Hierarchical Risk Parity
Jun 13, 2025
The Quanted Round-up is a curated summary that covers relevant research on key topics in quantitative financial decision-making.
Highlights
This edition looks at how hierarchy can inform portfolio construction, helping reduce estimation error and improve robustness. Rather than relying on flat covariance estimates, the papers use structure such as graphs and recursive methods to produce more stable interpretable allocations across different asset classes.
Hierarchical Minimum Variance Portfolios: A Theoretical and Algorithmic Approach
Gamal Mograby
We introduce a novel approach to portfolio optimization that leverages hierarchical graph structures and the Schur complement method to systematically reduce computational complexity while preserving full covariance information. Inspired by Lopez de Prados hierarchical risk parity and Cottons Schur complement methods, our framework models the covariance matrix as an adjacency- like structure of a hierarchical graph. We demonstrate that portfolio optimization can be recursively reduced across hierarchical levels, allowing optimal weights to be computed efficiently by inverting only small submatrices regardless of portfolio size. Moreover, we translate our results into a recursive algorithm that constructs optimal portfolio allocations. Our results reveal a transparent and mathematically rigorous connection between classical Markowitz mean-variance optimization, hierarchical clustering, and the Schur complement method.
Schur Complementary Allocation: A Unification of Hierarchical Risk Parity and Minimum Variance Portfolios
Peter Cotton
Despite many attempts to make optimization-based portfolio construction in the spirit of Markowitz robust and approachable, it is far from universally adopted. Meanwhile, the collection of more heuristic divide-and- conquer approaches was revitalized by Lopez de Prado where Hierarchical Risk Parity (HRP) was introduced. This paper reveals the hidden connection between these seemingly disparate approaches.
Beyond risk parity – A machine learning-based hierarchical risk parity approach on cryptocurrencies
Tobias Burggraf
It has long been known that estimating large empirical covariance matrices can lead to very unstable solutions, with estimation errors more than offsetting the benefits of diversification. In this study, we employ the Hierarchical Risk Parity approach, which applies state-of-the-art mathematics including graph theory and unsupervised machine learning to a large portfolio of cryptocurrencies. An out-of-sample comparison with traditional risk- minimization methods reveals that Hierarchical Risk Parity outperforms in terms of tail risk-adjusted return, thereby working as a potential risk management tool that can help cryptocurrency investors to better manage portfolio risk. The results are robust to different rebalancing intervals, covariance estimation windows and methodologies.
Overcoming Markowitz's Instability with the Help of the Hierarchical Risk Parity (HRP): Theoretical Evidence
Alexandre Antonov, Alex Lipton & Marcos Lopez de Prado
In this paper we compare two methods of portfolio allocation: the classical Markowitz one and the hierarchical risk parity (HRP) approach. We derive analytical values for the noise of allocation weights coming from the estimated covariance. We demonstrate that the HRP is indeed less noisy (and thus more robust) w.r.t. the classical Markowitz. The second part of the paper is devoted to a detailed analysis of the optimal portfolio variance for which we derive analytical formulas and theoretically demonstrate the superiority of the HRP w.r.t to the Markowitz optimization. We also address practical outcomes of our analytics. The first one is a fast estimation of the confidence level of the optimization weights calculated for a single (real-life) scenario. The second practical usefulness of the analytics is an HRP portfolio construction criterion which selects assets and clusters minimizing the analytical portfolio variance. We confirm our theoretical results with numerous numerical experiments. Our calculation technique can be also used in other areas of portfolio optimization.
Network Risk Parity: graph theory- based portfolio construction
Vito Ciciretti & Alberto Pallotta
This study presents network risk parity, a graph theory- based portfolio construction methodology that arises from a thoughtful critique of the clustering-based approach used by hierarchical risk parity. Advantages of network risk parity include: the ability to capture one-to-many relationships between securities, overcoming the one-to- one limitation; the capacity to leverage the mathematics of graph theory, which enables us, among other things, to demonstrate that the resulting portfolios is less concentrated than those obtained with mean-variance; and the ability to simplify the model specification by eliminating the dependency on the selection of a distance and linkage function. Performance-wise, due to a better representation of systematic risk within the minimum spanning tree, network risk parity outperforms hierarchical risk parity and other competing methods, especially as the number of portfolio constituents increases.
