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

In 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.     

On the discounted aggregate claim costs until ruin in dependent Sparre Andersen risk processes

In this paper, a dependent Sparre Andersen risk process in which the joint density of the interclaim time and the resulting claim severity satisfies the factorization as in Willmot and Woo is considered. We study a generalization of the Gerber–Shiu function (i) whose penalty function further depends on the surplus level immediately after the second last claim before ruin; and (ii) which involves the moments of the discounted aggregate claim costs until ruin. The generalized discounted density with a moment-based component proposed in Cheung plays a key role in deriving recursive defective renewal equations. We pay special attention to the case where the marginal distribution of the interclaim times is Coxian, and the required components in the recursion are obtained. A reverse type of dependency structure, where the claim severities follow a combination of exponentials, is also briefly discussed, and this leads to a nice explicit expression for the expected discounted aggregate claims until ruin. Our results are applied to generate some numerical examples involving (i) the covariance of the time of ruin and the discounted aggregate claims until ruin; and (ii) the expectation, variance and third central moment of the discounted aggregate claims until ruin.     

Authors’ Reply: Corporate Hedging in the Insurance Industry: The Use of Financial Derivatives by U.S. Insurers - Discussion by L. Lee Colquitt; Arlette C. Wilson; Gray G. Venter; Morton Lane; Joan Lamm-Tennant,January 1997

Abstract

This paper studies the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. The time of ruin is analyzed in terms of its Laplace transforms, which can naturally be interpreted as discounting. Hence the classical risk theory model is generalized by discounting with respect to the time of ruin. We show how to calculate an expected discounted penalty, which is due at ruin and may depend on the deficit at ruin and on the surplus immediately before ruin. The expected discounted penalty, considered as a function of the initial surplus, satisfies a certain renewal equation, which has a probabilistic interpretation. Explicit answers are obtained for zero initial surplus, very large initial surplus, and arbitrary initial surplus if the claim amount distribution is exponential or a mixture of exponentials. We generalize Dickson’s formula, which expresses the joint distribution of the surplus immediately prior to and at ruin in terms of the probability of ultimate ruin. Explicit results are obtained when dividends are paid out to the stockholders according to a constant barrier strategy.     

The Time of Recovery and the Maximum Severity of Ruin in a Sparre Andersen Model

Abstract

Phase-type distributions are one of the most general classes of distributions permitting a Markovian interpretation. Sparre Andersen risk models with phase-type claim interarrival times or phase-type claims can be analyzed using Markovian techniques, and results can be expressed in compact matrix forms. Computations involved are readily programmable in practice.

This paper studies some quantities associated with the first passage time and the time of ruin in a Sparre Andersen risk model with phase-type interclaim times. In an earlier discussion the present author obtained a matrix expression for the Laplace transform of the first time that the surplus process reaches a given target from the initial surplus. Using this result, we analyze (1) the Laplace transform of the recovery time after ruin, (2) the probability that the surplus attains a certain level before ruin, and (3) the distribution of the maximum severity of ruin. We also give a matrix expression for the expected discounted dividend payments prior to ruin for the Sparre Andersen model in the presence of a constant dividend barrier.     

Moments of the Dividend Payments and Related Problems in a Markov-Modulated Risk Model

Abstract

In this paper we derive some results on the dividend payments prior to ruin in a Markovmodulated risk process in which the rate for the Poisson claim arrival process and the distribution of the claim sizes vary in time depending on the state of an underlying (external) Markov jump process {J(t); t ≥ 0}. The main feature of the model is the flexibility in modeling the arrival process in the sense that periods with very frequent arrivals and periods with very few arrivals may alternate, and that the states of {J(t); t ≥ 0} could describe, for example, epidemic types in health insurance or weather conditions in car insurance. A system of integro-differential equations with boundary conditions satisfied by the nth moment of the present value of the total dividends prior to ruin, given the initial environment state, is derived and solved. We show that the probabilities that the surplus process attains a dividend barrier from the initial surplus without first falling below zero and the Laplace transforms of the time that the surplus process first hits a barrier without ruin occurring can be expressed in terms of the solution of the above-mentioned system of integro-differential equations. In the two-state model, explicit results are obtained when both claim amounts are exponentially distributed.     

