We apply simple geometrical arguments to show that well-known approaches to determine the premium in insurance contract minimize a weighted squared differences both between the individual premiums and the individual claims and between the total premiums for classes of homogeneous risks and total claims from these blocks of business.

We gıve a complete parametrıc solutıon of the followıng problem: Fınd a claım sıze dıstrıbutıon F on the fınıte ınterval [ο, ω], maxımizıng the stop-loss premıum correspondıng to a gıven excess e, under the constraınts that the fırst moment of F be at most equal to μ and the second at most equal to ν The method used ıs the dualıty technıque ın semı-contınuous lınear programmıng descrıbed in De Vylder (1978) Thıs technıque ıs summarızed, wıthout proofs, ın the fırst part of the paper.

A simple upper bound for the variance of the frequency estimates in a multivariate tariff using class criteria is deduced. This upper bound is based exclusively on univariate statistics and can, therefore, be calculated before a GLM analysis is carried out. It can be used to estimate the number of claims that will be needed for a tariff calculation depending on the number of tariff criteria and the number of levels of each criterion.

The article is a revised version of a paper presented at the XXXIst ASTIN Colloquium in Porto Cervo.

The paper develops a methodology to enable microscopic models of transportation systems to be accessible for a statistical study of traffic accidents. Our approach is intended to permit an understanding not only of historical losses but also of incidents that may occur in altered, potential future systems. Through such a counterfactual analysis, it is possible, from an insurance, but also from an engineering perspective, to assess the impact of changes in the design of vehicles and transport systems in terms of their impact on road safety and functionality.

Structurally, we characterize the total loss distribution approximatively as a mean-variance mixture. This also yields valuation procedures that can be used instead of Monte Carlo simulation. Specifically, we construct an implementation based on the open-source traffic simulator SUMO and illustrate the potential of the approach in counterfactual case studies.

We propose a flexible lattice model to evaluate the fair value of insurance contracts embedding both financial and actuarial risk factors. Flexibility relies on the ability of the model to manage different specifications of the correlated processes governing interest rate, mortality, and fund dynamics, thus allowing the insurer to make the most appropriate choices. The model is also able to handle additional guarantees like a surrender opportunity for which explicit formulae are not available being it similar to an American derivative. The model discretizes mortality and interest rate dynamics through two different binomial lattices and then combines them into a bivariate tree characterized by the presence of four branches for each node. The probability of each branch is defined to replicate the correlation affecting the two processes. The bivariate model is useful to compute the value of survival zero coupon bond. When adding another source of risk, such as the fund dynamics for evaluating fund-linked insurance products, we model it through a bivariate tree that captures the influence of the interest rate on its drift term. Then, the mortality risk is embedded by defining a trivariate tree presenting eight branches emanating from each node with probabilities defined in order to capture the correlations of the processes. Extensive numerical experiments assess the model accuracy by considering some stylized policies, but the model application is not limited to them being it able to manage different contract specifications.

The purpose of this paper is to describe a technical procedure, which enables one to compute values of the generalised Poisson distribution function, with an accuracy considered sufficient for insurance companies and with satisfactory speed. The procedure requires a fast medium sized computer.

The computation of values of the generalised Poisson distribution function has become a timely problem in Finland, because of the introduction by the Supervisory Service of more stringent requirements in determining limits of the so called equalisation reserves, which have their theoretical basis in the random fluctuations of claims amounts. The question has also been discussed in papers submitted by Dr. Pentikainen [3] and Dr. Pesonen [4] to this Colloquium. Because the practical computation is a further problem, the Federation of Finnish Insurance Companies set up a committee in 1962 to gather and work up the necessary statistics from various branches of insurance and to develop the computational methods ready for use in practice. The committee has almost completed its work, and one of the results, a procedure to compute values of the generalised Poisson function with a mixed method, is described below. The method is referred to briefly in [4]. The programming and further planning work has been done by Dr. Loimaranta and M. Sc. Porn.

The aim of this study is to find suitable methods for utilizing available statistical information on the risk development of terminated insurances. This information may comprise not only data respecting sex, age, insurance tariff, etc., but also the points of time when certain events—disability, injuries or damages—have occurred as well as their durability and cause. The method study here presented is based on statistical material gathered from long-term disability insurances which have ceased to be valid, either in consequence of the expiration of the term of insurance, or the insured's death. We endeavour to study the point of time when the first event (disability) occurred and the relation between subsequent events, searching suitable methods for assessing risks after the occurrence of the primary event.

The influence of passed events upon the future risk process may be explained by either a direct dependence between actual events—actual damage or disability may have an impairment for the future of the risk—or by heterogeneity a priori. In the former case, we may speak of a contagious process (Polya-Eggenberger), and, in the latter case, the occurrence or non-occurrence of an event may, according to Lexis and Newbold, act as a risk differentiating factor providing a theoretical basis for a technique of experience rating.

