A simple Bayesian state-space approach to the collective risk models

Abstract

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

Keywords:

• Dependence
• posterior ratemaking
• dynamic random effects
• conjugate-prior
• local-level models
• three-part model

JEL Classification:

• C300

Acknowledgments

The authors warmly thank two anonymous referees for their numerous constructive comments that greatly helped to improve the paper compared to its initial version.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

1 To our knowledge, only the model of Lee & Shi (2019) and Oh et al. (2021b) can be applied to panel data.

2 Note that $\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{o}(b,p,q)$ is a special case of the generalized beta of the second kind (McDonald 1984, Cummins et al. 1990), $\mathrm{G}\mathrm{B}2(a,b,p,q)$, with a = 1 where the density function of $\mathrm{G}\mathrm{B}2(a,b,p,q)$ is given by $f_{\mathrm{G}\mathrm{B}2}(y;a,b,p,q)=\frac{|a|y^{ap-1}}{b^{ap}B(p,q){(1+(y/b{)}^a)}^{p+q}},y>0.$

3 We similarly define $\underset{\_}{B_t}:=(B_1,\dots ,B_t),\underset{\_}{{\theta }_t}:=({\theta }_1,\dots ,{\theta }_t)\text{and}\underset{\_}{{\lambda }_t}:=({\lambda }_1,\dots ,{\lambda }_t).$

4 These covariates are sometimes also called features. As usual, they are assumed exogenous, that is, their dynamics do not depend on the count process $\left(n_t\right)$.

5 If we replace the assumption in (1(1) ${\theta }_1\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}({\alpha }_0,{\text{β}}_0),$(1) ) with (4) ${\theta }_1\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(p{\alpha }_0,p{\text{β}}_0),$(4) the updated parameters in the filtering distribution in (3(3) ${\theta }_t|\underset{\_}{n_t}\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}({\alpha }_t,{\text{β}}_t),$(3) ) is given by ${\alpha }_t:=p{\alpha }_{t-1}+n_t>0\text{and}{\text{β}}_t:=p{\text{β}}_{t-1}+{\lambda }_t>0$for $t=1,2,3,\dots$. While the assumption in (4(4) ${\theta }_1\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(p{\alpha }_0,p{\text{β}}_0),$(4) ) results in much simpler updated formula compared to the recursion in (5(5) ${\alpha }_t:=\{\begin{array}{ll}p{\alpha }_{t-1}+n_t>0,& t=2,3,\dots ;\\ {\alpha }_0+n_1,& t=1;\end{array}\text{and}{\text{β}}_t:=\{\begin{array}{ll}p{\text{β}}_{t-1}+{\lambda }_t>0,& t=2,3,\dots ;\\ {\text{β}}_0+{\lambda }_1,& t=1;\end{array}$(5) ), we stick to the original assumption in (1(1) ${\theta }_1\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}({\alpha }_0,{\text{β}}_0),$(1) ) for the simplicity of the model setting.

6 Indeed, since the predictive distribution in the HF model is gamma, both the predictive expectation (for prediction purposes) and the likelihood function (for model estimation purpose) are highly tractable, compared to usual parameter-driven models.

7 In these latter models, the filtering distribution of ${\theta }_t$ given past counts and exogenous covariates is not gamma. Hence the increased complexity.

8 Recall that $\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(\frac{1}{\psi },\frac{1}{{\theta }_t{\lambda }_t\psi })$ is the Gamma distribution with mean and variance are given by ${\theta }_t{\lambda }_t\text{and}{\theta }_t^2{\lambda }_t^2\psi ,$respectively. In other words, ψ is the dispersion parameter of this gamma distribution.

9 Here, we assume that ${\alpha }_{t-1}^{\left[2\right]}$, ${\text{β}}_{t-1}^{\left[2\right]}$ are positive deterministic functions of past observations $(\underset{\_}{n_{t-1}},\underset{\_}{y_{t-1}})$ and $(\underset{\_}{{\lambda }_{t-1}^{\left[1\right]}},\underset{\_}{{\lambda }_{t-1}^{\left[2\right]}})$ as well as past values of exogenous covariates.

10 Recall that $\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(\frac{n_t}{\psi },\frac{1}{{\theta }_t^{\left[2\right]}{\lambda }_t^{\left[2\right]}\exp \left(\eta n_t\right)\psi })$ is the Gamma distribution with mean and variance $n_t{\lambda }_t^{\left[2\right]}{\theta }_t^{\left[2\right]}\exp \left(\eta n_t\right)\text{and}{n}_t({\theta }_t^{\left[2\right]}{\lambda }_t^{\left[2\right]}{)}^2{\psi }^{\left[2\right]}\exp (2\eta n_t),$ respectively.

11 By (A1(A1) $q_t:=\frac{q({\alpha }_{t-1}-1)+1}{{\alpha }_{t-1}}\text{and}{q}_t^{\ast }:=q,\text{for}q\in (0,1).$(A1) ), this is equivalent to $q({\alpha }_0-1)>1.$

Additional information

Funding

Jae Youn Ahn was partly supported by a National Research Foundation of Korea (NRF) grant funded by the Korean Government [grant number 2022R1F1A1064048] and Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government(MSIT) [grant number RS-2022-00155966]. Yang Lu thanks NSERC through a discovery grant [grant numbers RGPIN-2021-04144, DGECR-2021-00330]. Himchan Jeong was supported by the Simon Fraser University New Faculty Start-up Grant (NFSG).