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A study of one-factor copula models from a tail dependence perspective

Published online by Cambridge University Press:  04 October 2024

Nariankadu Shyamalkumar Open the ORCID record for Nariankadu Shyamalkumar [Opens in a new window]  and
Siyang Tao Open the ORCID record for Siyang Tao [Opens in a new window]
Nariankadu Shyamalkumar
Affiliation:
Department of Statistics and Actuarial Science, The University of Iowa, Iowa City, 52242, IA, USA
Siyang Tao*
Affiliation:
Department of Mathematical Sciences, Ball State University, Muncie, 47306, IN, USA
*
Corresponding author: Siyang Tao; Email: [email protected]
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Abstract

Modeling multivariate dependence in high dimensions is challenging, with popular solutions constructing multivariate copula as a composition of lower dimensional copulas. Pair-copula constructions do so by using bivariate linking copulas, but their parametrization, in size, being quadratic in the dimension, is not quite parsimonious. Besides, the number of regular vines grows super-exponentially with the dimension. One parsimonious solution is factor copulas, and in particular, the one-factor copula is touted for its simplicity – with the number of parameters linear in the dimension – while being able to cater to asymmetric non-linear dependence in the tails. In this paper, we add nuance to this claim from the point of view of a popular measure of multivariate tail dependence, the tail dependence matrix (TDM). We focus on the one-factor copula model with the linking copula belonging to the BB1 family, pointing out later the applicability of our results to a wider class of linking copulas. For this model, we derive tail dependence coefficients and study their basic properties as functions of the parameters of the linking copulas. Based on this, we study the representativeness of the class of TDMs supported by this model with respect to the class of all possible TDMs. We establish that since the parametrization is linear in the dimension, it is no surprise that the relative volume is zero for dimensions greater than three, and hence, by necessity, we present a novel manner of evaluating the representativeness that has a combinatorial flavor. We formulate the problem of finding the best representative one-factor BB1 model given a target TDM and suggest an implementation along with a simulation study of its performance across dimensions. Finally, we illustrate the results of the paper by modeling rainfall data, which is relevant in the context of weather-related insurance.

Keywords

Tail dependence coefficienttail dependence matrixcorrelation polytopeBB1 family
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 54 , Issue 3 , September 2024 , pp. 679 - 711
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The International Actuarial Association

