HERC2
A simplified variant of Hierarchical Equal Risk Contribution that uses equal weighting within clusters, combined with the same hierarchical inter-cluster risk split as HERC.
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 HERC's naive risk parity. Equal weights estimate nothing inside a cluster, at the cost of ignoring the differences in volatility between its members.
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 argument of DeMiguel, Garlappi and Uppal (2009) that fewer estimated parameters reduce exposure to estimation noise. By estimating nothing within each cluster, HERC2 removes one source of noise from the 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 into terminal clusters. is chosen from the data by a two-difference gap rule, and is at most 15 and at most one less than the number of assets. The method needs at least four assets. With three assets, this configuration always forms one cluster and gives exactly equal weights, so FolioLab does not offer HERC2 below four assets. For a 1/N portfolio, use Equal Weight. HERC still runs on three assets. If an asset has zero variance over the window, its Spearman correlation is undefined and the method does not run.
Step 2: Inter-Cluster Allocation
The inter-cluster allocation is the same pairwise risk split as HERC. Each of the top nodes of the dendrogram splits the capital held at that node between its two sides in inverse proportion to the risk on each side, measured here by CVaR.
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 | Pairwise risk split at each dendrogram node | The same pairwise risk split, same code path |
| Intra-cluster (the allocation difference) | Naive risk parity, closed-form inverse risk | Equal weighting (1/n within cluster) |
| Codependence | Pearson | Spearman |
| Risk measure | Variance (MV) | CVaR |
| Linkage | Ward | Complete |
| Cluster-count ceiling (max_k) | 10 | 15 |
| Computation | Closed-form at every stage, no numerical solve | Closed-form at every stage, no numerical solve |
Neither method runs a numerical risk-parity solve, and neither equalizes risk exactly across the clusters. Both descend the same dendrogram with the same pairwise risk split. The only allocation rule that differs is the terminal one: HERC sizes members inside a cluster by naive risk parity, HERC2 weights them equally. The other four rows are estimator settings pinned per method, and they change the tree the descent walks, so two runs on the same universe can differ for reasons that have nothing to do with the terminal rule.
Advantages & Limitations
Advantages
- Simplicity: Equal weighting within clusters eliminates the need for intra-cluster optimization, making implementation straightforward.
- No intra-cluster estimate: The terminal rule reads no volatility or covariance inside a cluster.
- Fewer estimated parameters: Less exposure to estimation error inside each cluster.
- 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.
- Group sums are unsupported: Per-name weight bounds and asset exclusions are carried into the descent. Equal weights fix the ratio between the members of a cluster, so a bound limits the capital its cluster can take, and a bound the fixed shares cannot satisfy makes the method refuse the run. A group sum, such as a sector cap, is not supported in this release.
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.
Not investment advice. Past performance is not indicative of future results.