GIRO 2004, Killarney Individual Large Claims Reserving

1. GIRO 2004, KillarneyIndividual Large Claims Reserving 13 October 2004

2. Overview Overview • Current methods of “netting down” gross results • Current methods of analysing variability of reserves • A new method • Examples • Development uncertainty • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

3. Overview Why are we doing this? • 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 • Given the importance of large claims within the historic run-off of motor claims, we wanted to do something a little bit better than simply applying aggregate methods • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

4. Netting Down of Reserves Netting Down • How do you net down gross reserves? • Could assume reinsurance ultimate reserves = reinsurance current reserves • Prudent if deficiencies in reserves • Prudent if no allowance for IBNR claims • Optimistic if redundancies • One option – 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 • Losses Occurring During vs Risks Attaching bases • Another option – from gross and capped results, get average deficiency of excess claims – ie IBNER on those above cap. Apply average IBNER loading to open claims to get ultimate • need to split out IBNR and IBNER from claims below the cap • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

5. Netting Down of Reserves Netting Down • Example: excess IBNER of £0.5m, two claims of incurred of £250k, and retention of £500k • Gross-up claims to ultimate of £500k each • Calculate reinsurance recoveries: 500k-500k = 0 – no reinsurance recoveries • Net reserves = gross reserves • Because of the one-sided nature of reinsurance (max(ultimate-retention, 0)), 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 • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

6. Variability Variability of Reserves • 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 • No information about the distribution of individual claims – will have same problems of netting down gross results as deterministic methods • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

7. Individual Large Claim Simulation – the Murphy & McLennan Method New Method • Our methodology simulates large claims individually • Separately projects known claims and IBNR claims • Known claims: • 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 – eg a claim with incurred of £10m will behave differently to a claim with incurred of £1m – claims are banded into layers • Layers can vary by development period – e.g. 25% of claims go into each layer at each development period. • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

8. IBNER Individual Large Claim Simulation - IBNER • Development year 1 is the year that claim became large • In the above example, running off claim 4 involves simulating chain ladder factor and open status from historic claims for development year 2 and 3. • For example, it could simulate a development ratio of 300/775 = 0.39 from claim 3, and it remains open. Equally it could simulate from claims 1 or 2. • For development year 3, it then simulates chain ladder factor from open claims. • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

9. IBNER Individual Large Claim Simulation • Simulates from claim 3 for DY2 and claim 1 for DY3 • Simulates from claim 2 for DY2 and claim 2 for DY3 • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

10. IBNER Individual Large Claim Simulation - IBNER • Graph below shows distribution of one individual claim, current incurred 350k • 90% of the time, ultimate cost of claim doesn’t exceed 650k • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

11. IBNER Individual Large Claim Simulation - IBNER • But, • 2.5% of the time, ultimate cost of claim exceeds £1.1m • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

12. IBNER Individual Large Claim Simulation - IBNER • Graph below shows progression of one claim that has been large for 5 years, and is still open • Still significant variability in ultimate cost • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

13. IBNER 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 • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

14. IBNR Individual Large Claim Simulation - IBNR • 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 • Alternatively estimate distribution parameters from historic data and simulate from these (need to allow for possibility of claims becoming small again) • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

15. IBNR Individual Large Claim Simulation- IBNR • Data below shows the claim number triangle, and frequency of claims • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

16. IBNR Individual Large Claim Simulation - IBNR • Table below shows results for one simulation • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

17. Data Individual Large Claim Simulation – Data • 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 – eg all claims above £100k, but then calculate excess above £200k – allows for claims developing just below the layer • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

18. Variability of whole account 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 • Add net excess claims for total net results • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

19. Comparison of Results Comparison of Results • Next graph shows distribution of gross results for most recent accident year from simulation of whole account (in green) versus M&M method (in blue) • Similar distribution of outcomes • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

20. Results Results • Some years produce significantly different distributions • On investigation, this year has significant large claims, including two claims open at £5m each • Expect higher variability in results than “average” year at same point in development • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

21. Results Results • Despite significant volatility in gross results, net is much less variable as expected – variability is in layer above retention • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

22. Results Results • Large number of claims with small outstandings – less variable than having 1 or 2 claims open with large outstanding • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

23. Results Results • Graph below shows usage of a £5m aggregate deductible • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

24. Assumptions Assumptions • Historic claims provide the full distribution of possible development 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 • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

25. Conclusion Conclusion • Existing stochastic methods work well for homogenous data, but run-off of motor portfolios is dominated by small number of large claims • Removing these claims, or lessening the impact, from the portfolio allows existing methods to be used • Our individual claim simulation technique allows for variability in these large claims explicitly, and allows for net and gross results to be calculated consistently • It’s not a replacement of the aggregate gross methodology, but improvement on aggregate net methodology • Overview • Netting Down • Variability • New Method • IBNER • IBNR • Data • Variability of whole account • Comparison of Results • Results • Assumptions • Conclusion

26. Model, Parameter and Process Uncertainty

27. Definitions • Process uncertainty is the uncertainty in results due to random variation, given that the model and associated parameters are correct • Given a model structure, parameters are usually estimated from a sample of data • There is uncertainty associated with the values of the parameter estimates • Parameter uncertainty concerns the sensitivity of results from an analysis to the choice of parameter estimates, given that the model is correct • Model uncertainty concerns the sensitivity of results and decisions from an analysis to the choice of model structure

28. Model, Parameter and Process Uncertainty • Process uncertainty • Simulate using a given model for the data with given parameters • Parameter uncertainty • Simulate the parameters • Model uncertainty • Try alternative models (pragmatic)

29. Parameter Uncertainty • Simple methods • “Classical” statistics • Bootstrapping • Bayesian Methods

30. Parameter Uncertainty - “Classical” methods • “Classical” statistical inference often involves assumptions of asymptotic (multivariate) normality • Simulate parameters from a (multivariate) normal distribution • Is the assumption appropriate? • Are simulated values always valid? • Covariance matrix of parameter estimates needed

31. Parameter Uncertainty - Bootstrapping • Bootstrapping is a simple but effective way of obtaining a distribution of the parameters • The method involves creating many new data sets from which the parameters are estimated • The new data sets are created by sampling with replacement from the observed data • Results in a (“simulated”) distribution of the parameters

32. Simple Bootstrapping Example

33. Bootstrapping - Multiple Parameters • Create bootstrap data samples • For each sample, obtain maximum likelihood parameter estimates • Provides a joint distribution of the parameters

34. Model, Parameter, Development and Process Uncertainty

35. Development Uncertainty • Bootstrapping methods can be used to allow for development uncertainty • Large claims: • simulate 10,000 sets of ultimate individual large claims (e.g. using the Murphy & McLennan method) • for each simulated set, for which there are n claims, sample with replacement n times, fit a distribution and obtain the parameters • simulate 10,000 claim frequencies (again perhaps allowing for parameter/development uncertainty) • For the ith claim frequency simulation, use the ith set of severity parameters • Triangulated data: • simulate 10,000 sets of ultimate claims (perhaps using “traditional” stochastic techniques) • for each simulated set, generate statistic of interest, for example the cost per unit of exposure, or rate adjusted loss ratio for each origin period • sample with replacement from the origin periods to allow for parameter uncertainty • calculate average across this sampled data

36. Example • Take previous example • Allow for development uncertainty by multiplying by uniform random distribution between 0.75 and 1.25