Tsallis relative entropy , which is the generalization of Kullback-Leibler relative entropy  to non-extensive systems, is investigated as a possible risk measure in constructing risk optimal portfolios whose returns beat market returns. 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. Portfolios are constructed by binning the securities according to their risk values. The mean risk value and the mean return in excess of market returns for each bin is calculated to get the risk-return patterns of the portfolios. The investigations have been carried out for both long (~18 years) and shorter  (~9 years) terms that include the dot-com bubble and the 2008 crash periods . In all cases, a linear fit can be obtained for the risk and excess return profiles , both for long and shorter periods. For longer periods, the linear fits have a positive slope, with Tsallis relative entropy giving the best goodness of fit. For shorter periods, the risk-return profiles from Tsallis relative entropy show a more consistent behavior in terms of  goodness of fit  than the other three risk measures 
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 .
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