Claim Sizes Research Articles

Claims Modelling with Three-Component Composite Models

In this paper, we develop a number of new composite models for modelling individual claims in general insurance. All our models contain a Weibull distribution for the smallest claims, a lognormal distribution for the medium-sized claims, and a long-tailed distribution for the largest claims. They provide a more detailed categorisation of claims sizes when compared to the existing composite models which differentiate only between the small and large claims. For each proposed model, we express four of the parameters as functions of the other parameters. We fit these models to two real-world insurance data sets using both maximum likelihood and Bayesian estimation, and test their goodness-of-fit based on several statistical criteria. They generally outperform the existing composite models in the literature, which comprise only two components. We also perform regression using the proposed models.

Estimating a VaR-type ruin measure by Laguerre series expansion in classical compound Poisson risk model

Value at Risk (VaR) has been extensively applied within the banking and financial sectors. Regulators use it to reflect riskiness and determine capital requirements. In this paper, we propose an efficient method for estimating a VaR-type ruin measure under the classical risk model. We first use Laguerre series expansion to approximate the ruin probability and utilize a tabularization method to approximate the VaR-type ruin measure. A practical scenario where the distribution of individual claim size and the Poisson intensity are unknown has been considered. Based on observed data providing claim information, we illustrate the estimation of the VaR ruin measure. Moreover, the approximation convergence rates and statistical errors are studied, respectively. Various numerical examples are provided to demonstrate the performance of the proposed method.

On a time-changed variant of the generalized counting process

AbstractIn this paper, we time-change the generalized counting process (GCP) by an independent inverse mixed stable subordinator to obtain a fractional version of the GCP. We call it the mixed fractional counting process (MFCP). The system of fractional differential equations that governs its state probabilities is obtained using the Z transform method. Its one-dimensional distribution, mean, variance, covariance, probability generating function, and factorial moments are obtained. It is shown that the MFCP exhibits the long-range dependence property whereas its increment process has the short-range dependence property. As an application we consider a risk process in which the claims are modelled using the MFCP. For this risk process, we obtain an asymptotic behaviour of its finite-time ruin probability when the claim sizes are subexponentially distributed and the initial capital is arbitrarily large. Later, we discuss some distributional properties of a compound version of the GCP.

PEMODELAN DATA FREKUENSI KLAIM ASURANSI KENDARAAN BERMOTOR UNTUK CAKUPAN THIRD PARTY LIABILITY MENGGUNAKAN DISTRIBUSI POISSON-ARADHANA

The presence of insurance can provide coverage for all risks that occur for owners or motorists. Insurance will provide various forms of coverage in accordance with their respective products, so that vehicle owners or drivers will get protection, benefits or compensation for losses from various risks that occur above by buying an insurance policy of course. In Indonesia, motor vehicle insurance is included in the group of liability insurance. One of these insurance coverages is third-party legal liability. This is a protection against liability or third party legal responsibility or Third Party Liability, because the benefits provide protection against claims and losses experienced by third parties, in this case victims involved in accidents. Insurance premiums that can be provided by insurance companies to policyholders can be calculated based on the distribution model of claim frequency and claim size. Claim frequency is the number of insurance claims that occur in one period. Claim frequency data is often overdispersed. Poisson mixture distribution is often used as an alternative method for modeling claim frequency data when overdispersion occurs. This thesis will discuss the Poisson-Aradhana distribution modeling of motor vehicle insurance claim frequency data in Indonesia in 2019. The maximum likelihood estimation method is used to estimate the parameters of the Poisson-Aradhana distribution. The fit test used in this study is the Chi-Square test. The research material used is secondary data on the frequency of motor vehicle insurance claims recorded by PT. X from 8 categories and 3 regions in 2019. The results of the application of the Poisson-Aradhana mixed distribution on the frequency data of motor vehicle insurance claims for TPL coverage with comprehensive coverage at PT. X in 2019, most of the claim frequency data comes from a population with a Poison-Aradhana mixed distribution except for claim frequency data in category 1 region 3 and category 4 region 2

