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

Abstract

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, risk-parity portfolios include all investable assets, and idiosyncratic risk has little effect on weight magnitude. The algebraic forms for optimal asset weights derived in this article 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. These analytic solutions reveal precisely how various kinds of predicted risk affect the relative magnitude of security weights in each type of risk-based portfolio construction.