Tails of Archimedean Copulas tail dependence in risk management - - PowerPoint PPT Presentation

Arthur CHARPENTIER - tails of Archimedean copulas

Tails of Archimedean Copulas

tail dependence in risk management Arthur Charpentier

CREM-Universit´ e Rennes 1 (joint work with Johan Segers, UCLN)

http ://perso.univ-rennes1.fr/arthur.charpentier/

Colloque ´ Evaluation et couverture des risques extrˆ emes Universit´ e Paris-Dauphine & Chaire AXA de la Fondation du Risque, Juin 2008 1

Arthur CHARPENTIER - tails of Archimedean copulas

Tail behavior and risk management

In reinsurance (XS) pricing, use of Pickands-Balkema-de Haan’s theorem 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. 1 − F(x) ≈ (1 − F(u))

• 1 − Hξ,σ(u) (x − u)
• , for all x > u.

So, 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

Arthur CHARPENTIER - tails of Archimedean copulas

Pure premium of XS contract

Recall that πd = E((X − d)+) with d large, thus, πd = 1 P(X > d) ∞

d

1 − F(x)dx ≈ k n σ 1 − ξ

• 1 + ξ d − Xn−k:n

σ 1− 1

ξ

, i.e.

• πd = k

n

• σk

1 − ξk

• 1 +

ξk d − Xn−k:n

• σk

1− 1

• ξk

(see e.g. Beirlant et al. (2005). Possible to derive explicit formulas for any tail risk measure (VaR, TVaR...). 3

Arthur CHARPENTIER - tails of Archimedean copulas

Extending extreme value theory in higher dimension

univariate case bivariate case limiting distribution dependence structure of

• f 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

• f 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

Arthur CHARPENTIER - tails of Archimedean copulas

Tail dependence in risk management

• 1e+01

1e+03 1e+05 1e+01 1e+02 1e+03 1e+04 1e+05 Loss (log scale) Allocated Expenses

• ●
• ●
• ●
• 1

10 100 1000 10000 1e+00 1e+02 1e+04 1e+06 Car claims (log scale) Household claims

• Fig. 1 – Multiple risks issues.

5

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. 6

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

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) = ∂2C(u, v)

∂u∂v . 8

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

Arthur CHARPENTIER - tails of Archimedean copulas Gaussian copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

GAUSSIAN

• Fig. 4 – L and R cumulative curves.

10

Arthur CHARPENTIER - tails of Archimedean copulas Gumbel copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

GUMBEL

• Fig. 5 – L and R cumulative curves.

11

Arthur CHARPENTIER - tails of Archimedean copulas Clayton copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

CLAYTON

• Fig. 6 – L and R cumulative curves.

12

Arthur CHARPENTIER - tails of Archimedean copulas Student t copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

STUDENT (df=5)

• Fig. 7 – L and R cumulative curves.

13

Arthur CHARPENTIER - tails of Archimedean copulas Student t copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

STUDENT (df=3)

• Fig. 8 – L and R cumulative curves.

14

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,

• r 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,

• r 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

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

Arthur CHARPENTIER - tails of Archimedean copulas Gaussian copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Chi dependence functions

lower tails upper tails

GAUSSIAN

• Fig. 9 – χ functions.

17

Arthur CHARPENTIER - tails of Archimedean copulas Gumbel copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Chi dependence functions

lower tails upper tails

GUMBEL

• Fig. 10 – χ functions.

18

Arthur CHARPENTIER - tails of Archimedean copulas Clayton copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

• ●
• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Chi dependence functions

lower tails upper tails

CLAYTON

• Fig. 11 – χ functions.

19

Arthur CHARPENTIER - tails of Archimedean copulas Student t copula

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Chi dependence functions

lower tails upper tails

STUDENT (df=3)

• Fig. 12 – χ functions.

20

Arthur CHARPENTIER - tails of Archimedean copulas

Application in risk management : Loss-ALAE

• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Loss Allocated Expenses

• Fig. 13 – Losses and allocated expenses.

21

Arthur CHARPENTIER - tails of Archimedean copulas

Application in risk management : Loss-ALAE

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

• Gumbel copula
• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Chi dependence functions

lower tails upper tails

• Gumbel copula
• Fig. 14 – L and R cumulative curves, and χ functions.

