根据以上内容，以通俗易懂的语言讲解Skewness and kurtosis---**Textual Information:**
**Title:** Skewness
**Definition of Skewness:**
A distribution that is not symmetrical is termed skewed. A return distribution with positive skew has frequent small losses and a few extreme gains. A return distribution with negative skew has frequent small gains and a few extreme losses. Panel A of Exhibit 17 illustrates a continuous positively skewed distribution, which has a long tail on its right side; Panel B illustrates a continuous negatively skewed distribution, which has a long tail on its left side.
**Exhibit Title:** Exhibit 17: Properties of Skewed Distributions
**Description below charts:**
For a continuous positively skewed unimodal distribution, the mode is less than the median, which is less than the mean. For the continuous negatively skewed unimodal distribution, the mean is less than the median, which is less than the mode. For a given expected return and standard deviation, investors should be attracted by a positive skew because the mean return lies above the median. Relative to the mean return, positive skew amounts to limited, though frequent, downside returns compared with somewhat unlimited, but less frequent, upside returns.
Skewness is the name given to a statistical measure of skew. (The word "skewness" is also sometimes used interchangeably for "skew".) Like variance, skewness is computed using each observation's deviation from its mean. Skewness (sometimes
**Chart/Diagram Description:**
**Overall:** Exhibit 17 contains two panels, Panel A and Panel B, each showing a Density of Probability distribution curve. Both panels have a horizontal axis and a vertical axis labeled "Density of Probability".
**Panel A: A. Positively Skewed**
* **Type:** Density of Probability distribution chart (curve).
* **Main Elements:**
* A curved line forming a unimodal distribution shape.
* The curve is skewed to the right, with a long tail extending towards the right side of the chart.
* The peak of the curve is on the left.
* Three vertical lines are marked on the horizontal axis, labeled from left to right: Mode, Median, Mean.
* **Coordinate Axes:** Horizontal axis is unlabeled but represents the variable values (e.g., returns). Vertical axis is labeled "Density of Probability".
**Panel B: B. Negatively Skewed**
* **Type:** Density of Probability distribution chart (curve).
* **Main Elements:**
* A curved line forming a unimodal distribution shape.
* The curve is skewed to the left, with a long tail extending towards the left side of the chart.
* The peak of the curve is on the right.
* Three vertical lines are marked on the horizontal axis, labeled from left to right: Mean, Median, Mode.
* **Coordinate Axes:** Horizontal axis is unlabeled but represents the variable values (e.g., returns). Vertical axis is labeled "Density of Probability".
referred to as relative skewness) is computed as the average cubed deviation from the mean, standardized by dividing by the standard deviation cubed to make the measure free of scale.
Cubing, unlike squaring, preserves the sign of the deviations from the mean. If a distribution is positively skewed with a mean greater than its median, then more than half of the deviations from the mean are negative and less than half are positive. However, for the sum of the cubed deviations to be positive, the losses must be small and likely and the gains less likely but more extreme. Therefore, if skewness is positive, the average magnitude of positive deviations is larger than the average magnitude of negative deviations.
The approximation for computing sample skewness when n is large (100 or more) is:
Skewness ≈ (1/n) Σ_{i=1}^{n} (X_i - X̄)³ / s³ (8)
As you will learn later in the curriculum, different investment strategies may introduce different types and amounts of skewness into returns.
Kurtosis
Another way in which a return distribution might differ from a normal distribution is its relative tendency to generate large deviations from the mean. Most investors would perceive a greater chance of extremely large deviations from the mean as higher risk.
Kurtosis is a measure of the combined weight of the tails of a distribution relative to the rest of the distribution—that is, the proportion of the total probability that is outside of, say, 2.5 standard deviations of the mean. A distribution that has fatter tails than the normal distribution is referred to as leptokurtic or fat-tailed; a distribution that has thinner tails than the normal distribution is referred to as platykurtic or thin-tailed; and a distribution similar to the normal distribution as it concerns relative weight in the tails is called mesokurtic. A fat-tailed (thin-tailed) distribution tends to generate more frequent (less frequent) extremely large deviations from the mean than the normal distribution.
Exhibit 18 illustrates a distribution with fatter tails than the normal distribution. By construction, the fat-tailed and normal distributions in Exhibit 18 have the same mean, standard deviation, and skewness. Note that this fat-tailed distribution is more likely than the normal distribution to generate observations in the tail regions defined by the intersection of the distribution lines near a standard deviation of about ±2.5. This fat-tailed distribution is also more likely to generate observations that are near the mean, defined here as the region ±1 standard deviation around the mean. However, to ensure probabilities sum to 1, this distribution generates fewer observations in the regions between the central region and the two tail regions.
**Exhibit 18: Fat-Tailed Distribution Compared to the Normal Distribution**
**Chart Description:**
Type: Line chart representing probability density functions.
