论文标题
风险和高阶预期不足的概率等效价值水平
Probability equivalent level of Value at Risk and higher-order Expected Shortfalls
论文作者
论文摘要
我们研究了风险处的概率等效价值水平，以及$ n^{\ mathrm {th}} $ - 预期的预期短缺（称为pelve_n），可以将其视为由于Li和Wang（20222）的风险和预期短缺（称为PELVE）的概率等效价值水平的概念的变体。我们研究了pelve_n的有限性，独特性和几种特性，我们计算了一些值得注意的分布的pelve_n，pelve_2的随机变量的pelve_2具有概括性多余的分布，我们描述了定期变化的分布的pelve_2的渐近行为，因为该水平趋于0美元。还研究了$ n^{\ mathrm {th}} $的某些属性 - 预期的订单不足。除其他外，事实证明，与（加载）参数$λ\ geq 0 $相对应的GINI短缺$ p \ in [0,1）$ in [0,1）$是预期的不足$ p $的线性组合，$ 2^{\ 2^{\ mathrm {nd}} $ - 订单预期的$ p $ p $ p $ $ $ 1-2om $1-2λ$ 1-2om $ 1-2）。
We investigate the probability equivalent level of Value at Risk and $n^{\mathrm{th}}$-order Expected Shortfall (called PELVE_n), which can be considered as a variant of the notion of the probability equivalent level of Value at Risk and Expected Shortfall (called PELVE) due to Li and Wang (2022). We study the finiteness, uniqueness and several properties of PELVE_n, we calculate PELVE_n of some notable distributions, PELVE_2 of a random variable having generalized Pareto excess distribution, and we describe the asymptotic behaviour of PELVE_2 of regularly varying distributions as the level tends to $0$. Some properties of $n^{\mathrm{th}}$-order Expected Shortfall are also investigated. Among others, it turns out that the Gini Shortfall at some level $p\in[0,1)$ corresponding to a (loading) parameter $λ\geq 0$ is the linear combination of the Expected Shortfall at level $p$ and the $2^{\mathrm{nd}}$-order Expected Shortfall at level $p$ with coefficients $1-2λ$ and $2λ$, respectively.
