A natural approach to enhance portfolio diversification is to rely on factor-risk parity, which yields the portfolio whose risk is equally spread among a set of uncorrelated factors. The standard choice is to take the variance as risk measure, and the principal components (PCs) of asset returns as factors. Although PCs are unique and useful for dimension reduction, they are an arbitrary choice: any rotation of the PCs results in uncorrelated factors. This is problematic because we demonstrate that any portfolio is a factor-variance-parity portfolio for some rotation of the PCs. More importantly, choosing the PCs does not account for the higher moments of asset returns. To overcome these issues, we propose using the independent components (ICs) as factors, which are the rotation of the PCs that are maximally independent, and care about higher moments of asset returns. We demonstrate that using the IC-variance-parity portfolio helps to reduce the return kurtosis. We also show how to exploit the near independence of the ICs to parsimoniously estimate the factor-risk-parity portfolio based on value at risk. Finally, we empirically demonstrate that portfolios based on ICs outperform those based on PCs, and several state-of-the-art benchmarks.

Statistical modeling techniques—and factor models in particular—are extensively used in practice, especially in the insurance and finance industry, where many risks have to be accounted for. In risk management applications, it might be important to analyze the situation when fixing the value of a weighted sum of factors, for example to a given quantile. In this work, we derive the $(n-1)$-dimensional distribution corresponding to a n-dimensional i.i.d. standard Normal vector $\mathbf{Z}={(Z_1,Z_2,\dots ,Z_n)}^{\prime }$ subject to the weighted sum constraint ${\mathbf{w}}^{\prime }\mathbf{Z}=c$, where $\mathbf{w}={(w_1,w_2,\dots ,w_n)}^{\prime }$ and $w_i\ne 0$. This law is proven to be a Normal distribution, whose mean vector $\mathit{\mu }$ and covariance matrix $\mathsf{\Sigma }$ are explicitly derived as a function of $(\mathbf{w},c)$. The derivation of the density relies on the analytical inversion of a very specific positive definite matrix. We show that it does not correspond to naive sampling techniques one could think of. This result is then used to design algorithms for sampling $\mathbf{Z}$ under constraint that ${\mathbf{w}}^{\prime }\mathbf{Z}=c$ or ${\mathbf{w}}^{\prime }\mathbf{Z}\le c$ and is illustrated on two applications dealing with Value-at-Risk and Expected Shortfall.