Reserve Uncertainty

1. Reserve Uncertainty byRoger M. Hayne, FCAS, MAAAMilliman USACasualty Loss Reserve SeminarSeptember 8-9, 2003 Milliman USA

2. Reserves Are Uncertain? • Reserves are just numbers in a financial statement • What do we mean by “reserves are uncertain?” • Numbers are estimates of future payments • Not estimates of the average • Not estimates of the mode • Not estimates of the median • Not really much guidance in guidelines • Rodney Kreps has more to say on this subject Milliman USA

3. Let’s Move Off the Philosophy • Should be more guidance in accounting/actuarial literature • Not clear what number should be booked • Less clear if we do not know the distribution of that number • There may be an argument that the more uncertain the estimate the greater the “margin” • Need to know distribution first Milliman USA

4. “Traditional” Methods • Many “traditional” reserve methods are somewhat ad-hoc • Oldest, probably development factor • Fairly easy to explain • Subject of much literature • Not originally grounded in theory, though some have tried recently • Known to be quite volatile for less mature exposure periods Milliman USA

5. “Traditional” Methods • Bornhuetter-Ferguson • Overcomes volatility of development factor method for immature periods • Needs both development and estimate of the final answer (expected losses) • No statistical foundation • Frequency/Severity (Berquist, Sherman) • Also ad-hoc • Volatility in selection of trends & averages Milliman USA

6. “Traditional” Methods • Not usually grounded in statistical theory • Fundamental assumptions not always clearly stated • Often not amenable to directly estimate variability • “Traditional” approach usually uses various methods, with different underlying assumptions, to give the actuary a “sense” of variability Milliman USA

7. Basic Assumption • When talking about reserve variability primary assumption is: Given current knowledge there is a distribution of possible future payments (possible reserve numbers) • Keep this in mind whenever answering the question “How uncertain are reserves?” Milliman USA

8. Some Concepts • Baby steps first, estimate a distribution • Sources of uncertainty: • Process (purely random) • Parameter (distributions are correct but parameters unknown) • Specification/Model (distribution or model not exactly correct) • Keep in mind whenever looking at methods that purport to quantify reserve uncertainty Milliman USA

9. Why Is This Important? • Consider “usual” development factor projection method, Cikaccident year i, paid by age k • Assume: • There are development factors fisuch that E(Ci,k+1|Ci1, Ci2,…, Cik)= fkCik • {Ci1, Ci2,…, CiI}, {Cj1, Cj2,…, CjI} independent for i  j • There are constants ksuch that Var(Ci,k+1|Ci1, Ci2,…, Cik)= Cik k2 Milliman USA

10. Conclusions • Following Mack (ASTIN Bulletin, v. 23, No. 2, pp. 213-225) are unbiased estimates for the development factors fi • Can also estimate standard error of reserve Milliman USA

11. Conclusions • Estimate of mean squared error (mse) of reserve forecast for one accident year: Milliman USA

12. Conclusions • Estimate of mean squared error (mse) of the total reserve forecast: Milliman USA

13. Sounds Good -- Huh? • Relatively straightforward • Easy to implement • Gets distributions of future payments • Job done -- yes? • Not quite • Why not? Milliman USA

14. An Example • Apply method to paid and incurred development separately • Consider resulting estimates and errors • What does this say about the distribution of reserves? • Which is correct? Milliman USA

15. “Real Life” Example • Paid and Incurred as in handouts (too large for slide) • Results Milliman USA

16. Some Ranges • 95% Probability Interval • Paid $277 - $440 million • Incurred $161 - $214 million • Middle Third • Paid $347 - $376 million • Incurred $182 - $193 million Milliman USA

17. A “Real Life” Example Milliman USA

18. A “Real Life” Example Milliman USA

19. What Happened? • Conclusions follow unavoidably from assumptions • Conclusions contradictory • Thus assumptions must be wrong • Independence of factors? Not really (there are ways to include that in the method) • What else? Milliman USA

20. What Happened? • Obviously the two data sets are telling different stories • What is the range of the reserves? • Paid method? • Incurred method? • Extreme from both? • Something else? • Main problem -- the method addresses only one method under specific assumptions Milliman USA

21. What Happened? • Not process (that is measured by the distributions themselves) • Is this because of parameter uncertainty? • No, can test this statistically (from normal distribution theory) • If not parameter, what? What else? • Model/specification uncertainty Milliman USA

