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Practical volume computation of structured convex bodies, and an application to modeling portfolio dependencies and financial crises

Author

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  • Ludovic Cales
  • Apostolos Chalkis
  • Ioannis Z. Emiris
  • Vissarion Fisikopoulos

Abstract

We examine volume computation of general-dimensional polytopes and more general convex bodies, defined as the intersection of a simplex by a family of parallel hyperplanes, and another family of parallel hyperplanes or a family of concentric ellipsoids. Such convex bodies appear in modeling and predicting financial crises. The impact of crises on the economy (labor, income, etc.) makes its detection of prime interest. Certain features of dependencies in the markets clearly identify times of turmoil. We describe the relationship between asset characteristics by means of a copula; each characteristic is either a linear or quadratic form of the portfolio components, hence the copula can be constructed by computing volumes of convex bodies. We design and implement practical algorithms in the exact and approximate setting, we experimentally juxtapose them and study the tradeoff of exactness and accuracy for speed. We analyze the following methods in order of increasing generality: rejection sampling relying on uniformly sampling the simplex, which is the fastest approach, but inaccurate for small volumes; exact formulae based on the computation of integrals of probability distribution functions; an optimized Lawrence sign decomposition method, since the polytopes at hand are shown to be simple; Markov chain Monte Carlo algorithms using random walks based on the hit-and-run paradigm generalized to nonlinear convex bodies and relying on new methods for computing a ball enclosed; the latter is experimentally extended to non-convex bodies with very encouraging results. Our C++ software, based on CGAL and Eigen and available on github, is shown to be very effective in up to 100 dimensions. Our results offer novel, effective means of computing portfolio dependencies and an indicator of financial crises, which is shown to correctly identify past crises.

Suggested Citation

  • Ludovic Cales & Apostolos Chalkis & Ioannis Z. Emiris & Vissarion Fisikopoulos, 2018. "Practical volume computation of structured convex bodies, and an application to modeling portfolio dependencies and financial crises," Papers 1803.05861, arXiv.org.
  • Handle: RePEc:arx:papers:1803.05861
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    1. Ludovic Cales & Apostolos Chalkis & Ioannis Z. Emiris & Vissarion Fisikopoulos, 2018. "Practical volume computation of structured convex bodies, and an application to modeling portfolio dependencies and financial crises," Papers 1803.05861, arXiv.org.
    2. Dominique Guegan & Ludovic Calès & Monica Billio, 2011. "A Cross-Sectional Score for the Relative Performance of an Allocation," PSE-Ecole d'économie de Paris (Postprint) halshs-00646070, HAL.
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    4. Lo Duca, Marco & Koban, Anne & Basten, Marisa & Bengtsson, Elias & Klaus, Benjamin & Kusmierczyk, Piotr & Lang, Jan Hannes & Detken, Carsten & Peltonen, Tuomas, 2017. "A new database for financial crises in European countries," ESRB Occasional Paper Series 13, European Systemic Risk Board.
    5. Banerjee, Anurag & Hung, Chi-Hsiou, 2011. "Informed momentum trading versus uninformed "naive" investors strategies," Journal of Banking & Finance, Elsevier, vol. 35(11), pages 3077-3089, November.
    6. John M. Griffin & Xiuqing Ji & J. Spencer Martin, 2003. "Momentum Investing and Business Cycle Risk: Evidence from Pole to Pole," Journal of Finance, American Finance Association, vol. 58(6), pages 2515-2547, December.
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    Cited by:

    1. Apostolos Chalkis & Emmanouil Christoforou & Ioannis Z. Emiris & Theodore Dalamagas, 2020. "Modeling asset allocation strategies and a new portfolio performance score," Papers 2012.05088, arXiv.org, revised Sep 2021.
    2. Ludovic Cales & Apostolos Chalkis & Ioannis Z. Emiris & Vissarion Fisikopoulos, 2018. "Practical volume computation of structured convex bodies, and an application to modeling portfolio dependencies and financial crises," Papers 1803.05861, arXiv.org.
    3. Cyril Bachelard & Apostolos Chalkis & Vissarion Fisikopoulos & Elias Tsigaridas, 2024. "Randomized Control in Performance Analysis and Empirical Asset Pricing," Papers 2403.00009, arXiv.org.

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