It is difficult to decide whether the relationship noted in respect of a risk process is of the direct or the indirect kind. A direct dependence between one event and the following ones has no relevance to a study of the incidence of the first event and the time of its occurrence. We avoid in such a study the obstacles created by formal rules and regulations with regard to the problem of whether two or more disability periods occurring at short intervals should be considered as a single disability period or not. In order to measure the frequency of the first event we have—as in the case of measuring mortality—to limit the period exposed to risk until the occurrence of the first event.

An insurance company aims to increase the volume of its business.

The volume is measured by the premium income during one year.

The company makes use of N tariff groups.

pn = premium charged by the company per unit of insurance in tariff group n.

qn = premium per unit of insurance in tariff group n corresponding to the statistical experience in the said group.

Ln = The size of the market in tariff group n. If all insurances in the group pay the same premium and if the company is able to cover the whole of its potential market then Ln is equal to the number of policies. If the premium per policy is equal to pn times the sum insured and if the company is able to cover the whole of its potential market then Ln is equal to the total sum insured.

Fn(p) = the proportion of Ln that actually buys insurance in tariff group n if the company charges the premium p per unit of insurance.

In pricing insurance contracts based on the individual policyholder’s aggregate losses for non-life insurers, the literature has mainly focused on using detailed information from policies and closed claims. However, the information on open claims can reflect shifts in the distribution of the expected claim payments better than closed claims. Such shifts may be needed to be reflected in the ratemaking process earlier rather than later, especially when insurers are experiencing environmental changes. In practice, actuaries use ad hoc techniques to adjust data to current levels to determine premiums. This paper presents an intuitive ratemaking model, employing a marked Poisson process framework, which ensures that the multivariate risk analysis is done more routinely using all reported claims and makes an adjustment for Incurred But Not Reported claims. Utilizing data from the Wisconsin Local Government Property Insurance Fund, we find that by determining rates based on current data, the proposed ratemaking model leads to better alignment of premiums and provides insurers with a more financially sound portfolio.

Modelling loss reserve data in run-off triangles is challenging due to the complex but unknown dynamics in the claim/loss process. Popular loss reserve models describe the mean process through development year, accident year, and calendar year effects using the analysis of variance and covariance (ANCOVA) models. We propose to include in the mean function the persistence terms in the conditional autoregressive range model for modelling the persistence of claim across development years. In the ANCOVA model, we adopt linear trends for the accident and calendar year effects and a quadratic trend for the development year effect. We investigate linear or log-transformed mean functions and four distributions, namely generalised beta type 2, generalised gamma, Weibull, and exponential extension, with positive support to enhance the model flexibility. The proposed models are implemented using the Bayesian user-friendly package Stan running in the R environment. Results show that the models with log-transformed mean function and persistence terms provide better model fits. Lastly, the best model is applied to forecast partial loss reserve and calendar year reserve for three years.

The paper discusses the problem of estimating the mean of a long-tailed claim size distribution when the investigator's knowledge of the distribution is only vague.

One method of dealing with this problem, the method developed by Johnson and Hey, is examined and found to produce strongly biased estimators.

The situation in which a sufficient statistic (but nothing else) for the claim size distribution is known is examined, and an approximately unbiased estimator developed. This estimator is substantially more efficient than the arithmetic mean in some cases. It appears to be quite successful when the sufficient statistic is real-valued. It is of limited use when the sufficient statistic is vector-valued.

Solvency regulations require financial institutions to hold initial capital so that ruin is a rare event. An important practical problem is to estimate the regulatory capital so the ruin probability is at the regulatory level, typically with less than 0.1% over a finite-time horizon. Estimating probabilities of rare events is challenging, since naive estimations via direct simulations of the surplus process is not feasible. In this paper, we present a stratified sampling algorithm for estimating finite-time ruin probabilities. We further introduce a sequence of measure changes to remove the pathwise discontinuities of the estimator, and compute unbiased first and second-order derivative estimates of the finite-time ruin probabilities with respect to both distributional and structural parameters. We then estimate the regulatory capital and its sensitivities. These estimates provide information to insurance companies for meeting prudential regulations as well as designing risk management strategies. Numerical examples are presented for the classical model, the Sparre Andersen model with interest and the periodic risk model with interest to demonstrate the speed and efficacy of our methodology.

Three methods for fitting multiplicative models to observed, cross-classified risk data are compared. They are the method of Bailey–Simon, the method of marginal totals and a maximum likelihood method. The methods are applied to a number of risk data sets and compared with respect to balance and goodness-of-fit.