A Claim Size Change as a Case of Abuse of a Procedural Right

A Claim Size Change as a Case of Abuse of a Procedural Right

Asymptotic ruin probabilities for a two-dimensional risk model with dependent claims and stochastic return

We consider a continuous-time two-dimensional risk model, in which the claims from the two lines of insurance businesses satisfy an extensive asymptotic independence structure and the stochastic return is driven by a geometric Lévy process. Under a mild technical condition regarding the Laplace exponent of the Lévy process, we obtain explicit asymptotic expansions for both finite-time and infinite-time ruin probabilities when the claim sizes have regularly varying distributions.

Optimal reinsurance contract and investment strategy for multiple competitive-cooperative insurers and a reinsurer

Abstract Accepted by: Giorgio Consigli In this article, we consider a reinsurance contract design by taking into account the joint interests of multiple insurers and a reinsurer. The reinsurance contract consists of optimal reinsurance strategies and reinsurance prices. The former are chosen by the competitive–cooperative insurers and the latter are determined by a reinsurer. Both insurers and the reinsurer can invest in the common risk-free asset and one different risky asset. The optimal time-consistent reinsurance–investment strategies of the insurers and the optimal reinsurance prices and investment strategy of the reinsurer are derived analytically. Numerical experiments are carried out to illustrate the influences of model parameters on the optimal reinsurance contract and optimal investment strategies. We find that the establishment of the reinsurance contract is affected by the correlation between the claim sizes of insurers as well as that of claim numbers. The results reveal that competition and cooperation will lead to a decrease and increase of the reinsurance price, respectively, showing the importance of opting for cooperation among insurers. Both competition and cooperation are beneficial to insurers, especially for those with high risk aversion.

AWARENESS OF AGRO INPUT DEALERS ABOUT INSECTICIDE LABEL CLAIMS

Insecticide label claims are important sources of information for the farmers and the agro input dealers who sell them. They provide information about the active ingredients, mode of action, target pests, dosage, application method, safety precautions and environmental impact of the insecticides. However, the awareness level of the agro input dealers about these label claims is not well known. This study aimed to assess the awareness level of 120 agro-input dealers in Beed district of Maharashtra state, India, about the insecticide label claims and their toxicity labels. The study also examined the relationship between the profile of the agro input dealers and their awareness level, and identified the constraints and suggestions of the agro-input dealers regarding the insecticide label claims. The study used a structured questionnaire to collect data from the respondents through personal interview method. The data were analysed using frequency, percentage, mean, standard deviation and correlation coefficient. The results showed that the majority of the agro-input dealers had low to medium level of awareness about the insecticide label claims and their toxicity labels. The profile variables such as age, education, experience, training and membership in association had significant positive correlation with their awareness level. The major constraints faced by the agro input dealers were lack of training, inadequate display of label claims, complex language of label claims and low literacy level of farmers. The major suggestions given by the agro input dealers were to provide regular training, simplify the language of label claims, increase the font size and colour contrast of label claims and educate the farmers about the importance of label claims.

UKURAN RISIKO ATAS PERTANGGUNGAN DENGAN BESAR KLAIM YANG TERMODIFIKASI

Coverage modifications results in the amount of payment or reimbursement of claims not being the same as that proposed by the policyholder. Determination of the deductible causes claims below that value to be borne by the policyholder. Meanwhile, setting a policy limit results in a maximum value for claims borne by the insurance company (maximum covered loss). The size of risk as an important value in risk management, which in its determination depends on the distribution of losses in the form of claims, will also be affected by coverage modifications. In this study, risk measure was determined based on the standard deviation and value at risk that using quantile values. In addition, a risk measure is also determined in the form of a conditional tail expectation which is the average value of the loss over a certain quantile and the risk measure is based on a proportional hazard transformation. The size of the risk is determined by the coverage modified by the deductible and policy limit. Apart from that, it is also taken into account that there may be inflation, which of course will also have an impact on the distribution of claims. The observed claim size is known to have a Weibull distribution with certain parameter values, which heavy tailed. The results obtained indicate that modifications in the form of setting deductibles and policy limits have an impact on the magnitude of all risk measures obtained.

