VaR is not subadditivity, yet there is formula for diversified VaR?

[Guannn]

I have learned that one of the reasons why Expected shortfall is a better measurement of risk is because Expected shortfall is subadditivity while VaR is not.
Yet, there is formula for calculating diversified VaR for portfolio (I believe the formula is in the book of investment risk).
Wouldn't that be self-contradictory?

 

[David Harper CFA FRM]

Hi @Guannn the diversified portfolio VaR assumes the classic mean-variance framework (aka, MPT) which assumes returns are normally distributed (why? because it only cares about means, variances and correlations; not skew or kurtosis). The normal distribution is subadditive: mixing assets with normal returns rewards diversification benefits. Hence parametric normal VaR is subadditive. However, non-normal returns, especially heavy tailed returns, are likely to lack subadditivity.

Put another way, VaR is subadditive in our convenient academic setting (where we tend to assume normal returns) but is likely to lack this quality when we most need it (realistically where returns tend to have heavy tails)! In this way, VaR may or may not be subadditive. To quality a risk measure as coherent, the measure must meet all four criteria (including subadditivity) and it must meet them for all distributions (in all cases). If we can find an example where the measure's outcome is not subadditive, the measure is not subadditive; it must be necessarily subadditive. VaR is not subadditive because it is not always subadditive. ES is always subadditive.

 

[Guannn]

Hi @David Harper CFA FRM. Thank you for your explanation. That really helps solving my problem.