英文标题：
《Calculating CVaR and bPOE for Common Probability Distributions With
  Application to Portfolio Optimization and Density Estimation》
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作者：
Matthew Norton, Valentyn Khokhlov, Stan Uryasev
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最新提交年份：
2019
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英文摘要：
  Conditional Value-at-Risk (CVaR) and Value-at-Risk (VaR), also called the superquantile and quantile, are frequently used to characterize the tails of probability distribution\'s and are popular measures of risk. Buffered Probability of Exceedance (bPOE) is a recently introduced characterization of the tail which is the inverse of CVaR, much like the CDF is the inverse of the quantile. These quantities can prove very useful as the basis for a variety of risk-averse parametric engineering approaches. Their use, however, is often made difficult by the lack of well-known closed-form equations for calculating these quantities for commonly used probability distribution\'s. In this paper, we derive formulas for the superquantile and bPOE for a variety of common univariate probability distribution\'s. Besides providing a useful collection within a single reference, we use these formulas to incorporate the superquantile and bPOE into parametric procedures. In particular, we consider two: portfolio optimization and density estimation. First, when portfolio returns are assumed to follow particular distribution families, we show that finding the optimal portfolio via minimization of bPOE has advantages over superquantile minimization. We show that, given a fixed threshold, a single portfolio is the minimal bPOE portfolio for an entire class of distribution\'s simultaneously. Second, we apply our formulas to parametric density estimation and propose the method of superquantile\'s (MOS), a simple variation of the method of moment\'s (MM) where moment\'s are replaced by superquantile\'s at different confidence levels. With the freedom to select various combinations of confidence levels, MOS allows the user to focus the fitting procedure on different portions of the distribution, such as the tail when fitting heavy-tailed asymmetric data.
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中文摘要：
条件风险值（CVaR）和风险值（VaR），也称为超分位数和分位数，经常用于描述概率分布的尾部，是常用的风险度量。缓冲超越概率（bPOE）是最近引入的尾部表征，它是CVaR的倒数，很像CDF是分位数的倒数。这些数量可以证明是非常有用的，作为各种风险规避参数化工程方法的基础。然而，由于缺乏用于计算常用概率分布的这些量的众所周知的封闭式方程，它们的使用往往很困难。在本文中，我们推导了各种常见单变量概率分布的超分位数和bPOE公式。除了在单个参考中提供有用的集合外，我们使用这些公式将超分位数和bPOE合并到参数过程中。特别地，我们考虑两个方面：投资组合优化和密度估计。首先，当投资组合收益服从特定的分布族时，我们证明了通过bPOE最小化找到最优投资组合比超分位数最小化具有优势。我们证明，在给定固定阈值的情况下，对于整个类别的分布，单个投资组合是最小的bPOE投资组合。其次，我们将我们的公式应用于参数密度估计，并提出了超分位数（MOS）方法，这是矩（MM）方法的一个简单变体，其中矩在不同置信水平下被超分位数替换。由于可以自由选择置信水平的各种组合，MOS允许用户将拟合过程集中在分布的不同部分，例如拟合重尾非对称数据时的尾部。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Risk Management        风险管理
分类描述：Measurement and management of financial risks in trading, banking, insurance, corporate and other applications
衡量和管理贸易、银行、保险、企业和其他应用中的金融风险
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一级分类：Mathematics        数学
二级分类：Optimization and Control        优化与控制
分类描述：Operations research, linear programming, control theory, systems theory, optimal control, game theory
运筹学，线性规划，控制论，系统论，最优控制，博弈论
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一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Statistics        统计学
二级分类：Other Statistics        其他统计数字
分类描述：Work in statistics that does not fit into the other stat classifications
从事不适合其他统计分类的统计工作
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