184equation.0.1
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Copules, avenir de l'actuariat
ou gadget statistique?
Arthur Charpentier
http://perso.univ-rennes1.fr/arthur.charpentier/
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
... de la théorie à la pratique, considérations générales
Le paradoxe de Saint Petersbourg: le prix d'un jeu est le �produit scalaire des
probabilités et des gains�
< p,x >=n∑i=1
pixi = EP(X),
i.e. � l'espérance mathématique est donc le juste prix des chances� (Cournot A.
A. (1843). Exposition de la théorie des chances et des probabilités).
• réponse de Bernoulli, introduire une utilité morale de l'argent, i.e. distortion
sur les gains, Bernoulli, D. (1738). Exposition of a New Theory on the
Measurement of Risk. Econometrica, 22, 23-36.
• réponse de d'Alembert, introduire une distorsion sur les probabilités,
D�Alembert, J. (1768). Sur l'analyse des jeux. Opuscules mathématiques,
4, 80-92.
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
... de la théorie à la pratique, considérations générales
Problème d'allocation d'actifs et de frontière d'e�cience, deux articles en 1956:
• approche moyenne-variance, Markowitz, H. (1956). Portfolio Selection.
Journal of Finance 7, 1, 77-91.
• approche moyenne-quantile, Roy, A.D. (1956). Safety First and the Holding
of Assets. Econometrica, 20, 3, 431-449.
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
le problème de la dépendance... les lois elliptiques
première piste pour générer des lois multivariés dépendentes: les lois elliptiques,
extension de la loi normale multivariée.
1. on partir de deux vecteurs X et Y N (0, 1) indépendants
2. on fait subir une homothétie et une rotation
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9
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
le problème de la dépendance... les lois elliptiques
idée: si X = µ+AX0 où X0 ∼ N (0,1), alors X ∼ N (µ,Σ = A′A).
on passe ainsi de vecteurs sphériques aux vecteurs elliptiques.
marche pour le vecteur Student t multivarié.
permet de modéliser d'autres types de lois, e.g. des lois binomiales multivariées:
modèle probit multivarié (cf CreditMetrics)
10
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
le problème de la dépendance... les mélanges
deuxième piste pour générer des lois multivariés dépendentes: l'indépendance
conditionnelle et les modèles de frailty.
11
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0 5 10 15
05
1015
20
Conditional independence, two classes
!3 !2 !1 0 1 2 3
!3
!2
!1
01
23
Conditional independence, two classes
Figure 1: Deux classes de risques, (Xi, Yi) et (Φ−1(FX(Xi)),Φ−1(FY (Yi))).
12
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0 5 10 15 20 25 30
010
2030
40
Conditional independence, three classes
!3 !2 !1 0 1 2 3
!3
!2
!1
01
23
Conditional independence, three classes
Figure 2: Trois classes de risques, (Xi, Yi) et (Φ−1(FX(Xi)),Φ−1(FY (Yi))).
13
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0 20 40 60 80 100
020
4060
80100
Conditional independence, continuous risk factor
!3 !2 !1 0 1 2 3
!3
!2
!1
01
23
Conditional independence, continuous risk factor
Figure 3: Classes de risques continues, (Xi, Yi) et (Φ−1(FX(Xi)),Φ−1(FY (Yi))).
14
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
le problème de la dépendance... les chocs communs
troisième piste pour générer des lois multivariés dépendentes: les modèles à choc
commun
premier exemple, la loi de Poisson Soient X,Y, Z trois variables P(λX),P(λY ) et P(λZ) independentes. On pose alors
U = X + Z ∼ P(λX + λZ) et Z = Y + Z ∼ P(λY + λZ)
deuxième exemple, la loi exponentielle Soient X,Y, Z trois variables E(µX),E(µY ) et E(µZ) independentes. On pose alors
U = min{X,Z} et V = min{Y,Z}
15
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
comment comparer ces di�érentes dépendances
Pour comparer les structures de dépendance, il faut que les lois marginales soient
identiques,
• deux lois uniformes sur [0, 1]
• deux lois normales centrées réduites N (0, 1)
Si X a pour fonction de répartition F continue, F (x) = P(X ≤ x), alors F (X)suit une loi uniforme sur [0, 1]. Aussi, Φ−1(F (X)) suit une loi N (0, 1).
16
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
comment comparer ces di�érentes dépendances ?
