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  • 1. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Copulas, risk dependence and applications to Solvency II Arthur Charpentier CREM-Universit´e Rennes 1 http ://perso.univ-rennes1.fr/arthur.charpentier/ Summer School of the Groupe Consultatif Actuariel Europen : Enterprise Risk Management (ERM) and Solvency II, July 2008. 1
  • 2. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II On risk dependence in QIS’s http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 2
  • 3. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II On risk dependence in QIS’s http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 3
  • 4. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II On risk dependence in QIS’s http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 4
  • 5. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II On risk dependence in QIS’s http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 5
  • 6. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II On risk dependence in QIS’s http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 6
  • 7. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II How to capture dependence in risk models ? Is correlation relevant to capture dependence information ? Consider (see McNeil, Embrechts & Straumann (2003)) 2 log-normal risks, • X ∼ LN(0, 1), i.e. X = exp(X ) where X ∼ N(0, 1) • Y ∼ LN(0, σ2 ), i.e. Y = exp(Y ) where Y ∼ N(0, σ2 ) Recall that corr(X , Y ) takes any value in [−1, +1]. Since corr(X, Y )=corr(X , Y ), what can be corr(X, Y ) ? 7
  • 8. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II How to capture dependence in risk models ? 0 1 2 3 4 5 −0.50.00.51.0 Standard deviation, sigma Correlation Fig. 1 – Range for the correlation, cor(X, Y ), X ∼ LN(0, 1) ,Y ∼ LN(0, σ2 ). 8
  • 9. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II How to capture dependence in risk models ? 0 1 2 3 4 5 −0.50.00.51.0 Standard deviation, sigma Correlation Fig. 2 – cor(X, Y ), X ∼ LN(0, 1) ,Y ∼ LN(0, σ2 ), Gaussian copula, r = 0.5. 9
  • 10. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II What about regulatory technical documents ? 10
  • 11. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II What about regulatory technical documents ? 11
  • 12. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II What about regulatory technical documents ? 12
  • 13. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II What about regulatory technical documents ? 13
  • 14. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Motivations : dependence and copulas Definition 1. A copula C is a joint distribution function on [0, 1]d , with uniform margins on [0, 1]. Theorem 2. (Sklar) Let C be a copula, and F1, . . . , Fd be d marginal distributions, then F(x) = C(F1(x1), . . . , Fd(xd)) is a distribution function, with F ∈ F(F1, . . . , Fd). Conversely, if F ∈ F(F1, . . . , Fd), there exists C such that F(x) = C(F1(x1), . . . , Fd(xd)). Further, if the Fi’s are continuous, then C is unique, and given by C(u) = F(F−1 1 (u1), . . . , F−1 d (ud)) for all ui ∈ [0, 1] We will then define the copula of F, or the copula of X. 14
  • 15. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Copula density Level curves of the copula Fig. 3 – Graphical representation of a copula, C(u, v) = P(U ≤ u, V ≤ v). 15
  • 16. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Copula density Level curves of the copula Fig. 4 – Density of a copula, c(u, v) = ∂2 C(u, v) ∂u∂v . 16
  • 17. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Some very classical copulas • The independent copula C(u, v) = uv = C⊥ (u, v). The copula is standardly denoted Π, P or C⊥ , and an independent version of (X, Y ) will be denoted (X⊥ , Y ⊥ ). It is a random vector such that X⊥ L = X and Y ⊥ L = Y , with copula C⊥ . In higher dimension, C⊥ (u1, . . . , ud) = u1 × . . . × ud is the independent copula. • The comonotonic copula C(u, v) = min{u, v} = C+ (u, v). The copula is standardly denoted M, or C+ , and an comonotone version of (X, Y ) will be denoted (X+ , Y + ). It is a random vector such that X+ L = X and Y + L = Y , with copula C+ . (X, Y ) has copula C+ if and only if there exists a strictly increasing function h such that Y = h(X), or equivalently (X, Y ) L = (F−1 X (U), F−1 Y (U)) where U is U([0, 1]). 17
  • 18. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Some very classical copulas In higher dimension, C+ (u1, . . . , ud) = min{u1, . . . , ud} is the comonotonic copula. • The contercomotonic copula C(u, v) = max{u + v − 1, 0} = C− (u, v). The copula is standardly denoted W, or C− , and an contercomontone version of (X, Y ) will be denoted (X− , Y − ). It is a random vector such that X− L = X and Y − L = Y , with copula C− . (X, Y ) has copula C− if and only if there exists a strictly decreasing function h such that Y = h(X), or equivalently (X, Y ) L = (F−1 X (1 − U), F−1 Y (U)). In higher dimension, C− (u1, . . . , ud) = max{u1 + . . . + ud − (d − 1), 0} is not a copula. But note that for any copula C, C− (u1, . . . , ud) ≤ C(u1, . . . , ud) ≤ C+ (u1, . . . , ud) 18
