M.J. Goovaerts

The laplace transform of annuities certain with exponential time distribution

Abstract By means of Wiener processes, randomness in interest rates for annuities can be modelled. This paper wants to give an expression for the Laplace transform of annuities certain, when time is exponentially distributed.

Supermodular Ordering and Stochastic Annuities

In this paper, we consider several types of stochastic annuities, for which an explicit expression for the distribution function is not available. We will construct a random variable with the same mean and which is larger in stop-loss order, for which the distribution function can be obtained easily.

An analytical inversion of a laplace transform related to annuities certain

Abstract The present contribution deals with the Laplace inversion of a modified Bessel function with respect to the index in a straightforward analytical way. This kind of modified Bessel functions appears when annuities certain with a stochastic interest rate are considered, and also when evaluating the value of Asian options.

The compound Poisson approximation for a portfolio of dependent risks

Abstract A well-known approximation of the aggregate claims distribution in the individual risk theory model with mutually independent individual risks is the compound Poisson approximation. In this paper, we relax the assumption of independency and show that the same compound Poisson approximation will still perform well under certain circumstances.

Some further results on annuities certain with random interest

Abstract In a former contribution, the authors indicated how interest randomness in annuities can be modelled by means of Wiener processes or path-integrals. In particular, an expression was given for the Laplace transform when time is exponentially distributed. In this contribution we will derive the moment generating function and the moments as well as the distribution function for annuities certain.

Premium calculation in insurance

Opening session.- Invited address.- Invited lecture: Some major issues in economics and insurance developments.- Main lectures.- Risk convolution calculations.- Risk sharing, incentives and moral hazard.- State-dependent utility, the demand for insurance and the value of safety.- Separation of risk parameters.- Weighted Markov processes with an application to risk theory.- Practical models in credibility theory, including parameter estimation.- Rate making and the societys sense of fairness.- The impact of reinsurance on the insurers risk.- Chains of reinsurance.- Net stop-loss ordering and related ordering.- Limit theorems for risk processes.- Semi-Markov models in economics and insurance.- Loss distributions: estimation, large sample theory, and applications.- Rating of non proportional reinsurance treaties based on ordered claims.- Resistant line fitting in actuarial science.- Quantitative models of pension costs.- Credibility: estimation of structural parameters.- Short communications.- Population and social security projections for Bangladesh.- Stability of premium principles under maximum entropy perturbations.- Practical rating of variable accident excess-loss premiums.- Motor premium rating.- The mean square error of a randomly discounted sequence of uncertain payments.- Operational time: a short and simple existence proof.- Simulation in actuarial work. Some computational problems.- Bayesian sequential analysis of multivariate point processes.- The actuary in practice.- The influence of reinsurance limits on infinite time ruin probabilities.- Some Berry-Esseen theorems for risk processes.- Some notes on the methods of calculation of life assurance premiums in the United Kingdom.- A stochastic model for investment variables in the United Kingdom.- Inflationary effects on pension plans: wage and benefit patterns.

A straightforward analytical calculation of the distribution of an annuity certain with stochastic interest rate

Abstract Starting from the moment generating function of the annuity certain with stochastic interest rate written by means of a time discretization of the Wiener process as an n-fold integral, a straightforward evaluation of the corresponding distribution function is obtained by letting n tend to infinity. The advantage of the present method consists in the direct calculation technique of the n-fold integral, instead of using moment calculation or differential equations, and in the possible applicability of the present method to varying annuities which could be applied to IBNR results, as well as to pension fund calculations, etc.

Homogeneous risk models with equalized claim amounts

We consider an homogeneous risk modelon a fixed bounded time interval [0, t] and we denote by Nt the number of claims in that interval. The claim amounts are X1 ;X 2 ;::: ; XNt . The homogeneous model is an extension of the classical actuarial risk model with Nt not necessarily Poisson distributed. In the model with equalized claim amounts , each amount Xk is replaced with X k D .X1 C C XNt /=Nt . Let 9(t, u) be the ruin probability before t in the homogenous model, corresponding to the initial risk reserve u0 and let 9(t, u) be the corresponding ruin probability evaluated in the associated model with equalized claim amounts. The essence of the classical Prabhu formula is that 9(t ,0 )D9(t, 0). By rather systematic numerical investigations in the classical risk model, we verify that 9(t, u)9(t, u) for any value of u0 and that 9(t, u) is an excellent approximation of 9(t, u). Then these conclusions must be valid in any homogeneous model and this is an interesting observation because 9(t, u) can be calculated numerically, whereas no algorithms are yet available for the numerical evaluation of 9(t, u) in general homogeneous risk models.

A recursive scheme for perpetuities with random positive interest rates, Part 1: Analytical results

Abstract Recently, the authors showed how interest randomness in actuarial functions can· be described by means of Wiener processes using path integrals. This paper wants to present an extension of this kind of models, by investigating the situation of interest rates that cannot become negative. The case of an annuity certain and in particular that of a perpetuity will be dealt with in detail.

Convexity Inequalities for the Swiss premium

SummaryThe Swiss premium calculation principle, introduced by Bühlmann, Gagliardi, Gerber, StrÄub (1977), assigns to a given risk X (a random variable, mostly nonnegative in practice) a premium p, solution of the equation E f (X-z p) = f ((1-z) p), where f is a continuous strictly increasing function and z e [0,1]. The premium p = π (X, f, z) may depend on X, f, z.Let g also be continuous strictly increasing. Then we prove that π (X, f, z) ≦ π (X, g, z) for all X iff g is convex in f. This result can be extended to the case where g is not necessarily assumed to be continuous strictly increasing, but then it must be stated a bit differently since π (X, g, z) may be meaningless.The main application concerns sub-additivity. For fixed f, z the Swiss premium calculation principle is said to be sub-additive if1