Multivariate random sums appear in many scientific fields, most notably in actuarial science, where they model both the number of claims and their sizes. Unfortunately, they pose severe inferential problems. For example, their density function is analytically intractable, in the general case, thus preventing likelihood inference. In this paper, we address the problem by the method of moments, under the assumption that the claim size and the claim number have a multivariate skew-normal and a Poisson distribution, respectively. In doing so, we also derive closed-form expressions for some fundamental measures of multivariate kurtosis and highlight some limitations of both projection pursuit and invariant coordinate selection.