Bayesian Premium Research Articles

Effective experience rating for large insurance portfolios via surrogate modeling

Experience rating in insurance uses a Bayesian credibility model to upgrade the current premiums of a contract by taking into account policyholders' attributes and their claim history. Most data-driven models used for this task are mathematically intractable, and premiums must be obtained through numerical methods such as simulation via MCMC. However, these methods can be computationally expensive and even prohibitive for large portfolios when applied at the policyholder level. Additionally, these computations become “black-box” procedures as there is no analytical expression showing how the claim history of policyholders is used to upgrade their premiums. To address these challenges, this paper proposes a surrogate modeling approach to inexpensively derive an analytical expression for computing the Bayesian premiums for any given model, approximately. As a part of the methodology, the paper introduces a likelihood-based summary statistic of the policyholder's claim history that serves as the main input of the surrogate model and that is sufficient for certain families of distribution, including the exponential dispersion family. As a result, the computational burden of experience rating for large portfolios is reduced through the direct evaluation of such analytical expression, which can provide a transparent and interpretable way of computing Bayesian premiums.

Bayesian premium of a credibility model based on a heterogeneous SETINAR(2, 1) process

<abstract><p>In this paper, we propose a new credibility model based on heterogeneous integer-valued self-exciting threshold autoregressive time series, in which the SETINAR(2, 1) process is used to fit the claim numbers of policyholders for consecutive periods, and the unobservable heterogeneity is assumed to follow Gamma distribution. We obtain the Bayesian pricing formula for the proposed model and present some numerical examples to illustrate how the claim history affects the future premiums. We also apply the proposed model to a real panel dataset from the Wisconsin Local Government Property Insurance Fund. By comparing with some existing models, we find that our model can exploit the past information more efficiently and has better predictive performance.</p></abstract>

Exact credibility reference Bayesian premiums

In this paper, reference analysis, the tool provided by Berger et al. (2009), is used to obtain reference Bayesian premiums, which can be helpful when the practitioner has insufficient information to elicit a prior distribution. The Bayesian premiums thus obtained are based exclusively on prior distributions built from the model generated and from the available data. This mechanism produces an objective Bayesian inference, which appears to be the same as the robust Γ-minimax inference. In an informational-theoretical sense, the prior distribution used to make the inference is less informative. These Bayesian premiums are expected to approximate those which would have been obtained using proper priors describing a vague initial state of knowledge. Useful credibility expressions are readily derived by taking classes of priors involving restrictions on moments, i.e., restrictions on the collective or prior premium when the weighted squared-error loss function is used.

Bayesian credibility under a bivariate prior on the frequency and the severity of claims

In this paper, as opposed to the usual assumption of independence, we propose a credibility model in which the (unobservable) risk profiles of the claim frequency and the claim severity are dependent. Given the risk profiles, the (conditional) marginal distributions of frequency and severity are assumed to belong to the exponential family. A bivariate conjugate prior is proposed for the risk profiles, where the dependency is incorporated via a factorization structure of the joint density. The bivariate posterior is derived, and in turn the Bayesian premium for the aggregate claim is given along with some results on the predictive joint and marginal distributions involving the claim number and the aggregate claim in the next period. To demonstrate the generality of our proposed model, we provide four different examples of bivariate conjugate priors in relation to mixed Erlang, gamma mixture, Farlie-Gumbel-Morgenstern (FGM) copula, and bivariate beta, where each choice has different merits. In these examples, more explicit results can be obtained, and in particular the predictive variance and Value-at-Risk (VaR) of the aggregate claim certainly provide more information on the inherent risk than the Bayesian premium which is merely the predictive mean. Finally, numerical examples will be given to illustrate the effect of dependence on the results, including the use of a real data set that further takes observable risk factors into consideration under a regression setting.

Determination of risk-based for rice farmers using the nonparametric Bayesian method

The Paddy Farmers Business Insurance or Asuransi Usaha Tani Padi (AUTP) is one of the insurance programs organized by the Indonesian government to improve the welfare of rice farmers in Indonesia. In this study, we formulated a premium estimation model of risk-based AUTP insurance. The method used was the Bayes method with a nonparametric approach. The data used was the AUTP claim data in Aceh province. The determination of the premium has been done by providing a settlement algorithm based on past data for two periods. From the analysis conducted, it was found that the Bayesian premium obtained is still not in accordance with what is currently applied by the government, especially in the area of study.

