Slides euria-2

Arthur CHARPENTIER - École d'été EURIA.

mesures de risques et dépendance
Arthur Charpentier
Université de Rennes 1 & École Polytechnique
arthur.charpentier@univ-rennes1.fr
http://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/

1


3 0.9

V (rank of Y)
Y

0.4
-1
(X i,Y i)

(U i,V i)

-3 -1 1 3 0.2 0.5 0.8
X U (rank of X)

Density of the copula

Isodensity curves of the density

2


Agenda
• General introduction

Modelling correlated risks
• A short introduction to copulas

• Quantifying dependence

• Statistical inference

• Agregation properties

3


Agenda





4


Some references on large and correlated risks
Rank , J. (2006). Copulas: From Theory to Application in Finance. Risk Book ,

Nelsen , R. (1999,2006). An introduction to copulas. Springer Verlag ,

Cherubini , U., Luciano , E. & Vecchiato, W. (2004). Copula Methods in

Finance. Wiley,

Beirlant , J., Goegebeur , Y., Segers, J. & Teugels
, J. (2004). Statistics of

Extremes: Theory and Applications. Wiley,

McNeil , A. Frey , R., & Embrechts , P. (2005). Quantitative Risk

Management: Concepts, Techniques, and Tools. Princeton University Press,

5


Agenda





6


Copulas, an introduction (in dimension 2)
Denition 1. A copula C is a joint distribution function on [0, 1]2 , with uniform
margins on [0, 1].
Set C(u, v) = P(U ≤ u, V ≤ v), where (U, V ) is a random pair with uniform

margins.

C is a distribution function on [0, 1]2 , and thus C(0, v) = C(u, 0) = 0, C(1, 1) = 1.
Furthermore C is increasing : since P is a positive measure, for all u1 ≤ u2 and

v1 ≤ v2 ,
Copula, positive area

1.0
P(u1 U ≤ u2 , v1 V ≤ v2 ) ≥ 0,

0.8
0.6
thus
0.4
C(u2 , v2 ) − C(u1 , v2 )
0.2

−C(u2 , v1 ) + C(u1 , v1 ) ≥ 0.
0.0

0.0 0.2 0.4 0.6 0.8 1.0

7


C has uniform margins, and thus

C(u, 1) = P(U ≤ u, V ≤ 1) = P(U ≤ u) = u on [0, 1].

Proposition 2. C is a copula if and only if C(0, v) = C(u, 0) = 0, C(u, 1) = u
and C(1, v) = v for all u, v , with the following 2-increasingness property
C(u2 , v2 ) − C(u1 , v2 ) − C(u2 , v1 ) + C(u1 , v1 ) ≥ 0,

for any u1 ≤ u2 and v1 ≤ v2 .

8


Borders of the copula function

!0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

1.2
1.0
0.8
0.6
0.4
0.2
0.0
!0.2
!0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 1: Value of the copula on the border of the unit square.

9


Fonction de répartition à marges uniformes

Z

Y
X

Figure 2: Graphical representation of a copula.

10


If C is twice dierentiable, one can dene its density as

∂ 2 C(u, v)
c(u, v) = .
∂u∂v

11


Densité d’une loi à marges uniformes

z

x x

Figure 3: Density of a copula.

12


Fonction de répartition à marges uniformes Densité d’une loi à marges uniformes


Figure 4: Distribution functions and densities.

13




Figure 5: Distribution functions and densities.

14


Sklar's theorem
Theorem 3. (Sklar) Let C be a copula, and FX and FY two marginal
distributions, then F (x, y) = C(FX (x), FY (y)) is a bivariate distribution
function, with F ∈ F(FX , FY ).
Conversely, if F ∈ F(FX , FY ), there exists C such that
F (x, y) = C(FX (x), FY (y)). Further, if FX and FY are continuous, then C is
unique, and given by
C(u, v) = F (FX (u), FY (v)) for all (u, v) ∈ [0, 1] × [0, 1]
−1 −1

We will then dene the copula of F , or the copula of (X, Y ).
In that case, the copula of (X, Y ) is the distribution function of (FX (X), FY (Y )).
Proposition 4. If (X, Y ) has copula C , the copula of (g(X), h(Y )) is also C for
any increasing functions g and h.

15


Copulas, an introduction (in dimension d ≥ 2)
Denition 5. A copula C is a joint distribution function on [0, 1]d , with
uniform margins on [0, 1].
Let U = (U1 , ..., Ud ) denote a random pair with uniform margins.

C is a distribution function on [0, 1]d , and thus C(u) = 0 if ui = 0 for some

i ∈ {1, . . . , d}, and C(1) = 1.
Furthermore C satises some increasing property since P is a positive measure

(for all 0 ≤ u ≤ v ≤ 1, P(u U ≤ v) ≥ 0), thus

sign(z)C(z) ≥ 0,
z

where the sum is taken over all vertices of [u × v], and where sign(z) is +1 if

zi = ui for an even number of i (and −1 otherwise, see Figure 6). And nally C
has uniform margins, and thus

C(1, . . . , 1, ui , 1, . . . , 1) = ui on [0, 1].

16


Increasing functions in dimension 3

Figure 6: The notion of 3-increasing functions.

17


Theorem 6. (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 (F1 (u1 ), . . . , Fd (ud )) for all (ui ) ∈ [0, 1]
−1 −1

We will then dene the copula of F , or the copula of X .
In that case, the copula of (X = (X1 , . . . , Xd ) is the distribution function of

U = (F1 (X1 ), . . . , Fd (Yd )).
Again, if C is dierentiable, one can dene its density,

∂ d C(u1 , . . . , ud )
c(u1 , . . . , ud ) = .
∂u1 . . . ∂ud

18


Copulas in high dimension, a dicult problem
It is usually dicult to represent dependence in dimension d 2, and it is

usually studied by pairs.

In dimension d = 2, one can dene the following Fréchet class F(FX , FY , FZ )
dened by its marginal distributions. But it can also be interested to study

F(FXY , FXZ , FY Z ) dened by it paired distributions.

One of the problem that arises is the compatibility of marginals: one has to

verify that

CXY (x, y) = CX|Z (x|z)CY |Z (y|z)dz,

for instance.

19


1.0
0.8
1.0

0.6
0.8

p
0.6

0.4
1.0
0.4

0.8

0.2
0.6
0.2

0.4
0.2

0.0
0.0

0.0
0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Composante 1

1.0
1.0

0.8
0.8

0.6
0.6
p

p

0.4
0.4
0.2

0.2
0.0

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

Composante 2 Composante 3

Figure 7: Scatterplot in dimension 3 including projections.

20


Copulas and ranks
The copula of X = (X1 , . . . , Xd ) is the distribution function of

U = (F1 (X1 ), . . . , Fd (Yd )).
In practice, since marginal distributions are unknown, the idea is to substitute

empirical distribution function,

n
#{observations Xi,j 's lower than xi } 1
Fi (xi ) = = 1(Xi,j ≤ xi ).
#{observations } n j=1

Note that
n
#{observations Xi,j 's lower than Xi,j0 } 1 Ri,j0
Fi (Xi,j0 ) = = 1(Xi,j ≤ Xi,j0 ) = ,
#{observations } n j=1
n

where Ri,j0 denotes the rank of Xi,j0 within {Xi,1 , ..., Xi,n }.
On a statistical point of view, studying the copula means studying ranks.

21


Scatterplot of (X,Y) Scatterplot of the ranks of (X,Y)

9.0

20
8.5
8.0

15
Ranks of the Yi’s
7.5
Y (raw data)

10
7.0
6.5

5
6.0
5.5

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 5 10 15 20

X (raw data) Ranks of the Xi’s

Scatterplot of the ranks of (X,Y), divided by n Scatterplot o+ ,-,/0, t1e copula!t3pe tran+orm o+ ,6,70
1.0

1.0
0.8
0.8

Vi=Ranks of the Yi’s/n+1
Ranks of the Yi’s/n

0.6
0.6

0.4
0.4

0.2
0.2

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

Ranks of the Xi’s/n Ui=Ranks of the Xi’s/n+1

Figure 8: Copulas, ranks and parametric inference, from (Xi , Yi ) to (Ui , Vi ).

22


Some very classical copulas
• The independent copula C(u, v) = uv = C ⊥ (u, v).
The copula is standardly denoted Π, P or C ⊥, and an independent version of
L
(X, Y ) will be denoted (X ⊥ , Y ⊥ ). It is a random vector such that X⊥ = X and
L
Y⊥ =Y, with copula C ⊥.
In higher dimension, C ⊥ (u1 , . . . , ud ) = u1 × . . . × ud is the independent copula.

• The comonotonic copula C(u, v) = min{u, v} = C + (u, v).
The copula is standardly denoted M, or C +, and an comonotone version of
L
(X, Y ) will be denoted (X + , Y + ). It is a random vector such that X+ = X and
L
Y+ =Y, with copula C +.
(X, Y ) has copula C+ if and only if there exists a strictly increasing function h
L−1 −1
such that Y = h(X), or equivalently (X, Y ) = (FX (U ), FY (U )) where U is

U([0, 1]).

23


Note that for any u, v

P(U ≤ u, V ≤ v) = P({U ∈ [0, u]} ∩ {V ∈ [0, v]})
≤ min{P(U ∈ [0, u]), P(V ∈ [0, v])}

thus, C(u, v) ≤ min{u, v} = C + (u, v). Thus, C+ is an upper bound for the set of

copulas.

In higher dimension, C + (u1 , . . . , ud ) = min{u1 , . . . , ud } is the comonotonic

copula.

• The contercomotonic copula C(u, v) = max{u + v − 1, 0} = C − (u, v).
The copula is standardly denoted W, or C −, and an contercomontone version of
L
(X, Y ) will be denoted (X − , Y − ). It is a random vector such that X− = X and
L
Y− =Y, with copula C −.
(X, Y ) has copula C− if and only if there exists a strictly decreasing function h
L−1 −1
such that Y = h(X), or equivalently (X, Y ) = (FX (1 − U ), FY (U )) where U is

U([0, 1]).

24


Note that for any u, v ,

P(U ≤ u, V ≤ v) = P({U ∈ [0, u]} ∩ {V ∈ [0, v]})
= P(U ∈ [0, u]) + P(V ∈ [0, v]) − P({U ∈ [0, u]} ∪ {V ∈ [0, v]})

thus, C(u, v) ≥ u + v − 1 since P({U ∈ [0, u]} ∪ {V ∈ [0, v]}) ≤ 1, and since
C(u, v) ≥ 0, C(u, v) ≥ max{u + v − 1, 0} = C − (u, v). Thus, C − is a lower bound
for the set of copulas.

In higher dimension, C − (u1 , . . . , ud ) = max{u1 + . . . + ud − (d − 1), 0} is not a

copula: if (X, Y ) and (X, Z) are countercomonotonic, (Y, Z) is necessarily
comonotonic - it is not possible to have all component highly negatively

correlated.

Anyway, it is still the best pointwise lower bound.

25


1

1

0.8 1
nd
0.8

0.2 0.4 0.6 0.8
la

nd
Independence copu
Frechet lower bou

Frechet upper bou
0 0.2 0.4 0.6

0 0.2 0.4 0.6
0
0.8 0.8
0.8
0.6 0.6
0.8 0.8 0.6
0.8
u_ 0.4 0.6 u_ 0.4 0.6
2 2 u_ 0.4 0.6
2
0.4 0.4 u_1
0.2 u_1 0.2
0.2 0.4
u_1
0.2 0.2
0.2

Fréchet Lower Bound Independent copula Fréchet Upper Bound
1.0

1.0

1.0
0.8

0.8

0.8
0.6

0.6

0.6
0.4

0.4

0.4
0.2

0.2

0.2
0.0

0.0

0.0
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

Scatterplot, Lower Fréchet!Hoeffding bound Scatterplot, Indepedent copula random generation Scatterplot, Upper Fréchet!Hoeffding bound
1.0

1.0

1.0
0.8

0.8

0.8
0.6

0.6

0.6
0.4

0.4

0.4
0.2

0.2

0.2
0.0

0.0

0.0
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

Figure 9: Contercomontonce, independent, and comonotone copulas.

