Claim Severity Research Articles

A multi-year microlevel collective risk model

For a typical insurance portfolio, the claims process for a short period, typically one year, is characterized by observing frequency of claims together with the associated claims severities. The collective risk model describes this portfolio as a random sum of the aggregation of the claim amounts. In the classical framework, for simplicity, the claim frequency and claim severities are assumed to be mutually independent. However, there is a growing interest in relaxing this independence assumption which is more realistic and useful for the practical insurance ratemaking. While the common thread has been capturing the dependence between frequency and aggregate severity within a single period, the work of Oh et al. (2021) provides an interesting extension to the addition of capturing dependence among individual severities. In this paper, we extend these works within a framework where we have a portfolio of microlevel frequencies and severities for multiple years. This allows us to develop a factor copula model framework that captures various types of dependence between claim frequencies and claim severities over multiple years. It is therefore a clear extension of earlier works on one-year dependent frequency-severity models and on random effects model for capturing serial dependence of claims. We focus on the results using a family of elliptical copulas to model the dependence. The paper further describes how to calibrate the proposed model using illustrative claims data arising from a Singapore insurance company. The estimated results provide strong evidences of all forms of dependencies captured by our model.

Asymptotic Tail Probability of the Discounted Aggregate Claims under Homogeneous, Non-Homogeneous and Mixed Poisson Risk Model

In this paper, we derive a closed-form expression of the tail probability of the aggregate discounted claims under homogeneous, non-homogeneous and mixed Poisson risk models with constant force of interest by using a general dependence structure between the inter-occurrence time and the claim sizes. This dependence structure is relevant since it is well known that under catastrophic or extreme events the inter-occurrence time and the claim severities are dependent.

Bayesian credibility under a bivariate prior on the frequency and the severity of claims

In this paper, as opposed to the usual assumption of independence, we propose a credibility model in which the (unobservable) risk profiles of the claim frequency and the claim severity are dependent. Given the risk profiles, the (conditional) marginal distributions of frequency and severity are assumed to belong to the exponential family. A bivariate conjugate prior is proposed for the risk profiles, where the dependency is incorporated via a factorization structure of the joint density. The bivariate posterior is derived, and in turn the Bayesian premium for the aggregate claim is given along with some results on the predictive joint and marginal distributions involving the claim number and the aggregate claim in the next period. To demonstrate the generality of our proposed model, we provide four different examples of bivariate conjugate priors in relation to mixed Erlang, gamma mixture, Farlie-Gumbel-Morgenstern (FGM) copula, and bivariate beta, where each choice has different merits. In these examples, more explicit results can be obtained, and in particular the predictive variance and Value-at-Risk (VaR) of the aggregate claim certainly provide more information on the inherent risk than the Bayesian premium which is merely the predictive mean. Finally, numerical examples will be given to illustrate the effect of dependence on the results, including the use of a real data set that further takes observable risk factors into consideration under a regression setting.

PEMODELAN KLAIM ASURANSI MENGGUNAKAN PENDEKATAN BAYESIAN DAN MARKOV CHAIN MONTE CARLO

The determination of the correct prediction of claims frequency and claims severity is very important in the insurance business to determine the outstanding claims reserve which should be prepared by an insurance company. One approach which may be used to predict a future value is the Bayesian approach. This approach combines the sample and the prior information The information is used to construct the posterior distribution and to determine the estimate of the parameters. However, in this approach, integrations of functions with high dimensions are often encountered. In this Thesis, a Markov Chain Monte Carlo (MCMC) simulation is used using the Gibbs Sampling algorithm to solve the problem. The MCMC simulation uses ergodic chain property in Markov Chain. In Ergodic Markov Chain, a stationary distribution, which is the target distribution, is obtained. The MCMC simulation is applied in Hierarchical Poisson Model. The OpenBUGS software is used to carry out the tasks. The MCMC simulation in Hierarchical Poisson Model can predict the claims frequency.

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

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 Ohet al.[(2020). Bonus-Malus premiums under the dependent frequency-severity modeling.Scandinavian Actuarial Journal2020(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.

mvClaim : an R package for multivariate general insurance claims severity modelling

Abstract The mvClaim package in R provides flexible modelling frameworks for multivariate insurance claim severity modelling. The current version of the package implements a parsimonious mixture of experts (MoE) model family with bivariate gamma distributions, as introduced in Hu et al., and a finite mixture of copula regressions within the MoE framework as in Hu & O’Hagan. This paper presents the modelling approach theory briefly and the usage of the models in the package in detail. This package is hosted on GitHub at https://github.com/senhu/.