EM estimation for the mixed Pareto regression model for claim severities

ABSTRACT This article presents a mixed Pareto model and examines its suitability for modeling non life insurance claim severity data sets which exhibit peculiar characteristics that cannot be captured by the Pareto distribution. Furthermore, we introduce regression specifications for both the mean and the dispersion parameters of the mixed Pareto model. Our main achievement is that we develop a novel Expectation-Maximization (EM) algorithm for finding the maximum likelihood (ML) estimates of the parameters of the mixed Pareto regression model which is used for demonstration purposes. Finally, a real-data application based on motor insurance claim size data is provided.

Large deviations of aggregate amount of claims in compound risk model with arbitrary dependence between claim sizes and waiting times

Large deviations of aggregate amount of claims in compound risk model with arbitrary dependence between claim sizes and waiting times

Bridging the gap between pricing and reserving with an occurrence and development model for non-life insurance claims

AbstractDue to the presence of reporting and settlement delay, claim data sets collected by non-life insurance companies are typically incomplete, facing right censored claim count and claim severity observations. Current practice in non-life insurance pricing tackles these right censored data via a two-step procedure. First, best estimates are computed for the number of claims that occurred in past exposure periods and the ultimate claim severities, using the incomplete, historical claim data. Second, pricing actuaries build predictive models to estimate technical, pure premiums for new contracts by treating these best estimates as actual observed outcomes, hereby neglecting their inherent uncertainty. We propose an alternative approach that brings valuable insights for both non-life pricing and reserving. As such, we effectively bridge these two key actuarial tasks that have traditionally been discussed in silos. Hereto, we develop a granular occurrence and development model for non-life claims that tackles reserving and at the same time resolves the inconsistency in traditional pricing techniques between actual observations and imputed best estimates. We illustrate our proposed model on an insurance as well as a reinsurance portfolio. The advantages of our proposed strategy are most compelling in the reinsurance illustration where large uncertainties in the best estimates originate from long reporting and settlement delays, low claim frequencies and heavy (even extreme) claim sizes.

Ruin Analysis on a New Risk Model with Stochastic Premiums and Dependence Based on Time Series for Count Random Variables

In this paper, we propose a new discrete-time risk model of an insurance portfolio with stochastic premiums, in which the temporal dependence among the premium numbers of consecutive periods is fitted by the first-order integer-valued autoregressive (INAR(1)) process and the temporal dependence among the claim numbers of consecutive periods is described by the integer-valued moving average (INMA(1)) process. To measure the risk of the model quantitatively, we study the explicit expression for a function whose solution is defined as the Lundberg adjustment coefficient and give the Lundberg approximation formula for the infinite-time ruin probability. In the case of heavy-tailed claim sizes, we establish the asymptotic formula for the finite-time ruin probability via the large deviations of the aggregate claims. Two numerical examples are provided in order to illustrate our theoretical findings.

Joint discrete and continuous matrix distribution modeling

In this paper, we introduce a bivariate distribution on arising from a single underlying Markov jump process. The marginal distributions are phase-type and discrete phase-type distributed, respectively, which allow for flexible behavior for modeling purposes. We show that the distribution is dense in the class of distributions on and derive some of its main properties, all explicit in terms of matrix calculus. Furthermore, we develop an effective EM algorithm for the statistical estimation of the distribution parameters. In the last part of the paper, we apply our methodology to an insurance dataset, where we model the number of claims and the mean claim sizes of policyholders, which is seen to perform favorably. An additional consequence of the latter analysis is that the total loss size in the entire portfolio is captured substantially better than with independent phase-type models.

