Claims Prediction with Dependence using Copula Models Yeo Keng - PowerPoint PPT Presentation

Claims Prediction with Dependence using Copula Models Yeo Keng Leong and Emiliano A. Valdez School of Actuarial Studies Faculty of Commerce and Economics University of New South Wales Sydney, AUSTRALIA 11 November 2005 Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 1/28

Scope ■ Introduction, Motivation ● Scope Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 2/28

Scope ■ Introduction, Motivation ● Scope Introduction, Motivation ■ Setting and Notations Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 2/28

Scope ■ Introduction, Motivation ● Scope Introduction, Motivation ■ Setting and Notations Setting and Notations ■ Modelling Time Dependence with Copulas Modelling Time Dependence with Copulas ◆ Assumptions and Model Applications ◆ Conditional Expectation Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 2/28

Scope ■ Introduction, Motivation ● Scope Introduction, Motivation ■ Setting and Notations Setting and Notations ■ Modelling Time Dependence with Copulas Modelling Time Dependence with Copulas ◆ Assumptions and Model Applications ◆ Conditional Expectation Choice of Copula ■ Application Illustrating the Copula Density Ratio ◆ Gaussian and Student- t Copulas Conclusion ◆ Archimedean Copulas - Cook-Johnson Copula ◆ FGM Copulas ◆ Remarks on Choice of Copula ◆ Illustrations of Copula Density Ratio Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 2/28

Scope ■ Introduction, Motivation ● Scope Introduction, Motivation ■ Setting and Notations Setting and Notations ■ Modelling Time Dependence with Copulas Modelling Time Dependence with Copulas ◆ Assumptions and Model Applications ◆ Conditional Expectation Choice of Copula ■ Application Illustrating the Copula Density Ratio ◆ Gaussian and Student- t Copulas Conclusion ◆ Archimedean Copulas - Cook-Johnson Copula ◆ FGM Copulas ◆ Remarks on Choice of Copula ◆ Illustrations of Copula Density Ratio ■ Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 2/28

● Scope Introduction, Motivation ● Introduction ● Motivation Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 3/28

Introduction ■ Experience Rating ● Scope Introduction, Motivation ● Introduction ◆ Rating process that takes into account, at least partially, ● Motivation the individual risk’s experience Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 4/28

Introduction ■ Experience Rating ● Scope Introduction, Motivation ● Introduction ◆ Rating process that takes into account, at least partially, ● Motivation the individual risk’s experience Setting and Notations Modelling Time Dependence with Copulas ■ Credibility Theory Applications Choice of Copula Premium = Z · Own Experience +(1 − Z ) · Group Experience Illustrating the Copula Density Ratio where Z ∈ [0 , 1] is known as “credibility factor” Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 4/28

Motivation ■ Traditional practice in many credibility models to assume ● Scope independence of claims Introduction, Motivation ● Introduction ● Motivation ◆ either across time for individual risk or between individuals Setting and Notations Modelling Time Dependence with Copulas ◆ Gerber and Jones (1975) and Frees, et al. (1999) are Applications examples of the former case Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 5/28

Motivation ■ Traditional practice in many credibility models to assume ● Scope independence of claims Introduction, Motivation ● Introduction ● Motivation ◆ either across time for individual risk or between individuals Setting and Notations Modelling Time Dependence with Copulas ◆ Gerber and Jones (1975) and Frees, et al. (1999) are Applications examples of the former case Choice of Copula Illustrating the Copula Density Ratio ■ We offer additional insight into modelling dependence of Conclusion claims across time periods for a fixed individual by considering use of copulas Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 5/28

● Scope Introduction, Motivation Setting and Notations ● Setting and Notations Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 6/28

Setting and Notations ■ T time periods ● Scope Introduction, Motivation Setting and Notations ● Setting and Notations Modelling Time Dependence with Copulas Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 7/28

Setting and Notations ■ T time periods ● Scope Introduction, Motivation ■ Claims amount for one individual at time period t , X t , Setting and Notations ● Setting and Notations t = 1 , 2 , ..., T Modelling Time Dependence with Copulas ◆ realisation denoted by x t Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 7/28

Setting and Notations ■ T time periods ● Scope Introduction, Motivation ■ Claims amount for one individual at time period t , X t , Setting and Notations ● Setting and Notations t = 1 , 2 , ..., T Modelling Time Dependence with Copulas ◆ realisation denoted by x t Applications Choice of Copula Illustrating the Copula Density ■ Individual’s claim vector: X T = ( X 1 , X 2 , ..., X T ) ′ Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 7/28

● Scope Introduction, Motivation Setting and Notations Modelling Time Dependence Modelling Time Dependence with with Copulas ● Assumptions and model Copulas ● Conditional expectation Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 8/28

Assumptions and model ■ Multivariate distribution and density functions, ● Scope H T ( x 1 , . . . , x T ) and h T ( x 1 , . . . , x T ) Introduction, Motivation Setting and Notations Modelling Time Dependence with Copulas ● Assumptions and model ● Conditional expectation Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 9/28

Assumptions and model ■ Multivariate distribution and density functions, ● Scope H T ( x 1 , . . . , x T ) and h T ( x 1 , . . . , x T ) Introduction, Motivation Setting and Notations ■ Marginal distribution and density functions, F t ( x t ) and Modelling Time Dependence f t ( x t ) , t = 1 , 2 , ..., T with Copulas ● Assumptions and model ● Conditional expectation Applications Choice of Copula Illustrating the Copula Density Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 9/28

Assumptions and model ■ Multivariate distribution and density functions, ● Scope H T ( x 1 , . . . , x T ) and h T ( x 1 , . . . , x T ) Introduction, Motivation Setting and Notations ■ Marginal distribution and density functions, F t ( x t ) and Modelling Time Dependence f t ( x t ) , t = 1 , 2 , ..., T with Copulas ● Assumptions and model ● Conditional expectation ■ Copula function, C T ( F 1 ( x 1 ) , . . . , F T ( x T )) , thus Applications H T ( x 1 , . . . , x T ) = C T ( F 1 ( x 1 ) , . . . , F T ( x T )) Choice of Copula Illustrating the Copula Density = C T ( u 1 , . . . , u T ) Ratio Conclusion Claims Predictions with Dependence using Copula Models UNSW Actuarial Research Symposium 2005 - p. 9/28