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.

This paper develops a discrete time model for valuing treasury bills and either forward or futures contracts written against them. It provides formulae for bill prices, forward prices, futures prices, and their conditional variances and risk premiums. The interest rate process is described by a multiplicative binomial random walk whose features conform to some principal characteristics of observed processes. Initial forward rates are constrained to match initially observed term structure data.

Earthquakes and Windstorm can range in size from small easily insurable instances up to full scale Natural Disasters. This note is particularly concerned with the problem of forecasting, rating and insuring earthquakes and windstorm at the natural disaster end of this scale.

Before we tackle the peculiar problems of natural disaster insurance it is worth having a look at why insurance works so well in practice.

Is it because actuaries and statisticians have precisely measured the statistical risks involved, and have developed a detailed and complex mathematical approach to insurance, a true “Technical Basis” for Insurance? Or is it really quite simple?

A quotation (the first of many) from R. Heller [1] is relevant.

“The great and fabled business empires with hardly an exception were built… on irresistable ideas of elemental simplicity.”

Insurance is one of these great empires, and some of the simple ideas which may explain why insurance works so well are suggested below.

Age and period are the most widely used parameters for forecasting mortality rates. Empirical mortality rate differences in multiple populations often show strong geometric patterns on the two-dimensional age–period plane. The idea of this paper is to take these geometric patterns as the starting point for the development of forecasts. A parametric approach is presented and discussed which uses simple techniques from spatial statistics. The proposed model is statistically parsimonious and yields forecasts that are consistent with the historical data and coherent for multiple populations.

Customer churn, which insurance companies use to describe the non-renewal of existing customers, is a widespread and expensive problem in general insurance, particularly because contracts are usually short-term and are renewed periodically. Traditionally, customer churn analyses have employed models which utilise only a binary outcome (churn or not churn) in one period. However, real business relationships are multi-period, and policyholders may reside and transition between a wider range of states beyond that of the simply churn/not churn throughout this relationship. To better encapsulate the richness of policyholder behaviours through time, we propose multi-state customer churn analysis, which aims to model behaviour over a larger number of states (defined by different combinations of insurance coverage taken) and across multiple periods (thereby making use of readily available longitudinal data). Using multinomial logistic regression (MLR) with a second-order Markov assumption, we demonstrate how multi-state customer churn analysis offers deeper insights into how a policyholder’s transition history is associated with their decision making, whether that be to retain the current set of policies, churn, or add/drop a coverage. Applying this model to commercial insurance data from the Wisconsin Local Government Property Insurance Fund, we illustrate how transition probabilities between states are affected by differing sets of explanatory variables and that a multi-state analysis can potentially offer stronger predictive performance and more accurate calculations of customer lifetime value (say), compared to the traditional customer churn analysis techniques.

We present novel cross-sectional and longitudinal claim count models for vehicle insurance built upon the combinedd actuarial neural network (CANN) framework proposed by Wüthrich and Merz. The CANN approach combines a classical actuarial model, such as a generalized linear model, with a neural network. This blending of models results in a two-component model comprising a classical regression model and a neural network part. The CANN model leverages the strengths of both components, providing a solid foundation and interpretability from the classical model while harnessing the flexibility and capacity to capture intricate relationships and interactions offered by the neural network. In our proposed models, we use well-known log-linear claim count regression models for the classical regression part and a multilayer perceptron (MLP) for the neural network part. The MLP part is used to process telematics car driving data given as a vector characterizing the driving behavior of each insured driver. In addition to the Poisson and negative binomial distributions for cross-sectional data, we propose a procedure for training our CANN model with a multivariate negative binomial specification. By doing so, we introduce a longitudinal model that accounts for the dependence between contracts from the same insured. Our results reveal that the CANN models exhibit superior performance compared to log-linear models that rely on manually engineered telematics features.

Le “Contrôle actuariel dans la comptabilité des branches élémentaires” constitue le 2e thème de ce colloque, tel que l'annonce le programme officiel. Ce thème comprend une ière partie intitulée: “Problèmes relatifs aux sinistres”, qui est le sujet de cette séance.

Les notes préliminaires de notre Président, parues dans notre dernier bulletin, suggéraient d'orienter les travaux vers les applications pratiques de l'important travail théorique déjà réalisé dans le cadre d'Astin. Plus particulièrement en ce qui concerne les sinistres dans les assurances de responsabilité civile, ces notes attiraient l'attention sur une certaine analogie que la longue dureé de règlement des sinistres pouvait créer entre l'estimation des réserves pour sinistres à régler et l'évaluation des réserves mathématiques en branche Vie et elles soulevaient dès lors assez naturellement la question des méthodes d'estimation collective. En fait, les communications qui ont été rattachèes à cette rère partie du 2e thème interprètent le sujet de manière extrêmement large. Ne voyez dans cette réflexion initiale nul reproche aux auteurs, mais la simple constatation d'un fait dont mon rapport introductif doit tenir compte: il ne sera pas une synthese dans les directions proposées par notre Président.

Une partie des communications se rattache au fond à la théiorie du risque. Pour mieux dégager l'apport de ces notes, je me propose de le dépouiller autant que possible de son aspect purement mathématique et de le situer dans l'évolution des idées et des recherches en la matière.

J'indiquerai ensuite l'apport des communications au sujet de l'estimation des réserves pour sinistres à régler.

Enfin, je résumerai les travaux qui se rapportent à la tarification du risque automobile.

In January 1997, Winterthur Insurance, together with Credit Suisse First Boston (CSFB), issued the first listed CAT bond. The annual “WINCAT0 coupons of this three-year convertible bond are knocked out if any single storm event damages more than 6,000 vehicles insured by Winterthur Insurance in Switzerland.

