The main goal of insurance company management is to increase shareholders’ value and implement a strategy that promotes sustainable growth of the company. Well-known possible measures intended to achieve that goal are as follows: share price, economic value, market capitalisation, gross premiums earned and solvency ratio. These measures include efficient capital management as capital expenses could be a major cost position depending on risk appetite and the extent of capital needed to support it. This research focuses on non-life insurers for reserve risk modelling. In the current study, a more accurate risk quantification model has been developed than the standard model provided by the EU regulator under the Solvency II framework. The proposed model provides capital cost gains as well. A case study based on non-life real data set with underwriting in the Baltic countries is discussed with inclusion of pandemic trends that had an impact on economies and customer behaviours. The study considers different non-life reserve distributions for each insurance business line, risk aggregation and the way of choosing the most appropriate type of copula model for non-life reserve risk. Adequate capital is calculated by applying value at risk at 99.5%, which is mandatory in the EU market. The study considers which selected tests have to be implemented in order to choose the most appropriate copula model for reserve risk.
1. Aas, K., Czado, C., Frigessi, A. and Bakken, H. 2009. Pair-copula constructions of multiple dependence. Insur. Math. Econ., 44(2), 182–198.
https://doi.org/10.1016/j.insmatheco.2007.02.001
2. Ashe, F. 1986. An essay at measuring the variance of estimates of outstanding claim payments. ASTIN Bull., 16(S1), S99–S113.
https://doi.org/10.1017/S0515036100011685
3. Boumezoued, A., Yoboua, A., Devineau, L. and Boisseau, J. 2011. One-year reserve risk including a tail factor: closed formula and bootstrap approaches.
https://doi.org/10.48550/arXiv.1107.0164
4. Cherubini, U., Luciano, E. and Vecchiato, W. 2004. Copula Methods in Finance. John Wiley & Sons Press, Hoboken.
https://doi.org/10.1002/9781118673331
5. Claeskens, G. and Hjort, N. L. 2011. Model selection and model averaging. Psychometrika, 76, 507–509.
https://doi.org/10.1007/s11336-011-9219-3
6. Dacorogna, M. 2018. A change of paradigm for the insurance industry. Ann. Actuarial Sci., 12(2), 11–22.
https://doi.org/10.1017/S1748499518000040
7. Diers, D. 2008. Stochastic re-reserving in multi-year internal models – An approach based on simulations. In Proceedings of the ASTIN Colloquium, Helsinki, 2008.
https://www.actuaries.org.uk/system/files/field/document/S4_11_Diers.pdf (accessed 2022-08-19).
8. Efron, B. 1979. Bootstrap methods: another look at the jackknife. Ann. Stat., 7(1), 1–26.
https://doi.org/10.1214/aos/1176344552
9. Efron, B. and Tibshirani, R. J. 1993. An Introduction to the Bootstrap. Chapman&Hall/CRC, New York.
https://doi.org/10.1007/978-1-4899-4541-9
10. EIOPA. 2014. Delegated Regulation (EU) 2015/35 of 10 October 2014 supplementing Directive 2009/138/EC of the European Parliament and of the Council on the taking-up and pursuit of the business of Insurance and Reinsurance (Solvency II).
https://www.eiopa.europa.eu/browse/solvency-2_en (accessed 2022-08-19).
11. England, P. and Verrall, R. 1999. Analytic and bootstrap estimates of prediction errors in claims reserving. Insur. Math. Econ., 25(3), 281–293.
https://doi.org/10.1016/S0167-6687(99)00016-5
12. England, P. and Verrall, R. 2002. Stochastic Claims Reserving in General Insurance. British Actuarial J., 8(3), 443–518.
https://doi.org/10.1017/S1357321700003809
13. Fermanian, J.-D. 2005. Goodness-of-fit tests for copulas. J. Multivar. Anal., 95(1), 119–152.
https://doi.org/10.1016/j.jmva.2004.07.0044
14. Fersini, P. and Melisi, G. 2016. Stochastic model to evaluate the fair value of motor third-party liability under the direct reimbursement scheme and quantification of the capital requirement in a solvency II perspective. Insur. Math. and Econ., 68, 27–44.