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References

Aas, K., Czado, C., Frigessi, A. and Bakken, H. (2009) Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics, 44 (2), 182–198.Google Scholar
Asquith, W. (2020) lmomco-package. URL https://rdrr.io/cran/lmomco/man/lmomco-package.html.Google Scholar
Bassamboo, A., Juneja, S. and Zeevi, A. (2008) Portfolio credit risk with extremal dependence: Asymptotic analysis and efficient simulation. Operations Research, 56 (3), 593–606.CrossRefGoogle Scholar
Bedford, T. and Cooke, R.M. (2001) Probability density decomposition for conditionally dependent random variables modeled by vines. Annals of Mathematics and Artificial intelligence, 32, 245–268.CrossRefGoogle Scholar
Bedford, T. and Cooke, R.M. (2002) Vines–a new graphical model for dependent random variables. The Annals of Statistics, 30 (4), 1031–1068.CrossRefGoogle Scholar
Berend, D. and Tassa, T. (2010) Improved bounds on Bell numbers and on moments of sums of random variables. Probability and Mathematical Statistics, 30 (2), 185–205.Google Scholar
Blanchet, J. and Davison, A.C. (2011) Spatial modeling of extreme snow depth. Annals of Applied Statistics, 5 (3), 1699–1725.CrossRefGoogle Scholar
Brechmann, E.C., Czado, C. and Aas, K. (2012) Truncated regular vines in high dimensions with application to financial data. Canadian Journal of Statistics, 40 (1), 68–85.CrossRefGoogle Scholar
Capéraà, P., Fougères, A.-L. and Genest, C. (1997) A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika, 84 (3), 567–577.CrossRefGoogle Scholar
Chen, H., MacMinn, R. and Sun, T. (2015) Multi-population mortality models: A factor copula approach. Insurance: Mathematics and Economics, 63, 135–146.Google Scholar
Chen, S., Tong, Z. and Yang, Y. (2023) Portfolio losses driven by idiosyncratic risk. Mimeo, Available at SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4626393.CrossRefGoogle Scholar
Czado, C. and Nagler, T. (2022) Vine copula based modeling. Annual Review of Statistics and Its Application, 9, 453–477.CrossRefGoogle Scholar
Davis, R.A. and Mikosch, T. (2009) The extremogram: A correlogram for extreme events. Bernoulli, 15 (4), 977–1009.CrossRefGoogle Scholar
De Luca, G., Ruscone, M.N. and Amati, V. (2023) The use of conditional copula for studying the influence of economic sectors. Expert Systems with Applications, 120582.CrossRefGoogle Scholar
Donnelly, C. and Embrechts, P. (2010) The devil is in the tails: Actuarial mathematics and the subprime mortgage crisis. ASTIN Bulletin: The Journal of the IAA, 40 (1), 1–33.CrossRefGoogle Scholar
Durante, F. (2006) A new class of symmetric bivariate copulas. Journal of Nonparametric Statistics, 18 (7-8), 499–510. doi: 10.1080/10485250701262242.CrossRefGoogle Scholar
Embrechts, P., Frey, R. and McNeil, A. (2011) Quantitative Risk Management. Princeton, New Jersey: Princeton University Press.CrossRefGoogle Scholar
Embrechts, P., Hofert, M. and Wang, R. (2016) Bernoulli and tail-dependence compatibility. Annals of Applied Probability, 26 (3), 1636–1658.CrossRefGoogle Scholar
Fiebig, U.-R., Strokorb, K. and Schlather, M. (2017) The realization problem for tail correlation functions. Extremes, 20 (1), 121–168.CrossRefGoogle Scholar
Frahm, G., Junker, M. and Schmidt, R. (2005) Estimating the tail-dependence coefficient: Properties and pitfalls. Insurance: Mathematics and Economics, 37 (1), 80–100.Google Scholar
Frees, E.W. and Valdez, E.A. (1998) Understanding relationships using copulas. North American Actuarial Journal, 2 (1), 1–25.CrossRefGoogle Scholar
Frobenius, F.G. (1900) Über die charactere der symetrischen gruppe, s’ber. Akad. Wiss. Berlin, pp. 516–534.Google Scholar
Gao, L. (2022) Spatial models of insurance losses for claims management. Ph.D. Thesis, University of Wisconsin, Madison.Google Scholar
Gawrilow, E. and Joswig, M. (2000) Polymake: A framework for analyzing convex polytopes. In Polytopes–Combinatorics and Computation, pp. 43–73. Springer.CrossRefGoogle Scholar
Hosking, J.R.M. (2019) lmom-package. URL https://rdrr.io/cran/lmom/man/lmom-package.html.Google Scholar
Hosking, J.R.M. (1990) L-moments: Analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society: Series B (Methodological), 52 (1), 105–124.Google Scholar