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0.0 0.2 0.4 0.6 0.8 1.0
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1.0
Figure 4: Le modèle Gaussien, N (0, 1) et U([0, 1]).
17
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
comment comparer ces di�érentes dépendances ?
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
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1.0
Figure 5: Le modèle Student t, N (0, 1) et U([0, 1]).
18
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
comment comparer ces di�érentes dépendances ?
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0.0 0.2 0.4 0.6 0.8 1.0
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Figure 6: Le modèle Student t, N (0, 1) et U([0, 1]).
19
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
comment comparer ces di�érentes dépendances ?
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0.0 0.2 0.4 0.6 0.8 1.0
0.0
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1.0
Figure 7: Le modèle Gamma, N (0, 1) et U([0, 1]).
20
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
comment comparer ces di�érentes dépendances ?
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0.0 0.2 0.4 0.6 0.8 1.0
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1.0
Figure 8: Le modèle α-stable, N (0, 1) et U([0, 1]).
21
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
comment comparer ces di�érentes dépendances ?
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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
Figure 9: Le modèle min-exponentielles, N (0, 1) et U([0, 1]).
22
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Copulas, an introduction (in dimension 2)
De�nition 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,
P(u1 < U ≤ u2, v1 < V ≤ v2) ≥ 0,
thus
C(u2, v2)− C(u1, v2)
−C(u2, v1) + C(u1, v1) ≥ 0.0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
Copula, positive area
23
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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.
24
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Borders of the copula function
!0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4!0.2
0.0 0.2
0.4 0.6
0.8 1.0
1.2 1.4
!0.2 0.0
0.2 0.4
0.6 0.8
1.0 1.2
Figure 10: Value of the copula on the border of the unit square.
25
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
une courte histoire des copules, et de la dépendance
Hoeffding, W. (1940). Masstabinvariante Korrelationstheorie. Schriften des
Matematischen Instituts für Angewandte Matematik der Universitat Berlin. 5,
181-233.
26
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
une courte histoire des copules, et de la dépendance
Hoeffding, W. (1940). Masstabinvariante Korrelationstheorie. Schriften des
Matematischen Instituts für Angewandte Matematik der Universitat Berlin. 5,
181-233.
27
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
XY
Z
Fonction de répartition à marges uniformes
Figure 11: Graphical representation of a copula.
28
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
If C is twice di�erentiable, one can de�ne its density as
c(u, v) =∂2C(u, v)∂u∂v
.
29
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
xx
z
Densité d’une loi à marges uniformes
Figure 12: Density of a copula.
30
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
une courte histoire des copules, et de la dépendance
Hoeffding, W. (1940). Masstabinvariante Korrelationstheorie. Schriften des
Matematischen Instituts für Angewandte Matematik der Universitat Berlin. 5,
181-233.
31
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Fonction de répartition à marges uniformes Densité d’une loi à marges uniformes
Fonction de répartition à marges uniformes Densité d’une loi à marges uniformes
Figure 13: Distribution functions and densities.
32
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Fonction de répartition à marges uniformes Densité d’une loi à marges uniformes
Fonction de répartition à marges uniformes Densité d’une loi à marges uniformes
Figure 14: Distribution functions and densities.
33
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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 (F−1X (u), F−1
Y (v)) for all (u, v) ∈ [0, 1]× [0, 1]
We will then de�ne 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.
34
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Copulas, an introduction (in dimension d ≥ 2)
De�nition 5. A copula C is a joint distribution function on [0, 1]d, withuniform 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 satis�es some increasing property since P is a positive measure
(for all 0 ≤ u ≤ v ≤ 1, P(u < U ≤ v) ≥ 0), thus∑z
sign(z)C(z) ≥ 0,
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 15). And �nally C
has uniform margins, and thus
C(1, . . . , 1, ui, 1, . . . , 1) = ui on [0, 1].
35
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Increasing functions in dimension 3
Figure 15: The notion of 3-increasing functions.
36
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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 (F−11 (u1), . . . , F−1
d (ud)) for all (ui) ∈ [0, 1]
We will then de�ne 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 di�erentiable, one can de�ne its density,
c(u1, . . . , ud) =∂dC(u1, . . . , ud)∂u1 . . . ∂ud
.
37
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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,
F̂i(xi) =#{observations Xi,j 's lower than xi}
#{observations }=
1n
n∑j=1
1(Xi,j ≤ xi).
