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?

  • $\begingroup$ Simple is better sometimes, isn't it? The identity can't lie. Moreover, how would you control for the identity? Volatility seems to be an intrinsic property of both investments and geometric means. $\endgroup$ Jul 17, 2012 at 22:38
  • $\begingroup$ @EduardoSahione: Have a look at the article from Falkenstein. You could for example test the CAPM for different timeframes and mean bases and see whether and when it holds. You could do the same tests for low vol strategies. $\endgroup$
    – vonjd
    Jul 18, 2012 at 6:06
  • $\begingroup$ Do you mean long-term? Cuz volatility for higher-returning stocks in the short-term is way higher. $\endgroup$
    – user3232
    Dec 1, 2012 at 18:28

1 Answer 1


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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