Sponsor:
CAE Fall Meeting 2013
Dependencies in a Risky World
Pedro Fonseca
Head Risk Analytics & Reporting at SIX
München, 27 September 2013
Agenda
• Acknowledgements
• The Risky World of SIX
• Why Model Dependencies?
• Modeling Dependencies
• Compound Poisson Processes and Lévy-Copulas
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Acknowledgements
I have benefited from discussion with:
– Frank Cuypers (Prime Re Services)
– Hansjörg Albrecher (University of Lausanne)
– Andreas Troxler (Solen Versicherungen)
– Marc Sarbach (Deloitte)
– Fabian Qazimi (SIX)
– Carlos Arocha (Arocha & Associates)
– Christoph Hummel (Secquaero)
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The Risky World of SIX
CHF 4 billion
daily turnover
Per second: 120
card transactions;
CHF 5 million
interbank
payments
Infrastructure for the
Swiss financial center
• User-owned
Trading & Indices
• AA- Rating
Cash Transactions
• > 3’000 people
• 25 countries
1’500 corporate
actions daily
Handling of Securities
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Data on 7 million
financial
instruments
• Predominantly
Operational and
Counterparty Risks
• Many Dependencies
Financial Data
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“The Swiss Fort-Knox”
Olten
• CHF 2,5 trillion assets
• 800 tons of gold & silver = CHF 10b
Seen from the outside
Seen from the inside
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The Risky World of SIX
Systemic Risk
“Too big to
fail”
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“Too important
to fail”
“Too big to
fail”
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The Crash of 2:45
(aka Flash Crash)
Systemic Risk
Probability of
INformed Trading*
O’Hara et al, 1996
Incentive to develop internal risk models
* Probability that informed traders adversely select uninformed traders.
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Questions
•
How to best model dependencies?
•
Causality behind copulas?
•
Causality behind Lévy-copulas?
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Why Model Dependencies?
• Dependencies create riskier worlds
• In particular, positive dependencies create:
– Fatter-tails
– Less diversification
– Higher frequency of “rare” events
– Increased Value-at-Risk
– Increased Tail-Value-at-Risk
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Dependencies Create Riskier Worlds
Example – Mergers/Consolidation
Company
1
Company
2
Company
3
Company
4
Company
5
Merged Companies:
New fat-tail risks due to
new strong dependencies
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Why Model Dependencies?
• Regulation:
o Swiss Solvency Test and Solvency II (for Re- / Insurers)
Dependencies
modeled at the
level of risk drivers
Dependencies accounted by standard
correlations between risk types (e.g.
market, counterparty, life, health, nonlife) :
Solvency Capital Requirement =
𝑖,𝑗 𝜌𝑖𝑗 𝑆𝐶𝑅𝑖 𝑆𝐶𝑅𝑗
• Based on linear dependence
• Might not reflect the specific situation
of the insurer
Incentive to develop risk internal models
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Why Model Dependencies?
• Regulation:
o Swiss Solvency Test and Solvency II (for Re- / Insurers)
o Basel II/III with FINMA “Swiss finish” (for Banks)
• Reserving / Risk Adjusted Capital
• Pricing
• Capital Allocation
To Improve Strategy
(Profitability, Survival, …)
• Business Planning
• Portfolio and Risk Management
• For the fun of modeling
• …
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Bottom Line
• Modeling dependencies is important.
• “Theorem”: To model independence there is only one choice. To
model dependence there are infinitely many choices.
• “Lemma”: Given three experts on modeling correlations, there is only
one thing that two of them agree on… And that is, that the third one is
doing something wrong.
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Modeling Dependencies
Many Complex Levels
Reality is made up of:
Particles
Atoms
Molecules
Proteins
Cells
Organs
Organisms
...
