The relation between volatility and the predicted average returns of portfolios in excess of market returnsis investigated using four risk measures: 1) Tsallis q-Gaussian relative entropy, 2) Kullback-Leibler relative entropy, 3) the parameter 'beta' of the Capital Asset Pricing Model (CAPM), and 4) 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 pattern than those from the other three risk measures .
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 
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