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- 11. 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
- 21. 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
- 24. 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
- 25. 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
- 27. 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
- 28. 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
- 30. 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
- 31. 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
- 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
1[φ1(u1)+φ1(u2)+φ1(u3)+φ1(u4)],
which,ifφiisparametrizedwithparameterαi,canbesummarizedthrough
A=
1α2α4α4
α21α4α4
α4α41α3
alpha4α4α31
33
- 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
34
- 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
alpha4α4α31
35
- 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
2(φ2(u1)+φ2(u2))
�
+φ4
�
φ−1
3(φ3(u3)+φ3(u4))
�
),
which,ifφiisparametrizedwithparameterαi,canbesummarizedthrough
A=
1α2α4α4
α21α4α4
α4α41α3
α4α4α31
36
- 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
37
- 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
38
- 39. 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
- 41. 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
- 42. 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
- 43. 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
- 45. 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
- 63. 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
- 64. 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
- 74. 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
- 79. 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
- 83. 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
- 89. 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
- 90. 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
- 92. 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
- 97. 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
- 98. 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
- 117. 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).
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