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In the auto insurance industry, 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 the 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. [(2020). Bonus-Malus premiums under the dependent frequency-severity modeling. Scandinavian Actuarial Journal 2020(3): 172–195] claimed that the frequency-driven 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. [ABSTRACT FROM AUTHOR]

Under the Solvency Capital Requirement (SCR) formula of European Solvency II, the required capital for an insurer is calculated as the Value-at-Risk of the aggregate loss, utilizing a modular approach with a set of fixed correlations among business lines. This square-root method, while well-aligned with the multivariate normal model, does not easily extend to general loss distributions which are often right-skewed or heavy-tailed. To address this limitation, Sandström (Solvency II: Calibration for skewness, SAJ, 2007(2)) introduced a skewness adjustment term in the tail measure to improve SCR estimation. However, estimating the portfolio skewness remains challenging as it requires third-order cross-moments among the risks, which are generally impossible to derive from the prescribed correlations. In this paper, we propose a method to estimate portfolio skewness using the Normal Power approximation. By leveraging high-order cross-moments of the standard normal variable, our proposed method effectively calibrates portfolio skewness, while maintaining the mandated correlations. Additionally, we apply extreme value theory to estimate the Expected Shortfall within our framework. Our method is consistent with the current SCR framework and well-suited for insurance supervision that requires a balance between precision and simplicity. Our simulation study shows that the new approach performs significantly better than existing alternatives. [ABSTRACT FROM AUTHOR]