We investigate the relation between various alternative risk measures and future daily returns using a sample of firms over the 1988-2009 time period. Previous research indicates that returns are not normally distributed and that investors seem to care more about downside risk than total risk. Motivated by these findings and mixed empirical evidence supporting theoretical positive risk-return relationship, we model the relation between future returns and risk measures and investigate the following questions: (1) Are investors compensated for total risk and/or asymmetric measures of risk? (2) How does the degree of risk aversion in the lower tail of the return distribution impact the predictability of future returns? (3) Is upside risk or downside risk a better predictor of future returns? We find that, although investors seem to be compensated for total risk, measures of downside risk, such as the lower partial moment, are better at explaining future returns. Further, when comparing downside risk to upside risk, we find that investors are more concerned about downside risk. That is, downside risk is a better predictor of future returns. Our results are robust to the addition of traditional control variables, including size, book-to-market ratio of equity (B/M), leverage, and market risk measures, including beta, downside beta and co-skewness. Our findings are an important contribution to the literature as we document a positive risk-return relationship, using both total and asymmetric measures of risk.