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To examine the asymmetry of financial data in detail, we have considered both the tail dependence with diverse copulas and Jung et al.'s (8) directional dependence by copula. From the empirical study in this paper, we have found that the tail dependence by Patton's (11) modified symmetrized Joe-Clayton copula function did not show the asymmetry property sufficiently because there is no tail dependence for the currency exchange rates of Republic of Korea and Japan against the US dollar after the 2008 financial crisis. However, by using the copula approach for directional dependence, we showed that there exists the asymmetry of the currency exchange rates of Republic of Korea and Japan against the US dollar before and after the 2008 financial crisis because the directional dependence only when existing an asymmetry of data can be found. We conclude that the copula approach for directional dependence can be supplemented to explain the asymmetry property in addition to the tail dependence approach.