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A note on the computation of sharp numerical bounds for the distribution of the sum, product or ratio of dependent risks

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  • Cossette, Hélène
  • Côté, Marie-Pier
  • Mailhot, Mélina
  • Marceau, Etienne

Abstract

In this paper, an approximation method for computing numerically the cumulative distribution function of the sum of d random variables is developed. The method leads to numerical bounds for the distribution of the sum of dependent risks. The bounds are fast to compute and converge to the exact value if the joint probability density function exists. They also allow to evaluate sharp numerical bounds on the Value-at-Risk measure. Moreover, the fact that the approximation is deterministic, hence without uncertainty on the resulting values, is an advantage over MC simulation techniques. Applications in actuarial science and finance illustrate the accuracy of the procedure. We also present analogous bounds for the distribution of the product or the ratio of two random variables, which can be useful for actuarial or financial applications.

Suggested Citation

  • Cossette, Hélène & Côté, Marie-Pier & Mailhot, Mélina & Marceau, Etienne, 2014. "A note on the computation of sharp numerical bounds for the distribution of the sum, product or ratio of dependent risks," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 1-20.
    • Handle: RePEc:eee:jmvana:v:130:y:2014:i:c:p:1-20
    DOI: 10.1016/j.jmva.2014.04.023
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    References listed on IDEAS

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    1. Marius Hofert & Matthias Scherer, 2011. "CDO pricing with nested Archimedean copulas," Quantitative Finance, Taylor & Francis Journals, vol. 11(5), pages 775-787.
    2. Genest, Christian & Marceau, Etienne & Mesfioui, Mhamed, 2003. "Compound Poisson approximations for individual models with dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 32(1), pages 73-91, February.
    3. Lindskog, Filip & McNeil, Alexander J., 2003. "Common Poisson Shock Models: Applications to Insurance and Credit Risk Modelling," ASTIN Bulletin, Cambridge University Press, vol. 33(2), pages 209-238, November.
    4. Hofert, Marius, 2011. "Efficiently sampling nested Archimedean copulas," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 57-70, January.
    5. Mathieu Bargès & Hélène Cossette & Etienne Marceau, 2009. "TVaR-based capital allocation with copulas," Working Papers hal-00431265, HAL.
    6. Bargès, Mathieu & Cossette, Hélène & Marceau, Étienne, 2009. "TVaR-based capital allocation with copulas," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 348-361, December.
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    Cited by:

    1. Hélène Cossette & Etienne Marceau & Quang Huy Nguyen & Christian Y. Robert, 2019. "Tail Approximations for Sums of Dependent Regularly Varying Random Variables Under Archimedean Copula Models," Methodology and Computing in Applied Probability, Springer, vol. 21(2), pages 461-490, June.
    2. Cuberos A. & Masiello E. & Maume-Deschamps V., 2015. "High level quantile approximations of sums of risks," Dependence Modeling, De Gruyter, vol. 3(1), pages 1-18, October.
    3. Chaoubi, Ihsan & Cossette, Hélène & Gadoury, Simon-Pierre & Marceau, Etienne, 2020. "On sums of two counter-monotonic risks," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 47-60.

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