1.  Stochastic Reserving in General Insurance Peter England, PhD EMB Younger Members’ Convention 03 December 2002

2. Aims nTo provide an overview of stochastic reserving models, using England and Verrall (2002, BAJ) as a basis. nTo demonstrate some of the models in practice, and discuss practical issues

3. Why Stochastic Reserving? nComputer power and statistical methodology make it possible nProvides measures of variability as well as location (changes emphasis on best estimate) nCan provide a predictive distribution nAllows diagnostic checks (residual plots etc) nUseful in DFA analysis nUseful in satisfying FSA Financial Strength proposals

4. Actuarial Certification nAn actuary is required to sign that the reserves are “at least as large as those implied by a ‘best estimate’ basis without precautionary margins” nThe term ‘best estimate’ is intended to represent “the expected value of the distribution of possible outcomes of the unpaid liabilities”

5. Conceptual Framework

6. Example

7. Prediction Errors

9. Stochastic Reserving Model Types n“Non-recursive” nOver-dispersed Poisson nLog-normal nGamma n“Recursive” nNegative Binomial nNormal approximation to Negative Binomial nMack’s model

10. Stochastic Reserving Model Types nChain ladder “type” nModels which reproduce the chain ladder results exactly nModels which have a similar structure, but do not give exactly the same results nExtensions to the chain ladder nExtrapolation into the tail nSmoothing nCalendar year/inflation effects nModels which reproduce chain ladder results are a good place to start

11. Definitions Assume that the data consist of a triangle of incremental claims: The cumulative claims are defined by: and the development factors of the chain-ladder technique are denoted by

12. Basic Chain-ladder

13. Over-Dispersed Poisson

14. What does Over-Dispersed Poisson mean? nRelax strict assumption that variance=mean nKey assumption is variance is proportional to the mean nData do not have to be positive integers nQuasi-likelihood has same form as Poisson likelihood up to multiplicative constant

15. Predictor Structures (Chain ladder type) (Hoerl curve) (Smoother)

16. Chain-ladder Other constraints are possible, but this is usually the easiest. This model gives exactly the same reserve estimates as the chain ladder technique.

17. Excel nInput data nCreate parameters with initial values nCalculate Linear Predictor nCalculate mean nCalculate log-likelihood for each point in the triangle nAdd up to get log-likelihood nMaximise using Solver Add-in

18. Recovering the link ratios In general, remembering that

19. Variability in Claims Reserves nVariability of a forecast nIncludes estimation variance and process variance nProblem reduces to estimating the two components

20. Prediction Variance Prediction variance=process variance + estimation variance

21. Prediction Variance (ODP) Individual cell Row/Overall total

22. Bootstrapping nUsed where standard errors are difficult to obtain analytically nCan be implemented in a spreadsheet nEngland & Verrall (BAJ, 2002) method gives results analogous to ODP nWhen supplemented by simulating process variance, gives full distribution

23. Bootstrapping - Method nRe-sampling (with replacement) from data to create new sample nCalculate measure of interest nRepeat a large number of times nTake standard deviation of results nCommon to bootstrap residuals in regression type models

24. Bootstrapping the Chain Ladder (simplified) 1.Fit chain ladder model 2.Obtain Pearson residuals 3.Resample residuals 4.Obtain pseudo data, given 5.Use chain ladder to re-fit model, and estimate future incremental payments

25. Bootstrapping the Chain Ladder 6.Simulate observation from process distribution assuming mean is incremental value obtained at Step 5 7.Repeat many times, storing the reserve estimates, giving a predictive distribution 8.Prediction error is then standard deviation of results

26. Log Normal Models nLog the incremental claims and use a normal distribution nEasy to do, as long as incrementals are positive nDeriving fitted values, predictions, etc is not as straightforward as ODP

27. Log Normal Models

28. nSame range of predictor structures available as before nNote component of variance in the mean on the untransformed scale nCan be generalised to include non- constant process variances

29. Prediction Variance Individual cell Row/Overall total

30. Over-Dispersed Negative Binomial

32. Derivation of Negative Binomial Model from ODP nSee Verrall (IME, 2000) nEstimate Row Parameters first nReformulate the ODP model, allowing for fact that Row Parameters have been estimated nThis gives the Negative Binomial model, where the Row Parameters no longer appear

