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

TL;DR

This paper introduces a new family of f-Beta metrics derived from f-divergence induced risk measures, enhancing portfolio performance evaluation under market perturbations, with practical experiments demonstrating their effectiveness.

Contribution

The paper develops a novel f-Beta metric based on f-divergence risk measures, extending traditional Betas to incorporate market measure perturbations and drawdown considerations.

Findings

Hellinger-Beta provides new insights compared to Standard Beta.

Hellinger-Drawdown Beta extends the approach to drawdown risk.

Metrics are sensitive to risk aversion parameters.

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.