Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T23:05:45.980Z Has data issue: false hasContentIssue false
  • English
  • Français

Calculating Continuous Time Ruin Probabilities for a Large Portfolio with Varying Premiums

Published online by Cambridge University Press:  09 August 2013

Lourdes B. Afonso ,
Alfredo D. Egídio dos Reis  and
Howard R. Waters
Lourdes B. Afonso
Affiliation:
Depart. de Matemática and CMA, Faculdade Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal, E-mail: lbafonso@fct.unl.pt
Alfredo D. Egídio dos Reis
Affiliation:
Depart. of Mathematics, CEMAPRE and ISEG, Technical University of Lisbon, Rua do Quelhas 6, 1200-781 Lisboa, Portugal, E-mail: alfredo@iseg.utl.pt
Howard R. Waters
Affiliation:
Depart. of Actuarial Mathematics and Statistics andThe Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Riccarton Edinburgh EH14 4AS, Scotland, E-mail: H.R.Waters@ma.hw.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we present a method for the numerical evaluation of the ruin probability in continuous and finite time for a classical risk process where the premium can change from year to year. A major consideration in the development of this methodology is that it should be easily applicable to large portfolios. Our method is based on the simulation of the annual aggregate claims and then on the calculation of the ruin probability for a given surplus at the start and at the end of each year. We calculate the within-year ruin probability assuming a translated gamma distribution approximation for aggregate claim amounts.

We illustrate our method by studying the case where the premium at the start of each year is a function of the surplus level at that time or at an earlier time.

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 39 , Issue 1 , May 2009 , pp. 117 - 136
Copyright
Copyright © International Actuarial Association 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Afonso, L.B. (2008) Evaluation of ruin probabilities for surplus processes with credibility and surplus dependent premiums. PhD thesis, ISEG, Lisbon.Google Scholar
Bohman, H. and Esscher, F. (1963) Studies in risk theory with numerical illustrations concerning distribution functions and stop loss premiums. Part I. Skandinavisk Aktuarietidskrift, 1963: 173–225.Google Scholar
Cardoso, R.M.R. and Waters, H.R. (2005) Calculation of finite time ruin probabilities for some risk models. Insurance: Mathematics and Economics, 37(2): 197–215.Google Scholar
Davidson, Å. (1969) On the ruin problem in the collective theory of risk under the assumption of variable safety loading. Skandinavisk Aktuarietidskrift, 1969(3-4 Suppl): 70–83.Google Scholar
Daykin, C.D., Pentikäinen, T. and Pesonen, M. (1996) Practical Risk Theory for Actuaries. Chapman and Hall, London.Google Scholar
De Vylder, F. (1978) A practical solution to the problem of ultimate ruin probability. Scandinavian Actuarial Journal, 1978(2): 114–119.Google Scholar
Dickson, D.C.M. and Waters, H.R. (1993) Gamma processes and finite time survival probabilities. ASTIN Bulletin, 23(2): 259–272.CrossRefGoogle Scholar
Dickson, D.C.M. and Waters, H.R. (2006) Optimal dynamic reinsurance. ASTIN Bulletin, 36(2): 415–432.Google Scholar
Dickson, D.C.M. (1991) The probability of ultimate ruin with a variable premium loading – a special case. Scandinavian Actuarial Journal, 1991(1): 75–86.Google Scholar
Dufresne, F. and Gerber, H.U. (1989) Three methods to calculate the probability of ruin. ASTIN Bulletin, 19(1): 71–90.Google Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. Huebner Foundation for Insurance Education University of Pennsylvania, Philadelphia, Pa 19104 USA.Google Scholar
Jasiulewicz, H. (2001) Probability of ruin with variable premium rate in a Markovian environment. Insurance: Mathematics and Economics, 29(2): 291–296.Google Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2004) Loss models: From data to decisions. John Wiley and Sons, Inc., 2nd edition.Google Scholar
Malinovskii, V.K. (2008) Risk theory insight into a zone-adaptive control strategy. Insurance: Mathematics and Economics, 42(2): 656–667.Google Scholar
Michaud, F. (1996) Estimating the probabilities of ruin for variable premiums by simulation. ASTIN Bulletin, 26(1): 93–105.Google Scholar
Petersen, S.S. (1989) Calculation of ruin probabilities when the premium depends on the current reserve. Scandinavian Actuarial Journal, 1989(3): 147–159.Google Scholar
Seal, H.L. (1978a) From aggregate claims distribution to probability of ruin. ASTIN Bulletin, 10(1): 47–53.CrossRefGoogle Scholar
Seal, H.L. (1978b) Survival probabilities, the goal of risk theory. Wiley, New York.Google Scholar
Taylor, G.C. (1980) Probability of ruin with variable premium rate. Scandinavian Actuarial Journal, 1980(1): 57–76.Google Scholar
Wikstad, N. (1971) Exemplification of ruin probabilities. ASTIN Bulletin, 6(2): 147–152.Google Scholar