Slides erm

Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
Copulas, risk dependence
and applications to Solvency II
Arthur Charpentier
CREM-Universit´e Rennes 1
http ://perso.univ-rennes1.fr/arthur.charpentier/
Summer School of the Groupe Consultatif Actuariel Europen :
Enterprise Risk Management (ERM) and Solvency II, July 2008.
1

On risk dependence in QIS’s
http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
2

3

4

5

6

How to capture dependence in risk models ?
Is correlation relevant to capture dependence information ?
Consider (see McNeil, Embrechts & Straumann (2003)) 2 log-normal risks,
• X ∼ LN(0, 1), i.e. X = exp(X ) where X ∼ N(0, 1)
• Y ∼ LN(0, σ2
), i.e. Y = exp(Y ) where Y ∼ N(0, σ2
)
Recall that corr(X , Y ) takes any value in [−1, +1].
Since corr(X, Y )=corr(X , Y ), what can be corr(X, Y ) ?
7

0 1 2 3 4 5
−0.50.00.51.0
Standard deviation, sigma
Correlation
Fig. 1 – Range for the correlation, cor(X, Y ), X ∼ LN(0, 1) ,Y ∼ LN(0, σ2
).
8

0 1 2 3 4 5
−0.50.00.51.0
Standard deviation, sigma
Correlation
Fig. 2 – cor(X, Y ), X ∼ LN(0, 1) ,Y ∼ LN(0, σ2
), Gaussian copula, r = 0.5.
9

What about regulatory technical documents ?
10

11

12

13

Motivations : dependence and copulas
Deﬁnition 1. A copula C is a joint distribution function on [0, 1]d
, with
uniform margins on [0, 1].
Theorem 2. (Sklar) Let C be a copula, and F1, . . . , Fd be d marginal
distributions, then F(x) = C(F1(x1), . . . , Fd(xd)) is a distribution function, with
F ∈ F(F1, . . . , Fd).
Conversely, if F ∈ F(F1, . . . , Fd), there exists C such that
F(x) = C(F1(x1), . . . , Fd(xd)). Further, if the Fi’s are continuous, then C is
unique, and given by
C(u) = F(F−1
1 (u1), . . . , F−1
d (ud)) for all ui ∈ [0, 1]
We will then deﬁne the copula of F, or the copula of X.
14

Copula density Level curves of the copula
Fig. 3 – Graphical representation of a copula, C(u, v) = P(U ≤ u, V ≤ v).
15

Copula density Level curves of the copula
Fig. 4 – Density of a copula, c(u, v) =
∂2
C(u, v)
∂u∂v
.
16

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−1
X (U), F−1
Y (U)) where U is
U([0, 1]).
17

Some very classical 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−1
X (1 − U), F−1
Y (U)).
In higher dimension, C−
(u1, . . . , ud) = max{u1 + . . . + ud − (d − 1), 0} is not a
copula.
But note that for any copula C,
C−
(u1, . . . , ud) ≤ C(u1, . . . , ud) ≤ C+
(u1, . . . , ud)
18

0.2
0.4
0.6
0.8
u_10.2
0.4
0.6
0.8
u_2
00.20.40.60.81
Frechetlowerbound
0.2
0.4
0.6
0.8
u_10.2
0.4
0.6
0.8
u_2
00.20.40.60.81Independencecopula
0.2
0.4
0.6
0.8
u_10.2
0.4
0.6
0.8
u_2
00.20.40.60.81
Frechetupperbound
Fréchet Lower Bound
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Independent copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Fréchet Upper Bound
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Scatterplot, Lower Fréchet!Hoeffding bound
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Scatterplot, Indepedent copula random generation
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Scatterplot, Upper Fréchet!Hoeffding bound
Fig. 5 – Contercomontonce, independent, and comonotone copulas.
19

