Estimation and Application of Ranges of Reasonable Estimates Charles L. McClenahan, FCAS, MAAA

1. 2003 Casualty Loss Reserve Seminar Estimation and Application of Ranges of Reasonable EstimatesCharles L. McClenahan, FCAS, MAAA

2. Introduction • “Range of Reasonable Estimates” • Recent Development • Once was informal ± 5% • 5% of what was flexible • 1973 Robert Anker review described three ranges • Absolute Range = Lowest indication to Highest indication • Likely Range = Lowest selected to Highest Selected • Best Estimate Range 2003 Casualty Loss Reserve Seminar 2

3. Introduction (continued) • 1988 Statement of Principles • Principle 3 – “The uncertainty inherent in the estimation of required provisions for unpaid losses or loss adjustment expenses implies that a range of reserves can be actuarially sound. The true value of the liability for losses or loss adjustment expenses at any accounting date can only be known when all attendant claims have been settled.” • Principle 4 – “The most appropriate reserve within a range of actuarially sound estimates depends on both the relative likelihood of estimates within the range and the financial context in which the reserve will be presented.”

4. Introduction (continued) • AAA Committee on Property and Liability Financial Reporting • “a reserve makes a ‘reasonable provision’ if it is within the range of reasonable estimates of the actual outstanding loss and loss adjustment expense obligations.” • the “range of reasonable estimates is a range of estimates that would be produced by alternative sets of assumptions that the actuary judges to be reasonable, considering all information reviewed by the actuary.”

5. Introduction (continued) • Actuarial Standards Board – ASOP No. 36 – Statements of Actuarial Opinion Regarding Property/Casualty Loss and Loss Adjustment Expense Reserves • range of reasonable estimates is “a range of estimates that could be produced by appropriate actuarial methods or alternative sets of assumptions that the actuary judges to be reasonable.”

6. Introduction (continued) • Goals of this paper • Discuss concept of “Range of Reasonable Estimates” • Describe methods for determining range • Demonstrate a sound method for aggregation of line/year ranges • Recommend a basis for application of range to individual decisions

7. Range of Reasonable Estimates • “Reasonable” was unfortunate choice • implies estimates outside range are “unreasonable” • circularity in ASOP No. 36 • would have preferred: • reasonable assumptions • appropriate methodology • actuarially sound estimates

8. Range of Reasonable Estimates (continued) • Range arises from uncertainty associated with estimates • Range reflects both process and parameter variance • Statement of Principles focuses on process variance • ASOP No. 36 focuses on methods and assumptions

9. Range of Reasonable Estimates (continued) • Range does not contain all possibilities • Range may not contain most likely result • Example: • .01 probability of $1 million IBNR • .99 probability of $0 IBNR • Expected IBNR = $10,000 • Actuary sets range at $10,000 to $50,000 • Range excludes mode ($0) and median ($0)

10. Financial Condition and the Range of Reasonable Estimates • Materiality and potential impact influence what is “reasonable” • Return to our $0 or $1 million example • Assume $1 billion surplus • $0 reserve may be reasonable due to immateriality of $1 million loss • $1 million reserve would be unreasonable • Range (?) $0 - $20,000 • Assume $1 million surplus • $0 reserve not reasonable • $1 million reserve may be reasonable due to impact (insolvency) • Range (?) $10,000 - $1,000,000

11. Methods for Estimating Ranges • Assumed Allowable Deviations • Alternative Methods • Alternative Assumptions • Method of Convolutions

12. Methods for Estimating Ranges – Assumed Allowable Deviations • Example ±5% of Total Needed Reserve (TNR) • Assume TNR as follows: • Lognormal • mean = $1,000,000 (µ = 13.469) • c.v. = 1.0 ( = .83255)

13. Methods for Estimating Ranges – Assumed Allowable Deviations

14. Methods for Estimating Ranges – Assumed Allowable Deviations • Range established as ±5% of Total Needed Reserve (TNR) Low = $950,000, High = $1,000,000

15. Methods for Estimating Ranges – Assumed Allowable Deviations • Problems with method • Deviations should vary by line • Calculation of deviation equivalent to calculating range • Best estimate forced to midpoint

16. Methods for Estimating Ranges – Alternative Methods • Most common method in practice today • Run multiple methods and use results to estimate range

17. Methods for Estimating Ranges – Alternative Methods

18. Methods for Estimating Ranges – Alternative Methods • Where methods are independent this is reasonable approach • Adding Bornhuetter-Ferguson to loss development and loss ratio methods provides no additional insight – only weight. • Line by line review essential to check for underlying changes (e.g. case reserve adequacy)

19. Methods for Estimating Ranges – Alternative Assumptions • Actuary picks low (optimistic) and high (pessimistic) factors for each assumption • Results determine range

