IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v89y2019icp140-156.html

Explicit moments for a class of micro-models in non-life insurance

Author

Listed:
  • Wahl, Felix

Abstract

This paper considers properties of the micro-model analysed in Antonio and Plat (2014). The main results are analytical expressions for the moments of the outstanding claims payments subdivided into IBNR claims and individual RBNS claims. These moments are possible to compute explicitly using the discretisation scheme for estimation and simulation used in Antonio and Plat (2014) since the expressions then do not involve any integrals that, typically, would require numerical solutions. Other aspects of the model that are investigated are properties of the maximum likelihood estimators of the model parameters, such as bias and consistency, and a way of computing prediction uncertainty in terms of the mean squared error of prediction that does not require simulations. Moreover, a brief discussion is given on how to compute moments or risk-measures of the claims development result (CDR) using simulations, which based on the results of the present paper can be done without any nested simulations. Based on this it is straightforward to compute the one-year Solvency Capital Requirement, which corresponds to the 99.5% Value-at-Risk of the one-year CDR. A brief numerical illustration is used to show the theoretical performance of the maximum likelihood estimators of the parameters in the claims development process under this model using a realistic set-up based on the case-study of Antonio and Plat (2014). Additionally, the paper ends with a short numerical illustration discussing the model’s robustness under violations of an independence assumption.

