Bootstrap Estimation of the Predictive Distributions of Reserves Using Paid and Incurred Claims

1. Bootstrap Estimation of the Predictive Distributions of Reserves Using Paid and Incurred Claims Huijuan Liu Cass Business School Lloyd’s of London 11/07/2007

2. Introduction • Munich chain ladder (MCL) • Bootstrap MCL • Numerical Example • Conclusion Outline

3. Modelling Paid and Incurred • Issue with existing models • Munich chain ladder model • Uncertainty of the model • Approach – the bootstrap method Introduction

4. Munich chain ladder Paid Claims Incurred Claims In order to observe the dependency, the development year effect needs to be removed. It means the mean and variance for each development year is deducted and scaled, respectively, for all the cells from the same dev year. In the other word, the Pearson residuals are considered for modelling the correlation between paid and incurred claims.

5. Residual Plot Paid dev. factors vs. I/P Residual Plot Incurred dev. factors vs. P/I

6. By using the correlation, the MCL model adjusts each CL development factor to the ratios which affect different accident years. In the other word, the traditional development factors become more specific in the MCL model. Plot of the CL dev. Factors and the link ratios

7. Plot of the MCL ratios

8. ‘Level 3’ Predictive Distribution ‘Level 2’ Prediction Variance / Variability Motivation ‘Level 1’ Best Estimate / Mean

9. Bootstrap MCL Paid dev factor residuals Incurred dev factors residuals I / P residuals P / I residuals Sample of grouped residuals Re-sampling, with replacement + Simulation Pseudo samples of grouped residuals Re-sampled Paid dev factor residuals Re-sampled Incurred dev factors residuals Re-sampled I / P residuals Re-sampled P / I residuals

10. Numerical Example Table 1. Paid Claim Data from Quarg and Mack (2004) Table 2. Incurred Claim Data from Quarg and Mack (2004)

11. Results Table 3. Bootstrap Reserves and MCL Reserves Figure 1. Predictive Distributions – A comparison between CL and MCL Table 4. Bootstrap Prediction Errors of CL and MCL Model

12. Conclusion • Bootstrapping is well-suited for these purposes the ideal candidate from a the practical point of view, since it avoids the complicated theoretical calculations and is easily to be implemented by in a simple spreadsheet. • When bootstrapping the method, the dependence observed in the data is taken into account and maintained by re-sampling pairwised. • However, the MCL model does not always produce superior results to the straightforward chain ladder model. As a consequence, we believe that it is important for the data to be carefully checked to test whether the dependency assumptions of the MCL model are valid for each data set before it is applied.

13. Thank You For Your Attention!