Scaling Parametric Value at Risk Across Time Horizons
Summary
The document clarifies whether a parametric value-at-risk estimate should be scaled by changing asset standard deviations or by scaling the final VaR. Under a Gaussian model with zero mean, VaR is a quantile multiplier times the portfolio standard deviation. If returns over successive days are independent and variance grows proportionally with time, the portfolio variance grows with the horizon, so the standard deviation and VaR scale by the square root of that horizon.
The answer corrects a variance-versus-standard-deviation mix-up in the question and shows why scaling individual standard deviations consistently also scales covariance terms when correlations are unchanged. It cautions that this equivalence depends on the Gaussian and time-scaling assumptions. In non-Gaussian models, a quantile need not vary linearly with standard deviation, and the multi-day return distribution may need to be modeled directly rather than inferred by mechanically scaling a one-day estimate.
Key ideas
- Gaussian parametric VaR is a quantile multiplier times portfolio standard deviation.
- If variance grows linearly with time, standard deviation and VaR grow with the square root of the horizon.
- Scaling each asset’s standard deviation also scales covariance consistently when correlations are held fixed.
- The equivalence may fail when returns are not Gaussian or the horizon distribution is not captured by the scaling assumptions.
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Full text
# Is scaling the standard deviations in the VaR formula (parametric) equivalent to scaling the VaR figure at the end? # Is scaling the standard deviations in the VaR formula (parametric) equivalent to scaling the VaR figure at the end? 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 by piterbarg (score 2) https://quant.stackexchange.com/a/61782 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?
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