The goal of integrated risk management in a financial institution is to measure and manage risk and capital across a range of diverse business activities. This requires an approach for aggregating risk types (market, credit, and operational) whose distributional shapes vary considerably. In this paper, we use the method of copulas to construct the joint risk distribution for a typical large, internationally active bank. This technique allows us to incorporate realistic marginal distributions that capture some of the essential empirical features of these risks - such as skewness and fat tails - while allowing for a rich dependence structure. We explore the impact of business mix and inter-risk correlations on total risk, whether measured by value at risk or expected shortfall. We find that given a risk type, total risk is more sensitive to differences in business mix or risk weights than it is to differences in inter-risk correlations. A complex relationship between volatility and fat tails exists in determining the total risk: whether they offset or reinforce each other will depend on the setting. The choice of copula (normal versus student-t), which determines the level of tail dependence, has a more modest effect on risk. We then compare the copula-based method with several conventional approaches to computing risk, each of which may be thought of as an approximation. One easily implemented approximation, which uses empirical correlations and quantile estimates, tracks the copula approach surprisingly well. In contrast, the additive approximation, which assumes no diversification benefit, typically overestimates risk by about 30 to 40 percent.