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

Financial Portfolios based on Tsallis Relative Entropy as the Risk Measure