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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.
![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))
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![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.
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![Arthur CHARPENTIER - tails of Archimedean copulas
Some examples of Archimedean copulas
φ(t) range θ
(1) 1
θ
(t−θ − 1) [−1, 0) ∪ (0, ∞) Clayton, Clayton (1978)
(2) (1 − t)θ [1, ∞)
(3) log
1−θ(1−t)
t
[−1, 1) Ali-Mikhail-Haq
(4) (− log t)θ [1, ∞) Gumbel, Gumbel (1960), Hougaard (1986)
(5) − log e−θt−1
e−θ−1
(−∞, 0) ∪ (0, ∞) Frank, Frank (1979), Nelsen (1987)
(6) − log{1 − (1 − t)θ} [1, ∞) Joe, Frank (1981), Joe (1993)
(7) − log{θt + (1 − θ)} (0, 1]
(8) 1−t
1+(θ−1)t
[1, ∞)
(9) log(1 − θ log t) (0, 1] Barnett (1980), Gumbel (1960)
(10) log(2t−θ − 1) (0, 1]
(11) log(2 − tθ) (0, 1/2]
(12) ( 1
t
− 1)θ [1, ∞)
(13) (1 − log t)θ − 1 (0, ∞)
(14) (t−1/θ − 1)θ [1, ∞)
(15) (1 − t1/θ)θ [1, ∞) Genest & Ghoudi (1994)
(16) ( θ
t
+ 1)(1 − t) [0, ∞)
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![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
.
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![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
.
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![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
.
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![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).
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![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).
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![Arthur CHARPENTIER - tails of Archimedean copulas
upper tail lower tail
φ(t) range θ −φ (1) θ1 φ(0) θ0 κ
(1) 1
θ
(t−θ − 1) [−1, ∞) 1 1 1
(−θ)∨0
θ ∨ 0 ·
(2) (1 − t)θ [1, ∞) 1(θ = 1) θ 1 0 ·
(3) log
1−θ(1−t)
t
[−1, 1) 1 − θ 1 ∞ 0 0
(4) (− log t)θ [1, ∞) 1(θ = 1) θ ∞ 0 1 − 1
θ
(5) − log e−θt−1
e−θ−1
θ
eθ−1
1 ∞ 0 0
(6) − log{1 − (1 − t)θ} [1, ∞) 1(θ = 1) θ ∞ 0 0
(7) − log{θt + (1 − θ)} (0, 1] θ 1 − log(1 − θ) 0 ·
(8) 1−t
1+(θ−1)t
[1, ∞) 1
θ
1 1 0 ·
(9) log(1 − θ log t) (0, 1] θ 1 ∞ 0 −∞
(10) log(2t−θ − 1) (0, 1] 2θ 1 ∞ 0 0
(11) log(2 − tθ) (0, 1/2] θ 1 log 2 0 ·
(12) ( 1
t
− 1)θ [1, ∞) 1(θ = 1) θ ∞ θ ·
(13) (1 − log t)θ − 1 (0, ∞) θ 0 ∞ 0 1 − 1
θ
(14) (t−1/θ − 1)θ [1, ∞) 1(θ = 1) θ ∞ 1 ·
(15) (1 − t1/θ)θ [1, ∞) 1(θ = 1) θ 1 0 ·
(16) ( θ
t
+ 1)(1 − t) [0, ∞) 1 + θ 1 ∞ 1 ·
(17) − log
(1+t)−θ−1
2−θ−1
θ
2(2θ−1)
1 ∞ 0 0
(18) eθ/(t−1) [2, ∞) 0 ∞ e−θ 0 ·
(19) eθ/t − eθ (0, ∞) θeθ 1 ∞ ∞ ·
(20) et−θ
− e (0, ∞) θe 1 ∞ ∞ ·
(21) 1 − {1 − (1 − t)θ}1/θ [1, ∞) 1(θ = 1) θ 1 0 ·
(22) arcsin(1 − tθ) (0, 1] θ 1 π/2 0 ·
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![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.
