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  • 1. Arthur CHARPENTIER, Distortion in actuarial sciences Distorting probabilities in actuarial sciences Arthur Charpentier Université Rennes 1 arthur.charpentier@univ-rennes1.fr http ://freakonometrics.blog.free.fr/ Univeristé Laval, Québec, Avril 2011 1
  • 2. Arthur CHARPENTIER, Distortion in actuarial sciences1 Decision theory and distorted risk measuresConsider a preference ordering among risks, such that1. ˜ L ˜ L ˜ ˜ is distribution based, i.e. if X Y , ∀X = X Y = Y , then X Y ; hence, we can write FX FY2. is total, reflexive and transitive,3. is continuous, i.e. ∀FX , FY and FZ such that FX FY FZ , ∃λ, µ ∈ (0, 1) such that λFX + (1 − λ)FZ FY µFX + (1 − µ)FZ .4. satisfies an independence axiom, i.e. ∀FX , FY and FZ , and ∀λ ∈ (0, 1), FX FY =⇒ λFX + (1 − λ)FZ λFY + (1 − λ)FZ .5. satisfies an ordering axiom, ∀X and Y constant (i.e. P(X = x) = P(Y = y) = 1, FX FY =⇒ x ≤ y. 2
  • 3. Arthur CHARPENTIER, Distortion in actuarial sciencesTheorem1Ordering satisfies axioms 1-2-3-4-5 if and only if ∃u : R → R, continuous, strictlyincreasing and unique (up to an increasing affine transformation) such that ∀FX and FY : FX FY ⇔ u(x)dFX (x) ≤ u(x)dFY (x) R R ⇔ E[u(X)] ≤ E[u(Y )].But if we consider an alternative to the independence axiom4’. satisfies an dual independence axiom, i.e. ∀FX , FY and FZ , and ∀λ ∈ (0, 1), −1 −1 −1 −1 FX FY =⇒ [λFX + (1 − λ)FZ ]−1 [λFY + (1 − λ)FZ ]−1 .we (Yaari (1987)) obtain a dual representation theorem,Theorem2Ordering satisfies axioms 1-2-3-4’-5 if and only if ∃g : [0, 1] → R, continuous, strictlyincreasing such that ∀FX and FY : FX FY ⇔ g(F X (x))dx ≤ g(F Y (x))dx R R 3
  • 4. Arthur CHARPENTIER, Distortion in actuarial sciencesStandard axioms required on risque measures R : X → R, L– law invariance, X = Y =⇒ R(X) = R(Y )– increasing X ≥ Y =⇒ R(X) ≥ R(Y ),– translation invariance ∀k ∈ R, =⇒ R(X + k) = R(X) + k,– homogeneity ∀λ ∈ R+ , R(λX) = λ · R(X),– subadditivity R(X + Y ) ≤ R(X) + R(Y ),– convexity ∀β ∈ [0, 1], R(βλX + [1 − β]Y ) ≤ β · R(X) + [1 − β] · R(Y ).– additivity for comonotonic risks ∀X and Y comonotonic, R(X + Y ) = R(X) + R(Y ),– maximal correlation (w.r.t. measure µ) ∀X, R(X) = sup {E(X · U ) where U ∼ µ} ˜ ˜ ˜ L– strong coherence ∀X and Y , sup{R(X + Y )} = R(X) + R(Y ), where X = X ˜ L and Y = Y . 4
  • 5. Arthur CHARPENTIER, Distortion in actuarial sciencesProposition1If R is a monetary convex fonction, then the three statements are equivalent,– R is strongly coherent,– R is additive for comonotonic risks,– R is a maximal correlation measure.Proposition2A coherente risk measure R is additive for comonotonic risks if and only if there exists adecreasing positive function φ on [0, 1] such that 1 R(X) = φ(t)F −1 (1 − t)dt 0where F (x) = F(X ≤ x).see Kusuoka (2001), i.e. R is a spectral risk measure. 5
  • 6. Arthur CHARPENTIER, Distortion in actuarial sciencesDefinition1A distortion function is a function g : [0, 1] → [0, 1] such that g(0) = 0 and g(1) = 1.For positive risks,Definition1Given distortion function g, Wang’s risk measure, denoted Rg , is ∞ ∞ Rg (X) = g (1 − FX (x)) dx = g F X (x) dx (1) 0 0Proposition1Wang’s risk measure can be defined as 1 1 −1 Rg (X) = FX (1 − α) dg(α) = VaR[X; 1 − α] dg(α). (2) 0 0 6