X-axis: Labeled "Standard Deviation", ranging from -5 to 5 in increments of 1.
Y-axis: Labeled "Density of Probability", ranging from 0 to 0.6 in increments of 0.1.
Lines:
- A solid green line representing "Normal Distribution". It is bell-shaped and centered at 0, with the highest density around 0 and tails tapering off towards -5 and 5.
- A dotted blue line representing "Fat-Tailed Distribution". It is also bell-shaped and centered at 0, with a higher peak than the normal distribution at 0, but the tails are higher than the normal distribution's tails for values further from 0.
Legend: Indicates the solid green line is "Normal Distribution" and the dotted blue line is "Fat-Tailed Distribution".
**Textual Information:**
The calculation for kurtosis involves finding the average of deviations from the mean raised to the fourth power and then standardizing that average by dividing by the standard deviation raised to the fourth power. A normal distribution has kurtosis of 3.0, so a fat-tailed distribution has a kurtosis above 3.0 and a thin-tailed distribution has a kurtosis below 3.0.
Excess kurtosis is the kurtosis relative to the normal distribution. For a large sample size (n = 100 or more), sample excess kurtosis (K_E) is approximately as follows:
Formula (9):
K_E ≈ ( (1/n) * Σ_{i=1}^n (x_i - x̄)^4 / s^4 ) - 3.
As with skewness, this measure is free of scale. Many statistical packages report estimates of sample excess kurtosis, labeling this as simply "kurtosis."
Excess kurtosis thus characterizes kurtosis relative to the normal distribution. A normal distribution has excess kurtosis equal to 0. A fat-tailed distribution has excess kurtosis greater than 0, and a thin-tailed distribution has excess kurtosis less than 0. A return distribution with positive excess kurtosis—a fat-tailed return distribution—has more frequent extremely large deviations from the mean than a normal distribution.
**Exhibit 19: Summary of Kurtosis**
**Table Content:**
| If kurtosis is ... | then excess kurtosis is ... | Therefore, the distribution is ... | And we refer to the distribution as being ... |
| :----------------- | :-------------------------- | :------------------------------------------------ | :------------------------------------------ |
| above 3.0 | above 0 | fatter-tailed than the normal distribution. | fat-tailed (leptokurtic) |
| equal to 3.0 | equal to 0 | similar in tails to the normal distribution. | mesokurtic |
| less than 3.0 | less than 0 | thinner-tailed than the normal distribution. | thin-tailed (platykurtic) |
Most equity return series have been found to be fat-tailed. If a return distribution is fat-tailed and we use statistical models that do not account for that distribution, then we will underestimate the likelihood of very bad or very good outcomes. Example 6 revisits the EAA Equity Index from the earlier Example 1 and quantifies the shape of it return distribution.
EXAMPLE 6
Skewness and Kurtosis of EAA Equity Index Daily Returns
Consider the statistics in Exhibit 20 for the EAA Equity Index:
Exhibit 20: Properties of Skewed Distributions
| | Daily Return (%) |
| :--------------------- | :--------------- |
| Arithmetic mean | 0.0347 |
| Standard deviation | 0.8341 |
| Measure of Symmetry | |
| Skewness | -0.4260 |
| Excess kurtosis | 3.7962 |
The returns reflect negative skewness, which is illustrated in Exhibit 21 by comparing the distribution of the daily returns with a normal distribution with the same mean and standard deviation.
Exhibit 21: Negative Skewness
Number of Observations
Chart Description:
* Type: Histogram with overlaid density curves.
* X-axis: Labeled "Standard Deviation", scale from -5 to 5 with integer markers.
* Y-axis: Labeled "Number of Observations", with tick marks but no specific values visible on the axis itself. The bars represent frequency counts.
* Series:
* Orange Bars: Labeled "EAA Daily Returns" in the legend. Represents the frequency distribution of the daily returns. The bars are tallest around 0 standard deviations, slightly skewed towards the left.
* Teal Line: Labeled "Normal Distribution" in the legend. Represents a bell-shaped curve of a normal distribution centered around 0, narrower than the EAA Daily Returns distribution.
* Overall: The histogram of EAA Daily Returns is peaked near 0 but has longer tails, especially visible on the positive side, compared to the normal distribution curve which is centered at 0 and is more symmetric. The distribution appears somewhat skewed to the left due to a few bars on the extreme left, but the main bulk and the peak relative to the mean suggest the description below is key. The peak of the EAA Daily Returns histogram is slightly to the right of 0.
Using both the statistics and the graph, we see the following:
* The distribution is negatively skewed, as indicated by the negative calculated skewness of -0.4260 and the influence of observations below the mean of 0.0347 percent.
* The highest frequency of returns occurs within the 0.0 to 0.5 standard deviations from the mean (i.e., the mode is greater than the mean as the positive returns are offset by extreme negative deviations).
