Multivariate Frequency-Severity Regression Models in Insurance

Abstract

Abstract:In insurance and related industries including healthcare, it is common to have several outcome measures that the analyst wishes to understand using explanatory variables. For example, in automobile insurance, an accident may result in payments for damage to one's own vehicle, damage to another party's vehicle, or personal injury. It is also common to be interested in the frequency of accidents in addition to the severity of the claim amounts. This paper synthesizes and extends the literature on multivariate frequency-severity regression modeling with a focus on insurance industry applications. Regression models for understanding the distribution of each outcome continue to be developed yet there now exists a solid body of literature for the marginal outcomes. This paper contributes to this body of literature by focusing on the use of a copula for modeling the dependence among these outcomes; a major advantage of this tool is that it preserves the body of work established for marginal models. We illustrate this approach using data from the Wisconsin Local Government Property Insurance Fund. This fund offers insurance protection for (i) property; (ii) motor vehicle; and (iii) contractors' equipment claims. In addition to several claim types and frequency-severity components, outcomes can be further categorized by time and space, requiring complex dependency modeling. We find significant dependencies for these data; specifically, we find that dependencies among lines are stronger than the dependencies between the frequency and average severity within each line.

Bonus-Malus premiums under the dependent frequency-severity modeling

“…Although, a series of recent studies shows that this unwarranted assumption can be relaxed in the collective risk model, and, thus, further in the Bayesian premium and BMS. In fact, the dependence between the frequency and severity in auto insurance is shown to be statistically significant in recent empirical studies such as Frees et al (2016) and Garrido et al (2016).…”

“…However, a series of recent empirical studies have shown that the dependence between frequency and severity in auto insurance is statistically significant (Frees et al (2016); Garrido et al (2016)). This phenomenon invalidates the practice of using frequency-driven BMS and highlights the need to extend the classical collective risk model by allowing some dependence structure between frequency and severity.…”

Generalized linear models for dependent frequency and severity of insurance claims

“…Thus, modeling individual claim amounts is equivalent to modeling the average severity only when N is included as a weight in the model forȲ . For more discussion, see Frees et al (2016).…”

Bivariate credibility bonus–malus premiums distinguishing between two types of claims

“…The proposed dependence model has some connections with aggregate claims models based on copulas (see for example Côté and Genest (2015)) and also with dependent GLM models (see Czado et al (2012)) or Frees et al (2016) and references therein. In addition, we incorporate a bivariate prior distribution which is conjugated with the likelihood of a claim being made.…”