Hierarchical Risk Parity (HRP)

HRP begins by computing a distance matrix from the correlation matrix, transforming correlations into a metric space. A hierarchical clustering algorithm (typically single-linkage or Ward's method) then groups assets into a dendrogram. This tree structure captures the nested economic relationships between assets, ensuring that highly correlated assets—like Coca-Cola and PepsiCo—are placed close together on the branches before any capital allocation occurs.

The clustering output is used to reorder the covariance matrix into a quasi-diagonal form. This step places large covariances near the diagonal, effectively grouping similar assets together. Unlike the raw matrix, this reordered structure is highly intuitive and eliminates the need to invert a noisy, ill-conditioned matrix. The seriation process ensures that the subsequent allocation respects the natural economic hierarchy of the assets.

Capital is allocated top-down through the dendrogram using inverse-variance weighting at each split. At every node, the algorithm:

• Splits the cluster into two sub-clusters
• Assigns weight to each branch inversely proportional to its variance
• Recursively repeats until reaching single assets This ensures that risk is diversified at every level of the hierarchy, preventing concentrated bets in highly correlated clusters.

Traditional Mean-Variance Optimization (MVO) requires inverting the covariance matrix, which becomes numerically unstable when assets are highly correlated or when the number of observations is less than the number of assets. HRP completely bypasses this step. By using the dendrogram structure and recursive bisection, HRP produces robust, stable weights that do not suffer from the extreme concentration and high turnover plaguing MVO portfolios in high-dimensional settings.

Empirical studies demonstrate that HRP portfolios consistently achieve lower out-of-sample variance and lower turnover than MVO, Risk Parity, and equally-weighted benchmarks. Because HRP does not rely on precise return forecasts or matrix inversion, its allocations are less sensitive to estimation errors. This makes HRP particularly effective for large universes—such as the S&P 500 constituents—where the number of assets far exceeds the number of meaningful independent observations.

The clustering step can be customized with alternative distance measures beyond standard Pearson correlation. Practitioners can incorporate:

• Tail-dependence coefficients for downside risk clustering
• Mutual information for non-linear dependencies
• Dynamic conditional correlations for time-varying relationships This flexibility allows HRP to adapt to specific market regimes or asset classes, such as clustering cryptocurrencies by on-chain metrics rather than price correlations.