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f-Betas and Portfolio Optimization with f-Divergence induced Risk Measures

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  • Rui Ding

Abstract

In this paper, we build on using the class of f-divergence induced coherent risk measures for portfolio optimization and derive its necessary optimality conditions formulated in CAPM format. We derive a new f-Beta similar to the Standard Betas and also extended it to previous works in Drawdown Betas. The f-Beta evaluates portfolio performance under an optimally perturbed market probability measure, and this family of Beta metrics gives various degrees of flexibility and interpretability. We conduct numerical experiments using selected stocks against a chosen S\&P 500 market index as the optimal portfolio to demonstrate the new perspectives provided by Hellinger-Beta as compared with Standard Beta and Drawdown Betas. In our experiments, the squared Hellinger distance is chosen to be the particular choice of the f-divergence function in the f-divergence induced risk measures and f-Betas. We calculate Hellinger-Beta metrics based on deviation measures and further extend this approach to calculate Hellinger-Betas based on drawdown measures, resulting in another new metric which is termed Hellinger-Drawdown Beta. We compare the resulting Hellinger-Beta values under various choices of the risk aversion parameter to study their sensitivity to increasing stress levels.

Suggested Citation

  • Rui Ding, 2023. "f-Betas and Portfolio Optimization with f-Divergence induced Risk Measures," Papers 2302.00452, arXiv.org, revised May 2023.
    • Handle: RePEc:arx:papers:2302.00452
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    References listed on IDEAS

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    1. Paul Dommel & Alois Pichler, 2020. "Convex Risk Measures based on Divergence," Papers 2003.07648, arXiv.org, revised Mar 2020.
    2. Markowitz, Harry M, 1991. "Foundations of Portfolio Theory," Journal of Finance, American Finance Association, vol. 46(2), pages 469-477, June.
    3. Mohammad Reza Tavakoli Baghdadabad & Fauzias Mat Nor & Izani Ibrahim, 2013. "Mean-drawdown risk behavior: drawdown risk and capital asset pricing," Journal of Business Economics and Management, Taylor & Francis Journals, vol. 14(sup1), pages 447-469, June.
    4. Rockafellar, R.T. & Royset, J.O. & Miranda, S.I., 2014. "Superquantile regression with applications to buffered reliability, uncertainty quantification, and conditional value-at-risk," European Journal of Operational Research, Elsevier, vol. 234(1), pages 140-154.
    5. Zabarankin, Michael & Pavlikov, Konstantin & Uryasev, Stan, 2014. "Capital Asset Pricing Model (CAPM) with drawdown measure," European Journal of Operational Research, Elsevier, vol. 234(2), pages 508-517.
    6. William F. Sharpe, 1964. "Capital Asset Prices: A Theory Of Market Equilibrium Under Conditions Of Risk," Journal of Finance, American Finance Association, vol. 19(3), pages 425-442, September.
    7. Rui Ding & Stan Uryasev, 2020. "CoCDaR and mCoCDaR: New Approach for Measurement of Systemic Risk Contributions," JRFM, MDPI, vol. 13(11), pages 1-18, November.
    8. Matteo Rossi, 2016. "The capital asset pricing model: a critical literature review," Global Business and Economics Review, Inderscience Enterprises Ltd, vol. 18(5), pages 604-617.
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    Cited by:

    1. Yoshioka, Hidekazu, 2024. "Modeling stationary, periodic, and long memory processes by superposed jump-driven processes," Chaos, Solitons & Fractals, Elsevier, vol. 188(C).

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