We utilize the copula function methodology to separate out the components which describe the marginal behavior of the return processes and the dependence structure between the random variables from the joint density. In order to reflect the non-ellipticity of the joint distribution and heavy tails in the extreme quantile of the marginal distributions of asset returns, we use the generalized Pareto distribution (GPD) as the margins and a variety of parametric copula functions along with a nonparametric copula function in the analysis. We select the optimal copulas from a variety of non-nested copulas based on the model selection criteria. In calculating the risk measures, we assume that the returns are jointly distributed to the parametric copulas as well as to the empirical copula. We then compare the result with that from the bivariate normal distribution. The results show that the VaR and ES computed from the copula function which takes the complicated and possibly nonlinear dependence structure into account performs better than the one based on the linear correlation-based normality assumption.