Claim Amount Distribution Research Articles

A semi-parametric claims reserving model with monotone splines

AbstractStochastic reserving models used in the insurance industry are usually based on an assumed distribution of claim amounts. Despite their popularity, such models may unavoidably be affected by the misspecification issue given that it is likely that the underlying distribution will be different from that assumed. In this paper, we incorporate monotone splines to ensure the expected monotonically increasing pattern of cumulative development factors (CDFs) to develop a new semi-parametric reserving model that does not require a density assumption. To allow the maximum utilization of available information, we also propose an enhanced sampling approach that greatly increases the size of unbiased CDFs, particularly in later development periods. Based on the enhanced samples, a bootstrap technique is employed in the estimation of monotone splines, from which incurred-but-not-reported (IBNR) reserves and prediction errors can be obtained. Associated technical features, such as the consistency of estimator, are discussed and demonstrated. Our simulation studies suggest that the new model improves the accuracy of IBNR reserving, compared with a range of classic competing models. A real data analysis produces many consistent findings, thus supporting the usefulness of the monotone spline model in actuarial and insurance practice.

Inequalities on the ruin probability for light-tailed distributions with some restrictions

When there are some restrictions on the random variables of insurance risk model, it is impossible to calculate the exact value of ruin probabilities. For these cases, even finding a suitable approximation, is very important from a practical point of view. In the present paper, we consider the renewal insurance surplus model with light-tailed claim amount distributions and try to find some inequalities on the infinite time ruin probability depending on the amount of initial reserve using statistical and mathematical approaches if the assumption of net profit does not hold but there exist some other restrictions on the mathematical functions of random variables of model. The assertions depend on the amount of initial reserve, distribution of nonnegative claim occurrences times and successive claim amounts are obtained. Finally, to show the application and effectiveness of given theorems two examples are presented. Through these examples, the infinite time ruin probabilities are estimated using Monte Carlo simulation and give an intuitive way to understand the nature of ruin.

Learning inter-annual flood loss risk models from historical flood insurance claims

Flooding is a natural hazard that causes substantial loss of lives and livelihoods worldwide. Developing predictive models for flood-induced financial losses is crucial for applications such as insurance underwriting. This research uses the National Flood Insurance Program (NFIP) dataset between 2000 and 2020 to evaluate the predictive skill of past data in predicting near-future flood loss risk. Our approach applies neural networks (Conditional Generative Adversarial Networks), decision trees (Extreme Gradient Boosting), and kernel-based regressors (Gaussian Processes) to estimate pointwise losses. It aggregates them over intervals using a bias-corrected Burr–Pareto distribution to predict risk. The regression models help identify the most informative predictors and highlight crucial factors influencing flood-related financial losses. Applying our approach to quantify the county-level coastal flood loss risk in eight US Southern states results in an R2=0.807, substantially outperforming related work using stage-damage curves. More detailed testing on 11 counties with significant claims in the NFIP dataset reveals that Extreme Gradient Boosting yields the most favorable results, and bias correction significantly improves the similarity between the predicted and reference claim amount distributions. Our experiments also show that, despite the already experienced climate change, the difference in future short-term risk predictions of flood-loss amounts between historical shifting or expanding training data windows is insignificant.

Calculation of the aggregate claim amount distribution using a recursive method with R and Actuar

Calculation of the aggregate claim amount distribution using a recursive method with R and Actuar

Data Analysis and Modelingof Claim Amounts of Car Insurance using Big Data: A Study for Pakistan

Modelling of data of claim amount is of paramount importance to manage risk reserve for payment of claims. Actuaries model uncertainty using probability distributions.
 In this research paper claim amount distribution of the data of an insurance concern has been estimated and analysis was performed on big-data of claim amounts for better understanding and fitting of various probability distribution using R.
 It was noticed that the claim amounts distribution is highly positive skewed, therefore we have studied Exponential distribution, Gamma distribution and Weibull distribution as possible candidates for modelling the claim amount data. Chi Square test has been used as goodness of fit technique to decide suitable statistical model to representing the claim amounts under study.
 Exponential distribution is found suitable for modelling the data under study.
 Proposed model is usefulto estimate claim amount on aggregate for insurance concern when total loss is required to be computed to manage the risk reserve for the payments of claims.

The moments of the time of ruin in Sparre Andersen risk models

AbstractWe derive formulae for the moments of the time of ruin in both ordinary and modified Sparre Andersen risk models without specifying either the inter-claim time distribution or the individual claim amount distribution. We illustrate the application of our results in the special case of exponentially distributed claims, as well as for the following ordinary models: the classical risk model, phase-type(2) risk models, and the Erlang( $\mathscr{n}$ ) risk model. We also show how the key quantities for modified models can be found.

