The tail behavior of sums of dependent risks was considered by Wüthrich (2003) and by Alink et al. (2004, 2005) in the case where the variables are exchangeable and connected through an Archimedean copula model. It is shown here how their result can be extended to a broader class of dependence structures using multivariate extreme-value theory. An explicit form is given for the asymptotic probability of extremal events, and the behavior of the latter is studied as a function of the indices of regular variation of both the copula and the common distribution of the risks.

In this paper we present a stable recursive algorithm for the calculation of the probability of ultimate ruin in the classical risk model. We also present stable recursive algorithms for the calculation of the joint and marginal distributions of the surplus prior to ruin and the severity of ruin. In addition we present bounds for these distributions.

This paper explains securitization of insurance risk by describing its essential components and its economic rationale. We use examples and describe recent securitization transactions. We explore the key ideas without abstract mathematics. Insurance-based securitizations improve opportunities for all investors. Relative to traditional reinsurance, securitizations provide larger amounts of coverage and more innovative contract terms.

In Mack (1993), a formula for the standard error or chain ladder reserve estimates has been derived. In the present communication, a very intuitive and easily programmable recursive way of calculating the formula is given. Moreover, this recursive way shows how a tail factor can be implemented in the calculation of the standard error.

We develop a novel approach for pricing cyber insurance contracts. The considered cyber threats, such as viruses and worms, diffuse in a structured data network. The spread of the cyber infection is modeled by an interacting Markov chain. Conditional on the underlying infection, the occurrence and size of claims are described by a marked point process. We introduce and analyze a new polynomial approximation of claims together with a mean-field approach that allows to compute aggregate expected losses and prices of cyber insurance. Numerical case studies demonstrate the impact of the network topology and indicate that higher order approximations are indispensable for the analysis of non-linear claims.

A simple recursion for the n-fold convolution of an arithmetic distribution with itself is developed and its relation to Panjer's algorithm for compound distributions is shown.

Section A of this paper uses the Canadian experience for private passenger automobiles to show (1) that merit rating is almost as effective as the class plan in separating the better risks from the poorer risks, (2) that both merit rating and class rating leave unanalyzed a considerable amount of variation among risks and (3) that certain available evidence supports the conclusion that annual mileage, which has long been felt to be an important measure of hazard, is a very significant cause of this unanalyzed variation among risks.

Section B presents a method for obtaining relativities among groups on which a multiple classification system has been imposed. The customary method of calculating class relativities uses the total experience for each class with all subdivisions within the classes added together. With the customary method it is difficult to make a completely accurate adjustment for different distributions by territory or merit rating, because any change in the class relativities disturbs the other sets of relativities and conversely. It is shown that even if such an adjustment were made, the customary method of calculating relativities one set at a time does not reflect the relative credibility of each subgroup and does not produce the best fit to the actual data. Moreover it produces differences between the actual data and the fitted values which are far too large to be caused by chance. In addition, for private passenger automobile insurance in Canada, it is shown that two sets of relativities which are multiplied together cannot produce the best fit to the actual data, and some of the consequences of trying to do so are explained. Some methods are advanced whereby all sets of relativities for classes, merit ratings, territories, and so forth, can be calculated simultaneously, which will overcome all the deficiencies in the customary method. These improved methods use the technique of minimizing a measure (technically known as the chi-square test) of the differences between the actual data and the fitted values. Some applications to other lines of insurance are mentioned.

Let me begin with some practical examples of experience rating.

a) Swiss Automobile Tariff 1963

— Within each tariff-position there are 22 grades:

— The new owner of a car starts at grade 9

— The basic premium is determined on the basis of objective characteristics of the risk and essentially depends on the horse-power classification of the car

— The 22 grades are experience-rated as follows: For each accident one rises three grades and for each accident-free year one falls one grade. A driver who has I accident in every 4 years hence remains within four adjacent grades.

b) Sliding Scale Premiums in Reinsurance

Excess of Loss Contracts often stipulate that:

The rate of premium to be applied to the subject premium volume is determined at the end of the cover period as follows:

subject to a minimum of 0,04

and a maximum of 0,08

c) Participation in Mortality Profit in Group Life Insurance

A group life insurance covers the members of the group on a one year term basis. It is often agreed that at the end of the year mortality profits are given back to the group according to the formula

refund = x% gross premiums — y% claims (where x < y)

All these examples fall under the heading “Experience Rating”. What do they have in common?

Definition: A system by which the premium of the individual risk depends upon the claims experience of this same individual risk.

In this paper, we will cover the bonus-malus system in automobile insurance. Bonus-malus systems are based on the distribution of the number of car accidents. Therefore, the modelling and fitting of that distribution are considered. Fitting of data is done using the Poisson inverse Gaussian distribution, which shows a good fit. Building the bonus system is done by minimizing the insurer's risk, according to Lemaire's (1985) bonus system.

A new multivariate distribution possessing arbitrarily parametrized and positively dependent univariate Pareto margins is introduced. Unlike the probability law of Asimit et al. (2010), the structure in this paper is absolutely continuous with respect to the corresponding Lebesgue measure. The distribution is of importance to actuaries through its connections to the popular frailty models, as well as because of the capacity to describe dependent heavy-tailed risks. The genesis of the new distribution is linked to a number of existing probability models, and useful characteristic results are proved. Expressions for, e.g., the decumulative distribution and probability density functions, (joint) moments and regressions are developed. The distributions of minima and maxima, as well as, some weighted risk measures are employed to exemplify possible applications of the distribution in insurance.

