On finite-time ruin probabilities for classical risk models

Abstract

This paper examines 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 & Lefèvre (1997) 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

Acknowledgements

We are grateful to the Referee for helpful comments and suggestions. We also thank Professor F. Avram (Université de Pau) for useful remarks. This work was partly conducted while Cl. Lefèvre was visiting the Institut de Science Financière et d'Assurances, Université Claude Bernard Lyon 1. Lefèvre sincerely thanks all the members of the ISFA for their warm welcome and many interesting discussions.