Note that
F̂i(Xi,j0) =#{observations Xi,j 's lower than Xi,j0}
#{observations }=
1n
n∑j=1
1(Xi,j ≤ Xi,j0) =Ri,j0n
,
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.
38
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
5.56.0
6.57.0
7.58.0
8.59.0
Scatterplot of (X,Y)
X (raw data)
Y (raw
data)
5 10 15 20
510
1520
Scatterplot of the ranks of (X,Y)
Ranks of the Xi’s
Ranks
of the
Yi’s
0.2 0.4 0.6 0.8 1.0
0.20.4
0.60.8
1.0
Scatterplot of the ranks of (X,Y), divided by n
Ranks of the Xi’s/n
Ranks
of the
Yi’s/n
0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
Scatterplot o+ ,-,/0, t1e copula!t3pe tran+orm o+ ,6,70
Ui=Ranks of the Xi’s/n+1
Vi=Ra
nks of
the Yi
’s/n+1
Figure 16: Copulas, ranks and parametric inference, from (Xi, Yi) to (Ui, Vi).
39
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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
(X,Y ) will be denoted (X⊥, Y ⊥). It is a random vector such that X⊥L= X and
Y ⊥L= 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
(X,Y ) will be denoted (X+, Y +). It is a random vector such that X+ L= X and
Y + L= Y , with copula C+.
(X,Y ) has copula C+ if and only if there exists a strictly increasing function h
such that Y = h(X), or equivalently (X,Y ) L= (F−1X (U), F−1
Y (U)) where U is
U([0, 1]).
40
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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
(X,Y ) will be denoted (X−, Y −). It is a random vector such that X−L= X and
Y −L= Y , with copula C−.
(X,Y ) has copula C− if and only if there exists a strictly decreasing function h
such that Y = h(X), or equivalently (X,Y ) L= (F−1X (1− U), F−1
Y (U)) where U is
U([0, 1]).
41
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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 acopula: if (X,Y ) and (X,Z) are countercomonotonic, (Y,Z) is necessarilycomonotonic - it is not possible to have all component highly negatively
correlated.
Anyway, it is still the best pointwise lower bound.
42
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0.2
0.40.6
0.8
u_10.2
0.4
0.6
0.8
u_2
00.
20.
40.
60.
81
Frec
het lo
wer b
ound
0.2
0.4
0.6
0.8
u_10.2
0.4
0.6
0.8
u_2
00.
20.
40.
60.
81
Inde
pend
ence
copu
la
0.2
0.40.6
0.8
u_10.2
0.4
0.6
0.8
u_2
00.
20.
40.
60.
81
Frec
het u
pper
bou
nd
Fréchet Lower Bound
0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
Independent copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
Fréchet Upper Bound
0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
Scatterplot, Lower Fréchet!Hoeffding bound
0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
Scatterplot, Indepedent copula random generation
0.0 0.2 0.4 0.6 0.8 1.0
0.00.2
0.40.6
0.81.0
Scatterplot, Upper Fréchet!Hoeffding bound
Figure 17: Contercomontonce, independent, and comonotone copulas.
43
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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,
(3, 2, 3) with probability 1/4,
(3, 1, 3) with probability 1/4,
then X and Y are independent, and Y and Z are independent, but X = Z.
44
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0 1 2 3 4
01
23
4
X and Y independent
Component X
Comp
onent Y
0 1 2 3 40
12
34
Y and Z independent
Component Y
Comp
onent Z
0 1 2 3 4
01
23
4
X and Z comonotonic
Component X
Comp
onent Z
Figure 18: Mixing independence and comonotonicity.
45
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Pitfalls on independence and comonotonicity
The following proposition is false,
Uncorrect Proposition 8. If X and Y are comonotonic, if Y and Z are
comonotonic, then X and Z are comonotonic.
If
(X,Y, Z) = (1, 1, 1) with probability 1/4,
(1, 2, 3) with probability 1/4,
(3, 2, 1) with probability 1/4,
(3, 3, 3) with probability 1/4,
then X and Y are comonotonic, and Y and Z are comonotonic, but X and Z are
independent.
46
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0 1 2 3 4
01
23
4
X and Y comonotonic
Component X
Comp
onent Y
0 1 2 3 40
12
34
Y and Z comonotonic
Component Y
Comp
onent Z
0 1 2 3 4
01
23
4
X and Z independent
Component X
Comp
onent Z
Figure 19: Mixing independence and comonotonicity.
