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.

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.

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.

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.

A simple technique for continuity estimation for ruin probability in the compound Poisson risk model is proposed. The approach is based on the contractive properties of operators involved in the integral equations for the ruin probabilities. The corresponding continuity inequalities are expressed in terms of the Kantorovich and weighted Kantorovich distances between distribution functions of claims. Both general and light-tailed distributions are considered.

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.

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.

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.

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.

In the classical Bayesian approach to credibility the claims are conditionally independent and identically distributed random variables, with common density f(x, ϑ). The unknown parameter ϑ is a realization of a random variable Θ having initial (prior) density u(ϑ). Let

The initial pure premium is

The premium for X t + 1, given X1 +, …, Xt, is the conditional expectation

A central question is for which pairs f(x, ϑ) and u (ϑ) this expression is linear, i.e. of the form

where is the observed average. This is indeed the case for about half a dozen famous examples. Jewell (1974) has found an elegant and general approach to unify these examples, see also Goovaerts and Hoogstad (1987, chapter 2). The classical examples can be retrieved as special cases; however a preliminary reparameterization has to be performed on a case by case basis. The purpose of this note is to propose an alternative (but of course strongly related) formulation of the general model, from which the classical examples can be retrieved in a straightforward way.

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.

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.

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.

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.

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.

The precise wording of the heading is deliberate. The Canadian system is not a “bonus” nor is it properly described as “no claim”; it is a sub-classification of insureds according to whether they have been accident-free during a term of years ending at the inception of the insurance period. Specific rates are quoted in the Tariff for each sub-class, based broadly on the relative experiences; as a matter of arithmetical convenience, premium rates are calculated by applying discounts or surcharges to a “key” rate, but the percentage discounts or surcharges are not quoted in the Tariff, the Policy or the renewal papers.

In the early 1930's a “no claim bonus” on British lines was part of the Canadian automobile insurance system. A percentage reduction or “bonus” was allowed from the premium specified in the Policy, provided the insured had owned and operated a private passenger car, free of accident or revocation of driving license, for a period of up to three years before the inception of the Policy. At that time the Automobile Insurance Market was profitable and highly competitive, both as between Companies and as between Agents. Because of inter-Company competition, confirmation of entitlement to the bonus was rarely attempted. Often it was not possible, as most Companies did not maintain an alphabetical index of insureds; an effort to set up an inter-Company claim-recording bureau broke down through sheer weight of numbers of claims.

It is rare that the severity loss distribution for a specific line of business can be derived from first principles. One such example is the use of generalised Pareto distribution for losses above a large threshold (or more accurately: asymptotically), which is dictated by extreme value theory. Most popular distributions, such as the lognormal distribution or the Maxwell-Boltzmann-Bose-Einstein-Fermi-Dirac (MBBEFD), are convenient heuristics with no underlying theory to back them. This paper presents a way to derive a severity distribution for property losses based on modelling a property as a weighted graph, that is, a collection of nodes and weighted arcs connecting these nodes. Each node v (to which a value can also be assigned) corresponds to a room or a unit of the property where a fire can occur, while an arc (v, v′; p) between vertices v and v′ signals that the probability of the fire propagating from v to v′ is p. The paper presents two simple models for fire propagation (the random graph approach and the random time approach) and a model for explosion risk that allow one to calculate the loss distribution for a given property from first principles. The MBBEFD model is shown to be a good approximation for the simulated distribution of losses based on property graphs for both the random graph and the random time approach.

This paper studies optimal risk redistribution between firms, such as institutional investors, banks or insurance companies. We consider the case where every firm uses dual utility (also called a distortion risk measure) to evaluate risk. We characterize optimal risk redistributions via four properties that need to be satisfied jointly. The characterized risk redistribution is unique under three conditions. Whereas we characterize risk redistributions by means of properties, we can also use some results to study competitive equilibria.We characterize uniqueness of the competitive equilibrium in markets with dual utilities. Finally, we identify two conditions that are jointly necessary and sufficient for the case that there exists a trade that is welfare-improving for all firms.

High-cardinality categorical features are pervasive in actuarial data (e.g., occupation in commercial property insurance). Standard categorical encoding methods like one-hot encoding are inadequate in these settings.

In this work, we present a novel Generalised Linear Mixed Model Neural Network (“GLMMNet”) approach to the modelling of high-cardinality categorical features. The GLMMNet integrates a generalised linear mixed model in a deep learning framework, offering the predictive power of neural networks and the transparency of random effects estimates, the latter of which cannot be obtained from the entity embedding models. Further, its flexibility to deal with any distribution in the exponential dispersion (ED) family makes it widely applicable to many actuarial contexts and beyond. In order to facilitate the application of GLMMNet to large datasets, we use variational inference to estimate its parameters—both traditional mean field and versions utilising textual information underlying the high-cardinality categorical features.

We illustrate and compare the GLMMNet against existing approaches in a range of simulation experiments as well as in a real-life insurance case study. A notable feature for both our simulation experiment and the real-life case study is a comparatively low signal-to-noise ratio, which is a feature common in actuarial applications. We find that the GLMMNet often outperforms or at least performs comparably with an entity-embedded neural network in these settings, while providing the additional benefit of transparency, which is particularly valuable in practical applications.

Importantly, while our model was motivated by actuarial applications, it can have wider applicability. The GLMMNet would suit any applications that involve high-cardinality categorical variables and where the response cannot be sufficiently modelled by a Gaussian distribution, especially where the inherent noisiness of the data is relatively high.

We study a fully funded, collective defined contribution (DC) pension system with multiple overlapping generations. We investigate whether the welfare of participants can be improved by intergenerational risk sharing (IRS) implemented with a realistic investment strategy (e.g., no borrowing) and without an outside entity (e.g., shareholders) that helps finance the pension fund. To implement IRS, the pension system uses an automatic adjustment rule for the indexation of individual accounts, which adapts to the notional funding ratio of the pension system. The pension system has two parameters that determine the investment strategy and the strength of the adjustment rule, which are optimized by expected utility maximization using Bayesian optimization. The volatility of the retirement benefits and that of the funding ratio are analyzed, and it is shown that the trade-off between them can be controlled by the optimal adjustment parameter to attain IRS. Compared with the optimal individual DC benchmark using the life cycle strategy, the studied pension system with IRS is shown to improve the welfare of risk-averse participants, when the financial market is volatile.