I am currently analyzing the idiosyncratic volatility (IVOL) puzzle. (Ang, Hodrick, Xing, & Zhang (2006) found that idiosyncratic volatility (IVOL) and next-month cross-sectional returns are negatively related).

I find in a single sort that the higher the IVOL the lower the return (as expected).

Additionally I do a double sort, first sorting on skew and then on IVOL. I find that after controlling for skew the IVOL puzzle disappears. After that I do another double sort, first sorting on return of the previous month and then on IVOL. Here also, the IVOL puzzle disappears.

Now I am wondering if it is the lottery preferences (investors seeking highly skewed stocks) that explain the IVOL puzzle or the return reversal?

Would it be now correct to do a Fama/MacBeth Regression $ret \sim IVOL + RET_{t-1}$? The result of this approach is, that only $RET_{t-1}$ has a significantly negative influence on returns.

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    $\begingroup$ What's the economic behavior ultimately behind any of the observed variation in the cross-section of expected returns? We have ideas and evidence, but these questions don't have clear, settled answers. $\endgroup$ Commented Sep 19, 2018 at 14:46

1 Answer 1



The empirical finding of a strong negative cross-sectional relation between idiosyncratic volatility and future stock returns is highly inconsistent with the predictions of all theoretical models and therefore known as the idiosyncratic volatility (IVOL) puzzle. This phenomenon is also persuasive in international stock markets, as shown in Ang et al. (2009).

The IVOL puzzle is very strong when using value-weighted portfolio analysis and substantially weaker and, depending on the measure of IVOL used (underlying risk-model, regression time-horizon, etc.), nonexistent, in equal-weighted portfolios.

What drives the IVOL puzzle?

This is answered by the papers of Bali et al. (2011) for the US and Walkshäusl (2014) for non-US markets with regard to the variable MAX, which is the maximum daily return over the previous month of a stock.

Bali et al. (2011):

Motivated by existing evidence of a preference among investors for assets with lottery-like payoffs and that many investors are poorly diversified, we investigate the significance of extreme positive returns in the cross-sectional pricing of stocks. [...] Of particular interest, including MAX reverses the puzzling negative relation between returns and idiosyncratic volatility recently shown in Ang et al., 2006, Ang et al., 2009.

Walkshäusl (2014):

[...] Once it is controlled for MAX in the cross-section of average returns, the puzzling negative idiosyncratic volatility-return relation disappears. Consistent with the assumption that MAX is the true effect, for which idiosyncratic volatility is just a proxy, we find that MAX can be traced back to firm fundamentals in the manner of idiosyncratic volatility.

How about the skewness of a stock?

The link between the above mentioned variable MAX and the idiosyncratic skewness of a stock is quite mechanical: The higher the maximum return of a stock in the previous month, the higher the idio. skewness of this stock. So your findings should not be surprising, as the correlation of MAX and idio. skewness of a stock is very high.

You are right, that lottery-like stocks drive the IVOL puzzle. Kumar (2009) uses demographic data to demonstrate that investors who are more likely to play the lottery are also more likely to invest in lottery-like stocks, where lottery-like stocks are defined as stocks with los prices (below \$5) whose returns exhibit high idio. volatility and high idio. skewness.

Fama/MacBeth regression

Your specification is right in general, but for robustness, you have to consider much more specifications!

The equal-weighted cross-sectional regression in table 4 in Walkshäusl (2014) reports for the specification $ret \sim IVOL + RET_{t-1}$ (with $RET_{t-1}$ as the short-term reversal variable REV) a coefficient for IVOL (t-stat in parentheses) with a value of -0.687 (-2.26). However, if one considers a full set of common risk variables (beta, size, book-to-market ratio, momentum, illiquidity, etc.), you observe regression coefficients for MAX of -0.09 (-6.32), skewness of 0.006 (0.25) and IVOL of 0.862 (2.29). For value-weighted portfolios, the coefficient for IVOL as single variable is -1.243 (-3.25) and -0.205 (-0.40) for the full specification, whereas the coefficient for MAX in the full specification is -0.068 (-4.16).

  • The IVOL puzzle disappears and becomes statistically insignificant in value weighted portfolio analysis after controlling for the variable MAX.

  • The IVOL puzzle disappears and also reverses - both statistically significant - in equal weighted portfolio analysis after controlling for the variable MAX.


Ang et al. (2006), The cross-section of volatility and expected returns, Journal of Finance 61(1).

Ang et al. (2009), High idiosyncratic volatility and low returns: International and further U.S. evidence, Journal of Financial Economics 91(1).

Bali/Engle/Murray (2016), Empirical asset pricing: the cross section of stock returns, John Wiley & Sons, 1. ed.

Bali et al. (2011), Maxing out: Stocks as lotteries and the cross-section of expected returns, Journal of Financial Economics 99(2).

Kumar (2009), Who gambles in the stock market?, Journal of Finance, 64(4).

Walkshäusl (2014), The MAX effect: European evidence, Journal of Banking & Finance 42.


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