Ruin-related Quantities Research Articles

Uniform asymptotics for a risk model with constant force of interest and a random number of delayed claims

Consider an insurance risk model with constant force of interest under the assumption that an immediate claim is accompanied by a random number of delayed claims. In the presence of heavy tails on claim distributions and dependence structures among modelling components, we study the uniform asymptotics for an insurer's ruin-related quantities, where the uniformity holds for all time horizons varying in a relevant interval.

On the dual risk model with Parisian implementation delays under a mixed dividend strategy

AbstractIn this paper, we consider a mixed dividend strategy in a dual risk model. The mixed dividend strategy is the combination of a threshold dividend and a Parisian implementation delays dividend under periodic observation. Given a series of discrete observation points, when the surplus level is larger than the predetermined bonus barrier at observation point, the Parisian implementation delays dividend is immediately carried out, and the threshold dividend is performed continuously during the delayed period. We study the Gerber-Shiu expected discounted penalty function and the expected discounted dividend payments before ruin in such a dual risk model. Numerical illustrations are given to study the influence of relevant parameters on the ruin-related quantities and the selection of the optimal dividend barrier for a given initial surplus level.

A refracted Lévy process with delayed dividend pullbacks

ABSTRACT The threshold dividend strategy, under which dividends are paid only when the insurer's surplus exceeds a pre-determined threshold, has received considerable attention in risk theory. However, in practice, it seems rather unlikely that an insurer will immediately pull back the dividend payments as soon as its surplus level drops below the dividend threshold. Hence, in this paper, we propose a refracted Lévy risk model with delayed dividend pullbacks triggered by a certain Poissonian observation scheme. Leveraging the extensive literature on fluctuation identities for spectrally negative Lévy processes, we obtain explicit expressions for two-sided exit identities of the proposed insurance risk process. Also, penalties are incorporated into the analysis of dividend payouts as a mechanism to penalize for the volatility of the dividend policy and account for an investor's typical preference for more stable cash flows. An explicit expression for the expected (discounted) dividend payouts net of penalties is derived. The criterion for the optimal threshold level that maximizes the expected dividend payouts is also discussed. Finally, several numerical examples are considered to assess the impact of dividend delays on ruin-related quantities. We numerically show that dividend strategies with more steady dividend payouts can be preferred (over the well-known threshold dividend strategy) when penalty fee become too onerous.

Applications of the classical compound Poisson model with claim sizes following a compound distribution

In this paper, we discuss a generalization of the classical compound Poisson model with claim sizes following a compound distribution. As applications, we consider models involving zero-truncated geometric, zero-truncated negative-binomial and zero-truncated binomial batch-claim arrivals. We also provide some ruin-related quantities under the resulting risk models. Finally, through numerical examples, we visualize the behavior of these quantities.

Moment and polynomial bounds for ruin-related quantities in risk theory

We establish a novel numerical quantification method based upon mathematical programming for the ruin-related quantities on collective risk models with fairly general features, such as reflecting, non-homogeneous and multidimensional dynamics with non-smooth coefficients. Rather than seeking approximations with a great deal of computing effort as in the existing numerical approximation methods, the proposed method aims to instantly provide deterministic upper and lower bounds for such quantities as linear combinations of moments or in the explicit polynomial form, without random number generation or sophisticated numerical algorithms. Given that closed-form solutions are rarely available especially in finite-time problems, the upper and lower bounds have great potential to be useful information that can complement approximate solutions obtained from existing numerical methods. We present numerical results throughout to justify the theoretical developments and convergence results so as to illustrate the effectiveness of the proposed method with low computational complexity. We also demonstrate further improvements of the bounds by scaling the problem domain, employing piecewise polynomial test functions and the exponential tempering of the polynomial bases.

On series expansions for scale functions and other ruin-related quantities

ABSTRACTIn this note, we consider a nonstandard analytic approach to the examination of scale functions in some special cases of spectrally negative Lévy processes. In particular, we consider the compound Poisson risk process with or without perturbation from an independent Brownian motion. New explicit expressions for the first and second scale functions are derived which complement existing results in the literature. We specifically consider cases where the claim size distribution is gamma, uniform or inverse Gaussian. Some ruin-related quantities will also be re-examined in light of the aforementioned results.

