  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.
