Bayesian and Hierarchical Methods for Ratemaking

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2 Bayesian and Hierarchical Methods for Ratemaking Chris Laws Ratemaking and Product Management Huntington Beach, CA March 11 13, 2013

3 Presentation Overview 1 2 Objective Case Study 3 Laws Bayesian and Hierarchical Methods for Ratemaking 2/34

4 Presentation Overview 1 2 Objective Case Study 3 The R package lossdev is available at: Laws Bayesian and Hierarchical Methods for Ratemaking 3/34

5 Background Times series models of loss development have been pioneered by Kremer [3], Verrall [6], Verrall [5], Zehnwirth [7], Barnett and Zehnwirth [1], and Jong [2] Such approaches attempt to explicitly model the data-generating process Laying out the equations for the data-generating process adds another layer of scrutiny to these models Such models can be evaluated by the adequacy of the equations that describe the data-generating process Estimating the equations of the data-generating process facilitates learning about the nature of loss triangles Laws Bayesian and Hierarchical Methods for Ratemaking 4/34

6 Triangle Dynamics For this architecture of triangle analysis, see Barnett and Zehnwirth [1] Laws Bayesian and Hierarchical Methods for Ratemaking 5/34

7 Estimation and Error Process The model is Bayesian and estimated using MCMC (Markov Chain Monte Carlo simulation) The model fits to the (natural) logarithms of incremental payments Both negative payments and payments of zero amounts are coded as missing values An auxiliary model can be used to account for the presence of payments of zero amounts The model uses a skewed Student-t likelihood with endogenous degrees of freedom The scale parameter of the t-distribution is allowed to vary in development time Laws Bayesian and Hierarchical Methods for Ratemaking 6/34

8 Assumed Underlying Data-Generating Process The calendar year effect is modeled around an expert prior for the rate of inflation The expert prior for future rates of inflation is simulated from an Ornstein-Uhlenbeck process calibrated to the observed expert prior inflation series The calendar year effect error may optionally follow a first-order autoregressive process The consumption path is estimated using a linear spline The number of knots is estimated with Reversible Jump MCMC A changepoint version of the model allows for a structural break in the consumption path The model endogenously determines the changepoint within a user-supplied interval Laws Bayesian and Hierarchical Methods for Ratemaking 7/34

9 User-Friendly Deployed as the R package lossdev R can be used for pre- and post-data processing Open source Vignette Walks the novice user through a series of examples Explains how the functions are intended to work together Function Documentation All functions are extensively documented Options allow for customization Laws Bayesian and Hierarchical Methods for Ratemaking 8/34

10 How Is It Bayesian? lossdev s Bayesian Roots Surface in Two Ways The use of priors The lossdev model consists of many parameters The selective use of a few informed priors (e.g., the calendar year effect) aids in parameter identification Also, judgment may formally be incorporated through the use of priors (for select parameters) Reflection of uncertainty A frequentist parameter estimate typically consists of a single point One must explicitly account for parameter uncertainty A Bayesian parameter estimate consists of a distribution of points The posterior predictive distribution endogenously accounts for parameter uncertainty Laws Bayesian and Hierarchical Methods for Ratemaking 9/34

11 Degrees of Freedom Laws Bayesian and Hierarchical Methods for Ratemaking 10/34

12 Incremental Payments Laws Bayesian and Hierarchical Methods for Ratemaking 11/34

13 Consumption Path Laws Bayesian and Hierarchical Methods for Ratemaking 12/34

14 Number of Knots Laws Bayesian and Hierarchical Methods for Ratemaking 13/34

15 Calendar Year Effect Laws Bayesian and Hierarchical Methods for Ratemaking 14/34

16 Calendar Year Effect Error Laws Bayesian and Hierarchical Methods for Ratemaking 15/34

17 Exposure Year Laws Bayesian and Hierarchical Methods for Ratemaking 16/34

18 Standard Deviation Laws Bayesian and Hierarchical Methods for Ratemaking 17/34

19 Standardized Residuals Development Year Laws Bayesian and Hierarchical Methods for Ratemaking 18/34

20 Skewness Parameter Laws Bayesian and Hierarchical Methods for Ratemaking 19/34

21 What About the Tail? Using a variant of lossdev to estimate two large workers compensation triangles, Schmid [4] suggests the following assumptions for the consumption path: Indemnity The rate of decay for mature development years should approach mortality Medical The rate of decay estimated for the final observed development year is assumed for more mature periods Laws Bayesian and Hierarchical Methods for Ratemaking 20/34

22 Presentation Overview Objective Case Study 1 2 Objective Case Study , eforum, forthcoming, The paper is available at Laws Bayesian and Hierarchical Methods for Ratemaking 21/34

