IV ExpertWeek 10 • Lesson 28Duration: 55 min

EVT Extreme Value Theory

Modeling the tails — because "rare" events aren't that rare

Learning Objectives

•Understand why normal distributions fail for financial tails
•Learn the Generalized Pareto Distribution for tail modeling
•Apply EVT to get realistic tail risk estimates
•Compare EVT-based VaR to normal VaR at extreme quantiles

Explain Like I'm 5

Normal statistics says extreme events are vanishingly rare. Markets disagree. "Once in a century" events happen every few years. Extreme Value Theory is a branch of statistics designed specifically for modeling these tails — the rare but devastating events that normal distributions pretend don't exist.

Think of It This Way

Normal distribution says 10-foot floods happen once per century. EVT says "let me look at the actual biggest floods and model those directly." Turns out 10-foot floods happen every 20 years and 15-foot floods every 50. EVT tells the truth about extreme events.

1Fat Tails — The Numbers

Financial returns have fat tails. Here's what that means in practice. Under a normal distribution: • 3σ event: once every ~3 years • 4σ event: once every ~126 years • 5σ event: once every ~13,932 years In actual markets: • 3σ moves happen multiple times per year • 4σ moves happen several times per decade • 5σ moves happen a few times per century The 2008 crisis produced moves that were "25-sigma events" under normal assumptions. That would never happen in the lifetime of the universe. It happened. If you're using normal-distribution VaR, you're massively underestimating tail risk. Mandelbrot, B. (1963). "The Variation of Certain Speculative Prices." Journal of Business.

2Normal vs Actual — The Gap

The green line is what a normal distribution predicts. The red bars are what actually happens. Look at the tails — far more extreme events than the bell curve expects. That gap between predicted and actual in the tails is where people blow up. They sized positions based on the green line and got hit by the red reality.

Normal vs Actual Market Returns (Tails Highlighted)

3The Generalized Pareto Distribution

GPD is the workhorse of EVT for risk management. 1. Pick a threshold u (e.g., 95th percentile of losses) 2. Look at all losses exceeding that threshold 3. Fit a GPD to those excess losses 4. Use the fitted GPD to estimate probabilities of even more extreme events The GPD has a shape parameter ξ (xi) that captures tail heaviness: • ξ = 0 → thin tail (like normal) • ξ > 0 → fat tail (like markets) — typically 0.1–0.4 for financial data • ξ < 0 → bounded tail (has a maximum possible loss) For financial data, ξ is almost always positive. That single parameter tells you everything about how dangerous your tail risk really is. Embrechts, P. et al. (1997). "Modelling Extremal Events." Springer.

4EVT VaR vs Normal VaR — The Divergence

At moderate confidence levels, normal and EVT VaR are similar. Push to extreme quantiles and EVT VaR becomes 2–3× larger. This matters because extreme quantiles are where survival decisions live. "What's my 99.9% worst case?" determines whether you deploy. Get this wrong and it's over.

EVT VaR vs Normal VaR at Increasing Confidence

5Practical Takeaways

EVT gets deep into the math quickly, but here's what matters for practitioners: 1. Don't use normal distributions for tail risk. It's provably wrong for financial data. Use EVT or at minimum Student-t distributions. 2. Your VaR is probably too optimistic. If you assumed normality, multiply by 1.5–2× for a rough reality check. 3. The shape parameter ξ matters. Fit a GPD to your returns' tails. If ξ > 0.3, you have very fat tails. Size accordingly. 4. More data isn't always better. EVT focuses on extremes. 100 tail events from 10,000 observations is plenty. 5. Tail risk changes over time. Re-estimate regularly. Crisis periods have different tails than calm ones.

Key Formulas

GPD Survival Function

Probability of exceeding x given you've already exceeded threshold u. ξ is the shape (tail heaviness), σ is scale. Higher ξ = fatter tail = more extreme events.

EVT-based VaR

VaR using EVT. Much more accurate at extreme quantiles (99.9%) than normal VaR. n = total observations, N_u = exceedances above threshold u.

Hands-On Code

EVT Tail Risk Analysis

python

import numpy as np
from scipy import stats

def evt_analysis(returns, threshold_pct=95):
    """Extreme Value Theory tail risk analysis."""
    losses = -returns[returns < 0]
    threshold = np.percentile(losses, threshold_pct)
    exceedances = losses[losses > threshold] - threshold
    
    shape, loc, scale = stats.genpareto.fit(exceedances, floc=0)
    
    print(f"Threshold: {threshold:.4f}")
    print(f"Exceedances: {len(exceedances)}")
    print(f"Shape (xi): {shape:.3f} {'(fat tail)' if shape > 0 else ''}")
    
    for conf in [0.95, 0.99, 0.999]:
        normal_var = stats.norm.ppf(conf) * returns.std()
        p = 1 - conf
        n_u = len(exceedances)
        n = len(returns)
        evt_var = threshold + (scale/shape) * ((n/n_u * p)**(-shape) - 1)
        print(f"  {conf:.1%}: Normal={normal_var:.4f}, EVT={evt_var:.4f} ({evt_var/normal_var:.1f}x)")

At 99.9% confidence, EVT VaR is typically 2–3× larger than normal VaR. That's the fat tail effect — and it's real.

Knowledge Check

Q1.A "25-sigma event" under normal assumptions happened in 2008. What does this mean?

Assignment

Compute VaR at 95%, 99%, and 99.9% using both normal distribution and EVT. Plot the ratio of EVT-VaR to Normal-VaR at each level. The ratio should increase dramatically at extreme quantiles.

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