In this paper we study a class of Mixed Bivariate Poisson Distributions by extending the Hofmann Distribution from the univariate case to the bivariate case.

We show how to evaluate the bivariate aggregate claims distribution and we fit some insurance portfolios given in the literature.

This study typically extends the use of the Bivariate Independent Poisson Distribution, the Mixed Bivariate Negative Binomial and the Mixed Bivariate Poisson Inverse Gaussian Distribution.

Recursions are derived for a class of compound distributions having a claim frequency distribution of the well known (a,b)-type. The probability mass function on which the recursions are usually based is replaced by the distribution function in order to obtain increasing iterates. A monotone transformation is suggested to avoid an underflow in the initial stages of the iteration. The faster increase of the transformed iterates is diminished by use of a scaling function. Further, an adaptive weighting depending on the initial value and the increase of the iterates is derived. It enables us to manage an arbitrary large portfolio. Some numerical results are displayed demonstrating the efficiency of the different methods. The computation of the stop-loss premiums using these methods are indicated. Finally, related iteration schemes based on the cumulative distribution function are outlined.

An iteration scheme is derived for calculating the aggregate claims distribution in the individual life model. The (exact) procedure is an efficient reformulation of De Pril's (1986) algorithm, considerably reducing both the number of arithmetic operations to be carried out and the number of data to be kept at each step of iteration. Scaling functions are used to stabilize the algorithm in case of a portfolio with a large number of policies. Some numerical results are displayed to demonstrate the efficiency of the method.

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.

In an earlier paper the author derived a recursion formula which permits the exact computation of the aggregate claims distribution in the individual life model. To save computing time he also proposed an approximative procedure based on the exact recursion.

In the present contribution the exact recursion formula and the related approximations are generalized to the individual risk theory model with arbitrary positive claims. Error bounds for the approximations are given and it is shown that they are smaller than those of the Kornya-type approximations.

We compare three modern methods for calculating the aggregate claims distribution with respect to their computation amount and accuracy: Panjer's algorithm, the approximation method of Kornya and the most recent, exact procedure of De Pril. They are compared numerically in the case of actual Life portfolios. The computation amount of De Pril's method is much greater than that of the two others, which do not differ substantially in this respect. The accuracy of Kornya's and Panjer's methods is remarkably high in the examples considered. However, as the accuracy of Kornya's approximation method can be determined easily in advance, this procedure turns out to be the most useful one for the problems arising from practical work.

Kornya-type higher order approximations are derived for the aggregate claims distribution and for stop loss premiums in the individual model with arbitrary positive claims. Absolute error bounds and error bounds based on concentration functions are given. In the Gerber portfolio containing 31 policies, second order approximations lead to an accuracy of 3 × 10−4, and third order approximations to 1.7 ×10−5.

A recursive expression is derived for computing exactly the distribution of aggregate claims of a portfolio of life insurance policies. The recursion generalizes a formula of White and Greville for the claim numbers distribution and improves Kornya's approximation method for the aggregate claims distribution. It can be seen as the counterpart in the individual model of Panjer's recursion formula for the collective model.