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

Minwoo Kim,Himchan Jeong,Dipak Dey

doi:10.3390/risks10030054

Minwoo Kim, Himchan Jeong + Show 1 more

Open Access

https://doi.org/10.3390/risks10030054

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Abstract

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.

Highlights

Credibility premium has been widely used in actuarial practice to capture unobserved heterogeneity of policyholders via historical claim experiences
As in the simulation study, the proposed model shows the best performance on the prediction results with the Local Government Property Insurance Fund (LGPIF) data in terms of both root-mean squared error (RMSE) and mean absolute error (MAE), and much less computation time compared to the BA model that uses Markov Chain Monte Carlo (MCMC) to estimate the individual unobserved heterogeneity
As insurance companies are interested in better risk classification and tarrification by incorporating prevalent features of claims data such as indication of zero-inflation and the unobserved heterogeneity, computational costs in model calibration and individual premium calculation have been obstacles of using complicated models

Summary

Introduction

Credibility premium has been widely used in actuarial practice to capture unobserved heterogeneity of policyholders via historical claim experiences. The Poissongamma random effects model has been used as a benchmark to model claim frequency with unobserved heterogeneity (Dionne and Vanasse 1989). It is well-known that the traditional Poisson-gamma random effects model can enjoy natural conjugacy between the underlying distribution and the prior distribution of the random effects so that both the posterior distribution of the random effects and predictive premiums are readily available in closed forms, which is quite effective to compute individual premium for millions of policyholders. In spite of the aforementioned benefits, the traditional Poisson-gamma random effects model can be too restrictive due to natural indication of zero-inflation in claim frequency, which has been shown in many empirical studies including, but not limited to, Shi and Zhao (2020), Zhang et al (2020), and Lee (2021). Note that credibility premiums with these models are less computationally tractable compared to the traditional models such as Poisson-gamma random effects models

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