A major attraction of the Black–Litterman approach for portfolio optimization is the potential for integrating subjective views on expected returns. In this article, the authors provide a new approach for deriving the views and their uncertainty using predictive regressions estimated in a Bayesian framework. The authors show that the Bayesian estimation of predictive regressions fits perfectly with the idea of Black–Litterman. The subjective element is introduced in terms of the investors’ belief about the degree of predictability of the regression. In this setup, the uncertainty of views is derived naturally from the Bayesian regression, rather than by using the covariance of returns. Finally, the authors show that this approach of integrating uncertainty about views is the main reason this method outperforms other strategies.

In this article, the author introduces the Hierarchical Risk Parity (HRP) approach to address three major concerns of quadratic optimizers, in general, and Markowitz’s critical line algorithm (CLA), in particular: instability, concentration, and underperformance. HRP applies modern mathematics (graph theory and machine-learning techniques) to build a diversified portfolio based on the information contained in the covariance matrix. However, unlike quadratic optimizers, HRP does not require the invertibility of the covariance matrix. In fact, HRP can compute a portfolio on an ill-degenerated or even a singular covariance matrix—an impossible feat for quadratic optimizers. Monte Carlo experiments show that HRP delivers lower out-ofsample variance than CLA, even though minimum variance is CLA’s optimization objective. HRP also produces less risky portfolios out of sample compared to traditional risk parity methods.