Book excerpt: Most theoretical models in finance measure risk as variance or covariance. However, many financial decision-makers seem to regard risk as the volatility of below-target returns and treat the volatility of above-target returns as a sweetener. Using simple metrics of downside risk and upside potential, constructed from conditional covariances, I test for the empirical content of this asymmetry. I introduce a new composite metric of semidimensional risks which reveals that the nonlinearity in the covariation of stock returns with bearish and bullish conditions of the market is priced in the cross-section of stock returns. In particular, I find that stocks that have concave characteristic regression lines against the market earn higher average returns than stocks that have convex characteristic regression lines. This new metric captures the relevant information in returns asymmetry or nonlinearity better than either coskewness or the square-coefficient from quadratic regression. I also present results that are consistent with a semidimensional risk-based explanation for the twin puzzles of return momentum and reversal. The primitive representation of the security pricing kernel as the negative of covariance between marginal utility of consumption and security returns lends theoretical support for semidimensional risks and provides a unifying perspective for seemingly disparate literature on semivariances, skewness and behavioral finance.