Hierarchical Equal Risk Contribution 2 (HERC2)

A simplified hierarchical approach with equal weights within clusters

Overview

Hierarchical Equal Risk Contribution 2 (HERC2) is an extension of the hierarchical clustering portfolio optimization methods that combines the benefits of clustering with simplicity in weight allocation. HERC2 uses hierarchical clustering to identify related assets but distributes weights equally within each cluster, rather than using complex risk parity calculations.

This method provides a middle ground between the sophistication of hierarchical risk-based methods (like HRP and HERC) and the simplicity of equal weighting. HERC2 preserves the natural market structure through clustering while simplifying the final allocation step.

Intuitive Explanation

Think of HERC2 as organizing a diverse project team:

Imagine you're managing a large project with many tasks. First, you identify natural groupings of related tasks (clustering). But instead of complex calculations to determine how much attention each task deserves, you simply divide your attention equally among tasks within each group, while still recognizing that some groups may need more overall resources than others.

In HERC2:

Assets are clustered based on their correlations or other similarity measures.
The portfolio's capital is allocated across clusters based on their risk characteristics.
Within each cluster, capital is distributed equally among constituent assets.

This approach maintains the hierarchical structure's benefits while simplifying the final asset allocation step, potentially reducing turnover and improving stability in the portfolio.

Example: Consider a portfolio with international stocks clustered by region. HERC2 would first determine how much to allocate to each region based on risk considerations, but within each region (e.g., European stocks), all stocks would receive equal allocations of that region's budget.

Detailed Mathematical & Algorithmic Explanation

HERC2 follows a similar approach to other hierarchical methods but with a distinct allocation strategy:

Step 1: Hierarchical Clustering

Like other hierarchical methods, HERC2 begins by clustering assets based on their correlations:

d_{i,j} = \sqrt{ \frac{1 - \rho_{i,j}}{2} }

where:

$d_{i,j}$ is the distance between assets $i$ and $j$ .
$\rho_{i,j}$ is the correlation between asset returns.

This step creates a hierarchical structure (dendrogram) that groups similar assets together.

Step 2: Determining Optimal Clusters

HERC2 uses various methods to determine the optimal number of clusters, including:

Gap Statistics: Comparing the within-cluster dispersion to that expected under a null reference distribution.
Silhouette Analysis: Measuring how similar an object is to its own cluster compared to other clusters.
Dendrogram Distance: Using a distance threshold to determine where to "cut" the dendrogram.

Step 3: Inter-Cluster Allocation

HERC2 allocates capital across clusters based on their risk characteristics. This can be done using several approaches:

Equal Risk Contribution: Each cluster contributes equally to overall portfolio risk.
Inverse Variance: Allocate in proportion to the inverse of cluster variance.
Minimum Variance: Optimize to minimize the overall portfolio variance.

The allocation to each cluster $c$ is determined by the chosen risk allocation method.

Step 4: Equal Weighting Within Clusters

This is where HERC2 differs from other hierarchical methods. Within each cluster, assets receive equal weights:

w_{i,c} = \frac{w_c}{n_c}

where:

$w_{i,c}$ is the weight of asset $i$ in cluster $c$ .
$w_c$ is the total weight allocated to cluster $c$ .
$n_c$ is the number of assets in cluster $c$ .

This equal weighting approach within clusters simplifies the allocation process while still respecting the hierarchical structure of asset relationships.

Risk Measures

Like other hierarchical methods, HERC2 can use various risk measures for the inter-cluster allocation, including:

Standard Deviation/Variance: Traditional volatility-based measures.
Conditional Value at Risk (CVaR): Expected loss in the worst scenarios.
Maximum Drawdown: Largest peak-to-trough decline.
Semi-Variance: Downside risk measures that focus only on negative returns.

Advantages and Limitations

Advantages

Simplicity: Easier to implement and explain compared to full risk parity methods.
Reduced Turnover: Equal weighting within clusters can reduce portfolio turnover.
Hierarchical Structure: Maintains the benefits of respecting market structure through clustering.
Computational Efficiency: Less computationally intensive than methods requiring optimization within clusters.

Limitations

Suboptimal Intra-Cluster Allocation: Equal weighting may not be optimal for risk management within clusters.
Sensitivity to Cluster Formation: Results depend heavily on the clustering approach and parameters.
Potential Over-Allocation: May over-allocate to smaller assets within a cluster compared to risk-based approaches.

References

Lopez de Prado, M. (2016). "Building Diversified Portfolios that Outperform Out of Sample." The Journal of Portfolio Management, 42(4), 59-69. doi:10.3905/jpm.2016.42.4.059
Raffinot, T. (2018). "The Hierarchical Equal Risk Contribution Portfolio." SSRN Electronic Journal. doi:10.2139/ssrn.3237540
Sjöstrand, D., & Behnejad, N. (2020). "Exploration of hierarchical clustering in long-only risk-based portfolio optimization." Master's thesis, Copenhagen Business School.
Riskfolio-Lib Documentation: Hierarchical Clustering Portfolio Optimization