IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v396y2021ics0096300320308225.html
   My bibliography  Save this article

Analysis of an aggregate loss model in a Markov renewal regime

Author

Listed:
  • Ramírez-Cobo, Pepa
  • Carrizosa, Emilio
  • Lillo, Rosa E.

Abstract

In this article we consider an aggregate loss model with dependent losses. The loss occurrence process is governed by a two-state Markovian arrival process (MAP2), a Markov renewal process that allows for (1) correlated inter-loss times, (2) non-exponentially distributed inter-loss times and, (3) overdisperse loss counts. Some quantities of interest to measure persistence in the loss occurrence process are obtained. Given a real OpRisk database, the aggregate loss model is estimated by fitting separately the inter-loss times and severities. The MAP2 is estimated via direct maximization of the likelihood function, and severities are modeled by the heavy-tailed, double-Pareto Lognormal distribution. In comparison with the fit provided by the Poisson process, the results point out that taking into account the dependence and overdispersion in the inter-loss times distribution leads to higher capital charges.

Suggested Citation

  • Ramírez-Cobo, Pepa & Carrizosa, Emilio & Lillo, Rosa E., 2021. "Analysis of an aggregate loss model in a Markov renewal regime," Applied Mathematics and Computation, Elsevier, vol. 396(C).
    • Handle: RePEc:eee:apmaco:v:396:y:2021:i:c:s0096300320308225
    DOI: 10.1016/j.amc.2020.125869
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300320308225
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2020.125869?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. Shao, Jia & Papaioannou, Apostolos D. & Pantelous, Athanasios A., 2017. "Pricing and simulating catastrophe risk bonds in a Markov-dependent environment," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 68-84.
    2. Paul Fearnhead & Chris Sherlock, 2006. "An exact Gibbs sampler for the Markov‐modulated Poisson process," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(5), pages 767-784, November.
    3. Bodrog, L. & Heindl, A. & Horváth, G. & Telek, M., 2008. "A Markovian canonical form of second-order matrix-exponential processes," European Journal of Operational Research, Elsevier, vol. 190(2), pages 459-477, October.
    4. Dalla Valle, L. & Giudici, P., 2008. "A Bayesian approach to estimate the marginal loss distributions in operational risk management," Computational Statistics & Data Analysis, Elsevier, vol. 52(6), pages 3107-3127, February.
    5. Cheung, Eric C.K. & Landriault, David, 2010. "A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 127-134, February.
    6. Ng, Andrew C.Y. & Yang, Hailiang, 2006. "On the joint distribution of surplus before and after ruin under a Markovian regime switching model," Stochastic Processes and their Applications, Elsevier, vol. 116(2), pages 244-266, February.
    7. Pavel V. Shevchenko, 2010. "Calculation of aggregate loss distributions," Papers 1008.1108, arXiv.org.
    8. Ariane Chapelle & Yves Crama & Georges Hubner & Jean-Philippe Peeters, 2004. "Basel II and Operational Risk: Implications for risk measurement and management in the financial sector," Working Paper Research 51, National Bank of Belgium.
    9. Stutzer, Michael, 2020. "Persistence of averages in financial Markov Switching models: A large deviations approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 553(C).
    10. Pfeifer, Dietmar & Nešlehová, Johana, 2004. "Modeling and Generating Dependent Risk Processes for IRM and DFA," ASTIN Bulletin, Cambridge University Press, vol. 34(2), pages 333-360, November.
    11. Pietro Fodra & Huy^en Pham, 2013. "High frequency trading and asymptotics for small risk aversion in a Markov renewal model," Papers 1310.1756, arXiv.org, revised Jan 2015.
    12. Pavel V. Shevchenko, 2010. "Implementing loss distribution approach for operational risk," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 26(3), pages 277-307, May.
