1. 1 Math 479 / 568 Casualty Actuarial Mathematics Fall 2014 University of Illinois at Urbana-Champaign Professor Rick Gorvett Session 13: Reinsurance II – Exposure Rating and Increased Limits Factors October 14, 2014

2. 2 Reinsurance Pricing Pro rata reinsurance –Straightforward pricing, since a percentage –Consider adequacy of primary pricing Excess reinsurance –Price is generally specified as a percentage of “ subject premium ” –Two traditional approaches to pricing: Experience rating Exposure rating

3. 3 Exposure Rating Allocation of primary premium to different “ layers ” of coverage Consider adequacy of primary premium Depends upon: –Attachment point and limit of reinsurance layer –Primary premiums at different policy limits –Increased limits factors (ILFs)

4. 4 ILFs Consider a loss severity (size per claim) distribution For example: exponential distribution:

5. 5 ILFs (cont.) Specifically, consider an exponential distribution with a mean of 20:

6. 6 ILFs (cont.) Suppose that the primary insurer writes a “ basic limits ” policy, with a limit of 25 per occurrence Notation (Y = insurer ’ s payment):

7. 7 ILFs (cont.) One item of interest: Another item: what is the expected size of a claim paid by the insurer?

8. 8 ILFs (cont.) Consider different policy limits written by the primary insurance company: Policy Limit E[Y] “ ILF ” 2514.271.000 5018.361.287 10019.871.392 Unlimited20.001.402

9. 9 Exposure Rating Suppose basic limits (BL) premium = $100. Policy LimitPremium 25 (BL) 100.00 50 128.70 100 139.20 Unlimited 140.20

10. 10 Exposure Rating (cont.) Consider 50 x/s 50 layer: –Worth 139.20 - 128.70 = 10.50 –But this value can also be determined via ILFs: 100 (1.392-1.287) = 10.50 Exposure rating involves using quantitative relationships between limits to allocate premium to different layers

11. 11 Examples What proportion of the primary premium on a 100 limit policy should go to the 50 x/s 50 reinsurance layer? Suppose you determined that primary premiums are, in general, 10% inadequate. What proportion of the 50 limit policy premium should go to the 25 x/s 25 layer?

12. 12 Caveat A big issue here is the nature of the ILFs. –How are expenses handled? –How is risk load handled? A reinsurer might want to use loss-only ILFs, without risk load, to determine the loss portion of the layer, then determine the final reinsurance premium by incorporating its own expenses and risk load. Thus, adjustments may need to be made to results associated with situations like the examples.