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The Quantitative Modeling of Operational Risk: Between G-and-H and EVT

Published online by Cambridge University Press:  17 April 2015

Matthias Degen
Affiliation:
Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
Paul Embrechts
Affiliation:
Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
Dominik D. Lambrigger
Affiliation:
Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
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Abstract

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Operational risk has become an important risk component in the banking and insurance world. The availability of (few) reasonable data sets has given some authors the opportunity to analyze operational risk data and to propose different models for quantification. As proposed in Dutta and Perry [12], the parametric g-and-h distribution has recently emerged as an interesting candidate.

In our paper, we discuss some fundamental properties of the g-and-h distribution and their link to extreme value theory (EVT). We show that for the g-and-h distribution, convergence of the excess distribution to the generalized Pareto distribution (GPD) is extremely slow and therefore quantile estimation using EVT may lead to inaccurate results if data are well modeled by a g-and-h distribution. We further discuss the subadditivity property of Value-at-Risk (VaR) for g-and-h random variables and show that for reasonable g and h parameter values, superadditivity may appear when estimating high quantiles. Finally, we look at the g-and-h distribution in the one-claim-causes-ruin paradigm.

Type
Articles
Copyright
Copyright © ASTIN Bulletin 2007

References

[1] Asmussen, S. (2001) Ruin Probabilities. World Scientific, Singapore.Google Scholar
[2] Böcker, K. and Klüppelberg, C. (2006) Multivariate models for operational risk. Preprint, TU Munich.Google Scholar
[3] Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004) Statistics of Extremes – Theory and Applications. Wiley, Chichester.CrossRefGoogle Scholar
[4] Beirlant, J., Matthys, G. and Dierckx, G. (2001) Heavy-tailed distributions and rating. ASTIN Bulletin 31, 3758.CrossRefGoogle Scholar
[5] Bühlmann, H. and Gisler, A. (2005) A Course in Credibility Theory and its Applications. Springer, Berlin.Google Scholar
[6] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) Regular Variation. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[7] Danielsson, J., Embrechts, P., Goodhart, C., Keating, C., Muennich, F., Renault, O. and Song Shin, H. (2001) An academic response to Basel II. Financial Markets Group, London School of Economics.Google Scholar
[8] Danielsson, J., Jorgensen, B.N., Samorodnitsky, G., Sarma, M. and De Vries, C.G. (2005) Subadditivity re-examined: the case for Value-at-Risk. FMG Discussion Papers, London School of Economics.Google Scholar
[9] Degen, M. and Embrechts, P. (2007) EVT-based estimation of risk capital and convergence of high quantiles. Preprint, ETH Zurich.Google Scholar
[10] Diebold, F., Schuermann, T. and Stroughair, J. (2001) Pitfalls and opportunities in the use of extreme value theory in risk management. In: Refenes, A.-P., Moody, J. and Burgess, A. (Eds.), Advances in Computational Finance, Kluwer Academic Press, Amsterdam, pp. 312, Reprinted from: Journal of Risk Finance, 1, 3036 (Winter 2000).Google Scholar
[11] Dominici, D. (2003) The inverse of the cumulative standard normal probability function. Integral Transforms and Special Functions 14, 281292.CrossRefGoogle Scholar
[12] Dutta, K. and Perry, J. (2006) A tale of tails: an empirical analysis of loss distribution models for estimating operational risk capital. Federal Reserve Bank of Boston, Working Paper No 06-13.Google Scholar
[13] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance. Springer, Berlin.CrossRefGoogle Scholar
[14] Embrechts, P. and Neslehova, J. (2006) Extreme value theory. Copulas. Two talks on the DVD Quantitative Financial Risk Management. Fundamentals, Models and Techniques , Henry Stewart Publications, London.Google Scholar
[15] Guillou, A. and Hall, P. (2001) A diagnostic for selecting the threshold in extreme value analysis. J.R. Statist. Soc. B 63, 293305.CrossRefGoogle Scholar
[16] Hoaglin, D.C., Mosteller, F. and Tukey, J.W. (1985) Exploring Data Tables, Trends, and Shapes. Wiley, New York.Google Scholar
[17] Lambrigger, D.D., Shevchenko, P.V. and Wüthrich, M.V. (2007) The quantification of operational risk using internal data, relevant external data and expert opinions. Journal of Operational Risk. To appear.CrossRefGoogle Scholar
[18] Makarov, M. (2006) Extreme value theory and high quantile convergence. Journal of Operational Risk 2, 5157.CrossRefGoogle Scholar
[19] Matthys, G. and Beirlant, J. (2000) Adaptive threshold selection in tail index estimation. In: Embrechts, P. (Ed.), Extremes and Integrated Risk Management, pp. 3748, Risk Books, London.Google Scholar
[20] 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, 271300.CrossRefGoogle Scholar
[21] McNeil, A.J., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton.Google Scholar
[22] McNeil, A.J. and Saladin, T. (1997) The peaks over thresholds method for estimating high quantiles of loss distributions. Proceedings of 28th International ASTIN Colloquium. Google Scholar
[23] McNeil, A.J. and Saladin, T. (2000) Developing scenarios for future extreme losses using the POT method. In: Embrechts, P. (Ed.), Extremes and Integrated Risk Management, pp. 253267, Risk Books, London.Google Scholar
[24] Morgenthaler, S. and Tukey, J.W. (2000) Fitting quantiles: doubling, HR, HH and HHH distributions. Journal of Computational and Graphical Statistics 9, 180195.Google Scholar
[25] Moscadelli, M. (2004) The modelling of operational risk: experiences with the analysis of the data collected by the Basel Committee. Bank of Italy, Working Paper No 517.Google Scholar
[26] Neslehova, J., Embrechts, P. and Chavez-Demoulin, V. (2006) Infinite mean models and the LDA for operational risk. Journal of Operational Risk 1, 325.CrossRefGoogle Scholar
[27] Panjer, H.H. (2006) Operational Risk: Modeling Analytics. Wiley, New York.CrossRefGoogle Scholar
[28] Raoult, J.-P. and Worms, R. (2003) Rate of convergence for the generalized Pareto approximation of the excesses. Advances in Applied Probability 35, 10071027.CrossRefGoogle Scholar
[29] Resnick, S.I. (1987) Extreme Values, Regular Variation and Point Processes. Springer, New York.CrossRefGoogle Scholar