Robustness analysis and convergence of empirical finite-time ruin probabilities and estimation risk solvency margin

Scandinavian Actuarial Journal

2008

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This paper is concerned with the problem of ruin in the classical compound binomial and compound Poisson risk models. Our primary purpose is to extend to those models an exact formula derived by Picard and Lefèvre [24] for the probability of (non-)ruin within finite time. First, a standard method based on the ballot theorem and an argument of Seal-type provides an initial (known) formula for that probability. Then, a concept of pseudo-distributions for the cumulated claim amounts, combined with some simple implications of the ballot theorem, leads to the desired formula. Two expressions for the (non-)ruin probability over an infinite horizon are also deduced as corollaries. Finally, an illustration within the framework of Solvency II is briefly presented.Keywords: ruin probability, finite and infinite horizon, compound binomial model, compound Poisson model, ballot theorem, pseudo-distributions, Solvency II, Value-at-Risk. IntroductionThe two classical models in risk theory are the compound binomial model, which is a discrete-time process where the claim amounts are usually assumed to be integer-valued random variables, and the compound Poisson model, which is a continuous-time analogue of this process where the claim amounts are generally assumed to have an absolutely continuous distribution. Although the continuous-time version is quite popular, a discretetime version can be appropriate for some applications and provides a better intuitive understanding.The present paper deals with the problem of evaluating, for both risk models, the probability of (non-)ruin within finite time. This problem has received much attention in the literature, and different algorithms have been proposed, with their own advantages and drawbacks; see, e.g., the analysis by Dickson [11] and the references therein.Recently, Picard and Lefèvre [24] derived an elegant explicit formula, called P.L. formula below, for the finite-time non-ruin probability in a compound Poisson model where the claim amounts are integer-valued. Such a case is important because in practice a et al. [17]. A refined discussion of its merits in comparison with other available formulas is given in Rullière and Loisel [28]. Besides, De Vylder and Goovaerts [10] proved that a similar formula holds too for a compound Poisson model with continuous claim amounts; see also Ignatov and Kaishev [18].1 discretization of the claim amounts is often required for numerical calculations (e.g., de Vylder and Goovaerts [9]). The importance of the P.L. formula has been pointed out by De Vylder [6], [7] and IgnatovSeveral methods are possible to establish the P.L. formula. In Picard and Lefèvre [24] and De Vylder and Goovaerts [10], the first step consists in stipulating for a total claim amount until some time t and conditioning on the last claim instant and amount before t. This yields an integral equation which is then solved by the former authors using the generalized Appell polynomials (in an extended framework), and by the latter authors using the convolution products and th...

Section: An Alternative Approachmentioning

confidence: 99%

Section: A Brief Illustrationmentioning

confidence: 99%

On finite-time ruin probabilities for classical risk models

Lefèvre

Scandinavian Actuarial Journal

2008

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“…In a recent paper, (Marceau and Rioux(2001)) provided an algorithm to compute the influence function of the eventual probability of ruin. We obtained in (Loisel, Mazza and Rullière(2007)) that for all u 0,…”

Section: Introductionmentioning

confidence: 99%

“…It would be of course much better to carry out a robustness analysis at the same time. This led us to define in an earlier paper (see (Loisel, Mazza and Rullière(2007))) reliable ruin probabilities as quantiles of empirical finite-time ruin probabilities, and the Estimation Risk Solvency Margin (ERSM) as the additional capital required to cover estimation risk:…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes

Insurance: Mathematics and Economics

Mazza

Rullière

2009

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Abstract. The classical risk model is considered and a sensitivity analysis of finite-time ruin probabilities is carried out. We prove the weak convergence of a sequence of empirical finite-time ruin probabilities. So-called partly shifted risk processes are introduced, and used to derive an explicit expression of the asymptotic variance of the considered estimator. This provides a clear representation of the influence function associated with finite time ruin probabilities, giving a useful tool to quantify estimation risk according to new regulations.

Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities

Lefèvre

Methodol Comput Appl Probab

2009

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International audienceThis paper is concerned with the compound Poisson risk model and two generalized models with still Poisson claim arrivals. One extension incorporates inhomogeneity in the premium input and in the claim arrival process, while the other takes into account possible dependence between the successive claim amounts. The problem under study for these risk models is the evaluation of the probabilities of (non-)ruin over any horizon of finite length. The main recent methods, exact or approximate, used to compute the ruin probabilities are reviewed and discussed in a unified way. Special attention is then paid to an analysis of the qualitative impact of dependence between claim amounts