Hsieh, M.-H., Tsai, C.J. and Wang, J.L. (2021) Mortality risk management under the factor copula framework–with applications to insurance policy pools. North American Actuarial Journal, 25 (sup1), S119–S131.CrossRefGoogle Scholar
Hua, L., Xia, M. and Basu, S. (2017) Factor copula approaches for assessing spatially dependent high-dimensional risks. North American Actuarial Journal, 21 (1), 147–160.CrossRefGoogle Scholar
Janßen, A., Neblung, S. and Stoev, S. (2023) Tail-dependence, exceedance sets, and metric embeddings. In Extremes, pp. 1–39.CrossRefGoogle Scholar
Jaworski, P., Durante, F., Hardle, W.K. and Rychlik, T. (2010) In Copula Theory and its Applications, Vol. 198. Springer.CrossRefGoogle Scholar
Joe, H. (1997) Multivariate Models and Dependence Concepts, Monographs on Statistics and Applied Probability, Vol. 73. London: Chapman & Hall.Google Scholar
Joe, H. (2011) Tail dependence in vine copulae. In Dependence Modeling, pp 165–187. Hackensack, NJ: World Scientific Publishing.Google Scholar
Joe, H. (2014) Dependence Modeling with Copulas. Boca Raton, Florida: CRC Press.CrossRefGoogle Scholar
Joe, H. and Hu, T. (1996) Multivariate distributions from mixtures of max-infinitely divisible distributions. Journal of Multivariate Analysis, 57, 240–265.CrossRefGoogle Scholar
Johnston, N. (2014) Counting the possible orderings of pairwise multiplication. URL http://www.njohnston.ca/2014/02/counting-the-possible-orderings-of-pairwise-multiplication/.Google Scholar
Klein Tank, A.M.G., Wijngaard, J.B., Können, G.P., Böhm, R., Demarée, G., Gocheva, A., Mileta, M., Pashiardis, S., Hejkrlik, L., Kern-Hansen, C., Heino, R., Bessemoulin, P., Müller-Westermeier, G., Tzanakou, M., Szalai, S., Pálsdóttir, T., Fitzgerald, D., Rubin, S., Capaldo, M., Maugeri, M., Leitass, A., Bukantis, A., Aberfeld, R., van Engelen, A.F.V., Forland, E., Mietus, M., Coelho, F., Mares, C., Razuvaev, V., Nieplova, E., Cegnar, T., Antonio López, J., Dahlström, B., Moberg, A., Kirchhofer, W., Ceylan, A., Pachaliuk, O., Alexander, L.V. and Petrovic, P. (2002) Daily dataset of 20th-century surface air temperature and precipitation series for the european climate assessment. International Journal of Climatology: A Journal of the Royal Meteorological Society, 22 (12), 1441–1453.CrossRefGoogle Scholar
Knuth, D.E. (1998) Sorting and searching. In The Art of Computer Programming, Vol. 3.Google Scholar
Kolman, M. (2014) A one-factor copula-based model for credit portfolios. Journal of Risk, 17 (2), 93–132.CrossRefGoogle Scholar
Kroese, D.P., Taimre, T. and Botev, Z.I. (2013) Handbook of Monte Carlo Methods, Vol. 706. Hoboken, New Jersey: John Wiley & Sons.Google Scholar
Krupskii, P. and Joe, H. (2013) Factor copula models for multivariate data. Journal of Multivariate Analysis, 12, 85–101.Google Scholar
Krupskii, P. and Joe, H. (2020) Flexible copula models with dynamic dependence and application to financial data. Econometrics and Statistics, 16, 148–167.CrossRefGoogle Scholar
Kularatne, T.D., Li, J. and Pitt, D. (2021) On the use of archimedean copulas for insurance modelling. Annals of Actuarial Science, 15 (1), 57–81.CrossRefGoogle Scholar
Lan, M., Shao, Y., Zhu, J., Lo, S. and Ng, S.T. (2021) A hybrid copula-fragility approach for investigating the impact of hazard dependence on a process facility’s failure. Process Safety and Environmental Protection, 149, 1017–1030.CrossRefGoogle Scholar
Lazoglou, G. and Anagnostopoulou, C. (2019) Joint distribution of temperature and precipitation in the mediterranean, using the copula method. Theoretical and Applied Climatology, 135, 1399–1411.CrossRefGoogle Scholar
Li, D.X. (2000) On default correlation: A copula function approach. The Journal of Fixed Income, 9 (4), 43–54.CrossRefGoogle Scholar
Li, J., Balasooriya, U. and Liu, J. (2021) Using hierarchical archimedean copulas for modelling mortality dependence and pricing mortality-linked securities. Annals of Actuarial Science, 15 (3), 505–518.CrossRefGoogle Scholar
Lu, J.C. and Bhattacharyya, G.K. (1990) Some new constructions of bivariate weibull models. Annals of the Institute of Statistical Mathematics, 42, 543–559.CrossRefGoogle Scholar
MacMahon, P.A. (1909) Memoir on the theory of the partitions of numbers.–part iv. Philosophical Transactions of the Royal Society of London. Series A, 209 (441–458), 153–175.Google Scholar
