Haberman and Sung (1994) have presented a dynamic model for a defined benefit occupational pension scheme which considered two types of risk: the “contribution rate” and the “solvency” risk. The current paper, extends this work by deriving optimal funding control procedures for determining the contribution rate for the case of a stochastic model with incomplete state information, making use of the separation principle. The stochastic inputs modelled are the investment returns and the benefit outgo.

Ce rapport n'a pas le but de résumer toute la littérature actuarielle sur les assurances non viagères: on se proposait de tracer le fil conducteur de quelques uns des développements d'idées où l'apport des mathématiques est plus profond et, en particulier, de ceux où il semble qu'il y aura encore à travailler longtemps. A ce compte, seulement quelques étapes pouvaient être explicitement mentionnées dans le texte, tandis qu'une bibliographic complète aurait dû suivre à la fin du rapport.

Maintes circonstances ont restreint le temps à disposition des rapporteurs et ont augmenté les difficultés d'un travail en commun à distance; sans doute quelques contributions ont échappé ou n'ont pas reçu le relief qu'il aurait fallu leur donner. Les auteurs prient de bien vouloir justifier les omissions par ces circonstances, et de les leur signaler.

L'assurance sur la vie et les assurances non viagères semblent présenter des différences essentielles, soit au point de vue de leur objet (indemnité d'une perte dans le deuxième cas, mais non dans le premier), soit au point de vue de leur méthode (théorie mathématique complétement développée dès longtemps pour le premier cas, mais non dans le deuxième).

Ya-t-il néanmoins une notion unique d'assurance, en général, qui comprenne toutes les deux? Est-il possible de concevoir d'une façon unique l'application du calcul des probabilités et des méthodes mathématiques qui le traduisent, dans le champ de l'assurance en général? Ou ya-t-il des limitations, ou des circonstances qui obligent à distinguer à cet effet les deux champs de l'assurance sur la vie et de l'assurance non viagère?

A functional approach is taken for the total claim amount distribution for the individual risk model. Various commonly used approximations for this distribution are considered, including the compound Poisson approximation, the compound binomial approximation, the compound negative binomial approximation and the normal approximation. These are shown to arise as zeroth order approximations in the functional set-up. By taking the derivative of the functional that maps the individual claim distributions onto the total claim amount distribution, new first order approximation formulae are obtained as refinements to the existing approximations. For particular choices of input, these new approximations are simple to calculate. Numerical examples, including the well-known Gerber portfolio, are considered. Corresponding approximations for stop-loss premiums are given.

In this study, we propose new risk measures from a regulator’s perspective on the regulatory capital requirements. The proposed risk measures possess many desired properties, including monotonicity, translation-invariance, positive homogeneity, subadditivity, nonnegative loading, and stop-loss order preserving. The new risk measures not only generalize the existing, well-known risk measures in the literature, including the Dutch, tail value-at-risk (TVaR), and expectile measures, but also provide new approaches to generate feasible and practical coherent risk measures. As examples of the new risk measures, TVaR-type generalized expectiles are investigated in detail. In particular, we present the dual and Kusuoka representations of the TVaR-type generalized expectiles and discuss their robustness with respect to the Wasserstein distance.

It is often found even today in Europe that for certain statistical investigations the conclusion is drawn that the extent of the available statistical data is not sufficient. Going to the root of this pretention, however, we notice that there is a want of clear conception about the extent that is in fact necessary in order that a valid conclusion may with greater probability be arrived at. This, for instance, is the case when obvious tariff reductions are shirked from by entrenching oneself behind the law of large numbers, which by its very nature can in actual practice be never accomplished in its inherent sense.

Apart from a proper understanding of the limits within which a set of statistical data may subject to certain assumptions be ascribed full measure of credence, there is further a lack of the necessary tools that would permit, on the basis of ascertainable values alone, far-reaching conclusions to be drawn or a maximum of useful information to be gathered from an investigation of which the scope is evidently not sufficient.

Credibility Theory, of which Prof. Mowbray [4] may be regarded as the initiator, was evolved in the U.S.A. about 50 years ago to fill this lacunae. The development of Credibility Theory may be considered as one of the most significant contributions of American actuarial science, and it is frankly astonishing that apart from certain specific realisations in the collective risk theory which are fairly closely related to Credibility Theory, it is only in recent years that this interesting topic has met with the required attention in Europe.

In many risk theoretical questions a central problem is the numerical evaluation of a convolution integral and much effort has been devoted over the years to mathematical and computational aspects. The paper presented to this colloquium by O. Thorin shows the subject to be topical but the present note stems from a recent paper by H. L. Seal, “Simulation of the ruin potential of non-life insurance companies”, published in the Transactions of the Society of Actuaries, Volume XXI page 563.

