EDHEC-Risk Institute October 2016

Multi-Dimensional Risk and Performance Analysis for Equity Portfolios — October 2016

1. Literature and Practice Reviews

for all k = 1,…, K .

statistical artefacts. The second shortcoming, particularly severe in the context of the design of a relative contribution 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 contribution of the remaining factors tends to be biased towards low values. The minimum linear torsion technique of Deguest, Martellini and Meucci (2013) avoids these problems and facilitates the interpretation of the orthogonalised factors. The minimum-torsion factors are defined as the factors that have the same variances as the original ones and minimise the sum of tracking errors with respect to them. Mathematically, they solve the program:

An explicit expression for the minimum torsion matrix t can be found in Carli, Deguest and Martellini (2014). Once factors have been made orthogonal, we estimate the exposures of a portfolio with respect to the uncorrelated factors, as well as the idiosyncratic return η t :

The portfolio variance now involves no covariance term:

so the contribution of each factor is:

,

(1.9)

subject to

Figure 4: Risk (Volatility) Decomposition with Orthogonalised Factors of Selected Mutual Funds with the Fama-French 4 Factor Model (2001-2015) Mutual fund returns are downloaded from Datastream and factor returns are from Ken French's library. Returns are monthly. We first orthogonalise the risk factors via the minimum linear torsion approach. We then regress for each mutual fund their excess returns against the orthogonalised factor returns for the period 2001-2015. The minimum-torsion factors have the same variance as the original factors. We measure historical factor covariance matrix over the same period and use Equation (1.9) to perform the risk attribution.

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