Predictive Distributions for Reserves which Separate True IBNR and IBNER Claims

1. Predictive Distributions for Reserves which Separate True IBNR and IBNER Claims Huijuan Liu Cass Business School Lloyd’s of London 30/05/2007

2. Introduction • The Schnieper’s Model (1991) • Extended Stochastic Models • Analytical Prediction Errors of the Reserves • Straightforward Bootstrapping Procedure for Estimating the Prediction Errors • The full Predictive Distribution of Reserves

3. The Schnieper’s Model According to when the claim occurs, we can separate Incremental Incurred into Incurred But Not Reported (IBNR) and Incurred But Not Enough Reported (IBNER) Incremental Incurred Development year j Development year j IBNR IBNER Accident year i + Accident year i Changes in Old Claims New Claims

4. Incurred IBNR IBNER

5. Questions from the Schnieper Model • Since the expected ultimate loss can be produced analytically, • what about the prediction variance? • Can the analytical result of the prediction variance be tested? • Is there a possibility to extend the limits of the model, which • is the model can not be applied to the data without exposure • and the claims details?

6. A Stochastic Model To derive a prediction distribution variance and test it, a stochastic model is necessary. A normal process distribution is the ideal candidate, i.e.

7. Prediction Variances of Overall Reserves Prediction Variance = Process Variance + Estimation Variance

8. Process Variances of Overall Total Estimation Variances of Overall Total Process Variances of Row Total Estimation Variance of Row Total Covariance between Estimated Row Total

9. Process / Estimation Variances of Row Total Recursive approach

10. Estimation Covariance between Row Totals Recursive approach Calculate correlation between estimates Correlation = 0 Calculate correlation using previous correlation

11. The Results

12. Bootstrap Original Data with size m Draw randomly with replacement, repeat n times Estimation Variance Pseudo Data with size m Bootstrap Prediction Variances Simulate with mean equal to corresponding Pseudo Data Original Data with size m Draw randomly with replacement, repeat n times Prediction Variance Simulated Data with size m Pseudo Data with size m Simulate with mean equal to corresponding Pseudo Data

13. Example Schnieper Data

16. Analytical & Bootstrap

17. Empirical Prediction Distribution Fig. 1 Empirical Predictive Distribution of Overall Reserves Fig. 1 Empirical Predictive Distribution of Overall Reserves

18. Further Work • Apply the idea of mixture modelling to other situation, such as paid and incurred data, which may have some practical appeal. • Bayesian approach can be extended from here. • To drop the exposure requirement, we can change the Bornheutter-Ferguson model for new claims to a chain-ladder model type.

19. The End