Claim Amount Distribution Research Articles

Wind storm risk management: sensitivity of return period calculations and spread on the territory

Modeling and forecasting damage from wind storms is a major issue for insurance companies. In this article, we focus on the sensitivity of estimations of return periods for extreme events with respect to modeling assumptions and the type of input data. Numerous variables play a role: the quality of data concerning the location of insured buildings and weather report homogeneity, missing updates for correcting non-stationarities concerning the insurance portfolio history, ground roughness or climate change, the evolution of the model after an unprecedented event such as the Lothar storm observed in 1999 in Europe, temporal aggregation of daily events over several days, where events could span over several days up to one week, and storm trajectories, which could change due to global warming or sweep larger areas. Our work explores three important aspects. First, we highlight the geographic heterogeneity of the spatial distribution of wind speeds and the resulting damages. Therefore, we propose to partition the French territory into 6 relatively homogeneous storm zones, based on the dependence among observed wind speeds and geographic distance. Second, we extend a storm index—defined in Mornet et al. (Risk Anal 35:2029–2056, 2015)—to take into account geographic heterogeneity, and we analyze its tail behavior to show the difficulties met to obtain reliable results on extreme events. Third, we explore the calculation of Solvency Capital Requirements based on a model that we propose for the annual claim amount distribution. The purpose of our analysis is to quantify and to point out the high level of uncertainty in the computation of return periods and of other quantities strongly influenced by extreme events.

Ruin problems in the generalized Erlang(n) risk model

For actuarial applications we consider the Sparre–Andersen risk model when the interclaim times are Generalized Erlang(n) distributed. Unlike the standard Erlang(n) case, the roots of the generalized Lundberg’s equation with positive real parts can be multiple. This has a significant impact in the formulae for ruin probabilities that have to be found. We start by addressing the problem of solving an integro–differential equation that is satisfied by the survival probability, as well as other probabilities related, and present a method to solve such equation. This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and then establishing a one-one relation between them and the solutions of the integro–differential equation mentioned above. We first study the cases when all the roots are single and when there are roots with higher multiplicity. Secondly, we show that it is possible to have double roots but no higher multiplicity. Also, we show that the number of double roots depend on the choice of the parameters of the generalized Erlang(n) distribution, with a maximum number depending on n being even or odd. Afterwards, we extend our findings above for the computation of the distribution of the maximum severity of ruin as well as, considering an interest force, to the study the expected discounted future dividends, prior to ruin. Our findings show an alternative and more general method to the one provided by Albrecher et al. (Insur Math Econ 37:324–334, 2005), by considering a general claim amount distribution.

Stochastic comparison of aggregate claim amounts between two heterogeneous portfolios and its applications

The aggregate claim amount in a particular time period is a quantity of fundamental importance for proper management of an insurance company and also for pricing of insurance coverages. In this paper, we show that the proportional hazard rates (PHR) model, which includes some well-known distributions such as exponential, Weibull and Pareto distributions, can be used as the aggregate claim amount distribution. We also present some conditions for the use of exponentiated Weibull distribution as the claim amount distribution. The results established here complete and extend the well-known result of Khaledi and Ahmadi (2008).

Markov-dependent risk model with multi-layer dividend strategy

In this paper, we propose a Markov-dependent risk model with multi-layer dividend strategy in which the claim occurrence and the claim amount are regulated by an external discrete time Markov chain. A system of piecewise integro-differential equations with boundary conditions satisfied by the Gerber–Shiu function, with given initial environment state, is derived. The closed form expression of the Gerber–Shiu function is obtained when all the claim amount distributions belong to the rational family. Also, we adopt the Chebyshev polynomial approach to find the approximate solution of the integro-differential equation. The numerical simulation results show the effectiveness of the proposed methods.

Analysis of ruin measures for two classes of risk processes with stochastic income

In this paper, we consider the ruin measures for two classes of risk processes. We assume that the claim number processes are independent Poisson and generalized Erlang(n) processes, respectively. Historically, it has been assumed that the premium size is a constant. In this contribution, the premium income arrival process is a Poisson process. In this framework, both the integro-differential equation and the Laplace transform for the expected discounted penalty function are established. Explicit expressions for the expected discounted penalty function are derived when the claim amount distributions belong to the rational family. Finally, Numerical examples are

The expected discounted penalty function for two classes of risk processes perturbed by diffusion with multiple thresholds

In this paper, we consider 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 with boundary conditions for the expected discounted penalty function. Explicit expressions are derived if the two classes claim amount distributions both belong to the rational family.

