Operational risk quantification using extreme value theory and copulas: from theory to practice (original) (raw)
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Introduction to the Extreme Value theory applied to operational risk
International Journal of Innovation and Applied Studies, 2013
This paper aims to present the main lines of the Extreme Value Theory applied to the operational risk. The idea is to present a methodology which allows to identify a threshold by type of risk, and to feign the losses below the threshold with the classical laws, and the losses above with a Generalized Pareto Distribution (GPD). The adequacy of the data to the law GPD allows to consider an extreme quantile, as minimal strategy, sensitive to the size of samples, and to plan random costs whose probability of occurrence is very low, but the choice of the threshold beyond of which the observation will be judged extreme, is a point to be handled with precaution, even if we propose a technique to quantify this threshold. Furthermore, the costs of extreme losses do not lend themselves to modeling; by definition this type of costs is rare, and the forecasts or the estimations must be often established with a big distrust, and outside the available data. The models must be used in a supple way, without believing completely to the limit. The adoption of this method could allow the risk managers to observe the extreme events with a certain objectivity, to check the hierarchical organization of the classes of operational risks, and in the other hand, establish reserves to face these risks.
Aggregation issues in operational risk
The Journal of Operational Risk, 2008
In this paper we study copula-based models for aggregation of operational risk capital across business lines in a bank. A commonly used method of summation of the value-at-risk (VaR) measures, that relies on a hypothesis of full correlation of losses, becomes inappropriate in the presence of dependence between business lines and may lead to over-estimation of the capital charge. The problem can be further aggravated by the persistence of heavy tails in operational loss data; in some cases, the subadditivity property of value-at-risk may fail and the capital charge becomes underestimated. We use α-stable heavy-tailed distributions to model the loss data and then apply the copula approach in which the marginal distributions are consolidated in the symmetric and skewed Student tcopula framework. In our empirical study, we compare VaR and conditional VaR estimates with those obtained under the full correlation assumption. Our results demonstrate significant reduction in capital when a t-copula is employed. However, the capital reduction is significantly smaller than in cases where a moderately heavy-tailed or thin-tailed distribution is calibrated to loss data. We also show that for confidence levels below 94% VaR exhibits the super-additivity property.
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Methodology and Computing in Applied Probability, 2009
The management of Operational Risk has been a difficult task due to the lack of data and the high number of variables. In this project, we treat operational risks as multivariate variables. In order to model them, copula functions are employed, which are a widely used tool in finance and engineering for building flexible joint distributions. The purpose of this research is to propose a new methodology for modelling Operational Risks and estimating the required capital. It combines the use of graphical models and the use of copula functions along with hyper-Markov law. Historical loss data of an Italian bank is used, in order to explore the methodology's behaviour and its potential benefits.
Semi-nonparametric Estimation of Operational Risk Capital with Extreme Loss Events
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Operational risk modeling using the parametric models can lead to a counter-intuitive estimate of value at risk at 99.9% as economic capital due to extreme events. To address this issue, a flexible semi-nonparametric (SNP) model is introduced using the change of variables technique to enrich the family of distributions that can be used for modeling extreme events. The SNP models are proved to have the same maximum domain of attraction (MDA) as the parametric kernels, and it follows that the SNP models are consistent with the extreme value theory peaks over threshold method but with different shape and scale parameters. By using the simulated datasets generated from a mixture of distributions with varying body-tail thresholds, the SNP models in the Fréchet and Gumbel MDAs are shown to fit the datasets satisfactorily through increasing the number of model parameters, resulting in similar quantile estimates at 99.9%. When applied to an actual operational risk loss dataset from a major ...
A Bayesian approach to extreme value estimation in operational risk modeling
The Journal of Operational Risk, 2013
We propose a new approach for estimating operational risk models under the loss distribution approach from historically observed losses. Our method is based on extreme value theory and, being Bayesian in nature, allows us to incorporate other external information about the unknown parameters by use of expert opinions via elicitation or external data sources. This additional information can play a crucial role in reducing the statistical uncertainty about both parameter and capital estimates in situations where observed data are insufficient to accurately estimate the tail behavior of the loss distribution. Challenges of and strategies for formulating suitable priors are discussed. A simulation study demonstrates the performance of the new approach.
Operational-Risk Dependencies and the Determination of Risk Capital
SSRN Electronic Journal, 2000
With the advent of Basel II, risk-capital provisions need to also account for operational risk. The specification of dependence structures and the assessment of their effects on aggregate risk-capital are still open issues in modeling operational risk. In this paper, we investigate the potential consequences of adopting the restrictive Basel's Loss Distribution Approach (LDA), as compared to strategies that take dependencies explicitly into account. Drawing on a real-world database, we fit alternative dependence structures, using parametric copulas and nonparametric tail-dependence coefficients, and discuss the implications on the estimation of aggregate risk capital. We find that risk-capital estimates may increase relative to that derived for the LDA when accounting explicitly for the presence of dependencies. This phenomenon is not only be due to the (fitted) characteristics of the data, but also arise from the specific Monte Carlo setup in simulation-based risk-capital analysis.
International Journal of Risk …, 2008
The management of Operational Risks has always been difficult due to the high number of variables to work with and their complex multivariate distribution. A Copula is a statistic tool which has been recently used in finance and engineering to build flexible joint distributions in order to model a high number of variables. The goal of this paper is to propose its use to model Operational Risks, by showing its benefits with an empirical example.
Flexible dependence modeling of operational risk losses and its impact on total capital requirements
Journal of Banking & Finance, 2014
Operational risk data, when available, are usually scarce, heavy-tailed and possibly dependent. In this work, we introduce a model that captures such real-world characteristics and explicitly deals with heterogeneous pairwise and tail dependence of losses. By considering flexible families of copulas, we can easily move beyond modeling bivariate dependence among losses and estimate the total risk capital for the seven-and eightdimensional distributions of event types and business lines. Using real-world data, we then evaluate the impact of realistic dependence modeling on estimating the total regulatory capital, which turns out to be up to 38% smaller than what the standard Basel approach would prescribe.