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Methods for estimating the optimal dividend barrier and the probability of ruin

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  • Gerber, Hans U.
  • Shiu, Elias S.W.
  • Smith, Nathaniel

Abstract

In applications of collective risk theory, complete information about the individual claim amount distribution is often not known, but reliable estimates of its first few moments may be available. For such a situation, this paper develops methods for estimating the optimal dividend barrier and the probability of ruin. In particular, two De Vylder approximations are explained, and the first and second order diffusion approximations are examined. For several claim amount distributions, the approximate values are compared numerically with the exact values. The De Vylder and diffusion approximations can be adapted to the more general situation where the aggregate claims process is a Lévy process with nonnegative increments.

Suggested Citation

  • Gerber, Hans U. & Shiu, Elias S.W. & Smith, Nathaniel, 2008. "Methods for estimating the optimal dividend barrier and the probability of ruin," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 243-254, February.
    • Handle: RePEc:eee:insuma:v:42:y:2008:i:1:p:243-254
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    References listed on IDEAS

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    1. Dickson, D. C. M. & Drekic, S., 2006. "Optimal Dividends Under a Ruin Probability Constraint," Annals of Actuarial Science, Cambridge University Press, vol. 1(2), pages 291-306, September.
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    3. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    4. Dufresne, François & Gerber, Hans U. & Shiu, Elias S. W., 1991. "Risk Theory with the Gamma Process," ASTIN Bulletin, Cambridge University Press, vol. 21(2), pages 177-192, November.
    5. Dickson,David C. M., 2005. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9780521846400.
    6. Gerber, Hans U. & Lin, X. Sheldon & Yang, Hailiang, 2006. "A Note on the Dividends-Penalty Identity and the Optimal Dividend Barrier," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 489-503, November.
    7. Chan, Beda, 1990. "Ruin Probability for Translated Combination of Exponential Claims," ASTIN Bulletin, Cambridge University Press, vol. 20(1), pages 113-114, April.
    8. Daniel Dufresne, 2007. "Fitting combinations of exponentials to probability distributions," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 23(1), pages 23-48, January.
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    Cited by:

    1. Liu, Zaiming & Li, Manman & Ameer, Sherbaz, 2009. "Methods for estimating optimal Dickson and Waters modification dividend barrier," Economic Modelling, Elsevier, vol. 26(5), pages 886-892, September.
    2. Florin Avram & Dan Goreac & Rim Adenane & Ulyses Solon, 2022. "Optimizing Dividends and Capital Injections Limited by Bankruptcy, and Practical Approximations for the Cramér-Lundberg Process," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2339-2371, December.
    3. Hu, Xiang & Duan, Baige & Zhang, Lianzeng, 2017. "De Vylder approximation to the optimal retention for a combination of quota-share and excess of loss reinsurance with partial information," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 48-55.
    4. Asaf Cohen & Virginia R. Young, 2019. "Rate of Convergence of the Probability of Ruin in the Cram\'er-Lundberg Model to its Diffusion Approximation," Papers 1902.00706, arXiv.org, revised Jun 2020.
    5. Liang, Xiaoqing & Liang, Zhibin & Young, Virginia R., 2020. "Optimal reinsurance under the mean–variance premium principle to minimize the probability of ruin," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 128-146.
    6. Florin Avram & Andras Horváth & Serge Provost & Ulyses Solon, 2019. "On the Padé and Laguerre–Tricomi–Weeks Moments Based Approximations of the Scale Function W and of the Optimal Dividends Barrier for Spectrally Negative Lévy Risk Processes," Risks, MDPI, vol. 7(4), pages 1-24, December.
    7. Yuen, Kam-Chuen & Zhou, Ming & Guo, Junyi, 2008. "On a risk model with debit interest and dividend payments," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2426-2432, October.
    8. Cohen, Asaf & Young, Virginia R., 2020. "Rate of convergence of the probability of ruin in the Cramér–Lundberg model to its diffusion approximation," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 333-340.
    9. Wang, Chunwei & Yin, Chuancun & Li, Erqiang, 2010. "On the classical risk model with credit and debit interests under absolute ruin," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 427-436, March.
    10. Gerber, Hans U. & Smith, Nathaniel, 2008. "Optimal dividends with incomplete information in the dual model," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 227-233, October.
    11. Chunwei Wang & Chuancun Yin, 2009. "Dividend payments in the classical risk model under absolute ruin with debit interest," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(3), pages 247-262, May.

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