Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-25T21:14:17.288Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal Dynamic Reinsurance

Published online by Cambridge University Press:  17 April 2015

David C.M. Dickson  and
Howard R. Waters
David C.M. Dickson
Affiliation:
Centre for Actuarial Studies, Department of Economics, University of Melbourne, Victoria 3010, Australia, E-mail: [email protected]
Howard R. Waters
Affiliation:
Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, Great Britain, Email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a classical surplus process where the insurer can choose a different level of reinsurance at the start of each year. We assume the insurer’s objective is to minimise the probability of ruin up to some given time horizon, either in discrete or continuous time. We develop formulae for ruin probabilities under the optimal reinsurance strategy, i.e. the optimal retention each year as the surplus changes and the period until the time horizon shortens. For our compound Poisson process, it is not feasible to evaluate these formulae, and hence determine the optimal strategies, in any but the simplest cases. We show how we can determine the optimal strategies by approximating the (compound Poisson) aggregate claims distributions by translated gamma distributions, and, alternatively, by approximating the compound Poisson process by a translated gamma process.

Keywords

Finite time ruinreinsurancedynamic strategytranslated gamma distributiontranslated gamma process
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 36 , Issue 2 , November 2006 , pp. 415 - 432
Copyright
Copyright © ASTIN Bulletin 2006

References

Centeno, M. de L. (1986) Measuring the effects of reinsurance by the adjustment coefficient. Insurance: Mathematics & Economics 5, 169–182.Google Scholar
Daykin, C.D., Pentikäinen, T. and Pesonen, M. (1993) Practical Risk Theory for Actuaries. Chapman and Hall, London.CrossRefGoogle Scholar
De Vylder, F. and Goovaerts, M.J. (1988) Recursive calculation of finite time ruin probabilities. Insurance: Mathematics & Economics 7, 1–8.Google Scholar
De Vylder, F. and Goovaerts, M.J. (1999) Explicit finite-time and infinite-time ruin probabilities in the continuous case. Insurance: Mathematics & Economics 24, 155–172.Google Scholar
Dickson, D.C.M. and Waters, H.R. (1993) Gamma processes and finite time survival probabilities. ASTIN Bulletin 23, 259–272.CrossRefGoogle Scholar
Dickson, D.C.M. and Waters, H.R. (1996) Reinsurance and ruin. Insurance: Mathematics & Economics 19, 61–80.Google Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation, Philadelphia, PA.Google Scholar
Hipp, C. and Vogt, M. (2003) Optimal dynamic XLreinsurance. ASTIN Bulletin 33, 193–207.CrossRefGoogle Scholar
Prabhu, N.U. (1961) On the ruin problem of collective risk theory. Annals of Mathematical Statistics 32, 757–764.CrossRefGoogle Scholar
Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (1992) Numerical Recipes in Fortran 77, 2nd edition. Cambridge University Press, Cambridge.Google Scholar
Promislow, S.D. and Young, V.R. (2005) Minimizing the probability of ruin when claims follow Brownian motion with drift. North American Actuarial Journal 9(3), 109–128.Google Scholar
Schmidli, H. (2001) Optimal proportional reinsurance policies in a dynamic setting. Scandinavian Actuarial Journal, 55–68.CrossRefGoogle Scholar
Schmidli, H. (2002) On minimising the ruin probability by investment and reinsurance. Annals of Applied Probability 12, 890–907.CrossRefGoogle Scholar
Seal, H.L. (1978) From aggregate claims distribution to probability of ruin. ASTIN Bulletin 10, 47–53.CrossRefGoogle Scholar
Wu, R., Wang, G., and Wei, L. (2003) Joint distributions of some actuarial random vectors containing the time of ruin. Insurance: Mathematics & Economics 33, 147–161.Google Scholar