Credibility Premium Research Articles

The generalized equity and exact credibility premiums with examples

We investigate a more generalized alternative to the conventional credibility premium that results from squared-error loss, namely, equitable credibility premium (Promislow and Young, 1999), based on exponential family, and we derive an accurate credibility for Bühlmann model with an example. We provide sufficient conditions for accurate credibility and expand a generalised formula for the credibility weight in specific circumstances. In this article, we expand upon the research conducted by Promislow and Young (2000) to include situations where an actuary is provided with data regarding a policyholder who has a more extensive exponential family. In the first section, we established our notation and reformulation of exact credibility for a Bühmann model. We derive the class of general equitable credibility premiums in the second section.

A new adaptive approach for bridging the gap in insurance premium estimation

This paper proposes a new adaptive approach called adaptive mean (ADM) that combines the strengths of trimming and winsorization to minimize the mean square error (MSE). ADM will contribute significantly to insurers by enabling them to mitigate risk exposure caused by inaccurate premium estimation and establish more precise premiums for their clients. Furthermore, the proposed mean offers several pros compared to the conventional mean. It facilitates the natural and intuitive calculation of risk loading, identifies and measures the impact of large claims, and analyzes premium susceptibility to skewed risk by capturing the tails of the relevant loss models. In addition, this paper reviews two established, robust credibility methods (trimming and winsorization). It compares them with our proposed method, resulting in a more reliable and robustly accurate method for credibility premium estimation. We presented and analyzed two real data sets from engineering insurance companies and the Wisconsin Office of the Insurance Commissioner to highlight the advances of ADM in minimizing the MSE and building more robust credibility models that are less vulnerable to outlier events and model uncertainty.

Quantile balanced credibility model

This paper introduces a model into the methodology of greatest accuracy approach in credibility theory, we extend Pitselis (2013) quantile credibility model by including a different type of losses which is called the weighted balanced loss function (WBLF henceforth) in order to establish novel credibility premiums under the balanced squared error loss function. More specifically, we further explain how quantiles under the WBLF can be incorporated into the classical Bühlmann’s (1967) credibility model. A new model of quantile credibility under the WBLF is presented, where the premiums are estimated using the non-parametric approach where premiums must be estimated based on data only, without postulating any type of distribution. Furthermore, we establish the credibility estimators and provide an application to real data of the results using R software to evaluate the effectiveness of our proposed model, and conclude with a comparison with the Pitselis (2013) model and the Bühlmann (1967) model to show the effectiveness of our premium.

Premium Estimation of General Insurance Contracts Using Buhlmann Credibility Model

Insurance coverage is reliant on appropriate premium determination, a focus addressed in this study through the application credibility theory. The aim was to compute short-term insurance premiums by leveraging authentic data from general insurance contracts as published in the IRA annual reports spanning from 2013 to 2021. The study employed the Buhlmann credibility model to derive net credible premiums for 13 non-life insurance contracts, assuming constant changes in business volumes. The estimation of credibility premiums was conducted by initially calculating within and between-portfolio variance with an aim of determining these premiums through linear estimation approach with the help of the credibility premium formula. The results revealed consistent credibility factors across all contracts, although other influencing factors were neglected. Therefore, it was highly recommended that an appropriate credibility model alongside other factors that influence premium estimation be considered to enhance the accuracy of premium estimates. Moreover, embracing heterogeneity over homogeneity is advised to better reflect individual risk profiles. The research underscores the importance of effective premium determination in insurance firms, as premiums constitute their primary revenue. Stakeholders, including insurers and regulators, are encouraged to integrate this approach and other relevant factors to enhance pricing accuracy and overall sector practices.

Policy Evaluation in Polarized Polities: The Case of Randomized Controlled Trials

This paper provides a political-economic analysis of policy evaluation. We focus on Randomized Controlled Trials (RCTs) as a subset of policy evaluations and argue that they are used instrumentally by decision-makers in order to improve perceptions of reforms and help secure policy legacy. We theorize that this ’credibility premium’ is more valuable for incumbents in politically polarized societies, which we empirically examine using two methods. First, we provide a series of vignettes of prominent randomized evaluations embedded by governments in policy roll-outs and a detailed case study of the Liberian government’s decision to commission a third-party RCT evaluation of a proposed primary school privatization reform. Second, we have compiled a unique cross-country panel data set on RCTs in development policy since 1996, with which we demonstrate that RCTs are more likely to occur in polarized societies, and that the effect is amplified by the degree of political competition.

