Classical Compound Poisson Risk Model Research Articles

On an insurance ruin model with a causal dependence structure and perturbation

The classical compound Poisson risk model and the Sparre-Andersen risk model for insurance businesses assume that the interclaim times and the claim amounts are independently distributed. To relax the independence assumption, we consider a continuous-time risk model under which the interclaim-time distribution depends on the size of the previously occurred claim. For practical purposes, the surplus process is further assumed to be perturbed by a Brownian motion to address small financial fluctuations or investment returns. To analyze the risk associated with the event when the insurer’s ruin occurs, explicit solutions for the Gerber–Shiu discounted penalty function may be derived through the defective renewal equations provided here when claim amounts follow an arbitrary distribution. Applications with Kn family claim amounts and the Laplace transform of the ruin time are discussed in detail. An illustrative numerical example is presented to assess the impact of different perturbations on the underlying dependent-structure surplus process.

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.

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.

A Deep Neural Network Approach to Solving for Seal’s Type Partial Integro-Differential Equation

In this paper, we study the problem of solving Seal’s type partial integro-differential equations (PIDEs) for the classical compound Poisson risk model. A data-driven deep neural network (DNN) method is proposed to calculate finite-time survival probability, and an alternative scheme is also investigated when claim payments are exponentially distributed. The DNN method is then extended to the numerical solution of generalized PIDEs. Numerical approximation results under different claim distributions are given, which show that the proposed scheme can obtain accurate results under different claim distributions.

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.

Refinements of bounds for tails of compound distributions and ruin probabilities

In this paper we derive lower bounds for right-tails of compound geometric distributions and ruin probabilities in the classical compound Poisson risk model in the heavy-tailed cases using the truncated Lundberg condition, which improve all the corresponding known lower bounds. Some upper bounds are also derived. Examples are given and numerical comparison for ruin probabilities when the adjustment coefficient does not exist are also considered, illustrating the effectiveness of the proposed new bounds. In addition, several bounds for tails of negative binomial distributions are obtained in terms of the tail of compound geometric distributions as well as bounds in the heavy-tailed cases. Also, some bounds for the stop-loss premium associated with compound negative binomial distributions are given. Using Chernoff's upper bounds we derive two-sided bounds for tails of compound Poisson distributions. Finally, two-sided bounds are given for tails of compound logarithmic distributions in the heavy-tailed cases.

Statistical estimation for some dividend problems under the compound Poisson risk model

In this paper, we consider some dividend problems in the classical compound Poisson risk model under a constant barrier dividend strategy. Suppose that the Poisson intensity for the claim number process and the distribution for the individual claim sizes are both unknown. We use the COS method to study the statistical estimation for the expected present value of dividend payments before ruin and the expected discounted penalty function. The convergence rates under large sample setting are derived. Some simulation results are also given to show effectiveness of the estimators under finite sample setting.

A cyclic approach on classical ruin model

The ruin problem has long since received much attention in the literature. Under the classical compound Poisson risk model, elegant results have been obtained in the past few decades. We revisit the finite-time ruin probability by using the idea of cycle lemma, which was used in proving the ballot theorem. The finite-time result is then extended to infinite-time horizon by applying the weak law of large numbers. The cycle lemma also motivates us to study the claim instants retrospectively, and this idea can be used to reach the ladder height distribution on the infinite-time horizon. The new proofs in this paper link the classical finite-time and infinite-time ruin results, and give an intuitive way to understand the nature of ruin.

Interval estimation of the ruin probability in the classical compound Poisson risk model

Estimating ruin probability is an important problem in insurance. Zhang et al. (2014) proposed a novel nonparametric estimation method for the ruin probability in the classical risk model with unknown claim size distribution, based on the Fourier transform and the kernel density estimation. However, asymptotic distributions of their estimators are unknown, which hampers statistical inference for the ruin probability. The authors establish asymptotic normal distributions of the estimators with known and with unknown intensity. Since the standard deviations of estimators are hard to estimate, a bootstrap method is advanced to estimate them. This allows one to construct a confidence interval estimate of the ruin probability. Furthermore, a new method is proposed to fast calculate the estimates, and the numerical results are stable and free of the “curse of large initial surplus” problem. Simulations are conducted to demonstrate nice finite sample performance of the estimators. A real dataset from a car insurance company is analyzed for illustrating the use of the proposed methodology.

