The purpose of this note is to draw attention to a semimartingale method which can be applied to very general types of risk models to obtain local martingales or martingales, which can then be used in the now classical way to evaluate ruin probabilities. Relations to the theory of exponential families of stochastic processes are also pointed out and utilized.

Two methods for approximating the limiting distribution of the present value of the benefits of a portfolio of identical endowment insurance contracts are suggested. The model used assumes that both future lifetimes and interest rates are random. The first method is similar to the one presented in Parker (1994b). The second method is based on the relationship between temporary and endowment insurance contracts.

We consider three classes of bivariate counting distributions and the corresponding compound distributions. For each class we derive a recursive algorithm for calculating the bivariate compound distribution.

Policyholders often decide to buy, renew, or cancel insurance based on the premium charged by the insurer compared with what they expect their claims will be. It is important for actuaries to consider the persistency of policyholders because the financial well-being of the insurer depends on spreading its risk over a large book of business. We use statistical decision theory to develop premium formulas that account for the past experience of a given policyholder, the experience of the entire collection of policyholders, and the likelihood of the policyholder renewing with or buying from a given insurer, that is, persistency.

We assume that the persistency of policyholders depends on the arithmetic difference between the premium charged and their anticipated claims. We extend the work of Taylor (1975) in which he obtains linear credibility formulas by minimizing loss functions that incorporate the persistency of policyholders. We consider Taylor's loss functions and other objective functions, including those that account for the amount of business the insurer writes or renews.

This paper examines a class of premium functionals which are (i) comonotonic additive and (ii) stochastic dominance preservative. The representation for this class is a transformation of the decumulative distribution function. It has close connections with the recent developments in economic decision theory and non-additive measure theory. Among a few elementary members of this class, the proportional hazard transform seems to stand out as being most plausible for actuaries.

There is a duality between the surplus process of classical risk theory and the single-server queue. It follows that the probability of ruin can be retrieved from a single sample path of the waiting time process of the single-server queue. In this paper, premiums are allowed to vary. It has been shown that the stationary distribution of a corresponding storage process is equal to the survival probability (with variable premiums). Thus by simulation of the corresponding storage process, the probability of ruin can be obtained. The special cases where the surplus earns interest and the premiums are charged by layers are considered and illustrated numerically.

The premiums for a bonus-malus system which stays in financial equilibrium over the years are calculated. This is done by minimizing a quadratic function of the difference between the premium for an optimal BMS with an infinite number of classes and the premium for a BMS with a finite number of classes, weighted by the stationary probability of being in a certain class, and by imposing various constraints on the system.

In the present note we consider the Hipp approximation to the convolution of a finite number of probability distributions on the non-negative integers. It is shown that the moment up to and including order r of the rth order approximation are equal to the corresponding moments of the exact distribution. We also give a relation between the exact and approximated (r + 1)th order moments and indicate how similar relations can be obtained for higher order moments.

Maximum likelihood estimation is derived for the Lagrangian Poisson distribution for a simple and a loglinear model and illustrated with real data.