Paper: arXiv 2604.23833

Authors: Bernd Johannes Wuebben

Abstract

Hierarchical Risk Parity (De Pardo) and the Schur-complement generalization of Cotton 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-$μ$ is a hierarchical allocator that accepts an arbitrary signal $μ$ and nests standard HRP when $γ= 0$ and $μ=\mathbf{1}$. It preserves the tree-based structure of HRP while extending it beyond the minimum-variance setting. HRP-$Σμ$ 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 $P_γw = μ$ with $P_γ= (1-γ)\operatorname{diag}(Σ) + γΣ$, so that $γ$ 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 $γ$ tuned for out-of-sample Sharpe rather than covariance-estimation loss. In Monte Carlo experiments across multiple covariance regimes and estimation ratios, HRP-$μ$ and HRP-$Σμ$ both outperform plain HRP with HRP-$Σμ$ consistently improving on HRP-$μ$. CRISP at intermediate $γ$ is the dominant method in both regimes, outperforming HRP, Cotton, Ledoit-Wolf shrinkage, direct Markowitz, and the signal-aware hierarchical methods.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10 (capped from 8.0: transaction costs not modelled; see penalty flags)
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical framework, including proofs of convergence, perturbation theory, and Neumann expansions, while simultaneously offering extensive Monte Carlo out-of-sample testing and fully reproducible open-source implementations.

Research Flowchart

  flowchart TD
    Goal[Research Goal: Incorporate Alpha signals into Hierarchical & Regularised Portfolios]
    Inputs[Inputs: Covariance Estimates, Alpha Forecasts μ, Shrinkage Factor γ]
    Methods[Methodology: Develop HRP-μ, HRP-Σμ, and CRISP]
    Computation[Computational Process: Recursive Clustering, Local Mean-Variance Optimization, Iterative Shrinkage]
    Validation[Validation: Monte Carlo Experiments across Covariance Regimes]
    Findings[Outcomes: HRP-Σμ improves on HRP-μ; CRISP dominates as the top performer]

    Goal --> Inputs
    Inputs --> Methods
    Methods --> Computation
    Computation --> Validation
    Validation --> Findings