There are two phases of difficulties in estimating a claims distribution. If we are going to estimate the claims distribution as accurately as possible, we should gather considerably long terms statistics. While economical and social environment will change. As a result the statistics gathered should be amended by a kind of trend value. One of difficulties here is the estimation of that trend value. Another difficulty is the estimation of claims distribution as being the stochastic distribution. In the case of considering claim size, the estimation becomes more difficult.

The intention of this paper is to propose an actual way of estimating stochastic claims distribution considering various kind of claim size by the use of a computer. Regarding the problem of amending claims distribution by a trend Value I will discuss at another time.

Several countries have made community rating mandatory for certain lines of insurance, particularly health insurance. This paper offers a theoretical solution to the problem of designing equalisation schemes to support community rating in a market where different insurers are selling different benefit plans. The criterion chosen is that an equalisation scheme should minimise the opportunities for arbitrage between insurers, which community rating otherwise would generate. Several possible measures of arbitrage are presented, and the optimal schemes are compared against data from Australian health insurers. Finally, the approach is extended to partial community rating, for example unisex rating.

Modern life is characterized by risks of different kind: some threatening all persons and some restricted to the owners of property, motor cars, etc., while still others are typical for some individuals or for special occupations. The corresponding accidents, losses or claims will occur suddenly and unexpectedly and may involve considerable financial loss. It is quite evident that modern life is a fit subject for risk theory, and that some results in the pure mathematic theory might have applications in the study of problems in real life.

In practice, however, we can identify risk theory with insurance risk theory or with the application of the theory of probability on insurance risk problems. This general definition has the advantage, that it covers a wide field of different risks and risk problems as specified in the insurance texts—and a great collection of risk situations ₌ claims occurred (with corresponding loss amounts) is available in the claims acts. In fact, I believe that any actuary or mathematician, starting his researches in risk theory or in risk statistics, should begin his studies by a series of actual claims acts.

Motor insurers in the United Kingdom are not subject to Government control over the rating structures they use or the levels of premiums they charge. The market is highly competitive, and each insurer therefore needs to make the best estimate he can of the premium required for each category of risk, to produce a given level of profit. He also needs to estimate the extent to which departures from such premium levels can be justified, for example in order to have a simple rating structure or to meet competition.

The purpose of this note is to give some examples of the statistical tabulations being produced in one insurance office in the United Kingdom. The statistical system which has been developed covers many aspects of motor insurance management, but in this note we shall direct attention to just two, namely

(i) continuous review of the variations in the claims experience from one risk category to another, as a guide to the relative premiums required for the different categories; and

(ii) continuous review of the changing composition of the portfolio and of the movements in and out, to try to assess the extent to which gains and losses of business can be attributed to pricing differences, marketing strategies, etc.

Unlike chain ladder, the loss ratio method requires volume measures. Typically, these volumes are assumed to be known. In practice, however, accurate volume measures are rarely available. We interpret the available volumes as estimators for the true volume measures and analyze the consequences for the loss ratio method. In particular, we calculate the mean squared error of prediction, including uncertainty of volume measures, and derive approximately optimal weights for the observed incremental loss ratios. We then introduce a generalization of the loss ratio method that is tailored to the situation of uncertain volume measures and calculate the prediction uncertainty of this generalized loss ratio method.

This paper examines the importance of pensions (employment and social security), taxes and government transfers for alternative retirement savings drawdown strategies (DS), compared to the conventional approach in published literature of using a gross income concept obtainable from retirement savings alone. Using a lifetime utility framework, our longitudinal dynamic micro-simulation model incorporates risk aversion, stochastic markets, stochastic mortality and the interactions among sources of retirement income within the complex Canadian tax and social benefit system, enabling us to rank commonly advocated DS and to ask whether incorporating pensions, taxes and transfers alters those rankings. Our findings show the importance of treating the evaluation of alternative DS as a comprehensive and integrated problem by including all sources of income — including pensions, taxes and government transfers. Using restricted income measures can potentially lead to simplistic, and possibly misleading, conclusions.

This article proposes a continuous time mortality model based on calendar years. Mortality rates belong to a mean-reverting random field indexed by time and age. In order to explain the improvement of life expectancies, the reversion level of mortality rates is the product of a deterministic function of age and of a decreasing jump-diffusion process driving the evolution of longevity. We provide a general closed-form expression for survival probabilities and develop it when the mean reversion level of mortality rates is proportional to a Gompertz–Makeham law. We develop an econometric estimation method and validate the model on the Belgian population.

