Claim Sizes Research Articles

On an Erlang(2) Process with Dependence Structure between Interclaim Arrivals and Claim Sizes

This paper considers an extension to the classical compound Poisson risk model for which an Erlang(2) process is utillized to the dependence structure between the claim sizes and interclaim times. In this framework, we derive the Lundberg generalised equation and the number of its roots, and the Laplace Transform(LT) of the expected discounted penalty function. We also show that the Gerber-Shiu function satisfies a defective renewal equation. Some explicit expressions are given to measure the impact of Erlang(2) dependence structure in the risk model on the ruin probability.

Integro-differential Equations for a Class of Delayed Renewal Risk Processes with Dependence

The Gerber-Shiu discounted penalty function is considered for a class of delayed renewal risk processes. In (Willmot 2004), special cases of the model include the stationary renewal risk model and the situation where the time until the first claim is exponentially distributed. In this paper, we consider a class of delayed and perturbed risk model with dependence between interclaim arrivals and claim sizes. The integro-differential equations for the Gerber-Shiu discounted penalty functions are derived.

Asymptotic behavior for sum ruin probability of a generalized bidimensional risk model with heavy-tailed claims

This paper considers a generalized bidimensional continuous-time risk model with subexponential claims, constant force of interest and Brownian perturbations, where the claim sizes from each line of business are dependent according to some dependence structure and the two components of each claim-inter-arrival-time vector are arbitrarily dependent. Some asymptotic presentations are shown for the finite-time sum ruin probability defined as the probability that the sum of two surplus processes generated by two lines of business goes below zero over a time horizon Particularly, the claim-number processes from different lines of business can be arbitrarily dependent when the claim sizes are long-tailed and dominatedly-varying-tailed.

Contributions Towards New Families of Distributions: An Investigation, Further Developments, Characterizations and Comparative Study

In the past couple of years, statistical models have been extensively used in applied areas for analyzing real data sets. However, in numerous situations, the traditional distributions are not flexible enough to cater to different aspects of the real phenomena. For example, (i) in the practice of reliability engineering and biomedical analysis, some distributions provide the best fit to the data having monotonic failure rate function, but fails to provide the best fit to the data having non-monotonic failure rate function, (ii) some statistical distributions provide the best fit for small insurance losses, but fails to provide an adequate fit to large claim size data, and (iii) some distributions do not have closed forms causing difficulties in the estimation process. To address the above issues, therefore, several methods have been suggested to improve the flexibility of the classical distributions. In this article, we investigate some of the former methods of generalizing the existing distributions. Further, we propose nineteen new methods of extending the classical distributions to obtain flexible models suitable for modeling data in applied fields. We also provide certain characterizations of the newly proposed families. Finally, we provide a comparative study of the newly proposed and some other existing well-known models via analyzing three real data sets from three different disciplines such as reliability engineering, medical, and financial sciences.

Two‐part D‐vine copula models for longitudinal insurance claim data

AbstractIn short‐term nonlife (e.g., car and homeowner) insurance, policies are renewed yearly. Insurance companies typically keep track of each policyholder's claims per year, resulting in longitudinal data. Efficient modeling of time dependence in longitudinal claim data will improve the prediction of future claims needed for routine actuarial practice, such as ratemaking. Insurance claim data usually follow a two‐part mixed distribution: a probability mass at zero corresponding to no claim and an otherwise positive claim from a skewed and long‐tailed distribution. This two‐part data structure leads to difficulties in applying established models for longitudinal data. In this paper, we propose a two‐part D‐vine copula model to study longitudinal mixed claim data. We build two stationary D‐vine copulas. One is used to model the time dependence in binary outcomes resulting from whether or not a claim has occurred. The other studies the dependence in the claim size given occurrence. Under the proposed model, the prediction of the probability of making claims and the quantiles of severity given occurrence is straightforward. We use our approach to investigate a dataset from the Local Government Property Insurance Fund in the state of Wisconsin.

Moderate deviations for a Hawkes-type risk model with arbitrary dependence between claim sizes and waiting times

Consider a risk model with consistently-varying-tailed claim sizes following a Hawkes-type arrival process. By allowing that the extended negatively dependent claim sizes and the claim inter-arrival (waiting) times are arbitrarily dependent, moderate deviation results for the aggregate amount of claims are obtained.

