Among the more notable anomalies in modern finance is the finding that the lowest-beta stocks have produced higher returns than the highest-beta stocks. Another anomaly is that idiosyncratic (diversifiable) volatility negatively predicts equity returns. In other words, stocks with the lowest idiosyncratic volatility outperform stocks with the highest idiosyncratic volatility.

These findings have spurred a large body of literature on what are referred to as “low-risk anomalies.” Such results are considered puzzling because higher risk should be rewarded with higher returns, but here we see just the opposite.

Paul Schneider, Christian Wagner and Josef Zechner—authors of the April 2015 paper “Low Risk Anomalies?”—add to our understanding of these anomalies by investigating the link between them and a higher moment of the return distribution, the skewness of returns. This is a link that standard measures of market risk and volatility ignore. We’ll begin with a definition.

**Defining Skewness**

Skewness measures the asymmetry of a distribution. In terms of the market, the historical pattern of returns does not resemble a normal distribution (also known as the familiar “bell curve”) and so demonstrates skewness. Negative skewness occurs when the values to the left of (less than) the mean are fewer but farther from it than values to the right of (greater than) the mean.

For example, the return series of -30%, 5%, 10% and 15% has a mean of 0%. There is only one return less than zero, and three that are higher. The single negative return is much farther from zero than the positive ones, so the return series has negative skewness. Positive skewness, on the other hand, occurs when values to the right of (greater than) the mean are fewer but farther from it than values to the left of (less than) the mean.

The question Schneider, Wagner and Zechner sought to answer is: Is there a link between skewness and returns? Said another way, do stocks with more negative skewness produce higher returns? Their theory is that investors require comparably lower (higher) expected equity returns for firms that are less (more) coskewed with the market.

Coskewness is a measure of the symmetry of a variable’s probability distribution in relation to the symmetry of another variable’s probability distribution. It’s calculated using a security’s historic price data as the first variable, and the market’s historic price data as the second. This provides an estimate of the security’s risk in relation to market risk.

All else equal, a positive coskewness means that the first variable’s probability distribution is skewed to the right of the second variable’s distribution. Investors prefer positive coskewness because this represents a higher probability for extreme positive returns in the security over market returns.

**The Study And Its Results**

The study’s database included about 5,000 U.S. firms for the period January 1996 to August 2014 and covered all CRSP firms for which data on common stocks and equity options was available.

Employing the options data, the authors computed ex-ante skewness from an options portfolio that took long positions in out-of-the-money (OTM) call options and short positions in OTM puts. This measure becomes more negative the more expensive put options become relative to call options (investors are willing to pay high premiums for protection against downside risk). Equity options are more expensive for firms with high compared with low credit risk. Thus, credit risk acts as a natural source of skewness.

Following is a summary of their findings:

- Corporate credit risk generates time-varying skewness in a firm’s equity returns, which in turn impacts the pricing of its stock. And credit risk matters for the shape of a firm’s equity return distribution—equity is an option on the underlying assets and, hence, its value can drop to zero.
- Ex-ante skew predicts the cross section of equity returns, and that conditioning on ex-ante skew affects the prevalence of low-risk anomalies.
- Firms with high (low) ex-ante skewness have the most (least) negative coskewness as measured by the covariance of their returns with the market’s return.
- The profitability of betting-against-beta strategies (buying low-beta stocks and shorting high-beta stocks) increases with firms’ downside risk.
- While betting against beta or volatility is profitable for firms with high downside risk, it generates losses among firms with less negative or positive ex-ante skewness.
- The return differential of implementing a long-high-skew-stocks and short-low-skew- stocks approach is economically large and most pronounced among firms with the most negative equity ex-ante skewness.
- The highest-skew firms generate a monthly four-factor alpha (controlling for the market, size, value and momentum factors) of about 0.82%, whereas the alpha of the lowest-skew firms is -0.54% per month. The difference, 1.36 percentage points, is statistically significant (with a t-stat of 4.6 using equal weighting). Using value weighting, the highest-skew firms generate a monthly alpha of 0.82% versus -0.34 for the lowest-skew firms (the t-stat of the difference is 3.4). The equity returns are associated with an identical pattern for realized skewness, with realized skewness monotonically declining from the high to the low ex-ante skewness portfolio.
- The negative relation between equity returns and idiosyncratic volatility (computed from CAPM or Fama-French factor model residuals) as well as ex-ante variance is most pronounced among firms with the most negative equity ex-ante skewness.

**Skewness Can Explain Low-Risk Anomalies**

The authors state: “All these findings suggest that low risk anomalies may not be as anomalous as they appear at first sight.” In other words, an investor preference for positive skewness—or avoidance of negative skewness and the accompanied larger downside risk—explains the pattern of returns.

Thus, “low risk anomalies may not necessarily pose asset pricing puzzles when accounting for higher moments of the return distribution.” The authors also noted: “Our results are consistent with findings that accounting for the lottery characteristics [positive skewness and excess kurtosis] of stocks reverses the relation between idiosyncratic volatility and equity returns.”

Schneider, Wagner and Zechner concluded: “Skewness appears to be a plausible candidate to provide insights for beta- and volatility-based low risk anomalies that receive considerable attention in the recent literature.”

**More Evidence**

This conclusion is supported by the findings from a paper by Diego Amaya, Peter Christoffersen, Kris Jacobs and Aurelio Vasquez, “Does Realized Skewness Predict the Cross-Section of Equity Returns?”, which was published in the October 2015 issue of the Journal of Financial Economics. In their study, the authors examined the higher moments of volatility, skewness and kurtosis to determine if they have provided incremental explanatory power in the cross section of stock returns.

Amaya, Christoffersen, Jacobs and Vasquez write: “Firm-specific realized volatility, skewness, and kurtosis all contain unique information about the cross-sectional distribution of equity returns.”

They also state that there’s “strong evidence of a negative cross-sectional relationship between realized skewness and future stock returns—stocks with negative skewness are compensated with high future returns for higher volatility. However, as skewness increases and becomes positive, the positive relation between volatility and returns turns into a negative relation. We conclude that investors may accept low returns and high volatility because they are attracted to high positive skewness.”

In other words, investors may buy highly volatile stocks that, on average, have poor prospects because they are trying to “hit the lottery”—a rare event, but one with very high upside.

The bottom line is that we now have a significant body of evidence that investor preference for positive skewness results in poor risk-adjusted returns. Those disciplined investors who accept the negative skewness associated with higher-returning assets can build more efficient portfolios.

*Larry Swedroe is the director of research for The BAM Alliance, a community of more than 140 independent registered investment advisors throughout the country.*