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ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures
XXXIIIrd International ASTIN Colloquium
Tail distribution and dependence measures
Arthur Charpentier, ACIA

ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 2
• Clayton’s copulas
C(u,v,θ)=(u-θ+v-θ -1)-1/θ pour θ ≥ 1
A short introduction to copulas

Sklar’s Theorem (1959)
• Let (X,Y) be a pair of random variables, with joint c.d.f.
FXY(x,y) = P(X≤x,Y≤y) and let FX(x) = P(X≤x) , FY(y) = P(Y≤y).
• There is a copula C such that FXY(x,y) = C(FX(x),FY(y))
• Conversely, C satisfies C(u,v) = FXY(FX
-1(u),FY
-1(v))
where -1 denotes the generalized inverse, g-1(t)=inf{s,g(s)=t}.

Archimedian copulas
• Let φ:[0,1]→[0,+∞] be a convex strictly decreaasing function,
such that φ(1)=0.
C(u,v)= φ-¹(φ(u)+ φ(v)) for any 0≤u,v ≤1
is a copula, and f is called the generator of the copula.
• Ex φ(t)=t-θ-1 Clayton’s copulas
φ(t)=exp(-t1/θ) Gumbel’s copulas
φ(t)=-log[(e -θt-1)/(e -θ-1)] Frank’s copulas

Survival copulas
• Let (X,Y) be a random pair with copula C, and U=FX(x), V=FY(y)
• C is the c.d.f. of (U,V), then, the c.d.f. of (1-U,1-V) is C * defined
as C*(u,v)=u+v-1+C(1-u,1-v)
• C * satisfies
• C * is called survival copula, or dual copula
( ) ( ) ( ) ( )( )yF,xFCy,xFyY,xXP YX
*
XY ==>>

Functional measures of dependence : G.Venter (2001)
Tail concentration functions
• L(z) = Pr(U<z,V<z)/z2 = C(z,z)/z2
• R(z) = Pr(U>z,V>z)/(1 – z)2 = [1 – 2z +C(z,z)]/(1 – z)2
Cumulative tau
1)v,u(dC)v,u(C4
]1,0[x]1,0[
−=τ
∫∫
( ) 1)v,u(dC)v,u(C
)z,z(C
4
zJ
]z,0[x]z,0[
2
−=
∫∫

The lower tail conditional copula
Lower tail conditional copula
• Let (X,Y) be a pair of random variables with copula C, and let
U=FX(X) et V=FY(Y), such that C(x,y) = P(U ≤ x,V ≤ y)
• The lower tail conditional copula with threshold u,v is the
copula of (U,V) given U≤u and V≤v, and is denoted Φ(C,u,v)
• Given u,v in [0,1], Φ(C,u,v) is the copula of
(X,Y) given X ≤ VaR(X,u) and Y ≤ VaR(Y,v)

• Clayton’s copula Φ(C,u,v) = C for any u,v in ]0,1]
Clayton’s copula is invariant by truncature
• Marshall-Olkin’s copula Cα,β(x,y)=min{x1-αy,xy1-β}
Φ(Cα,β, uβ,uα) = Cα,β
• Gumbel-Barnett’s copulas Cθ(x,y)=xyexp[-θlog(x)log(y)]
Φ(Cθ,u,v) = Cθ(u,v) where θ(u,v)=θ/[1+θlog(u)][1+θlog(v)]

• The joint c.d.f. of (U,V) given U≤u and V≤v is
• The marginal c.d.f.’s of (U,V) given U≤u and V≤v are
• The copula of (U,V) given U≤u and V≤v is (Sklar’s Theorem)
( ) ( ) ( )
( )v,uC
y,xC
vV,uUvV,xUPy,x)v,u,C(F =≤≤≤≤=
( ) ( ) ( )
( )v,uC
v,xC
vV,uUxUPx)v,u,C(FU =≤≤≤=
( )( ) ( ) ( )( )
( )v,uC
y)v,u,C(F,x)v,u,C(FC
y,xv,u,C
1
V
1
U
−−
=Φ

