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Copula based hierarchical risk aggregation through sample reordering


  • Arbenz, Philipp
  • Hummel, Christoph
  • Mainik, Georg


For high-dimensional risk aggregation purposes, most popular copula classes are too restrictive in terms of attainable dependence structures. These limitations aggravate with increasing dimension. We study a hierarchical risk aggregation method which is flexible in high dimensions. With this method it suffices to specify a low dimensional copula for each aggregation step in the hierarchy. Copulas and margins of arbitrary kind can be combined. We give an algorithm for numerical approximation which introduces dependence between originally independent marginal samples through reordering.

Suggested Citation

  • Arbenz, Philipp & Hummel, Christoph & Mainik, Georg, 2012. "Copula based hierarchical risk aggregation through sample reordering," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 122-133.
  • Handle: RePEc:eee:insuma:v:51:y:2012:i:1:p:122-133
    DOI: 10.1016/j.insmatheco.2012.03.009

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    References listed on IDEAS

    1. Laurent Devineau & Stéphane Loisel, 2009. "Risk aggregation in Solvency II: How to converge the approaches of the internal models and those of the standard formula?," Post-Print hal-00403662, HAL.
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    4. Jean-Philippe Bruneton, 2011. "Copula-based Hierarchical Aggregation of Correlated Risks. The behaviour of the diversification benefit in Gaussian and Lognormal Trees," Papers 1111.1113,, revised Nov 2011.
    5. Nikolay Nenovsky & S. Statev, 2006. "Introduction," Post-Print halshs-00260898, HAL.
    6. Philip M. Lurie & Matthew S. Goldberg, 1998. "An Approximate Method for Sampling Correlated Random Variables from Partially-Specified Distributions," Management Science, INFORMS, vol. 44(2), pages 203-218, February.
    7. Dirk Tasche, 2007. "Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle," Papers 0708.2542,, revised Jun 2008.
    8. Donnelly, Catherine & Embrechts, Paul, 2010. "The Devil is in the Tails: Actuarial Mathematics and the Subprime Mortgage Crisis," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 40(01), pages 1-33, May.
    9. Bürgi, Roland & Dacorogna, Michel M & Iles, Roger, 2008. "Risk aggregation, dependence structure and diversification benefit," MPRA Paper 10054, University Library of Munich, Germany.
    10. Paul Embrechts, 2009. "Copulas: A Personal View," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 76(3), pages 639-650.
    11. Christoph Hummel, 2009. "Shaping tail dependencies by nesting box copulas," Papers 0906.4853,, revised Aug 2009.
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    Cited by:

    1. repec:eee:jbfina:v:96:y:2018:i:c:p:236-248 is not listed on IDEAS
    2. repec:gam:jrisks:v:7:y:2019:i:2:p:54-:d:229383 is not listed on IDEAS
    3. Embrechts, Paul & Puccetti, Giovanni & Rüschendorf, Ludger, 2013. "Model uncertainty and VaR aggregation," Journal of Banking & Finance, Elsevier, vol. 37(8), pages 2750-2764.
    4. Zaks, Yaniv & Tsanakas, Andreas, 2014. "Optimal capital allocation in a hierarchical corporate structure," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 48-55.
    5. Joe, Harry & Sang, Peijun, 2016. "Multivariate models for dependent clusters of variables with conditional independence given aggregation variables," Computational Statistics & Data Analysis, Elsevier, vol. 97(C), pages 114-132.
    6. Guillén, Montserrat & Sarabia, José María & Prieto, Faustino, 2013. "Simple risk measure calculations for sums of positive random variables," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 273-280.
    7. Souhaib Ben Taieb & James W. Taylor & Rob J. Hyndman, 2017. "Coherent Probabilistic Forecasts for Hierarchical Time Series," Monash Econometrics and Business Statistics Working Papers 3/17, Monash University, Department of Econometrics and Business Statistics.
    8. repec:eee:riibaf:v:45:y:2018:i:c:p:208-218 is not listed on IDEAS
    9. Mainik, Georg, 2015. "Risk aggregation with empirical margins: Latin hypercubes, empirical copulas, and convergence of sum distributions," Journal of Multivariate Analysis, Elsevier, vol. 141(C), pages 197-216.
    10. Bernardi, Enrico & Falangi, Federico & Romagnoli, Silvia, 2015. "A hierarchical copula-based world-wide valuation of sovereign risk," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 155-169.
    11. Lee, Chien-Chiang & Chang, Chi-Hung & Chen, Mei-Ping, 2015. "Industry co-movements of American depository receipts: Evidences from the copula approaches," Economic Modelling, Elsevier, vol. 46(C), pages 301-314.

    More about this item


    IM12; IM22; IM43; IE43; IE46; Hierarchical risk aggregation; Copulas; High-dimensional dependence; Iman–Conover method;

    JEL classification:

    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies


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