Elliptical (Gaussian and t) copulas
The idea is to extend the multivariate probit model, X = (X1, . . . , Xd) with
marginal 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−1
ν (u)
−∞
t−1
ν (v)
−∞
1 +
x2
− 2αxy + y2
2(1 − α2)
−((ν+2)/2)
dxdy.
20

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 3. 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.
The frailty approach : 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) = ψ(ψ−1
(FX(x)) + ψ−1
(FY (y))),
where ψ denotes the Laplace transform of Θ, i.e. ψ(t) = E(e−tΘ
).
21

0 20 40 60 80 100
020406080100
Conditional independence, continuous risk factor
!3 !2 !1 0 1 2 3
!3!2!10123
Conditional independence, continuous risk factor
Fig. 6 – Continuous classes of risks, (Xi, Yi) and (Φ−1
(FX(Xi)), Φ−1
(FY (Yi))).
22

Some more examples of Archimedean copulas
ψ(t) range θ
(1) 1
θ
(t−θ − 1) [−1, 0) ∪ (0, ∞) Clayton, Clayton (1978)
(2) (1 − t)θ [1, ∞)
(3) log
1−θ(1−t)
t
[−1, 1) Ali-Mikhail-Haq
(4) (− log t)θ [1, ∞) Gumbel, Gumbel (1960), Hougaard (1986)
(5) − log e−θt−1
e−θ−1
(−∞, 0) ∪ (0, ∞) Frank, Frank (1979), Nelsen (1987)
(6) − log{1 − (1 − t)θ} [1, ∞) Joe, Frank (1981), Joe (1993)
(7) − log{θt + (1 − θ)} (0, 1]
(8) 1−t
1+(θ−1)t
[1, ∞)
(9) log(1 − θ log t) (0, 1] Barnett (1980), Gumbel (1960)
(10) log(2t−θ − 1) (0, 1]
(11) log(2 − tθ) (0, 1/2]
(12) ( 1
t
− 1)θ [1, ∞)
(13) (1 − log t)θ − 1 (0, ∞)
(14) (t−1/θ − 1)θ [1, ∞)
(15) (1 − t1/θ)θ [1, ∞) Genest & Ghoudi (1994)
(16) ( θ
t
+ 1)(1 − t) [0, ∞)
23

Extreme value copulas
• Extreme value copulas
C(u, v) = exp (log u + log v) A
log u
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 definition is the following : C is an extreme value copula if for all
z > 0,
C(u1, . . . , ud) = C(u
1/z
1 , . . . , u
1/z
d )z
.
Those copula are then called max-stable : define the maximum componentwise of
a sample X1, . . . , Xn, i.e. Mi = max{Xi,1, . . . , Xi,n}.
Remark more difficult to characterize when d ≥ 3.
24

On copula parametrization
• Gaussian, Student t (and elliptical) copulas
Focuses on pairwise dependence through the correlation matrix,







X1
X2
X3
X4







∼ N







0,
1 r12 r13 r14
r12 1 r23 r24
r13 r23 1 r34
r14 r24 r34 1







Dependence in [0, 1]d
←→ summarized in d(d + 1)/2 parameters,
25

• Archimedean copulas
Initially, dependence in [0, 1]d
←→ summarized in one functional parameters on
[0, 1]. But appears less ﬂexible because of exchangeability features.
It is possible to introduce hierarchical Archimedean copulas (see Savu & Trede
(2006) or McNeil (2007)). Let U = (U1, U2, U3, U4),
C(u1, u2, u3, u4) = φ−1
1 [φ1(u1) + φ1(u2) + φ1(u3) + φ1(u4)],
which, if φi is parametrized with parameter αi, can be summarized through
A =







1 α2 α4 α4
α2 1 α4 α4
α4 α4 1 α3
alpha4 α4 α3 1







26

C(u1, u2, u3, u4) = φ−1
4 (φ4 φ−1
2 (φ2(u1) + φ2(u2)) + φ4 φ−1
3 (φ3(u3) + φ3(u4)) ),
A =