Assessing the Performance of Random Forests for Modeling Claim Severity in Collision Car Insurance

For calculating non-life insurance premiums, actuaries traditionally rely on separate severity and frequency models using covariates to explain the claims loss exposure. In this paper, we focus on the claim severity. First, we build two reference models, a generalized linear model and a generalized additive model, relying on a log-normal distribution of the severity and including the most significant factors. Thereby, we relate the continuous variables to the response in a nonlinear way. In the second step, we tune two random forest models, one for the claim severity and one for the log-transformed claim severity, where the latter requires a transformation of the predicted results. We compare the prediction performance of the different models using the relative error, the root mean squared error and the goodness-of-lift statistics in combination with goodness-of-fit statistics. In our application, we rely on a dataset of a Swiss collision insurance portfolio covering the loss exposure of the period from 2011 to 2015, and including observations from 81 309 settled claims with a total amount of CHF 184 mio. In the analysis, we use the data from 2011 to 2014 for training and from 2015 for testing. Our results indicate that the use of a log-normal transformation of the severity is not leading to performance gains with random forests. However, random forests with a log-normal transformation are the favorite choice for explaining right-skewed claims. Finally, when considering all indicators, we conclude that the generalized additive model has the best overall performance.

ROBUST ESTIMATION OF LOSS MODELS FOR LOGNORMAL INSURANCE PAYMENT SEVERITY DATA

AbstractThe primary objective of this scholarly work is to develop two estimation procedures – maximum likelihood estimator (MLE) and method of trimmed moments (MTM) – for the mean and variance of lognormal insurance payment severity data sets affected by different loss control mechanism, for example, truncation (due to deductibles), censoring (due to policy limits), and scaling (due to coinsurance proportions), in insurance and financial industries. Maximum likelihood estimating equations for both payment-per-payment and payment-per-loss data sets are derived which can be solved readily by any existing iterative numerical methods. The asymptotic distributions of those estimators are established via Fisher information matrices. Further, with a goal of balancing efficiency and robustness and to remove point masses at certain data points, we develop a dynamic MTM estimation procedures for lognormal claim severity models for the above-mentioned transformed data scenarios. The asymptotic distributional properties and the comparison with the corresponding MLEs of those MTM estimators are established along with extensive simulation studies. Purely for illustrative purpose, numerical examples for 1500 US indemnity losses are provided which illustrate the practical performance of the established results in this paper.

Machine Learning Approaches for Auto Insurance Big Data

The growing trend in the number and severity of auto insurance claims creates a need for new methods to efficiently handle these claims. Machine learning (ML) is one of the methods that solves this problem. As car insurers aim to improve their customer service, these companies have started adopting and applying ML to enhance the interpretation and comprehension of their data for efficiency, thus improving their customer service through a better understanding of their needs. This study considers how automotive insurance providers incorporate machinery learning in their company, and explores how ML models can apply to insurance big data. We utilize various ML methods, such as logistic regression, XGBoost, random forest, decision trees, naïve Bayes, and K-NN, to predict claim occurrence. Furthermore, we evaluate and compare these models’ performances. The results showed that RF is better than other methods with the accuracy, kappa, and AUC values of 0.8677, 0.7117, and 0.840, respectively.

Dependence Modelling of Frequency-Severity of Insurance Claims Using Waiting Time for Claim

We propose a dependent frequency-severity model using a Gaussian copula. The copula links a latent variable of waiting time for the second claim with the claim severity. By assuming a log-normal distributed claim severity, we can analyze the effect of claim counts on the conditional expectation of severity. We propose a Monte Carlo simulation algorithm to simulate the predictive distribution of the aggregated claims amount. In an empirical example, we compare the proposed method with the conditional modeling by Garrido et al. (2016) and the mixed copula modeling by Czado et al. (2012).

