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

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

The Schnieper’s Model + Incremental Incurred IBNR IBNER 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

Incurred IBNR IBNER

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?

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.

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

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

Process / Estimation Variances of Row Total Recursive approach

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

The Results

Bootstrap Bootstrap Prediction Variances 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

Example X triangle 1 2 3 4 5 6 7 exposure 7.5 28.9 52.6 84.5 80.1 76.9 79.5 10224 1.6 14.8 32.1 39.6 55 60 12752 13.8 42.4 36.3 53.3 96.5 14875 2.9 14 32.5 46.9 17365 9.8 52.7 19410 1.9 29.4 17617 19.1 18129 Schnieper Data

N triangle 1 2 3 4 5 6 7 7.5 18.3 28.5 23.4 18.6 0.7 5.1 1.6 12.6 18.2 16.1 14 10.6 13.8 22.7 12.4 12.1 2.9 9.7 16.4 11.6 6.9 37.1 1.9 27.5 19.1

D Triangle 2 3 4 5 6 7 -3.1 4.8 -8.5 23 3.9 2.5 -0.6 0.9 8.6 -1.4 5.6 -5.9 10.1 -4.6 -31.1 -2.1 -2.8 -5.8

Analytical & Bootstrap   Reserves estimates Estimation errors Prediction errors prediction error % Analytical Bootstrap 2 4.4 3 4.8 5.2 6.0 9.5 9.8 196% 187% 4 32.5 32.1 13.6 13.2 27.2 30.3 84% 95% 5 61.6 60.0 21.8 20.9 39.0 41.5 63% 69% 6 78.6 77.2 22.3 21.3 41.7 45.8 53% 59% 7 105.4 104.4 26.7 25.5 47.6 50.3 45% 48% Total 287.3 283.3 77.1 80.3 110.9 112.4 39% 40%

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

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.

The End