Abstract
An important question in insurance is how to evaluate the probabilities of (non-) ruin of a company over any given horizon of finite length. This paper aims to present some (not all) useful methods that have been proposed so far for computing, or approximating, these probabilities in the case of discrete claim severities. The starting model is the classical compound Poisson risk model with constant premium and independent and identically distributed claim severities. Two generalized versions of the model are then examined. The former incorporates a non-constant premium function and a non-stationary claim process. The latter takes into account a possible interdependence between the successive claim severities. Special attention will be paid to a recursive computational method that enables us to tackle, in a simple and unified way, the different models under consideration. The approach, still relatively little known, relies on the use of remarkable families of polynomials which are of Appell or generalized Appell (Sheffer) types. The case with dependent claim severities will be revisited accordingly.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Asmussen S (2000) Ruin probabilities. World Scientific, Singapore
Biard R, Lefèvre C, Loisel S (2008) Impact of correlation crises in risk theory: asymptotics of finite-time ruin probabilities for heavy-tailed claim amounts when some independence and stationarity assumptions are relaxed. Insur Math Econ 43:412–421
Denuit M, Lefèvre C, Picard P (2003) Polynomial structures in order statistics distributions. J Stat Plan Inference 113:151–178
De Vylder FE (1996) Advanced risk theory: a self-contained introduction. Editions de l’Université Libre de Bruxelles, Bruxelles.
De Vylder FE (1997) La formule de Picard et Lefèvre pour la probabilité de ruine en temps fini. Bulletin Francais d’Actuariat 1:31–40
De Vylder FE (1999) Numerical finite-time ruin probabilities by the Picard-Lefèvre formula. Scand Actuar J 2:97–105
De Vylder FE, Goovaerts MJ (1988) Recursive calculation of finite-time ruin probabilities. Insur Math Econ 7:1–7
Dickson DCM (1999) On numerical evaluation of finite time survival probabilities. Br Actuar J 5:575–584
Dickson DCM (2005) Insurance risk and ruin. Cambridge University Press, Cambridge
Dickson DCM, Waters HR (1991) Recursive calculation of survival probabilities. ASTIN Bull 21:199–221
Gerber HU (1979) An introduction to mathematical risk theory. S.S. Huebner Foundation Monograph. University of Philadelphia, Philadelphia
Grandell J (1990) Aspects of risk theory. Springer, New York
Ignatov ZG, Kaishev VK (2000) Two-sided bounds for the finite time probability of ruin. Scand Actuar J 1:46–62
Ignatov ZG, Kaishev VK (2004) A finite-time ruin probability formula for continuous claim severities. J Appl Probab 41:570–578
Ignatov ZG, Kaishev VK, Krachunov RS (2001) An improved finite-time ruin probability formula and its mathematica implementation. Insur Math Econ 29:375–386
Kaas R, Goovaerts MJ, Dhaene J, Denuit M (2001) Modern actuarial risk theory. Kluwer, Dordrecht
Lefèvre C (2007) Discrete compound Poisson process with curved boundaries: polynomial structures and recursions. Methodol Comput Appl Probab 9:243–262
Lefèvre C, Loisel S (2008) On finite-time ruin probabilities for classical risk models. Scand Actuar J 1:41–60
Lefèvre C, Picard P (2006) A nonhomogeneous risk model for insurance. Comput Math Appl 51:325–334
Loisel S, Privault N (2008) Sensivity analysis and density estimation for finite-time ruin probabilities. Working paper, Cahiers de recherche de l’I.S.F.A. (W.P. 2041), Université de Lyon
Loisel S, Mazza C, Rullière D (2008) Robustness analysis and convergence of empirical finite-time ruin probabilities and estimation of risk solvency margin. Insur Math Econ 42:746–762
Niederhausen H (1981) Sheffer polynomials for computing exact Kolmogorov-Smirnov and Rényi type distributions. Ann Stat 9:923–944
Panjer HH, Willmot GE (1992) Insurance risk models. Society of Actuaries, Schaumburg
Picard P, Lefèvre C (1996) First crossing of basic counting processes with lower non-linear bundaries: a unified approach through pseudopolynomials (I). Adv Appl Probab 28:853–876
Picard P, Lefèvre C (1997) The probability of ruin in finite time with discrete claim size distribution. Scand Actuar J 1:58–69
Picard P, Lefèvre C (2003) Probabilité de ruine éventuelle dans un modèle de risque à temps discret. J Appl Probab 40:543–556
Picard P, Lefèvre C, Coulibaly I (2003a) Problèmes de ruine en théorie du risque à temps discret avec horizon fini. J Appl Probab 40:527–542
Picard P, Lefèvre C, Coulibaly I (2003b) Multirisks model and finite-time ruin probabilities. Methodol Comput Appl Probab 5:337–353
Rolski T, Schmidli H, Schmidt V, Teugels J (1999) Stochastic processes for insurance and finance. Wiley, Chichester
Rullière D, Loisel S (2004) Another look at the Picard-Lefèvre formula for finite-time ruin probabilities. Insur Math Econ 35:187–203
Seal HL (1969) The stochastic theory of a risk business. Wiley, New York
Shiu ESW (1988) Calculation of the probability of eventual ruin by Beekman’s convolution series. Insur Math Econ 7:41–47
Takács L (1967) Combinatorial methods in the theory of stochastic processes. Wiley, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lefèvre, C., Loisel, S. Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities. Methodol Comput Appl Probab 11, 425–441 (2009). https://doi.org/10.1007/s11009-009-9123-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-009-9123-9
Keywords
- Ruin probability
- Finite-time horizon
- Compound Poisson risk model
- Non-constant premium
- Non-stationary claim arrivals
- Interdependent claim severities
- Recursive computational methods
- Appell polynomials
- Generalized Appell (Sheffer) polynomials