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been reading about Drawdowns and respective returns to get back to breakeven as shown below:

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Many cite this as an evidence of the well publicized Vol Tax Formula (Geo Mean = Arithmetic Mean - 0.5*Variance)

While this makes sense intuitively since higher drawdowns tend to lead to a high variance which causes a further separation between the geometric and arithmetic mean.

Was wondering if theres a more explicit link between the two which can be shown mathematically?

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Assume the standard deviation of daily returns is $\sigma$. If the market is up the typical amount one day and down the typical amount next day, the 2 day return is $(1+\sigma)(1-\sigma)-1= -\sigma^2$. The one day return is $\sqrt{1-\sigma^2}-1$ or approximately $-\frac{1}{2}\sigma^2$, which is the estimate of volatility drag you most commonly see. It is a valid approximation of the square root provided $\sigma^2$ is sufficiently small compared to 1 (we are using $\sqrt{1\pm \epsilon}\approx 1\pm \frac{1}{2}\epsilon$ ).

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