Overview

1. Reserve Variability – Session I:Where Are We Today? Mark R. Shapland, FCAS, ASA, MAAACasualty Actuarial Society Spring MeetingSan Juan, Puerto RicoMay 7-10, 2006

2. Definitions of Terms Ranges vs. Distributions Methods vs. Models Types of Methods/Models What is “Reasonable?” Overview

3. Statements of Statutory Accounting Principles (SSAP): Definitions of Terms • Management’s Best Estimate – Management’s best estimate of its liabilities is to be recorded. This amount may or may not equal the actuary’s best estimate. • Ranges of Reserve Estimates – When management believes no estimate is better than any other within the range, management should accrue the midpoint. If a range can’t be determined, management should accrue the best estimate. Management’s range may or may not equal the actuary’s range. • Best Estimate by Line – Management should accrue its best estimate by line of business and in the aggregate. Recognized redundancies in one line of business cannot be used to offset recognized deficiencies in another line of business.

4. Actuarial Statement of Principles No. 36 (ASOP 36): Definitions of Terms • Risk Margin– An amount that recognizes uncertainty; also known as a provision for uncertainty. • Determination of Reasonable Provision– When the stated reserve amount is within the actuary’s range of reasonable reserve estimates, the actuary should issue a statement of actuarial opinion that the stated reserve amount makes a reasonable provision for the liabilities.

5. Actuarial Statement of Principles No. 36 (ASOP 36): Definitions of Terms • Range of Reasonable Reserve Estimates– The actuary may determine a range of reasonable reserve estimates that reflects the uncertainties associated with analyzing the reserves. A 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. The actuary may include risk margins in a range of reasonable estimates, but is not required to do so. A range of reasonable reserves, however, usually does not represent the range of all possible outcomes.

6. Current ASB Exposure Draft: Definitions of Terms • Actuarial Central Estimate– An estimate that represents a mean excluding remote or speculative outcomes that, in the actuary’s professional judgment, is neither optimistic nor pessimistic. An actuarial central estimate may or may not be the result of the use of a probability distribution or a statistical analysis. • Reasonableness – The actuary should assess the reasonableness of the unpaid claim estimate, using appropriate indicators or tests that, in the actuary’s professional judgment, provide a validation that the unpaid claim estimate is reasonable.

7. Current Principles Exposure Draft: Definitions of Terms • Actuarially Reasonable Loss Reserve – An actuarially reasonable loss reserve for a defined group of claims is an estimate, derived from reasonable assumptions and appropriate methods. • Probability Distribution Representation – The unpaid amounts required to settle a defined group of claims can be represented as a probability distribution for which neither the form nor the parameters are necessarily known. • Range of Reserves – The uncertainty in the estimate of the unpaid amounts required to settle a defined group of claims implies that a range of loss reserves can be actuarially reasonable.

8. Other Definitions : Definitions of Terms • Reserve – an amount carried in the liability section of a risk-bearing entity’s balance sheet for claims incurred prior to a given accounting date. • Liability – the actual amount that is owed and will ultimately be paid by a risk-bearing entity for claims incurred prior to a given accounting date. • Loss Liability – the expected value of all estimated future claim payments. • Risk (from the “risk-bearers” point of view) – the uncertainty (deviations from expected) in both timing and amount of the future claim payment stream.

9. Process Risk – the randomness of future outcomes given a known distribution of possible outcomes. Parameter Risk – the potential error in the estimated parameters used to describe the distribution of possible outcomes, assuming the process generating the outcomes is known. Model Risk – the chance that the model (“process”) used to estimate the distribution of possible outcomes is incorrect or incomplete. Other Definitions : Definitions of Terms

10. Measures of Risk from Statistics: • Variance, standard deviation, kurtosis, average absolute deviation, Value at Risk, Tail Value at Risk, etc. which are measures of dispersion. • Other measures useful in determining “reasonableness” could include: mean, mode, median, pain function, etc. • The choice for measure of risk will also be important when considering the “reasonableness” and “materiality” of the reserves in relation to the capital position. Definitions of Terms

11. A “Range” is not the same as a “Distribution” A 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. A Distribution is a statistical function that attempts to quantify probabilities of all possible outcomes. Ranges vs. Distributions

12. A Range, by itself, creates problems: • A range can be misleading to the layperson – it can give the impression that any number in that range is equally likely. • A range can give the impression that as long as the carried reserve is “within the range” anything is reasonable. Ranges vs. Distributions

13. A Range, by itself, creates problems: • There is currently no specific guidance on how to consistently determine a range within the actuarial community (e.g., +/- X%, +/- $X, using various estimates, etc.). • A range, in and of itself, needs some other context to help define it (e.g., how to you calculate a risk margin?) Ranges vs. Distributions