Suggested Citation

  • Wahl, Felix, 2019. "Explicit moments for a class of micro-models in non-life insurance," Insurance: Mathematics and Economics, Elsevier, vol. 89(C), pages 140-156.
    • Handle: RePEc:eee:insuma:v:89:y:2019:i:c:p:140-156
    DOI: 10.1016/j.insmatheco.2019.10.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S016766871930397X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2019.10.001?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. Ohlsson, Esbjörn & Lauzeningks, Jan, 2009. "The one-year non-life insurance risk," Insurance: Mathematics and Economics, Elsevier, vol. 45(2), pages 203-208, October.
    2. repec:cup:astinb:v:26:y:1996:i:02:p:139-164_00 is not listed on IDEAS
    3. Buchwalder, Markus & Bühlmann, Hans & Merz, Michael & Wüthrich, Mario V., 2006. "The Mean Square Error of Prediction in the Chain Ladder Reserving Method (Mack and Murphy Revisited)," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 521-542, November.
    4. Wahl, Felix & Lindholm, Mathias & Verrall, Richard, 2019. "The collective reserving model," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 34-50.
    5. Verrall, Richard & Nielsen, Jens Perch & Jessen, Anders Hedegaard, 2010. "Prediction of RBNS and IBNR Claims using Claim Amounts and Claim Counts," ASTIN Bulletin, Cambridge University Press, vol. 40(2), pages 871-887, November.
    6. Mathias Lindholm & Filip Lindskog & Felix Wahl, 2017. "Valuation of Non-Life Liabilities from Claims Triangles," Risks, MDPI, vol. 5(3), pages 1-28, July.
    7. Miranda, María Dolores Martínez & Nielsen, Bent & Nielsen, Jens Perch & Verrall, Richard, 2011. "Cash Flow Simulation for a Model of Outstanding Liabilities Based on Claim Amounts and Claim Numbers," ASTIN Bulletin, Cambridge University Press, vol. 41(1), pages 107-129, May.
    8. Buchwalder, Markus & Bühlmann, Hans & Merz, Michael & Wüthrich, Mario V., 2006. "The Mean Square Error of Prediction in the Chain Ladder Reserving Method – Final Remark," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 553-553, November.
    9. repec:cup:astinb:v:23:y:1993:i:01:p:95-115_00 is not listed on IDEAS
    10. Miranda, María Dolores Martínez & Nielsen, Jens Perch & Verrall, Richard, 2012. "Double Chain Ladder," ASTIN Bulletin, Cambridge University Press, vol. 42(1), pages 59-76, May.
    11. Norberg, Ragnar, 1999. "Prediction of Outstanding Liabilities II. Model Variations and Extensions," ASTIN Bulletin, Cambridge University Press, vol. 29(1), pages 5-25, May.
    12. repec:cup:astinb:v:19:y:1989:i:02:p:139-152_00 is not listed on IDEAS
    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. Wahl, Felix & Lindholm, Mathias & Verrall, Richard, 2019. "The collective reserving model," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 34-50.
    2. Lindholm, Mathias & Verrall, Richard, 2020. "Regression based reserving models and partial information," Insurance: Mathematics and Economics, Elsevier, vol. 94(C), pages 109-124.
    3. 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.
    4. Mammen, Enno & Martínez Miranda, María Dolores & Nielsen, Jens Perch, 2015. "In-sample forecasting applied to reserving and mesothelioma mortality," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 76-86.
    5. Hahn, Lukas, 2017. "Multi-year non-life insurance risk of dependent lines of business in the multivariate additive loss reserving model," Insurance: Mathematics and Economics, Elsevier, vol. 75(C), pages 71-81.
    6. Marcin Szatkowski, 2022. "Study of Actuarial Characteristics of One-Year and Ultimate Reserve Risk Distributions Based on Market Data," Central European Journal of Economic Modelling and Econometrics, Central European Journal of Economic Modelling and Econometrics, vol. 14(4), pages 225-262, December.
    7. Mathias Lindholm & Felix Wahl, 2020. "On the variance parameter estimator in general linear models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(2), pages 243-254, February.
    8. Diers, Dorothea & Linde, Marc & Hahn, Lukas, 2016. "Addendum to ‘The multi-year non-life insurance risk in the additive reserving model’ [Insurance Math. Econom. 52(3) (2013) 590–598]: Quantification of multi-year non-life insurance risk in chain ladder reserving models," Insurance: Mathematics and Economics, Elsevier, vol. 67(C), pages 187-199.
    9. Huang, Jinlong & Wu, Xianyi & Zhou, Xian, 2016. "Asymptotic behaviors of stochastic reserving: Aggregate versus individual models," European Journal of Operational Research, Elsevier, vol. 249(2), pages 657-666.
    10. Michel Denuit & Yang Lu, 2021. "Wishart‐gamma random effects models with applications to nonlife insurance," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 88(2), pages 443-481, June.
    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. Portugal, Luís & Pantelous, Athanasios A. & Verrall, Richard, 2021. "Univariate and multivariate claims reserving with Generalized Link Ratios," Insurance: Mathematics and Economics, Elsevier, vol. 97(C), pages 57-67.
    13. Steinmetz, Julia & Jentsch, Carsten, 2022. "Asymptotic theory for Mack's model," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 223-268.
    14. Yanto & Arwan Apriyono & Purwanto Bekti Santoso & Sumiyanto, 2022. "Landslide susceptible areas identification using IDW and Ordinary Kriging interpolation techniques from hard soil depth at middle western Central Java, Indonesia," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 110(2), pages 1405-1416, January.
    15. Marcin Szatkowski & Łukasz Delong, 2021. "One-Year and Ultimate Reserve Risk in Mack Chain Ladder Model," Risks, MDPI, vol. 9(9), pages 1-29, August.
    16. 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.
    17. Margraf, Carolin & Elpidorou, Valandis & Verrall, Richard, 2018. "Claims reserving in the presence of excess-of-loss reinsurance using micro models based on aggregate data," Insurance: Mathematics and Economics, Elsevier, vol. 80(C), pages 54-65.
    18. Baradel, Nicolas, 2026. "Continuous-time modeling and bootstrap for chain-ladder reserving," Insurance: Mathematics and Economics, Elsevier, vol. 126(C).
    19. Sukriye Tuysuz & Pervin Pekel, 2019. "A Comparison of Stochastic Claims Reserving Methods," European Journal of Business and Management Research, European Open Science, vol. 4(4), August.
    20. Nils Engler & Filip Lindskog, 2023. "Mack's estimator motivated by large exposure asymptotics in a compound Poisson setting," Papers 2310.12056, arXiv.org.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    JEL classification:

    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

    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:insuma:v:89:y:2019:i:c:p:140-156. 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: http://www.elsevier.com/locate/inca/505554 .

    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.