Modeling the Conditional Dependence between Discrete and Continuous Random Variables with Applications in Insurance

We jointly model amount of expenditure for outpatient visits and number of outpatient visits by considering both dependence and simultaneity by proposing a bivariate structural model that describes both variables, specified in terms of their conditional distributions. For that reason, we assume that the conditional expectation of expenditure for outpatient visits with respect to the number of outpatient visits and also, the number of outpatient visits expectation with respect to the expenditure for outpatient visits is related by taking a linear relationship for these conditional expectations. Furthermore, one of the conditional distributions obtained in our study is used to derive Bayesian premiums which take into account both the number of claims and the size of the correspondent claims. Our proposal is illustrated with a numerical example based on data of health care use taken from Medical Expenditure Panel Survey (MEPS), conducted by the U.S. Agency of Health Research and Quality.

Dynamic Bayesian Ratemaking: A Markov Chain Approximation Approach

We contribute to the non-life experience ratemaking literature by introducing a computationally efficient approximation algorithm for the Bayesian premium in models with dynamic random effects, where the risk of a policyholder is governed by an individual process of unobserved heterogeneity. Although intuitive and flexible, the biggest challenge of dynamic random effect models is that the resulting Bayesian premium typically lacks tractability. In this article, we propose to approximate the dynamics of the random effects process by a discrete (hidden) Markov chain and replace the intractable Bayesian premium of the original model by that of the approximate Markov chain model, for which concise, closed-form formula are derived. The methodology is general because it does not rely on any parametric distributional assumptions and, in particular, allows for the inclusion of both the cost and the frequency components in pricing. Numerical examples show that the proposed approximation method is highly accurate. Finally, a real data pricing example is used to illustrate the versatility of the approach.

Dynamic Bayesian Ratemaking: A Markov Chain Approximation Approach

We contribute to the non-life experience ratemaking literature by introducing a computationally efficient approximation algorithm for the Bayesian premium in models with dynamic random effect, where the risk of a policyholder is governed by an individual process of unobserved heterogeneity. Although intuitive and flexible, the biggest challenge of dynamic random effect models is that the resulting Bayesian premium typically lacks tractability. In this paper we propose to approximate the dynamics of the random effect process by a discrete (hidden) Markov chain, and replace the intractable Bayesian premium of the original model by that of the approximate Markov chain model, for which concise, closed form formula are derived. The methodology is general as it does not rely on any parametric distributional assumptions, and in particular allows for the inclusion of both the cost and the frequency components in pricing. Numerical examples show that the proposed approximation method is highly accurate. Finally, a real-data pricing example are utilized to illustrate the versatility of the approach.

Bayesian Premium Estimators for Mixture of Two Gamma Distributions Under Squared Error, Entropy and Linex Loss Functions: With Informative and Non Informative Priors

In this paper, we consider the Zeghdoudi distribution as the conditional distribution of Xn | θ, we focus on estimation of the Bayesian premium under three loss functions (squared error which is symmetric, Linex and entropy, which are asymmetric), using non-informative and informative priors (the extension of Jeffreys and Gamma priors) respectively. Because of its difficulty and non linearity, we use a numerical approximation for computing the Bayesian premium.

Bayesian ratemaking with common effects modeled by mixture of Polya tree processes

In classical models for Bayesian ratemaking, claims are usually assumed to be independent over risks. However, this assumption may be violated because there are situations that could derive possible dependence among the insured individuals. This paper aims to investigate the typical problem of experience ratemaking to account for a special type of dependence that is known as common effects in the literature. Polya tree processes are employed to model the common effects and, by means of an MCMC scheme, the corresponding Bayesian premiums are numerically computed. This provides a useful alternative to the well known results on Bayesian ratemaking with common effects.

