Alfredo D. Egídio dos Reis

How long is the surplus below zero

Abstract Assuming the classical compound Poisson continuous time surplus process, we consider the process as continuing if ruin occurs. Due to the assumptions presented, the surplus will go to infinity with probability one. If ruin occurs the process will temporarily stay below the zero level. The purpose of this paper is to find some features about how long the surplus will stay below zero. Using a martingale method we find the moment generating function of the duration of negative surplus, which can be multiple, as well as some moments. We also present the distribution of the number of negative surpluses. We further show that the distribution of duration time of a negative surplus is the same as the distribution of the time of ruin, given ruin occurs and initial surplus is zero. Finally, we present two examples, considering exponential and Gamma(2,β) individual claim amount distributions.

SOME STABLE ALGORITHMS IN RUIN THEORY AND THEIR APPLICATIONS

In this paper we present a stable recursive algorithm for the calculation of the probability of ultimate ruin in the classical risk model. We also present stable recursive algorithms for the calculation of the joint and marginal distributions of the surplus prior to ruin and the severity of ruin. In addition we present bounds for these distributions.

On the distribution of the duration of negative surplus

Abstract In the classical risk model we allow the surplus process to continue if the surplus falls below zero. We consider the distributions of the duration of a single period of negative surplus and of the total duration of negative surplus. We derive explicit results where possible and show how to approximate these distributions through the use of a discrete time risk model.

The effect of interest on negative surplus

Abstract In the classical continuous time surplus process, we allow the process to continue if the surplus falls below zero. When the surplus is below zero, we assume that the insurer borrows any sum of money required to pay claims, and pays interest on this borrowing. We use simulation to study moments and distributions of three quantities: the time to recovery to surplus level zero, the number of claims that occur when the surplus is below zero, and the maximum absolute value of the surplus process when it is below zero. We also show how simulation can be used to estimate the probability of absolute ruin.

On the moments of ruin and recovery times

Abstract In this paper we consider the calculation of moments of the time to ruin and the duration of the first period of negative surplus. We present a recursive method by considering a discrete time compound Poisson process used by Dickson et al. [Astin Bull. 25 (2) (1995) 153]. With this method we will also be able to calculate approximations for the corresponding quantities in the classical model. Furthermore, for the classical compound Poisson model we consider some asymptotic formulae, as initial surplus tends to infinity, for the severity of ruin, which allow us to find explicit formulae for the moments of the time to recovery.

How many claims does it take to get ruined and recovered

Abstract We consider in the classical surplus process the number of claims occurring up to ruin, by a different method presented by Stanford and Stroinski [Astin Bulletin 24 (2) (1994) 235]. We consider the computation of Laplace transforms (LTs) which can allow the computation of the probability function. Formulae presented are general. The method uses the computation of the probability function of the number of claims during a negative excursion of the surplus process, in case it gets ruined. When initial surplus is zero this probability function allows us to completely define the recursion for the transform above. This uses the fact that in this particular case, conditional time to ruin has the same distribution as the time to recovery, given that ruin occurs. We consider also the computation of moments of the number of claims during recovery time, which with initial surplus zero allows us to compute the moments of the number of claims up to ruin. We end this work by giving some insight on the shapes of the two types of probability functions involved.

Ruin problems and dual events

Abstract Dickson (1992) uses dual events to explain results relating to the distribution of the surplus immediately prior to ruin in the classical surplus process. In this paper we show that dual events can be used to explain other results in ruin theory. In particular we prove and explain the relationship between the density of the surplus immediately prior to ruin, and the joint density of the surplus immediately prior to ruin and the severity of ruin.

Recursive calculation of time to ruin distributions

Abstract In this paper we present a different approach on Dickson and Waters [Astin Bulletin 21 (1991) 199] and De Vylder and Goovaerts [Insurance: Mathematics and Economics 7 (1988) 1] methods to approximate time to ruin probabilities. By means of Markov chain application we focus on the direct calculation of the distribution of time to ruin, and we find that the above recursions appear to be less efficient, although giving the same approximation figures. We show some graphs of the time to ruin distribution for some examples, comparing the different shapes of the densities for different values of the initial surplus. Furthermore, we consider the presence of an upper absorbing barrier and apply the proposed recursion to find ruin probabilities in this case.

CALCULATING CONTINUOUS TIME RUIN PROBABILITIES FOR A LARGE PORTFOLIO WITH VARYING PREMIUMS

In this paper we present a method for the numerical evaluation of the ruin probability in continuous and finite time for a classical risk process where the premium can change from year to year. A major consideration in the development of this methodology is that it should be easily applicable to large portfolios. Our method is based on the simulation of the annual aggregate claims and then on the calculation of the ruin probability for a given surplus at the start and at the end of each year. We calculate the within-year ruin probability assuming a translated gamma distribution approximation for aggregate claim amounts. We illustrate our method by studying the case where the premium at the start of each year is a function of the surplus level at that time or at an earlier time.

Numerical Evaluation of Continuous Time Ruin Probabilities for a Portfolio with Credibility Updated Premiums

The probability of ruin in continuous and finite time is numerically evaluated in a classical risk process where the premium can be updated according to credibility models and therefore change from year to year. A major consideration in the development of this approach is that it should be easily applicable to large portfolios. Our method uses as a first tool the model developed by Afonso et al. (2009), which is quite flexible and allows premiums to change annually. We extend that model by introducing a credibility approach to experience rating. We consider a portfolio of risks which satisfy the assumptions of the BA¼hlmann (1967, 1969) or BA¼hlmann and Straub (1970) credibility models. We compute finite time ruin probabilities for different scenarios and compare with those when a fixed premium is considered.