High dimensional composite index makes experts’ preferences in set-ting weights a hard task. In the literature, one of the approaches to derive weights from a data set is Principal Component or Factor Analysis that, although conceptually different, they are similar in results when FA is based on Spectral Value Decomposition and rotation is not performed. This works motivates theoretical reasons to derive the weights of the elementary indicators in a composite index when multiple components are retained in the analysis. By Monte Carlo simulation it offers, moreover, the best strategy to identify the number of components to retain.