In various applications of heavy-tail modelling, the assumed Pareto behaviour is tempered ultimately in the range of the largest data. In insurance applications, claim payments are influenced by claim management and claims may, for instance, be subject to a higher level of inspection at highest damage levels leading to weaker tails than apparent from modal claims. Generalizing earlier results of Meerschaert et al. (2012) and Raschke (2020), in this paper we consider tempering of a Pareto-type distribution with a general Weibull distribution in a peaks-over-threshold approach. This requires to modulate the tempering parameters as a function of the chosen threshold. Modelling such a tempering effect is important in order to avoid overestimation of risk measures such as the value-at-risk at high quantiles. We use a pseudo maximum likelihood approach to estimate the model parameters and consider the estimation of extreme quantiles. We derive basic asymptotic results for the estimators, give illustrations with simulation experiments and apply the developed techniques to fire and liability insurance data, providing insight into the relevance of the tempering component in heavy-tail modelling.

The notion of ordering and danger of claim size distributions is extended to claim frequency distributions.

Some of the results obtained in an earlier paper entitled “An Invariance property of the Swiss premium calculation principle” by F. De Vylder and M. Goovaerts (1979) are generalized. For that purpose the notion of additivity and iterativity is extended. Some rather general characterization theorems for some premium calculation principles are obtained.

In this paper, a long-term equilibrium model of a local market is developed. Subject to minor qualifications, the model is arbitrage-free. The variables modelled are the prices of risk-free zero-coupon bonds – both index-linked and conventional – and of equities, as well as the inflation rate. The model is developed in discrete (nominally annual) time, but allowance is made for processes in continuous time subject to continuous rebalancing. It is based on a model of the market portfolio comprising all the above-mentioned asset categories. The risk-free asset is taken to be the one-year index-linked bond. It is assumed that, conditionally upon information at the beginning of a year, market participants have homogeneous expectations with regard to the forthcoming year and make their decisions in mean-variance space. For the purposes of illustration, a descriptive version of the model is developed with reference to UK data. The parameters produced by that process may be used to inform the determination of those required for the use of the model as a predictive model. Illustrative results of simulations of the model are given.

We view the retail non-life insurance decision from the perspective of the insured. We formalize different consumption–insurance problems depending on the flexibility of the insurance contract. For exponential utility and power utility we find the optimal flexible insurance decision or insurance contract. For exponential utility we also find the optimal position in standard contracts that are less flexible and therefore, for certain nonlinear pricing rules, lead to a welfare loss for the individual insuree compared to the optimal flexible insurance decision. For the exponential loss distribution, we quantify a significant welfare loss. This calls for product development in the retail insurance business.

When setting or modifying insurance rates on the basis of observed data, adjustments should be made for inflation between the average claim date of the observation period and the average claim date covered by policies to which the new rates will apply. In automobile insurance ratemaking in the United States, the adjustment is made as follows:

where Lα = trended losses and loss adjustment expenses incurred,

L0 = observed losses and loss adjustment expenses incurred during the experience period,

= the predicted rate of inflation per quarter, and

k = the number of quarters for which losses are being trended.

The inflation coefficient, β, is an exponential growth factor that is estimated using the following equation:

where = average paid claim costs for the year ending in quarter t,

t = time, in quarters, t = 1, 2, …, 12, and

εt = a random disturbance term.

The estimation period for the equation is the twelve quarters immediately preceding the first forecast period, and the estimation is conducted utilizing ordinary least squares. Annual averages of claim costs are used as the dependent variable to smooth random and seasonal fluctuations.

The chain ladder forecast of outstanding losses is known to be unbiased under suitable assumptions. According to these assumptions, claim payments in any cell of a payment triangle are dependent on those from preceding development years of the same accident year. If all cells are assumed stochastically independent, the forecast is no longer unbiased. Section 5 shows that, under rather general assumptions, it is biased upward. This result is linked to earlier work on some stochastic versions of the chain ladder.

In order to fix our ideas an illustration of the theory for (a) a general elementary random process, (b) a compound Poisson process and (c) a Polya process shall be given here below following Ove Lundberg (On Random Processes and Their Application to Accident and Sickness Statistics, Inaug. Diss., Uppsala 1940).

