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Infinitely stochastic micro reserving

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  • Maciak, Matúš
  • Okhrin, Ostap
  • Pešta, Michal

Abstract

Stochastic forecasting and risk valuation are now front burners in a list of applied and theoretical sciences. In this work, we propose an unconventional tool for stochastic prediction of future expenses based on the individual (micro) developments of recorded events. Considering a firm, enterprise, institution, or any entity, which possesses knowledge about particular historical events, there might be a whole series of several related subevents: payments or losses spread over time. This all leads to an infinitely stochastic process at the end. The aim, therefore, lies in predicting future subevent flows coming from already reported, occurred but not reported, and yet not occurred events. The emerging forecasting methodology involves marked time-varying Hawkes process with marks being other time-varying Hawkes processes. The estimated parameters of the model are proved to be consistent and asymptotically normal under simple and easily verifiable assumptions. The empirical properties are investigated through a simulation study. In the practical part of our exploration, we elaborate a specific actuarial application for micro claims reserving.

Suggested Citation

  • Maciak, Matúš & Okhrin, Ostap & Pešta, Michal, 2021. "Infinitely stochastic micro reserving," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 30-58.
    • Handle: RePEc:eee:insuma:v:100:y:2021:i:c:p:30-58
    DOI: 10.1016/j.insmatheco.2021.04.007
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    1. Bessy-Roland, Yannick & Boumezoued, Alexandre & Hillairet, Caroline, 2021. "Multivariate Hawkes process for cyber insurance," Annals of Actuarial Science, Cambridge University Press, vol. 15(1), pages 14-39, March.
    2. Davidson, James, 1994. "Stochastic Limit Theory: An Introduction for Econometricians," OUP Catalogue, Oxford University Press, number 9780198774037.
    3. Larsen, Christian Roholte, 2007. "An Individual Claims Reserving Model," ASTIN Bulletin, Cambridge University Press, vol. 37(1), pages 113-132, May.
    4. Pigeon, Mathieu & Antonio, Katrien & Denuit, Michel, 2014. "Individual loss reserving using paid–incurred data," LIDAM Reprints ISBA 2014024, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Haastrup, Svend & Arjas, Elja, 1996. "Claims Reserving in Continuous Time; A Nonparametric Bayesian Approach," ASTIN Bulletin, Cambridge University Press, vol. 26(2), pages 139-164, November.
    6. England, Peter & Verrall, Richard, 1999. "Analytic and bootstrap estimates of prediction errors in claims reserving," Insurance: Mathematics and Economics, Elsevier, vol. 25(3), pages 281-293, December.
    7. Dassios, Angelos & Zhao, Hongbiao, 2013. "Exact simulation of Hawkes process with exponentially decaying intensity," LSE Research Online Documents on Economics 51370, London School of Economics and Political Science, LSE Library.
    8. Francis Duval & Mathieu Pigeon, 2019. "Individual Loss Reserving Using a Gradient Boosting-Based Approach," Risks, MDPI, vol. 7(3), pages 1-18, July.
    9. Maximilien Baudry & Christian Y. Robert, 2019. "A machine learning approach for individual claims reserving in insurance," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 35(5), pages 1127-1155, September.
    10. Massimo De Felice & Franco Moriconi, 2019. "Claim Watching and Individual Claims Reserving Using Classification and Regression Trees," Risks, MDPI, vol. 7(4), pages 1-36, October.
    11. Badescu, Andrei L. & Lin, X. Sheldon & Tang, Dameng, 2016. "A marked Cox model for the number of IBNR claims: Theory," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 29-37.
    12. Weisberg, Herbert I & Tomberlin, Thomas J & Chatterjee, Sangit, 1984. "Predicting Insurance Losses under Cross-Classification: A Comparison of Alternative Approaches," Journal of Business & Economic Statistics, American Statistical Association, vol. 2(2), pages 170-178, April.
    13. Taylor, Greg & McGuire, Gráinne & Sullivan, James, 2008. "Individual Claim Loss Reserving Conditioned by Case Estimates," Annals of Actuarial Science, Cambridge University Press, vol. 3(1-2), pages 215-256, September.
    14. Bacry, E. & Delattre, S. & Hoffmann, M. & Muzy, J.F., 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2475-2499.
    15. Giesecke, Kay & Schwenkler, Gustavo, 2018. "Filtered likelihood for point processes," Journal of Econometrics, Elsevier, vol. 204(1), pages 33-53.
    16. Hesselager, Ole, 1994. "A Markov Model for Loss Reserving," ASTIN Bulletin, Cambridge University Press, vol. 24(2), pages 183-193, November.
    17. Jang, Jiwook & Dassios, Angelos, 2013. "A bivariate shot noise self-exciting process for insurance," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 524-532.
    18. Matúš Maciak & Michal Pešta & Barbora Peštová, 2020. "Changepoint in dependent and non-stationary panels," Statistical Papers, Springer, vol. 61(4), pages 1385-1407, August.