Estimations for aggregate amount of claims in a risk model with arbitrary dependence between claim sizes and inter-arrival times

<abstract><p>This paper considers a compound risk model, in which the individual claim sizes and their inter-arrival times can be arbitrarily dependent. We mainly investigate the claim sizes are extended negatively dependent. When the claim sizes have consistently-varying-tailed distributions, we obtain precise large deviations of the aggregate amount of claims in the above dependent compound risk model.</p></abstract>

Precise large deviations of aggregate claims in a nonstandard risk model with arbitrary dependence between claim sizes and waiting times

<abstract><p>Recently, Chen et al.<sup>[<xref ref-type="bibr" rid="b3">3</xref>]</sup> investigated the precise large deviations of aggregate claims in a renewal risk model with arbitrary dependence between claim sizes and their waiting times. In this paper, we extend their results to a nonstandard risk model in which various dependence structures are imposed on the modeling components, and obtain the asymptotic lower and upper bounds of precise large deviations for aggregate claims, which hold uniformly for all $ x $ in a $ t $-interval.</p></abstract>

Bayesian optimal investment and reinsurance with dependent financial and insurance risks

Abstract Major events like the COVID-19 crisis have impact both on the financial market and on claim arrival intensities and claim sizes of insurers. Thus, when optimal investment and reinsurance strategies have to be determined, it is important to consider models which reflect this dependence. In this paper, we make a proposal on how to generate dependence between the financial market and claim sizes in times of crisis and determine via a stochastic control approach an optimal investment and reinsurance strategy which maximizes the expected exponential utility of terminal wealth. Moreover, we also allow that the claim size distribution may be learned in the model. We give comparisons and bounds on the optimal strategy using simple models. What turns out to be very surprising is that numerical results indicate that even a minimal dependence which is created in this model has a huge impact on the optimal investment strategy.

Precise Large Deviations of Aggregate Claims in a Compound Risk Model with Arbitrary Dependence between Claim Sizes and Waiting Times

Precise Large Deviations of Aggregate Claims in a Compound Risk Model with Arbitrary Dependence between Claim Sizes and Waiting Times

The finite-time ruin probability of a risk model with a general counting process and stochastic return

<p style='text-indent:20px;'>This paper considers a general risk model with stochastic return and a Brownian perturbation, where the claim arrival process is a general counting process and the price process of the investment portfolio is expressed as a geometric Lévy process. When the claim sizes are pairwise strong quasi-asymptotically independent random variables with heavy-tailed distributions, the asymptotics of the finite-time ruin probability of this risk model have been obtained.</p>

Marine insurance claims analysis using the Weibull and log-normal models: Compensation for oil spill pollution due to tanker accidents

In marine insurance, the identification of the distribution of the claim size processes from given observations is of major importance. We evaluate the claim amounts due to accidental oil spill volume from tankers. Previous studies on oil spills were mainly concentrating on the relation between spill amounts and some accident factors. The trend of the spill claims is a major concern to the sustainability of insurance funds, tanker shipping and the whole oil industry. By analysing the real claim data series of IOPC (International Oil Pollution Compensation Funds), we identify the trends and reveal findings which enable policy-makers and insurers for the sufficient financial protection from oil spill damage. This study introduces the Weibull model and the log-normal model to the framework for oil spill claims. The Weibull model and the log-normal model are obtained with Markov chain Monte Carlo (MCMC) simulation method. The statistical properties of oil spill claims are determined in the time series of oil spill claims. Moreover, the relations among claim properties of different settings are examined. The log-normal of 3 groups of gross tonnage is found the best fit according to R-squared statistics and the mean squared errors.

On sequential maxima of geometric sample means, with extension to the ruin probability

In this paper, we consider the ruin probability of risk models with a geometric distribution of claim sizes. Since their probabilities can?t be calculated directly, we use exponential distribution to estimate its upper and lower bounds and asymptotic estimates based on the relationship between geometric distribution and exponential distribution. Finally, some numerical simulations are given to prove the superiority of our estimates.

Gerber-Shiu Function in a Discrete-time Risk Model with Dividend Strategy

In this paper, a discrete-time risk model with dividend strategy and a general premium rate is considered. Under such a strategy, once the insurer’s surplus hits a constant dividend barrier , dividends are paid off to shareholders at instantly. Using the roots of a generalization of Lundberg’s fundamental equation and the general theory on difference equations, two difference equations for the Gerber-Shiu discounted penalty function are derived and solved. The analytic results obtained are utilized to derive the probability of ultimate ruin when the claim sizes is a mixture of two geometric distributions. Numerical examples are also given to illustrate the applicability of the results obtained.