  • 19. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II 0.2 0.4 0.6 0.8 u_10.2 0.4 0.6 0.8 u_2 00.20.40.60.81 Frechetlowerbound 0.2 0.4 0.6 0.8 u_10.2 0.4 0.6 0.8 u_2 00.20.40.60.81Independencecopula 0.2 0.4 0.6 0.8 u_10.2 0.4 0.6 0.8 u_2 00.20.40.60.81 Frechetupperbound Fréchet Lower Bound 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Independent copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Fréchet Upper Bound 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Scatterplot, Lower Fréchet!Hoeffding bound 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Scatterplot, Indepedent copula random generation 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Scatterplot, Upper Fréchet!Hoeffding bound Fig. 5 – Contercomontonce, independent, and comonotone copulas. 19
  • 20. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Elliptical (Gaussian and t) copulas The idea is to extend the multivariate probit model, X = (X1, . . . , Xd) with marginal B(pi) distributions, modeled as Yi = 1(Xi ≤ ui), where X ∼ N(I, Σ). • The Gaussian copula, with parameter α ∈ (−1, 1), C(u, v) = 1 2π √ 1 − α2 Φ−1 (u) −∞ Φ−1 (v) −∞ exp −(x2 − 2αxy + y2 ) 2(1 − α2) dxdy. Analogously the t-copula is the distribution of (T(X), T(Y )) where T is the t-cdf, and where (X, Y ) has a joint t-distribution. • The Student t-copula with parameter α ∈ (−1, 1) and ν ≥ 2, C(u, v) = 1 2π √ 1 − α2 t−1 ν (u) −∞ t−1 ν (v) −∞ 1 + x2 − 2αxy + y2 2(1 − α2) −((ν+2)/2) dxdy. 20
  • 21. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Archimedean copulas • Archimedian copulas C(u, v) = φ−1 (φ(u) + φ(v)), where φ is decreasing convex (0, 1), with φ(0) = ∞ and φ(1) = 0. Example 3. If φ(t) = [− log t]α , then C is Gumbel’s copula, and if φ(t) = t−α − 1, C is Clayton’s. Note that C⊥ is obtained when φ(t) = − log t. The frailty approach : assume that X and Y are conditionally independent, given the value of an heterogeneous component Θ. Assume further that P(X ≤ x|Θ = θ) = (GX(x))θ and P(Y ≤ y|Θ = θ) = (GY (y))θ for some baseline distribution functions GX and GY . Then F(x, y) = ψ(ψ−1 (FX(x)) + ψ−1 (FY (y))), where ψ denotes the Laplace transform of Θ, i.e. ψ(t) = E(e−tΘ ). 21
  • 22. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II 0 20 40 60 80 100 020406080100 Conditional independence, continuous risk factor !3 !2 !1 0 1 2 3 !3!2!10123 Conditional independence, continuous risk factor Fig. 6 – Continuous classes of risks, (Xi, Yi) and (Φ−1 (FX(Xi)), Φ−1 (FY (Yi))). 22
  • 23. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Some more examples of Archimedean copulas ψ(t) range θ (1) 1 θ (t−θ − 1) [−1, 0) ∪ (0, ∞) Clayton, Clayton (1978) (2) (1 − t)θ [1, ∞) (3) log 1−θ(1−t) t [−1, 1) Ali-Mikhail-Haq (4) (− log t)θ [1, ∞) Gumbel, Gumbel (1960), Hougaard (1986) (5) − log e−θt−1 e−θ−1 (−∞, 0) ∪ (0, ∞) Frank, Frank (1979), Nelsen (1987) (6) − log{1 − (1 − t)θ} [1, ∞) Joe, Frank (1981), Joe (1993) (7) − log{θt + (1 − θ)} (0, 1] (8) 1−t 1+(θ−1)t [1, ∞) (9) log(1 − θ log t) (0, 1] Barnett (1980), Gumbel (1960) (10) log(2t−θ − 1) (0, 1] (11) log(2 − tθ) (0, 1/2] (12) ( 1 t − 1)θ [1, ∞) (13) (1 − log t)θ − 1 (0, ∞) (14) (t−1/θ − 1)θ [1, ∞) (15) (1 − t1/θ)θ [1, ∞) Genest & Ghoudi (1994) (16) ( θ t + 1)(1 − t) [0, ∞) 23
  • 24. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Extreme value copulas • Extreme value copulas C(u, v) = exp (log u + log v) A log u log u + log v , where A is a dependence function, convex on [0, 1] with A(0) = A(1) = 1, et max{1 − ω, ω} ≤ A (ω) ≤ 1 for all ω ∈ [0, 1] . An alternative definition is the following : C is an extreme value copula if for all z > 0, C(u1, . . . , ud) = C(u 1/z 1 , . . . , u 1/z d )z . Those copula are then called max-stable : define the maximum componentwise of a sample X1, . . . , Xn, i.e. Mi = max{Xi,1, . . . , Xi,n}. Remark more difficult to characterize when d ≥ 3. 24
  • 25. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II On copula parametrization • Gaussian, Student t (and elliptical) copulas Focuses on pairwise dependence through the correlation matrix,        X1 X2 X3 X4        ∼ N        0, 1 r12 r13 r14 r12 1 r23 r24 r13 r23 1 r34 r14 r24 r34 1        Dependence in [0, 1]d ←→ summarized in d(d + 1)/2 parameters, 25
  • 26. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II On copula parametrization • Archimedean copulas Initially, dependence in [0, 1]d ←→ summarized in one functional parameters on [0, 1]. But appears less flexible because of exchangeability features. It is possible to introduce hierarchical Archimedean copulas (see Savu & Trede (2006) or McNeil (2007)). Let U = (U1, U2, U3, U4), C(u1, u2, u3, u4) = φ−1 1 [φ1(u1) + φ1(u2) + φ1(u3) + φ1(u4)], which, if φi is parametrized with parameter αi, can be summarized through A =        1 α2 α4 α4 α2 1 α4 α4 α4 α4 1 α3 alpha4 α4 α3 1        26
