Properly Pricing the Catastrophe Exposure

1. Properly Pricing the Catastrophe Exposure David Chernick, FCAS, MAAA Michael Devine, FCAS, MAAA Sara Drexler, FCAS, MAAA CAS Ratemaking Seminar March, 2004

2. Agenda • Introduction • History of Methods • Terminology • Exposure Base • Capping • Discussion of Several Alternative Approaches • Traditional • Methods Using Cat/AIY – State Based • Methods Using Cat/AIY – Countrywide Based • Wrap-up

3. Introduction • The Panelists • The Data • The Issue

4. Introduction • The Panelists Introductions Sara Drexler Michael Devine David Chernick

5. Introduction • The Data The base data we are using in this presentation is included in the handouts. Real data Large Catastrophe in 1998 343.2% loss ratio 9.67 Ratio of Cat/AIY

6. Introduction • The Issue • Operating results • $19.1 Million profit prior to 1998 • $41.4 Million loss in 1998 • Statistics • Mean: • 1998 was 22.7 Standard Deviations from mean.

7. Introduction • The Issue • A rate should include all costs associated with the transfer of risk. • 20 or 30 or even 40 years of data is not sufficient to properly quantify the tail of the distribution • What is the true prospective average (mean) catastrophe provision?

8. Introduction • The Issue • Perspective of this presentation is from large insurers without reinsurance coverage. • Reinsurance covering some portion of the catastrophe exposure would most likely be an upper bound of the true mean. • What is the true prospective average (mean) catastrophe provision?

9. Terminology • AIY – Amount of Insurance years 1AIY=$1,000 of dwelling coverage • Losses/AIY – Damage Ratios or Cat/AIY

10. History of Methods

11. What Base to Relate Catastrophes To? • Premium: Cat provisions impacted by rate changes Trends in non-catastrophe loss & expense dictate cat provision

12. What Base to Relate Catastrophes To? • Premium: Cat provisions impacted by rate changes Trends in non-catastrophe loss & expense dictate cat provision • Non-Cat Loss/Ex-Wind Loss: Still heavily dictated by trends in Crime, Liability, etc. loss Ex-wind losses can include catastrophic losses

13. What Base to Relate Catastrophes To? • Premium: Cat provisions impacted by rate changes Trends in non-catastrophe loss & expense dictate cat provision • Non-Cat Loss/Ex-Wind Loss: Still heavily dictated by trends in Crime, Liability, etc. loss Ex-wind losses can include catastrophic losses • AIY or Amount of Insurance Years Definition:$1000 of Building Coverage in force for one year Inflation sensitive Direct measurement of exposure – incorporates policy growth and changes in building costs

14. Should Individual Catastrophes Be Capped?

15. Should Individual Catastrophes Be Capped? • Stabilizes provision • Can serve to more appropriately match experience period used with event return periods • Potentially more accurate estimate of expected value results

16. What Are Some Problems With Capping Individual Catastrophes?

17. What Are Some Problems With Capping Individual Catastrophes? • What criteria should be used? • The “unthinkable” is happening every year somewhere. • Is the result systematic underestimation of loss costs? • How do we really know appropriate event return periods?

18. Insurance Services Office (ISO)Excess Wind Procedure Basic Approach • Separate wind & non-wind losses • Examine wind/non-wind ratios • Years where wind/non-wind exceed 1.5 times median are “excess” • Average factor for excess wind • Factor developed for excess wind applied to non-wind, non-excess losses

19. Insurance Services Office (ISO)Excess Wind Procedure Basic Approach • Separate wind & non-wind losses • Examine wind/non-wind ratios • Years where wind/non-wind exceed 1.5 times median are “excess” • Average factor for excess wind • Factor developed for excess wind applied to non-wind, non-excess losses Characteristics • Straightforward application • Definition of “excess wind” can change as median changes • Assumes stable relationship between wind & non-wind losses • Doesn’t consider variability of wind losses • Doesn’t consider non-wind catastrophes

20. The Fix ‘Em Up Insurance GroupHomeownersThe State of Mich-con-ota20-Year Average Approach Provision (20-Year Average ) 0.8929

21. Confidence Interval Approach Step 1 – Establish Company Objective: • Factors include risk tolerance, surplus position/ availability of capital, reinsurance • Determine confidence demands for long-term companywide cat provision • Calculate companywide mean cat/aiy • Calculate standard deviation of mean cat/aiy

22. Confidence Interval Approach Step 1 – Establish Company Objective (Cont.): • Company has established that it would like to be 90% certain it has an adequate catastrophe provision over the long-term • The following have been calculated from the companywide catastrophe history: Mean Cat/AIY = .3151 Standard Deviation of Mean Cat/AIY = .0372