  • 7. Arthur CHARPENTIER, Distortion in actuarial sciencesMore generally (risks taking value in R)Definition2We call distorted risk measure 1 R(X) = F −1 (1 − u)dg(u) 0where g is some distortion function.Proposition3R(X) can be written +∞ 0 R(X) = g(1 − F (x))dx − [1 − g(1 − F (x))]dx. 0 −∞ 7
  • 8. Arthur CHARPENTIER, Distortion in actuarial sciences risk measures R distortion function g VaR g (x) = I[x ≥ p] Tail-VaR g (x) = min {x/p, 1} PH g (x) = xp 1/p Dual Power g (x) = 1 − (1 − x) Gini g (x) = (1 + p) x − px2 exponential transform g (x) = (1 − px ) / (1 − p) Table 1 – Standard risk measures, p ∈ (0, 1). 8
  • 9. Arthur CHARPENTIER, Distortion in actuarial sciencesHere, it looks like risk measures can be seen as R(X) = Eg◦P (X).Remark1Let Q denote the distorted measure induced by g on P, denoted g ◦ P i.e. Q([a, +∞)) = g(P([a, +∞))).Since g is increasing on [0, 1] Q is a capacity.Example1Consider function g(x) = xk . The PH - proportional hazard - risk measure is 1 ∞ R(X; k) = F −1 (1 − u)kuk−1 du = [F (x)]k dx 0 0If k is an integer [F (x)]k is the survival distribution of the minimum over k values.Definition2The Esscher risk measure with parameter h > 0 is Es[X; h], defined as E[X exp(hX)] d Es[X; h] = = ln MX (h). MX (h) dh 9
  • 10. Arthur CHARPENTIER, Distortion in actuarial sciences2 Archimedean copulasDefinition3Let φ denote a decreasing function (0, 1] → [0, ∞] such that φ(1) = 0, and such thatφ−1 is d-monotone, i.e. for all k = 0, 1, · · · , d, (−1)k [φ−1 ](k) (t) ≥ 0 for all t. Definethe inverse (or quasi-inverse if φ(0) < ∞) as   φ−1 (t) for 0 ≤ t ≤ φ(0) φ−1 (t) =  0 for φ(0) < t < ∞.The function C(u1 , · · · , un ) = φ−1 (φ(u1 ) + · · · + φ(ud )), u1 , · · · , un ∈ [0, 1],is a copula, called an Archimedean copula, with generator φ.Let Φd denote the set of generators in dimension d.Example2The independent copula C ⊥ is an Archimedean copula, with generator φ(t) = − log t. 10
  • 11. Arthur CHARPENTIER, Distortion in actuarial sciencesThe upper Fréchet-Hoeffding copula, defined as the minimum componentwise,M (u) = min{u1 , · · · , ud }, is not Archimedean (but can be obtained as the limit ofsome Archimedean copulas).Set λ(t) = exp[−φ(t)] (the multiplicative generator), then C(u1 , ..., ud ) = λ−1 (λ(u1 ) · · · λ(ud )), ∀u1 , ..., ud ∈ [0, 1],which can be written C(u1 , ..., ud ) = λ−1 (C ⊥ [λ(u1 ), . . . , λ(ud )]), ∀u1 , ..., ud ∈ [0, 1],Note that it is possible to get an interpretation of that distortion of theindependence.A large subclass of Archimedean copula in dimension d is the class ofArchimedean copulas obtained using the frailty approach.Consider random variables X1 , · · · , Xd conditionally independent, given a latent Θfactor Θ, a positive random variable, such that P (Xi ≤ xi |Θ) = Gi (x) whereGi denotes a baseline distribution function. 11
