This paper clarifies the distinctions between loan loss reserves （LLR）, expected loss （EL）, and loan loss provisions （LLP）. The paper also includes information on individual and collective impairment assessment ...This paper clarifies the distinctions between loan loss reserves （LLR）, expected loss （EL）, and loan loss provisions （LLP）. The paper also includes information on individual and collective impairment assessment of local commercial banks in Malaysia collected from their annual reports. Most banks have maintained collective assessment （CA） allowance ratio of lower than 1.2% of gross total loans.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
The environmental risk of chemical spills near coastal zones threatens abundant marine ecological resources. By appraising the ecological damage value of the environmental risk of marine chemical spills, we can facili...The environmental risk of chemical spills near coastal zones threatens abundant marine ecological resources. By appraising the ecological damage value of the environmental risk of marine chemical spills, we can facilitate decision-making for the development of a coastal zone and establishment of economic policy on coastal zone management. In this study, the ecological effect of the environmental risk of a chemical spill in the Haicang chemical industrial park in Xiamen was identified, after which its impact was forecasted and its ecological damage value was assessed. The information generated in this study will be useful in future studies evaluating marine ecological compensation based on environmental risk analysis.展开更多
金融稳定需要防范和化解金融市场之间的风险传染。与以往文献只是探究两个市场的风险传染不同,本文利用高维VAR for VaR模型将中国的汇市、债市、大宗商品、金融期货和股市等五个金融市场纳入统一框架,分析这5个金融市场在不同状态的风...金融稳定需要防范和化解金融市场之间的风险传染。与以往文献只是探究两个市场的风险传染不同,本文利用高维VAR for VaR模型将中国的汇市、债市、大宗商品、金融期货和股市等五个金融市场纳入统一框架,分析这5个金融市场在不同状态的风险溢出效应,这有助于捕捉冲击在不同金融市场之间传播而产生的间接影响。Wald检验和后验分析表明5个市场间只在危机或泡沫状态时存在明显的风险溢出效应。同时,本文利用压力测试发现单个市场的短期冲击影响会被其他金融市场如股市消化吸收,但4个金融市场都处于正常状态会明显降低其他金融市场如股市的左尾风险。此外,本文提出利用单个金融市场在同一时点的不同分位数计算每个金融市场在同一时点的预期收益、波动风险和崩盘风险,这种做法的好处在于结果更加稳健以及减轻极端值的影响。在此基础上,本文进一步探究金融市场间是否能够对冲彼此的波动风险和崩盘风险。结果显示大宗商品市场和金融期货市场能够有效地对冲其他金融市场的波动风险和崩盘风险,但汇市、债市和股市无法对冲其他金融市场的波动风险和崩盘风险。展开更多
International oil and gas projects feature high capital-intensity, high risks and contract diversity. Therefore, in order to help decision makers make more reasonable decisions under uncertainty, it is necessary to me...International oil and gas projects feature high capital-intensity, high risks and contract diversity. Therefore, in order to help decision makers make more reasonable decisions under uncertainty, it is necessary to measure the risks of international oil and gas projects. For this purpose, this paper constructs a probabilistic model that is based on the traditional economic evaluation model, and introduces value at risk(VaR) which is a valuable risk measure tool in finance, and applies Va R to measure the risks of royalty contracts, production share contracts and service contracts of an international oil and gas project. Besides, this paper compares the influences of different risk factors on the net present value(NPV) of the project by using the simulation results. The results indicate:(1) risks have great impacts on the project's NPV, therefore, if risks are overlooked, the decision may be wrong.(2) A simulation method is applied to simulate the stochastic distribution of risk factors in the probabilistic model. Therefore, the probability is related to the project's NPV, overcoming the inherent limitation of the traditional economic evaluation method.(3) VaR is a straightforward risk measure tool, and can be applied to evaluate the risks of international oil and gas projects. It is helpful for decision making.展开更多
Background Genotype-by-sequencing has been proposed as an alternative to SNP genotyping arrays in genomic selection to obtain a high density of markers along the genome.It requires a low sequencing depth to be cost ef...Background Genotype-by-sequencing has been proposed as an alternative to SNP genotyping arrays in genomic selection to obtain a high density of markers along the genome.It requires a low sequencing depth to be cost effective,which may increase the error at the genotype assigment.Third generation nanopore sequencing technology offers low cost sequencing and the possibility to detect genome methylation,which provides added value to genotype-by-sequencing.The aim of this study was to evaluate the performance of genotype-by-low pass nanopore sequencing for estimating the direct genomic value in dairy cattle,and the possibility to obtain methylation marks simultaneously.Results Latest nanopore chemistry(LSK14 and Q20)achieved a modal base calling accuracy of 99.55%,whereas previous kit(LSK109)achieved slightly lower accuracy(99.1%).The direct genomic