Emotions
Companies
Financial Markets
Societies
Models are made up of:
Typically used in
Processes & Controls
“explicit” models
Exposures
Events
Severities
Fitted Distributions
Networks/Hierarchies
Stochastic Processes
Typically used in
Regressions, GLM
“implicit” models
Copulas
Survival Copulas
Box-Copulas
Lévy-Copulas
Pareto-Copulas
Pareto-Lévy-Copulas
Semi-Martingale-Copulas
C. Hummel (2009)
arXiv:0906.4853
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Modeling Dependencies
Overview
Models
Explicit Models
also known as:
Causal Models,
Common-Factors Models
(or Common-Shocks)
Implicit Models
also known as:
Copula Models,
Covariance Models,
“No Common-Factors”
There is surely a copula
(Sklar’s theorem)
Causal interpretation?
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Modeling Dependencies
Very Schematic Description…
Common-Factors
Like making your own pants
Making sense out of the pieces
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“No Common-Factors”
Like buying pants
Making it fit
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Modeling Dependencies
Schematic Description
Common-Factors
Stochastic
Variable A
“No Common-Factors”
A
Stochastic
Variable A
A’
Stochastic
Common
Factor
Correlated
a priori
Stochastic
Variable B
B
Correlating
Algorithm
Correlated
a posteriori
B’
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Stochastic
Variable B
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Common-Factors
Example #1
Company
Property in
Germany
Property in
France
Windstorm
in France
Fire in
France
Windstorm
in Germany
Fire in
Germany
Common Windstorm
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Common-Factors
Example #2
Company
Property in
Germany
Property in
France
Windstorm
in France
Fire in
France
Windstorm
in Germany
Fire in
Germany
Common Clients
Common Event
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“No Common-Factors”
Example
Company
Copula 3
Property in
France
Property in
Germany
Copula 1
Copula 2
Windstorm
in France
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Fire in
France
Windstorm
in Germany
Fire in
Germany
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Modeling Dependencies
Common-Factors:
• Intuitive
• Potentially accurate
Pros
• Gives insight into business
• Demanding in terms of input
• Can lead to overly complicated
models and a false sense of
accuracy
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Cons
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Common-Factors
Avoid overly complicated models
Afghanistan war – social, political and economical risks:
‘
“When we understand that slide,
we'll have won the war”
Gen. Stanley McChrystal,
US and NATO force commander
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Modeling Dependencies
Common-Factors:
• Intuitive
• Potentially accurate
“No Common-Factors”:
Pros
• Many types of dependencies
• Explicit tail dependence
• Gives insight into business
• Demanding in terms of input
• Can lead to overly complicated
models and a false sense of
accuracy
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• Calibration can be complicated
Cons
• Causal interpretation?
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Causal Interpretation for Copulas?
Elliptical copulas: the Gaussian copula is about common-factors.
Archimedean copulas: 𝐶(𝑢1 , … , 𝑢𝑑 ) = 𝜓(𝜓 −1 𝑢1 + ⋯ + 𝜓 −1 𝑢𝑑 )
• Shared frailty models:
Given cumulative hazard functions Λ𝑖 , the Laplace transform of the
frailty distribution 𝑍 (evaluated at the sum of the Λ𝑖 ’s) is a survival
copula.
Suggested reading: 1) “Multivariate survival modelling (…) ”, P. Georges et
al, 2001; 2) “Types of Dependence (…) in Single Parameter Copula Models”,
Jaap Spreeuw, 2006.
• Simplex distributions: Given equitable allocation of resources 𝑅,
the Williamson transform of 𝑅 is the generator of a survival copula.
Suggested reading: “From Achimedean to Liouville Copulas”, A. McNeil and
J. Nešlehová, 2009.
Marshall-Olkin model: admits an elegant representation in terms of
survival copulas.
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Causal Interpretation for Copulas?
It seems to be all about survival!
Life
Medicine, Re/Insurance, …
Solvency II, Basel III …
Embrechts: “But why do we witness such an incredible growth in
[copula] papers published starting the end of the nineties?
Here I can give three reasons: finance, finance, finance.”
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Shared frailty models
(also known as proportional frailty models)
Consider the hazard functions (also known as failure rates)
𝜆𝑖 = −𝜕𝑖 ln 𝐹𝑖
with cumulative hazard functions Λ𝑖 . The survival functions are denoted
by 𝐹𝑖 = 1 − 𝐹𝑖 , where 𝐹𝑖 are the cumulative distributions.
