Beyond De Prado and Cotton: Hierarchical and Iterative Methods for General Mean-Variance Portfolios

Abstract

Hierarchical Risk Parity [López de Prado, 2016] and the Schur-complement generalisation of [Cotton, 2024] are among the most widely adopted regularised portfolio construction methods, yet both are signal-blind: they solve only the minimum-variance problem and cannot accommodate an arbitrary expected-return forecast. This paper introduces three methods that incorporate alpha signals into hierarchical and regularised portfolio construction.

HRP-$\mu$ is a hierarchical allocator that accepts an arbitrary signal $\mu$ and nests standard HRP when $\gamma = 0$ and $\mu=\mathbf{1}$. It preserves the tree-based structure of HRP while extending it beyond the minimum-variance setting. HRP-$\Sigma\mu$ strengthens this construction by replacing inverse-variance representatives with recursive local mean–variance optima, thereby using richer within-cluster covariance information at the same $O(N^2)$ asymptotic cost.

CRISP (Correlation-Regularised Iterative Shrinkage Portfolios) is an iterative solver for $P_\gamma w = \mu$ with $P_\gamma = (1-\gamma)\operatorname{diag}(\Sigma) + \gamma \Sigma$, so that $\gamma$ interpolates between a diagonal portfolio rule and full Markowitz. At convergence, CRISP is Markowitz applied to a variance-preserving shrunk covariance—diagonal variances unchanged, off-diagonal correlations shrunk—with $\gamma$ tuned for out-of-sample Sharpe rather than covariance-estimation loss.

In Monte Carlo experiments across multiple covariance regimes and estimation ratios, HRP-$\mu$ and HRP-$\Sigma\mu$ both outperform plain HRP—not only on general expected-return forecasts, where HRP is signal-blind, but also on the minimum-variance problem $\mu = \mathbf{1}$ that is HRP’s home territory—with HRP-$\Sigma\mu$ consistently improving on HRP-$\mu$. CRISP at intermediate $\gamma$ is the dominant method in both regimes, outperforming HRP, Cotton, Ledoit–Wolf shrinkage, direct Markowitz, and the signal-aware hierarchical methods at every sample size tested.

Keywords: portfolio optimization, hierarchical risk parity, Schur complement, signal-dependent allocation, covariance shrinkage, out-of-sample evaluation

JEL Classification: G11, C61, C63, C55

Code: All code, data, and reproduction scripts are available at https://github.com/bwuebben/beyond_hrp_and_cotton.