Bivariate Mixed Poisson and Normal Generalised Linear Models with Sarmanov Dependence—An Application to Model Claim Frequency and Optimal Transformed Average Severity

The aim of this paper is to introduce dependence between the claim frequency and the average severity of a policyholder or of an insurance portfolio using a bivariate Sarmanov distribution, that allows to join variables of different types and with different distributions, thus being a good candidate for modeling the dependence between the two previously mentioned random variables. To model the claim frequency, a generalized linear model based on a mixed Poisson distribution -like for example, the Negative Binomial (NB), usually works. However, finding a distribution for the claim severity is not that easy. In practice, the Lognormal distribution fits well in many cases. Since the natural logarithm of a Lognormal variable is Normal distributed, this relation is generalised using the Box-Cox transformation to model the average claim severity. Therefore, we propose a bivariate Sarmanov model having as marginals a Negative Binomial and a Normal Generalized Linear Models (GLMs), also depending on the parameters of the Box-Cox transformation. We apply this model to the analysis of the frequency-severity bivariate distribution associated to a pay-as-you-drive motor insurance portfolio with explanatory telematic variables.

Truncated, censored, and actuarial payment-type moments for robust fitting of a single-parameter Pareto distribution

With some regularity conditions maximum likelihood estimators (MLEs) always produce asymptotically optimal (in the sense of consistency, efficiency, sufficiency, and unbiasedness) estimators. But in general, the MLEs lead to non-robust statistical inference, for example, pricing models and risk measures. Actuarial claim severity is continuous, right-skewed, and frequently heavy-tailed. The data sets that such models are usually fitted to contain outliers that are difficult to identify and separate from genuine data. Moreover, due to commonly used actuarial “loss control strategies” in financial and insurance industries, the random variables we observe and wish to model are affected by truncation (due to deductibles), censoring (due to policy limits), scaling (due to coinsurance proportions) and other transformations. To alleviate the lack of robustness of MLE-based inference in risk modeling, here in this paper, we propose and develop a new method of estimation – method of truncated moments (MTuM) and generalize it for different scenarios of loss control mechanism. Various asymptotic properties of those estimates are established by using central limit theory. New connections between different estimators are found. A comparative study of newly-designed methods with the corresponding MLEs is performed. Detail investigation has been done for a single parameter Pareto loss model including a simulation study.

The relationships between vehicle characteristics and automobile accidents

AbstractIn this study, the relationships between vehicle characteristics and automobile accidents are examined based on physical damage insurance claims data from a major property insurance company in Taiwan. For the whole sample, we find that the car characteristics index (an estimate for the replacement cost associated with claims) is negatively correlated with the claim probability and positively correlated with claim severity. New cars and sport utility vehicles (SUVs) (compared with Sedans) are associated with higher claim probability. However, SUVs (compared with Sedans) are associated with lower claim severity. Small (compared with large) vehicles are associated with lower claim probability and lower claim severity. Compared with the imported cars from Germany, policyholders with the imported cars from Japan or the US are more likely to file a claim but to have lower claim severity. On subgroup analyses, the relationship between car characteristics index and claim probability is positive for domestic cars and negative for imported cars, implying that imported car policies subsidize domestic car policies. We also find the impacts of vehicle characteristics on claim probability and claim severity differ for domestic cars and imported cars in subsamples. Our empirical evidence provides important implications for premium pricing in the automobile physical damage insurance industry.

Classification Ratemaking Using Decision Tree in the Insurance Market of Bosnia and Herzegovina

Abstract This paper investigates the impact of risk classification on life insurance ratemaking with particular reference to Bosnia and Herzegovina (BiH). The research is based on a sample of over eighteen thousand insurance policies for passenger vehicles collected over the period 2015-2020. In our empirical investigation we develop a standard risk model based on the application of Poisson Generalized linear models (GLM) for claims frequency estimate and Gamma GLM for claim severity estimate. The analysis reveals that GLM does not provide a reliable parameter estimates for Multi-level factor (MLF) categorical predictors. Although GLM is widely used method to deter insurance premiums, improvements of GLM by using the data mining methods identified in this paper may solve practical challenges for the risk models. The popularity of applying data mining methods in the actuarial community has been growing in recent years due to its efficiency and precision. These models are recommended to be considered in BiH and South East European region in general.

IMPLEMENTASI METODE BAYES PADA PENGHITUNGAN PREMI ASURANSI KENDARAAN BERMOTOR

One type of general insurance is motor vehicle insurance. Premium pricing of general insurance can be calculated by some methods. In this study, Bayes method will be used. The distribution of claim frequency is Poisson distribution and the distribution of claim severity is Exponential distribution. The premium is calculated by multiplying the expectation of claim frequency and the expectation of claim severity. Based on the historical data analysis using the Bayes method, the highest pure premium of motor vehicle insurance in Indonesia is Hino brand and the lowest pure premium is Honda brand. The result of this premium pricing can be used as a reference for the insurance companies to manage their motor vehicle insurance reserves.

