Analysing systemic risk contribution using a closed formula for conditional value at risk through copula (original) (raw)

Measuring and Analysing Marginal Systemic Risk Contribution using CoVaR: A Copula Approach

This paper is devoted to the quantification and analysis of marginal risk contribution of a given single financial institution i to the risk of a financial system s. Our work expands on the CoVaR concept proposed by Adrian and Brunnermeier as a tool for the measurement of marginal systemic risk contribution. We first give a mathematical definition of CoVaR_{\alpha}^{s|L^i=l}. Our definition improves the CoVaR concept by expressing CoVaR_{\alpha}^{s|L^i=l} as a function of a state l and of a given probability level \alpha relative to i and s respectively. Based on Copula theory we connect CoVaR_{\alpha}^{s|L^i=l} to the partial derivatives of Copula through their probabilistic interpretation and definitions (Conditional Probability). Using this we provide a closed formula for the calculation of CoVaR_{\alpha}^{s|L^i=l} for a large class of (marginal) distributions and dependence structures (linear and non-linear). Our formula allows a better analysis of systemic risk using CoVaR in t...

Measuring systemic risk using vine-copula

We present an intuitive model of systemic risk to analyse the complex interdependencies between different borrowers. We characterise systemic risk by the way that financial institutions are interconnected. Using their probability of default, we classify different international financial institutions into five rating groups. Then we use the state-of-the-art canonical (C-) and D-vine copulae to investigate the partial correlation structure between the rating groups. Amongst many interesting findings, we discover that the second tier financial institutions pay a larger contribution to the systemic risk than the top tier borrowers. Further, we discuss an application of our methodology for pricing credit derivative swaps.

Integrating Systemic Risk and Risk Analysis Using Copulas

International Journal of Disaster Risk Science

Systemic risk research is gaining traction across diverse disciplinary research communities, but has as yet not been strongly linked to traditional, well-established risk analysis research. This is due in part to the fact that systemic risk research focuses on the connection of elements within a system, while risk analysis research focuses more on individual risk to single elements. We therefore investigate how current systemic risk research can be related to traditional risk analysis approaches from a conceptual as well as an empirical point of view. Based on Sklar's Theorem, which provides a one-to-one relationship between multivariate distributions and copulas, we suggest a reframing of the concept of copulas based on a network perspective. This provides a promising way forward for integrating individual risk (in the form of probability distributions) and systemic risk (in the form of copulas describing the dependencies among such distributions) across research domains. Copulas can link continuous node states, characterizing individual risks, with a gradual dependency of the coupling strength between nodes on their states, characterizing systemic risk. When copulas are used for describing such refined coupling between nodes, they can provide a more accurate quantification of a system's network structure. This enables more realistic systemic risk assessments, and is especially useful when extreme events (that occur at low probabilities, but have high impacts) affect a system's nodes. In this way, copulas can be informative in measuring and quantifying changes in systemic risk and therefore be helpful in its management. We discuss the advantages and limitations of copulas for integrative risk analyses from the perspectives of modeling, measurement, and management.

Systemic Risk Modeling with Lévy Copulas

Journal of Risk and Financial Management, 2021

We investigate a systemic risk measure known as CoVaR that represents the value-at-risk (VaR) of a financial system conditional on an institution being under distress. For characterizing and estimating CoVaR, we use the copula approach and introduce the normal tempered stable (NTS) copula based on the Lévy process. We also propose a novel backtesting method for CoVaR by a joint distribution correction. We test the proposed NTS model on the daily S&P 500 index and Dow Jones index with in-sample and out-of-sample tests. The results show that the NTS copula outperforms traditional copulas in the accuracy of both tail dependence and marginal processes modeling.

Measuring financial risks with copulas

International Review of Financial Analysis, 2004

This paper is concerned with the statistical modeling of the dependence structure of multivariate financial data using the concept of copulas. We select some special copulas and identify the type of dependency captured by each one. We fit copulas to daily returns and simulate from the fitted models. We compare the effect of the choice of copula on risk measures and assess the variability of one-step-ahead predictions of portfolio losses. We analyze extreme scenarios and fit extreme value copulas to the block maxima and minima from daily returns. The stress scenarios constructed are compared to those obtained using models from the extreme value theory. We illustrate the usefulness of the copula approach using two stock market indexes.

On multivariate extensions of the conditional Value-at-Risk measure

Insurance: Mathematics and Economics, 2015

CoVaR is a systemic risk measure proposed by Adrian and Brunnermeier [1] able to measure a financial institution's contribution to systemic risk and its contribution to the risk of other financial institutions. CoVaR stands for conditional Value-at-Risk, i.e. it indicates the Value at Risk for a financial institution that is conditional on a certain scenario. In this paper, two alternative extensions of the classic univariate Conditional Value-at-Risk are introduced in a multivariate setting. The two proposed multivariate CoVaRs are constructed from level sets of multivariate distribution functions (resp. of multivariate survival distribution functions). These vector-valued measures have the same dimension as the underlying risk portfolio. Several characterizations of these new risk measures are provided in terms of the copula structure and stochastic orderings of the marginal distributions. Interestingly, these results are consistent with existing properties on univariate risk measures. Furthermore, comparisons between existent risk measures and the proposed multivariate CoVaR are developed. Illustrations are given in the class of Archimedean copulas. Estimation procedure for the multivariate proposed CoVaRs is illustrated in simulated studies and insurance real data.

Using conditional copula to estimate value at risk

Journal of Data Science, 2006

Value at Risk (VaR) plays a central role in risk management. There are several approaches for the estimation of VaR, such as historical simulation, the variance-covariance (also known as analytical), and the Monte Carlo approaches. Whereas the first approach does not assume any distribution, the last two approaches demand the joint distribution to be known, which in the analytical approach is frequently the normal distribution. The copula theory is a fundamental tool in modeling multivariate distributions. It allows the definition of the joint distribution through the marginal distributions and the dependence between the variables. Recently the copula theory has been extended to the conditional case, allowing the use of copulae to model dynamical structures. Time variation in the first and second conditional moments is widely discussed in the literature, so allowing the time variation in the conditional dependence seems to be natural. This work presents some concepts and properties of copula functions and an application of the copula theory in the estimation of VaR of a portfolio composed by Nasdaq and S&P500 stock indices.

Bounds for value at risk – the approach based on copulas with homogeneous tails

The theory of copulas provides a useful tool for modeling dependence in risk management. In insurance and finance, as well as in other applications, dependence of extreme events is particularly important, hence there is a need for the detailed study of the tail behaviour of the multivariate copulas. In this paper we investigate the class of copulas having homogeneous lower tails. We show that having only such information on the structure of dependence of returns from assets is enough to get estimates on Value at Risk of the multiasset portfolio in terms of Value at Risk of one-asset portfolios.

Copula Methods and the Analysis of Credit Risk

SSRN Electronic Journal, 2000

Banks' internal credit risk assessment may take into account not only publicly available ratings but also internal measures of credit risk. We present a credit rating model of credit risk that permits banks to employ their own risk assessment in addition to public rating information.

Value-at-Risk Modeling with Conditional Copulas in Euclidean Space Framework

European Journal of Pure and Applied Mathematics

This paper aims to establish an analytic relation between a time-varying conditional copula and the value at risk modeled by the underlying. Specically, under the asumption that the space is euclidean we use scalar product to clarify a link between the conditional copula varying with time and norms. It is then established a new expression on the geometric yield