We consider a distribution system with one supplier and two retailers. For the two retailers, they face different demand and are both risk averse. We study a single period model which the supplier has ample goods and ...We consider a distribution system with one supplier and two retailers. For the two retailers, they face different demand and are both risk averse. We study a single period model which the supplier has ample goods and the retailers order goods separately. Market search is measured as the fraction of customers who unsatisfied with their "local" retailer due to stock-out, and search for the goods at the other retailer before leaving the system. We investigate how the retailers game for order quantity in a Conditional Value-at-Risk framework and study how risk averse degree, market search level, holding cost and backorder cost influence the optimal order strategies. Furthermore, we use uniform distribution to illustrate these results and obtain Nash equilibrium of order strategies.展开更多
This study investigates the volatility in daily stock returns for Total Nigeria Plc using nine variants of GARCH models:sGARCH,girGARCH,eGARCH,iGARCH,aGARCH,TGARCH,NGARCH,NAGARCH,and AVGARCH along with value at risk e...This study investigates the volatility in daily stock returns for Total Nigeria Plc using nine variants of GARCH models:sGARCH,girGARCH,eGARCH,iGARCH,aGARCH,TGARCH,NGARCH,NAGARCH,and AVGARCH along with value at risk estimation and backtesting.We use daily data for Total Nigeria Plc returns for the period January 2,2001 to May 8,2017,and conclude that eGARCH and sGARCH perform better for normal innovations while NGARCH performs better for student t innovations.This investigation of the volatility,VaR,and backtesting of the daily stock price of Total Nigeria Plc is important as most previous studies covering the Nigerian stock market have not paid much attention to the application of backtesting as a primary approach.We found from the results of the estimations that the persistence of the GARCH models are stable except for few cases for which iGARCH and eGARCH were unstable.Additionally,for student t innovation,the sGARCH and girGARCH models failed to converge;the mean reverting number of days for returns differed from model to model.From the analysis of VaR and its backtesting,this study recommends shareholders and investors continue their business with Total Nigeria Plc because possible losses may be overcome in the future by improvements in stock prices.Furthermore,risk was reflected by significant up and down movement in the stock price at a 99%confidence level,suggesting that high risk brings a high return.展开更多
This paper investigates methods of value-at-risk (VaR) estimation using extreme value theory (EVT). It compares two different estimation methods, 'two-step subsample bootstrap' based on moment estimation and m...This paper investigates methods of value-at-risk (VaR) estimation using extreme value theory (EVT). It compares two different estimation methods, 'two-step subsample bootstrap' based on moment estimation and maximum likelihood estimation (MLE), according to their theoretical bases and computation procedures. Then, the estimation results are analyzed together with those of normal method and empirical method. The empirical research of foreign exchange data shows that the EVT methods have good characters in estimating VaR under extreme conditions and 'two-step subsample bootstrap' method is preferable to MLE.展开更多
We consider risk minimization problems for Markov decision processes. From a standpoint of making the risk of random reward variable at each time as small as possible, a risk measure is introduced using conditional va...We consider risk minimization problems for Markov decision processes. From a standpoint of making the risk of random reward variable at each time as small as possible, a risk measure is introduced using conditional value-at-risk for random immediate reward variables in Markov decision processes, under whose risk measure criteria the risk-optimal policies are characterized by the optimality equations for the discounted or average case. As an application, the inventory models are considered.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
The real-time risk-averse dispatch problem of an integrated electricity and natural gas system(IEGS)is studied in this paper.It is formulated as a real-time conditional value-at-risk(CVaR)-based risk-averse dis-patch ...The real-time risk-averse dispatch problem of an integrated electricity and natural gas system(IEGS)is studied in this paper.It is formulated as a real-time conditional value-at-risk(CVaR)-based risk-averse dis-patch model in the Markov decision process framework.Because of its stochasticity,nonconvexity and nonlinearity,the model is difficult to analyze by traditional algorithms in an acceptable time.To address this non-deterministic polynomial-hard problem,a CVaR-based lookup-table approximate dynamic programming(CVaR-ADP)algo-rithm is proposed,and the risk-averse dispatch problem is decoupled into a series of tractable subproblems.The line pack is used as the state variable to describe the impact of one period’s decision on the future.This facilitates the reduction of load shedding and wind power curtailment.Through the proposed method,real-time decisions can be made according to the current information,while the value functions can be used to overview the whole opti-mization horizon to balance the current cost and future risk loss.Numerical simulations indicate that the pro-posed method can effectively measure and control the risk costs in extreme scenarios.Moreover,the decisions can be made within 10 s,which meets the requirement of the real-time dispatch of an IEGS.Index Terms—Integrated electricity and natural gas system,approximate dynamic programming,real-time dispatch,risk-averse,conditional value-at-risk.展开更多
