I took a historical dataset of ~2600 stocks and computed the 30-day returns for non-overlapping windows, for a 9 year period. For the returns, I computed the mean and variance. Then I plotted the mean vs the variance:

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I was rather surprised to see that there seems to be a negative correlation between return and variance. I was always in the belief that investments with higher risks should yield a higher expected return, otherwise no rational buyer would go with that option. I was wondering whether this could be correct at all and if so, what would be the explanation and why would anyone consider buying stocks from the lower right side of the plot.

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    $\begingroup$ What exactly do you plot here? The average of the average 30 day returns and variances for each stock (2600 data points)? Or each 30day value for each stock? $\endgroup$
    – AKdemy
    Commented Mar 21, 2023 at 15:50
  • $\begingroup$ @AKdemy, for each stock I take 30 days non-overlapping windows over which I compute the return. Then for each stock I compute the average return over all windows and the variance over all windows. I plot that average vs that variance. Each dot represents one individual stock. $\endgroup$
    – Botond
    Commented Mar 21, 2023 at 16:06
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    $\begingroup$ Botond: the idea that higher risk (which might or might not be proxied by high variance) stocks must have higher returns is sound in theory, but may fail empirically for very high risk (so-called lottery stocks that appeal to gambler type investors). It is a controversial issue, however, not everyone agrees. $\endgroup$
    – nbbo2
    Commented Mar 21, 2023 at 16:41
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    $\begingroup$ I don't think this question is a duplicate. Nor is it concerned with the failure with the CAPM directly. Echoing @nbbo2, I've seen work arguing that volatile stocks load on the (negative) variance risk premium, so you'd actually expect a lower expected return. There's a massive literature on whether volatility is priced, and if you which type of volatility. Also note that's econometrically not straight-forward because of all the measurement errors. $\endgroup$
    – Kevin
    Commented Mar 21, 2023 at 16:50
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    $\begingroup$ @Botond, thanks, this is good to know. $\endgroup$ Commented Mar 22, 2023 at 8:58