Bonus-Malus premiums under the dependent frequency-severity modeling

Abstract

In auto insurance, a Bonus-Malus System (BMS) is commonly used as a posteriori risk classification mechanism to set the premium for the next contract period based on a policyholder's claim history. Even though recent literature reports evidence of a significant dependence between frequency and severity, the current BMS practice is to use a frequency-based transition rule while ignoring severity information. Although Oh et al. (2019) claim that the frequencydriven BMS transition rule can accommodate the dependence between frequency and severity, their proposal is only a partial solution, as the transition rule still completely ignores the claim severity and is unable to penalize large claims. In this study, we propose to use the BMS with a transition rule based on both frequency and size of claim, based on the bivariate random effect model, which conveniently allows dependence between frequency and severity. We analytically derive the optimal relativities under the proposed BMS framework and show that the proposed BMS outperforms the existing frequency-driven BMS. Later numerical experiments are also provided using both hypothetical and actual datasets in order to assess the effect of various dependencies on the BMS risk classification and confirm our theoretical findings.

Designing a Bonus-Malus system reflecting the claim size under the dependent frequency–severity model

“…In our analysis, longitudinal data over the policy years from 2006 to 2010 with 497 governmental entities are used. There are two categorical variables identical to the dataset studied in Oh et al [17]: the entity type with six levels and the coverage with three levels, as shown in Table B.4. Following this, we model the frequency part as | ( [1] , [1] ) ∼ Poisson( [1] [1] ) with [1] = exp( [1] [1] ), and the individual severity part as , | ( [2] , [2] ) ∼ Gamma( [2] [2] , 1/ [2] ) with [2] = exp( [2] [2] ),…”

“…For the comparison of the quality of the various BMS systems, Oh et al [17] proposed the hypothetical mean square error (HMSE), a measure defined as the MSE between the aggregate severity and the BMS premium in the stationary state. For example, for the frequency-severity model in Section 2.2, we obtain HMSE( , −1/+ℎ) = E (E[ +1 | Λ [1] , Λ [2] , Θ [1] , Θ [2] ] − Λ [1] Λ [2] ( )) 1] )ℎ( [1] , [2] ) d [1] d [2] for a given relativity set = ( (0), .…”

“…To this extent, we study a BMS that depends on the history of both frequency and severity, with the bivariate random effect model as the underlying statistical model. Specifically, we generalize the approach of Oh et al [17] to propose a new BMS transition rule where the past claim's size as well as the number of claims affect the next year's BM level. Next, using the analogy between the Bayesian premium and the BMS premium, we analytically derive a set of optimal relativities that minimize the expected squared error in the presence of a frequency-severity dependence under this transition rule.…”

Frequency and Severity Dependence in the Collective Risk Model: An Approach Based on Sarmanov Distribution

“…However, studies on real data emphasized in several cases the existence of a certain dependence that should be taken into account because it can affect important actuarial quantities like premiums and ruin probabilities. Therefore, alternative approaches incorporate dependence between the number of claims and their average severity, see, for example, Erhardt and Czado [1], Czado et al [2], Krämer et al [3], Lee and Shi [4], or Oh et al [5]. In this paper, we propose the bivariate Sarmanov distribution to analyze the joint behavior of the number of claims and of each one of the individual claim amounts, instead of their average; i.e., we consider heterogeneity between the claim amounts associated with each number of claims.…”

Wishart‐gamma random effects models with applications to nonlife insurance

“…As a second application, a random effects model for correlated claim frequency and severity is worked out. 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).…”