Hierarchical Risk Parity: Accounting for Tail Dependencies in Multi-Asset Multi- Factor Allocations
Harald Lohre, Carsten Rother & Kilian Axel Schäfer
We investigate portfolio diversification strategies based on hierarchical clustering. These hierarchical risk parity strategies use graph theory and unsupervised machine learning to build diversified portfolios by acknowledging the hierarchical structure of the investment universe. In this chapter, we consider two dissimilarity measures for clustering a multi-asset multi-factor universe. While the Pearson correlation coefficient is a popular choice, we are especially interested in a measure based on the lower tail dependence coefficient. Such innovation is expected to achieve better tail risk management in the context of allocating to skewed style factor strategies. Indeed, the corresponding hierarchical risk parity strategies seem to have been navigating the associated downside risk better, yet come at the cost of high turnover. A comparison based on block-bootstrapping evidences alternative risk parity strategies along economic factors to be on par in terms of downside risk with those based on statistical clusters.
An Empirical Evaluation of Distance Metrics in Hierarchical Risk Parity Methods for Asset Allocation
Francisco Salas-Molina, David Pla-Santamaria, Ana Garcia-Bernabeu & Adolfo Hilario-Caballero
Hierarchical Risk Parity methods address instability, concentration, and underperformance in asset allocation by taking advantage of machine learning techniques to build a diversified portfolio. HRP methods produce a hierarchical structure to the correlation between assets by means of tree clustering that results in a reorganization of the covariance matrix of returns. However, HRP admits multiple variations in terms of clustering algorithms and distance metrics. In this paper, we evaluate the out-of-sample performance of alternative hierarchical distance metrics for clustering purposes using real stock markets in three different market scenarios: bull market, sideways trend, and bear market. We pay special attention to the mean-variance performance of the output portfolios as an estimation of the ability of alternative methods to estimate future return and risk. Our results show that correlation-based metrics provide better performance than non-correlation metrics. In addition, HRP methods outperform quadratic optimizers in two of the three stock market scenarios.
References
An Empirical Evaluation of Distance Metrics in Hierarchical Risk Parity Methods for Asset Allocation. February 2025.Salas-Molina, F., Pla-Santamaria, D., Garcia-Bernabeu, A. and Hilario-Caballero, A. Comput Econ (2025). Available at Springer Nature Link: https://doi.org/10.1007/s10614-025-10848-w
Beyond risk parity – A machine learning-based hierarchical risk parity approach on cryptocurrencies. January 2021. Burggraf, T. Finance Research Letters, 38(1). Available at
ScienceDirect: https://doi.org/10.1016/j.frl.2020.101523.
Hierarchical Minimum Variance Portfolios: A Theoretical and Algorithmic Approach. March 2025. Mograby, G. Available at arXiv: https://doi.org/10.48550/arXiv.2503.12328
Hierarchical Risk Parity: Accounting for Tail Dependencies in Multi-Asset Multi- Factor Allocations. November 2020. Lohre, H., Rother, C. and Schäfer, K. A. Chapter 9 in: Machine Learning and Asset Management, Emmanuel Jurczenko (ed.), Iste and Wiley, 2020, pp.
332-368. Available at SSRN: http://dx.doi.org/10.2139/ssrn.3513399
Network Risk Parity: graph theory-based portfolio construction. February 2024. Ciciretti, V. and Pallotta, A. J Asset Manag 25, 136–146 (2024). Available at Springer Nature Link: https://doi.org/10.1057/s41260-023-00347-8
Overcoming Markowitz's Instability with the Help of the Hierarchical Risk Parity (HRP): Theoretical Evidence. March 2024. Antonov, A., Lipton, A. and López de Prado, M. Available at SSRN: http://dx.doi.org/10.2139/ssrn.4748151
Schur Complementary Allocation: A Unification of Hierarchical Risk Parity and Minimum Variance Portfolios. October 2024. Cotton, P. Available at arXiv: https://doi.org/10.48550/arXiv.2411.05807
The Quanted Round-up is a curated summary that covers relevant research on key topics in quantitative financial decision-making.