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Slides ihp
- 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
- 5. Arthur CHARPENTIER - tails of Archimedean copulas Tail dependence in risk management q q qq q q q q q q q q q q q q q q q q q q q q qq qq q q qqq qqq qqq q qq qq q qq q q q q q q q q qq q q qq q q q q q q q q q q q qqqq q q q qqq qq qqqqqq qq q q q qqqqq qqq q q qq q q q q q q qq qq q q q q q q q q q q q q q qq q q qq q qq qqq qqq qqqqq 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 q q q q q q q qq q q q qq qqqqqqqq qq qqqqqq qqqqqq q q q q q qq q q q q q q q q q q q q q q q q q q qq q qq qqqq qq qqqqq qq qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq q q q q q q q q q qq q q q q q q q q qqqqqqqq q qqqqq q q q qqqq q q q q q q q q q q q q q q q q q q qq q qq qq qq qq q q q q q qq q qq q q q q q q q q q qq q q qqqqqq q q qq q q q q qq qq qq q q q qq q q q q q q q qqq q qq q qqqqq q q qqqq qqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqq q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q qqq q q qq q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q qqq qq qqq qq q qqq qqqqqq qqqqqqqq qqqqqq qqq q q q q q q q q qqq qq q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qqq q q q q q q q q qq q q q q qq q q q q q qqqqqqqqqq qqq qqqqqqq qqqqq qq qqqqqq qqqqqqqqqqqqqq qqqqq qqq qqq q q q q q q q q q qqqqq q q q q q q q q q q q q q q qqq q q q q q q q q q qq qqq q qq qqqqqq q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q qqqqq qqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqq qqq q 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 q q q q q q q q q q q qq q q q qq qq q q qq q q q qq q q q q q qq q q q qq q q q q q q qqq qqqqqqq q qqqqqqqqqqqq qq qqq q q q q q qqq q q q q qqq q q q q qq q q q q q q q q qq qq qqq q q q q qq q q q q q q q q qq qq qq q q q q q qqqqqqqqqq qqqq qqqqqqq qqqqq q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qqq qqq qqqqq q qq qq 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 q q q q q q q q qq q q q qq q q q q q q q q q q qq q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q qqq qq qq q q q qq q q q q qq q q q q q q q qq q qq qqqqqqq qq qq q qqqqqqqq qqq qq q q q q q q q q q q q qq q q q q q q q q qq q q qqqq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qqq qq qqqq qq q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qqq q qqqqqq 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 qq q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q qqq q q q q q q qq q q q q q q q q q q q qqq q q q q q q q q 1e+01 1e+03 1e+05 1e+011e+021e+031e+041e+05 Loss (log scale) AllocatedExpenses q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq 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 q q qq q q q q q q q q q q q q q q q q q q q q q 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 q q q q q 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 q q q q q q q q q 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 q qq q q q q q q q q q q qq 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 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 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 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 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 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 q q q q q q q q qq q q q q q q 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 q q q qq q q q q q q q q q q q q q q q q 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 qq q qq q q q q q q q q 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 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 q q q q q q qq q q q q q q q q q q q q 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 q q q q q q q qq q q 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 q q q q q q q q q q q q q q q q q 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 q q q q q q q q q q q q q q 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 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 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 q q q q q 1 10 100 1000 100001e+001e+021e+041e+06 Car claims (log scale) Householdclaims Fig. 1 – Multiple risks issues. 5
- 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
- 10. Arthur CHARPENTIER - tails of Archimedean copulas Gaussian copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 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 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 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 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 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 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 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 q q q q 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 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 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 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 q 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 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 q q q q q q q 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 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 q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q q q q q q q 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 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 q q q q q q q q q q q q q q q q q q q 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 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 q q q q q q q q q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q qq q q q q q q q q q q q q q q q 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 q qq q q q q q q q q q q qq q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 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 q 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 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 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 q q q q q q q q q q q q q q q q 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) GAUSSIAN q q Fig. 4 – L and R cumulative curves. 10
