More on Stochastic Reserving in General Insurance GIRO Convention, Killarney, October 2004 Peter England and Richard Verrall

Developments in Stochastic Reserving Overview The story so far Bootstrapping recursive models (including Mack's model) Working with incurred data Bayesian recursive models A comparison between bootstrapping and Bayesian methods Bootstrapping and the Bornhuetter-Ferguson Technique Including curve fitting for estimating tail factors

Developments in Stochastic Reserving Background England, P and Verrall, R (1999), Analytic and bootstrap estimates of prediction errors in claims reserving, Insurance: Mathematics and Economics 25, pp281-293. England, P (2002), Addendum to “Analytic and bootstrap estimates of prediction errors in claims reserving”, Insurance: Mathematics and Economics 31, pp461-466. England, PD and Verrall, RJ (2002), Stochastic Claims Reserving in General Insurance, British Actuarial Journal 8, III, pp443-544. + many other papers

Developments in Stochastic Reserving Conceptual Framework

Developments in Stochastic Reserving Example Developments in Stochastic Reserving

Prediction Errors – “Chain Ladder” Structure Developments in Stochastic Reserving

Over-Dispersed Poisson Developments in Stochastic Reserving Over-Dispersed Poisson

Example Predictor Structures Developments in Stochastic Reserving Example Predictor Structures Chain Ladder Hoerl Curve Smoother

Variability in Claims Reserves Developments in Stochastic Reserving Variability in Claims Reserves Variability of a forecast Includes estimation variance and process variance Problem reduces to estimating the two components

Developments in Stochastic Reserving Prediction Variance Individual cell Row/Overall total

EMBLEM Demo ODP Chain Ladder with constant scale parameter Developments in Stochastic Reserving

Developments in Stochastic Reserving Mack’s Model Mack, T (1993), Distribution-free calculation of the standard error of chain-ladder reserve estimates. ASTIN Bulletin, 22, 93-109 Specifies first two moments only:

Developments in Stochastic Reserving Mack’s Model

Developments in Stochastic Reserving Mack’s Model

ResQ Demo – Mack’s Model Developments in Stochastic Reserving

Parameter Uncertainty - Bootstrapping Developments in Stochastic Reserving Bootstrapping is a simple but effective way of obtaining a distribution of 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 (or residuals) The model is re-fitted to each new data set Results in a (“simulated”) distribution of the parameters

Reserving and Bootstrapping Developments in Stochastic Reserving Reserving and Bootstrapping Any model that can be clearly defined can be bootstrapped (see the England and Verrall papers for bootstrapping the ODP)

Bootstrapping Mack’s Model Developments in Stochastic Reserving Bootstrapping Mack’s Model

Bootstrapping Mack’s Model Developments in Stochastic Reserving Bootstrapping Mack’s Model

Recursive Models: Forecasting Developments in Stochastic Reserving Recursive Models: Forecasting With recursive models, forecasting proceeds one-step at a time: Move one-step ahead by multiplying the previous cumulative claims by the appropriate bootstrapped development factor Include the process error by sampling a single observation from the underlying process distribution, conditional on the mean given by the previous step Move to the next step Note that the process error is included at each step before proceeding

Igloo Demo 1 – Bootstrapping Chain Ladder Model Only Developments in Stochastic Reserving Igloo Demo 1 – Bootstrapping Chain Ladder Model Only ODP – with constant scale parameter ODP – with non-constant scale parameters Bootstrapping Mack’s Model

Negative Binomial Recursive Model Developments in Stochastic Reserving Negative Binomial Recursive Model This is a recursive equivalent to the ODP model

Developments in Stochastic Reserving Igloo Demo 2 – Bootstrapping Negative Binomial – Chain Ladder Model only Negative Binomial – with constant scale parameter Negative Binomial – with non-constant scale parameters Compare results with ODP and Mack shown earlier: ODP and Negative Binomial are very close Results with non-constant scale parameters are close to Mack’s method

Bootstrapping Recursive Models: Advantages Developments in Stochastic Reserving Bootstrapping Recursive Models: Advantages Consistent with traditional deterministic actuarial techniques Individual points can be weighted out for n-year volume weighted averages, exclude high/low etc Curve fitting can be incorporated Bootstrap version of Mack’s model can be used where negative incrementals are encountered For example: Incurred claims Bootstrapping incurred claims: Gives distribution of Ultimates and IBNR Can be combined with Paid to Date to give distribution of Outstanding claims Must be combined with (simulated) Paid to Incurred ratios to give distributions of payment cash flows

Reserving and Bayesian Methods Developments in Stochastic Reserving Reserving and Bayesian Methods Any model that can be clearly defined can be fitted as a Bayesian model

Excel Demo – Gibbs Sampling ODP - Chain Ladder Model Only Developments in Stochastic Reserving Excel Demo – Gibbs Sampling ODP - Chain Ladder Model Only

Developments in Stochastic Reserving Igloo Demo 3 – Bayesian Methods ODP, Negative Binomial and Mack’s model Comparison with bootstrapping Developments in Stochastic Reserving

Bayesian Stochastic Reserving: Advantages Developments in Stochastic Reserving Bayesian Stochastic Reserving: Advantages Overcomes some practical difficulties with bootstrapping Sets of pseudo-data are not required, therefore far less RAM hungry when simulating Arguably more statistically rigorous and theoretically appealing Flexible approach Informative priors Bayesian Bornhuetter-Ferguson Method (see latest NAAJ) Model uncertainty Individual claims and additional covariates

The Bornhuetter-Ferguson Method and Bootstrapping Developments in Stochastic Reserving The Bornhuetter-Ferguson Method and Bootstrapping Pseudo-development factors give simulated proportion of ultimate to emerge in each development year BF prior loss ratio gives prior Ultimate Adjust pseudo-data to take account of BF prior Ultimate and simulated proportions, before forecasting Forecast based on adjusted pseudo-data BUT, simulate the BF prior ultimate making assumptions about precision of prior Add simulated forecasts to historic Paid-to-Date to give distribution of Ultimate.

A Bayesian Bornhuetter-Ferguson Method Developments in Stochastic Reserving A Bayesian Bornhuetter-Ferguson Method A Bayesian framework is a natural candidate for a stochastic BF method Bayesian recursive models offer the best way forward Work in progress! Verrall, RJ (2004), A Bayesian Generalised Linear Model for the Bornhuetter-Ferguson Method of Claims Reserving, NAAJ, July 2004

Bootstrapping and Curve Fitting Developments in Stochastic Reserving Bootstrapping and Curve Fitting

Igloo Demo 4 – Bootstrapping Curve fitting and Tail Factors Developments in Stochastic Reserving Igloo Demo 4 – Bootstrapping Curve fitting and Tail Factors

Summary of Developments Developments in Stochastic Reserving Summary of Developments ODP with non-constant scale parameters Bootstrap version of Mack’s model Recursive version of ODP: Negative Binomial model Recursive models allow weighting out of points (exclude Max/Min, n-year volume weighted averages etc) Bootstrap version of the Negative Binomial model Curve fitting, tail factors and bootstrapping Bayesian stochastic reserving Any clearly defined model (ODP, Mack, NB, curve fitting etc) A stochastic Bornhuetter-Ferguson method