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: