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.

翻译：本文基于f-散度诱导的一致风险度量进行投资组合优化，推导了以资本资产定价模型（CAPM）形式表述的必要最优条件。我们提出了类似于标准贝塔的新型f-贝塔，并将其推广至先前关于回撤贝塔的研究。f-贝塔在最优扰动的市场概率测度下评估投资组合绩效，该贝塔指标族具有不同程度的灵活性与可解释性。我们选取特定股票组合，以标普500指数作为最优投资组合进行数值实验，展示Hellinger-贝塔相较于标准贝塔和回撤贝塔提供的新视角。实验中，我们选择平方Hellinger距离作为f-散度函数的特定形式，应用于f-散度诱导风险度量与f-贝塔。基于偏差度量计算Hellinger-贝塔指标后，我们进一步将方法扩展至基于回撤度量的Hellinger-贝塔计算，得到另一个新指标——Hellinger-回撤贝塔。通过比较不同风险厌恶参数下的Hellinger-贝塔值，研究其对递增压力水平的敏感性。