1. Testing Models on Simulated Data Presented at the Casualty Loss Reserve Seminar September 19, 2008 Glenn Meyers, FCAS, PhD ISO Innovative Analytics

2. The Application Estimating Loss Reserves Given a triangle of incremental paid losses –Ten years with 55 observations arranged by accident year and settlement lag Estimate the distribution of the sum of the remaining 45 accident years/settlement lags Loss reserve models typically have many parameters –Examples in this presentation – 9 parameters

3. Danger – Possible Overfitting! Model describes the sample, but not the population Understates the range of results The range is the goal! Is overfitting a problem with loss reserve models? If so, what do we do about it?

4. Outline Illustrate overfitting with a simple example –Example – fit a normal distribution with three observations –Illustrate graphically the effects overfitting Illustrate overfitting with a loss reserve model –Reasonably good loss reserve model –Show similar graphical effects as normal example

5. Normal Distribution MLE for parameters  and  n = 3 in these examples

6. Simulation Testing Strategy Select 3 observations at random  Population -  = 1000,  = 500 Predict a normal distribution using the maximum likelihood estimator Select 1,000 additional observations at random from the same population Compare distribution of additional observations with the predicted distribution

7. Simulated Fits

8. PP Plots Plot Predicted Percentiles x Uniform Percentiles If predicted percentiles are uniformly distributed, the plot should be a 45 o line.

9. PP Plots

10. Simulated Fits

11. View Maximum Likelihood As an Estimation Strategy If you estimate distributions by maximum likelihood repeatedly, how well do you do in the aggregate? Consider a space of possible parameters for a model Select parameters at random –Select a sample for estimation (training) –Select a sample for post-estimation (testing)

12. Continuing Prior Slide Select parameters at random –Select a sample for estimation (training) –Select a sample for post-estimation (testing) Fit a model for each training sample Calculate the predicted percentiles of the testing sample. Combine for all samples. In the aggregate, the predicted percentiles should be uniformly distributed.

13. PP Plots for Normal Distribution (n = 3) S-Shaped PP Plot - Tails are too light!

14. Problem – How to Fit Distribution? Proposed solution – Bayesian Analysis Likelihood = Pr{Data|Model} We need Pr{Model|Data} for each model in the prior

15. Bayesian Fits Predictive distributions are spread out more than the MLE’s. On individual fits, they do not always match the testing data.

16. Bayesian Fits as a Strategy Parameters of model were selected at random from the prior distribution Near perfect uniform distribution of predicted percentiles At least in this example, the Bayesian strategy does not overfit. Combined PP Plot for Bayesian Fitting Strategy

17. Analyze Overfitting in Loss Reserve Formulas Many candidate formulas - Pick a good one Paid Loss AY,Lag ~ Collective Risk Model Claim count distribution is negative binomial Claim severity distribution is Pareto –Claim severity increases with settlement lag Calculate likelihood using FFT

18. Simulation Testing Strategy Select triangles of data at random –Payment pattern at random –ELR at random –{Loss|Expected Loss} for each cell in the triangle from Collective Risk Model Randomly select outcomes using the same payment pattern and ELR Evaluate the Maximum Likelihood and Bayesian fitting methodology with PP plots.

19. Background on Formula Fit a Bayesian model to over 100 insurers and produced an “acceptable” combined PP plot on test data from six years later. This paper tests the approach to simulated data, rather than real data.

20. Prior Payout Patterns

21. Prior Probabilities for ELR

22. Selected Individual Estimates

23. Maximum Likelihood Fitting Methodology PP Plots for Combined Fits PP plot reveals the S-shape that characterizes overfitting. The tails are too light

24. Bayesian Fitting Methodology PP Plots for Combined Fits Nailed the Tails

25. Summary Examples illustrate the effect of overfitting Bayesian approach provides a solution These examples are based on simulated data, with the advantage that the “prior” is known. Previous paper extracted prior distributions from maximum likelihood estimates of similar claims of other insurers

26. Conclusion It is not enough to know if assumptions are correct. To avoid the light tails that arise from overfitting, one has to get information that is: Outside The Triangle