Improved Risk Reporting with Factor-Based Diversification Measures

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

2. Portfolio Diversification Measures

Extracting Implicit Factors via Minimal Linear Torsion (MLT) A competing approach to extracting uncorrelated factors from a basket of correlated constituents has been introduced in Meucci et al. (2013), who propose to look for the minimal uncorrelated linear transformation A in the sense that: •  the factors are as close as possible to the original constituents, i.e., is minimal; •  the factors have the same variances as the original constituents, i.e., σ F k = σ k for all k = 1, ..., N . By construction, the obtained factors are the closest uncorrelated representations of the original constituents. When the constituents are highly correlated, the factors can prove to be substantially distorted transformations of the original constituents, but at least they enjoy the desirable property of emanating from the least distorted transformation. We show in Appendix B.2 that the formal solution to this problem is to take A = P Λ −1 UV'D , where D is the diagonal matrix containing the original constituents’ volatilities, P and Λ the outputs of the PCA run on Σ , and U and V the matrices obtained from the singular value decomposition of Λ P'D = USV (where S is the diagonal matrix containing the singular values of Λ P'D ).

is a no-transformation in this case since the assets already were uncorrelated). In this case, the ENB will be equal to N , as the intuition suggests, as opposed to 1, which would be the misleading value obtained with the PCA approach. Using Explicit Factors - The Case of Fama-French-Carhart Four Factor Model In this case, we propose to consider explicit factors in order to perform our factor- based portfolio diversification analysis. In the following, we consider Fama-French- Carhart four factor model (FF in brief) and give a new framework to compute the variance contributions. We denote by r FF , the vector of the four-factor returns, and with r FF ⊥ , the vector of the orthogonalised four-factor returns. First, we build the orthogonalised factors from the minimal lineartransformation A FF such that: •  the orthogonalised factors are as close as possible to the original factors, i.e. is minimal; •  the orthogonalised factors have the same variances as the original factors, i.e. for all k = 1, ..., 4. Once we have the orthogonalised version of the four factors, we compute the exposure of any portfolio to each of the factors as:

, (2.9)

. (2.8)

where is a vector of residual that is uncorrelated with the orthogonalised four factors. Finally, since the factors are uncorrelated by construction, we can write the portfolio variance as:

One can easily check that starting with a set of N assets with the same volatility and a zero value for all pairwise correlations, the minimal linear torsion bets will be equal to the original assets (in other words, the minimal linear transformation

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An EDHEC-Risk Institute Publication