# Volatility Tax/Variance Drag and Drawdowns/Breakevens

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

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?

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 $$\sqrt{1\pm \epsilon}\approx 1\pm \frac{1}{2}\epsilon$$ ).