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GIRO 2004, Killarney Individual Large Claims Reserving

GIRO 2004, Killarney Individual Large Claims Reserving. 13 October 2004. 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

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GIRO 2004, Killarney Individual Large Claims Reserving

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  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

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