Let the continuous parameter t* be measured on an absolute scale from a given point of zero and consider the random function N* (t*) which takes only non-negative and integer values with N* (o) = o. This function constitutes a general elementary random process for which the conditional probability that N* (t*) = n relative to the hypothesis that shall be denoted , while the absolute probability that N* (t) = n i.e. shall be written If quantities of lower order than dt* are neglected, we may write for the conditional probability that N* (t* + dt*) = n + 1 relative to thehyp othesis that N* (t*) = n, i.e. is the intensity function of the process which is assumed to be a continuous function of t* (the condition of existence for the integral over the given interval of t* for every n > m may be substituted for the condition of continuity). The expectations for an arbitrary but fix value of t* of N* (t*) and p* (t*) will be denoted by the corresponding symbol with a bar so that

If is uniformly bounded for all n in the interval o ≤ t* < T*, where T* is an arbitrary but fix value of t*, we have i.a. that

In preparing the notes on the subjects for discussion at the 4th ASTIN Colloquium at Trieste [1] I used some of the material which formed the basis of a talk given to the Scandinavian Actuarial Societies in September 1962. The papers presented in Trieste have established on a firm mathematical footing the formula for the excess loss premium derived in my note but the discussions also showed that some of the other 1962 material would be of interest.

It will be appreciated that the question originally proposed, i.e. to calculate an excess of loss reinsurance premium when the only information available is the largest claim experienced in each of a succession of periods, was deliberately phrased in this form as being the most troublesome case likely to arise in practice. Clearly if other information is available it would not be rejected in arriving at a premium, but this immediately extends the problem to one of finding the best methods of combining information of different kinds. For example if, say, the largest 5 claims in each period are known, what technique will make the maximum use of the data? Such extensions of the problem are not discussed in this note. Furthermore, I am not unmindful of the valuable comment by Jung, “that there is a natural law which states that you can never get more out of a mincing machine than what you have put into it” [2].

In several different countries there has lately been a very unfavourable development as far as fire losses are concerned. There have been many and very bad big fires which have especially affected industry and trade. The facts which can be considered to have caused this development are well-known. Bigger and bigger and more concentrated units of production, business enterprises, and storing form an accumulation of values which in itself implies increasing risks. On account of production-technical and business-economical conditions it has not always been possible to divide the buildings into sections desirable from a fire service point of view. The new production methods have sometimes caused new fire risks.

The Northern Central Office for Fire Insurance Statistics from Mutual Companies (CNÖB) has made an investigation in order to see to what extent the international trend towards increasing fire losses has any equivalence in the Northern countries (Denmark, Finland, Norway, and Sweden). I.a. there was made a graphical representation of how the observed rate of net risk premium of the portfolio has developed during the period 1946–1966, the result of which may be seen from fig. 1. No increasing trend can be observed from this. After having been decreasing during the immediate postwar time, the observed rate of net risk premium has been fairly constant from the beginning of the fifties, perhaps even somewhat decreasing except during the later years. The conclusion is that in so far facts as mentioned above have influenced the development of claims of the portfolios or parts of these in a deteriorating way, there must in that case have been circumstances which have worked as neutralizers.

Japan is one of the most earthquake-prone countries in the world and a considerable volume of earthquake insurance is written by private insurers. Therefore, a report on earthquake insurance in Japan has been prepared for the 13th ASTIN Colloquium.

We will first summarise, from recent results of seismological studies, the matters which would form the basis for a study of earthquake insurance, and then explain how this insurance actually works in Japan.

We compute the optimal investment strategy for an individual who wishes to minimize her probability of lifetime ruin. The financial market in which she invests consists of two riskless assets. One riskless asset is a money market, and she consumes from that account. The other riskless asset is a bond that earns a higher interest rate than the money market, but buying and selling bonds are subject to proportional transaction costs. We consider the following three cases. (1) The individual is allowed to borrow from both riskless assets; ruin occurs if total imputed wealth reaches zero. Under the optimal strategy, the individual does not sell short the bond. However, she might wish to borrow from the money market to fund her consumption. Thus, in the next two cases, we seek to limit borrowing from that account. (2) We assume that the individual pays a higher rate to borrow than she earns on the money market. (3) The individual is not allowed to borrow from either asset; ruin occurs if both the money market and bond accounts reach zero wealth. We prove that the borrowing rate in case (2) acts as a parameter connecting the two seemingly unrelated cases (1) and (3).

