We analyse, in a probabilistic setting, Newcombe's (1981) life table method of estimating rates of onset of high-penetrance single-gene disorders, and extend this to a counting process model for individual life histories, including movement between risk groups arising from genetic testing and onset in relatives. A key result is that estimates of rates of onset at any age x must be conditioned only on information available when subjects were age x, even though their later life histories might be available to the investigator. This determines the data that must be included in pedigrees. We derive a Nelson-Aalen-type estimate of a function of the rate of onset, and show that when all that is known is that the persons in the study inherited a mutation with probability 1/2, the function estimated is bounded. In practice, the treatment of censored observations or the methods of ascertainment might cause the estimate to exceed this bound, which results in infinite estimates of the rate of onset but might be a useful diagnostic check on the presence of these features. We summarise the literature on mutations in the Presenilin-1 (PSEN-1) gene, associated with early-onset Alzheimer's disease (EOAD), and from published pedigrees we estimate rates of onset of EOAD.

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.

In the present paper, we study error bounds for approximations to multivariate distributions. In particular, we discuss some general versions of compound multivariate distributions and look at distributions of dependent random variables constructed by linear transforms of independent random variables or vectors. Special attention is paid to the case when the support of the original distribution is restricted. We also look at some applications with multivariate Bernoulli distributions.

This paper presents a universal framework for pricing financial and insurance risks. Examples are given for pricing contingent payoffs, where the underlying asset or liability can be either traded or not traded. The paper also outlines an application of the framework to prescribe capital allocations within insurance companies, and to determine fair values of insurance liabilities.

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.

Based on the notions of value-at-risk and conditional value-at-risk, we consider two functionals, abbreviated VaR and CVaR, which represent the economic risk capital required to operate a risky business over some time period when only a small probability of loss is tolerated. These functionals are consistent with the risk preferences of profit-seeking (and risk averse) decision makers and preserve the stochastic dominance order (and the stop-loss order). This result is used to bound the VaR and CVaR functionals by determining their maximal values over the set of all loss and profit functions with fixed first few moments. The evaluation of CVaR for the aggregate loss of portfolios is also discussed. The results of VaR and CVaR calculations are illustrated and compared at some typical situations of general interest.

In this paper a class of risk processes in which claims occur as a renewal process is studied. A clear expression for Laplace transform of the finite time ruin probability is well given when the claim amount distribution is a mixed exponential. As its consequence, a well-known result about ultimate ruin probability in the classical risk model is obtained.

For the Cramér-Lundberg risk model with phase-type claims, it is shown that the probability of ruin before an independent phase-type time H coincides with the ruin probability in a certain Markovian fluid model and therefore has an matrix-exponential form. When H is exponential, this yields in particular a probabilistic interpretation of a recent result of Avram & Usabel. When H is Erlang, the matrix algebra takes a simple recursive form, and fixing the mean of H at T and letting the number of stages go to infinity yields a quick approximation procedure for the probability of ruin before time T. Numerical examples are given, including a combination with extrapolation.

In this paper we use Fourier/Laplace transforms to evaluate numerically relevant probabilities in ruin theory as an application to insurance. The transform of a function is split in two: the real and the imaginary parts. We use an inversion formula based on the real part only, to get the original function.

By using an appropriate algorithm to compute integrals and making use of the properties of these transforms we are able to compute numerically important quantities either in classical or non-classical ruin theory. As far as the classical model is concerned the problems considered have been widely studied. In what concerns the non-classical model, in particular models based on more general renewal risk processes, there is still a long way to go. In either case the approach presented is an easy method giving good approximations for reasonable values of the initial surplus.

To show this we compute numerically ruin probabilities in the classical model and in a renewal risk process in which claim inter-arrival times have an Erlang(2) distribution and compare to exact figures where available. We also consider the computation of the probability and severity of ruin in the classical model.

