Stop Loss reinsurance has attracted the interest of ASTIN members for years. May I recall the paper of Borch [1] in which he demonstrates some optimality qualities of the stop loss reinsurance from the ceding company's point of view, the contribution of Kahn [2] and the paper of Pesonen [3]. I also mention the paper of Esscher [4] and Verbeek's contribution [5]. Going back to the pre-ASTIN days we find a paper of Dubois [6].

The rating problems have been dealt with by several authors. Let me recall the rating formula worked out by a group of Dutch Actuaries some 20 years ago. This was based on the assumption that the mean and the standard deviation were known. Based on Chebycheff's inequality an approximation formula was worked out which, of course, was heavily on the safe side.

Even younger members of ASTIN are probably familiar with the studies made in the early sixties by a group of Swedish Actuaries, the results of which were presented by Bohman at the Actuarial Congress in London in 1964. Partly based on this, Bühlmann worked out some tables which he used for rating purposes.

My present contribution to the subject may not justify the above reviews, particularly as I will deal with a very special retention situation which a practical underwriter will rightly not accept, namely a stop-loss point as low as equal to the mean value of the distribution.

My excuse for this is that the formula deduced is very handy and that it is of value to the underwriter to know the stop loss risk rate also at this low level.

In a note on the security loading of excess loss rates I am deducing a simple formula intended to replace some tedious calculations. In the beginning of that note I made the point that some authors recommend a loading proportional to the dispersion of the total claims amount of a treaty δ1 while others tend to favour .

I also stated that a loading proportional to δ1 or its estimate δ1* could be deduced from the statistical uncertainty in measuring the risk (section 4).

The question has been raised if and to what extent a loading system based on the dispersion is unduly punishing the smaller portfolios. This will be examined below.

The pricing concept will be analyzed from the point of view of a big dominating Reinsurer who wants to be fair in all directions. The conclusion of this study supports an affirmative answer to the question put above.

In a second part the loading is studied from a different angle bringing competition into the picture. The pricing or loading becomes a problem of operations research under the simplified assumption that profit is the only purpose of our activity. Not unexpectedly, the loading coming out from this aspect differs from those of part one.

Part two also deals with the question of how much of the loadings which we are aiming for, get lost in the competitive process. It is also shown that in most cases the harder the competition is, the higher loadings shall be used.

Part one and part two thus deal with the loading problem from different aspects, and illustrate the complexity of the problem. It is my hope that this note could stimulate further researches in this interesting and important area, also in a moment when some reinsurers are more concerned with the question of surviving than in fixing the loadings which should on the average and in the long run turn up as profits.

This paper investigates the effects of high or low fair-premium demand elasticity in an insurance market where risk classification is restricted. The effects are represented by the equilibrium premium, and the risk-weighted insurance demand or “loss coverage”. High fair-premium demand elasticity leads to a collapse in loss coverage, with an equilibrium premium close to the risk of the higher-risk population. Low fair-premium demand elasticity leads to an equilibrium premium close to the risk of the lower-risk population, and high loss coverage – possibly higher than under more complete risk classification. The demand elasticity parameters which are required to generate a collapse in coverage in the model in this paper appear higher than the values for demand elasticity which have been estimated in several empirical studies of various insurance markets. This offers a possible explanation of why some insurance markets appear to operate reasonably well under community rating, without the collapse in coverage which insurance folklore suggests.

Catastrophe insurance markets fail to provide sufficient protections against natural catastrophes, whereas they have the capacity to absorb the losses. In this paper, we assume the catastrophic risks are dependent and extremely heavy-tailed, and insurers have limited liability to cover losses up to a certain amount. We provide a comprehensive study to show that the diversification in the catastrophe insurance markets can be transited from suboptimal to preferred by increasing the number of insurers in the market. This highlights the importance of coordination among insurers and the government intervention in encouraging insurers to participate in the catastrophe insurance market to exploit risk sharing. Simulation studies are provided to illuminate the key findings of our results.

We consider a generalisation of a risk process under experience rating when the aggregation of claims up to time t is a Brownian motion (B.M.) with a drift. We prove that the distribution of ruin before time t is equivalent to the distribution of the first passage time of B.M. for parabolic boundary.