value accuracy from genotype-by-low pass sequencing ranged between 0.79 and 0.99,depending on the trait(milk,fat or protein yield),with a sequencing depth as low as 2×and using the latest chemistry(LSK114).Lower sequencing depth led to biased estimates,yet with high rank correlations.The LSK109 and Q20 achieved lower accuracies(0.57-0.93).More than one million high reliable methylated sites were obtained,even at low sequencing depth,located mainly in distal intergenic(87%)and promoter(5%)regions.Conclusions This study showed that the latest nanopore technology in useful in a LowPass sequencing framework to estimate direct genomic values with high reliability.It may provide advantages in populations with no available SNP chip,or when a large density of markers with a wide range of allele frequencies is needed.In addition,low pass sequencing provided nucleotide methylation status of>1 million nucleotides at≥10×,which is an added value for epigenetic studies.展开更多
This paper provides a comparative analysis of the stock investment value of Amazon,Microsoft,and Apple,three global companies that investors may consider when investing in stocks.The author analyzes each company’s bu...This paper provides a comparative analysis of the stock investment value of Amazon,Microsoft,and Apple,three global companies that investors may consider when investing in stocks.The author analyzes each company’s business diversity,financial situation,and industry competition.Amazon is a leading e-commerce company with strong domestic and international competitors in the field of e-commerce and cloud computing.Microsoft is a stable company with a dominant position in the personal computer operating system market and a growing market share in the field of cloud computing.Apple has a diverse product line,including hardware and software products,and faces competition from many competitors,but maintains its competitiveness in the market by investing in technological innovation.The paper concludes that the stock market is full of opportunities and challenges.And investors need to have a certain risk awareness and long-term investment vision.The analysis provided in this paper can help different investors select stocks that suit their preferences and investment objectives.The paper uses data from Yahoo Finance and company websites to provide an overview of each company’s performance,net sales,and stock prices compare the calculation results,and then select the stocks that investors with different preferences prefer.展开更多
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
With the application of phasor measurement units(PMU)in the distribution system,it is expected that the performance of the distribution system state estimation can be improved obviously with the PMU measurements into ...With the application of phasor measurement units(PMU)in the distribution system,it is expected that the performance of the distribution system state estimation can be improved obviously with the PMU measurements into consideration.How to appropriately place the PMUs in the distribution is therefore become an important issue due to the economical consideration.According to the concept of efficient frontier,a value-at-risk based approach is proposed to make optimal placement of PMU taking account of the uncertainty of measure errors,statistical characteristics of the pseudo measurements,and reliability of the measurement instrument.The reasonability and feasibility of the proposed model is illustrated with 12-node system and IEEE-33 node system.Simulation results indicated that uncertainties of measurement error and instrument fault result in more PMU to be installed,and measurement uncertainty is the main affect factor unless the fault rate of PMU is quite high.展开更多
The subset sum problem is a combinatorial optimization problem,and its complexity belongs to the nondeterministic polynomial time complete(NP-Complete)class.This problem is widely used in encryption,planning or schedu...The subset sum problem is a combinatorial optimization problem,and its complexity belongs to the nondeterministic polynomial time complete(NP-Complete)class.This problem is widely used in encryption,planning or scheduling,and integer partitions.An accurate search algorithm with polynomial time complexity has not been found,which makes it challenging to be solved on classical computers.To effectively solve this problem,we translate it into the quantum Ising model and solve it with a variational quantum optimization method based on conditional values at risk.The proposed model needs only n qubits to encode 2ndimensional search space,which can effectively save the encoding quantum resources.The model inherits the advantages of variational quantum algorithms and can obtain good performance at shallow circuit