Imre Kondor, Fabio Caccioli, Gábor Papp, Matteo Marsili
The contour map of estimation error of Expected Shortfall (ES) is constructed. It allows one to quantitatively determine the sample size (the length of the time series) required by the optimization under ES of large institutional portfolios for a given size of the portfolio, at a given confidence level and a given estimation error.
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The contour maps of the error of historical resp. parametric estimates for large random portfolios optimized under the risk measure Expected Shortfall (ES) are constructed. Similar maps for the sensitivity of the portfolio weights to small changes in the returns as well as the VaR of the ES-optimized portfolio are also presented, along with results for the distribution of portfolio weights over the random samples and for the out-of-sample and in-the-sample estimates for ES. The contour maps allow one to quantitatively determine the sample size (the length of the time series) required by the op…
The 2008 mortgage crisis is an example of an extreme event. Extreme value theory tries to estimate such tail risks. Modern finance practitioners prefer Expected Shortfall based risk metrics (which capture tail risk) over traditional approaches like volatility or even Value-at-Risk. This paper provides a quantum annealing algorithm in QUBO form for a dynamic asset allocation problem using expected shortfall constraint. It was motivated by the need to refine the current quantum algorithms for Markowitz type problems which are academically interesting but not useful for practitioners. The algorit…
It is well known that Expected Shortfall (also called Average Value-at-Risk) is a convex risk measure, i. e. Expected Shortfall of a convex linear combination of arbitrary risk positions is not greater than a convex linear combination with the same weights of Expected Shortfalls of the same risk positions. In this short paper we prove that Expected Shortfall is a concave risk measure with respect to probability distributions, i. e. Expected Shortfall of a finite mixture of arbitrary risk positions is not lower than the linear combination of Expected Shortfalls of the same risk positions (with …
The problem of estimation error of Expected Shortfall is analyzed, with a view of its introduction as a global regulatory risk measure.
Expectile bears some interesting properties in comparison to the industry wide expected shortfall in terms of assessment of tail risk. We study the relationship between expectile and expected shortfall using duality results and the link to optimized certainty equivalent. Lower and upper bounds of expectile are derived in terms of expected shortfall as well as a characterization of expectile in terms of expected shortfall. Further, we study the asymptotic behavior of expectile with respect to expected shortfall as the confidence level goes to $1$ in terms of extreme value distributions. We use …
The expectile can be considered as a generalization of 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 quantile. We provide its dual representation in terms of Bochner integral. Among other properties, we show that it is bounded from below in terms of convex combinations of expected shortfalls, and also from above by the smallest law invariant, coherent and comonotonic risk measure, for which we give the explicit formulation of the corresponding distortion function. As …
We introduce and study the main properties of a class of convex risk measures that refine Expected Shortfall by simultaneously controlling the expected losses associated with different portions of the tail distribution. The corresponding adjusted Expected Shortfalls quantify risk as the minimum amount of capital that has to be raised and injected into a financial position $X$ to ensure that Expected Shortfall $ES_p(X)$ does not exceed a pre-specified threshold $g(p)$ for every probability level $p\in[0,1]$. Through the choice of the benchmark risk profile $g$ one can tailor the risk assessment…
Digital currencies and cryptocurrencies have hesitantly started to penetrate the investors, and the next step will be the regulatory risk management framework. We examine the Value-at-Risk and Expected Shortfall properties for the major digital currencies, Bitcoin, Ethereum, Litecoin, and Ripple. The methodology used is GARCH modelling followed by Filtered Historical Simulation. We find that digital currencies are subject to a higher risk, therefore, to higher sufficient buffer and risk capital to cover potential losses.
Norms Based on Generalized Expected-Shortfalls and Applications
本文提出基于失真风险测度的广义期望短缺范数，构建风险量化统一框架，发展数学对偶理论支持投资组合优化，并应用于金融时间序列异常检测。
In this note, we comment on the relevance of elicitability for backtesting risk measure estimates. In particular, we propose the use of Diebold-Mariano tests, and show how they can be implemented for Expected Shortfall (ES), based on the recent result of Fissler and Ziegel (2015) that ES is jointly elicitable with Value at Risk.
