The theory of risk exchange is applied on the allocation of financial risk in capital markets. It is shown how the shape of individual payoff functions depends on risk tolerance and cautiousness. For the special case where the Neumann-Morgenstern utility functions of all individual investors belong to the HARA class and have non decreasing risk tolerance it is proved that generalized versions of “portfolio insurance”, “tactical asset allocation” and “collars” are the only strategies occurring in price equilibrium.

We consider compound distributions where the counting distribution has the property that the ratio between successive probabilities may be written as the ratio of two polynomials. We derive a recursive algorithm for the compound distribution, which is more efficient than the one suggested by Panjer & Willmot (1982) and Willmot & Panjer (1987). We also derive a recursive algorithm for the moments of the compound distribution. Finally, we present an application of the recursion to the problem of calculating the probability of ruin in a particular mixed Poisson process.

We show how ruin probabilities for the classical continuous time compound Poisson model can be approximated by ruin probabilities for a compound binomial model. We also discuss ruin related results for a compound binomial model with geometric claim amounts.

An approximation of the distribution of the present value of the benefits of a portfolio of temporary insurance contracts is suggested for the case where the size of the portfolio tends to infinity. The model used is the one presented in Parker (1922b) and involves random interest rates and future lifetimes. Some justifications of the approximation are given. Illustrations for limiting portfolios of temporary insurance contracts are presented for an assumed Ornstein-Uhlenbeck process for the force of interest.

The paper introduces an alternative approach to the traditional experience rating theory in automobile insurance. The approach is based on a simple theory of how high deductibles financed by loans maintain the risk differentiation in an automobile insurance arrangement. Thus the approach differs totally from the usual bonus-malus classes as well as from the credibility based experience rating ideas. The paper is of a theoretical nature and leads up to a mathematical description of how the approach may be optimalized within the framework of a risk model.

Holtan (1994) suggests to replace traditional bonus-malus systems by a high deductible financed by a short-term loan. Practical consequences of this proposal are investigated here. Simulation is used to evaluate the efficiency of the Taiwanese Bonus-malus system and the variability of premiums of an average policyholder. Holtan's high deductible system is analysed under a compound Poisson assumption, with truncated exponential claims. It is shown that the introduction of a high deductible would increase the variability of payments and the efficiency of the rating system for most policyholders.

According to the editorial rules of treating discussion situations in the ASTIN Bulletin the paper by J. Lemaire & H. Zi being somewhat a discussion on Holtan's paper was sent to the author of the original paper, who was given the opportunity to make an additional comment. The editors then received the following note by Jon Holtan:

In this note I want to give some general comments on the papers by Lemaire & Zi (1994) and Holtan (1994).

Interpret henceforth a bonus-malus (BM) principle as consisting of two basic components:

(a) The BM design.

(b) The BM tariff parameters.

Traditional actuarial literature has basically been preoccupied with component (b). Or more precisely, the tariff parameters of an initial accepted BM design have usually been mathematically optimalized within different criteria of succes like e.g. high efficiency and financial balance. In my opinion, however, this strategy seems to be too narrow if the aim is to construct a BM principle which is totally optimalized in favour of both the insurer and the insured. In our strive for maximizing BM advantages and minimizing BM disadvantages, actuarial BM research should instead simultaneously focus on both components (a) and (b). The construction of the High-Deductible System (HDS) in Holtan (1994) is an example of this strategy. However, as pointed out in Lemaire & Zi (1994) (see Section 1 and 4) and Holtan (1994) (see Section 3, 5 and 6), a HDS compared with existing BM systems both eliminates and generates important disadvantages which are linked to component (a). Based on some mathematical model assumptions, Lemaire & Zi moreover concludes (see Section 3 and 5) that this two-sided conclusion is in principle also valid within some mathematical criteria of success linked to component (b). These complex, and perhaps confusing, conclusions make it difficult for us to decide whether to prefer the existing BM systems or the HDS. However, the solution to this problem of decision seems to be naturally dependent on some strategic questions like: What kind of BM advantages and what kind of BM disadvantages will be the most important to focus on in the future automobile insurance market?

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.

Mortgage insurance indemnifies a mortage lender against loss on default by the borrower. The sequence of events leading to a claim under this type of insurance is relatively complex, depending not only on the credit worthiness of the borrower but also on a number of external economic factors.

Prominent among these external factors are the loan to valuation ratio of the insured loan, the disposable income of the borrower, and movements in property values. A broad theoretical model of the functional dependencies of claim frequency and average claim size on these variables is established in Sections 6 and 7. Section 8 fits these models, extended by other “internal” variables such as the geographic location of the mortgaged property, to a real data set.

Section 9 compares the fitted model with the data, and finds an acceptable fit despite extreme fluctuations in the claims experience recorded in the data set.

This paper gives a method for premium rating by postcode area. The method is based on spatial models in a Bayesian framework and uses the Gibbs sampler for estimation. A summary of the theory of Bayesian spatial methods is given and the data which was analysed by Taylor (1989) is reanalysed. An indication is given of the wide range of models within this class which would be suitable for insurance data. The aim of the paper is to introduce the models and to show how they can be utilised in an insurance setting.