The document discusses various topics related to modeling risk dependence and correlations, including:
1) The importance of tail distributions, tail correlations, and low-frequency high-severity events in risk management.
2) Different methods for capturing dependence in risk models, including correlation and copulas.
3) Examples of classical copulas like the independent, comonotonic, and countermonotonic copulas.
4) Specific copula families like elliptical, Archimedean, and extreme value copulas.
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how can I sell my pi coins for cash in a pi APPDOT TECH
You can't sell your pi coins in the pi network app. because it is not listed yet on any exchange.
The only way you can sell is by trading your pi coins with an investor (a person looking forward to hold massive amounts of pi coins before mainnet launch) .
You don't need to meet the investor directly all the trades are done with a pi vendor/merchant (a person that buys the pi coins from miners and resell it to investors)
I Will leave The telegram contact of my personal pi vendor, if you are finding a legitimate one.
@Pi_vendor_247
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#pi coins
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What website can I sell pi coins securely.DOT TECH
Currently there are no website or exchange that allow buying or selling of pi coins..
But you can still easily sell pi coins, by reselling it to exchanges/crypto whales interested in holding thousands of pi coins before the mainnet launch.
Who is a pi merchant?
A pi merchant is someone who buys pi coins from miners and resell to these crypto whales and holders of pi..
This is because pi network is not doing any pre-sale. The only way exchanges can get pi is by buying from miners and pi merchants stands in between the miners and the exchanges.
How can I sell my pi coins?
Selling pi coins is really easy, but first you need to migrate to mainnet wallet before you can do that. I will leave the telegram contact of my personal pi merchant to trade with.
Tele-gram.
@Pi_vendor_247
when will pi network coin be available on crypto exchange.DOT TECH
There is no set date for when Pi coins will enter the market.
However, the developers are working hard to get them released as soon as possible.
Once they are available, users will be able to exchange other cryptocurrencies for Pi coins on designated exchanges.
But for now the only way to sell your pi coins is through verified pi vendor.
Here is the telegram contact of my personal pi vendor
@Pi_vendor_247
how to sell pi coins effectively (from 50 - 100k pi)DOT TECH
Anywhere in the world, including Africa, America, and Europe, you can sell Pi Network Coins online and receive cash through online payment options.
Pi has not yet been launched on any exchange because we are currently using the confined Mainnet. The planned launch date for Pi is June 28, 2026.
Reselling to investors who want to hold until the mainnet launch in 2026 is currently the sole way to sell.
Consequently, right now. All you need to do is select the right pi network provider.
Who is a pi merchant?
An individual who buys coins from miners on the pi network and resells them to investors hoping to hang onto them until the mainnet is launched is known as a pi merchant.
debuts.
I'll provide you the Telegram username
@Pi_vendor_247
If you are looking for a pi coin investor. Then look no further because I have the right one he is a pi vendor (he buy and resell to whales in China). I met him on a crypto conference and ever since I and my friends have sold more than 10k pi coins to him And he bought all and still want more. I will drop his telegram handle below just send him a message.
@Pi_vendor_247
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Currently pi network is not tradable on binance or any other exchange because we are still in the enclosed mainnet.
Right now the only way to sell pi coins is by trading with a verified merchant.
What is a pi merchant?
A pi merchant is someone verified by pi network team and allowed to barter pi coins for goods and services.
Since pi network is not doing any pre-sale The only way exchanges like binance/huobi or crypto whales can get pi is by buying from miners. And a merchant stands in between the exchanges and the miners.
I will leave the telegram contact of my personal pi merchant. I and my friends has traded more than 6000pi coins successfully
Tele-gram
@Pi_vendor_247
What price will pi network be listed on exchangesDOT TECH
The rate at which pi will be listed is practically unknown. But due to speculations surrounding it the predicted rate is tends to be from 30$ — 50$.
So if you are interested in selling your pi network coins at a high rate tho. Or you can't wait till the mainnet launch in 2026. You can easily trade your pi coins with a merchant.