26


Pitfalls on independence and comonotonicity
The following proposition is false,

Uncorrect Proposition 7. If X and Y are independent, if Y and Z are
independent, then X and Z are independent.
If

(X, Y, Z) = (1, 1, 1) with probability 1/4,
(1, 2, 1) with probability 1/4,

then X and Y are independent, and Y and Z are independent, but X = Z.

27


X and Y independent Y and Z independent X and Z comonotonic
4

4

4
3

3

3
Component Y

Component Z

Component Z
2

2

2
1

1

1
0

0

0
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

Component X Component Y Component X

Figure 10: Mixing independence and comonotonicity.

28



Uncorrect Proposition 8. If X and Y are comonotonic, if Y and Z are
comonotonic, then X and Z are comonotonic.
If


then X and Y are comonotonic, and Y and Z are comonotonic, but X and Z are

independent.

29


X and Y comonotonic Y and Z comonotonic X and Z independent
4

4

4
3

3

3
Component Y

Component Z

Component Z
2

2

2
1

1

1
0

0

0
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4



30



Uncorrect Proposition 9. If X and Y are comonotonic, if Y and Z are
independent, then X and Z are independent.
If


then X and Y are comonotonic, and Y and Z are independent, but X and Z are

anticomonotonic.

31


If


then X and Y are comonotonic, and Y and Z are independent, but X and Z are

comonotonic.

32


X and Y comonotonic Y and Z independent X and Z comonotonic
4

4

4
3

3

3
Component Y

Component Z

Component Z
2

2

2
1

1

1
0

0

0
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4



33


Elliptical (Gaussian and t) copulas
The idea is to extend the multivariate probit model, Y = (Y1 , . . . , Yd ) with
marginal B(pi ) distributions, modeled as Yi = 1(Xi ≤ ui ), where X ∼ N (I, Σ).
• The Gaussian copula, with parameter α ∈ (−1, 1),
Φ−1 (u) Φ−1 (v)
1 −(x2 − 2αxy + y 2 )
C(u, v) = √ exp dxdy.
2π 1 − α2 −∞ −∞ 2(1 − α2 )

Analogously the t-copula is the distribution of (T (X), T (Y )) where T is the t-cdf,
and where (X, Y ) has a joint t-distribution.
• The Student t-copula with parameter α ∈ (−1, 1) and ν ≥ 2,
t−1 (u) t−1 (v) −((ν+2)/2)
1 ν ν x2 − 2αxy + y 2
C(u, v) = √ 1+ dxdy.
2π 1 − α2 −∞ −∞ 2(1 − α2 )

34


Archimedean copulas
Denition of Archimedean copulas

• Archimedian copulas C(u, v) = φ−1 (φ(u) + φ(v)), where φ is decreasing

convex (0, 1), with φ(0) = ∞ and φ(1) = 0.
Example 10. If φ(t) = [− log t]α , then C is Gumbel's copula, and if
φ(t) = t−α − 1, C is Clayton's. Note that C ⊥ is obtained when φ(t) = − log t.
How Archimedean copulas were introduced ?

1. The frailty approach ( Oakes (1989)).

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))

35


thus, since X and Y are conditionally independent,

F (x, y) = E(P(X ≤ x|Θ = θ) × P(Y ≤ y|Θ = θ))

and therefore

F (x, y) = 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 ) satises

F (x, y) = ψ(ψ −1 (FX (x)) + ψ −1 (FY (y))).

Example 11. If Θ is Gamma distributed, the associated copula is Clayton's. If
Θ has a stable distribution, the associated copula is Gumbel's.

36


Consider two risks, X and Y, such that

X|Θ = θG ∼ E(θG ) and Y |Θ = θG ∼ E(θG ) are independent,

X|Θ = θB ∼ E(θB ) and Y |Θ = θB ∼ E(θB ) are independent,

(unobservable good (G) and bad (B ) risks).

The following gures start from 2 classes of risks, then 3, and then a continuous

risk factor θ ∈ (0, ∞).

37


Conditional independence, two classes Conditional independence, two classes

20

3
2
15

1
10

0
!1
5

!2
!3
0

0 5 10 15 !3 !2 !1 0 1 2 3

Figure 13: Two classes of risks, (Xi , Yi ) and (Φ−1 (FX (Xi )), Φ−1 (FY (Yi ))).

38


Conditional independence, three classes Conditional independence, three classes

3
40

2
30

1
0
20

!1
10

!2
!3
0

0 5 10 15 20 25 30 !3 !2 !1 0 1 2 3

Figure 14: Three classes of risks, (Xi , Yi ) and (Φ−1 (FX (Xi )), Φ−1 (FY (Yi ))).

39


Conditional independence, continuous risk factor Conditional independence, continuous risk factor

100

3
2
80

1
60

0
40

!1
20

!2
!3
0

0 20 40 60 80 100 !3 !2 !1 0 1 2 3

Figure 15: Continuous classes of risks, (Xi , Yi ) and (Φ−1 (FX (Xi )), Φ−1 (FY (Yi ))).

40


2. The survival approach: assume that there is a convex survival function S,
with S(0) = 1, such that

P(X x, Y y) = S(x + y),

then the joint survival copula of (X, Y ) is

S(S −1 (u) + S −1 (v)).

Example 12. If S is the Pareto survival distribution, the associated copula is
Clayton's. If S is the Weibull survival distribution, the associated copula is
Gumbel's.

41


3. The use of Kendall's distribution function K(t) = P(C(U, V ) ≤ t) where

(U, V ) is a random pair with distribution function C .

Then, for Archimedean copulas,

φ (t)
K(t) = t − = t − λ(t),
φ(t)
which can be inverted easily in

1
1
φ(t) = φ(t0 ) exp dt ,
t0 λ(t)
for some 0 t0 1 and 0 ≤ u ≤ 1.

42


Some more examples of Archimedean copulas

ψ(t) range θ
(1) 1 (t−θ − 1)
θ
[−1, 0) ∪ (0, ∞) Clayton, Clayton (1978)

(2) (1 − t)θ [1, ∞)
1−θ(1−t)
(3) log [−1, 1) Ali-Mikhail-Haq

Gumbel Hougaard
t
(4) (− log t)θ [1, ∞) Gumbel, (1960), (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, ∞)
1+(θ−1)t
(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 − 1)θ [1, ∞)
t
(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 + 1)(1 − t) [0, ∞)
t

43


Some characterizations of Archimedean copula

L
• Frank copula is the only Archimedean such that (U, V ) = (1 − U, 1 − V )
(stability by symmetry),

• 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),

• 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),

44


Extreme value copulas
• Extreme value copulas

log u
C(u, v) = exp (log u + log v) A ,
log u + log v
where A is a dependence function, convex on [0, 1] with A(0) = A(1) = 1, et

max{1 − ω, ω} ≤ A (ω) ≤ 1 for all ω ∈ [0, 1] .

An alternative denition is the following: C is an extreme value copula if for all

z 0,
1/z 1/z
C(u1 , . . . , ud ) = C(u1 , . . . , ud )z .

Those copula are then called max-stable: dene the maximum componentwise of

a sample X 1 , . . . , Xn , i.e. Mi = max{Xi,1 , . . . , Xi,n }.

45


The joint distribution of M is

P(M ≤ x) = C(F1 (x1 , . . . , Fd (xd ))n ,

where C is the copula of the X i 's. Since P(Mi ≤ xi ) = Fi (xi )n , it can be written

P(M ≤ x) = C(P(M1 ≤ x1 )1/n , . . . , P(Md ≤ xd )1/n )n .
1/n 1/n
Thus, C(u1 , . . . , ud )n is the copula of the n maximum componentwise from a

sample with copula C .

Example 13. : If A is constant (1 on [0, 1]), then X and Y are independent,
and if A(ω) = max {ω, 1 − ω}, X and Y are comonotonic. Gumbel's copula is
obtained if
(
A(ω) = ((1 − ω)α + ω α + 1) 1/α),
for all 0 ≤ ω ≤ 1 and α ≥ 1.

46


Pickands dependence function A

1.0
0.9
0.8
0.7
0.6
0.5

0.0 0.2 0.4 0.6 0.8 1.0

Figure 16: Shape of Gumbel's dependence function A(ω).

47


How to construct much more copulas ?
Using geometric transformations

From a given copula C, cdf of random pair (U, V ), dene

• the copula of (U, 1 − V ),

C(U,1−V ) (u, v) = u − C(u, 1 − v)

• the copula of (1 − U, V ),

C(1−U,V ) (u, v) = v − C(1 − u, v)

• the copula of (1 − U, 1 − V ), the rotated or survival copula,
C(1−U,1−V ) (u, v) = C ∗ (u, v) = u + v − 1 + C(1 − u, 1 − v)

Note that if P(X ≤ x, Y ≤ y) = C(P(X ≤ x), P(Y ≤ y)), then

P(X x, Y y) = C ∗ (P(X x), P(Y y)).

48


0.8
0.8

0.4
0.4

0.0
0.0

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

Figure 17: Using geometric transformation to generate new copulas.

49


0.8

0.8
0.4

0.4
0.0
0.0

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


50


0.8
0.8

0.4
0.4

0.0
0.0

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


51


0.8

0.8
0.4

0.4
0.0
0.0

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


52


Using mixture of copulas

Lemma 14. The set of copulas is convex, i.e. if {Cθ , θ ∈ Ω} is a collection of
copulas,
C(u, v) = Cθ (u, v)dΠ(θ)
R
is a copula, where Π is a distribution on Ω
Thus C = αC1 + (1 − α)C2 denes a copula for all α ∈ [0, 1].
Example 15. Fréchet (1951) suggested a mixture of the lower and the upper
bound,
C(u, v) = αC − (u, v) + (1 − α)C + (u, v), for some α ∈ [0, 1].

Example 16. Mardia (1970) suggested a mixture of the lower, the upper
bound, and the independent copula
α2 − 2 ⊥ α2 +
C(u, v) = C (u, v) + (1 − α )C (u, v) + C (u, v), α ∈ [0, 1].
2 2

53


Using distortion functions

Denition 17. A distortion function is a function h : [0, 1] → [0, 1] strictly
increasing such that h(0) = 0 and h(1) = 1.
The set of distortion function will be denoted H.
Note that h∈H if and only if h−1 ∈ H. Given a copula C, dene

Ch (u, v) = h−1 (C(h(u), h(v))).

If h is convex, then Ch is a copula, called distorted copula.

Example 18. if h(x) = x1/n , the distorted copula is
Ch (u, v) = C n (u1/n , v 1/n ), for all n ∈ N, (u, v) ∈ [0, 1]2 .

if the survival copula of the (Xi , Yi )'s is C , then the survival copula of
(Xn:n , Yn:n ) = (max{X1 , ..., Xn }, max{Y1 , ..., Yn }) is Ch .

54


Example 19. if C(u, v) = uv = C ⊥ (u, v) (the independent copula), and
φ(·) = log h(·), then

Ch (u, v) = h−1 (h(u)h(v)) = φ−1 (φ(u) + φ(v)).

Example 20. if h(x) = [1 − e−αx ]/[1 − e−α ] (an exponential distortion), and if
C = C ⊥ , then
1 (e−αu − 1)(e−αv − 1)
Ch (u, v) = − log 1 + ,
α e−α − 1
which is Frank copula.

55


Distorted Frank copula, h(x) = x Distorted Frank copula, h(x) = x(1 2)

Distorted Frank copula, h(x) = x(1 3)
Distorted Frank copula, h(x) = x(1 4)

Figure 21: Distorted copula, from Frank copula.

56


Monte Carlo and copulas
Generation of independent variables can be done using a Random function.