Highlights
This edition looks at how hierarchy can inform portfolio construction, helping reduce estimation error and improve robustness. Rather than relying on flat covariance estimates, the papers use structure such as graphs and recursive methods to produce more stable interpretable allocations across different asset classes.
Hierarchical Minimum Variance Portfolios: A Theoretical and Algorithmic Approach
Gamal Mograby
We introduce a novel approach to portfolio optimization that leverages hierarchical graph structures and the Schur complement method to systematically reduce computational complexity while preserving full covariance information. Inspired by Lopez de Prados hierarchical risk parity and Cottons Schur complement methods, our framework models the covariance matrix as an adjacency- like structure of a hierarchical graph. We demonstrate that portfolio optimization can be recursively reduced across hierarchical levels, allowing optimal weights to be computed efficiently by inverting only small submatrices regardless of portfolio size. Moreover, we translate our results into a recursive algorithm that constructs optimal portfolio allocations. Our results reveal a transparent and mathematically rigorous connection between classical Markowitz mean-variance optimization, hierarchical clustering, and the Schur complement method.
Schur Complementary Allocation: A Unification of Hierarchical Risk Parity and Minimum Variance Portfolios
Peter Cotton
Despite many attempts to make optimization-based portfolio construction in the spirit of Markowitz robust and approachable, it is far from universally adopted. Meanwhile, the collection of more heuristic divide-and- conquer approaches was revitalized by Lopez de Prado where Hierarchical Risk Parity (HRP) was introduced. This paper reveals the hidden connection between these seemingly disparate approaches.
Beyond risk parity – A machine learning-based hierarchical risk parity approach on cryptocurrencies
Tobias Burggraf
It has long been known that estimating large empirical covariance matrices can lead to very unstable solutions, with estimation errors more than offsetting the benefits of diversification. In this study, we employ the Hierarchical Risk Parity approach, which applies state-of-the-art mathematics including graph theory and unsupervised machine learning to a large portfolio of cryptocurrencies. An out-of-sample comparison with traditional risk- minimization methods reveals that Hierarchical Risk Parity outperforms in terms of tail risk-adjusted return, thereby working as a potential risk management tool that can help cryptocurrency investors to better manage portfolio risk. The results are robust to different rebalancing intervals, covariance estimation windows and methodologies.
Overcoming Markowitz's Instability with the Help of the Hierarchical Risk Parity (HRP): Theoretical Evidence
Alexandre Antonov, Alex Lipton & Marcos Lopez de Prado
In this paper we compare two methods of portfolio allocation: the classical Markowitz one and the hierarchical risk parity (HRP) approach. We derive analytical values for the noise of allocation weights coming from the estimated covariance. We demonstrate that the HRP is indeed less noisy (and thus more robust) w.r.t. the classical Markowitz. The second part of the paper is devoted to a detailed analysis of the optimal portfolio variance for which we derive analytical formulas and theoretically demonstrate the superiority of the HRP w.r.t to the Markowitz optimization. We also address practical outcomes of our analytics. The first one is a fast estimation of the confidence level of the optimization weights calculated for a single (real-life) scenario. The second practical usefulness of the analytics is an HRP portfolio construction criterion which selects assets and clusters minimizing the analytical portfolio variance. We confirm our theoretical results with numerous numerical experiments. Our calculation technique can be also used in other areas of portfolio optimization.
Network Risk Parity: graph theory- based portfolio construction
Vito Ciciretti & Alberto Pallotta
This study presents network risk parity, a graph theory- based portfolio construction methodology that arises from a thoughtful critique of the clustering-based approach used by hierarchical risk parity. Advantages of network risk parity include: the ability to capture one-to-many relationships between securities, overcoming the one-to- one limitation; the capacity to leverage the mathematics of graph theory, which enables us, among other things, to demonstrate that the resulting portfolios is less concentrated than those obtained with mean-variance; and the ability to simplify the model specification by eliminating the dependency on the selection of a distance and linkage function. Performance-wise, due to a better representation of systematic risk within the minimum spanning tree, network risk parity outperforms hierarchical risk parity and other competing methods, especially as the number of portfolio constituents increases.
