Showing papers in "Scandinavian Actuarial Journal in 1999"
TL;DR: In this paper, both exponential and non-exponential upper bounds for ruin probabilities are obtained by using martingale inequalities, and similar results are obtained for the model with investment income.
Abstract: In this paper, we consider a discrete time risk model. First we discuss the classical model, both exponential and non-exponential upper bounds for ruin probabilities are obtained by using martingale inequalities. Then similar results are obtained for the model with investment income.
67 citations
TL;DR: In this paper, an extension of the Panjer recursions applied to the numerical evaluation of pseudo-compound distributions is proposed to obtain the numerical finite-time ruin probabilities in the classical actuarial risk model, which can most easily be obtained by a remarkable formula due to Picard and Lefevre.
Abstract: Numerical finite-time ruin probabilities in the classical actuarial risk model can most easily be obtained by a remarkable formula due to Picard and Lefevre (1997), via an obvious extension of the Panjer recursions applied to the numerical evaluation of pseudo-compound distributions.
42 citations
TL;DR: In this paper, the authors study the stochastic orderings defined by means of pointwise comparison of the Lorenz curves, or of the excess wealth transforms, of two risks, and the resulting order relations are presented in an actuarial context and put in relation with classical orderings.
Abstract: The purpose of this paper is to study the stochastic orderings defined by means of pointwise comparison of the Lorenz curves, or of the excess wealth transforms, of two risks. The resulting order relations are presented in an actuarial context and put in relation with classical stochastic orderings, namely the stochastic dominance, the stop-loss order, the convex order and the dispersive order. Several relevant applications in reinsurance theory are provided.
31 citations
TL;DR: In this article, the authors derived two-sided bounds for the ruin probability in the compound Poisson risk model when the adjustment coefficient of the individual claim size distribution does not exist, and these bounds also apply directly to the tails of compound geometric distributions.
Abstract: In this paper, we derive two-sided bounds for the ruin probability in the compound Poisson risk model when the adjustment coefficient of the individual claim size distribution does not exist. These bounds also apply directly to the tails of compound geometric distributions. The upper bound is tighter than that of Dickson (1994). The corresponding lower bound, which holds under the same conditions, is tighter than that of De Vylder and Goovaerts (1984). Even when the adjustment coefficient exists, the upper bound is, in some cases, tighter than Lundberg's bound. These bounds are applicable for any positive distribution function with a finite mean. Examples are given and numerical comparisons with asymptotic formulae for the ruin probability are also considered.
25 citations
TL;DR: This work describes three increasingly complex and realistic models for Bayesian evaluation of the accident proneness of insurants and describes their implementation in a forecasting system developed for an insurance company.
Abstract: As part of their resource allocation processes, insurance companies have to undertake various evaluation tasks concerning the accident proneness of their insurants. Bayesian methods are specially fit for that task since they allow for the coherent incorporation of all sources of information, including expert opinions and data. We describe three increasingly complex and realistic models for that purpose. For predictive and inference purposes, we have to rely on simulation methods. We illustrate the models with a real case and describe their implementation in a forecasting system developed for an insurance company.
24 citations
TL;DR: In this article, the authors introduced new classes of order relations for discrete bivariate random vectors that essentially compare the expectations of real functions of convex-type of the random vectors.
Abstract: New classes of order relations for discrete bivariate random vectors are introduced that essentially compare the expectations of real functions of convex-type of the random vectors. For the actuarial context, attention is focused on the so-called increasing convex orderings between discrete bivariate risks. First, various characterizations and properties of these orderings are derived. Then, they are used for comparing two similar portfolios with dependent risks and for constructing bounds on several multilife insurance premiums.
18 citations
TL;DR: Lejeune and Sarda (1992) and Jones (1993) introduced the principle of local linear estimation to nonparametric estimation of smooth densities on the positive real axes.
Abstract: Lejeune and Sarda (1992) and Jones (1993) introduced the principle of local linear estimation to nonparametric estimation of smooth densities on the positive real axes. This methodology results in the basic kernel smoother with Gasser and Muller (1979) type boundary kernels when estimating close to a boundary. This principle is extended to nonparametric multivariate density estimation with arbitrary boundary regions.