  • 27. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II On copula parametrization • Archimedean copulas Initially, dependence in [0, 1]d ←→ summarized in one functional parameters on [0, 1]. But appears less flexible because of exchangeability features. It is possible to introduce hierarchical Archimedean copulas (see Savu & Trede (2006) or McNeil (2007)). Let U = (U1, U2, U3, U4), C(u1, u2, u3, u4) = φ−1 4 (φ4 φ−1 2 (φ2(u1) + φ2(u2)) + φ4 φ−1 3 (φ3(u3) + φ3(u4)) ), which, if φi is parametrized with parameter αi, can be summarized through A =        1 α2 α4 α4 α2 1 α4 α4 α4 α4 1 α3 alpha4 α4 α3 1        27
  • 28. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II On copula parametrization • Archimedean copulas Initially, dependence in [0, 1]d ←→ summarized in one functional parameters on [0, 1]. But appears less flexible because of exchangeability features. It is possible to introduce hierarchical Archimedean copulas (see Savu & Trede (2006) or McNeil (2007)). Let U = (U1, U2, U3, U4), C(u1, u2, u3, u4) = φ−1 4 (φ4 φ−1 2 (φ2(u1) + φ2(u2)) + φ4 φ−1 3 (φ3(u3) + φ3(u4)) ), which, if φi is parametrized with parameter αi, can be summarized through A =        1 α2 α4 α4 α2 1 α4 α4 α4 α4 1 α3 alpha4 α4 α3 1        28
  • 29. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II On copula parametrization • Archimedean copulas Initially, dependence in [0, 1]d ←→ summarized in one functional parameters on [0, 1]. But appears less flexible because of exchangeability features. It is possible to introduce hierarchical Archimedean copulas (see Savu & Trede (2006) or McNeil (2007)). Let U = (U1, U2, U3, U4), C(u1, u2, u3, u4) = φ−1 4 (φ4 φ−1 2 (φ2(u1) + φ2(u2)) + φ4 φ−1 3 (φ3(u3) + φ3(u4)) ), which, if φi is parametrized with parameter αi, can be summarized through A =        1 α2 α4 α4 α2 1 α4 α4 α4 α4 1 α3 α4 α4 α3 1        29
  • 30. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II On copula parametrization • Archimedean copulas Initially, dependence in [0, 1]d ←→ summarized in one functional parameters on [0, 1]. But appears less flexible because of exchangeability features. It is possible to introduce hierarchical Archimedean copulas (see Savu & Trede (2006) or McNeil (2007)). Let U = (U1, U2, U3, U4), C(u1, u2, u3, u4) = φ−1 4 (φ4[φ−1 3 (φ3 φ−1 2 (φ2(u1) + φ2(u2)) + φ3(u3))] + φ4(u4)), which, if φi is parametrized with parameter αi, can be summarized through A =        1 α2 α3 α4 α2 1 α3 α4 α3 α3 1 α4 α4 α4 α4 1        30
  • 31. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II On copula parametrization • Archimedean copulas Initially, dependence in [0, 1]d ←→ summarized in one functional parameters on [0, 1]. But appears less flexible because of exchangeability features. It is possible to introduce hierarchical Archimedean copulas (see Savu & Trede (2006) or McNeil (2007)). Let U = (U1, U2, U3, U4), C(u1, u2, u3, u4) = φ−1 4 (φ4[φ−1 3 (φ3 φ−1 2 (φ2(u1) + φ2(u2)) + φ3(u3))] + φ4(u4)), which, if φi is parametrized with parameter αi, can be summarized through A =        1 α2 α3 α4 α2 1 α3 α4 α3 α3 1 α4 α4 α4 α4 1        31
  • 32. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II On copula parametrization • Archimedean copulas Initially, dependence in [0, 1]d ←→ summarized in one functional parameters on [0, 1]. But appears less flexible because of exchangeability features. It is possible to introduce hierarchical Archimedean copulas (see Savu & Trede (2006) or McNeil (2007)). Let U = (U1, U2, U3, U4), C(u1, u2, u3, u4) = φ−1 4 (φ4[φ−1 3 (φ3 φ−1 2 (φ2(u1) + φ2(u2)) + φ3(u3))] + φ4(u4)), which, if φi is parametrized with parameter αi, can be summarized through A =        1 α2 α3 α4 α2 1 α3 α4 α3 α3 1 α4 α4 α4 α4 1        32
  • 33. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II On copula parametrization • Extreme value copulas Here, dependence in [0, 1]d ←→ summarized in one functional parameters on [0, 1]d−1 . Further, focuses only on first order tail dependence. 33
  • 34. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Natural properties for dependence measures Definition 4. κ is measure of concordance if and only if κ satisfies • κ is defined for every pair (X, Y ) of continuous random variables, • −1 ≤ κ (X, Y ) ≤ +1, κ (X, X) = +1 and κ (X, −X) = −1, • κ (X, Y ) = κ (Y, X), • if X and Y are independent, then κ (X, Y ) = 0, • κ (−X, Y ) = κ (X, −Y ) = −κ (X, Y ), • if (X1, Y1) P QD (X2, Y2), then κ (X1, Y1) ≤ κ (X2, Y2), • if (X1, Y1) , (X2, Y2) , ... is a sequence of continuous random vectors that converge to a pair (X, Y ) then κ (Xn, Yn) → κ (X, Y ) as n → ∞. 34