1 α2 α4 α4
α2 1 α4 α4
α4 α4 1 α3
alpha4 α4 α3 1







27

C(u1, u2, u3, u4) = φ−1
4 (φ4 φ−1
2 (φ2(u1) + φ2(u2)) + φ4 φ−1
3 (φ3(u3) + φ3(u4)) ),
A =







1 α2 α4 α4
α2 1 α4 α4
α4 α4 1 α3
alpha4 α4 α3 1







28

C(u1, u2, u3, u4) = φ−1
4 (φ4 φ−1
2 (φ2(u1) + φ2(u2)) + φ4 φ−1
3 (φ3(u3) + φ3(u4)) ),
A =







1 α2 α4 α4
α2 1 α4 α4
α4 α4 1 α3
α4 α4 α3 1







29

C(u1, u2, u3, u4) = φ−1
4 (φ4[φ−1
3 (φ3 φ−1
2 (φ2(u1) + φ2(u2)) + φ3(u3))] + φ4(u4)),
A =







1 α2 α3 α4
α2 1 α3 α4
α3 α3 1 α4
α4 α4 α4 1







30

C(u1, u2, u3, u4) = φ−1
4 (φ4[φ−1
3 (φ3 φ−1
2 (φ2(u1) + φ2(u2)) + φ3(u3))] + φ4(u4)),
A =







1 α2 α3 α4
α2 1 α3 α4
α3 α3 1 α4
α4 α4 α4 1







31

C(u1, u2, u3, u4) = φ−1
4 (φ4[φ−1
3 (φ3 φ−1
2 (φ2(u1) + φ2(u2)) + φ3(u3))] + φ4(u4)),
A =







1 α2 α3 α4
α2 1 α3 α4
α3 α3 1 α4
α4 α4 α4 1







32

• Extreme value copulas
Here, dependence in [0, 1]d
[0, 1]d−1
.
Further, focuses only on ﬁrst order tail dependence.
33

Natural properties for dependence measures
Definition 4. κ is measure of concordance if and only if κ satisfies
• κ is defined for every pair (X, Y ) of continuous random variables,
• −1 ≤ κ (X, Y ) ≤ +1, κ (X, X) = +1 and κ (X, −X) = −1,
• κ (X, Y ) = κ (Y, X),
• if X and Y are independent, then κ (X, Y ) = 0,
• κ (−X, Y ) = κ (X, −Y ) = −κ (X, Y ),
• if (X1, Y1) P QD (X2, Y2), then κ (X1, Y1) ≤ κ (X2, Y2),
• 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 → ∞.
34

Natural properties for dependence measures
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 f almost
surely strictly decreasing (see Scarsini (1984)).
Rank correlations can be considered, i.e. Spearman’s ρ deﬁned as
ρ(X, Y ) = corr(FX(X), FY (Y )) = 12
1
0
1
0
C(u, v)dudv − 3
and Kendall’s τ deﬁned as
τ(X, Y ) = 4
1
0
1
0
C(u, v)dC(u, v) − 1.
35

Historical version of those coeﬃcients
Similarly Kendall’s tau was not deﬁned 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)).
Equivalently, τ(X, Y ) = 1 −
4Q
n(n2 − 1)
where Q is the number of inversions
between the rankings of X and Y (number of discordance).
36

!2.0 !1.5 !1.0 !0.5 0.0 0.5 1.0
!0.50.00.51.01.5 Concordant pairs
X
Y
!2.0 !1.5 !1.0 !0.5 0.0 0.5 1.0
!0.50.00.51.01.5
Discordant pairs
X
Y
Fig. 7 – Concordance versus discordance.
37