33. Prediction Errors Prediction variance = process variance + estimation variance Estimation variance is larger for ODP than NB but Process variance is larger for NB than ODP End result is the same

34. Est imation variance and process variance nThis is now formulated as a recursive model nWe require recursive procedures to obtain the estimation variance and process variance nSee Appendices 1&2 of England and Verrall (BAJ, 2002) for details

35. Normal Approximation to Negative Binomial

36. Joint modelling 1.Fit 1 st stage model to the mean, using arbitrary scale parameters (e.g. =1) 2.Calculate (Pearson) residuals 3.Use squared residuals as the response in a 2 nd stage model 4.Update scale parameters in 1 st stage model, using fitted values from stage 3, and refit 5.(Iterate for non-Normal error distributions)

37. Est imation variance and process variance nThis is also formulated as a recursive method nWe require recursive procedures to obtain the estimation variance and process variance nSee Appendices 1&2 of England and Verrall (BAJ, 2002) for details

38. Mack’s Model

41. Comparison nThe Over-dispersed Poisson and Negative Binomial models are different representations of the same thing nThe Normal approximation to the Negative Binomial and Mack’s model are essentially the same

42. The Bornhuetter-Ferguson Method nUseful when the data are unstable nFirst get an initial estimate of ultimate nEstimate chain-ladder development factors nApply these to the initial estimate of ultimate to get an estimate of outstanding claims

43. Estimates of outstanding claims To estimate ultimate claims using the chain ladder technique, you would multiply the latest cumulative claims in each row by f, a product of development factors. Hence, an estimate of what the latest cumulative claims should be is obtained by dividing the estimate of ultimate by f. Subtracting this from the estimate of ultimate gives an estimate of outstanding claims:

44. The Bornhuetter-Ferguson Method Let the initial estimate of ultimate claims for accident year i be The estimate of outstanding claims for accident year i is

45. Comparison with Chain-ladder replaces the latest cumulative claims for accident year i, to which the usual chain- ladder parameters are applied to obtain the estimate of outstanding claims. For the chain-ladder technique, the estimate of outstanding claims is

46. Multiplicative Model for Chain- Ladder

47. BF as a Bayesian Model Put a prior distribution on the row parameters. The Bornhuetter-Ferguson method assumes there is prior knowledge about these parameters, and therefore uses a Bayesian approach. The prior information could be summarised as the following prior distributions for the row parameters:

48. BF as a Bayesian Model nUsing a perfect prior (very small variance) gives results analogous to the BF method nUsing a vague prior (very large variance) gives results analogous to the standard chain ladder model nIn a Bayesian context, uncertainty associated with a BF prior can be incorporated

49. Stochastic Reserving and Bayesian Modelling nOther reserving models can be fitted in a Bayesian framework nWhen fitted using simulation methods, a predictive distribution of reserves is automatically obtained, taking account of process and estimation error nThis is very powerful, and obviates the need to calculate prediction errors analytically

50. Limitations nLike traditional methods, different stochastic methods will give different results nStochastic models will not be suitable for all data sets nThe model results rely on underlying assumptions nIf a considerable level of judgement is required, stochastic methods are unlikely to be suitable nAll models are wrong, but some are useful!

51. “I believe that stochastic modelling is fundamental to our profession. How else can we seriously advise our clients and our wider public on the consequences of managing uncertainty in the different areas in which we work?” - Chris Daykin, Government Actuary, 1995 “Stochastic models are fundamental to regulatory reform” - Paul Sharma, FSA, 2002

52. References England, PD and Verrall, RJ (2002) Stochastic Claims Reserving in General Insurance, British Actuarial Journal Volume 8 Part II (to appear). Verrall, RJ (2000) An investigation into stochastic claims reserving models and the chain ladder technique, Insurance: Mathematics and Economics, 26, 91-99. Also see list of references in the first paper.

53. G e n e r a l I n s u r a n c e A c t u a r i e s & C o n s u l t a n t s