I ran into a question about how VAR is smaller for shorter timer periods, which makes perfect sense to me, less time for things to go wrong, no need to discuss this.
But the conversion of VAR for time periods seems to contradict this:
If I have annual VAR computed with annual mean and annual variance, then I can compute the monthly VAR by doing this:
annual mean/12=monthly mean
annual std/sqrt(12)=monthly std
Now the mean becomes smaller as it is divided by 12, but the STD does only get divided by sqrt(12), a smaller number. That is, relatively speaking, the STD decreased by less than the mean.
If I apply the computation of the 5% VAR: monthly mean- 1.65 monthly std
This number should be more negative than the annual STD. So as the time period gets smaller, the VAR should increase? Am I missing something here?
You’re right. I’ve read in Schweser that “in most cases” daily mean returns are close to 0. Which reduces your scaling problem to just the square root of N, hence VAR stays proportional. This is for example the case in the Apollo Bank case in the 2018 AM (multiple choice) mock.
Does that mean that the intuitive answer that VAR for smaller periods is smaller is wrong?
Which Mock is that again? Schweser 2018?? I could not find that example in there. They had a VAR question where the VAR became smaller for smaller periods.
In absolute sense it’s always lower since your dividing by a larger number going from e.g. annual to daily. Relatively it will stay the same in the case of daily returns being (near) zero (the difference between mean divided by N vs. standard deviation divided by square root of N will be really small). Relatively it might decrease at a decreasing rate like Benjamin mentioned if daily returns are non-zero.
I agree with the point that if the mean is close or equal to zero, this should not be a problem. That is, the smaller time periods should have smaller VARs, since going to smaller time periods won’t change the mean (0 divided by something remains zero), but the STD will become smaller.
HOWEVER, regarding my original statement, I think I made a mistake (so you guys are right about the decreasing at a decreasing rate thing):
Consider this example:
annual mean is 12, annual STD is also 12. Now the 5% VAR is: 12-1.65*12=-7.8
monthly mean is 12/12=1
monthly STD is 12/sqrt(12)=-3.46
Monthly VAR is: 1-1.65*3.46=4.79
So the smaller time period VAR does indeed become smaller, maybe at a smaller rate, but it is still smaller, which is perfectly in line with what you would intuitively expect. Thanks for the help everyone.
Akthem: Thanks for your help but I think you misunderstood my question. I know how the conversion works, I did the same thing you show up there, my question was related to the generalization of the relationship between VARs of different time periods.