A number of recent studies finds two asymmetries in dependence structures in international equity markets; specifically, dependence tends to be high in both highly volatile markets and in bear markets. In this paper, a further investigation of asymmetric dependence structures in international equity markets is performed by using the Markov switching model and copula theory. Combining these two theories enables me to model dependence structures with sufficient flexibility. Using this flexible framework, I indeed find that there are two distinct regimes in the U.S.-U.K. market. I also show that for the U.S.-U.K. market the bear regime is better described by an asymmetric copula with lower tail dependence with clear rejection of the Markov switching multivariate normal model. In addition, I show that ignorance of this further asymmetry in bear markets is very costly for risk management. Lastly, I conduct a similar analysis for other G7 countries, where I find other cases in which the use of a Markov switching multivariate normal model would be inappropriate.