Tsallis Relative entropy from asymmetric distributions as a risk measure for financial portfolios

In an earlier study, we showed that Tsallis relative entropy (TRE), which is the generalization of Kullback-Leibler relative entropy (KLRE) to non-extensive systems, can be used as a possible risk measure in constructing risk optimal portfolios whose returns beat market returns. Over a long term (> 10 years), the risk-return profiles from TRE as the risk measure show a more consistent behavior than those from the commonly used risk measure 'beta' of the Capital Asset Pricing Model (CAPM). In these investigations, the model distributions derived from TRE are symmetric. However, observations show that distributions of the returns of financial markets and equities are in general asymmetric in positive and negative returns. In this work, we generalize TRE for the asymmetric case (ATRE) by considering the data distribution as a linear combination of two independent normalized distributions - one for negative returns and one for positive returns. Each of these two independent distributions are half q-Gaussians with different non-extensivity parameter q and temperature parameter b. The risk-return (in excess of market returns) patterns are investigated using ATRE as the risk measure. The results are compared with those from two other risk measures: TRE and the Tsallis relative entropy S- derived from the negative returns only. Tests on data, which include the dot-com bubble, the 2008 crash, and COVID periods, for both long (20 years) and shorter terms (10 years), show that a linear fit can be obtained for the risk-excess return profiles of all three risk measures. However, the fits for portfolios created during the chaotic market conditions (crashes) using S- as the risk show a much higher slope pointing to higher returns for a given risk value. Further, in this case, the excess returns of even short-term portfolios remain positive irrespective of the market behavior.