  • 35. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Natural properties for dependence measures If κ is measure of concordance, then, if f and g are both strictly increasing, then κ(f(X), g(Y )) = κ(X, Y ). Further, κ(X, Y ) = 1 if Y = f(X) with f almost surely strictly increasing, and analogously κ(X, Y ) = −1 if Y = f(X) with f almost surely strictly decreasing (see Scarsini (1984)). Rank correlations can be considered, i.e. Spearman’s ρ defined as ρ(X, Y ) = corr(FX(X), FY (Y )) = 12 1 0 1 0 C(u, v)dudv − 3 and Kendall’s τ defined as τ(X, Y ) = 4 1 0 1 0 C(u, v)dC(u, v) − 1. 35
  • 36. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Historical version of those coefficients Similarly Kendall’s tau was not defined using copulae, but as the probability of concordance, minus the probability of discordance, i.e. τ(X, Y ) = 3[P((X1 − X2)(Y1 − Y2) > 0) − P((X1 − X2)(Y1 − Y2) < 0)], where (X1, Y1) and (X2, Y2) denote two independent versions of (X, Y ) (see Nelsen (1999)). Equivalently, τ(X, Y ) = 1 − 4Q n(n2 − 1) where Q is the number of inversions between the rankings of X and Y (number of discordance). 36
  • 37. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II !2.0 !1.5 !1.0 !0.5 0.0 0.5 1.0 !0.50.00.51.01.5 Concordant pairs X Y !2.0 !1.5 !1.0 !0.5 0.0 0.5 1.0 !0.50.00.51.01.5 Discordant pairs X Y Fig. 7 – Concordance versus discordance. 37
  • 38. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Alternative expressions of those coefficients Note that those coefficients can also be expressed as follows ρ(X, Y ) = [0,1]×[0,1] C(u, v) − C⊥ (u, v)dudv [0,1]×[0,1] C+(u, v) − C⊥(u, v)dudv (the normalized average distance between C and C⊥ ), for instance. The case of the Gaussian random vector If (X, Y ) is a Gaussian random vector with correlation r, then (Kruskal (1958)) ρ(X, Y ) = 6 π arcsin r 2 and τ(X, Y ) = 2 π arcsin (r) . 38
  • 39. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II From Kendall’tau to copula parameters Kendall’s τ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Gaussian θ 0.00 0.16 0.31 0.45 0.59 0.71 0.81 0.89 0.95 0.99 1.00 Gumbel θ 1.00 1.11 1.25 1.43 1.67 2.00 2.50 3.33 5.00 10.0 +∞ Plackett θ 1.00 1.57 2.48 4.00 6.60 11.4 21.1 44.1 115 530 +∞ Clayton θ 0.00 0.22 0.50 0.86 1.33 2.00 3.00 4.67 8.00 18.0 +∞ Frank θ 0.00 0.91 1.86 2.92 4.16 5.74 7.93 11.4 18.2 20.9 +∞ Joe θ 1.00 1.19 1.44 1.77 2.21 2.86 3.83 4.56 8.77 14.4 +∞ Galambos θ 0.00 0.34 0.51 0.70 0.95 1.28 1.79 2.62 4.29 9.30 +∞ Morgenstein θ 0.00 0.45 0.90 - - - - - - - - 39
  • 40. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II From Spearman’s rho to copula parameters Spearman’s ρ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Gaussian θ 0.00 0.10 0.21 0.31 0.42 0.52 0.62 0.72 0.81 0.91 1.00 Gumbel θ 1.00 1.07 1.16 1.26 1.38 1.54 1.75 2.07 2.58 3.73 +∞ A.M.H. θ 1.00 1.11 1.25 1.43 1.67 2.00 2.50 3.33 5.00 10.0 +∞ Plackett θ 1.00 1.35 1.84 2.52 3.54 5.12 7.76 12.7 24.2 66.1 +∞ Clayton θ 0.00 0.14 0.31 0.51 0.76 1.06 1.51 2.14 3.19 5.56 +∞ Frank θ 0.00 0.60 1.22 1.88 2.61 3.45 4.47 5.82 7.90 12.2 +∞ Joe θ 1.00 1.12 1.27 1.46 1.69 1.99 2.39 3.00 4.03 6.37 +∞ Galambos θ 0.00 0.28 0.40 0.51 0.65 0.81 1.03 1.34 1.86 3.01 +∞ Morgenstein θ 0.00 0.30 0.60 0.90 - - - - - - - 40
  • 41. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Marges uniformes CopuledeGumbel !2 0 2 4 !2024 Marges gaussiennes Fig. 8 – Simulations of Gumbel’s copula θ = 1.2. 41
  • 42. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Marges uniformes CopuleGaussienne !2 0 2 4 !2024 Marges gaussiennes Fig. 9 – Simulations of the Gaussian copula (θ = 0.95). 42
  • 43. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Strong tail dependence Joe (1993) defined, in the bivariate case a tail dependence measure. Definition 5. Let (X, Y ) denote a random pair, the upper and lower tail dependence parameters are defined, if the limit exist, as λL = lim u→0 P X ≤ F−1 X (u) |Y ≤ F−1 Y (u) , = lim u→0 P (U ≤ u|V ≤ u) = lim u→0 C(u, u) u , and λU = lim u→1 P X > F−1 X (u) |Y > F−1 Y (u) = lim u→0 P (U > 1 − u|V ≤ 1 − u) = lim u→0 C (u, u) u . 43
  • 44. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Gaussian copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 L and R concentration functions L function (lower tails) R function (upper tails) GAUSSIAN q q Fig. 10 – L and R cumulative curves. 44
  • 45. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Gumbel copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 L and R concentration functions L function (lower tails) R function (upper tails) GUMBEL q q Fig. 11 – L and R cumulative curves. 45
  • 46. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Clayton copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 L and R concentration functions L function (lower tails) R function (upper tails) CLAYTON q q Fig. 12 – L and R cumulative curves. 46
