The quantitative modeling of operational risk: between \(g\)- and -\(h\) and EVT. (English) Zbl 1154.62077

Summary: 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 by K. Dutta and J. Perry [A tale of tails: an empirical analysis of loss distribution models for estimating operational risk capital. Fed. Res. Bank Boston. Working paper No. 06–13 (2006)], the parametric \(g\)- and \(h\)-distribution has recently emerged as an interesting candidate.
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 the data are well modeled by a \(g\)- and \(h\)-distribution. We further discuss the subadditivity property of the 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.


62P05 Applications of statistics to actuarial sciences and financial mathematics
62P20 Applications of statistics to economics
Full Text: DOI


[1] An academic response to Basel II (2001)
[2] EVT-based estimation of risk capital and convergence of high quantiles (2007)
[3] A Course in Credibility Theory and its Applications. (2005) · Zbl 1108.91001
[4] DOI: 10.2143/AST.31.1.993
[5] Statistics of Extremes – Theory and Applications. (2004) · Zbl 1070.62036
[6] Multivariate models for operational risk (2006)
[7] Ruin Probabilities. (2001) · Zbl 0977.62107
[8] Extremes and Integrated Risk Management pp 37– (2000)
[9] Journal of Operational Risk 2 pp 51– (2006)
[10] Exploring Data Tables, Trends, and Shapes (1985) · Zbl 0659.62002
[11] DOI: 10.1111/1467-9868.00286 · Zbl 0979.62039
[12] Extreme value theory. Copulas. Two talks on the DVD Quantitative Financial Risk Management. Fundamentals, Models and Techniques (2006)
[13] Modelling Extremal Events for Insurance and Finance. (1997) · Zbl 0873.62116
[14] A tale of tails: an empirical analysis of loss distribution models for estimating operational risk capital (2006)
[15] DOI: 10.1080/1065246031000081698 · Zbl 1035.60009
[16] Advances in Computational Finance 1 pp 3– (2001)
[17] Extreme Values, Regular Variation and Point Processes. (1987) · Zbl 0633.60001
[18] DOI: 10.1017/S0001867800012714
[19] Operational Risk: Modeling Analytics. (2006) · Zbl 1258.62101
[20] Journal of Operational Risk 1 pp 3– (2006)
[21] The modelling of operational risk: experiences with the analysis of the data collected by the Basel Committee (2004)
[22] Journal of Computational and Graphical Statistics 9 pp 180– (2000)
[23] Extremes and Integrated Risk Management pp 253– (2000)
[24] Subadditivity re-examined: the case for Value-at-Risk (2005)
[25] Quantitative Risk Management: Concepts, Techniques and Tools (2005) · Zbl 1089.91037
[26] DOI: 10.1016/S0927-5398(00)00012-8
[27] Regular Variation. (1987) · Zbl 0617.26001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.