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  1. 1. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measuresXXXIIIrd International ASTIN ColloquiumTail distribution and dependence measuresArthur Charpentier, ACIA
  2. 2. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 2• Clayton’s copulasC(u,v,θ)=(u-θ+v-θ -1)-1/θ pour θ ≥ 1A short introduction to copulas
  3. 3. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 3Sklar’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}.A short introduction to copulas
  4. 4. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 4A short introduction to copulasArchimedian 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 ≤1is 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
  5. 5. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 5A short introduction to copulasSurvival 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 * definedas 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 ==>>
  6. 6. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 6Functional 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)2Cumulative tau1)v,u(dC)v,u(C4]1,0[x]1,0[−=τ∫∫( ) 1)v,u(dC)v,u(C)z,z(C4zJ]z,0[x]z,0[2−=∫∫
  7. 7. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 7The lower tail conditional copulaLower tail conditional copula• Let (X,Y) be a pair of random variables with copula C, and letU=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 thecopula 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)
  8. 8. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 8The lower tail conditional copula
  9. 9. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 9The lower tail conditional copulaLower tail conditional copula• 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)]
  10. 10. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 10The lower tail conditional copulaLower tail conditional copula• 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,uCy,xCvV,uUvV,xUPy,x)v,u,C(F =≤≤≤≤=( ) ( ) ( )( )v,uCv,xCvV,uUxUPx)v,u,C(FU =≤≤≤=( )( ) ( ) ( )( )( )v,uCy)v,u,C(F,x)v,u,C(FCy,xv,u,C1V1U−−=Φ
  11. 11. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 11The upper tail conditional copulaUpper 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 dependencecould be done with the lower tail conditional copula Φ(C,u,v)• These copulas could be extended to higher dimension
  12. 12. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 12The lower tail conditional copulaLower tail conditional copula• Properties of the conditional copula : Dependence and taildistribution (IME 2003), especially limiting results, when u,vconverge towards 0• Archimedian copulas are stable : the conditional copulaobtained from an Archimedian copula, with an other generatorThe generator of Φ(C,u,v) is φ(C(u,v)t) - φ(C(u,v))
  13. 13. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 13The lower tail conditional copulaApplication in Credit Risk models• Let X and Y denote two times of defaults, and let C* denotethe survival copula at time 0• Assume that no default occurred at time t>0, then, the copulaof the times of default is not C* : it is the copula of (X,Y) givenX>t and Y>t• Ex : survival time are exponential (means µ and 2µ), and C* is aGumbel copula
  14. 14. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 14The lower tail conditional copula
  15. 15. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 15The lower tail conditional copula
  16. 16. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 16Functional measures and conditional copulasDependence measures based on copula functions• Kendall’s tau• Spearman’s rho – rank correlationcould be extended to any other dependence measure• Gini’s gamma or Blomqvist’s beta• Lp distance of Schweizer and Wolf
  17. 17. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 17Functional measures and conditional copulasLower / 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∫∫
  18. 18. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 18Functional measures and conditional copulasOther 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∫∫
  19. 19. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 19Functional measures and conditional copulasOther 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∫∫
  20. 20. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 20Functional measures and conditional copulas
  21. 21. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 21Functional measures and conditional copulas
  22. 22. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 22Functional measures and conditional copulas
  23. 23. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 23Functional measures and conditional copulas
  24. 24. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 24Functional measures and conditional copulas
  25. 25. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 25Functional measures and conditional copulas
  26. 26. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 26Functional measures and conditional copulas
  27. 27. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 27Functional measures and conditional copulas
  28. 28. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 28Functional measures and conditional copulas• Clayton’s copula
  29. 29. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 29Functional measures and conditional copulas• Survival Clayton’s copula
  30. 30. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 30Functional measures and conditional copulas• Gaussian copula
  31. 31. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 31Functional measures and conditional copulas• Gumbel copula
  32. 32. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 32Functional measures and conditional copulas• Student copula
  33. 33. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 33Functional measures and conditional copulas0.010.600.000.100.510.170.030.600.000.120.510.200.110.600.000.170.510.250.270.600.120.260.510.330.600.600.600.600.600.60ρ(X,Y)FrankSurvivalClaytonClaytonSurvivalGumbelGumbelGaussian( )%50ρ( )%75ρ( )%90ρ( )%95ρ
  34. 34. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 34Application : 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)
  35. 35. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 35Application : Loss-ALAE
  36. 36. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 36Application : Loss-ALAE
  37. 37. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 37Application : Loss-ALAE
  38. 38. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 38Application : Loss-ALAE
  39. 39. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 39Application : Loss-ALAE
  40. 40. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 40Application : Motor-Household• Dataset from Belguise,Levi (2003)• « the best candidate is the HRT copula » which is thesurvival/dual Clayton copula
  41. 41. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 41Application : Motor-Household
  42. 42. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 42Application : Motor-Household
  43. 43. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 43Application : Motor-Household
  44. 44. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 44• Dataset from the Society of Actuaries• CEBRIAN, A., DENUIT, M. and P. LAMBERT, Analysis ofbivariate tail dependence using extreme value copulas: anapplication to the SOA medical large claims database (2003)Application : Group Medical Large Claims
  45. 45. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 45Application : Group Medical Large Claims
  46. 46. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 46Application : Group Medical Large Claims
  47. 47. ARTHUR CHARPENTIER (ensae -CREST) Tail distribution and dependence measures 47Application : Group Medical Large Claims

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