Hierarchical Risk Parity

Various risk parity methodologies are a popular choice for the construction of better diversified and balanced portfolios. It is notoriously hard to predict the future performance of the majority of asset classes. Risk parity approach overcomes this shortcoming by building portfolios using only assets’ risk characteristics and correlation matrix. A new research paper written by Lohre, Rother and Schafer builds on the foundation of classical risk parity methods and presents hierarchical risk parity technique. Their method uses graph theory and machine learning to build a hierarchical structure of the investment universe. Such structure allows better division of assets/factors into clusters with similar characteristics without relying on classical correlation analysis. These portfolios then offer better tail risk management, especially for skewed assets and style factor strategies.

Authors: Lohre, Rother and Schafer

Title: Hierarchical Risk Parity: Accounting for Tail Dependencies in Multi-Asset Multi-Factor Allocations

Link: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3513399


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.

Notable quotations from the academic research paper:

“The recent literature has presented risk parity allocation paradigms guided by hierarchical clustering techniques|prompting Lopez de Prado (2016) to label the technique hierarchical risk parity (HRP). Given a set of asset class and style factor returns, the corresponding algorithm would cluster these according to some distance metric and then allocate equal risk budgets along these clusters. Such clusters might be deemed more natural building blocks than the aggregate risk factors in that they automatically pick up the dependence structure and form meaningful ingredients to aid portfolio diversi cation.

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The contribution of this chapter is to thoroughly examine the use and merits of hierarchical clustering techniques in the context of multi-asset multi-factor investing. In particular, it will contrast these techniques to several competing risk-based allocation paradigms, such as 1=N, minimum-variance, standard risk parity, and diversifi ed risk parity. A major innovation is to investigate HRP strategies based on tail dependence clustering as opposed to standard correlation-based clustering. Such an approach might be particularly relevant given the elevated tail risk of some style factors. Hierarchical risk parity strategies generally build on two steps: first, hierarchical clustering algorithms uncover a hierarchical structure of the considered investment universe, resulting in a tree-based representation. Second, the portfolio weights result from applying an allocation strategy along the hierarchical structure and thus the overall portfolio is expected to exhibit a meaningful degree of diversifi cation.

The traditional risk-based allocation strategies are first directly applied to the single assets and factors, and, second, to the eight aggregated factors resulting from the imposed risk model. These eight factors can be viewed as “economic” clusters, providing a benchmark for the “statistical” hierarchical
clustering. As for HRP, the allocation strategies used either within or across clusters are risk parity, either based on inverse volatility (IVP) or equal risk contributions (ERC). For hierarchical clustering, we use Ward’s method and the dissimilarity matrices, either based on the correlation matrix or the
LTDCs. For comparison purposes, two versions of Lopez De Prado’s HRP-strategy are considered, both based on recursive bisection: First, we replicate the original strategy, using single linkage and inverse variance allocation as described in Algorithm 1. Second, we consider a variation thereof described in Algorithm 2, using Ward’s method and IVP, enabling comparability with the original HRP-strategies of Lopez de Prado. An overview of the considered strategies can be found in Table 1.

We performed backtests of the investment strategies in the six-year period from January 2012 to December 2017.

Table 2 shows performance and risk statistics as well as the average strategy turnover. First, we note that the 1=N strategy across single assets and factors has the highest return across all strategies (see Panel A). At the same time, 1=N su ffers from the highest volatility as well as the highest maximum drawdown, rendering its risk-adjusted performance sub par. Notably, the underlying lack of diversi fication is not mitigated when considering economic factors as opposed to single assets and factors (Panel B); both variants hover around 3.5 bets averaged over time. Interestingly, minimum variance optimization enables to already double this number to 7.2 (or 6.7 for the MVP-variant based on economic factors). Unsurprisingly, these two portfolios exhibit the smallest portfolio volatilities in the sample period (0.84% and 0.90%, respectively). Of course, maximum drawdown figures and risk-adjusted returns are also improved relative to equal weighting.

Next, we examine the middle-ground solution in between 1=N and minimum-variance: risk parity. In that regard, inverse volatility (ignoring correlations) for single assets and factors shows the second highest return, yet comes at the cost of some diversi fication. With 6.06 bets over time, it is short around two bets relative to the ERC variant (accounting for correlations). Still, risk-adjusted performance is fairly comparable across the two strategies, and their characteristics are more aligned when switching to economic factors as building blocks. Actually, the corresponding strategy, ERC F, has fairly similar performance characteristics to the diversi fied risk parity (DRP) strategy that is designed to have maximum number of eight bets throughout time. Having investigated the risk-based strategies for economic factors, we are eager to learn how the approaches based on statistical clusters fare. We start o by examining the original strategy of Lopez de Prado (2016). Its gross return is slightly higher than the one of the risk parity variants, yet its turnover is more than three times higher (18% versus 5%). As a result, net returns are considerably smaller as well as net Sharpe ratios (0.94 versus 1.26 for ERC F). Interestingly, the two further correlation-based HRP allocations (H IVP C and H ERC C) have even higher turnover (23.21% and 21.93%, respectively), bringing net Sharpe ratios down to 0.50 and 0.55. Finally, we investigate the eff ect of replacing the correlation-based dissimilarity matrix by one that is driven by LTDCs. We would hope for improved tail risk statistics of related strategies, and indeed, we observe H IVP L and H ERC L to experience the two smallest maximum drawdowns over the sample period, -1.20% and -1.29%, respectively. These compare to the third smallest value for MVP F (-1.44%) and the fourth smallest value for IVP F (-1.51%). The two LTDC-based HRP strategies’ turnover is less elevated, rendering these two strategies more competitive relative to the alternative risk parity strategies in terms of (risk-adjusted) returns.”

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