Bayesian inference for CoVaR

Recent financial disasters emphasised the need to investigate the consequence associated with the tail co-movements among institutions; episodes of contagion are frequently observed and increase the probability of large losses affecting market participants' risk capital. Commonly used risk management tools fail to account for potential spillover effects among institutions because they provide individual risk assessment. We contribute to analyse the interdependence effects of extreme events providing an estimation tool for evaluating the Conditional Value-at-Risk (CoVaR) defined as the Value-at-Risk of an institution conditioned on another institution being under distress. In particular, our approach relies on Bayesian quantile regression framework. We propose a Markov chain Monte Carlo algorithm exploiting the Asymmetric Laplace distribution and its representation as a location-scale mixture of Normals. Moreover, since risk measures are usually evaluated on time series data and returns typically change over time, we extend the CoVaR model to account for the dynamics of the tail behaviour. Application on U.S. companies belonging to different sectors of the Standard and Poor's Composite Index (S&P500) is considered to evaluate the marginal contribution to the overall systemic risk of each individual institutio

Portfolio Optimization with Relative Tail Risk

This paper proposes analytic forms of portfolio CoVaR and CoCVaR on the normal tempered stable market model. Since CoCVaR captures the relative risk of the portfolio with respect to a benchmark return, we apply it to the relative portfolio optimization. Moreover, we derive analytic forms for the marginal contribution to CoVaR and the marginal contribution to CoCVaR. We discuss the Monte-Carlo simulation method to calculate CoCVaR and the marginal contributions of CoVaR and CoCVaR. As the empirical illustration, we show relative portfolio optimization with thirty stocks under the distress condition of the Dow Jones Industrial Average. Finally, we perform the risk budgeting method to reduce the CoVaR and CoCVaR of the portfolio based on the marginal contributions to CoVaR and CoCVaR

Measuring systemic risk in the Nordic countries - An application of CoVaR

Spillover effects and systemic risk contribution of institutions, as measured by their CoVaR and delta-CoVaR respectively, is one way of assessing risk both for an institution in isolation, as well as for regulators and the economy as a whole. CoVaR is the q%-VaR of an institution conditional on another institution already being at its q%-VaR level, whereas delta-CoVaR measures each institution’s marginal risk contribution. This essay applies the CoVaR methodology proposed by Adrian and Brunnermeier (2011) on the Nordic stock market (OMX Nordic 40 Index) in order to measure systemic risk contribution of 36 firms on this market, during the period January 2002 to March 2014. Publicly available stock market data is used to estimate abovementioned measures by applying quantile regression. The results, which are aggregated at sector level, suggest that systemic risk contribution is higher during times of financial distress and sectors generally show a similar pattern in how risky they are over time. VaR is further not positively correlated with CoVaR, i.e. even if a sector is considered risky in isolation as measured by its VaR, it is not necessarily the case that it spills over this risk to other sectors/institutions. However, there are some sectors that contribute more to systemic risk than they are risky in isolation, as measured by their delta-CoVaR and VaR. Sectors contributing the most to Nordic systemic risk are Forestry and Construction, as well as the European stock market as measured by the EuroStoxx50 Index. The banks included in the OMX Nordic 40 Index are also examined in a separate case study, finding Swedbank the most risky and Nordea the least risky in isolation, but the other way around when measuring risk contribution (delta-CoVaR) of these two banks, to other banks

Quantile Regression in Risk Calibration

Financial risk control has always been challenging and becomes now an even harder problem as joint extreme events occur more frequently. For decision makers and government regulators, it is therefore important to obtain accurate information on the interdependency of risk factors. Given a stressful situation for one market participant, one likes to measure how this stress affects other factors. The CoVaR (Conditional VaR) framework has been developed for this purpose. The basic technical elements of CoVaR estimation are two levels of quantile regression: one on market risk factors; another on individual risk factor. Tests on the functional form of the two-level quantile regression reject the linearity. A flexible semiparametric modeling framework for CoVaR is proposed. A partial linear model (PLM) is analyzed. In applying the technology to stock data covering the crisis period, the PLM outperforms in the crisis time, with the justification of the backtesting procedures. Moreover, using the data on global stock markets indices, the analysis on marginal contribution of risk (MCR) defined as the local first order derivative of the quantile curve sheds some light on the source of the global market risk.CoVaR, Value-at-Risk, quantile regression, locally linear quantile regression, partial linear model, semiparametric model

