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### Slides mc gill-v3

1. 1. Arthur CHARPENTIER, Distortion in actuarial sciences Distorting ... in actuarial sciences Arthur Charpentier Université Rennes 1 arthur.charpentier@univ-rennes1.fr http ://freakonometrics.blog.free.fr/ Montréal Seminar of Actuarial and Financial Mathematics, McGill, November 2010. 1
2. 2. Arthur CHARPENTIER, Distortion in actuarial sciences 1 Decision theory and distorted risk measures Consider a preference ordering among risks, such that 1. is distribution based, i.e. if X Y , ∀ ˜X L = X ˜Y L = Y , then ˜X ˜Y ; hence, we can write FX FY 2. 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. 3. Arthur CHARPENTIER, Distortion in actuarial sciences Théorème1 Ordering satisﬁes axioms 1-2-3-4-5 if and only if ∃u : R → R, continuous, strictly increasing and unique (up to an increasing afﬁne transformation) such that ∀FX and FY : FX FY ⇔ R u(x)dFX(x) ≤ R u(x)dFY (x) ⇔ E[u(X)] ≤ E[u(Y )]. But if we consider an alternative to the independence axiom 4’. satisﬁes an dual independence axiom, i.e. ∀FX, FY and FZ, and ∀λ ∈ (0, 1), FX FY =⇒ [λF−1 X + (1 − λ)F−1 Z ]−1 [λF−1 Y + (1 − λ)F−1 Z ]−1 . we (Yaari (1987)) obtain a dual representation theorem, Théorème2 Ordering satisﬁes axioms 1-2-3-4’-5 if and only if ∃g : [0, 1] → R, continuous, strictly increasing such that ∀FX and FY : FX FY ⇔ R g(FX(x))dx ≤ R g(FY (x))dx 3
4. 4. Arthur CHARPENTIER, Distortion in actuarial sciences Standard axioms required on risque measures R : X → R, – law invariance, X L = 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} – strong coherence ∀X and Y , sup{R( ˜X + ˜Y )} = R(X) + R(Y ), where ˜X L = X and ˜Y L = Y . 4
5. 5. Arthur CHARPENTIER, Distortion in actuarial sciences Proposition1 If 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. Proposition2 A coherente risk measure R is additive for comonotonic risks if and only if there exists a decreasing positive function φ on [0, 1] such that R(X) = 1 0 φ(t)F−1 (1 − t)dt where F(x) = F(X ≤ x). see Kusuoka (2001), i.e. R is a spectral risk measure. 5
6. 6. Arthur CHARPENTIER, Distortion in actuarial sciences Deﬁnition1 A distortion function is a function g : [0, 1] → [0, 1] such that g(0) = 0 and g(1) = 1. For positive risks, Deﬁnition1 Given distortion function g, Wang’s risk measure, denoted Rg, is Rg (X) = ∞ 0 g (1 − FX(x)) dx = ∞ 0 g FX(x) dx (1) Proposition1 Wang’s risk measure can be deﬁned as Rg (X) = 1 0 F−1 X (1 − α) dg(α) = 1 0 VaR[X; 1 − α] dg(α). (2) 6
7. 7. Arthur CHARPENTIER, Distortion in actuarial sciences More generally (risks taking value in R) Deﬁnition2 We call distorted risk measure R(X) = 1 0 F−1 (1 − u)dg(u) where g is some distortion function. Proposition3 R(X) can be written R(X) = +∞ 0 g(1 − F(x))dx − 0 −∞ [1 − g(1 − F(x))]dx. 7
8. 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 Dual Power g (x) = 1 − (1 − x) 1/p 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. 9. Arthur CHARPENTIER, Distortion in actuarial sciences Here, it looks like risk measures can be seen as R(X) = Eg◦P(X). Remark1 Let 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. Example1 Consider function g(x) = xk . The PH - proportional hazard - risk measure is R(X; k) = 1 0 F−1 (1 − u)kuk−1 du = ∞ 0 [F(x)]k dx If k is an integer [F(x)]k is the survival distribution of the minimum over k values. Deﬁnition2 The Esscher risk measure with parameter h > 0 is Es[X; h], deﬁned as Es[X; h] = E[X exp(hX)] MX(h) = d dh ln MX(h). 9