Using Wald identity for continuous time we give an explicit formula for this distribution. A connection is made with discounting risk model when the income process is a diffusion.

When the aggregation of claims is a mixture of B.M. and compound Poisson process, we give (using Gerber's result 1973) an upper bound for the distribution of finite time ruin probability.

This article presents an explicit formula for the value of a withdrawal benefit when the times of death and withdrawal are dependent. The derivation is based on an actuarial equivalence principle. As a special case, we show that in the fully continuous case, the withdrawal benefit is the reserve when the decrements are independent. We also present a definition of antiselection and prove that the withdrawal benefit will be smaller under antiselection.

We are proposing a parametric model to rate insurance for vehicles belonging to a fleet. The tables of premiums presented take into account past vehicle accidents, observable characteristics of the vehicles and fleets, and violations of the road-safety code committed by drivers and carriers. The premiums are also adjusted according to accidents accumulated by the fleets over time. The proposed model accounts directly for explicit changes in the various components of the probability of accidents. It represents an extension of bonus malus-type automobile insurance models for individual premiums (Lemaire, 1985; Dionne and Vanasse, 1989 and 1992; Pinquet, 1997 and 1998; Frangos and Vrontos, 2001; Purcaru and Denuit, 2003). The extension adds a fleet effect to the vehicle effect so as to account for the impact that the unobservable characteristics or actions of carriers can have on truck accident rates. This form of rating makes it possible to visualize what impact the behaviors of owners and drivers can have on the predicted rate of accidents and, consequently, on premiums. The results are compared to those of the semiparametric approach.

In the present note we deduce a class of bounds for the difference between the stop-loss transforms of two compound distributions with the same severity distribution. The class contains bounds of any degree of accuracy in the sense that the bounds can be chosen as close to the exact value as desired; the time required to compute the bounds increases with the accuracy.

This paper introduces a flexible risk decomposition method for life insurance contracts embedding several risk factors. Hedging can be naturally embedded in the framework. Although the method is applied to variable annuities in this work, it is also applicable in general to other insurance or financial contracts. The approach relies on applying an allocation principle to components of a Shapley decomposition of the gain and loss. The implementation of the allocation method requires the use of a stochastic on stochastic algorithm involving nested simulations. Numerical examples studying the relative impact of equity, interest rate and mortality risk for guaranteed minimal maturity benefit (GMMB) policies conclude our analysis.

Based on a recent contribution by Baione and others in this Journal, some consequences of the decrease of the mean merit coefficient for portfolios of Bonus-Malus policies, and some alternative ways to measure the “quality” of a Bonus-Malus system are discussed.

The purpose of this note is to draw attention to a semimartingale method which can be applied to very general types of risk models to obtain local martingales or martingales, which can then be used in the now classical way to evaluate ruin probabilities. Relations to the theory of exponential families of stochastic processes are also pointed out and utilized.

It is widely acknowledge that there has been a major breakthrough in the mathematical theory of option trading. This breakthrough, which is usually summarized by the Black–Scholes formula, has generated a lot of excitement and a certain mystique. On the mathematical side, it involves advanced probabilistic techniques from martingale theory and stochastic calculus which are accessible only to a small group of experts with a high degree of mathematical sophistication; hence the mystique. In its practical implications it offers exciting prospects. Its promise is that, by a suitable choice of a trading strategy, the risk involved in handling an option can be eliminated completely.

Since October 1987, the mood has become more sober. But there are also mathematical reasons which suggest that expectations should be lowered. This will be the main point of the present expository account. We argue that, typically, the risk involved in handling an option has an irreducible intrinsic part. This intrinsic risk may be much smaller than the a priori risk, but in general one should not expect it to vanish completely. In this more sober perspective, the mathematical technique behind the Black–Scholes formula does not lose any of its importance. In fact, it should be seen as a sequential regression scheme whose purpose is to reduce the a priori risk to its intrinsic core.