  • 47. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Student t copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 L and R concentration functions L function (lower tails) R function (upper tails) STUDENT (df=5) q q Fig. 13 – L and R cumulative curves. 47
  • 48. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Student t copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 L and R concentration functions L function (lower tails) R function (upper tails) STUDENT (df=3) q q Fig. 14 – L and R cumulative curves. 48
  • 49. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Estimating (strong) tail dependence From P ≈ P X > F−1 X (u) , Y > F−1 Y (u) P Y > F−1 Y (u) for u closed to 1, as for Hill’s estimator, a natural estimator for λ is obtained with u = 1 − k/n, λ (k) U = 1 n n i=1 1(Xi > Xn−k:n, Yi > Yn−k:n) 1 n n i=1 1(Yi > Yn−k:n) , hence λ (k) U = 1 k n i=1 1(Xi > Xn−k:n, Yi > Yn−k:n). λ (k) L = 1 k n i=1 1(Xi ≤ Xk:n, Yi ≤ Yk:n). 49
  • 50. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Asymptotic convergence, how fast ? 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 (Upper) tail dependence, Gaussian copula, n=200 Exceedance probability 0.001 0.005 0.050 0.500 0.00.20.40.60.81.0 Log scale, (lower) tail dependence Exceedance probability (log scale) Fig. 15 – Convergence of L and R functions, Gaussian copula, n = 200. 50
  • 51. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Asymptotic convergence, how fast ? 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 (Upper) tail dependence, Gaussian copula, n=200 Exceedance probability 0.001 0.005 0.050 0.500 0.00.20.40.60.81.0 Log scale, (lower) tail dependence Exceedance probability (log scale) Fig. 16 – Convergence of L and R functions, Gaussian copula, n = 2, 000. 51
  • 52. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Asymptotic convergence, how fast ? 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 (Upper) tail dependence, Gaussian copula, n=200 Exceedance probability 0.001 0.005 0.050 0.500 0.00.20.40.60.81.0 Log scale, (lower) tail dependence Exceedance probability (log scale) Fig. 17 – Convergence of L and R functions, Gaussian copula, n = 20, 000. 52
  • 53. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Weak tail dependence If X and Y are independent (in tails), for u large enough P(X > F−1 X (u), Y > F−1 Y (u)) = P(X > F−1 X (u)) · P(Y > F−1 Y (u)) = (1 − u)2 , or equivalently, log P(X > F−1 X (u), Y > F−1 Y (u)) = 2 · log(1 − u). Further, if X and Y are comonotonic (in tails), for u large enough P(X > F−1 X (u), Y > F−1 Y (u)) = P(X > F−1 X (u)) = (1 − u)1 , or equivalently, log P(X > F−1 X (u), Y > F−1 Y (u)) = 1 · log(1 − u). =⇒ limit of the ratio log(1 − u) log P(Z1 > F−1 1 (u), Z2 > F−1 2 (u)) . 53
  • 54. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Weak tail dependence Coles, Heffernan & Tawn (1999) defined Definition 6. Let (X, Y ) denote a random pair, the upper and lower tail dependence parameters are defined, if the limit exist, as ηL = lim u→0 log(u) log P(Z1 ≤ F−1 1 (u), Z2 ≤ F−1 2 (u)) = lim u→0 log(u) log C(u, u) , and ηU = lim u→1 log(1 − u) log P(Z1 > F−1 1 (u), Z2 > F−1 2 (u)) = lim u→0 log(u) log C (u, u) . 54
  • 55. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Gaussian copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Chi dependence functions lower tails upper tails GAUSSIAN q q Fig. 18 – χ functions. 55
  • 56. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Gumbel copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Chi dependence functions lower tails upper tails GUMBEL q q Fig. 19 – χ functions. 56
  • 57. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Clayton copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Chi dependence functions lower tails upper tails CLAYTON q q Fig. 20 – χ functions. 57
  • 58. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Student t copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Chi dependence functions lower tails upper tails STUDENT (df=3) q q Fig. 21 – χ functions. 58
  • 59. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Application in risk management : Loss-ALAE q q qq q q q q q q q q q q q q q q q q q q q q qq qq q q qqq qq q qq q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qqqq q q q qqq qq qq qqq q qq q q q qqqqq q q q q q qq q q q q q q qq qq q q q q q q q q q q q q q qq q q qq q q q q q q qqq q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq qq q q q qq qq qq qq qq q q qqq qqq qq q qq q q q q q q q q q q q q q q q q q q q q q qqqq q qqq qq qqqq q q qqq qq qq qq q qq q qqq qqqq qqq qqqqq qq qqqq qq qq q qq q q q q q q q q q q q q q q q q q q q qqqq qq q q q qq qqq q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q qq qq q qq q q qq q q q q q q qq q q q q qq q q q q q q q qq q q q q q q q qqqq qq q qqqqq q q qqqq qqqqq q qq q q q q qq qqqq qqqqqq q qqq qq