Practical Considerations for Catastrophe Aggregate Distribution Modeling

1. CARe Limited Attendance Seminar Some Practical Considerations for Catastrophe Aggregate Distribution Modeling Peter R. Martin, FCAS Axis Specialty

2. Two “Facts”: • All commercially available cat models have portfolio management modules • None of these models provide a true portfolio management system

3. What Do I Mean By This? • May not provide for consideration of some treaty types—e.g. retro, stop loss, per risk, etc. • No consideration of risk-attaching pro-rata treaties—potential losses above single treaty year event limit • No ability to easily model prospective portfolios, e.g January ‘03 in-force as at December ‘02 • Portfolio optimization routines difficult to run within the models • Difficult to test alternative frequency distributions for annual aggregate analysis, e.g. negative binomial for European Wind • Linking portfolio modeling to pricing selections cumbersome

4. Implication • Reinsurers need to develop their own portfolio management application, which: • Integrates with financial, pricing, and cat modeling systems • Is fast • Is flexible

5. Possible Process • For each program, create a vector of ground up losses for a specific, fixed set of events • Use any cat model • Record expected loss by peril for each contract (extract from pricing system) • Iteratively solve for scalars which (when applied to ground up loss vector) generate target expected loss cost to each layer by peril • Store vector and scalars—Contract vectors form matrix which, when multiplied by vector representing company participation produces a vector representing company cat loss distribution for each peril

6. Each Contract By Peril = 5 Events 5 Contracts Participations Loss Dist

7. Cat Model #1 Ground Up Vectors Cat Model #2 Ground Up Vectors Cat Model #3 Scaling Routine Accumulation Routine Frequency Assumption Convolution Method Pictorial Representation of Process Excel Pricing Application Data Store Annual Aggregate Distribution Severity Distribution

8. Hard Part—The Scaling Routine • Scaling one of the perils can lead to other perils getting out of balance (on limited reinstatement contracts) • Approach:

9. Peril1 Events x Scale1 Peril2 Events x Scale2 Periln Events x Scalen Layered EL Peril1 EL Peril2 EL Periln Target EL1 >EL1 Target EL1 <=EL1 Target EL2 >EL2 Target EL2 <=EL2 Target ELn <=ELn Target ELn >ELn Scale1 = Scale1 - D Scale1 = Scale1 + D Scale2 = Scale2 - D Scale2 = Scale2 + D Scalen = Scalen - D Scalen = Scalen + D D = D/2 Ground up vector Default Scale Factor Scaling

10. Advantages • Fast • Can scale, accumulate large portfolio in under 30 minutes using off the shelf database products and high spec processor • Quick enough to provide “real-time” incremental pricing analysis • Flexible • Most every type of cat exposed business can be analyzed, all you have to do is develop the vector of losses • Easy to add/remove contracts to analysis—perfect for optimization routines • Simple integration with DFA analysis • Can use FFT/Panjer/Simulation to develop annual aggregate distributions under various frequency assumptions • Consistent • Produces results that tie with pricing analysis

11. Disadvantages • Logic Flaws • Why apply a different scale factor to ground-up losses for different contracts in the same program? • Scaling can conceivably lead to higher layers having limit losses (for a given event) while lower layers have partials • Multiple Models • Doesn’t explicitly address the fact that different models have different shaped curves • Implicitly considered through blended selection of E(L) • Requires more “guesswork” from underwriters • Underwriters need to select expected loss by peril—is this reasonable? • May lead to abuse, e.g. understating hurricane E(L) to make incremental loss analysis less punitive • Scenario analysis more difficult