Extreme Value Theory for VaR and Historical Return Simulation

Summary

This question compares an extreme value theory approach to value-at-risk with simulation from empirical return distributions. The proposed EVT workflow models the center of returns with a Gaussian kernel and the tails with a generalized Pareto distribution, after filtering returns with a volatility or time-series model to address dependence and changing variance. A copula can then be used to generate correlated asset returns from the fitted marginal distributions. The simpler alternative uses empirical cumulative distributions with a copula.

The main potential advantage raised is that an empirical distribution cannot generate returns beyond the sample’s observed extremes, while a fitted tail model can extrapolate beyond them. That flexibility also creates model risk: tail estimates depend on data, filtering choices, and distributional assumptions, so extrapolated losses can be highly sensitive and implausible. The document poses the question but provides no empirical comparison, calibration guidance, or resolved recommendation. It frames EVT as a way to model unobserved tail outcomes, not as a guarantee of more reliable VaR estimates.

Key ideas

• EVT can model the body of returns and fit a generalized Pareto distribution to tail observations.
• Filtering with a time-series or volatility model is proposed to address changing variance and non-independent returns.
• A copula can connect fitted marginal distributions to simulate correlated asset returns.
• Empirical distributions are bounded by the historical observations, while EVT can extrapolate beyond them.
• Tail extrapolation is sensitive to modeling choices and can produce unrealistic outcomes.

Tags

# Extreme Value Theory for Value-at-Risk: Advantage versus historical simulation?


# Extreme Value Theory for Value-at-Risk: Advantage versus historical simulation?












I started researching this topic and it is well covered in the literature. My understanding is that it uses a historical time series of returns of a given set of stocks and represent such distributions by a combination of a Gaussian kernel and a generalized Pareto distribution (GPD) for the tails. This approach requires a filtering of the returns data via GARCH or ARIMA methods, which is apparently necessary to remove heteroskedasticity and other “bad” properties (non IID) that are otherwise out of the scope with the theory of EVT. I think these transformations must subsequently be undone to obtain “real” simulated returns. Lastly, to correlate the simulated variables, a T or elliptical copula can then be used on the individual CDFs calibrated as above. A more “naive” approach would be to simply use empirical CDF from each stock return series, and also apply a copula to generate correlated returns. So what is the value of the more complex EVT approach? I assume the main reason is that with historical, you cannot generate extreme returns that are lower(greater) than the minimum(maximum) of the historical returns for each stock. Whereas the parameters of the GPD can obviously be tweaked to produce more extremes, but how unrealistic can it become? I saw a short paper from Taleb where he shows that the impact on these methods can be (pun intended) extreme and impractical to use. So I am wondering what the main advantages are of implementing a somehow complex approach such as this?

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