Alternative expressions of those coeﬃcients
Note that those coeﬃcients can also be expressed as follows
ρ(X, Y ) =
[0,1]×[0,1]
C(u, v) − C⊥
(u, v)dudv
[0,1]×[0,1]
C+(u, v) − C⊥(u, v)dudv
(the normalized average distance between C and C⊥
), for instance.
The case of the Gaussian random vector
If (X, Y ) is a Gaussian random vector with correlation r, then (Kruskal (1958))
ρ(X, Y ) =
6
π
arcsin
r
2
and τ(X, Y ) =
2
π
arcsin (r) .
38

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 - - - - - - - -
39

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 - - - - - - -
40

0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Marges uniformes
CopuledeGumbel
!2 0 2 4
!2024
Marges gaussiennes
Fig. 8 – Simulations of Gumbel’s copula θ = 1.2.
41

0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Marges uniformes
CopuleGaussienne
!2 0 2 4
!2024
Marges gaussiennes
Fig. 9 – Simulations of the Gaussian copula (θ = 0.95).
42

Strong tail dependence
Joe (1993) defined, in the bivariate case a tail dependence measure.
Definition 5. Let (X, Y ) denote a random pair, the upper and lower tail
dependence parameters are defined, if the limit exist, as
λL = lim
u→0
P X ≤ F−1
X (u) |Y ≤ F−1
Y (u) ,
= lim
u→0
P (U ≤ u|V ≤ u) = lim
u→0
C(u, u)
u
,
and
λU = lim
u→1
P X > F−1
X (u) |Y > F−1
Y (u)
= lim
u→0
P (U > 1 − u|V ≤ 1 − u) = lim
u→0
C (u, u)
u
.
43

Gaussian copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
L and R concentration functions
L function (lower tails) R function (upper tails)
GAUSSIAN
q
q
Fig. 10 – L and R cumulative curves.
44

Gumbel copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
GUMBEL
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45

Clayton copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
CLAYTON
q
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46

Student t copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
STUDENT (df=5)
q
q
47

Student t copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
STUDENT (df=3)
q
q
48

Estimating (strong) tail dependence
From
P ≈
P X > F−1
X (u) , Y > F−1
Y (u)
P Y > F−1
Y (u)
for u closed to 1,
as for Hill’s estimator, a natural estimator for λ is obtained with u = 1 − k/n,
λ
(k)
U =
1
n
n
i=1 1(Xi > Xn−k:n, Yi > Yn−k:n)
1
n
n
i=1 1(Yi > Yn−k:n)
,
hence
λ
(k)
U =
1
k
n
i=1
1(Xi > Xn−k:n, Yi > Yn−k:n).
λ
(k)
L =
1
k
n
i=1
1(Xi ≤ Xk:n, Yi ≤ Yk:n).
49

Asymptotic convergence, how fast ?
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
(Upper) tail dependence, Gaussian copula, n=200
Exceedance probability
0.001 0.005 0.050 0.500
0.00.20.40.60.81.0
Log scale, (lower) tail dependence
Exceedance probability (log scale)
Fig. 15 – Convergence of L and R functions, Gaussian copula, n = 200.
50

0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
0.001 0.005 0.050 0.500
0.00.20.40.60.81.0
Fig. 16 – Convergence of L and R functions, Gaussian copula, n = 2, 000.
51

0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
0.001 0.005 0.050 0.500
0.00.20.40.60.81.0
Fig. 17 – Convergence of L and R functions, Gaussian copula, n = 20, 000.
52

Weak tail dependence
If X and Y are independent (in tails), for u large enough
P(X > F−1
X (u), Y > F−1
Y (u)) = P(X > F−1
X (u)) · P(Y > F−1
Y (u)) = (1 − u)2
,
or equivalently, log P(X > F−1
X (u), Y > F−1
Y (u)) = 2 · log(1 − u). Further, if X
and Y are comonotonic (in tails), for u large enough
P(X > F−1
X (u), Y > F−1
Y (u)) = P(X > F−1
X (u)) = (1 − u)1
,
or equivalently, log P(X > F−1
X (u), Y > F−1
Y (u)) = 1 · log(1 − u).
=⇒ limit of the ratio
log(1 − u)
log P(Z1 > F−1
1 (u), Z2 > F−1
2 (u))
.
53

