IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v22y2020i4d10.1007_s11009-020-09780-3.html
   My bibliography  Save this article

Finite-horizon general insolvency risk measures in a regime-switching Sparre Andersen model

Author

Listed:
  • Lesław Gajek

    (Institute of Mathematics Lodz University of Technology)

  • Marcin Rudź

    (Institute of Mathematics Lodz University of Technology)

Abstract

Insolvency risk measures play important role in the theory and practice of risk management. In this paper, we provide a numerical procedure to compute vectors of their exact values and prove for them new upper and/or lower bounds which are shown to be attainable. More precisely, we investigate a general insolvency risk measure for a regime-switching Sparre Andersen model in which the distributions of claims and/or wait times are driven by a Markov chain. The measure is defined as an arbitrary increasing function of the conditional expected harm of the deficit at ruin, given the initial state of the Markov chain. A vector-valued operator L, generated by the regime-switching process, is introduced and investigated. We show a close connection between the iterations of L and the risk measure in a finite horizon. The approach assumed in the paper enables to treat in a unified way several discrete and continuous time risk models as well as a variety of important vector-valued insolvency risk measures.

Suggested Citation

  • Lesław Gajek & Marcin Rudź, 2020. "Finite-horizon general insolvency risk measures in a regime-switching Sparre Andersen model," Methodology and Computing in Applied Probability, Springer, vol. 22(4), pages 1507-1528, December.
    • Handle: RePEc:spr:metcap:v:22:y:2020:i:4:d:10.1007_s11009-020-09780-3
    DOI: 10.1007/s11009-020-09780-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-020-09780-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-020-09780-3?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 search for a different version of it.

    References listed on IDEAS

    as
    1. Chen, An & Delong, Å ukasz, 2015. "Optimal Investment For A Defined-Contribution Pension Scheme Under A Regime Switching Model," ASTIN Bulletin, Cambridge University Press, vol. 45(2), pages 397-419, May.
    2. Chen, Xu & Xiao, Ting & Yang, Xiang-qun, 2014. "A Markov-modulated jump-diffusion risk model with randomized observation periods and threshold dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 76-83.
    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. Feng, Runhuan & Shimizu, Yasutaka, 2014. "Potential measures for spectrally negative Markov additive processes with applications in ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 11-26.
    5. Armelle Guillou & Stéphane Loisel & Gilles Stupfler, 2013. "Estimation of the parameters of a Markov-modulated loss process in insurance," Post-Print hal-00589696, HAL.
    6. Jiling Cao & Teh Raihana Nazirah Roslan & Wenjun Zhang, 2018. "Pricing Variance Swaps in a Hybrid Model of Stochastic Volatility and Interest Rate with Regime-Switching," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1359-1379, December.
    7. Kim, Bara & Kim, Hwa-Sung, 2007. "Moments of claims in a Markovian environment," Insurance: Mathematics and Economics, Elsevier, vol. 40(3), pages 485-497, May.
    8. Landriault, David & Li, Bin & Li, Shu, 2015. "Analysis of a drawdown-based regime-switching Lévy insurance model," Insurance: Mathematics and Economics, Elsevier, vol. 60(C), pages 98-107.
    9. 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.
    10. Centeno, Maria de Lourdes, 2002. "Excess of loss reinsurance and Gerber's inequality in the Sparre Anderson model," Insurance: Mathematics and Economics, Elsevier, vol. 31(3), pages 415-427, December.
    11. Guillou, Armelle & Loisel, Stéphane & Stupfler, Gilles, 2013. "Estimation of the parameters of a Markov-modulated loss process in insurance," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 388-404.
    12. Cheng, Jun & Zhan, Yang, 2020. "Nonstationary l2−l∞ filtering for Markov switching repeated scalar nonlinear systems with randomly occurring nonlinearities," Applied Mathematics and Computation, Elsevier, vol. 365(C).
    13. Ignatieva, Katja & Song, Andrew & Ziveyi, Jonathan, 2018. "Fourier Space Time-Stepping Algorithm For Valuing Guaranteed Minimum Withdrawal Benefits In Variable Annuities Under Regime-Switching And Stochastic Mortality," ASTIN Bulletin, Cambridge University Press, vol. 48(1), pages 139-169, January.
    14. Jae-Kyung Woo & Haibo Liu, 2018. "Discounted Aggregate Claim Costs Until Ruin in the Discrete-Time Renewal Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1285-1318, December.
    15. Paweł Lorek, 2017. "Generalized Gambler’s Ruin Problem: Explicit Formulas via Siegmund Duality," Methodology and Computing in Applied Probability, Springer, vol. 19(2), pages 603-613, June.
    16. Mark Brown & Shuangning Li, 2018. "Sharp Bounds for Exponential Approximations of NWUE Distributions," Methodology and Computing in Applied Probability, Springer, vol. 20(3), pages 875-896, September.
    17. Yue Liu & Nicolas Privault, 2018. "A Recursive Algorithm for Selling at the Ultimate Maximum in Regime-Switching Models," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 369-384, March.
    18. Guillou, Armelle & Loisel, Stéphane & Stupfler, Gilles, 2013. "Estimation of the parameters of a Markov-modulated loss process in insurance," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 388-404.
    19. Jin, Zhuo & Liu, Guo & Yang, Hailiang, 2020. "Optimal consumption and investment strategies with liquidity risk and lifetime uncertainty for Markov regime-switching jump diffusion models," European Journal of Operational Research, Elsevier, vol. 280(3), pages 1130-1143.
    Full references (including those not matched with items on IDEAS)

