Hierarchical Risk Parity (HRP) is a portfolio-allocation method introduced by Marcos López de Prado in 2016 that builds weights from the structure of the correlation matrix rather than by optimizing against its inverse. Its core tension is a deliberate trade-off: by refusing to invert the covariance matrix — the step that makes Markowitz mean-variance optimization mathematically optimal in-sample but wildly unstable out-of-sample ("Markowitz's curse") — HRP gives up theoretical optimality to gain robustness. It uses machine-learning clustering to group assets into a tree, then allocates capital top-down so that genuinely substitutable assets compete only against each other, not against the whole universe at once.

How it's calculated / formed

HRP runs in three sequential stages (López de Prado, JPM 2016):

1. Tree (hierarchical) clustering. Pairwise asset correlations ρ are converted to a distance via d(i,j) = √(½·(1 − ρ(i,j))), which maps ρ ∈ [−1, 1] to d ∈ [0, 1] (perfectly correlated → 0, perfectly anti-correlated → 1). A second-order Euclidean distance is then taken between assets' distance vectors, and agglomerative clustering merges assets into a binary dendrogram. The original paper used single linkage; common libraries (PortfolioLab, Riskfolio-Lib) also expose complete, average, and Ward linkage.

2. Quasi-diagonalization. The dendrogram's leaf order is used to permute the rows and columns of the covariance matrix so that similar assets sit adjacent. This concentrates large covariances near the diagonal without any change of basis (no eigenvectors, no inversion) — preserving interpretability while exposing the block structure.

3. Recursive bisection. Walking top-down, each cluster is split into two sub-clusters. Each sub-cluster's variance is estimated using an internal inverse-variance portfolio (valid because the quasi-diagonal block is treated as near-diagonal). Capital is then split between the two sub-clusters inversely to their variance: the lower-variance branch gets the larger share. The factor cascades down the tree until each leaf asset holds a final weight. Weights are naturally long-only and sum to 1.

No expected-return forecast enters anywhere — HRP is a risk-based allocator, like minimum-variance and equal-risk-contribution parity, not a return optimizer.

How it's used in practice

HRP is most attractive when the asset count is large relative to the lookback window — exactly the regime where the sample covariance matrix is ill-conditioned or singular and where quadratic optimizers blow up or pile weight onto a few estimation artifacts. Practitioners use it for cross-asset and multi-factor allocation, ETF baskets, and as a diversification overlay. Because it never inverts the matrix, it runs on a singular covariance matrix (more assets than observations) where Critical Line Algorithm (CLA) or analytic min-variance fail outright.

It is widely available: PyPortfolioOpt, Riskfolio-Lib, and Hudson & Thames' mlfinlab/PortfolioLab all ship reference implementations, which is a large part of why HRP spread quickly among quant practitioners. Variants exist — most notably Hierarchical Equal Risk Contribution (HERC), which replaces inverse-variance bisection with equal-risk-contribution within the dendrogram, and Nested Clustered Optimization (NCO), which applies mean-variance inside clusters after de-noising.

Adoption, debate & evidence

HRP is genuinely influential — it is taught in López de Prado's Advances in Financial Machine Learning, has a Wikipedia entry, and has multiple open-source implementations. But the evidence is mixed and contested, and the popular framing oversells it.

López de Prado's own headline result comes from Monte Carlo simulation, not live markets: across 10,000 simulations he reported out-of-sample variance of ~0.0671 for HRP versus ~0.0928 for inverse-variance (IVP) and ~0.1157 for CLA — i.e. CLA's variance running roughly 72% higher than HRP's (and IVP's ~38% higher), despite CLA explicitly minimizing variance. That is a strong claim, but it is generated under a covariance structure the author chose; the simulation design favors methods robust to estimation error.

Independent empirical studies are more sober. Peer-reviewed and working-paper comparisons (e.g. studies in Computational Economics, the Annals of Operations Research, and Journal of Economic Analysis on HRP) generally find HRP comparable to mean-variance and other risk-parity methods, with its main edge in lower drawdowns and more stable, diversified weights rather than higher returns. Multiple studies note HRP does not reliably beat the naïve 1/N (equal-weight) portfolio on out-of-sample return — which is the same humbling benchmark (DeMiguel, Garlappi & Uppal, 2009) that defeats most "optimized" methods. Results are also sensitive to choices the original paper underspecified: linkage method, distance metric, and estimation window all materially move outcomes. Honest summary: HRP's robustness advantage over naïve mean-variance is well supported; a broad claim that it "outperforms out of sample" in returns is not.

Strengths & limitations

Strengths. No matrix inversion, so it is numerically stable and works on singular/ill-conditioned covariance matrices; weights are intuitive and diversified rather than concentrated; it degrades gracefully when correlations are noisy; it is fast and parameter-light.

Limitations. It still relies entirely on the sample correlation matrix, so it inherits estimation error in the correlations themselves (clustering noisy correlations yields noisy trees). It ignores expected returns by construction, so it cannot express a view. Single-linkage clustering is prone to "chaining" and can produce unstable dendrograms; results are not invariant to the linkage/distance choice. The recursive-bisection weighting is a heuristic with no optimality proof — its theoretical justification is, as Wikipedia notes, "non-trivial." The #1 misuse is treating HRP as a return-enhancing alpha engine: it is a risk-diversification heuristic, and backtests that show it "winning" often do so on risk-adjusted or drawdown metrics, not raw return, and rarely against a fair 1/N benchmark.

Sources

• López de Prado, M. (2016). "Building Diversified Portfolios that Outperform Out of Sample." Journal of Portfolio Management 42(4): 59–69. jpm.pm-research.com · SSRN
• Wikipedia, "Hierarchical Risk Parity" (steps, distance formula, Monte Carlo figures, conflict-of-interest note). en.wikipedia.org
• Hudson & Thames, "Portfolio Optimisation with PortfolioLab: Hierarchical Risk Parity" (linkage methods, quasi-diagonalization, bisection). hudsonthames.org
• "An Empirical Evaluation of Distance Metrics in HRP," Computational Economics (2025). link.springer.com
• "Portfolio Optimisation: An Empirical Study of HRP and Mean-Variance Methods" (HRP comparable to MV; does not beat 1/N on return). researchgate.net
• DeMiguel, Garlappi & Uppal (2009) on the 1/N benchmark (general reference for the naïve-diversification challenge).

Disputes flagged: the 72%/CLA variance figure is from the author's own Monte Carlo simulation, not live data; independent studies find HRP roughly on par with MV and not a reliable 1/N-beater on returns; results vary with linkage/distance/window choices.