Risk Parity, Maximum Diversification, and Minimum Variance : An Analytic Perspective

Analytic solutions to Risk Parity, Maximum Diversification, and Minimum Variance portfolios provide useful perspectives about their construction and composition. Individual asset weights depend on both systematic and idiosyncratic risk in all three risk-based portfolios, but systematic risk eliminates many investable assets in long-only constrained Maximum Diversification and Minimum Variance portfolios. On the other hand, all investable assets are included in Risk Parity portfolios, and idiosyncratic risk has little impact on the magnitude of the weights. The algebraic forms for optimal asset weights derived in this paper yield generalizable properties of risk-based portfolios, in contrast to empirical simulations that employ a specific set of historical returns, proprietary risk models, and multiple constraints. The analytic solutions reveal precisely how the various kinds of predicted risk impact the relative magnitude of security weights under each type of risk-based portfolio construction