A merchant is someone who buys pi coins from miners and resell them to Investors looking forward to hold massive quantities till mainnet launch.
I will leave the telegram contact of my personal pi vendor to trade with.
@Pi_vendor_247
how to sell pi coins on Bitmart crypto exchangeDOT TECH
Yes. Pi network coins can be exchanged but not on bitmart exchange. Because pi network is still in the enclosed mainnet. The only way pioneers are able to trade pi coins is by reselling the pi coins to pi verified merchants.
A verified merchant is someone who buys pi network coins and resell it to exchanges looking forward to hold till mainnet launch.
I will leave the telegram contact of my personal pi merchant to trade with.
@Pi_vendor_247
The European Unemployment Puzzle: implications from population agingGRAPE
We study the link between the evolving age structure of the working population and unemployment. We build a large new Keynesian OLG model with a realistic age structure, labor market frictions, sticky prices, and aggregate shocks. Once calibrated to the European economy, we quantify the extent to which demographic changes over the last three decades have contributed to the decline of the unemployment rate. Our findings yield important implications for the future evolution of unemployment given the anticipated further aging of the working population in Europe. We also quantify the implications for optimal monetary policy: lowering inflation volatility becomes less costly in terms of GDP and unemployment volatility, which hints that optimal monetary policy may be more hawkish in an aging society. Finally, our results also propose a partial reversal of the European-US unemployment puzzle due to the fact that the share of young workers is expected to remain robust in the US.
how can i use my minded pi coins I need some funds.DOT TECH
If you are interested in selling your pi coins, i have a verified pi merchant, who buys pi coins and resell them to exchanges looking forward to hold till mainnet launch.
Because the core team has announced that pi network will not be doing any pre-sale. The only way exchanges like huobi, bitmart and hotbit can get pi is by buying from miners.
Now a merchant stands in between these exchanges and the miners. As a link to make transactions smooth. Because right now in the enclosed mainnet you can't sell pi coins your self. You need the help of a merchant,
i will leave the telegram contact of my personal pi merchant below. 👇 I and my friends has traded more than 3000pi coins with him successfully.
@Pi_vendor_247
how can i use my minded pi coins I need some funds.
Slides erm-cea-ia
1. Arthur CHARPENTIER - Extremes and correlation in risk management
Explain and demonstrate the importance
of the tails of the distributions,
tail correlations and
low frequency/high severity events
Arthur Charpentier
Universit´e de Rennes 1 & ´Ecole Polytechnique
http ://blogperso.univ-rennes1.fr/arthur.charpentier/
1
2. Arthur CHARPENTIER - Extremes and correlation in risk management
SCR and Solvency
2
3. Arthur CHARPENTIER - Extremes and correlation in risk management
SCR and Solvency
3
4. Arthur CHARPENTIER - Extremes and correlation in risk management
SCR and Solvency
4
5. Arthur CHARPENTIER - Extremes and correlation in risk management
SCR and Solvency
5
6. Arthur CHARPENTIER - Extremes and correlation in risk management
On risk dependence in QIS’s
http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
6
7. Arthur CHARPENTIER - Extremes and correlation in risk management
On risk dependence in QIS’s
http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
7
8. Arthur CHARPENTIER - Extremes and correlation in risk management
On risk dependence in QIS’s
http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
8
9. Arthur CHARPENTIER - Extremes and correlation in risk management
On risk dependence in QIS’s
http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
9
10. Arthur CHARPENTIER - Extremes and correlation in risk management
On risk dependence in QIS’s
http ://www.ceiops.eu/media/files/consultations/QIS/QIS3/QIS3TechnicalSpecificationsPart1.PDF
10
11. Arthur CHARPENTIER - Extremes and correlation in risk management
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 ) ?
11
12. Arthur CHARPENTIER - Extremes and correlation in risk management
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
).
