When Square-Root-of-Time Scaling Applies to VaR
Summary
The document examines the common practice of converting one-day value at risk to a ten-day estimate by multiplying by the square root of the horizon. It connects this scaling to variance growing in proportion to time, which requires returns with suitable independence and distributional properties. One answer states a lognormal-return and no-autocorrelation assumption; another argues that independent, identically distributed returns can suffice for variance scaling without requiring lognormality.
The discussion also cautions that financial returns may be heteroskedastic and non-independent, so the square-root rule can be inaccurate in practice. It offers no statistical test procedure or empirical analysis to quantify that error, despite the question asking how to test the assumptions. The answers differ in how they describe the distributional requirements, and the brief final response questions whether a one-day distribution can determine a ten-day one. Treat the exchange as a prompt to examine dependence and changing volatility in the data, rather than as a complete regulatory method or a settled account of VaR scaling.
Key ideas
- Square-root-of-time scaling relies on variance increasing proportionally with the horizon.
- Independence and stable return behavior matter when extending one-day risk estimates across multiple days.
- Heteroskedasticity and dependence can make the scaling inaccurate in real markets.
- The discussion raises the need for statistical checks but does not specify a testing procedure.
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Full text
# 1 day VaR vs 10 day VaR
# 1 day VaR vs 10 day VaR
Even while using historical simulation VaR, 1 day VaR is converted into 10 day VaR by multiplying 1 day VaR by Sqrt(10) for regulatory reporting purposes.
What are the underlying assumptions for doing this and how can those assumptions be tested statistically?
## Answer by amdopt (score 7, accepted)
https://quant.stackexchange.com/a/45147
> What are the underlying assumptions for doing this
Assumption: Historical returns are lognormally distributed with no autocorrelation.
> can those assumptions be tested statistically
Testing: $\sqrt{xy} = \sqrt{x} \sqrt{y}$
Substitute time $t$ and variance $\sigma^2$ for $x$ and $y$ respectively
$\sqrt{t\sigma^2} = \sqrt{t} \sqrt{\sigma^2} = \sigma\sqrt{t}$
Some links for you to check out if you would like to investigate further:
https://eprints.lse.ac.uk/24827/1/dp439.pdf
Square root of time
https://www.investopedia.com/articles/04/101304.asp
## Answer by Chris (score 3)
https://quant.stackexchange.com/a/45153
Practically, I can tell you the sqare root assumption doesn't actually hold in practice--vol is not actually homoskedastic as a result of underlying returns not being iid (the scale tends to fall just short of the square of 12 in equities as a result of heterskedasticity).
A quick google turned up this, which seems to walk through precisely what you're asking about. Would probably be as good as any place to start.
## Answer by Woodpecker (score 0)
https://quant.stackexchange.com/a/78636
Do we actually need lognormal returns as amdopt states? As long as returns are i.i.d., we have $\textrm{E}(r_tr_{t+1})=0$ and as a result $\textrm{Variance}(\sum_1^{10}r_t)=10\textrm{Variance}(r_t)$, so the VaR which is the threshold for a left tail weight of (say) $\alpha$ is scaled by $\sqrt{10}$.
## Answer by Zaur (score -3)
https://quant.stackexchange.com/a/78629
one-day VaR cannot be converted into ten-day VaR, as Z ~ N(r, sigma) should provide a different distribution over one-day, limited time horizon.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.