22

Arthur CHARPENTIER - tails of Archimedean copulas

Application in risk management : car-household

• ●
• ●
• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Car claims Household claims

• Fig. 15 – Motor and Household claims.

23

Arthur CHARPENTIER - tails of Archimedean copulas

Application in risk management : car-household

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

L and R concentration functions

L function (lower tails) R function (upper tails)

• Gumbel copula
• 0.0

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Chi dependence functions

lower tails upper tails

• Gumbel copula
• Fig. 16 – L and R cumulative curves, and χ functions.

24

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

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

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

Arthur CHARPENTIER - tails of Archimedean copulas

5 10 15 5 10 15 20

Conditional independence, two classes

!3 !2 !1 1 2 3 !3 !2 !1 1 2 3

Conditional independence, two classes

• Fig. 17 – Two classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).

28

Arthur CHARPENTIER - tails of Archimedean copulas

5 10 15 20 25 30 10 20 30 40

Conditional independence, three classes

!3 !2 !1 1 2 3 !3 !2 !1 1 2 3

Conditional independence, three classes

• Fig. 18 – Three classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).

29

Arthur CHARPENTIER - tails of Archimedean copulas

20 40 60 80 100 20 40 60 80 100

Conditional independence, continuous risk factor

!3 !2 !1 1 2 3 !3 !2 !1 1 2 3

Conditional independence, continuous risk factor

• Fig. 19 – Continuous classes of risks, (Xi, Yi) and (Φ−1(FX(Xi)), Φ−1(FY (Yi))).

30

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 is not Archimedean (but can be a limit of

Archimedean copulas).

0.2 0.4 0.6 0.8 u_1 0.2 0.4 0.6 0.8 u_2 0.2 0.4 0.6 0.8 1 Frechet lower bound 0.2 0.4 0.6 0.8 u_1 . 2 . 4 . 6 . 8 u _ 2 0.2 0.4 0.6 0.8 1 Independence copula 0.2 0.4 0.6 0.8 u_1 0.2 0.4 0.6 0.8 u_2 0.2 0.4 0.6 0.8 1 Frechet upper bound

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Scatterplot, Lower Fréchet!Hoeffding bound

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Scatterplot, Indepedent copula random generation

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Scatterplot, Upper Fréchet!Hoeffding bound

31

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

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. 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 independence (dependence in 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|−κy−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(·)). 34

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 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

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 1 < 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) . 36

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 1 < 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 · · · uk (t1 + · · · + tk)−(k−1)dt1 · · · dtk for integer k ≥ 2 and (u1, . . . , uk) ∈ (0, ∞)k. 37

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

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 · (3) log 1−θ(1−t) t [−1, 1) 1 − θ 1 ∞ (4) (− log t)θ [1, ∞) 1(θ = 1) θ ∞ 1 − 1 θ (5) − log e−θt−1 e−θ−1 θ eθ−1 1 ∞ (6) − log{1 − (1 − t)θ} [1, ∞) 1(θ = 1) θ ∞ (7) − log{θt + (1 − θ)} (0, 1] θ 1 − log(1 − θ) · (8) 1−t 1+(θ−1)t [1, ∞) 1 θ 1 1 · (9) log(1 − θ log t) (0, 1] θ 1 ∞ −∞ (10) log(2t−θ − 1) (0, 1] 2θ 1 ∞ (11) log(2 − tθ) (0, 1/2] θ 1 log 2 · (12) ( 1 t − 1)θ [1, ∞) 1(θ = 1) θ ∞ θ · (13) (1 − log t)θ − 1 (0, ∞) θ ∞ 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 · (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 ∞ (18) eθ/(t−1) [2, ∞) ∞ e−θ · (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 · (22) arcsin(1 − tθ) (0, 1] θ 1 π/2 ·

39

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

Arthur CHARPENTIER - tails of Archimedean copulas upper tail lower tail φα(t) range α φ′ α(1) θ1(α) φα(0) θ0(α) κ(α) (1) (φ(t))α (1, ∞) 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, ∞) αθ1 φ(0) θ0 κ (5) φ(αt) − φ(α) (0, 1) αφ′(α) 1 φ(0) − φ(α) θ0 κ

41