Insurance Risk Model Research Articles

Ruin and Deficit Under Claim Arrivals with the Order Statistics Property

We consider an insurance risk model with extended flexibility, under which claims arrive according to a point process with an order statistics (OS) property, their amounts may have any joint distribution and the premium income is accumulated following any non-decreasing, possibly discontinuous real valued function. We generalize the definition of an OS point process, assuming it is generated by an arbitrary cdf allowing jump discontinuities, which corresponds to an arbitrary (possibly discontinuous) claim arrival cumulative intensity function. The latter feature is appealing for insurance applications since it allows to consider clusters of claims arriving instantaneously. Under these general assumptions, a closed form expression for the joint distribution of the time to ruin and the deficit at ruin is derived, which remarkably involves classical Appell polynomials. Corollaries of our main result generalize previous non-ruin formulas e.g., those obtained by Ignatov and Kaishev (Scand Actuar J 2000(1):46–62, 2000; J Appl Probab 41(2):570–578, 2004; J Appl Probab 43:535–551, 2006) and Lefèvre and Loisel (Methodol Comput Appl Probab 11(3):425–441, 2009) for the case of stationary Poisson claim arrivals and by Lefèvre and Picard (Insurance Math Econom 49:512–519, 2011; Methodol Comput Appl Probab 16:885–905, 2014), for OS claim arrivals.

The Absolute Ruin Insurance Risk Model with a Threshold Dividend Strategy

The absolute ruin insurance risk model is modified by including some valuable market economic information factors, such as credit interest, debit interest and dividend payments. Such information is especially important for insurance companies to control risks. We further assume that the insurance company is able to finance and continue to operate when its reserve is negative. We investigate the integro-differential equations for some interest actuarial diagnostics. We also provide numerical examples to explain the effects of relevant parameters on actuarial diagnostics.

Extensions of Breiman’s Theorem of Product of Dependent Random Variables with Applications to Ruin Theory

We consider the tail behavior of the product of two dependent random variables X and $$\Theta $$ . Motivated by Denisov and Zwart (J Appl Probab 44:1031–1046, 2007), we relax the condition of the existing $$\alpha \,+\,\epsilon $$ th moment of $$\Theta $$ in Breiman’s theorem to the existing $$\alpha $$ th moment and obtain the similar result as Breiman’s theorem of the dependent product $$X \Theta $$ , while X and $$\Theta $$ follow a copula function. As applications, we consider a discrete-time insurance risk model with dependent insurance and financial risks and derive the asymptotic tail behaviors for the (in)finite-time ruin probabilities.

Joint moments of the total discounted gains and losses in the renewal risk model with two-sided jumps

This paper considers a renewal insurance risk model with two-sided jumps (e.g. Labbé et al., 2011), where downward and upward jumps typically represent claim amounts and random gains respectively. A generalization of the Gerber–Shiu expected discounted penalty function (Gerber and Shiu, 1998) is proposed and analyzed for sample paths leading to ruin. In particular, we shall incorporate the joint moments of the total discounted costs associated with claims and gains until ruin into the Gerber–Shiu function. Because ruin may not occur, the joint moments of the total discounted claim costs and gain costs are also studied upon ultimate survival of the process. General recursive integral equations satisfied by these functions are derived, and our analysis relies on the concept of ‘moment-based discounted densities’ introduced by Cheung (2013). Some explicit solutions are obtained in two examples under different cost functions when the distribution of each claim is exponential or a combination of exponentials (while keeping the distributions of the gains and the inter-arrival times between successive jumps arbitrary). The first example looks at the joint moments of the total discounted amounts of claims and gains whereas the second focuses on the joint moments of the numbers of downward and upward jumps until ruin. Numerical examples including the calculations of covariances between the afore-mentioned quantities are given at the end along with some interpretations.

