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. Outline 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

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

5. Schnieper’s Model Assumptions Independence between accident years

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

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

8. 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. Process / Estimation Variances of the Row Totals

9. where, and. 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

10. Process / Estimation Variances Approach Process Variance Estimation Variance

11. Covariance Approach Correlation = 0 Calculate correlation between estimates Calculate correlation using previous correlation 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.

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

14. X triangle1234567exposure 1 7.528.952.684.580.176.979.510224 2 1.614.832.139.6556012752 3 13.842.436.353.396.514875 4 2.91432.546.917365 5 2.99.852.719410 6 1.929.417617 7 19.118129 Example Schnieper Data

15. N triangle 1234567 1 7.518.328.523.418.60.75.1 2 1.612.618.216.11410.6 3 13.822.7412.412.1 4 2.99.716.411.6 5 2.96.937.1 6 1.927.5 7 19.1

16. D Triangle 234567 2 -3.14.8-8.5233.92.5 3 -0.60.98.6-1.45.6 4 -5.910.1-4.6-31.1 5 -1.4-2.1-2.8 6 0-5.8 7 0

17. Analytical & Bootstrap Reserves EstimatesPrediction errorsPrediction errors % AnalyticalBootstrapAnalyticalBootstrapAnalyticalBootstrap 2 4.4 9.58.8215%200% 3 4.8 14.313.5298%283% 4 32.532.629.830.891%94% 5 61.661.141.541.669%68% 6 78.678.144.944.058%56% 7 105.4104.651.549.849%48% Total 287.3285.7111.1114.539%40%

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

19. 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. Further Work

20. Thank You For Your Attention!