Means of estimating the tail index and the value at risk

Abstract

This article presents some methodologies to quantify risk when the loss distribution exhibits extreme events because the financial assets generally have high kurtosis. Thus, the main concept utilized in the document is the Value at Risk (VaR), a measure introduced by J. P. Morgan in 1995. From the statistical point of view, VaR is a quantile of a distribution function, but its value depends on the shape of the distribution that is used to fit the data loss. For this reason, in order to obtain a reliable measure of risk it is necessary to obtain a reliable shape parameter of the distribution of losses. The extreme value theory (EVT) is a statistical technique that has been used for this purpose. This document uses the EVT methodology, called peaks over threshold (POT), in which the shape parameter of the distribution of excesses is estimated by maximum likelihood. This estimation method is briefly reviewed in the document along with the weighted least squares method. The latter is used to quantify the Hill estimator and this value is used to calculate VaR for heavy tailed distributions. Finally, we compare the methods proposed in the article to measure VaR with two other methods that are; historical simulation and the assumption of normality through backtesting in two cases.