Hierarchical Risk Parity: Accounting for Tail Dependencies in Multi-Asset Multi- Factor Allocations
Harald Lohre, Carsten Rother & Kilian Axel Schäfer
We investigate portfolio diversification strategies based on hierarchical clustering. These hierarchical risk parity strategies use graph theory and unsupervised machine learning to build diversified portfolios by acknowledging the hierarchical structure of the investment universe. In this chapter, we consider two dissimilarity measures for clustering a multi-asset multi-factor universe. While the Pearson correlation coefficient is a popular choice, we are especially interested in a measure based on the lower tail dependence coefficient. Such innovation is expected to achieve better tail risk management in the context of allocating to skewed style factor strategies. Indeed, the corresponding hierarchical risk parity strategies seem to have been navigating the associated downside risk better, yet come at the cost of high turnover. A comparison based on block-bootstrapping evidences alternative risk parity strategies along economic factors to be on par in terms of downside risk with those based on statistical clusters.
An Empirical Evaluation of Distance Metrics in Hierarchical Risk Parity Methods for Asset Allocation
Francisco Salas-Molina, David Pla-Santamaria, Ana Garcia-Bernabeu & Adolfo Hilario-Caballero
Hierarchical Risk Parity methods address instability, concentration, and underperformance in asset allocation by taking advantage of machine learning techniques to build a diversified portfolio. HRP methods produce a hierarchical structure to the correlation between assets by means of tree clustering that results in a reorganization of the covariance matrix of returns. However, HRP admits multiple variations in terms of clustering algorithms and distance metrics. In this paper, we evaluate the out-of-sample performance of alternative hierarchical distance metrics for clustering purposes using real stock markets in three different market scenarios: bull market, sideways trend, and bear market. We pay special attention to the mean-variance performance of the output portfolios as an estimation of the ability of alternative methods to estimate future return and risk. Our results show that correlation-based metrics provide better performance than non-correlation metrics. In addition, HRP methods outperform quadratic optimizers in two of the three stock market scenarios.
References
An Empirical Evaluation of Distance Metrics in Hierarchical Risk Parity Methods for Asset Allocation. February 2025.Salas-Molina, F., Pla-Santamaria, D., Garcia-Bernabeu, A. and Hilario-Caballero, A. Comput Econ (2025). Available at Springer Nature Link: https://doi.org/10.1007/s10614-025-10848-w
Beyond risk parity – A machine learning-based hierarchical risk parity approach on cryptocurrencies. January 2021. Burggraf, T. Finance Research Letters, 38(1). Available at
ScienceDirect: https://doi.org/10.1016/j.frl.2020.101523.
Hierarchical Minimum Variance Portfolios: A Theoretical and Algorithmic Approach. March 2025. Mograby, G. Available at arXiv: https://doi.org/10.48550/arXiv.2503.12328
Hierarchical Risk Parity: Accounting for Tail Dependencies in Multi-Asset Multi- Factor Allocations. November 2020. Lohre, H., Rother, C. and Schäfer, K. A. Chapter 9 in: Machine Learning and Asset Management, Emmanuel Jurczenko (ed.), Iste and Wiley, 2020, pp.
332-368. Available at SSRN: http://dx.doi.org/10.2139/ssrn.3513399
Network Risk Parity: graph theory-based portfolio construction. February 2024. Ciciretti, V. and Pallotta, A. J Asset Manag 25, 136–146 (2024). Available at Springer Nature Link: https://doi.org/10.1057/s41260-023-00347-8
Overcoming Markowitz's Instability with the Help of the Hierarchical Risk Parity (HRP): Theoretical Evidence. March 2024. Antonov, A., Lipton, A. and López de Prado, M. Available at SSRN: http://dx.doi.org/10.2139/ssrn.4748151
Schur Complementary Allocation: A Unification of Hierarchical Risk Parity and Minimum Variance Portfolios. October 2024. Cotton, P. Available at arXiv: https://doi.org/10.48550/arXiv.2411.05807
The Quanted Round-up is a curated summary that covers relevant research on key topics in quantitative financial decision-making.
