Hierarchical Equal Risk Contribution (HERC)

An advanced portfolio optimization method combining clustering with risk parity

Overview

Hierarchical Equal Risk Contribution (HERC), introduced by Thomas Raffinot in 2017, extends the Hierarchical Risk Parity (HRP) approach by incorporating principles from Equal Risk Contribution (ERC) portfolios. HERC combines the benefits of hierarchical clustering for asset organization with the risk parity approach that equalizes risk contributions across portfolio components.

HERC addresses some limitations of HRP by ensuring that risk is allocated more evenly across the portfolio structure, potentially leading to better diversification and more robust performance across different market conditions.

Intuitive Explanation

Think of HERC as a balanced team organization strategy:

Imagine you're organizing a large project team. First, you group team members by their complementary skills (clustering). Then, instead of simply dividing work based on team size, you ensure each group bears an equal share of the project's risk and complexity, regardless of how many people are in each group.

In HERC:

Assets are clustered based on their relationships (typically correlation).
Each cluster is treated as a "risk unit" whose contribution should be equalized with other clusters.
Within clusters, risk is also distributed equally among constituent assets.

This multi-level risk equalization approach ensures both intra-cluster and inter-cluster risk parity, creating a hierarchical structure that is both balanced and intuitively organized.

Example: Consider a portfolio containing high-tech growth stocks, stable blue-chip stocks, bonds, and commodities. Traditional methods might struggle with precise risk estimation. HERC would first group these into logical clusters, then ensure each cluster contributes equally to overall portfolio risk, regardless of the number of assets in each group.

Detailed Mathematical & Algorithmic Explanation

HERC builds on the hierarchical clustering framework but implements a different allocation strategy:

Step 1: Hierarchical Clustering

Similar to HRP, HERC begins by grouping assets based on their correlations, using a distance measure:

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.

The clustering creates a hierarchical structure that groups similar assets together, forming a dendrogram.

Step 2: Quasi-Diagonalization

Assets are reordered according to the hierarchical structure to create a quasi-diagonal covariance matrix, where closely related assets appear adjacent to each other.

Step 3: Equal Risk Contribution Allocation

This is where HERC diverges from HRP. Instead of recursively bisecting based on inverse variance, HERC implements risk parity at each level of the hierarchy:

Inter-Cluster Allocation: Risk is allocated equally between clusters at the same hierarchical level.

RC_i = RC_j \quad \forall i,j \in \text{Clusters}

where $RC_i$ is the risk contribution of cluster $i$ .

Intra-Cluster Allocation: Within each cluster, weight is determined by equalizing risk contribution:

w_i \cdot \frac{\partial \sigma(w)}{\partial w_i} = w_j \cdot \frac{\partial \sigma(w)}{\partial w_j} \quad \forall i,j \in \text{Cluster}

where:

$w_i$ is the weight of asset $i$ .
$\sigma(w)$ is the portfolio volatility.
$\frac{\partial \sigma(w)}{\partial w_i}$ is the marginal contribution to risk.

This approach ensures that each asset's contribution to its cluster's risk is equal, and each cluster's contribution to the overall portfolio risk is also equal.

Risk Measures

HERC can utilize various risk measures to define risk contribution, including:

Standard Deviation/Variance: Traditional volatility-based measures.
Mean Absolute Deviation (MAD): Average of absolute deviations from the mean.
Conditional Value at Risk (CVaR): Expected loss in the worst scenarios.
Drawdown measures: Various metrics based on portfolio drawdowns.

The choice of risk measure allows HERC to be adapted to different market conditions and investor preferences.

Advantages and Limitations

Advantages

Improved Risk Distribution: More consistent risk allocation across the portfolio compared to HRP.
Hierarchical Structure: Maintains the benefits of clustered asset organization from HRP.
Flexibility in Risk Measures: Can be implemented with different risk definitions beyond variance.
Robust to Estimation Errors: Less sensitive to input errors than traditional mean-variance methods.

Limitations

Computational Complexity: Can be more complex to implement than HRP due to the risk parity calculations.
Parameter Sensitivity: Results can vary based on clustering method, linkage criteria, and risk measure choice.
No Expected Return Consideration: Like HRP, focuses primarily on risk without directly incorporating return expectations.

Comparison with Related Methods

HERC vs. HRP

Hierarchical Risk Parity (HRP) allocates based on inversely proportional risk at each split, which can lead to uneven risk contributions. HERC enhances this by explicitly equalizing risk contributions across clusters and within clusters, potentially providing better diversification.

HERC vs. Traditional Risk Parity

Traditional Risk Parity treats all assets as a flat structure, equalizing risk contributions across all assets. HERC respects the hierarchical structure of asset relationships, potentially leading to more intuitive allocations that better reflect market segments.

HERC vs. Mean-Variance Optimization

Mean-Variance Optimization depends heavily on expected return estimates and can produce concentrated portfolios.HERC focuses on risk diversification and is more robust to estimation errors, typically producing more balanced allocations.

Practical Applications and Implementation Considerations

HERC is particularly valuable in the following scenarios:

Large Asset Universes: When dealing with many assets where estimation errors can compound.
Risk-Focused Allocation: For investors primarily concerned with risk management rather than return maximization.
Multi-Asset Portfolios: When combining different asset classes with distinct risk characteristics.

Implementation considerations:

Linkage Method Selection: The choice of linkage method (single, complete, average, ward) can significantly impact clustering results.
Risk Measure Selection: Different risk measures (variance, CVaR, etc.) can lead to different allocations.
Rebalancing Frequency: Due to the hierarchical structure, HERC portfolios may require less frequent rebalancing than traditional methods.

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." 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.
López 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
Riskfolio-Lib Documentation: Hierarchical Clustering Portfolio Optimization