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On Exact Solutions for Dividend Strategies of Threshold and Linear Barrier Type in a Sparre Andersen Model*

Published online by Cambridge University Press:  17 April 2015

Hansjörg Albrecher
Affiliation:
Graz University of Technology, Steyrergasse 30, A-8010 Graz, Austria, and Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz, Austria
Jürgen Hartinger
Affiliation:
Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz, Austria
Stefan Thonhauser
Affiliation:
Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040 Linz, Austria
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Abstract

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For the classical Cramér-Lundberg risk model, a dividend strategy of threshold type has recently been suggested in the literature. This strategy consists of paying out part of the premium income as dividends to shareholders whenever the free surplus is above a given threshold level. In contrast to the well-known horizontal barrier strategy, the threshold strategy can lead to a positive infinite-horizon survival probability, with reduced profit in terms of dividend payments. In this paper we extend several of these results to a Sparre Andersen model with generalized Erlang(n)-distributed interclaim times. Furthermore, we compare the performance of the threshold strategy to a linear dividend barrier model. In particular, (partial) integro-differential equations for the corresponding ruin probabilities and expected discounted dividend payments are provided for both models and explicitly solved for n = 2 and exponentially distributed claim amounts. Finally, the explicit solutions are used to identify parameter sets for which one strategy outperforms the other and vice versa.

Type
Articles
Copyright
Copyright © ASTIN Bulletin 2007

Footnotes

*

Supported by the Austrian Science Fund Project P-18392.

References

[1] Albrecher, H., Claramunt, M.M. and Mármol, M. (2005) On the distribution of dividend payments in a Sparre Andersen model with generalized Erlang(n) interclaim times. Insurance: Mathematics & Economics, 37(2), 324334.Google Scholar
[2] Albrecher, H. and Hartinger, J. (2006) On the non-optimality of horizontal dividend barrier strategies in the Sparre Andersen model. Hermis J. Comp. Math. Appl., 7, 114.Google Scholar
[3] Albrecher, H., Hartinger, J. and Tichy, R.F. (2005) On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier. Scand. Actuar. J., 2, 103126.CrossRefGoogle Scholar
[4] Albrecher, H. and Kainhofer, R. (2002) Risk theory with a non-linear dividend barrier. Computing, 68(4), 289311.CrossRefGoogle Scholar
[5] Asmussen, S. (2000) Ruin probabilities. World Scientific, Singapore.CrossRefGoogle Scholar
[6] Azcue, P. and Muler, N. (2005) Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model. Math. Finance, 15(2), 261308.CrossRefGoogle Scholar
[7] Cheng, Y. and Tang, Q. (2003) Moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process. North American Actuarial Journal 7, 112.CrossRefGoogle Scholar
[8] Cohen, J. and Down, D. (1996) On the role of Rouché’s theorem in queueing analysis. Queueing Systems, 23, 281291.CrossRefGoogle Scholar
[9] Davis, M.H.A. (1984) Piecewise-deterministic Markov processes: a general class of nondif-fusion stochastic models. J. Roy. Statist. Soc. Ser. B, 46(3), 353388.Google Scholar
[10] Dickson, D.C.M. (1998) On a class of renewal risk process. North American Actuarial Journal, 2(3), 6073.CrossRefGoogle Scholar
[11] Dickson, D.C.M. (2005) Insurance Risk and Ruin. Cambridge University Press.CrossRefGoogle Scholar
[12] Dickson, D.C.M. and Hipp, C. (1998) Ruin probabilities for Erlang(2) risk process. Insurance Math. Econom., 22, 251262.CrossRefGoogle Scholar
[13] Dickson, D.C.M. and Hipp, C. (2001) On the time to ruin for Erlang(2) risk process. Insurance Math. Econom., 29, 333344.CrossRefGoogle Scholar
[14] Dickson, D. and Waters, H. (2004) Some optimal dividend problems. Astin Bulletin, 34(1), 4974.CrossRefGoogle Scholar
[15] Gerber, H.U. (1969) Entscheidungskriterien fuer den zusammengesetzten Poisson-Prozess. Schweiz. Aktuarver. Mitt., (1), 185227.Google Scholar
[16] Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. Huebner Foundation Monograph 8, Homewood, Ill.Google Scholar
[17] Gerber, H.U. (1981) On the probability of ruin in the presence of a linear dividend barrier. Scand. Actuar. J., (1), 105115.CrossRefGoogle Scholar
[18] Gerber, H.U. and Shiu, E. (2004) Optimal dividends: Analysis with Brownian motion. North American Actuarial Journal, 8(1), 120.CrossRefGoogle Scholar
[19] Gerber, H.U. and Shiu, E. (2005) The time value of ruin in a Sparre Andersen model. North American Actuarial Journal, 9(2), 4984.CrossRefGoogle Scholar
[20] Gerber, H.U. and Shiu, E. (2006) On optimal dividends: from reflection to refraction. J. Comput. Appl. Math., 126, 422.CrossRefGoogle Scholar
[21] Gerber, H.U. and Shiu, E. (2006) On optimal dividend strategies in the compound Poisson model. North American Actuarial Journal (to appear).CrossRefGoogle Scholar
[22] Hubalek, F. and Schachermayer, W. (2004) Optimizing expected utility of dividend payments for a Brownian risk process and a peculiar nonlinear ODE. Insurance Math. Econom., 34, 193225.CrossRefGoogle Scholar
[23] Li, S. and Garrido, J. (2004) On ruin for the Erlang(n) risk process. Insurance Math. Econom., 34(3), 391408.CrossRefGoogle Scholar
[24] Li, S. and Garrido, J. (2004) A class of renewal risk models with a constant dividend barrier. Insurance Math. Econom., 35(3), 691701.CrossRefGoogle Scholar
[25] Lin, X. and Pavlova, K. (2006) The compound poisson risk model with a threshold dividend strategy. Insurance Math. Econom. (to appear).CrossRefGoogle Scholar
[26] Lin, X.S., Willmot, G.E. and Drekic, S. (2003) The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function. Insurance Math. Econom., 33(3), 551566.Google Scholar
[27] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999) Stochastic processes for insurance and finance. Wiley Series in Probability and Statistics. John Wiley & Sons Ltd., Chichester.CrossRefGoogle Scholar
[28] Schmidli, H. (2006) Optimal control in insurance. Springer, Berlin (to appear).Google Scholar
[29] Siegl, T. and Tichy, R. (1996) Lösungsmethoden eines Risikomodells bei exponentiell fall-ender Schadensverteilung. Schweiz. Aktuarver. Mitt., (1), 85118.Google Scholar
[30] Sparre Andersen, E. (1957) On the collective theory of risk in the case of contagion between the claims. Transactions XVth Int. Congress of Actuaries, New York, (II), 219229.Google Scholar