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On the time value of absolute ruin with tax

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  • Ming, Rui-Xing
  • Wang, Wen-Yuan
  • Xiao, Li-Qun

Abstract

Consider a compound Poisson surplus process of an insurer with debit interest and tax payments. When the portfolio is in a profitable situation, the insurer may pay a certain proportion of the premium income as tax payments. When the portfolio is below zero, the insurer could borrow money at a debit interest rate to continue his/her business. Meanwhile, the insurer will repay the debts from his/her premium income. The negative surplus may return to a positive level except that the surplus is below a certain critical level. In the latter case, we say that absolute ruin occurs. In this paper, we discuss absolute ruin quantities by defining an expected discounted penalty function at absolute ruin. First, a system of integro-differential equations satisfied by the expected discounted penalty function is derived. Second, closed-form expressions for the expected discounted total sum of tax payments until absolute ruin and the Laplace-Stieltjes transform (LST) of the total duration of negative surplus are obtained. Third, for exponential individual claims, closed-form expressions for the absolute ruin probability, the LST of the time to absolute ruin, the distribution function of the deficit at absolute ruin and the expected accumulated discounted tax are given. Fourth, for general individual claim distributions, when the initial surplus goes to infinity, we show that the ratio of the absolute ruin probability with tax to that without tax goes to a positive constant which is greater than one. Finally, we investigate the asymptotic behavior of the absolute ruin probability of a modified risk model where the interest rate on a positive surplus is involved.

Suggested Citation

  • Ming, Rui-Xing & Wang, Wen-Yuan & Xiao, Li-Qun, 2010. "On the time value of absolute ruin with tax," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 67-84, February.
    • Handle: RePEc:eee:insuma:v:46:y:2010:i:1:p:67-84
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    References listed on IDEAS

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    1. He, Jingmin & Wu, Rong & Zhang, Huayue, 2009. "Total duration of negative surplus for the risk model with debit interest," Statistics & Probability Letters, Elsevier, vol. 79(10), pages 1320-1326, May.
    2. Embrechts, P. & Villasenor, J. A., 1988. "Ruin estimates for large claims," Insurance: Mathematics and Economics, Elsevier, vol. 7(4), pages 269-274, December.
    3. Dickson, David C. M. & Egidio dos Reis, Alfredo D., 1997. "The effect of interest on negative surplus," Insurance: Mathematics and Economics, Elsevier, vol. 21(1), pages 1-16, October.
    4. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    5. Wei, Li, 2009. "Ruin probability in the presence of interest earnings and tax payments," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 133-138, August.
    6. Gerber, Hans U. & Shiu, Elias S. W., 1997. "The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin," Insurance: Mathematics and Economics, Elsevier, vol. 21(2), pages 129-137, November.
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    Cited by:

    1. Ming, Ruixing & Wang, Wenyuan & Hu, Yijun, 2017. "On maximizing expected discounted taxation in a risk process with interest," Statistics & Probability Letters, Elsevier, vol. 122(C), pages 128-140.
    2. Yue He & Reiichiro Kawai & Yasutaka Shimizu & Kazutoshi Yamazaki, 2022. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Papers 2203.10680, arXiv.org, revised Dec 2022.
    3. Wang, Wenyuan & Hu, Yijun, 2012. "Optimal loss-carry-forward taxation for the Lévy risk model," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 121-130.
    4. Wang, Wenyuan & Ming, Ruixing, 2018. "Two-side exit problems for taxed Lévy risk process involving the general draw-down time," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 66-74.
    5. Wenyuan Wang & Zhimin Zhang, 2019. "Optimal loss-carry-forward taxation for L\'{e}vy risk processes stopped at general draw-down time," Papers 1904.08029, arXiv.org.
    6. Eric C. K. Cheung & David Landriault, 2012. "On a Risk Model with Surplus-dependent Premium and Tax Rates," Methodology and Computing in Applied Probability, Springer, vol. 14(2), pages 233-251, June.
    7. He, Yue & Kawai, Reiichiro & Shimizu, Yasutaka & Yamazaki, Kazutoshi, 2023. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 1-28.
    8. Wenyuan Wang & Xueyuan Wu & Cheng Chi, 2019. "Optimal implementation delay of taxation with trade-off for L\'{e}vy risk Processes," Papers 1910.08158, arXiv.org.

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