Dividend Payments in a Risk Model Perturbed by Diffusion with Multiple Thresholds

We consider a perturbed renewal risk model where the interclaim times are phase-type distributed and the dividend payment is a step function depending on the current surplus level. We obtain the integro-differential equations with boundary conditions for the moment-generating functions and the moments of arbitrary order of the present value of all dividend payments until ruin. Explicit expressions for the expected discounted dividend payments are derived if the claim amount distribution belongs to the rational family. Last, we present an example to show how the results are applied.

On an asymptotic rule [formula omitted] for ultimate ruin probabilities under dependence by mixing

The purpose of this paper is to point out that an asymptotic rule A+B/u for the ultimate ruin probability applies to a wide class of dependent risk processes, in continuous or discrete time. That dependence is incorporated through a mixing model in the individual claim amount distributions. Several special mixing distributions are examined in detail and some close-form formulas are derived. Claim tail distributions and the dependence structure are also investigated.

The finite time ruin probability in a risk model with capital injections

We consider a risk model with capital injections. We show that in the Sparre Andersen framework the density of the time to ruin for the model with capital injections can be expressed in terms of the density of the time to ruin in an ordinary Sparre Andersen risk process. In the special case of Erlang inter-claim times and exponential claims, we show that there exists a readily computable formula for the density of the time to ruin. When the inter-claim time distribution is exponential, we obtain an explicit solution for the density of the time to ruin when the individual claim amount distribution is Erlang(2), and we explain techniques to find the moments of the time to ruin. In the final section, we consider the related problem of the distribution of the duration of negative surplus in the classical risk model, and we obtain explicit solutions for the (defective) density of the total duration of negative surplus for two individual claim amount distributions.

The finite-time ruin probability under the compound binomial risk model

We study the compound binomial ruin model, which is considered to be the discrete analogue of the classical compound Poisson model. Our key result is a simple approach for inverting a generating function whose argument is the discount factor when we know the inverse of the same generating function, which this time has argument that is the solution to Lundberg’s equation. The main idea comes from a result in Dickson and Willmot (ASTIN Bulletin 35:45–60, 2005) who discuss the classical model. We are then able to derive the probability distribution of the time to ruin and to go beyond the results in Dickson and Willmot (ASTIN Bulletin 35:45–60, 2005) by deducing the distribution of the first hitting time of a specific level and the duration of the time when the surplus is negative. The paper contains several illustrative examples where specific claim-amount distributions are considered.

The phase-type risk model perturbed by diffusion under a threshold dividend strategy

This paper considers a perturbed renewal risk process in which the inter-claim times have a phasetype distribution under a threshold dividend strategy. Integro-differential equations with certain boundary conditions for the moment-generating function and the mth moment of the present value of all dividends until ruin are derived. Explicit expressions for the expectation of the present value of all dividends until ruin are obtained when the claim amount distribution is from the rational family. Finally, we present an example.

Optimal dividend payment and capital injection of the compound Poisson risk model with both proportional and xed costs

This paper deals with the optimal dividend payment and capital injection problem for the classical risk model. With each dividend payment and capital injection, there is a proportional cost and a fixed cost. It controls the timing and the amount of both dividends paid out and equity issuance. The objective of the corporation is to maximize the expected discounted dividends payout minus the equity issuance until the time of bankruptcy. Due to the presence of the fixed transaction costs with each dividend payment and capital injection, the problem is formulated as an impulse stochastic control problem. It turns out that the control problem is associated with qualitatively difierent optimal capital injection strategies, depending on the problem’s data. One allows for no capital injection and the other allows for capital injection. We solve this problem explicitly in the case of exponential claim amount distributions. It is shown that there can be essentially seven difierent solutions depending on the model’s parameters and the costs.

Impulse Stochastic Control for the Optimization of the Dividend Payments of the Compound Poisson Risk Model Perturbed by Diffusion

This article deals with the optimal dividend problem for the classical risk model perturbed by diffusion. With each dividend payment there is a proportional cost and a fixed cost. The objective of the corporation is to maximize the cumulative expected discounted dividends payout until the time of ruin. It controls the timing and the amount of dividends paid out to the shareholders. Due to the presence of a fixed transaction cost, the resulting mathematical problem becomes an impulse stochastic control problem. By solving the corresponding quasi-variational inequalities, we obtain the optimal dividend strategy. Furthermore, we solve this problem explicitly in the case of exponential claim amount distributions of this jump-diffusion model. It turns out that there can be essentially two different solutions depending on the model's parameters and the costs. Specifically, (1) whenever the reserve reaches an appropriate level x*, it is reduced to through a dividend payment, and the process continues; (2) whenever the reserve reaches an appropriate level x*, everything is paid out as dividends immediately, and the ruin occurs.