Optimal prevention of large risks with two types of claims

In this paper, we propose and study a risk model with two types of claims in which the insurer may invest into a prevention plan which decreases the intensity of large claims without impacting the small claims. We identify a necessary and sufficient condition for insurers to use prevention if there is no surplus. If, in addition, the severity of large claims dominates that of small claims by the harmonic mean residual life (HMRL) order, insurers invest more in prevention in the presence of a surplus. Finally, we characterize the asymptotic optimal prevention strategy when the initial surplus tends to infinity in the two main cases where both claim types are light-tailed and where one of them is light-tailed and the other one is heavy-tailed.

Analysis of two-dimensional warranty data considering global and local dependence of heterogeneous marginals

This paper conducts a two-dimensional warranty analysis on the data with heterogeneity in terms of both age and usage. Two models are developed to evaluate the severity of warranty claims. The first one uses a copula to describe the global dependence between the age and usage populations, whereas the second one further captures various local dependence in different warranty regions. Since the mixing proportions regarding heterogeneity are latent, the expectation maximization algorithm is proposed to estimate parameters and classify the warranty data. The likelihood ratio test is further utilized to determine the number of subpopulations. A comparison with two other models shows the superiority of the proposed one in terms of the Akaike information criterion. It is found that more than 85% of the claims are classified into normal failures, and local dependence structures might vary from a symmetric one to an asymmetric one. Meanwhile, as the sample size decreases, the heterogeneity of usage would vanish, and some local dependence structures would change.

Analyzing Workers' Compensation Claims and Payments Made Using Data from a Large Insurance Provider.

Background: All states in the USA have established Workers’ Compensation (WC) insurance systems/programs. WC systems address key occupational safety and health concerns. This effort uses data from a large insurance provider for the years 2011–2018 to provide estimates for WC payments, stratified by the claim severity, i.e., medical only, and indemnity. Methods: Besides providing descriptive statistics, we used generalized estimating equations to analyze the association between the key injury characteristics (nature, source, and body part injured) and total WC payments made. We also provide the overall cost burden for the former. Results: Out of the total 151,959 closed claims, 83% were medical only. The mean overall WC payment per claim for the claims that resulted in a payment was $1477 (SD: $7221). Adjusted models showed that mean payments vary by claim severity. For example, among medical only claims, the mean payment was the highest for amputations ($3849; CI: $1396, $10,608), and among disability and death related claims, ruptures cost the most ($14,285; $7772, $26,255). With frequencies taken into account, the overall cost burden was however the highest for strains. Conclusions: Workplace interventions should prioritize both the costs of claims on average and the frequency.

Stochastic gradient boosting frequency-severity model of insurance claims.

The standard GLM and GAM frequency-severity models assume independence between the claim frequency and severity. To overcome restrictions of linear or additive forms and to relax the independence assumption, we develop a data-driven dependent frequency-severity model, where we combine a stochastic gradient boosting algorithm and a profile likelihood approach to estimate parameters for both of the claim frequency and average claim severity distributions, and where we introduce the dependence between the claim frequency and severity by treating the claim frequency as a predictor in the regression model for the average claim severity. The model can flexibly capture the nonlinear relation between the claim frequency (severity) and predictors and complex interactions among predictors and can fully capture the nonlinear dependence between the claim frequency and severity. A simulation study shows excellent prediction performance of our model. Then, we demonstrate the application of our model with a French auto insurance claim data. The results show that our model is superior to other state-of-the-art models.

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

In actuarial mathematics, the claims of an insurance portfolio are often modeled using the collective risk model, which consists of a random number of claims of independent, identically distributed (i.i.d.) random variables (r.v.s) that represent cost per claim. To facilitate computations, there is a classical assumption of independence between the random number of such random variables (i.e., the claims frequency) and the random variables themselves (i.e., the claim severities). However, recent studies showed that, in practice, this assumption does not always hold, hence, introducing dependence in the collective model becomes a necessity. In this sense, one trend consists of assuming dependence between the number of claims and their average severity. Alternatively, we can consider heterogeneity between the individual cost of claims associated with a given number of claims. Using the Sarmanov distribution, in this paper we aim at introducing dependence between the number of claims and the individual claim severities. As marginal models, we use the Poisson and Negative Binomial (NB) distributions for the number of claims, and the Gamma and Lognormal distributions for the cost of claims. The maximum likelihood estimation of the proposed Sarmanov distribution is discussed. We present a numerical study using a real data set from a Spanish insurance portfolio.