A Method for Projecting Individual Large Claims

1. A Method for Projecting Individual Large Claims Casualty Loss Reserving Seminar 11-12 September 2006 Atlanta Karl Murphy and Andrew McLennan

2. Overview • Rationale for considering individual claims • Outline of methodology • Examples • Data Requirements • Assumptions • Whole account variability • Case Study • Conclusion

3. Rationale for Considering Individual Claims • Last few years has seen a significant change in requirements from actuaries in terms of understanding variability around results • Partially driven by a greater understanding by board members that things can go wrong, and partly by the increased use of DFA models • Much work done based on aggregate triangles, but very little on stochastic individual claims development • Weaknesses in methods for deriving consistent gross and net results

4. Traditional Netting Down Methods • How do you net down gross reserves? • Could assume reinsurance ultimate reserves = reinsurance current reserves • Prudent if deficiencies in reserves • Optimistic if redundancies • Analyse net data, and calculate net results from this • Disadvantages: • Retentions may change • look at data on consistent retention • lots of triangles! Ensuring consistency between gross and various nets difficult • Indexation of retention • need assumption of payment pattern • Aggregate deductibles • need assumption of ultimate position of individual claims • Another option – model excess claims above a threshold, and calculate average deficiency of excess claims – i.e. IBNER on those above threshold. Apply average IBNER loading to open claims to get ultimate

5. Deterministic Netting Down Methods Tend to Understand Effect of Reinsurance • Example: excess IBNER of £0.5m, two claims of incurred of £250k, and retention of £500k • Deterministic development factor of 2, so gross-up claims to ultimate of £500k each • Calculate reinsurance recoveries: 500k-500k = 0 – no reinsurance recoveries • Net reserves = gross reserves

6. Deterministic Netting Down Methods Tend to Understand Effect of Reinsurance • Because of the one-sided nature of reinsurance, this will understate the reinsurance recoveries: • Above example: • one claim settles for 250k, one for 750k • same gross result • Net reserves = gross reserves – 250k • Need method that allows for distribution of ultimate individual claims to allow for reinsurance correctly

7. Traditional Variability Methods • Traditional Methods: • Methods based on log(incremental data), i.e. lognormal models • Mack’s model – based on cumulative data • Provide mean and variance of outcomes only • Bootstrapping • Provides a full predictive distribution – not just first two moments • Bootstrap any well specified underlying model • Over-dispersed Poisson (England & Verrall) • Mack’s model • Characteristics • Usually applied to aggregate triangles • Works well with stable triangles • However, large claims can influence volatility unduly • Bayesian Methods: • Like Bootstrapping, provides a full predictive distribution • Ability to incorporate expert judgement with informative priors

8. Traditional Variability Methods • No allowance made for the number of large claims in an origin period, and no allowance made for the status (i.e. open/closed) • No linkage between variability of gross and net of reinsurance reserves • No information about the distribution of individual claims – will have same problems of netting down gross results as deterministic methods

9. Outline of Methodology • Our methodology simulates large claims individually • Separately simulate known claims (for IBNER) and IBNR claims • Consider dependencies between IBNER and IBNR claims • For non-large claims, use an aggregate “capped” triangle • when a individual claim reaches the capping level, ignore any development in excess of the capping • index the capping threshold at an appropriate level • use a “traditional” stochastic method • consider dependency between the run-off of capped and excess claims

10. Outline of Methodology: IBNER • Take latest incurred position and status of claim • Simulate next incurred position and status of claim based on movement of a similar historic claim • Allows for re-openings, to the extent they are in the historic data • Projects individual claims from the point they become “large” • Claims are considered “similar” by: • Status of claim (open / closed) • Number of years since a claim became large (development period) • Size of claim – e.g. a claim with incurred of £10m will behave differently to a claim with incurred of £1m – claims are banded into layers

11. Outline of Methodology: IBNR • IBNR large claims can be either genuine IBNR, or claims previously not reported as large • Apply “standard” stochastic methods to numbers triangles • Alternatively, simulate based on an assumed frequency per unit of exposure • For severity, can sample from the (simulated) known large claims, or simulate from an appropriately parameterised distribution

