Fitting bivariate cumulative returns with copulas

We propose a copula based statistical method of fitting joint cumulative returns between a market index and a stock from the index family to daily data. Modifying the method of inference functions for margins (IFM method), we perform two separate maximum likelihood estimations of the univariate marginal distributions, assumed to be normal inverse gamma mixtures with kurtosis parameter equal to 6, followed by a minimization of the bivariate chisquare statistic associated to an adequate bivariate version of the usual Pearson goodness-offit test. Our copula fitting results for daily cumulative returns between the Swiss Market Index and a stock in the index family for an approximate one-year period are quite satisfactory. The best overall fits are obtained for the new linear Spearman copula, as well as for the Frank and Gumbel-Hougaard copulas. Finally, a significant application to covariance estimation for the linear Spearman copula is discussed.