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.