We begin with a short introduction to the Black–Scholes formula in terms of currency options. Then we develop a general regression scheme in discrete time, first in an elementary two-period model, and then in a multiperiod model which involves martingale considerations and sets the stage for extensions to continuous time. Our method is based on the interpretation and extension of the Black–Scholes formula in terms of martingale theory. This was initiated by Kreps and Harrison; see, e.g. the excellent survey of Harrison and Pliska (1981,1983). The idea of embedding the Black-Scholes approach into a sequential regression scheme goes back to joint work of the first author with D. Sondermann. In continuous time and under martingale assumptions, this was worked out in Schweizer (1984) and Föllmer and Sondermann (1986). Schweizer (1988) deals with these problems in a general semimartingale model.

The notional defined contribution pension scheme combines pay-as-you-go financing and a defined contribution pension formula. The return on contributions is based on an index set by law, such as the growth rate of GDP, average wages or contribution payments. The volatility of this return compromises the system's pension adequacy and therefore guarantees may be needed. Here, we provide a minimum return guarantee to the pension contributions. The price is calculated in a utility indifference framework. We obtain a closed-form solution for a general dependence structure with exponential preferences and in presence of stochastic short interest rates.

Two premium calculation principles by negotiation. Using, as main tools,

the classical risk exchange model by Borch and

the bargaining models of Nash and Kalai-Smorodinsky,

we define two new premium calculation principles, whose main goal is to take explicitly into account the attitude towards risk of the policy-holders. Those principles are neither additive nor iterative, but they nevertheless possess several important properties: the premium is translation-invariant, it does not depend neither on the reserves nor on the portfolio of the company; it takes into account all the moments of the claim distribution; it is independent of the policy-holder's wealth but increases with his risk aversion.

Coalition against an insurance company. While computing the core of this risk exchange, we show that it can be of the policy-holder's interest to coalize in order to obtain premium cuts.

Le modèle d'échange de risques de Borch entre plusieurs compagnies d'assurances soucieuses d'améliorer leur situation en formant un pool de réassurance a fait l'objet de très nombreuses publications. Ce n'est que depuis quelques années cependant que l'on semble s'être aperçu que le même modèle pouvait être utilisé pour décrire toute économie d'échange, en particulier le contrat d'assurance simple entre un assuré et sa compagnie. Si l'on suppose que les préférences de l'assuré peuvent être décrites par une fonction d'utilité exponentielle, et que l'assureur est indifférent au risque en première approximation, les contrats Pareto-optimaux consistent en une couverture complète du risque, moyennant le paiement d'une prime que le critère de Pareto-optimalité ne permet pas de déterminer.

We give some actual possibilities for computing numerical values in the classical risk models both in transient and asymptotical cases by introducing the concept of normed model. Some recent approximations are tested on numerical examples.

We also emphasize the interest of these methods to compute waiting time distributions (transient and stationary cases) in queueing theory.

Credibility theory refers to the use of linear least-squares theory to approximate the Bayesian forecast of the mean of a future observation; families are known where the credibility formula is exact Bayesian. Second-moment forecasts are also of interest, for example, in assessing the precision of the mean estimate. For some of these same families, the second-moment forecast is exact in linear and quadratic functions of the sample mean. On the other hand, for the normal distribution with normal-gamma prior on the mean and variance, the exact forecast of the variance is a linear function of the sample variance and the squared deviation of the sample mean from the prior mean. Bühlmann has given a credibility approximation to the variance in terms of the sample mean and sample variance.

In this paper, we present a unified approach to estimating both first and second moments of future observations using linear functions of the sample mean and two sample second moments; the resulting least-squares analysis requires the solution of a 3 × 3 linear system, using 11 prior moments from the collective and giving joint predictions of all moments of interest. Previously developed special cases follow immediately. For many analytic models of interest, 3-dimensional joint prediction is significantly better than independent forecasts using the “natural” statistics for each moment when the number of samples is small. However, the expected squared-errors of the forecasts become comparable as the sample size increases.