The upper tail conditional copula
Upper tail conditional copula
• Given u,v in [0,1], Φ(C*,1-u,1-v) is the copula of
(X,Y) given X > VaR(X,u) and Y > VaR(Y,v)
where C* is the survival copula of (X,Y)
• Using this duality property, studying upper tail dependence
could be done with the lower tail conditional copula Φ(C,u,v)
• These copulas could be extended to higher dimension

• Properties of the conditional copula : Dependence and tail
distribution (IME 2003), especially limiting results, when u,v
converge towards 0
• Archimedian copulas are stable : the conditional copula
obtained from an Archimedian copula, with an other generator
The generator of Φ(C,u,v) is φ(C(u,v)t) - φ(C(u,v))

Application in Credit Risk models
• Let X and Y denote two times of defaults, and let C* denote
the survival copula at time 0
• Assume that no default occurred at time t>0, then, the copula
of the times of default is not C* : it is the copula of (X,Y) given
X>t and Y>t
• Ex : survival time are exponential (means µ and 2µ), and C* is a
Gumbel copula

Functional measures and conditional copulas
Dependence measures based on copula functions
• Kendall’s tau
• Spearman’s rho – rank correlation
could be extended to any other dependence measure
• Gini’s gamma or Blomqvist’s beta
• Lp distance of Schweizer and Wolf

Lower / upper tail rank correlation
• Lower tail rank correlation defined as Sperman’s rho of (X,Y)
given X ≤ VaR(X,u) and Y ≤ VaR(Y,u), 0 ≤ u ≤ 1.
• Upper tail rank correlation defined as Sperman’s rho of (X,Y)
given X > VaR(X,u) and Y > VaR(Y,u), 0 ≤ u ≤1
( ) ( )( ) 3dxdyy,xu,u,C12u
]1,0[x]1,0[
−Φ=ρ
∫∫
( ) ( )( ) 3dxdyy,xu1,u1,C12u
]1,0[x]1,0[
*
−−−Φ=ρ
∫∫

Other functional measures of dependence
• Upper tail Kendall’s tau defined as Kendall’s tau of (X,Y) given
X > VaR(X,u) and Y > VaR(Y,u), 0 ≤ u ≤1
( ) ( ) ( ) 1)y,x(u1,u1,Cd)y,x(u1,u1,C4u
]1,0[x]1,0[
**
−−−Φ−−Φ=τ
∫∫

Other functional measures of dependence
• The rank correlation of (X,Y) given X ≤ VaR(X,u) is
• Or, more generally, (X,Y) given X < h and Y < h,
( ) ( )( ) 3dxdyy,x1,u,C12u
]1,0[x]1,0[
−Φ=ρ
∫∫
( ) ( ) ( )( )( ) 3dxdyy,xhF,hF,C12u
]1,0[x]1,0[
YX −Φ=ρ
∫∫

• Clayton’s copula

• Survival Clayton’s copula

• Gaussian copula

• Gumbel copula

• Student copula

0.010.600.000.100.510.17
0.030.600.000.120.510.20
0.110.600.000.170.510.25
0.270.600.120.260.510.33
0.600.600.600.600.600.60ρ(X,Y)
Frank
Survival
Clayton
Clayton
Survival
Gumbel
GumbelGaussian
( )%50ρ
( )%75ρ
( )%90ρ
( )%95ρ

Application : Loss-ALAE
• Dataset used in Frees and Valdez (1997)
• (Loss,ALAE) given LOSS>VaR(Loss,u), ALAE>VaR(ALAE,u)
• (Loss,ALAE) given LOSS>VaR(Loss,u)

Application : Motor-Household
• Dataset from Belguise,Levi (2003)
• « the best candidate is the HRT copula » which is the
survival/dual Clayton copula

• Dataset from the Society of Actuaries
• CEBRIAN, A., DENUIT, M. and P. LAMBERT, Analysis of
bivariate tail dependence using extreme value copulas: an
application to the SOA medical large claims database (2003)
Application : Group Medical Large Claims

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