• In the absence of dependence: 𝑭𝒊 = 𝒆−𝜦𝒊 .
• In the shared multiplicative frailty model: 𝑭𝒊 = 𝒆−𝒁 𝜦𝒊 , where 𝑍 is
the shared frailty.
The joint survival function is then
𝐹 = E𝑍 𝑒 −𝑍 (Λ1+⋯+Λ𝑑)
A Laplace transform!
This is an Archimedean survival copula with generator 𝜓 = ℒ(𝑓),
where 𝑓 is the distribution function of 𝑍 and Λ𝑖 = 𝜓 −1 (𝐹𝑖 ).
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Shared frailty models
Clayton Model (1978)
• Suppose the shared frailty follows a gamma distribution with
mean and variance both equal to 1/𝛿, that is:
𝑍 ~ Γ 1/𝛿, 1
• Its Laplace transform yields the generator 𝜓(𝑡) = (1 + 𝑡)−1/𝛿 ,
and the joint survival function is the Clayton survival copula:
𝐹 = 𝑢1
−𝛿
+ 𝑢2
−𝛿
−1
−1/𝛿
, with 𝑢𝑖 = 𝐹𝑖
−1
• Cross ratio function (association between two lifetimes): the
ratio of one's failure risk at time 𝑡1 if the partner is known to
have failed versus survived at time 𝑡2 , is constant (!):
𝜆(𝑡1 𝑇2 = 𝑡2 )
𝜆(𝑡1 𝑇2 ≥ 𝑡2 )
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=1+𝛿
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Shared frailty models
Hougaard Model (1986)
• Suppose the shared frailty follows a stable distribution:
𝜋 𝛿
𝑍 ~ St 1/𝛿, 1, (cos ) , 0
2𝛿
• Its Laplace transform yields the generator 𝜓 = exp(−𝑡 1/𝛿 ), and
the joint survival function is the Gumbel-Hougaard survival
copula:
𝐹(𝑢1 , 𝑢2 ) = exp − (−ln 𝑢1
)𝛿 +(−ln 𝑢
2
)𝛿
1/𝛿
, with 𝑢𝑖 = 𝐹𝑖
−1
• Cross ratio function - the 2 lives become less dependent as they
age:
𝜆(𝑡1 𝑇2 = 𝑡2 )
𝛿−1
=1−
ln 𝐹(𝑡1 , 𝑡2 )
𝜆(𝑡1 𝑇2 ≥ 𝑡2 )
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Compound Processes
• There are many ways to introduce dependence:
𝑇1 (𝑡) =
𝑁1 (𝑡)
𝑘=0 𝑋1
and
𝑇2 (𝑡) =
𝑁2 (𝑡)
𝑘=0 𝑋2
• Between final outcomes:
o Copulas
o Survival copulas
o Regression
o Autoregressive models
o …
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Compound Processes
• There are many ways to introduce dependence:
𝑇1 (𝑡) =
𝑁1 (𝑡)
𝑘=0 𝑋1
and
𝑇2 (𝑡) =
𝜌=
𝜆𝐶
𝑁2 (𝑡)
𝑘=0 𝑋2
, where 𝜆𝐶 is the intensity
of common-jumps
𝜆1 𝜆2
• Between frequencies:
o Copulas
o Autoregressive models
o Superposition & splitting/thinning for Poisson processes
o Other common-factor models / causality
o …
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Compound Processes
• There are many ways to introduce dependence:
𝑇1 (𝑡) =
𝑁1 (𝑡)
𝑘=0 𝑋1
and
𝑇2 (𝑡) =
𝑁2 (𝑡)
𝑘=0 𝑋2
• Between severities:
o Copulas
o Survival copulas
o Regression
o Autoregressive models
o Common-factors / causality
o …
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Compound Processes
• There are many ways to introduce dependence:
𝑇1 (𝑡) =
𝑁1 (𝑡)
𝑘=0 𝑋1
and
𝑇2 (𝑡) =
𝑁2 (𝑡)
𝑘=0 𝑋2
• Between frequencies and severities within a same business unit:
o Copulas
o Regression
o Common-factors / causality
o …
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Compound Processes
• There are many ways to introduce dependence:
𝑇1 (𝑡) =
𝑁1 (𝑡)
𝑘=0 𝑋1
and
𝑇2 (𝑡) =
𝑁2 (𝑡)
𝑘=0 𝑋2
• Between frequencies and severities:
o Common-factors / causality
o Lévy copulas (for compound Poisson processes)
o …
Cont & Tankov, 2004
• They track losses originating from common events
(sort of “common-factor” model)
• They account for the dependence in both frequency and severity
(“entanglement)
Suggested reading: “Modelling and Measuring Multivariate Operational Risk
with Lévy Copulas”, K. Böcker & C. Klüppelberg, 2008 (München!)
