Determining Reserve Ranges and the Variability of Loss Reserves

1. Determining Reserve Ranges and the Variability of Loss Reserves CLRS 2001 by Rodney Kreps Guy Carpenter Instrat

2. Motivation: ASOP 36, 3.6.3 Expected Value Estimate – In evaluating the reasonableness of reserves, the actuary should consider one or more expected value estimates of the reserves, except when such estimates cannot be made based on available data and reasonable assumptions. Other statistical values such as the mode (most likely value) or the median (50th percentile) may not be appropriate measures for evaluating loss and loss adjustment expense reserves, such as when the expected value estimates can be significantly greater than these other estimates. Translation: use the mean. We don’t want low reserves.

3. ASOP 36, 3.6.3 (cont.) The actuary may use various methods or assumptions to arrive at expected value estimates. In arriving at such expected value estimates, it is not necessary to estimate or determine the range of all possible values, nor the probabilities associated with any particular values. Translation: distribution? I think I saw one once. Expected? This is what I expect will do me the most good.

4. ASOP 36, 3.6.4 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. Translation: how many ways can you get a mean? Without even introducing a distribution?

5. The real title of this talk is: Why the mean? OR Why the mean is almost surely not the best reserve measure. OR What is an economically rational basis for reserve measures? To answer this we need to go back to the beginning and consider fundamentals —

6. Why can’t you actuaries get the reserves right? Feel like a target?

7. What are Reserves? • Actual Dollars To Be Paid. • Distribution of Potential Actual Dollars To Be Paid. • Estimator of the Distribution of Potential Actual Dollars To Be Paid. • An esoteric mystery dependent on the whims of the CFO/CEO.

8. And the Right Answer - ALL of the above.

9. What are Reserves? • Actual Dollars To Be Paid.

10. Actual Dollars • Are only known after runoff. • Give a hindsight view. • Lie behind the question “Why can’t you get it right?” A changing estimate does not imply a mistake.

11. What are Reserves? • Actual Dollars To Be Paid. • Distribution of Potential Actual Dollars To Be Paid.

12. Distribution of Potential Actual Dollars To Be Paid • All planning quantities are distributions. • ALL planning quantities are distributions. • ALL planning quantities are DISTRIBUTIONS. • Basically, anything interesting in the future is a distribution.

13. Distributions are frequently characterized by spread and estimator • However, the choice of these is basically a subjective matter. • Mathematical convenience of calculation is not necessarily a good criterion for choice. • Neither is “Gramps did it this way.”

14. Measures of spread • Standard deviation • Usual confidence interval • Minimum uncertainty

15. Standard deviation • Simple formula. • Other spread measures often expressed as plus or minus so many standard deviations. • Familiar from (ab)normal distribution.

16. Usual confidence interval • Sense is, “How large an interval do I need to be reasonably comfortable that the value is in it?” • E.g., 90% confidence interval. Why 90%? • Why not 95%? 99%? 99.9%? • Statisticians’ canonical comfort level seems to be 95%. • Choice depends on situation and individual.

17. Minimum uncertainty • AKA “Intrinsic uncertainty,” Softness,” or “Slop.” • All estimates and most measurements have intrinsic uncertainty. • A stochastic variable is essentially not known to within its intrinsic uncertainty. • Sense is, “What is the smallest interval containing the value?”

18. Minimum uncertainty (2) • “How little can I include and not be too uncomfortable pretending that the value is inside the interval?” • Plausible choice: Middle 50%. • Personal choice: Middle third. • Clearly it depends on situation and individual.

19. E.g. Catastrophe PML • David Miller paper at May 1999 CAS meeting. • Treated only parameter uncertainty from limited data. • 95% confidence interval was factor of 2. • Minimum uncertainty was 30%.

20. What are Reserves? • Actual Dollars To Be Paid. • Distribution of Potential Actual Dollars To Be Paid. • Estimator of the Distribution of Potential Actual Dollars To Be Paid.

21. Estimator of the Distribution of Potential Actual Dollars Paid • Can’t book a distribution. • Need a estimator for the distribution. • Actuaries have traditionally used the mean. • WHY THE MEAN?

22. WHY THE MEAN? • It is simple to calculate. • It is encouraged by the CAS statement of principles. • It is safe and middle of the road.

23. Some Possible estimators • Mean • Mode • Median • Fixed percentile • Other ?!!

24. How to choose a relevant estimator? • Example: bet on one throw of a die whose sides are weighted proportionally to their values. • Obvious choice is 6. • This is the mode. • Why not the mean of 4.333? • Even rounded to 4?

25. What happened there? • Frame situation by a “pain” function. • Take pain as zero when the throw is our chosen estimator, and 1 when it is not. • This corresponds to doing a simple bet. • Minimize the pain over the distribution: • This leads to choosing the estimator as the most probable single value.

