Risk Dependency Research: A Progress Report

1
Risk Dependency ResearchA Progress Report

• Enterprise Risk Management Symposium
• Washington DC July 30, 2003

B. John Manistre FSA, FCIA, MAAA
2
Agenda

• Nature of the project
• Tool Development
• Risk Measures
• Special Results for Normal Risks
• Extreme Value Theory
• Copulas
• Formula Approximations
• Toward Real Application
• Literature Survey

3
Nature of the Project

• Response to SoAs Request for Proposal on RBC
  Covariance
• Broad Mandate determine the covariance and
  correlation among various insurance and
  non-insurance risks generally, particularly in
  the tail.
• Phase 1 Theoretical Framework/Literature Search
• Phase 2 Data Collection/Analysis - the practical
  element
• Project organized at University of Waterloo
• J Manistre (Aegon USA), H Panjer(U of W)
  graduate students J Rodriguez, V Vecchione

4
Phase 1 Theoretical Framework

• Tools
• Risk Measures
• Extreme Value Theory
• Copulas
• Formula Approximations to Risk Measures
• New results
• Formula Approximations suggest measures of tail
  covariance and correlation

5
Phase 1 Risk Measures

• Project focusing on risk measures defined by an
  increasing distortion function
• For a random variable X risk measure is given by
• where
• Capital is usually taken to be the excess of the
  risk measure over the mean

6
Phase 1 Risk Measures- Examples

• Project does not take a position on which risk
  measure is best
• Planning to work with the following
• Value at Risk
• Wang Transform
• Block Maximum
• Conditional Tail Expectation

7
Phase 1 Risk Measures

• For any Normal Risk X,
• Risk measure is mean plus a multiple of the std
  deviation
• Can use Kg as a tool to understand the risk
  measure

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Phase 1 Risk Measures
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Phase 1 Risk Measures - Aggregating Normal Risks

• Suppose all risks normal and
• Then
• For any g conclude
• This is An exact solution to an approximate
  problem.

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Phase 1Extreme Value Theory

• EVT applies when distribution of scaled maxima
  converge to a member of the three parameter EVT
  family
• Works for most standard distributions e.g.
  normal, lognormal, gamma, pareto etc.
• Key Result is the Peaks Over Thresholds
  approximation
• When EVT applies excess losses over a suitably
  high threshold have an approximate generalized
  pareto distribution
• Suggests that a generalized pareto distribution
  should be a reasonable model for the tail of a
  wide range of risks

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Phase 1Copulas

• A tool for modeling the dependency structure for
  a set of risks with known marginal distributions
• Technically a probability distribution on the
  unit n-cube
• Large academic literature
• Some sophisticated applications in PC
  reinsurance
• Project is concentrating on
• t- copulas
• Gumbel copulas
• Clayton copulas

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Phase 1Copulas
13
Phase 1Copulas
14
Phase 1Copulas
15
Phase 1Copulas
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Phase 1 Formula Approximations

• Simple Investment Problem. Let
• Fix the joint distribution of the Ui and consider
• Capital function is homogeneous of degree 1 in
  the exposure variables
• Choose a target mix of risks
• Put

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Phase 1 Formula Approximations

• Theoretical Result The first two derivatives
  are given by
• Some challenges in using these results to
  estimate derivatives. Second derivatives harder
  to estimate.
• Some risk measures easier to work with than
  others.
• Project team is working with a number of
  approaches.

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Phase 1 Formula Approximations

• Let ri be a vector such that
  then the homogeneous formula
  approximation
• agrees with the capital function and its first
  two derivatives at the target risk mix
  .
• If ri is a vector such that
  then a homogeneous formula
  approximation is

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Phase 1 Formula Approximation 1

• When ri 0
• Suggests definition of tail correlation.

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Phase 1 Formula Approximation 2

• Some simple choices
• ri 0
• ri Ci
• ri ciCg (Ui)
• When ri 0
• Exact for Normal Risks

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Phase 1 Formula Approximation 2

• When ri Ci formula is essentially first order
• Factors Ci lt ci already reflect
  diversification.
• Suggests many existing capital formulas are as
  good (or bad) as first order Taylor Expansions.

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Phase 1 Formula Approximation 3

• When ri ci we get
• Undiversified capital less an adjustment
  determined by inverse correlation

23
Phase 1 Formula Approximations

• Practical work so far suggests
• is a more robust approximation. In particular,
  when the risks are normal
• Other homogeneous approximations are possible.

24
Phase 1 Numerical Example Inputs

• Three Pareto Variates combined
  with t-copula

25
Phase 1 Numerical Example Results

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Phase 2 Real Application

• Phase 2 not yet begun
• Will not be totally objective
• Process
• Develop high level models for individual risks
• e.g. model C-1 losses with a pareto distn.
• Assume a copula consistent with expert opinion
• Adopt a measure of tail correlation and
  calculate
• Make subjective adjustments to final results as
  nec.

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Literature Survey Risk Measures

• Artzner, P., Delbaen, F., Thinking Coherently,
  Eber, J-M., Heath, D., Thinking Coherently,
  RISK (10), November 68-71.
• Artzner, P, Application of Coherent Risk
  Measures to Capital Requirements in Insurance,
  North American Actuarial Journal (3), April 1999.
• Wang,S.S., Young, V.R. , Panjer, H.H., Axiomatic
  Characterization of Insurance Prices, Insurance
  Mathematics and Economics (21) 171-183.
• Acerbi, C., Tasche, D., On the Coherence of
  Expected Shortfall, Preprint, 2001.

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Literature SurveyMeasures and Models of
Dependence (1)

• Frees, E.W., Valdez,E.A., Understanding
  Relationships Using Copulas, North American
  Actuarial Journal (2) 1998, pp 1-25.
• Embrechts, P., NcNeil, A., Straumann, D.,
  Correlation and Dependence in Risk Mangement
  Properties and Pitfalls, Preprint 1999
• Embrechts, P., Lindskog, F., McNeil, A.,
  Modelling Dependence with Copulas and
  Applications to Risk Management, Preprint 2001.
• McNeil, A., Rudiger, F., Modelling Dependent
  Defaults, Preprint 2001.

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Literature SurveyMeasures and Models of
Dependence (2)

• Lindskog, F., McNeil, A., Common Poisson Shock
  Models Applications to Insurance and Credit Risk
  Modelling, Preprint 2001.
• Joe, H, 1997 Multivariate Models and
  Dependence, Chapman-Hall, London
• Coles, S., Heffernan, J., Tawn, J. Dependence
  Measures for Extreme Value Analysis, Extremes
  24, 339-365, 1999.
• Ebnoether, S., McNeil, A., Vanini, P.,
  Antolinex-Fehr, P., Modelling Operational Risk,
  Preprint 2001.

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Literature SurveyExtreme Value Theory

• King, J.L., 2001 Operational Risk, John Wiley
  Sons UK.
• McNeil,A., Extreme Value Theory for Risk
  Managers, Preprint 1999.
• Embrechts, P. Kluppelberg, C., Mikosch, T.
  Modelling Extreme Events, Springer Verlag,
  Berlin, 1997.
• McNeil, A., Saladin, S., The Peaks over
  Thresholds Method for Estimating High Quantiles
  of Loss Distributions, XXVIIth International
  ASTIN Colloquim, pp 22-43.
• McNeil, A., On Extremes and Crashes, RISK,
  January 1998, London Risk Publications.

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Literature SurveyFormula Approximation

• Tasche, D.,Risk Contributions and Performance
  Measurement, Preprint 2000.