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Lévy Copulas
• Intro: A copula 𝐶 parameterizes the joint cumulative distribution:
𝑃 𝑋1 ≤ 𝑥1 ; 𝑋2 ≤ 𝑥2 = 𝐶 𝐹1 𝑥1 , 𝐹2 𝑥2
• A survival copula 𝐶 parameterizes the joint survival function, with :
𝑃 𝑋1 > 𝑥1 ; 𝑋2 > 𝑥2 = 𝐶 𝐹1 𝑥1 , 𝐹2 𝑥2
They are related: 𝐶 𝑢1 , 𝑢2 = 𝑢1 + 𝑢2 − 1 + 𝐶 1 − 𝑢1 , 1 − 𝑢2
• A Lévy copula 𝐶 Lévy parameterizes the joint tail integral, the expected
number of common losses above a certain value.
For compound Poisson processes with intensities 𝜆𝑖 and common
intensity 𝜆𝐶 , the copula for the tail integrals Π𝑖 = λ𝑖 𝐹𝑖 is:
𝜆𝐶 𝐹𝐶 𝑥1 , 𝑥2 = 𝜆𝐶 𝑃 𝑋1 > 𝑥1 ; 𝑋2 > 𝑥2 = 𝐶 Lévy λ1 𝐹1 𝑥1 , λ2 𝐹2 𝑥2
for common losses
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Lévy Copulas
• Examples of possible applications:
o Motor insurance, where an accident can result in both injury and
material claims.
o Work-related accidents, which could give rise to both medical and
allowance claims.
𝑇𝑖 =
𝑁𝐶
𝑘=0 𝑋𝑖,𝐶
Common claims
+
𝑁𝑖,𝑈
𝑘=0 𝑋𝑖,𝑈
Unique claims
• Dependent severities 𝑋𝑖,𝐶
• Independent 𝑁𝑖,𝑈 and 𝑋𝑖,𝑈
• Intensity: 𝜆𝐶 = 𝐶 Lévy 𝜆1 , 𝜆2
• Intensities: 𝜆𝑖,𝑈 = 𝜆𝑖 − 𝜆𝐶
• Survival functions:
𝜆𝐶 𝐹1,𝐶 𝑥1 = 𝐶
Lévy
𝜆1 𝐹1 𝑥1 , 𝜆2 ;
𝜆𝐶 𝐹2,𝐶 𝑥2 =
𝐶 Lévy 𝜆1 , 𝜆2 𝐹2 𝑥2 ;
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• Survival functions:
𝜆𝑖,𝑈 𝐹𝑖,𝑈 𝑥𝑖 = 𝜆𝑖 𝐹𝑖 𝑥𝑖 − 𝜆𝐶 𝐹𝑖,𝐶 𝑥𝑖
𝐶(𝐹1,𝐶 , 𝐹2,𝐶 ) =
1 Lévy
𝐶
𝜆𝐶
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𝜆1 𝐹1 , 𝜆2 𝐹2