On The Decomposition Of The Ruin Probability For A Jump-Diffusion Surplus Process Compounded By A Geometric Brownian Motion

Abstract

If one assumes that the surplus of an insurer follows a jump-diffusion process and the insurer would invest its surplus in a risky asset, whose prices are modeled by a geometric Brownian motion, the resulting surplus for the insurer is called a jump-diffusion surplus process compounded by a geometric Brownian motion. In this resulting surplus process, ruin may be caused by a claim or oscillation. We decompose the ruin probability in the resulting surplus process into the sum of two ruin probabilities: the probability that ruin is caused by a claim, and the probability that ruin is caused by oscillation. Integro-differential equations for these ruin probabilities are derived. When claim sizes are exponentially distributed, asymptotical formulas of the ruin probabilities are derived from the integro-differential equations, and it is shown that all three ruin probabilities are asymptotical power functions with the same orders and that the orders of the power functions are determined by the drift and volatility parameters of the geometric Brownian motion. It is known that the ruin probability for a jump-diffusion surplus process is an asymptotical exponential function when claim sizes are exponentially distributed. The results of this paper further confirm that risky investments for an insurer are dangerous in the sense that either ruin is certain or the ruin probabilities are asymptotical power functions, not asymptotical exponential functions, when claim sizes are exponentially distributed.     

On a renewal risk process with dependence under a Farlie–Gumbel–Morgenstern copula

In this article, we consider an extension to the renewal or Sparre Andersen risk process by introducing a dependence structure between the claim sizes and the interclaim times through a Farlie–Gumbel–Morgenstern copula proposed by Cossette et al. (2010) for the classical compound Poisson risk model. We consider that the inter-arrival times follow the Erlang(n) distribution. By studying the roots of the generalised Lundberg equation, the Laplace transform (LT) of the expected discounted penalty function is derived and a detailed analysis of the Gerber–Shiu function is given when the initial surplus is zero. It is proved that this function satisfies a defective renewal equation and its solution is given through the compound geometric tail representation of the LT of the time to ruin. Explicit expressions for the discounted joint and marginal distribution functions of the surplus prior to the time of ruin and the deficit at the time of ruin are derived. Finally, for exponential claim sizes explicit expressions and numerical examples for the ruin probability and the LT of the time to ruin are given.     

A generalised Gerber–Shiu measure for Markov-additive risk processes with phase-type claims and capital injections

In this paper we consider a risk reserve process where the arrivals (either claims or capital injections) occur according to a Markovian point process. Both claim and capital injection sizes are phase-type distributed and the model allows for possible correlations between these and the inter-claim times. The premium income is modelled by a Markov-modulated Brownian motion which may depend on the underlying phases of the point arrival process. For this risk reserve model we derive a generalised Gerber–Shiu measure that is the joint distribution of the time to ruin, the surplus immediately before ruin, the deficit at ruin, the minimal risk reserve before ruin, and the time until this minimum is attained. Numeral examples illustrate the influence of the parameters on selected marginal distributions.     

Analysis of a Generalized Penalty Function in a Semi-Markovian Risk Model

Abstract

In this paper an extension of the semi-Markovian risk model studied by Albrecher and Boxma (2005) is considered by allowing for general interclaim times. In such a model, we follow the ideas of Cheung et al. (2010b) and consider a generalization of the Gerber-Shiu function by incorporating two more random variables in the traditional penalty function, namely, the minimum surplus level before ruin and the surplus level immediately after the second last claim prior to ruin. It is shown that the generalized Gerber-Shiu function satisfies a matrix defective renewal equation. Detailed examples are also considered when either the interclaim times or the claim sizes are exponentially distributed. Finally, we also consider the case where the claim arrival process follows a Markovian arrival process. Probabilistic arguments are used to derive the discounted joint distribution of four random variables of interest in this risk model by capitalizing on an existing connection with a particular fluid flow process.     

