Skewness and Kurtosis

This reading summarizes a return distribution with measures of central tendency and location (mean, median, mode, trimmed and winsorized means, quantiles) and of dispersion (range, mean absolute deviation, variance and standard deviation), and it interprets skewness and kurtosis. It then calculates the covariance and correlation of two variables and measures downside risk and relative dispersion with the target downside deviation and the coefficient of variation.

LOS 5.a — Central tendency and location

A measure of central tendency summarizes where the "center" of a dataset lies, so a single number can stand for the typical or expected value. A measure of location (quantiles) describes where a given observation sits within the distribution.

Population vs. sample

A population is every element of interest (e.g., all monthly returns a fund has ever earned). A sample is a subset of it (e.g., the last 36 months). Sample statistics are used to make inferences about population parameters. Of all the measures of central tendency, analysts rely on the arithmetic mean most often.

The three classic measures

For continuous data such as returns, exact repeats are rare, so outcomes are grouped into intervals and the modal interval, the interval containing the most observations, is reported.

The geometric mean measures the compound growth rate of returns. Like the arithmetic mean, it uses every observation, so it is also pulled by extreme values.

• Mean $=36/6=6.0\%$.
• Sorted: −1, 2, 4, 4, 7, 20, so the median $=(4+4)/2=4.0\%$.
• Mode $=4\%$ (appears twice).
  The single 20% month pulls the mean above both the median and the mode. Replace 20% with 9% and the mean falls to 4.17%, while the median and mode stay at 4%.

Dealing with outliers

Outliers are unusually large or small observations. They can come from measurement, data entry or sampling errors, or simply from the real variability of the data. If they are errors, they distort the mean. If the data are measured correctly, keeping every value may be best, because deleting genuine observations throws away information about the distribution.

• Trimmed mean: drop a set percentage of the observations at the extremes, split equally between the two ends, then average what remains. Trimming identifies the outliers objectively but may remove useful information. A 4% trimmed mean drops the lowest 2% and the highest 2%. The denominator is the number of observations remaining.
• Winsorized mean: keep every observation but replace the extremes with percentile values. For a 96% winsorized mean, values below the 2nd percentile are set equal to the 2nd percentile and values above the 98th percentile are set equal to the 98th percentile; then take the mean of all $n$ values.

With only a small number of outliers, the winsorized mean is a better measure of the center than either the arithmetic mean or a trimmed mean. Both adjustments move the estimate away from the side where the extremes lie: a few huge positive values make the trimmed/winsorized mean lower than the arithmetic mean; a few huge negative values make it higher.

Quantiles

A quantile is a cut-off value: a given share of the observations lies at or below it. The median is the 50th percentile, which is also the second quartile and the fifth decile.

Because each quantile is named by the share of observations at or below it, the positions always run in the same order: first decile (10th percentile), first quintile (20th), first quartile (25th), median (50th). A lower position can equal but never exceed a higher one.

The interquartile range (IQR) runs from the first quartile (25th percentile) to the third quartile (75th percentile), so its width is $Q_3-Q_1$. It contains the central 50% of the data.

A box and whisker plot draws a box that spans the interquartile range, a line inside the box marks the median, and the whiskers extend to the smallest and largest observations (the full range). A whisker much longer on one side hints at outliers on that side.

Stationarity

Statistical inference works best when data are stationary: the mean and variance of the underlying distribution do not change over time. If they do change, the series is nonstationary and forecasts or tests based on it are less reliable. Price or index levels that trend upward are typically nonstationary; their periodic percentage changes are much more likely to be stationary.

LOS 5.a — Measures of dispersion

Dispersion is the variability of observations around their central tendency. In investments, central tendency measures reward and dispersion measures risk.

Range and mean absolute deviation

The range is simple, but because it uses only the two most extreme observations it says nothing about how the values between them are spread.

The mean absolute deviation (MAD) averages how far each observation lies from the arithmetic mean, ignoring the sign:

Absolute values are needed because the plain deviations from the mean always sum to zero. A MAD of 5% means that, on average, an observation lies 5 percentage points from the mean. The median absolute deviation, $\text{Med}(|X_i-\text{Med}(X)|)$, is a more robust alternative.

Variance and standard deviation

Variance averages the squared deviations from the mean; squaring removes the signs but magnifies outliers, so variance is sensitive to them. For asset returns, a higher variance indicates greater risk. Taking the positive square root of the variance gives the standard deviation, which is expressed in the same units as the data (variance is in squared units, e.g. "percent squared").

Using $n-1$ rather than $n$ in the sample variance is called Bessel's correction. Dividing by $n$ would systematically underestimate the population variance (especially in small samples), making it a biased estimator. Dividing by $n-1$ makes $s^2$ an unbiased estimator of ${\sigma }^2$. Exam convention: the sample standard deviation $s$ is also described as an unbiased estimator of $\sigma$. Current practice: $s$ is the standard estimate of $\sigma$, but the square root of an unbiased variance estimate is slightly biased downward; the effect is small except in very small samples.

• Mean $=24/4=6\%$. Deviations: 4, −2, −8, 6 (sum $=0$).
• MAD $=(4+2+8+6)/4=5.0\%$.
• Sum of squared deviations $=16+4+64+36=120$.
• Sample variance $s^2=120/3=40({\%}^2)$; sample standard deviation $s=\sqrt{40}=6.32\%$.
• Had these been the whole population: ${\sigma }^2=120/4=30$, $\sigma =5.48\%$.
• Range $=12-(-2)=14$ percentage points.

Calculator (TI BA II Plus): [2ND][DATA], enter X01 = 10, X02 = 4, X03 = −2, X04 = 12; [2ND][STAT] (1-V) and scroll: $\bar{x}=6$, $S_x=6.32$ (sample), ${\sigma }_x=5.48$ (population).

