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 10/07/2007

2. IBNR – Incurred But Not Reported claims. It is the aggregate claims, denoted as . • IBNER – Incurred But Not Enough Reported claims. It is the developed amount from the existing claims which have already been reported. It is regarded as the incremental old claims, denoted as . • Schnieper(1991) introduces a model that is designed for excess of loss cover in reinsurance business using IBNR claims. • Main approach – separating the aggregate IBNR run-off triangle into two more detailed run-off triangles according to the claims reported time, i.e. the new claims and the old claims. • The new claims, denoted as , are the claims that first reported in development year j, unknown in development year j-1. • The old claims (decrease), denoted as , are defined as the decrease in all claims that occurred between development year j and j-1, known in development year j-1. Background

3. The Schnieper Separation Approach for the best estimates • Theoretical Approach to the approximation of the Mean Squared Error of Prediction (MSEP) for the Schnieper model • Bootstrap and Simulation Approach for estimating the MSEP and the predictive distributions • A Numerical Example • Discussion and Further work Outline

4. The Schnieper’s Separation Approach Development year j According to when the claim is reported, we can separate the IBNR into the True IBNR and IBNER Incremental IBNR Accident year i Incremental True IBNR Incremental Decrease IBNER + Decrease from Old Claims New Claims

5. Schnieper’s Model Assumptions Independence between accident years

6. ‘Level 3’ Predictive Distribution ‘Level 2’ Prediction Variance / Variability Motivation for Estimating the MSEP ‘Level 1’ Best Estimate / Mean

7. Prediction Variance = Process Variance + Estimation Variance • Process Variance – the variability of the random • variables • Estimation Variance – the variability of the • fitted model Prediction Variance

8. Process / Estimation Variances of the Row Totals Process Variance – the squared error of the modelling process. It is straightforward to estimate, given the random variables are identically independent distributed. It is the sum of the process variances of each individual random variable from the underlying distribution. Estimation Variance – the squared error of the fitted underlying model. It is relatively more complicated to estimate, due to the same parameters involved for the loss claim predictions. Recursive Approach – to obtain the prediction variances of row totals (or ultimate losses) from different accident years. In the other word, the estimated process variances of the row totals are, recursively, calculated from the latest observed claims data (the leading diagonal) in the run-off triangle.

9. Therefore, the prediction variance of overall total loss claim is the approximate sum of process variance and estimation variance, which is written as follows, The above equation is expanded when considering the row totals, i.e. the prediction variance is approximated by the sum over process variances, estimation variances of row totals and all the covariance between any two of them. So we have where , and .

10. Process / Estimation Variances Approach Process Variance Estimation Variance

11. Covariance Approach Covariance between Row Totals (i.e. the ultimate losses) – is caused by the same parameter estimates in the row total predictions. It is also estimated recursively. Calculate correlation between estimates Correlation = 0 Calculate correlation using previous correlation

12. Results The prediction variance is estimated as follows, This looks complicated but is simple to implement in a spreadsheet, due to the recursive approach.

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

14. Example Schnieper Data

17. Analytical & Bootstrap

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

19. 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.

20. Thank You For Your Attention!