Highlights
This edition looks at how hierarchy can inform portfolio construction, helping reduce estimation error and improve robustness. Rather than relying on flat covariance estimates, the papers use structure such as graphs and recursive methods to produce more stable interpretable allocations across different asset classes.
Hierarchical Minimum Variance Portfolios: A Theoretical and Algorithmic Approach
Gamal Mograby
We introduce a novel approach to portfolio optimization that leverages hierarchical graph structures and the Schur complement method to systematically reduce computational complexity while preserving full covariance information. Inspired by Lopez de Prados hierarchical risk parity and Cottons Schur complement methods, our framework models the covariance matrix as an adjacency- like structure of a hierarchical graph. We demonstrate that portfolio optimization can be recursively reduced across hierarchical levels, allowing optimal weights to be computed efficiently by inverting only small submatrices regardless of portfolio size. Moreover, we translate our results into a recursive algorithm that constructs optimal portfolio allocations. Our results reveal a transparent and mathematically rigorous connection between classical Markowitz mean-variance optimization, hierarchical clustering, and the Schur complement method.
Schur Complementary Allocation: A Unification of Hierarchical Risk Parity and Minimum Variance Portfolios
Peter Cotton
Despite many attempts to make optimization-based portfolio construction in the spirit of Markowitz robust and approachable, it is far from universally adopted. Meanwhile, the collection of more heuristic divide-and- conquer approaches was revitalized by Lopez de Prado where Hierarchical Risk Parity (HRP) was introduced. This paper reveals the hidden connection between these seemingly disparate approaches.
Beyond risk parity – A machine learning-based hierarchical risk parity approach on cryptocurrencies
Tobias Burggraf
It has long been known that estimating large empirical covariance matrices can lead to very unstable solutions, with estimation errors more than offsetting the benefits of diversification. In this study, we employ the Hierarchical Risk Parity approach, which applies state-of-the-art mathematics including graph theory and unsupervised machine learning to a large portfolio of cryptocurrencies. An out-of-sample comparison with traditional risk- minimization methods reveals that Hierarchical Risk Parity outperforms in terms of tail risk-adjusted return, thereby working as a potential risk management tool that can help cryptocurrency investors to better manage portfolio risk. The results are robust to different rebalancing intervals, covariance estimation windows and methodologies.
Overcoming Markowitz's Instability with the Help of the Hierarchical Risk Parity (HRP): Theoretical Evidence
Alexandre Antonov, Alex Lipton & Marcos Lopez de Prado
In this paper we compare two methods of portfolio allocation: the classical Markowitz one and the hierarchical risk parity (HRP) approach. We derive analytical values for the noise of allocation weights coming from the estimated covariance. We demonstrate that the HRP is indeed less noisy (and thus more robust) w.r.t. the classical Markowitz. The second part of the paper is devoted to a detailed analysis of the optimal portfolio variance for which we derive analytical formulas and theoretically demonstrate the superiority of the HRP w.r.t to the Markowitz optimization. We also address practical outcomes of our analytics. The first one is a fast estimation of the confidence level of the optimization weights calculated for a single (real-life) scenario. The second practical usefulness of the analytics is an HRP portfolio construction criterion which selects assets and clusters minimizing the analytical portfolio variance. We confirm our theoretical results with numerous numerical experiments. Our calculation technique can be also used in other areas of portfolio optimization.
Network Risk Parity: graph theory- based portfolio construction
Vito Ciciretti & Alberto Pallotta
This study presents network risk parity, a graph theory- based portfolio construction methodology that arises from a thoughtful critique of the clustering-based approach used by hierarchical risk parity. Advantages of network risk parity include: the ability to capture one-to-many relationships between securities, overcoming the one-to- one limitation; the capacity to leverage the mathematics of graph theory, which enables us, among other things, to demonstrate that the resulting portfolios is less concentrated than those obtained with mean-variance; and the ability to simplify the model specification by eliminating the dependency on the selection of a distance and linkage function. Performance-wise, due to a better representation of systematic risk within the minimum spanning tree, network risk parity outperforms hierarchical risk parity and other competing methods, especially as the number of portfolio constituents increases.
