How does excess kurtosis (fat tails) affect VaR calculations, and what can you do about it?

Fat tails mean that extreme returns occur much more frequently than the normal distribution predicts. This directly undermines normal VaR, which relies on the assumption that returns are Gaussian.

Quantifying the Problem:

The normal distribution has a kurtosis of 3 (excess kurtosis = 0). Typical daily financial return series have excess kurtosis of 5-15. Here's what this means for tail probabilities:

At the 99% confidence level, normal VaR uses 2.326 standard deviations. But if returns have kurtosis of 8, the true 99th percentile might be at 2.8+ standard deviations — a 20% underestimation of risk.

Practical Fixes:

1. Cornish-Fisher Expansion:

Adjust the z-score for skewness (S) and excess kurtosis (K):

z_CF = z + (z^2-1)/6 x S + (z^3-3z)/24 x K - (2z^3-5z)/36 x S^2

For z = -2.326, S = -0.5, K = 4:

z_CF = -2.326 + (5.41-1)/6 x (-0.5) + (-12.57+6.98)/24 x 4 - ...

This gives a more negative z, producing higher VaR.

2. Student-t Distribution:

Fit returns to a Student-t with degrees of freedom estimated via MLE. Fewer degrees of freedom = fatter tails:

• nu = 30: nearly normal
• nu = 5: substantial fat tails
• nu = 3: very heavy tails (infinite kurtosis)

3. Extreme Value Theory (EVT):

Model only the tail using Generalized Pareto Distribution. This is the gold standard for estimating extreme quantiles.

4. Historical Simulation:

Use actual return data — inherently captures whatever tail behavior exists. But limited by sample size.

FRM Key Points:

• Normal VaR is a lower bound for the true risk when tails are fat
• Expected Shortfall (ES) is even more sensitive to fat tails than VaR
• The Basel framework now requires ES at 97.5% instead of VaR at 99% partly to address tail risk
• Always report which distributional assumption underlies your VaR

Deepen your understanding of tail risk in our FRM Part I Quantitative module.