The existence of second order moment or the finite variance is a commonly used assumption in financial time series analysis. We examine the validation of this condition for main stock index return series by applying the extreme value theory. We compare the performances of the adaptive Hill’s estimator and the Smith’s estimator for the tail index using Monte Carlo simulations for both i.i.d data and dependent data. The simulation results show that the Hill’s estimator with adaptive data-based truncation number performs better in both cases. It has not only smaller bias but also smaller MSE when the true tail index α is not more than 2. Moreover, the Hill’s estimator shows precise results for the hypothesis test of infinite variance. Applying the adaptive Hill’s estimator to main stock index returns over the world, we find that for most indices, the second moment does exist for daily, weekly and monthly returns. However, an additional test for the existence of the fourth moment shows that generally the fourth moment does not exist, especially for daily returns. And these results don’t change when a Gaussian-GARCH effect is removed from the original return series.