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- 1. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas On some classes of hierarchical Archimedean copulas and their use in actuarial science Arthur Charpentier and St´ephane Loisel 29/05/08 1 / 22
- 2. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Outline Introduction : some actuarial issues Solvency II and ruin probabilities A quick look at insurance data Dependence models for Solvency II Archimdedean copulas Pairwise Archimedean copulas Multivariate extension and link with frailty models Drawbacks of Archimedean copulas Nested Archimedean copulas Nested Archimedean copulas Nesting copulas of the same family Nesting copulas of diﬀerent families 2 / 22
- 3. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Outline Introduction : some actuarial issues Solvency II and ruin probabilities A quick look at insurance data Dependence models for Solvency II Archimdedean copulas Pairwise Archimedean copulas Multivariate extension and link with frailty models Drawbacks of Archimedean copulas Nested Archimedean copulas Nested Archimedean copulas Nesting copulas of the same family Nesting copulas of diﬀerent families 3 / 22
- 4. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Risk aggregation in Solvency II • Dependence between losses in diﬀerent lines of business (Liability, motor insurance, ﬁre insurance, ...) and new solvency regulations in Europe : Solvency II. • Typical problem : for some α ∈ [0.95, 0.999], compute V aRα(X1 + · · · + X25) or TV aRα(X1 + · · · + X25). • The Tail-Value-at-Risk is TV aRα(X) = 1 1−α 1 α V aRq(X)dq. • If FX is continuous, then TV aRα(X) = E [X | X > V aRα(X)] . • Why using TV aR instead of V aR ? TV aRα is sub-additive (and V aRα is not), comonotonic-additive and takes diversiﬁcation eﬀect into account : the worst case is when the Xk’s are comonotonic. 4 / 22
- 5. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Ruin probabilities with dependent claims • Dependence between successive claim amounts in risk theory. • Typical problem : compute or approximate the probability of ruin in ﬁnite or inﬁnite time with large initial reserve, when claim amounts are no longer independent. A typical risk process. 5 / 22
- 6. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas A quick look at very basic insurance data Some very basic insurance data : empirical Kendall’s τ of loss ratios of diﬀerent lines of business. The above risks are far from being exchangeable ! 6 / 22
- 7. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Hierarchical structure of Solvency II The standard formula for the Solvency Capital Requirement (SCR) uses a so-called bottom-up approach. 7 / 22
- 8. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Hierarchical structure of Solvency II The standard formula for the Solvency Capital Requirement (SCR) uses a so-called bottom-up approach. Each class of risks is subdivided into sub-classes of risks with a hierarchical model. Fig.: QIS 4 correlation parameters in non-life insurance Fig.: QIS 4 correlation parameters in life insurance 8 / 22
- 9. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Classical tools Average and large insurance companies currently develop internal models or partial internal models. To model dependence between risks, classical tools that are often used include : • Gaussian copulas (quite easy to implement and ﬂexibility of the correlation matrix, but often not adapted to real-world data) • Other elliptical copulas (Student for example) • Archimedean copulas (not adapted as risks are non-exchangeable) • Vine copulas (trees and conditional distributions). An alternative solution (among many others) in the spirit of the hierarchical standard approach : nested (or hierarchical) Archimedean copulas. 9 / 22
