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 1 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 2 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) 3 ILFs • Consider a loss severity (size per claim) distribution • For example: exponential distribution: f (x) 1 x e F (x) 1 e x E[ X ] 4 ILFs (cont.) • Specifically, consider an exponential distribution with a mean of 20: f (x) 1 x e 20 0 . 05 e 0 . 05 x 20 F (x) 1 e 0 . 05 x 5 ILFs (cont.) • Suppose that the primary insurer writes a “basic limits” policy, with a limit of 25 per occurrence • Notation (Y = insurer’s payment): Y X 25 min( X , 25 ) 6 ILFs (cont.) • One item of interest: P [ X 25 ] 1 e 25 20 1 . 25 1 e 0 . 713 • Another item: what is the expected size of a claim paid by the insurer? E [Y ] E [ X 25 ] 25 xf ( x )dx 0 (1 e 25 ) 20 (1 e 25 f ( x )dx 25 1 . 25 ) 14 . 27 7 ILFs (cont.) • Consider different policy limits written by the primary insurance company: Policy Limit E[Y] “ILF” 25 14.27 1.000 50 18.36 1.287 100 19.87 1.392 20.00 1.402 Unlimited 8 Exposure Rating • Suppose basic limits (BL) premium = $100. Policy Limit Premium 25 (BL) 100.00 50 128.70 100 139.20 Unlimited 140.20 9 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 10 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? 11 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 12 examples.