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.

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.

Let Sn be the set of motor claims Si, i = 1, 2, …, N occurred during a given year n, namely Sn represents the set of claims relevant to the generation (or cohort) n.

If Tn is a subset (even empty) of claims resulting without payement (that is the set of zero-claims), the set Pn = Sn − Tn shall denote the set of claims that should be settled.

For every si ∈ Pn, we can define the r.v. Xi which represents the period of time required for its settlement (namely the lag of settlement).

It is not sensible to deem that the r.v. Xi are equally distributed: as a matter of fact we know that the larger is the claim, the longer the lag of payement.

However, we can assume that in a subset U of Pn, the r.v. Xi(U) have the same distribution function, which will be denoted by FU(x) or in short F(x).

As F(x) represents the probability that a claim sj ∈ U is settled within a period o − x, the function 1 − F(x) = l(x) denotes the probability that the claim results unsettled after a lag x, that is the survival function of the claim.

In this study we intend to find an analytical expression of the function 1(x) on the basis of particular assumptions about the behaviour of the adjuster with regard to the settlement of claims.

The assumptions will be tested by fitting the function to some observed data.

One of the basic problems in life is: Given information (from the past), make decisions (that will affect the future). One of the classical actuarial examples is the adaptive ratemaking (or credibility) procedures; here the premium of a given risk is sequentially adjusted, taking into account the claims experience available when the decisions are made.

In some cases, the rates are fixed and the premiums cannot be adjusted. Then the actuary faces the question: Should a given risk be underwritten in the first place, and if yes, what is the criterion (in terms of claims performance) for cancellation of the policy at a later time?

Recently, Cozzolino and Freifelder [6] developed a model in an attempt to answer these questions. They assumed a discrete time, finite horizon, Poisson model. While the results lend themselves to straightforward numerical evaluation, their analytical form is not too attractive. Here we shall present a continuous time, infinite horizon, diffusion model. At the expense of being somewhat less realistic, this model is very appealing from an analytical point of view.

Mathematically, the cancellation of policies amounts to an optimal stopping problem, see [8], [4], or chapter 13 in [7], and (more generally) should be viewed within the framework of discounted dynamic programming [1], [2].

As we grapple with the problems of conducting non-life insurance in a world of high inflation, we should perhaps pause for a moment to look for historical analogies for our present predicament. Let us, then, look back at a distressing period many years ago in the history of the Kingdom of Carmania.

Inflation was by no means new to the citizens of Carmania. Year after year, all prices and incomes had been rising at a steady rate of 5% per annum. The economy functioned quite well, albeit in a rather uninteresting way, and the people were accustomed to the idea that the purchasing power of the Carmanian dollar would fall by roughly 5% every year. There were, however, some economists who argued that inflation was both undesirable and unnecessary, and prevailed upon the King to adopt some measures which, they assured him, would quickly reduce the rate of inflation to zero. The measures were adopted, and the following year the rate of inflation rose to 10%.

At this point a rival group of economists explained that this unfortunate development was simply what should have been expected, and they set out an alternative policy which would undoubtedly reverse the trend and bring inflation quickly under control. Convinced by the weight of their arguments, the King introduced what became known as Phase 2 of the counter-inflation policy. By the next year, the rate of inflation had risen to 20%.

A theorem of Borch characterizing Pareto-optimal treaties in a reinsurance market is extended to the case of non-differentiable utilities. Sufficient conditions for the existence of a solution to the equations are established. The problem is then shown to be identical to the determination of the value of a cooperative non-transferable m-person game. We show how to compute the Shapley value of this game, then we introduce a new value concept. An example illustrates both methods.

In motorcar insurance is widely used a merit rating system characterized by the fact that only the number of claims occurred (and not their amount) modifies the premium. This system induces the insured drivers to support themselves the cost of the cheap claims. We analyze this “hunger for bonus” and solve this decision problem by means of an algorithm related to dynamic programming. The method is then applied to the Belgian bonus system.

One of the most important properties of a distribution function is that it fits the data well enough for the decision-makers' or analysts' purposes. The statisticians' problem is to select a specific form for the distribution function and to determine its parameters from the available data. Various methods (graphical method, method of moments, maximum likelihood method) are available for that purpose.

In many real world situations a single distribution function, however, may not be appropriate over the entire range of the available data. This suggests that the underlying process changes over the range of the respective variable. This fact should be considered in curve fitting. A typical example of such a situation is given in Figure 1 representing third party liability losses for trucks.

It is interesting to speculate about the different raisons d'être (Seal [5]) for the observed discontinuity. It may be the result of out-of-court or in-court settlements or could stem from differences between bodily injury and property damages.

By experience rating the main problem is to estimate the credibilities. We have for the credibility αk the famous formula)

but it is often troublesome to find suitable estimates for the variances and . In the present paper a general method to estimate them from the actual statistics is given.