26. Generalization to continuous variables • Define an appropriate pain function. • Depends on business meaning of distribution. • Function of estimator and stochastic variable. • Choose the estimator so as to minimize the average pain over the distribution. • “Statistical Decision Theory” • Can be generalized many directions • Parallel to Hamiltonian Principle of Least Work

27. Mathematical representation • f(x) – the distribution density function • p(m,x) – the relative pain if x ≠ m • P(m) = ∫ p(m,x) f(x) dx – the average of the pain over the distribution • Choose the pain to represent business reality • Choose m so as to minimize the average pain.

28. Claim: All the usual estimators can be framed this way. Further claim: this gives us a way to see the relevance of different estimators in the given business context.

29. Example: Mean • Pain function is quadratic in x with minimum at the estimator: • p(m,X) = (X- m)^2 • Note that it is equally bad to come in high or low, and two dollars off is four times as bad as one dollar off.

30. Squigglies: Proof for Mean • Integrate the pain function over the distribution, and express the result in terms of the mean M and variance V of x. This gives Pain as a function of the estimator: • P(m) = V + (M- m)^2 • Clearly a minimum at m = M

31. Why the Mean? • Is there some reason why this symmetric quadratic pain function makes sense in the context of reserves? • Perhaps unfairly: ever try to spend a squared dollar?

32. Example: Mode • Pain function is zero in a small interval around the estimator, and 1 elsewhere. • The estimator is the most likely result. • Could generalize to any finite interval (and get differing results) • Corresponds to simple bet, no degrees of pain.

33. Example: Median • Pain function is the absolute difference of x and the estimator: • p(m,X) = Abs(m -L) • Equally bad on upside and downside, but linear: two dollars off is only twice as bad as one dollar off. • The estimator is the 50th percentile of the distribution.

34. Example: Arbitrary Percentile • Pain function is linear but asymmetric with different slope above and below the estimator: • p(m,X) = (m -X) for X< m and S*(X- m) for X> m • If S>1, then coming in high (above the estimator) is worse than coming in low. • The estimator is the S/(S+1) percentile. E.g., S=3 gives the 75th percentile.

35. What are Reserves? • Actual Dollars To Be Paid. • Distribution of Potential Actual Dollars To Be Paid. • Estimator of the Distribution of Potential Actual Dollars To Be Paid. • An esoteric mystery dependent on the whims of the CFO/CEO.

36. An esoteric mystery dependent on the whims of the CFO • What shape would we expect for the pain function? • Assume a CFO who is in it for the long term and has no perverse incentives. • Assume a stable underwriting environment. • Take the context, for example, of one-year reserve runoff.

37. Suggestion for pain function: The decrease in net economic worth of the company as a result of the reserve changes.

38. Some interested parties who affect the pain function: • policyholders • stockholders • agents • regulators • rating agencies • investment analysts • lending institutions

39. If the Losses come in lower than the stated reserves: • Analysts perceive company as strongly reserved. • Problems from the IRS. • Dividends could have been larger. • Slightly uncompetitive if underwriters talk to pricing actuaries and pricing actuaries talk to reserving actuaries.

40. If the Losses come in higher than the stated reserves: • Increasing problems from the regulators. • Start to trigger IRIS tests. • Credit rating suffers. • Analysts perceive company as weak. • Possible troubles in collecting Reinsurance, etc. • Renewals and reinsurance problematical.

41. Reserving Pain function • Climbs much more steeply on the high side than on the low. • Probably has steps as critical values are exceeded. • Is probably non-linear on the high side. • Has weak dependence on the low side

42. Reserving Pain function (cont.) • The pain function for the mean is quadratic and therefore symmetric. • It gives too much weight to the low side • Consequently, the estimate is almost surely too low.

43. Reserving Pain function (cont.) • Simplest form is linear on the low side and quadratic on the high: • p(m,X) = S*(m -X) for X< m and (X- m)^2 for X> m • S an “appropriate” constant

44. Made-up example: • Company has lognormally distributed reserves, with coefficient of variation of 10%. • Mean reserves are 3.5 and S = 0.1 (units of surplus). • Then 10% low is the same pain as 10% high, 16% high is the same pain as 25% low, and 25% high is the same pain as 63% low. • Estimator is 5.1% above the mean, at the 71st percentile level.

45. . . . ESSENTIALS . . . • All estimates are soft. • Sometimes shockingly so. • The uncertainty in the reserves is MUCH LARGER THAN the uncertainty in the reserve estimator. • The appropriate reserve estimate depends on the pain function. • The mean is almost surely not the correct estimator, since it comes from a symmetric pain function. • It is probably too low.