Value-at-Risk is not subadditive

Value-at-Risk violates subadditivity: there are positions whose combined VaR exceeds the sum of their individual VaRs, so under VaR the pooling of positions can appear to increase risk. A witness on 25 states: X loses 100 in state 0 alone and Y loses 100 in state 1 alone, each a single losing state in twenty-five, so at tolerance k=1 both have VaR 0; their sum loses in two states, exceeding the tolerance, giving VaR 100. Note that VaR here counts STATES and refers to no probability measure, per its own definition -- the state count 25 is not load-bearing (two states suffice) and reading k=1 as a 4% tail is an interpretation the formal statement does not make. This is the failure that moved the Basel market-risk standard from Value-at-Risk to Expected Shortfall.