Optimal Reinsurance and Investment Strategy for an Insurer in a Model with Delay and Jumps

This paper studies an optimal excess-of-loss reinsurance and investment problem in a model with delay and jumps for an insurer, who can purchase excess-of-loss reinsurance and invest his surplus in a risk-free asset and a risky asset whose price is governed by a jump-diffusion model. The insurer’s surplus is described by a diffusion model, which is an approximation of the classical compound Poisson risk model. In particular, the wealth process of the insurer is modeled by a stochastic differential delay equation via introducing the performance-related capital inflow or outflow. Under the criterion for maximizing the expected exponential utility of the combination of terminal wealth and average performance wealth, optimal strategy and the corresponding value function are obtained by using the dynamic programming approach. Finally, numerical examples are provided to show the effects of model parameters on the optimal strategies and illustrate the economic meaning.

Robust optimal proportional reinsurance and investment strategy for an insurer with defaultable risks and jumps

In this paper, we consider a robust optimal proportional reinsurance and investment problem in a model with default risks and jumps for an insurer whose objective is to maximize the expected exponential utility of terminal wealth. The surplus process of the insurer is assumed to follow a Brownian motion with drift (which is an approximation of the classical compound Poisson risk model). Assume that the insurer is allowed to purchase proportional reinsurance and invest in a financial market which consists of a risk-free asset, a defaultable bond and a risky asset whose price process is governed by a jump–diffusion model. Using stochastic control approach, we establish the robust Hamilton–Jacobi–Bellman equations for the post-default case and the pre-default case, respectively. Furthermore, we derive the closed-form expressions of the optimal strategies and the corresponding value functions in both cases. Finally, we provide numerical examples to illustrate the effects of some model parameters on the robust optimal strategies.

Finite time ruin probability and structural density properties in the presence of dependence in insurance risk model

ABSTRACTIn the present paper, we consider the classical compound Poisson risk model with dependence between claim sizes and claim inter-arrival time. We attempt to analyze the approximation of finite time ruin probability. The finite time ruin probabilities are plotted for fixed threshold value associated to the claim inter-arrival time and also for fixed dependence parameter in Nelsen (2006) copula separately. Additionally, a general form for joint density of the interclaim times and claim sizes is considered. With respect to the classical Gerber-Shiu's (1998) function, first some structural density properties of dependent collective risk model is obtained. Then the ladder height probability density function of claim sizes is computed and the dependency structure investigated for Erlang interclaim time. As the application, some dependent models of the interclaim times and claim sizes are studied.

A Note on the Convexity of Ruin Probabilities

Conditions for the convexity of compound geometric tails and compound geometric convolution tails are established. The results are then applied to analyze the convexity of the ruin probability and the Laplace transform of the time to ruin in the classical compound Poisson risk model with and without diffusion. An application to an optimization problem is given.

A note on the convexity of ruin probabilities

Conditions for the convexity of compound geometric tails and compound geometric convolution tails are established. The results are then applied to analyze the convexity of the ruin probability and the Laplace transform of the time to ruin in the classical compound Poisson risk model with and without diffusion. An application to an optimization problem is given.

