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?

2

Arthur CHARPENTIER - Copules, avenir de l'actuariat ou gadget statistique?

3

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

0.0

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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

0.2

0.4

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0.8

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

0.0

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1.0

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

0.2

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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

0.0

0.2

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0.8

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@4A48@48B4C0.21,8D31.7E0"$C0.21,8D31.7E0"2F7G-40D31.7E0"$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

89