47
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Pitfalls on independence and comonotonicity
The following proposition is false,
Uncorrect Proposition 9. If X and Y are comonotonic, if Y and Z are
independent, then X and Z are independent.
If
(X,Y, Z) = (1, 1, 3) with probability 1/4,
(2, 1, 1) with probability 1/4,
(2, 3, 3) with probability 1/4,
(3, 3, 1) with probability 1/4,
then X and Y are comonotonic, and Y and Z are independent, but X and Z are
anticomonotonic.
48
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
If
(X,Y, Z) = (1, 1, 1) with probability 1/4,
(2, 1, 3) with probability 1/4,
(2, 3, 1) with probability 1/4,
(3, 3, 3) with probability 1/4,
then X and Y are comonotonic, and Y and Z are independent, but X and Z are
comonotonic.
49
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0 1 2 3 4
01
23
4
X and Y comonotonic
Component X
Comp
onent Y
0 1 2 3 40
12
34
Y and Z independent
Component Y
Comp
onent Z
0 1 2 3 4
01
23
4
X and Z comonotonic
Component X
Comp
onent Z
Figure 20: Mixing independence and comonotonicity.
50
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Elliptical (Gaussian and t) copulas
The idea is to extend the multivariate probit model, Y = (Y1, . . . , Yd) withmarginal B(pi) distributions, modeled as Yi = 1(Xi ≤ ui), where X ∼ N (I,Σ).
• The Gaussian copula, with parameter α ∈ (−1, 1),
C(u, v) =1
2π√
1− α2
∫ Φ−1(u)
−∞
∫ Φ−1(v)
−∞exp
{−(x2 − 2αxy + y2)
2(1− α2)
}dxdy.
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,
C(u, v) =1
2π√
1− α2
∫ t−1ν (u)
−∞
∫ t−1ν (v)
−∞
(1 +
x2 − 2αxy + y2
2(1− α2)
)−((ν+2)/2)
dxdy.
51
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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.
52
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
ψ(t) range θ
(1) 1θ
(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−1e−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−t1+(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) ( 1t− 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) ( θt
+ 1)(1 − t) [0,∞)
53
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Copulas in �nance: options on multiple assets
Remark 11. Recall that Breeden & Litzenberger (1978) proved that the risk
neutral probability can be obtrained from option prices: consider the price of a call
Π(T,K) = e−rTEQ((ST −K)+). Since (ST −K)+ =∫∞K
1(ST > x)dx, one gets
Π(T,K) = e−rT∫ ∞K
Q(ST > x)dx,
hence
Q(ST ≤ x) = −e−rT ∂C∂K
(T, x), or Q(ST ≤ x) = −erT ∂P∂K
(T, x)
where P denotes the price of a put option.
Consider an option on 2 assets, with payo� h(S1T , S
2T ). The price at time 0 is
e−rTEQ(h(S1T , S
2T )).
54
Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Copulas in �nance: call on maximum
Here the payo� is h(S1T , S
2T ) = (max{S1
T , S2T } −K)+. The price is then
Π(T,K) = e−rTEQ((max{S1T , S
2T } −K)+)
= e−rTEQ
(∫ ∞K
1− 1(max{S1T , S
2T } ≤ x)dx
)= e−rT
∫ ∞K
1−Q(max{S1T , S
2T } ≤ x)︸ ︷︷ ︸
Q(S1T≤x,S2
T≤x)
dx,
hence, if (S1T , S
2T ) has copula C (under Q), then
Π(T,K) = e−rT∫ ∞K
1− C(erT
∂P 1
∂K(T, x), erT
∂P 2
∂K(T, x)
)dx.
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Copulas in �nance: call on spreads
Here the payo� is h(S1T , S
2T ) = ([S1
T − S2T ]−K)+. The price is then
Π(T,K) = e−rTEQ((S1T − S2
T −K)+) = e−rTEQ
(∫ ∞−∞
1(S2T +K ≤ x ≤ S1
T )dx)
= e−rT∫ ∞−∞
Q(K + S2T ≤ x)−Q(S2
T +K ≤ x, S1T ≤ x} ≤ x)︸ ︷︷ ︸
Q(S1T≤x,S2
T≤x+K)
dx,
hence, if (S1T , S
2T ) has copula C (under Q), then
Π(T,K) = e−rT∫ ∞−∞
erT∂P 2
∂K(T, x−K)−C
(erT
∂P 1
∂K(T, x), erT
∂P 2
∂K(T, x−K)
)dx.
