Diego Amaya, Peter Christoffersen, Kris Jacobs and Aurelio Vasquez, authors of the new paper, “Does Realized Skewness Predict the Cross-Section of Equity Returns?”, examined higher moments of volatility, skewness and kurtosis to determine if they have provided incremental explanatory power in the cross section of stock returns.

Before reviewing the authors’ findings, which appear in the October 2015 Journal of Financial Economics, we’ll provide some brief definitions and background.

**Skewness**

Skewness measures the asymmetry of a distribution. In terms of the market, the historical pattern of returns doesn’t resemble a normal distribution, 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 percent, 5 percent, 10 percent and 15 percent has a mean of 0 percent. 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.

Studies in behavioral finance have found that, in general, people like assets with positive skewness. This is evidenced by an investor’s willingness to accept low, or even negative, expected returns when an asset exhibits positive skewness. The classic example of positive skewness is a lottery ticket. Expected returns are -50 percent (the government only pays out about 50 percent of the sales proceeds) and the vast majority of tickets end up worthless, but investors hope to hit the big jackpot anyway. Some examples of assets that exhibit both positive skewness and poor returns are IPOs, “penny stocks,” stocks in bankruptcy and small-cap growth stocks with low profitability.

Alternatively, investors generally don’t like assets with negative skewness. High-risk asset classes (such as stocks) typically exhibit negative skewness. In addition, some investment vehicles, such as hedge funds, also exhibit negative skewness.

**Kurtosis**

Kurtosis measures the degree to which exceptional values, those much larger or much smaller than the average, occur more frequently (high kurtosis) or less frequently (low kurtosis) than in a normal (bell shaped) distribution.

High kurtosis results in exceptional values that are called “fat tails.” Fat tails indicate a higher percentage of very low and very high returns than would be expected with a normal distribution. Low kurtosis results in “thin tails” and a wide middle to the curve. In other words, more values are closer to the average than would be found in a normal distribution, and tails are thinner.

**The Study**

The authors analyzed every listed stock in the Trade and Quote (TAQ) database from January 4, 1993 through September 30, 2008. TAQ provides historical tick-by-tick data for all the stocks listed on the New York Stock Exchange, American Stock Exchange, Nasdaq National Market System and regional exchanges. Stocks with prices below $5 were excluded from the analysis. To ensure sufficient liquidity, a stock had to have at least 80 daily transactions. The average number of intraday transactions per day for a stock was more than 1,000.

The authors used data from the Center for Research and Security Prices database to obtain the daily returns of each company in order to calculate weekly returns. They used Compustat data to extract the Standard and Poor’s issuer credit ratings and book values to calculate book-to-market ratios. And from the Thomson Returns Institutional Brokers Estimate System (I/B/E/S), they obtained the number of analysts that follow each individual firm.

The authors aggregated daily realized moments to obtain weekly realized volatility, skewness and kurtosis measures for more than 2 million firm-week observations.

They then sorted stocks into deciles based on the current-week realized moment and computed the subsequent one-week return of a trading strategy that buys the portfolio of stocks with a high realized moment—volatility, skewness or kurtosis—and sells the portfolio of stocks with a low realized moment.

**Results**

Following is a summary of the authors’ findings:

- Realized volatility increases from 19 percent for the first decile to 145 percent for the highest decile.
- A positive relationship exists between realized volatility and historical skewness.
- Realized skewness has a negative relationship with realized volatility (reflecting that stocks with big drops in price are more volatile) and realized kurtosis shows an increasing pattern through the volatility deciles.
- Over time, realized volatility tends to be consistently highest for firms with small market caps, low book-to-market ratios and high market betas.
- Firms with a high degree of asymmetry, either positive or negative, are small, highly illiquid and followed by fewer analysts. In addition, the number of intraday transactions for these firms is lower.
- When sorting on realized volatility, the resulting portfolio return differences are not statistically significant.
- When sorting by realized skewness, the long (stocks with low skewness), short (stocks with high skewness) and value-weighted portfolio produced an average weekly return of 24 basis points with a t-statistic of 3.65, meaning it was highly significant.
- The Carhart four-factor (beta, size, value and momentum) risk-adjusted alpha for the long/short skewness portfolio is also 24 basis points per week, and significant at the 1 percent level—the Carhart four factors don’t fully explain returns.
- Realized skewness was highly significant in explaining the cross section of returns after controlling for all the factors the authors examined, including realized volatility and kurtosis, firm size, book-to-market ratios, market beta, historical skewness, the number of analysts that follow a firm, idiosyncratic volatility and illiquidity (as well as a few others). And it’s significant when looking only at NYSE stocks.
- When idiosyncratic volatility increases, low-skewness stocks are compensated with higher returns while high-skewness stocks are compensated with lower returns. This pattern is stronger for small stocks.
- Across deciles, the average kurtosis ranged from about 4 to roughly 17.
- There’s a positive relationship between realized kurtosis and subsequent stock returns, but the economic magnitude is smaller and the results are less statistically significant. The long/short portfolio generates a return of 14 basis points per week with a t-statistic of 2.12. The estimates for the Carhart four-factor alpha are smaller and less statistically significant compared with those for the raw returns. The value-weighted alpha for the long/short portfolio is 7 basis points, and isn’t significant at the 5 percent level.
- Realized kurtosis tends to be consistently high for small-caps, firms with high book-to-market ratios and low-beta firms. This result for low-beta firms is unexpected, but may help explain the historical returns to low-beta stocks.
- Realized volatility, historical skewness, illiquidity and maximum monthly return were among the firm characteristics positively related to realized kurtosis.
- Size, market beta, number of I/B/E/S analysts, stock price and number of intraday transactions were among the variables that had a negative relationship with realized kurtosis.

The authors concluded: “Firm-specific realized volatility, skewness, and kurtosis all contain unique information about the cross-sectional distribution of equity returns.”

They also stated 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.”

As mentioned earlier, this is consistent with previous findings in the literature that investments with lottery ticket-like distributions have poor returns (and are best avoided).

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