Finite-time ruin probabilities using bivariate Laguerre series

In this paper, we revisit the finite-time ruin probability in the classical compound Poisson risk model. Traditional general solutions to finite-time ruin problems are usually expressed in terms of infinite sums involving the convolutions related to the claim size distribution and their integrals, which can typically be evaluated only in special cases where the claims follow exponential or (more generally) mixed Erlang distribution. We propose to tackle the partial integro-differential equation satisfied by the finite-time ruin probability and develop a new approach to obtain a solution in terms of bivariate Laguerre series as a function of the initial surplus level and the time horizon for a large class of light-tailed claim distributions. To illustrate the versatility and accuracy of our proposed method which is easy to implement, numerical examples are provided for claim amount distributions such as generalized inverse Gaussian, Weibull and truncated normal where closed-form convolutions are not available in the literature.

Diffusion Approximations of the Ruin Probability for the Insurer–Reinsurer Model Driven by a Renewal Process

We introduce here a diffusion-type approximation of the ruin probability both in finite and infinite time for a two-dimensional risk process, where claims and premiums are shared with a predetermined proportion. This type of process is often called the insurer–reinsurer model. We assume that the flow of claims is governed by a general renewal process. A simple ruin probability formula for the model is known only in infinite time for the special case of the Poisson process and exponentially distributed claims. Therefore, there is a need for simple analytical approximations. In the literature, in the infinite-time case, for the Poisson process, a De Vylder-type approximation has already been introduced. The idea of the diffusion approximation presented here is based on the weak convergence of stochastic processes, which enables one to replace the original risk process with a Brownian motion with drift. By applying this idea to the insurer–reinsurer model, we obtain simple ruin probability approximations for both finite and infinite time. We check the usefulness of the approximations by studying several claim amount distributions and comparing the results with the De Vylder-type approximation and Monte Carlo simulations. All the results show that the proposed approximations are promising and often yield small relative errors.

Conditional mean risk sharing in the individual model with graphical dependencies

AbstractConditional mean risk sharing appears to be effective to distribute total losses amongst participants within an insurance pool. This paper develops analytical results for this allocation rule in the individual risk model with dependence induced by the respective position within a graph. Precisely, losses are modelled by zero-augmented random variables whose joint occurrence distribution and individual claim amount distributions are based on network structures and can be characterised by graphical models. The Ising model is adopted for occurrences and loss amounts obey decomposable graphical models that are specific to each participant. Two graphical structures are thus used: the first one to describe the contagion amongst member units within the insurance pool and the second one to model the spread of losses inside each participating unit. The proposed individual risk model is typically useful for modelling operational risks, catastrophic risks or cybersecurity risks.

Ruin Probability for the Insurer–Reinsurer Model for Exponential Claims: A Probabilistic Approach

In this paper, we consider a two-dimensional risk process in which the companies split each claim and premium in a fixed proportion. It serves as a classical framework of a quota-share reinsurance contract for a given business line. Such a contract reduces the insurer’s exposure to the liabilities created through its underwriting activities. For the analyzed model, we derive a joint infinite-time ruin probability formula for exponentially distributed claims. To this end, we apply a change of measure technique. We illustrate the admissible range of parameters of the risk process. We also justify our result using Monte Carlo simulations and compare it with Theorem 2 in Avram, Palmowski and Pistorius [Insurance: Mathematics and Economics 42 (2008) 227], which was obtained by explicitly inverting a Laplace transform of the ruin probability. Our formula leads to a correction of that result. Finally, we note that the obtained formula leads to efficient approximation of the ruin probability for other claim amount distributions using De Vylder’s idea.

Ranking the extreme claim amounts in dependent individual risk models

In risk theory, the distribution of extreme claim amounts of dependent risks is an essential element, since it provides valuable information to companies for developing risk reduction strategies. In this article, we obtain a representation of the distributions of the smallest and the largest claim amounts based on a new concept of semi-distorted distribution. This concept extends the well known concept of distorted distributions introduced previously in risk theory. Based on this representation, we obtain distribution-free comparisons between extreme claim amounts in the sense of different stochastic dominances when the risks have a fixed dependence structure. Several examples illustrate the application of the theoretical results.

Bayesian estimation of ruin probability based on NHPP claim arrivals and Inverse-Gaussian distributed claim aggregates

The purpose of this article is to estimate the ruin probability at a future time past a truncated time τ before which ruin has not occurred. It is assumed that claim arrivals are from a non-homogenous Poisson process (NHPP). The distribution of claim amount X is assumed to be heavy-tailed such as Inverse-Gaussian (IG). Gamma priors are used to find Bayes estimates of IG parameters as well as the parameter of the intensity function using an MCMC algorithm. Based on observed arrival times and claim amounts before the truncated time τ, all parameters associated with the aggregate risk process and the NHPP are estimated to compute the ruin probability. Simulation results are presented to assess the accuracy of Maximum likelihood and Bayes estimates of the ruin probability.