* The distribution is fat-tailed, as indicated by the positive excess kurtosis of 3.7962. In Exhibit 21, we can see fat tails, a concentration of returns around the mean, and fewer observations in the regions between the central region and the two-tail regions.
To understand the trading liquidity of a stock, investors often look at the distribution of the daily trading volume for a stock. Analyzing the daily volume can provide insights about the interest in the stock, what factors may drive interest in the stock as well as whether the market can absorb a large trade in the stock. The latter may be of interest to investors interested in either establishing or exiting a large position in the particular stock.
INTERPRETING SKEWNESS AND KURTOSIS
Consider the daily trading volume for a stock for one year, as shown in Exhibit 22. In addition to the count of observations within each bin or interval, the number of observations anticipated based on a normal distribution (given the sample arithmetic average and standard deviation) is provided as well. The average trading volume per day for the stock during the year was 8.6 million shares, and the standard deviation was 4.9 million shares.
Exhibit 22: Histogram of Daily Trading Volume for a Stock for One Year
Chart Description:
* **Type:** Histogram (with superimposed curve representing a normal distribution).
* **X-axis:** Labeled "Trading Volume Range of Shares (millions)". Shows bins/intervals: 3.1 to 4.6, 4.6 to 6.1, 6.1 to 7.7, 7.7 to 9.2, 9.2 to 10.7, 10.7 to 12.3, 12.3 to 13.8, 13.8 to 15.3, 15.3 to 16.8, 16.8 to 18.4, 18.4 to 19.9, 19.9 to 21.4, 21.4 to 23.0, 23.0 to 24.5, 24.5 to 26.0, 26.0 to 27.5, 27.5 to 29.1, 29.1 to 30.6, 30.6 to 32.1, 32.1 to 33.7.
* **Y-axis:** Labeled "Number of Trading Days". Scale from 0 to 70 in increments of 10.
* **Data Series 1 (Bars):** Green bars representing "Based on the Sample". The bars show the frequency (number of trading days) within each trading volume bin. The bars are taller on the left side, peaking around the 4.6-6.1 million range, and then decrease rapidly, extending to the right with decreasing height.
* **Data Series 2 (Curve):** Blue filled curve representing "Based on the Normal Distribution". This smooth curve peaks around the center of the range shown by the bars and is roughly symmetrical, representing the expected distribution if the data were normally distributed with the given mean and standard deviation.
* **Legend:** Located below the chart, indicating "Based on the Sample" (green bars) and "Based on the Normal Distribution" (blue curve).
1. Would the distribution be characterized as being skewed? If so, what could account for this situation?
Solution:
The distribution appears to be skewed to the right, or positively skewed.
This is likely due to: (1) no possible negative trading volume on a given trading day, so the distribution is truncated at zero; and (2) greater-than-typical trading occurring relatively infrequently, such as when there are company-specific announcements.
The actual skewness for this distribution is 2.1090, which supports this interpretation.
2. Does the distribution displays kurtosis? Explain.
Solution:
The distribution appears to have excess kurtosis, with a right-side fat tail and with maximum shares traded in the 4.6 to 6.1 million range, exceeding what
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Skewness is a statistical measure that tells us whether data is symmetrical or leans to one side. A symmetrical distribution looks like a balanced bell curve. Positive skew means the data has a long tail stretching to the right, with many small values and few large ones. Negative skew means the opposite - a long tail to the left, with many large values and few small ones.
For investment returns, skewness has important implications. Positive skew is generally good for investors because it means frequent small losses but occasional large gains. The mean return is higher than the median, which is higher than the mode. Negative skew is generally bad because it means frequent small gains but occasional large losses. Understanding skewness helps investors assess the risk-return profile of their investments.
Kurtosis measures how fat or thin the tails of a distribution are compared to a normal distribution. A normal distribution has kurtosis of 3, or excess kurtosis of 0. Fat-tailed distributions have higher kurtosis and are more likely to produce extreme values - both very good and very bad outcomes. Thin-tailed distributions have lower kurtosis and are less likely to produce extreme values. For investors, understanding kurtosis helps assess the probability of extreme market events.
Let's look at a real example: the EAA Equity Index daily returns. The data shows negative skewness of minus 0.43, meaning frequent small gains but occasional large losses. The excess kurtosis of 3.8 indicates fat tails - a much higher probability of extreme events than a normal distribution would predict. This combination is typical of many equity markets and explains why investors experience more frequent small positive days but occasional severe market crashes.
To summarize: Skewness and kurtosis are essential tools for understanding investment risk. Skewness tells us about asymmetry - positive skew is generally good with occasional large gains, while negative skew is risky with occasional large losses. Kurtosis measures tail thickness - high kurtosis means more extreme events are likely. Together, these measures help investors make better decisions by understanding the true risk-return profile of their investments beyond just mean and standard deviation.