  • 12. Arthur CHARPENTIER, Distortion in actuarial sciencesThe joint distribution function of X is given byFX (x1 , · · · , xd ) = E (P (X1 ≤ x1 , · · · , Xd ≤ Xd |Θ)) d d Θ = E P (Xi ≤ xi |Θ) =E Gi (xi ) i=1 i=1 d d = E exp [−Θ (− log Gi (xi ))] =ψ − log Gi (xi ) , i=1 i=1where ψ is the Laplace transform of the distribution of Θ, i.e.ψ (t) = E (exp (−tΘ)) . Because the marginal distributions are given respectivelyby Fi (xi ) = P(Xi ≤ xi ) = ψ (− log Gi (xi )) ,the copula of X is −1 −1 C (u) = FX F1 (u1 ) , · · · , Fd (ud ) = ψ ψ −1 (u) + · · · + ψ −1 (ud )This copula is an Archimedean copula with generator φ = ψ −1 (see e.g. Clayton(1978), Oakes (1989), Bandeen-Roche & Liang (1996) for more details). 12
  • 13. Arthur CHARPENTIER, Distortion in actuarial sciences3 Hierarchical Archimedean copulasIt is possible to look at C(u1 , · · · , ud ) defined as φ−1 [φ1 [φ−1 (φ2 [· · · φ−1 [φd−1 (u1 ) + φd−1 (u2 )] + · · · + φ2 (ud−1 ))] + φ1 (ud )] 1 2 d−1where φi are generators. C is a copula if φi ◦ φ−1 is the inverse of a Laplace i−1transform. This copula is said to be a fully nested Archimedean (FNA) copula.E.g. in dimension d = 5, we getφ1 [φ1 (φ−1 [φ2 (φ−1 [φ3 (φ−1 [φ4 (u1 ) + φ4 (u2 )]) + φ3 (u3 )]) + φ2 (u4 )]) + φ1 (u5 )]. −1 2 3 4It is also possible to consider partially nested Archimedean (PNA) copulas, e.g.by coupling (U1 , U2 , U3 ), and (U4 , U5 ),φ−1 [φ4 (φ−1 [φ1 (φ−1 [φ2 (u1 ) + φ2 (u2 )]) + φ1 (u3 )]) + φ4 (φ−1 [φ3 (u4 ) + φ3 (u5 )])] 4 1 2 3Again, it is a copula if φ2 ◦ φ−1 is the inverse of a Laplace transform, as well as 1φ4 ◦ φ−1 and φ4 ◦ φ−1 . 1 3 13
  • 14. Arthur CHARPENTIER, Distortion in actuarial sciences φ1 φ2 φ4 φ3 φ1 φ4 φ2 φ3 U1 U2 U3 U4 U5 U1 U2 U3 U4 U5Figure 1 – fully nested Archimedean copula, and partially nested Archimedeancopula. 14
  • 15. Arthur CHARPENTIER, Distortion in actuarial sciencesIt is also possible to consider φ−1 [φ3 (φ−1 [φ1 (u1 ) + φ1 (u2 ) + φ1 (u3 )]) + φ3 (φ−1 [φ2 (u4 ) + φ2 (u5 )])]. 3 1 2if φ3 ◦ φ−1 and φ3 ◦ φ−1 are inverses of Laplace transform. Or 1 2 φ−1 [φ3 (φ−1 [φ1 (u1 ) + φ1 (u2 )] + φ3 (u3 ) + φ3 (φ−1 [φ2 (u4 ) + φ2 (u5 )])]. 3 1 2 15
  • 16. Arthur CHARPENTIER, Distortion in actuarial sciences φ3 φ3 φ1 φ2 φ1 φ2 U1 U2 U3 U4 U5 U1 U2 U3 U4 U5Figure 2 – Copules Archimédiennes hiérarchiques avec deux constructions dif-férentes. 16
  • 17. Arthur CHARPENTIER, Distortion in actuarial sciencesExample3If φi ’s are Gumbel’s generators, with parameter θi , a sufficient condition for C to be aFNA copula is that θi ’s increasing. Similarly if φi ’s are Clayton’s generators.Again, an heuristic interpretation can be derived, see Hougaard (2000), with twofrailties Θ1 and Θ2 such that 17
