Generalized Linear Mixed Models for Dependent Compound Risk Models

Jeong, Himchan; Valdez, Emiliano A.; Ahn, Jae Youn; Park, Sojung

doi:10.2139/ssrn.3045360

Cited by 11 publications

(13 citation statements)

References 17 publications

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“…Although the independence assumption between the frequency and severity has been widely used due to its simplicity, recent research works in actuarial science show empirical evidences of dependence between the frequency and severity in various claim datasets. For the detailed approach, please see Garrido et al (2016) and Jeong et al (2020). We also have a good Bayesian interpretation in this case so that (2.4) can be interpreted as a special case of (2.5) with P(θ C i = θ) = 1 {θ=1} for all i.…”

Section: Severity Part Modelmentioning

confidence: 97%

“…For example, Frangos and Vrontos (2001) tried to incorporate the random effects in bonus-malus system for automobile insurance and obtained a closed form formula for credibility premiums on compound loss, assuming the independence between the frequency and severity components. As an extension of their work, recently, Jeong et al (2020) also explored a random effects model for auto insurance claims considering possible dependence between the frequency and severity components.…”

Section: Introductionmentioning

confidence: 99%

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Testing for Random Effects in Compound Risk Models via Bregman Divergence

Jeong¹

2020

ASTIN Bull.

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Abstract The generalized linear model (GLM) is a statistical model which has been widely used in actuarial practices, especially for insurance ratemaking. Due to the inherent longitudinality of property and casualty insurance claim datasets, there have been some trials of incorporating unobserved heterogeneity of each policyholder from the repeated observations. To achieve this goal, random effects models have been proposed, but theoretical discussions of the methods to test the presence of random effects in GLM framework are still scarce. In this article, the concept of Bregman divergence is explored, which has some good properties for statistical modeling and can be connected to diverse model selection diagnostics as in Goh and Dey [(2014) Journal of Multivariate Analysis, 124, 371–383]. We can apply model diagnostics derived from the Bregman divergence for testing robustness of a chosen prior by the modeler to possible misspecification of prior distribution both on the naive model, which assumes that random effects follow a point mass distribution as its prior distribution, and the proposed model, which assumes a continuous prior density of random effects. This approach provides insurance companies a concrete framework for testing the presence of nonconstant random effects in both claim frequency and severity and furthermore appropriate hierarchical model which can explain both observed and unobserved heterogeneity of the policyholders for insurance ratemaking. Both models are calibrated using a claim dataset from the Wisconsin Local Government Property Insurance Fund which includes both observed claim counts and amounts from a portfolio of policyholders.

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Section: Severity Part Modelmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Testing for Random Effects in Compound Risk Models via Bregman Divergence

Jeong¹

2020

ASTIN Bull.

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“…However, so far most of the aforementioned (regression, or copula-based) models are designed for cross-sectional data only 4 . Recently, random effects have been introduced in longitudinal, or panel CRM's (Jeong et al, 2021;Lu, 2019;Jeong and Valdez, 2020b;Cheung et al, 2021;Denuit and Lu, 2021;Oh et al, 2021c). The random effect creates serial correlation between observation of different periods, as well as (serial and cross-sectional) dependence between the frequency and severity processes.…”

Section: Introductionmentioning

confidence: 99%

A simple Bayesian state-space model for the collective risk model

Ahn¹,

Jeong²,

Lu³

2021

Preprint

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The collective risk model (CRM) for frequency and severity is an important tool for retail insurance ratemaking, macro-level catastrophic risk forecasting, as well as operational risk in banking regulation. This model, which is initially designed for cross-sectional data, has recently been adapted to a longitudinal context to conduct both a priori and a posteriori ratemaking, through the introduction of random effects. However, so far, the random effect(s) is usually assumed static due to computational concerns, leading to predictive premium that omit the seniority of the claims. In this paper, we propose a new CRM model with bivariate dynamic random effect process. The model is based on Bayesian state-space models. It is associated with the simple predictive mean and closed form expression for the likelihood function, while also allowing for the dependence between the frequency and severity components. Real data application to auto insurance is proposed to show the performance of our method.

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“…While the existence of such a correlation and its economic significance is well documented in the literature (see e.g., Park, Kim, & Ahn, 2018), existing multivariate random effects based models capable of capturing dependence between claim frequency and severity typically suffer from computational intractability (see, e.g., Baumgartner, Gruber, & Czado, 2015; Czado & Gschlossl, 2007; Oh, Shi, & Ahn, 2020). Some recent papers try to evade this burden by letting the number of claims enter as a covariate in the severity model, without introducing random effect for the severity component (see, e.g., Garrido, Genest, & Schulz, 2016; Jeong, Valdez, Ahn, & Park, 2017; Park et al, 2018). The downside of this approach is that by doing so, the predictive power of previous claim costs is lost.…”

Section: Introductionmentioning

confidence: 99%

Wishart‐gamma random effects models with applications to nonlife insurance

Denuit

2020

J of Risk & Insurance

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Random effects are particularly useful in insurance studies, to capture residual heterogeneity or to induce cross‐sectional and/or serial dependence, opening hence the door to many applications including experience rating and microreserving. However, their nonobservability often makes existing models computationally cumbersome in a multivariate context. In this paper, it is shown that the multivariate extension to the Gamma distribution based on Wishart distributions for random symmetric positive‐definite matrices (considering diagonal terms) is particularly tractable and convenient to model correlated random effects in multivariate frequency, severity and duration models. Three applications are discussed to demonstrate the versatility of the approach: (a) frequency‐based experience rating with several policies or guarantees per policyholder, (b) experience rating accounting for the correlation between claim frequency and severity components, and (c) joint modeling and forecasting of the time‐to‐payment and amount of payment in microlevel reserving, when both are subject to censoring.

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Generalized Linear Mixed Models for Dependent Compound Risk Models

Cited by 11 publications

References 17 publications

Testing for Random Effects in Compound Risk Models via Bregman Divergence

Testing for Random Effects in Compound Risk Models via Bregman Divergence

A simple Bayesian state-space model for the collective risk model

Wishart‐gamma random effects models with applications to nonlife insurance

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