Sensitivity of Risk Aggregation in
Practice
Christoph Hummel
Workshop at Centre De Recherches Mathématiques on
Risk Measurement and Regulatory Issues in Business
12 September 2017
1. Risk aggregation in practise & Solvency II
2. Motivation
3. Models for multiple tail dependencies
4. Applications in practise
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Solvency II: Risk aggregation
Source: CEIOPS, Solvency II Calibration Paper, 15. April 2010, §3.1325
Correlation factors
between risk factors
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Solvency II Calibration Paper:
Advise to consider tail dependencies
Source: CEIOPS, Solvency II Calibration Paper, 15. April 2010, § 3.1256
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12. September 2017 5Source: Secquaero Advisors
Copula models
𝐹𝑖 CDF for risk factor 𝑋𝑖
Target variable is a function 𝑉(𝑋1, … , 𝑉𝑛)
Uniform random variable 𝑈𝑖 with 𝑋𝑖 = 𝐹𝑖−1(𝑈𝑖) provides „ranking“
Copula = joint distribution of the 𝑈𝑖
Illustration of a 3-dim Copula by means of joint samples of 𝑈1, 𝑈2, 𝑈3 :
U1 vs U2U1 vs U3 U2 vs U3 U1,U2,U3
Challenges in practise1. Validation of the assumptions
2. Communication of assumptions and justification of copula
3. Need for an efficient Monte Carlo algorithm
Item 3 makes above popular, but impose challenges to 1. and 2.
Modelling CDFs of 𝑋1, … , 𝑋𝑛 de-centrally
Determine multivariate distribution by
choosing a Copula
Established copulas
- Clayton, Gumbel, Gauß,...
- Determined by Rank-correlations or tail
dependencies of the pairs (𝑋𝑖 , 𝑋𝑗)
Monte Carlo simulation
- Draw samples from the copula
- Apply inverse of corresponding CDF
- Obtain samples of 𝑋1, … , 𝑋𝑛
Use of copula in practise: Internal models
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1. Risk aggregation in practise & Solvency II
2. Motivation
3. Models for multiple tail dependencies
4. Applications in practise
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Multiple dependencies
Illustration in dimension 3
Pairwise dependencies determine the probabilities 𝑎, 𝑏, 𝑐 of the
projections of the 3-dim copula to the 2-dim faces
Probability 𝑡 of the corresponding box of the cube is a “free
parameter”
a t
c t
b t
t
c
a
b
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If this is the box where all
three risk factors have
unfavourable realisations,
then 𝑡 may be the “essential
parameter”
Choosing an established
copula model, 𝑡 is
determined implicitly from
𝑎, 𝑏, 𝑐.
Sierpinski tetrahedron:
Pairwise dependence measures reveal little
Pairwise independent but 3-fold dependency
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• Write 𝑈𝑖 = 𝑛=1∞ 𝐵𝑖𝑛 ⋅ 2
−𝑛 , i = 1,2,3 with 𝐵𝑖𝑛 ∈ 0,1 .• Then 𝑖≠𝑘𝐵𝑖𝑛 = 𝐵𝑘 mod 2.• In particular 𝑈1 ⊕𝑈2 = 𝑈3 where ⊕ means adding the digits mod 2.
A. Joffe published in 1971 the following general result under the affiliation to
McGill University and Université de Montréal :
Source: A. Joffe, On a sequence of almost deterministic pairwise independent
random variables, Proceedings of the AMS 29 (2), 1971, pp. 381 -382.
The proof is not constructive.
See also A. Joffe, On a set of almost deterministic k-independent random
variables, Annals of Probability 2 (1), pp. 161 – 162 (1974)
A reference
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1. Risk aggregation in practise & Solvency II
2. Motivation
3. Models for multiple tail dependencies
4. Applications in practise
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Multiple tail dependenciesPairwise tail dependence
Tail dependence
1
Subsets 1, , correspond to front faces of the cube [0,1]
projection to and copula for X , ,X , then := copula for , .
n
F n F F i
F n
F C C C X i F
Notation :
0
0
, has two elements
liminf ( | )
( , )liminf , and hence
F i js
F
s
F i j
P U s U s
C s s
s
0
0
1, , having 2 elements
( , , )inf liminf = ,
( , , )liminf and hence
F
FF
s
FF bs
F n
C s sb
s
C s sa
s
( , ) , as 0F FC s s s s ( , , ) , as 0FbF FC s s a s s
2tail characteristic of : ( , )F F F
C a b
| | 2pairwise tail dependencies ( )F F
The ( ) have
positive tail dependence if .
i i F
F
X
b F
Definition :
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Reference: E.g. A. McNeil et al, Quantitative
Methods in Risk Management, Princeton University
Press, 2005
~> ~>
A. Charpentier and J. Segers (2009), https://arxiv.org/pdf/0901.1521
have investigated the asymptotic behaviour of Archimedean copulas in
a general setting
Examples
- Gumbel Copula with parameter 𝜃 > 1: tchar = 1, 𝐹 1 𝜃
- Clayon Copula with parameter 𝜃 > 0: tchar = (|𝐹| −1 𝜃 , 1)
- Sierpinski Tetrahedron: tchar = (𝑎|𝐹|, 2) for 𝐹 = 2,3 with 𝑎2 = 1, 𝑎3 ≈ 0.706.
