The paper presents an extension of the classical Cramér–Lundberg ruin theory: the famous upper bound for the ruin probability with an infinite time horizon can be extended in a certain sense to the short and middle term case. Furthermore, a relation between the average values of lifetime and ruin amount is given.

We give an extension of the Economic Premium Principle treated in Astin Bulletin, Volume 11 where only exponential utility functions were admitted. The case of arbitrary risk averse utility functions leads to similar quantitative results. The role of risk aversion in the treatment is essential. It also permits an easy proof for the existence of equilibrium.

In Industrial Fire insurance an aggregate limit for the amount retained by the policyholder under a deductible policy has been agreed upon more frequently in recent times. This agreement is equivalent to a stop-loss cover on the retained loss amount. For the Poisson-lognormal model the corresponding stop-loss net premium is calculated using various methods (normal power, translated gamma, various discretisations) and the methods are compared. Finally, the influence of the model parameters is examined and it is demonstrated how a variety of parameter value combinations can be reduced to only a few rating curves.

A premium calculation principle π is called positively homogeneous if π(cX) = cπ(X) for all c > 0 and all random variables X. For all known principles it is shown that this condition is fulfilled if it is satisfied for two specific values of c only, say c = 2 and c = 3, and for only all two point random variables X. In the case of the Esscher principle one value of c suffices. In short this means that local homogeneity implies global homogeneity. From this it follows that in the case of the zero utility principle or Swiss premium calculation principle, the underlying utility function is of a very specific type.

A very general theorem on premium calculation principles which satisfy a weak continuity condition, is added. Among others the proof uses Kroneckers Theorem on Diophantine Approximations.

We consider a risk model in which the claim inter-arrivals and amounts depend on a markovian environment process. Semi-Markov risk models are so introduced in a quite natural way. We derive some quantities of interest for the risk process and obtain a necessary and sufficient condition for the fairness of the risk (positive asymptotic non-ruin probabilities). These probabilities are explicitly calculated in a particular case (two possible states for the environment, exponential claim amounts distributions).

Much of the risk theory literature deals with the total claims distribution where pk = the probability of k claims and S(x) is the distribution function of severity. Both pk and S(x) are univariate probability distributions. Thus, F(x) can be interpreted as a model of claims from one class of policies or as an aggregate model where pk and S(x) represent mixed probability distributions from a heterogeneous portfolio of policies. An alternative approach to modelling total claims in the latter case would be to recognize explicitly that total claims are the result of the interaction of multivariate processes. In the most general case, total claims arise from a multivariate accident process where each accident triggers multivariate claims frequency and severity processes.

The purpose of this article is to present a multivariate model of total claims and to develop the cumulant generating function of this distribution. Such a model is superior to the traditional model in two respects: (1) It permits explicit recognition of shifts in the overall portfolio composition. Applications of the traditional model, in contrast, rely on the assumption that the portfolio composition is relatively constant over time. (2) It facilitates the evaluation of the effects of reinsurance on the total claims distribution when the reinsurance arrangements are not the same in different segments of the portfolio.

In a previous paper, Jewell and Sundt showed how to approximate a distribution of total losses from a large, fixed, heterogeneous portfolio, using a recursive algorithm developed by Panjer for the distribution of a random sum of random variables (a single casualty contract). This paper extends the approximation procedure to large, dynamic heterogeneous portfolios, in order to model either a portfolio of correlated casualty contracts, or a future portfolio, whose composition is not known with certainty.

Probabilities of ruin are solutions of differential or integrodifferential equations. Solving such equations numerically can be performed by means of approximate quadrature formulae for the convolution part of the equation. In this contribution it is shown how applicable recursion formulae, giving results within a prescribed tolerance level, can be obtained. Some numerical results are displayed.

The allocation of operating costs among the lines of an insurance company is one'of the toughest problems of accounting; it is first shown that most of the methods used by the accountants fail to satisfy some natural requirements. Next it is proved that a cost allocation problem is identical to the determination of the value of a cooperative game with transferable utilities, and 4 new accounting methods that originate from game theory are proposed. One of those methods, the proportional nucleus, is recommended, due to its properties. Several practical examples are discussed throughout the paper.