qqqqqqqq qq qqqq q qqq q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q qq qqq qq qqq q q q q qq q qq qq q qq qq q qq q qqq qqq q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q qqq qq q q q qqqqqqq qqq q qq qqq q qqq qqq qq qq qq q qqq qqq qqqqqqqqqqq qqq qq qqq qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q qq qq qqqq q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q qq qq qqq qqqqqqqqq q qq qqq q qq qq qqqq qq q qq q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q qq qq q q q q q q q qqq qq qq qq qqq q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qq qq q q q q q q qq q qqqqq q qq q q qq q q qqq qqqqq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q qq q q q q q q q qq q q q qq qq q qq qq q q q q qq qqq qq q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq qq qqqq qqqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q qq q q q qq q q q q q q q q q q q q qq q q q qqqqq q q q q q q q q q q q q q q q q q q q qqq qq q q q q qq 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Loss AllocatedExpenses Fig. 22 – Losses and allocated expenses. 59
  • 60. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Application in risk management : Loss-ALAE 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 L and R concentration functions L function (lower tails) R function (upper tails) qqq qqq qqq qqqqqqqqq qqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqq qqqqqq qqqq qqq qqqqqqq qqq qqqqq qq qqq qq q qq q q q q q q q q q q q q q Gumbel copula q q 0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 Chi dependence functions lower tails upper tails q q q qqq qqq qqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqq qqqqq qq qqqqqqqqqqqqqq qqq q q q q Gumbel copula q q Fig. 23 – L and R cumulative curves, and χ functions. 60
  • 61. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Application in risk management : car-household q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Car claims Householdclaims Fig. 24 – Motor and Household claims. 61
  • 62. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Application in risk management : car-household 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 L and R concentration functions L function (lower tails) R function (upper tails) qqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq qqq q qq qq qqqqqqqqqqqqqqq qqqqqqqq qqqqqqqqqqqqqq qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqq qqqq q qqq qqqq qqq qqq qqqqq qqqqq qq qqqqqqq q qqqq qqq q qq q q q q q q q q q q q q q q q q q q q q q q q q q Gumbel copula q q 0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 Chi dependence functions lower tails upper tails qqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq qqq q qqqqqqqqqqqqqqqqqqq qqqqqqqq qqqqqqqqqqqqqq qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqq qqqqqqq qqqqqqq q q q q q q q q q q q q q Gumbel copula q q Fig. 25 – L and R cumulative curves, and χ functions. 62
  • 63. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Case of Archimedean copulas For an exhaustive study of tail behavior for Archimedean copulas, see Charpentier & Segers (2008). • upper tail : function of φ (1) and θ1 = − lim s→0 sφ (1 − s) φ(1 − s) , ◦ φ (1) < 0 : tail independence ◦ φ (1) = 0 and θ1 = 1 : dependence in independence ◦ φ (1) = 0 and θ1 > 1 : tail dependence • lower tail : function of φ(0) and θ0 = − lim s→0 sφ (s) φ(s) , ◦ φ(0) < ∞ : tail independence ◦ φ(0) = ∞ and θ0 = 0 : dependence in independence ◦ φ(0) = ∞ and θ0 > 0 : tail dependence 0.0 0.2 0.4 0.6 0.8 1.0 05101520 63
  • 64. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Measuring risks ? the pure premium as a technical benchmark Pascal, Fermat, Condorcet, Huygens, d’Alembert in the XVIIIth century proposed to evaluate the “produit scalaire des probabilit´es et des gains”, < p, x >= n i=1 pixi = n i=1 P(X = xi) · xi = EP(X), based on the “r`egle des parties”. For Qu´etelet, the expected value was, in the context of insurance, the price that guarantees a financial equilibrium. From this idea, we consider in insurance the pure premium as EP(X). As in Cournot (1843), “l’esp´erance math´ematique est donc le juste prix des chances” (or the “fair price” mentioned in Feller (1953)). Problem : Saint Peterburg’s paradox, i.e. infinite mean risks (cf. natural catastrophes) 64
  • 65. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II the pure premium as a technical benchmark For a positive random variable X, recall that EP(X) = ∞ 0 P(X > x)dx. q q q q q q q q q q q 0 2 4 6 8 10 0.00.20.40.60.81.0 Expected value Loss value, X Probabilitylevel,P Fig. 26 – Expected value EP(X) = xdFX(x) = P(X > x)dx. 65