CoVaR

We propose a measure for systemic risk: CoVaR, the value at risk (VaR) of the financial system conditional on institutions being under distress. We define an institution's contribution to systemic risk as the difference between CoVaR conditional on the institution being under distress and the CoVaR in the median state of the institution. From our estimates of CoVaR for the universe of publicly traded financial institutions, we quantify the extent to which characteristics such as leverage, size, and maturity mismatch predict systemic risk contribution. We also provide out of sample forecasts of a countercyclical, forward looking measure of systemic risk and show that the 2006Q4 value of this measure would have predicted more than half of realized covariances during the financial crisis.

On dependence consistency of CoVaR and some other systemic risk measures

This paper is dedicated to the consistency of systemic risk measures with respect to stochastic dependence. It compares two alternative notions of Conditional Value-at-Risk (CoVaR) available in the current literature. These notions are both based on the conditional distribution of a random variable Y given a stress event for a random variable X , but they use different types of stress events. We derive representations of these alternative CoVaR notions in terms of copulas, study their general dependence consistency and compare their performance in several stochastic models. Our central finding is that conditioning on X  ≥ VaR α ( X ) gives a much better response to dependence between X and Y than conditioning on X  = VaR α ( X ). We prove general results that relate the dependence consistency of CoVaR using conditioning on X  ≥ VaR α ( X ) to well established results on concordance ordering of multivariate distributions or their copulas. These results also apply to some other systemic risk measures, such as the Marginal Expected Shortfall (MES) and the Systemic Impact Index (SII). We provide counterexamples showing that CoVaR based on the stress event X  = VaR α ( X ) is not dependence consistent. In particular, if ( X ,  Y ) is bivariate normal, then CoVaR based on X  = VaR α ( X ) is not an increasing function of the correlation parameter. Similar issues arise in the bivariate t model and in the model with t margins and a Gumbel copula. In all these cases, CoVaR based on X  ≥ VaR α ( X ) is an increasing function of the dependence paramete

MENGUKUR RISIKO SISTEMIK PERBANKAN DI INDONESIA: APLIKASI MODEL CONDITIONAL VALUE-AT-RISK (ΔCoVaR)

Systemic risk is a risk of collapse of the financial system that would cause the financial system is not functioning properly. Systemic risk is generally triggered by the failure of a financial institution that will be transmitted to other financial institutions. Measurement of systemic risk in the financial institutions, especially banks is crucial because banks are highly vulnerable to financial crisis. Methods of measurement of systemic risk conditional value-at-risk (CoVaR) introduced by Adrian and Brunnermeier (2011) is a correspondence between the value-at-risk yields obtained conditional on some event observed from a financial institution. The aim of this study was to measure the systemic risk contribution by individual banks and analyze the relationship between risk individuals with systemic risks posed to the financial system when individual banks during distress conditions. In this study, to estimate the conditional value-at-risk (CoVaR) used quantile regression, where the amount of the quantile can represent CoVaR when distress and CoVaR when conditions are medians. The amount of contribution of a financial institution to systemic risk in the financial system is measured by using a marginal CoVaR (ΔCoVaR), which represents the difference between CoVaR in distress with a median condition. Samples in this study of nine bank has total assets of the largest in Indonesia. This study using purposive sampling method, during the period January 2005 to December 2014. Testing the correlation between VaR and ΔCoVaR in this study using Spearman correlation and Kendall's Tau. Based on the results of this study indicate that the contribution of high systemic risk during the study period was not owned by the bank that has the largest total assets among sample banks. There are five banks that have a significant correlation between VaR and ΔCoVaR, meanwhile four others banks in the sample did not have a significant correlation. However, the correlation coefficient is below 0,5, which indicates that there is a weak correlation between VaR and ΔCoVaR