Financial Portfolios based on Tsallis Relative Entropy as the Risk Measure

TL;DR

This study explores Tsallis relative entropy as a novel risk measure for portfolio construction, demonstrating its superior consistency and predictive power over traditional measures across different market periods.

Contribution

It introduces Tsallis relative entropy as a new risk measure for portfolios and compares its effectiveness with existing measures using extensive historical data.

Findings

Tsallis relative entropy provides more consistent risk-return profiles.

The risk measure outperforms traditional metrics in various market conditions.

Profiles from TRE show better goodness of fit and stability.

Abstract

Earlier studies have shown that stock market distributions can be well described by distributions derived from Tsallis entropy, which is a generalization of Shannon entropy to non-extensive systems. In this paper, Tsallis relative entropy (TRE), which is the generalization of Kullback-Leibler relative entropy (KLRE) to non-extensive systems, is investigated as a possible risk measure in constructing risk optimal portfolios whose returns beat market returns. Portfolios are constructed by binning the risk values and allocating the stocks to bins according to their risk values. The average return in excess of market returns for each bin is calculated to get the risk-return patterns of the portfolios. The results are compared with those from three other risk measures: 1) the commonly used 'beta' of the Capital Asset Pricing Model (CAPM), 2) Kullback-Leibler relative entropy, and 3) the relative standard deviation. Tests carried out for both long (~18 years) and shorter terms (~9 years), which include the dot-com bubble and the 2008 crash periods, show that a linear fit can be obtained for the risk-excess return profiles of all four risk measures. However, in all cases, the profiles from Tsallis relative entropy show a more consistent behavior in terms of both goodness of fit and the variation of returns with risk, than the other three risk measures.