10. 10. Arthur CHARPENTIER, Distortion in actuarial sciences 2 Archimedean copulas Deﬁnition3 Let φ denote a decreasing function (0, 1] → [0, ∞] such that φ(1) = 0, and such that φ−1 is d-monotone, i.e. for all k = 0, 1, · · · , d, (−1)k [φ−1 ](k) (t) ≥ 0 for all t. Deﬁne the inverse (or quasi-inverse if φ(0) < ∞) as φ−1 (t) =    φ−1 (t) for 0 ≤ t ≤ φ(0) 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. Example2 The independent copula C⊥ is an Archimedean copula, with generator φ(t) = − log t. 10
11. 11. Arthur CHARPENTIER, Distortion in actuarial sciences The upper Fréchet-Hoeffding copula, deﬁned as the minimum componentwise, M(u) = min{u1, · · · , ud}, is not Archimedean (but can be obtained as the limit of some 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 (Cp erp[λ(u1), . . . , λ(ud)]), ∀u1, ..., ud ∈ [0, 1], Note that it is possible to get an interpretation of that distortion of the independence. A large subclass of Archimedean copula in dimension d is the class of Archimedean copulas obtained using the frailty approach. Consider random variables X1, · · · , Xd conditionally independent, given a latent factor Θ, a positive random variable, such that P (Xi ≤ xi|Θ) = Gi (x) Θ where Gi denotes a baseline distribution function. 11
12. 12. Arthur CHARPENTIER, Distortion in actuarial sciences The joint distribution function of X is given by FX (x1, · · · , xd) = E (P (X1 ≤ x1, · · · , Xd ≤ Xd|Θ)) = E d i=1 P (Xi ≤ xi|Θ) = E d i=1 Gi (xi) Θ = E d i=1 exp [−Θ (− log Gi (xi))] = ψ − d i=1 log Gi (xi) , where ψ is the Laplace transform of the distribution of Θ, i.e. ψ (t) = E (exp (−tΘ)) . Because the marginal distributions are given respectively by Fi(xi) = P(Xi ≤ xi) = ψ (− log Gi (xi)) , the copula of X is C (u) = FX F−1 1 (u1) , · · · , F−1 d (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. 13. Arthur CHARPENTIER, Distortion in actuarial sciences 3 Hierarchical Archimedean copulas It is possible to look at C(u1, · · · , ud) deﬁned as φ−1 1 [φ1[φ−1 2 (φ2[· · · φ−1 d−1[φd−1(u1) + φd−1(u2)] + · · · + φ2(ud−1))] + φ1(ud)] where φi are generators. C is a copula if φi ◦ φ−1 i−1 is the inverse of a Laplace transform. This copula is said to be a fully nested Archimedean (FNA) copula. E.g. in dimension d = 5, we get φ−1 1 [φ1(φ−1 2 [φ2(φ−1 3 [φ3(φ−1 4 [φ4(u1) + φ4(u2)]) + φ3(u3)]) + φ2(u4)]) + φ1(u5)]. It is also possible to consider partially nested Archimedean (PNA) copulas, e.g. 13
14. 14. Arthur CHARPENTIER, Distortion in actuarial sciences by coupling (U1, U2, U3), and (U4, U5), φ−1 4 [φ4(φ−1 1 [φ1(φ−1 2 [φ2(u1) + φ2(u2)]) + φ1(u3)]) + φ4(φ−1 3 [φ3(u4) + φ3(u5)])] Again, it is a copula if φ2 ◦ φ−1 1 is the inverse of a Laplace transform, as well as φ4 ◦ φ−1 1 and φ4 ◦ φ−1 3 . 14
15. 15. Arthur CHARPENTIER, Distortion in actuarial sciences U1 U2 U3 U4 U5 φ4 φ3 φ2 φ1 U1 U2 U3 U4 U5 φ2 φ1 φ3 φ4 Figure 1 – fully nested Archimedean copula, and partially nested Archimedean copula. 15
