Imprecise credibility theory

The classical credibility theory is a cornerstone of experience rating, especially in the field of property and casualty insurance. An obstacle to putting the credibility theory into practice is the conversion of available prior information into a precise choice of crucial hyperparameters. In most real-world applications, the information necessary to justify a precise choice is lacking, so we propose an imprecise credibility estimator that honestly acknowledges the imprecision in the hyperparameter specification. This results in an interval estimator that is doubly robust in the sense that it retains the credibility estimator’s freedom from model specification and fast asymptotic concentration, while simultaneously being insensitive to prior hyperparameter specification.

A credibility-based yield forecasting model for crop reinsurance pricing and weather risk management

Purpose The purpose of this paper is to propose an improved reinsurance pricing framework, which includes a crop yield forecasting model that integrates weather variables and crop production information from different geographically correlated regions using a new credibility estimator, and closed form reinsurance pricing formulas. A yield restatement approach to account for changing crop mix through time is also demonstrated. Design/methodology/approach The new crop yield forecasting model is empirically analyzed based on detailed farm-level data from Manitoba, Canada, covering 216 crop varieties from 19,238 farms from 1996 to 2011. As well, corresponding weather data from 30 stations, including daily temperature and precipitation, are considered. Algorithms that combine screening regression, cross-validation and principal component analysis are evaluated for the purpose of achieving efficient dimension reduction and model selection. Findings The results show that the new yield forecasting model provides significant improvements over the classical regression model, both in terms of in-sample and out-of-sample forecasting abilities. Research limitations/implications The empirical analysis is limited to data from the province of Manitoba, Canada, and other regions may show different results. Practical implications This research is useful from a risk management perspective for insurers and reinsurers, and the framework may also be used to develop improved weather risk management strategies to help manage adverse weather events. Originality/value This is the first paper to integrate a credibility estimator for crop yield forecasting, and develop a closed form reinsurance pricing formula.

A CREDIBILITY APPROACH FOR COMBINING LIKELIHOODS OF GENERALIZED LINEAR MODELS

Generalized linear models are a popular tool for the modelling of insurance claims data. Problems arise with the model fitting if little statistical information is available. In case that related statistics are available, statistical inference can be improved with the help of the borrowing-strength principle. We present a credibility approach that combines the maximum likelihood estimators of individual canonical generalized linear models in a meta-analytic way to an improved credibility estimator. We follow the concept of linear empirical Bayes estimation, which reduces the necessary parametric assumptions to a minimum. The concept is illustrated by a simulation study and an application example from mortality modelling.

BAYESIAN CHAIN LADDER MODELS

The literature on Bayesian chain ladder models is surveyed. Both Mack and cross-classified forms of the chain ladder are considered. Both cases are examined in the context of error terms distributed according to a member of the exponential dispersion family. Tweedie and over-dispersed Poisson errors follow as special cases. Bayesian cross-classified chain ladder models may randomise row, column or diagonal parameters. Column and diagonal randomisation has been largely absent from the literature until recently. The present paper allows randomisation of row and column parameters. The Bayes estimator, the linear Bayes (credibility) estimator, and the MAP estimator are shown to be identical in the Mack case, and in the cross-classified case provided that the error terms are Tweedie distributed. In the Mack case the variance structure is generalised considerably from the existing literature. In the cross-classified case the model structure differs somewhat from the existing literature, and a comparison is made between the two. MAP estimators for the cross-classified case are often given by implicit equations that require numerical solution. Recursive formulas are given for these in the general case of error terms from the exponential dispersion family. The connection between the cross-classified case and Bornhuetter-Ferguson prediction is explored.

A credibility method for profitable cross-selling of insurance products

A method is presented for identifying an expected profitable set of customers, to offer them an additional insurance product, by estimating a customer specific latent risk profile, for the additional product, by using the customer specific available data for an existing insurance product of the specific customer. For the purpose, a multivariate credibility estimator is considered and we investigate the effect of assuming that one (of two) insurance products is inactive (without available claims information) when estimating the latent risk profile. Instead, available customer specific claims information from the active existing insurance product is used to estimate the risk profile and thereafter assess whether or not to include a specific customer in an expected profitable set of customers. The method is tested using a large real data set from a Danish insurance company and it is shown that sets of customers, with up to 36% less claims than a priori expected, are produced as a result of the method. It is therefore argued that the proposed method could be considered, by an insurance company, when cross-selling insurance products to existing customers.

Robust Bayesian Experience Rating

Different rating methods which allow for exceptional large claims are discussed. A robust Bayesian statistical model is proposed which can cope with non negative, skewed data. An example from fire insurance is analyzed. The performance of the posterior mean is compared to the performance of a robust credibility estimator.

Robust Bayesian Experience Rating

Different rating methods which allow for exceptional large claims are discussed. A robust Bayesian statistical model is proposed which can cope with non negative, skewed data. An example from fire insurance is analyzed. The performance of the posterior mean is compared to the performance of a robust credibility estimator.

Credibility Weighted Hazard Estimation

Credibility weighting is helpful in many insurance applications where sparse data crave information from other sources of data. In this paper we aim at estimating a hazard curve using the nonparametric kernel method, where a credibility weighting principle is used locally, so that areas of sparse data for one subgroup can be alleviated by available information from other subgroups. The credibility estimator is found through a Hilbert space projection formulation of Buhlmann-Straub's credibility approach.