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On the solution uniqueness in portfolio optimization and risk analysis

Author

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  • Bogdan Grechuk
  • Andrzej Palczewski
  • Jan Palczewski

Abstract

We consider the issue of solution uniqueness for portfolio optimization problem and its inverse for asset returns with a finite number of possible scenarios. The risk is assessed by deviation measures introduced by [Rockafellar et al., Mathematical Programming, Ser. B, 108 (2006), pp. 515-540] instead of variance as in the Markowitz optimization problem. We prove that in general one can expect uniqueness neither in forward nor in inverse problems. We discuss consequences of that non-uniqueness for several problems in risk analysis and portfolio optimization, including capital allocation, risk sharing, cooperative investment, and the Black-Litterman methodology. In all cases, the issue with non-uniqueness is closely related to the fact that subgradient of a convex function is non-unique at the points of non-differentiability. We suggest methodology to resolve this issue by identifying a unique "special" subgradient satisfying some natural axioms. This "special" subgradient happens to be the Stainer point of the subdifferential set.

Suggested Citation

  • Bogdan Grechuk & Andrzej Palczewski & Jan Palczewski, 2018. "On the solution uniqueness in portfolio optimization and risk analysis," Papers 1810.11299, arXiv.org, revised Oct 2020.
    • Handle: RePEc:arx:papers:1810.11299
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    References listed on IDEAS

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    1. Dimitris Bertsimas & Vishal Gupta & Ioannis Ch. Paschalidis, 2012. "Inverse Optimization: A New Perspective on the Black-Litterman Model," Operations Research, INFORMS, vol. 60(6), pages 1389-1403, December.
    2. R. Rockafellar & Stan Uryasev & Michael Zabarankin, 2006. "Generalized deviations in risk analysis," Finance and Stochastics, Springer, vol. 10(1), pages 51-74, January.
    3. Grechuk, Bogdan, 2015. "The center of a convex set and capital allocation," European Journal of Operational Research, Elsevier, vol. 243(2), pages 628-636.
    4. Grechuk, Bogdan & Zabarankin, Michael, 2016. "Inverse portfolio problem with coherent risk measures," European Journal of Operational Research, Elsevier, vol. 249(2), pages 740-750.
    5. Damir Filipović & Michael Kupper, 2008. "Equilibrium Prices For Monetary Utility Functions," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(03), pages 325-343.
    6. Grechuk, Bogdan & Zabarankin, Michael, 2014. "Inverse portfolio problem with mean-deviation model," European Journal of Operational Research, Elsevier, vol. 234(2), pages 481-490.
    7. Jianming Xia, 2004. "Multi-agent investment in incomplete markets," Finance and Stochastics, Springer, vol. 8(2), pages 241-259, May.
    8. SALINETTI, Gabriella & WETS, Roger J.-B., 1979. "On the convergence of sequences of convex sets in finite dimensions," LIDAM Reprints CORE 352, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    9. Grechuk, Bogdan & Zabarankin, Michael, 2018. "Direct data-based decision making under uncertainty," European Journal of Operational Research, Elsevier, vol. 267(1), pages 200-211.
    10. Bogdan Grechuk & Michael Zabarankin, 2017. "Synergy effect of cooperative investment," Annals of Operations Research, Springer, vol. 249(1), pages 409-431, February.
    11. Michael Kalkbrener, 2005. "An Axiomatic Approach To Capital Allocation," Mathematical Finance, Wiley Blackwell, vol. 15(3), pages 425-437, July.
    12. Rose‐Anne Dana, 2005. "A Representation Result For Concave Schur Concave Functions," Mathematical Finance, Wiley Blackwell, vol. 15(4), pages 613-634, October.
    13. Rockafellar, R. Tyrrell & Uryasev, Stan & Zabarankin, M., 2007. "Equilibrium with investors using a diversity of deviation measures," Journal of Banking & Finance, Elsevier, vol. 31(11), pages 3251-3268, November.
    14. A. Cherny, 2006. "Weighted V@R and its Properties," Finance and Stochastics, Springer, vol. 10(3), pages 367-393, September.
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