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.

This paper was written in connection with the preparation of Marché Commun regulations in the insurance sector and has been submitted to the Commission Technique pour l'étude d'un indice de solvabilité relatif aux entreprises d'assurances contre les dommages. It aims at explaining the scope of the problem to non-mathematicians and for that reason emphasizes its logical in contradistinction to its computational aspects.

The probability of the insurer's ruin has two aspects. The occurrence of the event ruin may be considered with respect to a fixed period but also with respect to a period of undetermined length. In both cases the period starts at a moment at which the insurer's capital (patrimoine) is known and it is intuitively clear that in both cases the probability of ruin will be the higher as the insurer's capital is smaller and the risk to which he is exposed heavier.

For some purposes more precise conclusions are required. This requirement gives rise to problems which will be considered here with respect to the probability of the occurrence of ruin in a period of undetermined length. It will be taken for granted that besides the artificial events occurring in games of chance there are other classes of uncertain events to which numerical probabilities can be assigned and that, as far as claims are concerned, the insurer's losses belong to one of the said classes. On this understanding it makes sense to consider a numerical probability of ruin depending on a fixed initial capital, random losses to which numerical probabilities are assigned, and other profits and losses.

1. At an earlier ASTIN Colloquium participants were invited to present notes on problems which they considered as important but unsolved. There was little response to this invitation, presumably because a problem, once it is well formulated, is almost solved.

In this Note I do not present any new problems. In stead I try to outline a framework which may be useful for analysing different risk problems and seeing them in their proper perspective. In my view, a framework of this kind is urgently needed to place today's actuarial work on a sound foundation.

2. In general an insurance contract will define two stochastic processes. We lose little by assuming that the processes are discrete, and describing them in the following manner:

(i) The payment process: x0, x1 … xt …, where xt is the amount which the company pays to settle claims in period t, or at time t.

(ii) The premium process: p0, p1 … pt …, where pt is the premium which the company receives in period t, or at time t.

If the contract is concluded at time t = o, the Principle of Equivalence requires that

For the typical short-term contract with premium payable in advance (i) will reduce to

3. For a long-term insurance contract one usually requires that the inequality

shall hold for all τ. This means that the company must never be a net creditor of its customer.

Cordeiro (2002a) has presented a multiple state model for Income Protection (formerly known as Permanent Health Insurance) which enables us to analyse claims by cause of disability. In that paper average claim durations conditioned on recovery have been calculated.

Since, when an Income Protection claim is reported, the insurance company never knows whether it will end in recovery, in death or in expiry, to analyse only average claim durations conditioned on recovery could be misleading, specially for the people responsible for the claims control process.

The main purpose of this paper is to calculate average claim durations conditioned on death and average claim durations not conditioned on any particular mode of claim termination. We calculate these claim durations for different deferred periods, causes of disability and ages at the beginning of sickness and we analyse 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.

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.

Variable annuities are products offered by pension funds and life offices that provide periodic future payments to the investor and often have ancillary benefits that guarantee survival benefits or sums insured on death. This paper extends the benchmark approach to value and hedge long-dated variable annuities using a combination of cash, bonds and equities under a variety of market models, allowing for dependence between financial and insurance markets. Under a simplified case of independence, the results show that when the discounted index is modelled as a time-transformed squared Bessel process, less-expensive valuation and reserving is achieved regardless of the short rate model or the mortality model.

The paper proposes a three parameter measure by which the risk of a portfolio can be assessed. The parameters are: the probability of ruin, the severity of ruin (i.e. the amount of the deficit when ruin occurs) and the time of ruin. This type of analysis does not lend itself to closed form solutions, but it can be easily carried out on a PC. The author presents some theoretical and practical examples.

1. Risks of a catastrophic nature, such as war or earthquake, can differ, in a statistical sense, from other ordinarily insured perils only in respect of the distribution of the probability function. For example, although the probability of a loss occurring may be small, its magnitude once it occurs may be very large. One reason for the latter feature is that the sum insured may itself be heavy. But still more important is the fact that there would be strong positive correlation between similar contiguously situated risks. This is the main reason for the loss taking on a catastrophic character.

2. The fact that the probability of a loss is ordinarily small, leads to the result that the expected loss and hence the pure premium for the risk is small. Equally, and for the same reason, the “loss strain at risk” is large. If the latter could be considerably reduced—which can only be done by charging impracticably high premiums—the risk becomes more easily insurable.

3. If there are n risks of this nature, 1, 2, 3 … n and νi(x) is the conditional frequency function of the ith risk we may write

While we may imagine this as a frequency curve, the complementary probability Ϸi of a loss not occurring can be represented as a vertical line at the origin—the different methods of representation of Ϸ and q are actually convenient. With this representation, the expansion of the binomial product

shows that the convolution of the n curves can be expressed as the sum of n + I functions

where

(ii) the term A1(x) comprises n separate curves of the form νi(x) each multiplied by its complementary probability, being of the form

(iii) the term A2(x) comprises terms each consisting of the convolution of the curves taken two at a time and multiplied by the complementary probability, being of the form

and so on.

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.

In a simple stationary setting with constant interest rate, we derive pricing formulas for defaultable bonds with stochastic recovery rate using a replication argument. Replication is done by using an insurance contract (i.e. a kind of credit default swap), the price of which is determined by a dynamic premium calculation principle. We consider two cases, a linear one, where pricing amounts to solving an inhomogeneous linear ODE, and a super-linear case where a Riccati ODE has to be solved.

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.

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.

In this paper the dynamic portfolio selection problem is studied for the first time in a dual utility theory framework. The Wang transform is used as distortion function and well diversified optimal portfolios result both with and without short sales allowed.

Quadratic programming means maximizing or minimizing a quadratic function of one or more variables subject to linear restrictions i.e. linear equations and/or inequalities.

Among the numerous insurance problems which can be formulated as quadratic programs we shall only discuss four, namely the Credibility, Retention, IBNR and the Cost Distribution problems.

Generally, there is no explicite solution to quadratic optimization problems, only statements about the existence of a solution can be made or some algorithm may be recommended in order to get exact or approximate numerical solutions. Restricting ourselves to typical problems of the above mentioned type, however, enables us to give an explicit solution in terms of general formulae for quite a number of cases, such as the onedimensional credibility problem, the retention problem and—under relatively week assumptions— for the IBNR-problem.

The results given here are by no means new. The only goal of this paper is to describe a few fundamental insurance problems from a common mathematical standpoint, namely that of quadratic programming and at the same time, to draw attention to a few special aspects and open questions in this field.