Two types of asymmetries have been found in the joint distribution of stock returns: skewness in individual stock returns, and "asymmetric dependence" between stock returns. An example of the latter is that stock returns are more dependent during market downturns than during market upturns. We examine the connection between these two types of asymmetries, and their implications for portfolio decisions. We consider the problem of a CRRA investor allocating wealth between a small-cap and a large-cap portfolio, using monthly data from 1954 to 1999. We employ models of the dependence structure (copula) that allow for greater dependence during bear markets than bull markets. We find evidence that a portfolio based on a model that captures asymmetries in the data significantly outperforms a portfolio based on the bivariate normal distribution, for various performance measures and levels of relative risk aversion.