Consider the set of functions

Obviously, π1(x) represents the net premium of the excess cover over the priority x, and the variance thereof.

If a distribution function F(x) = 1 − π0(x) is given, the set (1) can be generated by means of the recursion formulae

Let us study the special class of d.fs. F(x) which satisfy

where

If these conditions are met, the integrals (1) have the solution:

Aj(y) and Bj(y), respectively, are polynomials of rank jk and jk − 1. Their coefficients are determined by the equations:

The system (5) is obtained by differentiation of (4) with respect to y, and observing (2).

When insurance claims are governed by fat-tailed distributions considerable uncertainty about the value of the tail-index is often inescapable. In this paper, using the theory of risk aversion, a new premium principle (the power principle – analogous to the exponential principle for thin-tailed claims) is established and its properties investigated. Applied to claims arising from generalized Pareto distributions, the resultant premium is shown to be the ratio of the two largest expected claims, for which the ratio of the actual claims is an unbiased as well as a consistent estimator. Whereas thin-tailed claim premiums are determined largely by the first two moments of the claims distribution, fat-tailed claim premiums are determined by the first two extremes. The context of risk-aversion leads to a natural model for incorporating tail-index uncertainty into premiums, which nevertheless leaves the basic ratio structure unaltered. To illustrate the theory, possible ‘premiums’ for US hurricane data are examined, which utilize the consistent pattern of observed extremes.

This paper provides a method to assess the risk relief deriving from a foreign expansion by a life insurance company. We build a parsimonious continuous-time model for longevity risk that captures the dependence across different ages in domestic versus foreign populations. We calibrate the model to portray the case of a UK annuity portfolio expanding internationally toward Italian policyholders. The longevity risk diversification benefits of an international expansion are sizable, in particular when interest rates are low. The benefits are judged based on traditional measures, such as the Risk Margin or volatility reduction, and on a novel measure, the Diversification Index.

While driver telematics has gained attention for risk classification in auto insurance, scarcity of observations with telematics features has been problematic, which could be owing to either privacy concerns or favorable selection compared to the data points with traditional features.

To handle this issue, we apply a data integration technique based on calibration weights for usage-based insurance with multiple sources of data. It is shown that the proposed framework can efficiently integrate traditional data and telematics data and can also deal with possible favorable selection issues related to telematics data availability. Our findings are supported by a simulation study and empirical analysis in a synthetic telematics dataset.

We propose a new inference strategy for general population mortality tables based on annual population and death estimates, completed by monthly birth counts. We rely on a deterministic population dynamics model and establish formulas that link the death rates to be estimated with the observables at hand. The inference algorithm takes the form of a recursive and implicit scheme for computing death rate estimates. This paper demonstrates both theoretically and numerically the efficiency of using additional monthly birth counts for appropriately computing annual mortality tables. As a main result, the improved mortality estimators show better features, including the fact that previous anomalies in the form of isolated cohort effects disappear, which confirms from a mathematical perspective the previous contributions by Richards, Cairns et al., and Boumezoued.

Even though the trend in mortality improvements has experienced several permanent changes in the past, the uncertainty regarding future mortality trends is often left unmodeled when pricing longevity-linked securities. In this paper, we present a stochastic modeling framework for the valuation of longevity-linked securities which explicitly considers the risk of random future changes in the long-term mortality trend. We construct a set of meaningful probability distortions which imply equivalent risk-adjusted pricing measures under which the basic model structure is preserved. Inspired by risk-based capital requirements for (re)insurers, we also establish a cost-of-capital pricing approach which then serves as the appropriate reference framework for finding a reasonable range for the market price of longevity risk. In a numerical application, we demonstrate that our model produces plausible risk loadings and show that a greater proportion of the risk loading is allocated to longer maturities when the risk of random future mortality trend changes is adequately modeled.

Although loss reserving has been deeply studied in the literature, there are still practical issues that have not been addressed a lot. One of them is the estimation of reserves during the year, which is necessary for forecasts or closings during the year. We will study the following question: What can be done for forecasts and closings during the year that goes along with the reserving at year end? In order to make it not too complicated, we will focus on the Chain-Ladder method introduced by Mack (1993). We will describe several methods that are used in practice. We will discuss advantages and disadvantages of these methods based on a simple deterministic example. Roughly spoken, we will see that you may shift development or accident periods, or may split development periods, but should not split accident periods.