This was a completely new way of securing insurance risks. The main intention was to test the Swiss capital market for such products and to make investors acquainted with them. Thereby Winterthur, together with CSFB, set new standards in product transparency, fairness of pricing and investor education by making the historical data available via internet and by publishing a special brochure (CSFB (1997)), where the pricing and the mathematical modelling are described in detail. This is also a prerequisite to enable a scientific discussion on pricing aspects of such new financial products. The developers of the bond are therefore grateful to Mr. Schmock for this valuable scientific contribution which can be seen as a thorough and profound statistical analysis on the knock-out probability for the purpose of quantifying the model uncertainty.

In Section 2 we briefly summarize the whole pricing of the bond at the issue date and show that there were several risk premium elements in this pricing where the conservative estimation of the knock-out probability was just one of them. In Section 3 we consider the modelling of low frequency risks from a practitioner's standpoint and formulate some requirements from practice. In Section 4 we make some further comments on the modelling of the Wincat data. Section 5 is a short summary.

We start by describing how, in some cases, we can use variance-related premium principles in ratemaking, when the claim numbers and individual claim amounts are independent. We use quasi-likelihood generalized linear models, under the assumption that the variance function is a power function of the mean of the underlying random variable. We extend this approach to the cases where the claim numbers are correlated. Some alternatives to deal with dependent risks are presented, taking explicitly into account overdispersion. We present regression models covering the bivariate Poisson, the generalized bivariate negative binomial and the bivariate Poisson–Laguerre polynomial, which nest the bivariate negative binomial. We apply these models to a portfolio of the Portuguese insurance company Tranquilidade and compare the results obtained.

Estimation of rates of onset of rare, late-onset dominantly inherited genetic disorders is complicated by: (a) probable ascertainment bias resulting from the ‘recruitment’ of strongly affected families into studies; and (b) inability to identify the true ‘at risk’ population of mutation carriers. To deal with the latter, Gui & Macdonald (2002a) proposed a non-parametric (Nelson-Aalen) estimate (x) of a simple function Λ(x) of the rate of onset at age x. The function Λ(x) had a finite bound, which was an increasing function of the probability p that a child of an affected parent inherits the mutation and σ the life-time penetrance. However if (x) exceeds this bound, it explodes to infinity, and this can happen at quite low ages. We show that such ‘failure’ may in fact be a useful measure of ascertainment bias. Gui & Macdonald assumed that p = 1/2 and σ = 1, but ascertainment bias means that p > 1/2 and σ ≠ 1 in the sample. The maximum attained by (x) allows us to estimate a range for the product pσ, and therefore the degree of ascertainment bias that may be present, leading to bias-corrected estimates of rates of onset. However, we find that even classical independent censoring, prior to ascertainment, can introduce new bias. We apply these results to early-onset Alzheimer’s disease associated with mutations in the Presenilin-1 gene.

In the context of life insurance with profit participation, the future discretionary benefits (FDB), which are a central item for Solvency II reporting, are generally calculated by computationally expensive Monte Carlo algorithms. We derive analytic formulas to estimate lower and upper bounds for the FDB. This yields an estimation interval for the FDB, and the average of lower and upper bound is a simple estimator. These formulae are designed for real world applications, and we compare the results to publicly available reporting data.

Pour construire un tarif “Automobile” il est nécessaire d'étudier la charge des sinistres causés par les véhicules appartenant à des groupes homogènes divers: il est commode de procéder à un échantillonnage.

On peut construire un échantillon où figurent des proportions constantes ou connues de chacun des groupes constituant la population à étudier.

En pratique l'échantillonnage, doit, pour avoir une efficacité admissible, être particulièrement important dans les groupes à faible fréquence pour donner un nombre significatif de sinistres.

Or, comme pour déterminer la méthode de sélection des échantillons on ne connait pas le groupe des véhicules sélectionnés, on serait contraint de prendre un échantillon très important.

En France, le Groupement Technique Accidents fait des études sur la totalité des véhicules assurés par les Compagnies adhérentes à la statistique commune et qui représentent un tiers du Pare National.

L'inconvénient de la méthode actuelle est l'impossibilité d'étudier rapidement un nouveau critère — car il faut attendre les mouvements sur le portefeuille à l'occasion desquels on enregistre le critère à étudier; les mouvements, en outre, frappent probablement moins les véliicules à faible fréquence. Le délai nécessaire pour l'étude d'un nouveau critère se compte en années.

Il peut être souhaitable pour suivre l'évolution économique ou sociale d'étudier rapidement dans un délai chiffré en mois, l'influence de certains critères nouveaux, pour éviter sous la pression d'événements extérieurs d'être amené à. prendre des décisions dites “politiques” — cet euphémisme justifiant, dans le doute, des réductions de tarif explicables uniquement par le fait qu'un concurrent dangereux est peut-être sérieux — à ce moment il est inévitable de procéder à un sondage.

We explicitly calculate price equilibria for power and logarithmic utility functions which—together with the exponential utility functions—form the so-called HARA (Hyperbolic Absolute Risk Aversion) class.

A price equilibrium is economically admissible in the market which is a closed system. Furthermore it is on the one side individually optimal for each participant of the market (in the sense of maximal expected utility), on the other side it is a Pareto optimum and thus collectively optimal for the market as a whole.

The concept of economic equilibrium under uncertainty is applied to a model of insurance market where, in distinction to the classic Borch’s model of a reinsurance market, risk exchanges are allowed between the insurer and each insured only, not among insureds themselves. Conditions characterizing an equilibrium are found. A variant of the conditions, based on the Pareto optimality notion and involving risk aversion functions of the agents, is derived. An existence theorem is proved. Computation of the market premiums and optimal indemnities is illustrated by an example with exponential utility functions.