Scott P. Weinstein, FCAS, MAAA Ash Ruparelia 2007 Casualty Loss Reserve Seminar September 10, 2007

1. GLOBAL ACTUARIAL SERVICES PRACTICE Practical Applications of the MACK and BOOTSTRAPPING Methods When Estimating Reserve Ranges A D V I S O R Y Scott P. Weinstein, FCAS, MAAA Ash Ruparelia 2007 Casualty Loss Reserve SeminarSeptember 10, 2007

2. Understanding the Issues Uncertainty – Where Does It Come from? Stochastic Approaches Industry-Based Examples Technical Walkthrough Practical Considerations Diagnostic Graphs Q&A Presentation Overview

3. Understanding the Issues

4. Understanding the Issues • For non-life insurance companies, loss reserves (technical reserves) comprise the majority of their liabilities • Uncertainty involved in estimating liabilities pose considerable risk • Adverse reserve run-off has led to insurance company downfalls • Negative impact on shareholders, policyholders, employees and other insurers

5. Understanding the Issues • The nature or extent of this uncertainty is generally not well understood by decision-makers • Financial statement reporting requires that a single number represent the technical reserve • Potential investors and regulators generally recognize that the number may change over time • The magnitude of potential variation is generally not identified or quantified in an informative way

6. Understanding the Issues • Why is the quantification of uncertainty important? A number of internal and external pressures:

7. Understanding the Issues • Compliance Pressures • SEC increasingly requesting disclosures regarding reserve uncertainty, including potential financial statement impact • Australian Prudential Regulatory Authority already requires that technical reserves be determined as the present value of a central estimate, with risk margin to approximate the 75% confidence level • Impending regulatory guidance of International Financial Reporting Standard’s (IFRS) Phase II requires the present value of a central estimate with an explicit additional margin for bearing risk

8. Understanding the Issues • Compliance Pressures • Rating Agency Interest • Enterprise Risk Management • Capital Adequacy • Europe’s Solvency II Initiative • Market value of reserves based on expected present value of future cash flows, but will include a market value margin that meets the objectives of either third-party portfolio transfer or recapitalizing the company to ensure a proper run-off scenario

9. Understanding the Issues • Implications for Senior Management, Boards of Directors and Other Interested Parties • Economic Capital – a mechanism by which companies are measuring the risks of their business • Strong Risk Governance – allows for better recognition of the uncertainties inherent in claims liabilities. Enables informed decision-making and enhanced transparency with internal and external audiences • Merger or Acquisition Benefits • Reinsurance Strategy – Risk assumption and mitigation

10. Uncertainty – Where Does It Come from?

11. Uncertainty – Where Does It Come from? • Random nature of claims • Exposure to claims is uncertain • Number, size and timing of claims • Data • Homogeneity of data • Credibility of data • Other uncertainty • Model error – is the model reliable? • Change in underlying exposure over time • External influences / internal operational changes

12. Uncertainty – Where Does It Come from? Uncertainty from future process Uncertainty from estimation

13. Uncertainty – Where Does It Come from? • Claim triangles encompass uncertainty from the past • The variability of the estimated claim around the “true” value of the distribution we are trying to measure indicates the estimation error • BUT, the future payments will also have variance i.e. the projected value in each future cell depends on the possible future outcomes

14. Uncertainty – Where Does It Come from? • Therefore, must allow for the future variability – this is called process error • The prediction error measures the variability of the deviation C – Ĉ. • It can be shown that (approximately): Prediction variance = estimation variance + process variance

15. Stochastic Approaches

16. Stochastic Approaches • Deterministic actuarial techniques e.g. chain ladder • Ignores the random nature of the claims process • Stochastic approach • Statistical model to describe the assumptions of the underlying claim settlement process • Allows for the random nature of the claims process • Can test fit of model • Appreciate: • the potential variability in the reserves • the shape of the reserve distribution • Provides insight into the risk profile of the underlying business

17. Stochastic Approaches • The distribution produces a range of possibleoutcomes, not the range of reasonable outcomes • Management must interpret the results in light of the intended purposes of the modeling

18. Stochastic Approaches • Several stochastic reserving methods are gaining ground: • Mack • Bootstrap • Factorial • Generalized linear or other statistical models • Focus on results of Mack and Bootstrapping for the remainder of the presentation

19. Stochastic Approaches • Mach Method - Details • It is a statistical model underlying pure chain ladder • It has three explicit assumptions • The expected value of cumulative claims at development period, k, is equal to cumulative claims at k-1 multiplied by the development factor, E[Ci,k+1| Ci,1....... Ci,k] = Ci,k.fk • The cumulative claim amounts are independent between origin years for all development periods, i.e. {Ci,1....... Ci,n}, {Cj,1....... Cj,n}, i≠j are independent. • The variance of the cumulative claims at development period k, is proportional to the cumulative claims at k, i.e. Var[Ci,k+1| Ci,1....... Ci,k] = Ci,k.σ2k

20. Stochastic Approaches • Mack Method - Details • A major consequence of the first and second assumptions is that the estimates of the development factors are unbiased and uncorrelated • Also, the estimates of the ultimate claims and the reserves are unbiased • The estimates of the development factors are such that they are minimum variance estimators amongst all linear estimates of the development factors

21. Stochastic Approaches • Mach Method - Strengths • Stochastic chain ladder model • Easy to explain assumptions underlying model • No distribution assumption required • Can be applied to paid and incurred claims data • Sensible progression of standard error relative to mean reserve over the origin years • The development factors are minimum variance unbiased estimators of the true development factors • Can adjust model to exclude or weight specific development factors • Can incorporate a tail factor into the standard error calculation (Mack 1999)