Comments
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Bernoulli-Copulas
Decompose each edge of 𝑛-dim
cube 0,1 𝑛in two parts- „favourable“ & „unfavourable“
percentiles
Obtain decomposition of unit cube
into 2𝑛 boxes
assign to each box a uniform
measure
- control copula condition “uniform
margins” and total measure = 1
- 𝑛 + 1 linear conditions
- Probabilities are non-negative
1-1 correspondence with
multivariate Bernoulli-distributions
Examples 𝑛 = 2 and 3
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uniform
1
12
7
12
1 6 1 6
2
3
1
3
1 4 3 4
7 36 5 36
1 18 1 91
4
1 6 0⋰
⋯
2
3
1
3
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Shaping the tail in dimension 2 by nesting
Bernoulli copulas
2-dim Bernoulli copula with probabilities
𝑝1, 𝑞1, 𝑟1, 𝑠1 and decomposition of edges by (𝑢1, 𝑣1)
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𝑟1𝑞1
𝑠1
𝑣1
𝑢1
𝑝1
Shaping the tail in dimension 2 by nesting
Bernoulli copulas
2-dim Bernoulli copula with probabilities
𝑝1, 𝑞1, 𝑟1, 𝑠1 and decomposition of edges by (𝑢1, 𝑣1)
Refining a box my multiplication with another
Bernoulli-Copula
Copula properties are preserved
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𝑟1𝑞1
𝑠1
𝑣1
𝑢1
𝑞2𝑞2𝑝1 𝑟2𝑝1𝑟2
𝑠2𝑠2𝑝1𝑝2𝑝2𝑝1
𝑝1
Such techniques have been used by G. Fredricks et al., Copulas with fractal support, Insurance: Mathematics and
Economics 37 (2005)
Shaping the tail in dimension 2 by nesting
Bernoulli copulas
2-dim Bernoulli copula with probabilities
𝑝1, 𝑞1, 𝑟1, 𝑠1 and decomposition of edges by (𝑢1, 𝑣1)
Refining a box my multiplication with another
Bernoulli-Copula
Copula properties are preserved
Sucessive refinement possible
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Controlling tail asymptotic by
appropriate choice of sequence
𝑢𝑘 , 𝑣𝑘 , 𝑝𝑘 𝑘=1,2,3,…
𝑟1𝑞1
𝑠1
𝑣1
𝑢1
𝑞2𝑞2𝑝1 𝑟2𝑝1𝑟2
𝑠2𝑠2𝑝1𝑝2𝑝2𝑝1
𝑝3𝑝2𝑝1 ⋯
𝑝1
⋯⋯
Tail nesting in higher dimensions
Bernoulli copula BC1 Tail boxes 𝑆0, 𝑆1, 𝑆2, 𝑆3
- 𝑆0: all risks unfavourable
- 𝑆1, 𝑆2, 𝑆3: two out of three
unfavourble
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𝑆3
𝑆2
𝑆1 𝑆0
Tail nesting in higher dimensions
Bernoulli copula BC1 Tail boxes 𝑆0, 𝑆1, 𝑆2, 𝑆3
- 𝑆0: all risks unfavourable
- 𝑆1, 𝑆2, 𝑆3: two out of three
unfavourble
Nesting BC2 into each tail box
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𝑆3
𝑆2
𝑆1
Tail nesting in higher dimensions
Bernoulli copula BC1 Tail boxes 𝑆0, 𝑆1, 𝑆2, 𝑆3
- 𝑆0: all risks unfavourable
- 𝑆1, 𝑆2, 𝑆3: two out of three
unfavourble
Nesting BC2 into each tail box
- Refine only unfavourbale
dimensions
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𝑆3
𝑆2
𝑆1 𝑆0
Tail nesting in higher dimensions
Bernoulli copula BC1 Tail boxes 𝑆0, 𝑆1, 𝑆2, 𝑆3
- 𝑆0: all risks unfavourable
- 𝑆1, 𝑆2, 𝑆3: two out of three
unfavourble
Nesting BC2 into each tail box
- Refine only unfavourbale
dimensions
And so on by nesting BC3, BC4, … .. obtain a limit copula
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Monte-Carlo Algorithms by recursion
Construction is intuitive
- Successive refinement: unfavourable, very unfavourble given unfavourable, ...
