VaR is not coherent as it can fail the subadditivity criteria.
However, for normally distributed variables, VaR is subadditive. Hence for many practical purposes we can treat VaR as subadditive and hence use it to "add across" risks/portfolios etc.
We do need to be careful however with non-normal distributions, especially very low likelihood/high impact events. For example if a single McDonalds has a 1 in a 10,000 chance of burning down, the risk would be efectively ignored for a single restaurant under even a 99.9% VaR, and would not figure if we simply summed across the VaRs for the restaurants. However with over 30,000 restaurants the whole portfolio VaR would include this risk, so the portfolio VaR is greater than the sum of the individual VaRs.
Last edited: Feb 11, 2015