1. Latvijas Aktuāru Asociācija AN INTRODUCTION TO COPULAS Gaida Pettere Professor, Dr. Math. Riga Technical University, Chairmen of Latvian Actuarial Association

2. Latvijas Aktuāru Asociācija CONTENT 1.INTRODUCTION 2.TWO ARGUMENT COPULAS 3.ARCHIMEDEAN COPULAS 4.MULTIVARIATE COPULAS 5.USING COPULAS IN FINANCE 6.USING COPULAS IN INSURANCE

3. Latvijas Aktuāru Asociācija 1. INTRODUCTION

4. Latvijas Aktuāru Asociācija CONTINUOUS RANDOM VARIABLES Distribution function: Properties: 1) Values are changing from 0 to 1. 2) Is not decreasing.

5. Latvijas Aktuāru Asociācija CONTINUOUS RANDOM VARIABLES Graph:

6. Latvijas Aktuāru Asociācija CONTINUOUS RANDOM VARIABLES Density function of continuous random variables is or Properties: 1) Non negative. 2)

7. Latvijas Aktuāru Asociācija CONTINUOUS RANDOM VARIABLES Graph

8. Latvijas Aktuāru Asociācija CONTINUOUS RANDOM VARIABLES VAR(p%) – risk measure (Value at risk)

9. Latvijas Aktuāru Asociācija CONTINUOUS RANDOM VARIABLES Graph of multivariate distribution function:

10. Latvijas Aktuāru Asociācija CONTINUOUS RANDOM VARIABLES Graph of multivariate density function:

11. Latvijas Aktuāru Asociācija 2. TWO ARGUMENT COPULAS

12. Latvijas Aktuāru Asociācija TWO ARGUMENT COPULAS Informal definition: H(x,y)= C(u,v), u=F 1 (x) v=F 2 (y)

13. Latvijas Aktuāru Asociācija TWO ARGUMENT COPULAS Formal definition:

14. Latvijas Aktuāru Asociācija TWO ARGUMENT COPULAS A two-dimensional copula is function C with the following properties: 1. For every :. 2. For every : and. 3. For every with and :. A function that fulfils property 1 is also called grounded and a function that fulfils property 2 is also called 2-increasing.

15. Latvijas Aktuāru Asociācija TWO ARGUMENT COPULAS Sklar’s theorem (1959) Let H be a joint distribution function with margins F 1 and F 2. Then there exists a copula C with for every. If F 1 and F 2 are continuous, then C is unique. On the other hand, if C is a copula and F 1 and F 2 are univariate distribution functions, then the function H defined by previous equality, is a joint distribution function with margins F 1 and F 2.

16. Latvijas Aktuāru Asociācija TWO ARGUMENT COPULAS Multivariate density: h(x,y)=c(u,v)*f 1 (x)*f 2 (y) where c(u,v)=∂²C(u,v)/(∂u∂v) and u=F 1 (x), v=F 2 (y)

17. Latvijas Aktuāru Asociācija TWO ARGUMENT COPULAS There are two principal ways of using the copula idea:  We can extract copulas from well-known multivariate distribution functions.  We can also create new multivariate distribution functions by joining arbitrary marginal distributions together with copulas.

18. Latvijas Aktuāru Asociācija TWO ARGUMENT COPULAS Methods of constructing copulas:  The inversion method.  Geometric methods.  Algebraic methods. ☺ Nelsen R B (1999) An Introduction to Copulas. Springer-Verlag, New York.

19. Latvijas Aktuāru Asociācija TWO ARGUMENT COPULAS Examples of two argument copulas oThe Farlie-Gumbel-Morgenstern family of copulas (Morgenstern, D.(1956), Gumbel, E. J. (1958), Farlie, D. J. G. (1960)) C(u,v)=uv+ θuv(1-u)(1-v) oThe Ali-Mikhail-Hag copula C(u,v)=uv/[1-θ(1-u)(1-v)] θ belongs to [-1,1]

20. Latvijas Aktuāru Asociācija TWO ARGUMENT COPULAS The main questions: 1.How to find appropriate copula to data? 2.How to estimate copula parameters? ?

21. Latvijas Aktuāru Asociācija TWO ARGUMENT COPULAS Roberto de Matteis, “Fitting Copulas to Data”, Diploma Thesis, Univ. Zurich, 2001 Lindskog, F., “Modelling Dependence with Copulas”, Master Theses-MS-2000-06, Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden, 2000

22. Latvijas Aktuāru Asociācija 3. ARCHIMEDEAN COPULAS

23. Latvijas Aktuāru Asociācija ARCHIMEDEAN COPULAS Copulas, which are possible to write in the form or are called Archimedean copulas. The function is called a generator of the Archimedean copula.