- 11. Arthur CHARPENTIER - tails of Archimedean copulas Gumbel copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 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 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 q q q q q q q q q q q q q q q q q q q q q 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 q q q q q q q q q q q q q 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 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 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 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 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 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 qq q q q q q q q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 q 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 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 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 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 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 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 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 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 q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q 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 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 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 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 q q q q q q q q q q q q q q q q q q 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 q q q q q q q q q q q 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 q q q q 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 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 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 q q q q 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 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 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 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 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 q q q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 q q q q q 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 q q q q q q q q q q q q q q q q q q q q q 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) GUMBEL q q Fig. 5 – L and R cumulative curves. 11
- 12. Arthur CHARPENTIER - tails of Archimedean copulas Clayton copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 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 q q q q q q q q q q q q q q q 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 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 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 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 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 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 q q q q q q q q q q q q q q q 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 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 q q q q q q 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 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 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 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 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 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 q q q q q q q q q q q q q q q q q q q q q q q 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 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 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 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 q q q q q q q q q q q q 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 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 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 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 q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq 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 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q 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 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 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 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 q q q q q q q q q q q q q 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 q 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) CLAYTON q q Fig. 6 – L and R cumulative curves. 12
- 13. Arthur CHARPENTIER - tails of Archimedean copulas Student t copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 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 q q q q q q q q q q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q qq q q q q q q q qq q q 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 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 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 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 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 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 q q q q q q q q q q q q 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 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 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 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 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 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 q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q q q 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 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 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 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 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 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 q q q q q q q q q q q q q 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 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 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 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 q q q q q q q q q q q q q q q 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 q q q q q q q q 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 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 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 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 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 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 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 q q q q qq q q qq q q q q q q q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 q 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) STUDENT (df=5) q q Fig. 7 – L and R cumulative curves. 13
- 14. Arthur CHARPENTIER - tails of Archimedean copulas Student t copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 q q q q q q 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 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 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 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 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 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 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 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 q q q q q 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q q q q qq q q q q q q q 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q q q q q 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 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 q q q q q q q q q q q 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 q q q 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q q q q q q q 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) STUDENT (df=3) q q Fig. 8 – L and R cumulative curves. 14