22. Stochastic Approaches • Mack Method - Weaknesses • Distribution free – no automatic reserve ranges • Need to assume a distribution to produce ranges • Needs reasonable history to work sensibly – 10 years worth • History not necessarily a good guide to future • Assumptions may not be met in practice: • Non-independent origin years • Development factors correlated • Calendar year trends • Tail factor – left up to individual actuary’s judgement both to estimate the factor and the error components

23. Stochastic Approaches • Bootstrapping - Details • Basic principle is to create many pseudo data sets from actual data set • Relies on having sufficient observations in the data otherwise create overlaps in pseudo data • Recently popular as computing power and storage has improved

24. Stochastic Approaches • Bootstrapping - Details • Started as an ad hoc method of deriving variability and distribution of the reserves using the chain ladder method applied to past data • Ignored the modeling of the underlying claims process • Therefore the variability of the reserves was underestimated • Recent statistical models replicate chain ladder e.g. Over-dispersed Poisson model, Negative Binomial and Normal approximation • Bootstrap can be used with these statistical models to produce a fuller picture of the variance and distribution

25. Stochastic Approaches • Bootstrapping - Strengths • Easy to set up in a spreadsheet - does not use complex formula or specialist software • More than point estimate - Can derive the variance and the simulated distribution of reserve outcomes • Tail variability can be allowed for in a pragmatic way e.g. by fitting a tail to the pseudo data using curve fitting

26. Stochastic Approaches • Bootstrapping - Weaknesses • Over-dispersed Poisson model has constraints • Needs positive total development of claims for each development year • Will only model positive claim amounts • So, is not usually suitable for incurred claims where there are savings on case estimates • Need to simulate future payments according to model to obtain simulated distribution otherwise assume a distribution to produce ranges (if not simulating future claims increments) • Needs reasonable history to work sensibly – 10 years worth • History not necessarily a good guide to future

27. Industry-Based Examples

28. Industry-Based Examples • Applied Mack and Bootstrapping to industry-based homeowners, commercial auto (motor) and workers’ compensation claims payment data • Due to credibility gained by examining aggregated industry data, the claims process appears to be fairly stable • The central reserve estimate of each method are reasonably consistent and the errors or standard deviations, relative to the reserves, are quite low • The increase in percentage error as the accident year matures is due to the smaller volume of claims still open in older time periods

29. Industry-Based Examples

30. Industry-Based Examples

31. Industry-Based Examples

32. Industry-Based Examples

33. Industry-Based Examples • The preceeding graphs showcase the possible outcomes around a statistical mean assuming a specifically modelled distribution • This result is not the same as a range of reasonable low and high technical reserve estimates

34. Industry-Based Examples • Depending on the nature of the claims to which the company is exposed, the central estimate would likely fall somewhere near the statistical mean of the distribution • If the reserve estimation process occurs separately from the quantification of uncertainty, inconsistencies between the central estimate and the statistical mean could be the unintended consequences

35. Technical Walkthrough

36. Analysis and Interpretation of the Output • Analysis of model outputs • Deriving ranges from model output • Interpretation • Interplay between best estimates and ranges

37. Practical Considerations

38. Practical Considerations • Paid or incurred triangle?

39. Practical Considerations • Prediction error can only reflect estimation error and statistical (process) error, BUT NOT SPECIFICATION ERROR… • Model chosen can be wrong – chain ladder may not work well in cases where incremental payments are not dependent on previous cumulative payments. • When model assumptions are violated, prediction error may be significantly underestimated or be simply invalid

40. Practical Considerations • If persistent trends are identified through any of the diagnostic graphs, the user may wish to adjust the actual data set to remove outliers (e.g. individual data points, entire years of origin, entire diagonals) to remove the biases • But removal of data may dampen the variability of the remaining dataset. Hence it is important to investigate the reasons for any observed bias.

41. Practical Considerations • Other considerations: • Tail • Large losses • Diversification • Changes in exposure • Reinsurance • Converting from UY to AY • White noise • Hindsight testing

42. Diagnostic Graphs

43. Standardized Residuals by Origin • Scatter plot of residuals: [(actual incremental – expected incremental) / square root of expected incremental] • Each column represents the residuals relative to the selected development factor for a given accident year • Red line is the average trend line • Ideally, the average trend line should center around zero, with small random negative and positive fluctuations

44. Standardized Residuals by Development • Each column represents the residuals relative to the selected development factor at each maturity • Ideally, the average trend line should center very closely around zero, especially if the selected development pattern is based on an all-year average

45. Standardized Residuals by Origin • Each column represents the residuals relative to the selected development factor by diagonal • Trend line represents calendar year trend • Ideally, the average trend line should center around zero, with small random negative and positive fluctuations

46. Presenter’s contact details Scott Weinstein sweinstein@kpmg.com KPMG LLP (US) 404 222 3594 Asheet Ruparelia ash.ruparelia@kpmg.co.uk KPMG LLP (UK) +44 (0)20 7694 2244 The information contained hereinis of a general nature and is not intended to address thecircumstances of any particular individual or entity. Although we endeavor to provide accurate and timely information, there can be no guarantee that such information is accurate as of the date it is received or that it will continue to be accurate in the future. No one should act on such information without appropriate professional advice after a thorough examination of the particular situation. ©2007 KPMG LLP, a U.S. limited liability partnership and a member firm of the KPMG network of independent member firms affiliated with KPMG International, a Swiss cooperative. All rights reserved.