Computational Economics

, Volume 27, Issue 2–3, pp 207–228 | Cite as

An Application of Extreme Value Theory for Measuring Financial Risk

  • Manfred GilliEmail author
  • Evis këllezi


Assessing the probability of rare and extreme events is an important issue in the risk management of financial portfolios. Extreme value theory provides the solid fundamentals needed for the statistical modelling of such events and the computation of extreme risk measures. The focus of the paper is on the use of extreme value theory to compute tail risk measures and the related confidence intervals, applying it to several major stock market indices.


extreme value theory generalized pareto distribution generalized extreme value distribution quantile estimation risk measures maximum likelihood estimation profile likelihood confidence intervals 


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  1. Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.CrossRefGoogle Scholar
  2. Azzalini, A. (1996). Statistical Inference Based on the Likelihood. Chapman and Hall, London.Google Scholar
  3. Balkema, A.A. and de Haan, L. (1974). Residual life time at great age. Annals of Probability, 2, 792–804.Google Scholar
  4. Castillo, E. and Hadi, A. (1997). Fitting the Generalized Pareto Distribution to Data. Journal of the American Statistical Association, 92(440), 1609–1620.CrossRefGoogle Scholar
  5. Christoffersen, P. and Diebold, F. (2000). How relevant is volatility forecasting for financial risk management. Review of Economics and Statistics, 82, 1–11.CrossRefGoogle Scholar
  6. Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer.Google Scholar
  7. Dacorogna, M.M., Müller, U.A., Pictet, O.V., and de Vries, C.G. (1995). The distribution of extremal foreign exchange rate returns in extremely large data sets. Preprint, O&A Research Group.Google Scholar
  8. Danielsson, J. and de Vries, C. (1997). Beyond the Sample: Extreme Quantile and Probability Estimation. Mimeo.Google Scholar
  9. Danielsson, J. and de Vries, C. (2000). Value-at-Risk and Extreme Returns. Annales d'Economie et de Statistique, 60, 239–270.Google Scholar
  10. Danielsson, J., de Vries, C., de Haan, L., and Peng, L. (2001). Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation. Journal of Multivariate Analysis, 76(2), 226–248.CrossRefGoogle Scholar
  11. Diebold, F.X., Schuermann, T., and Stroughair, J.D. (1998). Pitfalls and opportunities in the use of extreme value theory in risk management. In Refenes, A.-P., Burgess, A., and Moody, J., editors, Decision Technologies for Computational Finance, 3–12. Kluwer Academic Publishers.Google Scholar
  12. Dupuis, D.J. (1998). Exceedances over high thresholds: A guide to threshold selection. Extremes, 1(3), 251–261.CrossRefGoogle Scholar
  13. Efron, B. and Tibshirani, R.J. (1993). An Introduction to the Bootstrap. Chapman & Hall, New York.Google Scholar
  14. Embrechts, P., Klüppelberg, C., and Mikosch, T. (1999). Modelling Extremal Events for Insurance and Finance. Applications of Mathematics. Springer. 2nd ed. (1st ed., 1997).Google Scholar
  15. Fisher, R. and Tippett, L.H.C. (1928). Limiting forms of the frequency distribution of largest or smallest member of a sample. Proceedings of the Cambridge Pthilosophical Society, 24, 180–190.CrossRefGoogle Scholar
  16. Gençay, R., Selçuk, F., and Ulugülyaĝci, A. (2003a). EVIM: a software package for extreme value analysis in Matlab. Studies in Nonlinear Dynamics and Econometrics, 5, 213–239.CrossRefGoogle Scholar
  17. Gençay, R. Selçuk, F., and Ulugülyaĝci, A. (2003b). High volatility, thick tails and extreme value theory in value-at-risk estimation. Insurance: Mathematics and Economics, 33, 337–356.CrossRefGoogle Scholar
  18. Gnedenko, B.V. (1943). Sur la distribution limite du terme d'une série aléatoire. Annals of Mathematics, 44, 423–453.CrossRefGoogle Scholar
  19. Grimshaw, S. (1993). Computing the Maximum Likelihood Estimates for the Generalized Pareto Distribution to Data. Technometrics, 35(2), 185–191.CrossRefGoogle Scholar
  20. Hosking, J.R.M. and Wallis, J.R. (1987). Parameter and quantile estimation for the generalised Pareto distribution. Technometrics, 29, 339–349.MathSciNetCrossRefGoogle Scholar
  21. Jenkinson, A.F. (1955). The frequency distribution of the annual maximum (minimum) values of meteorological events. Quarterly Journal of the Royal Meteorological Society, 81, 158–172.CrossRefGoogle Scholar
  22. Jondeau, E. and Rockinger, M. (1999). The tail behavior of stock returns: Emerging versus mature markets. Mimeo, HEC and Banque de France.Google Scholar
  23. Koedijk, K.G., Schafgans, M., and de Vries, C. (1990). The Tail Index of Exchange Rate Returns. Journal of International Economics, 29, 93–108.CrossRefGoogle Scholar
  24. Kuan, C.H. and Webber, N. (1998). Valuing Interest Rate Derivatives Consistent with a Volatility Smile. Working Paper, University of Warwick.Google Scholar
  25. Longin, F.M. (1996). The Assymptotic Distribution of Extreme Stock Market Returns. Journal of Business, 69, 383–408.CrossRefGoogle Scholar
  26. Loretan, M. and Phillips, P. (1994). Testing the covariance stationarity of heavy-tailed time series. Journal of Empirical Finance, 1(2), 211–248.CrossRefGoogle Scholar
  27. McNeil, A.J. (1999). Extreme value theory for risk managers. In Internal Modelling and CAD II, 93–113. RISK Books.Google Scholar
  28. McNeil, A.J. and Frey, R. (2000). Estimation of tail-related risk measures for heteroscedastic financial time series: An extreme value approach. Journal of Empirical Finance, 7(3–4), 271–300.CrossRefGoogle Scholar
  29. Neftci, S.N. (2000). Value at risk calculations, extreme events, and tail estimation. Journal of Derivatives, pages 23–37.Google Scholar
  30. Pickands, J.I. (1975). Statistical inference using extreme value order statistics. Annals of Statististics, 3, 119–131.Google Scholar
  31. Reiss, R.D. and Thomas, M. (1997). Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields. Birkhäuser Verlag, Basel.Google Scholar
  32. Rootzèn, H. and Klüppelberg, C. (1999). A single number can't hedge against economic catastrophes. Ambio, 28(6), 550–555.Google Scholar
  33. Shao, J. and Tu, D. (1995). The Jackknife and Bootstrap. Springer Series in Statistics. Springer.Google Scholar
  34. Straetmans, S. (1998). Extreme financial returns and their comovements. Ph.D. Thesis, Tinbergen Institute Research Series, Erasmus University Rotterdam.Google Scholar
  35. Tajvidi, N. (1996a). Confidence Intervals and Accuracy Estimation for Heavytailed Generalized Pareto Distribution. Thesis article, Chalmers University of Technology,
  36. Tajvidi, N. (1996b). Design and Implementation of Statistical Computations for Generalized Pareto Distributions. Technical Report, Chalmers University of Technology,
  37. von Mises, R. (1954). La distribution de la plus grande de n valeurs. In Selected Papers, Volume II, pages 271–294. American Mathematical Society, Providence, RI.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of EconometricsUniversity of Geneva and FAMEGenevaSwitzerland
  2. 2.Mirabaud & Cie, Boulevard du Théâtre 3GenevaSwitzerland

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