EDHEC-Risk Institute October 2016

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

1. Literature and Practice Reviews

Figure 3: Tracking Error Decomposition of Selected Mutual Funds with the Carhart Model (2001-2015) Mutual fund returns are downloaded from Datastream and factor returns are from Ken French's library. Returns are monthly. We regress for each mutual fund their returns in excess of the market return on factor returns for the period 2001-2015. Market returns are from Ken French’s library. We measure historical factor covariances matrix over the same period and use the formula (1.7) and (1.8).

In Figure 3, we apply the same method to the decomposition of tracking error with respect to the market factor. By subtracting market returns from the returns to long-only fund, the influence of the market factor is largely reduced, so the market factor explains much less of the tracking error than of the volatility of long-only funds. For the long-short fund, it is the opposite: the market factor has little impact on the fund’s returns, so taking excess returns reinforces this impact. The problem of attributing correlated components in the expression of volatility or tracking error would be avoided if factors were orthogonal. Hence, a second idea to perform a risk decomposition is to transform the original factors into uncorrelated factors by using some rotation technique. The new factors are linear combinations of the original ones, so they generate the same set of uncertainty and they explain exactly 1.2.2.3 Second Decomposition: Orthogonalising Factors

the same fraction of the returns to a given portfolio: the coefficient of determination (R-squared, i.e. the ratio of systematic variance to overall variance of the portfolio) is the same with the orthogonal factors than with the original ones. This approach "hides" the arbitrary decision of how to assign the correlated component with the somewhat arbitrary selection of an orthogonalisation methodology. the original factors, that are uncorrelated, and that are represented by a suitable m × m de-correlating torsion matrix t . One natural way to turn correlated asset returns into uncorrelated factor returns is to use principal component analysis (PCA). While useful in other contexts, the PCA approach suffers from a number of shortcomings when computing the factor relative risk contribution (Carli, Deguest and Martellini, 2014). The first shortcoming is the difficulty in interpreting the factors, which are pure There exist several alternative linear transformations , or torsions, of

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