Denition 21. Function Random should satisfy the following properties (i) for
all 0 ≤ a ≤ b ≤ 1,
P (Random ∈ ]a, b]) = b − a.
(ii) successive calls of function Random should generate independent draws, i.e.
0 ≤ a ≤ b ≤ 1, 0 ≤ c ≤ d ≤ 1

P (Random1 ∈ ]a, b] , Random2 ∈ ]c, d]) = (b − a) (d − c) ,

or more generally, dene k-uniformity for all 0 ≤ ai ≤ bi ≤ 1, i = 1, ..., k,
k
P (Random1 ∈ ]a1 , b1 ] , ..., Randomk ∈ ]ak , bk ]) = (bi − ai ) .
i=1

Thus, one can generate easily random vectors U = (U1 , ..., Ud ) with independent

component.

57


The idea to generate correlated vectors U = (U1 , ..., Ud ), the idea is to use rst

P(U1 ≤ u1 , . . . , Ud ≤ ud ) = P(Ud ≤ ud |U1 ≤ u1 , . . . , Ud−1 ≤ ud−1 )
×P(Ud−1 ≤ ud−1 |U1 ≤ u1 , . . . , Ud−2 ≤ ud−2 )
×...
×P(U3 ≤ u3 |U1 ≤ u1 , U2 ≤ u2 )
×P(U2 ≤ u2 |U1 ≤ u1 ) × P(U1 ≤ u1 ).

58


Starting from the end, P(U1 ≤ u1 ) = u1 since U1 is uniform, while

P(U2 ≤ u2 |U1 = u1 )
= P(U2 ≤ u2 , U3 ≤ 1, . . . Ud ≤ 1|U1 = u1 )
= lim P(U2 ≤ u2 , U3 ≤ 1, . . . Ud ≤ 1|U1 ∈ [u1 , u1 + h])
h→0
P(u1 ≤ U1 ≤ u1 + h, U2 ≤ u2 , U3 ≤ 1, . . . Ud ≤ 1)
= lim
h→0 P(U1 ∈ [u1 , u1 + h])
P(U1 ≤ u1 + h, U2 ≤ u2 , U3 ≤ 1, . . . Ud ≤ 1) − P(U1 ≤ u1 , U2 ≤ u2 , U3 ≤ 1, . . .
= lim
h→0 P(U1 ∈ [u1 , u1 + h])
C(u1 + h, u2 , 1, . . . , 1) − C(u1 , u2 , 1, . . . , 1) ∂C
= lim = C(u1 , u2 , 1, . . . , 1).
h→0 h ∂u1
and more generally,

∂ k−1
P(Uk ≤ uk |U1 = u1 , . . . , Uk−1 = uk−1 ) = C(u1 , . . . , uk , 1, . . . , 1).
∂u1 . . . ∂uk−1

59


Thus, U = (U1 , .., Un ) with copula C could be simulated using the following

algorithm,

• simulate U1 uniformly on [0, 1],

u1 ← Random1 ,

• simulate U2 from the conditional distribution ∂1 C(·|u1 ),

u2 ← [∂1 C(·|u1 )]−1 (Random2 ),

• simulate Uk from the conditional distribution ∂1,...,k−1 C(·|u1 , ..., uk−1 ),

uk ← [∂1,...,k−1 C(·|u1 , ..., uk−1 )]−1 (Randomk ),

...etc, where the Randomi 's are independent calls of a Random function.

This is the underlying idea when using Cholesky decomposition.

60


Example: for Clayton's copula, C(u, v) = (u−α + v −α − 1)−1/α , (U, V ) has joint

distribution C if and only if U is uniform on on [0, 1] and V |U = u has
conditional distribution

P(V ≤ v|U = u) = ∂2 C(v|u) = (1 + uα [v −α − 1])−1−1/α .

The algorithm to generate Clayton's copula is the

• simulate U1 uniformly on [0, 1],

u1 ← Random1 ,

• simulate U2 from the conditional distribution ∂2 C(·|u),

u2 ← [∂1 C(·|u1 )]−1 (Random2 ),

i.e.

u2 ← [(Random2 )−α/(1+α − 1]u−α + 1−1/α .
1

61


1.5
0.0 0.5 1.0 1.5 2.0

1.0
0.5
0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Distribution of v given u=0.3 Distribution of v given u=0.5

Generation of Clayton’s copula
0.8

0.0 0.5 1.0 1.5
0.4
0.0

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

Distribution of v given u=0.8

Figure 22: Simulation of Clayton's copula.

62


500

400
/

/
300
q

q

200
0 100

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

!i#tribution +e - !i#tribution +e 9

4
0.8

2
0
0.4

!4 !2
0.0

0.0 0.2 0.4 0.6 0.8 1.0 !4 !2 0 2 4

-ni:orm mar=in# Stan+ar+ ?au##ian mar=in#

Figure 23: Simulation of the independent copula.

63


400

400
/

/
.

.
200

200
0

0
010 012 014 014 015 610 010 012 014 014 015 610

!is$riu$i(n +e - !is$riu$i(n +e 7

4
015

2
0
014

!2
010

010 012 014 014 015 610 !2 0 2 4

-ni8(r9 9argins $an+ar+ =aussian 9argins

Figure 24: Simulation of the comontone copula.

64


400

400
/

/
q

q
200

200
0

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

Distribution de - Distribution de V
0.8

2
0
0.4

!2
0.0

!4
0.0 0.2 0.4 0.6 0.8 1.0 !2 0 2 4

-niform margins tandard =aussian margins

Figure 25: Simulation of the contercomonotone copula.

65


900
400
/

/

300
.

.
200

0 100
0

010 012 014 014 015 110 010 012 014 014 015 110

!istri'tion de - !istri'tion de 7

4
015

2
0
014

!4 !2
010

010 012 014 014 015 110 !4 !2 0 2 4

-ni:orm mr=ins Stndrd ?'ssin mr=ins

Figure 26: Simulation of the Gaussian copula.

66


400
400
y

y
.

.

200
200
0

0
010 012 014 014 015 110 010 012 014 014 015 110

D#$%u$on +e U D#$%u$on +e V

4
015

2
0
014

!2
010

010 012 014 014 015 110 !2 0 2 4

Unfo%m ma%;n# S$an+a%+ =au##an ma%;n#

Figure 27: Simulation of Clayton's copula.

67


400

400
y

y
q

q
200

200
0

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

Distribution de U Distribution de V

4
0.8

2
0
0.4

!4 !2
0.0

0.0 0.2 0.4 0.6 0.8 1.0 !4 !2 0 2 4

Uniform margins Standard Gaussian margins

Figure 28: Simulation of Clayton's survival copula.

68


400

400
y

y
q

q
200

200
0

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

Distribution de U Distribution de V

4
0.8

2
0
0.4

!4 !2
0.0

0.0 0.2 0.4 0.6 0.8 1.0 !4 !2 0 2

Uniform margins Standard Gaussian margins

Figure 29: Simulation of a copula mixture.

69


Copulas in nance: options on multiple assets
Remark 22. Recall that Breeden Litzenberger (1978) proved that the risk
neutral probability can be obtrained from option prices: consider the price of a call
∞
C(T, K) = e−rT EQ ((ST − K)+ ). Since (ST − K)+ = K 1(ST x)dx, one gets
∞
−rT
C(T, K) = e Q(ST x)dx,
K

hence
∂C ∂P
Q(ST ≤ x) = −e−rT (T, x), or Q(ST ≤ x) = −erT (T, x)
∂K ∂K
where P denotes the price of a put option.
1 2
Consider an option on 2 assets, with payo h(ST , ST ). The price at time 0 is

e−rT EQ (h(ST , ST )).
1 2

70


Copulas in nance: call on maximum
1 2 1 2
Here the payo is h(ST , ST ) = (max{ST , ST } − K)+ . The price is then

C(T, K) = e−rT EQ ((max{ST , ST } − K)+ )
1 2

∞
= e−rT EQ 1 2
1 − 1(max{ST , ST } ≤ x)dx
K
∞
−rT 1 2
= e 1 − Q(max{ST , ST } ≤ x) dx,
K
1 2
Q(ST ≤x,ST ≤x)

1 2
hence, if (ST , ST ) has copula C (under Q), then

∞
∂P 1 ∂P 2
C(T, K) = e−rT 1 − C erT (T, x), erT (T, x) dx.
K ∂K ∂K

71


Copulas in nance: call on spreads
1 2 1 2
Here the payo is h(ST , ST ) = ([ST − ST ] − K)+ . The price is then

∞
−rT 1 2 −rT 2 1
C(T, K) = e EQ ((ST − ST − K)+ ) = e EQ 1(ST + K ≤ x ≤ ST )dx
−∞
∞
= e−rT 2 2 1
Q(K + ST ≤ x) − Q(ST + K ≤ x, ST ≤ x} ≤ x) dx,
−∞
1 2
Q(ST ≤x,ST ≤x+K)

1 2
hence, if (ST , ST ) has copula C (under Q), then

∞
−rT rT ∂P 2 rT ∂P
1
rT ∂P
2
C(T, K) = e e (T, x−K)−C e (T, x), e (T, x − K) dx.
−∞ ∂K ∂K ∂K

72


Copulas in nance: bonds on option prices
Using Tchen's inequality, it is possible to derive bounds for options when the

payo is supermodular.

73


Agenda





74


Natural properties for dependence measures

Denition 23. κ is measure of concordance if and only if κ satises
1. κ is dened for every pair (X, Y ) of continuous random variables,
2. −1 ≤ κ (X, Y ) ≤ +1, κ (X, X) = +1 and κ (X, −X) = −1,
3. κ (X, Y ) = κ (Y, X),
4. if X and Y are independent, then κ (X, Y ) = 0,
5. κ (−X, Y ) = κ (X, −Y ) = −κ (X, Y ),
6. if (X1 , Y1 ) P QD (X2 , Y2 ), then κ (X1 , Y1 ) ≤ κ (X2 , Y2 ),
7. if (X1 , Y1 ) , (X2 , Y2 ) , ... is a sequence of continuous random vectors that
converge to a pair (X, Y ) then κ (Xn , Yn ) → κ (X, Y ) as n → ∞.

75


As pointed out in Scarsini (1984), most of the axioms are self-evident .
Ifκ is measure of concordance, then, if f and g are both strictly increasing, then
κ(f (X), g(Y )) = κ(X, Y ). Further, κ(X, Y ) = 1 if Y = f (X) with f almost
surely strictly increasing, and analogously κ(X, Y ) = −1 if Y = f (X) with f

almost surely strictly decreasing (see Scarsini (1984)).

76


Association measures: Kendall's τ and Spearman's ρ
Rank correlations can be considered, i.e. Spearman's ρ dened as

1 1
ρ(X, Y ) = corr(FX (X), FY (Y )) = 12 C(u, v)dudv − 3
0 0

and Kendall's τ dened as

1 1
τ (X, Y ) = 4 C(u, v)dC(u, v) − 1.
0 0

Historical version of those coecients

Spearman's rho was introduced in Spearman (1904) as

ρ(X, Y ) = 3[P((X1 − X2 )(Y1 − Y3 ) 0) − P((X1 − X2 )(Y1 − Y3 ) 0)],

where (X1 , Y1 ), (X2 , Y2 ) and (X3 , Y3 ) denote three independent versions of

(X, Y ) (see Nelsen (1999)).

77


Similarly Kendall's tau was not dened using copulae, but as the probability of

concordance, minus the probability of discordance, i.e.

τ (X, Y ) = 3[P((X1 − X2 )(Y1 − Y2 ) 0) − P((X1 − X2 )(Y1 − Y2 ) 0)],

where (X1 , Y1 ) and (X2 , Y2 ) denote two independent versions of (X, Y ) (see

Nelsen (1999)).

4Q
Equivalently, τ (X, Y ) = 1 − 2 − 1)
where Q is the number of inversions
n(n
between the rankings of X and Y (number of discordance).

78


1.5 Concordant pairs Discordant pairs

1.5
1.0

1.0
0.5

0.5
Y

Y
0.0

0.0
!0.5

!0.5
!2.0 !1.5 !1.0 !0.5 0.0 0.5 1.0 !2.0 !1.5 !1.0 !0.5 0.0 0.5 1.0

X X

Figure 30: Concordance versus discordance.

