The use of extreme value theory and time series analysis to estimate risk measures for extreme events.

Abstract: In this thesis the main purpose is to use extreme value theory and time series analysis to find modelsfor estimating the two risk measures for potential losses, value at risk and expected shortfall. Focus ison the time horizon needed to obtain predictions that are consistent with the actual outcome of anasset or a portfolio of assets. The extreme value based methods used are the Hill estimator and the peak over threshold method.The Hill estimator is also combined with a time series model. The time series model used is an AR(1)-GARCH(1,1) model. For extreme value theory based models the choice of threshold between the observations belongingto the tail and the observations belonging to the center of the distribution is crucial. In this study thethreshold is set to be 10% of the sample size, by conventional choice. There are additional methodsof choosing the threshold and some of them are presented in this paper. For each models different length of historical data is used when predictions of the risk measures aremade for different assets. The main result is that the best model and appropriate time horizon ofhistorical data to use for estimating value at risk and expected shortfall differs from dataset todataset. However, the methods that combine extreme value theory and time series models are themost flexible ones and those are the ones most likely to capture extreme events. The conditional Hillmethods with shorter time horizons seem preferable when estimating the risk measures for indices,while the Hill estimator with time horizon of three or four years is preferable for foreign exchangerates. In this study only models for single assets are evaluated, but the models could easily beimplemented on a time series of a portfolio. A multivariate case of the extreme value theory existsbut its complexity makes it disadvantageously to implement. So if for example the univariateextreme value models alone are considered inadequate to capture all the relations in a portfolio themodels could be used as a complement to the commonly used model based solely on historicalsimulation and thereby improve the risk analysis.