    13. Dimitrova, Dimitrina S. & Kaishev, Vladimir K. & Zhao, Shouqi, 2016. "On the evaluation of finite-time ruin probabilities in a dependent risk model," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 268-286.
    14. Rodríguez, Joanna & Lillo, Rosa E. & Ramírez-Cobo, Pepa, 2015. "Failure modeling of an electrical N-component framework by the non-stationary Markovian arrival process," Reliability Engineering and System Safety, Elsevier, vol. 134(C), pages 126-133.
    15. Maegebier, Alexander, 2013. "Valuation and risk assessment of disability insurance using a discrete time trivariate Markov renewal reward process," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 802-811.
    16. Cheung, Eric C.K. & Liu, Haibo & Willmot, Gordon E., 2018. "Joint moments of the total discounted gains and losses in the renewal risk model with two-sided jumps," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 358-377.
    17. Stefan Gerhold & Uwe Schmock & Richard Warnung, 2010. "A generalization of Panjer’s recursion and numerically stable risk aggregation," Finance and Stochastics, Springer, vol. 14(1), pages 81-128, January.
    18. Feria-Domínguez, José Manuel & Jiménez-Rodríguez, Enrique & Sholarin, Ola, 2015. "Tackling the over-dispersion of operational risk: Implications on capital adequacy requirements," The North American Journal of Economics and Finance, Elsevier, vol. 31(C), pages 206-221.
    19. Xu, Chi & Zheng, Chunling & Wang, Donghua & Ji, Jingru & Wang, Nuan, 2019. "Double correlation model for operational risk: Evidence from Chinese commercial banks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 516(C), pages 327-339.
    20. Lothar Breuer, 2002. "An EM Algorithm for Batch Markovian Arrival Processes and its Comparison to a Simpler Estimation Procedure," Annals of Operations Research, Springer, vol. 112(1), pages 123-138, April.
    21. Chavez-Demoulin, V. & Embrechts, P. & Neslehova, J., 2006. "Quantitative models for operational risk: Extremes, dependence and aggregation," Journal of Banking & Finance, Elsevier, vol. 30(10), pages 2635-2658, October.
    22. Hiroyuki Okamura & Tadashi Dohi, 2016. "Fitting Phase-Type Distributions and Markovian Arrival Processes: Algorithms and Tools," Springer Series in Reliability Engineering, in: Lance Fiondella & Antonio Puliafito (ed.), Principles of Performance and Reliability Modeling and Evaluation, pages 49-75, Springer.
    23. Vallois, Pierre & Tapiero, Charles S., 2009. "A claims persistence process and insurance," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 367-373, June.
    24. Brechmann, Eike & Czado, Claudia & Paterlini, Sandra, 2014. "Flexible dependence modeling of operational risk losses and its impact on total capital requirements," Journal of Banking & Finance, Elsevier, vol. 40(C), pages 271-285.
    25. Yera, Yoel G. & Lillo, Rosa E. & Ramírez-Cobo, Pepa, 2019. "Fitting procedure for the two-state Batch Markov modulated Poisson process," European Journal of Operational Research, Elsevier, vol. 279(1), pages 79-92.
    26. Avanzi, Benjamin & Wong, Bernard & Yang, Xinda, 2016. "A micro-level claim count model with overdispersion and reporting delays," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 1-14.
    27. Ren, Jiandong, 2012. "A multivariate aggregate loss model," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 402-408.
    28. Ahn, Soohan & Badescu, Andrei L., 2007. "On the analysis of the Gerber-Shiu discounted penalty function for risk processes with Markovian arrivals," Insurance: Mathematics and Economics, Elsevier, vol. 41(2), pages 234-249, September.
    29. Kim, Chesoong & Klimenok, V.I. & Dudin, A.N., 2017. "Analysis of unreliable BMAP/PH/N type queue with Markovian flow of breakdowns," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 154-172.
    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. Xu, Chi & Zheng, Chunling & Wang, Donghua & Ji, Jingru & Wang, Nuan, 2019. "Double correlation model for operational risk: Evidence from Chinese commercial banks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 516(C), pages 327-339.
