An important feature when working with financial data is the fact that the residuals of GARCH-models often have fatter tails than the tails of a normal distribution due to the large number of “outliers” in the data. This requires more detailed study. Kurtosis and quantile-based measure of heavy-tailedness were analyzed and compared in the article in relation to the problem of choosing the GARCH(1,1)-model specification. The data of indices of the Moscow Exchange were considered for the period from April 01, 2019 to February 22, 2022. Kurtosis values ​​ranged from 3 to 52. Empirical data showed that kurtosis was very sensitive to “outliers” in the data, which made it difficult to make assumptions about the distribution of model residuals. The approach considered in this paper based on the heavy-tailedness measure made it possible to justify the choice of degrees of freedom of the t-distribution for the model residuals to explain the fat tails in financial data. It was found that GARCH(1,1)-models with t(5)-distribution in the residuals are common.