This paper focuses on portfolio risk forecasting in an asymmetric framework. Risk is defined by two factors: the dependence structure and the volatility. In order to account for asymmetric dependencies, the return series' interdependence is estimated via a copula approach rather than the correlation matrix. This allows us to capture tightening dependence during periods of market turmoil and loose dependence during positive markets. In order to account for heteroscedasticity, volatility levels are forecasted via an EGARCH model. The combination of both methodologies allows precise estimates of the return series' joint distribution and their joint risk, respectively. Statistical analyses find that this model quantifies the distribution tails very accurately, resulting in precise value-at-risk estimates. We also construct minimum-risk portfolios on the basis of the volatility forecasts. The allocation procedures shift the portfolio shares according to predicted volatility levels. We find very low portfolio volatility levels and small downside risks, which emphasize the accuracy of our model's volatility forecasts.