This paper is concerned with the statistical modeling of the dependence structure of multivariate financial data using the copula, and the application of copula functions in VaR valuation. After the introduction of th...This paper is concerned with the statistical modeling of the dependence structure of multivariate financial data using the copula, and the application of copula functions in VaR valuation. After the introduction of the pure copula method and the maximum and minimum mixture copula method, authors present a new algorithm based on the more generalized mixture copula functions and the dependence measure, and apply the method to the portfolio of Shanghai stock composite index and Shenzhen stock component index. Comparing with the results from various methods, one can find that the mixture copula method is better than the pure Gaussian copula method and the maximum and minimum mixture copula method on different VaR level.展开更多
Bayesian inference method has been presented in this paper for the modeling of operational risk. Bank internal and external data are divided into defined loss cells and then fitted into probability distributions. The ...Bayesian inference method has been presented in this paper for the modeling of operational risk. Bank internal and external data are divided into defined loss cells and then fitted into probability distributions. The distribution parameters and their uncertainties are estimated from posterior distributions derived using the Bayesian inference. Loss frequency is fitted into Poisson distributions. While the Poisson parameters, in a similar way, are defined by a posterior distribution developed using Bayesian inference. Bank operation loss typically has some low frequency but high magnitude loss data. These heavy tail low frequency loss data are divided into several buckets where the bucket frequencies are defined by the experts. A probability distribution, as defined by the internal and external data, is used for these data. A Poisson distribution is used for the bucket frequencies. However instead of using any distribution of the Poisson parameters, point estimations are used. Monte Carlo simulation is then carried out to calculate the capital charge of the in- ternal as well as the heavy tail high profile low frequency losses. The output of the Monte Carlo simulation defines the capital requirement that has to be allocated to cover potential operational risk losses for the next year.展开更多
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
基金Supported by the National Natural Science Foundation of China (70471034, A0324666)
文摘We consider a distribution system with one supplier and two retailers. For the two retailers, they face different demand and are both risk averse. We study a single period model which the supplier has ample goods and the retailers order goods separately. Market search is measured as the fraction of customers who unsatisfied with their "local" retailer due to stock-out, and search for the goods at the other retailer before leaving the system. We investigate how the retailers game for order quantity in a Conditional Value-at-Risk framework and study how risk averse degree, market search level, holding cost and backorder cost influence the optimal order strategies. Furthermore, we use uniform distribution to illustrate these results and obtain Nash equilibrium of order strategies.
文摘This study investigates the volatility in daily stock returns for Total Nigeria Plc using nine variants of GARCH models:sGARCH,girGARCH,eGARCH,iGARCH,aGARCH,TGARCH,NGARCH,NAGARCH,and AVGARCH along with value at risk estimation and backtesting.We use daily data for Total Nigeria Plc returns for the period January 2,2001 to May 8,2017,and conclude that eGARCH and sGARCH perform better for normal innovations while NGARCH performs better for student t innovations.This investigation of the volatility,VaR,and backtesting of the daily stock price of Total Nigeria Plc is important as most previous studies covering the Nigerian stock market have not paid much attention to the application of backtesting as a primary approach.We found from the results of the estimations that the persistence of the GARCH models are stable except for few cases for which iGARCH and eGARCH were unstable.Additionally,for student t innovation,the sGARCH and girGARCH models failed to converge;the mean reverting number of days for returns differed from model to model.From the analysis of VaR and its backtesting,this study recommends shareholders and investors continue their business with Total Nigeria Plc because possible losses may be overcome in the future by improvements in stock prices.Furthermore,risk was reflected by significant up and down movement in the stock price at a 99%confidence level,suggesting that high risk brings a high return.
