Conditional Expected Shortfall Research Articles

Financial Risk Measurement and Spatial Spillover Effects Based on an Imported Financial Risk Network: Evidence from Countries along the Belt and Road

Using the financial market data of 35 countries along the Belt and Road (B&R), this paper constructs an imported financial risk network based on the conditional expected shortfall (CoES) to measure the systemic financial risk of the countries along the B&R. Furthermore, complex network theory is combined with spatial econometrics to construct a spatial, financial network panel model to measure the spatial spillover effects of imported financial risks and further explore the macroeconomic influences on systemic financial risks. The results show that among the countries along the B&R, the level of systemic financial risk in the European region is higher than that in the Asian region from the imported risk perspective. The spatial spillover effect of financial risk and the spatial spillover effect from the imported risk perspective have time-varying characteristics, with the spatial spillover effect increasing significantly during crisis periods. In addition, indicators of the three dimensions of economic openness, the institutional environment, and the external policy environment all have significant effects on systemic financial risk, but the effects differ across regions and periods.

SIMULTANEOUS CONFIDENCE BANDS FOR CONDITIONAL VALUE-AT-RISK AND EXPECTED SHORTFALL

Conditional value-at-risk (CVaR) and conditional expected shortfall (CES) are widely adopted risk measures which help monitor potential tail risk while adapting to evolving market information. In this paper, we propose an approach to constructing simultaneous confidence bands (SCBs) for tail risk as measured by CVaR and CES, with the confidence bands uniformly valid for a set of tail levels. We consider one-sided tail risk (downside or upside tail risk) as well as relative tail risk (the ratio of upside to downside tail risk). A general class of location-scale models with heavy-tailed innovations is employed to filter out the return dynamics. Then, CVaR and CES are estimated with the aid of extreme value theory. In the asymptotic theory, we consider two scenarios: (i) the extreme scenario that allows for extrapolation beyond the range of the available data and (ii) the intermediate scenario that works exclusively in the case where the available data are adequate relative to the tail level. For finite-sample implementation, we propose a novel bootstrap procedure to circumvent the slow convergence rates of the SCBs as well as infeasibility of approximating the limiting distributions. A series of Monte Carlo simulations confirm that our approach works well in finite samples.

Worst-case conditional value-at-risk and conditional expected shortfall based on covariance information

In this paper, we study the worst-case conditional value-at-risk (CoVaR) and conditional expected shortfall (CoES) in a situation where only partial information on the underlying probability distribution is available. In the case of the first two marginal moments are known, the closed-form solution and the value of the worst-case CoVaR and CoES are derived. The worst-case CoVaR and CoES under mean and covariance information are also investigated.

Risk of spillovers between the Chinese and international crude oil futures’ markets: A dynamic Copula-CoES approach

To overcome the limitations of conditional value-at-risk (CoVaR), the upside conditional expected shortfall (CoES) is proposed and used as a measure of the extreme risk spillover effect between the Chinese and international crude oil futures markets. Then, the calculation formula of the upside CoES is given under the semiparametric dynamic Copula model, which does not need to set the specific evolution equation of the Copula parameters. The empirical results show that there is a strong positive correlation between Chinese and international crude oil futures, and the correlation gradually increases when oil prices rise. There is a significant two-way risk spillover effect for both of them, and the average risk spillover intensity between Chinese crude oil futures (INE) and West Texas intermediate crude oil futures (WTI) is the highest. Moreover, INE is not always the recipient of the price fluctuation risk. In particular, INE has played the role of risk spillover most of the time after 2019. This study provides new ideas for the measurement of risk spillovers.

Stochastic orders and multivariate measures of risk contagion

Co-risk measures and risk contributions measures are used in portfolio risk analysis to assess and quantify the risk of contagion, given that one or more assets in the portfolio are in distress. In this paper, given two random vectors X and Y that represent two portfolios of n assets (n≥2) and exhibit some kind of positive dependence, we give sufficient conditions based on stochastic orders to compare the risk of contagion of the portfolios. The measures of risk contagion that we consider are the conditional value at risk (CoVaR), the conditional expected shortfall (CoES) and the recently introduced marginal mean excess (MME).

