Successfully reported this slideshow.
Upcoming SlideShare
×

# Slides astin

13,767 views

Published on

• Full Name
Comment goes here.

Are you sure you want to Yes No

Are you sure you want to  Yes  No
• Sex in your area is here: ❶❶❶ http://bit.ly/2Qu6Caa ❶❶❶

Are you sure you want to  Yes  No

### Slides astin

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