Alternating Returns and the Approximation for Volatility Drag

Summary

The document connects volatility drag to the compounding effect of a gain followed by a loss of equal magnitude. If daily returns alternate between positive and negative moves of size equal to daily volatility, their two-day compounded return is negative, even though the arithmetic average of the two returns is zero. This illustrates why arithmetic average returns can overstate compounded growth.

Taking the per-day equivalent of that two-day loss yields a geometric return of approximately minus one-half the return variance, matching the familiar small-variance estimate for the gap between arithmetic and geometric mean returns. The approximation comes from a first-order expansion of the square root and is reliable when variance is small relative to one. The answer offers a simple illustrative link, not a general mathematical identity between drawdowns and variance; it does not model arbitrary return paths or establish how often alternating moves occur.

Key ideas

• Equal-sized up and down returns compound to a loss over the two-period sequence.
• The resulting per-period geometric return is approximately negative one-half the return variance.
• The approximation uses a first-order expansion and requires variance to be small relative to one.
• The example illustrates compounding drag but does not establish a general formula relating every drawdown to variance.

Tags

# Volatility Tax/Variance Drag and Drawdowns/Breakevens


# 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?

## Answer by nbbo2 (score 2)

https://quant.stackexchange.com/a/77784

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

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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