HERC2
A simplified variant of Hierarchical Equal Risk Contribution that uses equal weighting within clusters combined with hierarchical inter-cluster risk allocation for a computationally efficient and robust portfolio.
Overview
HERC2 is a variant of the Hierarchical Equal Risk Contribution framework that simplifies the intra-cluster allocation step by using equal weighting within each cluster rather than solving for equal risk contribution. This makes HERC2 faster to compute and more robust to estimation errors within clusters, at the cost of not achieving perfectly equalized risk contributions at the individual asset level.
The rationale for this simplification is that within a cluster, assets are already highly correlated (by construction of the hierarchical clustering), so equal weighting provides a reasonable approximation to equal risk contribution. The primary diversification benefit comes from the inter-cluster allocation, where HERC2 maintains the full hierarchical risk-based allocation.
This approach follows the philosophy that simpler models often outperform more complex ones out of sample, particularly when estimation noise is significant. By reducing the number of parameters estimated within each cluster, HERC2 achieves greater stability in the resulting portfolio weights.
Mathematical Formulation
Step 1: Correlation Distance and Clustering
Like HRP and HERC, the algorithm begins by computing the correlation-based distance metric:
Agglomerative hierarchical clustering is performed on the distance matrix to produce a dendrogram. The dendrogram is then cut at an appropriate level to define clusters, or the full tree structure is used for recursive bisection.
Step 2: Inter-Cluster Allocation
The inter-cluster allocation follows the same hierarchical risk-based approach as HRP and HERC. At each node of the dendrogram, the portfolio is split between two sub-clusters based on their relative risk, using inverse-variance weighting or equal risk contribution depending on the implementation.
Each cluster receives a weight that reflects its risk characteristics relative to other clusters. The sum of all cluster weights equals one.
Step 3: Intra-Cluster Equal Weighting
The key simplification in HERC2: within each cluster containing assets, the cluster weight is divided equally among all assets:
where is the final weight of asset in cluster , is the total weight allocated to cluster , and is the number of assets in cluster .
Weight Properties
The final weights satisfy the budget constraint by construction:
All weights are non-negative since both the cluster weights and the equal within-cluster allocations are non-negative.
Comparison with HERC
| Aspect | HERC | HERC2 |
|---|---|---|
| Inter-cluster | Equal Risk Contribution | Hierarchical risk-based |
| Intra-cluster | Equal Risk Contribution (optimized) | Equal weighting (1/n within cluster) |
| Computation | Requires numerical optimization per cluster | Closed-form, no optimization needed |
| Risk equalization | Exact at both levels | Exact between clusters, approximate within |
| Estimation sensitivity | Moderate | Lower (fewer estimated parameters) |
Advantages & Limitations
Advantages
- Simplicity: Equal weighting within clusters eliminates the need for intra-cluster optimization, making implementation straightforward.
- Computational speed: No numerical optimization required within clusters, reducing computation time significantly for large portfolios.
- Robustness: Fewer estimated parameters means less sensitivity to estimation error and more stable out-of-sample performance.
- No matrix inversion: Like all hierarchical methods, avoids the numerical issues associated with covariance matrix inversion.
- Transparent: The equal-weight-within-cluster rule is easy to explain and audit.
Limitations
- Approximate risk parity: Does not achieve exact equal risk contribution at the individual asset level within clusters.
- Cluster size sensitivity: Clusters with more assets receive proportionally less weight per asset, which may not be optimal.
- Ignores within-cluster structure: Equal weighting does not differentiate between high and low volatility assets within a cluster.
- No return optimization: Like other hierarchical methods, cannot incorporate expected return views.
References
- Lopez de Prado, M. (2016). "Building Diversified Portfolios that Outperform Out of Sample." The Journal of Portfolio Management, 42(4), 59-69.
- Raffinot, T. (2018). "The Hierarchical Equal Risk Contribution Portfolio." Working paper, SSRN.
- DeMiguel, V., Garlappi, L., & Uppal, R. (2009). "Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy?" The Review of Financial Studies, 22(5), 1915-1953.