14. A Distribution provides: • Information about “all” possible outcomes. • Context for defining a variety of other measures (e.g., risk margin, materiality, risk based capital, etc.) Ranges vs. Distributions

15. Should we use the same: • criterion for judging the quality of a range vs. a distribution? • basis for determining materiality? risk margins? • selection process for which numbers are “reasonable” to chose from? Ranges vs. Distributions

16. A Method is an algorithm or recipe – a series of steps that are followed to give an estimate of future payments. The well known chain ladder (CL) and Bornhuetter-Ferguson (BF) methods are examples Methods vs. Models

17. A Model specifies statistical assumptions about the loss process, usually leaving some parameters to be estimated. Then estimating the parameters gives an estimate of the ultimate losses and some statistical properties of that estimate. Methods vs. Models

18. Many good probability models have been built using “Collective Risk Theory” • Each of these models make assumptions about the processes that are driving claims and their settlement values • None of them can ever completely eliminate “model risk” Methods vs. Models

19. Processes used to calculate liability ranges can be grouped into four general categories: 1) Multiple Projection Methods, 2) Statistics from Link Ratio Models, 3) Incremental Models, and 4) Simulation Models Methods vs. Models

20. Description: • Uses multiple methods, data, assumptions • Assume various estimates are a good proxy for the variation of the expected outcomes Multiple Projection Methods Primary Advantages: • Better than no range at all • Better than +/- X%

21. Problems: • It does not provide a measure of the density of the distribution for the purpose of producing a probability function • The “distribution” of the estimates is a distribution of the methods and assumptions used, not a distribution of the expected future claim payments. • Link ratio methods only produce a single point estimate and there is no statistical process for determining if this point estimate is close to the expected value of the distribution of possible outcomes or not. Multiple Projection Methods

22. Problems: • Since there are no statistical measures for these models, any overall distribution for all lines of business combined will be based on the addition of the individual ranges by line of business with judgmental adjustments for covariance, if any. Multiple Projection Methods

23. Uses: • Data limitations may prevent the use of more advanced models. • A strict interpretation of the guidelines in ASOP No. 36 seems to imply the use of this “method” to create a “reasonable” range Multiple Projection Methods

24. Description: • Calculate standard error for link ratios to calculate distribution of outcomes / range • Typically assume normality and use logs to get a skewed distribution • Examples: Mack, Murphy, Bootstrapping and others Statistics from Link Ratio Models Primary Advantages: • Significant improvement over multiple projections • Focused on a distribution of possible outcomes

25. Problems: • The expected value often based on multiple methods • Often assume link ratio errors are normally distributed and constant by (accident) year – this violates three criterion • Provides a process for calculating an overall probability distribution for all lines of business combined, still requires assumptions about the covariances between lines Statistics from Link Ratio Models

26. Uses: • If data limitations prevent the use of more sophisticated models Caveats: • Need to make sure statistical tests are satisfied. • ASOP No. 36 still applies to the expected value portion of the calculations Statistics from Link Ratio Models

27. Description: • Directly model distribution of incremental claims • Typically assume lognormal or other skewed distribution • Examples: Finger, Hachmeister, Zehnwirth, England, Verrall and others Incremental Models Primary Advantages: • Overcome the “limitations” of using cumulative values • Modeling of calendar year inflation (along the diagonal)

28. Problems: • Actual distribution of incremental payments may not be lognormal, but other skewed distributions generally add complexity to the formulations • Correlations between lines will need to be considered when they are combined (but can usually be directly estimated) • Main limitation to these models seems to be only when some data issues are present Incremental Models

29. Uses: • Usually, they allow the actuary to tailor the model parameters to fit the characteristics of the data. • An added bonus is that some of these models allow the actuary to thoroughly test the model parameters and assumptions to see if they are supported by the data. • They also allow the actuary to compare various goodness of fit statistics to evaluate the reasonableness of different models and/or different model parameters. Incremental Models

30. Description: • Dynamic risk model of the complex interactions between claims, reinsurance, surplus, etc., • Models from other groups can be used to create such a risk model Simulation Models Primary Advantage: • Can generate a robust estimate of the distribution of possible outcomes

31. Problems: • Models based on link ratios often exhibit statistical properties not found in the real data being modeled. • Usually overcome with models based on incremental values or with ground-up simulations using separate parameters for claim frequency, severity, closure rates, etc. • As with any model, the key is to make sure the model and model parameters are a close reflection of reality. Simulation Models

32. A Range, by itself, creates problems: • A range (arbitrary or otherwise) can be misleading to the layperson – it can give the impression that any number in that range is equally likely. • A range can also give a false sense of security to the layperson – it gives the impression that as long as the carried reserve is “within the range” anything is reasonable (and therefore in compliance) as long as it can be justified by other means. What is “Reasonable”?