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Sampling Lévy Copulas
1) Compute the intensities and number of events:
• Compute 𝜆𝐶 and draw 𝑁𝐶
• Compute 𝜆1,𝑈 and 𝜆2,𝑈 and draw 𝑁1,𝑈 and 𝑁2,𝑈
Common claims
Unique claims
• Dependent severities 𝑋𝑖,𝐶
• Independent 𝑁𝑖,𝑈 and 𝑋𝑖,𝑈
• Intensity: 𝜆𝐶 = 𝐶 Lévy 𝜆1 , 𝜆2
• Intensities: 𝜆𝑖,𝑈 = 𝜆𝑖 − 𝜆𝐶
• Survival functions:
𝜆𝐶 𝐹1,𝐶 𝑥1 = 𝐶
Lévy
𝜆1 𝐹1 𝑥1 , 𝜆2 ;
𝜆𝐶 𝐹2,𝐶 𝑥2 =
𝐶 Lévy 𝜆1 , 𝜆2 𝐹2 𝑥2 ;
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• Survival functions:
𝜆𝑖,𝑈 𝐹𝑖,𝑈 𝑥𝑖 = 𝜆𝑖 𝐹𝑖 𝑥𝑖 − 𝜆𝐶 𝐹𝑖,𝐶 𝑥𝑖
𝐶(𝐹1,𝐶 , 𝐹2,𝐶 ) =
1 Lévy
𝐶
𝜆𝐶
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𝜆1 𝐹1 , 𝜆2 𝐹2
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Sampling Lévy Copulas
2) Compute the unique claims:
• Draw independent 𝑢1 , 𝑢2 ~U 0,1 , as many times as dictated by 𝑁1,𝑈
and 𝑁2,𝑈 .
• Find 𝑥𝑖,𝑈 = 𝐹𝑖,𝑈 −1 𝑢𝑖 , with 𝐹𝑖,𝑈 = 1 − 𝐹𝑖,𝑈
Common claims
Unique claims
• Dependent severities 𝑋𝑖,𝐶
• Independent 𝑁𝑖,𝑈 and 𝑋𝑖,𝑈
• Intensity: 𝜆𝐶 = 𝐶 Lévy 𝜆1 , 𝜆2
• Intensities: 𝜆𝑖,𝑈 = 𝜆𝑖 − 𝜆𝐶
• Survival functions:
𝜆𝐶 𝐹1,𝐶 𝑥1 = 𝐶
Lévy
𝜆1 𝐹1 𝑥1 , 𝜆2 ;
𝜆𝐶 𝐹2,𝐶 𝑥2 =
𝐶 Lévy 𝜆1 , 𝜆2 𝐹2 𝑥2 ;
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• Survival functions:
𝜆𝑖,𝑈 𝐹𝑖,𝑈 𝑥𝑖 = 𝜆𝑖 𝐹𝑖 𝑥𝑖 − 𝜆𝐶 𝐹𝑖,𝐶 𝑥𝑖
𝐶(𝐹1,𝐶 , 𝐹2,𝐶 ) =
1 Lévy
𝐶
𝜆𝐶
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𝜆1 𝐹1 , 𝜆2 𝐹2
37
Sampling Lévy Copulas
3) Compute the common claims:
• Draw independent 𝑢1 , 𝑣2 ~U 0,1 , as many times as dictated by 𝑁𝐶 .
• Find 𝑢2 by solving 𝑣2 =
𝜕𝐶(𝑢1 ,𝑢2 )
.
