The three annual 2¼% interest coupons of the Winterthur Insurance convertible bond (face value Chf 4 700) will only be paid out if during their corresponding observation periods no major storm or hail storm on one single day damages at least 6 000 motor vehicles insured with Winterthur Insurance. Data for events, where storm or hail damaged more than 1 000 insured vehicles, are available for the last ten years. Using a constant-parameter model, the estimated discounted value of the three Wincat coupons together is Chf 263.29. A conservative evaluation, which accounts for the standard deviation of the estimate, gives a coupon value of Chf 238.25. However, fitting models which admit a trend or a change-point, leads to substantially higher knock-out probabilities of the coupons. The estimated discounted values of the coupons can drop below the above conservative value; a conservative evaluation as above leads to substantially lower values. Hence, already the model uncertainty is higher than the standard deviations of the used estimators. This shows the dominance of the model risk. Consistency, dispersion, robustness and sensitivity of the models are analysed by a simulation study.

We study a stochastic differential game problem between two insurers, who invest in a financial market and adopt reinsurance to manage their claim risks. Supposing that their reinsurance premium rates are calculated according to the generalized mean-variance principle, we consider the competition between the two insurers as a non-zero sum stochastic differential game. Using dynamic programming technique, we derive a system of coupled Hamilton–Jacobi–Bellman equations and show the existence of equilibrium strategies. For an exponential utility maximizing game and a probability maximizing game, we obtain semi-explicit solutions for the equilibrium strategies and the equilibrium value functions, respectively. Finally, we provide some detailed comparative-static analyses on the equilibrium strategies and illustrate some economic insights.

Two approaches used to model interest randomness are presented. They are the modeling of the force of interest accumulation function and the modeling of the force of interest. The expected value, standard deviation and coefficient of skewness of the present value of annuities-immediate are presented as illustrations. The implicit behavior of the force of interest under the two approaches is investigated by looking at a particular conditional expectation of the force of interest accumulation function.

For the infinite time ruin probability in the classical risk process, efficient estimators are proposed in cases in which the claim amount distribution is unknown. Confidence intervals are computed which are based on normal approximations or on the bootstrap method. The procedures are checked in a Monte-Carlo study.

Ratemaking is one of the most important tasks of non-life actuaries. Usually, the ratemaking process is done in two steps. In the first step, a priori ratemaking, an a priori premium is computed based on the characteristics of the insureds. In the second step, called the a posteriori ratemaking, the past claims experience of each insured is considered to the a priori premium and set the final net premium. In practice, for automobile insurance, this correction is usually done with bonus-malus systems, or variations on them, which offer many advantages. In recent years, insurers have accumulated longitudinal information on their policyholders, and actuaries can now use many years of informations for a single insured. For this kind of data, called panel or longitudinal data, we propose an alternative to the two-step ratemaking approach and argue this old approach should no longer be used. As opposed to a posteriori models of cross-section data, the models proposed in this paper generate premiums based on empirical results rather than inductive probability. We propose a new way to deal with bonus-malus systems when panel data are available. Using car insurance data, a numerical illustration using at-fault and non-at-fault claims of a Canadian insurance company is included to support this discussion. Even if we apply the model for car insurance, as long as another line of business uses past claim experience to set the premiums, we maintain that a similar approach to the model proposed should be used.

This paper describes the dynamic financial analysis model currently being used by a property catastrophe reinsurer to manage its business. The model is-an integral part of the day-to-day operations at the Company, and is used as a decision making tool in the underwriting, investment, and capital management processes. The paper begins by describing the framework that the Company uses for risk management. This includes a classification of the risks facing the Company, which is used to define and prioritize their implementation in the model. Also included is a description of the conceptual approach the Company takes to evaluate the tradeoff between risk and return. The paper then goes on to describe the structure and operation of the dynamic financial analysis model and provides examples of its use at the Company, along with illustrative examples of the various types of output it produces.

