Determining Reserve Ranges

1. Determining Reserve Ranges CLRS 1999 by Rodney Kreps Guy Carpenter Instrat

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

3. What are Reserves? • Actual Dollars Paid. • Distribution of Potential Actual Dollars Paid. • Locator of the Distribution of Potential Actual Dollars Paid. • An esoteric mystery dependent on the whims of the CFO.

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

5. Actual Dollars Paid • Only true after runoff. • Gives a hindsight view. • Lies behind the question “Why can’t you get it right?”

6. Distribution of Potential Actual Dollars Paid • All planning estimates are distributions. • ALL planning estimates are distributions. • ALL planning estimates are DISTRIBUTIONS. • Basically, anything interesting on a going-forward basis is a distribution

7. Distributions frequently characterized by locator and spread • 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.”

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

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

10. 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.

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

12. 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.

13. 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%.

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

15. WHY THE MEAN? • It is simple to calculate. • It is encouraged by the CAS statement of principles. • It is safe - “Nobody ever got fired for buying IBM.”

16. Some Possible locators • Mean • Mode • Median • Fixed percentile • Other ?!!

17. How to choose a relevant locator? • 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?

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

19. Generalization to continuous variables • Define an appropriate pain function. • Depends on business meaning of distribution. • Function of locator and stochastic variable. • Choose the locator 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

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

21. Example: Mean • Pain function is quadratic in x with minimum at the locator: • P(L,X) = (X-L)^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.

22. 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 Locator: • P(L) = V + (M-L)^2 • Clearly a minimum at L = M

23. 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?

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

25. Example: Median • Pain function is the absolute difference of x and the locator: • P(L,X) = Abs(X-L) • Equally bad on upside and downside, but linear: two dollars off is only twice as bad as one dollar off. • Generates the X corresponding to the 50th percentile.

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

27. 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.

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

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

30. If the Losses come in lower than the stated reserves: • Analysts perceive company as strongly reserved. • Not much problem from the IRS. • Dividends could have been larger. • Slightly uncompetitive if underwriters talk to pricing actuaries and pricing actuaries talk to reserving actuaries.

31. If the Losses come in higher than the stated reserves: • If only slightly higher, same as industry. • Otherwise, 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 problematical.

32. Generic Reserving Pain function • Climbs much more steeply on high side than low. • Probably has steps as critical values are exceeded. • Probably non-linear on high side. • Weak dependence on low side

33. Generic Reserving Pain function (2) • Simplest form is linear on the low side and quadratic on the high: • P(L,X) = S*(L-X) for X<L and (X-L)^2 for X>L

34. 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% high is as bad as 10% low, 16% high is as bad as 25% low, and 25% high is as bad as 63% low. • Locator is 5.1% above the mean, at the 71st percentile level.

35. . . . ESSENTIALS . . . • All estimates are soft. • Sometimes shockingly so. • The uncertainty in the reserves is NOT the uncertainty in the reserve estimator. • The appropriate reserve estimate depends on the pain function. • The mean is unlikely to be the correct estimator.