# Skewness and kurtosis relationship quotes

### Symmetry, Skewness and Kurtosis | Real Statistics Using Excel

ex-ante measures of volatility, skewness, and kurtosis derived from option prices are . we examine the relations between expected returns and the volatility, skew- .. 10The actual quote is “for the entire cross-section of stocks, these proxies. based on natural skewness functional are preferable to those related to Analogous kurtosis orderings are also discussed. Here To quote our dictionary (Web- . amples to any other inter-relationships among the orderings will appear in a. A recent study considers skewness, volatility and kurtosis relative to stock returns. The authors analyzed every listed stock in the Trade and Quote (TAQ) Realized skewness has a negative relationship with realized.

This is why this answer seems to respond to a differently worded question.

## Differences Between Skewness and Kurtosis

However much of the post should be relevant here. Within some distribution families perhaps, you can say it describes the shape, but more generally kurtosis doesn't tell you terribly much about the actual shape. Shape is impacted by many things, including things unrelated to kurtosis. If one does image searches for kurtosis, quite a few images like this one show up: For comparison, here's three normal densities I just drew using R with different standard deviations: As you can see, it looks almost identical to the previous picture.

These all have exactly the same kurtosis. By contrast, here's an example that is probably nearer to what the diagram was aiming for The green curve is both more peaked and heavier tailed though this display isn't well suited to seeing how much heavier the tail actually is.

This is usually what people mean when they talk about kurtosis indicating the shape of the density. However, kurtosis can be subtle -- it doesn't have to work like that. For example, at a given variance higher kurtosis can actually occur with a lower peak.

One must also beware the temptation and in quite a few books it's openly stated that zero excess kurtosis implies normality. The average number of intraday transactions per day for a stock was more than 1, 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.

## Symmetry, Skewness and Kurtosis

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.

Realized volatility increases from 19 percent for the first decile to 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.

### Keep Skewness In Perspective

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 skewnessshort stocks with high skewness and value-weighted portfolio produced an average weekly return of 24 basis points with a t-statistic of 3.

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.

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 The estimates for the Carhart four-factor alpha are smaller and less statistically significant compared with those for the raw returns. 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.

However, as skewness increases and becomes positive, the positive relation between volatility and returns turns into a negative relation.