Is scaling the standard deviations in the VaR formula (parametric) equivalent to scaling the VaR figure at the end?

Question

I have come across people calculating parametric VaR who scaled the standard deviations by say square root of 10 to scale up to a 10 day horizon. Elsewhere I have seen textbooks suggesting that it is the whole VaR figure which should be multiplied by the square root of 10.

Which is correct, and are both equivalent?

If we consider VaR = z-score * sqrt(portfolio variance) (assuming mean is zero)

I believe the formulas are not equivalent. Eg in a 2 asset portfolio, where SD stands for standard deviation, the portfolio variance can be written as (if we believe the standard deviations should be scaled up, rather than the overall VaR figure):

wA^2 * (SD A * sqrt 10)^2 + wB^2 * (SD B * sqrt 10)^2 + 2 cov(ab) * wA * wB

which can also be written as:

wA^2 * (SD A * sqrt 10)^2 + wB ^2 * (SD B * sqrt 10)^2 + 2 corr(ab) * (SD A * sqrt 10) * (SD B * sqrt 10) * wA * wB

is not the same as z-score * portfolio variance * sqrt 10 (as sqrt10^2 = 10 and (ab)^2 = a^2 b^2)

Which is the correct approach? Many thanks!

Answer 1

There are a couple of point here. This:

If we consider VaR = z-score * portfolio variance (assuming >mean is zero)

is not quite right. In a Gaussian model

(**) VaR = z-score * sqrt(portfolio variance) = z-score * (standard deviation of the portfolio)

So your math actually works out. And in a Gaussian model scaling VaR or scaling variances is the same thing

The other point is that this formula does not necessarily hold for models that are not Gaussian, in which case VaR (which is a given quantile of a distribution) is not linear in the standard deviation of the portfolio and scaling VaR vs scaling variances will give different results. The latter is (more) correct but in models that are more complicated than Gaussian you may have to consider the actual 10-day distribution rather than scale somehow a 1-day distribution

edit

With regards to your comment, I am not sure where the misunderstanding lies so here I am trying to spell it out:

VaR(10 day) = z_score x sqrt(variance(10 day)) = z_score x sqrt(variance(1 day)) x sqrt(10) = VaR(1 day) x sqrt(10)

Which of these steps are you not convinced about?