthatmodelinglowertaildependenceisbyfarthemostinterestingforfinancialpurposes.Weusethiscopulaasasubstituteforthemixed-Gumbelwhentwoassetspresentnegativedependenceorchangesofdependencewithinthesample(liketheEuro$andT-bondnotefutures,forexample).Itsdensityfunctionisthefollowing: 4)Mixed-Normalcopula: PDF:c(u1;u2;θ)=0.5*cNormal(u1,u2;ρ)+0.5*cRotatedGumbel(u1,u2;λL) wherethefunctionalformofcNormal(u1,u2;ρ)andcRotatedGumbel(u1,u2;λL)werestatedabove.Weexplicitlyconsidertimedynamicsbyallowingtheparametersofthecopulatoevolvethroughtime.FollowingPatton(2003)andDiasandEmbrechts(2003),wesuggestthesegeneralevolutionequationsasstartingpointsfortheafore-mentionedcopulae: Table3:Copulasparametersdynamicspecifications EllipticalcopulaeCopulaGeneralspecification6i=1 NormalcopulaT-copula ρt=Λω+α·ρt−1+ βi·|ut−i−vt−i| ArchimedeancopulaeCopulaGeneralspecificationMixed-GumbelMixed-Normal ρt=Λ ω+α·ρt−1+ 6i=1 βi·|ut−i−vt−i| ˜ω+β·ρt−1+λUαi·|ut−i−vt−i|t=Λ i=1 6L˜λt=Λω+β·ρt−1+αi·|ut−i−vt−i| i=16ρt=Λω+β·ρt−1+αi·|ut−i−vt−i| i=1 6L˜λt=Λω+β·ρt−1+αi·|ut−i−vt−i| 6i=1whereΛ(x)≡(1-e−x)/(1+e−x)isthemodifiedlogistictransformation,designedtokeepρtin(-1, ˜t≡1/(1+e−x)isthesimplelogistictransformation,usedtokeepλUand1)atalltimes,whileΛ λLin(0,1)atalltimes.Table3showsthatweassumecopulaparametersfollowanARMA(1,6)-typeprocess:weincludeρt−1asaregressortocaptureanypersistenceinthedependenceparameter,andthedifferencesbetweenut−jandvt−jovertheprevioussixobservationstocaptureanyvariationindependence.Wetriedsomevariantsforthelastregressor,butwedidn’tfindanysignificantimprovement,sowesticktothissimplespecification.6.2.2 Copulaestimationresults TheestimationresultsforeachpairofassetsandthebestcopulaspecificationarereportedinTable4–7(thebestmodelamongallcopulaeforaspecifiedpairofassets,ishighlightedinboldfonts).Lookingattables4–7,weseethatthebestresultsareachievedbytheT-copulainmostofthecasesandthemixed-Gumbelcopulainonecase.However,thenormalcopuladoesnotperformmuchworseandpresentstheimportantfeaturetobeveryeasytocompute!Themixed-normalpresentsbetterresultsthanthenormalcopulainsomecases,butitisalwaysworsethantheT-copula;moreoveritscomputationcanbequiteintensive. Wenowlookhowthesecopulaeperformbothintermsofout-of-sampleportfolioresultsandVaRback-testing. 22 Table4:NormalCopulaestimationresults NORMALcopula(3400observation–PentiumIV2400Mhz) Pairassets ofBestdynamicspecificationρtρtρtρtρtρtρtρtρtρt =Λ=Λ=Λ=Λ=Λ=Λ=Λ=Λ=Λ=Λ(ω+β·ρt−1+α1·|ut−1−vt−1|)(ω+β·ρt−1+α1·|ut−1−vt−1|)(ω+β·ρt−1+α1·|ut−1−vt−1|)(ω+β·ρt−1+α1·|ut−1−vt−1|)(ω+β·ρt−1+α1·|ut−1−vt−1|)(ω+β·ρt−1+α1·|ut−1−vt−1|)(ω+β·ρt−1+α1·|ut−1−vt−1|)(ω+β·ρt−1+α1·|ut−1−vt−1|)(ω+β·ρt−1+α1·|ut−1−vt−1|)(ω+β·ρt−1+α1·|ut−1−vt−1|)N.param.3333333333Log-likelihood 