Probability equivalent level of Value at Risk and higher-order Expected Shortfalls
本文研究了风险价值与n阶期望短缺的概率等价水平（PELVE_n）的有限性、唯一性及性质，计算了多种分布的PELVE_n并分析了渐近行为，同时揭示了基尼短缺与期望短缺及二阶期望短缺之间的线性组合关系。
An elementary proof of the dual representation of Expected Shortfall
本文在一般概率空间上利用分位数函数的基本性质，给出了Expected Shortfall对偶表示的初等证明，并由此推导出其次可加性的新证明。
On the coherence of Expected Shortfall
期望缺口（ExpectedShortfall，ES）作为风险价值（Value-at-Risk，VaR）的一种非一致性风险度量方法，被提出来作为弥补VaR不足的一种方法。事实上，ES的大多数定义在应用于连续损失分布时会导致相同的结果。当基础损失分布具有不连续性时，可能会出现差异。在这种情况下，甚至ES的相干性也会丢失，除非在其定义中注意到细节。我们比较了一些定义的预期短缺，指出有一个是强大的意义上产生一个连贯的风险度量，无论潜在的分布。此外，即使在通常的VaR估计失败的情况下，也可以有效地估计预期短缺。关键词：预期短缺;风险度量;最坏条件期望尾部一致性期望;风险价值（VaR）;条件风险价值;尾部均值;协整性;分位数;次可加性。
Credit Risk Contributions to Value-at-Risk and Expected Shortfall
本文给出了CreditRisk+方法中如何计算合理的风险价值（Value-at-Risk，VaR）和预期损失（ExpectedShortfall，ES）贡献的问题的解析解。通过ES贡献，ES本身可以在许多步骤中精确计算。数值算例说明了该方法的有效性。
Expected Shortfall: a natural coherent alternative to Value at Risk
本文讨论了作为金融风险度量的预期缺口的一致性性质。这一统计数据很自然地产生于对投资组合回报率样本中“100%最坏损失的平均值”的估计。这里p是某个固定的置信水平。我们还比较了ES的几种替代表示，结果更适合于某些目的。
Expected Shortfall as a Tool for Financial Risk Management
本文从财务风险管理的角度研究了预期亏损的性质。这种衡量方法-事实上，我们表明，与VaR不同，这个变量通常是次可加的，因此它是参考意义上的一致性风险度量（artzner）
Expected Shortfall and Beyond
金融机构必须分配所谓的“经济资本”，以保证其客户和交易对手的偿付能力。从数学上讲，任何分配资本的方法都是一种“风险度量”，即将随机变量映射到实数的函数。风险价值是目前最流行的风险度量方法，它被定义为随机变量的固定水平分位数。不幸的是，它未能奖励多样化，因为它不是“次加性”。在寻找一个合适的替代风险价值，“预期短缺”（或“条件风险价值”或“尾部风险价值”）已被描述为最小的“一致”和“法律不变”的风险措施，以主导风险价值。我们讨论了这些和其他一些性质的预期短缺，以及它的推广到一类相干的风险措施，可以将更高的时刻的影响。此外，我们提出了一个一般的方法，如何属性预期不足的“风险贡献”的组合成分。关键词：预期亏损;风险价值;谱风险度量;一致性;风险贡献
Value-at-Risk and expected shortfall for linear portfolios with elliptically distributed risk factors
本文将参数delta-VaR方法从风险因子服从正态分布的投资组合推广到风险因子服从椭圆分布的投资组合。我们处理的预期短缺和风险价值的投资组合。特别注意的是一个多元t分布的特殊情况。
A tail-shape actuarial index based on equal level relationships between Value at Risk and Expected Shortfall
本文提出θ指数，用于基于价值风险与预期短缺的相等水平关系，评估损失分布的尾部行为，并通过均值超额函数与广义帕累托模型建立联系，同时推导风险贡献并用于稳定的价值风险分配计算。
We propose a new backtesting framework for Expected Shortfall that could be used by the regulator. Instead of looking at the estimated capital reserve and the realised cash-flow separately, one could bind them into the secured position, for which risk measurement is much easier. Using this simple concept combined with monotonicity of Expected Shortfall with respect to its target confidence level we introduce a natural and efficient backtesting framework. Our test statistics is given by the biggest number of worst realisations for the secured position that add up to a negative total. Surprising…
Motivated by the Basel 3 regulations, recent studies have considered joint forecasts of Value-at-Risk and Expected Shortfall. A large family of scoring functions can be used to evaluate forecast performance in this context. However, little intuitive or empirical guidance is currently available, which renders the choice of scoring function awkward in practice. We therefore develop graphical checks (Murphy diagrams) of whether one forecast method dominates another under a relevant class of scoring functions, and propose an associated hypothesis test. We illustrate these tools with simulation exa…
Expected Shortfall (ES), the average loss above a high quantile, is the current financial regulatory market risk measure. Its estimation and optimization are highly unstable against sample fluctuations and become impossible above a critical ratio $r=N/T$, where $N$ is the number of different assets in the portfolio, and $T$ is the length of the available time series. The critical ratio depends on the confidence level $α$, which means we have a line of critical points on the $α-r$ plane. The large fluctuations in the estimation of ES can be attenuated by the application of regularizers. In this…