- 10. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Outline Introduction : some actuarial issues Solvency II and ruin probabilities A quick look at insurance data Dependence models for Solvency II Archimdedean copulas Pairwise Archimedean copulas Multivariate extension and link with frailty models Drawbacks of Archimedean copulas Nested Archimedean copulas Nested Archimedean copulas Nesting copulas of the same family Nesting copulas of diﬀerent families 10 / 22
- 11. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas 2-dimensional Archimedean copulas Deﬁnition : An Archimedean copula C with generator ϕ : [0, 1] → [0, +∞] is deﬁned by ∀(u, v) ∈ [0, 1]2 , C(u, v) = ϕ[−1] (ϕ(u) + ϕ(v)) , • where ϕ is convex, decreasing, and such that ϕ(1) = 0, • and where ϕ[−1] : [0, +∞] → [0, 1] is either • the reciprocal bijection ϕ−1 of ϕ if ϕ(0) = +∞ (strict Archimedean copulas), • or deﬁned by • ϕ[−1] (x) = ϕ−1 (x) for x ≤ ϕ(0), • ϕ[−1] (x) = 0 for x > ϕ(0) if ϕ(0) < +∞ (non-strict Archimedean copulas). 11 / 22
- 12. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Examples of Archimedean copulas Classical examples can be found in Nelsen (1999) : 21 parametric families, including Family ϕθ(t) Dom(θ) strict Clayton 1 θ t−θ − 1 [−1, +∞) {0} if θ > 0 Gumbel (− ln t) θ [−1, +∞) {0} yes Frank − ln e−θt −1 e−θ−1 [−1, +∞) {0} yes AMH ln 1−θ(1−t) t [−1, +∞) {0} yes 12 / 22
- 13. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Extensions to dimension n ≥ 3 • In dimension n ≥ 3, deﬁne an Archimedean copula Cn with generator ϕ by ∀(u1, . . . , un) ∈ [0, 1]n , C(u1, . . . , un) = ϕ[−1] (ϕ(u1) + · · · + ϕ(un)) . • Notation : set Ln = {f s.t. ∀0 ≤ k ≤ n, ∀x, (−1)k ∂k ∂xk f(x) ≥ 0}. • Cn exists for all n ≥ 2 if and only if ϕ is completely monotonic (i.e. ϕ ∈ L∞). • Cn exists for 2 ≤ n ≤ m if and only if ϕ ∈ Lm (Kimberling, 1974). • Bernstein’s theorem and link with frailty models : ϕ−1 completely monotone if and only if it ϕ−1 = LΘ for some non-negative random variable Θ. In that case U1, . . . , Um can be seen as being conditionally independent w.r.t. Θ. 13 / 22
- 14. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Drawbacks of Archimedean copulas • Exchangeable distributions : pairwise dependence is the same for every pair of risks • Basic example in insurance with three lines of business : zoom on QIS4 parameters for some life insurance risks. LoB’s Mortality Longevity Disability Mortality 1 Longevity 0 1 Disability 0.5 0 1 These values are just QIS4 parameters and do not correspond to any real-world linear correlation coeﬃcient or concordance measure. 14 / 22
- 15. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Drawbacks of Archimedean copulas • Exchangeable distributions : pairwise dependence is the same for every pair of risks • Basic example in insurance with three lines of business : zoom on QIS4 parameters for some life insurance risks. LoB’s Mortality Longevity Disability Mortality 1 Longevity 0 1 Disability 0.5 0 1 Negative dependence and positive dependence, with (probably) positive tail dependence index. 15 / 22
- 16. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Outline Introduction : some actuarial issues Solvency II and ruin probabilities A quick look at insurance data Dependence models for Solvency II Archimdedean copulas Pairwise Archimedean copulas Multivariate extension and link with frailty models Drawbacks of Archimedean copulas Nested Archimedean copulas Nested Archimedean copulas Nesting copulas of the same family Nesting copulas of diﬀerent families 16 / 22
