IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v259y2015icp1030-1045.html

Ruin probabilities and optimal investment when the stock price follows an exponential Lévy process

Author

Listed:
  • Li, Ping
  • Zhao, Wu
  • Zhou, Wei

Abstract

This paper investigates the infinite and finite time ruin probability under the condition that the company is allowed to invest a certain amount of money in some stock market, and the remaining reserve in the bond with constant interest force. The total insurance claim amount is modeled by a compound Poisson process and the price of the risky asset follows a general exponential Lévy process. Exponential type upper bounds for the ultimate ruin probability are derived when the investment is a fixed constant, which can be calculated explicitly. This constant investment strategy yields the optimal asymptotic decay of the ruin probability under some mild assumptions. Finally, we provide an approximation of the optimal investment strategy, which maximizes the expected wealth of the insurance company under a risk constraint on the Value-at-Risk.

Suggested Citation

  • Li, Ping & Zhao, Wu & Zhou, Wei, 2015. "Ruin probabilities and optimal investment when the stock price follows an exponential Lévy process," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 1030-1045.
    • Handle: RePEc:eee:apmaco:v:259:y:2015:i:c:p:1030-1045
    DOI: 10.1016/j.amc.2014.12.042
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300314016968
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2014.12.042?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Kalashnikov, Vladimir & Norberg, Ragnar, 2002. "Power tailed ruin probabilities in the presence of risky investments," Stochastic Processes and their Applications, Elsevier, vol. 98(2), pages 211-228, April.
    2. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    3. Browne, S., 1995. "Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin," Papers 95-08, Columbia - Graduate School of Business.
    4. Anna Frolova & Serguei Pergamenshchikov & Yuri Kabanov, 2002. "In the insurance business risky investments are dangerous," Finance and Stochastics, Springer, vol. 6(2), pages 227-235.
    5. Goll, Thomas & Kallsen, Jan, 2000. "Optimal portfolios for logarithmic utility," Stochastic Processes and their Applications, Elsevier, vol. 89(1), pages 31-48, September.
    6. Klüppelberg, Claudia & Kostadinova, Radostina, 2008. "Integrated insurance risk models with exponential Lévy investment," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 560-577, April.
    7. 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.
    8. Juan-Pedro Gómez & Tridib Sharma, 2006. "Portfolio delegation under short-selling constraints," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 28(1), pages 173-196, May.
    9. Sid Browne, 1995. "Optimal Investment Policies for a Firm With a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin," Mathematics of Operations Research, INFORMS, vol. 20(4), pages 937-958, November.
    10. Hipp, Christian & Plum, Michael, 2000. "Optimal investment for insurers," Insurance: Mathematics and Economics, Elsevier, vol. 27(2), pages 215-228, October.
    11. Almazan, Andres & Brown, Keith C. & Carlson, Murray & Chapman, David A., 2004. "Why constrain your mutual fund manager?," Journal of Financial Economics, Elsevier, vol. 73(2), pages 289-321, August.
    12. Susanne Emmer & Claudia Klüppelberg & Ralf Korn, 2001. "Optimal Portfolios with Bounded Capital at Risk," Mathematical Finance, Wiley Blackwell, vol. 11(4), pages 365-384, October.
    13. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
    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. Jorge González Cázares & Aleksandar Mijatović, 2022. "Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation," Finance and Stochastics, Springer, vol. 26(4), pages 671-732, October.
    2. Yanfei Bai & Zhongbao Zhou & Helu Xiao & Rui Gao & Feimin Zhong, 2019. "A hybrid stochastic differential reinsurance and investment game with bounded memory," Papers 1910.09834, arXiv.org.
    3. Bai, Yanfei & Zhou, Zhongbao & Xiao, Helu & Gao, Rui & Zhong, Feimin, 2022. "A hybrid stochastic differential reinsurance and investment game with bounded memory," European Journal of Operational Research, Elsevier, vol. 296(2), pages 717-737.
    4. Jorge Gonz'alez C'azares & Aleksandar Mijatovi'c, 2020. "Simulation of the drawdown and its duration in L\'{e}vy models via stick-breaking Gaussian approximation," Papers 2011.06618, arXiv.org, revised Mar 2021.
    5. Jorge Ignacio Gonz'alez C'azares & Aleksandar Mijatovi'c, 2021. "Monte Carlo algorithm for the extrema of tempered stable processes," Papers 2103.15310, arXiv.org, revised Dec 2022.