  • 18. Arthur CHARPENTIER, Distortion in actuarial sciences4 Distorting copulasGenest & Rivest (2001) extended the concept of Archimedean copulasintroducing the multivariate probability integral transformation (Wang, Nelsen &Valdez (2005) called this the distorted copula, while Klement, Mesiar & Pap(2005) or Durante & Sempi (2005) called this the transformed copula). Considera copula C. Let h be a continuous strictly concave increasing function[0, 1] → [0, 1] satisfying h (0) = 0 and h (1) = 1, such that Dh (C) (u1 , · · · , ud ) = h−1 (C (h (u1 ) , · · · , h (ud ))), 0 ≤ ui ≤ 1is a copula. Those functions will be called distortion functions.Example4A classical example is obtained when h is a power function, and when the power is theinverse of an integer, hn (x) = x1/n , i.e. Dhn (C) (u, v) = C n (u1/n , v 1/n ), 0 ≤ u, v ≤ 1 and n ∈ N.Then this copula is the survival copula of the componentwise maxima : the copula of 18
  • 19. Arthur CHARPENTIER, Distortion in actuarial sciences(max{X1 , · · · , Xn }, max{Y1 , · · · , Yn }) is Dhn (C), where {(X1 , Y1 ), · · · , (Xn , Yn )}is an i.i.d. sample, and the (Xi , Yi )’s have copula C.A max-stable copula is a copula C such that ∀n ∈ N, 1/n 1/n C n (u1 , · · · , ud ) = C(u1 , · · · , ud ).Example5Let φ denote a convex decreasing function on (0, 1] such that φ(1) = 0, and defineC(u, v) = φ−1 (φ(u) + φ(v)) = Dexp[−φ] (C ⊥ ). This function is an Archimedean copula.Example6A distorted version of the comonontonic copula is the comonotonic copula, h−1 [min{h(u1 ), · · · , h(ud )}] = min{u1 , · · · , ud }Example7Following the idea of Capéraà, Fougères & Genest (2000), it is possible to constructArchimax copulas as distortions of max-stable copulas. In dimension d = 2, max-stable 19
  • 20. Arthur CHARPENTIER, Distortion in actuarial sciencescopulas are characterized through a generator A such that log(u) C(u, v) = exp log(uv)A log(uv)Here consider φ an Archimedean generator, then Archimax copulas are defined as φ(u) C(u, v) = φ−1 [φ(u) + φ(v)]A φ(u) + φ(v)In the bivariate case, h need not be differentiable, and concavity is a sufficientcondition. 20
  • 21. Arthur CHARPENTIER, Distortion in actuarial sciencesWith nonconcave distortion function, distorted copulas are semi-copulas, fromBassan & Spizzichino (2001).Definition4Function S : [0, 1]d → [0, 1] is a semi-copula if 0 ≤ ui ≤ 1, i = 1, · · · , d, S(1, ..., 1, ui , 1, ..., 1) = ui , (3) S(u1 , ..., ui−1 , 0, ui+1 , ..., ud ) = 0, (4)and s → S(u1 , ..., ui−1 , s, ui+1 , ..., ud ) is increasing on [0, 1].Let Hd denote the set of continuous strictly increasing functions [0, 1] → [0, 1]such that h (0) = 0 and h (1) = 1, C ∈ C, Dh (C) (u1 , · · · , ud ) = h−1 (C (h (u1 ) , · · · , h (ud ))) , 0 ≤ ui ≤ 1is a copula, called distorted copula.Hd -copulas will be functions Dh (C) for some distortion function h and somecopula C.d-increasingness of function Dh (C) is obtained when h ∈ Hd , i.e. h is continuous, 21
  • 22. Arthur CHARPENTIER, Distortion in actuarial scienceswith h (0) = 0 and h (1) = 1, and such that h(k) (x) ≤ 0 for all x ∈ (0, 1) andk = 2, 3, · · · , d (see Theorem 2.6 and 4.4 in Morillas (2005)).As a corollary, note that if φ ∈ Φd , then h(x) = exp(−φ(x)) belongs to Hd .Further, observe that for h, h ∈ Hd , Dh◦h (C) (u1 , · · · , ud ) = (Dh ◦ Dh ) (C) (u1 , · · · , ud ) , 0 ≤ ui ≤ 1.Again, it is possible to get an intuitive interpretation of that distortion.Consider a max-stable copula C. Let X be a random vector such that X given Θ Θhas copula C and P (Xi ≤ xi |Θ) = Gi (xi ) , i = 1, · · · , d.Then, the (unconditional) joint distribution function of X is given by F (x) = E (P (X1 ≤ x1 , · · · , Xd ≤ xd |Θ)) = E (C (P (X1 ≤ xi |Θ) , · · · , P (Xd ≤ xd |Θ))) Θ Θ = E C G1 (x1 ) , · · · , Gd (xd ) = E C Θ (G1 (x1 ) , · · · , Gd (xd )) = ψ (− log C (G1 (x1 ) , · · · , Gd (xd ))) , 22