In this paper, a new three-parameter discrete family of distributions, the $r{\mathcal B}ell$ family, is introduced. The family is based on series expansion of the r-Bell polynomials. The proposed model generalises the classical Poisson and the recently proposed Bell and Bell–Touchard distributions. It exhibits interesting stochastic properties. Its probabilities can be computed by a recursive formula that allows us to calculate the probability function of the amount of aggregate claims in the collective risk model in terms of an integral equation. Univariate and bivariate regression models are presented. The former regression model is used to explain the number of out-of-use claims in an automobile insurance portfolio, by showing a good out-of-sample performance. The latter is used to describe the number of out-of-use and parking claims jointly. This family provides an alternative to other traditionally used distributions to describe count data such as the negative binomial and Poisson-inverse Gaussian models.

Pension funds and insurers face difficulties in hedging their longevity risk, which is the uncertainty of how long their clients will live. A possible solution could be using longevity-linked securities to transfer some of this risk to other parties. However, these securities may not match the actual mortality rates of the insurer’s clients, resulting in a potential loss due to basis risk. In this paper, we measure this basis risk through the pricing of a longevity derivative under Solvency II. We also compare this method with other common pricing methods in finance. We explore and evaluate different hedging strategies for insurers, using a multi-population model derived from a two-dimensional Hull and White model that captures the dynamics of mortality over time.

It is rather a pity that we received only three papers especially devoted to subject 2, “applications of methods of operations research and modern economic theory”. Maybe the time for real applications of operations research techniques in the insurance field has still to come. However, I am convinced that sooner or later such techniques—in connection with the use of the computor—will play a more or less important part in the management of insurance business. An impulse to this of course will be the appearance of —let me say—“textbooks” on this subject; such as the book written by Wolff, Methoden der Unternehmensforschung im Versicherungswesen, Springerverlag 1966, a copy of which I received a few days ago.

Meanwhile, the fact that only a few papers have been sent in gives me the opportunity to report on them in some detail and to make a few side remarks.

The studies presented are:

Borch. Dynamic decision problems in an insurance company, de Leve and Weeda. Driving with Markov-programming.

d'Hooge, Franckx and Gennart. Utilisation pratique de la méthode de simulation dans l'assurance “non-life”.

The paper Prof. Borch submitted, deals essentially with some problems related with random walk theory. Partly it can be considered to be the counterpart of the classical discontinuous ruin theory as its subject is the expected duration of life of the company and maximisation of the discounted value of dividend payments.

The question of calculating the expected duration of life and more specially the means the company has to influence its future lifetime involves reinsurance and utility theory to which topic in recent years Prof. Borch has devoted a number of very interesting publications.

La question est controversée de savoir si l'on doit tenir compte du revenu des placements — et éventuellement des bénéfices sur réalisations de valeurs — pour apprécier le résultat technique de la branche Automobiles.

Rappelons qu'en France, réglementairement, dans les états B3 qui font apparaître le résultat technique de la catégorie, on tient compte des frais généraux, ce qui est normal, mais non des intérêts.

En réalité, il semble bien qu'il soit difficile de faire abstraction d'un élément non négligeable de profits et pertes qui dérive de l'exercice même de l'industrie de l'Assurance, mais la question demeure de savoir dans quelle mesure il faut en tenir compte.

Nous allons montrer que la question est intimement liée à la conception même que l'on se fait du montant des réserves techniques à inscrire au passif du bilan.

On ne peut avoir une idée saine tant des résultats techniques que des revenus de placements que si la comptabilité est établie par exercice de souscription. C'est l'hypothèse que nous retiendrons dans ce qui va suivre).

Un tel exercice donne lieu à encaissement d'un certain montant de primes, au paiement de commissions et de frais généraux et enfin au paiement de sinistres s'échelonnant dans le temps, au delà de la clôture calendrier de l'exercice.

Pour plus de rigueur même — et l'on peut se référer au mode de raisonnement appliqué en Assurance sur la Vie — il faut considérer séparément les différentes échéances de primes, chacune se comportant comme une sorte d'exercice isolé: ensemble des assurances à échéance du Ier janvier, ensemble des assurances à échéance du 2 janvier, etc… jusqu'à celles à échéance du 31 décembre. (Il est indifférent de raisonner sur un ensemble ou „en valeur probable” selon le principe actuariel classique).

Discriminant analysis is an application of multivariate analysis, which may have its use in determining accident risk levels and premiums of industrial enterprises. This paper only aims to give some suggestions. The following questions will be considered.

1. Determining a discriminant function which makes it possible to discriminate between the risk levels of the industrial branches in an efficient way. The industrial branches comprise enterprises with comparable risk levels, hence they are to be considered as homogeneous groups. The function will at the same time serve as a means to classify seperate enterprises into one of these groups.

2. Fixing risk functions which enable us to rank the enterprises of an industrial branch to increasing risk on the ground of observations of a number of variates which characterize the risk situation.

