Modeling the Frequency and Severity of Auto Insurance Claims Using Statistical Distributions

Vehicle insurance companies in many countries use the Bonus-Malus System to determine the policyholder’s net premium. The determination of net premiums on the Bonus-Malus System is based solely on the frequency of claims and ignores the severity of claims. This is unfair to policyholders who have small claims. To overcome this problem, the net premium determination method in Bonus-Malus System was developed taking into account both the frequency and severity of claims. Frequency and severity be assumed to be independent. In determining the net premium, a posterior distribution of parameters of the frequency and severity distribution is required. In the case of frequency and severity independent, the determination of the posterior distribution for frequency and severity is performed separately. This thesis discusses the determination of net premium based on frequency distribution and severity distribution for frequency and severity independent.

This study aims to determine the copula-based model for estimating aggregate losses in traffic accident insurance. This research method is descriptive quantitative. Determination of the best model and the accuracy of the model is determined based on the Akaike Information Criterion (AIC), the smallest Root Mean Square Error (RMSE), and the Vuong Test. The results of the study, there is a dependency relationship or correlation between the frequency of claims and the severity of claims, so that copula is used to model the frequency of claims with a poison distribution and claim severity with a gamma distribution. In conclusion, the clayton copula model is the best model for estimating aggregate losses at the insurance company PT XYZ in the future.
 Keywords: Insurance, Loss, Correlation, Copula

Management or Avoidance of Medical Malpractice Crises?: Time to Choose

Modeling Frequency and Severity of Claims with the Zero-Inflated Generalized Cluster-Weighted Models

Modeling frequency and severity of claims with the zero-inflated generalized cluster-weighted models

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.

In this work, we propose a capital injection strategy which is periodically implemented based on the number of claims in the classical Poisson risk model. Especially, capital injection decisions are made at a predetermined accumulated number of claim instants, if the surplus is lower than a minimum required level. There appears to be a similar problem found in reliability theory such that preventive maintenance policies are performed at certain shock numbers. Assuming a combination of exponentials for the claim severities, we first derive an explicit expression for the discounted density of the surplus level after a certain number of claims if ruin has not yet occurred. Utilising this result, we study the expected total discounted capital injection until the first ruin time. To solve the differential equation associated with this quantity, we analyse an extended Lundberg’s fundamental equation. Similarly, an expression for the Laplace transform of the time to ruin is also explicitly found. Finally, we illustrate the applicability of the present capital injection strategy and methodologies through various numerical examples. In particular, for exponential claim severities, some optimal capital injection strategy which minimises the expected capital spending per unit time is numerically studied.

The insurance has often been involved to minimize financial losses. As the product providers, the insurance companies must effectively manage risks to prevent errors in risk measurement. The amount of risk or loss experienced by the policyholder refers to the claim amount. The Value at Risk (VaR) is commonly used to measure risk. The VaR is calculated from the probability function, which can be obtained by evaluating the distribution of claims data. Most claim frequencies are small, but occasionally, huge claims appear. Therefore, the appropriate distribution would be characterized by a heavy-tailed. Thus, this research aims to model and evaluate insurance claims data using exponential, Weibull, Pareto, and lognormal distributions to assess financial risk through VaR. The insurance claims data were collected from a single insurance company and include 1,326 claims. This research specifically examines variables such as gender, diabetic status, smoking status, the number of claims, and the level of confidence. The data were analysed using descriptive statistics, Maximum Likelihood Estimation for parameter estimation, and Goodness of Fit tests to determine the best-fitting distribution, along with VaR calculations based on the results. The suitability of the distribution model is assessed through the VaR and is analysed based on the appropriate distribution of insurance claims data. It is obtained that the Weibull and lognormal distributions appropriately model insurance claims data. The highest VaR is observed in the claim data for female non-diabetic smokers, with a level of confidence of 99.5%. The lowest VaR is obtained from the claim data for male diabetic non-smokers, with a level of confidence of 90%. This approach enhances the prediction of large potential losses for specific demographic groups, aiding more informed decision-making in premium pricing and risk management. The integration of heavy-tailed distributions in risk assessment, with a particular focus on demographic specificity, constitutes a substantial and novel contribution to this research.

The fragmented management of traditional construction projects, which mainly lacks the integration of project processes, results in schedule delays and cost overruns which often lead to client dissatisfaction, quality defects, and a raise in safety-related accidents. As a result, claims and disputes are most likely to arise between the contracting parties that can be extremely expensive and may severely impact the project performance. Numerous studies have investigated the impact of integrating sustainable management practices (SMPs) in construction projects on specific project performance objectives, such as cost, time, and quality; however, there is a need to investigate the effect of implementing SMPs on claims and dispute resolution. This research aims to fill the existing literature gap by identifying correlations between implementing SMPs and the frequency and severity of claims and disputes that may arise in construction projects. To achieve this goal, 25 SMP and 13 common construction claims were extracted based on an extensive literature review. A 5 point Likert-scale questionnaire was developed and administered to construction professionals to explore the aforementioned correlations. A total of 93 responses were received through online data collection. The research results demonstrate that empowering communication and collaboration among project stakeholders at early stages of the construction phase can significantly mitigate the occurrence of claims. This paper contributes in providing construction professionals with recommendations to improve construction sites’ performance and reduce claims.