One of the most critical problems in property/casualty insurance is to determine an appropriate reserve for incurred but unpaid losses. These provisions generally comprise most of the liabilities of a non-life insurance company. The global provisions are often determined under an assumption of independence between the lines of business. Recently, Shi and Frees (2011) proposed to put dependence between lines of business with a copula that captures dependence between two cells of two different runoff triangles. In this paper, we propose to generalize this model in two steps. First, by using an idea proposed by Barnett and Zehnwirth (1998), we will suppose a dependence between all the observations that belong to the same calendar year (CY) for each line of business. Thereafter, we will then suppose another dependence structure that links the CYs of different lines of business. This model is done by using hierarchical Archimedean copulas. We show that the model provides more flexibility than existing models, and offers a better, more realistic and more intuitive interpretation of the dependence between the lines of business. For illustration, the model is applied to a dataset from a major US property-casualty insurer, where a bootstrap method is proposed to estimate the distribution of the reserve.

A model for the claim number process is considered. The claim number process is assumed to be a weighted Poisson process with a three-parameter gamma distribution as the structure function. Fitting of this model to several data encountered in the literature is considered, and the model is compared with the two-parameter gamma model giving the negative binomial distribution. Some credibility theory formulae are also presented.

A practical method is developed for computing moments of insurance functions when interest rates are assumed to follow an autoregressive integrated moving average process.

Traditional Chain Ladder models are based on a few cells in an upper triangle and often give inaccurate projections of the reserve. Traditional stochastic models are based on the same few summaries and in addition are based on the often unrealistic assumption of independence between the aggregate incremental values. In this paper a set of stochastic models with weaker assumptions based on the individual claims development are described. These models can include information about settlement and can handle seasonal effects, changes in mix of business and claim types as well as changes in mix of claim size. It is demonstrated how the distribution of the process can be specified and especially how the distribution of the reserve can be determined. The method is illustrated with an example.

In this paper some results are given on the addivity of chain-ladder projections. Given two claims development triangles, when do their chain-ladder projections add up to the projections of the combined triangle, that is the triangle being the element-wise sum of the two given triangles?

Necessary and sufficient conditions for equality are given. These are of a fairly simply form and are directly connected to the ordinary chain-ladder calculations. In addition, sufficient conditions of the same form are given for inequality between the combined projection vector and the sum of the two original projections vectors.

The operation of a bonus-malus system, superimposed on a premium system involving a number of other rating variables, is considered. To the extent that good risks are rewarded in their base premiums, through the other rating variables, the size of the bonus they require for equity is reduced. This issue is discussed quantitatively, and a numerical example given.

The approximation of the individual risk model by a compound Poisson model plays an important role in computational risk theory. It is thus desirable to have sharp lower and upper bounds for the error resulting from this approximation if the aggregate claims distribution, related probabilities or stop-loss premiums are calculated.

The aim of this paper is to unify the ideas and to extend to a more general setting the work done in this connection by Bühlmann et al. (1977), Gerber (1984) and others. The quality of the presented bounds is discussed and a comparison with the results of Hipp (1985) and Hipp & Michel (1990) is made.

This paper studies optimal risk redistribution between firms, such as banks or insurance companies. The introduction of the Basel II regulation and the Swiss Solvency Test has increased the use of risk measures to evaluate financial or insurance risk. We consider the case where firms use a distortion risk measure (also called dual utility) to evaluate risk. The paper first characterizes all Pareto optimal redistributions. Thereafter, it characterizes all competitive equilibria. It presents three conditions that are jointly sufficient for existence of a unique equilibrium redistribution. This equilibrium's redistribution and prices are provided in closed form via a representative agent.

In a recent paper on the theory of demand for insurance Arrow [1] has proved that the optimal policy for an insurance buyer is one which gives complete coverage, beyond a fixed deductible. The result is proved under very general assumptions, but its content can be illustrated by the following simple example.

Assume that a person is exposed to a risk which can cause him a loss x, represented by a stochastic variable with the distribution F(x). Assume further that he by paying the premium P(y) can obtain an insurance contract which will guarantee him a compensation y(x), if his loss amounts to x. The problem of our person is to find the optimal insurance contract, i.e. the optimal function y(x), when the price is given by the functional P(y).

In order to give an operational formulation to the problem we have outlined, we shall assume that the person's attitude to risk can be represented by a Bernoulli utility function u(x), and we shall write S for his “initial wealth”. His problem will then be to maximize

when the functional P(y) is given, and y(x) є Y. The set Y can be interpreted as the set of insurance policies available in the market. It is, natural to assume that o ≤ y(x) ≤ x, but beyond this there is no need for assuming additional restrictions on the set Y.