1854.63951.531561.76123.49134.96107.65183.96210.57204.5859.41AIC-1.089-0.558-0.917-0.071-0.078-0.062-0.106-0.122-0.119-0.033BIC-1.084-0.553-0.912-0.065-0.072-0.056-0.101-0.117-0.113-0.028SP/DJND/DJSP/NDDJ/ECSP/ECND/ECDJ/USSP/USND/USEC/USTimetocompute6sec4sec5sec7sec7sec7sec7sec8sec7sec6secTable5:T-Copulaestimationresults T-copula(3400observation–PentiumIV2400Mhz) Pairassets ofBestdynamicspecificationρt=Λω+β·ρt−1+α1·ut−1−vt−1+α2·ut−3−vt−3 ρt=Λω+β·ρt−1+α1·ut−1−vt−1+α2·ut−3−vt−3 ρt=Λω+α1·ut−1−vt−1+α2·ut−3−vt−3 N.par.Log-likelihood 44333333331875.21975.441588.75125.55140.15111.66203.20239.40226.1760.82AIC-1.100-0.571-0.932-0.072-0.080-0.063-0.117-0.138-0.131-0.033BIC-1.091-0.562-0.923-0.064-0.073-0.056-0.110-0.131-0.123-0.026SP/DJND/DJSP/NDDJ/ECSP/ECND/ECDJ/USSP/USND/USEC/USρtρtρtρtρtρtρt =Λ=Λ=Λ=Λ=Λ=Λ=Λ(ω+β·ρt−1+α1·|ut−1−vt−1|)(ω+β·ρt−1+α1·|ut−1−vt−1|)(ω+β·ρt−1+α1·|ut−1−vt−1|)(ω+β·ρt−1+α1·|ut−1−vt−1|)(ω+β·ρt−1+α1·|ut−1−vt−1|)(ω+β·ρt−1+α1·|ut−1−vt−1|)(ω+β·ρt−1+α1·|ut−1−vt−1|)Timetocompute49sec42sec27sec37sec38sec32sec31sec40sec28sec37secTable6:Mixed-GumbelCopulaestimationresults(positivedependenceonly) Mixed-Gumbelcopula(3400observations–PentiumIV2400Mhz) AssetsSP/DJND/DJSP/ND BestdynamicspecificationλLt ˜(ω+β·ρt−1+α1·|ut−1−vt−1|)λUt=Λ ˜(ω+β·ρt−1+α1·|ut−1−vt−1|)=Λ ˜(ω+β·ρt−1+α1·|ut−1−vt−1|)=Λ ˜(ω+β·ρt−1+α1·|ut−1−vt−1|)=Λ ˜(ω+β·ρt−1+α1·|ut−1−vt−1|)=Λ N.par.Log-likelihood 766 1851.12960.911589.23 AIC-1.085-0.562-0.932 BIC-1.075-0.551-0.923 Timeto comput λUtλLtλUtλLt ˜ω+β·ρt−1+α1·ut−1−vt−1+α2·ut−3−vt−3=Λ 40sec21sec80sec 23 Table7:Mixed-NormalCopulaestimationresults Mixed-Normalcopula(3400observations–PentiumnIV2400Mhz) AssetsDJ/ECSP/ECND/ECDJ/USSP/USND/USEC/USBestdynamicspecificationρt=Λ(ω+β·ρt−1+α1·|ut−1−vt−1|)˜λLt=Λ(ω) ρt=Λ(ω+β·ρt−1+α1·|ut−1−vt−1|)˜λLt=Λ(ω) ρt=Λ(ω+β·ρt−1+α1·|ut−1−vt−1|)˜λLt=Λ(ω) ρt=Λ(ω+β·ρt−1+α1·|ut−1−vt−1|)˜λLt=Λ(ω+α1·|ut−1−vt−1|) ρt=Λ(ω+β·ρt−1+α1·|ut−1−vt−1|)˜λLt=Λ(ω+α1·|ut−1−vt−1|) ρt=Λ(ω+β·ρt−1+α1·|ut−1−vt−1|)ρt=Λ(ω+β·ρt−1+α1·|ut−1−vt−1|)˜λLt=Λ(ω+α1·|ut−1−vt−1|) λLtN.par.Log-likelihood 4445565114.56133.39104.00187.27219.02206.3858.82AIC-0.065-0.076-0.059-0.107-0.126-0.118-0.032BIC-0.058-0.069-0.052-0.098-0.117-0.107-0.023Timeto comput 12sec40sec10sec22sec91sec92sec14sec˜(ω+α1·|ut−1−vt−1|+α2·|ut−3−vt−3|)=Λ 6.3Portfoliomanagementresults Weconsideraportfoliocomposedofthefivefutures,andweuseaniterativeprocedurewhereAR(6)-GARCH(1,1)andCopulaemodelsareestimatedtopredictone-day-aheadvariancesσi,t+1,corre-lationsρi,t+1andKendall’sTauτi,t+1.Webuildthevariance-covariancematrixΣt+1andwethen estimatetheglobalminimumvarianceportfoliominimizing(5)under(5a).Thefirstestimationsam-pleisgivenbythefirst2000observations.Thepredictedoptimalvectorofshareswt+1isthenusedtoestimatetherealportfolioreturnandthisresultisrecordedforlaterassessmentusingrisk-returnsmeasures.Atthej-thiterationwherejgoesfrom2000to3399(foratotalof1400observations),theestimationsampleisaugmentedtoincludeonemoreobservation,andtheforecastedoptimalvectorwt+jandobservedportfolioreturnRp,t+j=wt+j’Rt+jarerecorded.Werepeatthisprocedureuntilalldayshavebeenincludedintheestimationsample.6.3.1 