1. Applications of Resampling Methods in Actuarial Practice by Dr. Richard Derrig Automobile Insurers Bureau of Massachusetts Dr. Krzysztof Ostaszewski Illinois State University Dr. Grzegorz Rempala University of Louisville CAS Annual Meeting Washington, DC November 12-15, 2000

2. Actuarial Modeling Processes uDistributions of variables determined uParametric, with parameters estimated from data uMonte Carlo simulation of variables uCash flow testing, sensitivity analysis and profit testing uIntegrated in the company DFA model

3. What can go wrong? uIgnoring uncertainties:  Maybe common in early models  Hardly the case in modern methodologies uOverfitting: the data must submit to prescribed distributions  May work in practice but not in theory  There is nothing more impractical than the wrong theory

4. Loss Distributions uData clustered around certain values uData truncated from below or censored from above uMixtures of distributions possible uData may simply not fit the desired parametric distribution

5. The Concept of Bootstrap uRandom sample of size n from an unknown distribution F. uCreate empirical distribution uGenerate an IID random sequence (resample) from empirical distribution uUse it to estimate parameters or characteristics of the original distribution

6. Overview of this Work uBasics of bootstrap (including estimating standard errors and confidence intervals) uApply bootstrap to two empirical data sets uCompare bootstrap to traditional estimates uSmoothing bootstrap estimate uClustered data, data censoring, inflation adjustment

7. Plug-in Principle  Given a parameter of interest  depending on CDF F, estimate it by replacing F by its empirical counterpart obtained from the observed data. uThis is referred to as the bootstrap estimate of the parameter

8. Bootstrap Methodology uEfron (1979) uBickel and Freedman (1981): conditions for consistency, quantile processes, multiple regression, and stratified sampling uSingh (1981): for many statistics bootstrap is asymptotically equivalent to the one-term Edgeworth expansion

9. Boostrap SE estimate (Efron)

10. Bootstrap Standard Error Estimate  Rarely practical to calculate standard error  directly uInstead approximate with multiple resamples uEfron’s BESE, by the Law of Large Numbers, approximates the theoretical standard error in the limit uShould take about 250 resamples

11. The Method of Percentiles  Bootstrap estimate of , let G* be the distribution function  Bootstrap percentiles method uses inverse images of  and 1-  under G* as the bounds for the confidence intervals uIn practice, these bounds are taken from multiple resamples, empirical percentiles

12. Application to Wind Losses: Quantiles uHogg & Klugman (1984): data on 40 losses due to wind-related catastrophes in 1977 uStandard approach to confidence intervals: normal approximation to the sample quantiles uHogg and Klugman obtain: (9,32) uUsing the bootstrap method of percentiles we obtain the interval (8,27), considerably shorter

13. Smoothed Boostrap: Excess Losses uEstimate the probability that wind loss will exceed a $29,500,000 threshold, i.e., 1 - F(29.5). Plug in: 1 - F(29.5) = 0.05. uBut relative frequency is constant on an interval containing 29.5, and data is rounded off. uHogg and Klugman use MLE to fit truncated exponential and truncated Pareto distributions.

14. Solid line: exponential, Dashed line: Pareto

15. Smoothed Boostrap using the three term moving average smoother

16. Clustered Data: Massachusetts Auto Bodily Injury Liability Data u432 closed losses, bodily injury liability in Boston territory for 1995, as of mid-1997 uPolicy limits capped 16 out of 432 losses, data is right censored uOverwhelming presence of suspected fraud and buildup claims. This causes some numerical values to have unusually high frequencies.

17. Clustered Data: Massachusetts Auto Bodily Injury Liability Data

18. Approximation to empirical CDF adjusted for clustering. Also zoomed at (3.5, 5)

19. Bootstrap Estimates for Loss Elimination Ratio uStandard approach to reinsurance purchase: loss elimination ratio uCan use plug-in bootstrap estimate (empirical loss elimination ratio) uBetter: smoothed empirical loss elimination ratio uResult in the following figure:

20. SELER

21. Policy Limits and Deductibles. Bootstrapping Censored Data uWe use Kaplan-Meier estimator uCan be viewed as a generalization of usual empirical CDF adjusted for the fact of censoring losses. uNext figure shows Kaplan-Meier vs. SELER, first censoring point at 20

22. Kaplan-Meier estimator

23. Kaplan-Meier vs. SELER

24. Some Conclusions uThese ideas can be extended to all modeled variables uThey should be extended uMost interesting for interest rates and capital assets in general uTime series and dependence of variables most challenging uLong Tails may be problematic