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Michael Brenndoerfer

Probability Theory Fundamentals › Higher Moments: Skewness and Kurtosis

Michael BrenndoerferOctober 19, 20258 min read

Quantitative FinanceAll chapters

Part 1 of 15Mathematical Foundations

Chapter 2Probability Theory Fundamentals › Higher Moments: Skewness and Kurtosis

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Mean and variance describe the first two moments of a distribution, but they don't fully characterize it. Two distributions can have identical means and variances yet differ dramatically in their shapes. Higher moments capture asymmetry and tail behavior, which are both critical for understanding financial risk. Investors care about average outcomes, volatility, and the likelihood of extreme gains versus extreme losses.

SkewnessLink Copied

Skewness

When $\sigma>0$ and the third absolute central moment is finite, skewness measures asymmetry around the mean:

\text{Skewness} = E\left[\left(\frac{X - \mu}{\sigma}\right)^3\right] = \frac{E[(X - \mu)^3]}{\sigma^3}

where:

$X$ : random variable
$\mu = E[X]$ : mean of $X$
$\sigma$ : standard deviation of $X$
$E[\cdot]$ : expected value operator
The cubic power captures asymmetry: positive deviations contribute positively ( $+^3 = +$ ) while negative deviations contribute negatively ( $-^3 = -$ ). Skewness is positive when the probability-weighted cubic contributions above the mean exceed the magnitude of those below it
Dividing by $\sigma^3$ makes skewness dimensionless and invariant to positive fixed rescaling; negative rescaling reverses its sign

The choice of the cubic power is not arbitrary. It's the lowest odd power that captures asymmetry. The first power (plain deviations) would sum to zero by definition of the mean. The second power (squared deviations) gives variance but loses sign information. The third power preserves the sign of each deviation, allowing positive and negative extremes to contribute differently. Their probabilities matter as well as their sizes: positive skewness requires the positive weighted cubes to dominate, and negative skewness requires the negative weighted cubes to dominate. The length of one tail or the largest deviation alone does not determine the sign.

A distribution's skewness reveals the direction and magnitude of its asymmetry:

Positive skewness: In the plotted skew-normal example, the right tail is longer and rare positive values dominate the third moment.
Negative skewness: In the plotted skew-normal example, the left tail is longer and rare negative values dominate the third moment. Some equity-return samples and horizons show this pattern, but it is not universal across assets or regimes.
Zero skewness: The third standardized moment is zero. A symmetric distribution with a finite third moment has zero skewness, but zero skewness does not by itself imply symmetry.

Out[15]:

Visualization

Three probability density curves showing negative, zero, and positive skewness. — Distributions with different skewness. Negative skewness (left) has a longer left tail; positive skewness (right) has a longer right tail.

Skewness can matter even when two investments have the same mean and variance. In the plotted examples, positive skewness reflects a longer right tail and negative skewness a longer left tail. More generally, skewness measures signed deviations from the mean; its sign alone does not establish the probability or size of absolute gains or losses. Whether an investor prefers one profile depends on the investor's utility, constraints, and the rest of the return distribution; skewness alone does not determine the choice.

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Kurtosis

When $\sigma>0$ and the fourth central moment is finite, kurtosis is the fourth standardized moment:

\text{Kurtosis} = E\left[\left(\frac{X - \mu}{\sigma}\right)^4\right] = \frac{E[(X - \mu)^4]}{\sigma^4}

where:

$X$ : random variable
$\mu = E[X]$ : mean of $X$
$\sigma$ : standard deviation of $X$
The fourth power emphasizes extreme deviations because raising to the fourth power magnifies large values much more than small ones (e.g., $2^4 = 16$ but $4^4 = 256$ ), making kurtosis highly sensitive to tail behavior
Unlike the cubic power in skewness, the fourth power is nonnegative: deviations of either sign contribute equally, while zero contributes zero

Excess kurtosis subtracts 3 (the kurtosis of a normal distribution): $\text{Excess Kurtosis} = \text{Kurtosis} - 3$ . This normalization sets the normal distribution as the baseline with excess kurtosis of zero.

Kurtosis uses fourth powers and therefore requires a finite fourth moment as a population quantity. A Student's t distribution with four degrees of freedom has no finite population fourth moment, so its population kurtosis is undefined; a finite sample still produces a numerical sample statistic, but that statistic is not estimating a finite t(4) kurtosis parameter.

Excess kurtosis compares a distribution's fourth standardized moment with the normal benchmark:

Excess kurtosis > 0 (leptokurtic): A fourth standardized moment above the normal benchmark. Many financial-return samples exhibit positive excess kurtosis, but this alone does not guarantee larger tail probability at every cutoff.
Excess kurtosis = 0 (mesokurtic): Equal to the normal distribution's fourth-moment benchmark; this does not imply the distribution is normal.
Excess kurtosis < 0 (platykurtic): A fourth standardized moment below the normal benchmark; it does not guarantee uniformly lighter tails.

Many financial-return series have tails that a fitted normal model understates. Positive excess kurtosis is one warning sign, but tail probabilities should be checked directly because a fourth moment is not a complete description of tail behavior. Market moves during stressed periods can therefore be much more frequent than a normal calibration suggests without supporting a universal "once-in-a-century" label.

In[16]:

Code

from scipy.stats import kurtosis, t

# Generate samples from normal and t-distributions
np.random.seed(42)
n_samples = 100000

normal_samples = np.random.randn(n_samples)
t_samples = t.rvs(df=4, size=n_samples) * np.sqrt(
    (4 - 2) / 4
)  # t(4) scaled to unit variance

# Calculate finite-sample excess-kurtosis statistics; t(4) has no finite population kurtosis
normal_kurtosis = kurtosis(normal_samples)
t_kurtosis = kurtosis(t_samples)

Out[17]:

Console

Normal distribution excess kurtosis: -0.008
Student's t (df=4) finite-sample excess-kurtosis statistic: 17.487

Estimated sample frequency of |X| > 4:
Normal distribution: 0.0050%
Student's t (df=4): 0.4960%

Out[18]:

Visualization

Line plot comparing standard normal density to unit-variance Student's t density with df=4 over x from -6 to 6. — Standard normal vs. unit-variance Student's t (df=4) density across the full range.

Line plot focused on x > 2 where the standard-normal and unit-variance t(4) densities cross, after which the t density is higher farther into the tail. — Tail-region zoom showing the density crossover and the t-distribution's higher density farther into the tail.

After scaling both models to unit variance, the t-distribution with 4 degrees of freedom has about 4.9 times the standard normal's upper-tail probability at $x=3$ . The densities cross within the displayed tail region, so the t density is not higher at every threshold. This is a comparison between two specified models, not an estimate of a single "true" market probability or proof of a particular loss outcome. For risk management, the practical lesson is narrower: a model with lighter tails can assign a much smaller exceedance probability than a heavier-tailed alternative sufficiently far into the tail.