The Capital Asset Pricing Model arises in an economy where agents have exponential utility functions and aggregate consumption is normally distributed, and gives the prices of assets with payoffs which are jointly normal with consumption. Such assets have normal marginal distributions and have dependence with consumption characterised by a normal copula. Wang has derived a transform which extends the CAPM by allowing pricing of assets in such an economy which have non-normal marginal distributions but still are normal-copula with consumption.

Here we set out the stochastic discount factors corresponding to this version of the CAPM and to Wang’s transform, and show how to calculate stochastic discount factors and hence asset prices for assets whose dependence with consumption is non-normal. We show that the impact of dependency structure on asset prices is significant.

The present paper is concerned with optimal premium plans for a reinsurance contract with reinstatements. In the reinsurance contract considered here, constant reinstatement premiums are due when the reinsurer’s loss exceeds certain bounds. For this reinsurance contract we examine the existence of a premium plan which minimizes the expected squared difference between the loss and the total premium income of the reinsurer. We show that an optimal premium plan exists, that it is unique, and that it satisfies the net premium principle.

Risk Theory for Life Insurance is simplified by the fact that the distribution Ψ (x) of claim amounts x approximately coincides with the distribution of “Risk sums” (not exactly, owing to differaences in the claim frequency with age and actual state of health), ond this distribution is comparatively stable.—The dependence on the claim frequency is eliminated by the introduction of a new time variable, and the system reduced to a (stationary) Poisson Process, which should be valid at least for large risk systems and for the total Life branch for a moderate sequence of years.

In almost all non-life branches, partial claims will dominate and Ψ (x) can only be determined by risk statistics, leaving a certain space of indetermination, in particular for large claims and for mediumsized statistical risk groups.

In my previous analyses, in particular New York 1957, interest has been concentrated on traffic and motor car insurance, where the risk depends on cars insured and on the meeting traffic (including road conditions). In one year the same car can be involved in many accidents and double claims (=collisions) are rather frequent.—According to my experience, this system is best represented by a sequence of single and double risk situations in time (for individual cars or for risk groups).

Analysis is simplified for Fire Insurance (and many other non-life branches), because the risk system is composed of mostly independent insurances (or risk objects), which are best described by the ordinary Individual Risk Theory.

We present a model for pricing credit risk protection for a limited liability non-life insurance company. The protection is typically provided by a guaranty fund. In the case of continuous monitoring, i.e., where the market values of the company's assets and liabilities are continuously observable, and where the market values of assets and liabilities follow continuous processes, regulators can liquidate the insurance company at the instant the market value of its assets equals the market value of its liabilities, implying that the credit protection is worthless. When jumps are included in the claims process, the protection provided by the guaranty fund has a strictly positive market value. The ability to continuously monitor asset prices with continuous sample paths eliminates economic losses from default. Our analysis suggests that economic losses from default stem from jumps in continuously observed asset prices and/or that asset prices are not continuously observed.

In this paper, we present closed-type formulas for some multivariate compound distributions with multivariate Sarmanov counting distribution and independent Erlang distributed claim sizes. Further on, we also consider a type-II Pareto dependency between the claim sizes of a certain type. The resulting densities rely on the special hypergeometric function, which has the advantage of being implemented in the usual software. We numerically illustrate the applicability and efficiency of such formulas by evaluating a bivariate cumulative distribution function, which is also compared with the similar function obtained by the classical recursion-discretization approach.

Les colloques successifs d'Astin ent eu le mérite de clarifier certaines idées théoriques importantes. En invitant au colloque d'Arnhem les actuaires á penser aux méthodes de Monte Carlo, les organisateurs ont forcément orienté les recherches vers le domaine des applications. Car simuler, c'est á. dire établir des résultats conformes á un processus aléatoire bien déterminé, suppose á priori que les données numériques d'introduction (ou de définition) ont été fixées.

Nous ne reprendrons pas les principes de la méthode de simulation, ils sont donnés de maniére magistrale dans “Notes on operations research 1959” du centre de recherche opérationnelle du Massachusetts Institute of Technology. Le chapitre II, portant titre „Simulation of Random processes” rédigé par Herbert P. Galliher donne un exposé complet du probléme. Nous renvoyons á cette référence.

Parmi les idées que l'on doit retenir, car elle est d'application dans tous les domaines, nous citons „Never simulate a single numerical problem;always simulate several of a class of problems”.

