This document discusses copulas and their use in modeling risk dependence. It introduces copulas as joint distribution functions with uniform margins that can be used to fully characterize dependence between random variables. Several classical copulas are described, including the independent, comonotonic, and countermonotonic copulas. Elliptical copulas like the Gaussian and Student t copulas are presented. Archimedean and extreme value copulas are also discussed. The document explores how copulas can capture dependence information that may not be reflected in correlation alone. Copulas provide flexible tools for modeling multivariate risks and dependencies.
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Using data from the S&P 500 stocks from 1990 to 2015, we address the uncertainty of distribution of assets’ returns in Conditional Value-at-Risk (CVaR) minimization model by applying multidimensional mixed Archimedean copula function and obtaining its robust counterpart. We implement a dynamic investing viable strategy where the portfolios are optimized using three different length of rolling calibration windows. The out-of-sample performance is evaluated and compared against two benchmarks: a multidimensional Gaussian copula model and a constant mix portfolio. Our empirical analysis shows that the Mixed Copula-CVaR approach generates portfolios with better downside risk statistics for any rebalancing period and it is more profitable than the Gaussian Copula-CVaR and the 1/N portfolios for daily and weekly rebalancing. To cope with the dimensionality problem we select a set of assets that are the most diversified, in some sense, to the S&P 500 index in the constituent set. The accuracy of the VaR forecasts is determined by how well they minimize a capital requirement loss function. In order to mitigate data-snooping problems, we apply a test for superior predictive ability to determine which model significantly minimizes this expected loss function. We find that the minimum average loss of the mixed Copula-CVaR approach is smaller than the average performance of the Gaussian Copula-CVaR.
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The Bayesian framework offers an intuitive approach towards uncertainty quantification. We present general Bernstein-von-Mises theorems for several survival objects, and discuss how these lead to relevant uncertainty quantification in practice.
The lattice Boltzmann equation: background, boundary conditions, and Burnett-...Tim Reis
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Talk at the modcov19 CNRS workshop, en France, to present our article COVID-19 pandemic control: balancing detection policy and lockdown intervention under ICU sustainability
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According to TechSci Research report, “India Orthopedic Devices Market -Industry Size, Share, Trends, Competition Forecast & Opportunities, 2030”, the India Orthopedic Devices Market stood at USD 1,280.54 Million in 2024 and is anticipated to grow with a CAGR of 7.84% in the forecast period, 2026-2030F. The India Orthopedic Devices Market is being driven by several factors. The most prominent ones include an increase in the elderly population, who are more prone to orthopedic conditions such as osteoporosis and arthritis. Moreover, the rise in sports injuries and road accidents are also contributing to the demand for orthopedic devices. Advances in technology and the introduction of innovative implants and prosthetics have further propelled the market growth. Additionally, government initiatives aimed at improving healthcare infrastructure and the increasing prevalence of lifestyle diseases have led to an upward trend in orthopedic surgeries, thereby fueling the market demand for these devices.
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1. 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
2. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
On risk dependence in QIS’s
http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
2
3. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
On risk dependence in QIS’s
http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
3
4. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
On risk dependence in QIS’s
http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
4
5. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
On risk dependence in QIS’s
http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
5
6. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
On risk dependence in QIS’s
http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
6
7. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
8. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
How to capture dependence in risk models ?
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
9. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
How to capture dependence in risk models ?
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
10. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
What about regulatory technical documents ?
10
11. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
What about regulatory technical documents ?
11
12. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
What about regulatory technical documents ?
12
13. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
What about regulatory technical documents ?
13
14. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
15. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
Copula density Level curves of the copula
Fig. 3 – Graphical representation of a copula, C(u, v) = P(U ≤ u, V ≤ v).
15
16. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
Copula density Level curves of the copula
Fig. 4 – Density of a copula, c(u, v) =
∂2
C(u, v)
∂u∂v
.
16
17. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
18. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
20. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
21. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
22. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
24. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
25. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
26. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
On copula parametrization
• 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
27. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
On copula parametrization
• 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
4 (φ4 φ−1
2 (φ2(u1) + φ2(u2)) + φ4 φ−1
3 (φ3(u3) + φ3(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
27
28. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
On copula parametrization
• 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
4 (φ4 φ−1
2 (φ2(u1) + φ2(u2)) + φ4 φ−1
3 (φ3(u3) + φ3(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
28
29. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
On copula parametrization
• 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
4 (φ4 φ−1
2 (φ2(u1) + φ2(u2)) + φ4 φ−1
3 (φ3(u3) + φ3(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
α4 α4 α3 1
29
30. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
On copula parametrization
• 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
4 (φ4[φ−1
3 (φ3 φ−1
2 (φ2(u1) + φ2(u2)) + φ3(u3))] + φ4(u4)),
which, if φi is parametrized with parameter αi, can be summarized through
A =
1 α2 α3 α4
α2 1 α3 α4
α3 α3 1 α4
α4 α4 α4 1
30
31. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
On copula parametrization
• 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
4 (φ4[φ−1
3 (φ3 φ−1
2 (φ2(u1) + φ2(u2)) + φ3(u3))] + φ4(u4)),
which, if φi is parametrized with parameter αi, can be summarized through
A =
1 α2 α3 α4
α2 1 α3 α4
α3 α3 1 α4
α4 α4 α4 1
31
32. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
On copula parametrization
• 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
4 (φ4[φ−1
3 (φ3 φ−1
2 (φ2(u1) + φ2(u2)) + φ3(u3))] + φ4(u4)),
which, if φi is parametrized with parameter αi, can be summarized through
A =
1 α2 α3 α4
α2 1 α3 α4
α3 α3 1 α4
α4 α4 α4 1
32
33. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
On copula parametrization
• Extreme value copulas
Here, dependence in [0, 1]d
←→ summarized in one functional parameters on
[0, 1]d−1
.
