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 have derived a new f-Beta similar to the Standard Betas and 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 conducted numerical experiments using DOW 30 stocks against a chosen market portfolio as the optimal portfolio to demonstrate the new perspectives provided by Hellinger-Beta as compared with Standard Beta and Drawdown Betas, based on choosing square Hellinger distance to be the particular choice of f-divergence function in the general f-divergence induced risk measures and f-Betas. We calculated Hellinger-Beta metrics based on deviation measures and further extended this approach to calculate Hellinger-Betas based on drawdown measures, resulting in another new metric which we termed Hellinger-Drawdown Beta. We compared the resulting Hellinger-Beta values under various choices of the risk aversion parameter to study their sensitivity to increasing stress levels.

翻译：本文在f-散度诱导的一致风险测度框架下构建投资组合优化方法，并导出其CAPM形式下的必要最优性条件。我们提出了一种新型f-Beta指标，该指标类似于标准Beta及先前工作中的回撤Beta。f-Beta在最优扰动的市场概率测度下评估投资组合表现，该族Beta指标具有不同程度的灵活性与可解释性。我们以道琼斯30指数成分股为样本，选择最优投资组合作为市场代理组合进行数值实验，通过选取平方Hellinger距离作为通用f-散度函数的具体实现，将Hellinger-Beta与标准Beta及回撤Beta进行对比，揭示了全新视角。基于偏差度量计算了Hellinger-Beta指标，并进一步将该方法扩展至基于回撤度量的Hellinger-Beta计算，从而衍生出另一新指标——Hellinger-回撤Beta。我们比较了不同风险厌恶参数选择下的Hellinger-Beta值，以研究其对递增压力水平的敏感性。