Uniform Asymptotic Estimate for the Ruin Probability in a Renewal Risk Model with Cox–Ingersoll–Ross Returns

Consider an insurance risk model with arbitrary dependence structures between the claim sizes. Suppose that the risky investment in the insurer can be established by the Cox–Ingersoll–Ross model. When the claim-size distribution is heavy-tailed, a uniform asymptotic formula for ruin probability is obtained.

Precise large deviations in a non stationary risk model with arbitrary dependence between subexponential claim sizes and waiting times

Consider a risk model in which the claims follow a non stationary arrival process that satisfies a large deviation principle. Supposing that the claim sizes form a sequence of i.i.d. random variables with subexponential tail, precise large deviations for the aggregate claims are obtained, by allowing the claim-sizes and claim inter-arrival (waiting) times to be arbitrarily dependent.

Deep quantile and deep composite triplet regression

A main difficulty in actuarial claim size modeling is that covariates may have different effects on the body of the conditional distribution and on its tail. To cope with this problem, we introduce a deep composite regression model whose splicing point is given in terms of a quantile of the conditional claim size distribution (rather than a constant). This allows us to simultaneously fit different regression models in the two different parts of the conditional distribution function. To facilitate M-estimation for such models, we introduce and characterize the class of strictly consistent scoring functions for the triplet consisting of a quantile, as well as the lower and upper expected shortfall beyond that quantile. In a second step, this elicitability result is applied to fit deep neural network regression models. We demonstrate the applicability of our approach and its superiority over classical approaches on a real data set from accident insurance.

Uniform asymptotics for ruin probabilities of a time-dependent bidimensional renewal risk model with dependent subexponential claims

This paper considers a continuous-time bidimensional renewal risk model with subexponential claims. In this model, the claim size vectors and their inter-arrival times form a sequence of independent and identically distributed random vectors following some general dependence structures introduced by [T. Jiang, Y. Wang, Y. Chen and H. Xu, Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model, Insur. Math. Econom. 64 (2015), pp. 45–53.]. We not only extend the results of [T. Jiang, Y. Wang, Y. Chen and H. Xu, Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model, Insur. Math. Econom. 64 (2015), pp. 45–53.] under some weak conditions, but we also obtain an asymptotic estimate of the finite-time sum-ruin probability. Furthermore, when the distributions of claim sizes have nonzero lower Karamata indices, some explicit asymptotic formulas are established for the infinite-time ruin probabilities.

Dependence modeling of frequency-severity of insurance claims using waiting time

Mixed copula approach has been used to jointly model discrete variable of claim counts and continuous variable of claim amounts. We propose to use a copula to link two continuous variables of the waiting time for the second claim and the average claim size. The frequency-severity dependence can be derived using the relationship between the waiting time and the counts of a Poisson process. Assuming a Gaussian copula and a log-normal distributed average claim size, we can investigate the effect of claim counts on the conditional claim severity analytically, which would be difficult in the mixed copula approach. We propose a Monte Carlo algorithm to simulate from the predictive distribution of the aggregated claims amount. In an empirical example, we illustrate the proposed method and compare with other competing methods. It shows that our proposed method provides quite competitive results.

Asymptotic estimates for finite-time ruin probabilities in a generalized dependent bidimensional risk model with CMC simulations

<p style='text-indent:20px;'>This paper studies ruin probabilities of a generalized bidimensional risk model with dependent and heavy-tailed claims and additional net loss processes. When the claim sizes have long-tailed and dominated-varying-tailed distributions, precise asymptotic formulae for two kinds of finite-time ruin probabilities are derived, where the two claim-number processes from different lines of business are almost arbitrarily dependent. Under some extra conditions on the independence relation of claim inter-arrival times, the class of the claim-size distributions is extended to the subexponential distribution class. In order to verify the accuracy of the obtained theoretical result, a simulation study is performed via the crude Monte Carlo method.</p>