Tweedie multivariate semi-parametric credibility with the exchangeable correlation

This article proposes a framework for determining credibility premiums for multiple coverages in a compound risk model with Tweedie distribution. The framework builds upon previous results on credibility premium and provides an explicit multivariate credibility premium formula that is applicable to the Tweedie family assuming that the unobserved heterogeneity for the multiple coverage have the common correlation. The practical applicability of the proposed framework is evaluated through simulation and empirical analysis using the LGPIF dataset, which includes claims and policy characteristics data for various types of coverages observed over time. The findings suggest that the proposed framework can be useful in ratemaking practice by incorporating a non-trivial dependence structure among the multiple types of claims.

Bühlmann's Credibility Model with Claims of Negative Binomial and 2-Poisson Distribution

One of the premium determination techniques is to use credibility theory. In this study, a credibility premium determination model was derived with the best accuracy approach in the form of Bühlmann’s credibility premium. The claim data is assumed to have a Negative Binomial and 2-Poisson distribution. Bühlmann's credibility premium formula is given explicitly for these two data distributions. The obtained model is also applied to the correct data following these distributions. From the simulation results, it is obtained that the premium values are very close in value so that both models can be applied to the data and have a high level of credibility because they have a high credibility factor value.

Approximation of Zero-Inflated Poisson Credibility Premium via Variational Bayes Approach

While both zero-inflation and the unobserved heterogeneity in risks are prevalent issues in modeling insurance claim counts, determination of Bayesian credibility premium of the claim counts with these features are often demanding due to high computational costs associated with a use of MCMC. This article explores a way to approximate credibility premium for claims frequency that follows a zero-inflated Poisson distribution via variational Bayes approach. Unlike many existing industry benchmarks, the proposed method enables insurance companies to capture both zero-inflation and unobserved heterogeneity of policyholders simultaneously with modest computation costs. A simulation study and an empirical analysis using the LGPIF dataset were conducted and it turned out that the proposed method outperforms many industry benchmarks in terms of prediction performances and computation time. Such results support the applicability of the proposed method in the posterior ratemaking practices.

Some results on quadratic credibility premium using the balanced loss function

PurposeThis paper generalizes the quadratic framework introduced by Le Courtois (2016) and Sumpf (2018), to obtain new credibility premiums in the balanced case, i.e. under the balanced squared error loss function. More precisely, the authors construct a quadratic credibility framework under the net quadratic loss function where premiums are estimated based on the values of past observations and of past squared observations under the parametric and the non-parametric approaches, this framework is useful for the practitioner who wants to explicitly take into account higher order (cross) moments of past data.Design/methodology/approachIn the actuarial field, credibility theory is an empirical model used to calculate the premium. One of the crucial tasks of the actuary in the insurance company is to design a tariff structure that will fairly distribute the burden of claims among insureds. In this work, the authors use the weighted balanced loss function (WBLF, henceforth) to obtain new credibility premiums, and WBLF is a generalized loss function introduced by Zellner (1994) (see Gupta and Berger (1994), pp. 371-390) which appears also in Dey et al. (1999) and Farsipour and Asgharzadhe (2004).FindingsThe authors declare that there is no conflict of interest and the funding information is not applicable.Research limitations/implicationsThis work is motivated by the following: quadratic credibility premium under the balanced loss function is useful for the practitioner who wants to explicitly take into account higher order (cross) moments and new effects such as the clustering effect to finding a premium more credible and more precise, which arranges both parts: the insurer and the insured. Also, it is easy to apply for parametric and non-parametric approaches. In addition, the formulas of the parametric (Poisson–gamma case) and the non-parametric approach are simple in form and may be used to find a more flexible premium in many special cases. On the other hand, this work neglects the semi-parametric approach because it is rarely used by practitioners.Practical implicationsThere are several examples of actuarial science (credibility).Originality/valueIn this paper, the authors used the WBLF and a quadratic adjustment to obtain new credibility premiums. More precisely, the authors construct a quadratic credibility framework under the net quadratic loss function where premiums are estimated based on the values of past observations and of past squared observations under the parametric and the non-parametric approaches, this framework is useful for the practitioner who wants to explicitly take into account higher order (cross) moments of past data.

