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This research investigates four tail relationships between the U.S. dollar and S&amp;P 500 returns by a dynamic mixture copula model with a link between the dependence structure and dependence intensity. The empirical results find that four tail relationships are evident and time-varying. Regarding the negative dependence, the tail relationship is prominently stronger in the situation where the U.S. dollar depreciates and the stock index increases than in the situation where the U.S. dollar appreciates and the stock index declines. Regarding the positive dependence, the situation of two returns increase occurs slightly more frequently than the situation of two returns decrease.

Four types of tail dependence structures between U.S. dollar index and S&P 500 stock returns:1990-2019