Interpreting Drawdowns Relative to Annualized Volatility

Summary

The document addresses whether a drawdown observed over several years should be compared directly with annualized volatility or first rescaled to an annual horizon. It explains that maximum drawdown accumulates over time, so a multi-year drawdown should not be interpreted as though it were a one-year move. For a driftless Brownian motion, the cited relationship makes expected maximum drawdown proportional to volatility times the square root of elapsed time.

Applying that relationship to the example, the accepted answer infers volatility from the observed drawdown and reports a value higher than the stated annualized volatility. It also cautions that fat-tailed return distributions can produce larger maximum drawdowns than a volatility-based Brownian model predicts. The calculation is model-dependent: it assumes a driftless Brownian setting, and the answer notes that applications may use volatility, drawdown, or their joint information depending on the purpose.

Key ideas

• A multi-year drawdown should be assessed over its full time horizon.
• Under a driftless Brownian model, expected maximum drawdown scales with volatility and the square root of time.
• Drawdown-based volatility estimates can differ from estimates based on observed volatility.
• Fat tails can make maximum drawdowns larger than the Brownian model predicts.

Tags

# Sigma moves - annualize return or no?


# Sigma moves - annualize return or no?












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?

## Answer by onlyvix.blogspot.com (score 3, accepted)

https://quant.stackexchange.com/a/40089

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.