While many of the prevalent stochastic mortality models provide adequate short- to medium-term forecasts, only few provide biologically plausible descriptions of mortality on longer horizons and are sufficiently stable to be of practical use in smaller populations. Among the very first to address the issue of modelling adult mortality in small populations was the SAINT model, which has been used for pricing, reserving and longevity risk management by the Danish Labour Market Supplementary Pension Fund (ATP) for more than a decade. The lessons learned have broadened our understanding of desirable model properties from the practitioner’s point of view and have led to a revision of model components to address accuracy, stability, flexibility, explainability and credibility concerns. This paper serves as an update to the original version published 10 years ago and presents the SAINT model with its modifications and the rationale behind them. The main improvement is the generalization of frailty models from deterministic structures to a flexible class of stochastic models. We show by example how the SAINT framework is used for modelling mortality at ATP and make comparisons to the Lee-Carter model.

When an insurance company accepts new insurances or when the premiums of earlier accepted insurances have to be changed on renewal the company has to

— search for the factors that influence the premium and

— calculate the premium according to the values of these factors.

In order to calculate the premiums the company gathers data consisting of factors eventually influencing the amount of claims. On the basis of these data the company calculates the tariff which has to fulfill the following general principles:

1. The tariff has to be as correct as possible in relation to different risk groups.

2. The structure of the tariff has to be such that the calculation of the insurance premium is quite straightforward. With this in mind the factors influencing the tariff have to be few enough and the structure of the tariff has to be the simple (e.g. linear or multiplicative) function of the factors or it should be rather easy to put them into tabular form.

These principles are partly contradictory. If the premium is correct, the structure of the tariff is not usually simple.

This paper considers a particular credibility model for the claim numbers N1, N2, …, Nn, … of a single risk within a collective in successive periods 1, 2, …, n, … In the terminology of Jewell (1975) the model is an evolutionary credibility model, which means that the underlying risk parameter Λ is allowed to vary in successive periods (the structure function is allowed to be time dependent). Evolutionary credibility models for claim amounts have been studied by Bühlmann (1969, pp. 164–165), Gerber and Jones (1975), Jewell (1975, 1976), Taylor (1975), Sundt (1979, 1981, 1983) and Kremer (1982). Again in Jewell's terminology the considered model is on the other hand stationary, in the sense that the conditional distribution of Ni given the underlying risk parameter does not vary with i.

The computation of the credibility estimate of Nn+1 involves the considerable labor of inverting an n × n covariance matrix (n is the number of observations). The above mentioned papers have therefore typically looked for model structures for which this inversion is unnecessary and instead a recursive formula for the credibility forecast can be obtained. Typically nth order stationary a priori sequences (e.g., ARMA (p, q)-processes) lead to an nth order recursive scheme. In this paper we impose the restriction that the conditional distribution of Ni is Poisson (which by the way leads to a model identical to the so called “doubly stochastic Poisson sequences” considered in the theory of stochastic point processes). What we gain is a recursive formula for the coefficients of the credibility estimate (not for the estimate itself!) in case of an arbitrary weakly stationary a priori sequence. In addition to this central result the estimation of the structural parameters is considered in this case and some more special models are analyzed. Among them are EARMA-processes (which are positive-valued stationary sequences possessing exponentially distributed marginals and the same autocorrelation structure as ARMA-processes) as a priori sequence and models which can be considered as (discrete) generalizations of the Pólya process.

The no-claim bonus problem has given rise to a considerable amount of discussion throughout the whole world. There is quite a difference of opinion among the actuaries and other experts concerned in this field and several exchanges of view have taken place the last few years. The ASTIN section of the Permanent Committee has been well aware of this fact and it has devoted one Colloquium to this subject and discussed it at others.

In 1959 the major part of the La Baule meeting was dedicated to this subject and attention was focussed on this problem once again at Rattvik in 1961. Nevertheless controversies on this subject still continue. Almost every conference where the bonus problem is discussed is marked by a widespread difference of opinion.

As is well known, the claim frequencies under insurance policies show a considerable heterogeneity, especially in the early years. It is not possible to get homogeneous sub-groups by means of a continuous subdivision; what may be gained in homogeneity, is lost in credibility. It seems therefore that a subsequent adjustment of premiums according to the past claim record may well be a suitable way of obtaining a fair premium.

Those who are in favour of a rating procedure granting a bonus at a careful driver will stress that criticism is useless as long as no better solution is available, whereas actuaries who reject such a rating system argue that the unfairness of a flat rate is not at all eliminated by means of a bonus.