Before revision:  It turns out that the minimal tail dependence measure coincides with the tail dependence coefficient, and the maximal tail dependencemeasure overestimates the degree of extreme co-movements.
After revision:  It turns out that the minimal tail dependence measure coincides with the tail dependence coefficient, and thus the classical notion always underestimates the degree of tail dependence. A formula for the supremum TCM is derived and shown to overestimate the degree of extreme co-movements.