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

Outline 1.Introduction 2.Munich chain ladder (MCL) 3.Bootstrap MCL 4.Numerical Example 5.Conclusion

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

Munich chain ladder Paid ClaimsIncurred 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.

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

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

Plot of the MCL ratios

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

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

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

Results Bootstrap ReservesMCL Reserves PaidIncurredPaidIncurred Year Year Year Year Year Year Year 75,5125,6555,5055,606 Total 6,8937,1756,8467,163 Table 3. Bootstrap Reserves and MCL Reserves Bootstrap CL Prediction Error % Bootstrap MCL Prediction Error % PaidIncurredPaidIncurred Year 1-0%- Year 245%9%14%5% Year 333%96%42%52% Year 421%38%21%26% Year 518%62%24%35% Year 631%47%31% Year 722%14%13%12% Total16%13%11% Table 4. Bootstrap Prediction Errors of CL and MCL Model Figure 1. Predictive Distributions – A comparison between CL and MCL

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. Conclusion

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