Volatility has evolved as an attractive new asset class of its own. The most common instruments for trading volatility are variance swaps. Mean returns of DAX and ESX variance swaps over the time period of 1995 to 2004 are strongly negative, and only part of the negative premium can be explained by the negative correlation of variance swap returns with stock market indices. We analyze the implications of this observation for optimal portfolio composition. Mean-variance efficient portfolios are characterized by sizable short positions in variance swaps. Typically, the stock index is also sold short to achieve a better portfolio diversification. To capture heterogeneous preferences for higher moments, we use a variant of the polynomial goal programming method. We assume that investors strive for a high Sharpe ratio, high skewness, and low kurtosis. Our analysis reveals that it is often not possible to achieve a balanced tradeoff between Sharpe ratio and skewness. Investors are advised to hold the extreme portfolios (Sharpe ratio driven, skewness driven, or kurtosis driven) and avoid the middle ground. This “all-or-nothing” characteristic is reflected in jumps of asset weights when certain thresholds of preference parameters are crossed. These empirical findings can explain why many investors are so reluctant to implement option-based short-selling strategies.