Bergische Universität Wuppertal
Fachbereich Mathematik und Naturwissenschaften
Angewandte Mathematik - Numerische Analysis


Assessment of the Applicability of the Peaks over Threshold (POT) Method for the Implementation of Stress-Testing on the Market Risk in the IBB

Bachelorarbeit Wirtschaftsmathematik



This bachelor thesis studies and practices the employment of the extreme value theory (EVT) into risk management, as a method of modeling and assessing the extreme risks. Without loss of generality, a sequence of procedures utilizing the peaks over threshold (POT) model in stress-testing of the market risks is performed. Risk management is concerned with the identification, assessment and prioritization of various risks, so as to minimize and control the probability or impact of unfavorable events. Since years, the operation reforms from subjective recognition by leadership fellows, to risk quantification through statistical models. Specifically, the EVT has been studied and applied widely in dealing with low probability high consequence events, which usually possess the character of a fat-tailed distribution pattern.

Thus, the practice of tail-fitting approach of the EVT is of great importance. Hereby, the POT approach, based on modeling excess values of a sample set over a threshold within a time period, is focused on and it provides a straightforward tool for estimating measures of tail risks at high quantiles. Generalized Pareto distribution (GPD) is used to approximate the distribution of excess amounts over sufficiently high thresholds. The thresholds are optimized through the combined graphic approaches of the ME-plot and the Hill-plot. Both methods are conducted to estimate the parameters for the GPD. With the aid of parameter assessment methods and criteria, parameter pairs with appropriate quality can be determined.

This procedure is performed and used in stress-testing of the market risks in the Investionsbank Berlin (IBB). The risk-free interest rate premium for each quantile is thereupon acquired for the stress-testing. The results imply that the POT method performs effectively for the stress-testing.


Peaks over Threshold method, generalized Pareto distribution


  1. E. Brodin, C. Klüppelberg, Extreme Value Theory in Finance, Chalmers University of Technology, Munich University of Technology, 2007.
  2. J. Choi, Estimation of the Tail Behavior of Mutual Fund Returns: an EVT-based Approach, Master's thesis, The University of Chicago (2011).
  3. J. Cotter, K. Dowd, Extreme Spectral Risk Measures: an Application to Futures Clearinghouse Margin Requirements, University library of Munich, 2005.
  4. P. Embrechts, C. Klüppelberg, T. Mikosch, Modelling Extremal Events, Springer-Verlag, 2004.
  5. P. Embrechts, A. J. McNeil, R. Frey, Quantitative Risk Management, Princeton University, 2005.
  6. M. Gilli, E. Kellezi, An Application of Extreme Value Theory for Measuring Risk, University of Geneva and FAME, 2003.
  7. W. Härdle, S. Borak, Y. Chen, B. Choros, Measuring Statistical Risk Extremes, Joint Extremes and Copulae, Humboldt - Universität Berlin, 2010.
  8. D. S. Mapa, P. J. A. Cayton, M. T. Lising, Estimating Value-at-Risk (VaR) using TiVEx-POT Models, University Library of Munich, Germany, 2009.
  9. A. J. McNeil, T. Saladin, The Peaks over Thresholds Method for Estimating High Quantiles of Loss Distributions, ETH Zurich, 1997.
  10. A. J. McNeil, Extreme Value Theory for Risk Managers, ETH Zentrum, 1999.
  11. G. Matthys, J. Beirlant, Estimating the Extreme Value Index and High Quantiles with Exponential Regression Models, Katholieke Universiteit Leuven, 2003.
  12. D. Schirmacher, E. Schirmacher, N. Thandi, Stochastic Excess-of-Loss Pricing within a Financial Framework, Springer-Verlag, 2005.
  13. Q. Xie, The Application of the EVT for the Extreme Loss Incidents, Nankai University, 2008.
  14. What is a bank stress test?, IMF Survey Magazin: Policy (2010).

University of Wuppertal
Faculty of Mathematics and Natural Sciences
Department of Mathematics
Applied Mathematics & Numerical Analysis Group

Last modified:   Disclaimer