Measuring non-exchangeable tail dependence using tail copulas

Abstract

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.

Dependence on Tail Copula

“…Standard linear dependence measures prove inadequate in such cases, necessitating the exploration of alternative statistical models, such as the Gaussian copula, which serves as a convenient tool for modeling dependence near the mean of multivariate distributions [1]. However, it is essential to recognize that the Gaussian copula lacks the capability to measure dependence at the tails, underscoring the need to explore alternative methodologies in extreme situations [4,5].…”

“…Quantifying the degree of (tail) non-exchangeable dependence [6] poses a significant challenge in insurance and risk management literature. Tail-dependence coefficients often underestimate this degree and fail to capture non-exchangeable tail dependence as they assess the limiting tail probability solely along the diagonals [5].…”

“…While widely used, the tail dependence coefficient tends to underestimate the extent of tail dependence. This is due to its focus on measuring the rate of decline of joint tail probability exclusively along the main diagonal of Λ [5]. Furthermore, the tail dependence coefficient falls short in capturing non-exchangeable tail dependence situations, where γ(Λ) = γ( Λ) holds true, and Λ represents the copula of the random variables X 2 , X 1 [1,6].…”