This paper finds payoff frequency distributions for valuing European and American fixed strike average options on a discrete time, recombining multiplicative binomial asset price process. In comparison to other discrete valuation methods the distributions, obtained analytically from a generating function, greatly reduce the computational requirements needed for accurate valuation. Less data are needed to value geometric than arithmetic averages, but the magnitude of calculations is similar for both instruments. Calculations of order T3 are needed to value European instruments, of order T4 to value their American counterparts. A frequency distribution of a quantity called path sums is used to value geometric average options, and a joint distribution of path sums and realized prices is used to value arithmetic average options. The frequency distributions give an exact value for geometric average, an approximate value for arithmetic average instruments. The method obtains additional information from the generating function to estimate approximation errors relative to the exact binomial solution. If the errors are significant they can be reduced using still further detail from the generating function. Error reduction can be performed selectively to minimize additional calculation.

The problem of determining optimal retention levels for a non-life portfolio consisting of a number of independent sub-portfolios was first discussed by de Finetti (1940). He considered retention levels as optimal if they minimised the variance of the insurer's profit from the portfolio subject to the constraint of a fixed level of expected profit. In this paper we consider a similar problem, changing the criterion for optimality to minimising the probability of ruin, either in discrete or continuous time. We investigate this problem through a series of case studies based on a real portfolio.

This paper is concerned with two methods to estimate the parameters of the Poisson-Goncharov distribution introduced recently by Lefevre and Picard (1996). These methods are applied to fit, inter alia, the six observed claims distributions, from automobile insurance third party liability portfolios, studied by Gossiaux and Lemaire (1981) and analysed afterwards by several authors.

In the present paper we discuss error bounds for approximations to aggregate claims distributions. We consider approximations to convolutions by approximating each of the distributions and taking the convolution of these approximations. For compound distributions we consider two classes of approximations. In the first class we approximate the counting distribution, but keep the severity distribution unchanged, whereas in the second class we approximate the severity distribution, but keep the counting distribution unchanged. We finally look at some examples.

When claims in the compound Poisson risk model are from a heavy-tailed distribution (such as the Pareto or the lognormal), traditional techniques used to compute the probability of ultimate ruin converge slowly to desired probabilities. Thus, faster and more accurate methods are needed. Product integration can be used in such situations to yield fast and accurate estimates of ruin probabilities because it uses quadrature weights that are suited to the underlying distribution. Tables of ruin probabilities for the Pareto and lognormal distributions are provided.

To use Bayesian analysis to model insurance losses, one usually chooses a parametric conditional loss distribution for each risk and a parametric prior distribution to describe how the conditional distributions vary across the risks. A criticism of this method is that the prior distribution can be difficult to choose and the resulting model may not represent the loss data very well. In this paper, we apply techniques from nonparametric density estimation to estimate the prior. We use the estimated model to calculate the predictive mean of future claims given past claims. We illustrate our method with simulated data from a mixture of a lognormal conditional over a lognormal prior and find that the estimated predictive mean is more accurate than the linear Bühlmann credibility estimator, even when we use a conditional that is not lognormal.

This note generalizes Jewell's theorem on exact credibility from the classical Bühlmann model to the (weighted) Bühlmann-Straub model.

We consider the classical risk model with subexponential claim size distribution. Three methods are presented to simulate the probability of ultimate ruin and we investigate their asymptotic efficiency. One, based upon a conditional Monte Carlo idea involving the order statistics, is shown to be asymptotically efficient in a certain sense. We use the simulation methods to study the accuracy of the standard Embrechts-Veraverbeke [16] approximation for the ruin probability and also suggest a new one based upon ideas of Hogan [21].

The operation of a bonus-malus system, superimposed on a premium system involving a number of other rating variables, is considered. To the extent that good risks are rewarded in their base premiums, through the other rating variables, the size of the bonus they require for equity is reduced. This issue is discussed quantitatively, and a numerical example given.

This paper presents a new reinsurance product, called ‘Adaptive Pivot Smoothing’ (APS). It is designed to reduce the variance of the risk reinsured without affecting the mean. Investment theories have provided the idea for the product.

This paper describes the dynamic financial analysis model currently being used by a property catastrophe reinsurer to manage its business. The model is-an integral part of the day-to-day operations at the Company, and is used as a decision making tool in the underwriting, investment, and capital management processes. The paper begins by describing the framework that the Company uses for risk management. This includes a classification of the risks facing the Company, which is used to define and prioritize their implementation in the model. Also included is a description of the conceptual approach the Company takes to evaluate the tradeoff between risk and return. The paper then goes on to describe the structure and operation of the dynamic financial analysis model and provides examples of its use at the Company, along with illustrative examples of the various types of output it produces.