Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T04:06:19.875Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite-time ruin probabilities under large-claim reinsurance treaties for heavy-tailed claim sizes

Part of: Mathematical economics Limit theorems Mathematical modeling, applications of mathematics

Published online by Cambridge University Press:  16 July 2020

Hansjörg Albrecher ,
Bohan Chen ,
Eleni Vatamidou  and
Bert Zwart
Hansjörg Albrecher*
Affiliation:
University of Lausanne and Swiss Finance Institute
Bohan Chen*
Affiliation:
Centrum Wiskunde & Informatica (CWI)
Eleni Vatamidou*
Affiliation:
University of Lausanne
Bert Zwart*
Affiliation:
CWI and Eindhoven University of Technology
*
*Postal address: Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, 1015 Lausanne, Switzerland.
**Postal address: Centrum Wiskunde & Informatica (CWI), P.O. Box 94079, 1090 GB Amsterdam, The Netherlands.
*Postal address: Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, 1015 Lausanne, Switzerland.
**Postal address: Centrum Wiskunde & Informatica (CWI), P.O. Box 94079, 1090 GB Amsterdam, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the probability that an insurance portfolio gets ruined within a finite time period under the assumption that the r largest claims are (partly) reinsured. We show that for regularly varying claim sizes the probability of ruin after reinsurance is also regularly varying in terms of the initial capital, and derive an explicit asymptotic expression for the latter. We establish this result by leveraging recent developments on sample-path large deviations for heavy tails. Our results allow, on the asymptotic level, for an explicit comparison between two well-known large-claim reinsurance contracts, namely LCR and ECOMOR. Finally, we assess the accuracy of the resulting approximations using state-of-the-art rare event simulation techniques.

Keywords

finite-time ruin probabilitieslarge deviationsheavy tailslarge-claim reinsuranceECOMOR

MSC classification

Primary: 91B30: Risk theory, insurance
Secondary: 60F10: Large deviations 97M30: Financial and insurance mathematics
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 2 , June 2020 , pp. 513 - 530
Copyright
© Applied Probability Trust 2020

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

Albrecher, H., Robert, C. andTeugels, J. (2014). Joint asymptotic distributions of smallest and largest insurance claims. Risks 2, 289–314.CrossRefGoogle Scholar
Albrecher, H., Teugels, J. L. andBeirlant, J. (2017). Reinsurance: Actuarial and Statistical Aspects. John Wiley, Chichester.CrossRefGoogle Scholar
Ammeter, H. (1964). The rating of “Largest Claim” reinsurance covers. Quarterly letter from the Algemeene Reinsurance Companies Jubilee, 5–17.Google Scholar
Asmussen, S. andAlbrecher, H. (2010). Ruin Probabilities, 2nd edn (Adv. Ser. Statist. Sci. Appl. Prob. 14). World Scientific, Singapore.10.1142/7431CrossRefGoogle Scholar
Asmussen, S. andGlynn, P. (2007). Stochastic Simulation: Algorithms and Analysis (Stoch. Model. Appl. Prob. 57). Springer, New York.CrossRefGoogle Scholar
Asmussen, S. andKlüppelberg, C. (1996). Large deviations results for subexponential tails, with applications to insurance risk. Stoch. Process. Appl. 64, 103–125.10.1016/S0304-4149(96)00087-7CrossRefGoogle Scholar
Berliner, B. (1972). Correlations between excess of loss reinsurance covers and reinsurance of the n largest claims. ASTIN Bull. 6, 260–275.CrossRefGoogle Scholar
Castaño-Martìnez, A., Pigueiras, G. andSordo, M. (2019). On a family of risk measures based on largest claims. Insurance Math. Econom. 86, 92–97.CrossRefGoogle Scholar
Chen, B., Blanchet, J., Rhee, C.-H. andZwart, B. (2019). Efficient rare-event simulation for multiple jump events in regularly varying random walks and compound poisson processes. Math. Operat. Res. 44, 919–942.CrossRefGoogle Scholar
Embrechts, P., Goldie, C. M. andVeraverbeke, N. (1979). Subexponentiality and infinite divisibility. Prob. Theory Relat. Fields 49, 335–347.Google Scholar
Embrechts, P., Klüppelberg, C. andMikosch, T. (1997). Modelling Extremal Events: for Insurance and Finance (Appl. Math. 33). Springer, New York.Google Scholar
Hashorva, E. andLi, J. (2013). ECOMOR and LCR reinsurance with gamma-like claims. Insurance Math. Econom. 53, 206–215.10.1016/j.insmatheco.2013.05.004CrossRefGoogle Scholar
Jiang, J. andTang, Q. (2008). Reinsurance under the LCR and ECOMOR treaties with emphasis on light-tailed claims. Insurance Math. Econom. 43, 431–436.10.1016/j.insmatheco.2008.08.005CrossRefGoogle Scholar
Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Ladoucette, S. A. andTeugels, J. L. (2006). Reinsurance of large claims. J. Comput. Appl. Math. 186, 163–190.10.1016/j.cam.2005.03.069CrossRefGoogle Scholar
Li, J. (2015). Asymptotics for large claims reinsurance in a time-dependent renewal risk model. Scand. Actuar. J. 2015, 172–183.CrossRefGoogle Scholar
Peng, L. (2014). Joint tail of ECOMOR and LCR reinsurance treaties. Insurance Math. Econom. 58, 116–120.CrossRefGoogle Scholar
Rhee, C.-H.et al. (2019). Sample path large deviations for Lévy processes and random walks with regularly varying increments. Ann. Prob. 47, 3551–3605.10.1214/18-AOP1319CrossRefGoogle Scholar
Thépaut, A. (1950). Une nouvelle forme de réassurance: Le traité d’excédent du coût moyen relatif (ECOMOR). Bulletin Trimestriel de l’Institut des Actuaires Français, 273–343.Google Scholar
Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and their Application to Queues. Springer, New York.CrossRefGoogle Scholar