Standard, PCA-based factor analysis suffers from a number of well known problems due to the random nature of pairwise correlations of asset returns. We analyse an alternative based on ICA, where factors are identified based on their non-Gaussianity, instead of their variance. Generalizations of portfolio construction to the ICA framework leads to two semi-optimal portfolio construction methods: a fat-tailed portfolio, which maximises return per unit of non-Gaussianity, and the hybrid portfolio, which asymptotically reduces variance and non-Gaussianity in parallel. For fat-tailed portfolios, the portfolio weights scale like performance to the power of $1/3$, as opposed to linear scaling of Kelly portfolios; such portfolio construction significantly reduces portfolio concentration, and the winner-takes-all problem inherent in Kelly portfolios. For hybrid portfolios, the variance is diversified at the same rate as Kelly PCA-based portfolios, but excess kurtosis is diversified much faster than in Kelly, at the rate of $n^{-2}$ compared to Kelly portfolios $n^{-1}$ for increasing number of components $n$.
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