https://doi.org/10.1016/j.insmatheco.2016.02.002
15. Genest, C., and Rémillard, B. 2008. Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models. Ann. I. H. Poincaré–Pr.,44(6), 1096–1127.
https://doi.org/10.1214/07-AIHP148
16. Genest, C., Rémillard, B. and Beaudoin, D. 2009. Goodness-of-fit tests for copulas: a review and a power study. Insur. Math. Econ., 44(2), 199–213.
https://doi.org/10.1016/j.insmatheco.2007.10.005
17. Gesmann, M., Murphy, D., Zhang, Y., Carrato, A. et al. 2015. ChainLadder: Statistical Methods and Models for Claims Reserving in General Insurance.
https://mages.github.io/ChainLadder/
18. Grønneberg, S. and Hjort, N. L. 2014. The copula information criteria. Scan. J. of Statist., 41, 436–459.
https://doi.org/10.1111/sjos.12042
19. Hindley, D. 2017. Claims Reserving in General Insurance. International Series on Actuarial Science. Cambridge University Press, Cambridge.
https://doi.org/10.1017/9781139924696.002
20. Hofert, M. and Maechler, M. 2011. Nested Archimedean copulas meet R: The nacopula package. J. Stat. Softw., 39(9), 1–20.
https://doi.org/10.18637/jss.v039.i09
21. Hofert, M., Kojadinovic, I., Mächler, M. and Yan, J. 2018. Elements of Copula Modelling with R. Springer, Cham.
https://doi.org/10.1007/978-3-319-89635-9
22. Hofert, M., Kojadinovic, I., Maechler, M., Yan, J. 2020. Copula: Multivariate Dependence with Copulas. R package version 1.0-0.
https://CRAN.R-project.org/package=copula
23. International Actuarial Association. 2016. ASTIN Report on Non-life Reserving Practices.
http://www.actuaries.org/ASTIN/Documents/ASTIN_WP_NL_Reserving_Report1.0_2016-06-15.pdf
24. Jordanger, L. A. and Tjøstheim, D. 2014. Model selection of copulas: AIC versus a cross validation copula information criterion. Stat. Probab. Lett., 92, 249–255.
https://doi.org/10.1016/j.spl.2014.06.006
25. Kojadinovic I. and Yan, J. 2010. Modeling multivariate distributions with continuous margins using the copula R package. J. Stat. Softw., 34(9), 1–20.
https://www.jstatsoft.org/v34/i09/
https://doi.org/10.18637/jss.v034.i09
26. Kremer, E. 1982. IBNR claims and the two way model of ANOVA. Scand. Actuar. J., 1, 47–55.
https://doi.org/10.1080/03461238.1982.10405432
27. Mack, T. 1993a. Distribution-free calculation of the standard error of chain-ladder reserve estimates. ASTIN Bull., 23(2), 213–225.
https://doi.org/10.2143/AST.23.2.2005092
28. Mack, T. 1993b. Measuring the variability of chain-ladder reserve estimates. Casualty Actuarial Society Spring Forum, May 1993. CAS, 101–182.
29. Mack, T. 1994. Which stochastic model is underlying the chain-ladder method? Insur. Math. Econ., 15(2–3), 133–138.
https://doi.org/10.1016/0167-6687(94)90789-7
30. Markowitz, H. M. 1952. Portfolio Selection. J. Finance, 7(1), 77–91.
https://doi.org/10.2307/2975974
31. Marshall, K., Collings, S., Hodson, M., and O’Dowd, C. 2008. A framework for assessing risk margins. In Proceedings of the Institute of Actuaries of Australia 16th General Insurance Seminar, 9–12 November 2008. IAA, 1–43.
32. McCullagh, P. and Nelder, J. A. 1989. Generalized Linear Models. 2nd ed. Routledge, New York.
https://doi.org/10.1007/978-1-4899-3242-6
33. McNeil, A., Frey, R. and Embrechts, P. 2005. Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton University Press, Princeton, NJ.