23. Confidence Interval Approach Step 1 – Establish Company Objective (Cont.): • The long-run companywide benchmark cat provision is established as follows: Provision (Cat/AIY) = Mean + (t) x (Standard Deviation) = .3151 + (1.323) x (.0372) = .3643 Where : Mean = average cat/aiy companywide 1.323 = t – stat for 90% and (N-1) degrees of freedom .0372 = standard deviation of the mean

24. Confidence Interval Approach Step 2 – Establish State Level Objective: • Goal period becomes interval rates are in effect • Need to be reasonably certain provision is adequate • Desire to use cap on individual cats to limit volatility • Largest 5% of companywide cats exceeded .65/AIY • Establish required confidence for state capped average

25. Non-Hurricane Catastrophe Provisions

26. Confidence Interval Approach Step 2 – Establish State Level Objective (Cont.): • It’s determined that 65% confidence is required • Calculate state mean cat/aiy (capped) • Calculate state standard deviation of cat/aiy (capped)

27. Confidence Interval Approach Step 3 – Calculate State Provision: • The following was calculated from the capped state level catastrophe history: Mean Cat/AIY = .3912 Standard Deviation of Cat/AIY = .5450 • The short-run state cat provision is established as follows: Provision (Cat/AIY) = Mean + (t) x (standard deviation) = (.3912) + (.389) x (.5450) = .6032 Where: Mean = average capped cat/aiy for Mich-con-ota .389 = t – stat for 65% and (N-1) degrees of freedom .5450 = standard deviation of the annual capped cat/aiy

28. The Fix ‘Em Up Insurance GroupHomeownersThe State of Mich-con-otaConfidence Interval Approach Provision (Confidence Interval Approach) 0.6032

29. Issues With Confidence Interval Approach Pluses • Recognizes individual state variability • Stable provision • Provides means to assure companywide sufficiency

30. Issues With Confidence Interval Approach Pluses • Recognizes individual state variability • Stable provision • Provides means to assure companywide sufficiency Drawbacks • Not particularly responsive to distributional changes, coverage changes, etc. (data back to 1971) • Capping can result in less responsiveness • Recognition of variability interpreted as risk margin

31. The Fix ‘Em Up Insurance GroupHomeownersThe State of Mich-con-otaExtreme Events Adjustment 20 Year Average 0.0871 0.4576 Provision (Extreme plus Non-Extreme) 0.5446

32. Extreme Events Adjustment Pluses • Relatively stable • As opposed to censoring, reflects events fully Drawbacks • Accurate determination of event return period difficult • Can be viewed as arbitrary and difficult to explain

33. 95% / 5% Trended Approach: Methodology: • All years used • Exponential smoothing • Trend factor applied – recognizes static cat definition • 10% annual cap to change in provision

34. The Fix ‘Em Up Insurance GroupHomeownersThe State of Mich-con-ota95/5 Trended Provision (95/5 Trended) 0.4002

35. 95% / 5% Trended Approach: Advantages: • Not volatile, yet responsive for non-extreme events • Simple to understand • Trend factor to compensate for static definition of cats • Reduced data complications

36. Summary of Results So Far:

37. Pivotal Question: Can Countrywide or Regional Data Help Quantify the True Prospective Mean Catastrophe Loss in a Given State?

38. Pivotal Question: Can Countrywide or Regional Data Help Quantify the True Prospective Mean Catastrophe Loss in a Given State? Issues: • Provisions need to reflect adequacy and stability • All company surplus is generally available and at risk • Are regional or sub state provisions appropriate? • Perceived cost sharing will be scrutinized

39. Goals of Relativity Method • Develop accurate, stable results by state that results in an appropriate provision on a countrywide basis • Systematic approach to handle extreme events so a single outlying year does not drive the cat provision for a state • Appropriate application of credibility procedure • Provide result that is responsive to recent demographic and cat definition shifts

40. Issues Addressed • How to be responsive to changes in risk due to population shifts or cat definition changes while still including an appropriate number of years • How does one define an outlying event • Individual state vs. countrywide • How to incorporate credibility

41. State Relativity Weighted with Countrywide Complement – General Outline • Develop State Damage Ratios • Calculate Countywide Damage Ratios • Calculate State Relativities • Cap State Relativities • Average Capped Relativities • Credibility Weight with CW Average of 1.000 • Balance Back to CW Average of 1.000 • Calculate Statewide Catastrophe Provision

42. Develop State Damage Ratios • Calculate Countrywide Damage Ratios • Calculate State Relativities • Cap State Relativities • Average Capped Relativities • Credibility Weight with CW Average of 1.000 • Balance Back to CW Average of 1.000 • Calculate Statewide Catastrophe Provision State Relativity Weighted with Countrywide Complement • Develop each state’s damage ratios for years 1981-2000 • State Damage Ratios – Losses/AIY • Only use years 1981 forward. Data for years 1971 through 1980 is sparse as evidenced by yearly variance.

