Efficient Estimation of Distributional Tail Shape and the Extremal Index with Applications to Risk Management

Abstract

Abundant evidence indicates that financial asset returns are thicker-tailed than a normal distribution would suggest. The most negative outcomes which carry the potential to wreak financial disaster also tend to be the most rare and may fall outside the scope of empirical observation. The difficulty of modelling these rare but extreme events has been greatly reduced by recent advances in extreme value theory (EVT). The tail shape parameter and the extremal index are the fundamental parameters governing the extreme behavior of the distribution, and the effectiveness of EVT in forecasting depends upon their reliable, accurate estimation. This study provides a comprehensive analysis of the performance of estimators of both key parameters. Five tail shape estimators are examined within a Monte Carlo setting along the dimensions of bias, variability, and probability prediction performance. Five estimators of the extremal index are also examined using Monte Carlo simulation. A recommended best estimator is selected in each case and applied within a Value at Risk context to the Wilshire 5000 index to illustrate its usefulness for risk measurement.