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**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.

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.

### MSC:

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

62P20 | Applications of statistics to economics |

### Keywords:

extreme value theory; Hill estimator; LDA; operational risk; peaks over threshold; second order regular variation; subadditivity; value-at-risk
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\textit{M. Degen} et al., ASTIN Bull. 37, No. 2, 265--291 (2007; Zbl 1154.62077)

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