23 Objective Formalizing Decision Making Under Uncertainty Objective Case Study Provide a tool for decision making under uncertainty within the existing NCCI framework of trend analysis and selection The development of a new forecasting model is out of scope The time series used in trend selection are extremely short c) Copyright 2013 National Council on Compensation Insurance, Inc. All Rights Reserved. Laws Bayesian and Hierarchical Methods for Ratemaking 22/34

24 Trend Analysis Models Considered Objective Case Study Results from three exponential trend (ET) models are typically considered when performing a trend analysis at NCCI 1 5-point ET 2 8-point ET 3 15-point ET Depending on the nature of the data-generating process, one model may be theoretically superior The Bayesian Trend Selection (BTS) formalizes the process of deciding among the various results in the presence of uncertainty Laws Bayesian and Hierarchical Methods for Ratemaking 23/34

25 Trend Analysis Considering the Data-Generating Process Objective Case Study The statistical quality of an estimate can be quantified in terms of its Bias: How relevant is the answer? Variance: How reliable is the answer? If the data-generating process is unchanged, All observations are relevant Incorporating more observations in the regression will reduce the variance of the estimate If the data-generating process has recently changed, Older observations fundamentally differ from newer observations Incorporating these fundamentally different observations in the regression will increase the bias of the estimate Laws Bayesian and Hierarchical Methods for Ratemaking 24/34

26 Model Estimation Objective Case Study The BTS considers how well each of the three ET models performed in the recent past for each series (in isolation) using the three most recent NCCI ratemaking data sets Each data set is split into a training and holdout set The holdout period consists of three years, which parallels the typical trend period Each of the three ETs are estimated on each of the training sets Nine estimates in total The three estimates from each ET are compared to the three compound annual growth rates observed in the respective holdout periods Laws Bayesian and Hierarchical Methods for Ratemaking 25/34

27 The Bayesian Trend Selection The BTS Estimate Objective Case Study The BTS estimate is the weighted average of the three estimates originating from the competing models The weights correspond to the posterior probability that (observed) data from the recent past is a realization from the associated competing model In this way, the BTS estimate is more akin to model averaging than model selection In model selection, one only allows the chosen model to influence the prediction In model averaging, one allows each model to influence the prediction relative to the probability it is the true model Laws Bayesian and Hierarchical Methods for Ratemaking 26/34

28 Indemnity Loss Ratio Growth Rates From a Select State Objective Case Study Laws Bayesian and Hierarchical Methods for Ratemaking 27/34

29 Indemnity Loss Ratio Posterior ET Probabilities From a Select State Objective Case Study Laws Bayesian and Hierarchical Methods for Ratemaking 28/34

30 Medical Loss Ratio Growth Rates From a Select State Objective Case Study Laws Bayesian and Hierarchical Methods for Ratemaking 29/34

31 Medical Loss Ratio Posterior ET Probabilities From a Select State Objective Case Study Laws Bayesian and Hierarchical Methods for Ratemaking 30/34

32 The lossdev R package is a highly customizable, easy-to-use platform for statistical loss development The BTS model seeks to formalize decision making under uncertainty for trend analysis Laws Bayesian and Hierarchical Methods for Ratemaking 31/34

33 References [1] Glen Barnett and Ben Zehnwirth. Best Estimates for Reserves. In: Casualty Actuarial Society Proceedings (2000), pp Url: http: // [2] Piet de Jong. Forecasting Runoff Triangles. In: North American Actuarial Journal 10 (2006), pp Url: http: // [3] Erhard Kremer. A Class of Autoregressive Models for Predicting the Final Claim Amount. In: Insurance Mathematics and Economics 3 (1984), pp Laws Bayesian and Hierarchical Methods for Ratemaking 32/34

34 References (cont.) [4] Frank Schmid. The Workers Compensation Tails. In: Variance 6.1 (2012), pp Url: http: // [5] Richard Verrall. A State Space Representation of the Chain Ladder Linear Model. In: Journal of the Institute of Actuaries 116 (1989), pp [6] Richard Verrall. Modelling Claims Run-Off Triangles with Two-Dimensional Time Series. In: Scandinavian Actuarial Journal (1989), pp Laws Bayesian and Hierarchical Methods for Ratemaking 33/34

35 References (cont.) [7] Ben Zehnwirth. Probabilistic Development Factor Models with Applications to Loss Reserve Variability. Prediction Intervals and Risk Based Capital. In: Casualty Actuarial Society Forum (1994), pp Url: casact.org/pubs/forum/94spforum/94spf447.pdf. Laws Bayesian and Hierarchical Methods for Ratemaking 34/34