20. Methods for Estimating Ranges – Alternative Assumptions

21. Methods for Estimating Ranges – Alternative Assumptions • This method tends to produce ranges which are too wide. • Individual age-to-age factors are not successively independent • Combination of many optimistic or pessimistic assumptions produces unreasonably low or high aggregations • There is a way to overcome problems…

22. Methods for Estimating Ranges – Method of Convolutions • Consider a standard 5x5 development triangle

23. Methods for Estimating Ranges – Method of Convolutions • Which gives rise to a 4x4 triangle of development factors

24. Methods for Estimating Ranges – Method of Convolutions • Assume all claims settled by age 60 • Use “Chinese menu” method (“One from column A, …) • 4! (24) combinations for 2001 year • 3! (6) combinations for 2000 year • 2! (2) combinations for 1999 year • 1! (1) combination for 1998 year • 24 x 6 x 2 x 1 = 288 combinations for aggregate ultimate loss

25. Methods for Estimating Ranges – Method of Convolutions • Produces Aggregate IBNR Distribution

26. Methods for Estimating Ranges – Method of Convolutions • Best Estimate (from average factors) between 53rd and 54th percentiles

27. Methods for Estimating Ranges – Method of Convolutions • Example – select range from 10th to 90th percentile as reasonable

28. Methods for Estimating Ranges – Method of Convolutions • In practice, several methods are convoluted • Each method separately • Results combined into single distribution • Since different methods have different numbers of convolutions, must be careful with weighting – e.g. loss ratio method • AY 2002 .680, .690, .700, .710, .720 • AY 2001 .675, .680, .685, .690 • AY 2000 .645, .650, .655 • AY 1999 .678, .680 • 5 x 4 x 3 x 2 = 120 convolutions – must be doubled to roughly equal weight of 288 development factor convolutions

29. Methods for Estimating Ranges – Method of Convolutions • Number of convolutions escalates quickly! • individual values from a k x k development factor triangle (k+1 by k+1 loss triangle) • 4 x 4 triangle: 288 • 8 x 8 triangle: 5,056,584,744,960,000

30. Methods for Estimating Ranges – Method of Convolutions • Limiting Number of Convolutions – One Method • Convolute “youngest” 4x4 triangle and use average for remainder • Example AY 7 as of age 3 • Convolute 3 to 7 development (4x4) and multiply by average 7-ult

31. Methods for Estimating Ranges – Method of Convolutions • Limiting Number of Convolutions – One Method (continued) • Method reduces convolutions for the 8x8 triangle to: • 1! x 2! x 3! x 4! x 4! x 4! x 4! x 4! = 95,551,488 • Reasonable number for computer analysis

32. Aggregation of Ranges • Recall that we are dealing with reasonable estimates, not possibilities • Lows, highs of component estimates cannot be added • Example: Four lines, four open accident years for each line • Assume two reasonable estimates for each (“loway,l” and “highay,l”) • Assume pr(loway,l) = pr(highay,l) = 50% • Sum of reasonable lows is not a reasonable estimate

33. Aggregation of Ranges • A Probability Approach • Toss of ten true coins • Estimate number of “heads” • Reasonable range contains about 90% • Range = 3 to 7 heads • 89% probability

34. Aggregation of Ranges • Consider 10 groups of 10 coins

35. Reasonable (90%) range for number of heads in 100 coins • 42 to 58 heads (91% probability) • If we used the 3 to 7 range 10 times • 30 to 70 heads (99.997% probability)

36. Aggregation of Ranges • A Proposed Method • Assume accident year selections are independent • Assume line of business selections are independent • Not strictly true, but reasonable when applied to most methods • Assume width of range is k (where  is standard deviation of estimates) • Width of aggregate range is square root of sum of squares of individual widths • Aggregate best estimate placement weighted average

37. Aggregation of Ranges • Example

38. Aggregation of Ranges • Example (continued)

39. Application of Ranges • ASOP No. 36 • “When the stated reserve amount is within the actuary’s range of reasonable estimates the actuary should issue a statement of actuarial opinion that the stated reserve amount makes a reasonable provision for the liabilities associated with the specified reserves.” • Statement of Principles • Actuary should consider “both the relative likelihood of estimates within the range and the financial reporting context in which the reserve will be presented.”

40. Application of Ranges • Where company has established the reserve independently of the opining actuary’s analysis (“untutored” reserve) • ASOP No. 36 “stated reserve” language applies • Where company establishes reserve based upon opining actuary’s analysis • Opining actuary now “owns” the estimate and the Statement of Principles language requires the reserves be at or above the opining actuary’s best estimate • Note that this is my opinion, not established doctrine.

41. Conclusion • We must guard against the use of the concept of a “range of reasonable estimates” as justification for carrying reserves which we expect will be inadequate.