Sundt and Jewell have shown that a nondegenerate claim number distribution Q = {qn}nϵN0 satisfies the recursion

for all n≥0 if and only if Q is a binomial, Poisson or negativebinomial distribution. This recursion is of interest since it yields a recursion for the aggregate claims distribution in the collective model of risk theory when the claim size distribution is integer-valued as well. A similar characterization of claim number distributions satisfying the above recursion for all n ≥ 1 has been obtained by Willmot. In the present paper we extend these results and the subsequent recursion for the aggregate claims distribution to the case where the recursion holds for all n ≥ k with arbitrary k. Our results are of interest in catastrophe excess-of-loss reinsurance.

We study the distribution of the time to ruin in the classical risk model. We consider some methods of calculating this distribution, in particular by using algorithms to calculate finite time ruin probabilities. We also discuss calculation of the moments of this distribution.

A method for analysing the risk of taking a too low reserve level by the use of Chain Ladder method is developed. We give an answer to the question of how much safety loading in terms of the Chain Ladder standard error has to be added to the Chain Ladder reserve in order to reach a specified security level in loss reserving. This is an important question in the framework of integrated risk management of an insurance company. Furthermore we investigate the relative bias of Chain Ladder estimators. We use Monte Carlo simulation technique as well as the collective model of risk theory in each cell of run-off table. We analyse deviation between Chain Ladder reserves and Monte Carlo simulated reserves statistically. Our results document dependency on claim number and claim size distribution types and parameters.

Haberman and Sung (1994) have presented a dynamic model for a defined benefit occupational pension scheme which considered two types of risk: the “contribution rate” and the “solvency” risk. The current paper, extends this work by deriving optimal funding control procedures for determining the contribution rate for the case of a stochastic model with incomplete state information, making use of the separation principle. The stochastic inputs modelled are the investment returns and the benefit outgo.

Based on a recent contribution by Baione and others in this Journal, some consequences of the decrease of the mean merit coefficient for portfolios of Bonus-Malus policies, and some alternative ways to measure the “quality” of a Bonus-Malus system are discussed.

The purpose of this paper is to obtain approximations to the transition intensities defined for a multiple state model for Permanent Health Insurance (PHI) which enables us to analyse PHI claims by cause of disability.

The approximations to the transition intensities are obtained using a set of PHI data classified by 18 sickness categories and the graduations of the transition intensities defined for a simpler model proposed in Continuous Mortality Investigation Reports, 12 (1991).

In order to derive the approximations to the recovery and mortality of the sick intensities for our model, we carry out tests of hypotheses based on the distributions of average sickness durations. The approximations to the sickness intensities are obtained by estimating a statistical model for the number of claim inceptions, which can be formulated as a generalized linear model.

We reconsider the problem of producing fair and accurate tariffs based on aggregated insurance data giving numbers of claims and total costs for the claims. Jørgensen and de Souza (Scand Actuarial J., 1994) assumed Poisson arrival of claims and gamma distributed costs for individual claims. Jørgensen and de Souza (1994) directly modelled the risk or expected cost of claims per insured unit, μ say. They observed that the dependence of the likelihood function on μ is as for a linear exponential family, so that modelling similar to that of generalized linear models is possible. In this paper we observe that, when modelling the cost of insurance claims, it is generally necessary to model the dispersion of the costs as well as their mean. In order to model the dispersion we use the framework of double generalized linear models. Modelling the dispersion increases the precision of the estimated tariffs. The use of double generalized linear models also allows us to handle the case where only the total cost of claims and not the number of claims has been recorded.

BMS in force show a progressive reduction of the observed average premium, which causes a financial imbalance in the system (see Lemaire (1995)). As a onsequence, frequent premium adjustments become necessary and result in discrepancy between the reduction defined in the policy contract and the ffective discount applied to the driver. Most policyholders are not aware of his “lack of transparency”. This paper deals with the problem of designing an optimal tariff structure so that the designed BMS is adequate and satisfies both transparency and financial balance conditions.