Bayesian Premium Estimators for Mixture of Two Gamma Distributions Under Squared Error, Entropy and Linex Loss Functions: With Informative and Non Informative Priors

Bayesian Premium Estimators for Mixture of Two Gamma Distributions Under Squared Error, Entropy and Linex Loss Functions: With Informative and Non Informative Priors

On Bayesian Premium Estimators for Gamma Lindley Model under Squared Error Loss Function and Linex Loss Function

We consider the Gamma Lindley distribution (GaL) as the conditional distribution of (X|θ,γ), we focus on the estimation of the Bayesian premium under squared error loss function (symmetric) and linear-exponential (Linex) loss function (asymmetric), using informative priors (the Gamma prior). Because of its difficulty and non-linearity, we use a numerical approximation for computing the Bayesian premium. Finally, a simulation and comparative study with varying sample sizes are given.

Bayesian and Bühlmann credibility for phase-type distributions with a univariate risk parameter

Credibility theory is a statistical tool to calculate the premium for the next period based on past claims experience and the manual rate. Each contract is characterized by a risk parameter. A phase-type (or PH) random variable, which is defined as the time until absorption in a continuous-time Markov chain, is fully characterized by two sets of parameters from that Markov chain: the initial probability vector and transition intensity matrix. In this article, we identify an interpretable univariate risk parameter from amongst the many candidate parameters, by means of uniformization. The resulting density form is then expressed as an infinite mixture of Erlang distributions. These results are used to obtain a tractable likelihood function by a recursive formula. Then the best estimator for the next premium, i.e. the Bayesian premium, as well as its approximation by the Bühlmann credibility premium are calculated. Finally, actuarial calculations for the Bühlmann and Bayesian premiums are investigated in the context of a gamma prior, and illustrated by simulated data in a series of examples.

On bounds for ratios of posterior expectations under asymmetric loss functions

AbstractWe compute the bounds of the Bayesian premium when the prior distribution belongs to the ε‐contaminated class of priors and the loss function is weighted asymmetric Linex, by applying a result of Gomenz–Deniz et al [1]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Bayesian Premium Rating with Latent Structure

We propose a fully Bayesian approach to non-life risk premium rating, based on hierarchical models with latent variables for both claim frequency and claim size. Inference is based on the joint posterior distribution and is performed by Markov Chain Monte Carlo. Rather than plug-in point estimates of all unknown parameters, we take into account all sources of uncertainty simultaneously when the model is used to predict claims and estimate risk premiums. Several models are fitted to both a simulated dataset and a small portfolio regarding theft from cars. We show that interaction among latent variables can improve predictions significantly. We also investigate when interaction is not necessary. We compare our results with those obtained under a standard generalized linear model and show through numerical simulation that geographically located and spatially interacting latent variables can successfully compensate for missing covariates. However, when applied to the real portfolio data, the proposed models are not better than standard models due to the lack of spatial structure in the data.

Robust Bayesian Premium Principles in Actuarial Science

The term premium relates to the purchase price of an insurance contract. Bayesian models in credibility theory require a complete specification of the model (basically, the prior) and it is difficult to justify any one particular choice. According to robust Bayesian methodology, uncertainty in the prior can be modelled by specifying a class Γ of priors instead of a single prior. We examine the ranges of Bayesian premiums when the priors belong to such a class. Most robust Bayesian procedures include measures of sensitivity of quantities which can be expressed in terms of a posterior expectation (e.g. the mean, variance and probability of sets). Nevertheless, a significant difference that appears in the actuarial context is considered here. The expression for some Bayes premiums in credibility theory suggests that the quantity of interest can be expressed in terms of the ratio of posterior expectations. Appropriate techniques to do this are considered here. Two models and two situations are presented for a non-compound collective model. Even though the model is very robust, a consideration of unimodality significantly reduces the sensitivity of the Bayesian premium arising from a base prior π0. Therefore, unimodality is very convenient for modelling subjective beliefs about the risk parameter.

The Esscher premium principle in risk theory: a Bayesian sensitivity study

In this paper the Esscher premium calculation principle is applied to the non-compound collective model in a robust Bayesian context. We consider that uncertainty with regard to the prior distribution can be represented by the assumption that the unknown prior distribution belongs to a class of distributions Γ and examine the ranges of the Bayesian premium when the priors belong to such a class. The assessment of the influence of the prior is termed sensitivity analysis or robustness analysis.