Hierarchical PCA and Applications to Portfolio Management
Trading and Investing / Education
Principal Components Analysis (PCA) and random matrix theory (RMT) have become widespread tools for data analysis. PCA provides a mathematical and objective approach to extract economic information from the correlation matrices of asset returns. In this approach, the analyst extracts common risk factors from the eigenvectors and eigenvalues of the correlation matrix.
It is widely known that the common risk-factors derived from PCA beyond the first eigen portfolio are generally difficult to interpret and thus to use in practical portfolio management. Professor Marco Avellaneda explores an alternative approach (HPCA) which makes strong use of the partition of the market into sectors.
The HPCA spectrum has much higher eigenvalues (separated from zero) than PCA, since they are bounded from below by the lowest eigenvalue from all the sectors. Thus, the HPCA matrix is better conditioned than the full empirical matrix.
This provides an interpretation for common risk factors, as “portfolios of long-only sector portfolios”.
The HPCA defines the factors into “sector-sector” and “long-short intra-sector”. PCA eigenvectors, in contrast, become increasingly difficult to interpret as simple sector-sector interactions or intra-sector interactions.
Furthermore, Professor Avellaneda shows that this approach leads to no loss of information with respect to PCA in the case of equities (constituents of the S&P 500) and also that the associated common factors admit simple interpretations.
The model can also be used in markets in which the sectors have asynchronous price information. Such as single-name credit default swaps, generalizing the works of Cont and Kan (2011) and Ivanov (2016).