  • 66. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II from pure premium to expected utility principle Ru(X) = u(x)dP = P(u(X) > x))dx where u : [0, ∞) → [0, ∞) is a utility function. Example with an exponential utility, u(x) = [1 − e−αx ]/α, Ru(X) = 1 α log EP(eαX ) , i.e. the entropic risk measure. See Cramer (1728), Bernoulli (1738), von Neumann & Morgenstern (1944), Rochet (1994)... etc. 66
  • 67. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Distortion of values versus distortion of probabilities q q q q q q q q q q q 0 2 4 6 8 10 0.00.20.40.60.81.0 Expected utility (power utility function) Loss value, X Probabilitylevel,P Fig. 27 – Expected utility u(x)dFX(x). 67
  • 68. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Distortion of values versus distortion of probabilities q q q q q q q q q q q 0 2 4 6 8 10 0.00.20.40.60.81.0 Expected utility (power utility function) Loss value, X Probabilitylevel,P Fig. 28 – Expected utility u(x)dFX(x). 68
  • 69. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II from pure premium to distorted premiums (Wang) Rg(X) = xdg ◦ P = g(P(X > x))dx where g : [0, 1] → [0, 1] is a distorted function. Example • if g(x) = I(X ≥ 1 − α) Rg(X) = V aR(X, α), • if g(x) = min{x/(1 − α), 1} Rg(X) = TV aR(X, α) (also called expected shortfall), Rg(X) = EP(X|X > V aR(X, α)). See D’Alembert (1754), Schmeidler (1986, 1989), Yaari (1987), Denneberg (1994)... etc. Remark : Rg(X) might be denoted Eg◦P. But it is not an expected value since Q = g ◦ P is not a probability measure. 69
  • 70. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Distortion of values versus distortion of probabilities q q q q q q q q q q q 0 2 4 6 8 10 0.00.20.40.60.81.0 Distorted premium beta distortion function) Loss value, X Probabilitylevel,P Fig. 29 – Distorted probabilities g(P(X > x))dx. 70
  • 71. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Distortion of values versus distortion of probabilities q q q q q q q q q q q 0 2 4 6 8 10 0.00.20.40.60.81.0 Distorted premium beta distortion function) Loss value, X Probabilitylevel,P Fig. 30 – Distorted probabilities g(P(X > x))dx. 71
  • 72. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II some particular cases a classical premiums The exponential premium or entropy measure : obtained when the agent as an exponential utility function, i.e. π such that U(ω − π) = EP(U(ω − S)), U(x) = − exp(−αx), i.e. π = 1 α log EP(eαX ). Esscher’s transform (see Esscher ( 1936), B¨uhlmann ( 1980)), π = EQ(X) = EP(X · eαX ) EP(eαX) , for some α > 0, i.e. dQ dP = eαX EP(eαX) . Wang’s premium (see Wang ( 2000)), extending the Sharp ratio concept E(X) = ∞ 0 F(x)dx and π = ∞ 0 Φ(Φ−1 (F(x)) + λ)dx 72
  • 73. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Risk measures The two most commonly used risk measures for a random variable X (assuming that a loss is positive) are, q ∈ (0, 1), • Value-at-Risk (VaR), V aRq(X) = inf{x ∈ R, P(X > x) ≤ α}, • Expected Shortfall (ES), Tail Conditional Expectation (TCE) or Tail Value-at-Risk (TVaR) TV aRq(X) = E (X|X > V aRq(X)) , Artzner, Delbaen, Eber & Heath (1999) : a good risk measure is subadditive, TVaR is subadditive, VaR is not subadditive (in general). 73
  • 74. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Risk measures and diversification Any copula C is bounded by Frchet-Hoeffding bounds, max d i=1 ui − (d − 1), 0 ≤ C(u1, . . . , ud) ≤ min{u1, . . . , ud}, and thus, any distribution F on F(F1, . . . , Fd) is bounded max d i=1 Fi(xi) − (d − 1), 0 ≤ F(x1, . . . , xd) ≤ min{F1(x1), . . . , Ff (xd)}. Does this means the comonotonicity is always the worst-case scenario ? Given a random pair (X, Y ), let (X− , Y − ) and (X+ , Y + ) denote contercomonotonic and comonotonic versions of (X, Y ), do we have R(φ(X− , Y − )) ? ≤ R(φ(X, Y ) ) ? ≤ R(φ(X+ , Y + )). 74
  • 75. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Tchen’s theorem and bounding some pure premiums If φ : R2 → R is supermodular, i.e. φ(x2, y2) − φ(x1, y2) − φ(x2, y1) + φ(x1, y1) ≥ 0, for any x1 ≤ x2 and y1 ≤ y2, then if (X, Y ) ∈ F(FX, FY ), E φ(X− , Y − ) ≤ E (φ(X, Y )) ≤ E φ(X+ , Y + ) , as proved in Tchen (1981). Example 7. the stop loss premium for the sum of two risks E((X + Y − d)+) is supermodular. 75
  • 76. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Example 8. For the n-year joint-life annuity, axy:n = n k=1 vk P(Tx > k and Ty > k) = n k=1 vk kpxy. Then a− xy:n ≤ axy:n ≤ a+ xy:n , where a− xy:n = n k=1 vk max{kpx + kpy − 1, 0}( lower Frchet bound ), a+ xy:n = n k=1 vk min{kpx, kpy}( upper Frchet bound ). 76
  • 77. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II Makarov’s theorem and bounding Value-at-Risk In the case where R denotes the Value-at-Risk (i.e. quantile function of the P&L distribution), R− ≤ R(X− + Y − )≤R(X + Y )≤R(X+ + Y + ) ≤ R+ , where e.g. R+ can exceed the comonotonic case. Recall that R(X + Y ) = VaRq[X + Y ] = F−1 X+Y (q) = inf{x ∈ R|FX+Y (x) ≥ q}. Proposition 9. Let (X, Y ) ∈ F(FX, FY ) then for all s ∈ R, τC− (FX, FY )(s) ≤ P(X + Y ≤ s) ≤ ρC− (FX, FY )(s), where τC(FX, FY )(s) = sup x,y∈R {C(FX(x), FY (y)), x + y = s} and, if ˜C(u, v) = u + v − C(u, v), ρC(FX, FY )(s) = inf x,y∈R { ˜C(FX(x), FY (y)), x + y = s}. 77