16. 16. Arthur CHARPENTIER, Distortion in actuarial sciences It is also possible to consider φ−1 3 [φ3(φ−1 1 [φ1(u1) + φ1(u2) + φ1(u3)]) + φ3(φ−1 2 [φ2(u4) + φ2(u5)])]. if φ3 ◦ φ−1 1 andφ3 ◦ φ−1 2 are inverses of Laplace transform. Or φ−1 3 [φ3(φ−1 1 [φ1(u1) + φ1(u2)] + φ3(u3) + φ3(φ−1 2 [φ2(u4) + φ2(u5)])]. 16
17. 17. Arthur CHARPENTIER, Distortion in actuarial sciences U1 U2 U3 U4 U5 φ1 φ3 φ2 U1 U2 U3 U4 U5 φ1 φ3 φ2 Figure 2 – Copules Archimédiennes hiérarchiques avec deux constructions dif- férentes. 17
18. 18. Arthur CHARPENTIER, Distortion in actuarial sciences Example3 If φi’s are Gumbel’s generators, with parameter θi, a sufﬁcient condition for C to be a FNA copula is that θi’s increasing. Similarly if φi’s are Clayton’s generators. Again, an heuristic interpretation can be derived. 18
19. 19. Arthur CHARPENTIER, Distortion in actuarial sciences 4 Distorting copulas Genest & Rivest (2001) extended the concept of Archimedean copulas introducing 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). Consider a 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 ≤ 1 is a copula. Those functions will be called distortion functions. Example4 A classical example is obtained when h is a power function, and when the power is the inverse of an integer, hn(x) = x1/n , i.e. Dhn (C) (u, v) = Cn (u1/n , v1/n ), 0 ≤ u, v ≤ 1 and n ∈ N. Then this copula is the survival copula of the componentwise maxima : the copula of 19
20. 20. 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, Cn (u 1/n 1 , · · · , u 1/n d ) = C(u1, · · · , ud). Example5 Let φ denote a convex decreasing function on (0, 1] such that φ(1) = 0, and deﬁne C(u, v) = φ−1 (φ(u) + φ(v)) = Dexp[−φ](C⊥ ). This function is an Archimedean copula. In the bivariate case, h need not be diﬀerentiable, and concavity is a suﬃcient condition. 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 ≤ 1 is a copula, called distorted copula. Hd-copulas will be functions Dh (C) for some distortion function h and some 20
21. 21. Arthur CHARPENTIER, Distortion in actuarial sciences copula C. d-increasingness of function Dh (C) is obtained when h ∈ Hd, i.e. h is continuous, with h (0) = 0 and h (1) = 1, and such that h(k) (x) ≤ 0 for all x ∈ (0, 1) and k = 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. 21
22. 22. Arthur CHARPENTIER, Distortion in actuarial sciences 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 has copula C and P (Xi ≤ xi|Θ) = Gi (xi) Θ , 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)) = E C G1 (x1) Θ , · · · , Gd (xd) Θ = E CΘ (G1 (x1) , · · · , Gd (xd)) = ψ (− log C (G1 (x1) , · · · , Gd (xd))) , where 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. C G1 (x1) Θ , · · · , Gd (xd) Θ = CΘ (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) . 22
23. 23. Arthur CHARPENTIER, Distortion in actuarial sciences Note that since ψ−1 is completly montone, then h belongs to Hd. 23
24. 24. Arthur CHARPENTIER, Distortion in actuarial sciences 5 Application to aging problems Let T = (T1, · · · , Td) denote remaining lifetime, at time t = 0. Consider the conditional distribution (T1, · · · , Td) given T1 > t, · · · , Td > t for 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 < F1(t), · · · , Ud < Fd(t) where (U1, · · · , Ud) has distribution C , and where Fi is the distribution of Ti Let C be a copula and let U be a random vector with joint distribution function C. Let u ∈ (0, 1]d be such that C(u) > 0. The lower tail dependence copula of C 24
