The determination of optimal rules for sharing risks and constructing reinsurance treaties has important practical and theoretical interest. Medolaghi, de Finetti, and Ottaviani developed the first linear reciprocal reinsurance treaties based upon minimizing individual and aggregate variance of risk. Borch then used the economic concept of utility to justify choosing Pareto-optimal forms of risk exchange; in many cases, this leads to familiar linear quota-sharing of total pooled losses, or to stop-loss arrangements. However, this approach does not give a unique, risk-sharing agreement, and may lead to substantial fixed side payments. Gerber showed how to constrain a Pareto-optimal risk exchange to avoid invasion of reserves.

To these ideas, the authors have added the actuarial concept of long-run fairness to each participant in the risk exchange; the result is a unique, Pareto-optimal risk pool, with “quota-sharing-by-layers” of the total losses. There are many interesting special cases, especially when all individual utility functions are of exponential form, giving linear quota-sharing-by-layers. Algorithms and numerical examples are given.

This paper concerns the Borch model of a reinsurance market seen as a model of an economy under uncertainty.

In a market of this type the goods traded are unit coverings contingent to a particular state of nature (n-tuple of claims).

Our idea is to regard the probability of a state of nature as a sort of intrinsic value of the related contingent covering. From this point of view we examine the role of the reinsurance market in modifying values in market equilibrium prices and other questions, related to this classical economic problem, in the particular case of a quadratic utility function for all companies.

We compute a merit-rating system for automobile third party liability insurance by two different ways, both with the help of an exponential utility function.

(i) We apply the principle of zero utility to exponential utilities.

(ii) We break the symmetry between the overcharges and the undercharges by weighting them differently through the introduction of a utility function, in order to penalize the overcharges.

The results are applied to the portfolio of a Belgian company and compared to the premium system provided by the expected value principle.

Deux méthodes différentes, basées sur l'emploi de fonctions d'utilité exponentielles nous permettent de définir un système bonus-malus en assurance automobile:

(i) le principe de l'utilité nulle;

(ii) la pénalisation des injustices de la compagnie, obtenue en pondérant les erreurs de prime au moyen d'une fonction d'utilité de manière à briser la symétrie entre les primes trop élevées et les primes trop basses.

Les résultats théoriques sont appliqués au portefeuille d'une compagnie belge et comparés aux primes fournies par le principe de l'espérance mathématique.

This paper examines the balancing of the investment and liability portfolios of a (re)insurance firm operating in the international market. The model captures two effects which are ignored by traditional analysts.

(a) According to the Interest Parity Theorem, the expected changes in the exchange rates should already be reflected in the expected rate of return on foreign investments. Therefore, an insurer operating in a perfect market should be indifferent to the currency denomination of its financial assets.

(b) A second effect which has often been ignored is related to the additional risk resulting from the unexpected fluctuations of the exchange rates.

The multi-index model suggested in this paper is capable of capturing these effects. The model can be used to examine and analyze alternative policies of the firm operating in international markets. For example, the model can be used to examine whether an insurer should take positions in certain currencies, or rather take a “full-hedge” policy.

The problem of rating the discount for an excess is considered under the condition where premium rates for excess-free policies are calculated according to a multiplicative (or equivalently, points rating) model.

Section 3 obtains some inequalities on the manner in which the discount should vary over the various risk-classes in the portfolio and compares the results with the current practical situation in Australia.

Section 4 derives a formula for rating the discount and gives some reasonable and practical approximations.

Consider a claim size distribution with complementary d.f. H(.). Let E(x) denote the average claim payment under a policy subject to this claim size d.f. but with a deductible of x. That is

where E is the expectation operator and X is the random variable claim size before application of the excess.

It is well-known—see Benktander and Segerdahl (1960, p. 630)—that:

It was shown by them that E(x) is a constant for all x ≥ o if and only if the claim size d.f. is negative exponential:

This property of the negative exponential distribution is closely related to the fact that it is the only distribution with constant failure rate. See Kaufmann (1969, pp. 20-22).

The constancy of average excess claim size with varying deductible can be useful in practice. For example, if the distribution of motor vehicle (property damage) claim sizes can be assumed roughly exponential, which will often be reasonable, then a variation in the deductible will not induce any variation in the average size of claims paid by the insurer, i.e. after application of the deductible. This will be a particularly useful piece of information if for example one is examining trends in average claim size over a period during which a change in deductible occurred.

The purpose of this statement is to provide general guidelines for the use of actuaries engaged in the establishment and review of loss and loss adjustment expense reserves. It is a statement of the Casualty Actuarial Society's Committee on loss Reserves. The statement consists of three parts.

1. Definitions

2. Considerations

3. Procedures

The balance sheet of the Fire and Casualty Annual Statement includes liabilities that are not subject to precise valuation. The reserves for unpaid losses (line 1, page 3 of the Annual Statement) and the reserve for unpaid loss adjustment expense (line 2, page 3 of the Annual Statement) cannot be precisely determined in advance. These reserves must be estimated. Because of their relative size and the difficulty in achieving accurate estimates of their values, these liabilities are vitally important balance sheet items. It is important that proper actuarial and statistical procedures be employed in order to improve the likelihood of reliable reserve estimates. Without reliable reserve estimates, an accurate evaluation of the financial condition of a fire and casualty insurer cannot be accomplished.

Loss reserving involves the current financial evaluation of costs associated with future contingent events, a matter of fundamental interest to actuaries. The contingencies involved are those factors which affect the cost of future payments on insured events which have already occurred. As ratemaking is another application of the same estimating process, the actuarial methodology is similar in many respects.

The definitions in the next section apply to both loss reserves and loss adjustment expense reserves.

For the purpose of this statement the terms “loss” and “claim” will be used interchangeably.

A numerical procedure is described to evaluate the stop-loss premium in case the risk process is a compound Poisson process. The method is mainly based on an algorithm of R. Piessens and M. Branders for the numerical evaluation of Fourier transforms.

Gerber (1974) has given a characterization of the exponential utility function by proving the fact that a premium calculation principle of zero utility is iterative iff the underlying utility function is linear or exponential. In the present note we prove the more general result that the premium calculation principle introduced by Bühlmann, Gagliardi, Gerber and Straub (1977) is iterative iff the underlying function v is linear or exponential or when the principle is a mean value principle.

It has been known for generations that the fire risk rate increases with the size of the insured object in a similar way as the death rate increases with the age.

Professor d'Addario and other Italian mathematicians have shown that statistical data often can be graduated by the formulas where S denotes the sum insured:

In 1940, d'Addario) in a practical case found the values α = 0.78, β = 0.44. In 1956, Blandin and Depoid have used the same formulas. Although these formulas in practical cases often lead to good approximations, one can hardly say that a proper mathematical or physical model describing the behaviour of fires lies behind.

About 1950, I worked with the statistics of a group of Danish fire insurance companies, in particular covering farm buildings. This investigation was organized by Gunnar Benktander. Our data confirmed the increase of the fire risk rate with the size of the buildings.

In one special group: Farm houses with thatched roofs, this increase was so important that the risk premium was approximately proportionate to the square of the sum insured. The statistics fully justified the tariffing and we tried to construct a model describing and explaining the observed facts.

The risk group in question was characterized by the overwhelming importance of total or practically total losses. Only a few per cent of the damages went to minor fires. When a fire breaks out in such a building, and reaches a certain slight extent, then it is not possible to save the building from total destruction. With sufficient approximation, we may say that only total losses occur (as in life assurance).