12
13. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
13
14. Arthur CHARPENTIER - Extremes and correlation in risk management
What about official actuarial documents ?
14
15. Arthur CHARPENTIER - Extremes and correlation in risk management
What about official actuarial documents ?
15
16. Arthur CHARPENTIER - Extremes and correlation in risk management
What about official actuarial documents ?
16
17. Arthur CHARPENTIER - Extremes and correlation in risk management
What about regulatory technical documents ?
17
18. Arthur CHARPENTIER - Extremes and correlation in risk management
What about regulatory technical documents ?
18
19. Arthur CHARPENTIER - Extremes and correlation in risk management
What about regulatory technical documents ?
19
20. Arthur CHARPENTIER - Extremes and correlation in risk management
What about regulatory technical documents ?
20
21. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
21
22. Arthur CHARPENTIER - Extremes and correlation in risk management
Copula density Level curves of the copula
Fig. 3 – Graphical representation of a copula, C(u, v) = P(U ≤ u, V ≤ v).
22
23. Arthur CHARPENTIER - Extremes and correlation in risk management
Copula density Level curves of the copula
Fig. 4 – Density of a copula, c(u, v) =
∂2
C(u, v)
∂u∂v
.
23
24. Arthur CHARPENTIER - Extremes and correlation in risk management
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]).
24
25. Arthur CHARPENTIER - Extremes and correlation in risk management
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)
25
27. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
27
28. Arthur CHARPENTIER - Extremes and correlation in risk management
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
).
28
31. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
31
32. Arthur CHARPENTIER - Extremes and correlation in risk management
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,
32
33. Arthur CHARPENTIER - Extremes and correlation in risk management
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
33
34. Arthur CHARPENTIER - Extremes and correlation in risk management
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
34
35. Arthur CHARPENTIER - Extremes and correlation in risk management
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
35
36. Arthur CHARPENTIER - Extremes and correlation in risk management
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
36
37. Arthur CHARPENTIER - Extremes and correlation in risk management
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
37
38. Arthur CHARPENTIER - Extremes and correlation in risk management
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
38
39. Arthur CHARPENTIER - Extremes and correlation in risk management
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
39
40. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
40
41. Arthur CHARPENTIER - Extremes and correlation in risk management
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 → ∞.
41
42. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
42
43. Arthur CHARPENTIER - Extremes and correlation in risk management
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).
43
44. Arthur CHARPENTIER - Extremes and correlation in risk management
!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.
44
45. Arthur CHARPENTIER - Extremes and correlation in risk management
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) .
45
48. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
48
49. Arthur CHARPENTIER - Extremes and correlation in risk management
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).
49
50. Arthur CHARPENTIER - Extremes and correlation in risk management
Tail correlation and Solvency II
50
51. Arthur CHARPENTIER - Extremes and correlation in risk management
Tail correlation and Solvency II
51
52. Arthur CHARPENTIER - Extremes and correlation in risk management
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
.
52
58. Arthur CHARPENTIER - Extremes and correlation in risk management
Estimation of tail dependence
58
59. Arthur CHARPENTIER - Extremes and correlation in risk management
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).
59
60. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
60
61. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
61
62. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
62
63. Arthur CHARPENTIER - Extremes and correlation in risk management
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))
.
63
64. Arthur CHARPENTIER - Extremes and correlation in risk management
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)
.
64
72. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
72
73. Arthur CHARPENTIER - Extremes and correlation in risk management
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
73
74. Arthur CHARPENTIER - Extremes and correlation in risk management
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)
74
75. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
75
76. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
76
77. Arthur CHARPENTIER - Extremes and correlation in risk management
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).
77
78. Arthur CHARPENTIER - Extremes and correlation in risk management
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).
78
79. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
79
80. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
80
81. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
81
82. Arthur CHARPENTIER - Extremes and correlation in risk management
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
82
83. Arthur CHARPENTIER - Extremes and correlation in risk management
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).
