The Logjam Session

1. The Logjam Session Glenn Meyers Insurance Services Office CAS Annual Meeting November, 2004 Full write-up and spreadsheet at: http://www.casact.org/cotor/round2.htm

2. You want to fit a distribution to 250 Claims • Knee jerk first reaction, plot a histogram.

3. This will not do! Take logs • And fit some standard distributions.

4. Still looks skewed. Take double logs. • And fit some standard distributions.

5. Still looks skewed. Take triple logs. • Still some skewness. • Lognormal and gamma fits look somewhat better.

6. Candidate #1Quadruple lognormal

7. Candidate #2Triple loggamma

8. Candidate #3Triple lognormal

9. All three cdf’s are within confidence interval for the quadruple lognormal.

10. Elements of Solution • Three candidate models • Quadruple lognormal • Triple loggamma • Triple lognormal • Parameter uncertainty within each model • Construct a series of models consisting of • One of the three models. • Parameters within a broad confidence interval for each model. • 7803 possible models

11. Steps in Solution • Calculate likelihood (given the data) for each model. • Use Bayes’ Theorem to calculate posterior probability for each model • Each model has equal prior probability.

12. Steps in Solution • Calculate layer pure premium for 5 x 5 layer for each model. • Expected pure premium is the posterior probability weighted average of the model layer pure premiums. • Second moment of pure premium is the posterior probability weighted average of the model layer pure premiums squared.

13. CDF of Layer Pure Premium Probability that layer pure premium ≤ x equals Sum of posterior probabilities for which the model layer pure premium is ≤ x

14. Numerical Results

15. Histogram of Predictive Pure Premium

16. A Guess at the “True” Solution • -0.3 within two standard errors of mu. • 0.1 within two standard errors of sigma. • Stop at quadruple logs. • Guess is quadruple lognormal mu=-0.3 and sigma = 0.1. • Layer cost = 4081.