The bounds for risk measures of a portfolio when its components have known marginal distributions but the dependence among the risks is unknown are often too wide to be useful in practice. Moreover, availability of additional dependence information, such as knowledge of some higher‐order moments, makes the problem significantly more difficult. We show that replacing knowledge of the marginal distributions with knowledge of the mean of the portfolio does not result in significant loss of information when estimating bounds on value‐at‐risk. These results are used to assess the margin by which total capital can be underestimated when using the Solvency II or RBC capital aggregation formulas.