1. CARe Seminar ILF estimation Oliver Bettis 15 th September 2009

2. 2 What is an Increased Limit Factor? The increased limit factor allows excess layer premiums to be calculated from underlying layer premiums. 21/02/2016TITLE i.e. ILF = LEV(increased limit)/LEV(base limit) ILF does not depend on the frequency of losses.

3. 3 Example ILF Curve 21/02/2016TITLE ILFs must always: Monotonically increase Increase at a decreasing rate. (The difference between successive ILFs approaches 0.)

4. 4 Empirical ILF versus ILF from fitting a probability distribution to the claim sizes On higher excess layers there may be no observed losses. Empirical ILF gives zero premium. Smoothing. Empirical ILF is subject to random variation. It might give same premium for many different layers e.g. 10m xs 50m and 10m xs 60m might pay the same amount using an empirical ILF. 21/02/2016TITLE

5. 5 Example of claim size pdfs that can be used to use for ILFs 21/02/2016TITLE TypeThickness of tail Result ParetoThickLayer with double limit and attachment of underlying pays fixed factor of underlying premium. e.g. 20x20 pays 1.5 times 10x10 layer 40x40 pays 1.5 times 20x20 layer LognormalMediumFaster drop off in premium with increasing attachment point than Pareto. Two parameters so quite flexible. ExponentialThinIncrease attachment by fixed amount and layer pays fixed factor of underlying. e.g. 10x20 pays 75% of 10x10 layer 10x30 pays 75% of 10x20 layer

6. 6 21/02/2016TITLE Example method for ILF estimation: T/P liability claims First step - collect claim data & categorise claims by business sector Circa 150 claims in this data set, collected over 1 year. Sample of claims below:

7. 7 21/02/2016TITLE Group business sectors by severity category E.g. by quartile of mean claim size Category 1 = business sectors least exposed to large claims Category 4 = business sectors most exposed to large claims

8. 8 Analyse claims within each severity category Arrange claims in order within each category to get empirical percentiles 21/02/2016TITLE

9. 9 Decide on pdf to use for loss distribution Lognormal may be useful for casualty lines Obtain best fit curve e.g. by MLE method 21/02/2016TITLE

10. 10 Calculate percentiles from best fit lognormal curve 21/02/2016TITLE

11. 11 Compare percentiles from claim data to percentiles from best fit distribution 21/02/2016TITLE

12. 12 Use different fitting methods e.g. percentile fitting Or judgementally adjust parameters to get a better fit ILF curve on part of curve most at interest 21/02/2016TITLE

13. 13 Calculate empirical risk premium to layer by slicing FGU losses into layers 21/02/2016TITLE

14. 14 Plot premiums obtained from fitted ILFs vs empirical ILFs to show difference between excess layer premiums 21/02/2016TITLE Chart of risk premium for a $10 million layer at various attachment points, where primary $10 million layer pays $100

15. 15 Plot relative premiums for various size primary layers Compare fitted ILFs vs empirical ILFs 21/02/2016TITLE Chart of risk premium for various primary layers ($2.5m, $5m, $7.5m etc), where primary $10 million layer pays $100.

16. Thank you very much for your attention.