Normal distribution is a good measure for central tendency of financial market asset returns

Very often financial time series are fat tailed -> normal distribution does not adequately capture the probability of losses that happen rarely but are severe

EVT describes the distribution that characterizes losses specifically in the tail.

blocks <- losses$date[seq(1, nrow(losses), by=100)]

threshold <- quantile(losses$log_losses, probs = 0.95)

The standard cumulative distribution function (cdf) is defined by:

\[\begin{gather*} G_{\xi,\beta}(x)\begin{cases} 1-(1+\frac{\xi x}{\beta})^{-\frac{1}{\xi}} & \text{for }\xi \neq 0,\\ 1-e^{-\frac{x}{\beta}} & \text{for }\xi =0 . \end{cases} \end{gather*}\]

\[F_u(y) = P(X-u)\leq y | X > u)\]

• EVT models are compared to conventional models such as GARCH, historical simulation and filtered historical
• Oil markets are naturally vulnerable to significant negative volatility

Conclusion:

• Conditional Extreme Value Theory and Filtered Historical Simulation procedures offer a major improvement over the traditional methods.

The advantages:

• Both theoretical and computational Tools are available

• Extrapolation: can produce confidence intervals e.g. beyond 99% VaR; a "dangerous job but someone has to do it"

• Complements VaR model

• (my personal opinion) Expected Shortfall

The pitfalls:

• Convergence of (Maximum Likelihood) estimated parameters is not guaranteed

• EVT is not a panacea for risk management, there are several theoretical issues that are unresolved

• Might require Monte Carlo simulations when applied to portfolios