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Nobel Prize winner Harry Markowitz described diversification, with its ability to enhance portfolio returns while reducing risk, as the “only free lunch” in investing (Markowitz [1952]). Yet, diversifying a portfolio in real life is easier said than done.
Investors are aware of the benefits of diversification but form portfolios without giving proper consideration to the correlations (Goetzmann and Kumar [2008]). Moreover, modern and complex portfolio optimization methods are optimal in sample but often provide rather poor out-of-sample forecast performance. For instance, DeMiguel et al. [2009] demonstrate that the equal-weighted allocation, which gives the same importance to each asset, beats the entire set of commonly used portfolio optimization techniques. In fact, optimized portfolios depend on expected returns and risks, but even small estimation errors can result in large deviations from optimal allocations in an optimizer’s result (Michaud [1989]).
To overcome this issue, academics and practitioners have developed risk-based portfolio optimization techniques (minimum variance, equal-risk contribution, risk budgeting, etc.) that do not rely on return forecasts (Roncalli [2013]). However, these still require the inversion of a positive-definite covariance matrix, which leads to errors of such magnitude that they entirely offset the benefits of diversification (López de Prado [2016b]).
Exploring a new way of capital allocation, López de Prado [2016a] introduces a portfolio diversification technique called hierarchical risk parity (HRP). One of the main advantages of HRP is in computing a portfolio on an ill-degenerated or even a singular covariance matrix. Lau et al. [2017] apply HRP to different cross-asset universes consisting of many tradable risk premia indexes and confirm that HRP delivers superior risk-adjusted returns. Alipour et al. [2016] propose a quantum-inspired version of HRP, which outperforms HRP and thus other conventional methods.
The starting point of HRP is that a correlation matrix is too complex to be properly analyzed and understood. If you have N assets of interest, there are [Formula omitted: see PDF.] pairwise correlations among them and that number grows quickly. For example, there are as many as 4,950 correlation coefficients between stocks of the FTSE 100 and 124,750 between stocks of the S&P 500. More importantly, correlation matrices lack the notion of hierarchy. Actually, Nobel Prize laureate Herbert Simon has argued that complex systems can be arranged in a natural hierarchy comprising...