Ranges of Reasonable Estimates Charles L. McClenahan, FCAS, MAAA

1. Iowa Actuaries Club, February 9, 2004 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) • Statement of Actuarial Opinion on P&C Loss Reserves • Year-end 2004 • Actuary must include either a point estimate or a range of reasonable estimates for gross and net reserves in the Actuarial Report • Year-end 2005 • Confidential Actuarial Opinion Summary (AOS) • Range (gross and net) • Point-estimate (gross and net)

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

8. 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

9. 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

10. 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)

11. Some Methods for Estimating Ranges • Assumed Allowable Deviations • Alternative Methods • Alternative Assumptions

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 = $1,050,000

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 • Calculates all (or most) combinations of observed development factors • Application to latest diagonal gives rise to distribution of ultimate losses • First documented application circa 1992 by C. K. Stan Khury • Methodology discussed in my Fall 2003 CAS Forum paper

23. 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

24. 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

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

26. Aggregation of Ranges • 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)

27. 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

28. Aggregation of Ranges • Example

29. Aggregation of Ranges • Example (continued)

30. Problems With the Application of Ranges Confusion about the concept Some view “range of reasonable estimates” in the context of an individual actuary – Different assumptions, different methods, distribution of TNR

31. Problems With the Application of Ranges

32. Problems With the Application of Ranges Confusion about the concept Some view “range of reasonable estimates” in the context of the universe of actuaries – Different assumptions, different methods, distribution of “best estimates”

33. Problems With the Application of Ranges

34. Problems With the Application of Ranges Consider the following hypothetical: Appointed Actuary uses individual actuary interpretation

35. Problems With the Application of Ranges Consider the following hypothetical: Appointed Actuary uses individual actuary interpretation Appointed Actuary is also “Low” Actuary

36. Problems With the Application of Ranges Minimum “clean opinion” reserve becomes low actuary’s low

37. Problems With the Application of Ranges Many reserves judged reasonable by other actuaries outside range

38. Application of Ranges in Individual Situations • 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.”

39. Application of Ranges in Individual Situations • A “Modest Proposal” (apologies to Jonathan Swift) • Profession should agree that “range of reasonable estimates” be interpreted as a range of alternative “best estimates” (multiple actuaries approach) • Concept of range should be used only in reviewing estimates of others, not in establishing liabilities

40. Application of Ranges in Individual Situations • 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.

42. For More Information • Fall 2003 CAS Forum • Probabilistic Framework for Evaluating Materiality and Variability in Loss Reserve Estimates – Bass & Khury • Measurement of Reserve Variability - Hayne • Estimation and Application of Ranges of Reasonable Estimates - McClenahan • Loss Reserve Estimates: A Statistical Approach for Determining “Reasonableness” – Shapland