IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v87y2019icp34-50.html
   My bibliography  Save this article

The collective reserving model

Author

Listed:
  • Wahl, Felix
  • Lindholm, Mathias
  • Verrall, Richard

Abstract

This paper sets out a model for analysing claims development data, which we call the collective reserving model (CRM). The model is defined on the individual claim level and it produces separate IBNR and RBNS reserve estimators at the collective level without using any approximations. The CRM is based on ideas from a paper by Verrall, Nielsen and Jessen (VNJ) from 2010 in which a model is proposed that relies on a claim giving rise to a single payment. This is generalised by the CRM to the case of multiple payments per claim. All predictors of outstanding claims payments for the VNJ model are shown to hold for this new model. Moreover, the quasi-Poisson GLM estimation framework will be applicable as well, but without using an approximation. Furthermore, analytical expressions for the variance of the total outstanding claims payments are given, with a subdivision on IBNR and RBNS claims. To quantify the effect of allowing only one payment per claim, the model is related and compared to the VNJ model, in particular by looking at variance inequalities. The double chain ladder (DCL) method is discussed as an estimation method for this new model and it is shown that both the GLM- and DCL-based estimators are consistent in terms of an exposure measure. Lastly, both of these methods are shown to asymptotically reproduce the regular chain ladder reserve estimator when restricting predictions to the lower right triangle without the tail, motivating the chain ladder technique as a large-exposure approximation of this model.

Suggested Citation

  • Wahl, Felix & Lindholm, Mathias & Verrall, Richard, 2019. "The collective reserving model," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 34-50.
    • Handle: RePEc:eee:insuma:v:87:y:2019:i:c:p:34-50
    DOI: 10.1016/j.insmatheco.2019.04.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668718304128
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2019.04.003?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. Schnieper, R., 1991. "Separating True IBNR and IBNER Claims1," ASTIN Bulletin, Cambridge University Press, vol. 21(1), pages 111-127, April.
    2. 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.
    3. Mack, Thomas, 1991. "A Simple Parametric Model for Rating Automobile Insurance or Estimating IBNR Claims Reserves," ASTIN Bulletin, Cambridge University Press, vol. 21(1), pages 93-109, April.
    4. 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.
    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. Mack, Thomas, 1993. "Distribution-free Calculation of the Standard Error of Chain Ladder Reserve Estimates," ASTIN Bulletin, Cambridge University Press, vol. 23(2), pages 213-225, November.
    7. 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.
    8. 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.
    9. repec:eme:jrfpps:v:14:y:2013:i:2:p:353-377 is not listed on IDEAS
    10. Renshaw, A.E. & Verrall, R.J., 1998. "A Stochastic Model Underlying the Chain-Ladder Technique," British Actuarial Journal, Cambridge University Press, vol. 4(4), pages 903-923, October.
    11. 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.
    12. Paulo J. R. Pinheiro & João Manuel Andrade e Silva & Maria De Lourdes Centeno, 2003. "Bootstrap Methodology in Claim Reserving," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 70(4), pages 701-714, December.
    13. Norberg, Ragnar, 1993. "Prediction of Outstanding Liabilities in Non-Life Insurance1," ASTIN Bulletin, Cambridge University Press, vol. 23(1), pages 95-115, May.
    14. Diers, Dorothea & Linde, Marc, 2013. "The multi-year non-life insurance risk in the additive loss reserving model," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 590-598.
    15. Dorothea Diers & Martin Eling & Christian Kraus & Marc Linde, 2013. "Multi-year non-life insurance risk," Journal of Risk Finance, Emerald Group Publishing, vol. 14(4), pages 353-377, August.
    16. Norberg, Ragnar, 1999. "Prediction of Outstanding Liabilities II. Model Variations and Extensions," ASTIN Bulletin, Cambridge University Press, vol. 29(1), pages 5-25, May.
    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. 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.
    2. 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.
    3. Lindholm, Mathias & Verrall, Richard, 2020. "Regression based reserving models and partial information," Insurance: Mathematics and Economics, Elsevier, vol. 94(C), pages 109-124.
    4. Daniel J. Geiger & Akim Adekpedjou, 2022. "Analysis of IBNR Liabilities with Interevent Times Depending on Claim Counts," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 815-829, June.

    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. Lindholm, Mathias & Verrall, Richard, 2020. "Regression based reserving models and partial information," Insurance: Mathematics and Economics, Elsevier, vol. 94(C), pages 109-124.
    2. 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.
    3. Liivika Tee & Meelis Käärik & Rauno Viin, 2017. "On Comparison of Stochastic Reserving Methods with Bootstrapping," Risks, MDPI, vol. 5(1), pages 1-21, January.
    4. 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.
    5. England, P.D. & Verrall, R.J. & Wüthrich, M.V., 2019. "On the lifetime and one-year views of reserve risk, with application to IFRS 17 and Solvency II risk margins," Insurance: Mathematics and Economics, Elsevier, vol. 85(C), pages 74-88.
    6. 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.
    7. 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.
    8. Gian Paolo Clemente & Nino Savelli & Diego Zappa, 2019. "Modelling Outstanding Claims with Mixed Compound Processes in Insurance," International Business Research, Canadian Center of Science and Education, vol. 12(3), pages 123-138, March.
    9. Eduardo Ramos-P'erez & Pablo J. Alonso-Gonz'alez & Jos'e Javier N'u~nez-Vel'azquez, 2020. "Stochastic reserving with a stacked model based on a hybridized Artificial Neural Network," Papers 2008.07564, arXiv.org.
    10. Steinmetz, Julia & Jentsch, Carsten, 2024. "Bootstrap consistency for the Mack bootstrap," Insurance: Mathematics and Economics, Elsevier, vol. 115(C), pages 83-121.
    11. 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 ladde," Insurance: Mathematics and Economics, Elsevier, vol. 67(C), pages 187-199.
    12. 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.
    13. Klaus Schmidt, 2012. "Loss prediction based on run-off triangles," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 96(2), pages 265-310, June.
    14. Stephan M. Bischofberger, 2020. "In-Sample Hazard Forecasting Based on Survival Models with Operational Time," Risks, MDPI, vol. 8(1), pages 1-17, January.
    15. Steinmetz, Julia & Jentsch, Carsten, 2022. "Asymptotic theory for Mack's model," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 223-268.
    16. Eduardo Ramos-P'erez & Pablo J. Alonso-Gonz'alez & Jos'e Javier N'u~nez-Vel'azquez, 2022. "Mack-Net model: Blending Mack's model with Recurrent Neural Networks," Papers 2205.07334, arXiv.org.
    17. Verdonck, T. & Debruyne, M., 2011. "The influence of individual claims on the chain-ladder estimates: Analysis and diagnostic tool," Insurance: Mathematics and Economics, Elsevier, vol. 48(1), pages 85-98, January.
    18. Nils Engler & Filip Lindskog, 2023. "Mack's estimator motivated by large exposure asymptotics in a compound Poisson setting," Papers 2310.12056, arXiv.org.
    19. Huang, Jinlong & Qiu, Chunjuan & Wu, Xianyi & Zhou, Xian, 2015. "An individual loss reserving model with independent reporting and settlement," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 232-245.
    20. Francis Duval & Mathieu Pigeon, 2019. "Individual Loss Reserving Using a Gradient Boosting-Based Approach," Risks, MDPI, vol. 7(3), pages 1-18, July.

    More about this item

    Keywords

    Stochastic claims reserving; Risk; Solvency; Chain ladder; Discrete time Poisson process;
    All these 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:87:y:2019:i:c:p:34-50. 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.