The first method, essentially due to GOOVAERTS and DE VYLDER, uses the connection between the probability of ruin and the maximal aggregate loss random variable, and the fact that the latter has a compound geometric distribution. For the second method, the claim amount distribution is supposed to be a combination of exponential or translated exponential distributions. Then the probability of ruin can be calculated in a transparent fashion; the main problem is to determine the nontrivial roots of the equation that defines the adjustment coefficient. For the third method one observes that the probability, of ruin is related to the stationary distribution of a certain associated process. Thus it can be determined by a single simulation of the latter. For the second and third methods the assumption of only proper (positive) claims is not needed.

In the recent actuarial literature, several proofs have been given for the fact that if a random vector (X1X2, …, Xn) with given marginals has a comonotonic joint distribution, the sum X1 + X2 + … + Xn is the largest possible in convex order. In this note we give a lucid proof of this fact, based on a geometric interpretation of the support of the comonotonic distribution.

In this paper we present algorithms to calculate the probability and severity of ruin in both finite and infinite time for a discrete time risk model. We show how the algorithms can be applied to give approximate values for the same quantities in the classical continuous time risk model.

We consider three classes of bivariate counting distributions and the corresponding compound distributions. For each class we derive a recursive algorithm for calculating the bivariate compound distribution.

A recursive expression is derived for computing exactly the distribution of aggregate claims of a portfolio of life insurance policies. The recursion generalizes a formula of White and Greville for the claim numbers distribution and improves Kornya's approximation method for the aggregate claims distribution. It can be seen as the counterpart in the individual model of Panjer's recursion formula for the collective model.

In this paper, we investigate asymptotic properties of the tail probabilities of the maxima of partial sums of independent random variables. For some large classes of heavy-tailed distributions, we show that the tail probabilities of the maxima of the partial sums asymptotically equal to the sum of the tail probabilities of the individual random variables. Then we partially extend the result to the case of random sums. Applications to some commonly used risk processes are proposed. All heavy-tailed distributions involved in this paper are supposed on the whole real line.

Using risk-reducing properties of conditional expectations with respect to convex order, Denuit and Dhaene [Denuit, M. and Dhaene, J. (2012). Insurance: Mathematics and Economics 51, 265–270] proposed the conditional mean risk sharing rule to allocate the total risk among participants to an insurance pool. This paper relates the conditional mean risk sharing rule to the size-biased transform when pooled risks are independent. A representation formula is first derived for the conditional expectation of an individual risk given the aggregate loss. This formula is then exploited to obtain explicit expressions for the contributions to the pool when losses are modeled by compound Poisson sums, compound Negative Binomial sums, and compound Binomial sums, to which Panjer recursion applies. Simple formulas are obtained when claim severities are homogeneous. A couple of applications are considered: first, to a peer-to-peer insurance scheme where participants share the first layer of their respective risks while the higher layer is ceded to a (re)insurer; second, to survivor credits to be shared among surviving participants in tontine schemes.

The premium calculation principle is one of the main objectives of study for actuaries. There seems to be full agreement among the leading theoreticians in the field that the insurance premium should reflect both the expected claims and certain loadings. This is true for policy, risk or portfolio. There are three types of positive loadings: a) a loading to cover commissions, administrative costs and claim-settlement expenses; b) a loading to cover some profit (a cost-plus approach); and c) a loading for the risk taken by the insurer when underwriting the policy. The administrative costs can be considered a part of “expected gross claims”. Thus, the insurer's ratemaking decision depends on his ability to estimate expected claims (including costs) and on the selection of a fair risk loading.

The main concern in the literature is the appropriate measurement of the risk and the exact loading formula. Bühlmann [1970, ch. 5] and others identified four possible principles of risk loading, namely, the expected value principle, the standard deviation loading, the variance loading, and the loading according to the principle of constant utility. Various studies point to the advantages and disadvantages of these principles and also examine some additional loading forms—semi-variance, skewness, etc. (e.g., Bühlmann [1970], Benktander [1971], Berger [1972], Burness [1972], Berliner [1974], Berliner and Benktander [1976], Bohman [1976], Cooper [1974], Gerber [1975] and others). Despite different preferences in choosing the appropriate loading calculation principle, all seem to agree that the risk loading must be positive, since, otherwise, the firm would just have to wait for its ruin, that is bound to come sooner or later, according to risk theory.

The purpose of this article is to re-examine the appropriate principle of premium calculation in light of the recent developments in the theory of finance and especially in the theory of capital market equilibrium. These developments may suggest a new point of view and raise a few questions regarding the loading rules.

The run off-pattern of long-term reinsurance treaties is described by means and standard deviations of logarithmic increments of premiums and loss ratios in a normal distribution. From this description forecasts of ultimate claims and current IBNR-reserves are derived, with associated distributions and confidence limits. Aggregation from individual treaties to portfolio level is proposed by normal approximation. Security loading of IBNR-reserves is proposed by a contingency reserve at portfolio level.

In the dual model, the surplus of a company is a Lévy process with sample paths that are skip-free downwards. In this paper, the aggregate gains process is the sum of a shifted compound Poisson process and an independent Wiener process. By means of Laplace transforms, it is shown how the expectation of the discounted dividends until ruin can be calculated, if a barrier strategy is applied, and how the optimal dividend barrier can be determined. Conditions for optimality are discussed and several numerical illustrations are given. Furthermore, a family of models is analysed where the individual gain amount distribution is rescaled and compensated by a change of the Poisson parameter.