Portfolioresults:Assetswithpositivedependenceonly Wefirstconsideraportfoliocomposedoffutureswhichpresentpositivedependenceonly,thatistheDowJones,Nasdaq,andS&P500.Bydoingthis,weareabletocomparetheperformancesoftheMixed-GumbelcopulaagainsttheNormalandT-Copula.Besidestheusualstatistics,wepresentheretwootherimportantmeasuresofrisk,the1%ValueatRisk(VaR)andthe1%ExpectedShortfall(ES).The1%VaRisdefinedasthenegativeofthefirstempiricalpercentileoftherealised −1(0.01),whereFistheempiricaldistributionfunctionportfolioreturns,VaR(X;0.01)=–Fnn ofportfolioreturnsX,whereweusethenout-of-sampleobservations.WhileVaRpresentsomeadvantagesovertraditionalriskmeasures,ithasbeenshowedthatitisnotacoherentmeasureofrisk(Artzneretal.,1999);moreoverVaRmayunderestimatetheriskofsecuritieswithfat-tailedpropertiesandahighpotentialforlargelosses,anditmaydisregardthetaildependenceofassetreturns(YamaiandYoshiba,2002).AnalternativetoVaRthathasgainedsomeattentionrecentlyistheExpectedShortfallofaportfolio(Rockafellar,andUryasev1999,Rockafellar,andUryasev2001,Acerbi,Nordio,Sirtori2001,AcerbiandTasche2002).The1%expectedshortfallisdefinedasthenegativeofaveragereturnonaportfoliogiventhatthereturnhasexceededits1%VaR,thatisES(X;0.01)≡En[X|X≤VaR(X;0.01),whereEnisthesampleaverage.YamaiandYoshiba(2002)showthatExpectedShortfallsufferslessindisregardingassets’fattailsandtaildependencethanVaRdoes.Wepresenttheresultsrelativetoanunconditionalportfoliotoo,whichiscomposedofequallyweightedfutures,thatis[1/3,1/3,1/3]. Empiricalresultsandrisk-returnsmeasuresassociatedwithobservedportfoliosarereportedinTable8. 24 Table8:Portfolioreturnssummarystatistics(positivedependenceonly) NORMALcopula ConditionalRHO ConditionalTAU T–Copula ConditionalRHO ConditionalTAU Mixed–Gumbel ConditionalTAU UnconditionalPortfolio-(1/3each) Meanreturn Cumulatedret.Std.deviationSkewnessKurtosisSharperatioVaR(1%)ES(1%)2.99e-070.0004180.001013-0.4800478.357190-0.0005250.0024400.003938-1.96e-06-0.0027430.001092-0.4460888.513731-0.0025540.0028420.0042381.19e-060.0016700.000978-0.5024997.2953400.0003710.0026300.003789-5.85e-08-0.0000820.001028-0.4780878.442339-0.0008640.0025380.0040011.90e-060.0026670.001000-0.4914238.3747640.0010750.0024380.003896-1.09e-05-0.0159530.001855-0.1653996.979274-0.0062930.0046940.006604Table8showsthattheKendall’sTau–mixedGumbelportfoliooutperformsallotherportfoliowiththehighestcumulatedreturnandSharpeRatio,andthelowestValueatRisk;however,theT-copulapresentsthelowestExpectedShortfallanditsresultsarenotfardistantfromtheMixed-Gumbel(likeinparagraph6.2.2wheretheAICandBICofthetwoareveryclose).6.3.2 