The objective of this paper is to present an analysis of a bonus-malus system (BMS) within the framework of the theory of ergodic Markov set-chains. It is shown that this type of Markov chains enables the evaluation of BMS, even in steady-state, under the assumption that transition probabilities change in a definite range. We introduce a model that allows the determination of the consequences of changes in the claim frequency of a policyholder. In a numerical example we examine the BMS employed by one of the Polish insurance companies.

Estimation of pure premiums for alternative rate classes using regression methods requires the choice of a functional form for the statistical model. Common choices include linear and log-linear models. This paper considers maximum likelihood estimation and testing for functional form using the power transformation suggested by Box and Cox. The linear and log-linear models are special cases of this transformation. Application of the procedure is illustrated using auto insurance claims data from the state of Massachusetts and from the United Kingdom. The predictive accuracy of the method compares favorably to that for the linear and log-linear models for both data sets.

This paper investigates risk aggregation and capital allocation problems for an insurance portfolio consisting of several lines of business. The class of multivariate INAR(1) processes is proposed to model different sources of dependence between the number of claims of the portfolio. The total capital required for the whole portfolio is evaluated under the TVaR risk measure, and the contribution of each line of business is derived under the TVaR-based allocation rule. We provide the risk aggregation and capital allocation formulas in the general case of continuous and strictly positive claim sizes and then in the case of mixed Erlang claim sizes. The impact of both time dependence and cross-dependence on the behavior of risk aggregation and capital allocation is numerically illustrated.

The need and extent of reinsurance of third party motor insurance depends fundamentally on the risk limits prescribed in the legislation of the country in question (and on the other hand the legal limits of the compulsory insurance may have been fixed with regard to the reasonable possibilities of the insurers getting reinsurance). There are two kinds of risk limits which are applied in different countries: total limits and individual limits. The former defines the maximum joint indemnity for an accident, paid to all claiments together, and the latter defines the maximum indemnity paid to each claiment separately. From the social point of view limits of this sort are not expedient, especially in regard to physical injuries. Owing to this total limit the indemnity for a single claiment can depend on the number of other claiments, which is quite inadequate from the point of view of the actual need to get insurance cover for injuries. The individual lump sum limit allows full compensation for slight injuries but can cut down the compensation for serious ones, which is an irrational method of settling an indemnity system. Owing to these risk limits motor car drivers may also be held responsible for the extra claims personally on the basis of civil (or criminal) law, which compels them to take an extra third party liability insurance (which often also has risk limits).

By a general theorem the necessary and sufficient condition for a function φ0 (σ) being completely monotonic for σ lying in the right semi-plane, i.e. that the nth derivative with respect to σ has the sign of (— 1)n, is that the function may be represented by the Laplace-Stieltjes integral , where U (v) is a non-decreasing function of v, independent of σ and bounded in every finite interval, σ a real or complex variable represented in the right semi-plane, s a real constant ≤ the real part of σ. By the notation φn (σ) we designate , which for U (v) being independent of σ, as assumed above, is equal to .

Definition 1. A compound Poisson process (in the narrow sense) is a process for which the probability distribution of the number of changes in the random function Y (t), constituting the process, occurring while the parameter, which is represented on the positive real axis, is in the interval (o, t) for every value of t, is defined by the following relation

the function φn (σ) being defined by the integral given above and subject to the condition that φ0 (σ) tends to unity, when σ tends to s. The function U (v) in the integrand of φn (σ) is, then, a distribution function which defines the risk distribution, in this case said to be t-independent.

Drs. H. Bohman and F. Esscher have reported in a recent paper) an extensive research performed in Sweden on the different methods of calculation of the distribution function of the total amount of claims. In the present paper certain methods are discussed in so far as they are different from those presented in the above quoted paper. The consideration is restricted to the generalised Poisson function even though some results can be easily extended. The author has already commented on some of the results represented in the sequel at a special meeting of the 17th International Congress of Actuaries in Edinburgh.

I. Lemma. Let be the generalised Poisson function under investigation. If aiSi(x), where Σ ai = 1 (the functions Si need not be distribution functions, neither must the constants ai be real numbers of interval [0,1]), then

F(x; n, S) = F(.; a1n, S1) * …*F(.; arn, Sr) (x),

as is easily verified by the use of characteristic functions. This component representation is repeatedly used in the sequel.

