This document summarizes a presentation on analyzing the tail behavior of Archimedean copulas. It discusses how extreme value theory can be used to approximate risk measures like Value-at-Risk and Tail Value-at-Risk by modeling the distribution of losses exceeding a high threshold using generalized Pareto distributions. The presentation also examines how to extend this univariate extreme value analysis to higher dimensions to model the dependence structure between losses in the tails.
1. Arthur CHARPENTIER - tails of Archimedean copulas
Tails of Archimedean Copulas
Arthur Charpentier
CREM-Universit´e Rennes 1
(joint work with Johan Segers, UCLN)
http ://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/
Workshop Dynamic and Multivariate Risk measures
Institut Henri Poincar´e, October 2008
1
2. Arthur CHARPENTIER - tails of Archimedean copulas
Tail behavior and risk management
Pickands-Balkema-de Haan’s theorem describes tail behavior (in dimension 1),
Theorem 1. F ∈ MDA (Gξ) if and only if
lim
u→xF
sup
0<x<xF
Pr (X − u ≤ x|X > u) − Hξ,σ(u) (≤ x) = 0,
for some positive function σ (·), where Hξ,σ (x) =
1 − (1 + ξx/σ)
−1/ξ
, ξ = 0
1 − exp (−x/σ) , ξ = 0.
Consider a i.i.d. sample {X1, · · · , Xn}, then recall that
1 − F(x) ≈ (1 − F(u)) 1 − Hξ,σ(u) (x − u) , for all x > u.
Hence, if u = Xk:n, then
1 − F(x) ≈ (1 − F(Xk:n))
≈1−Fn(Xk:n)=k/n
1 − Hξ,σ(Xk:n) (x − Xk:n) , for all x > Xk:n,
2
3. Arthur CHARPENTIER - tails of Archimedean copulas
Deriving tail approximations for risk measures
If the distribution exceeding u = Xn−k:n can be approximated by a Generalized
Pareto distribution alors
V aR(X, p) ≈ Xn−k:n +
σk
ξk
n
k
(1 − p)
−ξk
− 1 ,
and
TV aR(X, p) = E(X|X > V aR(X, p))
≈ V aR(X, p)
1
1 − ξk
+
σk − ξkXn−k:n
1 − ξkV aR(X, p)
3
4. Arthur CHARPENTIER - tails of Archimedean copulas
Extending extreme value theory in higher dimension
univariate case bivariate case
limiting distribution dependence structure of
of Xn:n (G.E.V.) componentwise maximum
when n → ∞, i.e. Hξ (Xn:n, Yn:n)
(Fisher-Tippet)
dependence structure of
limiting distribution (X, Y ) |X > x, Y > y
of X|X > x (G.P.D.) when x, y → ∞
when x → ∞, i.e. Gξ,σ dependence structure of
(Balkema-de Haan-Pickands) (X, Y ) |X > x
when x → ∞
4
6. Arthur CHARPENTIER - tails of Archimedean copulas
Motivations : dependence and copulas
Definition 2. A copula C is a joint distribution function on [0, 1]d
, with
uniform margins on [0, 1].
Theorem 3. (Sklar) Let C be a copula, and F1, . . . , Fd be d marginal
distributions, then F(x) = C(F1(x1), . . . , Fd(xd)) is a distribution function, with
F ∈ F(F1, . . . , Fd).
Conversely, if F ∈ F(F1, . . . , Fd), there exists C such that
F(x) = C(F1(x1), . . . , Fd(xd)). Further, if the Fi’s are continuous, then C is
unique, and given by
C(u) = F(F−1
1 (u1), . . . , F−1
d (ud)) for all ui ∈ [0, 1]
We will then define the copula of F, or the copula of X.
Let C denote the survival copula,
P(X1 ≤ x1, · · · , Xd ≤ xd) = C(P(X1 ≤ x1), · · · , P(Xd ≤ xd))
P(X1 > x1, · · · , Xd > xd) = C (P(X1 > x1), · · · , P(Xd > xd))
6
7. Arthur CHARPENTIER - tails of Archimedean copulas
X
Y
Z
Fonction de répartition à marges uniformes
Fig. 2 – Graphical representation of a copula, C(u, v) = P(U ≤ u, V ≤ v).
7
8. Arthur CHARPENTIER - tails of Archimedean copulas
x
x
z
Densité d’une loi à marges uniformes
Fig. 3 – Density of a copula, c(u, v) =
∂2
C(u, v)
∂u∂v
.