The Effects of Largest Claim and Excess of Loss Reinsurance on a Company’s Ruin Time and Valuation

We compare two types of reinsurance: excess of loss (EOL) and largest claim reinsurance (LCR), each of which transfers the payment of part, or all, of one or more large claims from the primary insurance company (the cedant) to a reinsurer. The primary insurer’s point of view is documented in terms of assessment of risk and payment of reinsurance premium. A utility indifference rationale based on the expected future dividend stream is used to value the company with and without reinsurance. Assuming the classical compound Poisson risk model with choices of claim size distributions (classified as heavy, medium and light-tailed cases), simulations are used to illustrate the impact of the EOL and LCR treaties on the company’s ruin probability, ruin time and value as determined by the dividend discounting model. We find that LCR is at least as effective as EOL in averting ruin in comparable finite time horizon settings. In instances where the ruin probability for LCR is smaller than for EOL, the dividend discount model shows that the cedant is able to pay a larger portion of the dividend for LCR reinsurance than for EOL while still maintaining company value. Both methods reduce risk considerably as compared with no reinsurance, in a variety of situations, as measured by the standard deviation of the company value. A further interesting finding is that heaviness of tails alone is not necessarily the decisive factor in the possible ruin of a company; small and moderate sized claims can also play a significant role in this.

APPROXIMATING THE DENSITY OF THE TIME TO RUIN VIA FOURIER-COSINE SERIES EXPANSION

AbstractIn this paper, the density of the time to ruin is studied in the context of the classical compound Poisson risk model. Both one-dimensional and two-dimensional Fourier-cosine series expansions are used to approximate the density of the time to ruin, and the approximation errors are also obtained. Some numerical examples are also presented to show that the proposed method is very efficient.

The Maximum Surplus Before Ruin for Dependent Risk Models Through Farlie-Gumbel-Morgenstern Copula

We extend the classical compound Poisson risk model to consider the distribution of the maximum surplus before ruin where the claim sizes depend on inter-claim times via the Farlie-Gumbel-Morgenstern copula. We derive an integro-differential equation with certain boundary conditions for this distribution, of which the Laplace transform is provided. We obtain the renewal equation and explicit expressions for this distribution are derived when the claim amounts are exponentially distributed. Finally, we present numerical examples.

The maximum surplus before ruin for dependent risk models through Farlie–Gumbel–Morgenstern copula

We extend the classical compound Poisson risk model to consider the distribution of the maximum surplus before ruin where the claim sizes depend on inter-claim times via the Farlie–Gumbel–Morgenstern copula. We derive an integro-differential equation with certain boundary conditions for this distribution, of which the Laplace transform is provided. We obtain the renewal equation and explicit expressions for this distribution are derived when the claim amounts are exponentially distributed. Finally, we present numerical examples.

On a ruin model with both interclaim times and premiums depending on claim sizes

Under the classical compound Poisson risk model and the Sparre-Andersen risk model, one crucial assumption is that the interclaim times and the claim sizes are independent. However, this assumption might be inappropriate in practice. In this paper, we consider a continuous-time risk process where the interclaim-time distribution and premium rate both depend on the size of the previous claim. Explicit solutions for the Gerber–Shiu discounted penalty function with arbitrary claim-size distribution are derived utilizing the roots of a generalized Lundberg’s equation. Applications with exponential thresholds and -family claim sizes are presented. A numerical example is provided.

Analysis of the discounted sum of ascending ladder heights

Within the Sparre Andersen risk model, the ruin probability corresponds to the survival function of the maximal aggregate loss. It is well known that the maximum aggregate loss follows a compound geometric distribution, in which the summands consist of the ascending ladder heights. In the present paper, we propose to investigate the distribution of the discounted sum of ascending ladder heights over a finite or an infinite-time intervals. In particular, the moments of the discounted sum of ascending ladder heights over a finite and an infinite-time intervals are derived in both the classical compound Poisson risk model and the Sparre Andersen risk model with exponential claims. The application of a particular Gerber–Shiu functional is central to the derivation of these results, as is the mixed Erlang distributional assumption. Finally, we define VaR and TVaR risk measures in terms of the discounted sum of ascending ladder heights. We use a moment-matching method to approximate the distribution of the discounted sum of ascending ladder heights allowing the computation of the VaR and TVaR risk measures. We conclude this paper with a numerical example illustrating different topics discussed in the paper.