Modelling of Dependencies
Hans Waszink
Waszink Actuarial Advisory
30 March 2011
Palisade Conference Amsterdam
Introduction
Content
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Introduction
What is Risk
Risk & Models
Dependencies
Conclusions
30 March 2011
Palisade Conference Amsterdam
Introduction
About Hans Waszink
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Mathematician & Actuary
15 years of experience in modelling Insurance & Financial Risks
Led various implementation projects for Risk Models in insurance
companies in UK & European Continent.
Explored use of different dependence structures in a number of
published articles and client applications.
30 March 2011
Palisade Conference Amsterdam
What is Risk
We are at Risk!
By Risk I mean: the occurrence of an unexpected event that adversely
affects me.
A rainy day is not a risk for me, but a stock market crash is.
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There are risks we know
Risks we know we don’t know
Risks we don’t know we don’t know
The less you know, the more extreme the risk.
30 March 2011
Palisade Conference Amsterdam
Risk & Models
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We can not change the course of events.
We can not predict the future.
But we can build models for risk
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The models are not right or wrong in themselves, they are
simplifications and abstractions of reality that help us
understand real phenomena.
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Models help us, amongst other things, to understand the
dependencies between risks.
30 March 2011
Palisade Conference Amsterdam
Example
– Lower interest rates make the effects of other
financial risks more pronounced.
• Why? The present value of a liability is inversely
proportional to the discount rate.
• For example, for a Pension Fund, longevity risk has a
bigger impact when interest rates are low. Models help you
understand and quantify this dynamic.
30 March 2011
Palisade Conference Amsterdam
Dependence
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Many phenomena in the real world are related to each other, e.g they
have a common cause, one causes the other, one sometimes causes
the other etc.
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The extreme risk IS the dependence:
– One mortgage default does not affect us.
– One failing bank does not affect us.
– But if they all do….
Warren Buffet in 2002 called CDOs financial weapons of mass destruction - since he
believes, contrary to the philosophy behind CDOs, that default risk is correlated and
cannot be diversified away. CDOs and other such debt-related derivatives have been
blamed for the 2007 credit crisis.
30 March 2011
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Dependence
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Modelling of dependencies is an integral and extremely important aspect
of risk modelling. In Solvency II, the impact of correlations can be as much as
half the sum of the stand-alone risks.
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Dependencies in the financial world are not static, they are changeable
and capricious.
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Unfortunately, they tend to aggravate in extreme conditions. Historic
data are not always a good indicator of future dependencies.
30 March 2011
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How to model dependencies
– The Normal option: correlation.
Correlation (X,Y) =
E[XY]-E[X]E[Y].
σ(X) σ(Y)
When all your distributions are Normal, the correlation fully defines the
dependency.
N.B.
– Correlation is a measure of overall dependence, not of tail dependence.
– It does not distinguish between cause and effect.
– No correlation does not mean independence - interest and longevity may not be
correlated, but the former increases the impact of the latter.
30 March 2011
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Say ‘know’ to correlation
– Suppose you define two variables X and Y.
– Choose a correlation between them.
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What is the distribution of X+Y?
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The distribution of X + Y is not fully defined;
The linear correlation coefficient specified may not exist in
combination with the marginal distribution functions.
Additional assumptions are required to fully specify the model
and perform a simulation.
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30 March 2011
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Say ‘know’ to correlation
How to resolve these issues?
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Apply the correlations to Normal distributions.
This fully defines a set of dependent distributions.
Then the ‘Normals’ are transformed to the specified
distributions. For example, the Lognormal distribution is the
exponential of the Normal distribution. The linear correlations
changes as a result, but the dependence remains.
30 March 2011
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Say ‘know’ to correlation
• The procedure just described is also called the
‘Normal Copula’, and is used by @Risk.
• It can be applied to any set of distributions, Normal
or not.
• It has certain characteristics that may be appropriate
or not.
30 March 2011
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Tail correlation?
Normal copula with 90% correlation. When zooming in on the left lower
corner, the remaining correlation is only 50%!
The correlation drops off under extreme circumstances!!
30 March 2011
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Tail Correlation
A minor extension to the Normal copula, called the ‘T-copula’ can increase the
tail correlation
Normal Copula
30 March 2011
The linear correlation is 0 for both
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t-copula 1 df
Tail Correlation
What if all of your data is non-extreme?
Beware of using historic data alone for parameterisation!
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Is this relevant?
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In normal daily life, maybe not so much
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But in Solvency II and Basel III the focus is on extreme
outcomes: 99.5% and 99.9% confidence levels of all risks in
aggregate.
30 March 2011
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Case Study
6 Risks with different distributions:
– Normal, Gamma, Lognormal, Uniform, Weibull, Poisson
– All with Stdev around 2.
– Correlations varying from 0 to 0.75
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Case Study
We look at the aggregate under 4 different calculation methods:
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Simulation with t-copula:
– Use the distributions as specified with maximum tail correlations.
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Simulation with Normal copula
– Use the distributions as specified without correlations.
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No simulation: Percentile & Correlation
– Only use one point of each distribution.
– Assume Normality of all risks, and Normal Copula for dependence.
– Used for Solvency II, Basel III standard models
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Full dependence = No diversification
– Found by adding the 99.5%/ 99.9% levels of all individual risks.
30 March 2011
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Case Study
Comparison of Results
Confidence
level
Simulation
T-copula
Simulation
Normal Copula
Solvency II
Standard Mdl
Method 4
99.5%
28.7
23.5
20.8
41.1
Diversification
Benefit @99.9
30%
43%
49%
0%
99.9%
42.0
29.6
27.4
53.0
Diversification
Benefit @99.9
21%
44%
48%
0%
30 March 2011
Palisade Conference Amsterdam
Conclusions
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The dependence structure beyond the correlation can have a
material impact on the tail of the distribution.
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The T-copula is an extension of the Normal copula that can be
used in combination with @Risk in a straightforward manner.
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Considering different types of dependence enriches the
analysis, and therefore strengthens the overall risk
management framework.
30 March 2011
Palisade Conference Amsterdam