Insights into Robust Portfolio Optimization : Decomposing Robust Portfolios into Mean-Variance and Risk-Based Portfolios

For a number of different formulations of robust portfolio optimization, quadratic and absolute, we show that a) in the limit of low uncertainty in estimated asset mean returns the robust portfolio converges towards the mean-variance portfolio obtained with the same inputs; and b) in the limit of high uncertainty the robust portfolio converges towards a risk-based portfolio, which is a function of how the uncertainty in estimated asset mean returns is defined. We give examples in which the robust portfolio converges toward the minimum variance, the inverse variance, the equal-risk budget and the equally weighted portfolio in the limit of sufficiently large uncertainty in asset mean returns. At intermediate levels of uncertainty we find that a weighted average of the mean-variance portfolio and the respective limiting risk-based portfolio offer a good representation of the robust portfolio, in particular in the case of the quadratic formulation. The results remain valid even in the presence of portfolio constraints, in which case the limiting portfolios are the corresponding constrained mean-variance and constrained risk-based portfolios. We believe our results are important in particular for risk-based investors who wish to take into account expected returns to gently tilt away from their current allocations, e.g. risk parity or minimum variance