A joint conditional autoregressive expectile and Expected Shortfall framework is proposed. The framework is extended through incorporating a measurement equation which models the contemporaneous dependence between the realized measures and the latent conditional expectile. Nonlinear threshold specification is further incorporated into the proposed framework. A Bayesian Markov Chain Monte Carlo method is adapted for estimation, whose properties are assessed and compared with maximum likelihood via a simulation study. One-day-ahead VaR and ES forecasting studies, with seven market indices, provi…
Asymptotic Error Analysis of Multilevel Stochastic Approximations for the Value-at-Risk and Expected Shortfall
本文研究多级随机近似算法在计算价值-at-风险和预期短缺时的渐近误差，建立中心极限定理并验证了数值例子。
CAESar: Conditional Autoregressive Expected Shortfall
提出CAESar模型，结合VaR与ES的联合可激励性，通过三步优化（CAViaR拟合VaR、ES自回归过程、联合估计并施加单调性约束）灵活处理动态模式和异方差性，回测表现优于现有方法。
We study a non-concave optimization problem in which a financial company maximizes the expected utility of the surplus under a risk-based regulatory constraint. For this problem, we consider four different prevalent risk constraints (Expected Shortfall, Expected Discounted Shortfall, Value-at-Risk, and Average Value-at-Risk), and investigate their effects on the optimal solution. Our main contributions are in obtaining an analytical solution under each of the four risk constraints, in the form of the optimal terminal wealth. We show that the four risk constraints lead to the same optimal solut…
This paper introduces novel backtests for the risk measure Expected Shortfall (ES) following the testing idea of Mincer and Zarnowitz (1969). Estimating a regression framework for the ES stand-alone is infeasible, and thus, our tests are based on a joint regression for the Value at Risk and the ES, which allows for different test specifications. These ES backtests are the first which solely backtest the ES in the sense that they only require ES forecasts as input parameters. As the tests are potentially subject to model misspecification, we provide asymptotic theory under misspecification for …
Value at risk and expected shortfall are increasingly popular tail risk measures in the financial risk management field. Both academia and financial institutions are working to improve tail risk forecasts in order to meet the requirements of the Basel Capital Accord; it states that one purpose of risk management and measuring risk accuracy is, since extreme movements cannot always be avoided, financial institutions can prepare for these extreme returns by capital allocation, and putting aside the appropriate amount of capital so as to avoid default in times of extreme price or index movements.…
We introduce new forecast encompassing tests for the risk measure Expected Shortfall (ES). The ES currently receives much attention through its introduction into the Basel III Accords, which stipulate its use as the primary market risk measure for the international banking regulation. We utilize joint loss functions for the pair ES and Value at Risk to set up three ES encompassing test variants. The tests are built on misspecification robust asymptotic theory and we investigate the finite sample properties of the tests in an extensive simulation study. We use the encompassing tests to illustra…
Lambda Expected Shortfall
本文提出Lambda预期短缺作为ES的推广，展示其作为最小准凸且法律不变风险度量的性质，并探讨其对偶表示和优化问题。
Optimal nonparametric estimation of the expected shortfall risk
本文提出了一种鲁棒的非参数估计方法，用于最优估计预期短差风险，且在数据被恶意修改时仍能保持优良性能。
