David C. M. Dickson

On the time to ruin for Erlang(2) risk processes

Abstract In this paper, we consider a Sparre Andersen risk process for which the claim inter-arrival distribution is Erlang(2). Our purpose is to find expressions for moments of the time to ruin, given that ruin occurs. To do this, we define an auxiliary function φ along the lines of Gerber and Shiu [N. Am. Actu. J. 2 (1998) 48] and Gerber and Landry [Ins.: Math. Econ. 22 (1998) 263]. Our method of solution differs from that of Willmot and Lin [Ins.: Math. Econ. 25 (1999) 570; Ins.: Math. Econ. 27 (2000) 19] who consider this problem for the classical risk model, in that we first solve for the auxiliary function φ .

Some Optimal Dividends Problems

We consider a situation originally discussed by De Finetti (1957) in which a surplus process is modified by the introduction of a constant dividend barrier. We extend some known results relating to the distribution of the present value of dividend payments until ruin in the classical risk model and show how a discrete time risk model can be used to provide approximations when analytic results are unavailable. We extend the analysis by allowing the process to continue after ruin.

Insurance Risk and Ruin

Based on the authors experience of teaching final-year actuarial students in Britain and Australia, and suitable for a first course in insurance risk theory, this book focuses on the two major areas of risk theory - aggregate claims distributions and ruin theory. For aggregate claims distributions, detailed descriptions are given of recursive techniques that can be used in the individual and collective risk models. For the collective model, different classes of counting distribution are discussed, and recursion schemes for probability functions and moments presented. For the individual model, the three most commonly applied techniques are discussed and illustrated. Care has been taken to make the book accessible to readers who have a solid understanding of the basic tools of probability theory. Numerous worked examples are included in the text and each chapter concludes with exercises, which have answers in the book and full solutions available for instructors from www.cambridge.org/9780521846400.

On the distribution of the surplus prior to ruin

Abstract The distribution of the surplus immediately prior to ruin in the classical compound Poisson risk model was considered in a paper by Dufresne and Gerber (1988). The main purpose of this paper is to present a simple method for finding the distribution function and the density function of this surplus. We show that explicit solutions for these functions can be found when we know the probability and severity of ruin, denoted G ( u, y ). We also derive an algorithm which can be used to give approximate values of the distribution function when explicit solutions cannot be found. We also briefly consider the joint density of the surplus immediately prior to ruin and the deficit at the time of ruin.

Recursive Calculation of Survival Probabilities

AbstractIn this paper we present an algorithm for the approximate calculation of finite time survival probabilities for the classical risk model. We also show how this algorithm can be applied to the calculation of infinite time survival probabilities. Numerical examples are given and the stability of the algorithms is discussed.

On the expected discounted penalty function at ruin of a surplus process with interest

Abstract In this paper, we study the expected value of a discounted penalty function at ruin of the classical surplus process modified by the inclusion of interest on the surplus. The ‘penalty’ is simply a function of the surplus immediately prior to ruin and the deficit at ruin. An integral equation for the expected value is derived, while the exact solution is given when the initial surplus is zero. Dickson’s [Insurance: Mathematics and Economics 11 (1992) 191] formulae for the distribution of the surplus immediately prior to ruin in the classical surplus process are generalised to our modified surplus process.

Ruin probabilities for Erlang(2) risk processes

Abstract In this paper we consider a risk process in which claim inter-arrival times have an Erlang(2) distribution. We consider the infinite time survival probability as a compound geometric random variable and give expressions from which both the survival probability from initial surplus zero and the ladder height distribution can be calculated. We consider explicit solutions for the survival/ruin probability in the case where the individual claim amount distribution is phase-type, and show how the survival/ruin probability can be calculated for other individual claim amount distributions.

Some Comments on the Compound Binomial Model

We show how ruin probabilities for the classical continuous time compound Poisson model can be approximated by ruin probabilities for a compound binomial model. We also discuss ruin related results for a compound binomial model with geometric claim amounts.

On a Class of Renewal Risk Processes

In this paper I show how methods that have been applied to derive results for the classical risk process can be adapted to derive results for a class of risk processes in which claims occur as a renewal process. In particular, claims occur as an Erlang process. I consider the problem of finding the survival probability for such risk processes and then derive expressions for the probability and severity of ruin and for the probability of absorption by an upper barrier. Finally, I apply these results to consider the problem of finding the distribution of the maximum deficit during the period from ruin to recovery to surplus level 0.

Reinsurance and ruin

Abstract We study the effect of reinsurance on the probability of ultimate ruin in the classical surplus process and consider a retention level as optimal if it minimises the ruin probability. We show that optimal retention levels can be found when the reinsurers premium loading depends on the retention level. We also show that when the aggregate claims process is approximated by a translated Gamma process, very good approximations to both optimal retention levels and ruin probabilities can be obtained. Finally, we discuss the effect of reinsurance on the probability of ruin in finite time.