证券组合Shortfall风险度量方法研究

金融资产的风险度量是投资决策中的一个重要问题,本文提出了一种新的风险度量(标杆Shortfall),并证明了它为弱一致性风险度量,从而弱化了A rtzner(1999)提出的公理性条件。同时本文还研究了基于新风险度量下证券组合有效边界的性质,最后通过对标杆Shortfall风险度量的灵敏度分析,给出了一些有效的风险投资策略。

Calculation of Expected Shortfall for Measuring Risk and Its Applications

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.

A note on calculating expected shortfall for discrete time stochastic volatility models

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.

Three Methods to Calculate the Financial Risk Measurement: Value- At-Risk and Expected Shortfall

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.

Uncertainty Comparison Between Value-at-Risk and Expected Shortfall

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.

Change-point detection for expected shortfall in time series

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.

A New Robust Risk Measure:Quantile Shortfall

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.

Dual representation of expectile-based expected shortfall and its properties

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.

Rural Shortfall

The widening income gap between farmers and urban residents takes the shine off China’s emerging

资产相关结构对投资组合风险测度的影响分析

文章从分析金融资产收益率的统计特征入手,以GARCH模型为基础,用非对称幂分布描述组合资产中各金融资产收益率的边缘分布函数,在多种Copula函数情形下计算组合资产的风险值VaR及ES。结果表明:基于由多元Clayton Copula和多元Gumbel Copula组成的混合Copula函数较好地刻画了多只股票的相关结构,而且ES比VaR能够较准确地估计组合资产的尾部风险。

一种新的证券组合风险度量方法

在尾部条件期望(TCE)基础上,考虑投资者的真实风险感受,研究了一种新的风险度量方法———Shortfall风险度量,并在一致性公理下研究了它的一些统计性质,最后在多元椭球分布下得到了证券组合的Shortfall风险,还在多元t分布下得到了证券组合的Shortfall风险的数值结果。

基于Copula-SV-t模型的开放式基金投资组合优化

现代投资组合理论要求投资者研究不同金融资产间非线性非正态的分布特征和相关关系,进而优化相应的资产组合。结合SV-t模型和Copula函数,对单支开放式基金收益率分布的厚尾特征和不同基金间非线性的相关关系进行描述,并用ES方法度量不同权重下基金组合间的风险。实证表明:t-Copula函数下的模型更能反映出组合间的尾部相关性,此时最优组合下ES较正态Copula函数下的结果要小,说明t-Copula函数下的Copula-SV-t模型在实际中更具应用价值。

SV-t模型下人民币汇率VaR和ES的动态度量与分析

人民币汇率的风险度量是汇率风险管理的关键环节。本文首先用SV-t模型对人民币对美元、日元及欧元汇率收益的波动率进行分析,在此基础上分别运用VaR和ES方法对其风险进行动态的度量与分析比较。结果表明:SV-t模型能够刻画出人民币汇率收益率序列的波动特征;人民币对美元的VaR基本反映出其最大可能的损失;人民币对日元及欧元的Va R则低估了实际的风险水平,相应的ES虽保守却比较准确地估计出尾部风险。文章最后也对中国汇率市场的风险管理及风险监管提出了一些政策建议。

Repairable Configuration Optimization Model under Two-Echelon Inventory

Spares inventory configuration optimization is an effective way to improve readiness and reduce life cycle cost of equipment.Through analyzing two-echelon spares support system,the METRIC model basic theory was used.An inventory configuration optimization model of two-echelon spares support system was proposed which took the spares expected shortfall as the object and made the minimum repairable parts expected shortfall instead of the maximum spares supportability as the objective function.Marginal efficiency analysis algorithm was applied to optimizing the spares configuration and generating a rational spares inventory configuration.Finally,several examples are given to verify the model.