Worst-case bounds on the expected shortfall risk given only limited information on the distribution of the random variables has been studied extensively in the literature. In this paper, we develop a new worst-case bound on the expected shortfall when the univariate marginals are known exactly and additional expert information is available in terms of bivariate marginals. Such expert information allows for one to choose from among the many possible parametric families of bivariate copulas. By considering a neighborhood of distance $ρ$ around the bivariate marginals with the Kullback-Leibler di…
Combining a Large Pool of Forecasts of Value-at-Risk and Expected Shortfall
本文研究了在大量候选预测可用时，如何结合价值-at-风险和预期缺口预测，提出多种新方法，包括简单方法和正则化技术，实验证明混合均值和性能加权方法效果显著。
Quantile-based modeling of scale dynamics in financial returns for Value-at-Risk and Expected Shortfall forecasting
本文提出一种半参数方法，通过限制分位数回归建模金融收益率的条件尺度，用于预测风险价值和预期短缺，展现其在极端损失估计和捕捉偏态、厚尾和杠杆效应方面的优势。
A new semi-parametric Expected Shortfall (ES) estimation and forecasting framework is proposed. The proposed approach is based on a two-step estimation procedure. The first step involves the estimation of Value-at-Risk (VaR) at different quantile levels through a set of quantile time series regressions. Then, the ES is computed as a weighted average of the estimated quantiles. The quantiles weighting structure is parsimoniously parameterized by means of a Beta weight function whose coefficients are optimized by minimizing a joint VaR and ES loss function of the Fissler-Ziegel class. The proper…
Although quantile regression to calculate risk measures has been widely established in the financial literature, when considering data observed at mixed--frequency, an extension is needed. In this paper, a model is suggested built on a mixed--frequency quantile regression to directly estimate the Value--at--Risk (VaR) and the Expected Shortfall (ES) measures. In particular, the low--frequency component incorporates information coming from variables observed at, typically, monthly or lower frequencies, while the high--frequency component can include a variety of daily variables, like market ind…
On approximations of Value at Risk and Expected Shortfall involving kurtosis
本文提出了在正偏度和超额峰度下风险价值与期望损失的新近似方法，基于正态幂近似的扩展，并在多种分布中验证了其精度与性能。
Combining Value-at-Risk and Expected Shortfall forecasts via the Model Confidence Set
本文提出基于模型置信集方法的VaR和ES预测组合策略，通过严格一致的损失函数识别超优模型并进行加权组合，实证分析显示其在预测尾部风险指标方面表现稳健。
Backtesting Expected Shortfall: Accounting for both duration and severity with bivariate orthogonal polynomials
提出结合违规间隔持续期与损失严重程度的二元正交多项式 ES 回测框架及 Wald 检验，可分别诊断模型问题并具有良好有限样本表现。
Tail Risk and Systemic Risk Estimation of Cryptocurrencies: an Expectiles and Marginal Expected Shortfall based approach
本文针对加密货币尾部风险量化问题，提出基于期望分位数和GARCH残差建模的动态尾部风险估计方法，并引入边际期望损失（MES）度量单个加密货币对市场系统性风险的贡献。
Under the Fundamental Review of the Trading Book (FRTB) capital charges for the trading book are based on the coherent expected shortfall (ES) risk measure, which show greater sensitivity to tail risk. In this paper it is argued that backtesting of expected shortfall - or the trading book model from which it is calculated - can be based on a simultaneous multinomial test of value-at-risk (VaR) exceptions at different levels, an idea supported by an approximation of ES in terms of multiple quantiles of a distribution proposed in Emmer et al. (2015). By comparing Pearson, Nass and likelihood-rat…
We propose forecast encompassing tests for the Expected Shortfall (ES) jointly with the Value at Risk (VaR) based on flexible link (or combination) functions. Our setup allows testing encompassing for convex forecast combinations and for link functions which preclude crossings of the combined VaR and ES forecasts. As the tests based on these link functions involve parameters which are on the boundary of the parameter space under the null hypothesis, we derive and base our tests on nonstandard asymptotic theory on the boundary. Our simulation study shows that the encompassing tests based on our…
