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!