Hélène Cossette

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.

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.

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.

Modeling Catastrophes and their Impact on Insurance Portfolios

Abstract The authors propose a general individual catastrophe risk model that allows damage ratios to be random functions of the catastrophe intensity. They derive some distributional properties of the insured risks and of the aggregate catastrophic loss under this model. Through the model and ruin probability calculations, they formally illustrate the well-known fact that the catastrophe risk cannot be diversified through premium collection alone, as is the case with the usual “day-to-day” risk, even for an arbitrary large portfolio. They also derive some risk orderings between different catastrophe portfolios and show that the risk level of a realistic portfolio falls between that of a portfolio of comonotonic risks and that of a portfolio of independent risks. Finally, the authors illustrate their findings with a numerical example inspired from earthquake insurance.

On the Moments of Aggregate Discounted Claims with Dependence Introduced by a FGM Copula

In this paper, we investigate the computation of the moments of the compound Poisson sums with discounted claims when introducing dependence between the interclaim time and the subsequent claim size. The dependence structure between the two random variables is defined by a Farlie-Gumbel-Morgenstern copula. Assuming that the claim distribution has finite moments, we give expressions for the first and the second moments and then we obtain a general formula for any mth order moment. The results are illustrated with applications to premium calculation, moment matching methods, as well as inflation stress scenarios in Solvency II.

Discrete-time Risk Models Based on Time Series for Count Random Variables

In this paper, we consider various specifications of the general discrete-time risk model in which a serial dependence structure is introduced between the claim numbers for each period. We consider risk models based on compound distributions assuming several examples of discrete variate time series as specific temporal dependence structures: Poisson MA(1) process, Poisson AR(1) process, Markov Bernoulli process and Markov regime-switching process. In these models, we derive expressions for a function that allow us to find the Lundberg coefficient. Specific cases for which an explicit expression can be found for the Lundberg coefficient are also presented. Numerical examples are provided to illustrate different topics discussed in the paper.

Generalized least squares estimators for covariance parameters for credibility regression models with moving average errors

Abstract Weighted least squares methods are developed for the estimation of variance–covariance parameters of credibility regression models with moving average dependent errors. The estimators proposed are shown to be useful for constructing empirical Bayes estimators and credibility type estimators. Numerical examples are included to illustrate the proposed methods.