“Relative Importance of Risk Sources in Insurance Systems”, Edward W. Frees,April 1998

Abstract

This paper considers a Sparre Andersen collective risk model in which the distribution of the interclaim time is that of a sum of n independent exponential random variables; thus, the Erlang(n) model is a special case. The analysis is focused on the function φ(u), the expected discounted penalty at ruin, with u being the initial surplus. The penalty may depend on the deficit at ruin and possibly also on the surplus immediately before ruin. It is shown that the function φ(u) satisfies a certain integro-differential equation and that this equation can be solved in terms of Laplace transforms, extending a result found in Lin (2003). As a consequence, a closed-form expression is obtained for the discounted joint probability density of the deficit at ruin and the surplus just before ruin, if the initial surplus is zero. For this formula and other results, the roots of Lundberg’s fundamental equation in the right half of the complex plane play a central role. Also, it is shown that φ(u) satisfies Li’s (2003) renewal equation. Under the assumption that the penalty depends only on the deficit at ruin and that the individual claim amount density is a combination of exponential densities, a closed-form expression for φ(u) is derived. In this context, known results of the Cauchy matrix are useful. Surprisingly, certain results are best expressed in terms of divided differences, a topic deleted from the actuarial examinations at the end of last century.     

Direct Derivation of Finite-Time Ruin Probabilities in the Discrete Risk Model with Exponential or Geometric Claims

Abstract

Growing research interest has been shown in finite-time ruin probabilities for discrete risk processes, even though the literature is not as extensive as for continuous-time models. The general approach is through the so-called Gerber-Shiu discounted penalty function, obtained for large families of claim severities and discrete risk models. This paper proposes another approach to deriving recursive and explicit formulas for finite-time ruin probabilities with exponential or geometric claim severities. The proposed method, as compared to the general Gerber-Shiu approach, is able to provide simpler derivation and straightforward expressions for these two special families of claims.     

On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier

In the framework of classical risk theory we investigate a model that allows for dividend payments according to a time-dependent linear barrier strategy. Partial integro-differential equations for Gerber and Shiu's discounted penalty function and for the moment generating function of the discounted sum of dividend payments are derived, which generalizes several recent results. Explicit expressions for the nth moment of the discounted sum of dividend payments and for the joint Laplace transform of the time to ruin and the surplus prior to ruin are derived for exponentially distributed claim amounts.     

Extremes on the discounted aggregate claims in a time dependent risk model

This paper presents an extension of the classical compound Poisson risk model for which the inter-claim time and the forthcoming claim amount are no longer independent random variables (rv's). Asymptotic tail probabilities for the discounted aggregate claims are presented when the force of interest is constant and the claim amounts are heavy tail distributed rv's. Furthermore, we derive asymptotic finite time ruin probabilities, as well as asymptotic approximations for some common risk measures associated with the discounted aggregate claims. A simulation study is performed in order to validate the results obtained in the free interest risk model.     

The expected discounted penalty function: from infinite time to finite time

In this paper we study the finite-time expected discounted penalty function (EDPF) and its decomposition in the classical risk model perturbed by diffusion. We first give the solution to a class of second-order partial integro-differential equations (PIDEs) with certain boundary conditions. We then show that the finite-time EDPFs as well as their decompositions satisfy this specific class of PIDEs so that their explicit expressions are obtained. Furthermore, we demonstrate that the finite-time EDPF may be expressed in terms of its ordinary counterpart (infinite-time) under the same risk model. Especially, the finite-time ruin probability due to oscillations and the finite-time ruin probability caused by a claim may also be expressed in terms of the corresponding quantities under the infinite-time horizon. Numerical examples are given when claims follow an exponential distribution.     