The standard theory of coherent risk measures fails to consider individual institutions as part of a system which might itself experience instability and spread new sources of risk to the market participants. In compliance with an approach adopted by Shapley and Shubik (1969), this paper proposes a cooperative market game where agents and institutions play the same role can be developed. We take into account a multiple institutions framework where some of them jointly experience distress events in order to evaluate their individual and collective impact on the remaining institutions in the mar…
We present the Shortfall Deviation Risk (SDR), a risk measure that represents the expected loss that occurs with certain probability penalized by the dispersion of results that are worse than such an expectation. SDR combines Expected Shortfall (ES) and Shortfall Deviation (SD), which we also introduce, contemplating two fundamental pillars of the risk concept, the probability of adverse events and the variability of an expectation, and considers extreme results. We demonstrate that SD is a generalized deviation measure, whereas SDR is a coherent risk measure. We achieve the dual representatio…
A new realized conditional autoregressive Value-at-Risk (VaR) framework is proposed, through incorporating a measurement equation into the original quantile regression model. The framework is further extended by employing various Expected Shortfall (ES) components, to jointly estimate and forecast VaR and ES. The measurement equation models the contemporaneous dependence between the realized measure (i.e., Realized Variance and Realized Range) and the latent conditional ES. An adaptive Bayesian Markov Chain Monte Carlo method is employed for estimation and forecasting, the properties of which …
We consider a multi-step algorithm for the computation of the historical expected shortfall such as defined by the Basel Minimum Capital Requirements for Market Risk. At each step of the algorithm, we use Monte Carlo simulations to reduce the number of historical scenarios that potentially belong to the set of worst scenarios. The number of simulations increases as the number of candidate scenarios is reduced and the distance between them diminishes. For the most naive scheme, we show that the L p-error of the estimator of the Expected Shortfall is bounded by a linear combination of the probab…
Dynamic Semiparametric Models for Expected Shortfall (and Value-at-Risk)
本文提出动态半参数模型用于预期损失和价值-at-风险，通过统计决策理论结果解决预期损失的可 elicibility 问题，并通过实证研究验证模型在国际股市日回报预测中的优越性。
We introduce a novel regression framework which simultaneously models the quantile and the Expected Shortfall (ES) of a response variable given a set of covariates. This regression is based on a strictly consistent loss function for the pair quantile and ES, which allows for M- and Z-estimation of the joint regression parameters. We show consistency and asymptotic normality for both estimators under weak regularity conditions. The underlying loss function depends on two specification functions, whose choice affects the properties of the resulting estimators. We find that the Z-estimator is num…
Value-at-Risk (VaR) and Expected Shortfall (ES) are widely used in the financial sector to measure the market risk and manage the extreme market movement. The recent link between the quantile score function and the Asymmetric Laplace density has led to a flexible likelihood-based framework for joint modelling of VaR and ES. It is of high interest in financial applications to be able to capture the underlying joint dynamics of these two quantities. We address this problem by developing a hybrid model that is based on the Asymmetric Laplace quasi-likelihood and employs the Long Short-Term Memory…
