Using Copulas to Model Extreme Events

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Using Copulas to Model Extreme Events

• by Donald F. Behan and Sam Cox
• Georgia State University

2
Overview

• Research sponsored by the Society of Actuaries
• Paper to be posted on SoA web site
• A tool for learning about and applying copulas to
  the modeling of extreme events

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Executive Summary

• All multivariate distributions may be analyzed as
  marginal distributions and a copula.
• Allows focus on dependence relationship for known
  individual distributions.
• A tool for implementing this model in Excel or
  Mathematica is to be distributed.

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Copulas

• A copula captures the dependence relationship in
  a multivariate distribution.
• Given the marginal distributions of a
  multivariate distribution, the distribution is
  completely determined by its copula.
• This process may be carried out for any
  multivariate distribution, without any
  assumptions about the nature of the distribution.

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Copula Example

• Start with an arbitrary bivariate distribution
  F(u,v) we chose a standard bivariate normal
  distribution with correlation coefficient 0.6.
• Let the cumulative marginal distributions be X
  and Y.
• Define C(X(u),Y(v)) F(u,v).
• C is the copula associated with F.

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Continuous and Discrete Distributions

• Copulas are most easily visualized for continuous
  distributions.
• Discrete distributions can be viewed as limits of
  continuous distributions.
• If the marginals are not continuous, the copula
  is not unique, but it is unique for continuous
  distributions.

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Numerical Examples

• As an example, let F be the standard bivariate
  normal distribution with ? 0.6. Then F(0,0)
  0.352, i.e. the probability that u and v are both
  less than zero is 0.352.
• Under these assumptions, X and Y are standard
  normal, so X(0) Y(0) 0.5. Therefore,
  C(0.5,0.5) 0.352.
• F is reconstructed by the formula
  F(u,v)C(X(u),Y(v)).

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Mathematical Foundation

• Sklars Theorem (bivariate case) For any
  bivariate distribution F(u,v) with marginal
  distributions X(u) and Y(v) there exists a copula
  C such that F(u,v) C(X(u),Y(v)).
• C is a copula if and only if it is a bivariate
  probability distribution on 0,1?0,1 with
  uniform marginals.

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Dependence Structure

• The marginal distributions can be changed while
  retaining the copula. This retains the
  dependence structure.
• Example For (u,v) e 0,1?0,1 let
  C(u,v)max(u,v). This is the x y copula.
• If the marginal distributions are equal and
  non-trivial, the correlation is 1.0.

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Dependence vs. Correlation

• In the preceding example (xy copula) let X be
  uniform, but let Y be the exponential
  distribution Y(v) 1 e-x. Then the
  correlation coefficient is 0.866.
• Conclusion The correlation coefficient is not
  uniquely determined by the dependence.

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Measures of Dependence

• The copula uniquely determines the rank
  correlation.
• Spearmans rho is an example of a measure of rank
  correlation. This is used in our examples.

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Structure of the Example

• To provide an example of the use of copulas to
  study dependence of extreme events, we assume
  that the marginals are known.
• We assume that the dependence structure is known
  for common events, but not for extreme events.

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Excel Tool

• The Excel tool is intended as a learning vehicle,
  and as the basis to develop simulations.
• The tool includes an assumed structure to
  facilitate learning.
• The structure can be modified to simulate
  variables of interest.

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Excel Tool, Continued

• For illustration the marginal distributions are
  assumed to be a normal distribution and a
  lognormal distribution. Both have mean 1,000,000
  and standard deviation 50,000.
• For events between the 5th and 95th percentile
  the assumed dependence structure is normal with
  correlation 0.6.

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Excel Tool, Continued

• The copula is displayed on a square grid.
• The number of cells was kept small to facilitate
  manipulation rather than accurate modeling, and
  can be expanded.
• The example provides for manipulation of the
  dependence structure.

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Simulation Tool

• Sim Tools, a free download provided by Roger
  Myerson at the University of Chicago, is used to
  simulate the results of the assumed dependence.
  _at_risk could be used as well.
• Examples of alternative copulas and of
  manipulation of the given copula are provided in
  the Excel workbook.

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Linearity of the Constraints

• Constraints generated from the properties of a
  copula are linear.
• Spearmans rho depends on the copula in a linear
  manner.
• We have used the simplex method to compute
  various extreme configurations that are
  illustrated in the workbook and paper.

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Conclusion

• A tool consisting of an Excel workbook and an
  explanatory paper will be posted on the Society
  of Actuaries website.
• While that is pending, the draft documents are
  available at my website www.behan.ws