15 citations
TL;DR: In this paper, the authors derived recursions for direct evaluation of O t f when f itself satisfies a certain sort of recursion, which can be applied for exact and approximate evaluation of distribution functions and stop-loss transforms of probability distributions.
Abstract: For any function f on the non-negative integers, we can evaluate the cumulative function o f given by o f ( s )= ~ s x=0 f ( x ) from the values of f by the recursion o f ( s )= o f ( s -1)+ f ( s ). Analogously we can use this procedure t times to evaluate the t -th order cumulative function o t f . As an alternative, in the present paper we shall derive recursions for direct evaluation of o t f when f itself satisfies a certain sort of recursion. We shall also derive recursions for the t -th order tails v t f where v f ( s )= ~ X x=s+1 f ( x ). The recursions can be applied for exact and approximate evaluation of distribution functions and stop-loss transforms of probability distributions. The class of recursions for f includes the classes discussed by Sundt (1992), incorporating the class studied by Panjer's (1981). We discuss in particular convolutions and compound functions.
13 citations
TL;DR: In this paper, the Sparre-Andersen model in the collective risk theory is investigated, assuming that the claim size is heavy-tailed, say subexponential, and the second order asymptotic behaviour of ruin probabilities is obtained.
Abstract: The Sparre-Andersen model in the collective risk theory is investigated. Assuming that the claim-size is heavy-tailed, say subexponential, the second order asymptotic behaviour of ruin probabilities is obtained.
12 citations
TL;DR: In this paper, a generalized sequential credibility formula is proposed and an optimal stepwise gain sequence is derived for the location Dispersion Family (LDF) for which no simple credibility formula exists.
Abstract: Stochastic approximation is a powerful tool for sequential estimation of zero points of a function. This methodology is defined and is shown to be related to a broad class of credibility formulae derived for the Exponential Dispersion Family (EDF). We further consider a Location Dispersion Family (LDF) which is rich enough and for which no simple credibility formula exists. For this case, a Generalized Sequential Credibility Formula is suggested and an optimal stepwise gain sequence is derived.
11 citations
TL;DR: In this paper, the authors generalize this method to the renewal model, first introduced by Andersen (1957), and calculate the approximations in case of phase type distributed claims and interarrival times.
Abstract: In Asmussen (1984), 'corrected diffusion approximations' for finite horizon ruin probabilities in the classical Cramer-Lundberg model is given. In this paper we generalize this method to the renewal model, first introduced by Andersen (1957). We calculate the approximations in case of phase type distributed claims and interarrival times. A comparison with simulated values indicates a good fit in situations of main interest in risk theory.
TL;DR: In this paper, an alternative solution to the problem of negative interest rates is presented, where the randomness is modelled by means of an ordinary Wiener process, and as a consequence negative interest rate is possible.
Abstract: In some former contributions, the authors investigated actuarial quantities with stochastic interest rates. In a first model, the randomness is modelled by means of an ordinary Wiener process, and as a consequence negative interest rates are possible. A second model provides a tool to avoid these negative interest rates, which can be necessary in particular situations. This paper wants to present an alternative solution to the problem of negative interest rates. This new model will be implemented to the case of an annuity certain and of a perpetuity.
TL;DR: In this paper, it was shown that increases in costs of the type considered by Borch are not necessarily followed by increases in loadings, since in equilibrium insurance may be Giffen, and loadings do not increase with deductibles, because the only viable equilibrium is a Stackelberg one.
Abstract: In a recently reprinted paper Borch wonders whether an increase in insurance loadings, together with the consequent increase in customers' deductibles, may be the start of a vicious circle, in which higher deductibles produce higher loadings and vice versa, ad infinitum. This paper rules out the possibility of a vicious circle, in a model a la Borch. First of all, increases in costs of the type considered by Borch are not necessarily followed by increases in loadings. Second, increases in loadings are not necessarily followed by increases in deductibles, since in equilibrium insurance may be Giffen. Last but not least, loadings do not increase with deductibles, because the only viable equilibrium is a Stackelberg one.