    2. Yera, Yoel G. & Lillo, Rosa E. & Ramírez-Cobo, Pepa, 2019. "Fitting procedure for the two-state Batch Markov modulated Poisson process," European Journal of Operational Research, Elsevier, vol. 279(1), pages 79-92.
    3. Yera, Yoel G. & Lillo, Rosa E. & Nielsen, Bo F. & Ramírez-Cobo, Pepa & Ruggeri, Fabrizio, 2021. "A bivariate two-state Markov modulated Poisson process for failure modeling," Reliability Engineering and System Safety, Elsevier, vol. 208(C).
    4. 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.
    5. Lu Wei & Jianping Li & Xiaoqian Zhu, 2018. "Operational Loss Data Collection: A Literature Review," Annals of Data Science, Springer, vol. 5(3), pages 313-337, September.
    6. Eckert, Christian & Gatzert, Nadine, 2017. "Modeling operational risk incorporating reputation risk: An integrated analysis for financial firms," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 122-137.
    7. Iñaki Aldasoro & Leonardo Gambacorta & Paolo Giudici & Thomas Leach, 2020. "Operational and cyber risks in the financial sector," BIS Working Papers 840, Bank for International Settlements.
    8. Lu, Zhaoyang, 2011. "Modeling the yearly Value-at-Risk for operational risk in Chinese commercial banks," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(4), pages 604-616.
    9. Kley, Oliver & Klüppelberg, Claudia & Paterlini, Sandra, 2020. "Modelling extremal dependence for operational risk by a bipartite graph," Journal of Banking & Finance, Elsevier, vol. 117(C).
    10. Gareth W. Peters & Aaron D. Byrnes & Pavel V. Shevchenko, 2010. "Impact of Insurance for Operational Risk: Is it worthwhile to insure or be insured for severe losses?," Papers 1010.4406, arXiv.org, revised Nov 2010.
    11. Dong-Young Lim, 2021. "A Neural Frequency-Severity Model and Its Application to Insurance Claims," Papers 2106.10770, arXiv.org, revised Feb 2024.
    12. Casale, Giuliano & Sansottera, Andrea & Cremonesi, Paolo, 2016. "Compact Markov-modulated models for multiclass trace fitting," European Journal of Operational Research, Elsevier, vol. 255(3), pages 822-833.
    13. Guillén, Montserrat & Sarabia, José María & Prieto, Faustino, 2013. "Simple risk measure calculations for sums of positive random variables," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 273-280.
    14. Peters, Gareth W. & Byrnes, Aaron D. & Shevchenko, Pavel V., 2011. "Impact of insurance for operational risk: Is it worthwhile to insure or be insured for severe losses?," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 287-303, March.
    15. Cheung, Eric C.K. & Feng, Runhuan, 2013. "A unified analysis of claim costs up to ruin in a Markovian arrival risk model," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 98-109.
    16. Mark Bentley & Alec Stephenson & Peter Toscas & Zili Zhu, 2020. "A Multivariate Model to Quantify and Mitigate Cybersecurity Risk," Risks, MDPI, vol. 8(2), pages 1-21, June.
    17. 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.
    18. Silvia Figini & Lijun Gao & Paolo Giudici, 2013. "Bayesian operational risk models," DEM Working Papers Series 047, University of Pavia, Department of Economics and Management.
    19. Yera Mora, Yoel Gustavo & Lillo Rodríguez, Rosa Elvira & Ramírez-Cobo, Pepa, 2017. "Findings about the two-state BMMPP for modeling point processes in reliability and queueing systems," DES - Working Papers. Statistics and Econometrics. WS 24622, Universidad Carlos III de Madrid. Departamento de Estadística.
    20. Wang, Zongrun & Wang, Wuchao & Chen, Xiaohong & Jin, Yanbo & Zhou, Yanju, 2012. "Using BS-PSD-LDA approach to measure operational risk of Chinese commercial banks," Economic Modelling, Elsevier, vol. 29(6), pages 2095-2103.

    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:396:y:2021:i:c:s0096300320308225. 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.