Risk & Portfolio

How often should a five-standard-deviation down day happen? Under a normal distribution the probability is 2.9 in 10 million — about once every 13,800 years of trading. Under a Student-t with 3 degrees of freedom, scaled to the same volatility, it is 0.16% — roughly once every 2.5 years. Same volatility, a different tail, and a different world.

• The tail shape is hard to estimate: by definition there are few tail observations, and the choice of model (t, GPD threshold) drives the answer.
• Fat tails partly come from changing volatility; scaling returns by a GARCH estimate makes them look much less extreme (McNeil & Frey 2000).
• Kurtosis is a noisy statistic dominated by a handful of days; one crash can double it.

from scipy import stats
k = 5                                                      # a five-standard-deviation down day
p_normal = stats.norm.cdf(-k)
p_fat = stats.t.cdf(-k * np.sqrt(3), df=3)                 # Student-t with 3 d.o.f., scaled to the same volatility
years_normal, years_fat = 1 / (p_normal * 252), 1 / (p_fat * 252)   # how often, in trading years

Assumes import numpy as np and import pandas as pd. Each snippet carries its own example numbers; the comments say which are made up or simulated.