79


The case of the Gaussian random vector

If (X, Y ) is a Gaussian random vector with correlation r, Kruskal
then ( (1958))

6 r 2
ρ(X, Y ) = arcsin and τ (X, Y ) = arcsin (r) .
π 2 π

80


Link between Kendall's tau and Spearman's rho

Note that Kendall's tau and Spearman's are linked: it is impossible to have at

the same time τ ≥ 0.4 and ρ = 0.
Hence ρ and τ satisfy

3τ − 1 1 + 2τ − τ 2
≤ρ≤ if τ ≥0
2 2
τ 2 + 2τ − 1 1 + 3τ
≤ρ≤ if τ ≤ 0.
2 2
which yield the area given below.

81


1.0

0.5
Rho de Spearman

0.0

-0.5

-1.0
-1.0 -0.5 0.0 0.5 1.0
Tau de Kendall

Figure 31: Admissible region of ρ and τ.

82


From Kendall'tau to copula parameters
Kendall's τ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Gaussian θ 0.00 0.16 0.31 0.45 0.59 0.71 0.81 0.89 0.95 0.99 1.00

Gumbel θ 1.00 1.11 1.25 1.43 1.67 2.00 2.50 3.33 5.00 10.0 +∞
Plackett θ 1.00 1.57 2.48 4.00 6.60 11.4 21.1 44.1 115 530 +∞
Clayton θ 0.00 0.22 0.50 0.86 1.33 2.00 3.00 4.67 8.00 18.0 +∞
Frank θ 0.00 0.91 1.86 2.92 4.16 5.74 7.93 11.4 18.2 20.9 +∞
Joe θ 1.00 1.19 1.44 1.77 2.21 2.86 3.83 4.56 8.77 14.4 +∞
Galambos θ 0.00 0.34 0.51 0.70 0.95 1.28 1.79 2.62 4.29 9.30 +∞
Morgenstein θ 0.00 0.45 0.90 - - - - - - - -

83


From Spearman's rho to copula parameters
Spearman's ρ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Gaussian θ 0.00 0.10 0.21 0.31 0.42 0.52 0.62 0.72 0.81 0.91 1.00

Gumbel θ 1.00 1.07 1.16 1.26 1.38 1.54 1.75 2.07 2.58 3.73 +∞
A.M.H. θ 1.00 1.11 1.25 1.43 1.67 2.00 2.50 3.33 5.00 10.0 +∞
Plackett θ 1.00 1.35 1.84 2.52 3.54 5.12 7.76 12.7 24.2 66.1 +∞
Clayton θ 0.00 0.14 0.31 0.51 0.76 1.06 1.51 2.14 3.19 5.56 +∞
Frank θ 0.00 0.60 1.22 1.88 2.61 3.45 4.47 5.82 7.90 12.2 +∞
Joe θ 1.00 1.12 1.27 1.46 1.69 1.99 2.39 3.00 4.03 6.37 +∞
Galambos θ 0.00 0.28 0.40 0.51 0.65 0.81 1.03 1.34 1.86 3.01 +∞
Morgenstein θ 0.00 0.30 0.60 0.90 - - - - - - -

84


Alternative expressions of those coecients

Note that those coecients can also be expressed as follows

[0,1]×[0,1]
C(u, v) − C ⊥ (u, v)dudv
ρ(X, Y ) = (1)
[0,1]×[0,1]
C + (u, v) − C ⊥ (u, v)dudv

(the normalized average distance between C and C ⊥ ), for instance.

A dependence measure in higher dimension ?

From equations 1 and ??, it is possible to obtain a natural mutlidimensional
extention (see Wolf (1980), Joe (1990) or Nelsen (1996)),

[0,1]d
C(u) − C ⊥ (u)du d+1
ρ(X) = = 2d C(u)du − 1
[0,1]×[0,1]
C + (u) − C ⊥ (u)du 2d − (d + 1) [0,1]d
(2)

and similarly

1
τ (X) == d−1 2d C(u)dCu − 1 (3)
2 − 1) [0,1]d

85


Note that a lower bound for τ is then −1/(2d−1 − 1), while it is

(2d − (d + 1)!)/(d!(2d − (d + 1))).
In dimension 3, Kendall's τ is the average of the three 2-dimensional Kendall's τ,
1
τ (X, Y, Z) = (τ (X, Y ) + τ (X, Z) + τ (Y, Z)).
3

86


Tail concentration functions
Venter (2002) suggest to use several Tail Concentration Functions

Denition 24. For lower tails, dene
L(z) = P(U z, V z)/z = C(z, z)/z = P r(U z|V z) = P r(V z|U z),

and for upper tails,
R(z) = P(U z, V z)/(1 − z) = P r(U z|V z).

Joe (1990) uses the term upper tail dependence parameter for

R = R(1) = limz→1 R(z), and lower tail dependence parameter for

L = L(0) = limz→0 L(z).

87


Functional correlation measures
1 1
Consider also Kendall's tau, dened as −1 + 4 0 0
C(u, v)dC(u, v).
Denition 25. The cumulative tau can be dened as
z z
J(z) = −1 + 4 C(u, v)dC(u, v)/C(z, z)2 .
0 0

88


1.0 Gaussian copula L and R concentration functions

1.0
q q q q qqqqqq
q q q q q q q qq q qq q
q q q
q q
q q q q q
q q q qq q qq
q q
q q q qq q q qq q q
q
q q qq q q q q q q qq
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 qq
q qq q
q q q qq q qq q qq qq q q
q q q q q q q qq
q q
q q q q
q qq q q q q
q
q qq
q q q q q q
q qq
q
q q qqq q q
qq q q
q q
q q q
q qq
qq qq q q q
q q qq q q q q qq q
q q qq q q q
q q qq qq q q
q q q q
qq q q q q
q q q q qq q qq q
q q q qq 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 qq q q q
q q q q q q qq q q q
q
q qq q
q q
qq
qq q q qq
q
q
q q qq qq qq q qq
q q q q q q q q
q q qq qq qq q
0.8

q

0.8
q q q q qq q q qq q
qqq q q q q qq
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 qq q qq
q q q q q q qq q qq q q q
q q q
q
q
q q q q q qq q
qq
q
qq 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 qq q q q qq
q q q q
q q qq q q q q q qq q
q q
q q q q q q
qq qq qq
q qq q q q q qq qq q
q
q qq
q q
q q q qq
q q q
q q
q
q

q
q
qq
q q
q

qq qq q
qq
q
q q
q q q
q
q
q
q
qq
q q
qq
q
q q qq
q q
q
q
q q q q
q q q
q
GAUSSIAN
q q qq 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 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 qq qq q q q
q q q q qq qqq qq qq q qq qq q q q
q q q
q qq q q
q q q q qqq
q
q q q q q q
q q qq q q q
q
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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 function (lower tails) R function (upper tails)

Figure 32: L and R cumulative curves.

89


1.0 Gumbel copula L and R concentration functions

1.0
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90


1.0 Clayton copula L and R concentration functions

1.0
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91


1.0 Student t copula L and R concentration functions

1.0
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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 q q qqq 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 qq q q q
q
q qq q q q q q q
q qq
q q
q qq q qq q
q
qq q
q qq q q
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0.6

0.6
q q qq q q q q q
q q q
qq 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 qq q q

q
qq q
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q
q
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q
q
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q
q q
q
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q
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q qq
q qqq
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q q q q qq
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qq q q q
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q q q q
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qq q q q q qq q qqq q q q
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q q q qq
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q q q q q qq
q qq
q
qq q q q q q q
q q q q q q q q q
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q q q qq q
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q q q q q q q q
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q
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q q q
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qq q qq q q
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qq
q q q q qq q
q
qq q q q q q q q
q q q q q q q q
qq q q q q
q qq q qq q qq q q
0.4

0.4
q q q q
q qq q q q q q q
q
qq qqq q q q q
q q q q q qq qq q q qq q
q
qq q q q q q q q q
q q
q qqq q q q q q qq q q q
q q q q q
q q q q q
qq q qq q q q q q qq qq
q
q q q q q qq q q q q q
q qqqq q
q
q q q
q q q q
q q q q q qq
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q
qq
q q q q q
q
q q qq
q qq q q q q
q q
q q q
q q
q q qq q q
q q qqq q q q q q
q
q q q q q
q qq q q q q
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q q q q q q q
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q q q q
q q q q q
q q q qq
q q q q qq q
q q q q q qq
q q q q
q q qq q q q q
q q q q
qq qq q
q qq q q q q q q
q q q q
qq q q q
q qq q q
q q qq qq q q q q q
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q q q q q q
q q
q qq q q q q q
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q qq
q q
qq q qq
qq q qq 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
0.2

0.2
q q q q q qq q q q
q
q q qq q q q
q q q q q qq qq q q q q
q qq qq qq q q
q q q q q q q
q q
qq q qq q q q q
q qq q q q q qq q q q q qq q q qq
q q q q q qq q
q q q q q q q
q q qq qq
q q qq
q q q q q
q
q q q
q
q q q q
qq q q q
qq q qq qq q
qq q q qq q q q
q q q q q q q q q
q q q q q q qqq q q q qq qq q
q
q q q qq q q q q q
q q q q qq qq q q q q q q q q q q qq q
qq q qq q q q q qq q q q
q qq q qq q q
qq qq
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
qq q q q
q
q qq qq q qq q q q q
q q q q qq q qq q
qq qq q
q qq
q q q q q q q q q q qq
q qqq q q q
q q q q q q
q qq
qq q q q q q qq q
q q
q qq qq q q
qq q q q
q qq
q qq q q
q q q qq
q q q qq qq q
qq q qq q q
q
qq q q q
q q q
0.0

q

0.0
q
qqq
qq
q
q
qq
qq q q q q
q q
qq
q
q
qq q
qq q q

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



92


Student t copula L and R concentration functions
1.0

1.0
q q q q q q
q
q
q
q q qq q q q qq qq
q q qq q
qq
q q q q q q q q q qq qq q qqq
q q q q q q qqq
q qq q qq q qq q q qq q q
q q q q q
q q q qq q qq q qq
q q q q
q q qq q q q
q q qq 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 qq 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 qq q
q
qq
q q q q
q q qq q q
q
qq q q q q qq q q q q q q
q q
q q q q q qq
q q q q q qq qq q
q q
q q q q q q
qq q q q q q q qqq
q
q q q q
q qq q
q
q q qq q qq
qq q q q
q
q q q qq q
q qq q q q
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q q
q q q q q q q q q qq q q q q
q
q qqq
q q q
qq q qq
q qq qq q
q q q q q q q qq q
q q q q
q
q q q q qq qq q
q q q q
0.8

0.8
q q qq 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 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 qq qq q
qq 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 q q qq q q qq
qq
q q q
q qq
q q q q q q
q qqq
q qq

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 qq
q qqq
q qq
q qq
q
q q q
q
q q q
q
q
q STUDENT (df=3)
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 qq q qq q q q q q q
qq 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 qq q qq qq q q
q qq q q qq
q qq q q q
qq qq qq q qq q q
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q q q q q qq q q q
q
q qq q q
q q qq q qq q q q qq q
q
q q
q
qq q
q
q
q
q
q q q q
qq
q
q qq
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q
q q
q
q
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q
q q
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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
0.6

0.6
q q q qq q qq
q q q
q q q q q
qq 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
qq
q
q q
q
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q
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q q q q q q q qq q q q qq q
q qq q
q q q
q
q q q
q
q q qq qq qq
q q q
q q qq q q q q q q
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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 qq q q q
q q qq qq q q q q
q q q q q qq
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
qq q q
q q qq q q
q q q q
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q
q q
q q q q q qq q q q qq
q q qq q qq q q q q q qq q
q q q q q q
qq q q qq q q q
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q q qq qq q q q q q q q q
q q q qq q q qq 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 qqq q q q
0.4

