Hierarchical Equal Risk Contribution (HERC): combining hierarchical clustering with equal risk allocation

Hierarchical Risk Parity (HRP) groups assets by similarity and allocates within and between clusters to spread risk. Hierarchical Equal Risk Contribution (HERC) is a refinement that imposes equal risk contribution at each level of the cluster tree, producing portfolios that are more uniformly diversified than standard HRP while retaining the robustness to estimation error that makes hierarchical methods appealing.

What HERC is

HERC was developed by Raffinot (2018) in The Hierarchical Equal Risk Contribution Portfolio as an extension of López de Prado's HRP framework. The standard HRP procedure first clusters assets by their pairwise distance (typically computed from the correlation matrix), then allocates capital recursively: at each split in the cluster tree, capital is divided between the two sub-clusters in inverse proportion to their cluster volatilities, and the same procedure is applied recursively within each sub-cluster.

HERC modifies the second step. At each split, instead of inverse-volatility-weighted allocation, HERC imposes equal risk contribution between the two sub-clusters: each sub-cluster contributes equally to the portfolio's overall risk at that level. The resulting portfolio is more uniformly diversified by construction, particularly in cases where one sub-cluster contains many highly-correlated assets and another contains a single uncorrelated asset.

The algorithm preserves HRP's main computational advantages: it does not require inverting the covariance matrix, scales gracefully to large universes, and produces stable allocations in the presence of estimation error. The trade-off is a modest increase in computational complexity for the equal-risk-contribution step at each cluster level.

How it works

The HERC procedure has three stages. First, assets are clustered using a hierarchical clustering algorithm (typically single-linkage or average-linkage) applied to a distance matrix derived from the correlation matrix. The result is a binary tree where the leaves are individual assets and the internal nodes represent groups of related assets.

Second, the clusters are allocated capital recursively from the root of the tree downward. At each split, the two sub-clusters' weights are chosen so that each contributes equally to the variance of the combined sub-portfolio. Mathematically, this is the equal-risk-contribution problem applied to a two-asset portfolio: weight α for sub-cluster A and weight 1−α for sub-cluster B, with α determined by σ_B / (σ_A + σ_B) where σ are the cluster volatilities.

Third, within each leaf cluster, the same equal-risk-contribution principle is applied across the underlying assets. The result is a complete portfolio where, by construction, no single cluster or asset dominates the portfolio's risk profile.

Compared with the standard HRP allocation (which uses inverse-volatility weights at each split), HERC produces portfolios with smaller maximum-asset weights, more even risk distribution across clusters, and slightly higher turnover at rebalancing. The empirical performance differences are small but typically favour HERC in samples with substantial estimation error.

What the evidence shows

Raffinot (2018) compared HRP, HERC, and equal-weight portfolios across multi-asset universes over 2002–2017 and found that HERC produced higher Sharpe ratios than HRP in approximately 60% of evaluation windows, with the largest gaps concentrated in periods where the underlying covariance structure shifted materially (2008–2009, 2014 commodity crash). The methodology's edge is attributable to its better diversification properties when the cluster structure changes, not to a fundamentally better expected-return forecast.

Subsequent work by Raffinot (2018b) and others has extended the comparison to single-asset-class universes (US equities, global fixed income), with similar but smaller effects. The HERC advantage tends to be largest in heterogeneous multi-asset universes and smallest in single-asset-class equity contexts.

The robustness advantage over standard mean-variance optimisation is large and consistent across all studies. Mean-variance optimisation underperforms hierarchical methods (both HRP and HERC) by 30–50% in turnover and 10–20% in realised drawdown across most evaluation windows, with the gap largest precisely in the regimes where variance estimation is most challenging.

Limitations and trade-offs

HERC depends on the choice of clustering algorithm and the distance metric used to build the cluster tree. Different choices (single vs average linkage, correlation-based vs return-based distance) produce different cluster structures and therefore different portfolios. The choice is not arbitrary, but it is also not universally optimal—some choices work better in some universes than others.

The methodology is also computationally heavier than standard HRP. The equal-risk-contribution allocation at each cluster level requires solving a small numerical optimisation problem, multiplied by the number of internal nodes in the cluster tree. For very large universes (1000+ assets), the additional cost is meaningful relative to HRP's near-instantaneous allocation.

HERC is also not a forecasting tool. Like HRP, the methodology produces a sensible allocation given the historical covariance structure but does not anticipate regime changes. Out-of-sample performance depends on the stability of the cluster structure across rebalancing periods; rapid regime changes can produce reallocation patterns that look reasonable retrospectively but do not capture the underlying shift fast enough.

HERC in pfolio

HERC is not currently a built-in option in pfolio. Hierarchical Risk Parity, the closest related approach, is available alongside mean-variance optimisation and equal weight as one of the three optimisation methods.

Related articles

• Hierarchical Risk Parity (HRP): a robust alternative to mean-variance optimisation
• Risk budgeting: allocating capital by risk contribution rather than by weight
• Risk parity investing: how to allocate by risk contribution rather than capital
• Covariance estimation in portfolio optimisation: look-back periods, shrinkage, and stability