Weak tail dependence
Coles, Heffernan & Tawn (1999) defined
Definition 6. Let (X, Y ) denote a random pair, the upper and lower tail
dependence parameters are defined, if the limit exist, as
ηL = lim
u→0
log(u)
log P(Z1 ≤ F−1
1 (u), Z2 ≤ F−1
2 (u))
= lim
u→0
log(u)
log C(u, u)
,
and
ηU = lim
u→1
log(1 − u)
log P(Z1 > F−1
1 (u), Z2 > F−1
2 (u))
= lim
u→0
log(u)
log C (u, u)
.
54

Gaussian copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Chi dependence functions
lower tails upper tails
GAUSSIAN
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Fig. 18 – χ functions.
55

Gumbel copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
GUMBEL
q
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56

Clayton copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
CLAYTON
q
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57

Student t copula
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
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0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
STUDENT (df=3)
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58

Application in risk management : Loss-ALAE
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0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Loss
AllocatedExpenses
Fig. 22 – Losses and allocated expenses.
59

Application in risk management : Loss-ALAE
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
qqq
qqq
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q
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qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
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Gumbel copula
q
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0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0
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qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqq
qqqqq
qq
qqqqqqqqqqqqqq
qqq
q
q
q
q
Gumbel copula
q
q
Fig. 23 – L and R cumulative curves, and χ functions.
60

Application in risk management : car-household
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0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
Car claims
Householdclaims
Fig. 24 – Motor and Household claims.
61

Application in risk management : car-household
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
qqqqqqqqqqqqqqqqqqqqq
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Gumbel copula
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0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0
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Gumbel copula
q
q
Fig. 25 – L and R cumulative curves, and χ functions.
62

Case of Archimedean copulas
For an exhaustive study of tail behavior for Archimedean copulas, see
Charpentier & Segers (2008).
• upper tail : function of φ (1) and θ1 = − lim
s→0
sφ (1 − s)
φ(1 − s)
,
◦ φ (1) < 0 : tail independence
◦ φ (1) = 0 and θ1 = 1 : dependence in independence
◦ φ (1) = 0 and θ1 > 1 : tail dependence
• lower tail : function of φ(0) and θ0 = − lim
s→0
sφ (s)
φ(s)
,
◦ φ(0) < ∞ : tail independence
◦ φ(0) = ∞ and θ0 = 0 : dependence in independence
◦ φ(0) = ∞ and θ0 > 0 : tail dependence
0.0 0.2 0.4 0.6 0.8 1.0
05101520
63

Measuring risks ?
the pure premium as a technical benchmark
Pascal, Fermat, Condorcet, Huygens, d’Alembert in the XVIIIth century
proposed to evaluate the “produit scalaire des probabilités et des gains”,
< p, x >=
n
i=1
pixi =
n
i=1
P(X = xi) · xi = EP(X),
based on the “règle des parties”.
For Quételet, the expected value was, in the context of insurance, the price that
guarantees a financial equilibrium.
From this idea, we consider in insurance the pure premium as EP(X). As in
Cournot (1843), “l’espérance mathématique est donc le juste prix des chances”
(or the “fair price” mentioned in Feller (1953)).
Problem : Saint Peterburg’s paradox, i.e. infinite mean risks (cf. natural
catastrophes)
64

the pure premium as a technical benchmark
For a positive random variable X, recall that EP(X) =
∞
0
P(X > x)dx.
q
q
q
q
q
q
q
q
q
q
q
0 2 4 6 8 10
0.00.20.40.60.81.0
Expected value
Loss value, X
Probabilitylevel,P
Fig. 26 – Expected value EP(X) = xdFX(x) = P(X > x)dx.
65

from pure premium to expected utility principle
Ru(X) = u(x)dP = P(u(X) > x))dx
where u : [0, ∞) → [0, ∞) is a utility function.
Example with an exponential utility, u(x) = [1 − e−αx
]/α,
Ru(X) =
1
α
log EP(eαX
) ,
i.e. the entropic risk measure.
See Cramer (1728), Bernoulli (1738), von Neumann & Morgenstern
(1944), Rochet (1994)... etc.
66