- 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
- 17. Arthur CHARPENTIER - tails of Archimedean copulas Gaussian copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q 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 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 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 q q q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Chi dependence functions lower tails upper tails GAUSSIAN q q Fig. 9 – χ functions. 17
- 18. Arthur CHARPENTIER - tails of Archimedean copulas Gumbel copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q q q q q 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 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 q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 q 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 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 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 q q q q q q q q q q q q q q q q q q q q 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 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 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 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 q q q q q q q q q q q 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 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 q q q q q q q q q q q q q q q 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 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 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 q q q q q q q q q q q q 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q q q q q 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 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 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 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 q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Chi dependence functions lower tails upper tails GUMBEL q q Fig. 10 – χ functions. 18
- 19. Arthur CHARPENTIER - tails of Archimedean copulas Clayton copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 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 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 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 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 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 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 q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q q q q q q q 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 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 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 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 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 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 q q q q q q q q q q q q qq q q q 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 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 q q q q q q q 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 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 q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 q q q q q q q qq q q q 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 q q q q q q q q q q q q q q q q q q 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 q q q q q 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 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 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 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 q q q q q q q q q q q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Chi dependence functions lower tails upper tails CLAYTON q q Fig. 11 – χ functions. 19
- 20. Arthur CHARPENTIER - tails of Archimedean copulas Student t copula 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 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 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 q q q q q q q q q q q 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 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 q q q q q q 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 q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q 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 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 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 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 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 q q q q q q 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q q q q q q q q q q q q q q q 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 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 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 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 q q q q q q q q q q q q q q q q q q 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 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 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 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 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 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 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 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 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 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 q q q q q q q q q 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 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 q q q q q q q qq q q q 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 q q q q q q q q q q q q q q q q q q 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 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 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 q q q q q q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Chi dependence functions lower tails upper tails STUDENT (df=3) q q Fig. 12 – χ functions. 20
