Étienne Marceau

On a risk model with dependence between interclaim arrivals and claim sizes

We consider an extension to the classical compound Poisson risk model for which the increments of the aggregate claim amount process are independent. In Albrecher and Teugels (2006), an arbitrary dependence structure among the interclaim time and the subsequent claim size expressed through a copula is considered and they derived asymptotic results for both the finite and infinite-time ruin probabilities. In this paper, we consider a particular dependence structure among the interclaim time and the subsequent claim size and we derive the defective renewal equation satisfied by the expected discounted penalty function. Based on the compound geometric tail representation of the Laplace transform of the time to ruin, we also obtain an explicit expression for this Laplace transform for a large class of claim size distributions. The ruin probability being a special case of the Laplace transform of the time to ruin, explicit expressions are therefore obtained for this particular ruin related quantity. Finally, we measure the impact of the various dependence structures in the risk model on the ruin probability via the comparison of their Lundberg coefficients.

Stochastic bounds on sums of dependent risks

There is a growing concern in the actuarial literature for the effect of dependence between individual risks Xi on the distribution of the aggregate claim S=X1+⋯+Xn. Recent work by Dhaene and Goovaerts (Dhaene, J., Goovaerts, M.J., 1996. ASTIN Bulletin 26, 201–212; Dhaene, J., Goovaerts, M.J., 1997. Insurance: Mathematics and Economics 19, 243–253) and Muller (Muller, A., 1997a. Insurance: Mathematics and Economics 21, 219–223; Muller, A., 1997b. Advances in Applied Probability 29, 414–428) has led, among other things, to the identification of the portfolio yielding the smallest and largest stop-loss premiums and hence to bounds on E{φ(S)} for arbitrary non-decreasing, convex functions φ in situations of dependence between the Xi’s. This paper extends these results by showing how to compute bounds on P(S>s) and more generally on E{φ(S)} for monotone, but not necessarily convex functions φ. Special attention is paid to the numerical implementation of the results and examples of application are provided.

The discrete-time risk model with correlated classes of business

Abstract The discrete-time risk model with correlated classes of business is examined. Two different relations of dependence are considered. The impact of the dependence relation on the finite-time ruin probabilities and on the adjustment coefficient is also studied. Numerical examples are presented.

On two dependent individual risk models

Abstract In this paper, we propose two constructions which allow dependence between the risks of an insurance portfolio in the individual risk model. In the first construction, each risk’s experience is influenced by an individual and a collective risk factor, as well as a class factor if the portfolio is divided into different classes. The second construction uses copulas. The impact on the cumulative distribution function of the aggregate claim amount and on the stop-loss premium is presented via numerical examples.

Ruin Probabilities in the Compound Markov Binomial Model

In this paper, we present a compound Markov binomial model which is an extension of the compound binomial model proposed by Gerber (1988a, b) and further examined by Shiu (1989) and Willmot (1993). The compound Markov binomial model is based on the Markov Bernoulli process which introduces dependency between claim occurrences. Recursive formulas are provided for the computation of the ruin probabilities over finite- and infinite-time horizons. A Lundberg exponential bound is derived for the ruin probability and numerical examples are also provided.

Analysis of ruin measures for the classical compound Poisson risk model with dependence

In this paper, we consider an extension to the classical compound Poisson risk model. Historically, it has been assumed that the claim amounts and claim inter-arrival times are independent. In this contribution, a dependence structure between the claim amount and the interclaim time is introduced through a Farlie–Gumbel–Morgenstern copula. In this framework, we derive the integro-differential equation and the Laplace transform (LT) of the Gerber–Shiu discounted penalty function. An explicit expression for the LT of the discounted value of a general function of the deficit at ruin is obtained for claim amounts having an exponential distribution.

Compound Poisson approximations for individual models with dependent risks

Abstract This paper shows how compound Poisson distributions can be used to approximate the distribution of the total claim amount in the context of single- or multi-class individual risk models where dependence between the contracts arises through mixtures. Some of these models are generated by Archimedean copulas, and others are seen to fall under the purview of a general multi-class shock model whose structure is both intuitive and easily tractable. A numerical study is used to illustrate the quality of the approximation as a function of the heterogeneity and the dependence in the portfolio. A theoretical result is also provided which helps to explain the effect of dependence on the total claim amount when the contracts are linked through an Archimedean copula model.

On life insurance reserves in a stochastic mortality and interest rates environment

Abstract The calculation of the reserves in a stochastic mortality and interest rates environment for a general portfolio of life insurance policies is examined. The first two moments of the prospective loss random variable for the general portfolio are derived. A Monte Carlo simulation method is used to estimate the distribution of this random variable. Another approximation of the prospective loss random variable which is based on the assumption of a large portfolio is also considered. In the numerical examples, a discrete-time model for the stochastic interest rates is assumed.

Classical numerical ruin probabilities

Abstract Finite and infinite-time classical ruin probabilities can be approximated in Gerbers elementary binomial risk model. In order to obtain good results, rather fine discretizations may be necessary and then the computing times may be much too long. Here we show how rather rough discretizations provide approximations of excellent quality when a new claimsize distribution (with one negative probability mass!!!) is adopted and when a new security loading is introduced.

Pension Plan Valuation and Mortality Projection

Abstract It is now well documented that human mortality globally declined during the course of the twentieth century. These mortality improvements pose a challenge for pricing and reserving in life insurance and for the management of public pension regimes. Assuming a further continuation of the stable pace of mortality decline, a Poisson log-bilinear projection model is applied to population mortality data to forecast future death rates. Then a relational model embedded in a Poisson regression approach is used to merge a dynamic mortality table based on data of a large population (in this case the Canadian province of Quebec) to mortality data of a given pension plan (here the Régie des Rentes du Québec) to create another dynamic mortality table, which can be used to make any assessments on the total costs of the pension plan. We provide at the end numerical examples that illustrate the impact of mortality improvements on a pension plan.