The joint distribution of ruin related quantities in the classical risk model

Ruin theory plays a crucial role in risk measurement and risk regulation. Bankruptcy claim is a major focus of ruin theory and the distribution of the aggregate amount of claim can well describe the risk of insurance portfolio. According to the distribution characteristics, we can adopt such means as capital injection and premium re-adjustment to regulate risk. In the classical risk model, the priorities are given to four ruin-related variables: the time of ruin, the aggregate claim amount up to ruin, the total number of claims up to ruin and the deficit at ruin. In this paper, we mainly consider the joint probability density function of the aggregate claim amount up to ruin with other ruin related quantities. The explicit expressions are given for the joint densities when the individual claim follows exponential distribution. In addition, when the individual claim follows a particular decomposition form, the joint density can also be obtained in a decomposition form.

Potential measures and expected present value of operating costs until ruin in renewal risk models with general interclaim times

ABSTRACTIn this paper, a Sparre Andersen risk process with arbitrary interclaim time distribution is considered. We analyze various ruin-related quantities in relation to the expected present value of total operating costs until ruin, which was first proposed by Cai et al. [(2009a). On the expectation of total discounted operating costs up to default and its applications. Advances in Applied Probability 41(2), 495–522] in the piecewise-deterministic compound Poisson risk model. The analysis in this paper is applicable to a wide range of quantities including (i) the insurer's expected total discounted utility until ruin; and (ii) the expected discounted aggregate claim amounts until ruin. On one hand, when claims belong to the class of combinations of exponentials, explicit results are obtained using the ruin theoretic approach of conditioning on the first drop via discounted densities (e.g. Willmot [(2007). On the discounted penalty function in the renewal risk model with general interclaim times. Insurance: Mathematics and Economics 41(1), 17–31]). On the other hand, without any distributional assumption on the claims, we also show that the expected present value of total operating costs until ruin can be expressed in terms of some potential measures, which are common tools in the literature of Lévy processes (e.g. Kyprianou [(2014). Fluctuations of L'evy processes with applications: introductory lectures, 2nd ed. Berlin Heidelberg: Springer-Verlag]). These potential measures are identified in terms of the discounted distributions of ascending and descending ladder heights. We shall demonstrate how the formulas resulting from the two seemingly different methods can be reconciled. The cases of (i) stationary renewal risk model and (ii) surplus-dependent premium are briefly discussed as well. Some interesting invariance properties in the former model are shown to hold true, extending a well-known ruin probability result in the literature. Numerical illustrations concerning the expected total discounted utility until ruin are also provided.

Discounted Aggregate Claim Costs Until Ruin in the Discrete-Time Renewal Risk Model

In this paper, we focus on analyzing the relationship between the discounted aggregate claim costs until ruin and ruin-related quantities including the time of ruin. To facilitate the evaluation of quantities of our interest as an approximation to the ones in the continuous case, discrete-time renewal risk model with certain dependent structure between interclaim times and claim amounts is considered. Furthermore, to provide explicit expressions for various moment-based joint probabilities, a fairly general class of distributions, namely the discrete Coxian distribution, is used for the interclaim times. Also, we assume a combination of geometrics claim size with arbitrary interlciam time distribution to derive a nice expression for the Gerber-Shiu type function involving the discounted aggregate claims until ruin. Consequently, the results are applied to evaluate some interesting quantities including the covariance between the discounted aggregate claim costs until ruin and the discounted claim causing ruin given that ruin occurs.

Densities of Ruin-Related Quantities in the Cramér-Lundberg Model with Pareto Claims

In this paper, we consider the classical yet widely applicable Cramer-Lundberg risk model with Pareto distributed claim sizes. Building on the previously known expression for the ruin probability we derive distributions of different ruin-related quantities. The results rely on the theory of scale functions and are intended to illustrate the simplicity and effectiveness of the theory. A particular emphasis is put on the tail behavior of the distributions of ruin-related quantities and their tail index value is established. Numerical illustrations are provided to show the influence of the claim sizes distribution tail index on the tails of the ruin-related quantities distribution.