基金the National Natural Science Foundation of China (No. 79970041).
文摘This paper investigates methods of value-at-risk (VaR) estimation using extreme value theory (EVT). It compares two different estimation methods, 'two-step subsample bootstrap' based on moment estimation and maximum likelihood estimation (MLE), according to their theoretical bases and computation procedures. Then, the estimation results are analyzed together with those of normal method and empirical method. The empirical research of foreign exchange data shows that the EVT methods have good characters in estimating VaR under extreme conditions and 'two-step subsample bootstrap' method is preferable to MLE.
文摘We consider risk minimization problems for Markov decision processes. From a standpoint of making the risk of random reward variable at each time as small as possible, a risk measure is introduced using conditional value-at-risk for random immediate reward variables in Markov decision processes, under whose risk measure criteria the risk-optimal policies are characterized by the optimality equations for the discounted or average case. As an application, the inventory models are considered.
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supported by State Key Laboratory of HVDC under Grant SKLHVDC-2021-KF-09.
文摘The real-time risk-averse dispatch problem of an integrated electricity and natural gas system(IEGS)is studied in this paper.It is formulated as a real-time conditional value-at-risk(CVaR)-based risk-averse dis-patch model in the Markov decision process framework.Because of its stochasticity,nonconvexity and nonlinearity,the model is difficult to analyze by traditional algorithms in an acceptable time.To address this non-deterministic polynomial-hard problem,a CVaR-based lookup-table approximate dynamic programming(CVaR-ADP)algo-rithm is proposed,and the risk-averse dispatch problem is decoupled into a series of tractable subproblems.The line pack is used as the state variable to describe the impact of one period’s decision on the future.This facilitates the reduction of load shedding and wind power curtailment.Through the proposed method,real-time decisions can be made according to the current information,while the value functions can be used to overview the whole opti-mization horizon to balance the current cost and future risk loss.Numerical simulations indicate that the pro-posed method can effectively measure and control the risk costs in extreme scenarios.Moreover,the decisions can be made within 10 s,which meets the requirement of the real-time dispatch of an IEGS.Index Terms—Integrated electricity and natural gas system,approximate dynamic programming,real-time dispatch,risk-averse,conditional value-at-risk.
基金Supported by Research Projects of Humanities and Social Sciences Foundation of Ministry of Education
文摘This paper is concerned with the statistical modeling of the dependence structure of multivariate financial data using the copula, and the application of copula functions in VaR valuation. After the introduction of the pure copula method and the maximum and minimum mixture copula method, authors present a new algorithm based on the more generalized mixture copula functions and the dependence measure, and apply the method to the portfolio of Shanghai stock composite index and Shenzhen stock component index. Comparing with the results from various methods, one can find that the mixture copula method is better than the pure Gaussian copula method and the maximum and minimum mixture copula method on different VaR level.
文摘Bayesian inference method has been presented in this paper for the modeling of operational risk. Bank internal and external data are divided into defined loss cells and then fitted into probability distributions. The distribution parameters and their uncertainties are estimated from posterior distributions derived using the Bayesian inference. Loss frequency is fitted into Poisson distributions. While the Poisson parameters, in a similar way, are defined by a posterior distribution developed using Bayesian inference. Bank operation loss typically has some low frequency but high magnitude loss data. These heavy tail low frequency loss data are divided into several buckets where the bucket frequencies are defined by the experts. A probability distribution, as defined by the internal and external data, is used for these data. A Poisson distribution is used for the bucket frequencies. However instead of using any distribution of the Poisson parameters, point estimations are used. Monte Carlo simulation is then carried out to calculate the capital charge of the in- ternal as well as the heavy tail high profile low frequency losses. The output of the Monte Carlo simulation defines the capital requirement that has to be allocated to cover potential operational risk losses for the next year.
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.