Scaling with time

Under the root of time law, variance grows linearly with the length of the period when returns are independent: if daily variance is ${\sigma }^2$, the variance over $T$ days is $T{\sigma }^2$, so the standard deviation is $\sigma \sqrt{T}$. In practice, long-horizon variance is often lower than this rule predicts.

LOS 5.b — Skewness and kurtosis

Skewness

A symmetrical distribution looks the same on both sides of its mean; its mean, median and mode are equal. With symmetry, a loss interval (e.g., −6% to −4% around a zero mean) occurs as often as the matching gain interval (+4% to +6%). Skewness measures the lack of symmetry.

The mean is pulled toward the long tail because it is the measure most affected by outliers. In a skewed unimodal distribution, the median lies between the mode and the mean. For returns, positive skew means that more than half of the periods earn less than the mean (the median is below it), while a few periods bring very large gains. Negative skew is the mirror image: most periods earn more than the mean, and occasional large losses pull the mean down. Sample skewness averages the cubed standardized deviations, where $s$ is the sample standard deviation (the LOS asks for interpretation, not calculation):

Right-skewed data have positive sample skewness, since upside deviations from the mean are bigger on average. A quick way to judge the direction of skew is to compare the distance from the mode (peak) to each end of the range: the longer side is the tail.

Kurtosis

Kurtosis measures how peaked a distribution is and how thick its tails are, relative to a normal distribution (kurtosis $=3$). Excess kurtosis $=$ kurtosis $-3$. Kurtosis uses the fourth power of the standardized deviations:

Fat tails matter for risk: a leptokurtic return distribution produces extreme outcomes (both large gains and large losses) more often than a normal model predicts. Actual asset returns tend to show both skewness and excess kurtosis, so risk models that assume normality understate the chance of extreme losses. In general, more negative skew and greater excess kurtosis both mean greater risk.

LOS 5.c — Covariance and correlation

Scatter plots

A scatter plot graphs paired observations of two variables, one on each axis. It shows at a glance whether the variables move together, whether the relationship is roughly a straight line, and whether there is a nonlinear relationship that a correlation coefficient would miss.

Covariance

Covariance measures how two variables move together:

(The population version uses the population means and divides by $N$.) A positive covariance means the variables tend to be above (or below) their means at the same time; a negative covariance means they tend to move in opposite directions.

1. Deviations from the means (percentage points): fund X gives 2, −6, 6, −2; fund Y gives 1, −3, 5, −3.
2. Multiply each pair and add: $2+18+30+6=56$.
3. Divide by $n-1=3$: ${\text{Cov}}_{XY}=56/3=18.67$ (%²).

Every cross-product is positive, so the two funds were above or below their means in the same years.

Covariance is hard to interpret: its size depends on the units of the data, and its units are the product of the two variables' units (e.g. percent squared). A covariance of 30 shows only the direction of co-movement; it says nothing about how strong it is. Covariance has no upper or lower bound. The covariance of a variable with itself is its variance: ${\text{Cov}}_{XX}=s_X^2$.

Correlation

The correlation coefficient (correlation) standardizes covariance by the two standard deviations:

Here $s_X$ and $s_Y$ are the sample standard deviations. The population version uses ${\sigma }_X$ and ${\sigma }_Y$ in the same way.

Correlation has no units and always lies between −1 and +1. It is not a probability: $\rho =-0.6$ does not mean "a 60% chance the variables move in opposite directions." Because the covariance is divided by the two standard deviations, the correlation moves further from zero when the standard deviations shrink and the covariance stays the same.

This is a positive but modest linear association. The variances must be converted to standard deviations first.

For the four years of fund returns in the covariance example, $s_X=\sqrt{80/3}=5.16\%$ and $s_Y=\sqrt{44/3}=3.83\%$, so $\rho =18.67/(5.16\times 3.83)=0.94$, a strong positive linear association.

Limits of correlation

• Correlation is not causation. A strong correlation does not show that one variable drives the other, or which way any causality runs. Say the variables show a positive (or negative) association and investigate cause separately.
• Outliers can create or destroy a correlation; check whether removing them changes the result materially and whether they are information or noise.
• Spurious correlation is correlation that arises by chance, or because both variables are related to a third variable (e.g., two series that both trend upward with inflation or population), with no causal link between them.

LOS 5.d — Downside risk and relative dispersion

Semivariance and target downside deviation

Variance and standard deviation treat deviations above and below the mean alike. In some situations it is more useful to count only outcomes below the mean or below a chosen target, which measures downside risk.

• Semivariance: a variance built only from the outcomes that fall below the mean. The squared deviations of the below-mean observations are summed and divided by the sample size:

Its square root is the semideviation, a downside counterpart of the standard deviation.

• Target downside deviation (also target semideviation): like a standard deviation, but only outcomes below a chosen target $B$ contribute; the denominator is still the full sample size minus one:

Coefficient of variation

Comparing standard deviations of assets with very different mean returns can mislead. A relative dispersion measure scales risk by the mean. The coefficient of variation (CV) is:

It measures risk per unit of mean return, so a lower CV is better (less risk per unit of return). Because it is a ratio of two numbers in the same units, it is unit-free and can be quoted as a decimal or a percentage. The Sharpe ratio (Reading 8) turns the CV around: it divides the excess return over the risk-free rate by the standard deviation. With a zero risk-free rate, the CV is simply the inverse of the Sharpe ratio. When a Sharpe ratio and a risk-free rate are given, first recover $s=(\bar{X}-R_f)/\text{Sharpe ratio}$, then divide by the mean.