Improved Risk Reporting with Factor-Based Diversification Measures

Improved Risk Reporting with Factor-Based Diversification Measures — February 2014

2. Portfolio Diversification Measures

α ≥ 0, α ≠ 1.

factor. In that case, we simply set A = P . Hence, an allocation to the N constituents can be regarded as an allocation to the principal factor portfolios, with weights

(2.5)

Note thatMeucci (2009a) defines the effective number of uncorrelated bets using the entropy metric for the dispersion of the factor contributions:

F = A −1 = P'

and covariance matrix

Σ F = Λ 2 . This leads to the following variance contributions:

(2.7)

(2.6)

There are a number of shortcomings, however, with the PCA approach. The first shortcoming is the difficulty in interpreting the factors, which are pure statistical artifacts. The second shortcoming, particularly severe in the context of the design of a diversification measure, is that by construction principal components are defined so as to achieve the highest possible explanatory power. As a result, the contribution of the first few factors is often overwhelmingly large with respect to the contribution of other factors, and the portfolio diversification measure empirically tends to be biased towards low values (see Section 3). This approach may lead to counter-intuitive results. In particular, Meucci et al. (2013) show that the ENB measure is equal to 1 for an equally-weighted portfolio based on a universe of assets with equal volatility and pairwise correlation values, regardless of the correlation value, while the intuition would suggest that for a vanishing constant correlation value such a portfolio should have a number of bets equal to the number of assets. This counter-intuitive full-concentration effect follows because the equally-weighted portfolio is in this case fully exposed to the first principal component and not exposed to any other principal component (for more details, see Appendix A2 in Meucci et al. (2013)).

In order to remain coherent with the choice made for the ENC measure, we will follow Meucci (2009a) and use the entropy metric denoted as ENB 1 . There are many ways to decompose the portfolio into N contributions of uncorrelated factors, i.e., to compute a valid matrix A . In the following, we will explore two competing approaches to extract N implicit uncorrelated factors from the constituents. We also consider an approach based on an explicit factor model obtained by orthogonalisation of the Fama-French market, value and size factors (Fama and French (1992)) together with the momentum factor (Carhart (1997)). Extracting Implicit Factors via Principal Component Analysis (PCA) Principal co-moment analysis is a standard procedure for extracting uncorrelated factors from a basket of correlated constituents (see for instance Meucci (2009a), Frahm and Wiechers (2011), Lohre et al. (2011), and Deguest et al. (2013)). Formally, it is based on the diagonalisation of the covariance matrix Σ of asset returns, Σ = P Λ P' , with Λ 2 being the diagonal matrix of eigenvalues , ..., of Σ and P the matrix of eigenvectors. Each eigenvector can be interpreted as the vector of weights for a “principal factor portfolio”, while each eigenvalue represents the variance of a

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An EDHEC-Risk Institute Publication