Why Arithmetic-Geometric Return Differences Do Not Explain the Low-Volatility Anomaly

Summary

The document asks whether the low-volatility anomaly—the empirical pattern that lower-volatility stocks can earn higher returns than riskier stocks—could be explained by the difference between arithmetic and geometric average returns. It points to the relationship between these averages and variance, which implies that volatility reduces compounded growth relative to arithmetic average return, and asks whether studies control for this effect.

The response says this does not explain the anomaly. It argues that statistical tests of the low-beta effect use models based on arithmetic mean returns, with the volatility adjustment accounted for in the relationship between arithmetic and geometric means. Thus the compounding penalty alone cannot account for the observed difference. The exchange offers a concise conceptual answer, but provides no study citations, data, test design, or detailed treatment of the distinction between low-beta and low-volatility portfolios; it is not a full review of competing explanations.

Key ideas

• The low-volatility anomaly describes higher returns among lower-volatility stocks than the standard risk-return intuition predicts.
• Geometric average returns reflect a volatility-related compounding penalty relative to arithmetic averages.
• The response states that anomaly tests use arithmetic mean returns.
• The arithmetic-geometric return relationship therefore does not, by itself, explain the low-beta anomaly.
• The discussion gives no empirical details or comparison of alternative explanations.

Tags

# Is Arithmetic Return Bias Basis of Low Vol Anomaly?


# Is Arithmetic Return Bias Basis of Low Vol Anomaly?












An observation in capital markets is that the connection between return and risk (measured as volatility) is not that straightforward (at least not as modern portfolio theory assumes). One interesting instance is the so called low-volatility anomaly:

It turns out empirically that stocks that exhibit low volatility show higher returns than high-volatility stocks.

I stumbled upon some articles which try to explain this anomaly away with the simple relationship between geometric and arithmetic means with continuous compounding:$$GM=AM-\frac{\sigma^2}{2},$$ see e.g. here and here.

My question Can it be that easy? It looks almost insultingly simple to use this well known identity as the basis for the anomaly (which wouldn't be an anomaly after all). Do you know of any low-vol studies that control for that effect?

## Answer by Bryce (score 2, accepted)

https://quant.stackexchange.com/a/4827

No, the "low-beta" anomaly is not the result of the difference between arithmetic and geometric mean returns.

Statistical tests verifying the existence of the anomaly rely on models employing the arithmetic mean returns, $$\mu_a = \mu_g + \frac{\sigma^2}{2}$$, hence the penalty excess volatility incurs when compounding returns over time does not explain the difference.

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