In this paper, amongst other topics, Seal has presented some simulations of ruin probabilities over a finite time interval. Some years ago (Journal Institute of Actuaries Students' Society, Volume 15, 1959). I pointed out that the form of ruin probability could be expressed as a successive product of values of a distribution function. To my knowledge no one has attempted to see if this product form was capable of development and the numerical values in Seal's paper prompted me to spend a little time on the problem. In the time I had available it has not been possible to do more than experiment, but the conclusions reached may be of interest to other workers in this field. They showed that calculation is feasible but laborious. However, the knowledge that it can be done may suggest methods of improving the techniques.

Seal's first simulation example is the calculation of ruin probabilities when the distribution of the interval of time between the claims is negative exponential and the individual claim distribution is also negative exponential. Instead of following Seal's method we can determine the “gain per interval” and find that if λ is the security loading the frequency function for the gain z is

i.e. a Laplace distribution.

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.

Dans plusieurs études [1] Ammeter a démontré que la distribution binomiale négative généralisée peut être interprétée de différentes manières, à savoir comme

a. une contagion des probabilités (Pólya-Eggenberger);

b. une distribution de Poisson basée sur des probabilités fondamentales variables;

c. une distribution de Poisson où la probabilité fondamentale de sinistres n'est pas suffisamment connue;

d. une distribution de Poisson par rapport aux événements, où un seul événement peut comporter plusieurs prétentions de sinistres;

e. une distribution de Poisson basée sur une modification de la répartition des montants de sinistres.

On a, en effet, les relations suivantes:

a. la contagion des probabilités

où F(x, P, h, p(z)) représente la probabilité que le sinistre est au plus égal à x dans un intervalle de temps où le sinistre probable est P. x et P y sont exprimés en le sinistre moyen par événement. p(z)dz est la fonction de fréquence du sinistre par événement.

Nous posons est la convolution de p(z) avec p(z), faite r fois, tandis que h est la notation du paramètre qui figure dans la distribution binomiale négative. Si h est égal à ∞ nous obtenons comme fonction de répartition pour le nombre d'événements survenus la distribution de Poisson.

In this paper an attempt is made to find an answer to the question, “What is the most advantageous size for the retention limit of a risk portfolio, given the fact that a certain stability requirement is to be satisfied?”

This problem will be approached from the viewpoint of an insurer who wishes to obtain a certain degree of stability at lowest cost.

It is assumed that in his choice of reinsurance methods, the insurer restricts himself to either a surplus treaty, a stop loss treaty or a combination of both these types.

Moreover it is assumed that “stability” can be adequately measured by the variance of the risks retained for own account.

We start to consider a reinsurance policy based on the surplus system where the amount of risk in excess of a retention limit u is ceded.

By thus limiting the potential loss on each risk individually, the variance is kept at a certain level, but at the expense of an amount of premium payable to a reinsurer.

The insurer could, of course, reduce the reinsurance cost by increasing his retention but he then is bound to incur a higher variance in his portfolio, which would mean a loss of stability.

One might ask, however, whether a suitably chosen stop loss coverage could bring the variance down again to the proper level at lesser cost than the profit obtained by taking a higher retention. A reduction in reinsurance cost would then have been effected.

The question leads to an optimization problem, which in a more general setting, has been discussed by K. Borch.

Il est un fait bien connu que la technique des assurances, qu'elles soient privées ou sociales, qui versent une indemnité en cas de maladie, se base premièrement sur la connaissance des fréquences et des durées moyennes de maladie, ou bien des coefficients de morbidité. Pour résoudre des problèmes particuliers, des instruments plus raffinés sont toutefois nécessaires, savoir:

a) la distribution des assurés d'après le nombre de cas dont ils ont été frappés au cours d'une année;

b) la distribution des cas d'après la durée en jours;

c) la distribution des assurés d'après les jours de maladie dont ils ont été frappés au cours d'une annèe.

Nous avons eu la possibilité d'elaborer les fiches de 44.829 travailleurs salariés (36.134 hommes et 8.695 femmes âgés de 15 à 65 ans) résidant à Rome et assurés obligatoirement auprès de l'Institut National d'assurance-maladie (Istituto Nazionale per l'assicurazione contro le malattie — INAM), et de relever les cas de maladie qui les ont frappés au cours de l'année 1960, en même temps que les durées correspondantes.).

Puisqu'il s'agit, comme on le verra, de données très détaillées, il nous a paru utile d'étudier les caractéristiques principales des trois distributions en question et les relations existant entre elles: l'exposé de certains résultats obtenus dans ce sens constitue l' objet de la présente note.

We propose a new approach to mortality prediction under survival energy hypothesis (SEH). We assume that a human is born with initial energy, which changes stochastically in time and the human dies when the energy vanishes. Then, the time of death is represented by the first hitting time of the survival energy (SE) process to zero. This study assumes that SE follows a time-inhomogeneous diffusion process and defines the mortality function, which is the first hitting time distribution function of the SE process. Although SEH is a fictitious construct, we illustrate that this assumption has the potential to yield a good parametric family of cumulative probability of death, and the parametric family yields surprisingly good predictions for future mortality rates.