Actuaries have always been in search of ways to determine premiums which match the risks insured as closely as possible. They do this by differentiating between them on the basis of observable risk factors. In practice, many examples of such risk factors are being used: age and sex for life insurance; location, type of building etc. for fire insurance. Motor insurance is perhaps the most characteristic branch with respect to this phenomenon: in tariffs we find factors like weight, price or cylinder capacity of the car, age of the driver, area of residence, past claims experience (Bonus/Malus), annual mileage etc.

Outsiders may not always be very positive about such a refined premium differentiation. The basis of insurance, they say, should be solidarity among insureds; premium differentiation is basically opposed to this. Another statement heard in the field is: “Premium differentiation ultimately results in letting every individual pay his own claims, it is the end of insurance”.

Much confusion arises during discussions about this subject, especially between actuaries and non-actuaries. We will therefore first give a mathematical definition of solidarity, (Section 2), followed by a brief description of certain trends in society which might bring insurers to deliberately drop certain risk factors from their tariffs in order to increase solidarity (Section 3). The consequences of doing so are examined and it is shown that increased solvency requirements will in the end prove to be ineffective. A possible solution is a voluntary transfer of premium between companies (Section 4). The situation is illustrated by an example of health insurance in the Netherlands, where proposals to arrive at such transfeis are presently being discussed.

In non-life insurance, it is nearly always assumed that the expense loading is a fraction of the risk premium. This may deeply affect the fairness of a tariff, as illustrated in the case of the Belgian bonus-malus system.

French insurance companies usually classify their agents according to their results by branch and to this effect they calculate their respective claim ratios.

As regards motor insurance many agents have argued against this practice as they believed it was unfair to be adversely classified when they had the misfortune to incur a large claim.

In this article, the merits of various statistical systems for attenuating the effects of large claims on the portfolio are considered in order to come to the most equitable solution possible, i.e., to ascertain the best method and system for attenuating the effects of large claims. The conclusion is that the excess of loss technique is best suited to this effect.

To arrive at this conclusion the author of this article relates in detail the various stages of the study, the various systems envisaged and the tests which have allowed to select the most appropriate system. The main results obtained are also given together with their respective drawbacks.

At the end of the article the author mentions the still fairly limited bibliography which deals with this question.

Dans le but d'améliorer les résultats, les compagnies d'assurance ont l'habitude en France de publier un classement de leurs Agents Généraux basé sur le ratio S/P (sinistres/primes) et de tenir compte de ce classement pour l'attribution de certains avantages commerciaux.

Or, en assurance automobile, un malaise profond s'était emparé d'un certain nombre d'entre eux, et ils avaient mis l'accent sur un point qu'ils estimaient injuste:

“Nous ne pouvons pas être pénalisés” disaient-ils, “si un très gros sinistre, atteignant 500 ou 1000 fois la prime annuelle, frappe un de nos clients, car il s'agit alors d'un phénomène aléatoire dont nous ne sommes pas responsables”.

Telle est l'origine de la politique de l'écrêtement.

Planning is, or ought to be, an ever-developing process in which virtually each member of the company has to be involved. Planning without controlling, i.e., without feedback, planning on its own, is useless.

In the first part of the present note, general aspects of planning are briefly described inasfar as they are relevant to possible treatment by actuarial methods.

The circles in the above figure represent what could be called the hardware of planning consisting of three sets of figures, namely

• Actual figures describing the most recent reality. Most of these figures are part of what is usually called the Earnings Statement (EST).

• Forecasts for the near future, say the next three years. Most of these figures are contained in the Planning Budget (PLB).

• Control figures or signals in the sense of a “bread-line” expressing, for example, how much the company should earn as a minimum in order to remain self-financing. Such figures are calculated on the basis of so-called Return on Equity considerations and they are found in a corresponding ROE-document.

By software we mean everything done with the hardware described above, i.e., primarily the comparison of figures from different domains of the hardware on different levels within the organisation of the company, the analysis, for example, of deviations between actual and planned, the conclusions to be taken from such analyses and the planning of new actions as a consequence.

In the above graph, circles symbolize the hardware and arrows indicate the software.

Also non-numeric planning instruments (e.g., project planning, action plans, assessment and decisions) are considered to be part of the planning software.