Further, focuses only on first order tail dependence.
33
34. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
35. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
36. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
37. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
!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
38. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
41. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
42. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
43. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
49. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
50. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
51. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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. 16 – Convergence of L and R functions, Gaussian copula, n = 2, 000.
51
52. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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. 17 – Convergence of L and R functions, Gaussian copula, n = 20, 000.
52
53. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
54. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
62. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
L and R concentration functions
L function (lower tails) R function (upper tails)
qqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qq
qqq
q
qq
qq
qqqqqqqqqqqqqqq
qqqqqqqq
qqqqqqqqqqqqqq
qqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqq
q
qqq
qqqq
qqq
qqq
qqqqq
qqqqq
qq
qqqqqqq
q
qqqq
qqq
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
Gumbel copula
q
q
0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0
Chi dependence functions
lower tails upper tails
qqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qq
qqq
q
qqqqqqqqqqqqqqqqqqq
qqqqqqqq
qqqqqqqqqqqqqq
qqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqq
qqqqqqq
qqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
Gumbel copula
q
q
Fig. 25 – L and R cumulative curves, and χ functions.
62
63. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
64. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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 probabilit´es et des gains”,
< p, x >=
n
i=1
pixi =
n
i=1
P(X = xi) · xi = EP(X),
based on the “r`egle des parties”.
For Qu´etelet, 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’esp´erance math´ematique 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
65. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
66. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
67. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
Distortion of values versus distortion of probabilities
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 utility (power utility function)
Loss value, X
Probabilitylevel,P
Fig. 27 – Expected utility u(x)dFX(x).
67
68. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
Distortion of values versus distortion of probabilities
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 utility (power utility function)
Loss value, X
Probabilitylevel,P
Fig. 28 – Expected utility u(x)dFX(x).
68
69. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
70. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
Distortion of values versus distortion of probabilities
q
q
q
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
71. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
Distortion of values versus distortion of probabilities
q
q
q
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. 30 – Distorted probabilities g(P(X > x))dx.
71
72. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
73. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
74. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
75. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
76. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
77. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
78. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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
79. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
0.90 0.92 0.94 0.96 0.98 1.00
0123456
Bornes de la VaR d’un portefeuille
Somme de 2 risques Gaussiens
Fig. 32 – Value-at-Risk for 2 Gaussian risks N(0, 1).
79
80. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
0.0 0.2 0.4 0.6 0.8 1.0
05101520
Bornes de la VaR d’un portefeuille
Somme de 2 risques Gamma
Fig. 33 – Value-at-Risk for 2 Gamma risks G(3, 1).
80
81. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
0.90 0.92 0.94 0.96 0.98 1.00
05101520
Bornes de la VaR d’un portefeuille
Somme de 2 risques Gamma
Fig. 34 – Value-at-Risk for 2 Gamma risks G(3, 1).
81
82. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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 Fr´echet 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 M¨uller & Stoyen (2002) for some possible extensions.
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83. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
The more correlated, the more risky ?
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.
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84. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
The more correlated, the more risky ?
Fig. 35 – 99.75% TVaR (or expected shortfall) for Gaussian copulas.
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85. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
The more correlated, the more risky ?
Fig. 36 – 99% TVaR (or expected shortfall) for Gaussian copulas.
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86. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
The more correlated, the more risky ?
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)).
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87. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
The more correlated, the more risky ?
Fig. 37 – 99.75% VaR for Gaussian copulas.
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88. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
The more correlated, the more risky ?
Fig. 38 – 99% VaR for Gaussian copulas.
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89. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
The more correlated, the more risky ?
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.
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90. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
The more correlated, the more risky ?
Fig. 39 – 99.75% TVaR (or expected shortfall) for Student t-copulas.
90
91. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
The more correlated, the more risky ?
Fig. 40 – 99% TVaR (or expected shortfall) for Student t-copulas.
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92. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
The more correlated, the more risky ?
Fig. 41 – 99.75% VaR for Student t-copulas.
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93. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
The more correlated, the more risky ?
Fig. 42 – 99% VaR for Student t-copulas.
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94. Arthur CHARPENTIER - Copulas, risk dependence and applications to Solvency II
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).
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