Exact Credibility and Tweedie Models

We consider the chain ladder reserving method in a Bayesian set up, which allows for combining the information from a specific claims development triangle with the information from a collective. That is, for instance, to consider simultaneously own company specific data and industry-wide data to estimate the own company's claims reserves. We derive Bayesian estimators and credibility estimators within this Bayesian framework. We show that the credibility estimators are exact Bayesian in the case of the exponential dispersion family with its natural conjugate priors. Finally, we make the link to the classical chain ladder method and we show that using non-informative priors we arrive at the classical chain ladder forecasts. However, the estimates for the mean square error of prediction differ in our Bayesian set up from the ones found in the literature. Hence, the paper also throws a new light upon the estimator of the mean square error of prediction of the classical chain ladder forecasts and suggests a new estimator in the chain ladder method.

“…• p = 2: Gamma distribution This family of models was also considered in Ohlsson and Johansson [16] in connection with calculating risk premiums in a multiplicative tariff structure.…”

Section: Definition 61 a Distribution G ‡ Is Said To Be Of The Expomentioning

confidence: 99%

Section: Definition 61 a Distribution G ‡ Is Said To Be Of The Expomentioning

confidence: 99%

Credibility for the Chain Ladder Reserving Method

Gisler

Wüthrich

2008

“…Indeed, if f W U,i disappears on the boundary of the canonical parameter space of the EDF with cumulant b U , for any Jewell, 1974;Bühlmann and Gisler, 2005). If, in addition to the above hypothesis, the first derivative of f W U,i disappears on the boundary of the canonical parameter space, then Ohlsson and Johansson, 2006). Similar considerations apply to W V, j .…”

Section: A4 Distributional Assumptions For the Risk Parametersmentioning

confidence: 91%

“…Within regression models, the techniques of GLMs are combined with those of credibility theory (e.g. Nelder and Verrall, 1997;Ohlsson and Johansson, 2006;Ohlsson, 2008) or the Generalized Linear Mixed Models are used (Antonio et al, 2006;Antonio and Beirlant, 2007). To estimate GLMs with random effects, Lee and Nelder (1996), Lee and Nelder (2001), Lee et al (2006) suggest the hierarchical or h-likelihood approach.…”

Section: Introductionmentioning

confidence: 99%

A Mixture Model for Payments and Payment Numbers in Claims Reserving

Gigante¹,

Picech

Sigalotti³

2016

We consider a Tweedie's compound Poisson regression model with fixed and random effects, to describe the payment numbers and the incremental payments, jointly, in claims reserving. The parameter estimates are obtained within the framework of hierarchical generalized linear models, by applying the h-likelihood approach. Regression structures are allowed for the means and also for the dispersions. Predictions and prediction errors of the claims reserves are evaluated. Through the parameters of the distributions of the random effects, some external information (e.g. a development pattern of industry wide-data) can be incorporated into the model. A numerical example shows the impact of external data on the reserve and prediction error evaluations

“…are related to the moments of the risk parameter U. Specifically, ψ U,i = E(U i ) (see e.g., Jewell, 1974;Bühlmann and Gisler, 2005) and, in Tweedie models with variance function Ohlsson and Johansson, 2006). Similar considerations apply to W V ,i+j .…”

Section: (A3) Structural Assumptions For the Response Variablesmentioning

confidence: 99%

Calendar Year Effect Modeling for Claims Reserving in HGLM

Gigante¹,

Picech²,

Sigalotti³

2019

Claims reserving models are usually based on data recorded in run-off tables, according to the origin and the development years of the payments. The amounts on the same diagonal are paid in the same calendar year and are influenced by some common effects, for example, claims inflation, that can induce dependence among payments. We introduce hierarchical generalized linear models (HGLM) with risk parameters related to the origin and the calendar years, in order to model the dependence among payments of both the same origin year and the same calendar year. Besides the random effects, the linear predictor also includes fixed effects. All the parameters are estimated within the model by the h-likelihood approach. The prediction for the outstanding claims and an approximate formula to evaluate the mean square error of prediction are obtained. Moreover, a parametric bootstrap procedure is delineated to get an estimate of the predictive distribution of the outstanding claims. A Poisson-gamma HGLM with origin and calendar year effects is studied extensively and a numerical example is provided. We find that the estimates of the correlations can be significant for payments in the same calendar year and that the inclusion of calendar effects can determine a remarkable impact on the prediction uncertainty.