VaR (X) + VaR (Y) + VaR (Z),
and that the three portfolios are subsequently merged into one by the method
VaR (X + Y + Z).
Of course, coherence would require that
VaR (X + Y + Z) = VaR (X) + VaR (Y) + VaR (Z).
However, VaR is neither subadditive nor distributive, and depending on the composition of portfolios X, Y, and Z, the post-merger VaR can exceed the summation of VaR for the segregated portfolios, resulting in
VaR (X + Y + Z) ≥ VaR (X) + VaR (Y) + VaR (Z).
The point of this exercise is to caution that merging portfolios can increase VaR. Yet, our nation’s financial regulators and leadership continue to defend and justify such capital formations (e.g., bank mergers) as necessary and expedient within the rubric of "globalization," "too big to fail," or some other convolution of fear mongering. Given the risk calculus above, I maintain that financial institutions that become "too big to fail" are likewise too risky to keep around. Bank mergers and acquisitions that expand VaR make no sense in today's business climate.