0.4
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 qq q q q q q
q qq q q qq q q q
q q q qq q q q q q
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q
q q q q q
q
q q q q q q q q
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q qq q qq q qq q q qq q q q q
q qq
q q q q q qq q q q
q q qq qq q qq q 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 qq q q q
q q q qq q q q q
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q q q qq
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q q q q q qq q
q
qq q q q q q q qq q q q qq
q
qq q q q q q q q q
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qq
q q q qq q q
qq qq qq
q q q q q q
q qq q q qq
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
qq q
qq qq
q qqq qqqq q q
q q qq q q
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q
q q
q qq qq q qq
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q
q q q q
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q qq q q q q q q
qq q q q q q q q q q q q
q q q
0.2

0.2
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 qq q qq
q q q q q q q q
q q q qq
qq q qq
q q q qq q q q
q qq q
q q q q q
q q q qq q
q q q q
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q qq q q qq q q
q qq q qq q q qq q
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q
q q q
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q qq q
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q q q q q q q qqq q q q
q q
q q
qq q qq q
q q q q q q qq qq q q q q q q qq q q
q q q qq q q
q q q qq q
q q q qq q q qq q q q
q q
q q q
q
q q
qq q q q q q q
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q q q q q q
q
q qqq q
q q q q q q q q q q qq q q q
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q q q q q q
q qq q q q
q q q q q q q qq qq q q q q
q
qqq q q
qq qq q
q q q q q q q
q qq q q q q
q
q qqqq
q q qq q q
q
q q q
q qq
q q q
q q qqq
qq
q q
q
q q q q q q q
q
qq q q qq q qq q q q
qqqq q q
q
0.0

0.0
qqq qq
q qq
qq
q qq q q q q q q q
qq
qq
qq q q
q
q

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



93


Dependence in independence
Coles, Heffernan Tawn (1999) propose another function,

2 log(1 − z)
χ(z) = −1
log C(z, z)
Then set η = (1 + limz→1 χ(z))/2

94


1.0 Gaussian copula Chi dependence functions

1.0
q qq q q
q q qq q q qqq qq q
q q
q
q q q q q
q qq q q q q q q q qq q q
qq
q
qq
q
q qq q q q q q qq q q
q q q
q
q qq qq qq q
q q q
q q
q
qq qq q q
q q
qq q q qq q q
qq
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 qq q
q q
q q q q q
q qq q
qq
q
q q
q q qq q q q
q q q q q qq
q q q q q q q qq
qq
q q q q
q q
q q q q
q q
q q q qq q q q q q q qq q q qq q q
q
q
q q qq q qq q
q q q q q q
qq
q q q qq q q
q qq
q
qq q q q q qq 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 qq q q q q q q
q q q q q q q qq q
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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 qq qq q q q
q q
qq qqq q q q q
q q q q q
q q q q q q qq
0.8

0.8
q q q q qq q q q q
qq q
q q qq q q q q q q qq q q
q
q q q q qq q
q q q q q q
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q q qq q q qq q q
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q
q
qq
q
q
q q q q
q q
q q qq q
qq
qq qq q q q qq
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 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 qq
qq
qq
q q q qq q q q
qq q q q qq
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 q q q qq
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q
q
q q q
q
q
q q q q q
q qq q q
q q qq q
q q qqq q q q
q q q qq q 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 q q
q q q q q q
q
q q qq
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q q q q q qq q q q q q qq
q q q qq q q q q q
q q q q
q q q
qq q q qq
q
qq
q
q q q
q qq
q
q
q q qqq
qq qq qq
q q qq
q
q qq q q qq
q q q
q q
q
q GAUSSIAN
0.6

0.6
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
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qq q q q q q
q
q
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q
q q q q q q
q q q q q q q
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q qq q q q q
q q qq q
q qq q q
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q
q
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q qq qq qqq q q q q
q
q
q q q q qq q
q qq q q q q qq q q q qq
q
q q q qq q q q q q
q q q q q qq q qq q q q q
qq q
q q q qq q qq
q qq q q q qq qq q
q q qq q
q q q q qq q
q q q
qqq q q q q q
q
q
q q q qq q qq q q q qq q
q qqq
qq q q q
q
qqqq
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
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q
q
q
q q
q q qq qq
q
q q
q
q
q
qq qq q
q
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q
q
q q
q
q
q q q q qq
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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 qqq q q q
q q qq
qq q q q q q q q q q q q
0.4

0.4
q q q q q q qq q q qq 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 qq 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 qq 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 qq q
q q q q q q qq q q
q q q q q q q
q q qq
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 q q q q q q
qq
q q q qq q q q
q q q q
q qq qq q
q q q q
q qq q q qq q q q q
qq q q q q q qq q q qq
q
q q q q
qq q q q q
qq q q qq q q q q q q
q
q q
qq q
q
q q q
qqqq
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 qq q q q q q
q q q q qq q q q q qq qq q q q
q qqq q q q q q q qq q q
q
q q
q
q
q q q q q q q
qq qqq 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
qq q qq q q q q q
q
0.2

0.2
qq q qq 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
qq q q q q q q
q qq qq
q q qq
q q q qq q
q q q qq q
q q q qq q q
q
qq q q q q qq
q q q q q
q
q q
qqq qq q q q q q q
q q q q q q q
q qq q q q
qq q q
q
qq
q
q
q q qq
q q q q q
q q qq q q q
qq
q q
q qq q q q q qq q q
q q q q
q q q q qq q q
qq
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
qqq
q qq q
q qq 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 qq q q q
q q
q q q q
q q q
q q
q
qqq q qqq 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 qq q
q
q q qq qq
q q q q q
q
q q qq q
q q q q qq q
q
q
q
q qq
qq q q q
q
qq q q q q q
qq
q
q q qq q q q q q qq
q
q
0.0

0.0
q
q q q qq q
qq
qq q qq qq
q q q
qq q q q q
q q q
q
q q q
qq q

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

lower tails upper tails

Figure 37: χ functions.

95


1.0 Gumbel copula Chi dependence functions

1.0
q
qq qq qq qq
qq
q
q
q qq q qq q
q qq qq qq
q
qq
q qq qq q q
q q qq q q
qq
q q qq q qqqq q
q qq q
qq q qq qq q
qq q q
q qqq q q
q
q qq qq q
qq
q q q q q q qq q
q q qq q q q q qq q q qq
q q
q
q
qq
q q qqq q qqq q qq
q qq q q qq
q q q q
q q qq q q qq q qq
q q
q q
q q q
q
q
q q q q q q qq q q qq q
q qq 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 q q q q qq
q
q
q qq
qq
qq q q qq q q q q qq q qq q q
q q
q q q q qq q q q q qq q
qq q q qq q q q q
q q
q q qq qqq
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 qq q q q qq q
q q q
q qq q q
q q q
q q q q qq q
q q qq qq q q q q q
q q q q qq q q qq q q
0.8

0.8
q q q q q q q qq
q q qqq 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 qq q q q q qq
q q qq q q q qq q q
q qq qq
q q q q
q q
q
qq qq q q
q q
q
q
q q qq 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 qq q q qq q q qq
q q q q q q q qq
qq q q q qq qqq q qq q qq
q qq q q q q 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 qq
q q
q q q q q q q q
q q q q qq q
q q qq q qq
q q
q q q
q
q
q q q
q q q q qq qq q
q qq q q q q q q qq q
q q q q q qq qq
q qq 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 qq q qq q q
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q q
qqq q
q q
q
q
q q q q q q q q q q qqq qq qq q
q q q q q
q
q
q q q q q q q q q
q q qq
0.6

0.6
q q q q q q q q
q q q
q q q q q q q q
qq qq qq
qq qq
q q q q q q q q
q q q
q qq q q qq qq q q
q q q q q q qqq 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 qq q q q q q q q
q q qqq 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 qqq q q
q
q q
q q
q GUMBEL
q q
q q q q q
q q q q
q qq q q
qq
q
q q
qq q qq qq q
qq q 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 qq
q q
qqq
q
q
q
qq qq
qq
q qq
q q
q q
qq q
q
q q q
qq q qq 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 qq 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 qq q q q q q
q qq q qq q q
qq q q q
q q q qq qq q q q q
qq q
q q q q q q q q q
q q qq q qq
q q q qq q q
q q q q q qq q q
q
0.4

0.4
q qq q
q
q q q q
q q q q q q q q q q qq q q q
qq q q qq 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 qq qq q q
q q q q q
q q q q qq q q
q q
q q
q q q qq q qq q q qq q q q
q
qq 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 qq q q q
q
q q q qq
q q q q q qq
q q q q q q q qq qq q q q q
q qqqq
q q qq q
qq
q qq q
q
q
q q q q q q q q q
qq qq qqqq
q
q q 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 q
q q q qq q
q qqq q qq
qq q q q q qq qq q qq q q q qq q
q qq
q q q q qq q q q q
qq q qq q qq q q qq q
q q
q
q
qq q q q q q
q qq q qq q
q
q q
q
q qq 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 qq
q q qq q q
q q q qq q q q
q q qq q q qq q q
q qq q q q q
qq q q
0.2

0.2
qq q q
q q q q qq q q q q q q qq
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 q q qq q q q
q q q qqq q
q q q q
qq q qq q q q
q
q q q
q qq
q qq qq
qq
q
q q q q qq q q
q q q
q q q
q qq q q qq qqq q q
q q qq q q q q qq q q qq
qq q q q
q q q q q q
q
q
q qq q q
q
q q q qq q q q q qq 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 qq q
q q q
q qq 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
q
qq q q
qq
qq
q
q q
qq qq
q q qq
q q
q
qq q q q q qq qq q
q q q qq q q q
q q q qq qq 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 q
q q qq q q q q
q q q q qq qq q q q
qq q
q qq qqq
q q q q q q q q
qqq q
q q q q q
0.0

0.0
q q q qqq 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

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



96


1.0 Clayton copula Chi dependence functions

1.0
q q
qq q q q q qq q q
q q
q q q
q q q qq
q q q q q q qq q
q q q q q q q
q q q qq qq q q q q q q q q
q qq q q q q q q qq
q q q q qq q q
q q
q q qq q
q q q q q q q q q
qq q qqq q q qq
q
q
q q
q qq
q
q q q qq q
q q
q
q q q q q qq
q q q q q
q q qq q
qq q q q q
q q q
qq q q
q q q q q
q q q qq q
q q qq
q q
q q
qq q q q
q q q qq q qqq
qq q
q
q
q
q q q qq qq q q
q
q q
q
qq q q q qq q q q q
qq q
q qq
q
q qq q q
qq 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
qq q q q qq q q q q qq
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 qq q q q q q q q q qqq
q q q q q q q
q q q q qq
q
0.8

0.8
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
qq q q
qq q
q q
qq q
qq
q q
q
q
qq
q q qq
q
qq
q q
qq q
CLAYTON
q q q q q q q q
q q qq 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 qq q q
q q q
q q q qq q qq q q qqqq q q q
q q q qq q
q q qq q q
q
q q q q
qq q qq qq q q q
q
q q q q q q qq qq 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 qq q q
qq q q
q q qq q qq
q q
q q q qq q q q q q
q q q q
q qq qq q q qq q
q
q q q q q q
q q q qq
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 q
qq
q qq
q q q qq
q
q q
q
qq
q
q q
qq
qq q
q
q
qq
q
q
qq 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
qq
q
q q q q q q q q q
q q q q qq
0.6

0.6
q q q q q q
q q qq q q q q q
q qq
q q q q q qqq q
q q qq q qq q
qq qq qq 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 qq q
q q q q q q q qq
q q qq q q
q q q qq qq q q q q
q q q q qq 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 qqq
q q
q
q
q
q q
qq q q q q
q q q q
qq
q q q q qq q
q q
q q q
qq q q
q qqq q q q
q q q qq q q
q q qq q q qq qq
q q
q q
q
q q q q qq q q q q qq qq q q qq q
q q q q qq qq q q qq
q qq q
q q
q q q q q q qq q
q qq q
q q q q qq q q q q
q qq q q q
q q
q q q q q q q qq q
q qq q qq qq q q
q
q q q qq qq 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
qq q qq q q q q q qq q q
q q q q q q q
qq q qqq q q q q q q q qq q
q qq q q q
q
q q qq
q qq q
qqq qq
q
q qq q qq qq q
0.4

