ABSTRACT

Substantial changes in the financial markets and insurance companies have needed the development of the structure of the risk benchmark, which is the challenge addressed in this paper. We propose a theorem that expands the tail conditional moment (TCM) measure from elliptical distributions to wider classes of skew-elliptical distributions. This family of distributions is suitable for modeling asymmetric phenomena. We obtain the analytical formula for the $n^{\text{th}}$ TCM for skew-elliptical distributions to help well to figure out the risk behavior along the tail of loss distributions. We derive four significant results and generalize the tail conditional skewness (TCS) and the tail conditional kurtosis (TCK) measures for generalized skew-elliptical distributions, which are used to determine the skewness and the kurtosis in the tail of loss distributions. The proposed TCM measure has been applied to well-known families of generalized skew-elliptical distributions. We also provide a practical example of a portfolio problem by calculating the proposed TCM measure for the weighted sum of generalized skew-elliptical distributions.

摘要

金融市场和保险公司的重大变化需要风险基准结构的发展，这是本文要解决的挑战。我们提出了一个定理，将尾条件矩 (TCM) 度量从椭圆分布扩展到更广泛的斜椭圆分布类别。这一系列分布适用于对不对称现象进行建模。我们得到解析公式$n^{\text{th}}$用于偏斜椭圆分布的 TCM 有助于很好地找出沿损失分布尾部的风险行为。我们得出了四个重要结果，并概括了广义偏斜椭圆分布的尾部条件偏度 (TCS) 和尾部条件峰度 (TCK) 度量，用于确定损失分布尾部的偏度和峰度。所提出的 TCM 度量已应用于众所周知的广义斜椭圆分布家族。我们还通过计算广义偏斜椭圆分布的加权和的建议 TCM 度量来提供投资组合问题的实际示例。