The aging of population is perhaps the most important problem that developed countries must face in the near future. Dependency can be seen as a consequence of the process of gradual aging. In a health context, this contingency is defined as a lack of autonomy in performing basic activities of daily living that requires the care of another person or significant help. In Europe in general and in Spain in particular, this phenomena represents a problem with economic, political, social and demographic implications. The prevalence of dependency in the population, as well as its intensity and evolution over the course of a person’s life are issues of greatest importance that should be addressed. The aim of this work is the estimation of life expectancy free of dependency (LEFD) based on functional trajectories to enhance the regular estimation of health expectancy. Using information from the Spanish survey EDAD 2008, we estimate the number of years spent free of dependency for disabled people according to gender, dependency degree (moderate, severe, major) and the earlier or later onset of dependency compared to a central trend. The main findings are as follows: first, we show evidence that to estimate LEFD ignoring the information provided by the functional trajectories may lead to non-representative LEFD estimates; second, in general, dependency-free life expectancy is higher for women than for men. However, its intensity is higher in women with later onset on dependency; Third, the loss of autonomy is higher (and more abrupt) in men than in women. Finally, the diversity of patterns observed at later onset of dependency tends to a dependency extreme-pattern in both genders.

The Benktander (1976) and Neuhaus (1992) credibility claims reserving methods are reconsidered in the framework of a credible loss ratio reserving method. As a main contribution we provide a simple and practical optimal credibility weight for combining the chain-ladder or individual loss ratio reserve (grossed up latest claims experience of an origin period) with the Bornhuetter-Ferguson or collective loss ratio reserve (experience based burning cost estimate of the total ultimate claims of an origin period). The obtained simple optimal credibility weights minimize simultaneously the mean squared error and the variance of the claims reserve. We note also that the standard Chain-Ladder, Cape Cod and Bornhuetter-Ferguson methods can be reinterpreted in the credible context and extended to optimal credible standard methods. The new approach is inspired from Mack (2000). Two advantages over the Mack method are worthwhile to be mentioned. First, a pragmatic estimation of the required parameters leads to a straightforward calculation of the optimal credibility weights and mean squared errors of the credible reserves. An advantage of the collective loss ratio claims reserve over the Bornhuetter-Ferguson reserve in Mack (2000) is that different actuaries come always to the same results provided they use the same actuarial premiums.

We put one of the predictions of adverse-selection models to the test, using data from the Danish automobile insurance market: that there is a positive correlation between claims risk and insurance coverage. We can find a statistically significant insurance coverage--risk correlation when coverage is expressed relative to the insurance premium, but not when it is expressed in monetary terms.

In the present discussion we point out the relation of some results in Dickson & Waters (1999) to similar results in Sundt (1998a, b).

We shall need some notation. For a positive integer m let m be the set of all m × 1 vectors with positive integer-valued elements and m+ = m ~ {0}. A vector will be denoted by a bold-face letter and each of its elements by the corresponding italic with a subscript denoting the number of the elements; the subscript · denotes the sum of the elements. Let m0 be the class of probability functions on m with a positive probability at 0 and m+ the class of probability functions on m+. For j = 1,…, m we introduce the m × 1 vector ej where the jth element is one and all the other elements zero. We make the convention that summation over an empty range is equal to zero.

Let g ∈ m0 be the compound probability function with counting distribution with probability function v ∈ 10 and severity distribution with probability function h ∈ m+; we shall denote this compound probability function by v V h. Sundt (1998a) showed that

where φv denotes the De Pril transform of v, given by

Motivated by (2) Sundt (1998a) defined the De Pril transform φg of g by

This defines the De Pril transform for all probability functions in m0. Insertion of (2) in (3) gives

and by solving φg(X) we obtain

Sundt (1998a) studies the De Pril transform defined in this way and found in particular that it is additive for convolutions.

