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Stochastic Loss ReservingIntroduction to Stochastic Loss Reserving for General Insurance Valuations

IntroductionLoss reserving covers Incurred but Not Reported (IBNR) claims. These reserves are usually estimated based on historical claims paid based on accident/reporting patterns.

In the past a point estimator for the reserves was sufficient. New regulatory requirements (such as Solvency II and other risk management regimes) foster stochastic methods for loss reserving.

Stochastic Reserving has been applied on real but completely anonymized data using statistical software R.

Lag-Factor Practice of IBNR Reserving

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IBNR reserve is shown here as calculated by LAG ANALYSIS. Countdown to Lag Analysis Practice

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Current Reserving PracticeHere IBNR Reserve was based on Lag analysis. One of the major aspects for unpaid claims estimation is the appropriate assessment of reporting lag. We have analyzed the lag between the incident date and the reported date where the data indicated the following reporting lags (by accident year):

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Current Reserving Practice

This can be illustrated as follows:

>>0 >>1 >> 2 >> 3 >> 4 >>Current Reserving PracticeBased on the lag analysis, Gross IBNR was determined as follows:

>>0 >>1 >> 2 >> 3 >> 4 >>Chain Ladder-Improvement on Basic Lag Analysis

Chain Ladder can be readily applied on and the following diagram elaborates on the methodology behind triangulation:

Chain Ladder-Improvement on Basic Lag AnalysisThe Paid Claims triangle on accident quarter to reported quarter basis can be shown below (Due to the magnitude, the triangle will be shown in 2 parts):

Chain Ladder-Improvement on Basic Lag Analysis

Chain Ladder-Improvement on Basic Lag AnalysisThis claim development triangle can be visualized as follows:

Chain Ladder-Improvement on Basic Lag AnalysisApplying the basic Chain Ladder methodology on this triangle, the total Gross IBNR comes out to be AED 64, 577,062.

Stochastic Loss ReservingWelcome to the Future!

The Future begins NOW!

Why Stochastic Loss Reserving?The following delivers reasons for why we should adopt stochastic reserving:Computer power and statistical methodology make it possibleProvides measures of variability as well as location (changes emphasis on best estimate)Can provide a predictive distributionAllows diagnostic checks (residual plots etc)Useful in Dynamic Financial Analysis (DFA)Useful in satisfying Own Risk Solvency Assessment (ORSA) under different risk management regimesAs per Richard Verrall of CASS Business School, who is the leading authority in stochastic loss reserving: In recent years, a lot more attention has been given to the possible differences between the reserves and the actual outcome in terms of the claims paid. And also, there is considerable interest in the variability of the reserves from year to year: as more information becomes available, the estimates of what will have to be paid in claims may change. This is very important because it influences the amount of capital that companies are required to hold to ensure their continuing solvency. The capital that is required also shows to investors and management how profitable the business is (or different parts of the business are). For all these reasons, a lot more attention has been given to the likely variability of the claims reserves, as well as the actual amount held in these reserves.

Why Stochastic Reserving?Stochastic reserving does not just give us the best estimate, it also provides us with the variability in claims reserve. The variability of a forecast is given as prediction error or standard error, where prediction error is:Prediction Error= (Process Variance + Estimation Variance)^1/2The following stochastic loss reserving models have been applied:Mack ModelBootstrap ModelGeneralized Linear Model (GLM) utilizing Over Dispersed Poisson Kindly note that Mack produces results that are very close to Chain Ladder and GLM using Over Dispersed Poisson distribution produces same results as Mack.

Mack ModelThomas Mack published in 1993 a method which estimates the standard errors of the chain-ladder forecast without assuming any particular distribution.The Mack-chain-ladder model gives an unbiased estimator for IBNR (Incurred But Not Reported) claims. The Mack-chain-ladder model can be regarded as a weighted log-linear regression through the origin for each development period. Prediction Error is the total variability in the projection of future losses by the Mack Method.

Mack Model-Results The total results are depicted below:Prediction Error helps us to estimate that we are 95% confident that the true IBNR value lies around between AED 40 million and AED 89 million.

Mack Model- Model DiagnosticsThe residual plots show the standardized residuals against fitted values, origin period, calendar period and development period. All residual plots should show no pattern or direction other than broadly converging towards zero. Slight deviation is expected and is not a cause for concern. Kindly see next slide for the residual plots.

Mack Model- Model Diagnostics

Bootstrap ModelThe Bootstrap model uses a two-stage bootstrapping/simulation approach following the paper by England and Verrall [2002]. In the first stage an ordinary chain-ladder methods is applied to the cumulative claims triangle. From this we calculate the scaled Pearson residuals which we bootstrap R times to forecast future incremental claims payments via the standard chain-ladder method. In the second stage we simulate the process error with the bootstrap value as the mean and using the process distribution assumed. The set of reserves obtained in this way forms the predictive distribution, from which summary statistics such as mean, prediction error or quantiles can be derived.

Bootstrap Model

Bootstrap uses Gamma Distribution in arriving at its estimates of IBNR reserves.The results of simulating bootstrap replicates to generate estimates for IBNR can be depicted as a histogram of frequency as below:

Bootstrap Model

The mean ultimate claims that have been simulated can be made evident as follows:

Bootstrap Model- ResultsThe total results can be displayed as follows:Through utilizing prediction error, we can be 95% confident that the true IBNR value lies between around AED 58 million and AED 104 million.

Generalized Linear Model Although GLM is mainly used in pricing of general insurance premiums, GLMS have found their way into reserving as well. The GLM takes an insurance loss triangle, converts it to incremental losses internally if necessary, and fits the resulting loss data with a Generalised Linear Model where the mean structure includes both the accident year and the development lag effects. The GLM also provides analytical methods to compute the associated prediction errors, based on which the uncertainty measures such as predictive intervals can be computed.Only the Tweedie family of distributions are allowed, that is, the exponential family that admits a power variance function V (u) = up. For the purposes of stochastic loss reserving, the distribution of Over-Dispersed Poisson was utilized, especially since this choice of distribution will lead to same mean figure for IBNR as provided in Mack.

Generalized Linear Model Total results can be displayed as follows:

For the purposes of stochastic loss reserving, the distribution of Over-Dispersed Poisson was utilized, especially since this choice of distribution will lead to same mean figure for IBNR as provided in Mack.

This shows, amongst others, that we are 95% confident that the true IBNR reserve value lies around between AED 32 million and AED 98 million.

Conclusions for our BusinessStochastic Modeling is the way forward!

Conclusions- How to maximize ability to introduce these models in emerging markets

Mathematical IntegrityUsing powerful software like RUser-Friendly, Succinct and Results-Oriented ReportingContinuous MonitoringCustomer

ConclusionsApplying triangulation method of chain ladder does not just improve deterministic reserving but also enables stochastic reserving to be introduced through comparison of the results of Chain Ladder.

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

ReferencesStochastic Claims Reserving in General Insurance England, PD and Verrall, RJ (2002) Claims Reserving in R-Markus GesmannClaims Reserving with R: package Vignette-Markus Gesmann, Dan Murphy and Wayne Zhang

Thank You for Your Attention!

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