# Modelli statistici per la gestione dei rischi operativi

**Author(s):**

Ziller, Stefano

**Title:**

Modelli statistici per la gestione dei rischi operativi

**Date:**

Tuesday 20th December 2011

**Advisor:**

Paganoni, A.M.

**Advisor II:**

**Abstract:**

This work presents a statistical model for operational risk management.
Such risk includes losses deriving from natural disaters, system failures,
human errors or frauds. All financial institutions have to set a provision
up, in order to face such losses. The thesis is focused on AMA (Advanced
Measurement Approach) models, aimed at computing the capital at risk.
These statistical models are based on the analysis of operational losses
time series. First of all, seven operational ET (Event Types) can be distinguished, according to the different causes. The idea of the model is to fit each risk class separately and then aggregate them to obtain a single distribution. Hence, the provision can be computed through the VaR (Value at Risk) indicator, defined as the 99.9% quantile of the aggregated distribution. The approach proposed is an actuarial one: the probability of event occurrence (the frequency distribution) and the economic impact of the single event (the severity distribution) are treated separately, and then an aggregated distribution is obtained through convolution of frequency and severity, for each ET. A first problem arises since losses with a small economical impact are often neglected, hence they can rarely be trusted. Thus, the severity distribution is fitted with truncated distribution, above a threshold, which is fixed by the bank. Moreover, due to the sensibility of the capital at risk with respect to high level quantiles, the right tail of the severity distribution, which includes losses above a certain threshold, which has to be estimated, is fitted with the GPD (Generalized Pareto Distribution), which is the most appropriate in extreme values theory.
On the other hand, the frequency distribution is modeled with Poisson
distribution, considering only losses above the lower threshold for the estimation. Thus, according to the actuarial approach, each ET aggregated
annual loss distribution is obtained through convolution, , via Monte Carlo simulation, under the appropriate independence hypothesis. Finally, the ETs multivariate distribution, which the VaR is computed on, is obtained exploiting copulas, which allows to aggregate marginal distributions maintaining the desired dependence structure. The work is divided into five main Chapters. In Chapter 1, operational risk and Bank of Italy main regulations are introduced, then the considered model is described, highlighting the main steps of the analysis, which are deeply discussed in the next few Chapters. Chapter 2 presents truncated distribution, focusing on lognormal and Weibull distributions, presenting the relative properties and proposing algorithms for parameters estimate. In Chapter 3 the theory of extreme values is presented, with particular attention devoted to the threshold selection and the estimation of GPD parameters. Chapter 4 is devoted to the modeling of multivariate distributions with copulas: after a brief theoretical discussion, the attention is set on dependence structures and on the comparison of the main copulas families (elliptical and Archimedean). Finally, in Chapter 5, we present the results obtained by applying the model to the dataset of one of the major Italian banks, anonymous due to the sensitivity of the data. The conclusion is the determination of the capital at risk for that bank. The fundamental result of this thesis is not only the study and the integration of advanced statistical techniques in a single model, but also the innovation in some of its critical aspects, including the determination of MLEs for particular truncated distributions and methods to choose the correct threshold for extreme values.