Using Copulas to Model Extreme Events

1. Using Copulas to Model Extreme Events by Donald F. Behan and Sam Cox Georgia State University Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

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 Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

3. 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. Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

4. 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. Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

5. 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. Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

6. 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. Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

7. Numerical Examples • As an example, let F be the standard bivariate normal distribution with ρ = 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)). Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

8. Mathematical Foundation • Sklar’s 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. Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

9. Dependence Structure • The marginal distributions can be changed while retaining the copula. This retains the dependence structure. • Example: For (u,v) ε [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. Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

10. Dependence vs. Correlation • In the preceding example (x=y 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. Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

11. Measures of Dependence • The copula uniquely determines the rank correlation. • Spearman’s rho is an example of a measure of rank correlation. This is used in our examples. Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

12. 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. Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

13. 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. Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

14. 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. Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

15. 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. Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

16. 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. @risk could be used as well. • Examples of alternative copulas and of manipulation of the given copula are provided in the Excel workbook. Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

17. Linearity of the Constraints • Constraints generated from the properties of a copula are linear. • Spearman’s 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. Donald F. Behan Society of Actuaries Meeting Phoenix, AZ

18. 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 Donald F. Behan Society of Actuaries Meeting Phoenix, AZ