Insurance companies and pension funds must value liabilities using mortality rates that are appropriate for their portfolio. These can only be estimated in a reliable way from a sufficiently large historical dataset for such portfolios, which is often not available. We overcome this problem by introducing a model to estimate portfolio-specific mortality simultaneously with population mortality. By using a Bayesian framework, we automatically generate the appropriate weighting for the limited statistical information in a given portfolio and the more extensive information that is available for the whole population. This allows us to separate parameter uncertainty from uncertainty due to the randomness in individual deaths for a given realization of mortality rates. When we apply our method to a dataset of assured lives in England and Wales, we find that different prior specifications for the portfolio-specific factors lead to significantly different posterior distributions for hazard rates. However, in short-term predictive distributions for future numbers of deaths, individual mortality risk turns out to be more important than parameter uncertainty in the portfolio-specific factors, both for large and for small portfolios.

A method of inverting the Laplace transform based on the integration between zeros technique and a simple acceleration algorithm is presented. This approach was designed to approximate ultimate ruin probabilities for Γ-convolutions claim sizes, but it can be also used with other distributions. The stable algorithm obtained yields interval approximations (lower and upper bounds) to any desired degree of accuracy even for very large values of u (1,000,000), initial reserves, without increasing the number of computations. This last fact can be considered an interesting property compared with other recursive methods previously used in actuarial literature or other methods of inverting Laplace transforms.

In this paper we use Fourier/Laplace transforms to evaluate numerically relevant probabilities in ruin theory as an application to insurance. The transform of a function is split in two: the real and the imaginary parts. We use an inversion formula based on the real part only, to get the original function.

By using an appropriate algorithm to compute integrals and making use of the properties of these transforms we are able to compute numerically important quantities either in classical or non-classical ruin theory. As far as the classical model is concerned the problems considered have been widely studied. In what concerns the non-classical model, in particular models based on more general renewal risk processes, there is still a long way to go. In either case the approach presented is an easy method giving good approximations for reasonable values of the initial surplus.

To show this we compute numerically ruin probabilities in the classical model and in a renewal risk process in which claim inter-arrival times have an Erlang(2) distribution and compare to exact figures where available. We also consider the computation of the probability and severity of ruin in the classical model.

The importance of funded private or occupational old-age provision is expected to increase due to demographic changes and the resulting problems for government-run pay-as-you-go systems. Clients and advisors therefore need reliable methodologies to match offered products with clients' needs and risk appetite. In Graf et al. (2012), the authors have introduced a methodology based on stochastic modeling to properly assess the risk-return profiles — i.e. the probability distribution of future benefits — of various old-age provision products. In this paper, we additionally consider the impact of inflation on the risk-return profile of old-age provision products. In a model with stochastic interest rates, stochastic inflation and equity returns including stochastic equity volatility, we derive risk-return-profiles for various types of existing unit-linked products with and without embedded guarantees and especially focus on the difference between nominal and real returns. We find that typical “rule of thumb” approximations for considering inflation risk are inappropriate and further show that products that are considered particularly safe by practitioners because of nominal guarantees may bear significant inflation risk. Finally, we propose product designs suitable to reduce inflation risk and investigate their risk-return profile in real terms.

Some comments are given on a recent paper by de Wit and Kastelijn (1980) and alternative methods for analysing loss ratios are proposed in connection with the determination of the necessary solvency margins of non-life insurance companies. The methods are illustrated by a numerical example.

This paper addresses the task of modeling severity losses using segmentation when the data distribution does not fall into the usual regression frameworks. This situation is not uncommon in lines of business such as third-party liability insurance, where heavy-tails and multimodality often hamper a direct statistical analysis. We propose to use regression models based on phase-type distributions, regressing on their underlying inhomogeneous Markov intensity and using an extension of the expectation–maximization algorithm. These models are interpretable and tractable in terms of multistate processes and generalize the proportional hazards specification when the dimension of the state space is larger than 1. We show that the combination of matrix parameters, inhomogeneity transforms, and covariate information provides flexible regression models that effectively capture the entire distribution of loss severities.

In Kim and Hardy (2007) the exact bootstrap was used to estimate certain risk measures including Value at Risk and the Conditional Tail Expectation. In this paper we continue this work by deriving the influence function of the exact-bootstrapped quantile risk measure. We can use the influence function to estimate the variance of the exact-bootstrap risk measure. We then extend the result to the L-estimator class, which includes the conditional tail expectation risk measure. The resulting formula provides an alternative way to estimate the variance of the bootstrapped risk measures, or the whole L-estimator class in an analytic form. A simulation study shows that this new method is comparable to the ordinary resampling-based bootstrap method, with the advantages of an analytic approach.

