An Optimal Tail Selection in Risk Measurement

Abstract

The appropriate choice of a threshold level, which separates the tails of the probability distribution of a random variable from its middle part, is considered to be a very complex and challenging task. This paper provides an empirical study on various methods of the optimal tail selection in risk measurement. The results indicate which method may be useful in practice for investors and financial and regulatory institutions. Some methods that perform well in simulation studies, based on theoretical distributions, may not perform well when real data are in use. We analyze twelve methods with different parameters for forty-eight world indices using returns from the period of 2000–Q1 2020 and four sub-periods. The research objective is to compare the methods and to identify those which can be recognized as useful in risk measurement. The results suggest that only four tail selection methods, i.e., the Path Stability algorithm, the minimization of the Asymptotic Mean Squared Error approach, the automated Eyeball method with carefully selected tuning parameters and the Hall single bootstrap procedure may be useful in practical applications.

Improvements in the estimation of the Weibull tail coefficient: A comparative study

“…However, the adaptive choice of an optimal value of $$$ k $$$ is still an open problem. Additional details can be found in previous studies [32–34]. Note that the simulated optimal sample fraction $$$ {\hat{k}}_{p,0}/n $$$ is provided only in Appendix .…”

“…In Tables 1 and 2, we present the simulated values of the mean value (E) and RMSE, both as given in (34), of the class of estimators under study. For each model, the mean value closest to the target value 𝜃 and the smallest RMSE are highlighted in bold.…”