PENERAPAN METODE BAYES DALAM MENGESTIMASI PREMI KREDIBILITAS PADA ASURANSI UMUM

Determination of insurance premiums is very important the calculation must be done carefully so that there is experience losses. The purpose of this research is to find out the application of empirical Bayes credibility theory Model 1 and estimate of the credibility premium on general insurance. A method that can help in overcoming these problems, that is empirical Bayes credibility theory Model 1, results of the estimated credibility premium credibility (in Euros) for insurance companies Alianz, Csob, Generali, Koop, Unisqa, and Wusten respectively as follows: 46.774811, 7.801307, 10.368991, 58.812250, 6.703035, and 5.091605. These results, the average claim is greater than the credibility premium, so that insurance companies can reserve premiums for the future.

On the risk of credibility premium rules

ABSTRACT A discrete-time risk process is considered where the full distribution of the claim size X is not completely known to the insurance company. Rather, it assumes that the distribution of X given is where Z is some structural random variable for which a prior is available. The main emphasis of the paper is the unconditional ruin probability in this setting where the premium is either updated according to incoming information about the claim distribution or computed by the expected value principle. This is in turn studied via the conditional ruin probability , for which large deviations estimates are available. Rigorous proofs are given only for the case of the forming a scale parameter family, including the classical case of gamma claims with a gamma prior. However, the analysis readily suggests what should be the behaviour of in different models for the claims.

Multi-peril frequency credibility premium via shared random effects

It is not uncommon to observe that an insurance policy consists of multiple coverages which cover different perils. In order to capture possible dependence among the claims from multiple coverages and reflect it in a posteriori ratemaking for property and casualty insurance, a shared random effects model is proposed in this article which not only captures the unobserved heterogeneity in risks but also induces natural dependence structure among the claims from multiple perils. The proposed model has a nice interpretation and comes with closed forms of joint likelihood and credibility premium which are desirable for ease of implementation in practice. Both the proposed and naive methods are calibrated using a public insurance dataset provided by Wisconsin Local Government Property Insurance Fund, where one can observe both policy characteristics and historical claims information and it turns out that the proposed method outperforms the methods which are industry benchmarks in terms of out-of-sample validation.

Quantile Credibility Models with Common Effects

Different from classical Bühlmann and Bühlmann Straub credibility models in which independence between different risks are assumed, this paper takes dependence between risks into consideration and extends the classical Bühlmann model by introducing a common stochastic shock element. What is more, instead of relying on complete information of historical data, we aim to derive the premium using quantile of the available data. By the method of linear regression, we manage to obtain the quantile credibility premium with common effects. Our result is the generalization of existing results in credibility theory. Both quantile credibility model proposed by Pitselis (2013) and credibility premium for models with dependence induced by common effects obtained by Wen et al. (2009) are special cases of our model. Numerical simulations are also presented to illustrate the impact of quantile credibility with common effect.

Predictive compound risk models with dependence

The two-part regression model, which subdivides aggregate claims into frequency and severity components, is a widespread tool in actuarial practice for predicting pure premium. The assumption of independence between frequency and severity is conventional but there is increased interest in advancing models to capture the possible dependence as is done in Garrido et al. (2016). This paper extends the work of Garrido et al. (2016) and explores the benefits of using random effects for predicting insurance claims observed longitudinally, or over a period of time, within a two-part framework relaxing the assumption of independence. More specifically, we introduce a generalized formula for credibility premium of compound sum with dependence, which extends and integrates previous work in both credibility premium of compound sums and dependent two-part compound risk models. In this generalized formula of credibility premium of compound sum, we are able to derive a dependence function, DN(γ), that offers an informative measure of the strength and direction of the association between frequency and severity. This function is easy to interpret and allows for practical implementation useful for actuarial ratemaking. Our model calibration, based on longitudinal claims from a Singapore automobile insurance company, shows that there is a strong negative dependence between frequency and severity.