A practical method is developed for computing moments of insurance functions when interest rates are assumed to follow an autoregressive integrated moving average process.

A variable annuity is a modern life insurance product that offers its policyholders participation in investment with various guarantees. To address the computational challenge of valuing large portfolios of variable annuity contracts, several data mining frameworks based on statistical learning have been proposed in the past decade. Existing methods utilize regression modeling to predict the market value of most contracts. Despite the efficiency of those methods, a regression model fitted to a small amount of data produces substantial prediction errors, and thus, it is challenging to rely on existing frameworks when highly accurate valuation results are desired or required. In this paper, we propose a novel hybrid framework that effectively chooses and assesses easy-to-predict contracts using the random forest model while leaving hard-to-predict contracts for the Monte Carlo simulation. The effectiveness of the hybrid approach is illustrated with an experimental study.

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 cost of adverse selection in the life and critical illness (CI) insurance markets, brought about by restrictions on insurers' use of genetic test information, has been studied for a variety of rare single-gene disorders. Only now do we have a study of a common disorder (breast cancer) that accounts for the risk associated with multiple genes. Such a collection of genes is called a polygene. We take two approaches to modelling the severity of adverse selection which may result from insurers being unable to take account of tests for polygenes as well as major genes. First, we look at several genetic testing scenarios, with a corresponding range of possible insurance-buying behaviours, in a market model for CI insurance. Because a relatively large proportion of the population is exposed to adverse polygenic risk, the costs of adverse selection are potentially much greater than have been associated with rare single genes. Second, we use utility models to map out when adverse selection will appear, and which risk groups will cause it. Levels of risk aversion consistent with some empirical studies do not lead to significant adverse selection in our model, but lower levels of risk aversion could effectively eliminate the market.

Cash balance pension plans with crediting rates linked to long bond yields are relatively common in the United States, but their liabilities are proving very challenging to hedge. In this paper, we consider dynamic hedge strategies using the one-factor and two-factor Hull White models, based on results for the liability valuation from Hardy et al. (2014). The strategies utilise simple hedge portfolios combining one or two zero-coupon bonds, and a money market account. We assess the effectiveness of the strategies by considering how accurately each one would have hedged a 5-year cash balance liability through the past 20 years, using real-world returns and crediting rates, and assuming parameters calibrated using the information available at the time. We show that there is considerable impact of model and parameter uncertainty, with additional, less severe impact from discrete hedging error and transactions costs. Despite this, the dynamic hedge strategies do manage to stabilize surplus substantially, even through the turbulence of the past two decades.

Claims reserving models are usually based on data recorded in run-off tables, according to the origin and the development years of the payments. The amounts on the same diagonal are paid in the same calendar year and are influenced by some common effects, for example, claims inflation, that can induce dependence among payments. We introduce hierarchical generalized linear models (HGLM) with risk parameters related to the origin and the calendar years, in order to model the dependence among payments of both the same origin year and the same calendar year. Besides the random effects, the linear predictor also includes fixed effects. All the parameters are estimated within the model by the h-likelihood approach. The prediction for the outstanding claims and an approximate formula to evaluate the mean square error of prediction are obtained. Moreover, a parametric bootstrap procedure is delineated to get an estimate of the predictive distribution of the outstanding claims. A Poisson-gamma HGLM with origin and calendar year effects is studied extensively and a numerical example is provided. We find that the estimates of the correlations can be significant for payments in the same calendar year and that the inclusion of calendar effects can determine a remarkable impact on the prediction uncertainty.

Principal component analysis is employed to construct a new formula defining ‘sports cars’, a classification variable commonly used by Belgian insurers in motor insurance. Five hundred and eighty-one different car models were used in the design of the formula. It is based solely on the cars' technical characteristics and hence does not rely on the subjective opinion of experts; the resulting classification is independent of units of measurement employed. Thus the definition is suitable for application world-wide.

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

Information relating to the expected number of losses is of importance in automobile insurance systems. The distribution of risks by number of losses per year may be based on the following model

with λ representing the average number of losses per year. This distribution is the Poisson distribution. Tests of this model versus actual observations often indicate significant deviation. This discrepency can result from the constancy of λ which makes the model appropriate for an individual but would require an isohazardous population when applied to a group of individuals. In reality, however, λ will vary from individual to individual. A model accounting for this spread in λ is given in

where z(λ) is a distribution describing the spread of λ. The results of model (2) certainly will depend on the form of z(λ). It has been hypothesized that z(λ) can be represented by (3)

which is a Pearson Type III [1, 5, 7]. With this assumption model (2) becomes the negative binominal distribution with a mean of and a variance of

If the observed mean is and the observed variance it is possible to determine a and b by solving the above equations for mean and variance. Thus and

The results indicate an improved fit to actual observations [1, 5, 6].