Fat tails, VaR and subadditivity

Financial institutions rely heavily on Value-at-Risk (VaR) as a risk measure, even though it is not subadditive. First, we theoretically show that the VaR portfolio measure is subadditive in the relevant tail region if asset returns are multivariate regularly varying, thus allowing for dependent returns. Second, we note that VaR estimated from historical simulations may lead to violations of subadditivity. This upset of the theoretical VaR subadditivity in the tail arises because the coarseness of the empirical distribution can affect the apparent fatness of the tails. Finally, we document a dramatic reduction in the frequency of subadditivity violations, by using semi--parametric extreme value techniques for VaR estimation instead of historical simulations.