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Univariate and multivariate claims reserving with Generalized Link Ratios

Author

Listed:
  • Portugal, Luís
  • Pantelous, Athanasios A.
  • Verrall, Richard

Abstract

In actuarial practice, it is important to select an adequate claims reserving method for each line of business, however it might not always be appropriate to apply the same method for all triangles involved in the portfolio of business. In this regard, we develop a versatile univariate and multivariate Generalized Link Ratios framework, inside the same triangle, that includes some existing methods (such as the chain-ladder, vector projection, and simple average) as special cases, and calculates the prediction errors analytically. Our methodology allows us to simultaneously estimate the loss development factors, the reserves, and the prediction errors over all the regressions, without utilizing recursive formulas. Further, the criterion employed for the model selection, which is based on the lowest prediction error, also estimates a key parameter that corresponds to a certain level of heteroscedasticity. Finally, several numerical examples with irregular, regular, and real data illustrate the applicability of our treatment and check the assumptions made in the paper.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:insuma:v:97:y:2021:i:c:p:57-67
    DOI: 10.1016/j.insmatheco.2020.11.011
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    References listed on IDEAS

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    1. Piet de Jong, 2006. "Forecasting Runoff Triangles," North American Actuarial Journal, Taylor & Francis Journals, vol. 10(2), pages 28-38.
    2. 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.
    3. 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.
    4. Taylor, Greg, 2011. "Maximum Likelihood and Estimation Efficiency of the Chain Ladder," ASTIN Bulletin, Cambridge University Press, vol. 41(1), pages 131-155, May.
    5. White, Halbert, 1980. "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, Econometric Society, vol. 48(4), pages 817-838, May.
    6. 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.
    7. Brydon, D. & Verrall, R. J., 2009. "Calendar Year Effects, Claims Inflation and the Chain-Ladder Technique," Annals of Actuarial Science, Cambridge University Press, vol. 4(2), pages 287-301, September.
    8. Leung, Melvern & Li, Youwei & Pantelous, Athanasios A. & Vigne, Samuel A., 2021. "Bayesian Value-at-Risk backtesting: The case of annuity pricing," European Journal of Operational Research, Elsevier, vol. 293(2), pages 786-801.
    9. Mario Wüthrich, 2010. "Accounting Year Effects Modeling in the Stochastic Chain Ladder Reserving Method," North American Actuarial Journal, Taylor & Francis Journals, vol. 14(2), pages 235-255.
    10. Verrall, R. J., 2000. "An investigation into stochastic claims reserving models and the chain-ladder technique," Insurance: Mathematics and Economics, Elsevier, vol. 26(1), pages 91-99, February.
    11. Michael Merz & Mario Wüthrich, 2008. "Prediction Error of the Multivariate Chain Ladder Reserving Method," North American Actuarial Journal, Taylor & Francis Journals, vol. 12(2), pages 175-197.
    12. Luís Portugal & Athanasios A. Pantelous & Hirbod Assa, 2018. "Claims Reserving with a Stochastic Vector Projection," North American Actuarial Journal, Taylor & Francis Journals, vol. 22(1), pages 22-39, January.
    13. Braun, Christian, 2004. "The Prediction Error of the Chain Ladder Method Applied to Correlated Run-off Triangles," ASTIN Bulletin, Cambridge University Press, vol. 34(2), pages 399-423, November.
    14. 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.
    15. de Alba, Enrique & Nieto-Barajas, Luis E., 2008. "Claims reserving: A correlated Bayesian model," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 368-376, December.
    16. Pesaran, M. Hashem, 2015. "Time Series and Panel Data Econometrics," OUP Catalogue, Oxford University Press, number 9780198759980.
    17. Hess, Klaus Th. & Schmidt, Klaus D. & Zocher, Mathias, 2006. "Multivariate loss prediction in the multivariate additive model," Insurance: Mathematics and Economics, Elsevier, vol. 39(2), pages 185-191, October.
    18. Mack, Thomas, 1994. "Which stochastic model is underlying the chain ladder method?," Insurance: Mathematics and Economics, Elsevier, vol. 15(2-3), pages 133-138, December.
    19. Shi, Peng & Frees, Edward W., 2011. "Dependent Loss Reserving using Copulas," ASTIN Bulletin, Cambridge University Press, vol. 41(2), pages 449-486, November.
    20. Taylor, G. C. & Ashe, F. R., 1983. "Second moments of estimates of outstanding claims," Journal of Econometrics, Elsevier, vol. 23(1), pages 37-61, September.
    21. Shi, Peng, 2014. "A Copula Regression For Modeling Multivariate Loss Triangles And Quantifying Reserving Variability," ASTIN Bulletin, Cambridge University Press, vol. 44(1), pages 85-102, January.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Reserving; Multivariate regression; Homoscedastic and heteroscedastic errors; Seemingly unrelated regression; Prediction errors;
    All these keywords.

    JEL classification:

    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C35 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions

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