    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. Lesław Gajek & Marcin Rudź, 2020. "Finite-Horizon Ruin Probabilities in a Risk-Switching Sparre Andersen Model," Methodology and Computing in Applied Probability, Springer, vol. 22(4), pages 1493-1506, December.
    2. Gajek, Lesław & Rudź, Marcin, 2018. "Banach Contraction Principle and ruin probabilities in regime-switching models," Insurance: Mathematics and Economics, Elsevier, vol. 80(C), pages 45-53.
    3. Gajek, Lesław & Rudź, Marcin, 2017. "A generalization of Gerber’s inequality for ruin probabilities in risk-switching models," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 236-240.
    4. Lesław Gajek & Marcin Rudź, 2018. "Risk-switching insolvency models," Collegium of Economic Analysis Annals, Warsaw School of Economics, Collegium of Economic Analysis, issue 51, pages 129-146.
    5. Benjamin Avanzi & Greg Taylor & Bernard Wong & Alan Xian, 2020. "Modelling and understanding count processes through a Markov-modulated non-homogeneous Poisson process framework," Papers 2003.13888, arXiv.org, revised May 2020.
    6. 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.
    7. Dutang, C. & Lefèvre, C. & Loisel, S., 2013. "On an asymptotic rule A+B/u for ultimate ruin probabilities under dependence by mixing," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 774-785.
    8. Jiaen Xu & Chunwei Wang & Naidan Deng & Shujing Wang, 2023. "Numerical Method for a Risk Model with Two-Sided Jumps and Proportional Investment," Mathematics, MDPI, vol. 11(7), pages 1-22, March.
    9. Woo, Jae-Kyung & Cheung, Eric C.K., 2013. "A note on discounted compound renewal sums under dependency," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 170-179.
    10. F. Baltazar-Larios & Luz Judith R. Esparza, 2022. "Statistical Inference for Partially Observed Markov-Modulated Diffusion Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 571-593, June.
    11. Avanzi, Benjamin & Taylor, Greg & Wong, Bernard & Xian, Alan, 2021. "Modelling and understanding count processes through a Markov-modulated non-homogeneous Poisson process framework," European Journal of Operational Research, Elsevier, vol. 290(1), pages 177-195.
    12. Ahn, Soohan & Badescu, Andrei L. & Cheung, Eric C.K. & Kim, Jeong-Rae, 2018. "An IBNR–RBNS insurance risk model with marked Poisson arrivals," Insurance: Mathematics and Economics, Elsevier, vol. 79(C), pages 26-42.
    13. Choi, Michael C.H. & Cheung, Eric C.K., 2014. "On the expected discounted dividends in the Cramér–Lundberg risk model with more frequent ruin monitoring than dividend decisions," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 121-132.
    14. Zhimin Zhang & Eric C. K. Cheung, 2016. "The Markov Additive Risk Process Under an Erlangized Dividend Barrier Strategy," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 275-306, June.
    15. 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.
    16. Zan Yu & Lianzeng Zhang, 2024. "Computing the Gerber-Shiu function with interest and a constant dividend barrier by physics-informed neural networks," Papers 2401.04378, arXiv.org.
    17. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078.
    18. Shihao Zhu & Jingtao Shi, 2019. "Optimal Reinsurance and Investment Strategies under Mean-Variance Criteria: Partial and Full Information," Papers 1906.08410, arXiv.org, revised Jun 2020.
    19. Wu, Rong & Wang, Guojing & Zhang, Chunsheng, 2005. "On a joint distribution for the risk process with constant interest force," Insurance: Mathematics and Economics, Elsevier, vol. 36(3), pages 365-374, June.
    20. Young, Virginia R., 2017. "Purchasing casualty insurance to avoid lifetime ruin," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 133-142.

    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:spr:metcap:v:22:y:2020:i:4:d:10.1007_s11009-020-09780-3. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.