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Asymptotic behaviors of stochastic reserving: Aggregate versus individual models

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  • Huang, Jinlong
  • Wu, Xianyi
  • Zhou, Xian

Abstract

In this paper, we investigate the asymptotic behaviors of the loss reservings computed by individual data method and its aggregate data versions by Chain-Ladder (CL) and Bornhuetter–Ferguson (BF) algorithms. It is shown that all deviations of the three reservings from the individual loss reserve (the projection of the outstanding liability on the individual data) converge weakly to a zero-mean normal distribution at the nrate. The analytical forms of the asymptotic variances are derived and compared by both analytical and numerical examples. The results show that the individual method has the smallest asymptotic variance, followed by the BF algorithm, and the CL algorithm has the largest asymptotic variance.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:ejores:v:249:y:2016:i:2:p:657-666
    DOI: 10.1016/j.ejor.2015.09.039
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    References listed on IDEAS

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    1. 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).
    2. Pigeon, Mathieu & Antonio, Katrien & Denuit, Michel, 2013. "Individual Loss Reserving With The Multivariate Skew Normal Framework," ASTIN Bulletin, Cambridge University Press, vol. 43(3), pages 399-428, September.
    3. Alai, D. H. & Merz, M. & Wüthrich, M. V., 2009. "Mean Square Error of Prediction in the Bornhuetter–Ferguson Claims Reserving Method," Annals of Actuarial Science, Cambridge University Press, vol. 4(1), pages 7-31, March.
    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. Jinlong Huang & Chunjuan Qiu & Xianyi Wu, 2015. "Stochastic Loss Reserving in Discrete Time: Individual vs. Aggregate Data Models," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 44(10), pages 2180-2206, May.
    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. Norberg, Ragnar, 1993. "Prediction of Outstanding Liabilities in Non-Life Insurance1," ASTIN Bulletin, Cambridge University Press, vol. 23(1), pages 95-115, May.
    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. Norberg, Ragnar, 1999. "Prediction of Outstanding Liabilities II. Model Variations and Extensions," ASTIN Bulletin, Cambridge University Press, vol. 29(1), pages 5-25, May.
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    Cited by:

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    2. Benjamin Avanzi & Gregory Clive Taylor & Melantha Wang & Bernard Wong, 2020. "SynthETIC: an individual insurance claim simulator with feature control," Papers 2008.05693, arXiv.org, revised Aug 2021.
    3. Avanzi, Benjamin & Taylor, Greg & Wang, Melantha & Wong, Bernard, 2021. "SynthETIC: An individual insurance claim simulator with feature control," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 296-308.
    4. 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.
    5. Greg Taylor, 2019. "Loss Reserving Models: Granular and Machine Learning Forms," Risks, MDPI, vol. 7(3), pages 1-18, July.
    6. Lindholm, Mathias & Verrall, Richard, 2020. "Regression based reserving models and partial information," Insurance: Mathematics and Economics, Elsevier, vol. 94(C), pages 109-124.
    7. Benjamin Avanzi & Gregory Clive Taylor & Bernard Wong & Xinda Yang, 2020. "On the modelling of multivariate counts with Cox processes and dependent shot noise intensities," Papers 2004.11169, arXiv.org, revised Dec 2020.
    8. Denuit, Michel & Trufin, Julien, 2016. "Collective Loss Reserving with Two Types of Claims in Motor Third Party Liability Insurance," LIDAM Discussion Papers ISBA 2016029, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

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