Rank-based methods for modeling dependence between loss triangles. (English) Zbl 1394.91205

Summary: In order to determine the risk capital for their aggregate portfolio, property and casualty insurance companies must fit a multivariate model to the loss triangle data relating to each of their lines of business. As an inadequate choice of dependence structure may have an undesirable effect on reserve estimation, a two-stage inference strategy is proposed in this paper to assist with model selection and validation. Generalized linear models are first fitted to the margins. Standardized residuals from these models are then linked through a copula selected and validated using rank-based methods. The approach is illustrated with data from six lines of business of a large Canadian insurance company for which two hierarchical dependence models are considered, i.e., a fully nested Archimedean copula structure and a copula-based risk aggregation model.


91B30 Risk theory, insurance (MSC2010)
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62P05 Applications of statistics to actuarial sciences and financial mathematics


TwoCop; NSM3; QRM; nacopula
Full Text: DOI


[1] Abdallah, A; Boucher, JP; Cossette, H, Modeling dependence between loss triangles with hierarchical Archimedean copulas, ASTIN Bull, 45, 577-599, (2015) · Zbl 1390.91154
[2] Ajne, B, Additivity of chain-ladder projections, ASTIN Bull, 24, 311-318, (1994)
[3] Andrews, DW, Inconsistency of the bootstrap when a parameter is on the boundary of the parameter space, Econometrica, 68, 399-405, (2000) · Zbl 1015.62044
[4] Arbenz, P; Hummel, C; Mainik, G, Copula based hierarchical risk aggregation through sample reordering, Insur Math Econ, 51, 122-133, (2012) · Zbl 1284.91198
[5] Bargès, M; Cossette, H; Marceau, E, Tvar-based capital allocation with copulas, Insur Math Econ, 45, 348-361, (2009) · Zbl 1231.91141
[6] Ben Ghorbal, N; Genest, C; Nešlehová, J, On the ghoudi, khoudraji, and rivest test for extreme value dependence, Can J Stat, 37, 534-552, (2009) · Zbl 1191.62083
[7] Braun, C, The prediction error of the chain ladder method applied to correlated run-off triangles, ASTIN Bull, 34, 399-434, (2004) · Zbl 1274.62689
[8] Brehm, P, Correlation and the aggregation of unpaid loss distributions, Casualty Actuar Soc Forum (Fall), 2, 1-23, (2002)
[9] Bürgi R, Dacorogna MM, Iles R (2008) Risk aggregation, dependence structure and diversification benefit. Stress Testing for Financial Institutions. https://ssrn.com/abstract=1468526
[10] Côté MP (2014) Copula-based risk aggregation modelling. Master’s thesis, McGill University, Montréal, Québec, Canada · Zbl 1390.91154
[11] Côté, MP; Genest, C, A copula-based risk aggregation model, Can J Stat, 43, 60-81, (2015) · Zbl 1310.62075
[12] Jong, P, Modeling dependence between loss triangles, N Am Actuar J, 16, 74-86, (2012)
[13] Embrechts, P; Lindskog, F; McNeil, AJ; Rachev, S (ed.), Modelling dependence with copulas and applications to risk management, (2003), Amsterdam
[14] Genest, C; Favre, AC, Everything you always wanted to know about copula modeling but were afraid to ask, J Hydrol Eng, 12, 347-368, (2007)
[15] Genest, C; Ghoudi, K; Rivest, LP, A semiparametric estimation procedure of dependence parameters in multivariate families of distributions, Biometrika, 82, 543-552, (1995) · Zbl 0831.62030
[16] Genest, C; MacKay, RJ, Copules archimédiennes et familles de lois bidimensionnelles dont LES marges sont données, Can J Stat, 14, 145-159, (1986) · Zbl 0605.62049