0.4
q q q q qq 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 qq q
q qq qq
qq 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
qqq q q q
q qqq 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 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
qq 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 qq
q q q qq
q
q q q q
q qq
q
q q
q q qq 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 qq q qq qq q q q qq
q q q q qq q q q
q qq qq q
q qq q q q q q qq qq 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 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
qq
q q
q q q
q q q q q qq q qq q qq q q q q
q q qq q qq q q qq 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 qq q qq qqq
q q
0.2

0.2
q q q q q q q q
qq 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
q qq q q
q
qqq
qq q qq q q
q q q
q q q q q qq
q q qq 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 qq q
q qq qq
q q q q qq 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 qqq qq qq qq q q
q q q qq q qq 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 qq q q
qq q q q q q q qq q
qq q qqq q q
qq q
q q q qq qq qqq q q
q q qq qq q qq
qq 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 qq
q
q q q q qq qq q
qq
q q q
qqq q qq q q q
qq q q
q
qqq q q q q
q q q q q q
qq q qq
qqq q q
qq q q q
qq
qqq qq
q q
q qq q q
q qq
q q
q qq q
0.0

0.0
qq q
qq
q
qqq
q
qq q
qq
qqq
qq
q

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



97


Student t copula Chi dependence functions
1.0

1.0
qq q q q q q q qq
q
q
q q q q qqq
qq qqq
qq
q q q qq
q q q qqq
q
q q q
q
qq qqq q
qq
q q q q q qq q q
q q q q q q q q qqq
qq
q q q q q q q qq qq qq q q
q q q q qq q
q qq
q q q q qq q q q q qq q q
q q q qq q
q
q q q q
q q qq q q
q q q q
q q q
qq
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 qq qq q
q q q qq q q q
q
q q q q q q q q q qq q q qq q
qq q q
q q q
q q q qq q q
qq 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 q
q qq q
q q q q qq qq qq
q q q q q q q qq q qq q q q
q q
q
q q qq q
q qq q q q qq q qq q qq q qq
qq q q q q q
qq q
qq
qq q q q q
q q qq q
q q q q
q q qq q q
q
0.8

0.8
q q
q q q q
q q qq
q q q q qq 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 qq q
q q
q qq
qqq q q qq q
qqq
q q qq qq 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 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
qq
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 qq q
q qq q qq q q
q q qq qq q qq
q qq
q qq
q qq
q q q q
qq q q
q q q q q q qq q q
q q
q q q q qq qq
qq q q
q q qq qq qq q qq q q q qq
q q
q q
q q q q q qq q q q qq qq qq q q q
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 q
q q q q
q q
q
q
STUDENT (df=3)
q q qq q q q qq q qq 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 qq
q q qq q q q qq
0.6

0.6
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 qq q qqq q q
q q q q
q q q q q q qq q q
q q q q q qq q qq q q q q q
q q q
qq q q
q
q q qq q q
qq qq 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 qqq q q q q q q
q q q qq q q qq qq q q
q
q q
q
q q q
q
q q qq 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 qq
q
q
q
q q q
q qq
q q q
q
q q qq
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
q q
q
qqq qq
q
q
q qq
qqq
qq 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
qq
qq
qq q
q
qq q
q qq qqqq
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 qq q q q
q q
q q
q qq q qq q q q q
q
q
q
q q qq q q q qq
q qq q q q q qq q q q
q q q q q q
0.4

0.4
q q q
qq q q qq q q
q q q qq q q q q q
q q qq q q
q q q q qq q q qq q
q q qq
q
qq
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
qq q q q
q qq qq qq 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 qq 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 qq qq q q q q qq q q q q
q q
q q q
qq q q q qq q qq q
q q
qq
qq
q qq q
q
q q q q q q qqq qq q qq q
q
q q qq q
q q q qq q q
q q q q q q q
qq qq q q
q q q qq qqq 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
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 qq q qq q q q q q
q q qq q q q
q q
q
q qq q q q qq
q q qq q q
q q q q q
q q qq q q
q q
q q qq
q q qq
q q q q q
0.2

0.2
q q q q q qq qq q q
q q qq q q
q q qq q q q
q q qq q q
q q q q q q
q q q
q
q qq qq
q q qq q q q
q
q
q q q
q q q q q q q qq
q qq qq q
qq q q q q q
q q q q
q qq q qq q qq q q q
q qq q qq
qq qq q q qq q qq
qq q qq q q q q q q q
q q qqq q qq 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
qq qq q q q q
q q
q q q q q
q q q q q qq q
q qq 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 qq q q
q q qq q q q q q q q q q
q qq q
q q qq qq q q q qq q
q q q
qq qq q q q q q q
q q qq
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 qq
q qq q q q
q
q qq q q qq q q
q qq q q q
qq qq q q qq
qq q
q q q
0.0

0.0
qq qqq
qq q q q
q qq q q
q
q
q
q
q
q qq
q q

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



98


(Strong) tail dependence measure
Joe (1993) dened, in the bivariate case a tail dependence measure.

Denition 26. Let (X, Y ) denote a random pair, the upper and lower tail
dependence parameters are dened, if the limit exist, as
−1 −1
λL = lim P X ≤ FX (u) |Y ≤ FY (u) ,
u→0

and
−1 −1
λU = lim P X FX (u) |Y FY (u) .
u→1

As mentioned in Fougères (2004), this coecient can be obtained dierently:

set
log P(max{X, Y } ≤ x)
θ(x) = .
log P(X ≤ x)
Then

λU = 2 − lim θ(x),
x→∞

99


since when x→∞
log P(max{X, Y } ≤ x) P(X x, Y x)
2− ∼ = P(Y x|X x).
log P(X ≤ x) 1 − P(X x)

Note that these coecient can be expressed only through the copula,

Proposition 27. Let (X, Y ) denote a random pair with copula C , the upper and
lower tail dependence parameters are dened, if the limit exist, as
C(u, u) C ∗ (u, u)
λL = lim and λU = lim .
u→0 u u→1 1 − u

Does λ=0 implies that extremal events are independent ?

Example 28. If (X, Y ) has a Gaussian copula with parameter θ 1, then λ = 0.
Hence, visually, dependence is weaker than any Gumbel's copula (even with θ is

rather small), but are extremal events independent ?

100


1.0

4
0.8

2
Copule de Gumbel

0.6
0.4

0
0.2

!2
0.0

0.0 0.2 0.4 0.6 0.8 1.0 !2 0 2 4

Marges uniformes Marges gaussiennes

Figure 41: Simulations of Gumbel's copula θ = 1.2.

101


1.0

4
0.8
Copule Gaussienne

2
0.6
0.4

0
0.2

!2
0.0

0.0 0.2 0.4 0.6 0.8 1.0 !2 0 2 4

Marges uniformes Marges gaussiennes

Figure 42: Simulations of the Gaussian copula (θ = 0.95).

102


Example 29. Consider the case of Archimedean copulas, then
1 − φ−1 (2x) φ−1 (2φ(x)) φ−1 (2x)
λU = 2 − lim and λL = lim = lim −1 .
x→0 1 − φ−1 (x) x→0 x x→∞ φ (x)

Further, properties can be derived for distorted generators, φα,β (·) = φ(·α )β ,
upper and lower tails coecients are respectively

1/α
λU and λL for φα,1 (·) = φ(·α )

and
1/β
2 − (2 − λU )1/β and λL for φ1,β (·) = φ(·)β

(Weak) tail dependence measure
Ledford Tawn (1996) propose the following model to study tail dependence.
L
Consider a random vector with identically distributed marginals, X =Y.
• under independence, P(X t, Y t) = P(X t) × P(Y t) = P(X t)2 ,
• under comonotonicity, P(X t, Y t) = P(X t) = P(X t)1 ,

103


Assume that P(X t, Y t) ∼ P(X t)1/η as t → ∞, where η ∈ (0, 1] will be

called coecient of tail dependence

More precisely,

• η = 1, perfect positive dependence (tail comontonicity),

• 1/2 η 1, more dependent than independence, but asymptotically

independent,

• η = 1/2, tail independence

• 0 η 1/2 less dependent than independence.

104


One can then dene upper tail coecient ηU and lower tail coecient ηL .
Example 30. : If (X, Y ) has Gumbel copula,
P(X ≤ x, Y ≤ y) = exp(−(x−α + y −α )1/α ), α ≥ 0

then ηU = 1. Further, ηL = 1/2α .
Example 31. : If (X, Y ) has a Clayton copula, then ηU = 1/2 while ηL = 1.
Example 32. : If (X, Y ) has a Gaussian copula, then ηU = ηL = (1 + r)/2.

105


Agenda





106


Estimation of copulas
−1 −1
Since C(u, v) = F (FX (u), FY (v)), copula has be estimated only after

estimating marginal distribution.

Margins Copula

Parametric Fα and Fβ C
θ
Nonparametric FX and FY C

(Fully) parametric estimation of copulas
Step 1: t the 2 univariate marginal cdf 's FX and FY with the help of the
observations{x1 , x2 , . . . , xn } and {y1 , y2 , . . . , yn } respectively; let α and β be the

corresponding MLE's of α and β .

Step 2: estimate θ with the parameters α=α and β=β xed at the estimated

values from Step 1; i.e. on pseudo-observations (Ui , Vi )'s, where

Ui = Fα (Xi ) and Vi = Fβ (Yi ),

107


let the result be θ.
Step 3: using α, β and θ as starting values, determine the global MLE's α, β and

θ of the parameters α, β and θ.

Parametric estimation of copulas
An alternative is to use nonparametric estimation of margins, FX and FY .
Step 1: estimate θ based on pseudo-observations (Ui , Vi )'s, where

Ui = FX (Xi ) and Vi = FY (Yi ),

let the result be θ.

Nonparametric estimation of copulas
Given an estimation of marginal distributions (parametric or nonparametric), the

108


idea is to consider the empirical copula C, dened as

#{i such that Ui ≤ u and Vi ≤ v}
C(u, v) =
#{i}
n
1
= 1(FX (Xi ) ≤ u) × 1(FY (Yi ) ≤ v).
n i=1

109


Example Loss-ALAE: consider the following dataset, were the Xi 's are loss
amount (paid to the insured) and the Yi 's are allocated expenses. Denote by Ri
and Si the respective ranks of ˆ
Xi and Yi . Set Ui = Ri /n = FX (Xi ) and
ˆ
Vi = Si /n = FY (Yi ).
Figure 43 shows the log-log scatterplot (log Xi , log Yi ), and the associate copula

based scatterplot (Ui , Vi ).
Figure 44 is simply an histogram of the (Ui , Vi ), which is a nonparametric

estimation of the copula density.

Note that the histogram suggests strong dependence in upper tails (the

interesting part in an insurance/reinsurance context).

110


Log!log scatterplot, Loss!ALAE Copula type scatterplot, Loss!ALAE

1.0
5

0.8
Probability level ALAE
4

0.6
log(ALAE)

0.4
3

0.2
2

0.0
1

1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0

log(LOSS) Probability level LOSS

Figure 43: Loss-ALAE, scatterplots (log-log and copula type).

111


Figure 44: Loss-ALAE, histogram of copula type transformation.

112


The basic idea to get an estimator of the density at some point x is to count how
many observation are in the neighborhood of x (e.g. in [x − h, x + h) for some
h 0).
Therefore, consider the moving histogram or naive estimator as suggested by

Rosenblatt (1956),

n
1
f (x) = I(Xi ∈ [x − h, x + h)).
2nh i=1

Note that this can be easily extended using other denitions of the neighborhood

of x,
n
1 x − Xi
f (x) = K ,
nh i=1
h
where K is a kernel function (e.g. K(ω) = I(|ω| ≤ 1)/2).

113


Estimation of Frank copula

1.0
5

0.8
4

0.6
3

0.4
2
0.8
0.6

0.2
1 0.4

0.2

0.0
0
0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8 1.0

Figure 45: Theoretical density of Frank copula.

114



1.0
5

0.8
4

0.6
3

0.4
2
0.8
0.6

0.2
1 0.4

0.2

0.0
0
0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8 1.0

Figure 46: Estimated density of Frank copula, using standard Gaussian (indepen-

dent) kernels, h = h∗ .

