Fe De Vylder

Recursive Calculation of Finite-time Ruin Probabilities

Abstract We develop a simple algorithm for the numerical calculation of finite-time ruin probabilities in a general discrete-time risk process model. These probabilities can be used for the calculation of approximations for the finite-time ruin probabilities in the classical actuarial risk model.

Analytical Best Upper-bounds On Stop-loss Premiums

Abstract We provide a list of best upper bounds on the stop-loss premium E(X−t)+ corresponding to the risk X and the retention limit t. Various information (moments, unimodality, symmetry,…) on the distribution F of X is taken into account.

Estimation of Ibnr Claims By Credibility Theory

Abstract We develop a stochastic multiplicative model for the forecasting of IBNR claims. The factor depending on the accident year is credibility adjusted. The title of this note also suits for the papers by Straub (1971) and Kramreiter and Straub (1973). We made stronger assumptions simplifying drastically the numerical calculations and the parameter estimation problem. As showed in the numerical illustrations, the developed method is also applicable in case of scarce irregular data.

Best Upper-bounds for Integrals With Respect To Measures Allowed To Vary Under Conical and Integral Constraints

Abstract We consider the general problem sup мϵM ∫ƒdм|∫ƒdм=z 1 (i=1…n) , where the integrals are over an abstract space Ω, the functions ƒ i ( i =0) …..n ) are defined on that space, and where μm varies in a cone M of measures defined on the space. The integral on the left of the bar has to be maximized. The equalities on the right of the bar are further constraints on μ. The solution of this primal problem goes via the solution of an associated dual problem. The particular cases where M is the cone of positive measures and the cone of positive unimodal measures with fixed mode are investigated in more detail. Only two simple illustrations are given, but several actuarial applications are planned by De Vylder, Goovaerts and Haezendonck.

Classical numerical ruin probabilities

Abstract Finite and infinite-time classical ruin probabilities can be approximated in Gerbers elementary binomial risk model. In order to obtain good results, rather fine discretizations may be necessary and then the computing times may be much too long. Here we show how rather rough discretizations provide approximations of excellent quality when a new claimsize distribution (with one negative probability mass!!!) is adopted and when a new security loading is introduced.

Bounds for Classical Ruin Probabilities

Abstract We derive upper and lower bounds for the ruin probability over infinite time in the classical actuarial risk model (usual independence and equidistribution assumptions; the claim-number process is Poisson). Our starting point is the renewal equation for the ruin probability, but no renewal theory is used, except for the elementary facts proved in the note. Some bounds allow a very simple new proof of an asymptotic result akin to heavy-tailed claim-size distributions.

Practical Credibility Theory with Emphasis on Optimal Parameter Estimation

We develop Hachemeisters regression model 111 credlbihty theory (without proofs) and indicate how the involved structural parameters can be estimated from the observable variables (with proofs for the simple results and those not yet published). Large famlhes of unbiased estinaators are available. From the practical viewpoint this is rather a handicap because it creates the problem to decide what estimators actually to use. In order to fix optimal estimators, we adopt the small-sample critermn of minimum-variance But in the research for general solutmns three kinds of difficulties arise.

Numerical best bounds on stop-loss preminus

Abstract The determination of the maximum or minimum of the stop-loss premium E(X − t), (t = retention limit) under various constraints on the distribution of the risk X, leads to linear programs with an infinite number of linear inequality constraints. Retaining a properly chosen increasing finite number of the constraints such a program can be approximated as a sequence of usual finite-dimensional linear programs.

Martingales and ruin in a dynamical risk process

Abstract The classical Lundbergs model is based on the assumptions: (i) The claim number process is a Poisson process. (ii) The premium income process is deterministic. (iii) The claim amounts are i.i.d. variables independent of the claim number process.

Maximization, under equality constraints, of a functional of a probability distribution

Let F be a family of probability distributions. Let O, C1…Cn be real functions on F. Let z1…zn be real numbers. Then we consider the problem of maximization of the object function O(F)(FϵF) under the equality constraints C1(F)=z1(i=1,…,n) . The theory is developed in order to solve problems of the following kind: Find the maximal variance of a stop-loss reinsured risk under partial information on the risk such as its range and two first moments.