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On Finite-Time Ruin Probabilities for Classical Risk Models

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Author Info

  • Claude Lefèvre

    ()
    (Département de Mathématique - Université Libre de Bruxelles)

  • Stéphane Loisel

    ()
    (SAF - Laboratoire de Sciences Actuarielle et Financière - Université Claude Bernard - Lyon I : EA2429)

Abstract

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 (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.

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Bibliographic Info

Paper provided by HAL in its series Post-Print with number hal-00168958.

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Date of creation: Jan 2008
Date of revision:
Publication status: Published, Scandinavian Actuarial Journal, 2008, 2008, 1, 41-60
Handle: RePEc:hal:journl:hal-00168958

Note: View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/hal-00168958/en/
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Related research

Keywords: ruin probability; finite and infinite horizon; compound binomial model; compound Poisson model; ballot theorem; pseudo-distributions; Solvency II; Value-at-Risk.;

References

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  1. Cheng, Shixue & Gerber, Hans U. & Shiu, Elias S. W., 2000. "Discounted probabilities and ruin theory in the compound binomial model," Insurance: Mathematics and Economics, Elsevier, vol. 26(2-3), pages 239-250, May.
  2. De Vylder, F. Etienne & Goovaerts, Marc J., 1999. "Explicit finite-time and infinite-time ruin probabilities in the continuous case," Insurance: Mathematics and Economics, Elsevier, vol. 24(3), pages 155-172, May.
  3. Stéphane Loisel & Christian Mazza & Didier Rullière, 2008. "Robustness analysis and convergence of empirical finite-time ruin probabilities and estimation risk solvency margin," Post-Print hal-00168714, HAL.
  4. Willmot, Gordon E., 1993. "Ruin probabilities in the compound binomial model," Insurance: Mathematics and Economics, Elsevier, vol. 12(2), pages 133-142, April.
  5. Rulliere, Didier & Loisel, Stephane, 2004. "Another look at the Picard-Lefevre formula for finite-time ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 187-203, October.
  6. Cardoso, Rui M. R. & R. Waters, Howard, 2003. "Recursive calculation of finite time ruin probabilities under interest force," Insurance: Mathematics and Economics, Elsevier, vol. 33(3), pages 659-676, December.
  7. De Vylder, F. & Goovaerts, M. J., 1988. "Recursive calculation of finite-time ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 7(1), pages 1-7, January.
  8. Ignatov, Zvetan G. & Kaishev, Vladimir K. & Krachunov, Rossen S., 2001. "An improved finite-time ruin probability formula and its Mathematica implementation," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 375-386, December.
  9. Konstantopoulos, Takis, 1995. "Ballot theorems revisited," Statistics & Probability Letters, Elsevier, vol. 24(4), pages 331-338, September.
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Citations

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Cited by:
  1. Dutang, C. & Lefèvre, C. & Loisel, S., 2013. "On an asymptotic rule A+B/u for ultimate ruin probabilities under dependence by mixing," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 774-785.
  2. Christophe Dutang & Claude Lefèvre & Stéphane Loisel, 2013. "On an asymptotic rule A+B/u for ultimate ruin probabilities under dependence by mixing," Post-Print hal-00746251, HAL.
  3. Loisel, Stéphane & Mazza, Christian & Rullière, Didier, 2009. "Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 374-381, December.
  4. Romain Biard & Stéphane Loisel & Claudio Macci & Noel Veraverbeke, 2010. "Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation," Post-Print hal-00372525, HAL.
  5. Julien Trufin & Stéphane Loisel, 2013. "Ultimate ruin probability in discrete time with Bühlmann credibility premium adjustments," Post-Print hal-00426790, HAL.
  6. Gathy, Maude & Lefèvre, Claude, 2010. "On the Lagrangian Katz family of distributions as a claim frequency model," Insurance: Mathematics and Economics, Elsevier, vol. 47(1), pages 76-83, August.
  7. repec:hal:wpaper:hal-00746251 is not listed on IDEAS
  8. Irmina Czarna & Zbigniew Palmowski, 2009. "De Finetti's dividend problem and impulse control for a two-dimensional insurance risk process," Papers 0906.2100, arXiv.org, revised Feb 2011.
  9. Stéphane Loisel & Claude Lefèvre, 2009. "Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities," Post-Print hal-00201377, HAL.

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