24. Latvijas Aktuāru Asociācija For Archimedean copula is true that for any in, function is the univariate distribution function

25. Latvijas Aktuāru Asociācija It is natural to compare univariate distribution function with its nonparametric estimator where

26. Latvijas Aktuāru Asociācija ARCHIMEDEAN COPULAS Nelsen R B (1999) An Introduction to Copulas. Springer-Verlag, New York. Genest, C., and Rivest, L., “Statistical Inference Procedures for Bivariate Archimedean Copulas”, Journal of the American Statistical Association 88: 1993 - p. 1034-1043.

27. Latvijas Aktuāru Asociācija ARCHIMEDEAN COPULAS

28. Latvijas Aktuāru Asociācija Frank’s copula

29. Latvijas Aktuāru Asociācija Frank’s copula with uniform margins:

30. Latvijas Aktuāru Asociācija Gumbel copula

31. Latvijas Aktuāru Asociācija Gumbel copula with uniform margins

32. Latvijas Aktuāru Asociācija Clayton’s copula

33. Latvijas Aktuāru Asociācija Clayton’s copula with uniform margins

34. Latvijas Aktuāru Asociācija Frank’s copula with beta and normal margins

35. Latvijas Aktuāru Asociācija Gumbel copula with beta and normal margins

36. Latvijas Aktuāru Asociācija Clayton’s copula with beta and normal margins

37. Latvijas Aktuāru Asociācija Algorithm for Archimedean copula Generate two independent uniform variates and ; Set, where denotes the quasi-inverse of the distribution function Set and The desired pair is

38. Latvijas Aktuāru Asociācija 4. MULTIVARIATE COPULAS

39. Latvijas Aktuāru Asociācija MULTIVARIATE COPULAS Definition: multivariate copula is function C from with following properties: 1) for every u from C(u)=0, if at least one coordinate of u is 0, 2) if all u coordinates equal 1 except one, then C(u) equals to that coordinate, 3) n-th order differences is not negative. Sklar’s theorem proofs the same what was proved in two dimensional case.

40. Latvijas Aktuāru Asociācija MULTIVARIATE COPULAS Examples: 1.The Archimedean copulas: Definition a) Clayton family b) Gumbel-Hougaard family

41. Latvijas Aktuāru Asociācija MULTIVARIATE COPULAS Examples: 2. The Farlie-Gumbel-Morgenstern copula 3. Two argument copula

42. Latvijas Aktuāru Asociācija MULTIVARIATE COPULAS Kollo, T., Pettere, G. “Multivariate skew t- copula” sent for publication in journal „Statistical Papers”

43. Latvijas Aktuāru Asociācija MULTIVARIATE COPULAS

44. Latvijas Aktuāru Asociācija MULTIVARIATE COPULAS

45. Latvijas Aktuāru Asociācija 5. USING COPULAS IN FINANCE

46. Latvijas Aktuāru Asociācija USING COPULAS IN FINANCE Cherubini U, Luciano E, Vecchiato W (2004) Copula Methods in Finance, Wiley, New York. Breymann W., Dias A., Embrechts P. (2003) Dependence Structures for Multivariate High-Frequency Data I Finance, Quantitative Finance 3(1) 1-16. Embrechts P., Hoing A., Juri A.(2003) Using Copulae to bound the value-at-Risk for functions of dependent risks, Finance & Stochastics 7(2) 145-167. Embrechts P., Hoing A., Juri A.(2003) Using Copulae to bound the value-at-Risk for functions of dependent risks, Finance & Stochastics 7(2) 145-167 Fantazzini D. (2004) Copula’s Conditional Dependence Measures for Portfolio Management and value at Risk, RTN MicFinMa Summer School, Economics and Economtrics of MarketMicrostructure, Constance, June 7-11

47. Latvijas Aktuāru Asociācija 6. USING COPULAS IN INSURANCE

48. Latvijas Aktuāru Asociācija USING COPULAS IN INSURANCE Frees, E. W., and Valdez, E. A., “Understanding Relationships Using Copulas”, North American Actuarial Journal, V 2, No 1: p. 1-25. Bagarry M. (2006) Economic capital: a plea for the Student copula. 28th International Congress of Actuaries, http://papers.ica2006.com/3014.html.http://papers.ica2006.com/3014.html

49. Latvijas Aktuāru Asociācija USING COPULAS IN INSURANCE Pettere G., Kollo T. Modeling claim size in time via copulas, Transactions of 28th International Congress of Actuaries, N 206, 2006, p. 1-10. Pettere G., Jansons V. Stochastic modelling of insurance liabilities, returned for second review in Scandinavian Actuarial Journal

50. Latvijas Aktuāru Asociācija USING COPULAS IN INSURANCE

51. Latvijas Aktuāru Asociācija THANKS ABOUT ATTENTION