  • 23. Arthur CHARPENTIER, Distortion in actuarial scienceswhere ψ is the Laplace transform of the distribution of Θ, i.e.ψ (t) = E (exp (−tΘ)), since C is a max-stable copula, i.e. Θ Θ C G1 (x1 ) , · · · , Gd (xd ) = C Θ (G1 (x1 ) , · · · , Gd (xd )) .The unconditional marginal distribution functions are Fi (xi ) = ψ (− log Gi (xi )),and therefore CX (x1 , · · · , xd ) = ψ − log C exp −ψ −1 (x) , exp −ψ −1 (y) .Note that since ψ −1 is completly montone, then h belongs to Hd . 23
  • 24. Arthur CHARPENTIER, Distortion in actuarial sciencesRemark2It is possible to use distortion to obtain stronger tail dependence (with results that can berelated to C & Segers (2007)). Recall that C(u, u) 1 − C(u, u) λL = lim and λU = lim . u→0 u u→1 1−uIf h−1 is regularly varying in 0 with exponent α > 0, i.e. h−1 (t) ∼ L0 tα in 0, thenλL (Dh (C)) = [λL (C)]α .If h−1 is regularly varying in 1 with exponent β > 0, i.e. 1 − h−1 (t) ∼ L0 [1 − t]β in 1,then λU (Dh (C)) = 2 − [2 − λU (C)]β . 24
  • 25. Arthur CHARPENTIER, Distortion in actuarial sciences5 Application to multivariate risk measureWang (1996) proposed the risk measure based on distortion functiong(t) = Φ(Φ−1 (t) − λ), with λ ≥ 0 (to be convex).Valdez (2009) suggested a multivariate distortion. 25
  • 26. Arthur CHARPENTIER, Distortion in actuarial sciences6 Application to aging problemsLet T = (T1 , · · · , Td ) denote remaining lifetime, at time t = 0. Consider theconditional distribution (T1 , · · · , Td ) given T1 > t, · · · , Td > tfor some t > 0.Let C denote the survival copula of T , P(T1 > t1 , · · · , Td > td ) = C(P(T1 > t1 ), · · · , P(T1 > tc )).The survival copula of the conditional distribution is the copula of (U1 , · · · , Ud ) given U1 < F 1 (t) , · · · ,underbraceF d (t)ud u1where (U1 , · · · , Ud ) has distribution C , and where Fi is the distribution of Ti 26
  • 27. Arthur CHARPENTIER, Distortion in actuarial sciencesLet C be a copula and let U be a random vector with joint distribution functionC. Let u ∈ (0, 1]d be such that C(u) > 0. The lower tail dependence copula of Cat level u is defined as the copula, denoted Cu , of the joint distribution of Uconditionally on the event {U ≤ u} = {U1 ≤ u1 , · · · , Ud ≤ ud }.6.1 Aging with Archimedean copulasIf C is a strict Archimedean copula with generator φ (i.e. φ(0) = ∞), then thelower tail dependence copula relative to C at level u is given by the strictArchimedean copula with generator φu defined by φu (t) = φ(t · C(u)) − φ(C(u)), 0 ≤ t ≤ 1,where C(u) = φ−1 [φ(u1 ) + · · · + φ(ud )] (see Juri & Wüthrich (2002) or C & Juri(2007)). 27
