Richard Verrall

STOCHASTIC CLAIMS RESERVING IN GENERAL INSURANCE

This paper considers a wide range of stochastic reserving models for use in general insurance, beginning with stochastic models which reproduce the traditional chain-ladder reserve estimates. The models are extended to consider parametric curves and smoothing models for the shape of the development run-off, which allow extrapolation for the estimation of tail factors. The Bornhuetter-Ferguson technique is also considered, within a Bayesian framework, which allows expert opinion to be used to provide prior estimates of ultimate claims. The primary advantage of stochastic reserving models is the availability of measures of precision of reserve estimates, and in this respect, attention is focused on the root mean squared error of prediction (prediction error). Of greater interest is a full predictive distribution of possible reserve outcomes, and different methods of obtaining that distribution are described. The techniques are illustrated with examples throughout, and the wider issues discussed, in particular, the concept of a ‘best estimate’; reporting the variability of claims reserves; and use in dynamic financial analysis models.

Analytic and bootstrap estimates of prediction errors in claims reserving

Abstract We consider an appropriate residual definition for use in a bootstrap exercise to provide a computationally simple method of obtaining reserve prediction errors for a generalised linear model which reproduces the reserve estimates of the chain ladder technique (under certain restrictions which are specified in the paper). We show how the bootstrap prediction errors can be computed easily in a spreadsheet, without the need for statistical software packages. The bootstrap prediction errors are compared with their analytic equivalent from other stochastic reserving models, and also compared with other methods commonly used, including Mack’s distribution free approach (Mack, 1993. ASTIN Bulletin 23 (2), 213–225) and methods based on log-linear models.

A Stochastic Model Underlying the Chain-Ladder Technique

This paper presents a statistical model underlying the chain-ladder technique. This is related to other statistical approaches to the chain-ladder technique which have been presented previously. The statistical model is cast in the form of a generalised linear model, and a quasi-likelihood approach is used. It is shown that this enables the method to process negative incremental claims. It is suggested that the chain-ladder technique represents a very narrow view of the possible range of models.

Predictive Distributions of Outstanding Liabilities in General Insurance

ABSTRACT This paper extends the methods introduced in England & Verrall (2002), and shows how predictive distributions of outstanding liabilities in general insurance can be obtained using bootstrap or Bayesian techniques for clearly defined statistical models. A general procedure for bootstrapping is described, by extending the methods introduced in England & Verrall (1999), England (2002) and Pinheiro et al. (2003). The analogous Bayesian estimation procedure is implemented using Markov-chain Monte Carlo methods, where the models are constructed as Bayesian generalised linear models using the approach described by Dellaportas & Smith (1993). In particular, this paper describes a way of obtaining a predictive distribution from recursive claims reserving models, including the well known model introduced by Mack (1993). Macks model is useful, since it can be used with data sets which exhibit negative incremental amounts. The techniques are illustrated with examples, and the resulting predictive distributions from both the bootstrap and Bayesian methods are compared.

An investigation into parametric models for mortality projections, with applications to immediate annuitants’ and life office pensioners’ data

Abstract This paper investigates the use of parametric models for projecting mortality rates. The basic framework used is that of generalised linear and non-linear models and can be considered as an extension of the Gompertz–Makeham models [Forfar et al., J. Inst. Actuaries 115 (1988) 1; Trans. Faculty Actuaries 41 (1988) 97] to include calendar period. The data considered are the CMI ultimate experience for immediate annuitants (male and female) over the period 1958–1994, and for life office pensioners (male and female) over the period 1983–1996. The modelling structure suggested by Renshaw et al. [British Actuarial J. 2 (II) (1996) 449] is used to investigate the data sets pertaining to the ultimate experiences, and to determine a range of suitable models, analysing the data by age and calendar period. The properties of these models are investigated and recommendations are made on which models are appropriate for use in projections. The select experience for immediate annuitants’ is modelled using the structure suggested by Renshaw and Haberman [Insurance: Math. Econ. 19 (2) (1997) 105]. Projected forces of mortality using the recommended model are given for each experience. These are compared with the CMI projected mortality rates.

A Bayesian Generalized Linear Model for the Bornhuetter-Ferguson Method of Claims Reserving

Abstract This paper shows how Bayesian models within the framework of generalized linear models can be applied to claims reserving. The author demonstrates that this approach is closely related to the Bornhuetter-Ferguson technique. Benktander (1976) and Mack (2000) previously studied the Bornhuetter-Ferguson technique and advocated using credibility models. The present paper uses a Bayesian parametric model within the framework of generalized linear models.

Modeling Operational Risk with Bayesian Networks

Bayesian networks is an emerging tool for a wide range of risk management applications, one of which is the modeling of operational risk. This comes at a time when changes in the supervision of financial institutions have resulted in increased scrutiny on the risk management of banks and insurance companies, thus giving the industry an impetus to measure and manage operational risk. The more established methods for risk quantification are linear models such as time series models, econometric models, empirical actuarial models, and extreme value theory. Due to data limitations and complex interaction between operational risk variables, various nonlinear methods have been proposed, one of which is the focus of this article: Bayesian networks. Using an idealized example of a fictitious on line business, we construct a Bayesian network that models various risk factors and their combination into an overall loss distribution. Using this model, we show how established Bayesian network methodology can be applied to: (1) form posterior marginal distributions of variables based on evidence, (2) simulate scenarios, (3) update the parameters of the model using data, and (4) quantify in real-time how well the model predictions compare to actual data. A specific example of Bayesian networks application to operational risk in an insurance setting is then suggested.

CREDIBILITY THEORY AND GENERALIZED LINEAR MODELS

This paper shows how credibility theory can be encompassed within the theory of Hierarchical Generalized Linear Models. It is shown that credibility estimates are obtained by including random effects in the model. The framework of Hierarchical Generalized Linear Models allows a more extensive range of models to be used than straightforward credibility theory. The model fitting and testing procedures can be carried out using a standard statistical package. Thus, the paper contributes a further range of models which may be useful in a wide range of actuarial applications, including premium rating and claims reserving.

On the estimation of reserves from loglinear models

Abstract This paper considers the estimation of claims reserves and outstanding claims when a loglinear model is applied. Unbiased estimators are derived and these are compared with maximum likelihood and actuarial estimators. The set of loglinear models considered includes the commonly used chain ladder model. An example is given illustrating the results for the chain ladder model.

Bayes and Empirical Bayes Estimation for the Chain Ladder Model

The subject of predicting outstanding claims on a porfolio of general insurance policies is approached via the theory of hierarchical Bayesian linear models. This is particularly appropriate since the chain ladder technique can be expressed in the form of a linear model. The statistical methods which are applied allow the practitioner to use different modelling assumptions from those implied by a classical formulation, and to arrive at forecasts which have a greater degree of inherent stability. The results can also be used for other linear models. By using a statistical structure, a sound approach to the chain ladder technique can be derived. The Bayesian results allow the input of collateral information in a formal manner. Empirical Bayes results are derived which can be interpreted as credibility estimates. The statistical assumptions which are made in the modelling procedure are clearly set out and can be tested by the practitioner. The results based on the statistical theory form one part of the reserving procedure, and should be followed by expert interpretation and analysis. An illustration of the use of Bayesian and empirical Bayes estimation methods is given.