MathWorks (2021) fmincon - nonlinear constrained optimization (optimization toolbox).Google Scholar
Mazo, G., Girard, S. and Forbes, F. (2016) A flexible and tractable class of one-factor copulas. Statistics and Computing, 26, 965–979.CrossRefGoogle Scholar
McNeil, A. J., Frey, R. and Embrechts, P. (2015) Quantitative Risk Management. Princeton Series in Finance. Princeton, NJ: Princeton University Press, revised edition. Concepts, techniques and tools.Google Scholar
Morales-Napoles, O. (2010) Counting vines. In Dependence Modeling: Vine Copula Handbook, pp. 189–218. World Scientific.CrossRefGoogle Scholar
Nelsen, R.B. (2006) An Introduction to Copulas. New York, NY: Springer.Google Scholar
Nikoloulopoulos, A.K., Joe, H. and Li, H. (2012) Vine copulas with asymmetric tail dependence and applications to financial return data. Computational Statistics & Data Analysis, 56 (11), 3659–3673.CrossRefGoogle Scholar
Oh, D.H. and Patton, A.J. (2017) Modeling dependence in high dimensions with factor copulas. Journal of Business & Economic Statistics, 35 (1), 139–154.CrossRefGoogle Scholar
Oh, R., Ahn, J.Y. and Lee, W. (2021) On copula-based collective risk models: From elliptical copulas to vine copulas. Scandinavian Actuarial Journal, 2021 (1), 1–33.CrossRefGoogle Scholar
Othus, M. and Li, Y. (2010) A gaussian copula model for multivariate survival data. Statistics in Biosciences, 2, 154–179.CrossRefGoogle ScholarPubMed
Da Silva, P.P., Rebelo, P.T. and Afonso, C. (2014) Tail dependence of financial stocks and cds markets – evidence using copula methods and simulation-based inference. Economics, 8 (1), 20140039.CrossRefGoogle Scholar
Renard, B. and Lang, M. (2007) Use of a gaussian copula for multivariate extreme value analysis: Some case studies in hydrology. Advances in Water Resources, 30 (4), 897–912.CrossRefGoogle Scholar
Righi, M.B. and Ceretta, P.S. (2013) Analyzing the dependence structure of various sectors in the brazilian market: A pair copula construction approach. Economic Modelling, 35, 199–206.CrossRefGoogle Scholar
Robbins, H. (1955) A remark on stirling’s formula. The American Mathematical Monthly, 62 (1), 26–29.CrossRefGoogle Scholar
Salmon, F. (2009) Recipe for disaster: The formula that killed wall street. Wired Magazine, 17 (3), 17–03.Google Scholar
Salvadori, G., De Michele, C., Kottegoda, N.T. and Rosso, R. (2007) Extremes in Nature: An Approach Using Copulas, Vol. 56. Dordrecht, The Netherlands: Springer Science & Business Media.CrossRefGoogle Scholar
Serinaldi, F. (2008) Analysis of inter-gauge dependence by kendall’s $\tau$ k, upper tail dependence coefficient, and 2-copulas with application to rainfall fields. Stochastic Environmental Research and Risk Assessment, 22 (6), 671–688.Google Scholar
Shimizu, K. (1993) A bivariate mixed lognormal distribution with an analysis of rainfall data. Journal of Applied Meteorology and Climatology, 32 (2), 161–171.Google Scholar
Shorack, G.R. (2017) Probability for Statisticians. Cham, Switzerland: Springer.CrossRefGoogle Scholar
Shyamalkumar, N.D. and Tao, S. (2020) On tail dependence matrices. Extremes, 23 (2), 245–285.CrossRefGoogle Scholar
Shyamalkumar, N.D. and Tao, S. (2022) t-copula from the viewpoint of tail dependence matrices. Journal of Multivariate Analysis, 105027.CrossRefGoogle Scholar
Sklar, M. (1959) Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de statistique de l’Universite Paris, 8, 229–231.Google Scholar
Song, P.X.-K., Li, M. and Yuan, Y. (2009) Joint regression analysis of correlated data using gaussian copulas. Biometrics, 65 (1), 60–68.CrossRefGoogle ScholarPubMed
Strokorb, K. (2013) Characterization and construction of max-stable processes. Ph.D. Thesis. URL http://hdl.handle.net/11858/00-1735-0000-0001-BB44-9.Google Scholar
Tang, Q., Tang, Z. and Yang, Y. (2019) Sharp asymptotics for large portfolio losses under extreme risks. European Journal of Operational Research, 276 (2), 710–722.CrossRefGoogle Scholar
Thrall, R.M. (1952) A combinatorial problem. Michigan Mathematical Journal, 1 (1), 81–88.CrossRefGoogle Scholar
Van de Vyver, H. and Van den Bergh, J. (2018) The gaussian copula model for the joint deficit index for droughts. Journal of Hydrology, 561, 987–999.CrossRefGoogle Scholar