In recent times, the subject of deductibles in Industrial Fire insurance has gained significance to an increasing extent. In fact, up to a short time ago it was by no means common to apply deductibles in Industrial Fire insurance in Europe. The situation is entirely different in the USA where deductibles are the usual thing and are even obligatory with individual risk categories (e.g. petroleum refineries), hazards (e.g. explosion), or special types of companies (e.g. factory mutuals).

Today, however, there is a definite increase in the demand for deductibles in Industrial Fire insurance in Europe, too.

This trend may be explained by the fact that in order to reorganize their business successfully, insurers have been forced to increase their premiums by a considerable amount. It seems to be an unwritten law that when premiums in general are increased considerably, the point will some time or other be reached when the insured do not accept any further premium increases. Thus, even if such increases are completely justified in a view of claims experience, the insured will demand that premiums are reduced either by the insurer granting rebates for appropriate loss prevention measures or by the introduction of deductibles. In the USA, for example, such a situation arose around 1960 when considerable premium increases very quickly led to the introduction of substantial deductibles in Industrial business. In some European countries we are currently experiencing a similar development.

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.

The rapid economic growth in the last decade and the fierce competition have forced industry to raise its output, to develop new manufacturing methods and, where possible, to lower the fixed costs per unit of output. Consequently bigger factories and warehouses have been and are being built. Furthermore increasing labour costs have speeded up rationalisation and the introduction of efficient machinery.

Often, in the course of this development, too little attention has been paid to safety. This is reflected in the increasing number of Fire and Consequential Loss claims which have become so costly that the premium income has proved inadequate. Insurers have therefore adjusted their tariffs and increased their rates.

Of course, this is no solution to the problem. Insurers have to insist on adequate fire prevention and fire protection measures. Progress in the right direction can certainly be expedited by realistic tariffs which take account of all the positive features (sprinklers, inspection reports, etc.) and the negative features of a given risk (in commerce or industry).

Rating experts say that the tariffs of industrialized countries do not always point to the actual risk and claims fluctuations; one very important factor here, the size of the risk or of the building (i.e. its insured value, volume, surface area), brings me to the main theme of my today's talk.

In this paper, we consider the problem of Pareto optimal allocation in a general framework, involving preference functionals defined on a general real vector space. The optimization problem is equivalent to a modified sup-convolution of the different agents’ preference functionals. The results are then applied to a multi-period setting and some further characterization of Pareto optimality for an allocation is obtained for expected utility for processes.

On peut penser que la dimension des sinistres est la résultante d'un très grand nombre de causes indépendantes à effets positifs.

Soient

les variables aléatoires indépendantes qui représentent les facteurs élémentaires exerçant leur action dans l'ordre indiqué par les indices.

Si Xν est l'ampleur du sinistre dû aux facteurs ξ1, ξ2, …., ξν nous pouvons supposer que l'augmentation causée par ξν+1 soit proportionelle à ξν+1 et à une certaine fonction, g(Xν) de la dimension Xν.

C'est à dire, nous supposons qu'il y ait les relations:

Si chaque facteur n'apporte qu'un faible contribut à l'augmentation du sinistre, nous pouvons alors poser approximativement aussi:

Comme X varie de Xo à Xn (et que X = Xn désigne la dimension extrème du sinistre) le second membre de la (2) décrit une variable aléatoire, fonction de X.

Puisque par hypothèse ξ1,, ξ2, …., ξn sont des variables aléatoires indépendante, si n est suffisament grand, d'après le théorème de la limite des probabilités il s'insuit que la variable aléatoire au second membre de la (2), est, à la limite, distribuée suivant la loi normale. Si l'on pose g(t) = t − c, la variable aléatoire normale qui décrit la dimension du sinistre pourra être posée dans la forme In(X − c).

La question posée au colloque constitue un cas particulier du problème de la chance et de la malchance. Il appartient à une catégorie plus large de problèmes qui touche aussi bien à la statistique qu'à la recherche opérationnelle.

Pour une catégorie de risque déterminée, il faut définir deux éléments essentiels.

a) pour une période de temps de référence conventionnelle — la semaine par exemple, il faut disposer de la loi de survenance des sinistres c'est-à-dire que pour la période de temps considérée, il faut disposer de la loi qui constate qu'il se présente

la fonction génératrice de cette loi de survenance, toujours définie pour o ≤ s ≤ 1.

b) pour la catégorie de risques, nous considérons d'autre part la fonction de distribution des sinistres survenus.

S désignant la variable aléatoire du montant du sinistre, m la somme payée.