Portfolioresults:allassetsconsidered Wenowpresenttheresultsrelativetoportfolioscomposedofallfivefutures,thusincludingtheoneswithnegativedependence.BesidesNormalandT-copulaportfolios,asthemixed-Gumbelcopulacanmodelonlypositivedependence,wedecidetobuildaportfoliobyusingtheforecastedmixed-GumbelKendall’sTauτi,t+1forpositivedependenceassetsandNormalorMixed-NormalorT-copulaτi,t+1fortheremainingassets.Moreoverwepresenttheresultsrelativetoanunconditionalportfoliotoo,whichiscomposedofequallyweightedfutures,thatis[1/5,1/5,1/5,1/5,1/5]. Empiricalresultsandrisk-returnsmeasuresassociatedwithobservedportfoliosarereportedinTable9. Table9:Portfolioreturnssummarystatistics(allassets) NORMALcopula ConditionalRHO1.44E-050.0201810.0003920.3734656.7661490.0346570.0009180.001131 ConditionalTAU1.39E-050.0194960.0003930.3365166.7412710.0333500.0009210.001145 T−Copula ConditionalRHO1.13E-050.0158620.0003970.3758876.8135940.0264710.0009740.001190 ConditionalTAU1.44E-050.0201110.0003900.3796096.8830770.0346820.0009060.001125 Mix−Gum+NormalConditionalTAU1.47E-050.0205530.0003940.3656476.5985020.0351830.0009310.001128 MeanreturnCumulatedret.Std.DeviationSkewnessKurtosisSharperatioVaR(1%)ES(1%)Mix−GumMix−GumUnc.Port+T−copula+Mix−Nor(1/5 each) ConditionalConditionalTAUTAU1.45e-051.48E-051.68e-060.0202620.0206890.0023490.0003900.0003970.0010670.3846510.379876-0.060156.9248606.5198786.4977700.0349900.0351720.0007950.0008960.000913-0.0027910.0011180.001126-0.003714 Aseasilyexpected,thethreemixed-Gumbelbasedportfoliosarethebestandallthreehave analogousresults.ThebiggestdifferencewhenconsideringallassetsaretheunsatisfactoryresultsfromcorrelationbasedportfolioswithrespecttoKendall’sTau:whilethelatterpresentsimilarresultsamongallcopulae,theformerseemtobeverysensitivetotheparticularcopulausedforestimation.Look,forexample,atFigure3whichreportstheconditionallinearcorrelationcoefficientρt+iandKendall’sTauτt+iforEuro$andT-bondNote,bothestimatedwiththeNormalandtheT-copula:Kendall’sTauestimationappearstobemorestableamongdifferentcopulas. 25 Figure3:Euro-dollar/T-bondNoteconditionalρandτ 6.3.3Testsforsuperiorportfolioperformance Wenowwanttotestwhethertheperformancesamongportfoliosbuiltbyusingdifferentcopulaeanddependencemeausuresinparagragh6.3.2arestatisticallysignificant.WepresenttheresultsoftherealitycheckofWhite(2000),asmodifiedbyHansen(2001).Thistestcomparesallmodelsjointly,andtestwhetheragivenbenchmarkportfolioperformsaswellasthebestcompetingalternativemodel,chosenamongmanyalternatives.Wereportthethreeestimatesofthep-valuesdiscussedinHansen(2001),andfocusontheconsistentp-valueestimates.Werejectthenullhypothesisthatthebenchmarkmodel(builtwithaspecifiedcopula)performsaswellasthebestcompetingalternativemodel,wheneverthep-valueislessthan0.10.Theperformancemeasureusedisthesamplemeanoftherealisedreturn(arejectionofthenullispresentedinboldfonts): Table10:RealitycheckP-values BenchmarkportfolioNormalcopula(RHO)Normalcopula(Kendall)T-copula(RHO)T-copula(Kendall)Mixed-Gumbel+T-copula(Kendall)Mixed-Gumbel+Normalc.