A necessary and sufficient condition for the subexponentiality of the product convolution

Abstract Let X and Y be two independent and nonnegative random variables with corresponding distributions F and G. Denote by H the distribution of the product XY, called the product convolution of F and G. Cline and Samorodnitsky (1994) proposed sufficient conditions for H to be subexponential, given the subexponentiality of F. Relying on a related result of Tang (2008) on the long-tail of the product convolution, we obtain a necessary and sufficient condition for the subexponentiality of H, given that of F. We also study the reverse problem and obtain sufficient conditions for the subexponentiality of F, given that of H. Finally, we apply the obtained results to the asymptotic study of the ruin probability in a discrete-time insurance risk model with stochastic returns.

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.

An IBNR–RBNS insurance risk model with marked Poisson arrivals

Inspired by the claim reserving problem in non-life insurance, this paper proposes to study the insurer’s surplus process under a micro-level framework, with particular focus on modeling the Incurred But Not Reported (IBNR) and the Reported But Not Settled (RBNS) claims. It is assumed that accidents occur according to a Poisson point process, and each accident is accompanied by a claim developmental mark that contains the reporting time, the settlement time, and the size of (possibly multiple) payments between these two times. Under exponential reporting and settlement delays, we show that our model can be represented as a Markovian risk process with countably infinite number of states. This can in turn be transformed to an equivalent fluid flow model when the payments are phase-type distributed. As a result, classical measures such as ruin probability or more generally the Gerber–Shiu expected discounted penalty function follow directly. The joint Laplace transform and the pairwise joint moments involving the ruin time and the aggregate payments of different types (with and without claim settlement) are further derived. Numerical illustrations are given at the end, including the use of a real insurance dataset.

Two-Sided Exit Problems in the Ordered Risk Model

The insurance risk model in the presence of two horizontal absorbing barriers is considered. The lower barrier is the usual ruin barrier while the upper one corresponds to the dividend barrier. The distribution of two first-exit times of the risk process from the strip between the two horizontal lines is under study. The claim arrival process is governed by an Order Statistic Point Process (OSPP) which enables the derivation of formulas in terms of the joint distribution of the order statistics of a sample of uniform random variables.

ON THE COMPOUND POISSON RISK MODEL WITH PERIODIC CAPITAL INJECTIONS

AbstractThe analysis of capital injection strategy in the literature of insurance risk models (e.g. Pafumi, 1998; Dickson and Waters, 2004) typically assumes that whenever the surplus becomes negative, the amount of shortfall is injected so that the company can continue its business forever. Recently, Nie et al. (2011) has proposed an alternative model in which capital is immediately injected to restore the surplus level to a positive level b when the surplus falls between zero and b, and the insurer is still subject to a positive ruin probability. Inspired by the idea of randomized observations in Albrecher et al. (2011b), in this paper, we further generalize Nie et al. (2011)'s model by assuming that capital injections are only allowed at a sequence of time points with inter-capital-injection times being Erlang distributed (so that deterministic time intervals can be approximated using the Erlangization technique in Asmussen et al. (2002)). When the claim amount is distributed as a combination of exponentials, explicit formulas for the Gerber–Shiu expected discounted penalty function (Gerber and Shiu, 1998) and the expected total discounted cost of capital injections before ruin are obtained. The derivations rely on a resolvent density associated with an Erlang random variable, which is shown to admit an explicit expression that is of independent interest as well. We shall provide numerical examples, including an application in pricing a perpetual reinsurance contract that makes the capital injections and demonstration of how to minimize the ruin probability via reinsurance. Minimization of the expected discounted capital injections plus a penalty applied at ruin with respect to the frequency of injections and the critical level b will also be illustrated numerically.

Optimal reinsurance and investment in a jump-diffusion financial market with common shock dependence

In this paper, we study the optimal reinsurance and investment problem in a financial market with jump-diffusion risky asset. It is assumed that the insurance risk model is modulated by a compound Poisson process, and that the jumps in both the risky asset and insurance risk process are correlated through a common shock. Under the criterion of maximizing the expected exponential utility, we adopt a nonstandard approach to examine the existence and uniqueness of the optimal strategy. Using the technique of stochastic control theory, closed-form expressions for the optimal strategy and the value function are derived not only for the expected value principle but also for the variance premium principle. Also, we investigate the effect of the common shock parameter as well as some other important parameters on the optimal strategies. In particular, a numerical example shows that the optimal investment strategy decreases as the degree of common shock dependence increases but the optimal insurance retention level does not behave the same.