8
9. Arthur CHARPENTIER - tails of Archimedean copulas
Strong tail dependence
Joe (1993) defined, in the bivariate case a tail dependence measure.
Definition 4. Let (X, Y ) denote a random pair, the upper and lower tail
dependence parameters are defined, if the limit exist, as
λL = lim
u→0
P X ≤ F−1
X (u) |Y ≤ F−1
Y (u) ,
= lim
u→0
P (U ≤ u|V ≤ u) = lim
u→0
C(u, u)
u
,
and
λU = lim
u→1
P X > F−1
X (u) |Y > F−1
Y (u)
= lim
u→0
P (U > 1 − u|V ≤ 1 − u) = lim
u→0
C (u, u)
u
.
9
15. Arthur CHARPENTIER - tails of Archimedean copulas
Weak tail dependence
If X and Y are independent (in tails), for u large enough
P(X > F−1
X (u), Y > F−1
Y (u)) = P(X > F−1
X (u)) · P(Y > F−1
Y (u)) = (1 − u)2
,
or equivalently, log P(X > F−1
X (u), Y > F−1
Y (u)) = 2 · log(1 − u). Further, if X
and Y are comonotonic (in tails), for u large enough
P(X > F−1
X (u), Y > F−1
Y (u)) = P(X > F−1
X (u)) = (1 − u)1
,
or equivalently, log P(X > F−1
X (u), Y > F−1
Y (u)) = 1 · log(1 − u).
=⇒ limit of the ratio
log(1 − u)
log P(Z1 > F−1
1 (u), Z2 > F−1
2 (u))
.
15
16. Arthur CHARPENTIER - tails of Archimedean copulas
Weak tail dependence
Coles, Heffernan & Tawn (1999) defined
Definition 5. Let (X, Y ) denote a random pair, the upper and lower tail
dependence parameters are defined, if the limit exist, as
ηL = lim
u→0
log(u)
log P(Z1 ≤ F−1
1 (u), Z2 ≤ F−1
2 (u))
= lim
u→0
log(u)
log C(u, u)
,
and
ηU = lim
u→1
log(1 − u)
log P(Z1 > F−1
1 (u), Z2 > F−1
2 (u))
= lim
u→0
log(u)
log C (u, u)
.
16
24. Arthur CHARPENTIER - tails of Archimedean copulas
Application in risk management : car-household
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
L and R concentration functions
L function (lower tails) R function (upper tails)
qqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qq
qqq
q
qq
qq
qqqqqqqqqqqqqqq
qqqqqqqq
qqqqqqqqqqqqqq
qqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqq
q
qqq
qqqq
qqq
qqq
qqqqq
qqqqq
qq
qqqqqqq
q
qqqq
qqq
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
Gumbel copula
q
q
0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0
Chi dependence functions
lower tails upper tails
qqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qq
qqq
q
qqqqqqqqqqqqqqqqqqq
qqqqqqqq
qqqqqqqqqqqqqq
qqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqq
qqqqqqq
qqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
Gumbel copula
q
q
Fig. 16 – L and R cumulative curves, and χ functions.
24
25. Arthur CHARPENTIER - tails of Archimedean copulas
Archimedean copulas
Definition 6. A copula C is called Archimedean if it is of the form
C(u1, · · · , ud) = φ−1
(φ(u1) + · · · + φ(ud)) ,
where the generator φ : [0, 1] → [0, ∞] is convex, decreasing and satisfies φ(1) = 0.
A necessary and sufficient condition is that φ−1
is d-monotone.
25
27. Arthur CHARPENTIER - tails of Archimedean copulas
Why Archimedean copulas ?
Assume that X and Y are conditionally independent, given the value of an
heterogeneous component Θ. Assume further that
P(X ≤ x|Θ = θ) = (GX(x))θ
and P(Y ≤ y|Θ = θ) = (GY (y))θ
for some baseline distribution functions GX and GY . Then
F(x, y) = P(X ≤ x, Y ≤ y) = E(P(X ≤ x, Y ≤ y|Θ = θ))
= E(P(X ≤ x|Θ = θ) × P(Y ≤ y|Θ = θ))
= E (GX(x))Θ
× (GY (y))Θ
= ψ(− log GX(x) − log GY (y))
where ψ denotes the Laplace transform of Θ, i.e. ψ(t) = E(e−tΘ
). Since
FX(x) = ψ(− log GX(x)) and FY (y) = ψ(− log GY (y))
and thus, the joint distribution of (X, Y ) satisfies
F(x, y) = ψ(ψ−1
(FX(x)) + ψ−1
(FY (y))).
