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Reserve Uncertainty and Traditional Methods in Casualty Loss Reserves

Explore the uncertainty of reserves in casualty loss, traditional reserve methods, and the importance of estimating reserve variability. This analysis reveals contradictory conclusions and highlights the need to consider distribution and model/specification uncertainty in reserve calculations.

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Reserve Uncertainty and Traditional Methods in Casualty Loss Reserves

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  1. Reserve Uncertainty byRoger M. Hayne, FCAS, MAAAMilliman USACasualty Loss Reserve SeminarSeptember 23-24, 2002 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. A “Real Life” Example Milliman USA

  17. A “Real Life” Example Milliman USA

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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

  29. 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

  30. 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

  31. 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

  32. 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

  33. An Example(Continued) Milliman USA

  34. 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

  35. An Example(Continued) Milliman USA

  36. 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

  37. An Example(Continued) Milliman USA

  38. An Example(Continued) Milliman USA

  39. 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

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