Dans le domaine non life, cet aspect est fondamental. Ce qui y fait défaut, ce sont des algorithmes généraux de résolution numériques. Une des difficultés majeures rencontrées par les actuaires a été et est encore le calcul effectif des convolutions. Ceux qui ont établi cette note se sont posé la question de savoir si la méthode de Monte Carlo ne pouvait fournir un procédé:

— évitant les convolutions.

— suffisamment général pour être applicable à une classe de risques arbitraire.

— donnant comme sous-produit des éléments intéressant aussi bien les théoriciens que les praticiens.

It is fantastic how the computer has changed our attitude to numerical problems. In the old days when our numerical tools were paper, pencil, desk calculator and logarithm tables we had to stay away from formulas and methods which led to too lengthy calculations. A consequence is that we have a tendency to think of numerical analysis in terms of the classical tools. If we go back to the results of earlier writers it seems, however, very likely that many results and formulas developed by them which had earlier a theoretical interest only could nowadays be applied successfully in numerical analysis.

As an example I take the ruin probability ψ(x). The Laplace transform of ψ(x) is given by the following expression

where c > 1. In fact (c — 1) is equal to the “security loading”. The function p(y) is equal to the Laplace transform of the claim distribution. We assume that the mean claim amount is equal to one, i.e. p′(0) = — 1.

In his book from 1955 [1] Cramer points out that this formula will be more easy to handle if the claim distribution is an exponential polynomial. In this case we have

where

Cramér's results are given on pages 81-83 in his book. We reproduce them here with a slight change of notations only.

Plusieurs communications sont présentées au Colloque sur les Mathématiques de l'Assurance Automobile. Afin de pouvoir mieux situer chacune d'elles dans ce vaste sujet, il peut être bon de schématiser l'ensemble de la question. Il comprend:

1. la survenance des accidents

2. le tarif et la mesure de son efficacité par rapport à l'estimation optimale.

Groupons dans une classe les propriétaires de véhicules ayant un même ensemble de caractéristiques extérieures de risques: même modèle de voiture, même zone de garage habituel, même usage ou profession du propriétaire, etc. Les risques s de survenance d'accident ne sont pas identiques pour tous les propriétaires de la classe, mais sont distribués selon une fonction de répartition F(s).

Désignons par P(s′ ∣ s) la probabilité de survenance de s′ sinistres pour un assuré dont le risque individuel est s (des études antérieures ont montré que P(s′ ∣ s) est représentable par une loi de probabilité de Poisson). La probabilité qu'un assuré, pris au hasard dans la classe, ait s′ sinistres est:

s0 étant le risque minimum de la classe.

Désignons par g(X) la densité de probabilité de la distribution des couts X des sinistres. On peut admettre qu'à l'intérieur de la classe, au moins en première approximation, g(X) est indépendant de s, d'où la probabilité de survenance d'un sinistre de coût X est:

By the term actual run-off triangle we shall mean the two-way tabulation—according to year of origin and year of payment—of claims paid to date, which has the following form:

where Cij is the amount paid during development year j in respect of claims whose year of origin is i.

The information relating to the area below and/or to the right of this triangle is unknown since it represents the future development of various cohorts of claims.

Now in seeking to use this triangle as a basis for projection of claims in future development years for each of the years of origin 0, 1, 2, etc., we must recognise that the entries Cij in the above triangle, being random variables, contain random deviations from their expected values uij. It is the corresponding triangle of these expected values in which we are interested, and which shall be called the expected run-off triangle.

It has been argued in previous studies that the expected utility decision criterion provides useful insights for certain insurance problems, such as underwriting, reinsurance and portfolio optimization problems. In this study three new models are developed which extend and generalize previous results. The first model analyses modified stop-loss reinsurance. The second model analyses risk pooling where both inward and outward reinsurance occur. Expected utility calculations can be used to provide insight on the attractiveness of competing reinsurance and risk pooling options. The third model is for strategic planning, where risk/reward tradeoffs for all the insurer's business activities (underwriting, investment, reinsurance) can be considered in aggregate. The simpler models can often be solved analytically however the strategic planning model is relatively complex and uses Monte Carlo techniques to determine retained earnings distributions. The expected utility approach has been found to be powerful, flexible and comprehensive as a decision aiding mechanism. From a normative viewpoint, this approach accounts very well for all the important decision elements. Recent developments in decision support systems will allow these models to be made available to practitioners in user friendly forms.