Arthur CHARPENTIER - Extremes and correlation in risk management Extremes and dependence in the context of Solvency II for insurance companies Arthur Charpentier e de Rennes 1 & ´ Universit´ Ecole Polytechnique http ://blogperso.univ-rennes1.fr/arthur.charpentier/ 1
Arthur CHARPENTIER - Extremes and correlation in risk management On risk dependence in QIS’s http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 2
Arthur CHARPENTIER - Extremes and correlation in risk management On risk dependence in QIS’s http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 3
Arthur CHARPENTIER - Extremes and correlation in risk management On risk dependence in QIS’s http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 4
Arthur CHARPENTIER - Extremes and correlation in risk management On risk dependence in QIS’s http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF 5
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 , σ 2 ), i.e. Y = exp( Y ⋆ ) where Y ⋆ ∼ N (0 , σ 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
Arthur CHARPENTIER - Extremes and correlation in risk management How to capture dependence in risk models ? 1.0 0.5 Correlation 0.0 −0.5 0 1 2 3 4 5 Standard deviation, sigma Fig. 1 – Range for the correlation, cor ( X, Y ), X ∼ LN (0 , 1) , Y ∼ LN (0 , σ 2 ). 7
Arthur CHARPENTIER - Extremes and correlation in risk management How to capture dependence in risk models ? 1.0 0.5 Correlation 0.0 −0.5 0 1 2 3 4 5 Standard deviation, sigma Fig. 2 – cor ( X, Y ), X ∼ LN (0 , 1) , Y ∼ LN (0 , σ 2 ), Gaussian copula, r = 0 . 5. 8
Arthur CHARPENTIER - Extremes and correlation in risk management What about official actuarial documents ? 9
Arthur CHARPENTIER - Extremes and correlation in risk management What about official actuarial documents ? 10
Arthur CHARPENTIER - Extremes and correlation in risk management What about official actuarial documents ? 11
Arthur CHARPENTIER - Extremes and correlation in risk management What about regulatory technical documents ? 12
Arthur CHARPENTIER - Extremes and correlation in risk management What about regulatory technical documents ? 13
Arthur CHARPENTIER - Extremes and correlation in risk management What about regulatory technical documents ? 14
Arthur CHARPENTIER - Extremes and correlation in risk management What about regulatory technical documents ? 15
Arthur CHARPENTIER - Extremes and correlation in risk management 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 F 1 , . . . , F d be d marginal distributions, then F ( x ) = C ( F 1 ( x 1 ) , . . . , F d ( x d )) is a distribution function, with F ∈ F ( F 1 , . . . , F d ) . Conversely, if F ∈ F ( F 1 , . . . , F d ) , there exists C such that F ( x ) = C ( F 1 ( x 1 ) , . . . , F d ( x d )) . Further, if the F i ’s are continuous, then C is unique, and given by C ( u ) = F ( F − 1 ( u 1 ) , . . . , F − 1 d ( u d )) for all u i ∈ [0 , 1] 1 We will then define the copula of F , or the copula of X . 16
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
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 ) . ∂u∂v 18
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 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 ⊥ ( u 1 , . . . , u d ) = u 1 × . . . × u d 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 L = ( F − 1 X ( U ) , F − 1 such that Y = h ( X ), or equivalently ( X, Y ) Y ( U )) where U is U ([0 , 1]). 19
Arthur CHARPENTIER - Extremes and correlation in risk management Some very classical copulas In higher dimension, C + ( u 1 , . . . , u d ) = min { u 1 , . . . , u d } 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 L = ( F − 1 X (1 − U ) , F − 1 such that Y = h ( X ), or equivalently ( X, Y ) Y ( U )). In higher dimension, C − ( u 1 , . . . , u d ) = max { u 1 + . . . + u d − ( d − 1) , 0 } is not a copula. But note that for any copula C , C − ( u 1 , . . . , u d ) ≤ C ( u 1 , . . . , u d ) ≤ C + ( u 1 , . . . , u d ) 20
Arthur CHARPENTIER - Extremes and correlation in risk management 1 1 1 d 8 a 8 d n . l . 8 u 0 u 0 n . u 0 o p o o b c b 6 6 r . e . 6 e 0 0 r c e . w 0 n p o e p l 4 d 4 u 4 t . 0 n . 0 e e t . e 0 h p h c e e 2 d 2 c 2 r . n 0 . e F 0 . I r 0 F 0 0 0 0.8 0.8 0.8 0.6 0.6 0.8 . 8 0 0.6 . 8 0 0.6 0 . 6 0.4 u_2 0.4 6 u_2 0.4 0 . u_2 4 0.4 0 . 1 u_1 _ 4 u 0 . 0.2 0.2 _ 1 0.2 u 2 0.2 0 . 2 0 . Fréchet Lower Bound Independent copula Fréchet Upper Bound 1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Scatterplot, Lower Fréchet ! Hoeffding bound Scatterplot, Indepedent copula random generation Scatterplot, Upper Fréchet ! Hoeffding bound 1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 5 – Contercomontonce, independent, and comonotone copulas. 21
Arthur CHARPENTIER - Extremes and correlation in risk management Elliptical (Gaussian and t ) copulas The idea is to extend the multivariate probit model, X = ( X 1 , . . . , X d ) with i ≤ u i ), where X ⋆ ∼ N ( I , Σ ). marginal B ( p i ) distributions, modeled as Y i = 1 ( X ⋆ • The Gaussian copula, with parameter α ∈ ( − 1 , 1), � − ( x 2 − 2 αxy + y 2 ) � � Φ − 1 ( u ) � Φ − 1 ( v ) 1 √ C ( u, v ) = exp dxdy. 2(1 − α 2 ) 1 − α 2 2 π −∞ −∞ 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, � � − (( ν +2) / 2) � t − 1 � t − 1 1 + x 2 − 2 αxy + y 2 ( u ) ( v ) 1 ν ν √ C ( u, v ) = dxdy. 2(1 − α 2 ) 1 − α 2 2 π −∞ −∞ 22
Arthur CHARPENTIER - Extremes and correlation in risk management 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 ) = ( G X ( x )) θ and P ( Y ≤ y | Θ = θ ) = ( G Y ( y )) θ for some baseline distribution functions G X and G Y . Then F ( x, y ) = ψ ( ψ − 1 ( F X ( x )) + ψ − 1 ( F Y ( y ))) , where ψ denotes the Laplace transform of Θ, i.e. ψ ( t ) = E ( e − t Θ ). 23
Arthur CHARPENTIER - Extremes and correlation in risk management Conditional independence, continuous risk factor Conditional independence, continuous risk factor 100 3 2 80 1 60 0 40 ! 1 20 ! 2 ! 3 0 0 20 40 60 80 100 ! 3 ! 2 ! 1 0 1 2 3 Fig. 6 – Continuous classes of risks, ( X i , Y i ) and (Φ − 1 ( F X ( X i )) , Φ − 1 ( F Y ( Y i ))). 24
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