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Risk, VaR, CVaR and their associated Portfolio Optimizations when Asset Returns have a Multivariate Student T Distribution

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  • William T. Shaw

Abstract

We show how to reduce the problem of computing VaR and CVaR with Student T return distributions to evaluation of analytical functions of the moments. This allows an analysis of the risk properties of systems to be carefully attributed between choices of risk function (e.g. VaR vs CVaR); choice of return distribution (power law tail vs Gaussian) and choice of event frequency, for risk assessment. We exploit this to provide a simple method for portfolio optimization when the asset returns follow a standard multivariate T distribution. This may be used as a semi-analytical verification tool for more general optimizers, and for practical assessment of the impact of fat tails on asset allocation for shorter time horizons.

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  • William T. Shaw, 2011. "Risk, VaR, CVaR and their associated Portfolio Optimizations when Asset Returns have a Multivariate Student T Distribution," Papers 1102.5665, arXiv.org.
    • Handle: RePEc:arx:papers:1102.5665
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    References listed on IDEAS

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    1. William T. Shaw, 2010. "Monte Carlo Portfolio Optimization for General Investor Risk-Return Objectives and Arbitrary Return Distributions: a Solution for Long-only Portfolios," Papers 1008.3718, arXiv.org.
    2. Xavier Gabaix & Parameswaran Gopikrishnan & Vasiliki Plerou & H. Eugene Stanley, 2003. "A theory of power-law distributions in financial market fluctuations," Nature, Nature, vol. 423(6937), pages 267-270, May.
    3. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
    4. W. Breymann & D. R. Lüthi & E. Platen, 2009. "Empirical behavior of a world stock index from intra-day to monthly time scales," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 71(4), pages 511-522, October.
    5. Kevin Fergusson & Eckhard Platen, 2006. "On the Distributional Characterization of Daily Log-Returns of a World Stock Index," Applied Mathematical Finance, Taylor & Francis Journals, vol. 13(1), pages 19-38.
    6. Shaw, W.T. & Lee, K.T.A., 2008. "Bivariate Student t distributions with variable marginal degrees of freedom and independence," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1276-1287, July.
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    Cited by:

    1. Balbás, Alejandro & Balbás, Beatriz & Balbás, Raquel, 2016. "VaR as the CVaR sensitivity : applications in risk optimization," INDEM - Working Paper Business Economic Series id-16-01, Instituto para el Desarrollo Empresarial (INDEM).
    2. Ahmed, Dilan & Soleymani, Fazlollah & Ullah, Malik Zaka & Hasan, Hataw, 2021. "Managing the risk based on entropic value-at-risk under a normal-Rayleigh distribution," Applied Mathematics and Computation, Elsevier, vol. 402(C).

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