Dependent frequency–severity modeling of insurance claims

Abstract

Standard ratemaking techniques in non-life insurance assume independence between the number and size of claims. Relaxing the independence assumption, this article explores methods that allow for the correlation among frequency and severity components for micro-level insurance data. To introduce granular dependence, we rely on a hurdle modeling framework where the hurdle component concerns the occurrence of claims and the conditional component looks into the number and size of claims given occurrence. We propose two strategies to correlate the number of claims and the average claim size in the conditional component. The first is based on conditional probability decomposition and treats the number of claims as a covariate in the regression model for the average claim size, the second employed a mixed copula approach to formulate the joint distribution of the number and size of claims. We perform a simulation study to evaluate the performance of the two approaches and then demonstrate their application using a U.S. auto insurance dataset. The hold-out sample validation shows that the proposed model is superior to the industry benchmarks including the Tweedie and the two-part generalized linear models.

Introduction

Insurance claims modeling is a critical actuarial task in property–casualty insurance. A direct output–the predictive distribution of claims–serves as a foundation in various actuarial decision-making process. At individual level, predictive models are used for risk classification and to determine the premium and loadings for each policyholder. At aggregate level, predictive models quantify the risk of a portfolio or a block of business, which helps insurers choose the appropriate level of risk capital and treaty or facultative reinsurance arrangements.

The function of insurance as a risk management tool relies on the law of large numbers, i.e. the insurer can spread the risk of an individual among a pool of homogeneous policyholders. This risk pooling mechanism determines the unique semi-continuous feature of insurance claims data. Specifically, when examining the claims from a random sample of policyholders, one often observes a significant fraction of zeros associated with a relatively small percentage of positive claim amounts, also known as zero-inflated data in the literature. The zeros correspond to the policyholders without any claim during the policy year and usually account for the majority of observations. The phenomenon of zero inflation is not surprising when imaging the odds of car accidents or hail damage to a real property.

Modeling insurance claims is commonly done within the generalized linear model GLM framework (de Jong and Heller, 2008). Popular GLM-based approaches to handling zero inflation include the frequency–severity model (see, for example,  Frees (2014)) and the Tweedie compound Poisson model (see, for example,  Jørgensen and de Souza (1994)). The former, also known as two-part model, decomposes the cost of claims into two pieces, the frequency part examines whether or not a claim occurs (a logit regression) or the number of claims (a Poisson regression), and the severity part looks into the amount of claims conditional on occurrence (a gamma or inverse Gaussian regression). The latter is defined as a Poisson sum of i.i.d. gamma variables and a mass probability at zero is naturally incorporated into an otherwise continuous distribution through the compound Poisson process. The resulting Tweedie distribution belongs to the exponential family and hence the inference procedure for GLMs directly applies. There is a vast literature extending the two classes of approaches so as to capture different features of the insurance data. For example, hurdle and zero-inflated models are employed to accommodate the overdispersion and zero inflation in claim counts (see  Boucher et al. (2007)), various fat-tailed regression techniques have been proposed to model the claim size (see  Shi (2014)), and dispersion model is introduced for the Tweedie model in the context of double GLMs (see  Smyth and Jørgensen (2002)). Each method has its own advantages. For instance, the frequency–severity model is more flexible in the modeling of the occurrence and the size of insurance claims. In contrast, with a more parsimonious specification, the Tweedie model simplifies the variable selection process.

However, both methods assume an independent relationship between the frequency and severity of claims. In the Tweedie compound Poisson model, the derivation of the Tweedie distribution is based on the independence between the Poisson and gamma models in the underlying stochastic process. In the two-part framework, the explicit effect of claim frequency on claim severity is usually ignored, although, technically speaking, the decomposition is based on conditional probability and does not require the independence assumption between the two components. However, the number and the size of insurance claims could be correlated and this is the case for the automobile claims data in this study. In the spirit of estimating simultaneous equations, failure to account for the dependence among a set of regression models could bias the parameter estimates and thus the predictive distribution. Despite of the development in the recent actuarial literature, the effort toward relaxing the independence assumption is still sparse. We note that a few papers are in line with our study. In the spatial modeling of frequency and severity of automobile insurance claims,  Gschlößl and Czado (2007) used claims counts as a predictor for the claim size. This conditional probability approach has also been used in modeling health care expenditures, for example, see  Frees et al. (2011) and  Erhardt and Czado (2012).  Czado et al. (2012) and  Krämer et al. (2013) employed parametric copulas to jointly model the number and average size of claims for aggregated car insurance data.

Motivated by the above observations, this work aims to design the claims modeling framework that could accommodate the statistical association between the number and the size of insurance claims. We focus on the modeling strategy for cross-sectional data. We assume that the insurance dataset contains at least policy level information, including whether there is any accident, the number and the amount of claims if one or more accidents occur during the observation period, as well as a set of predictors. To introduce granular dependence, we propose a hurdle modeling framework where the hurdle component examines the probability of at least one claim occurring, and the conditional component models the number of claims and their size, given that at least one claim has occurred. We employ two strategies to correlate the number of claims and the average claim size in the conditional model. The first is based on conditional probability decomposition and treats the number of claims as a covariate in the regression model for the average claim size, the second employed a mixed copula approach to formulate the joint distribution of the number and size of claims. We evaluate the performance of both methods through a simulation study and then demonstrate their application using a U.S. auto insurance dataset.

Our work differs from and contributes to the existing literature in the sense that we extend the traditional two-part model to a three-part framework so as to incorporate the association between the frequency and severity component for micro-level insurance claims data. First, we emphasize the inference for granular observations because aggregating claims data could result in significant information loss. Second, the proposed framework preserves the flexibility of capturing some unique features in the insurance claims data, such as the zero inflation in claim counts and the heavy tails in claim severity. Note that, unlike existing studies, one does not need to limit to the GLM framework for the marginal models. Third, for model comparison, we emphasize hold-out sample validation in addition to the goodness-of-fit, which is of more practical importance for prediction purposes.

The rest of the article is structured as follows: Section  2 introduces the hurdle modeling framework under which two different approaches are proposed to accommodate the dependence between the frequency and severity of insurance claims. Section  3 presents a simulation study to investigate the performance of the two approaches. Section  4 summarizes the application of Massachusetts automobile insurance, including the characteristics of the dataset, inference and model comparison, and prediction of the proposed methods. Section  5 concludes the paper.