Quantifying tail dependence is an important issue in insurance and risk management. The prevalent tail dependence coefficient (TDC), however, is known to underestimate the degree of tail dependence and it does not capture non-exchangeable tail dependence since it evaluates the limiting tail probability only along the main diagonal. To overcome these issues, two novel tail dependence measures called the maximal tail concordance measure (MTCM) and the average tail concordance measure (ATCM) are proposed. Both measures are constructed based on tail copulas and possess clear probabilistic interpretations in that the MTCM evaluates the largest limiting probability among all comparable rectangles in the tail, and the ATCM is a normalized average of these limiting probabilities. In contrast to the TDC, the proposed measures can capture non-exchangeable tail dependence. Analytical forms of the proposed measures are also derived for various copulas. A real data analysis reveals striking tail dependence and tail non-exchangeability of the return series of stock indices, particularly in periods of financial distress.
翻译:量化尾鱼依赖性是保险和风险管理中的一个重要问题,但众所周知,普遍尾尾鱼依赖性系数(TDC)低估了尾鱼依赖性的程度,它没有捕捉不可交换的尾鱼依赖性,因为它只评估沿主要对角线的极限尾巴概率;为解决这些问题,提出了两个新的尾尾鱼依赖性新措施,称为最大尾鱼协调性措施(MTCM)和平均尾鱼协调性措施(ATCM),这两项措施都是根据尾巴合金制定,并具有明确的概率解释,因为MTCM评估了尾鱼所有可比较的矩形的最大限制概率,而适应措施是这些限制性概率的正常平均数;与TDC不同的是,拟议措施可以捕捉不可交换的尾鱼依赖性;还从各种对拟议措施的分析形式中得出各种对尾巴依赖性和尾鱼群指数返回系列的尾巴不可交换性,特别是在财政困难时期。