Kurtosis — Fat Tails and Extreme Events

Kurtosis

Kurtosis measures the tailedness of a distribution — how much probability mass is in the tails relative to a normal distribution.

Excess Kurtosis = (4th standardized moment) − 3

Normal distribution: excess kurtosis = 0 (baseline)

import numpy as np
from scipy import stats

np.random.seed(42)
n = 5000
normal  = np.random.normal(0, 1, n)
t3      = np.random.standard_t(df=3, size=n)    # fat tails
uniform = np.random.uniform(-3, 3, n)            # thin tails

for name, data in [("Normal", normal), ("t(df=3) Fat-tail", t3), ("Uniform Thin-tail", uniform)]:
    ek = stats.kurtosis(data, fisher=True)   # Fisher's: normal=0
    print(f"{name:<22}: Excess Kurtosis = {ek:+.4f}")

Fat Tails in Finance

# Simulate daily stock returns — compare crash frequency
normal_rets = np.random.normal(0, 0.01, 2520)
fat_rets    = np.random.standard_t(5, 2520) * 0.01

threshold = 0.03  # 3% daily move
print(f"Normal model >3% moves: {np.sum(np.abs(normal_rets)>threshold)}/2520")
print(f"Fat-tail model >3% moves: {np.sum(np.abs(fat_rets)>threshold)}/2520")
print("Fat tails vastly increase extreme event frequency!")

Jarque-Bera Test for Normality

# Combines skewness and kurtosis into a single normality test
for name, data in [("Normal", normal), ("t(df=3)", t3)]:
    jb_stat, p = stats.jarque_bera(data)
    reject = "YES — not normal" if p < 0.05 else "NO — cannot reject normality"
    print(f"{name}: JB={jb_stat:.2f}, p={p:.6f} → Reject normality? {reject}")

Key Takeaways

1. Excess kurtosis = 0 for normal; > 0 = fat tails; < 0 = thin tails
2. Leptokurtic distributions produce more extreme outliers than normal theory predicts
3. Financial returns are almost always leptokurtic — risk models must account for this
4. SciPy's kurtosis() uses Fisher's definition (excess kurtosis, normal=0) by default
5. Jarque-Bera test formally tests normality using both skewness and kurtosis
6. Fat tails = underestimated risk — Value at Risk models using normality are dangerous