On the dual risk model with Parisian implementation delays in dividend payments

In this paper, we study the dual compound Poisson risk process, which is suitable for a business that pays expenses at a constant rate over time and earns random amount of income at random times. In contrast to the usual dividend barrier strategy (e.g. Avanzi, Gerber, and Shiu (2007)) in which any overshoot over a pre-specified barrier is paid immediately to the company’s shareholders as a dividend, it is assumed that dividend is payable only when the process has stayed above the barrier continuously for a certain amount of time d (known as the ‘Parisian implementation delay’ in Dassios and Wu (2009)). Under such a modification, the Laplace transform of the time of ruin and the expected discounted dividends paid until ruin are derived. Motivated by the ‘Erlangization’ technique (e.g. Asmussen, Avram, and Usabel (2002)) of approximating a fixed time using an Erlang distribution, we also analyze the case where the delay d is replaced by an Erlang random variable. Numerical illustrations are given to study the effect of Parisian implementation delays on ruin-related quantities and to demonstrate the good performance of Erlangization. Interestingly, unlike the traditional barrier strategy, it is found that the optimal dividend barrier maximizing the expected discounted dividends does depend on the initial surplus level.

Gerber–Shiu analysis of a risk model with capital injections

We consider the risk model with capital injections studied by Nie et al. (Ann Actuar Sci 5:195–209, 2011; Scand Actuar J 2015:301–318, 2015). We construct a Gerber–Shiu function and show that whilst this tool is not efficient for finding the ultimate ruin probability, it provides an effective way of studying ruin related quantities in finite time. In particular, we find a general expression for the joint distribution of the time of ruin and the number of claims until ruin, and find an extension of Prabhu’s (Ann Math Stat 32:757–764, 1961) formula for the finite time survival probability in the classical risk model. We illustrate our results in the case of exponentially distributed claims and obtain some interesting identities. In particular, we generalise results from the classical risk model and prove the identity of two known formulae for that model.

Analysis of some ruin-related quantities in a Markov-modulated risk model

ABSTRACTIn this paper, we study the joint Laplace transform and probability generating function of some random quantities that occur in each environment state by the time of ruin in a Markov-modulated risk process. These quantities include the duration spent in each state, the number of claims and the aggregate amount of claims that occurred in each state by the time of ruin. Explicit formulae for the joint transforms, given the initial surplus, and the initial and terminal environment states, are expressed in terms of a matrix version of the scale function. Moments and covariances of these ruin-related quantities are obtained and numerical illustrations are presented. The joint transform of the duration spent in each state, the number of claims, and the aggregate amount of claims that occurred in each state by the time the surplus attains a certain level are also investigated.

On the Joint Analysis of the Total Discounted Payments to Policyholders and Shareholders: Dividend Barrier Strategy

In the compound Poisson insurance risk model under a dividend barrier strategy, this paper aims to analyze jointly the aggregate discounted claim amounts until ruin and the total discounted dividends until ruin, which represent the insurer’s payments to its policyholders and shareholders, respectively. To this end, we introduce a Gerber–Shiu-type function, which further incorporates the higher moments of these two quantities. This not only unifies the individual study of various ruin-related quantities, but also allows for new measures concerning covariances to be calculated. The integro-differential equation satisfied by the generalized Gerber–Shiu function and the boundary condition are derived. In particular, when the claim severity is distributed as a combination of exponentials, explicit expressions for this Gerber–Shiu function in some special cases are given. Numerical examples involving the covariances between any two of (i) the aggregate discounted claims until ruin, (ii) the discounted dividend payments until ruin and (iii) the time of ruin are presented along with some interpretations.

On the analysis of ruin-related quantities in the delayed renewal risk model

This paper first explores the Laplace transform of the time of ruin in the delayed renewal risk model. We show that Ḡδd(u), the Laplace transform of the time of ruin in the delayed model, also satisfies a defective renewal equation and use this to study the Cramer–Lundberg asymptotics and bounds of Ḡδd(u). Next, the paper considers the stochastic decomposition of the residual lifetime of maximal aggregate loss and more generally Lδd in the delayed renewal risk model, using the framework equation introduced in Kim and Willmot (2011) and the defective renewal equation for the Laplace transform of the time of ruin. As a result of the decomposition, we propose a way to calculate the mean and the moments of the proper deficit in the delayed renewal risk model. Lastly, closed form expressions are derived for the Gerber–Shiu function in the delayed renewal risk model with the distributional assumption of time until the first claim and simulation results are included to assess the impact of different distributional assumptions on the time until the first claim.