33. A Range, by itself, creates problems: • There is currently no specific guidance on how to consistently determine a range within the actuarial community (e.g., +/- X%, +/- $X, using various estimates, etc.). • A Range, in and of itself, has insufficient meaning without some other context to help define it. What is “Reasonable”?

34. What is “Reasonable”? $16M $11M

35. We should define a “reasonable” range based on probabilities of the distribution of possible outcomes. This can be translated into a range of liabilities that correspond to those probabilities. Premise: What is “Reasonable”?

36. A probability range has several advantages: • The “risk” in the data defines the range. • Adds context to other statistical measures. • A “reserve margin” can be defined more precisely. • Can be related to risk of insolvency and materiality issues. • Others can define what is reasonable for them. What is “Reasonable”?

37. A probability range has several advantages: • The “risk” in the data defines the range. • Adds context to other statistical measures. • A “reserve margin” can be defined more precisely. • Can be related to risk of insolvency and materiality issues. • Others can define what is reasonable for them. What is “Reasonable”?

38. Relatively Stable LOB More Volatile LOB Method Low EV High Low EV High Expected +/- 20% 80 100 120 80 100 120 50th to 75th Percentile 97 100 115 90 100 150 Comparison of “Reasonable” Reserve Ranges by Method What is “Reasonable”?

39. A probability range has several advantages: • The “risk” in the data defines the range. • Adds context to other statistical measures. • A “reserve margin” can be defined more precisely. • Can be related to risk of insolvency and materiality issues. • Others can define what is reasonable for them. What is “Reasonable”?

40. Comparison of “Normal” vs. “Skewed” Liability Distributions “Normally” Distributed Liabilities “Skewed” Liability Distribution What is “Reasonable”? 50th Percentile 50th Percentile Probability Probability Mode Expected Value Expected Value, Mode & Median Median Value Value

41. Comparison of Aggregate Liability Distributions What is “Reasonable”?

42. A probability range has several advantages: • The “risk” in the data defines the range. • Adds context to other statistical measures. • A “reserve margin” can be defined more precisely. • Can be related to risk of insolvency and materiality issues. • Others can define what is reasonable for them. What is “Reasonable”?

43. 75th Percentile th 50 Percentile Reasonable & Prudent Margin What is “Reasonable”? Probability Reasonable & Conservative Margin Expected Value Mode Median Liability Estimate

44. A probability range has several advantages: • The “risk” in the data defines the range. • Adds context to other statistical measures. • A “reserve margin” can be defined more precisely. • Can be related to risk of insolvency and materiality issues. • Others can define what is reasonable for them. What is “Reasonable”?

45. Corresponding Surplus Depending on Situation Loss Reserves Situation A Situation B Situation C Amount Prob. Amount Prob. Of Ins. Amount Prob. Of Ins. Amount Prob. Of Ins. 100 50% 80 40% 120 15% 160 1% 110 75% 70 40% 110 15% 150 1% 120 90% 60 40% 100 15% 140 1% Comparison of “Reasonable” Reserve Ranges with Probabilities of Insolvency “Low” Reserve Risk What is “Reasonable”?

46. Corresponding Surplus Depending on Situation Loss Reserves Situation A Situation B Situation C Amount Prob. Amount Prob. Of Ins. Amount Prob. Of Ins. Amount Prob. Of Ins. 100 50% 80 60% 120 40% 160 10% 120 75% 60 60% 100 40% 140 10% 140 90% 40 60% 80 40% 120 10% Comparison of “Reasonable” Reserve Ranges with Probabilities of Insolvency “Medium” Reserve Risk What is “Reasonable”?

47. Corresponding Surplus Depending on Situation Loss Reserves Situation A Situation B Situation C Amount Prob. Amount Prob. Of Ins. Amount Prob. Of Ins. Amount Prob. Of Ins. 100 50% 80 80% 120 50% 160 20% 150 75% 30 80% 70 50% 110 20% 200 90% -20 80% 20 50% 60 20% Comparison of “Reasonable” Reserve Ranges with Probabilities of Insolvency “High” Reserve Risk What is “Reasonable”?

48. A probability range has several advantages: • The “risk” in the data defines the range. • Adds context to other statistical measures. • A “reserve margin” can be defined more precisely. • Can be related to risk of insolvency and materiality issues. • Others can define what is reasonable for them. What is “Reasonable”?

49. Satisfying Different Constituents: • Principle of Greatest Common Interest – the “largest amount” considered “reasonable” when a variety of constituents share a common goal or interest, such that all common goals or interests are met; and the • Principle of Least Common Interest – the “smallest amount” considered “reasonable” when a variety of constituents share a common goal or interest, such that all common goals or interests are met. What is “Reasonable”?

50. What is “Reasonable”? $16M $11M