27
28. Arthur CHARPENTIER - tails of Archimedean copulas
0 5 10 15
05101520
Conditional independence, two classes
!3 !2 !1 0 1 2 3
!3!2!10123
Conditional independence, two classes
Fig. 17 – Two classes of risks, (Xi, Yi) and (Φ−1
(FX(Xi)), Φ−1
(FY (Yi))).
28
29. Arthur CHARPENTIER - tails of Archimedean copulas
0 5 10 15 20 25 30
010203040
Conditional independence, three classes
!3 !2 !1 0 1 2 3
!3!2!10123
Conditional independence, three classes
Fig. 18 – Three classes of risks, (Xi, Yi) and (Φ−1
(FX(Xi)), Φ−1
(FY (Yi))).
29
31. Arthur CHARPENTIER - tails of Archimedean copulas
Properties of Archimedean copulas
• the countercomonotonic copula C−
is Archimedean, φ(t) = 1 − t,
• the independent copula C⊥
is Archimedean, φ(t) = − log(t),
• the comonotonic copula C+
is not Archimedean (but can be a limit of
Archimedean copulas).
0.2
0.4
0.6
0.8
u_10.2
0.4
0.6
0.8
u_2
00.20.40.60.81
Frechetlowerbound
0.2
0.4
0.6
0.8
u_10.2
0.4
0.6
0.8
u_2
00.20.40.60.81Independencecopula
0.2
0.4
0.6
0.8
u_10.2
0.4
0.6
0.8
u_2
00.20.40.60.81
Frechetupperbound
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Scatterplot, Lower Fréchet!Hoeffding bound
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Scatterplot, Indepedent copula random generation
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Scatterplot, Upper Fréchet!Hoeffding bound
31
32. Arthur CHARPENTIER - tails of Archimedean copulas
Properties of Archimedean copulas
• Frank copula is the only Archimedean such that (U, V )
L
= (1 − U, 1 − V )
(stability by symmetry),
• Gumbel copula is the only Archimedean such that (U, V ) has the same copula
as (max{U1, ..., Un}, max{V1, ..., Vn}) for all n ≥ 1 (max-stability),
• Clayton copula is the only Archimedean such that (U, V ) has the same copula
as (U, V ) given (U ≤ u, V ≤ v) (stability by truncature).
32
33. Arthur CHARPENTIER - tails of Archimedean copulas
Lower tails of Archimedean copulas
Study regular variation property of φ at 0,
lim
s→0
φ(st)
φ(s)
= t−θ0
, t ∈ (0, ∞) ⇐⇒ θ0 = − lim
s→0
sφ (s)
φ(s)
.
If θ0 > 0 : asymptotic dependence
Proposition 7. If 0 < θ0 < ∞, then for every ∅ = I ⊂ {1, . . . , d}, every
(xi)i∈I ∈ (0, ∞)|I|
and every (y1, . . . , yd) ∈ (0, ∞)d
,
lim
s↓0
Pr[∀i = 1, . . . , d : Ui ≤ syi | ∀i ∈ I : Ui ≤ sxi]
= i∈Ic y−θ0
i + i∈I(xi ∧ yi)−θ0
i∈I x−θ0
i
−1/θ0
This is Clayton’s copula. Further, λL = 2−1/θ0
.
33
34. Arthur CHARPENTIER - tails of Archimedean copulas
Lower tails of Archimedean copulas
Study regular variation property of φ at 0,
lim
s→0
φ(st)
φ(s)
= t−θ0
, t ∈ (0, ∞) ⇐⇒ θ0 = − lim
s→0
sφ (s)
φ(s)
.
If θ0 = 0 : asymptotic independence for strict generators (φ(0) = ∞)
Proposition 8. If θ0 = 0 and φ(0) = ∞, for every ∅ = I ⊂ {1, . . . , d}, every
(xi)i∈I ∈ (0, ∞)|I|
and every (y1, . . . , yd) ∈ (0, ∞)d
,
lim
s↓0
Pr[∀i ∈ I : Ui ≤ syi; ∀i ∈ Ic
: Ui ≤ χs(yi) | ∀i ∈ I : Ui ≤ sxi]
=
i∈I
yj
xj
∧ 1
|I|−κ
i∈Ic
exp −|I|−κ
y−1
i ,
where χs(·) = φ−1
(−sφ (s)/·), and κ is the index of regular variation of ψ, with
ψ(·) = −φ−1
(·)φ (φ−1
(·)).