Largest claims reinsurance covers are reconsidered. Allowing the original claims sizes to be not necessarily independent, a new, upper premium bound is derived and explored.

It is commonly thought that the characteristic function (Fourier transform) of the Pareto distribution has no known functional form (e.g. Seal, 1978, pp. 14, 40, 57). This is quite untrue. Nevertheless the characteristic function of the Pareto density is conspicuously absent from standard reference works even when the Pareto distribution itself receives substantial comment (e.g. Haight, 1961; Johnson and Kotz, 1970, Ch. 19; Patel, Kapadia and Owen, 1976, § 1. 5).

The Pareto density may be written

with distribution function

mean = p/(v − 1) and variance = b2v(v − 1)2(v−2). These are infinite when v≤1 and v≤2, respectively. Its Laplace transform (s= c + iu)

where E is the generalized exponential integral (Pagurova, 1961) and can be written in terms of incomplete gamma or confluent hypergeometric functions (Slater, 1960, Sec. 5.6). When s = − it β(s) becomes the characteristic function (see Appendix I).

As Benktander (1970) tells us, the Pareto distribution has been particularly successful at representing the distribution of the larger claim amounts. In earlier years it was employed to represent the distribution of life insurance sums assured but more recently it has been used for the claim distributions of fire and automobile insurance. Table 1 provides the v-values we have been able to locate. Note that the variance of the distribution is infinite when v≤2 and if it were not for the anomalous v-values of Andersson (1971) we would have ventured the opinion that modern claim data encourage the assumption that v>2. In our numerical work we have used v = 2.7 and smaller values might change some of the computer rules we have proposed in Appendix II.

A simple model for IBNR claims is presented. Estimates for the loss reserves and for the ultimate claims rate are derived. Approximations to the mean square error of the estimators are produced. A more specific parametric model is suggested for the case that we deal with claim numbers instead of claim amounts. The general method is illustrated by a practical application to the pricing of a casualty excess of loss cover.

We compute a merit-rating system for automobile third party liability insurance by two different ways, both with the help of an exponential utility function.

(i) We apply the principle of zero utility to exponential utilities.

(ii) We break the symmetry between the overcharges and the undercharges by weighting them differently through the introduction of a utility function, in order to penalize the overcharges.

The results are applied to the portfolio of a Belgian company and compared to the premium system provided by the expected value principle.

Deux méthodes différentes, basées sur l'emploi de fonctions d'utilité exponentielles nous permettent de définir un système bonus-malus en assurance automobile:

(i) le principe de l'utilité nulle;

(ii) la pénalisation des injustices de la compagnie, obtenue en pondérant les erreurs de prime au moyen d'une fonction d'utilité de manière à briser la symétrie entre les primes trop élevées et les primes trop basses.

Les résultats théoriques sont appliqués au portefeuille d'une compagnie belge et comparés aux primes fournies par le principe de l'espérance mathématique.

Suppose that an automobile insurance plan is characterized by a double classification. The risks are thus divided into classes, i = 1, 2, …, p (e.g. defined by use of car and age of operator), and groups j = 1, 2, …, q (e.g. defined by licence and by accidents during the last three years). The experience of the company is described by the observed “relative loss ratios” rij and some measure of exposure nij. A general model, often used, is that the rij: s are observations of random variables with the expected values gij = g(αi, βj), where the relativities αi are parameters representing the classes i and the relativities βj represent the influence of the groups j. One of the ratemaker's problems is to find a realistic function g(α, β) and to obtain estimates ai of αi and bj of βj.

In their paper “Two Studies in Automobile Insurance Rate-making” (ASTIN Bulletin Vol. I, Part IV, page 192-217) Robert Bailey and LeRoy Simon have thoroughly analyzed this problem for private passenger automobiles in Canada. They have principally studied three different types of the function g(α β), namely g(α β) = αβ (Method 2), g(α β) = α + β (Method 3) and g(α β) = 3αβ— 2 (Method 4). The authors show in an appendix, that the variance of rij is approximately g(αi βj)/Knij where K ≅ 0.005 for the Canadian data. They estimate the relativities αi and βj; by making χ2 = K. Σnij(rij — gij)2/gij a minimum. For the Canadian material, the “method 4” agrees best with the observations. This method gives an observed χ2 value of about 8 for 11 degrees of freedom.