On Portfolio VAR (FRM 2002 exam question No. 3)

[Liming]

Dear David,
I’ve have struggling with the following question from FRM practice and past exams. Appreciate your kind help on this!

1) On Portfolio VAR (FRM 2002 exam question No. 3)

Portfolio A has a 1-day , 95% VAR of $5 million, portfolio B has a 1-day,95% VAR of $7 million. Is it possible for the 1-day, 95% VAR of the combined portfolio (A+B) to be greater than $12 million? Answer provided: This is possible but unusual. It does remind us that we usually need the whole distribution and not just a single point to aggregate VARs.

My question: I don’t understand why the combined portfolio can have VAR greater than the sum of its components. Because I thought portfolio VAR should always be smaller than the sum (when correlation is positive and less than one) or equal to one (when correlation is equal to one). I think the only possible explanation for the answer provided is that the two assets are negatively correlated, which is unusual. Is my thought correct?

Thank you for your enlightenment and correction!
Cheers
Liming
10/11/09

 

[David Harper CFA FRM]

Hi Liming,

When you say, "I thought portfolio VaR should always be smaller than the sum," this is indeed true (i) the correlation is less than 1.0 (incl. negative) and (ii) if the distribution is normal (elliptical is the superclass). In learning VaR (e.g., Jorion's diversified VaR), we are often assuming normality out of convenience, and if normal, you are correct.

However, VaR is not sub-additive (i.e., is not coherent). See http://www.bionicturtle.com/learn/article/illustration_of_vars_failure_to_meet_coherence/
This means that, under some distriubtions, VaR (A+B) > VaR(A) + VaR(B)
...this is a genuine, significant drawback of VaR

if you'd like to see an example of lack of sub-additive, we can get it with (Bernoulli) bond default.
See http://www.bionicturtle.com/premium/spreadsheet/5.d.1._expected_shortfall/
...where the 95% VaR of a single bond is 0, but the 95% VaR of a portfolio of three such bonds is $100

David