MAD risk parity portfolios (Q6549614)

A time honored exercise in portfolio theory is the construction of portfolios in which each assets contributes the the overall risk in exactly the same way. In order for this to be possible one needs to compute analytically the marginal contribution of each asset, which is possible whenever the risk measure adopted is differentiable and positively homogeneous. In this paper the risk measure adopted is the \(L^1\) deviation from the mean, a convex, positively homogeneous but not monotonic functional (and thus not coherent). Moreover, such risk function will not be differentiable and is actually additive only for pairs of variable whose deviations from the mean are comonotonic. Nevertheless, the authors are able to derive risk parity portfolios using subdifferentials. In particular they prove, under mild conditions, the existence and uniqueness of such portfolios and show that the existence problem is actually related to a dual problem involving the minimization of the mean absolute deviations under logarithmic constraints.