Calculating Premium Credibility Using the Buhlmann-Straub Modelwith Nonparametric Assessment

When an insurance company calculates the premium it will divides the policy holders into groups. The division is considered based on risk level in each group. The problem is then to devise a way of combining the experience risk of the group with the experience of the individual risk to calculate the premium, so then Credibility Theory provides a solution to this problem.This script discuss about calculation of credibility premium use Buhlmann-Straub Model with nonparametric estimation to the aggregate claim amount data set within few years observation in some group of policy holders in general insurance. By using credibility theory we can calculate the value of credibility factor and credibility premium or future premium. The value of premium credibility is calculated from only one group of policyholders from the previous year's data. For better value of premium credibility, data with more experience years and the policyholder group better reflect the total loss value during the observation year.The result of this calculation are credibility factor per group, average credibility premium per members in group and credibility premium total for the last year for each group. We can obtain total losses and total premium which surprisingly equal.

Bayesian credibility premium with GB2 copulas

Abstract For observations over a period of time, Bayesian credibility premium may be used to predict the value of a response variable for a subject, given previously observed values. In this article, we formulate Bayesian credibility premium under a change of probability measure within the copula framework. Such reformulation is demonstrated using the multivariate generalized beta of the second kind (GB2) distribution. Within this family of GB2 copulas, we are able to derive explicit form of Bayesian credibility premium. Numerical illustrations show the application of these estimators in determining experience-rated insurance premium. We consider generalized Pareto as a special case.

Bühlmann & Bühlmann-Straub Credibility for Extreme Claims Applied on Non-Life Egyptian Insurance Market: An Actuarial Approach

Credibility theory is an actuarial approach used to calculate the short term insurance premiums. The aim of this article is to calculate the credibility premium based on real data from non-life Egyptian insurance industry during 10 years period (2006 to 2015), taking into consideration the amount of incurred claims from six insurance branches and the number of extreme losses in each branch. The analysis was based on the assumptions of Bühlmann and Bühlmann-Straub credibility models in order to estimate the net credible premium for the upcoming year as a linear function of the prior claims and the number of extreme events.

Regression Tree Credibility Model

This article applies machine learning techniques to credibility theory and proposes a regression-tree-based algorithm to integrate covariate information into credibility premium prediction. The recursive binary algorithm partitions a collective of individual risks into mutually exclusive subcollectives and applies the classical Bühlmann-Straub credibility formula for the prediction of individual net premiums. The algorithm provides a flexible way to integrate covariate information into individual net premiums prediction. It is appealing for capturing nonlinear and/or interaction covariate effects. It automatically selects influential covariate variables for premium prediction and requires no additional ex ante variable selection procedure. The superiority in prediction accuracy of the proposed algorithm is demonstrated by extensive simulation studies. The proposed method is applied to the U.S. Medicare data for illustration purposes.

Regression Tree Credibility Model

This paper applies machine learning techniques to credibility theory and proposes a regression-tree-based algorithm to integrate covariate information into credibility premium prediction. The recursive binary algorithm partitions a collective of individual risks into mutually exclusive sub-collectives, and applies the classical Buhlmann-Straub credibility formula for the prediction of individual net premiums. The algorithm provides a flexible way to integrate covariate information into individual net premiums prediction. It is appealing for capturing non-linear and/or interaction covariate effects. It automatically selects influential covariate variables for premium prediction and requires no additional ex-ante variable selection procedure. The superiority in the prediction accuracy of the proposed algorithm is demonstrated by extensive simulation studies. The proposed method is applied to the U.S. Medicare data for illustration purposes.

Bayesian credibility for GLMs

We revisit the classical credibility results of Jewell (1974) and Bühlmann (1967) to obtain credibility premiums for a GLM using a modern Bayesian approach. Here the prior distribution can be chosen without restrictions to be conjugate to the response distribution. It can even come from out-of-sample information if the actuary prefers.Then we use the relative entropy between the “true” and the estimated models as a loss function, without restricting credibility premiums to be linear. A numerical illustration on real data shows the feasibility of the approach, now that computing power is cheap, and simulations software readily available.