Why can we scale VaR using the square root of time, and when does it fail?

Question

AcadiFi · Accepted Answer

Square-root-of-time scaling rests on three assumptions: returns are independently distributed, returns have constant volatility, and the mean is negligible over the horizon. Under these conditions the variance of an n-period return equals n times the one-period variance, so standard deviation scales with the square root. If daily 99% VaR is $1M, ten-day VaR equals $1M times sqrt(10) roughly $3.16M. The rule fails whenever returns show autocorrelation (trending markets inflate true 10-day variance), mean reversion (compresses it), volatility clustering (GARCH effects mean a quiet day is followed by another quiet day), or heavy tails that compound differently than Gaussian tails. Liquidity risk also breaks it: a one-day VaR assumes immediate exit, but a ten-day horizon during a crisis may require fire-sale discounts that do not scale with sqrt(n). Regulators historically accepted sqrt-time scaling for the 10-day market-risk capital charge because it is simple, but stressed VaR and the FRTB Expected Shortfall framework abandon the shortcut in favor of directly modeled horizons. For FRM exam purposes, cite the three IID assumptions, then list autocorrelation, mean reversion, volatility clustering, and liquidity as the main failure modes.

Why can we scale VaR using the square root of time, and when does it fail?

Master Part I with our FRM Course

Related Questions

Practice Questions