In recent years the mean-semivariance has been proposed in place of the mean-variance as an alternative approach to portfolio analysis since different investors assign a lower weight to positive deviations from the mean than to negative ones. The present work investigates empirically the relationship between risk and return in a downside risk framework and in a regular risk framework by utilizing returns of securities traded on the London Stock Exchange and Paris Stock Exchange . The results reveal that in many cases the downside risk measures are equivalent or better in explaining mean returns than the regular risk measures. The paper also introduces a new risk-return relation that holds when the distribution of security returns are normal and the market index lies inside the semi-deviation-expected return efficient frontier. The existence of this model may provide a possible explanation of the empirical results included in this work. Finally, it is argued that for skewed distributions of security returns it may be better to employ a three parameter asset pricing than the mean – semivariance risk-return relation.