This article develops on the link between the asymmetry of asset return distributions and the concentration of portfolios. We start by recalling the rationale behind the theory of diversification, in order to let appear that this theory relies on a reduction of risk viewed only at order 2 and on the related application of the theory of errors, as developed during the XVIIIth century. We also expose the controversy opened by E. Fama in 1965 on this theory of errors, in order to let appear that a change in the type of underlying randomness can lead to the concentration and not the diversification of portfolios. Then, we examine how the inclusion in the optimization program of the asymmetry between gains and losses can lead to a propensity to concentrate.We recall the main aspects of the Mitton and Vorkink (2007) model, and then we propose a new approach in the spirit of this model. We end up with an illustration of the latter framework on American data, letting appear important differences between the performance obtained with a classically diversified port- folio, a portfolio concentrated along existing models, and a portfolio concentrated along the model that we propose.