VaR: subadditive

[Kavita.bhangdia]

Hi All,

We know that VaR is not a coherent measure of risk because it is not SUBADDITIVE.

In P2.T8. Jorion chapter 7, David mentions that

"Note that portfolio risk must be lower than the sum of individual i.e
VaR(p) <VaR1 + VaR2.

so are we assuming here that the VaR is subadditive for simplification?

And if VaR is subadditive, then the true representation should be

VaR(p) <= VaR1+VaR2.

Please advice.

Thanks,
Kavita

 

[ShaktiRathore]

Hi,
please see: https://forum.bionicturtle.com/threads/var-not-subadditive-coherent-risk-measure.7117/
thanks

 

[David Harper CFA FRM]

Hi @Kavita.bhangdia It is true that analytical portfolio VaR in Jorion's Chapter 7 is sub-additive; i.e., under those conditions, VaR (a+b) ≤ VaR(a) + VaR(b). But he's using a delta-normal model such that all returns are assumed normally distributed (this is the consequence of "stopping" at the second moment, the covariance matrix, you are implicitly assuming normality anyway). When we say "VaR is not sub-additive" we mean "VaR is not necessarily sub-additive." Here is Dowd:

"We can only ‘make’ the VaR subadditive if we impose restrictions on the form of the P/L distribution. It turns out, in fact, that we can only ‘make’ the VaR subadditive by imposing the severe restriction that the P/L distribution is elliptically distributed [ and this is of limited consolation because in the real world non-elliptical distributions are the norm rather than the exception.." --- Dowd page 34

please not that normal distributions are elliptical, as elliptical generalizes mulitivariate normal https://en.wikipedia.org/wiki/Elliptical_distribution