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Natural properties for dependence measures
De�nition 12. κ is measure of concordance if and only if κ satis�es
1. κ is de�ned 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) �PQD (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→∞.
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
As pointed out in Scarsini (1984), most of the axioms are �self-evident�.
If κ 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 falmost surely strictly decreasing (see Scarsini (1984)).
Association measures: Kendall's τ and Spearman's ρ
Rank correlations can be considered, i.e. Spearman's ρ de�ned as
ρ(X,Y ) = corr(FX(X), FY (Y )) = 12∫ 1
0
∫ 1
0
C(u, v)dudv − 3
and Kendall's τ de�ned as
τ(X,Y ) = 4∫ 1
0
∫ 1
0
C(u, v)dC(u, v)− 1.
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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 - - - - - - - -
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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 - - - - - - -
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Les copules, un gadget ?
copulas: tales and facts, Mikosch, T. (2006)
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Les copules, un gadget ?
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Les copules, un gadget ?
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Marges uniformes
Cop
ule
Gau
ssie
nne
!2 0 2 4!
20
24
Marges gaussiennes
Figure 21: Simulations of the Gaussian copula (θ = 0.95).
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Marges uniformes
Cop
ule
de G
umbe
l
!2 0 2 4!
20
24
Marges gaussiennes
Figure 22: Simulations of Gumbel's copula θ = 1.2.
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Les copules, un gadget ?
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Les copules, un gadget ?
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Les copules, un gadget ?
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Les copules, un gadget ?
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Les copules, un gadget ?
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Les copules, un gadget ?
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Risk measures and diversi�cation
Any copula C is bounded by Fréchet-Hoe�ding bounds,
max
{d∑i=1
ui − (d− 1), 0
}≤ C(u1, . . . , ud) ≤ min{u1, . . . , ud},
and thus, any distribution F on F(F1, . . . , Fd) is bounded
max
{d∑i=1
Fi(xi)− (d− 1), 0
}≤ F (x1, . . . , xd) ≤ min{F1(x1), . . . , Ff (xd)}.
Does this means the comonotonicity is always the worst-case scenario ?
Given a random pair (X,Y ), let (X−, Y −) and (X+, Y +) denotecontercomonotonic and comonotonic versions of (X,Y ), do we have
R(φ(X−, Y −))?
≤ R(φ(X ,Y ))?
≤ R(φ(X+, Y +)).
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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 13. the stop loss premium for the sum of two risks E((X + Y − d)+) is
supermodular.
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Example 14. For the n-year joint-life annuity,
axy:nq =n∑k=1
vkP(Tx > k and Ty > k) =n∑k=1
vkkpxy.
Then
a−xy:nq ≤ axy:nq ≤ a+xy:nq,
where
a−xy:nq =n∑k=1
vk max{kpx + kpy − 1, 0}( lower Fréchet bound ),
a+xy:nq =
n∑k=1
vk min{kpx, kpy}( upper Fréchet bound ).
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Example 15. For the n-year last-survivor annuity,
axy:nq =n∑k=1
vkP(Tx > k or Ty > k) =n∑k=1
vkkpxy,
where kpxy = P(Tx > k or Ty > k) = kpx + kpy − kpxy.
Then
a−xy:nq ≤ axy:nq ≤ a+xy:nq,
where
a−xy:nq =n∑k=1
vk (1−min{kqx, kqy}) ( upper Fréchet bound ),
a+xy:nq =
n∑k=1
vk (1−max{kqx + kqy − 1, 0}) ( lower Fréchet bound ).
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Example 16. For the widow's pension annuity,
ax|y = ay − axy =∞∑k=1
vkkpy −∞∑k=1
vkkpxy.
Then
a−x|y ≤ ax|y ≤ a+x|y,
where
a−x|y = ay − axy =∞∑k=1
vkkpy −∞∑k=1
vk min{kpx, kpy}.( upper Fréchet bound ),
a+x|y = ay−axy =
∞∑k=1
vkkpy−∞∑k=1
vk max{kpx+kpy−1, 0}.( lower Fréchet bound ).
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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 in�nite
variance i.e. X ∈ Lp with p < 1),
• X and Y are N (0, 1) with (special) tail dependence.
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Makarov's theorem and bounding Value-at-Risk
In the case where R denotes the Value-at-Risk (i.e. quantile function of the P&L
distribution),
R− ≤ R(X− + Y −)6≤R(X + Y ) 6≤R(X+ + Y +) ≤ R+,
where e.g. R+ can exceed the comonotonic case. Recall that
R(X + Y ) = VaRq[X + Y ] = F−1X+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 logα− β log β.