A disadvantage of the method is that good estimates require relatively extensive statistical material. If one of the variances is known, the method can be easily modified to give the other variance from statistics of moderate size.

The method is based on the Maximum Likelihood principle and leads to a system of non-linear equations. The equations can be solved by an iterative process, easily programmable for computers.

The mathematical model underlying the experience rating problem differs in our case lightly from the usual one.

This note is an attempt to put the problems referring to the reserves for outstanding claims into a simple understandable form in order to facilitate the discussion of the difficult questions. In that purpose I have taken up some of Harald Bohman's ideas of the subject). I find it convenient to start with the simplest case where the liability consists of index-regulated payments at fixed epochs. My presentation is restricted to reserves of incurred and reported claims.

It seems that there are people who are prepared to accept what the numerical analyst would regard as a shockingly poor approximation to F (x, t), the distribution function of aggregate claims in the interval of time (o, t), provided it can be quickly produced on a desk or pocket computer with the use of standard statistical tables. The so-called NP (Normal Power) approximation has acquired an undeserved reputation for accuracy among the various possibilities and we propose to show why it should be abandoned in favour of a simple gamma function approximation.

Discounting encomiums on the NP method such as Bühlmann's (1974): “Everybody known to me who has worked with it has been surprised by its unexpectedly good accuracy”, we believe there are only three sources of original published material on the approximation, namely Kauppi et al (1969), Pesonen (1969) and Berger (1972). Only the last two authors calculated values of F(x, t) by the NP method and compared them with “true” four or five decimal values obtained by inverting the characteristic function of F(x, t) on an electronic computer.

In recent years, as a result of more concentrated research together with the ravages wrought upon some insurers by inflation, the fundamental significance of the so-called run-off triangle in the calculation of provisions for outstanding claims has been increasingly recognised. The run-off triangle, which is a two-way tabulation—according to year of origin and year of payment—of claims paid to date, has the following form, where Cij is the amount paid by the end of development year j in respect of claims whose year of origin is i, i.e. Cij is the total amount paid in year of origin i and the following j years.

The information relating to the area below this triangle is unknown since it represents the future development of the various cohorts of claims.

In this paper some ruin probabilities are calculated for an example of a lognormal claim distribution. For that purpose it is shown that the lognormal distribution function, Λ(y), may be written in the form

where V(x) is absolutely continuous and without being a distribution function preserves some useful properties of such a function.

An attempt is also made to give an approximant Λα(y) to Λ(y) such that Λα(y) is a linear combination of a low number of exponential distributions. For comparison, ruin probabilities are also calculated for two examples of Λα(y).

In the considered numerical cases it is assumed that the occurrence of claims follows a Poisson process.

The paper considers the problem of finding an upper bound for the Stop loss premium.

We will start with a brief sketch of the practical context in which this problem is relevant.

If it is reasonable to assume, that the accumulated claims variable of the underlying risk can be represented by a Compound Poisson Process, the following data are needed for fixing the Stop loss premium:

— the claims intensity,

— the distribution of the claimsizes (jump-size variable).

In practical situations it is usually possible to find a reasonable estimate for the claims intensity (expected number of claims in a given period).

Generally speaking, however, it is not so easy to get sufficient data on the claimsize distribution. Ordinarily only its mean is known. This deficiency in information can of course be offset by assuming the unknown distribution to be one of the familiar types, such as Exponential, Gamma or Pareto.

Stop loss premiums are however very sensitive to variations in the type of claimsize distribution and consequently it can make a lot of difference in the result what particular choice is made.

To gain some insight into the consequences of a specific assumption, it is useful to know within what range the premium can move for varying distributional suppositions. This means establishing an upper bound and a lower bound. The lower bound is trivially obtained if the mass of the claimsize distribution is solely concentrated at its mean. The upper bound on the other hand should correspond to the “worst” possible claimsize distribution. This means, that we have to look for a distribution which maximizes the Stop loss premium.

Many studies concerning the frequency of claims by size in health insurance are not generally known). A possible explanation of this circumstance could be the fact that in most countries this line of insurance has been brought entirely within the ambit of social insurance. Also from the side of the social insurance very few investigations have been published).

In this paper we will analyse the claim experience (relating, to the calendar year 1972) of a private health insurance business. The data have been subdivided according to three levels of coverage (in increasing order of benefits these are: class III, class IIb and class IIa). The claim payments comprise nursing costs, auxiliary costs and the fees for specialist treatment in and out of the hospital.

We will use the following notations:

Si: claim amount paid for the insured i in one year,

n: number of claims,

v: number of risks (policies insured).

In many instances the premium is simply determined as a level premium. In other words each insured pays the premium p, calculated as follows:

.

Actually we make the assumption that the claims are normally distributed, the parameters of which can be estimated as follows:

which permits the calculation of the premium according to:

.