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A Black-Litterman asset allocation model under Elliptical distributions

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  • Yugu Xiao
  • Emiliano A. Valdez

Abstract

In optimal portfolio allocation, Black and Litterman [ Financ. Anal. J. , 1992, 48 , 28-43] provide for a pioneering framework of allowing to incorporate investors' views based on a prior distribution to derive a posterior distribution of portfolio returns and optimal asset allocations. Meucci [ Risk and Asset Allocation , 2005] rephrases the model in terms of investors' views on the market, rather than just the market parameters as in the original Black and Litterman [ Financ. Anal. J. , 1992, 48 , 28-43]. This market-based version is believed to be much more parsimonious and allows for a more natural extension to directly input views in a non-Normal market. This paper extends Meucci's market-based version of the Black-Litterman model to the case when returns in the market fall within the class of Elliptical distributions, while also importantly preserving the equilibrium-based assumption in the model. Here, within this class for which the Normal distribution is a special case, we develop the explicit form of the posterior distribution after considering proper conditional conjugate-type prior distributions. This resulting posterior allows us to obtain solutions to optimization problems of asset allocation based on a variety of risk measures (e.g. Mean-Variance, Mean-VaR, Mean-Conditional VaR). Elliptical models of portfolio returns have recently crept into the financial literature because of their greater flexibility to accommodate larger tails. As numerical demonstrations, we examine how these principles work in a portfolio with international stock indices and show why models with more flexible tails can affect the asset allocation.

Suggested Citation

  • Yugu Xiao & Emiliano A. Valdez, 2015. "A Black-Litterman asset allocation model under Elliptical distributions," Quantitative Finance, Taylor & Francis Journals, vol. 15(3), pages 509-519, March.
  • Handle: RePEc:taf:quantf:v:15:y:2015:i:3:p:509-519
    DOI: 10.1080/14697688.2013.836283
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    1. Felipe Aparicio & Javier Estrada, 2001. "Empirical distributions of stock returns: European securities markets, 1990-95," The European Journal of Finance, Taylor & Francis Journals, vol. 7(1), pages 1-21.
    2. Arellano-Valle, R.B. & del Pino, G. & Iglesias, P., 2006. "Bayesian inference in spherical linear models: robustness and conjugate analysis," Journal of Multivariate Analysis, Elsevier, vol. 97(1), pages 179-197, January.
    3. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    4. Rosella Giacometti & Marida Bertocchi & Svetlozar T. Rachev & Frank J. Fabozzi, 2007. "Stable distributions in the Black-Litterman approach to asset allocation," Quantitative Finance, Taylor & Francis Journals, vol. 7(4), pages 423-433.
    5. Owen, Joel & Rabinovitch, Ramon, 1983. "On the Class of Elliptical Distributions and Their Applications to the Theory of Portfolio Choice," Journal of Finance, American Finance Association, vol. 38(3), pages 745-752, June.
    6. Zinoviy Landsman & Emiliano Valdez, 2003. "Tail Conditional Expectations for Elliptical Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 55-71.
    7. Arellano-Valle, Reinaldo B. & Bolfarine, Heleno, 1995. "On some characterizations of the t-distribution," Statistics & Probability Letters, Elsevier, vol. 25(1), pages 79-85, October.
    8. Akihiro Fukushima, 2011. "Hybrid forecasting models for S&P 500 index returns," Journal of Risk Finance, Emerald Group Publishing, vol. 12(4), pages 315-328, August.
    9. Bawa, Vijay S., 1975. "Optimal rules for ordering uncertain prospects," Journal of Financial Economics, Elsevier, vol. 2(1), pages 95-121, March.
    10. Landsman, Zinoviy, 2010. "On the Tail Mean-Variance optimal portfolio selection," Insurance: Mathematics and Economics, Elsevier, vol. 46(3), pages 547-553, June.
    11. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
    12. N. H. Bingham & Rudiger Kiesel, 2002. "Semi-parametric modelling in finance: theoretical foundations," Quantitative Finance, Taylor & Francis Journals, vol. 2(4), pages 241-250.
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    3. Yuanyuan Zhang & Xiang Li & Sini Guo, 2018. "Portfolio selection problems with Markowitz’s mean–variance framework: a review of literature," Fuzzy Optimization and Decision Making, Springer, vol. 17(2), pages 125-158, June.
    4. Shushi, Tomer, 2018. "Stein’s lemma for truncated elliptical random vectors," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 297-303.
    5. Palczewski, Andrzej & Palczewski, Jan, 2019. "Black–Litterman model for continuous distributions," European Journal of Operational Research, Elsevier, vol. 273(2), pages 708-720.
    6. Erindi Allaj, 2020. "The Black–Litterman model and views from a reverse optimization procedure: an out-of-sample performance evaluation," Computational Management Science, Springer, vol. 17(3), pages 465-492, October.

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