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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).
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0.0 0.2 0.4 0.6 0.8 1.0
!4!2
02
4
Bornes de la VaR d’un portefeuille
Somme de 2 risques Gaussiens
Figure 23: Value-at-Risk for 2 Gaussian risks N (0, 1).
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0.90 0.92 0.94 0.96 0.98 1.00
01
23
45
6
Bornes de la VaR d’un portefeuille
Somme de 2 risques Gaussiens
Figure 24: Value-at-Risk for 2 Gaussian risks N (0, 1).
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0.0 0.2 0.4 0.6 0.8 1.0
05
1015
20
Bornes de la VaR d’un portefeuille
Somme de 2 risques Gamma
Figure 25: Value-at-Risk for 2 Gamma risks G(3, 1).
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0.90 0.92 0.94 0.96 0.98 1.00
05
1015
20
Bornes de la VaR d’un portefeuille
Somme de 2 risques Gamma
Figure 26: Value-at-Risk for 2 Gamma risks G(3, 1).
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
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 in�mal convolutions,
Fsup (FX , FY ) (z) = sup {C (FX (x) , FY (y)) , ψ (x, y) = z} (1)
Finf (FX , FY ) (z) = inf {C (FX (x) , FY (y)) , ψ (x, y) = z} (2)
Williamson (1991) and Embrechts, Hoing & Juri (2002) proposednumerical 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) = supx∈R
max{P(X < x) + P(Y < s− x)− 1, 0}
and
P(Smin ≤ s) = infx∈R
min{P(X ≤ x) + P(Y ≤ s− x), 1}.
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
Note that those bounds can also be written as follows,
Proposition 17. 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) = supx,y∈R
{C(FX(x), FY (y)), x+ y = s}
and, if C̃(u, v) = u+ v − C(u, v),
ρC(FX , FY )(s) = infx,y∈R
{C̃(FX(x), FY (y)), x+ y = s}.
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
Density of the sum of two uniform random variablesD
ensi
ty
IndependenceClayton, tau=0.5Clayton, tau=0.2Gumbel, tau=0.5Gumbel, tau=0.2Comonotonicity
0"#0 0"#$ 0"%0 0"%$ 1"00
1"2
1"(
1")
1"#
2"0
Quantile of the sum of two uniform random variables
*+,-.-/0/12304540
67.81/0439:.074!.1!;/<=>
?8@[email protected],8D31.7E0"$C0.21,8D31.7E0"2F7G-40D31.7E0"$F7G-40D31.7E0"2C,G,8,1,8/B/12
Figure 27: Sum to two U([0, 1]) random variables.
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Density of the sum of three uniform random variablesD
ensi
ty
IndependenceClayton, tau=0.5Clayton, tau=0.2Gumbel, tau=0.5Gumbel, tau=0.2Comonotonicity
0"#0 0"#$ 0"%0 0"%$ 1"00
1"$
2"0
2"$
3"0
Quantile of the sum of three uniform random variables
)ro,-,i/it1 /e4e/
56-
nti/e
89
-/6e!
-t!
:is
<=
>ndependenceB/-1ton, t-6D0"$B/-1ton, t-6D0"2E6m,e/, t-6D0"$E6m,e/, t-6D0"2Bomonotonicit1
Figure 28: Sum to three U([0, 1]) random variables.
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
0 1 2 3 4 5
0.0
0.1
0.2
0.3
0.4
0.5
Density of the sum of five uniform random variablesD
ensi
ty
IndependenceClayton, tau=0.5Clayton, tau=0.2Gumbel, tau=0.5Gumbel, tau=0.2Comonotonicity
!"#! !"#$ !"%! !"%$ &"!!
'"!
'"$
4"!
4"$
$"!
Quantile of the sum of five uniform random variables
)robabilit1 l343l
5ua
ntil3
(9
alu3!
at!
:is
<)
>nd3p3nd3nc3Bla1tonC tauD!"$Bla1tonC tauD!"2Fumb3lC tauD!"$Fumb3lC tauD!"2Bomonotonicit1
Figure 29: Sum to �ve U([0, 1]) random variables.
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Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?
To go further
http://perso.univ-rennes1.fr/arthur.charpentier/slides-edf-2.pdf
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