IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1710.11065.html
   My bibliography  Save this paper

On Fair Reinsurance Premiums; Capital Injections in a Perturbed Risk Model

Author

Listed:
  • Zied Ben Salah
  • Jos'e Garrido

Abstract

We consider a risk model where deficits after ruin are covered by a new type of reinsurance contract that provides capital injections. To allow the insurance company's survival after ruin, the reinsurer injects capital only at ruin times caused by jumps larger than a chosen retention level. Otherwise capital must be raised from the shareholders for small deficits. The problem here is to determine adequate reinsurance premiums. It seems fair to base the net reinsurance premium on the discounted expected value of any future capital injections. Inspired by the results of Huzak et al. (2004) and Ben Salah (2014) on successive ruin events, we show that an explicit formula for these reinsurance premiums exists in a setting where aggregate claims are modeled by a subordinator and a Brownian perturbation. Here ruin events are due either to Brownian oscillations or jumps and reinsurance capital injections only apply in the latter case. The results are illustrated explicitly for two specific risk models and in some numerical examples.

Suggested Citation

  • Zied Ben Salah & Jos'e Garrido, 2017. "On Fair Reinsurance Premiums; Capital Injections in a Perturbed Risk Model," Papers 1710.11065, arXiv.org, revised Jun 2018.
  • Handle: RePEc:arx:papers:1710.11065
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1710.11065
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. Dufresne, Francois & Gerber, Hans U., 1991. "Risk theory for the compound Poisson process that is perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 10(1), pages 51-59, March.
    3. Biffis, Enrico & Kyprianou, Andreas E., 2010. "A note on scale functions and the time value of ruin for Lévy insurance risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 85-91, February.
    4. Furrer, H. J. & Schmidli, H., 1994. "Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 15(1), pages 23-36, October.
    5. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Eisenberg, Julia & Fabrykowski, Lukas & Schmeck, Maren Diane, 2021. "Optimal Surplus-dependent Reinsurance under Regime-Switching in a Brownian Risk Model," Center for Mathematical Economics Working Papers 648, Center for Mathematical Economics, Bielefeld University.
    2. Julia Eisenberg & Lukas Fabrykowski & Maren Diane Schmeck, 2021. "Optimal Surplus-Dependent Reinsurance under Regime-Switching in a Brownian Risk Model," Risks, MDPI, vol. 9(4), pages 1-25, April.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ben Salah, Zied & Garrido, José, 2018. "On fair reinsurance premiums; Capital injections in a perturbed risk model," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 11-20.
    2. Morales, Manuel, 2007. "On the expected discounted penalty function for a perturbed risk process driven by a subordinator," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 293-301, March.
    3. Zied Ben-Salah & H'el`ene Gu'erin & Manuel Morales & Hassan Omidi Firouzi, 2014. "On the Depletion Problem for an Insurance Risk Process: New Non-ruin Quantities in Collective Risk Theory," Papers 1406.6952, arXiv.org.
    4. Willmot, Gordon E. & Lin, Xiaodong, 1996. "Bounds on the tails of convolutions of compound distributions," Insurance: Mathematics and Economics, Elsevier, vol. 18(1), pages 29-33, May.
    5. Kolkovska, Ekaterina T. & Martín-González, Ehyter M., 2016. "Gerber–Shiu functionals for classical risk processes perturbed by an α-stable motion," Insurance: Mathematics and Economics, Elsevier, vol. 66(C), pages 22-28.
    6. Kam C. Yuen & Yuhua Lu & Rong Wu, 2009. "The compound Poisson process perturbed by a diffusion with a threshold dividend strategy," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(1), pages 73-93, January.
    7. Christensen, Bent Jesper & Parra-Alvarez, Juan Carlos & Serrano, Rafael, 2021. "Optimal control of investment, premium and deductible for a non-life insurance company," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 384-405.
    8. Schlegel, Sabine, 1998. "Ruin probabilities in perturbed risk models," Insurance: Mathematics and Economics, Elsevier, vol. 22(1), pages 93-104, May.
    9. Cai, Jun, 2004. "Ruin probabilities and penalty functions with stochastic rates of interest," Stochastic Processes and their Applications, Elsevier, vol. 112(1), pages 53-78, July.
    10. Chiu, S. N. & Yin, C. C., 2003. "The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 33(1), pages 59-66, August.
    11. Wang, Guojing & Wu, Rong, 2008. "The expected discounted penalty function for the perturbed compound Poisson risk process with constant interest," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 59-64, February.
    12. Diko, Peter & Usábel, Miguel, 2011. "A numerical method for the expected penalty-reward function in a Markov-modulated jump-diffusion process," Insurance: Mathematics and Economics, Elsevier, vol. 49(1), pages 126-131, July.
    13. Zhang, Chunsheng & Wang, Guojing, 2003. "The joint density function of three characteristics on jump-diffusion risk process," Insurance: Mathematics and Economics, Elsevier, vol. 32(3), pages 445-455, July.
    14. Gerber, Hans U. & Landry, Bruno, 1998. "On the discounted penalty at ruin in a jump-diffusion and the perpetual put option," Insurance: Mathematics and Economics, Elsevier, vol. 22(3), pages 263-276, July.
    15. Schmidli, Hanspeter, 2010. "Conditional law of risk processes given that ruin occurs," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 281-289, April.
    16. Claude Lefèvre & Philippe Picard, 2013. "Ruin Time and Severity for a Lévy Subordinator Claim Process: A Simple Approach," Risks, MDPI, vol. 1(3), pages 1-21, December.
    17. Danijel Grahovac, 2018. "Densities of Ruin-Related Quantities in the Cramér-Lundberg Model with Pareto Claims," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 273-288, March.
    18. 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.
    19. Liu, Peng & Zhang, Chunsheng & Ji, Lanpeng, 2017. "A note on ruin problems in perturbed classical risk models," Statistics & Probability Letters, Elsevier, vol. 120(C), pages 28-33.
    20. Biffis, Enrico & Morales, Manuel, 2010. "On a generalization of the Gerber-Shiu function to path-dependent penalties," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 92-97, February.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1710.11065. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.