Distortion of values versus distortion of probabilities
q
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q
q
q
q
q
0 2 4 6 8 10
0.00.20.40.60.81.0
Expected utility (power utility function)
Loss value, X
Probabilitylevel,P
Fig. 27 – Expected utility u(x)dFX(x).
67

q
q
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q
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q
q
q
q
0 2 4 6 8 10
0.00.20.40.60.81.0
Expected utility (power utility function)
Loss value, X
Probabilitylevel,P
Fig. 28 – Expected utility u(x)dFX(x).
68

from pure premium to distorted premiums (Wang)
Rg(X) = xdg ◦ P = g(P(X > x))dx
where g : [0, 1] → [0, 1] is a distorted function.
Example
• if g(x) = I(X ≥ 1 − α) Rg(X) = V aR(X, α),
• if g(x) = min{x/(1 − α), 1} Rg(X) = TV aR(X, α) (also called expected
shortfall), Rg(X) = EP(X|X > V aR(X, α)).
See D’Alembert (1754), Schmeidler (1986, 1989), Yaari (1987), Denneberg
(1994)... etc.
Remark : Rg(X) might be denoted Eg◦P. But it is not an expected value since
Q = g ◦ P is not a probability measure.
69

q
q
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q
q
q
q
q
q
q
q
0 2 4 6 8 10
0.00.20.40.60.81.0
Distorted premium beta distortion function)
Loss value, X
Probabilitylevel,P
Fig. 29 – Distorted probabilities g(P(X > x))dx.
70

q
q
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q
q
q
q
q
0 2 4 6 8 10
0.00.20.40.60.81.0
Distorted premium beta distortion function)
Loss value, X
Probabilitylevel,P
Fig. 30 – Distorted probabilities g(P(X > x))dx.
71

some particular cases a classical premiums
The exponential premium or entropy measure : obtained when the agent
as an exponential utility function, i.e.
π such that U(ω − π) = EP(U(ω − S)), U(x) = − exp(−αx),
i.e. π =
1
α
log EP(eαX
).
Esscher’s transform (see Esscher ( 1936), B¨uhlmann ( 1980)),
π = EQ(X) =
EP(X · eαX
)
EP(eαX)
,
for some α > 0, i.e.
dQ
dP
=
eαX
EP(eαX)
.
Wang’s premium (see Wang ( 2000)), extending the Sharp ratio concept
E(X) =
∞
0
F(x)dx and π =
∞
0
Φ(Φ−1
(F(x)) + λ)dx
72

Risk measures
The two most commonly used risk measures for a random variable X (assuming
that a loss is positive) are, q ∈ (0, 1),
• Value-at-Risk (VaR),
V aRq(X) = inf{x ∈ R, P(X > x) ≤ α},
• Expected Shortfall (ES), Tail Conditional Expectation (TCE) or Tail
Value-at-Risk (TVaR)
TV aRq(X) = E (X|X > V aRq(X)) ,
Artzner, Delbaen, Eber & Heath (1999) : a good risk measure is
subadditive,
TVaR is subadditive, VaR is not subadditive (in general).
73

Risk measures and diversiﬁcation
Any copula C is bounded by Frchet-Hoeﬀding 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 +
) denote
contercomonotonic and comonotonic versions of (X, Y ), do we have
R(φ(X−
, Y −
))
?
≤ R(φ(X,
Y )
)
?
≤ R(φ(X+
, Y +
)).
74

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 7. the stop loss premium for the sum of two risks E((X + Y − d)+) is
supermodular.
75