25. 25. Arthur CHARPENTIER, Distortion in actuarial sciences at level u is deﬁned as the copula, denoted Cu, of the joint distribution of U conditionally on the event {U ≤ u} = {U1 ≤ u1, · · · , Ud ≤ ud}. 5.1 Aging with Archimedean copulas If C is a strict Archimedean copula with generator φ (i.e. φ(0) = ∞), then the lower tail dependence copula relative to C at level u is given by the strict Archimedean copula with generator φu deﬁned by φu(t) = φ(t · C(u)) − φ(C(u)), 0 ≤ t ≤ 1, (3) where C(u) = φ−1 [φ(u1) + · · · + φ(ud)] (see Juri & Wüthrich (2002) or C & Juri (2007)). Example6 Gumbel copulas have generator φ (t) = [− ln t] θ where θ ≥ 1. For any u ∈ (0, 1]d , the corresponding conditional copula has generator φu (t) = M1/θ − ln t θ − M where M = [− ln u1] θ + · · · + [− ln ud] θ . 25
26. 26. Arthur CHARPENTIER, Distortion in actuarial sciences Example7 Clayton 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 unique up to a multiplicative constant, φu is also the generator of Clayton copula, with parameter θ. Théorème3 Consider X with Archimedean copula, having a factor representation, and let ψ denote the Laplace transform of the heterogeneity factor Θ. Let u ∈ (0, 1]d , then X given X ≤ F−1 X (u) (in the pointwise sense, i.e. X1 ≤ F−1 1 (u1), · · · ., Xd ≤ F−1 d (ud)) is an Archimedean copula with a factor representation, where the factor has Laplace transform ψu (t) = ψ t + ψ−1 (C(u)) C(u) . 26
27. 27. Arthur CHARPENTIER, Distortion in actuarial sciences 5.2 Aging with distorted copulas copulas class of Hd-copulas, 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, conditionally on Θ, random vector X = (X1, · · · , Xd) has copula C, which does not depend on Θ. Assume moreover that C is in extreme value copula, or max-stable copula (see e.g. Joe (1997)) : C xh 1 , · · · , xh d = Ch (x1, · · · , xd) for all h ≥ 0. The following result holds, Lemma1 Let Θ be a random variable with Laplace transform ψ, and consider a random vector X = (X1, · · · , Xd) such that X given Θ has copula C, an extreme value copula. Assume that, for all i = 1, · · · , d, P (Xi ≤ xi|Θ) = Gi (xi) Θ where the Gi’s are 27
28. 28. Arthur CHARPENTIER, Distortion in actuarial sciences distribution 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 (·) . Théorème4 Let X be a random vector with an Hd-copula with a factor representation, let ψ denote the Laplace transform of the heterogeneity factor Θ, C denote the underlying copula, and Gi’s the marginal distributions. Let u ∈ (0, 1]d , then, the copula of X given X ≤ F−1 X (u) is CX,u (x) = ψu − log Cu exp −ψ−1 u (x1) , · · · , exp −ψ−1 u (xd) = Dhu (Cu)(x), where hu(·) = exp −ψ−1 u (·) , and where – ψu is the Laplace transform deﬁned as ψu (t) = ψ (t + α) /ψ (α) where α = − log (C (u∗ )), u∗ i = exp −ψ−1 (ui) for all i = 1, · · · , d. Hence, ψu is the Laplace transform of Θ given X ≤ F−1 X (u), 28
29. 29. Arthur CHARPENTIER, Distortion in actuarial sciences – P Xi ≤ xi|X ≤ F−1 X (u) , Θ = Gi (xi) Θ for all i = 1, · · · , d, where Gi (xi) = C (u∗ 1, u∗ 2, · · · , Gi (xi) , · · · , u∗ d) C (u∗ 1, u∗ 2, · · · , u∗ i , · · · , u∗ d) , – and Cu is the following copula Cu (x) = C G1 G1 −1 (x1) , · · · , Gd Gd −1 (xd) C G1 F−1 1 (u1) , · · · , Gd F−1 d (ud) . 29