115


Problem of nonparametric estimation with kernel: bias on the borders.

Let K denote a symmetric kernel with support [−1, 1]. Note that

1
E(f (0, h) = f (0) + O(h)
2

116


1.2 Kernel based estimation of the uniform density on [0,1] Kernel based estimation of the uniform density on [0,1]

1.2
1.0

1.0
0.8

0.8
Density

Density
0.6

0.6
0.4

0.4
0.2

0.2
0.0

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

Figure 47: Density estimation of an uniform density on [0, 1].

117


Several techniques have been introduce to get a better estimation on the border,

• boundary kernel ( Müller (1991))

• mirror image modication ( Deheuvels Hominal (1989), Schuster
(1985))

• transformed kernel ( Devroye Györfi (1981), Wand, Marron
Ruppert (1991))

• Brown Chen
Beta kernel ( Chen
(1999), (1999, 2000)),

see Charpentier, Fermanian Scaillet (2006) for a survey with

application on copulas.

118


Consider the kernel estimator of the density of the

(Xi , Yi ) = (G−1 (Ui ), G−1 (Vi ))'s, where G is a strictly increasing distribution

function R → [0, 1], with a dierentiable density.

119


Since density f of (X, Y ) is continuous, twice dierentiable, and bounded above,

for all (x, y) ∈ R2 , consider
n
1 x − Xi y − Yi
f (x, y) = K K .
nh2 i=1
h h

Since

f (x, y) = g(x)g(y)c[G(x), G(y)]. (4)

can be inverted in

f (G−1 (u), G−1 (v))
c(u, v) = −1 (u))g(G−1 (v))
, (u, v) ∈ [0, 1] × [0, 1], (5)
g(G

one gets, substituting f in (5)

n
1 G−1 (u) − G−1 (Ui ) G−1 (v) − G−1 (Vi )
c(u, v) = K , ,
nh · g(G−1 (u)) · g(G−1 (v)) i=1
h h
(6)

120



5

0.8
4

0.6
3

0.4
2
0.8
0.6

0.2
1 0.4

0.2

0
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8

Figure 48: Estimated density of Frank copula, using a Gaussian kernel, after a

Gaussian normalization.

121


The Beta-kernel based estimator of the copula density at point (u, v), is obtained

using product beta kernels, which yields

n
1 u 1−u v 1−v
c(u, v) = K Xi , + 1, + 1 · K Yi , + 1, +1 ,
n i=1
b b b b

where K(·, α, β) denotes the density of the Beta distribution with parameters α
and β.

122


Beta (independent) bivariate kernel , x=0.0, y=0.0 Beta (independent) bivariate kernel , x=0.2, y=0.0 Beta (independent) bivariate kernel , x=0.5, y=0.0



Figure 49: Shape of bivariate Beta kernels K(·, x/b + 1, (1 − x)/b + 1) × K(·, y/b +
1, (1 − y)/b + 1) for b = 0.2.
123


Estimation of the copula density (Beta kernel, b=0.1) Estimation of the copula density (Beta kernel, b=0.1)

1.0
3.0

0.8
2.5

0.6
2.0

1.5

0.4
0.8
1.0
0.6

0.2
0.4
0.5
0.2

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

Figure 50: Estimated density of Frank copula, Beta kernels, b = 0.1

124


Estimation of the copula density (Beta kernel, b=0.05) Estimation of the copula density (Beta kernel, b=0.05)

1.0
3.0

0.8
2.5

0.6
2.0

1.5

0.4
0.8
1.0
0.6

0.2
0.4
0.5
0.2

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

Figure 51: Estimated density of Frank copula, Beta kernels, b = 0.05

125


Standard Gaussian kernel estimator, n=100 Standard Gaussian kernel estimator, n=1000 Standard Gaussian kernel estimator, n=10000
4

4

4
3

3

3
Density of the estimator


2

2

2
1

1

1
0

0

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

Estimation of the density on the diagonal Estimation of the density on the diagonal Estimation of the density on the diagonal

Figure 52: Density estimation on the diagonal, standard kernel.

126


Transformed kernel estimator (Gaussian), n=100 Transformed kernel estimator (Gaussian), n=1000 Transformed kernel estimator (Gaussian), n=10000
4

4

4
3

3

3


2

2

2
1

1

1
0

0

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


Figure 53: Density estimation on the diagonal, transformed kernel.

127


Beta kernel estimator, b=0.05, n=100 Beta kernel estimator, b=0.02, n=1000 Beta kernel estimator, b=0.005, n=10000
4

4

4
3

3

3


2

2

2
1

1

1
0

0

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


Figure 54: Density estimation on the diagonal, Beta kernel.

128


Tail dependence and statistical inference
Consider an i.i.d. sample (X1 , Y1 ) , ...., (Xn , Yn ).
Consider unit Pareto transformation of margins: set

1 1
T = ∧ .
1 − FX (X) 1 − FY (Y )

Observe that the survival distribution function of T, FT, is regularly varying

with parameter η. But because FX and FY are unknown, dene the pseudo

observations Ti 's as

1 1n+1 n+1
Ti = ∧ = ∧ ,
1 − FX,n (Xi ) 1 − FY,n (Yi ) n + 1 − Ri n + 1 − Si
where Ri and Si denote the ranks of the Xi 's and Yi 's. Hill estimator can then

be used, based on the k+1 largest values of the Ti 's,
k
1 Tn−i+1:n
ηHill = log .
k i=1
Tn−k:n

129


Estimation of η (Hill's estimator)
Proposition 33. Assume that (X, Y ) has upper tail dependence, with tail index
√
η , with additional regularity conditions, then k (ηHill − η) is asymptotically
normally distributed, with mean 0 and variance
2 2 ∂c (1, 1) ∂c (1, 1)
σ = η (1 − l) 1 − 2 .
∂x ∂y

Remark 34. From this Proposition, a test for asymptotic dependence (i.e.
η = 1) can de dened: asymptotic dependence is accepted if
1 − ηHill
≤ Φ−1 (95%)
σ (η = 1)

130


Estimation of η (Peng's estimator)
Set
n
Sn (k) = I (Xi Xn−k:n , Yi Yn−k:n )
i=1
and
−1
1 Sn (k)
ηPeng = log .
log 2 Sn ( k/2 )
Proposition 35. Assume that (X, Y ) has upper tail √
dependence, with tail index
η , and the same technical assumption as before, then k (ηPeng − η) is
asymptotically normally distributed, with mean 0.

131


Estimation of λ (Huang's estimator)
It is also possible to estimate λU : Huang-estimator of is based on the denition

of the upper tail index.

n k k
λHuang = P (Ui , Vi ) ∈ 1− × 1−
k n n
n
1
= I (Ri n − k, Si n − k)
k i=1
n
1
= I (Xi Xn−k:n , Yi Yn−k:n ) .
k i=1

132


Eta (Hill estimate), Gaussian copula, tau=0.3 Eta (Peng estimate), Gaussian copula, tau=0.3 Lambda (Huang estimate), Gaussian copula, tau=0.3

1.1 1.1
1.0

0.8
0.9 0.9

0.6

0.7 0.7 0.4

0.2

0.5 0.5

0.0

0 100 200 300 400 0 100 200 300 400 0 100 200 300 400

Eta (Hill estimate), Clayton copula, tau=0.3 Eta (Peng estimate), Clayton copula, tau=0.3 Lambda (Huang estimate), Clayton copula, tau=0.3

1.1 1.1
1.0

0.8
0.9 0.9

0.6

0.7 0.7 0.4

0.2

0.5 0.5

0.0

0 100 200 300 400 0 100 200 300 400 0 100 200 300 400

Figure 55: Estimation of η and λ for Gaussian and Clayton copulas, with Kendall's
tau equal to 0.3

133


Eta (Hill estimate), survival Clayton copula, tau=0.3 Eta (Peng estimate), survival Clayton copula, tau=0.3 Lambda (Huang estimate), survival Clayton copula, tau=0.3

1.1 1.1
1.0

0.8
0.9 0.9

0.6

0.7 0.7 0.4

0.2

0.5 0.5

0.0

0 100 200 300 400 0 100 200 300 400 0 100 200 300 400

Eta (Hill estimate), Gumbel copula, tau=0.3 Eta (Peng estimate), Gumbel copula, tau=0.3 Lambda (Huang estimate), Gumbel copula, tau=0.3

1.1 1.1
1.0

0.8
0.9 0.9

0.6

0.7 0.7 0.4

0.2

0.5 0.5

0.0

0 100 200 300 400 0 100 200 300 400 0 100 200 300 400

Figure 56: Estimation of η and λ for survival Clayton and Gumbel copulas, with

Kendall's tau equal to 0.3

134


Eta (Hill estimate), Gaussian copula, tau=0.7 Eta (Peng estimate), Gaussian copula, tau=0.7 Lambda (Huang estimate), Gaussian copula, tau=0.7

1.1 1.1
1.0

0.8
0.9 0.9

0.6

0.7 0.7 0.4

0.2

0.5 0.5

0.0

0 100 200 300 400 0 100 200 300 400 0 100 200 300 400

Eta (Hill estimate), Clayton copula, tau=0.7 Eta (Peng estimate), Clayton copula, tau=0.7 Lambda (Huang estimate), Clayton copula, tau=0.7

1.1 1.1
1.0

0.8
0.9 0.9

0.6

0.7 0.7 0.4

0.2

0.5 0.5

0.0

0 100 200 300 400 0 100 200 300 400 0 100 200 300 400

Figure 57: Estimation of η and λ for Gaussian and Clayton copulas, with Kendall's
tau equal to 0.7

135


Eta (Hill estimate), survival Clayton copula, tau=0.7 Eta (Peng estimate), survival Clayton copula, tau=0.7 Lambda (Huang estimate), survival Clayton copula, tau=0.7

1.1 1.1
1.0

0.8
0.9 0.9

0.6

0.7 0.7 0.4

0.2

0.5 0.5

0.0

0 100 200 300 400 0 100 200 300 400 0 100 200 300 400

Eta (Hill estimate), Gumbel copula, tau=0.7 Eta (Peng estimate), Gumbel copula, tau=0.7 Lambda (Huang estimate), Gumbel copula, tau=0.7

1.1 1.1
1.0

0.8
0.9 0.9

0.6

0.7 0.7 0.4

0.2

0.5 0.5

0.0

0 100 200 300 400 0 100 200 300 400 0 100 200 300 400

Figure 58: Estimation of η and λ for survival Clayton and Gumbel copulas, with

Kendall's tau equal to 0.7

136


Agenda





137


Risk measures and diversication
Any copula C is bounded by Fréchet-Hoeding bounds,

d
max ui − (d − 1), 0 ≤ C(u1 , . . . , ud ) ≤ min{u1 , . . . , ud },
i=1

and thus, any distribution F on F(F1 , . . . , Fd ) is bounded

d
max Fi (xi ) − (d − 1), 0 ≤ F (x1 , . . . , xd ) ≤ min{F1 (x1 ), . . . , Ff (xd )}.
i=1

Does this means the comonotonicity is always the worst-case scenario ?

Given a random pair (X, Y ), let (X − , Y − ) and (X + , Y + ) denote
contercomonotonic and comonotonic versions of (X, Y ), do we have

− −
? ?
R(φ(X , Y )) ≤ R(φ(X Y ) ≤ R(φ(X + , Y + )).
, )

138


Tchen's theorem and bounding some pure premiums
If φ : R2 → R is supermodular, i.e.

φ(x2 , y2 ) − φ(x1 , y2 ) − φ(x2 , y1 ) + φ(x1 , y1 ) ≥ 0,

for any x1 ≤ x2 and y1 ≤ y2 , then if (X, Y ) ∈ F(FX , FY ),

E φ(X − , Y − ) ≤ E (φ(X, Y )) ≤ E φ(X + , Y + ) ,

as proved in Tchen (1981).

Example 36. the stop loss premium for the sum of two risks E((X + Y − d)+ ) is
supermodular.

