Showing papers in "Scandinavian Actuarial Journal in 1989"

Risk theory in a Markovian environment

Abstract: We consider risk processes t t⩾0 with the property that the rate β of the Poisson arrival process and the distribution of B of the claim sizes are not fixed in time but depend on the state of an underlying Markov jump process {Zt } t⩾0 such that β=β i and B=Bi when Zt=i . A variety of methods, including approximations, simulation and numerical methods, for assessing the values of the ruin probabilities are studied and in particular we look at the Cramer-Lundberg approximation and diffusion approximations with correction terms. The mathematical framework is Markov-modulated random walks in discrete and continuous time, and in particular Wiener-Hopf factorisation problems and conjugate distributions (Esscher transforms) are involved.

The total claims distribution under inflationary conditions

Abstract: The total claims distribution over a fixed period of time with time dependent claim amounts is considered. A representation for the associated density function is found under certain conditions, including the important case with Poisson or mixed Poisson claim number processes and constant inflation. Methods of evaluation of this density are considered, and the cases with exponential claim sizes and regular variation of the tail are discussed in more detail.

On bonus systems with credibility scales

Abstract: In the present paper we study bonus systems with linear premium scales in a set-up presented in a paper by Borgan, Hoem & Norberg. Some numerical examples are given.

Calculation of Ruin Probabilities when the Premium Depends on the Current Reserve

Abstract: The purpose of this paper is to show how the ruin probability can be found for a compound Poisson risk process with a general premium rate p(r) depending on the reserve r, and it is illustrated how the probability of ruin can be calculated using a simple numerical method.

Experience Rating in Group Life Insurance

Abstract: Methods for experience rating of group life contracts are obtained as empirical Bayes or linear Bayes solutions in heterogeneity models. Each master contract is assigned a latent random quantity representing unobservable risk characteristics, which comprise mortality and possibly also age distribution and distribution of the sums insured, depending on the information available about the group. Hierarchical extensions of the set-up are discussed. An application of the theory to data from an authentic portfolio of groups revealed substantial between-group risk variations, hence experience rating could be statistically justified.

On evaluation of the Delaporte distribution and related distributions

Abstract: In the present paper we develop recursive algorithms for evaluation of the Delaporte distribution, the compound Delaporte distribution, and convolutions of compound Delaporte distributions. Some asymptotic results are given. We discuss how the approach can sometimes be generalized to other classes of compound mixed Poisson distributions when the mixing distribution is a shifted infinitely divisible distribution.

Best upper bounds on risks altered by deductibles under incomplete information

Abstract: By means of deductibles the original risk X is transformed into a risk W=f(X, d). In the present contribution best upper bounds are derived for E(W) in case of incomplete information on the original risk X.

Modelling Claims Runoff Triangles with Two-dimensional Time Series

Abstract: 1. Introduction This paper deals with the problem of forecasting the claims runoff of ordinary insurance business. Insurance companies are faced with the problem of making adequate provision for notified claims, and claims which may occur, on policies which they write. They have to do this to the satisfaction of policyholders, shareholders and the regulatory bodies and it is thus important that a soundly-based method is used to establish claims reserves. It is also important that such a method should be easily understood and that underlying the method is a theoretical basis which is known to give satisfactory results. Therefore, we shall take the simplest method of calculating claims reserves as the starting point and build into this more sophisticated estimation and forecasting techniques, without losing sight of the original model.

On Posterior Probabilities and Moments in Mixed Poisson Processes

Abstract: In this paper basic expressions for posterior probabilities and moments in mixed Poisson processes are reviewed. The associated compound distribution representing the total claims is discussed. Asymptotic formulae are derived, and it is shown quite generally that for poor risks the posterior distribution is approximately negative binomial. The effect of a shifted mixing distribution is considered.

A Class of Conjugate Hierarchical Priors for Gammoid Likelihoods

Abstract: The class of gamma distributions is known to be the natural conjugate family of priors for gammoid likelihoods like e.g. the Poisson or exponen­ tial. This result is extended to the hierarchical case where the observations are stratified in a nested pattern. A suitatile class of priors is a hierarchical system of finite gamma mixtures with fixed scale parameters and random shape parameters. Algorithms for posterior calculations are worked out for the two-stage hierarchy with simple gamma priors.

A Model of Annuities with Adverse Selection

Abstract: A model is developed where risks of various types are lumped together by annuity contract issuing companies due to asymmetry of information between the companies and their customers. If firms behave nonstrategically, an equilibrium exists with the firms charging uniform price to all customers. In such an equilibrium, I study how changes in (a) the fractions of various groups of individuals, (b) survival probabilities of various types and (c) attitudes towards risks of customers affect the equilibrium price of the annuity contracts.

A note on the computation in Whittaker-Henderson graduation

Abstract: This note considers the computation of graduated values with the Whittaker- Henderson graduation method. It is shown that the fundamental matrices are banded and thus solution of the linear equations can be done very efficiently. Numerical timings are presented to support this contention.

On Gerber's fun

Abstract: This paper presents an alternative derivation for some surprising probability results recently presented by H. U. Gerber.

Random number generators for compound distributions

Abstract: Modem digital computers have made it possible to apply more complex models for stochastic phenomena by using random number simulation when it would be impossible, or impractical, to construct and explicitly solve equations describing the phenomena in question. Compound random variables often play a central role in insurance risk processes. A well-known difficulty in generating random numbers with a compound distribution is that, in order to make them accurate enough, the computer time needed tends to grow too much. In this paper a random number generator for compound distributions is introduced, which is, in principle, exact or highly accurate for a wide class of distributions of the size of one claim, and yet generates random numbers in a rate essentially independent of the number of claims.

Numerical calculation of the Cramér-Lundberg approximation

Abstract: We consider the classical risk process, i.e. the claims occur in a Poisson process with intensity λ and are independent with common distribution F, where F(0)=0, and mean µ. The premium rate is c.