Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-18T04:21:45.014Z Has data issue: false hasContentIssue false
  • English
  • Français

FROM RUIN TO BANKRUPTCY FOR COMPOUND POISSON SURPLUS PROCESSES

Published online by Cambridge University Press:  18 June 2013

Hansjörg Albrecher  and
Volkmar Lautscham
Hansjörg Albrecher
Affiliation:
Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, UNIL-Dorigny, CH-1015 Lausanne, and Swiss Finance Institute, Switzerland E-Mail: [email protected]
Volkmar Lautscham*
Affiliation:
Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, UNIL-Dorigny, CH-1015 Lausanne, Switzerland
*
E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In classical risk theory, the infinite-time ruin probability of a surplus process Ct is calculated as the probability of the process becoming negative at some point in time. In this paper, we consider a relaxation of the ruin concept to the concept of bankruptcy, according to which one has a positive surplus-dependent probability to continue despite temporary negative surplus. We study the resulting bankruptcy probability for the compound Poisson risk model with exponential claim sizes for different bankruptcy rate functions, deriving analytical results, upper and lower bounds as well as an efficient simulation method. Numerical examples are given and the results are compared with the classical ruin probabilities. Finally, it is illustrated how the analysis can be extended to study the discounted penalty function under this relaxed ruin criterion.

Keywords

Classical risk processOmega modelruin probabilitydiscounted penalty functionbankruptcy rate function
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 43 , Issue 2 , May 2013 , pp. 213 - 243
Copyright
Copyright © ASTIN Bulletin 2013 

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

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions, Applied Mathematics Series 55. Washington, DC: US State Department of Commerce.Google Scholar
Albrecher, H., Cheung, E.C.K. and Thonhauser, S. (2011a) Randomized observation times for the compound Poisson risk model: Dividends. ASTIN Bulletin, 41 (2), 645–672.Google Scholar
Albrecher, H., Cheung, E.C.K. and Thonhauser, S. (2011b) Randomized observation times for the compound Poisson risk model: The discounted penalty function. Scandinavian Actuarial Journal, in press, 10.1080/03461238.2011.624686.Google Scholar
Albrecher, H., Gerber, H.U. and Shiu, E.S.W. (2011) The optimal dividend barrier in the Gamma–Omega model. European Actuarial Journal, 1 (1), 43–56.CrossRefGoogle Scholar
Asmussen, S. and Albrecher, H. (2010) Ruin Probabilities, 2nd ed. World Scientific, Hackensack, New Jersey.CrossRefGoogle Scholar
Chen, N., Glasserman, P., Nouri, B. and Pelger, M. (2012) CoCos, bail-in and tail risk. Office of Financial Research Working Paper No. 4 (2013), New York.CrossRefGoogle Scholar
Dassios, A. and Embrechts, P. (1989) Martingales and insurance risk. Stochastic Models, 5, 181–217.CrossRefGoogle Scholar
Dickson, D.C.M. (2005) Insurance Risk and Ruin. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Drekic, S., Dickson, D.C.M., Stanford, D.A. and Willmot, G.E. (2004) On the distribution of the deficit at ruin when claims are phase-type. Scandinavian Actuarial Journal, 2, 105–120.CrossRefGoogle Scholar
Gerber, H.U. (1971) Der Einfluss von Zins auf die Ruinwahrscheinlichkeit. Mitteilungen der Schweizerischen Vereinigung der Versicherungsmatematiker, 71 (1), 63–70.Google Scholar
Glasserman, P. and Nouri, B. (2010) Contingent capital with a capital-ratio trigger. SSRN report (abstract 1669686).CrossRefGoogle Scholar
Insurance Information Institute (2012) Insolvencies/Guaranty Funds. Available at: www.iii.org.Google Scholar
Landriault, D., Renaud, J.-F. and Zhou, X. (2011) Occupation times of spectrally negative Lévy processes with applications. Stochastic Processes and Their Application, 121, 2629–2641.CrossRefGoogle Scholar
Maes, S. and Schoutens, W. (2010) Contingent capital: An in-depth discussion. SSRN report (abstract 1653877).CrossRefGoogle Scholar
Oxera (2007) Insurance guarantee schemes in the EU. Commissioned by the European Commission (DG Internal Market and Services). Available at: ec.europa.eu/internal_market/insurance/.Google Scholar
Schacht, J. W. and Hepler, L.P. (2007) Insolvencies in-depth series. Risk and Insurance, Jan–Apr 2007 (in four parts). Available at: http://www.riskandinsurance.com/indepthdetail.jsp?storyId=13404484Google Scholar
Zhu, J. and Yang, H. (2008) Estimates for the absolute ruin probability in the compound Poisson model with credit and debit interest. Journal of Applied Probability, 45, 818–830.CrossRefGoogle Scholar