论文标题
测量尾巴风险
Measuring Tail Risks
论文作者
论文摘要
风险（VAR）和预期短缺（ES）的价值是金融法规和风险管理中采用的常见高分子风险措施。在本文中，我们提出了一项基于最大风险事件（MPMR）最大大小（MPMR）的尾巴风险度量，该措施可能会在一段时间内发生。 MPMR强调了尾巴风险对风险管理时间范围的依赖性。与VAR和ES不同，MPMR不需要指定置信度。我们分析了几个众所周知的分布的风险度量。特别是，对于风险事件的大小遵循功率定律或帕累托分布的情况，我们表明mpmr还通过电源法的观测值$ n $（或同等的时间间隔）来扩展，$ \ \ \ text {mpmr}（mpmr}（n）\ propto n^η^η$，其中$η$是$η$的标准量表。根据更可靠的短期风险估计，规模不变性可以合理地估计长期风险。缩放关系还产生了尾巴索引（Ti）$ξ$的强大且低偏置的估计器，$ξ= 1/η$。我们证明了这种风险措施在金融市场中描述尾巴风险以及与自然危害（地震，海啸和过多降雨）相关的风险。
Value at risk (VaR) and expected shortfall (ES) are common high quantile-based risk measures adopted in financial regulations and risk management. In this paper, we propose a tail risk measure based on the most probable maximum size of risk events (MPMR) that can occur over a length of time. MPMR underscores the dependence of the tail risk on the risk management time frame. Unlike VaR and ES, MPMR does not require specifying a confidence level. We derive the risk measure analytically for several well-known distributions. In particular, for the case where the size of the risk event follows a power law or Pareto distribution, we show that MPMR also scales with the number of observations $n$ (or equivalently the length of the time interval) by a power law, $\text{MPMR}(n) \propto n^η$, where $η$ is the scaling exponent. The scale invariance allows for reasonable estimations of long-term risks based on the extrapolation of more reliable estimations of short-term risks. The scaling relationship also gives rise to a robust and low-bias estimator of the tail index (TI) $ξ$ of the size distribution, $ξ= 1/η$. We demonstrate the use of this risk measure for describing the tail risks in financial markets as well as the risks associated with natural hazards (earthquakes, tsunamis, and excessive rainfall).