𝜕𝑢1
• Find 𝑥𝑖,𝐶 = 𝐹𝑖,𝐶 −1 𝑢𝑖 , with 𝐹𝑖,𝐶 = 1 − 𝐹𝑖,𝐶
Common claims
Unique claims
• Dependent severities 𝑋𝑖,𝐶
• Independent 𝑁𝑖,𝑈 and 𝑋𝑖,𝑈
• Intensity: 𝜆𝐶 = 𝐶 Lévy 𝜆1 , 𝜆2
• Intensities: 𝜆𝑖,𝑈 = 𝜆𝑖 − 𝜆𝐶
• Survival functions:
𝜆𝐶 𝐹1,𝐶 𝑥1 = 𝐶
Lévy
𝜆1 𝐹1 𝑥1 , 𝜆2 ;
𝜆𝐶 𝐹2,𝐶 𝑥2 =
𝐶 Lévy 𝜆1 , 𝜆2 𝐹2 𝑥2 ;
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• Survival functions:
𝜆𝑖,𝑈 𝐹𝑖,𝑈 𝑥𝑖 = 𝜆𝑖 𝐹𝑖 𝑥𝑖 − 𝜆𝐶 𝐹𝑖,𝐶 𝑥𝑖
𝐶(𝐹1,𝐶 , 𝐹2,𝐶 ) =
1 Lévy
𝐶
𝜆𝐶
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𝜆1 𝐹1 , 𝜆2 𝐹2
38
A few Lévy Copulas
Clayton:
𝐶 Lévy 𝑢1 , 𝑢2 = 𝑢1 −𝛿 + 𝑢2 −𝛿
Archimedean I:
𝐶 Lévy
𝑢1 , 𝑢2
−1/𝛿
1
1 − 𝑒 −𝛿(𝑢1+𝑢2)
= ln −𝛿 𝑢
1 − 2 𝑒 −𝛿 𝑢1 +𝑢2 + 𝑒 −𝛿 𝑢2
𝛿
𝑒
Archimedean II:
𝐶 Lévy
𝑢1 , 𝑢2 = ln
𝑒 𝑢1
−1
−𝛿
+
𝑒 𝑢2
−1
−𝛿
−1/𝛿
+1
• In all examples above 𝛿 > 0.
• The Clayton and “Archimedean I” have upper-tail dependency.
• The “Archimedean II” has both lower- / upper-tail dependencies and
allows negative dependency between severities.
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Clayton-Lévy Copula
𝐶 Lévy 𝑢1 , 𝑢2 = 𝑢1 −𝛿 + 𝑢2 −𝛿
−1/𝛿
• Survival copula for common jumps:
𝐶 𝑢1 , 𝑢2 = 𝑢1 −𝛿 + 𝑢2 −𝛿 − 1
• Time invariant:
𝐶𝑇
Lévy
−1/𝛿
The Clayton
copula!
For the Clayton-Lévy copula
𝑢1 , 𝑢2 = 𝑇
𝐶 Lévy
𝑢1 𝑢2
,
= 𝐶 Lévy 𝑢1 , 𝑢2
𝑇 𝑇
• Intensity for common jumps: 𝜆𝐶 = 𝜆1
• Frequency correlation:
𝜌=
CAE Fall Meeting 2013
𝜆1
−𝛿
+ 𝜆2
−𝛿
+ 𝜆2
−𝛿 −1/𝛿
−𝛿 −1/𝛿
𝜆1 𝜆2
München, 27 September 2013
40
Lévy Copula at Work
Fit to claims arising from accidents in the construction sector.
It features 2’249 medical claims and 1’099 daily allowance claims.
Logarithm of claims
Logarithm of claims
CAE Fall Meeting 2013
Logarithm of claims
Tail Integral
Tail Integral
Tail Integral
Logarithm of claims
Avanzi et al,
2011
Tail Integral
Tail Integral
Tail Integral
•
•
Logarithm of claims
Logarithm of claims
München, 27 September 2013
41
Closing Remark #1
Solvency II on Internal Model Approval:
1.
Senior management shall be able to demonstrate understanding of
the internal model and how this fits with their business model.
2.
Senior management shall be able to demonstrate understanding of
the limitations of the internal model and that they account of it in
their decisions.
3.
The timely calculation of results is essential.
(From “Use Test”, Section 3 of the CEIOPS Paper)
CAE Fall Meeting 2013
München, 27 September 2013
42
Closing Remark #2
• Bad Governance influences model risk, among other things.
• Example: the head of risk management tells you: “quantitative risk
management is useless, because I can twist the knobs in such a way
that the number being outputted is the number I want”.
• A “dependence” between that person and the model parameters
would be bad governance.
CAE Fall Meeting 2013
München, 27 September 2013
43
Closing Remark #2
Standard safety rules for good governance should include these:
Cut along
dotted line
Stochastic area
Authorized
Personnel only
Thank you very much for your attention
CAE Fall Meeting 2013
München, 27 September 2013
44
Casualty European Association – Fall 2013 Meeting
Fell free to send me your questions as well as contact me for
potential colaborations:
Pedro Fonseca
pedro.fonseca@six-group.com
Head Risk Analytics & Reporting