Every VaR and ES formula derived under the normal distribution (or any fixed parametric distribution) produces a number that is only as reliable as the tail model. Stylised fact (6) established that daily return series are leptokurtic — heavier-tailed than the normal. This note quantifies how badly that matters for high-confidence risk measures. The normal model's tail decays exponentially; equity returns' tail decays like a power law. At the 99th percentile the difference is already significant; at the 99.9th percentile (the Basel Advanced Approaches threshold) the discrepancy grows so large that a normal-based VaR can understate the true risk by a factor of two or more. This is the fundamental motivation for extreme value theory: we need a framework that makes universal statements about tail behaviour without committing to a globally wrong distributional assumption.
Learning objectives. By the end of this note you should be able to:
- describe what leptokurtosis means and demonstrate it from the Jarque–Bera test;
- state the two types of tail decay and explain what "power-law tail" means;
- compute a normal-based VaR and explain why it systematically underestimates at extreme quantiles;
- identify which distributions lead to which EVT limits (Fréchet/Gumbel/Weibull) using the tail decay criterion;
- state the question that EVT answers.
In this lesson
-
- The normal assumption and its failure
-
- Power-law tails vs exponential-type tails
-
- The question EVT answers
-
- Why the normal model cannot be patched
Before you start
- Value-at-Risk: Definition, Properties, and the Three Computation Methods
- Expected Shortfall and Other Tail Risk Measures
- Stylised Facts of Financial Return Series
The full lesson, with worked examples and exercise solutions, is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or browse the Quantitative Risk Management syllabus.