Paper: arXiv 2302.00452

Authors: 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.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 5.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a mathematically rigorous extension of CAPM beta metrics using f-divergence and coherent risk measures, specifically introducing Hellinger-Beta and Hellinger-Drawdown Beta. While the theoretical formulation is strong and novel, the empirical section relies primarily on basic numerical experiments with a small stock subset rather than an extensive multi-asset backtest.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Formulate CAPM-style optimality using<br>f-divergence coherent risk measures"] --> B["Inputs:<br>S&P 500 Index & Selected Stock Data"]
    
    B --> C["Theoretical Framework:<br>Derive f-Beta & extend to<br>f-Drawdown Beta under perturbed measures"]
    
    C --> D["Computational Experiments:<br>Apply Squared Hellinger Distance to calculate<br>Hellinger-Beta & Hellinger-Drawdown Beta"]
    
    D --> E["Sensitivity Analysis:<br>Evaluate metrics across varying<br>risk aversion parameters (stress levels)"]
    
    E --> F["Outcomes:<br>New flexible Beta metrics providing enhanced<br>risk perspectives vs. Standard & Drawdown Betas"]