Penerapan Metode Limited-Fluctuation Credibility dalam Menentukan Premi Murni pada Asuransi Kendaraan Bermotor di PT XYZ

<p><strong>Abstract. </strong>This paper discusses the method of limited-fluctuation credibility, also known as classic credibility. Credibility theory is a technique for predicting future premium rates based on past experience data. Limited fluctuation credibility consists of two credibility, namely full credibility if Z = 1 and partial credibility if Z <1. Full credibility is achieved if the amount of recent data is sufficient for prediction, whereas if the latest data is insufficient then the partial credibility approach is used. Calculations for full and partial credibility standards are used for loss measures such as frequency of claims, size of claims, aggregate losses and net premiums. The data used in this paper is secondary data recorded by the company PT. XYZ in 2014. This data contains data on the frequency of claims and the size of the policyholder's partial loss claims for motor vehicle insurance products category 4 areas 1. Based on the results of the application, the prediction of pure premiums for 2015 cannot be fully based on insurance data for 2014 because the credibility factor value is less than 1. So based on the limited-fluctuation credibility method, the prediction of pure premiums for 2015 must be based on manual values for pure premiums as well as insurance data for 2014. If manual values for pure premium is 2,000,000 rupiah, then the prediction of pure premium for 2015 is 1,849,342 rupiah.</p><p><strong>Keywords</strong><strong>: </strong>limited fluctuation credibility, full credibility, partial credibility and partial loss</p>

Ruin probabilities for risk process in a regime-switching environment

In this paper we give a few expressions and asymptotics of ruin probabilities for a Markov modulated risk process for various regimes of a time horizon, initial reserves and a claim size distribution. We also consider a few versions of the ruin time.

Uniform asymptotics for ruin probabilities of a time-dependent renewal risk model with dependence structures and stochastic returns

In this article, we study a renewal risk model with compound dependence structures and stochastic returns. An insurance company is allowed to make risk-free and risky investments, where the price process of the investment portfolio follows an exponential Lévy process. Assume that claim sizes follow a one-sided linear process with independent and identically distributed steps sizes, and the step sizes and the corresponding inter-arrival times form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure. By restricting the distribution of the step sizes to the class of extended regular variation (), we obtain some asymptotic estimates, which holds uniformly for all time horizons. Finally, we show the accuracy of the derived asymptotic formula by numerical simulations.

Contributions from Spatial Models to Non-Life Insurance Pricing: An Empirical Application to Water Damage Risk

This paper explores the application of spatial models to non-life insurance data focused on the multi-risk home insurance branch. In the pricing modelling and rating process, spatial information should be considered by actuaries and insurance managers because frequencies and claim sizes may vary by region and the premium should be different considering this rating variable. In addition, it is relevant to examine the spatial dependence due to the fact that the frequency of claims in neighbouring regions is often expected to be more closely related than those in regions far from each other. In this paper, a comparison between spatial models, such as spatial autoregressive models (SAR), the spatial error model (SEM), and the spatial Durbin model (SDM), and a non-spatial model has been developed. The data used for this analysis are for a home insurance portfolio located in Spain, from which we have selected peril of water coverage.

Ruin probability of a continuous-time model with dependence between insurance and financial risks caused by systematic factors

Ruin problems of continuous-time models with various dependence structures under risky investment are extensively studied in the literature. Those studies focus on dependence structures arising from insurance businesses themselves and pay few attentions to dependence between insurance risk and investment risk. This study considers a continuous-time Poisson risk model with insurance claims and investment return jumps shocked by common systematic risk factors. Systematic risk factors cause immediate jumps of investment returns but there are often random delays for insurers to settle incurred claims. The study further assumes that the arrival times of claims are delayed by a common random time to the ones of their corresponding investment return jumps. When claim sizes are heavy-tailed distributed, a uniform asymptotic estimate for ruin probabilities of the model is established and a numerical study is given to show the accuracy of the result. The result captures the effect of the dependence structures caused by systematic risk factors on the asymptotic behaviors of ruin probabilities and allows arbitrary dependence structures between claim sizes and their corresponding investment return jumps.

How Much We Gain by Surplus-Dependent Premiums—Asymptotic Analysis of Ruin Probability

In this paper, we generate boundary value problems for ruin probabilities of surplus-dependent premium risk processes, under a renewal case scenario, Erlang (2) claim arrivals, and a hypoexponential claims scenario, Erlang (2) claim sizes. Applying the approximation theory of solutions of linear ordinary differential equations, we derive the asymptotics of the ruin probabilities when the initial reserve tends to infinity. When considering premiums that are linearly dependent on reserves, representing, for instance, returns on risk-free investments of the insurance capital, we firstly derive explicit solutions of the ordinary differential equations under considerations, in terms of special mathematical functions and integrals, from which we can further determine their asymptotics. This allows us to recover the ruin probabilities obtained for general premiums dependent on reserves. We compare them with the asymptotics of the equivalent ruin probabilities when the premium rate is fixed over time, to measure the gain generated by this additional mechanism of binding the premium rates with the amount of reserve owned by the insurance company.