- 17. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Nested Archimedean copulas • A four-dimensional fully nested Archimedean copula : C(u1, . . . , u4) = ϕ [−1] 3 ϕ3 ϕ [−1] 2 ϕ2 ϕ [−1] 1 (ϕ1(u1) + ϕ1(u2)) + ϕ2(u3) + ϕ3(u4) Uniform variates (U1, . . . , U4) with copula C are such that • (U1, U2) has copula C1 with generator ϕ1, • (U1, U3) and (U2, U3) both have copula C2 with generator ϕ2, • and the couples (U1, U4), (U2, U4) and (U3, U4) all have copula C3 with generator ϕ3. • A four-dimensional partially nested Archimedean copula : C(u1, . . . , u4) = ϕ [−1] 0 ϕ0 ϕ [−1] 12 (ϕ12(u1) + ϕ12(u2)) + ϕ0 ϕ [−1] 34 (ϕ34(u3) + ϕ34(u4)) Uniform variates (U1, . . . , U4) with copula C are such that • (U1, U2) has copula C12 with generator ϕ12, • (U3, U4) has copula C34 with generator ϕ34, • and the couples (U1, U3), (U2, U3), (U1, U4) and (U2, U4) all have copula C0 with generator ϕ0. 17 / 22
- 18. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Sufficient conditions • Suﬃcient conditions to have a copula in the fully nested case : C(u1, . . . , u4) = ϕ [−1] 3 ϕ3 ϕ [−1] 2 ϕ2 ϕ [−1] 1 (ϕ1(u1) + ϕ1(u2)) + ϕ2(u3) + ϕ3(u4) • ϕ−1 1 , ϕ−1 2 and ϕ−1 3 completely monotonic • (ϕ−1 2 ◦ ϕ1) and (ϕ−1 3 ◦ ϕ2) completely monotonic • Suﬃcient conditions to have a copula in the partially nested case : C(u1, . . . , u4) = ϕ [−1] 0 ϕ0 ϕ [−1] 12 (ϕ12(u1) + ϕ12(u2)) + ϕ0 ϕ [−1] 34 (ϕ34(u3) + ϕ34(u4)) • ϕ−1 0 , ϕ−1 12 and ϕ−1 34 completely monotonic • (ϕ−1 0 ◦ ϕ12) and (ϕ−1 0 ◦ ϕ34) completely monotonic 18 / 22
- 19. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Nesting copulas of the same family • Many ordered families can be nested (Clayton, AMH, Cook-Johnson, Gumbel, ...) • Useful transitivity theorem (see McNeil (2007) e.g.) : if (ϕ−1 2 ◦ ϕ1) and (ϕ−1 3 ◦ ϕ2) are completely monotonic, so does (ϕ−1 3 ◦ ϕ1) . • Simulation methods using the Laplace transform approach. • Some classical references (among others) : Joe (1997), McNeil (2007), Savu and Trede (2006), Embrechts et al. (2003), Hofert (2007), ... 19 / 22
- 20. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Nesting copulas of different families Hofert (2007) characterizes the set of pairs of copulas of Nelsen’s list (1999) that can be nested, in the case where generator inverses are completely monotonic, and where each involved (ϕ−1 r ◦ ϕs) is completely monotonic. In particular, these 7 combinations include : • AMH-Clayton (with some parameter restrictions) • Clayton-Families 12, 14, 19, 20 (with some parameter restrictions) Limitation : only positive dependence is possible with the same or diﬀerent families if one considers the completely monotonic case. Can we get nested Archimedean copulas involving some negative dependence in low and reasonable dimensions (3-15) ? 20 / 22
- 21. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Nesting copulas and negative dependence • It is possible to build models for the three risks (mortality, longevity, disability) ? LoB’s Mortality Longevity Disability Mortality (1) Longevity (2) - Disability (3) ++ - : comonotonic, - : Negative dependence and ++ : positive dependence, with (probably) positive tail dependence index. • One possible solution (among many others) : C(u1, u2, u3) = ϕ [−1] 2 ϕ2 ϕ [−1] 1 (ϕ1(u1) + ϕ1(u3)) + ϕ2(u2) , with • ϕ1 generator of a Gumbel copula with parameter θ1 = 3, • and ϕ2 generator of a Frank copula with parameter θ2 = −2. 21 / 22
- 22. Introduction : some actuarial issues Archimdedean copulas Nested Archimedean copulas Nested Archimedean copulas with negative dependence • This kind of models can be useful for partial internal models in Solvency II. • Suﬃcient conditions can be adapted from completely monotonic conditions depending on the dimension. • Existence of such nested Archimedean copulas is of course limited to low dimensions. • Simulation methods are more complicated as we do not have anymore the frailty model representation. 22 / 22

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