Slides astin

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 copulasφ(t)=exp(-t1/θ) Gumbel’s copulasφ(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

4zJ

]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,xCvV,uUvV,xUPy,x)v,u,C(F =≤≤≤≤=

( ) ( ) ( )( )v,uC

v,xCvV,uUxUPx)v,u,C(FU =≤≤≤=

( )( ) ( ) ( )( )( )v,uC

y)v,u,C(F,x)v,u,C(FCy,xv,u,C

1V

1U

−−=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) givenX > 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)

FrankSurvival Clayton

ClaytonSurvival 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 ofbivariate tail dependence using extreme value copulas: an application to the SOA medical large claims database (2003)

Application : Group Medical Large Claims







Slides astin

Technology