Example 8. For the n-year joint-life annuity,
axy:n =
n
k=1
vk
P(Tx > k and Ty > k) =
n
k=1
vk
kpxy.
Then
a−
xy:n ≤ axy:n ≤ a+
xy:n ,
where
a−
xy:n =
n
k=1
vk
max{kpx + kpy − 1, 0}( lower Frchet bound ),
a+
xy:n =
n
k=1
vk
min{kpx, kpy}( upper Frchet bound ).
76

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 −
)≤R(X + Y )≤R(X+
+ Y +
) ≤ R+
,
where e.g. R+
can exceed the comonotonic case. Recall that
R(X + Y ) = VaRq[X + Y ] = F−1
X+Y (q) = inf{x ∈ R|FX+Y (x) ≥ q}.
Proposition 9. 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
x,y∈R
{C(FX(x), FY (y)), x + y = s}
and, if ˜C(u, v) = u + v − C(u, v),
ρC(FX, FY )(s) = inf
x,y∈R
{ ˜C(FX(x), FY (y)), x + y = s}.
77

0.0 0.2 0.4 0.6 0.8 1.0
!4!2024
Bornes de la VaR d’un portefeuille
Somme de 2 risques Gaussiens
Fig. 31 – Value-at-Risk for 2 Gaussian risks N(0, 1).
78

0.90 0.92 0.94 0.96 0.98 1.00
0123456
Somme de 2 risques Gaussiens
Fig. 32 – Value-at-Risk for 2 Gaussian risks N(0, 1).
79

0.0 0.2 0.4 0.6 0.8 1.0
05101520
Somme de 2 risques Gamma
Fig. 33 – Value-at-Risk for 2 Gamma risks G(3, 1).
80

0.90 0.92 0.94 0.96 0.98 1.00
05101520
Somme de 2 risques Gamma
Fig. 34 – Value-at-Risk for 2 Gamma risks G(3, 1).
81

The more correlated, the more risky ?
Will the risk of the portfolio increase with correlation ?
Recall the following theoretical result :
Proposition 10. Assume that X and X are in the same Fréchet space (i.e.
Xi
L
= Xi), and define
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) =⇒ TV aR(S, p) ≤ TV aR(S , p), for all p ∈ (0, 1).
See Müller & Stoyen (2002) for some possible extensions.
82

Consider
• d lines of business,
• simply a binomial distribution on each line of business, with small loss
probability (e.g. π = 1/1000).
Let



1 if there is a claim on line i
0 if not
, and S = X1 + · · · + Xd.
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.
83

Fig. 35 – 99.75% TVaR (or expected shortfall) for Gaussian copulas.
84

Fig. 36 – 99% TVaR (or expected shortfall) for Gaussian copulas.
85

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)).
86

Fig. 37 – 99.75% VaR for Gaussian copulas.
87

Fig. 38 – 99% VaR for Gaussian copulas.
88

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.
89

Fig. 39 – 99.75% TVaR (or expected shortfall) for Student t-copulas.
90

Fig. 40 – 99% TVaR (or expected shortfall) for Student t-copulas.
91

Fig. 41 – 99.75% VaR for Student t-copulas.
92

Fig. 42 – 99% VaR for Student t-copulas.
93

A possible conclusion
• (standard) correlation is deﬁnitively not an appropriate tool to describe
dependence features,
◦ in order to fully describe dependence, use copulas,
◦ since major focus in risk management is related to extremal event, focus on
tail dependence meausres,
• which copula can be appropriate ?
◦ Elliptical copulas oﬀer a nice and simple parametrization, based on pairwise
comparison,
◦ Archimedean copulas might be too restrictive, but possible to introduce
Hierarchical Archimedean copulas,
• Value-at-Risk might yield to non-intuitive results,
◦ need to get a better understanding about Value-at-Risk pitfalls,
◦ need to consider alternative downside risk measures (namely TVaR).
94

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