Extreme Value Theory: An Empirical Analysis of Equity Risk for Shanghai Stock Market

Abstract

Prediction of the frequency of extreme events is of primary importance in many financial applications such as Value-at-Risk (VaR) analysis. We provide an overview of the role of extreme value theory (EVT) in risk management, as a method for modeling and measuring extreme risks. Extreme value theory models the tails of the return distribution rather than the whole distribution, which is more meaningful during the volatile market conditions, under which the distribution of returns almost has a fat tail. In particular, we concentrate on the peaks-over-threshold (POT) model and emphasize the generality of this approach. According to extreme value theory, the POT is a generalized Pareto distribution (GPD) with two parameters, which is widely used for modeling exceedances of a random variable over a high threshold and has proven to be one of the best ways to apply extreme value theory in practice. But the main problem is the selection of the threshold. Extreme value theory tells us that threshold should be high in order to satisfy theorem conditions, however the higher the threshold the less observations are left for the estimation of the parameters of the tail distribution function. The issue of determining the fraction of data belonging to the tail is treated by mean excess function. Tools from exploratory data analysis prove helpful in approaching this problem. Moreover we concentrate on two measures which attempt to describe the tail of a loss distribution-VaR and expected shortfall. VaR is a high quantile of the distribution of losses, typically the 95 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> or 99 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> percentile. It provides a kind of upper bound for a loss that only exceeded on a small proportion of occasions. Expected shortfall-the tail conditional expectation is used to estimate the expected size of a loss that exceeds VaR. Finally, the application of EVT is illustrated by Shanghai stock market data. We conclude that EVT is an useful complement to traditional VaR methods