Dividend Payments in a Risk Model Perturbed by Diffusion with Multiple Thresholds

We consider a perturbed renewal risk model where the inter-claim times are phase-type distributed and the dividend payment is a step function depending on the current surplus level. We obtain the integro-differential equations with boundary conditions for the moment-generating functions and the moments of arbitrary order of the present value of all dividend payments until ruin. Explicit expressions for the expected discounted dividend payments are derived if the claim amount distribution belongs to the rational family. Last, we present an example to show how the results are applied.

The ruin probabilities for the two-dimensional risk model based on Copula dependence

During the process of applying the Copulas in the two-dimensional risk models, with the consideration of the dependence relation between the claim amounts, the author chooses the FGM Copula to be put into application. For the compound Poisson model, a partial integro-differential equation satisfied by the survival probability is derived. The recursive formula of the finite-time survival or ruin probabilities are then derived separately in the situation of continuous and discrete distribution of claim amount, for the compound binomial model. Furthermore, the uncertainty for the Copulas under the discrete case is also illustrated.

The discounted penalty function with multi-layer dividend strategy in the phase-type risk model

This paper considers a Sparre Andersen model in which the inter-claim times have a phase-type distribution and the premium rate is a step function depending on the current surplus level. We derive the system of piecewise integro-differential equations for the Gerber–Shiu discounted penalty functions and obtain the closed form expressions of the Gerber–Shiu functions if the claim amount distribution belongs to the rational family. We provide a recursive approach to calculate Gerber–Shiu functions and present an example.

The Discounted Penalty Function with Multi-Layer Dividend Strategy in the Phase-Type Risk Model

This paper considers a Sparre Andersen model in which the inter-claim times have a phase-type distribution and the premium rate is a step function depending on the current surplus level. We derive the system of piecewise integro-differential equations for the Gerber-Shiu discounted penalty functions and obtain the closed form expressions of the Gerber-Shiu functions if the claim amount distribution belongs to the rational family. We provide a recursive approach to calculate Gerber-Shiu functions and present an example.

Optimal Dividend Strategies for a Compound Poisson Process Under Transaction Costs and Power Utility

We characterize the value function of maximizing the total discounted utility of dividend payments for a compound Poisson insurance risk model when strictly positive transaction costs are included, leading to an impulse control problem. We illustrate that well known simple strategies can be optimal in the case of exponential claim amounts. Finally we develop a numerical procedure to deal with general claim amount distributions.

A Direct Approach to the Discounted Penalty Function

This paper provides a new and accessible approach to establishing certain results concerning the discounted penalty function. The direct approach consists of two steps. In the first step, closed-form expressions are obtained in the special case in which the claim amount distribution is a combination of exponential distributions. A rational function is useful in this context. For the second step, one observes that the family of combinations of exponential distributions is dense. Hence, it suffices to reformulate the results of the first step to obtain general results. The surplus process has downward and upward jumps, modeled by two independent compound Poisson processes. If the distribution of the upward jumps is exponential, a series of new results can be obtained with ease. Subsequently, certain results of Gerber and Shiu [H. U. Gerber and E. S. W. Shiu, North American Actuarial Journal 2(1): 48–78 (1998)] can be reproduced. The two-step approach is also applied when an independent Wiener process is added to the surplus process. Certain results are related to Zhang et al. [Z. Zhang, H. Yang, and S. Li, Journal of Computational and Applied Mathematics 233: 1773–1 784 (2010)], which uses different methods.

On the expected discounted penalty function in a Markov-dependent risk model with constant dividend barrier

This article considers a Markov-dependent risk model with a constant dividend barrier. A system of integro-differential equations with boundary conditions satisfied by the expected discounted penalty function, with given initial environment state, is derived and solved. Explicit formulas for the discounted penalty function are obtained when the initial surplus is zero or when all the claim amount distributions are from rational family. In two state model, numerical illustrations with exponential claim amounts are given.