Subadditivity of Value-at-Risk for Bernoulli random variables

Necessary and sufficient conditions for the subadditivity of Value-at-Risk (V aRαV aRα) for portfolios of bonds are presented under various dependence assumptions. For sufficiently large αα, V aRαV aRα is subadditive. However, for any αα one can construct portfolios for which V aRαV aRα is superadditive.