(Kendall)Mixed-Gumbel+Mixed-Normalc.(Kendall)Lower0.7440.5040.07350.69250.7870.82350.6295Consistent0.85750.54550.07350.7570.8470.92350.7865Upper0.85750.54550.07350.85050.9470.92350.7865Theseresultssupportwhatwe’vehaveseeninthepreviousparagraph:differentcopulaegivesimilarresultsandtheirperformancesarestatisticallyjustasgoodastheirbestalternative.Whatreallymattersistheuseddependencemeasure,instead:fortheportfoliobuiltwithlinearcorrelationsestimatedbyusingtheT-copula,weareabletorejectthenullthatitperformsaswellasthebestalternative. 6.4ValueatRiskresults Weconsidertheportfoliocomposedofthefivefutures,andweusethepreviousiterativeproceduretopredictone-stepaheadvariancesσi,t+1,correlationsρij,t+1,Kendall’sTauτij,t+1andlowerTaildependencesλUij,t+1.Thelatterareestimatedbyusingthemixed-Gumbelcopulaforpositivede-pendentassetsandtheT-copulafortheremainingpairsofassets:inthislastcasetheyarezero 26 inalmostallcasesexceptfortheEuro$-TbondNote.TheseforecastsarenowusedtoestimatetheportfolioValueatRisk,bothwiththemodifiedRiskMetricsmultiplepositionsmodel(11a)(11b)(11c)andwiththeMultivariatecopulamodel(12a)and(12b).Thepredictedone-step-aheadVaRisthencomparedwiththeobservedreturnandbothresultsarerecordedforlaterassessmentusingaback-testingprocedure.Themodelisre-estimatedforeachobservationrangingfrom2000to3399,asexplainedinparagraph6.3,andtheprocedureisiterateduntilalldayshavebeenincludedintheestimationsample. WeassesstheperformanceofthemodelsbycomputingKupiec’s(1995)LRtestsontheempir-icalfailurerates:thistestisbasedonbinomialtheoryandteststhedifferencebetweentheobservedandexpectednumberofVaRexceedancesoftheeffectiveportfoliolosses.AsVaRisbasedonaconfidencelevel1-p,whenweobserveNlossesinexcessofVaRoutofTobservations,henceweobserveN/Tproportionofexcessivelosses:theKupiec’stestanswersthequestionwhetherN/Tisstatisticallysignificantlydifferentfromp. Followingbinomialtheory,theprobabilityofobservingNfailuresoutofTobservationsisT(1-p)−NpN,sothatthetestofthenullhypothesisH0:p=p∗isgivenbyalikelihoodratioteststatistic: LR=2·ln[(1-p∗)T−Np∗N]+2·ln[(1–T/N)T−N(N/T)N] whichisdistributedasχ2(1)underH0.Itiswellknownthatthepowerofthistest,thatistheabilitytorejectabadmodel,riseswithT:asweareworkingwith1400observations,thistestshouldworkwell. TherealVaRexceedancesandtheKupiectestp-valuesarereportedinTable10–11(whenH0isnotrejectedatthe5%level,thenumbersarereportedinboldfont): Table11:VaRback-testinganalysis(modifiedRiskMetricsmultiplepositionsmodel) Normalcopula(RHO)Normalcopula(Kendall)T-copula(RHO)T-copula(Kendall)Mix.