43. Develop State Damage Ratios • Calculate Countrywide Damage Ratios • Calculate State Relativities • Cap State Relativities • Average Capped Relativities • Credibility Weight with CW Average of 1.000 • Balance Back to CW Average of 1.000 • Calculate Statewide Catastrophe Provision State Relativity Weighted with Countrywide Complement • Each year’s Countrywide damage ratio is calculated as the weighted average of state damage ratios using latest year AIYs as weights • Eliminates distortion of state distributional shifts over time • Countrywide catastrophe provision is the arithmetic average of the most recent 10 years of damage ratios

44. Figure 1 YearsLinear trend 1971-1978 0.006 1979-1989 0.000 1990-1999 -0.019 1990-2000 -0.010

45. Develop State Damage Ratios • Calculate Countrywide Damage Ratios • Calculate State Relativities • Cap State Relativities • Average Capped Relativities • Credibility Weight with CW Average of 1.000 • Balance Back to CW Average of 1.000 • Calculate Statewide Catastrophe Provision State Relativity Weighted with Countrywide Complement • Calculate state relativities as the ratio of state damage ratios to countrywide damage ratios • Relativities should be more stable than damage ratios • Trend should not be a problem so we can use more years of data than the Countrywide Catastrophe Provision

46. Develop State Damage Ratios • Calculate Countrywide Damage Ratios • Calculate State Relativities • Cap State Relativities • Average Capped Relativities • Credibility Weight with CW Average of 1.000 • Balance Back to CW Average of 1.000 • Calculate Statewide Catastrophe Provision State Relativity Weighted with Countrywide Complement • Any relativity greater than the mean plus three standard deviations is capped to the next lowest relativity (not the cap number) • Intuitively we are replacing a once in a hundred year event with a once in 20 • Benefit of capping process • Represents a systematic approach to dealing with extreme events • Cap is dynamic and is allowed to shift if exposure in a state is changing over time • Censoring at the cap would not have much impact and therefore would not result in increased stability

47. Develop State Damage Ratios • Calculate Countrywide Damage Ratios • Calculate State Relativities • Cap State Relativities • Average Capped Relativities • Credibility Weight with CW Average of 1.000 • Balance Back to CW Average of 1.000 • Calculate Statewide Catastrophe Provision State Relativity Weighted with Countrywide Complement • Calculate arithmetic average of 1981-2000 capped relativities • Using a arithmetic average is simple • No benefit of weighting relativities has been shown since relationship of variability to exposure level is unclear • Arithmetic average relativity does not differ significantly from an AIY weighted average

48. Develop State Damage Ratios • Calculate Countrywide Damage Ratios • Calculate State Relativities • Cap State Relativities • Average Capped Relativities • Credibility Weight with CW Average of 1.000 • Balance Back to CW Average of 1.000 • Calculate Statewide Catastrophe Provision State Relativity Weighted with Countrywide Complement • Uses Buhlmann credibility factor: n/(n+k) • n = number of years of relativities in average • We use number of years rather than exposures because exposures not independent, especially past a certain threshold where exposure concentration increases • k = average process variance/variance of hypothetical means • The process variance and variance of hypothetical means are calculated using all available years of capped relativities across all states. • Complement of credibility of 1.000 is not appropriate when there is a wide spread of average relativities • Solution lies in balancing process described on next slide

49. Develop State Damage Ratios • Calculate Countrywide Damage Ratios • Calculate State Relativities • Cap State Relativities • Average Capped Relativities • Credibility Weight with CW Average of 1.000 • Balance Back to CW Average of 1.000 • Calculate Statewide Catastrophe Provision State Relativity Weighted with Countrywide Complement • At this point, the individual state relativities result in a countrywide relativity of less than 1.000. Relativities are adjusted to achieve an overall adequate level as follows: • Determined on a countrywide basis what our expected losses would be based on the countrywide selected catastrophe factor • Sum the pre-balanced expected losses across all states • We distribute the difference between 1 and 2 in proportion to each state’s standard deviation measured in latest year expected losses. • Using this approach has several benefits: • Results in an appropriate provision countrywide • It compensates for high (low) relativity states being underestimated (overestimated) by the use of a 1.000 complement of credibility. • Each state’s resulting cat load is a function of its own size and variability

50. Develop State Damage Ratios • Calculate Countrywide Damage Ratios • Calculate State Relativities • Cap State Relativities • Average Capped Relativities • Credibility Weight with CW Average of 1.000 • Balance Back to CW Average of 1.000 • Calculate Statewide Catastrophe Provision State Relativity Weighted with Countrywide Complement • Statewide catastrophe provision is calculated by multiplying capped, credibility weighted, balanced relativity by the countrywide catastrophe provision • Benefits of method: • Allows use of long term data to determine relativity while using more responsive data for countrywide provision • Adjustments to data are determined objectively with each state’s characteristics used to determine both capping and balancing