  • 78. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II 0.0 0.2 0.4 0.6 0.8 1.0 !4!2024 Bornes de la VaR d’un portefeuille Somme de 2 risques Gaussiens Fig. 31 – Value-at-Risk for 2 Gaussian risks N(0, 1). 78
  • 79. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II 0.90 0.92 0.94 0.96 0.98 1.00 0123456 Bornes de la VaR d’un portefeuille Somme de 2 risques Gaussiens Fig. 32 – Value-at-Risk for 2 Gaussian risks N(0, 1). 79
  • 80. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II 0.0 0.2 0.4 0.6 0.8 1.0 05101520 Bornes de la VaR d’un portefeuille Somme de 2 risques Gamma Fig. 33 – Value-at-Risk for 2 Gamma risks G(3, 1). 80
  • 81. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II 0.90 0.92 0.94 0.96 0.98 1.00 05101520 Bornes de la VaR d’un portefeuille Somme de 2 risques Gamma Fig. 34 – Value-at-Risk for 2 Gamma risks G(3, 1). 81
  • 82. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II The more correlated, the more risky ? Will the risk of the portfolio increase with correlation ? Recall the following theoretical result : Proposition 10. Assume that X and X are in the same Fr´echet space (i.e. Xi L = Xi), and define S = X1 + · · · + Xn and S = X1 + · · · + Xn. If X X for the concordance order, then S T V aR S for the stop-loss or TVaR order. A consequence is that if X and X are exchangeable, corr(Xi, Xj) ≤ corr(Xi, Xj) =⇒ TV aR(S, p) ≤ TV aR(S , p), for all p ∈ (0, 1). See M¨uller & Stoyen (2002) for some possible extensions. 82
  • 83. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II The more correlated, the more risky ? Consider • d lines of business, • simply a binomial distribution on each line of business, with small loss probability (e.g. π = 1/1000). Let    1 if there is a claim on line i 0 if not , and S = X1 + · · · + Xd. Will the correlation among the Xi’s increase the Value-at-Risk of S ? Consider a probit model, i.e. Xi = 1(Xi ≤ ui), where X ∼ N(0, Σ), i.e. a Gaussian copula. Assume that Σ = [σi,j] where σi,j = ρ ∈ [−1, 1] when i = j. 83
  • 84. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II The more correlated, the more risky ? Fig. 35 – 99.75% TVaR (or expected shortfall) for Gaussian copulas. 84
  • 85. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II The more correlated, the more risky ? Fig. 36 – 99% TVaR (or expected shortfall) for Gaussian copulas. 85
  • 86. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II The more correlated, the more risky ? What about other risk measures, e.g. Value-at-Risk ? corr(Xi, Xj) ≤ corr(Xi, Xj) V aR(S, p) ≤ V aR(S , p), for all p ∈ (0, 1). (see e.g. Mittnik & Yener (2008)). 86
  • 87. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II The more correlated, the more risky ? Fig. 37 – 99.75% VaR for Gaussian copulas. 87
  • 88. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II The more correlated, the more risky ? Fig. 38 – 99% VaR for Gaussian copulas. 88
  • 89. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II The more correlated, the more risky ? What could be the impact of tail dependence ? Previously, we considered a Gaussian copula, i.e. tail independence. What if there was tail dependence ? Consider the case of a Student t-copula, with ν degrees of freedom. 89
  • 90. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II The more correlated, the more risky ? Fig. 39 – 99.75% TVaR (or expected shortfall) for Student t-copulas. 90
  • 91. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II The more correlated, the more risky ? Fig. 40 – 99% TVaR (or expected shortfall) for Student t-copulas. 91
  • 92. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II The more correlated, the more risky ? Fig. 41 – 99.75% VaR for Student t-copulas. 92
  • 93. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II The more correlated, the more risky ? Fig. 42 – 99% VaR for Student t-copulas. 93
  • 94. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II A possible conclusion • (standard) correlation is definitively not an appropriate tool to describe dependence features, ◦ in order to fully describe dependence, use copulas, ◦ since major focus in risk management is related to extremal event, focus on tail dependence meausres, • which copula can be appropriate ? ◦ Elliptical copulas offer a nice and simple parametrization, based on pairwise comparison, ◦ Archimedean copulas might be too restrictive, but possible to introduce Hierarchical Archimedean copulas, • Value-at-Risk might yield to non-intuitive results, ◦ need to get a better understanding about Value-at-Risk pitfalls, ◦ need to consider alternative downside risk measures (namely TVaR). 94

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