-Gum.+T-cop.(Kendall)Mix.-Gum.+Normal(Kendall)Mix-Gum+Mix-Norm.(Kendall)Mix-Gum.+T-cop.(TailDep.)1%VaRRealExceed.(%)KupiecTest:p-val.0.640.1500.500.0372.640.0000.640.1500.710.2580.430.0150.290.0020.290.0025%VaRRealExceed.(%)KupiecTest:p-val.5.210.7144.860.8127.790.0005.360.5355.430.4685.070.9024.210.1663.930.056Table12:VaRback-testinganalysis(Multivariatecopulamodelling) 1%VaRNormalquantileT-st.quan.(10df)RealKup.RealKup.Exc.TestExc.Test1.210.4361.001.0001.070.7910.570.0803.790.0003.070.0001.360.2031.140.5991.570.0471.140.5991.001.0000.710.2580.710.2580.360.0055%VaRNormalquantileT-st.quan.(10df)RealKup.RealKup.Exc.TestExc.Test4.570.4564.710.6214.430.3174.570.4567.000.0017.290.0004.790.7115.001.0004.710.6214.790.7114.210.1664.350.2603.710.0213.790.030Normalcopula(RHO)Normalcopula(Kendall)T-copula(RHO)T-copula(Kendall)Mix.-Gum.+T-cop.(Kendall)Mix.-Gum+Normal(Kendall)Mix-Gum+Mix-Norm.(Kendall)TheVaRresultsconfirmwhatwe’vealreadyseeninparagraph6.3forportfoliomanagement,thatisthelinearcorrelationmeasureisverysensitivetothecopulausedandcangivepoorVaRforecasts:aswecanseefromtables10–11,thelinearcorrelationbasedVaRistheonlycasewheretheKupiectestrejectthemodelbecauseitistooaggressive,whileallotherrejectionsaredueto 27 tooconservativeestimates.AslinearcorrelationestimationbasedontheT-copulaisbecomingquitepopular,practitionersshouldbeawarethatthisdependencemeasureisverysensitivetocopulamisspecification,whilethisisnotthecaseforKendall’sTau.Taildependence,ontheotherhand,tendtobequiteconservativeinstead. WhenwecomparethemodifiedRiskMetricsmodelandthemultivariatecopulamodelweseethelatterasbeinggenerallymorepreciseandpresentingfewerrejections:thisisnotsurprising,astheRiskMetricsmodelassumesthatlog-returnsofeachassetfollowarandom-walkIGARCH(1,1)withnomeandynamics,whichisamorerestrictiveframeworkthanthemultivariatecopulamodel. Finally,wenotethatwithinthemultivariatecopulamodel,theresultsamongnormalquantileandstandardizedT-studentquantilearequitesimilarexceptforonecase:thisfactpointsoutthatthemajorbenefitwhenestimatingVaRcomesfromconditionaldependencemodeling.Thechoiceof10degreesoffreedomhasbeendonebyobservingthatportfolioreturnsestimatedwiththemodelspresentedinsection6.3,exhibitaT-studentdistributionwith10degreesoffreedominalmostallcases.Needlesstosay,loweringthed.o.f.wouldhavedeterminedmoreconservativeestimates. 