Forecasting and Backtesting Gradient Allocations of Expected Shortfall
本研究聚焦预期短缺（ES）的梯度分配（ESCs），基于联合可识别性和多目标可导出性提出回测框架，扩展Murphy图，并构建新型半参数模型以预测动态ESCs，实证证明其性能优越。
Risk budgeting is a portfolio strategy where each asset contributes a prespecified amount to the aggregate risk of the portfolio. In this work, we propose an efficient numerical framework that uses only simulations of returns for estimating risk budgeting portfolios. Besides a general cutting planes algorithm for determining the weights of risk budgeting portfolios for arbitrary coherent distortion risk measures, we provide a specialised version for the Expected Shortfall, and a tailored Stochastic Gradient Descent (SGD) algorithm, also for the Expected Shortfall. We compare our algorithm to s…
The present article deals with intra-horizon risk in models with jumps. Our general understanding of intra-horizon risk is along the lines of the approach taken in Boudoukh, Richardson, Stanton and Whitelaw (2004), Rossello (2008), Bhattacharyya, Misra and Kodase (2009), Bakshi and Panayotov (2010), and Leippold and Vasiljević (2019). In particular, we believe that quantifying market risk by strictly relying on point-in-time measures cannot be deemed a satisfactory approach in general. Instead, we argue that complementing this approach by studying measures of risk that capture the magnitude of…
We show that coherent risk measures are ineffective in curbing the behaviour of investors with limited liability or excessive tail-risk seeking behaviour if the market admits statistical arbitrage opportunities which we term $ρ$-arbitrage for a risk measure $ρ$. We show how to determine analytically whether such $ρ$-arbitrage portfolios exist in complete markets and in the Markowitz model. We also consider realistic numerical examples of incomplete markets and determine whether expected shortfall constraints are ineffective in these markets. We find that the answer depends heavily upon the pro…
Static marginal expected shortfall: Systemic risk measurement under dependence uncertainty
本文针对单个公司风险分布已知但依赖结构未知的情况，推导边际期望损失（MES）的边界，并基于部分依赖信息和线性风险关系提出改进边界，实证表明其对从业者和政策制定者具有实际意义。
A Multilevel Stochastic Approximation Algorithm for Value-at-Risk and Expected Shortfall Estimation
本文提出了一种多级随机逼近算法，用于计算金融损失的价值-at-风险和预期短缺，通过模拟在未来的风险因素实现，并展示了该算法在精度和执行时间上的优化性能。
Submodular risk measures
本文研究了对称不变函数的亚模性，证明凸风险度量的亚模性条件，并通过实证发现ES无违规，而VaR和AES存在不同程度的违规。
ReSGA: A Large Tail Risk Model for Learning Value-at-Risk and Expected Shortfall
提出检索增强自分组自编码器（ReSGA），利用数百万参数捕捉资产横截面依赖和长期时间动态，在1926-2023年美国股票数据上优于12种基准模型，并通过新规模增强左尾动量策略实现经济收益。
E-Backtesting Expected Shortfall without Value-at-Risk Forecasts under Time-series Models
该研究针对时间序列模型，开发了两种仅用观测损失和ES预测的电子回溯测试方法，可控制任意停止时间的错误拒绝率，通过模拟和纳斯达克损失应用验证了其性能。
We present a general framework for a comparative theory of variability measures, with a particular focus on the recently introduced one-parameter families of inter-Expected Shortfall differences and inter-expectile differences, that are explored in detail and compared with the widely known and applied inter-quantile differences. From the mathematical point of view, our main result is a characterization of symmetric and comonotonic variability measures as mixtures of inter-Expected Shortfall differences, under a few additional technical conditions. Further, we study the stochastic orders induce…
On Joint Marginal Expected Shortfall and Associated Contribution Risk Measures
本文提出新型系统性风险测度联合边际期望短缺（JMES）及两种贡献测度，研究其性质与比较条件，并通过数值示例和股市风险传染实证验证其性能。
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