Markov Chain Monte Carlo Methods for Estimating Systemic Risk Allocations

In this paper, we propose a novel framework for estimating systemic risk measures and risk allocations based on Markov Chain Monte Carlo (MCMC) methods. We consider a class of allocations whose jth component can be written as some risk measure of the jth conditional marginal loss distribution given the so-called crisis event. By considering a crisis event as an intersection of linear constraints, this class of allocations covers, for example, conditional Value-at-Risk (CoVaR), conditional expected shortfall (CoES), VaR contributions, and range VaR (RVaR) contributions as special cases. For this class of allocations, analytical calculations are rarely available, and numerical computations based on Monte Carlo (MC) methods often provide inefficient estimates due to the rare-event character of the crisis events. We propose an MCMC estimator constructed from a sample path of a Markov chain whose stationary distribution is the conditional distribution given the crisis event. Efficient constructions of Markov chains, such as the Hamiltonian Monte Carlo and Gibbs sampler, are suggested and studied depending on the crisis event and the underlying loss distribution. The efficiency of the MCMC estimators is demonstrated in a series of numerical experiments.

Dependence and risk spillovers between mainland China and London stock markets before and after the Stock Connect programs

This paper investigates the variations in dependence and risk spillover between mainland China and London stock markets before and after two Stock Connect programs launched in China, by measuring the Conditional Value-at-Risk (CoVaR) and Conditional Expected Shortfall (CoES) using realized volatility which is calculated with high-frequency data and Copula model. The empirical results show that, first, the Shanghai–Hong Kong Stock Connect program enhances the dependence between mainland China and London stock markets, while the overall dependence decreases slightly after the Shenzhen–Hong Kong Stock Connect program. More specifically, the lower-tail dependence between mainland China and London stock markets continues to increase after the second Stock Connect scheme. The upper-tail dependence, however, rises up after the first Stock Connect scheme but decreases sharply after the second Connect scheme. Second, the Stock Connect programs strength the downside risk spillovers between the mainland China and London markets in most cases. Finally, the equal tests on risk spillovers show significant asymmetric features, indicating that the mainland China stock markets are more likely affected by the London stock market.

Confidence Intervals for Conditional Tail Risk Measures in ARMA–GARCH Models

ABSTRACT ARMA–GARCH models are widely used to model the conditional mean and conditional variance dynamics of returns on risky assets. Empirical results suggest heavy-tailed innovations with positive extreme value index for these models. Hence, one may use extreme value theory to estimate extreme quantiles of residuals. Using weak convergence of the weighted sequential tail empirical process of the residuals, we derive the limiting distribution of extreme conditional Value-at-Risk (CVaR) and conditional expected shortfall (CES) estimates for a wide range of extreme value index estimators. To construct confidence intervals, we propose to use self-normalization. This leads to improved coverage vis-à-vis the normal approximation, while delivering slightly wider confidence intervals. A data-driven choice of the number of upper order statistics in the estimation is suggested and shown to work well in simulations. An application to stock index returns documents the improvements of CVaR and CES forecasts.

Measuring bank downside systemic risk in Taiwan

In this study, both systemic and individual bank risks across domestic and international banks in Taiwan are analyzed given a risk event breakout, using the conditional value-at-risk (CoVaR) and the conditional expected shortfall (CoES) by estimating the bivariate quantile autoregression with asymmetric downside risk adjustment to the daily stock returns of banks. The estimation results reveal the significant external asymmetric downside risk in individual bank systemic risk in contrast to the internal asymmetric downside risk in system itself. The individual bank systemic risk contribution is idiosyncratic and incoherent, but the system risk impact on banks is consistent and co-integrated toward all banks. The empirical results show that the larger size or leverage of the individual banks causes higher system risk impact, as they are hit and go down jointly in shocking events. The foreign banks incur and spread more symmetric external risk, whereas the domestic banks suffer and release more asymmetric external risk. Accordingly, the banks in North America and Europe not only take system risk but contribute the most systemic risk during the subprime crisis and Greek debt crisis, respectively, leading to the episode of asymmetric volatility externality effect on the Taiwanese banks in Asia.

Stochastic orders and co-risk measures under positive dependence

Conditional risk measures (or co-risk measures) and risk contribution measures are increasingly used in actuarial portfolio analysis to evaluate the systemic risk, which is related to the risk that the failure or loss of a component spreads to another component or even to the whole portfolio: while co-risk measures are risk-adjusted versions of measures usually employed to assess isolate risks, risk contribution measures quantify how a stress situation for a component affects another one. In this paper, we provide sufficient conditions under which two random vectors could be compared in terms of CoVaR (conditional value-at-risk), CoES (conditional expected shortfall) and different risk contribution measures. Conditions are given in terms of the increasing convex order, the dispersive order and the excess wealth order of the marginals under some assumptions of positive dependence.