  • 28. Arthur CHARPENTIER, Distortion in actuarial sciencesExample8 θGumbel copulas have generator φ (t) = [− ln t] where θ ≥ 1. For any u ∈ (0, 1]d , thecorresponding conditional copula has generator θ 1/θ θ θ φu (t) = M − ln t − M where M = [− ln u1 ] + · · · + [− ln ud ] .Example9Clayton copulas C have generator φ (t) = t−θ − 1 where θ > 0. Hence,φu (t) = [t·C(u)]−θ −1−φ(C(u)) = t−θ ·C(u)−θ −1−[C(u)−θ −1] = C(u)−θ ·[t−θ −1],hence φu (t) = C(u)−θ · φ(t). Since the generator of an Archimedean copula is uniqueup to a multiplicative constant, φu is also the generator of Clayton copula, withparameter θ.Theorem3Consider X with Archimedean copula, having a factor representation, and let ψ denotethe Laplace transform of the heterogeneity factor Θ. Let u ∈ (0, 1]d , then X given −1 −1 −1X ≤ FX (u) (in the pointwise sense, i.e. X1 ≤ F1 (u1 ), · · · ., Xd ≤ Fd (ud )) is an 28
  • 29. Arthur CHARPENTIER, Distortion in actuarial sciencesArchimedean copula with a factor representation, where the factor has Laplace transform ψ t + ψ −1 (C(u)) ψu (t) = . C(u)6.2 Aging with distorted copulas copulasRecall that Hd -copulas are defined as Dh (C)(u1 , · · · , ud ) = h−1 (C(h(u1 ), · · · , h(ud ))), 0 ≤ ui ≤ 1,where C is a copula, and h ∈ Hd is a d-distortion function.Assume that there exists a positive random variable Θ, such that, conditionallyon Θ, random vector X = (X1 , · · · , Xd ) has copula C, which does not depend onΘ. Assume moreover that C is in extreme value copula, or max-stable copula (seee.g. Joe (1997)) : C xh , · · · , xh = C h (x1 , · · · , xd ) for all h ≥ 0. The following 1 dresult holds,Lemma1 29
  • 30. Arthur CHARPENTIER, Distortion in actuarial sciencesLet Θ be a random variable with Laplace transform ψ, and consider a random vectorX = (X1 , · · · , Xd ) such that X given Θ has copula C, an extreme value copula. ΘAssume that, for all i = 1, · · · , d, P (Xi ≤ xi |Θ) = Gi (xi ) where the Gi ’s aredistribution functions. Then X has copula CX (x1 , · · · , xd ) = ψ − log C exp −ψ −1 (x1 ) , · · · , exp −ψ −1 (xd ) ,whose copula is of the form Dh (C) with h(·) = exp −ψ −1 (·) .Theorem4Let X be a random vector with an Hd -copula with a factor representation, let ψ denotethe Laplace transform of the heterogeneity factor Θ, C denote the underlying copula, andGi ’s the marginal distributions. −1Let u ∈ (0, 1]d , then, the copula of X given X ≤ FX (u) is −1 −1CX,u (x) = ψu − log Cu exp −ψu (x1 ) , · · · , exp −ψu (xd ) = Dhu (Cu )(x), −1where hu (·) = exp −ψu (·) , and where– ψu is the Laplace transform defined as ψu (t) = ψ (t + α) /ψ (α) where α = − log (C (u∗ )), u∗ = exp −ψ −1 (ui ) for all i = 1, · · · , d. Hence, ψu is the i 30
  • 31. Arthur CHARPENTIER, Distortion in actuarial sciences −1 Laplace transform of Θ given X ≤ FX (u), −1 Θ– P Xi ≤ xi |X ≤ FX (u) , Θ = Gi (xi ) for all i = 1, · · · , d, where C (u∗ , u∗ , · · · , Gi (xi ) , · · · , u∗ ) 1 2 d Gi (xi ) = , C (u∗ , u∗ , · · · , u∗ , · · · , u∗ ) 1 2 i d– and Cu is the following copula C G1 G1 −1 (x1 ) , · · · , Gd Gd −1 (xd ) Cu (x) = −1 −1 . C G1 F1 (u1 ) , · · · , Gd Fd (ud ) 31

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