2. A Modified Esscher Method. The Esscher method is based on an observation that the well-known Edgeworth expansion is more advantageously applicable to a conveniently modified distribution function instead of the original generalised Poisson function. Let us assume that the value of F(x) is required at a point

We define and study in this work a simple model designed for managing long-term market risk of financial institutions with long-term commitments. It allows the assessment of solvency capital requirements and the allocation of risk budgets. This model allows one to avoid over-assessment of solvency capital requirements specifically after market disruptions. It relies on a dampener component in charge of refining risk assessment after market failures. Rather than aiming at a realistic and thus complex description of equity prices movements, this model concentrates on minimal features enabling accurate computation of capital requirements. It is defined both in a discrete and continuous fashion. In the latter case, we prove the existence, uniqueness and stability of the solution of the stochastic functional differential equation that specifies the model. One difficulty is that the proposed underlying stochastic process has neither stationary nor independent increments. We are however able to perform statistical analyses in view of its validation. Numerical experiments show that our model outperforms more elaborate ones of common use as far as medium-term (between 6 months and 5 years) risk assessment is concerned.

Longevity risk is putting more and more financial pressure on governments and pension plans worldwide due to pensioners’ increasing trend of life expectancy and the growing numbers of people reaching retirement age. Lee and Carter (1992, Journal of the American Statistical Association, 87(419), 659–671.) applied a one-factor dynamic factor model to forecast the trend of mortality improvement, and the model has since become the field’s workhorse. It is, however, well known that their model is subject to the limitation of overlooking cross-dependence between different age groups. We introduce Factor-Augmented Vector Autoregressive (FAVAR) models to the mortality modelling literature. The model, obtained by adding an unobserved factor process to a Vector Autoregressive (VAR) process, nests VAR and Lee–Carter models as special cases and inherits both frameworks’ advantages. A Bayesian estimation approach, adapted from the Minnesota prior, is proposed. The empirical application to the US and French mortality data demonstrates our proposed method’s efficacy in both in-sample and out-of-sample performance.

In this paper, we study the tail risk measures for several commonly used multivariate aggregate loss models where the claim frequencies are dependent but the claim sizes are mutually independent and independent of the claim frequencies. We first develop formulas for the moment (or size biased) transforms of the multivariate aggregate losses, showing their relationship with the moment transforms of the claim frequencies and claim sizes. Then, we apply the formulas to compute some popular risk measures such as the tail conditional expectation and tail variance of the multivariate aggregated losses and to perform capital allocation analysis.

In a deregulated insurance market, insurance carriers have an incentive to be innovative in their pricing decisions by segmenting their portfolios and designing new bonus-malus systems (BMS). This paper examines the evolution of market shares and claim frequencies in a two-company market, when one insurer breaks off the existing stability by introducing a super-discount class in its BMS. Several assumptions concerning policyholders and insurers behavior are tested. Diffusion theory is used to model the spread of the information concerning the new BMS among prospective customers. A wide variety of market outcomes results: one company may take over the market or the two may survive with equal or unequal market shares, each specializing in a specific niche of the market. Before engaging in an aggressive competitive behavior, insurers should consequently be reasonably confident in their assumptions concerning the reactions of their policyholders to the new BMS.

This paper provides an optimal asset allocation strategy to enhance risk management performance in the face of a financial crisis; this strategy entails constructing a good asset model – a multivariate jump-diffusion (MJD) model which includes idiosyncratic and systematic jumps simultaneously – and choosing suitable asset allocations and objective functions for fund management. This study also provides the dependence structure for the MJD model. The empirical implementation demonstrates that the proposed MJD model provides more detailed information about the financial crisis, allowing fund managers to determine an appropriate asset allocation strategy that enhances investment performance during the crisis.

The paper analyses the question: Should an insurance customer carry an occurred loss himself, or should he make a claim to the insurance company? This question is important within bonus-malus contracts with individual experience adjustments of the premium. The analysis model includes a bonus hunger strategy where the customers prefer the most profitable financial alternative, that is, the alternative which represents the lowest rate of interest. Hence the loss of bonus after a claim is calculated as a rate of interest paid from the customer to the insurer. Within this model the paper outlines the existence of a true compensation function and a relative cost function for each customer. A set of properties for bonus-malus contracts are presented and discussed. A concrete example of a bonus-malus system and an insurance compensation function illustrates the theoretical framework in a practical manner.

In this article, we study parameter uncertainty and its actuarial implications in the context of economic scenario generators. To account for this additional source of uncertainty in a consistent manner, we cast Wilkie’s four-factor framework into a Bayesian model. The posterior distribution of the model parameters is estimated using Markov chain Monte Carlo methods and is used to perform Bayesian predictions on the future values of the inflation rate, the dividend yield, the dividend index return and the long-term interest rate. According to the US data, parameter uncertainty has a significant impact on the dispersion of the four economic variables of Wilkie’s framework. The impact of such parameter uncertainty is then assessed for a portfolio of annuities: the right tail of the loss distribution is significantly heavier when parameters are assumed random and when this uncertainty is estimated in a consistent manner. The risk measures on the loss variable computed with parameter uncertainty are at least 12% larger than their deterministic counterparts.