Moment generating functions of compound renewal sums with discounted claims

Léveillé & Garrido (2001a, 2001b) have obtained recursive formulas for the moments of compound renewal sums with discounted claims, which incorporate both, Andersen's (1957) generalization of the classical risk model, where the claim number process is an ordinary renewal process, and Taylor's (1979), where the joint effect of the claims cost inflation and investment income on a compound Poisson risk process is considered.

In this paper, assuming certain regularity conditions, we improve the preceding results by examining more deeply the asymptotic and finite time moment generating functions of the discounted aggregate claims process. Examples are given for claim inter-arrival times and claim severity following phase-type distributions, such as the Erlang case.     

Ruin probabilities in classical risk models with gamma claims

In this paper, we provide three equivalent expressions for ruin probabilities in a Cramér–Lundberg model with gamma distributed claims. The results are solutions of integro-differential equations, derived by means of (inverse) Laplace transforms. All the three formulas have infinite series forms, two involving Mittag–Leffler functions and the third one involving moments of the claims distribution. This last result applies to any other claim size distributions that exhibits finite moments.     

Authors’ Reply: Credit Standing and the Fair Value of Liabilities: A Critique - Discussion by Marsha Wallace

Abstract

As investment plays an increasingly important role in the insurance business, ruin analysis in the presence of stochastic interest (or stochastic return on investments) has become a key issue in modern risk theory, and the related results should be of interest to actuaries. Although the study of insurance risk models with stochastic interest has attracted a fair amount of attention in recent years, many significant ruin problems associated with these models remain to be investigated. In this paper we consider a risk process with stochastic interest in which the basic risk process is the classical risk process and the stochastic interest process (or the stochastic return-on-investmentgenerating process) is a compound Poisson process with positive drift. Within this framework, we first derive an integro-differential equation for the Gerber-Shiu expected discounted penalty function, and then obtain an exact solution to the equation. We also obtain closed-form expressions for the expected discounted penalty function in some special cases. Finally, we examine a lower bound for the ruin probability of the risk process.     

On the severity of ruin in a Markov-modulated risk model

We consider a Markov-modulated risk model in which the claim inter-arrivals, amounts and premiums are influenced by an external Markovian environment process. A system of Laplace transforms of the probabilities of the severity of ruin, given the initial environment state, is established from a system of integro-differential equations derived by Snoussi [The severity of ruin in Markov-modulated risk models Schweiz Aktuarver. Mitt., 2002, 1, 31–43]. In the two-state model, explicit formulas for probabilities of the severity of ruin are derived, when the initial reserve is zero or when both claim amount distributions are from the rational family. Numerical illustrations are also given.     

Moments of Discounted Dividends for a Threshold Strategy in the Compound Poisson Risk Model

Abstract

We consider a compound Poisson risk model in which part of the premium is paid to the shareholders as dividends when the surplus exceeds a specified threshold level. In this model we are interested in computing the moments of the total discounted dividends paid until ruin occurs. However, instead of employing the traditional argument, which involves conditioning on the time and amount of the first claim, we provide an alternative probabilistic approach that makes use of the (defective) joint probability density function of the time of ruin and the deficit at ruin in a classical model without a threshold. We arrive at a general formula that allows us to evaluate the moments of the total discounted dividends recursively in terms of the lower-order moments. Assuming the claim size distribution is exponential or, more generally, a finite shape and scale mixture of Erlangs, we are able to solve for all necessary components in the general recursive formula. In addition to determining the optimal threshold level to maximize the expected value of discounted dividends, we also consider finding the optimal threshold level that minimizes the coefficient of variation of discounted dividends. We present several numerical examples that illustrate the effects of the choice of optimality criterion on quantities such as the ruin probability.     

A note on deficit analysis in dependency models involving Coxian claim amounts

In this paper, we consider a fairly large class of dependent Sparre Andersen risk models where the claim sizes belong to the class of Coxian distributions. We analyze the Gerber–Shiu discounted penalty function when the penalty function depends on the deficit at ruin. We show that the system of equations needed to solve for this quantity is surprisingly simple. Various applications of this result are also considered.