3. Using these risk functions to calculate premiums. The classification-question mentioned under I was the reason to consider the technique of the discriminant analysis. By virtue of the Dutch Industrial Accidents Act every five years a tariffdecree is being published. This decree contains the premiums per wage-unit for the industrial branches. However, there are enterprises e.g. large compound enterprises which do not fall under these regulations. These enterprises ought to be classified according to their own experience. That means we need the knowledge of the risk levels of these particular enterprises in relation to the fixed risk levels of the industrial branches. As mentioned, this is a problem inherent to the typical Dutch situation. It seems, however, probable that such problems and the techniques we intend to sketch have a wider and more general meaning for the accident insurance.

We consider the holder of an individual tontine retirement account, with maximum and minimum withdrawal amounts (per year) specified. The tontine account holder initiates the account at age 65 and earns mortality credits while alive, but forfeits all wealth in the account upon death. The holder wants to maximize total withdrawals and minimize expected shortfall at the end of the retirement horizon of 30 years (i.e., it is assumed that the holder survives to age 95). The holder controls the amount withdrawn each year and the fraction of the retirement portfolio invested in stocks and bonds. The optimal controls are determined based on a parametric model fitted to almost a century of market data. The optimal control algorithm is based on dynamic programming and the solution of a partial integro differential equation (PIDE) using Fourier methods. The optimal strategy (based on the parametric model) is tested out of sample using stationary block bootstrap resampling of the historical data. In terms of an expected total withdrawal, expected shortfall (EW-ES) efficient frontier, the tontine overlay dramatically outperforms an optimal strategy (without the tontine overlay), which in turn outperforms a constant weight strategy with withdrawals based on the ubiquitous four per cent rule.

We present a dynamic pay-how-you-drive pricing scheme for motor insurance using telematics signals. More specifically, our approach allows the insurer to apply penalties to a baseline premium on the occurrence of events such as hard acceleration or braking. In addition, we incorporate a bonus-malus system (BMS) adapted for telematics data, providing a credibility component based on past telematics signals to the claim frequency predictions. We purposefully consider a weekly setting for our ratemaking approach to benefit from the signal’s high-frequency rate and to encourage safe driving via dynamic premium corrections. Moreover, we provide a detailed structure that allows our model to benefit from historical records and detailed telematics data collected weekly through an onboard device. We showcase our results numerically in a case study using data from an insurance company.

The response to the invitation for the 5th Colloquium has been, especially for subject 1, very gratifying; a large number of meritorious papers have been sent in. Fortunately, it took not much time in this case to find a good guide for all this work. R. E. Beard presents in his article “Some Actuarial Aspects of Non-Life Insurance Company Management”) a broad cross-section of the many problems that confront the non-life insurer. It was therefore regarded as not quite unwise to follow his steps and to discuss the other papers at suitable points.

In the first half of his work R. E. Beard deals with a practical example from motor insurance, which is fully calculated with the help of numerical data. The question relates to the estimation of the value of outstanding claims, a quantity which is of decisive importance for the evaluation of the results of a portfolio. Apart from chance fluctuations which might affect the statistics from year to year, there are three major factors that render the task of estimation difficult:

(a) the natural growth of the portfolio

(b) the fact that bigger claims require longer time for settlement than the smaller ones

(c) the change in the value of money with time.

How does a change in the risk-free interest-rate affect the value of a non-life insurance company or portfolio? Risk managers typically argue that there should be little impact as long as assets and liabilities are properly matched. However, the risk-management perspective focuses on existing assets and liabilities, while neglecting the value of future business potential. This paper argues that interest-rate changes can have a significant impact on the appraisal value of a non-life insurance company, even if assets and liabilities are matched. This impact can be positive as well as negative, depending on the underlying parameters. Relevant parameters include reserving intensity, combined ratio, business growth-rate, asset allocation, risk-capital relative to premium income and the correlation between interest-rate and technical insurance results.

We develop a semi-parametric predictor of the IBNR reserve in a macro-model when the claim amount for a certain accident and development year can be expressed in a loglinear form composed of a deterministic part and a random error. We need to make assumptions only on the first two moments of the error, without any specified parametric assumption on its distribution. We give its properties, present its advantages and compare the estimates obtained with various predictors of the IBNR reserve, parametric and non-parametric, using a data set.

In this paper, we study the optimal VIX-linked target benefit (TB) pension design. By applying the dynamic programming approach, we show the optimal risk-sharing structure for the benefit payment exhibits a linear form that consists of three components: (1) a model-robust performance adjustment, (2) a counter-cyclical volatility adjustment that depends on the VIX index, and (3) a TB level that is partially indexed to the cost-of-living adjustment. Differences between our results and the previous literature are highlighted via both theoretical derivations and numerical illustrations.