1.1. — In this paper we shall consider some of the decisions which have to be made in the normal course of business in an insurance company. We shall see that the “right” decisions can be found only when the problems are analysed in their proper dynamic context.

As examples of the decision problems which we shall study, we can mention the following:

(i) What premium rates should be quoted on the insurance contracts, which the company offers to the public?

(ii) How much should the company spend to promote the sale of its policies?

(iii) When should the company refuse to underwrite a proposed insurance contract?

(iv) How shall the company reinsure its portfolio of insurance contracts?

(v) What reserve funds should an insurance company keep?

(vi) How shall the company's funds be invested?

Any actuary will be familiar with such problems, and he will probably feel that these problems cannot be satisfactorily solved with the methods offered by the classical actuarial theory.

1.2. — In some earlier papers [I] and [2] it has been argued that such problems can best be solved in the frame work of utility theory. As an illustration we shall take Problem (iii) in the preceding paragraph, and consider an insurance company in the following situation:

(i) The company has a capital S.

(ii) The company holds a portfolio of insurance contracts which will lead to a total payment of x to settle claims. F1(x) is the distribution of the variate x.

En matière de détermination de la prime de réassurance en “excédent de sinistres”, nous nous trouvons devant le problème de la distribution des valeurs extrêmes, duquel nous avons déjà une littérature vaste et bien connue, plus particulièrement en ce qui concerne l'allure asymptotique de la distribution qui nous intéresse. Dans cette note on veut indiquer, dans ses grandes lignes, un procédé largement dégagé des résultats de la théorie mentionnée, qui permet un examen de l'influence réciproque entre les hypothèses et la signification des résultats expérimentaux qui est dans l'esprit du procédé par induction.

Pour fixer l'attention sur l'allure asymptotique de la distribution, sans nous occuper de sa forme d'ensemble, nous n'examinerons que la distribution des sinistres qui dépassent une certaine valeur ξ. En particulier, nous pouvons choisir pour cette valeur la limite assignée pour une certaine forme de réassurance; d'ailleurs, on peut penser à un choix arbitraire, ce qui se fera également dans cette note, où elle sera supposée indéterminée à priori, afin que l'on puisse la choisir sur la base des résultats obtenus.

Soit done ξ l'extrême inferieur des sinistres, exprimés en termes monétaires, dont la distribution est à l'étude; et soit F(x;ξ = P(X ≤ x | X ˃ ξ) la fonction de répartition correspondante. En pratique, nous pourrons assumer comme F(x;ξ) une fonction, opportunément simple, qui respecce nos hypothèses dans un voisinage droit suffisamment grand de ξ. Cette précision sera sousentendu par la suite, de sorte que, lorsque nous considérerons le simple exemple concret exposé plus bas, nous devrons tenir compte du fait que la foncion F(x; ξ) considérée n'est qu'une approximation, valide dans un voisinage droit de ξ, de la distribution effective.

In this paper we propose a discrete-time model with fixed maximum time to maturity of traded bonds. At each trading time, a bond matures and a new bond is introduced in the market, such that the number of traded bonds is constant. The entry price of the newly issued bond depends on the prices of the bonds already traded and a stochastic term independent of the existing bond prices. Hence, we obtain a bond market model for the reinvestment risk, which is present in practice, when hedging long term contracts. In order to determine optimal hedging strategies we consider the criteria of super-replication and risk-minimization.

To some extent, this paper provides theoretical back-up for Taylor's (1980b) see-saw method of analysis of outstanding claims.

Section 3 deals with the payments per claim finalized (PPCF) method. This assumes PPCF to be a function of development year only.

Section 4 carries out a theoretical investigation of the validity of this assumption. It is found that the assumption is not justified in general; that PPCF can (and usually will) be sensitive to changes in speed of finalization of claims; and that this sensitivity will depend on the convexity properties of PPCF as a function of development year. The analysis is made on the basis of the so-called hypothesis of invariant order.

Section 5 develops the see-saw method from this theoretical basis, although Section 7 is at pains to point to situations in which that method remains applicable despite violation of the hypothesis of invariant order.

Section 6 provides a numerical example.