Hierarchical Risk Parity: Accounting for Tail Dependencies in Multi-Asset Multi- Factor Allocations
Harald Lohre, Carsten Rother & Kilian Axel Schäfer
We investigate portfolio diversification strategies based on hierarchical clustering. These hierarchical risk parity strategies use graph theory and unsupervised machine learning to build diversified portfolios by acknowledging the hierarchical structure of the investment universe. In this chapter, we consider two dissimilarity measures for clustering a multi-asset multi-factor universe. While the Pearson correlation coefficient is a popular choice, we are especially interested in a measure based on the lower tail dependence coefficient. Such innovation is expected to achieve better tail risk management in the context of allocating to skewed style factor strategies. Indeed, the corresponding hierarchical risk parity strategies seem to have been navigating the associated downside risk better, yet come at the cost of high turnover. A comparison based on block-bootstrapping evidences alternative risk parity strategies along economic factors to be on par in terms of downside risk with those based on statistical clusters.
An Empirical Evaluation of Distance Metrics in Hierarchical Risk Parity Methods for Asset Allocation
Francisco Salas-Molina, David Pla-Santamaria, Ana Garcia-Bernabeu & Adolfo Hilario-Caballero
Hierarchical Risk Parity methods address instability, concentration, and underperformance in asset allocation by taking advantage of machine learning techniques to build a diversified portfolio. HRP methods produce a hierarchical structure to the correlation between assets by means of tree clustering that results in a reorganization of the covariance matrix of returns. However, HRP admits multiple variations in terms of clustering algorithms and distance metrics. In this paper, we evaluate the out-of-sample performance of alternative hierarchical distance metrics for clustering purposes using real stock markets in three different market scenarios: bull market, sideways trend, and bear market. We pay special attention to the mean-variance performance of the output portfolios as an estimation of the ability of alternative methods to estimate future return and risk. Our results show that correlation-based metrics provide better performance than non-correlation metrics. In addition, HRP methods outperform quadratic optimizers in two of the three stock market scenarios.
References
An Empirical Evaluation of Distance Metrics in Hierarchical Risk Parity Methods for Asset Allocation. February 2025.Salas-Molina, F., Pla-Santamaria, D., Garcia-Bernabeu, A. and Hilario-Caballero, A. Comput Econ (2025). Available at Springer Nature Link: https://doi.org/10.1007/s10614-025-10848-w
Beyond risk parity – A machine learning-based hierarchical risk parity approach on cryptocurrencies. January 2021. Burggraf, T. Finance Research Letters, 38(1). Available at
ScienceDirect: https://doi.org/10.1016/j.frl.2020.101523.
Hierarchical Minimum Variance Portfolios: A Theoretical and Algorithmic Approach. March 2025. Mograby, G. Available at arXiv: https://doi.org/10.48550/arXiv.2503.12328
Hierarchical Risk Parity: Accounting for Tail Dependencies in Multi-Asset Multi- Factor Allocations. November 2020. Lohre, H., Rother, C. and Schäfer, K. A. Chapter 9 in: Machine Learning and Asset Management, Emmanuel Jurczenko (ed.), Iste and Wiley, 2020, pp.
332-368. Available at SSRN: http://dx.doi.org/10.2139/ssrn.3513399
Network Risk Parity: graph theory-based portfolio construction. February 2024. Ciciretti, V. and Pallotta, A. J Asset Manag 25, 136–146 (2024). Available at Springer Nature Link: https://doi.org/10.1057/s41260-023-00347-8
Overcoming Markowitz's Instability with the Help of the Hierarchical Risk Parity (HRP): Theoretical Evidence. March 2024. Antonov, A., Lipton, A. and López de Prado, M. Available at SSRN: http://dx.doi.org/10.2139/ssrn.4748151
Schur Complementary Allocation: A Unification of Hierarchical Risk Parity and Minimum Variance Portfolios. October 2024. Cotton, P. Available at arXiv: https://doi.org/10.48550/arXiv.2411.05807