    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. Jostein Paulsen, 2008. "Ruin models with investment income," Papers 0806.4125, arXiv.org, revised Dec 2008.
    2. Kostadinova, Radostina, 2007. "Optimal investment for insurers when the stock price follows an exponential Lévy process," Insurance: Mathematics and Economics, Elsevier, vol. 41(2), pages 250-263, September.
    3. Tang, Qihe & Wang, Guojing & Yuen, Kam C., 2010. "Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 362-370, April.
    4. Henrik Hult & Filip Lindskog, 2011. "Ruin probabilities under general investments and heavy-tailed claims," Finance and Stochastics, Springer, vol. 15(2), pages 243-265, June.
    5. Xu, Lin & Zhang, Liming & Yao, Dingjun, 2017. "Optimal investment and reinsurance for an insurer under Markov-modulated financial market," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 7-19.
    6. Schmidli, Hanspeter, 2005. "On optimal investment and subexponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 36(1), pages 25-35, February.
    7. Emms, P. & Haberman, S., 2007. "Asymptotic and numerical analysis of the optimal investment strategy for an insurer," Insurance: Mathematics and Economics, Elsevier, vol. 40(1), pages 113-134, January.
    8. Klüppelberg, Claudia & Kostadinova, Radostina, 2008. "Integrated insurance risk models with exponential Lévy investment," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 560-577, April.
    9. Azcue, Pablo & Muler, Nora, 2009. "Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 26-34, February.
    10. Bilel Jarraya & Abdelfettah Bouri, 2013. "A Theoretical Assessment on Optimal Asset Allocations in Insurance Industry," International Journal of Finance & Banking Studies, Center for the Strategic Studies in Business and Finance, vol. 2(4), pages 30-44, October.
    11. Tatiana Belkina & Christian Hipp & Shangzhen Luo & Michael Taksar, 2011. "Optimal Constrained Investment in the Cramer-Lundberg model," Papers 1112.4007, arXiv.org.
    12. Wang, Nan, 2007. "Optimal investment for an insurer with exponential utility preference," Insurance: Mathematics and Economics, Elsevier, vol. 40(1), pages 77-84, January.
    13. Xiang Lin, 2009. "Ruin theory for classical risk process that is perturbed by diffusion with risky investments," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(1), pages 33-44, January.
    14. Junna Bi & Junyi Guo, 2013. "Optimal Mean-Variance Problem with Constrained Controls in a Jump-Diffusion Financial Market for an Insurer," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 252-275, April.
    15. Wang, Zengwu & Xia, Jianming & Zhang, Lihong, 2007. "Optimal investment for an insurer: The martingale approach," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 322-334, March.
    16. Landriault, David & Li, Bin & Loke, Sooie-Hoe & Willmot, Gordon E. & Xu, Di, 2017. "A note on the convexity of ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 1-6.
    17. Zhou, Qing, 2009. "Optimal investment for an insurer in the Lévy market: The martingale approach," Statistics & Probability Letters, Elsevier, vol. 79(14), pages 1602-1607, July.
    18. Korn, Ralf, 2005. "Worst-case scenario investment for insurers," Insurance: Mathematics and Economics, Elsevier, vol. 36(1), pages 1-11, February.
    19. Zhao, Hui & Rong, Ximin & Zhao, Yonggan, 2013. "Optimal excess-of-loss reinsurance and investment problem for an insurer with jump–diffusion risk process under the Heston model," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 504-514.
    20. Bayraktar, Erhan & Young, Virginia R., 2007. "Minimizing the probability of lifetime ruin under borrowing constraints," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 196-221, July.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    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:eee:apmaco:v:259:y:2015:i:c:p:1030-1045. 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: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.