Sometimes I am thinking about VaR calculation and always trying to protect what I am working on. But I have to admit I feel there exists certain big drawbacks of VaR. One thing I feel kind of difficult to give the way to VaR.

VaR(A+B) > VaR(A) + VaR(B)

As a portfolio, composed by A and B investment, should be more diversed than the single investment itself. This is easy to understand, but VaR computation does not give us this theory.

Let us assume, risk of A is 1 million and the chance is 4%. And the same is to B. Well, the VaR’s cut off is set to 5% (means 95% confidence level), so VaR(A) and VaR(B) both are zero. Then let us see A+B. The chance is 1 - (1-4%)(1-4%) = 7.84%, so the VaR(A+B) is 2 million. Well, VaR(A+B) >> VaR(A) + VaR(B).

Subset of VaR’s summerization won’t be maintained. So when we do the portfolio VaR computation, we have to be very careful on this. A supplementary stress test will be quite necessary. And the case always happened as the extreme time and easy to be ignored since it is rare.