Expected shortfall(ES) is a new method to measure market risk. In this paper, an example was presented to illustrate that the ES is coherent but value-at-risk(VaR) is not coherent. Three formulas for calculating the E...Expected shortfall(ES) is a new method to measure market risk. In this paper, an example was presented to illustrate that the ES is coherent but value-at-risk(VaR) is not coherent. Three formulas for calculating the ES based on historical simulation method, normal method and GARCH method were derived. Further, a numerical experiment on optimizing portfolio using ES was provided.展开更多
In this paper we consider the problem of estimating expected shortfall(ES)for discrete time stochastic volatility(SV)models.Specifically,we develop Monte Carlo methods to evaluate ES for a variety of commonly used SV ...In this paper we consider the problem of estimating expected shortfall(ES)for discrete time stochastic volatility(SV)models.Specifically,we develop Monte Carlo methods to evaluate ES for a variety of commonly used SV models.This includes both models where the innovations are independent of the volatility and where there is dependence.This dependence aims to capture the well-known leverage effect.The performance of our Monte Carlo methods is analyzed through simulations and empirical analyses of four major US indices.展开更多
This paper analyzes the relationship between the risk factor of each stock and the portfolio’s risk based on a small portfolio with four U.S.stocks,and the reason why these risk factors can be regarded as a market in...This paper analyzes the relationship between the risk factor of each stock and the portfolio’s risk based on a small portfolio with four U.S.stocks,and the reason why these risk factors can be regarded as a market invariant.Then,it evaluates the properties of the convex and coherent risk indicators of the capital requirement index composed of VaR and ES,and use three methods(the historical estimation method,boudoukh’s mixed method and Monte Carlo method)to estimate the risk measurement indicators VaR and ES respectively based on the assumption of multivariate normal distribution’risk factors and multivariate student t-copula distribution’s one,finally it figures out that these three calculation results are very close.展开更多
Expected shortfall(ES),which conveys information regarding potential exceedances beyond the value-at-risk(VaR),is an important measure to characterize the properties of the tails of distribution.In this article,we stu...Expected shortfall(ES),which conveys information regarding potential exceedances beyond the value-at-risk(VaR),is an important measure to characterize the properties of the tails of distribution.In this article,we study two two-step estimation procedures for ES regression with censored responses.Considering the potential dependence between the censoring variable and the covariates,two locally weighted estimation algorithms are proposed based on local Kaplan–Meier estimation and the joint elicitability ofVaRand ES.The potential applications of thiswork aremanifold,especially survival analysis,pharmacodynamic analysis,and sociological investigations.The resulting estimators are shown to be consistent.Extensive simulations demonstrate that the proposed method performs quite well in finite samples,with regard to estimation bias and mean squared errors.Last,the analysis of a real dataset illustrates the usefulness of our developed methodologies.展开更多
Expected shortfall(ES)is a popular risk measure and plays an important role in risk and portfolio management.Recently,change-point detection of risk measures has been attracting much attention in finance.Based on the ...Expected shortfall(ES)is a popular risk measure and plays an important role in risk and portfolio management.Recently,change-point detection of risk measures has been attracting much attention in finance.Based on the self-normalized CUSUM statistic in Fan,Glynn and Pelger(2018)and the Wild Binary Segmentation(WBS)algorithm in Fryzlewicz(2014),this paper proposes a variant WBS procedure to detect and estimate change points of ES in time series.The strengthened Schwarz information criterion is also introduced to determine the number of change points.Monte Carlo simulation studies are conducted to assess the finite-sample performance of our variant WBS procedure about ES in time series.An empirical application is given to illustrate the usefulness of our procedure.展开更多
Among recent measures for risk management,value at risk(VaR)has been criticized because it is not coherent and expected shortfall(ES)has been criticized because it is not robust to outliers.Recently,[Math.Oper.Res.,38...Among recent measures for risk management,value at risk(VaR)has been criticized because it is not coherent and expected shortfall(ES)has been criticized because it is not robust to outliers.Recently,[Math.Oper.Res.,38,393–417(2013)]proposed a risk measure called median shortfall(MS)which is distributional robust and easy to implement.In this paper,we propose a more generalized risk measure called quantile shortfall(QS)which includes MS as a special case.QS measures the conditional quantile loss of the tail risk and inherits the merits of MS.We construct an estimator of the QS and establish the asymptotic normality behavior of the estimator.Our simulation shows that the newly proposed measures compare favorably in robustness with other widely used measures such as ES and VaR.展开更多
Value-at-Risk(VaR)and expected shortfall(ES)are two key risk measures in financial risk management.Comparing these two measures has been a hot debate,and most discussions focus on risk measure properties.This paper us...Value-at-Risk(VaR)and expected shortfall(ES)are two key risk measures in financial risk management.Comparing these two measures has been a hot debate,and most discussions focus on risk measure properties.This paper uses independent data and autoregressive models with normal or t-distribution to examine the effect of the heavy tail and dependence on comparing the nonparametric inference uncertainty of these two risk measures.Theoretical and numerical analyses suggest that VaR at 99%level is better than ES at 97.5%level for distributions with heavier tails.展开更多