139


Example 37. For the n-year joint-life annuity,
n n
axy:n = v k P(Tx k and Ty k) = v k k pxy .
k=1 k=1

Then
a− +
xy:n ≤ axy:n ≤ axy:n ,

where
n
a−
xy:n = v k max{k px + k py − 1, 0}( lower Fréchet bound ),
k=1

n
a+
xy:n = v k min{k px , k py }( upper Fréchet bound ).
k=1

140


Example 38. For the n-year last-survivor annuity,
n n
axy:n = v k P(Tx k or Ty k) = v k k pxy ,
k=1 k=1

where k pxy = P(Tx k or Ty k) = k px + k py − k pxy .
Then
a−
xy:n
+
≤ axy:n ≤ axy:n ,
where
n
a−
xy:n = v k (1 − min{k qx , k qy }) ( upper Fréchet bound ),
k=1
n
a+
xy:n = v k (1 − max{k qx + k qy − 1, 0}) ( lower Fréchet bound ).
k=1

141


Example 39. For the widow's pension annuity,
∞ ∞
ax|y = ay − axy = v k k py − v k k pxy .
k=1 k=1

Then
a− ≤ ax|y ≤ a+ ,
x|y x|y

where
∞ ∞
a− = ay − axy =
x|y v k k py − v k min{k px , k py }.( upper Fréchet bound ),
k=1 k=1

∞ ∞
a+ = ay −axy =
x|y v k k py − v k max{k px + k py −1, 0}.( lower Fréchet bound ).
k=1 k=1

142


Value-at-Risk fails to be subadditive
One of the axiom of coherence is subadditivity, i.e. R(X + Y ) ≤ R(X) + R(Y ).
But Value-at-Risk is not coherent: one can have

VaR(X + Y, α)≥VaR(X, α) + VaR(Y, α) e.g. (see Embrechts (2007))

• X and Y are independent but very skew (see e.g. credit risk, with default

probabilities α),
• X and Y are independent but very heavy-tailed (see e.g. the case of innite

variance i.e. X ∈ Lp with p 1),
• X and Y are N (0, 1) with (special) tail dependence.

143


Makarov's theorem and bounding Value-at-Risk
In the case where R denotes the Value-at-Risk (i.e. quantile function of the PL

distribution),

R− ≤ R(X − + Y − )≤R(X + Y )≤R(X + + Y + ) ≤ R+ ,
where e.g. R+ can exceed the comonotonic case. Recall that

−1
R(X + Y ) = VaRq [X + Y ] = FX+Y (q) = inf{x ∈ R|FX+Y (x) ≥ q}.

If X ∼ E(α) and Y ∼ E(β),
P(X x) = exp(−x/α), P(Y y) = exp(−y/β) and x ∈ R+ .
The inequalities

exp(−x/ max{α, α}) ≤ Pr[X + Y x] ≤ exp(−(x − ξ)+ /(α + β))
hold for all x ∈ R+ , whatever the dependence between X and Y, where

ξ = (α + β) log(α + β) − α log α − β log β.

144


Therefore, the inequalities

− max{α, β} log(1 − q) ≤ VaRq [X + Y ] ≤ ξ − (α + β) log(1 − q)

hold for all q ∈ (0, 1).
Recall that in the independence case independence X + Y ∼ G(2, 1) and under

perfect positive dependence X + Y ∼ E(2).

145


Bornes de la VaR d’un portefeuille

4
2
0
!2
!4

0.0 0.2 0.4 0.6 0.8 1.0

Somme de 2 risques Gaussiens

Figure 59: Value-at-Risk for 2 Gaussian risks N (0, 1).

146



6
5
4
3
2
1
0

0.90 0.92 0.94 0.96 0.98 1.00

Somme de 2 risques Gaussiens

Figure 60: Value-at-Risk for 2 Gaussian risks N (0, 1).

147



20
15
10
5
0

0.0 0.2 0.4 0.6 0.8 1.0

Somme de 2 risques Gamma

Figure 61: Value-at-Risk for 2 Gamma risks G(3, 1).

148



20
15
10
5
0

0.90 0.92 0.94 0.96 0.98 1.00

Somme de 2 risques Gamma

Figure 62: Value-at-Risk for 2 Gamma risks G(3, 1).

149


Bounding Value-at-Risk for a sum of 2 risks
In a general (and theoretical) context, Schweizer Sklar (1981), studied the

distribution of ψ(X, Y ) for some R2 → R function ψ, where (X, Y ) ∈ F(FX , FY ),
using the concept of supremal and inmal convolutions,

Fsup (FX , FY ) (z) = sup {C (FX (x) , FY (y)) , ψ (x, y) = z} (7)

Finf (FX , FY ) (z) = inf {C (FX (x) , FY (y)) , ψ (x, y) = z} (8)

Williamson (1991) and Embrechts, Hoing Juri (2002) proposed

numerical algorithm to calculate those bounds.

The idea is to observe that bounds for the distribution of the sum of X and Y is

given by the distribution of Smin and Smax , where

P(Smax s) = sup max{P(X x) + P(Y s − x) − 1, 0}
x∈R

and

P(Smin ≤ s) = inf min{P(X ≤ x) + P(Y ≤ s − x), 1}.
x∈R

150


Note that those bounds can also be written as follows,

Proposition 40. Let (X, Y ) ∈ F(FX , FY ) then for all s ∈ R,
τC − (FX , FY )(s) ≤ P(X + Y ≤ s) ≤ ρC − (FX , FY )(s),

where
τC (FX , FY )(s) = sup {C(FX (x), FY (y)), x + y = s}
x,y∈R

and, if C(u, v) = u + v − C(u, v),
˜

˜
ρC (FX , FY )(s) = inf {C(FX (x), FY (y)), x + y = s}.
x,y∈R

151


Density of the sum of two uniform random variables Quantile of the sum of two uniform random variables

20
0.8

1#
67.81/0439:.074!.1!;/=
0.6
Density

1)
0.4

1(
0.2

?8@4A48@48B4
Independence C0.21,8D31.7E0$
Clayton, tau=0.5 C0.21,8D31.7E02
Clayton, tau=0.2 F7G-40D31.7E0$
Gumbel, tau=0.5 F7G-40D31.7E02
Gumbel, tau=0.2 C,G,8,1,8/B/12
0.0

12
Comonotonicity

0.0 0.5 1.0 1.5 2.0 0#0 0#$ 0%0 0%$ 100

*+,-.-/0/12304540

Figure 63: Sum to two U([0, 1]) random variables.

152


0.7 Density of the sum of three uniform random variables Quantile of the sum of three uniform random variables

30
0.6
0.5

2$
56-nti/e 89-/6e!-t!:is=
0.4
Density

20
0.3
0.2

ndependence
Independence B/-1ton, t-6D0$
Clayton, tau=0.5 B/-1ton, t-6D02
Clayton, tau=0.2 E6m,e/, t-6D0$

1$
0.1

Gumbel, tau=0.5 E6m,e/, t-6D02
Gumbel, tau=0.2 Bomonotonicit1
Comonotonicity
0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0#0 0#$ 0%0 0%$ 100

)ro,-,i/it1 /e4e/

Figure 64: Sum to three U([0, 1]) random variables.

153


0.5 Density of the sum of five uniform random variables Quantile of the sum of five uniform random variables

$!
0.4

4$
5uantil3 (9alu3!at!:is)
0.3

4!
Density

0.2

'$
Independence
Clayton, tau=0.5 nd3p3nd3nc3
0.1

Clayton, tau=0.2 Bla1tonC tauD!$
Gumbel, tau=0.5 Bla1tonC tauD!2
Gumbel, tau=0.2 Fumb3lC tauD!$

'!
Comonotonicity Fumb3lC tauD!2
Bomonotonicit1
0.0

0 1 2 3 4 5 !#! !#$ !%! !%$ !!

)robabilit1 l343l

Figure 65: Sum to ve U([0, 1]) random variables.

154


Bounding Value-at-Risk for a sum of d ≥ 3 risks
The standard bounds (derived in dimension 2) can be extended, and still holds in

arbitrary dimensions,

VaR(X1 + ... + Xd , α) ≤ G−1 (α)

where

G(s) = sup(x1 ,...,xd−1 ) C − (F1 (x1 ) + ... + Fd−1 (xd−1 ) + Fd (s − x1 − ... − xd−1 ). But

when d ≥ 3, it fails to be sharp.

Embrechts Puccetti (2006) suggested to use the dual optmization

problem, and thus

VaR(X1 + ... + Xd , α) ≤ H −1 (α)
s−(d−1)r
1
where H(s) = 1 − d inf [1 − F (x)]dx, if X1 , ..., Xd have
r∈[0,s/d) s − rd r
identical marginal distributions (denotes F ).

155


Numerical implications
Consider 3 risks, and try to get the VaR of the sum of log-normal risks,

α independence comonotonicity dual standard
bound bound
0.90 7.54 8.85 14.44 15.38
0.95 9.71 12.73 19.50 20.63
0.99 16.06 25.16 35.31 37.03
0.999 29.78 53.99 69.98 73.81
Table 1: Range for VaR(X1 + X2 + X3 ) for a LN (−0.2, 1) portfolio.

156


Numerical implications
In the context of operational risks, Moscadelli (2004) considered 8 business
lines

α comonotonic dual standard
(Basel II) bound bound
99% 2.8924 ×10 4
1.4778 ×10
5
2.6950 ×10 5

99.5% 6.7034 ×10 4
3.3922 ×10
5
6.1114 ×10 5

99.9% 4.8347 ×10 5
2.3807 ×10
6
4.1685 ×10 6

99.99% 8.7476 ×10 6
4.0740 ×10
7
6.7936 ×10 7

Table 2: Range for VaR(X1 + ... + X8 ).

Remark 41. In banks, Basel II suggests to use the comonotonic case as a worst
case scenario.

157


The more correlated, the more risky ?
Will the risk of the portfolio increase with correlation ?

Recall the following theoretical result:

Proposition 42. Assume that X and X are in the same Fréchet space (i.e.
L
Xi = Xi ), and dene

S = X1 + · · · + Xn and S = X1 + · · · + Xn .

If X X for the concordance order, then S T V aR S for the stop-loss or
TVaR order.
A consequence is that if X and X are exchangeable,

corr(Xi , Xj ) ≤ corr(Xi , Xj ) =⇒ T V aR(S, p) ≤ T V aR(S , p), for all p ∈ (0, 1).

See Müller Stoyen (2002) for some possible extensions.

158


Consider

• d lines of business,

• simply a binomial distribution on each line of business, with small loss

probability (e.g. π = 1/1000).

 1 if there is a claim on line i
Let , and S = X1 + · · · + Xd .
 0 if not

Will the correlation among the Xi 's increase the Value-at-Risk of S ?

Consider a probit model, i.e. Xi = 1(Xi ≤ ui ), where X ∼ N (0, Σ), i.e. a

Gaussian copula.

Assume that Σ = [σi,j ] where σi,j = ρ ∈ [−1, 1] when i = j.

159



Figure 66: 99.75% TVaR (or expected shortfall) for Gaussian copulas.

160



Figure 67: 99% TVaR (or expected shortfall) for Gaussian copulas.

161


What about other risk measures, e.g. Value-at-Risk ?

corr(Xi , Xj ) ≤ corr(Xi , Xj ) V aR(S, p) ≤ V aR(S , p), for all p ∈ (0, 1).

(see e.g. Mittnik Yener (2008)).

162



Figure 68: 99.75% VaR for Gaussian copulas.

163



Figure 69: 99% VaR for Gaussian copulas.

164


What could be the impact of tail dependence ?

Previously, we considered a Gaussian copula, i.e. tail independence. What if

there was tail dependence ?

Consider the case of a Student t-copula, with ν degrees of freedom.

165



Figure 70: 99.75% TVaR (or expected shortfall) for Student t-copulas.

166



Figure 71: 99% TVaR (or expected shortfall) for Student t-copulas.

167



Figure 72: 99.75% VaR for Student t-copulas.

168



Figure 73: 99% VaR for Student t-copulas.

169

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