The Bayesian Roots of Risk Parity in a Mean-Risk World

Abstract

Risk parity has been considered a heuristic asset allocation method. In this paper, we show that, to the contrary, risk parity is a special case of a mean-risk type of a portfolio optimization problem with log-regularization to constrain weights. We show that log-regularization leads to a fund separation type property where the optimal solution lies between an unconstrained mean-risk portfolio and a risk parity (or risk budgeted in general) portfolio. We also demonstrate in a Bayesian framework that these log-regularizations are actually equivalent to putting prior distributions on the optimal portfolio weights. While leading to non-convex solutions, we further provide applications of these log-regularized mean-risk models in long/short asset allocation so as to generalize risk parity beyond long-only investments. We provide numerical examples on a typical institutional asset universe, and finally, we prove that there are efficient risk parity portfolios not necessarily satisfying equal Sharpe and constant correlation conditions, which have been widely mistaken as the only efficiency conditions for risk parity portfolios in the industry.