Systematic Tail Risk Management with Extreme Value Theory

Empirical distributions of financial data are frequently observed to have “heavy tails”, i.e. a greater probability of extreme events than can be accounted for by the standard normal distribution. Non-normal modeling of the tails is therefore a topic of active research in risk management. This area draws heavily from a branch of mathematics known as “Extreme Value Theory” (EVT). In this article we apply results from EVT to returns of major US stock indices to show that the tails should follow a power-law form i.e. f(x) ~ |x|^(-α) for large negative returns “x”. We also develop a methodology to systematically estimate the tail index,α, which is a measure of how quickly the probability diminishes as returns get more extreme. We also provide a systematic way of defining the tail region of a distribution. Applications to calculation of tail risk metrics such as VaR and Expected Tail Loss are also discussed