On the Probability and Severity of Ruin

Abstract

In the usual model of the collective risk theory, we are interested in the severity of ruin, as well as its probability. As a quantitative measure, we propose G (u, y), the probability that for given initial surplus u ruin will occur and that the deficit at the time of ruin will be less than y, and the corresponding density g (u, y). First a general answer in terms of the transform is obtained. Then, assuming that the claim amount distribution is a combination of exponential distributions, we determine g; here the roots of the equation that defines the adjustment coefficient play a central role. An explicit answer is also given in the case in which all claims are of constant size. KEYWORDSRuin; severity of ruin.

Approximations for the Gerber-Shiu expected discounted penalty function in the compound poisson risk model

“…This is true for any choice of δ, that is, only the coefficients of these terms depend on δ. It then follows from standard results in ruin theory (see, for instance, Gerber et al (1987)) that m δ (u) has the form…”

“…The family of combinations of exponential distributions, on the other hand, has several attractive features and we choose our input claim size distribution, h 0 , to be a member of that family. More specifically, a combination of exponential distributions studied by, inter alios, and in the context of ruin theory, Gerber et al (1987) has a density with the general form…”

Actuariele Statistiek - Verleden en Toekomst

“…Ter gelegenheid van een bezoek aan de Universiteit van Amsterdam van Hans Gerber verscheen het artikel Gerber et al (1987) daarover, dat nog veelvuldig geciteerd wordt. Ook is er een expliciet algoritme als de claims discreet zijn en slechts weinig mogelijke waarden hebben.…”

Calculating multivariate ruin probabilities via Gaver–Stehfest inversion technique

“…The probability of ultimate ruin with initial reserves u, parameter λ and severity of ruin less than y and surplus less than x + u is defined, We will now use the results obtained by Dufresne and Gerber (1988) and a similar renewal argument as in Frey and Schmidt (1996) and Gerber et al (1987) in order to express the former probability with the following defective renewal equation: for x ∈ (0, ∞) and y ∈ (0, ∞).…”