The mean chance of ultimate ruin time in random fuzzy insurance risk model

In this paper, we study a modified risk model in which both the claim amount and premium are assumed to be random fuzzy variables. In this risk model, some new theorems concerning the mean chance of ultimate ruin time are proved in two cases where the initial surplus is zero and nonzero. Finally, a numerical example is mentioned to illustrate the method.

A note on a Lévy insurance risk model under periodic dividend decisions

In this paper, we consider a spectrally negative Levy insurance risk process with a barrier-type dividend strategy. In contrast to the traditional barrier strategy in which dividends are payable to the shareholders immediately when the surplus process reaches a fixed level b (as long as ruin has not yet occurred), it is assumed that the insurer only makes dividend decisions at some discrete time points in the spirit of [ 1 ]. Under such a dividend strategy with Erlang inter-dividend-decision times, expressions for the Gerber-Shiu expected discounted penalty function proposed in [ 24 ] and the moments of total discounted dividends payable until ruin are derived. The results are expressed in terms of the scale functions of a spectrally negative Levy process and an embedded spectrally negative Markov additive process. Our analyses rely on the introduction of a potential measure associated with an Erlang random variable. Numerical illustrations are also given.

Time-consistent mean-variance reinsurance-investment in a jump-diffusion financial market

This paper study an optimal time-consistent reinsurance-investment strategy selection problem in a financial market with jump-diffusion risky asset, where the insurance risk model is modulated by a compound Poisson process. The aggregate claim process and the price process of risky asset are correlated by a common Poisson process. The objective of the insurer is to choose an optimal time-consistent reinsurance-investment strategy so as to maximize the expected terminal surplus while minimizing the variance of the terminal surplus. Since this problem is time-inconsistent, we attack it by placing the problem within a game theoretic framework and looking for subgame perfect Nash equilibrium strategy. We investigate the problem using the extended Hamilton–Jacobi–Bellman dynamic programming approach. Closed-form solutions for the optimal reinsurance-investment strategy and the corresponding value functions are obtained. Numerical examples and economic significance analysis are also provided to illustrate how the optimal reinsurance-investment strategy changes when some model parameters vary.

A revisit to ruin probabilities in the presence of heavy-tailed insurance and financial risks

Recently, Sun and Wei (2014) studied the finite-time ruin probability under a discrete-time insurance risk model, in which the one-period insurance and financial risks are assumed to be independent and identically distributed copies of a random pair (X,Y). For the heavy-tailed case, under a restriction on the dependence structure of (X,Y), they established an asymptotic formula for the finite-time ruin probability. In this paper we make an effort to remove this restriction as it excludes the cases with asymptotically dependent X and Y. We also extend the study to the infinite-time ruin probability. Employing a multivariate regular variation framework, we simplify the formula so that it shows in a transparent way how the ruin probabilities are affected by the tail dependence of (X,Y).

A queueing/inventory and an insurance risk model

AbstractWe study an M/G/1-type queueing model with the following additional feature. The server works continuously, at fixed speed, even if there are no service requirements. In the latter case, it is building up inventory, which can be interpreted as negative workload. At random times, with an intensity ω(x) when the inventory is at level x>0, the present inventory is removed, instantaneously reducing the inventory to 0. We study the steady-state distribution of the (positive and negative) workload levels for the cases ω(x) is constant and ω(x) = ax. The key tool is the Wiener–Hopf factorization technique. When ω(x) is constant, no specific assumptions will be made on the service requirement distribution. However, in the linear case, we need some algebraic hypotheses concerning the Laplace–Stieltjes transform of the service requirement distribution. Throughout the paper, we also study a closely related model arising from insurance risk theory.