7Conclusions Theaimofthispaperhasbeentoshowthemainshortcomingsoftraditionalcorrelationanalysisandpresentspossiblesolutionswhichhastheadvantagetokeeprisk-returnsmodelstractablebutatthesametimeconsiderthenon-lineardependencyamongtheconsideredvariables. Traditionalportfoliotheorybasedonmultivariatenormaldistributionassumesthatinvestorscanbenefitfromdiversificationbyinvestinginassetswithlowercorrelations.Howeverthisisnotwhathappensinreality,asoneoftenfindsfinancialmarketswithdifferentcorrelationsbutalmostthesamenumberofmarketcrashes(ifwedefinemarketcrashaswhenreturnsareintheirlowestqthpercentile). Inordertoovercometheseproblemswecanresorttocopulatheory,sincecopulaecapturethosepropertiesofthejointdistributionwhichareinvariantunderstrictlyincreasingtransformation.AcommondependencemeasurethatcanbeexpressedasafunctionofcopulaparametersandisscaleinvariantisKendall’sTau.Itsatisfiesmostofthedesiredpropertiesthatadependencemeasuremusthaveanditmeasurestheconcordancebetweentworandomvariables.Itisforthisreasonthatitcandetectnon-linearassociationthatcorrelationcannotsee. WeshowedhowtouseKendall’sTauwithinthetraditionalmean-varianceframework,intheplaceofthecorrelationcoefficients:Theempiricalresultsconfirmthatlinearcorrelationcangivepoorout-of-sampleportfolioperformancesanditisverysensitivetotheparticularcopulaused.ThebestcopulatomodelpositivedependencewasdeterminedtobetheMixed-Gumbel,whileformoregeneraldependencetheT-copulaprovedtobeagoodsolution(whenusingKendall’sTau).Thenormalcopulapresentedthemainadvantageatbeingveryeasytocompute,withresultsverysimilartotheprevioustwo. Inasimilarfashion,recentempiricalstudiesshowthatinvolatileperiodsfinancialmarketstendtobecharacterizedbydifferentlevelsofdependencethanoccursinquietperiods.Inordertotakethisrealityintoaccount,weproposedtousetheconceptofTaildependence,whichreferstothedependencethatarisesbetweenrandomvariablesfromextremeobservations.Weconsideredthismeasurewithinthewell-knownRiskMetricsVaRmodelattheplaceoflinearcorrelationcoefficientsandwedidthesamewithKendall’sTau,too.Again,wefoundthatlinearcorrelationisverysensitivetothecopulausedandcangivepoorVaRforecasts:aslinearcorrelationestimationbasedontheT-copulaisbecomingquitepopular,practitionersshouldbeawarethatthisdependencemeasureisverysensitivetocopulamisspecification,whilethisisnotthecaseforKendall’sTau.Taildependence,ontheotherhand,gaveafairlyconservativeresult,whileKendall’sTauexhibitedthebestoutcomes. WealsoproposedamoregeneralframeworkthantheRiskMetricsmodel,whereweconsideredaparametriccopula,aparametricjointdistribution,butweexcludethemarginals.ThischoiceismostlyjustifiedbythefactthatasfarasweareconcernedwithportfolioallocationandVaR,dependencesacrossassetsplaythemainroleintoday’sfinancialindustry:moreover,themisspeci-28 ficationofmarginalscanleadtodangerousbiasesindependencemeasuresestimation.ThismodelwasgenerallymorepreciseandpresentedfewerrejectionsthantheRiskMetricsmodelwhenusingback-testingprocedures:thisisnotasurprise,asthelatterassumesthatlog-returnsofeachassetfollowarandom-walkIGARCH(1,1)withnomeandynamics,whichisamorerestrictiveframeworkthanthemultivariatecopulamodel. 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