83
84. Arthur CHARPENTIER - Extremes and correlation in risk management
Risk measures : a pratitionner (mis)understanding
84
85. Arthur CHARPENTIER - Extremes and correlation in risk management
Risk measures : a pratitionner (mis)understanding
85
86. Arthur CHARPENTIER - Extremes and correlation in risk management
Risk measures : a pratitionner (mis)understanding
86
87. Arthur CHARPENTIER - Extremes and correlation in risk management
Risk measures : a pratitionner (mis)understanding
87
88. Arthur CHARPENTIER - Extremes and correlation in risk management
Risk measures : a pratitionner (mis)understanding
88
89. Arthur CHARPENTIER - Extremes and correlation in risk management
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 +
)).
89
90. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
90
91. Arthur CHARPENTIER - Extremes and correlation in risk management
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 ).
91
92. Arthur CHARPENTIER - Extremes and correlation in risk management
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}.
92
93. Arthur CHARPENTIER - Extremes and correlation in risk management
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).
93
94. Arthur CHARPENTIER - Extremes and correlation in risk management
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).
94
95. Arthur CHARPENTIER - Extremes and correlation in risk management
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).
95
96. Arthur CHARPENTIER - Extremes and correlation in risk management
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).
96
97. Arthur CHARPENTIER - Extremes and correlation in risk management
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.
97
98. Arthur CHARPENTIER - Extremes and correlation in risk management
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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99. Arthur CHARPENTIER - Extremes and correlation in risk management
The more correlated, the more risky ?
Fig. 35 – 99.75% TVaR (or expected shortfall) for Gaussian copulas.
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100. Arthur CHARPENTIER - Extremes and correlation in risk management
The more correlated, the more risky ?
Fig. 36 – 99% TVaR (or expected shortfall) for Gaussian copulas.
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101. Arthur CHARPENTIER - Extremes and correlation in risk management
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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102. Arthur CHARPENTIER - Extremes and correlation in risk management
The more correlated, the more risky ?
Fig. 37 – 99.75% VaR for Gaussian copulas.
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103. Arthur CHARPENTIER - Extremes and correlation in risk management
The more correlated, the more risky ?
Fig. 38 – 99% VaR for Gaussian copulas.
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104. Arthur CHARPENTIER - Extremes and correlation in risk management
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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105. Arthur CHARPENTIER - Extremes and correlation in risk management
The more correlated, the more risky ?
Fig. 39 – 99.75% TVaR (or expected shortfall) for Student t-copulas.
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106. Arthur CHARPENTIER - Extremes and correlation in risk management
The more correlated, the more risky ?
Fig. 40 – 99% TVaR (or expected shortfall) for Student t-copulas.
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107. Arthur CHARPENTIER - Extremes and correlation in risk management
The more correlated, the more risky ?
Fig. 41 – 99.75% VaR for Student t-copulas.
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108. Arthur CHARPENTIER - Extremes and correlation in risk management
The more correlated, the more risky ?
Fig. 42 – 99% VaR for Student t-copulas.
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109. Arthur CHARPENTIER - Extremes and correlation in risk management
The more correlated, the more risky ?
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110. Arthur CHARPENTIER - Extremes and correlation in risk management
On the CEIPS recommendations
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111. Arthur CHARPENTIER - Extremes and correlation in risk management
On the CEIPS recommendations
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112. Arthur CHARPENTIER - Extremes and correlation in risk management
On the CEIPS recommendations
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113. Arthur CHARPENTIER - Extremes and correlation in risk management
On the CEIPS recommendations
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114. Arthur CHARPENTIER - Extremes and correlation in risk management
On the CEIPS recommendations
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115. Arthur CHARPENTIER - Extremes and correlation in risk management
On the CEIPS recommendations
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116. Arthur CHARPENTIER - Extremes and correlation in risk management
A first conclusion
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117. Arthur CHARPENTIER - Extremes and correlation in risk management
Another 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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