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Hierarchical Equal Risk Contribution

An extension of Hierarchical Risk Parity that combines hierarchical clustering with equal risk contribution principles, ensuring balanced risk allocation both between and within clusters.

Overview

Hierarchical Equal Risk Contribution (HERC), proposed by Thomas Raffinot in 2017, bridges two powerful portfolio construction ideas: the hierarchical clustering approach of HRP and the risk budgeting framework of Equal Risk Contribution (ERC). While HRP allocates weights using simple inverse-variance bisection, HERC enforces that each cluster contributes equally to the total portfolio risk.

The method operates in two stages. First, it uses hierarchical clustering to identify natural groupings of assets based on their correlation structure. Second, it allocates risk equally across clusters (inter-cluster allocation) and then within each cluster ensures that every asset contributes equally to the cluster's risk (intra-cluster allocation).

This dual-level risk equalization produces portfolios that are well-diversified both across asset groups and within them, addressing a common criticism of naive risk parity approaches that can still produce concentrated exposures when assets within a group are highly correlated.

Mathematical Formulation

Correlation Distance

As in HRP, the algorithm begins by computing a distance metric from the correlation matrix:

This distance is used to perform agglomerative hierarchical clustering, producing a dendrogram that reveals the natural grouping structure of the assets.

Inter-Cluster Allocation: Equal Risk Contribution

At the top level, HERC allocates risk equally across the identified clusters. If there are clusters, the risk contribution of each cluster must be equal:

where denotes the total risk contribution of cluster to the portfolio. This means each cluster contributes exactly of the total portfolio risk, regardless of the number of assets within the cluster or their individual volatilities.

Intra-Cluster Allocation: Equal Risk Contribution

Within each cluster, the weights are determined such that every asset contributes equally to the cluster's risk. The equal risk contribution condition requires:

Expanding the partial derivative for portfolio volatility :

This simplifies to requiring that is equal for all assets within the cluster. This is typically solved using numerical optimization since no closed-form solution exists for the general case.

Final Weights

The final portfolio weight for asset in cluster is:

where is the weight allocated to cluster (from the inter-cluster ERC) and is asset 's weight within cluster (from the intra-cluster ERC).

Risk Measures

HERC is flexible in the choice of risk measure used for the equal risk contribution allocation. Common options include:

  • Variance: The standard portfolio variance .
  • Standard deviation: The portfolio volatility .
  • Conditional Value at Risk (CVaR): The expected loss beyond the VaR threshold, providing a tail-risk-sensitive allocation.
  • Maximum drawdown: The maximum peak-to-trough loss, aligning the allocation with drawdown-based risk management.

Advantages & Limitations

Advantages

  • Dual-level diversification: Ensures risk balance both across clusters and within clusters, avoiding hidden concentrations.
  • Flexible risk measures: Can use any risk measure (variance, CVaR, drawdown) for the allocation, adapting to different risk management needs.
  • No matrix inversion: Like HRP, avoids the numerical instability of covariance matrix inversion.
  • No return estimates: Purely risk-based allocation that does not depend on expected return forecasts.

Limitations

  • Cluster sensitivity: The number of clusters and the clustering method can significantly affect the allocation.
  • Computational cost: The intra-cluster ERC requires numerical optimization for each cluster, adding computational overhead.
  • No return optimization: Cannot incorporate views on expected returns, potentially missing opportunities.
  • Heuristic elements: The hierarchical decomposition is not derived from a single global optimization problem.

References

  • Raffinot, T. (2017). "Hierarchical Clustering-Based Asset Allocation." The Journal of Portfolio Management, 44(2), 89-99.
  • Raffinot, T. (2018). "The Hierarchical Equal Risk Contribution Portfolio." Working paper, SSRN.
  • Lopez de Prado, M. (2016). "Building Diversified Portfolios that Outperform Out of Sample." The Journal of Portfolio Management, 42(4), 59-69.
  • Maillard, S., Roncalli, T., & Teiletche, J. (2010). "The Properties of Equally Weighted Risk Contribution Portfolios." The Journal of Portfolio Management, 36(4), 60-70.