Measuring systemic risk in the European banking sector: a copula CoVaR approach

ABSTRACTWe propose a new methodology based on copula functions to estimate CoVaR, the Value-at-Risk (VaR) of the financial system conditional on an institution being under financial distress. Our Copula CoVaR approach provides simple, closed-form expressions for various definitions of CoVaR for a broad range of copula families and allows the CoVaR of an institution to have time-varying exposure to its VaR. We extend this approach to estimate other ‘co-risk’ measures such as Conditional Expected Shortfall (CoES). We focus on a portfolio of large European banks and examine the existence of common market factors triggering systemic risk episodes. Further, we analyse the extent to which bank-specific characteristics such as size, leverage, and equity beta are associated with institutions' contribution to systemic risk and highlight the importance of liquidity risk at the outset of the financial crisis in summer 2007. Finally, we investigate the link between macroeconomy and systemic risk and find that changes in major macroeconomic variables can contribute significantly to systemic risk.

NONPARAMETRIC ESTIMATION OF CONDITIONAL VALUE-AT-RISK AND EXPECTED SHORTFALL BASED ON EXTREME VALUE THEORY

We propose nonparametric estimators for conditional value-at-risk (CVaR) and conditional expected shortfall (CES) associated with conditional distributions of a series of returns on a financial asset. The return series and the conditioning covariates, which may include lagged returns and other exogenous variables, are assumed to be strong mixing and follow a nonparametric conditional location-scale model. First stage nonparametric estimators for location and scale are combined with a generalized Pareto approximation for distribution tails proposed by Pickands (1975,Annals of Statistics3, 119–131) to give final estimators for CVaR and CES. We provide consistency and asymptotic normality of the proposed estimators under suitable normalization. We also present the results of a Monte Carlo study that sheds light on their finite sample performance. Empirical viability of the model and estimators is investigated through a backtesting exercise using returns on future contracts for five agricultural commodities.

Nonlinear expectile regression with application to Value-at-Risk and expected shortfall estimation

This paper considers nonlinear expectile regression models to estimate conditional expected shortfall (ES) and Value-at-Risk (VaR). In the literature, the asymmetric least squares (ALS) regression method has been widely used to estimate expectile regression models. However, no literatures rigorously investigated the asymptotic properties of the ALS estimates in nonlinear models with heteroscedasticity. Motivated by this aspect, this paper studies the consistency and asymptotic normality of the ALS estimates and conditional VaR and ES in those models. To illustrate, a simulation study and real data analysis are conducted.

Interconnected Risk Contributions: A Heavy-Tail Approach to Analyze U.S. Financial Sectors

This paper investigates the dynamic evolution of tail risk interdependence among U.S. banks, financial services and insurance sectors. Life and non-life insurers have been considered separately to account for their different characteristics. The tail risk interdependence measurement framework relies on the multivariate Student-t Markov switching (MS) model and the multiple-conditional value-at-risk (CoVaR) (conditional expected shortfall (CoES)) risk measures introduced in Bernardi et al. (2013), accounting for both the stylized facts of financial data and the contemporaneous multiple joint distress events. The Shapley value methodology is then applied to compose the puzzle of individual risk attributions, providing a synthetic measure of tail interdependence. Our empirical investigation finds that banks appear to contribute more to the tail risk evolution of all of the remaining sectors, followed by the financial services and the insurance sectors, showing that the insurance sector significantly contributes as well to the overall risk. We also find that the role of each sector in contributing to other sectors’ distress evolves over time according to the current predominant financial condition, implying different interdependence strength.

Weighted Nadaraya-Watson Estimation of Conditional Expected Shortfall

This paper addresses the problem of nonparametric estimation of the conditional expected shortfall (CES) which has gained popularity in nancial risk management. We propose a new nonparametric estimator of the CES. The proposed estimator is dened as a conditional counterpart of the sample average estimator of the unconditional expected shortfall, where the empirical distribution function is replaced by the weighted Nadaraya-Watson estimator of the conditional distribution function. We establish asymptotic normality of the proposed estimator under an -mixing condition. The asymptotic results reveal that the proposed estimator has a good bias property. Simulation results illustrate the usefulness of the proposed estimator.