Financial portfolios based on Tsallis relative entropy as the risk measure

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) KLRE, and (3) the ratio of standard deviations. Tests carried out for both long (similar to 18 years) and shorter terms (similar to 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 TRE 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.