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CAPM形式表述的必要最优条件。我们提出了新的f-Beta指标，其构建方法与标准Beta及先前研究中的回撤Beta类似。f-Beta在最优扰动市场概率测度下评估投资组合绩效，该系列Beta度量具有不同程度的灵活性和可解释性。我们选取平方Hellinger距离作为一般f散度诱导风险度量和f-Beta中的特定f散度函数，利用道琼斯30指数成分股进行数值实验，以所选市场组合作为最优投资组合，验证了Hellinger-Beta相较于标准Beta和回撤Beta的新视角。基于偏差度量计算Hellinger-Beta指标后，我们扩展该方法以基于回撤度量计算Hellinger-Beta，从而得到另一个新指标——Hellinger-回撤Beta。通过在不同风险厌恶参数下比较所获Hellinger-Beta值，我们研究了其对递增压力水平的敏感性。