AdvancedQuantitative MethodsPython

Run this module

cd "Quantitative Methods - Extreme Value Theory"
python "extreme_value_theory.py"

Extreme Value Theory (EVT) for Tail Risk¶

Most risk models assume returns are normally distributed. They are not — markets crash far more often than a bell curve allows. Extreme Value Theory fixes this by modelling the tail of the loss distribution directly, instead of forcing one distribution to fit both the calm middle and the violent edge.

This module implements the Peaks-Over-Threshold (POT) method and uses it to compute Value-at-Risk and Expected Shortfall that respect fat tails.

Functions¶

Function	Description
`fit_gpd_mom(excesses)`	Fit a Generalised Pareto Distribution to threshold excesses (method of moments)
`pot_var_es(losses, confidence, threshold_quantile)`	EVT-based VaR and Expected Shortfall
`hill_estimator(data, k)`	Tail-index estimate from the top `k` order statistics
`mean_excess(losses, thresholds)`	Mean-excess function `e(u)` for threshold-selection diagnostics

The idea¶

The Pickands–Balkema–de Haan theorem says that for a wide class of distributions, the excesses over a high threshold u converge to a Generalised Pareto Distribution (GPD):

G(y) = 1 - (1 + xi * y / beta)^(-1/xi),   y = loss - u > 0

xi (shape / tail index): xi > 0 ⇒ heavy tail (power-law). Equity losses typically sit around xi ≈ 0.2–0.4.
beta (scale): sets the spread of the excesses.

Once the tail is fitted, VaR and ES follow in closed form (McNeil, Frey & Embrechts):

VaR_q = u + (beta/xi) * [ (n/Nu * (1 - q))^(-xi) - 1 ]
ES_q  = VaR_q / (1 - xi) + (beta - xi*u) / (1 - xi)

where n is the sample size and Nu the number of excesses above u.

Example¶

import numpy as np
from extreme_value_theory import pot_var_es, hill_estimator

returns = ...                 # your return series
losses = -np.asarray(returns) # POT works on positive losses

evt = pot_var_es(losses, confidence=0.99, threshold_quantile=0.90)
print(evt["var"], evt["es"], evt["xi"])

# Heavy-tail diagnostic
print(hill_estimator(losses, k=200))

Why it matters¶

At the 99.5% level a Gaussian model and an EVT model can disagree by a wide margin — and the EVT number is almost always the larger, more realistic one. The __main__ demo fits both to fat-tailed Student-t returns so you can see the gap the normal assumption hides.

Practical notes¶

Threshold choice is the art. Too low and you contaminate the tail with the body; too high and you have too few points to fit. The mean-excess plot (mean_excess) should be roughly linear above a good threshold; the 0.90 quantile default is a sensible starting point.
Expected Shortfall needs xi < 1. If the fitted shape exceeds 1 the tail has no finite mean and ES is infinite — a sign the data (or threshold) needs another look.
EVT complements the empirical and parametric methods in Value at Risk (VaR) and Finance - Expected Shortfall; use it specifically for the deep tail (99%+).
The Hill estimator is simple but sensitive to k; plot xi_hat against k and look for a stable region.

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