depths while being robust to noise,and it is convenient to be deployed in the Noisy Intermediate Scale Quantum era.We investigate the effects of the scalability,the variational ansatz type,the variational depth,and noise on the model.Moreover,we also discuss the performance of the model under different conditional values at risk.Through computer simulation,the scale can reach more than nine qubits.By selecting the noise type,we construct simulators with different QVs and study the performance of the model with them.In addition,we deploy the model on a superconducting quantum computer of the Origin Quantum Technology Company and successfully solve the subset sum problem.This model provides a new perspective for solving the subset sum problem.展开更多
Value at risk (VaR) is adopted to measure the risk level in the electricity market. To estimate VaR at higher accuracy and reliability, the wavelet variance decomposed approach for value at risk estimates (WVDVaR) is ...Value at risk (VaR) is adopted to measure the risk level in the electricity market. To estimate VaR at higher accuracy and reliability, the wavelet variance decomposed approach for value at risk estimates (WVDVaR) is proposed. Empirical studies conduct in five Australian electricity markets, which evaluate the performances of both the proposed approach and the traditional ARMA-GARCH approach using the Kupiec backtesting procedure. Experimental results suggest that the proposed approach measures electricity market risks at higher accuracy and reliability than the bench mark ARMA-GARCH approach, as indicated by the higher p values during the Kupiec backtesting procedure. In addition, the new approach also provides more insight into the risk evolution process over time and helps in adjusting VaR estimates to the time horizons that best suit investor interests. The distribution of risk according to investor preferences is shown by decomposing VaR across different time horizons. This also provides important information for the appropriate aggregation of risk measures based on investor investment preferences.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
A new stochastic volatility（SV）method to estimate the conditional value at risk（CVaR）is put forward.Firstly,it makes use of SV model to forecast the volatility of return.Secondly,the Markov chain Monte Carlo（MCMC...A new stochastic volatility（SV）method to estimate the conditional value at risk（CVaR）is put forward.Firstly,it makes use of SV model to forecast the volatility of return.Secondly,the Markov chain Monte Carlo（MCMC）simulation and Gibbs sampling have been used to estimate the parameters in the SV model.Thirdly,in this model,CVaR calculation is immediate.In this way,the SV-CVaR model overcomes the drawbacks of the generalized autoregressive conditional heteroscedasticity value at risk（GARCH-VaR）model.Empirical study suggests that this model is better than GARCH-VaR model in this field.展开更多
In order to study the effect of different risk measures on the efficient portfolios （fron- tier） while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper ...In order to study the effect of different risk measures on the efficient portfolios （fron- tier） while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivariate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios （frontier） based on risk measures such as variance, value at risk （VaR）, and expected shortfall （ES） is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance （MV） efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.展开更多
文摘This paper clarifies the distinctions between loan loss reserves （LLR）, expected loss （EL）, and loan loss provisions （LLP）. The paper also includes information on individual and collective impairment assessment of local commercial banks in Malaysia collected from their annual reports. Most banks have maintained collective assessment （CA） allowance ratio of lower than 1.2% of gross total loans.
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.
基金supported by Marine Science Fund of SOA for Young Scholars (No.2008121)
文摘The environmental risk of chemical spills near coastal zones threatens abundant marine ecological resources. By appraising the ecological damage value of the environmental risk of marine chemical spills, we can facilitate decision-making for the development of a coastal zone and establishment of economic policy on coastal zone management. In this study, the ecological effect of the environmental risk of a chemical spill in the Haicang chemical industrial park in Xiamen was identified, after which its impact was forecasted and its ecological damage value was assessed. The information generated in this study will be useful in future studies evaluating marine ecological compensation based on environmental risk analysis.