An expectile can be considered a generalization of a quantile.While expected shortfall is a quantile-based risk measure,we study its counterpart-the expectile-based expected shortfall-where expectile takes the place o...An expectile can be considered a generalization of a quantile.While expected shortfall is a quantile-based risk measure,we study its counterpart-the expectile-based expected shortfall-where expectile takes the place of a quantile.We provide its dual representation in terms of a Bochner integral.Among other properties,we show that it is bounded from below in terms of the convex combination of expected shortfalls,and also from above by the smallest law invariant,coherent,and comonotonic risk measures,for which we give the explicit formulation of the corresponding distortion function.As a benchmark to the industry standard expected shortfall,we further provide its comparative asymptotic behavior in terms of extreme value distributions.Based on these results,we finally explicitly compute the expectile-based expected shortfall for selected classes of distributions.展开更多
This study investigates the return dynamics,volatility structure,and risk characteristics of five representative S&P 500 stocks:Johnson&Johnson,Microsoft,NVIDIA,Coca-Cola,and Home Depot,using ARMA-GARCH models...This study investigates the return dynamics,volatility structure,and risk characteristics of five representative S&P 500 stocks:Johnson&Johnson,Microsoft,NVIDIA,Coca-Cola,and Home Depot,using ARMA-GARCH models.Descriptive statistics and diagnostic tests confirm non-normality,negative skewness,fat tails,and volatility clustering,providing strong justification for conditional mean-variance modelling.Optimal model specifications are selected via the Bayesian Information Criterion,with EGARCH frameworks generally outperforming alternative GARCH variants in capturing asymmetric volatility responses.Rolling-window forecasts for 2024Q1 show that the models generate stable and reliable volatility predictions for low-volatility stocks(JNJ,KO),while performance is weaker for highly volatile stocks(NVDA),highlighting structural limitations under extreme market shifts.To evaluate risk management implications,one percent Value-at-Risk and expected shortfall were computed and backtested.Results indicated conservative tail-risk forecasts,with violation rates well within acceptable thresholds.Portfolio applications are further explored by constructing the Global Minimum Variance Portfolio(GMVP)and the Maximum Sharpe Ratio(Max SR)portfolio using rolling covariance estimates.Out-of-sample backtesting demonstrated that the GMVP delivered low volatility but modest returns,whereas the Max SR portfolio achieved significantly higher performance,consistent with the risk-return trade-off.Overall,the findings confirm that ARMA-GARCH models are effective tools for modelling conditional volatility and informing dynamic asset allocation.However,their limited adaptability to jump risk and nonlinear structural breaks underscores the need for more advanced modelling approaches in high-volatility environments.展开更多
文摘Expected shortfall(ES) is a new method to measure market risk. In this paper, an example was presented to illustrate that the ES is coherent but value-at-risk(VaR) is not coherent. Three formulas for calculating the ES based on historical simulation method, normal method and GARCH method were derived. Further, a numerical experiment on optimizing portfolio using ES was provided.
文摘In this paper we consider the problem of estimating expected shortfall(ES)for discrete time stochastic volatility(SV)models.Specifically,we develop Monte Carlo methods to evaluate ES for a variety of commonly used SV models.This includes both models where the innovations are independent of the volatility and where there is dependence.This dependence aims to capture the well-known leverage effect.The performance of our Monte Carlo methods is analyzed through simulations and empirical analyses of four major US indices.
文摘This paper analyzes the relationship between the risk factor of each stock and the portfolio’s risk based on a small portfolio with four U.S.stocks,and the reason why these risk factors can be regarded as a market invariant.Then,it evaluates the properties of the convex and coherent risk indicators of the capital requirement index composed of VaR and ES,and use three methods(the historical estimation method,boudoukh’s mixed method and Monte Carlo method)to estimate the risk measurement indicators VaR and ES respectively based on the assumption of multivariate normal distribution’risk factors and multivariate student t-copula distribution’s one,finally it figures out that these three calculation results are very close.
基金supported by the National Natural Science Foundation of China(Grant No.71973133)the Anhui Provincial Natural Science Foundation(Grant No.2208085J41).
文摘Expected shortfall(ES),which conveys information regarding potential exceedances beyond the value-at-risk(VaR),is an important measure to characterize the properties of the tails of distribution.In this article,we study two two-step estimation procedures for ES regression with censored responses.Considering the potential dependence between the censoring variable and the covariates,two locally weighted estimation algorithms are proposed based on local Kaplan–Meier estimation and the joint elicitability ofVaRand ES.The potential applications of thiswork aremanifold,especially survival analysis,pharmacodynamic analysis,and sociological investigations.The resulting estimators are shown to be consistent.Extensive simulations demonstrate that the proposed method performs quite well in finite samples,with regard to estimation bias and mean squared errors.Last,the analysis of a real dataset illustrates the usefulness of our developed methodologies.