On dividends in the phase–type dual risk model

The dual risk model assumes that the surplus of a company decreases at a constant rate over time, and grows by means of upward jumps which occur at random times with random sizes. In the present work, we study the dual risk renewal model when the waiting times are phase-type distributed. Using the roots of the fundamental and the generalized Lundberg’s equations, we get expressions for the ruin probability and the Laplace transform of the time of ruin for an arbitrary single gain distribution. Then, we address the calculation of expected discounted future dividends particularly when the individual common gains follow a phase-type distribution. We further show that the optimal dividend barrier does not depend on the initial reserve. As far as the roots of the Lundberg equations and the time of ruin are concerned, we address the existing formulae in the corresponding Sparre-Andersen insurance risk model for the first hitting time, and we generalize them to cover also the situations where we have multiple roots. We do that working a new approach and technique, approach we also use for working the dividends, unlike others, it can be also applied for every situation.

First crossing time, overshoot and Appell–Hessenberg type functions

We consider a general insurance risk model with extended flexibility under which claims arrive according to a point process with independent increments, their amounts may have any joint distribution and the premium income is accumulated following any non-decreasing, possibly discontinuous, real valued function. Point processes with independent increments are in general non-stationary, allowing for an arbitrary (possibly discontinuous) claim arrival cumulative intensity function which is appealing for insurance applications. Under these general assumptions, we derive a closed form expression for the joint distribution of the time to ruin and the deficit at ruin, which is remarkable, since as we show, it involves a new interesting class of what we call Appell–Hessenberg type functions. The latter are shown to coincide with the classical Appell polynomials in the Poisson case and to yield a new class of the so called Appell–Hessenberg factorial polynomials in the case of negative binomial claim arrivals. Corollaries of our main result generalize previous ruin formulas e.g. those obtained for the case of stationary Poisson claim arrivals.

The ruin probabilities of a discrete-time risk model with dependent insurance and financial risks

Following the work of Sun and Wei (2014) [7], we investigate the ruin probabilities of a discrete-time insurance risk model with dependent insurance and financial risks. Assume that the one-period net insurance losses and discount factors form a sequence of independent and identically distributed copies of a random pair (X,θ). When the product Xθ is heavy tailed, we establish an asymptotic formula for the finite-time ruin probability without any restriction on the dependence structure of (X,θ) and extend the result to the infinite time ruin probability.

Optimal mean–variance reinsurance and investment in a jump-diffusion financial market with common shock dependence

In this paper, we study the optimal reinsurance-investment problems in a financial market with jump-diffusion risky asset, where the insurance risk model is modulated by a compound Poisson process, and the two jump number processes are correlated by a common shock. Moreover, we remove the assumption of nonnegativity on the expected value of the jump size in the stock market, which is more economic reasonable since the jump sizes are always negative in the real financial market. Under the criterion of mean–variance, based on the stochastic linear–quadratic control theory, we derive the explicit expressions of the optimal strategies and value function which is a viscosity solution of the corresponding Hamilton–Jacobi–Bellman equation. Furthermore, we extend the results in the linear–quadratic setting to the original mean–variance problem, and obtain the solutions of efficient strategy and efficient frontier explicitly. Some numerical examples are given to show the impact of model parameters on the efficient frontier.

Time-consistent investment-reinsurance strategy for mean-variance insurers with a defaultable security

This paper considers an optimal investment and reinsurance problem involving a defaultable security for an insurer under the mean-variance criterion in a jump-diffusion risk model. The insurer can purchase proportional reinsurance or acquire new insurance business and invest in a financial market consisting of a risk-free asset, a stock and a defaultable bond. In particular, the correlation between the insurance risk model and the financial market is also considered. From a game theoretic perspective, the extended Hamilton–Jacobi–Bellman systems of equations are established for the post-default case and the pre-default case, respectively. In both cases, closed-form expressions for the optimal time-consistent investment-reinsurance strategies and the corresponding value functions are derived. Moreover, some properties of optimal strategies, value functions and efficient frontiers are discussed either analytically or numerically.