文摘金融稳定需要防范和化解金融市场之间的风险传染。与以往文献只是探究两个市场的风险传染不同,本文利用高维VAR for VaR模型将中国的汇市、债市、大宗商品、金融期货和股市等五个金融市场纳入统一框架,分析这5个金融市场在不同状态的风险溢出效应,这有助于捕捉冲击在不同金融市场之间传播而产生的间接影响。Wald检验和后验分析表明5个市场间只在危机或泡沫状态时存在明显的风险溢出效应。同时,本文利用压力测试发现单个市场的短期冲击影响会被其他金融市场如股市消化吸收,但4个金融市场都处于正常状态会明显降低其他金融市场如股市的左尾风险。此外,本文提出利用单个金融市场在同一时点的不同分位数计算每个金融市场在同一时点的预期收益、波动风险和崩盘风险,这种做法的好处在于结果更加稳健以及减轻极端值的影响。在此基础上,本文进一步探究金融市场间是否能够对冲彼此的波动风险和崩盘风险。结果显示大宗商品市场和金融期货市场能够有效地对冲其他金融市场的波动风险和崩盘风险,但汇市、债市和股市无法对冲其他金融市场的波动风险和崩盘风险。
基金supported by the Young Fund of Shanxi University of Finance and Economics(No.QN-2018002)National Natural Science Foundation of China(No.71774105)the Fund for Shanxi Key Subjects Construction(FSKSC)and Shanxi Repatriate Study Abroad Foundation(No.2016-3)
文摘International oil and gas projects feature high capital-intensity, high risks and contract diversity. Therefore, in order to help decision makers make more reasonable decisions under uncertainty, it is necessary to measure the risks of international oil and gas projects. For this purpose, this paper constructs a probabilistic model that is based on the traditional economic evaluation model, and introduces value at risk(VaR) which is a valuable risk measure tool in finance, and applies Va R to measure the risks of royalty contracts, production share contracts and service contracts of an international oil and gas project. Besides, this paper compares the influences of different risk factors on the net present value(NPV) of the project by using the simulation results. The results indicate:(1) risks have great impacts on the project's NPV, therefore, if risks are overlooked, the decision may be wrong.(2) A simulation method is applied to simulate the stochastic distribution of risk factors in the probabilistic model. Therefore, the probability is related to the project's NPV, overcoming the inherent limitation of the traditional economic evaluation method.(3) VaR is a straightforward risk measure tool, and can be applied to evaluate the risks of international oil and gas projects. It is helpful for decision making.
文摘Background Genotype-by-sequencing has been proposed as an alternative to SNP genotyping arrays in genomic selection to obtain a high density of markers along the genome.It requires a low sequencing depth to be cost effective,which may increase the error at the genotype assigment.Third generation nanopore sequencing technology offers low cost sequencing and the possibility to detect genome methylation,which provides added value to genotype-by-sequencing.The aim of this study was to evaluate the performance of genotype-by-low pass nanopore sequencing for estimating the direct genomic value in dairy cattle,and the possibility to obtain methylation marks simultaneously.Results Latest nanopore chemistry(LSK14 and Q20)achieved a modal base calling accuracy of 99.55%,whereas previous kit(LSK109)achieved slightly lower accuracy(99.1%).The direct genomic value accuracy from genotype-by-low pass sequencing ranged between 0.79 and 0.99,depending on the trait(milk,fat or protein yield),with a sequencing depth as low as 2×and using the latest chemistry(LSK114).Lower sequencing depth led to biased estimates,yet with high rank correlations.The LSK109 and Q20 achieved lower accuracies(0.57-0.93).More than one million high reliable methylated sites were obtained,even at low sequencing depth,located mainly in distal intergenic(87%)and promoter(5%)regions.Conclusions This study showed that the latest nanopore technology in useful in a LowPass sequencing framework to estimate direct genomic values with high reliability.It may provide advantages in populations with no available SNP chip,or when a large density of markers with a wide range of allele frequencies is needed.In addition,low pass sequencing provided nucleotide methylation status of>1 million nucleotides at≥10×,which is an added value for epigenetic studies.
文摘This paper provides a comparative analysis of the stock investment value of Amazon,Microsoft,and Apple,three global companies that investors may consider when investing in stocks.The author analyzes each company’s business diversity,financial situation,and industry competition.Amazon is a leading e-commerce company with strong domestic and international competitors in the field of e-commerce and cloud computing.Microsoft is a stable company with a dominant position in the personal computer operating system market and a growing market share in the field of cloud computing.Apple has a diverse product line,including hardware and software products,and faces competition from many competitors,but maintains its competitiveness in the market by investing in technological innovation.The paper concludes that the stock market is full of opportunities and challenges.And investors need to have a certain risk awareness and long-term investment vision.The analysis provided in this paper can help different investors select stocks that suit their preferences and investment objectives.The paper uses data from Yahoo Finance and company websites to provide an overview of each company’s performance,net sales,and stock prices compare the calculation results,and then select the stocks that investors with different preferences prefer.