基金supported in part by the NSFC(Nos.71973077 and 11771239)the Tsinghua University Initiative Scientific Research Program(No.2019Z07L01009).
文摘Expected shortfall(ES)is a popular risk measure and plays an important role in risk and portfolio management.Recently,change-point detection of risk measures has been attracting much attention in finance.Based on the self-normalized CUSUM statistic in Fan,Glynn and Pelger(2018)and the Wild Binary Segmentation(WBS)algorithm in Fryzlewicz(2014),this paper proposes a variant WBS procedure to detect and estimate change points of ES in time series.The strengthened Schwarz information criterion is also introduced to determine the number of change points.Monte Carlo simulation studies are conducted to assess the finite-sample performance of our variant WBS procedure about ES in time series.An empirical application is given to illustrate the usefulness of our procedure.
基金Supported by the National Natural Science Foundation of China(Grant No.11571263)Fundamental Research Funds for the Central Universities(Grant No.2042018kf0243)We thank the referees for their time and comments.
文摘Among recent measures for risk management,value at risk(VaR)has been criticized because it is not coherent and expected shortfall(ES)has been criticized because it is not robust to outliers.Recently,[Math.Oper.Res.,38,393–417(2013)]proposed a risk measure called median shortfall(MS)which is distributional robust and easy to implement.In this paper,we propose a more generalized risk measure called quantile shortfall(QS)which includes MS as a special case.QS measures the conditional quantile loss of the tail risk and inherits the merits of MS.We construct an estimator of the QS and establish the asymptotic normality behavior of the estimator.Our simulation shows that the newly proposed measures compare favorably in robustness with other widely used measures such as ES and VaR.
文摘Value-at-Risk(VaR)and expected shortfall(ES)are two key risk measures in financial risk management.Comparing these two measures has been a hot debate,and most discussions focus on risk measure properties.This paper uses independent data and autoregressive models with normal or t-distribution to examine the effect of the heavy tail and dependence on comparing the nonparametric inference uncertainty of these two risk measures.Theoretical and numerical analyses suggest that VaR at 99%level is better than ES at 97.5%level for distributions with heavier tails.
基金This research is supported by National Science Foundation of China(Grant No.11971310,11671257)“Assessment of Risk and Uncertainty in Finance”(Grant No.AF0710020)from Shanghai Jiao Tong University.
文摘An expectile can be considered a generalization of a quantile.While expected shortfall is a quantile-based risk measure,we study its counterpart-the expectile-based expected shortfall-where expectile takes the place of a quantile.We provide its dual representation in terms of a Bochner integral.Among other properties,we show that it is bounded from below in terms of the convex combination of expected shortfalls,and also from above by the smallest law invariant,coherent,and comonotonic risk measures,for which we give the explicit formulation of the corresponding distortion function.As a benchmark to the industry standard expected shortfall,we further provide its comparative asymptotic behavior in terms of extreme value distributions.Based on these results,we finally explicitly compute the expectile-based expected shortfall for selected classes of distributions.
文摘This study investigates the return dynamics,volatility structure,and risk characteristics of five representative S&P 500 stocks:Johnson&Johnson,Microsoft,NVIDIA,Coca-Cola,and Home Depot,using ARMA-GARCH models.Descriptive statistics and diagnostic tests confirm non-normality,negative skewness,fat tails,and volatility clustering,providing strong justification for conditional mean-variance modelling.Optimal model specifications are selected via the Bayesian Information Criterion,with EGARCH frameworks generally outperforming alternative GARCH variants in capturing asymmetric volatility responses.Rolling-window forecasts for 2024Q1 show that the models generate stable and reliable volatility predictions for low-volatility stocks(JNJ,KO),while performance is weaker for highly volatile stocks(NVDA),highlighting structural limitations under extreme market shifts.To evaluate risk management implications,one percent Value-at-Risk and expected shortfall were computed and backtested.Results indicated conservative tail-risk forecasts,with violation rates well within acceptable thresholds.Portfolio applications are further explored by constructing the Global Minimum Variance Portfolio(GMVP)and the Maximum Sharpe Ratio(Max SR)portfolio using rolling covariance estimates.Out-of-sample backtesting demonstrated that the GMVP delivered low volatility but modest returns,whereas the Max SR portfolio achieved significantly higher performance,consistent with the risk-return trade-off.Overall,the findings confirm that ARMA-GARCH models are effective tools for modelling conditional volatility and informing dynamic asset allocation.However,their limited adaptability to jump risk and nonlinear structural breaks underscores the need for more advanced modelling approaches in high-volatility environments.
