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Applications of Resampling Methods in Actuarial Practice

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.

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Applications of Resampling Methods in Actuarial Practice

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  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 • Distributions of variables determined • Parametric, with parameters estimated from data • Monte Carlo simulation of variables • Cash flow testing, sensitivity analysis and profit testing • Integrated in the company DFA model

  3. What can go wrong? • Ignoring uncertainties: • Maybe common in early models • Hardly the case in modern methodologies • Overfitting: 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 • Data clustered around certain values • Data truncated from below or censored from above • Mixtures of distributions possible • Data may simply not fit the desired parametric distribution

  5. The Concept of Bootstrap • Random sample of size n from an unknown distribution F. • Create empirical distribution • Generate an IID random sequence (resample) from empirical distribution • Use it to estimate parameters or characteristics of the original distribution

  6. Overview of this Work • Basics of bootstrap (including estimating standard errors and confidence intervals) • Apply bootstrap to two empirical data sets • Compare bootstrap to traditional estimates • Smoothing bootstrap estimate • Clustered 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. • This is referred to as the bootstrap estimate of the parameter

  8. Bootstrap Methodology • Efron (1979) • Bickel and Freedman (1981): conditions for consistency, quantile processes, multiple regression, and stratified sampling • Singh (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 • Instead approximate with multiple resamples • Efron’s BESE, by the Law of Large Numbers , approximates the theoretical standard error in the limit • Should 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 • In practice, these bounds are taken from multiple resamples, empirical percentiles

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

  13. Smoothed Boostrap: Excess Losses • Estimate 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. • But relative frequency is constant on an interval containing 29.5, and data is rounded off. • Hogg 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 • 432 closed losses, bodily injury liability in Boston territory for 1995, as of mid-1997 • Policy limits capped 16 out of 432 losses, data is right censored • Overwhelming 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 • Standard approach to reinsurance purchase: loss elimination ratio • Can use plug-in bootstrap estimate (empirical loss elimination ratio) • Better: smoothed empirical loss elimination ratio • Result in the following figure:

  20. SELER

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

  22. Kaplan-Meier estimator

  23. Kaplan-Meier vs. SELER

  24. Some Conclusions • These ideas can be extended to all modeled variables • They should be extended • Most interesting for interest rates and capital assets in general • Time series and dependence of variables most challenging • Long Tails may be problematic

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