𝑆3
𝑆2
𝑆1 𝑆0
Copulas with multiple tail dependencies
Monotonicity of probabilities:
If bF = bF’ holds somewhere, monotonicity conditions apply to the aF
'' F FF F b b
2
Let 0, 1 for front faces of the cube [0,1] with F 2.
Assume that is strictly increasing in . Then we can construct a copula C
with tail characteristic ( , ) and provide a Monte Carlo a
nF F
F
F F F
a b F
b F
a b
lgorithm
for sampling from
. C
Proposition
Tail characteristic
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Remark: For a more general version if bF is not strictly increasing see arxiv.org/abs/0906.4853
Proof of Proposition
Idea: For C with given tail characteristic
go along the diagonal by
choosing Qk=[0,sk]n
General construction
- Choose nested sequence (Qk) of tail
boxes in [0,1]n collapsing to the origin
- In constructing C = tnest((zk)), use 2.) in
order to control tail probabilities
simultaneously for each F. To this end,
choose zk using 1.)
Shaping asymptotic behaviour of tail nested
copulas
the set of front faces of [0,1]
tp( ) the probability of the tail box of ,
that is the box containing the origin
n
z z
F
There is a one-to-one correspondence
between Bernoulli copulas and
sequences tp in [0,1]
satisfying tp 1 and the
monotonicity condition for probabilities.
F F
z
z
z
F
1,2, 1,2,
Projections to faces and tnest commute:
tnest (z ) tnest ( z )F k k F k k
F
Notation
2.) Projections
1.) Specifying Bernoulli copulas
o
1FQ
F kQ
2FQ
tail of
( )F C
( ) ( , ) as F F k i kC Q P U Q i F k
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1
C
( ) tp( )
F F
k
F F k F j
j
C
C Q z
( , ) F F Fa bF
1. Risk aggregation in practise & Solvency II
2. Motivation
3. Models for multiple tail dependencies
4. Applications in practise
12. September 2017 Source: Secquaero Advisors 24
TNCs provide insight into various (tail-)dependence structures,
have “easy & efficient” Monte Carlo algorithms,
Parameters are intuitive and hence TNC are suitable for calibration by
expert opinion and sensitivity / stress testing
Remarks on tail nested copulas (TNC)
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{1,2} {1,3} {2,3} {1,2,3}
Example
There is a smooth
transition
Positive 3-fold
tail-dependency
Negative 3-fold
tail-dependency
All 2-dimensional boundary
distributions are identical.
Positive 2-fold tail-dependency
Vary 3-fold dependencies
while fixing the 2-folds
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Case B: positive 2-fold tail dep.
Case study: Impact of multiple dependency
Set up
Case A: pairwise independent
𝑋1, … , 𝑋𝑛 identically distributed, exchangeable
−lognormal distributed with coefficient of variation 7%
fix all 2-dim boundary distributions, vary higher order dependencies
Study tail risk measure 𝑅 = TVaR1% for 𝑉 = 𝑋1 +⋯+ 𝑋𝑛
corr(Xi,Xj)=25%
P(Ui<1/256 | Uj<1/256)=10.3%
corr(Xi,Xj)=0%
P(Ui<1/256 | Uj<1/256)=1/256
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Case study: Impact of multiple dependency
Results
Multiple dependencies are crucial for estimating the solvency capital.
2 3 4 5 6 7 8 90
0.2
0.4
0.6
0.8
1
Anzahl Risiken
RB
C n
orm
iert
2 3 4 5 6 7 8 90
0.2
0.4
0.6
0.8
1
Anzahl Risiken
RB
C n
orm
iert
2 3 4 5 6 7 8 90
0.2
0.4
0.6
0.8
1
Anzahl Risiken
RB
C n
orm
iert
Fall A
Fall B
Überschneidung
2-fold tail-dependencies &
high multiple dependencies
2-fold tail-dependencies &
low multiple dependencies
All risks independent
Pairwise independent &
high multiple dependencies
Normalised risk =diversified risk
undiversified risk=
𝑅(𝑋1+⋯+𝑋𝑛)
𝑅 𝑋1 +⋯+𝑅 𝑋𝑛as a function of 𝑛
Norm
alis
ed R
isk
Number of risk factors
Case A
Case B
overlap
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This presentation is intended to be for information purposes only and for the sole and the exclusive use of the recipient. The
views and opinions contained herein are those of the presenter, which may change without notice and which may not
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Issued by Secquaero Advisors AG, Central 2, CH-8001 Zurich.
Disclaimer
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Appendix
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