12. Example Data

13. Claim D • Need to simulate into development period 3 • Open status as at development period 2 • Similar to claims B and C, with development factors of 0.53 and 1.5

14. Claim D: Simulations

15. Claim E • Closed status as at development period 2 • Similar to claim A, with no development

16. Claim F • Open status as at development period 1 • For development into year 2, can consider any of A to E • Consider also the status

17. Claim F Simulations to Year 2

18. Claim F Simulations to Year 3

19. IBNR Claims • Two sources of IBNR claims: • True IBNR claims • Known claims which are not yet large • Triangle of claims that ever become large • Calculate frequency of large claims in development period • Simulate number of large claims going forward • Simulate IBNR claim costs from historic claims that became large in that period

20. IBNR • Data below shows the claim number triangle, and frequency of claims

21. IBNR • Result for one simulation

22. Data Requirements • Individual large claim information: • Full incurred and payment history • Historic open status of claims • Claims that were ever large, not just currently large • Accident year exposure • Definition of “large” depends on: • Historic retentions • Number of claims above threshold • Consider having two thresholds – e.g. all claims above $100k, but then calculate excess above $200k – allows for claims developing just below the layer

23. Assumptions • Historic claims provide the full distribution of possible chain ladder factors for claims • Development by year is independent • No significant changes to case estimation procedures • Can allow for this by standardising the historic chain ladder factors, as is done in aggregate modelling • Historic reopening and settlement experience is representative of future • Method cannot be applied blindly – it is not a replacement of gross aggregate best estimate modelling, rather a tool to analyse variability around the aggregate modelling, and netting down of results

24. Variability of Whole Account • Simulate variability of small claims via “capped” triangle, using existing methods • Capped triangles preferred to triangles which totally exclude large claims • if claims are taken out once they become large, we see negative development • if history of claim is taken out, then triangles change from analysis to analysis • becomes difficult to allow for IBNR large claims • Add gross excess claims from individual simulations for total gross results, with appropriate dependency structure • Add net excess claims for total net results

25. Case Study • UK auto account • 16 years of data • Individual claims > £100k • 2 layers used to simulate IBNER claims, 80% in lower layer, 20% in upper layer

26. IBNER • Distribution of one individual claim, current incurred £125k • Expected ultimate of £300k • 90% of the time, ultimate cost of claim doesn’t exceed £700k

27. IBNER • Occasionally the claim can grow very large, however

28. IBNER • Progression of one claim that has been large for 4 years, and is still open • Still significant variability in ultimate cost

29. Ultimate Loss Development Factors • Graph shows ultimate LDF (ultimate / latest incurred) for “big” and “little” claim from same point in development • Probability of observe an large LDF (>4) 60% higher for small claim than large claim • Average LDF for small claim 1.1, for big claim 0.87

30. Distribution of Capped Reserve

31. Comparison with Mack Method

32. 2003 Distribution • Higher proportion of large claims • One claim of £6m • Greater uncertainty than implied by aggregate projection

33. 2004 and 2005 Distributions • Distributions from individual claims distributions slightly heavier tailed than aggregate method • Caused by increase in large claims proportions over time, not adequately allowed for in aggregate methods

34. Netting Down

35. Reinsurance Structures • Even simple portfolios can have reinsurance structures that are difficult to model • Aggregate Deductibles • Loss Occurring During vs Risk Attaching coverages • Partial Placements • Indexation Clauses • By having individual claims, can explicitly allow for any structure

36. Example: Aggregate Deductible • Graph shows percentile chart of the usage of a £2.25m aggregate deductible attaching to layer £400k XS £600k

37. Conclusion • Existing stochastic methods work well for homogenous data, but some lines of business are dominated by small number of large claims • Treating these claims separately allows existing methods to be used on the attritional claims, with our individual claims simulation technique allowing for variability in these large claims explicitly • This allows net and gross results to be calculated on a consistent basis, allowing explicitly for any reinsurance structures in place