Sensitivity of Risk Aggregation in Practice · 2017. 9. 29. · -In constructing C = tnest((z k)), use 2.) in order to control tail probabilities simultaneously for each F. To this

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

12. September 2017 Source: Secquaero Advisors 2

Solvency II: Risk aggregation

Source: CEIOPS, Solvency II Calibration Paper, 15. April 2010, §3.1325

Correlation factors

between risk factors

12. September 2017 3

Solvency II Calibration Paper:

Advise to consider tail dependencies

Source: CEIOPS, Solvency II Calibration Paper, 15. April 2010, § 3.1256


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



2. Motivation




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


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


• 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



2. Motivation




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 :


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


https://arxiv.org/pdf/0901.1521

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

12. September 2017

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

14Source: Secquaero Advisors

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)


𝑟1𝑞1

𝑠1

𝑣1

𝑢1

𝑝1


Bernoulli copulas



Refining a box my multiplication with another

Bernoulli-Copula

Copula properties are preserved


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)


Bernoulli copulas



Refining a box my multiplication with another

Bernoulli-Copula

Copula properties are preserved

Sucessive refinement possible


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


𝑆3

𝑆2

𝑆1 𝑆0





unfavourble

Nesting BC2 into each tail box


𝑆3

𝑆2

𝑆1





unfavourble


- Refine only unfavourbale

dimensions


𝑆3

𝑆2

𝑆1 𝑆0





unfavourble


- Refine only unfavourbale

dimensions

And so on by nesting BC3, BC4, … .. obtain a limit copula


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


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


1

C

( ) tp( )

F F

k

F F k F j

j

C

C Q z

( , ) F F Fa bF


2. Motivation




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)


{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


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


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


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

necessarily represent views expressed or reflected in other Schroders communications or strategies. The material is not

intended as an offer or solicitation for the purchase or sale of any financial instrument and should therefore not be relied on for

accounting, legal or tax advice, or investment recommendations. Reliance should not be placed on the views and information

in this document when taking individual investment and/or strategic decisions. Information herein is believed to be reliable but

Secquaero Advisors AG does not warrant its completeness or accuracy. Some information quoted was obtained from external

sources we consider to be reliable. No responsibility can be accepted for errors of fact obtained from third parties, and this

data may change with market conditions.

Issued by Secquaero Advisors AG, Central 2, CH-8001 Zurich.

Disclaimer


Appendix


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Sensitivity of Risk Aggregation in Practice · 2017. 9. 29. · -In constructing C = tnest((z k)), use 2.) in order to control tail probabilities simultaneously for each F. To this