Ces deux éléments essentiels peuvent être donnés soit sur leur forme mathématique (fonctions spécifiées) sont sur leur forme expérimentale (image statistique d'un échantillon).

One can find in the literature three main sets of estimators for the variance components in the hierarchical credibility model. This paper presents these estimators in a unified notation, studies some of their properties important for numerical evaluation and compares their relative performance by simulation. The paper also demonstrates how function cm of the R package actuar can be used to fit hierarchical models to insurance data.

The collective risk model for the insurance claims is considered. The objective is to estimate a premium which is defined as a functional H specified up to an unknown parameter θ (the expected number of claims). Four principles of calculating a premium are applied. The Bayesian methodology, which combines the prior knowledge about a parameter θ with the knowledge in the form of a random sample is adopted. Two loss functions (the square-error loss function and the asymmetric loss function LINEX) are considered. Some uncertainty about a prior is assumed by introducing classes of priors. Considering one of the concepts of robust procedures the posterior regret Γ-minimax premiums are calculated, as an optimal robust premiums. A numerical example is presented.

In this paper, we propose an approach for modeling claim dependence, with the assumption that the claim numbers and the aggregate claim amounts are mutually and serially dependent through an underlying hidden state and can be characterized by a hidden finite state Markov chain using bivariate Hidden Markov Model (BHMM). We construct three different BHMMs, namely Poisson–Normal HMM, Poisson–Gamma HMM, and Negative Binomial–Gamma HMM, stemming from the most commonly used distributions in insurance studies. Expectation Maximization algorithm is implemented and for the maximization of the state-dependent part of log-likelihood of BHMMs, the estimates are derived analytically. To illustrate the proposed model, motor third-party liability claims in Istanbul, Turkey, are employed in the frame of Poisson–Normal HMM under a different number of states. In addition, we derive the forecast distribution, calculate state predictions, and determine the most likely sequence of states. The results indicate that the dependence under indirect factors can be captured in terms of different states, namely low, medium, and high states.

Reinsurance forms can roughly be classified into proportional and non-proportional. The authors of this paper had planned to investigate the “efficiency” of two different reinsurance forms, one from each of these categories. Efficiency is here understood as reduction in the variance of the annual results of the risk business achieved per unit of ceded reinsurance risk premium. This investigation may be carried out in full later.

This note will only deal with the interplay between surplus and excess of loss reinsurance; more specifically the effect of changes in the volume of surplus cessions on the excess of loss risk premium.

The study came out of a practical Fire Reinsurance rating problem and will be carried through under very simplified assumptions. Thus we will ignore the conflagration hazard and the possibility of a wrongly taxed PML. This means that if amounts above a PML of M are ceded on a surplus basis the highest loss per event will be M, and an excess cover above a priority m will never pay more than M-m per event.

The following notations will be used

R(M) ceded risk premium volume on surplus basis, the PML retention being M.

πM(m) excess of loss risk premium if priority is m and surplus cessions are made above a PML of M.

The formulation of generalized linear models in Klugman, Panjer and Willmot (2004) is a bit more general than is often seen, in that the residuals are not restricted to following a member of the exponential family. Some of the distributions this allows have potentially useful applications. The cost is that there is no longer a single form for the likelihood function, so each has to be fit directly. Here the use of loss distributions (frequency, severity and aggregate) in generalized linear models is addressed, along with a few other possibilities.

La solvabilité d'une entreprise est généralement référée à des vérifications d'ordre patrimonial et financier, à des considérations de nature économique (relatives à l'entreprise en elle-même, à son avenir, au milieu dans lequel elle opère, au marché dont elle constitue une unité élémentaire), à des relèvements, constatations, informations même de caractère subjectif et personnel. En d'autres termes, elle se définit sur la base d'un ensemble d'éléments “réels”, mais aussi de facteurs diversement constitués et diversement déterminés qui consentent, dans leur ensemble, de formuler un jugement, positif ou négatif, dont dérive l'attitude des “tiers” face à la possibilité, plus ou moins large, d'établir ou de développer ou de réduire ou de trancher des rapports contractuels.

Dans le domaine des entreprises d'assurances — à l'égard desquelles l'élément “confiance” de la part du grand public a un poids fondamental, s'agissant d'engagements qui se réalisent dans le temps en fonction de certaines probabilités — le critère de solvabilité n'est pas uniquement basé sur l'ensemble des facteurs d'appréciation indiqués ci-haut, mais trouve un point de répère solide, sur la consistance des “Réserves techniques” (ou, au moins, d'une partie de ces réserves) et sur le contrôle que, dans presque tous les Pays, est exercé par l'Etat en sauvegarde de la “masse des assurés”.