The Cotor Challenge, Round 2

1. Matthew Flynn (860) 633-4119 x8764 Matt.Flynn@sas.com The Cotor Challenge, Round 2

4. A little EDA … Proc GCHART;The data are dominated by single large claim, dashed horizontal lines are at the 95% and 99% percentiles $10M $5M 99th pctile 95th pctile

5. A little EDA … Proc BOXPLOTThe data are dominated by single large claim

6. A little EDA continued… Proc UNIVARIATE;

7. A little EDA … Proc UNIVARIATE; Loss Histogram – very, very long tail

8. A little EDA … Proc UNIVARIATE; Losses verses Exponential distribution Large loss (upper right)

9. A little EDA continued … Proc UNIVARIATE - logLoss; overall fits are unlikely to fit tails well.

10. A little EDA … Proc GCHART;The data are dominated by single large claim, vertical lines are at $5m and $10m

11. A little EDA continued … Proc UNIVARIATE - logLoss;

12. A little EDA continued… - logLoss;Top loss = 60% of total dollars, 90% of all dollars are in the top 25 (or 1%) losses

13. Sample Mean Excess Distribution The sample mean excess distribution is the sum of the excesses over the threshold u divided by the number of data points, n − k + 1, which exceed the threshold u. The sample mean excess function describes the expected excess of a threshold given that exceedance occurs and is an empirical estimate of the mean excess function; e(u) = E [x − u|x > u]. If a graph of the sample mean excess function is horizontal, the tail is exponential. An upward sloping graph is said to be ‘fat-tailed’, relative to an exponential.

16. Extreme Value Theory – “Peaks Over Threshold” and the Generalized Pareto distribution Next, fitting A GPD fit the tail of the loss distribution via SAS Proc NLMIXED. proc nlmixed data=Cotor(where=(logLoss>11.9)); parms sigma=1 xi=0.3; bounds sigma >= 0; if (1 + xi * ((logLoss – 11.9) / sigma)) <= 0 then lnlike = 11.9 ** 6; else lnlike = -log(sigma) - (1 + (1 / xi))*log(1 + xi * ((logLoss – 11.9) / sigma)); model logLoss ~ general(lnlike); run;

17. Quantile or Tail Estimator – VaR (Value at Risk) See: McNeil, Alexander J. The Peaks over Thresholds Method for Estimating High Quantiles of Loss Distributions, ASTIN Colloquium, 1997, equation 5, page 10.

18. Expected Shortfall – Tail VaR – Conditional Tail Expectation If things go bad, how bad is bad? Expected value of a layer from r to R

20. GPD Model Fit – Parameter estimates

21. GPD Model Fit – additional estimates – estimated percentiles, expected shortfall

23. Sensitivity analysis – expected shortfall, varying size of single largest loss $5M xs $5M layer price estimate $2,364

25. The entire analysis can be run directly from Excel

26. The entire analysis can be run directly from Excel

27. The entire analysis can be run directly from Excel

28. The entire analysis can be run directly from Excel

29. The entire analysis can be run directly from Excel

30. Bibliography/Resources Beirlant, Jan and Gunther Matthys, G., Heavy Tailed Distributions and Rating, ASTIN Bulletin, 2001, v.31, n.1, p.37-58, http://www.casact.org/library/astin/vol31no1/37.pdf Cebrian, Ana C. , Michel Denuit, and Philippe Lambert, Generalized Pareto Fit to the Society Of Actuaries Large Claims Database, North American Actuarial Journal, 2003, v.7, n.3, p.18-36, http://www.soa.org/bookstore/naaj03_07.html#generalized Chavez-Demoulin, Valerie and Paul Embrechts, Smooth Extremal Models in Finance and Insurance, Journal of Risk and Insurance, 2001, v. 71, n. 2, p. 183-199, http://statwww.epfl.ch/people/chavez/ Coles, Stuart, An Introduction to Statistical Modeling of Extreme Values , Springer, 2001, http://www.maths.bris.ac.uk/~masgc/ismev/summary.html Corradin, Stefano, Economic Risk Capital and Reinsurance: an Application to Fire Claims of an Insurance Company, WP, 2001, http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/StefanoCorradin.pdf

31. Bibliography/Resources, cont. Cummins, J. David, Christopher M. Lewis and Richard D. Phillips, Pricing excess-of-loss reinsurance contracts against catastrophic loss, Wharton WP, 1998, n. 98-9, http://fic.wharton.upenn.edu/fic/papers/98/9809.pdf Joossens, Elisabeth and Johan Segers, Modeling large 3rd party claims in car insurance with an extended GPD, WP, June 2004, http://dad.ulyssis.org/~bettie/motorfleet/tekst.pdf McNeil, Alexander J., Estimating the Tails of Loss Severity Distributions using Extreme Value Theory, ASTIN Bulletin, 1997, v. 27, n. 1, p. 117-137, http://www.casact.org/library/astin/vol27no1/117.pdf McNeil, Alexander J., The Peaks over Thresholds Method for Estimating High Quantiles of Loss Distributions, ASTIN Colloquium, 1997

32. Bibliography/Resources, cont. Reiss, Rolf-Dieter and Michael Thomas , Statistical analysis of extreme values, extended 2nd edition with applications to insurance, finance, hydrology and other fields, Birkhauser, 2001, http://www.xtremes.math.uni-siegen.de/ Smith, Richard L., Statistics of extremes, with applications in environmental science, insurance and finance, U. North Carolina, Statistics WP, July 2002, http://www.stat.unc.edu/postscript/rs/semstatrls.ps See also: SAS Online Docs – Proc NLMIXED

33. Matt Flynn (860) 633-4119 x8764 Matt.Flynn@sas.com