34. McNeil, A. J., Frey, R. and Embrechts, P. 2015. Quantitative Risk Management: Concepts, Techniques and Tools. Revised ed. Princeton University Press, Princeton, NJ.
35. Merz, M. and Wüthrich, M. V. 2008. Modelling the claims development result for solvency purposes. ASTIN Colloquium, Manchester, UK, July 2008.Casualty Actuarial Society.
36. Merz, M. and Wüthrich, M. V. 2014. Claims Run-Off Uncertainty: The Full Picture. Swiss Finance Institute Research Paper, 14-69.
https://www.ssrn.com/abstract=2524352.2014
https://doi.org/10.2139/ssrn.2524352
37. Patton, A. J. 2006. Modelling Asymmetric Exchange Rate Dependence. Int. Econ. Rev., 47(2), 527–556.
https://doi.org/10.1111/j.1468-2354.2006.00387.x
38. Quessy, J.-F. 2005. Méthodologie et application des copules: tests d’adéquation, tests d’indépendance, et bornes pour la valeur-à-risque. PhD thesis. Université Laval, Québec, Canada.
39. Renshaw, A. E. 1989. Chain-ladder and interactive modelling. (Claims reserving and GLIM). J. of the Institute of Actuaries, 116(3), 559–587.
https://doi.org/10.1017/S0020268100036702
40. Renshaw, A. E. 1994. On the second moment properties and the implementation of certain GLIM based stochastic claims reserving models. Actuarial Research Paper, 65, City University, London.
41. Rüschendorf, L. 2009. On the distributional transform, Sklar’s theorem, and the empirical copula process. J. Stat. Plan. Inference, 139(11), 3921–3927.
https://doi.org/10.1016/j.jspi.2009.05.030
42. Shi, P., and Frees, E. 2011. Dependent loss reserving using copulas. ASTIN Bull., 41(2), 449–486.
https://doi.org/10.2143/AST.41.2.2136985
43. Sklar, A. 1959. Fonctions de répartition à n dimensions et leurs marges. Publications del’Institutde Statistique de l’Université de Paris, 8, 229–231.
44. Sklar, A. 1996. Random variables, distribution functions, and copulas – a personal look backward and forward. In Distributions with Fixed Marginals and Related Topics. IMS Lecture Notes –Monograph series, 28, 1–14.
https://doi.org/10.1214/lnms/1215452606
45. Spearman, C. 1904. The proof and measurement of association between two things. American J. Psychology. 15(1), 72–101.
https://doi.org/10.2307/1412159. JSTOR 1412159
46. Tarbell, T. F. 1934. Incurred but not reported claim reserves. Proceedings of the Casualty Actuarial Society, 20, 275–280.
47. Taylor, G. C. and Ashe, F. R. 1983. Second moments of estimates of outstanding claims. J. Econom., 23(1), 37–61.
https://doi.org/10.1016/0304-4076(83)90074-X
48. Venables, W. N. and Ripley, B. D. 2002. Modern Applied Statistics with S. 4th ed. Springer, New York.
https://www.stats.ox.ac.uk/pub/MASS4/
https://doi.org/10.1007/978-0-387-21706-2
49. Yan, J. 2007. Enjoy the joy of copulas: with a package copula. J. Stat. Softw., 21(4), 1–21.
https://doi.org/10.18637/jss.v021.i04
50. Zarina, I., Voronova, I. and Pettere, G. 2019. Internal model for insurers: possibilities and issues. In Proceedings of the International Scientific Conference on Contemporary Issues in Business, Management and Economics Engineering, Vilnius, 9–10 May 2019. Vilnius Gediminas Technical University, 255–265.
https://doi.org/10.3846/cibmee.2019.026
51. Zhang, S., Okhrin, O., Zhou, Q. M. and Song, P. X.-K. 2016. Goodness-of-fit test for specification of semiparametric copula dependence models. J. Econom., 193(1), 215–233.
https://doi.org/10.1016/j.jeconom.2016.02.017
