Actuarial Applications of a Hierarchical Insurance Claims Model

Abstract

This paper demonstrates actuarial applications that can be performed when modern statistical methods are applied to detailed, micro-level automobile insurance records. We consider 1993-2001 data consisting of policy, claims and payment files from a major Singapore insurance company. A hierarchical statistical model, developed in prior work Frees and Valdez (2008), is fit using the micro-level data. This model allows us to study the accident frequency, loss type and severity jointly and to incorporate individual characteristics such as age, gender and driving history that explain heterogeneity among policyholders.Based on this hierarchical model, one can analyze the risk profile of either a single policy (micro-level) or a portfolio of business (macro-level). This paper investigates three types of actuarial applications. First, we demonstrate the calculation of the predictive mean of losses for individual risk rating. This allows the actuary to differentiate prices based on policyholder characteristics. The nonlinear effects of coverage modifications such as deductibles, policy limits and coinsurance are quantified. Moreover, our flexible structure allows us to "unbundle" contracts and price more primitive elements of the contract, such as coverage type. The second application concerns the predictive distribution of a portfolio of business. We demonstrate the calculation of various risk measures, including value at risk and conditional tail expectation, that are useful in determining economic capital for insurance companies. Third, we examine the effects of several reinsurance treaties. Specifically, we show the predictive loss distributions for both the insurer and reinsurer under quota share and excess-of-loss reinsurance agreements. In addition, we present an example of portfolio reinsurance, in which the combined effect of reinsurance agreements on the risk characteristics of ceding and reinsuring company are described. *

Modelling Dependence in a Ratemaking Procedure with Multivariate Poisson Regression Models

“…as in Frees et al (2009). With the Bayesian approach to multivariate Poisson models through MCMC, we can easily derive posterior summaries for several quantities of interest to account for uncertainty.…”

“…When this assumption is relaxed, it is interesting to see how the tariff system is affected. In Frees and Valdez (2008) and Frees et al (2009) a hierarchical statistical model is fitted using microlevel data. In Bermúdez (2009), the interpretation of a number of bivariate Poisson models was illustrated in the context of motor insurance claims and the conclusion was that using a bivariate Poisson model leads to an a priori ratemaking that presents larger variances and, hence, larger loadings than those obtained under the independence assumption.…”

Testing Adverse Selection with Two-Dimensional Information: Evidence from Singapore Auto Insurance Market

“…Copula regression begins in the social science literature with Wang (2005, 2006). In insurance, the most recent work on copula regression has been on predictive modeling, for example, see Frees and Valdez (2008), Frees et al (2009), and Shi and Frees (2010). In spite of the aforementioned work, the applications for discrete data are scanty.…”

Gaussian Copula Regression in R

“…This paper focuses on Gaussian copula regression method where dependence is conveniently expressed in the familiar form of the correlation matrix of a multivariate Gaussian distribution (Song 2000;Pitt, Chan, and Kohn 2006;Masarotto and Varin 2012). Gaussian copula regression models have been successfully employed in several complex applications arising, for example, in longitudinal data analysis (Frees and Valdez 2008;Sun, Frees, and Rosenberg 2008;Shi and Frees 2011;Song, Li, and Zhang 2013), genetics (Li, Boehnke, Abecasis, and Song 2006;He, Li, Edmondson, Raderand, and Li 2012), mixed data (Song, Li, and Yuan 2009;de Leon and Wu 2011;Wu and de Leon 2014;Jiryaie, Withanage, Wu, and de Leon Well-known limits of the Gaussian copula approach are the impossibility to deal with asymmetric dependence and the lack of tail dependence. These limits may impact the use of Gaussian copulas to model forms of dependence arising, for example, in extreme environmental events or in financial data.…”