Operational risk assessment of third-party payment platforms:a case study of China

Operational risk events have severely impacted the development of third-party payment(TPP)platforms,and have even led to a discussion on the operational risk capital charge settlement by relevant international regulators.However,prior studies have mostly focused on qualitative mechanism analysis,and have rarely examined quantitative risk assessment based on actual operational risk events.Therefore,this study attempts to assess the operational risk on TPP platforms in China by constructing a systematic framework incorporating database construction and risk modeling.First,the operational risk database that covers 202 events between Q1,2014,and Q2,2020 is constructed.Then,specific causes are clarified,and the characteristics are analyzed from both the trend and loss severity perspectives.Finally,the piecewise-defined severity distribution based-Loss Distribution Approach(PSD-LDA)with double truncation is utilized to assess the operational risk.Two main conclusions are drawn from the empirical analysis.First,legal risk and external fraud risk are the two main causes of operational risk.Second,the yearly Value at Risk and Expected Shortfall are 724.46 million yuan and 1081.98 million yuan under the 99.9%significance level,respectively.Our results are beneficial for both TPP platform operators and regulators in managing and controlling operational risk.

Econometric modeling of risk measures: A selective review of the recent literature

Since the financial crisis in 2008, the risk measures which are the core of risk management, have received increasing attention among economists and practitioners. In this review, the concentration is on recent developments in the estimation of the most popular risk measures, namely, value at risk (VaR), expected shortfall (ES), and expectile. After introducing the concept of risk measures, the focus is on discussion and comparison of their econometric modeling. Then, parametric and nonparametric estimations of tail dependence are investigated. Finally, we conclude with insights into future research directions.

STUDY ON THE INTERRELATION OF EFFICIENT PORTFOLIOS AND THEIR FRONTIER UNDER t DISTRIBUTION AND VARIOUS RISK MEASURES

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.

On Bounds of Value-at-Risk and Convex Risk Measure of Portfolio of Weighted Dependent Risks

This note analytically derives lower and upper bounds for Value-at-Risk and convex risk measures of a portfolio of weighted risks in the context of positive dependence.The bounds serve as extensions of the corresponding ones due to Bignozzi et al.(2015).Also, DU-spread of value-at-risk and expected shortfall of Bignozzi et al.(2015) are also improved in some particular cases.

Further Results about Calibration of Longevity Risk for the Insurance Business

In life insurance business, longevity risk, i.e. the risk that the insured population lives longer than the expected, represents the heart of the risk assessment, having significant impact in terms of solvency capital requirements (SCRs) needed to front the firm obligations. The credit crisis has shown that systemic risk as longevity risk is relevant and that for many insurers it is actually the dominant risk. With the adoption of the Solvency II directive, a new area for insurance in terms of solvency regulation has been opened up. The international guidelines prescribe a market consistent valuation of balance sheets, where the solvency capital requirements to be set aside are calculated according to a modular structure. By mapping the main risk affecting the insurance portfolio, the capital amount able to cover the liabilities corresponds to each measured risk. In Solvency II, the longevity risk is included into underwriting risk module. In particular, the rules propose that companies use a standard model for measuring the SCRs. Nevertheless, the legislation under consideration allows designing tailor-made internal models. As regards the longevity risk assessment, the regulatory standard model leads to noteworthy inconsistencies. In this paper, we propose a stochastic volatility model combined with a so-called coherent risk measure as the expected shortfall for measuring the SCRs according to more realistic assumptions on future evolution of longevity trend. Finally empirical evidence is provided.

Stress Testing of Liquidity Maturity Transformation Risk in Banks

One of the main causes of the past crisis was the inability of financial institutions to acquire funding at appropriate costs.The importance of applying a good liquidity risk measurement system becomes apparent.The present paper provides an approach to the measurement of liquidity maturity transformation risk within a stress testing framework,for middle-sized banks.The costs of liquidity arising due to a downturn in refinancing conditions are calculated by using modern risk measures.The forward-looking way is based on a liquidity gap report,where the consideration of the counterbalancing capacity enables to gain an insight into the real liquidity needs.The measurement of both,the portfolio-value in the respective time bucket and liquidity costs,is possible.Applying the expected shortfall can easily be included into the calculation.The results show that by using historical simulation,if no sufficient data are available,expected shortfall delivers an approximate value.Still,it can serve as an indicator of insurance against extreme events.The present approach combines a scenario-based view to a possible distress with a quantitative risk measurement.Therewith,it contributes to the bank’s wide stress testing as required by the regulatory authorities.