Tails of Archimedean Copulas tail dependence in risk management - PowerPoint PPT Presentation

Arthur CHARPENTIER - tails of Archimedean copulas Tails of Archimedean Copulas tail dependence in risk management Arthur Charpentier CREM-Universit´ e Rennes 1 (joint work with Johan Segers, UCLN) http ://perso.univ-rennes1.fr/arthur.charpentier/ Colloque ´ Evaluation et couverture des risques extrˆ emes Universit´ e Paris-Dauphine & Chaire AXA de la Fondation du Risque, Juin 2008 1

Arthur CHARPENTIER - tails of Archimedean copulas Tail behavior and risk management In reinsurance (XS) pricing, use of Pickands-Balkema-de Haan’s theorem Theorem 1. F ∈ MDA ( G ξ ) if and only if �� � �� � Pr ( X − u ≤ x | X > u ) − H ξ,σ ( u ) ( ≤ x ) lim sup = 0 , u → x F 0 <x<x F  1 − (1 + ξx/σ ) − 1 /ξ , ξ � = 0  for some positive function σ ( · ) , where H ξ,σ ( x ) =  1 − exp ( − x/σ ) , ξ = 0 . � � 1 − F ( x ) ≈ (1 − F ( u )) 1 − H ξ,σ ( u ) ( x − u ) , for all x > u. So, if u = X k : n , then � � 1 − F ( x ) ≈ (1 − F ( X k : n )) 1 − H ξ,σ ( X k : n ) ( x − X k : n ) , for all x > X k : n , � �� � ≈ 1 − � F n ( X k : n )= k/n 2

Arthur CHARPENTIER - tails of Archimedean copulas Pure premium of XS contract Recall that π d = E (( X − d ) + ) with d large, thus, � ∞ 1 π d = 1 − F ( x ) dx P ( X > d ) d � � 1 − 1 k σ 1 + ξ d − X n − k : n ξ ≈ , n 1 − ξ σ i.e. � � 1 − 1 π d = k σ k d − X n − k : n � � ξk 1 + � ξ k � 1 − � n σ k � ξ k (see e.g. Beirlant et al. (2005) . Possible to derive explicit formulas for any tail risk measure (VaR, TVaR...). 3

Arthur CHARPENTIER - tails of Archimedean copulas Extending extreme value theory in higher dimension univariate case bivariate case limiting distribution dependence structure of of X n : n (G.E.V.) componentwise maximum when n → ∞ , i.e. H ξ ( X n : n , Y n : n ) (Fisher-Tippet) dependence structure of limiting distribution ( X, Y ) | X > x, Y > y of X | X > x (G.P.D.) when x, y → ∞ when x → ∞ , i.e. G ξ,σ dependence structure of ( X, Y ) | X > x (Balkema-de Haan-Pickands) when x → ∞ 4