Then ηL = 2κ−1
.
34
35. Arthur CHARPENTIER - tails of Archimedean copulas
Upper tails of Archimedean copulas
Study regular variation property of φ at 1,
lim
s→0
φ(1 − st)
φ(1 − s)
= tθ1
, t ∈ (1, ∞) ⇐⇒ θ1 = − lim
s→0
sφ (1 − s)
φ(1 − s)
.
If θ1 > 1 : asymptotic dependence, and λU = 2 − 21/θ1
,
Proposition 9. If 1 < θ0 < ∞, then for every ∅ = I ⊂ {1, . . . , d}, every
(xi)i∈I ∈ (0, ∞)|I|
and every (y1, . . . , yd) ∈ (0, ∞)d
,
lim
s↓0
Pr[∀i = 1, . . . , d : Ui ≥ 1 − syi | ∀i ∈ I : Ui ≥ 1 − sxi] =
rd(z1, . . . , zd; θ1)
r|I|((xi)i∈I; θ1)
where zi = xi ∧ yi for i ∈ I and zi = yi for i ∈ Ic
and
rk(u1, . . . , uk; θ1) =
∅=J⊂{1,...,k}
(−1)|J|−1
i∈J
uθ1
j
1/θ1
for integer k ≥ 1 and (u1, . . . , uk) ∈ (0, ∞)k
.
35
36. Arthur CHARPENTIER - tails of Archimedean copulas
Upper tails of Archimedean copulas
Study regular variation property of φ at 1,
lim
s→0
φ(1 − st)
φ(1 − s)
= tθ1
, t ∈ (1, ∞) ⇐⇒ θ1 = − lim
s→0
sφ (1 − s)
φ(1 − s)
.
If θ1 = 1 and φ (1) < 0 : asymptotic independence, or near independence
Proposition 10. If θ1 = 1 and φ (1) < 0, then for all (xi)i∈I ∈ (0, ∞)|I|
and
(y1, . . . , yd) ∈ (0, 1]d
,
lim
s↓0
Pr[∀i ∈ I : Ui ≥ 1 − syi; ∀i ∈ Ic
: Ui ≤ yi | ∀i ∈ I : Ui ≥ 1 − sxi]
=
i∈I
yj ·
(−D)|I|
φ−1
( i∈Ic φ(yi))
(−D)|I|φ−1(0)
.
In that case, ηU = 1/2 (near independence).
36
37. Arthur CHARPENTIER - tails of Archimedean copulas
Upper tails of Archimedean copulas
If θ = 1 and φ (1) = 0 : asymptotic independence, dependence in independence
Proposition 11. If θ1 = 1 and φ (1) = 0, if I ⊂ {1, . . . , d} and |I| ≥ 2, then for
every (xi)i∈I ∈ (0, ∞)|I|
and every (y1, . . . , yd) ∈ (0, ∞)d
,
lim
s↓0
Pr[∀i = 1, . . . , d : Ui ≥ 1 − syi | ∀i ∈ I : Ui ≥ 1 − sxi] =
rd(z1, . . . , zd)
r|I|((xi)i∈I)
where zi = xi ∧ yi for i ∈ I and zi = yi for i ∈ Ic
and
rk(u1, . . . , uk) :=
∅=J⊂{1,...,k}
(−1)|J|
(
J
uj) log(
J
uj)
= (k − 2)!
u1
0
· · ·
uk
0
(t1 + · · · + tk)−(k−1)
dt1 · · · dtk
for integer k ≥ 2 and (u1, . . . , uk) ∈ (0, ∞)k
.
In that case, ηU = 1 (near asymptotic dependence).
37
38. Arthur CHARPENTIER - tails of Archimedean copulas
Tails of Archimedean copulas
• upper tail : calculate φ (1) and θ1 = − lim
s→0
sφ (1 − s)
φ(1 − s)
,
◦ φ (1) < 0 : asymptotic independence
◦ φ (1) = 0 et θ1 = 1 : dependence in independence
◦ φ (1) = 0 et θ1 > 1 : asymptotic dependence
• lower tail : calculate φ(0) and θ0 = − lim
s→0
sφ (s)
φ(s)
,
◦ φ(0) < ∞ : asymptotic independence
◦ φ(0) = ∞ et θ0 = 0 : dependence in independence
◦ φ(0) = ∞ et θ0 > 0 : asymptotic dependence
38
40. Arthur CHARPENTIER - tails of Archimedean copulas
How to extend to more general dependence structures ?
