# Square root of time

I am writing about VaR and I am wondering about the following: We can scale the VaR to different time horizons by using the square root of time, which means, that the volatility is adjusted by square root of the time horizon. So e.g. we have the daily volatility then the weekly volatility (for 5 trading days) is given by

$\sqrt{5}*$ daily volatility

Now my question is the following:

Does this hold only for log returns or also for simple returns?

I googled it but I could not find a proof for it. So where can I find a proof for this in terms of log returns and either also a proof for the case of simple returns or an estimation of the error I will be doing, if I use the square root of time while using simple returns?

And finally considering the VaR: Does this need the normal distribution as an assumption?

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It doesn't hold for simple returns, i.e. $S_{t_2} - S_{t_1}$ is not normally distributed under assumption that $S$ is a Geometric Brownian Motion. –  Alexey Kalmykov Mar 11 '13 at 17:00

Scaling volatility as you do is often leading to inaccurate results which is over-estimating volatility especially when you scale daily volatility to even longer periods. Please see the following for more:

http://economics.sas.upenn.edu/~fdiebold/papers/paper18/dsi.pdf

The above paper also explains why scaling the way you did does not properly account for the correct volatility when returns are not normally distributed/ prices are not log-normally distributed. But I like the following explanation better:

Scaling volatility as you did it only is mathematically correct when the underlying price model is driven by Geometric Brownian motion which implies that prices are log normally distributed and returns are normally distributed. The reason for that is that the driving Brownian motion accumulates variation at rate one per unit of time. The proof of that is pretty well published but my favorite source is Steven Shreve, Stochastic Calculus for Finance II, page 101-107 (2004 edition). So, volatility scales with the square root of time only when the underlying process is driven by a geometric Brownian motion.

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Who is less wrong? Using 1% daily returns, over h = 10 days. 2.99% Goldstein/Taleb, 3.16% Dibold/Hickman/Inoue/Schuermann. We Don't Quite Know What We are Talking About When We Talk About Volatility papers.ssrn.com/sol3/papers.cfm?abstract_id=970480 –  montyhall Mar 11 '13 at 16:07
@montyhall, its a well known fact that volatility is overestimated when scaled over long periods of time without a change of model to estimate such "long-term" volatility. Its a basic question in many quant interviews (Falcon Crack et.al.), its a well known fact among market practitioners (vol traders) and its nothing that Taleb contradicts. –  Matt Wolf Mar 11 '13 at 16:16
“without a change of model”, I agree, and a simple example would be to change your scale to a 10 day bar + a probability weight. “P. Samuelson and it is the fact that the best forecast for the future price of an asset is its present price”. Analysis of short term price trends in daily stock-market index data. arxiv.org/pdf/1211.3060v1.pdf –  montyhall Mar 11 '13 at 16:59
Well, I guess you should then stick to a career as analyst, no need to run complicated models, just current price tells you everything. Traders think slightly different and fortunately vol trading is just a little more complex than you suggested. This is the second time you completely move off topic here. Please offer your own answer if you have insight that differs from mine unless it is directly related to my answer. –  Matt Wolf Mar 11 '13 at 17:04

for the square-root rule: it holds for log-returns, if you assume the same variance and no autocorrelation. Because then: $$Var[r_1 + \cdots + r_d] = Var[r_1] + \cdots + Var[r_d] = d Var[r_1]$$ and thus $$\sqrt{Var[r_1 + \cdots + r_d] } = \sqrt{d} \sqrt{Var[r_1]}.$$ This is mathematically true for any distribution that fulfills the assumptions. For the case with autocorrelations you can look here: Scaling portfolio volatility and calculating risk contributions in the presence of serial cross-correlations.

Of course this holds in the ideal world of mathematics but it is at least a benchmark for anything better and more realistic.

Concerning VaR: VaR is a quantile as you can read on wikipedia. Very often a normal distribution is assumed but this is not the only possibility. Another one is a t-distribution, just to name one. This book deals a lot with VaR.

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