On Portfolio VAR (FRM handbook 5th edition, Example 17.3, page 407)

[Liming]

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

2) On Portfolio VAR (FRM handbook 5th edition, Example 17.3, page 407)

A relative value hedge fund manager holds a long position in Asset A and a short position in Asset B of roughly equal principal amounts. Asset A currently has a correlation with Asset B of 0.97. Te risk manager decides to overwrite this correlation assumption in the variance-covariance based VAR model to a level of 0.30. What effect will this change have on the resulting VAR measure?
a. It increases VAR
b. It decreases VAR
c. It has no effect on VAR, but changes profit or loss of strategy
d. Do not have enough information to answer.
Answer provided: a: It increases VAR

My question: I’m wondering if the portfolio VAR formula for a two-asset portfolio should be adjusted in this scenario here. The one we are used to: VAR(portfolio) = Va^2 + Vb^2+2Cov(Va, Vb) is only suitable for such a portfolio that consists of two long assets or two short assets; and in the context of long and short assets, the VAR formula should be adjusted to : VAR(portfolio) = Va^2 + Vb^2-2Cov(Va, Vb), due to the long and short “netting” effect. Is this a generalization I can make for all similar cases?

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

 

[David Harper CFA FRM]

Hi Liming,

Please note I just answered similar query here: http://forum.bionicturtle.com/viewthread/2237/
and, btw, this XLS is good for "proofs:" http://www.bionicturtle.com/premium/spreadsheet/8.b.1_correlated_var/

if we use the variance (not VaR) formula:
weight(A)^2*sigma(A)^2 + weight(B)^2*sigma(B)^2 + 2*weight(A)*weight(B)*COV(A,B)

note an elegance of the formula, it already handles short position if we enter the short with negative weight.
say position(B) is a short position, then weight(B) = -30% (e.g.):
weight(A)^2*sigma(A)^2 + weight(B)^2*sigma(B)^2: remains positive due to squaring
2*weight(A)*weight(B)*COV(A,B): negative (assuming positive correlation)

so if weight(B) is negative, then
weight(A)^2*sigma(A)^2 + weight(B)^2*sigma(B)^2 + 2*weight(A)*weight(B)*COV(A,B)
will subtract the third term anyway!

In regard to: VAR(portfolio) = Va^2 + Vb^2-2Cov(Va, Vb)
maybe the notion is throwing me ...?

from 8b page 9, I have (if we are using VaR directly)
VaR(portfolio) = SQRT (VaR(A)^2 + VaR(B)^2 + 2* VaR(A)*VaR(B)*correlation)
i.e., i cannot seem to reconcile either of mine with yours?

...but note this also works with a short position, as long as we input the short with a -VaR.

David