• mixtures of generators, since convex sums of generators defines a generator,
• the α − β transformations in Nelsen (1999), i.e.
φα(t) = φ(tα
) and φβ(t) = [φ(t)]β
, where α ∈ (0, 1) and β ∈ (1, ∞).
• other transformations, e.g.
◦ exp(αφ(t)) − 1, α ∈ (0, ∞),
◦ φ(1 − [1 − t]α
), α ∈ (1, ∞),
◦ φ(αt) − φ(α), α ∈ (0, 1),
=⇒ can be related to distortion of Archimedean copulas.
40
41. Arthur CHARPENTIER - tails of Archimedean copulas
How to extend to more general dependence structures ?
upper tail lower tail
φα(t) range α φα(1) θ1(α) φα(0) θ0(α) κ(α)
(1) (φ(t))α (1, ∞) 0 αθ1 (φ(0))α αθ0
κ
α
+ 1 − 1
α
(2) eαφ(t)−1
α
(0, ∞) αφ (1) θ1
αφ(0)−1
α
∗ ∗
(3) φ(tα) (0, 1) αφ (1) θ1 φ(0) αθ0 κ
(4) φ(1 − (1 − t)α) (1, ∞) 0 αθ1 φ(0) θ0 κ
(5) φ(αt) − φ(α) (0, 1) αφ (α) 1 φ(0) − φ(α) θ0 κ
see Charpentier & Segers (2006, 2007, 2008) for more details on tail
behavior for Archimedean copulas, and Charpentier & Juri (2006) for some
extensions in the general case.
41
42. Arthur CHARPENTIER - tails of Archimedean copulas
From tails of copulas to tails of sums
Assume that X1, · · · , Xd have identical marginal distributions, with tail index ξ,
lim
t→∞
P(Xi > xt)
P(Xi > t)
= x−α
, where α =
1
ξ
.
From Feller (1971),
lim
t→∞
P(S > t)
P(Xi > t)
=
P(X1 + · · · + Xd > t)
P(Xi > t)
= d,
when the Xi’s are mutually independent (regularly varying functions are
subexponential, when ξ > 0).
Alink, L¨owe & W¨uthrich (2004) obtained the expression of the limit,
assuming that X has an Archimedean surival copula, where φ is regularly
varying at 0 (see also Albrecher, Asmussen and Kortschak (2006),
Barbe, Fougres and Genest (2006) or Kortschak and Albrecher
(2008) for additional results).
42
43. Arthur CHARPENTIER - tails of Archimedean copulas
References
Albrecher, H., Asmussen, S. and Kortschak, D. (2006). Tail asymptotics for the sum of two heavy-tailed dependent
risks. Extremes, 9, 107-130.
Alink, S., L¨owe, M. and W¨uthrich, M.V. (2004). Diversification of aggregate dependent risks. Insurance : Mathematics and
Economics, 35, 77-95.
Barbe, P., Foug`eres, A.-L. and Genest, C. (2006). On the tail behavior of sums of dependent risks. ASTIN Bulletin, 36,
361-373.
Coles, S., Heffernan, J.E., and Tawn, J.A. (1999). Dependence measures for extreme value analyses. Extremes, 2,
339-365.
Charpentier, A. and Juri, A. (2006). Limiting dependence structures for tail events, with applications to credit
derivatives. Journal of Applied Probability, 44, 563-586.
Charpentier, A. and Segers, J. (2007). Lower tail dependence for Archimedean copulas : Characterizations and pitfalls.
Insurance : Mathematics and Economics, 40, 525-532.
Charpentier, A. and Segers, J. (2008). Convergence of Archimedean copulas. Proba- bility & Statistical Letters, 78, 412-419.
Charpentier, A. and Segers, J. (2007). Tails of Archimedean copulas. submited.
Joe, H. (1993). Multivariate dependence measures and data analysis. Computational Statistics & Data Analysis, 16, 279-297.
Feller, W. (1971). An introduction to probability theory and its applications, vol. 2. Wiley.
Kortschak, D. and Albrecher, H. (2008). Asymptotic results for the sum of dependent non-identically distributed
random variables. Methodology and Computing in Applied Probability, to appear.
Nelsen, R. (1999). An introduction to copulas. Springer-Verlag.
W¨uthrich, M. (2003). Asymptotic value-at-risk estimates for sums of dependent random variables. ASTIN Bulletin, 33,
75-92.
43