In this paper, a new continuous probability distribution is proposed for fitting real data using Biweight kernel function and the exponential distribution. The suggested distribution is named the Biweight-exponential distribution (BiEd). Some statistical properties of this distribution are derived and illustrated mathematically. The probability density function and the cumulative distribution function are derived. Some reliability analysis functions are defined. The moments and moment generating function are derived. Re’nyi entropy is derived. The maximum likelihood method of estimation is used to derive the parameter estimates. The Bonferroni and Lorenz curves and Gini index equations are derived. The distribution of the order statistic and the quantile function are derived as well. The mean and median absolute deviations of the new distribution are derived. A numerical study was conducted to the quantile equation. An application to real data set is conducted to investigate the usefulness of the suggested distribution. In the real data application, the values of Cramer-von misses (W), Anderson Darling statistic (A), Kolmogorov-Smirnov (D) statistic, the p-value, the maximum likelihood estimates (MLE), Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC), Hannan Quinn information criterion (HQIC) and minimum value of the log-likelihood function are obtained. The distribution is compared with the exponential (base) distribution based on these criteria. The results showed that the BiED fits better than the exponential distribution. This means that the suggested distribution can replace the exponential distribution for analysis of some real data sets.

A novel flexible extension of the Chen distribution is defined and studied in this paper. Relevant statistical properties of the novel model are derived. For the actuarial risk analysis and evaluation, the maximum likelihood, weighted least squares, ordinary least squares, Cramer–von Mises, moments, and Anderson–Darling methods are utilized. For actuarial purposes, a comprehensive simulation study is presented using various combinations to evaluate the performance of the six methods in analyzing insurance risks. These six methods are used in evaluating actuarial risks using insurance claims data. Two applications on bimodal data are presented to highlight the flexibility and relevance of the new distribution. The new distribution is compared to several competing distributions. Actuarial risks are analyzed and evaluated using actuarial data, and the ability to disclose actuarial risks is compared by a comprehensive simulation study, through which actuarial disclosure models are compared using a wide range of well-known models.

Medical Malpractice in Gastroenterology

There is little literature describing if and how payers are utilizing patient-reported outcomes to predict future costs. This study assessed if Patient-reported Outcomes Measurement Information System (PROMIS) domain scores, collected in routine practice at neurology clinics, improved payer predictive models for unplanned care utilization and cost. Retrospective cohort analysis of private Health Plan-insured patients with visits at 18 Health Plan-affiliated neurology clinics. PROMIS domains (Anxiety v1.0, Cognitive Function Abilities v2.0, Depression v1.0, Fatigue v1.0, Pain Interference v1.0, Physical Function v2.0, Sleep Disturbance v1.0, and Ability to Participate in Social Roles and Activities v2.0) are collected as part of routine care. Data from patients' first PROMIS measures between June 27, 2018 and April 16, 2019 were extracted and combined with claims data. Using (1) claims data alone and (2) PROMIS and claims data, we examined the association of covariates to utilization (using a logit model) and cost (using a generalized linear model). We evaluated model fit using area under the receiver operating characteristic curve (for unplanned care utilization), akaike information criterion (for unplanned care costs), and sensitivity and specificity in predicting top 15% of unplanned care costs. Area under the receiver operating curve values were slightly higher, and akaike information criterion values were similar, for PROMIS plus claims covariates compared with claims alone. The PROMIS plus claims model had slightly higher sensitivity and equivalent specificity compared with claims-only models. One-time PROMIS measure data combined with claims data slightly improved predictive model performance compared with claims alone, but likely not to an extent that indicates improved practical utility for payers.

This study presents a comparative analysis of the Inverse Weibull and Inverse Lognormal distributions using both simulated and real-world data with Maximum Likelihood Estimation (MLE).Model selection criteria included Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Anderson-Darling (AD), and Kolmogorov-Smirnov (KS) tests.Six simulated sample sizes (50, 100, 150, 200, 250, and 500) were used, with 1000 replications each.Results showed the Inverse Lognormal distribution consistently had lower AIC and BIC values at small to moderate sample sizes.Furthermore, real-world stock price data (sample size 100) from the Nigerian Stock Exchange was analyzed.Descriptive statistics and goodness-of-fit tests favored the Inverse Lognormal model.These findings support the utility of AIC, BIC, AD, and KS in lifetime data modeling and model selection.

Since Nelder & Verrall (1997), the connection between Generalized Linear Models (GLM’s) and credibility theory has been recognized in actuarial science. Specifically, the credibility construction amounts to add a random effect on the same scale as the fixed effects to model unexplained heterogeneity. The present paper aims to examine the dependence existing between future claims (severity or frequency components) and the Buhlmann linear credibility premium. As expected, future claims are shown to increase with the amount of Buhlmann premium.