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This might be a very simple dumb question. But when you look at a security's annualized volatility over a 3 year period, assuming the security has an annualized vol of 5% and the drawdown over three year period is -15%. Is it fair to say that it is a 3 sigma move or should you annualize that -15% and claim its ~ 1 sigma move?

What is the appropriate way to do this?

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    $\begingroup$ Neither, it's a 1.73 sigma move. $\endgroup$ – Raskolnikov May 27 '18 at 11:50
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    $\begingroup$ @Raskolnikov how did you get 1.73? $\endgroup$ – QFqs May 27 '18 at 13:51
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    $\begingroup$ Volatilities scale as $\sqrt{t}$. Hence, when you annualize a three year volatility, you should divide by $\sqrt{3}$. So you have $15/(5\sqrt{3})=\sqrt{3}\approx 1.73$. $\endgroup$ – Raskolnikov May 27 '18 at 14:15
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    $\begingroup$ You are asking about "annualizing return" but aren't you really asking about annualizing maximum drawdown? A different thing IMHO. $\endgroup$ – Alex C May 27 '18 at 14:46
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    $\begingroup$ You have to pay attention. To be able to use the square root rule explained by @raskolnikov above, the returns need to be iid (independant and identically distributed). If they exhibit autocorrelation or mean reversion it's not correct anymore! $\endgroup$ – byouness May 30 '18 at 11:25
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This is hardly a simple dumb question. Drawdown of BM with no drift is

$ 2 \sqrt \frac{\pi}{8} \sigma \sqrt T $

see Magdon-Ismail: On the Maximum Drawdown of a Brownian Motion, Eqn. (16)

or in your case $ 0.15 \approx 1.25 \sigma \sqrt 3 $, implying $\sigma$ of 6.9% using information from the drawdown alone. We often see this in various markets - if a distribution has fat tails, it's MDD will be larger than what is predicted by volatility.

Depending on your application, you may want to use one calculation method, or another, maybe even using a joint information of observing a 5% vol and 15% drawdown, to calculate $\sigma$.

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