- 21. Arthur CHARPENTIER - tails of Archimedean copulas Application in risk management : Loss-ALAE q q qq q q q q q q q q q q q q q q q q q q q q qq qq q q qqq qq q qq q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qqqq q q q qqq qq qq qqq q qq q q q qqqqq q q q q q qq q q q q q q qq qq q q q q q q q q q q q q q qq q q qq q q q q q q qqq q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq qq q q q qq qq qq qq qq q q qqq qqq qq q qq q q q q q q q q q q q q q q q q q q q q q qqqq q qqq qq qqqq q q qqq qq qq qq q qq q qqq qqqq qqq qqqqq qq qqqq qq qq q qq q q q q q q q q q q q q q q q q q q q qqqq qq q q q qq qqq q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q qq qq q qq q q qq q q q q q q qq q q q q qq q q q q q q q qq q q q q q q q qqqq qq q qqqqq q q qqqq qqqqq q qq q q q q qq qqqq qqqqqq q qqq qq qqqqqqqq qq qqqq q qqq q qq q q q q q q q q q 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 qqq 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 q q q q qq q q q qq q qq qqq qq qqq q q q q qq q qq qq q qq qq q qq q qqq qqq q qq q q q q q q q q q q 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 q q q q qq q q q q q q q q q qq q q q qqq qq q q q qqqqqqq qqq q qq qqq q qqq qqq qq qq qq q qqq qqq qqqqqqqqqqq qqq qq qqq qqq 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 q q q q q q q q q q q q q q q q q qqq q qq qq qqqq q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q qq qq qqq qqqqqqqqq q qq qqq q qq qq qqqq qq q qq q qq q 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 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 q q q q q q q q q q q q q q q q q q q q q q qqq q q qq qq q q q q q q q qqq qq qq qq qqq q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qq qq q q q q q q qq q qqqqq q qq q q qq q q qqq qqqqq 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 q q q q q q q q qqq qq q q q q q q q q q q q 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 q q q q q q q q qq q q q qq q q q q q q q q q 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 q q q q q q q q qq q q q qq q q q q q q q q q qq q q q q q q q qq q q q qq qq q qq qq q q q q qq qqq qq q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q 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 q q q q q q q q q q qq qq qqqq qqqq 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 q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q qq q q q qq q q q q q q q q q q q q qq q q q qqqqq q q q q q q q q q q q q q q q q q q q qqq qq q q q q qq 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Loss AllocatedExpenses Fig. 13 – Losses and allocated expenses. 21
- 22. Arthur CHARPENTIER - tails of Archimedean copulas Application in risk management : Loss-ALAE 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) qqq qqq qqq qqqqqqqqq qqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqq qqqqqq qqqq qqq qqqqqqq qqq qqqqq qq qqq qq q qq 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 q q q qqq qqq qqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqq qqqqq qq qqqqqqqqqqqqqq qqq q q q q Gumbel copula q q Fig. 14 – L and R cumulative curves, and χ functions. 22
- 23. Arthur CHARPENTIER - tails of Archimedean copulas Application in risk management : car-household q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq 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 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 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 q q q q q q q q 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 q q q q q q q q q 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 q q q q q q q q q q q q q q q 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 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 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 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 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 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 q q q q q q q 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 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 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 q q q q q q q 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 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 q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q 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 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 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 q q q q q q 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 q q q q q q q q q q q q q q 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 q q q q q q q q q q q q 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 q q q q q q q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Car claims Householdclaims Fig. 15 – Motor and Household claims. 23
- 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
- 26. Arthur CHARPENTIER - tails of Archimedean copulas Some examples of Archimedean copulas φ(t) range θ (1) 1 θ (t−θ − 1) [−1, 0) ∪ (0, ∞) Clayton, Clayton (1978) (2) (1 − t)θ [1, ∞) (3) log 1−θ(1−t) t [−1, 1) Ali-Mikhail-Haq (4) (− log t)θ [1, ∞) Gumbel, Gumbel (1960), Hougaard (1986) (5) − log e−θt−1 e−θ−1 (−∞, 0) ∪ (0, ∞) Frank, Frank (1979), Nelsen (1987) (6) − log{1 − (1 − t)θ} [1, ∞) Joe, Frank (1981), Joe (1993) (7) − log{θt + (1 − θ)} (0, 1] (8) 1−t 1+(θ−1)t [1, ∞) (9) log(1 − θ log t) (0, 1] Barnett (1980), Gumbel (1960) (10) log(2t−θ − 1) (0, 1] (11) log(2 − tθ) (0, 1/2] (12) ( 1 t − 1)θ [1, ∞) (13) (1 − log t)θ − 1 (0, ∞) (14) (t−1/θ − 1)θ [1, ∞) (15) (1 − t1/θ)θ [1, ∞) Genest & Ghoudi (1994) (16) ( θ t + 1)(1 − t) [0, ∞) 26
- 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
- 30. Arthur CHARPENTIER - tails of Archimedean copulas 0 20 40 60 80 100 020406080100 Conditional independence, continuous risk factor !3 !2 !1 0 1 2 3 !3!2!10123 Conditional independence, continuous risk factor Fig. 19 – Continuous classes of risks, (Xi, Yi) and (Φ−1 (FX(Xi)), Φ−1 (FY (Yi))). 30
- 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
- 39. Arthur CHARPENTIER - tails of Archimedean copulas upper tail lower tail φ(t) range θ −φ (1) θ1 φ(0) θ0 κ (1) 1 θ (t−θ − 1) [−1, ∞) 1 1 1 (−θ)∨0 θ ∨ 0 · (2) (1 − t)θ [1, ∞) 1(θ = 1) θ 1 0 · (3) log 1−θ(1−t) t [−1, 1) 1 − θ 1 ∞ 0 0 (4) (− log t)θ [1, ∞) 1(θ = 1) θ ∞ 0 1 − 1 θ (5) − log e−θt−1 e−θ−1 θ eθ−1 1 ∞ 0 0 (6) − log{1 − (1 − t)θ} [1, ∞) 1(θ = 1) θ ∞ 0 0 (7) − log{θt + (1 − θ)} (0, 1] θ 1 − log(1 − θ) 0 · (8) 1−t 1+(θ−1)t [1, ∞) 1 θ 1 1 0 · (9) log(1 − θ log t) (0, 1] θ 1 ∞ 0 −∞ (10) log(2t−θ − 1) (0, 1] 2θ 1 ∞ 0 0 (11) log(2 − tθ) (0, 1/2] θ 1 log 2 0 · (12) ( 1 t − 1)θ [1, ∞) 1(θ = 1) θ ∞ θ · (13) (1 − log t)θ − 1 (0, ∞) θ 0 ∞ 0 1 − 1 θ (14) (t−1/θ − 1)θ [1, ∞) 1(θ = 1) θ ∞ 1 · (15) (1 − t1/θ)θ [1, ∞) 1(θ = 1) θ 1 0 · (16) ( θ t + 1)(1 − t) [0, ∞) 1 + θ 1 ∞ 1 · (17) − log (1+t)−θ−1 2−θ−1 θ 2(2θ−1) 1 ∞ 0 0 (18) eθ/(t−1) [2, ∞) 0 ∞ e−θ 0 · (19) eθ/t − eθ (0, ∞) θeθ 1 ∞ ∞ · (20) et−θ − e (0, ∞) θe 1 ∞ ∞ · (21) 1 − {1 − (1 − t)θ}1/θ [1, ∞) 1(θ = 1) θ 1 0 · (22) arcsin(1 − tθ) (0, 1] θ 1 π/2 0 · 39
- 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