[17] Genest, C; Nešlehová, J; El-Shaarawi, AH (ed.); Piegorsch, WW (ed.), Copulas and copula models, (2012), Chichester
[18] Genest, C; Nešlehová, J; Ben Ghorbal, N, Estimators based on kendall’s tau in multivariate copula models, Aust N Z J Stat, 53, 157-177, (2011) · Zbl 1274.62367
[19] Genest, C; Rémillard, B; Beaudoin, D, Goodness-of-fit tests for copulas: a review and a power study, Insur Math Econ, 44, 199-213, (2009) · Zbl 1161.91416
[20] Hofert, M; Jaworski, P (ed.); Durante, F (ed.); Härdle, WK (ed.); Rychlik, T (ed.), Construction and sampling of nested Archimedean copulas, No. 198, 147-160, (2010), Berlin
[21] Hofert, M, Efficiently sampling nested Archimedean copulas, Comput Stat Data Anal, 55, 57-70, (2011) · Zbl 1247.62132
[22] Hofert, M; Mächler, M, Nested Archimedean copulas meet \(\textsf{R}\): the \(\texttt{nacopula}\) package, J Stat Softw, 39, 1-20, (2011)
[23] Hollander M, Wolfe DA, Chicken E (2014) Nonparametric statistical methods, 3rd edn. Wiley, Hoboken · Zbl 1279.62006
[24] Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall, London · Zbl 0990.62517
[25] Mainik, G, Risk aggregation with empirical margins: Latin hypercubes, empirical copulas, and convergence of sum distributions, J Multivar Anal, 141, 197-216, (2015) · Zbl 1327.62274
[26] McNeil, AJ, Sampling nested Archimedean copulas, J Stat Comput Simul, 78, 567-581, (2008) · Zbl 1221.00061
[27] McNeil AJ, Frey R, Embrechts P (2015) Quantitative risk management: concepts, techniques and tools, 2nd edn. Princeton University Press, Princeton · Zbl 1337.91003
[28] Merz, M; Wüthrich, M, Prediction error of the multivariate chain ladder reserving method, N Am Actuar J, 12, 175-197, (2008)
[29] Merz, M; Wüthrich, M; Hashorva, E, Dependence modelling in multivariate claims run-off triangles, Ann Actuar Sci, 7, 3-25, (2013)
[30] Nelsen RB (2006) An introduction to copulas. Springer, Berlin · Zbl 1152.62030
[31] Pröhl C, Schmidt K (2005) Multivariate chain-ladder. ASTIN Colloquium 2005, ETH Zürich, Switzerland · Zbl 1284.91198
[32] Rémillard, B; Scaillet, O, Testing for equality between two copulas, J Multivar Anal, 100, 377-386, (2009) · Zbl 1157.62401
[33] Savu, C; Trede, M, Hierarchies of Archimedean copulas, Quant Finance, 10, 295-304, (2010) · Zbl 1270.91086
[34] Schmidt K (2006) Optimal and additive loss reserving for dependent lines of business. Casualty Actuarial Society Forum (fall):319-351
[35] SCOR (2008) From principle-based risk management to solvency requirements. Technical report, SCOR, Switzerland. https://www.scor.com/images/stories/pdf/scorpapers/sstbook_second_edition_final.pdf. Accessed 22 June 2016 · Zbl 1231.91258
[36] Shi, P; Basu, S; Meyers, G, A Bayesian log-normal model for multivariate loss reserving, N Am Actuar J, 16, 29-51, (2012) · Zbl 1291.91126
[37] Shi, P; Frees, E, Dependent loss reserving using copulas, ASTIN Bull, 41, 449-486, (2011)
[38] Tasche D (1999) Risk contributions and performance measurement. Working paper, Technische Universität München, Germany · Zbl 1231.91141
[39] Taylor, G; McGuire, G, A synchronous bootstrap to account for dependencies between lines of business in the estimation of loss reserve prediction error, N Am Actuar J, 11, 70-88, (2007)
[40] Whelan, N, Sampling from Archimedean copulas, Quant Finance, 4, 339-352, (2004) · Zbl 1409.62108
[41] Zhang, Y, A general multivariate chain ladder model, Insur Math Econ, 46, 588-599, (2010) · Zbl 1231.91258
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.