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• 1. Arthur CHARPENTIER - Extremes and correlation in risk management Extremes and dependence in the context of Solvency II for insurance companies Arthur Charpentier Universite de Rennes 1 & Ecole Polytechnique http ://blogperso.univ-rennes1.fr/arthur.charpentier/ 1
• 2. Arthur CHARPENTIER - Extremes and correlation in risk management On risk dependence in QISs http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 2
• 3. Arthur CHARPENTIER - Extremes and correlation in risk management On risk dependence in QISs http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 3
• 4. Arthur CHARPENTIER - Extremes and correlation in risk management On risk dependence in QISs http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 4
• 5. Arthur CHARPENTIER - Extremes and correlation in risk management On risk dependence in QISs http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 5
• 6. Arthur CHARPENTIER - Extremes and correlation in risk management 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, 2 ), i.e. Y = exp(Y ) where Y N(0, 2 ) Recall that corr(X , Y ) takes any value in [1, +1]. Since corr(X, Y )=corr(X , Y ), what can be corr(X, Y ) ? 6
• 7. Arthur CHARPENTIER - Extremes and correlation in risk management 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, 2 ). 7
• 8. Arthur CHARPENTIER - Extremes and correlation in risk management 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, 2 ), Gaussian copula, r = 0.5. 8
• 9. Arthur CHARPENTIER - Extremes and correlation in risk management What about ocial actuarial documents ? 9
• 10. Arthur CHARPENTIER - Extremes and correlation in risk management What about ocial actuarial documents ? 10
• 11. Arthur CHARPENTIER - Extremes and correlation in risk management What about ocial actuarial documents ? 11
• 12. Arthur CHARPENTIER - Extremes and correlation in risk management What about regulatory technical documents ? 12
• 13. Arthur CHARPENTIER - Extremes and correlation in risk management What about regulatory technical documents ? 13
• 14. Arthur CHARPENTIER - Extremes and correlation in risk management What about regulatory technical documents ? 14
• 15. Arthur CHARPENTIER - Extremes and correlation in risk management What about regulatory technical documents ? 15
• 16. Arthur CHARPENTIER - Extremes and correlation in risk management Motivations : dependence and copulas Denition 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 Fis are continuous, then C is unique, and given by C(u) = F(F1 1 (u1), . . . , F1 d (ud)) for all ui [0, 1] We will then dene the copula of F, or the copula of X. 16
• 17. Arthur CHARPENTIER - Extremes and correlation in risk management Copula density Level curves of the copula Fig. 3 Graphical representation of a copula, C(u, v) = P(U u, V v). 17
• 18. Arthur CHARPENTIER - Extremes and correlation in risk management Copula density Level curves of the copula Fig. 4 Density of a copula, c(u, v) = 2 C(u, v) uv . 18
• 19. Arthur CHARPENTIER - Extremes and correlation in risk management Some very classical copulas The independent copula C(u, v) = uv = C (u, v). The copula is standardly denoted , 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 = (F1 X (U), F1 Y (U)) where U is U([0, 1]). 19
• 20. Arthur CHARPENTIER - Extremes and correlation in risk management 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 = (F1 X (1 U), F1 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) 20
• 21. Arthur CHARPENTIER - Extremes and correlation in risk management 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 Frchet 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 Frchet 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 Frchet!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 Frchet!Hoeffding bound Fig. 5 Contercomontonce, independent, and comonotone copulas. 21
• 22. Arthur CHARPENTIER - Extremes and correlation in risk management 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, ). The Gaussian copula, with parameter (1, 1), C(u, v) = 1 2 1 2 1 (u) 1 (v) exp (x2 2xy + y2 ) 2(1 2) 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 (1, 1) and 2, C(u, v) = 1 2 1 2 t1 (u) t