We have printed the foregoing paper by Mr. Munden as submitted because we think that the detailed analysis will be of interest to all who have interests in the field of motor insurance rating. Of necessity, the data does not lend itself to analysis with respect to some of the known variables and we are conscious that some of the conclusions are controversial; some factors have also emerged from the discussions within ASTIN on motor insurance and it is therefore hoped that the following comments will be of value in relation to the paper.

It is of the utmost importance that a clear distinction is drawn between the concept of accident proneness and the heterogeneity shown from observations of claim frequencies under insurance policies. As the discussion at La Baule brought out, the first conclusion to be derived when a compound Poisson distribution emerged is that there is a degree of heterogeneity in the data. This might be due to differences in accident probabilities of the underlying risks, but it could be due, for example, to different exposures of similar risks. Lanteli's paper to the Rättvik colloquium showed a substantial variation of claims experience with annual mileage and thus without an analysis controlled with respect to mileage the conclusion that a proneness factor is solely involved must be suspect.

This paper finds payoff frequency distributions for valuing European and American fixed strike average options on a discrete time, recombining multiplicative binomial asset price process. In comparison to other discrete valuation methods the distributions, obtained analytically from a generating function, greatly reduce the computational requirements needed for accurate valuation. Less data are needed to value geometric than arithmetic averages, but the magnitude of calculations is similar for both instruments. Calculations of order T3 are needed to value European instruments, of order T4 to value their American counterparts. A frequency distribution of a quantity called path sums is used to value geometric average options, and a joint distribution of path sums and realized prices is used to value arithmetic average options. The frequency distributions give an exact value for geometric average, an approximate value for arithmetic average instruments. The method obtains additional information from the generating function to estimate approximation errors relative to the exact binomial solution. If the errors are significant they can be reduced using still further detail from the generating function. Error reduction can be performed selectively to minimize additional calculation.

In many estimation problems, incomplete as well as complete samples are available for Bayesian prediction. After developing the theory for a special, but useful family of distributions, examples are given in life testing, renewal risk processes, life contingencies, and the problem of estimating a defective distribution.

In non-life insurance, it is nearly always assumed that the expense loading is a fraction of the risk premium. This may deeply affect the fairness of a tariff, as illustrated in the case of the Belgian bonus-malus system.

Un nouvel examen des normes qui régissent l'octroi des prestations d'assistance pharmaceutique semble s'imposer. En effet, l'introduction continuelle sur le marché de nouveaux produits pharmaceutiques — dont certains sont capables de juguler rapidement des formes morbides naguère lentes à guérir — et la confiance progressivement accrue que met le grand public dans des ordonnances toujours plus variées exigent, de la part des Instituts d'assurance sociale contre les maladies, des solutions permettant de répondre en même temps à des nécessités diverses: il s'agit en particulier de pouvoir autoriser l'utilisation des produits pharmaceutiques les plus modernes, capables d'accélérer la guérison des malades, même s'ils sont très coûteux, tout en évitant par ailleurs l'expansion anormale des ordonnances et en maintenant le coût de l'assistance dans les limites permises par les possibilités de financement.

Le choix de la solution la meilleure est done subordonné à un ensemble de considérations sur la nécessité et la possibilité des mesures à envisager; les aspects économiques de la question ne sont pas des moindres et leur étude doit de toute évidence reposer sur des bases techniques appropriées.

Il semblerait normal que ces bases techniques soient établies d'après des observations effectuées là où ces mêmes bases techniques doivent être employées; ceci n'est généralement pas possible et c'est pourquoi des éléments provenant d'autres expériences revêtent une importance particulière, ne serait-ce qu'à titre indicatif.

The finite and infinite horizon time probability of ruin are important parameters in the study of actuarial risk theory. This paper introduces procedures for directly estimating these key parameters from a random sample of observations without assumptions as to the parametric form of the distribution from which the observations arise. The estimators introduced apply to most of the classical models in which ruin probabilities are used and also apply to a much broader class of models. The procedures are based on the concept of sample reuse, an old idea in statistics which is becoming more popular due to the widespread availability of high speed computers. In this paper, the almost sure consistency of the estimators is established. Further, finite sample properties of the estimators are investigated in a simulation study.