Slides mc gill-v3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(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

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

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

Note that since ψ−1
is completly montone, then h belongs to Hd.
23

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

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

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

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

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

– 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

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