22. Why Talk About This? • Most papers in reserve distributions consider • Only one method • Applied to one data set • Only conclusion: distribution of results from a single method • Not distribution of reserves Milliman USA

23. Discussion • Some proponents of some statistically-based methods argue analysis of residuals the answer • Still does not address fundamental issue; model and specification uncertainty • At this point there does not appear much (if anything) in the literature with methods addressing multiple data sets Milliman USA

24. Moral of Story • Before using a method, understand underlying assumptions • Make sure what it measures what you want it to • The definitive work may not have been written yet • Casualty liabilities very complex, not readily amenable to simple models Milliman USA

25. All May Not Be Lost • Not presenting the definitive answer • More an approach that may be fruitful • Approach does not necessarily have “single model” problems in others described so far • Keeps some flavor of “traditional” approaches • Some theory already developed by the CAS (Committee on Theory of Risk, Phil Heckman, Chairman) Milliman USA

26. Collective Risk Model • Basic collective risk model: • Randomly select N, number of claims from claim count distribution (often Poisson, but not necessary) • Randomly select N individual claims, X1, X2, …, XN • Calculate total loss as T = Xi • Only necessary to estimate distributions for number and size of claims • Can get closed form expressions for moments (under suitable assumptions) Milliman USA

27. Adding Parameter Uncertainty • Heckman & Meyers added parameter uncertainty to both count and severity distributions • Modified algorithm for counts: • Select  from a Gamma distribution with mean 1 and variance c (“contagion” parameter) • Select claim counts N from a Poisson distribution with mean  • If c < 0, N is binomial, if c > 0, N is negative binomial Milliman USA

28. Adding Parameter Uncertainty • Heckman & Meyers also incorporated a “global” uncertainty parameter • Modified traditional collective risk model • Select  from a distribution with mean 1 and variance b • Select N and X1, X2, …, XN as before • Calculate total as T = Xi • Note  affects all claims uniformly Milliman USA

29. Why Does This Matter? • Under suitable assumptions the Heckman & Meyers algorithm gives the following: • E(T) = E(N)E(X) • Var(T)= (1+b)E(X2)+2(b+c+bc)E2(X) • Notice if b=c=0 then • Var(T)= E(X2) • Average, T/N will have a decreasing variance as E(N)= is large (law of large numbers) Milliman USA

30. Why Does This Matter? • If b 0 or c 0 the second term remains • Variance of average tends to (b+c+bc)E2(X) • Not zero • Otherwise said: No matter how much data you have you still have uncertainty about the mean • Key to alternative approach -- Use of b and c parameters to build in uncertainty Milliman USA

31. If It Were That Easy … • Still need to estimate the distributions • Even if we have distributions, still need to estimate parameters (like estimating reserves) • Typically estimate parameters for each exposure period • Problem with potential dependence among years when combining for final reserves Milliman USA

32. An Example • Consider the data set included in the handouts • This is hypothetical data but based on a real situation • It is residual bodily injury liability under no-fault • Rather homogeneous insured population Milliman USA

33. An Example(Continued) • Applied several “traditional” actuarial methods • Usual development factor • Berquist/Sherman • Hindsight reserve method • Adjustments for • Relative case reserve adequacy • Changes in closing patterns Milliman USA

34. An Example(Continued) Milliman USA

35. An Example(Continued) • Now review underlying claim information • Make selections regarding the distribution of size of open claims for each accident year • Based on actual claim size distributions • Ratemaking • Other • Use this to estimate contagion (c) value Milliman USA

36. An Example(Continued) Milliman USA

37. An Example(Continued) • Thus variation among various forecasts helps identify parameter uncertainty for a year • Still “global” uncertainty that affects all years • Measure this by “noise” in underlying severity Milliman USA

38. An Example(Continued) Milliman USA

39. An Example(Continued) • Without Uncertainty • 95% Probability $195 - $210 million • Middle third $201 - $204 million • With Uncertainty • 95% Probability $141 - $283 million • Middle third $184 - $214 million Milliman USA

40. An Example(Continued) Milliman USA

41. CAS To The Rescue • Still assumed independence • CAS Committee on Theory of Risk commissioned research into • Aggregate distributions without independence assumptions • Aging of distributions over life of an exposure year • Will help in reserve variability • Sorry, do not have all the answers yet Milliman USA