Recursive calculation of finite time ruin probabilities under interest force

Cardoso, Rui M. R.; Waters, Howard R.

doi:10.1016/j.insmatheco.2003.09.008

Cited by 15 publications

(19 citation statements)

References 11 publications

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“…This model plays a prominent role in risk theory and a considerable literature is devoted to its study. The reader is referred to the books by Gerber [12], Grandell [15], Panjer and Willmot [23], Asmussen [2] and Kaas et al [19]); see also, e.g., Gerber and Shiu [16], Albrecher et al [1] and Cardoso and Waters [3] for recent results on the ruin problem.…”

Section: Compound Poisson Risk Modelmentioning

confidence: 99%

On finite-time ruin probabilities for classical risk models

Lefèvre

Loisel

2008

Scandinavian Actuarial Journal

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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...

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Section: Compound Poisson Risk Modelmentioning

confidence: 99%

On finite-time ruin probabilities for classical risk models

Lefèvre

Loisel

2008

Scandinavian Actuarial Journal

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“…Much attention has been paid to a compound Poisson model with constant interest on the surplus of the company (see, e.g., [5][6][7][8][9][10]). Note that stochastic interest effect has also been incorporated in the model, e.g., in [11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

A nonhomogeneous risk model for insurance

Lefèvre

Picard

2006

Computers & Mathematics with Applications

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A discrete-time risk model is proposed that describes the temporal evolution of the surplus of an insurance company at fixed dates. The novelty of the model comes from its total nonhomogeneity in the sense that as in our previous paper [1], the premium income process is deterministic and nonuniform, but in addition, in the present work, the successive claim amounts are independent and nonidentically distributed. Our purpose is to evaluate for this extended model the probability of ruin over any finite time horizon. Rather surprisingly, the methodology which was developed in [1] can be generalized to the ease of nonstationary claim amounts. The key mathematical tool for that is a theory of pseudopolynomials of Appell type. Furthermore, it is shown that a similar approach can be applied to a multirisks version of the model, as well as to a continuous-time version.

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“…The numerical algorithms used in Sections 3 and 4 are based on algorithms developed by Cardoso and Waters (2003). In this earlier paper the authors consider a risk model which could be regarded as the special case of the model defined by (1) and (2) with ∆ = 0, so that interest is earned on any positive surplus.…”

Section: Introductionmentioning

confidence: 99%

“…In this earlier paper the authors consider a risk model which could be regarded as the special case of the model defined by (1) and (2) with ∆ = 0, so that interest is earned on any positive surplus. In Section 2 we summarise the methodology used by Cardoso and Waters (2003).…”

Section: Introductionmentioning

confidence: 99%

“…The models studied in Cardoso and Waters (2003) and in Sections 3 and 4 can be extended by allowing one or more of the key parameters or distributions defining the model, for example the claim size distribution or the premium rate, to change at pre-determined time points. In Section 5 we explain how our algorithms can be adapted to incorporate this feature and illustrate this numerically in relation to the model in Cardoso and Waters (2003).…”

Section: Introductionmentioning

confidence: 99%

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