Slides université Laval, Actuariat, Avril 2011

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


1 Decision theory and distorted risk measures
Consider a preference ordering among risks, such that
1. ˜ L ˜ L ˜ ˜
is distribution based, i.e. if X Y , ∀X = X Y = 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


Theorem1
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 ⇔ 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 axiom
4’. 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,
Theorem2
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 ⇔ g(F X (x))dx ≤ g(F Y (x))dx
R R

3


Standard 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


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
1
R(X) = φ(t)F −1 (1 − t)dt
0

where F (x) = F(X ≤ x).
see Kusuoka (2001), i.e. R is a spectral risk measure.

5


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) = g (1 − FX (x)) dx = g F X (x) dx (1)
0 0

Proposition1
Wang’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


More generally (risks taking value in R)
Deﬁnition2
We call distorted risk measure
1
R(X) = F −1 (1 − u)dg(u)
0

where g is some distortion function.
Proposition3
R(X) can be written
+∞ 0
R(X) = g(1 − F (x))dx − [1 − g(1 − F (x))]dx.
0 −∞

7


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


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
1 ∞
R(X; k) = F −1 (1 − u)kuk−1 du = [F (x)]k dx
0 0

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
E[X exp(hX)] d
Es[X; h] = = ln MX (h).
MX (h) dh

9


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) 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.
Example2
The independent copula C ⊥ is an Archimedean copula, with generator φ(t) = − log t.

10


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 (C ⊥ [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
Θ
factor Θ, a positive random variable, such that P (Xi ≤ xi |Θ) = Gi (x) where
Gi denotes a baseline distribution function.

11


The joint distribution function of X is given by
FX (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=1

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
−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


3 Hierarchical Archimedean copulas
It 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−1

where φi are generators. C is a copula if φi ◦ φ−1 is the inverse of a Laplace
i−1
transform. This copula is said to be a fully nested Archimedean (FNA) copula.
E.g. in dimension d = 5, we get

φ1 [φ1 (φ−1 [φ2 (φ−1 [φ3 (φ−1 [φ4 (u1 ) + φ4 (u2 )]) + φ3 (u3 )]) + φ2 (u4 )]) + φ1 (u5 )].
−1
2 3 4

It 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 3

Again, it is a copula if φ2 ◦ φ−1 is the inverse of a Laplace transform, as well as
1
φ4 ◦ φ−1 and φ4 ◦ φ−1 .
1 3

13


φ1

φ2 φ4

φ3 φ1

φ4 φ2 φ3

U1 U2 U3 U4 U5 U1 U2 U3 U4 U5

Figure 1 – fully nested Archimedean copula, and partially nested Archimedean
copula.

14


It 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 2

if 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


φ3 φ3

φ1 φ2 φ1 φ2

U1 U2 U3 U4 U5 U1 U2 U3 U4 U5

Figure 2 – Copules Archimédiennes hiérarchiques avec deux constructions dif-
férentes.

16


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, see Hougaard (2000), with two
frailties Θ1 and Θ2 such that

17


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) = 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


(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 ).

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.
Example6
A distorted version of the comonontonic copula is the comonotonic copula,

h−1 [min{h(u1 ), · · · , h(ud )}] = min{u1 , · · · , ud }

Example7
Following the idea of Capéraà, Fougères & Genest (2000), it is possible to construct
Archimax copulas as distortions of max-stable copulas. In dimension d = 2, max-stable

19


copulas 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 sufficient
condition.

20


With nonconcave distortion function, distorted copulas are semi-copulas, from
Bassan & Spizzichino (2001).
Deﬁnition4
Function 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 ≤ 1
is a copula, called distorted copula.
Hd -copulas will be functions Dh (C) for some distortion function h and some
copula C.
d-increasingness of function Dh (C) is obtained when h ∈ Hd , i.e. h is continuous,

21


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.

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
Θ
has 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


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.
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


Remark2
It is possible to use distortion to obtain stronger tail dependence (with results that can be
related to C & Segers (2007)). Recall that
C(u, u) 1 − C(u, u)
λL = lim and λU = lim .
u→0 u u→1 1−u
If h−1 is regularly varying in 0 with exponent α > 0, i.e. h−1 (t) ∼ L0 tα in 0, then
λL (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


5 Application to multivariate risk measure
Wang (1996) proposed the risk measure based on distortion function
g(t) = Φ(Φ−1 (t) − λ), with λ ≥ 0 (to be convex).
Valdez (2009) suggested a multivariate distortion.

25


6 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 < F 1 (t) , · · · ,underbraceF d (t)ud
u1

where (U1 , · · · , Ud ) has distribution C , and where Fi is the distribution of Ti

26


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
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 }.

6.1 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,

where C(u) = φ−1 [φ(u1 ) + · · · + φ(ud )] (see Juri & Wüthrich (2002) or C & Juri
(2007)).

27


Example8
θ
Gumbel copulas have generator φ (t) = [− ln t] where θ ≥ 1. For any u ∈ (0, 1]d , the
corresponding conditional copula has generator
θ
1/θ θ θ
φu (t) = M − ln t − M where M = [− ln u1 ] + · · · + [− ln ud ] .

Example9
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.
Theorem3
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
−1 −1 −1
X ≤ FX (u) (in the pointwise sense, i.e. X1 ≤ F1 (u1 ), · · · ., Xd ≤ Fd (ud )) is an

28


Archimedean 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 copulas
Recall 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, 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 , · · · , xh = C h (x1 , · · · , xd ) for all h ≥ 0. The following
1 d
result holds,
Lemma1

29


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.
Θ
Assume that, for all i = 1, · · · , d, P (Xi ≤ xi |Θ) = Gi (xi ) where the Gi ’s are
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 (·) .
Theorem4
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.
−1
Let u ∈ (0, 1]d , then, the copula of X given X ≤ FX (u) is
−1 −1
CX,u (x) = ψu − log Cu exp −ψu (x1 ) , · · · , exp −ψu (xd ) = Dhu (Cu )(x),
−1
where 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


−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

Slides université Laval, Actuariat, Avril 2011

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