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The Basel II Accord requires participating banks to quantify operational risk according to a matrix of business lines and event types. Proper modeling of univariate loss distributions and dependence structures across those categories of operational losses is critical for proper assessment of overall annual operational loss distributions. We illustrate our proposed methodology using Loss Data Collection Exercise 2004 (LDCE 2004) data on operational losses across five loss event types. We estimate a multivariate likelihood-based statistical model, which illustrates the benefits and risks of using extreme value theory (EVT) in modeling univariate tails of event type loss distributions. We find that abandoning EVT leads to unacceptably low estimates of risk capital requirements, while indiscriminate use of EVT to all data leads to unacceptably high ones. The judicious middle approach is to use EVT where dictated by data, and after separating clear outliers that need to be modeled via probabilistic scenario analysis. We illustrate all computational steps in estimation of marginal distributions and copula with an application to one bank's data (disguising magnitudes to ensure that bank's anonymity). The methods we use to overcome heretofore unexplored technical problems in estimation of codependence across risk types scales easily to larger models, encompassing not only operational, but also other types of risks.
ADVANCES IN HEAVY TAILED RISK MODELING A cutting-edge guide for the theories, applications, and statistical methodologies essential to heavy tailed risk modeling Focusing on the quantitative aspects of heavy tailed loss processes in operational risk and relevant insurance analytics, Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk presents comprehensive coverage of the latest research on the theories and applications in risk measurement and modeling techniques. Featuring a unique balance of mathematical and statistical perspectives, the handbook begins by introducing the motivation for heavy tailed risk processes. A companion with Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk, the handbook provides a complete framework for all aspects of operational risk management and includes: Clear coverage on advanced topics such as splice loss models, extreme value theory, heavy tailed closed form loss distribution approach models, flexible heavy tailed risk models, risk measures, and higher order asymptotic approximations of risk measures for capital estimation An exploration of the characterization and estimation of risk and insurance modeling, which includes sub-exponential models, alpha-stable models, and tempered alpha stable models An extended discussion of the core concepts of risk measurement and capital estimation as well as the details on numerical approaches to evaluation of heavy tailed loss process model capital estimates Numerous detailed examples of real-world methods and practices of operational risk modeling used by both financial and non-financial institutions Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk is an excellent reference for risk management practitioners, quantitative analysts, financial engineers, and risk managers. The handbook is also useful for graduate-level courses on heavy tailed processes, advanced risk management, and actuarial science.
Extreme Value Modeling and Risk Analysis: Methods and Applications presents a broad overview of statistical modeling of extreme events along with the most recent methodologies and various applications. The book brings together background material and advanced topics, eliminating the need to sort through the massive amount of literature on the subje
"Extreme-value theory is concerned with the tail behaviour of probability distributions. In recent years, it has found many applications in areas as diverse as hydrology, actuarial science, and finance, where complex phenomena must often be modelled from a small number of observations.Extreme-value theory can be used to assess the risk of rare events either through the block maxima or peaks-over-threshold method. The choice of threshold is both influential and delicate, as a balance between the bias and variance of the estimates is required. At present, this threshold is often chosen arbitrarily, either graphically or by setting it as some high quantile of the data.Bayesian inference is an alternative to deal with this problem by treating the threshold as a parameter in the model. In addition, a Bayesian approach allows for the incorporation of internal and external observations in combination with expert opinion, thereby providing a natural probabilistic framework to evaluate risk models.This thesis presents a Bayesian inference framework for extremes. We focus on a model proposed by Behrens et al. (2004), where an analysis of extremes is performed using a mixture model that combines a parametric form for the centre and a Generalized Pareto Distribution (GPD) for the tail of the distribution. Our approach accounts for all the information available in making inference about the unknown parameters from both distributions, the threshold included. A Bayesian analysis is then performed by using expert opinions to determine the parameters for prior distributions; posterior inference is carried out through Markov Chain Monte Carlo methods. We apply this methodology to operational risk data to analyze its performance.The contributions of this thesis can be outlined as follows:-Bayesian models have been barely explored in operational risk analysis. In Chapter 3, we show how these models can be adapted to operational risk analysis using fraud data collected by different banks between 2007 and 2010. By combining prior information to the data, we can estimate the minimum capital requirement and risk measures such as the Value-at-Risk (VaR) and the Expected Shortfall (ES) for each bank.-The use of expert opinion plays a fundamental role in operational risk modelling. However, most of time this issue is not addressed properly. In Chapter 4, we consider the context of the problem and show how to construct a prior distribution based on measures that experts are familiar with, including VaR and ES. The purpose is to facilitate prior elicitation and reproduce expert judgement faithfully.-In Section 4.3, we describe techniques for the combination of expert opinions. While this issue has been addressed in other fields, it is relatively recent in our context. We examine how different expert opinions may influence the posterior distribution and how to build a prior distribution in this case. Results are presented on simulated and real data.-In Chapter 5, we propose several new mixture models with Gamma and Generalized Pareto elements. Our models improve upon previous work by Behrens et al. (2004) since the loss distribution is either continuous at a fixed quantile or it has continuous first derivative at the blend point. We also consider the cases when the scaling is arbitrary and when the density is discontinuous.-Finally, we introduce two nonparametric models. The first one is based on the fact that the GPD model can be represented as a Gamma mixture of exponential distributions, while the second uses a Dirichlet process prior on the parameters of the GPD model." --
Operational Risk While operational risk has long been regarded as a mere part of "other" risks—outside the realm of credit and market risk—it has quickly made its way to the forefront of finance. In fact, with implementation of the Basel II Capital Accord already underway, many financial professionals—as well as those preparing to enter this field—must now become familiar with a variety of issues related to operational risk modeling and management. Written by the experienced team of Anna Chernobai, Svetlozar Rachev, and Frank Fabozzi, Operational Risk: A Guide to Basel II Capital Requirements, Models, and Analysis will introduce you to the key concepts associated with this discipline. Filled with in-depth insights, expert advice, and innovative research, this comprehensive guide not only presents you with an abundant amount of information regarding operational risk, but it also walks you through a wide array of examples that will solidify your understanding of the issues discussed. Topics covered include: The main challenges that exist in modeling operational risk The variety of approaches used to model operational losses Value-at-Risk and its role in quantifying and managing operational risk The three pillars of the Basel II Capital Accord And much more
The aggregation of event types (ETs) is a crucial step for operational risk management techniques. Basel II requires the computation of a 99.9% VaR for each ET, and their aggregation via a simple sum if the dependence among ETs is not specified. Such a procedure assumes perfect positive dependence and therefore involves the implementation of the most conservative aggregation model. We propose a methodology that uses extreme-value theory to model the loss severities, copulas to model their dependence and a general Poisson shock model to capture the dependencies among ETs. We show that this approach allows the allocation of capital and hedge operational risk in a more efficient way than the standard approach.
In this paper we point out several pitfalls of the standard methodologies for quantifying operational losses. Firstly, we use Extreme Value Theory to model real heavy-tailed data. We show that using the Value-at-Risk as a risk measure may lead to a mis-estimation of the capital requirements. In particular, we examine the issues of stability and coherence and relate them to the degree of heavy-tailedness of the data. Secondly, we introduce dependence between the business lines using Copula Theory. We show that standard economic thinking about diversification may be inappropriate when infinite-mean distributions are involved.