On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps

This paper studies the Parisian ruin problem first proposed by Dassios and Wu (2008a,b), where the Parisian ruin time is defined to be the first time when the surplus process has stayed below zero continuously for a pre-specified time length d. Both the insurance risk process and the dual model will be considered under exponential distributional assumption on the jump sizes while keeping the inter-arrival times arbitrary. In these two models, the Laplace transform of the Parisian ruin time is derived by extending the excursion techniques in Dassios and Wu (2008a) and taking advantage of the memoryless property of exponential distributions. Our results are represented in integral forms, which are expressed in terms of the (joint) densities of various ruin-related quantities that are available in the literature or obtainable using the Lagrange’s expansion theorem. As a by-product, we also provide the joint distribution of the numbers of periods of negative surplus that are of duration more than d and less than d, which can be obtained using some of our intermediate results. The case where the Parisian delay period d is replaced by a random time is also discussed, and it is applied to find the Laplace transform of the occupation time when the surplus is negative. Numerical illustrations concerning an Erlang(2) insurance risk model are given at the end.

On the depletion problem for an insurance risk process: new non-ruin quantities in collective risk theory

The field of risk theory has traditionally focused on ruin-related quantities. In particular, the socalled Expected Discounted Penalty Function has been the object of a thorough study over the years. Although interesting in their own right, ruin related quantities do not seem to capture path-dependent properties of the reserve. In this article we aim at presenting the probabilistic properties of drawdowns and the speed at which an insurance reserve depletes as a consequence of the risk exposure of the company. These new quantities are not ruin related yet they capture important features of an insurance position and we believe it can lead to the design of a meaningful risk measures. Studying drawdowns and speed of depletion for Levy insurance risk processes represent a novel and challenging concept in insurance mathematics. In this paper, all these concepts are formally introduced in an insurance setting. Moreover, using recent results in fluctuation theory for Levy processes, we derive expressions for the distribution of several quantities related to the depletion problem. Of particular interest are the distribution of drawdowns and the Laplace transform for the speed of depletion. These expressions are given for some examples of Levy insurance risk processes for which they can be calculated, in particular for the classical Cramer-Lundberg model.

On a risk model with claim investigation

In this paper, a queue-based claims investigation mechanism is considered to model an insurer’s claim processing practices. The resulting risk model may be viewed as a first step in developing models with more realistic claim investigation mechanisms. Related to claim investigations, claim settlement delays and time dependent payments have been studied in a ruin context by, e.g. Taylor (1979), Cai and Dickson (2002), and Trufin et al. (2011). However, little has been done on queue-based investigation mechanisms. We first demonstrate the impact of a particular claim investigation system on some common ruin-related quantities when claims arrive according to a compound Poisson process, and investigation times are of a combination of exponential form. Probabilistic interpretations for the defective renewal equation components are also provided. Finally, via numerical examples, we explore various risk management questions related to this problem such as how claim investigation strategies can help an insurer control its activities within its risk appetite.

A risk model with varying premiums: Its risk management implications

In this paper, we consider a risk model which allows the insurer to partially reflect the recent claim experience in the determination of the next period’s premium rate. In a ruin context, similar mechanisms to the one proposed in this paper have been studied by, e.g., Tsai and Parker (2004), Afonso et al. (2009) and Loisel and Trufin (2013). In our proposed risk model, we assume that the effective premium rate is determined based on the surplus increments between successive random review times. When review times are distributed as a combination of exponentials and claim arrivals follow a compound Poisson process, we derive a matrix-form defective renewal equation for the Gerber–Shiu function, and provide an explicit expression for the discounted joint density of the surplus prior to ruin and the deficit at ruin. Numerical examples are later considered to numerically evaluate certain ruin-related quantities. A comparison with their counterparts in a constant premium rate model is also presented.

Potential measures for spectrally negative Markov additive processes with applications in ruin theory

The Markov additive process (MAP) has become an increasingly popular modeling tool in the applied probability literature. In many applications, quantities of interest are represented as functionals of MAPs and potential measures, also known as resolvent measures, have played a key role in the representations of explicit solutions to these functionals. In this paper, closed-form solutions to potential measures for spectrally negative MAPs are found using a novel approach based on algebraic operations of matrix operators. This approach also provides a connection between results from fluctuation theoretic techniques and those from classical differential equation techniques. In the end, the paper presents a number of applications to ruin-related quantities as well as verification of known results concerning exit problems.