This paper develops a discrete time model for valuing treasury bills and either forward or futures contracts written against them. It provides formulae for bill prices, forward prices, futures prices, and their conditional variances and risk premiums. The interest rate process is described by a multiplicative binomial random walk whose features conform to some principal characteristics of observed processes. Initial forward rates are constrained to match initially observed term structure data.

For some time now, the convenient and fast calculability of collective risk models using the Panjer-algorithm has been a well-known fact, and indeed practitioners almost always make use of collective risk models in their daily numerical computations. In doing so, a standard link has been preferred for relating such calculations to the underlying heterogeneous risk portfolio and to the approximation of the aggregate claims distribution function in the individual risk model. In this procedure until now, the approximation quality of the collective risk model upon which such calculations are based is unknown.

It is proved that the approximation error which arises does not converge to zero if the risk portfolio in question continues to grow. Therefore, necessary and sufficient conditions are derived in order to obtain well-adjusted collective risk models which supply convergent approximations. Moreover, it is shown how in practical situations the previous natural link between the individual and the collective risk model can easily be modified to improve its calculation accuracy. A numerical example elucidates this.

Based on a representation of the aggregate claims random variable as linear combination of counting random variables, a linear multivariate Bayesian model of risk theory is defined. In case of the classical risk theoretical assumptions, that is conditional Poisson likelihood counting variates and Gamma structural density, the model is shown to identify with a Bayesian version of the collective model of risk theory. An interesting multivariate credibility formula for the predictive mean is derived. A new type of recursive algorithm, called three-stage nested recursive scheme, allows to evaluate the predictive density and associated predictive stop-loss premiums in an effective way.

This paper attempts to give an overview of the pricing of risks in a pure exchange economy, where trade takes place at time zero and where uncertainty is revealed at time one. An economic equilibrium model under uncertainty is formulated, where conditions characterizing a Pareto optimal exchange equilibrium are derived. We present two sets of sufficient conditions for the existence of an equilibrium, and demonstrate how equilibria can be characterized through several examples. Uniqueness of equilibrium is also discussed. Special attention is given to the principal components that the premiums in a reinsurance market must depend upon. We also apply the general theory to the risk exchange problem between a policyholder and an insurer, and in particular we compute market premiums of the resulting optimal contracts.

It is emphasized throughout how the formulation of a competitive equilibrium, rather than merely a general risk exchange formulation, is of particular interest in deriving a well-defined and unique set of equilibrium premiums in an insurance market. The theory is put into a framework which is fruitful for extensions beyond the one-period case.

We consider the problem of forecasting the total cost of claims in excess-of-loss reinsurance. The number of claims reported to the direct insurer is assumed to follow a Poisson law, and the claim severities are modelled by a Pareto distribution. The Poisson frequency as well as the Pareto parameter will be considered as random parameters in a Bayesian setting. We derive the class of conjugate joint prior distributions, which turn out to specify a (prior) dependence between the two parameters. The use of conjugate priors facilitates the mathematical analysis, and it also makes it easy to interpret the parameters of the prior distribution.

A fully time-continuous approach is taken to the problem of predicting the total liability of a non-life insurance company. Claims are assumed to be generated by a non-homogeneous marked Poisson process, the marks representing the developments of the individual claims. A first basic result is that the total claim amount follows a generalized Poisson distribution. Fixing the time of consideration, the claims are categorized into settled, reported but not settled, incurred but not reported, and covered but not incurred. It is proved that these four categories of claims can be viewed as arising from independent marked Poisson processes. By use of this decomposition result predictors are constructed for all categories of outstanding claims. The claims process may depend on observable as well as unobservable risk characteristics, which may change in the course of time, possibly in a random manner. Special attention is given to the case where the claim intensity per risk unit is a stationary stochastic process. A theory of continuous linear prediction is instrumental.

A distribution-free formula for the standard error of chain ladder reserve estimates is derived and compared to the results of some parametric methods using a numerical example.

Based on recurrence equation theory and relative error (rather than absolute error) analysis, the concept and criterion for the stability of a recurrence equation are clarified. A family of recursions, called congruent recursions, is proved to be strongly stable in evaluating its non-negative solutions. A type of strongly unstable recursion is identified. The recursive formula discussed by Panjer (1981) is proved to be strongly stable in evaluating the compound Poisson and the compound Negative Binomial (including Geometric) distributions. For the compound Binomial distribution, the recursion is shown to be unstable. A simple method to cope with this instability is proposed. Many other recursions are reviewed. Illustrative numerical examples are given.

Outlier observations caused by big claims or by an event producing a series of claims are a special problem in ratemaking and in tariff calculation. The authors believe that combining credibility and robust statistics is the right answer to this problem. The main idea is to robustify the individual claims experience by using a robust estimator Ti instead of the individual mean and to look at the credibility estimator based on the robust statistics {Ti: i = 1, 2, …} . Choosing a particular influence function leads to datatrimming with an observation-dependent trimming point.

A practical method of allowing for covariates in compound Poisson modelling distributions is discussed.

In this paper we derive formulae for finite time survival probabilities when the aggregate claims process is a Gamma process. We illustrate how a compound Poisson process can be approximated by a Gamma process and by a process defined as a translated Gamma process. We also show how survival probabilities for a compound Poisson process can be approximated by those for a Gamma process or a translated Gamma process.

Weighted mortality rates are commonly used in actuarial work, but the inter-relationship between the weights used and the underlying mortality rates seems not to have been widely investigated.

Calculation of the ratio of weighted mortality rates to conventional mortality rates provides a simple means for an insurance company to track changes in the underlying mortality of its portfolio over time, and acts as an early warning system for possible deterioration of underwriting results. Asymptotic distributions are found for this ratio, and for the mortality rates themselves. It is suggested that insurance companies commence to gather data for the calculation of this ratio for the insurance sector as a whole, for the main annuity and assurance classes.

Consider a group of n independent lives age x where each life puts § 1 in a fund at time 0. The fund earns interest at rate i, and at the end of t years the accumulated value of the fund is divided equally among the survivors. The traditional approach to calculating the expected lump sum benefit per survivor from the initial group of n lives is based on the concept of a deterministic survivorship group. This approach ignores the stochastic nature of the survivorship process. In reality, the benefit per survivor is actually a random variable with an expected value which depends on the first inverse moment of a positive binomial random variable. Using Grab's and Savage's (1954) recursive formula for the first inverse moment, it is shown that the traditional approach yields a fairly accurate approximation to the solution even when one assumes a random number of survivors.

The incidence of risk under a credit insurance policy depends on the original term of the policy and the policy duration at which the incidence of risk is considered. Section 3 of the paper describes the procedure used to fit a bivariate function to this incidence. Section 4 gives the numerical detail of this model. Section 5 makes a comparison of the model with the data from which it was developed. Section 6 adds some general comments.