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On the distribution of sums of random variables with copula-induced dependence

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  • Gijbels, Irène
  • Herrmann, Klaus

Abstract

We investigate distributional properties of the sum of d possibly unbounded random variables. The joint distribution of the random vector is formulated by means of an absolutely continuous copula, allowing for a variety of different dependence structures between the summands. The obtained expression for the distribution of the sum features a separation property into marginal and dependence structure contributions typical for copula approaches. Along the same lines we obtain the formulation of a conditional expectation closely related to the expected shortfall common in actuarial and financial literature. We further exploit the separation to introduce new numerical algorithms to compute the distribution and quantile function, as well as this conditional expectation. A comparison with the most common competitors shows that the discussed Path Integration algorithm is the most suitable method for computing these quantities. In our example, we apply the theory to compute Value-at-Risk forecasts for a trivariate portfolio of index returns.

Suggested Citation

  • Gijbels, Irène & Herrmann, Klaus, 2014. "On the distribution of sums of random variables with copula-induced dependence," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 27-44.
    • Handle: RePEc:eee:insuma:v:59:y:2014:i:c:p:27-44
    DOI: 10.1016/j.insmatheco.2014.08.002
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    References listed on IDEAS

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    1. Hafner, Christian M. & Reznikova, Olga, 2010. "Efficient estimation of a semiparametric dynamic copula model," Computational Statistics & Data Analysis, Elsevier, vol. 54(11), pages 2609-2627, November.
    2. Cherubini, Umberto & Mulinacci, Sabrina & Romagnoli, Silvia, 2011. "On the distribution of the (un)bounded sum of random variables," Insurance: Mathematics and Economics, Elsevier, vol. 48(1), pages 56-63, January.
    3. Huang, Jen-Jsung & Lee, Kuo-Jung & Liang, Hueimei & Lin, Wei-Fu, 2009. "Estimating value at risk of portfolio by conditional copula-GARCH method," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 315-324, December.
    4. Christian Meyer, 2009. "The Bivariate Normal Copula," Papers 0912.2816, arXiv.org.
    5. Filippo Domma & Sabrina Giordano, 2012. "A stress–strength model with dependent variables to measure household financial fragility," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 21(3), pages 375-389, August.
    6. Filippo Domma & Sabrina Giordano, 2013. "A copula-based approach to account for dependence in stress-strength models," Statistical Papers, Springer, vol. 54(3), pages 807-826, August.
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    Cited by:

    1. Montes, Ignacio & Salamanca, Juan Jesús & Montes, Susana, 2020. "A modified version of stochastic dominance involving dependence," Statistics & Probability Letters, Elsevier, vol. 165(C).
    2. Genest Christian & Scherer Matthias, 2023. "When copulas and smoothing met: An interview with Irène Gijbels," Dependence Modeling, De Gruyter, vol. 11(1), pages 1-16, January.

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