    19. Šárka Hudecová & Jana Klicnarová & Miroslav Šiman, 2020. "Incomplete interdirections and lift-interdirections," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 32(1), pages 93-108, January.
    20. Aït-Sahalia, Yacine & Laeven, Roger J.A. & Pelizzon, Loriana, 2014. "Mutual excitation in Eurozone sovereign CDS," Journal of Econometrics, Elsevier, vol. 183(2), pages 151-167.
    21. Badescu, Andrei L. & Chen, Tianle & Lin, X. Sheldon & Tang, Dameng, 2019. "A Marked Cox Model For The Number Of Ibnr Claims: Estimation And Application," ASTIN Bulletin, Cambridge University Press, vol. 49(3), pages 709-739, September.
    22. Michal Gerthofer & Michal Pešta, 2017. "Stochastic Claims Reserving in Insurance Using Random Effects," Prague Economic Papers, Prague University of Economics and Business, vol. 2017(5), pages 542-560.
    23. Jewell, William S., 1990. "Predicting IBNYR Events and Delays II. Discrete Time," ASTIN Bulletin, Cambridge University Press, vol. 20(1), pages 93-111, April.
    24. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
    25. Zhao, Xiao Bing & Zhou, Xian & Wang, Jing Long, 2009. "Semiparametric model for prediction of individual claim loss reserving," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 1-8, August.
    26. England, P.D. & Verrall, R.J., 2002. "Stochastic Claims Reserving in General Insurance," British Actuarial Journal, Cambridge University Press, vol. 8(3), pages 443-518, August.
    27. Michal Pešta & Martin Wendler, 2020. "Nuisance-parameter-free changepoint detection in non-stationary series," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(2), pages 379-408, June.
    28. Zhao, XiaoBing & Zhou, Xian, 2010. "Applying copula models to individual claim loss reserving methods," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 290-299, April.
    29. Jewell, William S., 1989. "Predicting Ibnyr Events and Delays: I. Continuous Time," ASTIN Bulletin, Cambridge University Press, vol. 19(1), pages 25-55, April.
    30. Pigeon, Mathieu & Antonio, Katrien & Denuit, Michel, 2014. "Individual loss reserving using paid–incurred data," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 121-131.
    31. Franz Konecny, 1987. "The asymptotic properties of maximum likelihood estimators for marked poisson processes with a cyclic intensity measure," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 34(1), pages 143-155, December.
    32. Aït-Sahalia, Yacine & Cacho-Diaz, Julio & Laeven, Roger J.A., 2015. "Modeling financial contagion using mutually exciting jump processes," Journal of Financial Economics, Elsevier, vol. 117(3), pages 585-606.
    33. Pešta, Michal & Okhrin, Ostap, 2014. "Conditional least squares and copulae in claims reserving for a single line of business," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 28-37.
    34. Els Godecharle & Katrien Antonio, 2015. "Reserving by Conditioning on Markers of Individual Claims: A Case Study Using Historical Simulation," North American Actuarial Journal, Taylor & Francis Journals, vol. 19(4), pages 273-288, October.
    35. Norberg, Ragnar, 1993. "Prediction of Outstanding Liabilities in Non-Life Insurance1," ASTIN Bulletin, Cambridge University Press, vol. 23(1), pages 95-115, May.
    36. Lopez, Olivier & Milhaud, Xavier & Thérond, Pierre-E., 2019. "A Tree-Based Algorithm Adapted To Microlevel Reserving And Long Development Claims," ASTIN Bulletin, Cambridge University Press, vol. 49(3), pages 741-762, September.
    37. Emmanuel Bacry & Sylvain Delattre & Marc Hoffmann & Jean-François Muzy, 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Post-Print hal-01313994, HAL.
    38. Rasmus Waagepetersen & Yongtao Guan, 2009. "Two‐step estimation for inhomogeneous spatial point processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(3), pages 685-702, June.
    39. Norberg, Ragnar, 1999. "Prediction of Outstanding Liabilities II. Model Variations and Extensions," ASTIN Bulletin, Cambridge University Press, vol. 29(1), pages 5-25, May.
    40. Arjas, Elja, 1989. "The Claims Reserving Problem in Non-Life Insurance: Some Structural Ideas," ASTIN Bulletin, Cambridge University Press, vol. 19(2), pages 139-152, November.
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    Cited by:

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    2. Rodi Lykou & George Tsaklidis, 2021. "Particle Filtering: A Priori Estimation of Observational Errors of a State-Space Model with Linear Observation Equation," Mathematics, MDPI, vol. 9(12), pages 1-16, June.

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    More about this item

    Keywords

    Stochastic prediction; Marked point process; Hawkes process; Time-varying model; Dynamic panel data; Consistency; Risk valuation; Micro claims reserving;
    All these keywords.

    JEL classification:

    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • C33 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Models with Panel Data; Spatio-temporal Models
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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