The recursive algorithm of Hesselager (1994) is extended to a more general class of counting distributions, which includes Sundt's (1992) class as well as all the mixed Poisson distributions discussed by Willmot (1993).

Two approaches used to model interest randomness are presented. They are the modeling of the force of interest accumulation function and the modeling of the force of interest. The expected value, standard deviation and coefficient of skewness of the present value of annuities-immediate are presented as illustrations. The implicit behavior of the force of interest under the two approaches is investigated by looking at a particular conditional expectation of the force of interest accumulation function.

The claims generating process for a non-life insurance portfolio is modelled as a marked Poisson process, where the mark associated with an incurred claim describes the development of that claim until final settlement. An unsettled claim is at any point in time assigned to a state in some state-space, and the transitions between different states are assumed to be governed by a Markovian law. All claims payments are assumed to occur at the time of transition between states. We develop separate expressions for the IBNR and RBNS reserves, and the corresponding prediction errors.

The method of Esscher transforms is a tool for valuing options on a stock, if the logarithm of the stock price is governed by a stochastic process with stationary and independent increments. The price of a derivative security is calculated as the expectation, with respect to the risk-neutral Esscher measure, of the discounted payoffs. Applying the optional sampling theorem we derive a simple, yet general formula for the price of a perpetual American put option on a stock whose downward movements are skip-free. Similarly, we obtain a formula for the price of a perpetual American call option on a stock whose upward movements are skip-free. Under the classical assumption that the stock price is a geometric Brownian motion, the general perpetual American contingent claim is analysed, and formulas for the perpetual down-and-out call option and Russian option are obtained. The martingale approach avoids the use of differential equations and provides additional insight. We also explain the relationship between Samuelson's high contact condition and the first order condition for optimality.

Credibility theory is closely related to Kalman filtering. As a consequence, methods proposed for robustifying the Kalman filter can often be specialised to obtain robust credibility rating procedures. The application of one such method to several classical credibility models is shown in this paper.

Finite time ruin methods typically rely on diffusion approximations or discretization. We propose a new method by looking at the surplus process embedded at claim instants and develop a recursive scheme for calculating ruin probabilities. It is assumed that claim sizes follow a phase-type distribution. The proposed method is exact. The application of the method reveals where in the future the relative vulnerability to the company lies.

Goovaerts and Kaas (1991) present a recursive scheme, involving Panjer's recursion, to compute the compound generalized Poisson distribution (CGPD). In the present paper, we study the CGPD in detail. First, we express the generating functions in terms of Lambert's W function. An integral equation is derived for the pdf of CGPD, when the claim severities are absolutely continuous, from the basic principles. Also we derive the asymptotic formula for CGPD when the distribution of claim severity satisfies certain conditions. Then we present a recursive formula somewhat different and easier to implement than the recursive scheme of Goovaerts and Kaas (1991), when the distribution of claim severity follows an arithmetic distribution, which can be used to evaluate the CGPD. We illustrate the usage of this formula with a numerical example.

An overview of the potential of Generalized Linear Models as a means of modelling the salient features of the claims process in the presence of rating factors is presented. Specific attention is focused on the rich variety of modelling distributions which can be implemented in this context.

The automobile third party insurance merit-rating systems of 22 countries are simulated and compared, using as main tools the stationary average premium level, the variability of the policyholders' payments, their elasticity with respect to the claim frequency, and the magnitude of the hunger for bonus. Principal components analysis is used to define an “Index of Toughness” for all systems.

In this paper some results are given on the addivity of chain-ladder projections. Given two claims development triangles, when do their chain-ladder projections add up to the projections of the combined triangle, that is the triangle being the element-wise sum of the two given triangles?

Necessary and sufficient conditions for equality are given. These are of a fairly simply form and are directly connected to the ordinary chain-ladder calculations. In addition, sufficient conditions of the same form are given for inequality between the combined projection vector and the sum of the two original projections vectors.

The calculation of mean claim sizes, in the presence of a deductible, is usually achieved through numerical integration. In case of a Lognormal or Gamma distribution, the quantities of interest can easily be expressed as functions of the cumulative distribution function, with modified parameters. This also applies to the F-distribution, where the incomplete Beta function enters the scene; see for instance the appendix in Hogg and Klugman (1984).

The purpose of this paper is to derive an explicit formula for the first two moments of the Inverse Gaussian distribution, in the presence of censoring. For reasons of completeness we also consider truncation of the Inverse Gaussian distribution by an upper limit.

The tractability of the derivation depends in a crucial way on two properties of the Inverse Gaussian distribution. Firstly, the cumulative distribution function of the Inverse Gaussian can be written as a simple function using the Normal probability integral. Secondly, the moment generating function of a censorized Inverse Gaussian distribution boils down to an expression containing the cumulative Inverse Gaussian distribution. This manifests itself most clearly in case of life insurance where the quantity of interest is the expectation of a present value. In case of non-life insurance, where the dimension of the Inverse Gaussian random variable is money instead of time, a further step is required: differentiation of the moment generating function.

So, a natural order of this paper is to address ourselves first to the derivation for the life case and afterwards tackling the more laborious derivation for the non-life case.

This paper considers the application of state space modelling to the chain ladder linear model in order to allow the run-off parameters to vary with accident year. In the usual application of the chain ladder technique, the development factors are assumed to be the same for each accident year. This implies that the run-off shape does not alter with accident year. This paper shows how this assumption can be relaxed in order to allow a recursive smooth model to be applied, or for large changes in the shape of the run-off curve. It is possible for these changes to be modelled using external inputs, or for a multiprocess model to be used to detect changes in the run-off shape.