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.
基金The author Min Liu received the grant of the National Natural Science Foundation of China(http://www.nsfc.gov.cn/)(51967004).
文摘With the application of phasor measurement units(PMU)in the distribution system,it is expected that the performance of the distribution system state estimation can be improved obviously with the PMU measurements into consideration.How to appropriately place the PMUs in the distribution is therefore become an important issue due to the economical consideration.According to the concept of efficient frontier,a value-at-risk based approach is proposed to make optimal placement of PMU taking account of the uncertainty of measure errors,statistical characteristics of the pseudo measurements,and reliability of the measurement instrument.The reasonability and feasibility of the proposed model is illustrated with 12-node system and IEEE-33 node system.Simulation results indicated that uncertainties of measurement error and instrument fault result in more PMU to be installed,and measurement uncertainty is the main affect factor unless the fault rate of PMU is quite high.
基金supported by the National Key R&D Program of China(Grant No.2019YFA0308700)the Innovation Program for Quantum Science and Technology(Grant No.2021ZD0301500)。
文摘The subset sum problem is a combinatorial optimization problem,and its complexity belongs to the nondeterministic polynomial time complete(NP-Complete)class.This problem is widely used in encryption,planning or scheduling,and integer partitions.An accurate search algorithm with polynomial time complexity has not been found,which makes it challenging to be solved on classical computers.To effectively solve this problem,we translate it into the quantum Ising model and solve it with a variational quantum optimization method based on conditional values at risk.The proposed model needs only n qubits to encode 2ndimensional search space,which can effectively save the encoding quantum resources.The model inherits the advantages of variational quantum algorithms and can obtain good performance at shallow circuit depths while being robust to noise,and it is convenient to be deployed in the Noisy Intermediate Scale Quantum era.We investigate the effects of the scalability,the variational ansatz type,the variational depth,and noise on the model.Moreover,we also discuss the performance of the model under different conditional values at risk.Through computer simulation,the scale can reach more than nine qubits.By selecting the noise type,we construct simulators with different QVs and study the performance of the model with them.In addition,we deploy the model on a superconducting quantum computer of the Origin Quantum Technology Company and successfully solve the subset sum problem.This model provides a new perspective for solving the subset sum problem.
基金The National Social Science Foundation of China (No.07AJL005)the Foundation of City University of Hong Kong (No.9610058)
文摘Value at risk (VaR) is adopted to measure the risk level in the electricity market. To estimate VaR at higher accuracy and reliability, the wavelet variance decomposed approach for value at risk estimates (WVDVaR) is proposed. Empirical studies conduct in five Australian electricity markets, which evaluate the performances of both the proposed approach and the traditional ARMA-GARCH approach using the Kupiec backtesting procedure. Experimental results suggest that the proposed approach measures electricity market risks at higher accuracy and reliability than the bench mark ARMA-GARCH approach, as indicated by the higher p values during the Kupiec backtesting procedure. In addition, the new approach also provides more insight into the risk evolution process over time and helps in adjusting VaR estimates to the time horizons that best suit investor interests. The distribution of risk according to investor preferences is shown by decomposing VaR across different time horizons. This also provides important information for the appropriate aggregation of risk measures based on investor investment preferences.
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.
基金Sponsored by the National Natural Science Foundation of China(70571010)
文摘A new stochastic volatility（SV）method to estimate the conditional value at risk（CVaR）is put forward.Firstly,it makes use of SV model to forecast the volatility of return.Secondly,the Markov chain Monte Carlo（MCMC）simulation and Gibbs sampling have been used to estimate the parameters in the SV model.Thirdly,in this model,CVaR calculation is immediate.In this way,the SV-CVaR model overcomes the drawbacks of the generalized autoregressive conditional heteroscedasticity value at risk（GARCH-VaR）model.Empirical study suggests that this model is better than GARCH-VaR model in this field.
基金Supported by the NNSF of China (10571141) the Key Project of the NNSF of China (70531030).
文摘In order to study the effect of different risk measures on the efficient portfolios （fron- tier） while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivariate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios （frontier） based on risk measures such as variance, value at risk （VaR）, and expected shortfall （ES） is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance （MV） efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.