A Note on Laplace Transforms of Some Common Distributions Used in Counting Processes Analysis

An important problem of actuarial risk management is the calculation of the probability of ruin. Using probability theory and the definition of the Laplace transform one obtains expressions, in the classical risk model, for survival probabilities in a finite time horizon. Then explicit solutions are found with the inversion of the double Laplace transform; using algebra, the Laplace complex inversion formula and Matlab, for the exponential claim amount distribution.

The stability of the probability of ruin

This article provides a computational formula for the the stability of the probability of ruin of the compound Poisson risk process. This stability is the infinitesimal standardized variation of the probability of ruin resulting from pointwise perturbation of the distribution of the individual claim amounts. The proposed formula is obtained from the saddlepoint approximation to the distribution of the geometric sum, where the distribution of the summands, which is the tail integration of the individual claim amounts distribution, is perturbed by the uniform distribution. A numerical illustration with the compound Poisson risk process with Weibull distributed individual claim amounts is provided.

On the total claim amount for marked Poisson cluster models

AbstractWe study the asymptotic distribution of the total claim amount for marked Poisson cluster models. The marks determine the size and other characteristics of the individual claims and potentially influence the arrival rate of future claims. We find sufficient conditions under which the total claim amount satisfies the central limit theorem or, alternatively, tends in distribution to an infinite-variance stable random variable. We discuss several Poisson cluster models in detail, paying special attention to the marked Hawkes process as our key example.

Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments

In this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard a Black–Scholes type, that is, it comprises a single risk-free asset that earns interest at a constant rate and a single risky asset whose price process is modelled by a geometric Brownian motion. Additionally, the company is allowed to purchase noncheap proportional reinsurance priced via the expected value principle. Using the Hamilton–Jacobi–Bellman (HJB) approach, we derive a second-order Volterra integrodifferential equation which we transform into a linear Volterra integral equation of the second kind. We proceed to solve this integral equation numerically using the block-by-block method for the optimal reinsurance retention level that minimizes the ultimate ruin probability. The numerical results based on light- and heavy-tailed individual claim amount distributions show that proportional reinsurance and investments play a vital role in enhancing the survival of insurance companies. But the ruin probability exhibits sensitivity to the volatility of the stock price.

Two Classes of Risk Model with Diffusion and Multiple Thresholds: the Discounted Dividends

In this paper, we consider the present value of total dividends until ruin in a perturbed risk model with two independent classes of risks under multiple thresholds, in which both of the two inter-claim times have phase-type distributions. We obtain the integro-differential equations for the moment-generating function and the $r$th moment of discounted dividend payments. Explicit expressions for the expectation of discounted dividend payments are derived if the two classes claim amount distributions both belong to the rational family.

An internal model for measuring premium risk when determining solvency of non-life insurers

Under contemporary dynamic approaches the solvency of insurance companies is determined by measuring the risks that threaten their business. This paper presents an internal model for measuring premium risk when evaluating the solvency of non-life insurers. The solvency capital requirement is calculated on the basis of a compound distribution of insurance portfolio aggregate claim amount, resulting from combining separately modelled claim frequency and severity distributions, with prior verification of earned technical premium sufficiency. The practical application of the model is illustrated by a case study of a specific non-life insurance company in Serbia. The research findings show that the dynamic model of premium risk measurement results in larger capital requirement and contributes to a more reliable assessment of insurers? solvency than the static model. This proves the inadequacy of the existing fixed ratio model and stresses the need for changes in the current methodology of determining the solvency of insurance companies in Serbia.

The maximum surplus before ruin for two classes of perturbed risk model

In this paper, we consider the distribution of the maximum surplus before ruin in a perturbed risk model with two independent classes of risks, in which both of the two inter-claim times have phase-type distributions. We obtain the integro-differential equations for the distribution of the maximum surplus before ruin. Explicit expressions are derived if the two classes claim amount distributions both belong to the rational family.

On some aspects of Maximum Severity of Ruin

The authors of this article engage ruin theory as a mathematical basis for quantifying the financial risks in insurance industry. Considering a classical risk model with dividend barrier, it is calibrated to obtain the maximum probability of ruin when the claim amount distribution is either exponential or Erlangian. It is to be noted that for numerical evaluation, the premium loading factor is taken to be 20 per cent in both the cases. In order to ensure fair comparison, exponential and Erlangian parameters have been chosen in such a way that their mean and the expected total claims are same for both the distributions over a given time interval. Ultimately, it is generalized that the classical risk model by considering a renewal risk model can be used to find an expression for the maximum severity of ruin in the insurance industry.