EDHEC-Risk Institute October 2016

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

1. Literature and Practice Reviews

As in the previous methods, we first estimate the factors’ covariance matrix and the beta vector , and we then solve the minimum-torsion optimisation so as to obtain the minimal torsion transformation t . Next, we perform the variance decomposition according to Equation (1.9). Figure 4 shows the result of this procedure for the four mutual funds. It is very close to Figure 2, because the long-short factors have low correlations between themselves and with the market, so that making them perfectly orthogonal has only a small impact on the results. Most of the ex-post risk of the long/only funds in Figure 4 comes from their market exposure and from specific risk. The long/short fund has the highest specific variance and the lowest market risk. The only change comes from the momentum factor, which has a higher risk contribution with the MLT method because of its negative correlation with the market factor over the period. 1.2.2.4 Other Methods for Variance Decomposition Several other methods can be considered to decompose the risk of a portfolio. This subsection presents two of them, which can be applied to volatility or to tracking error. Disregarding Correlated Components Another approach consists in keeping the previous marginal contribution of factor definition but we overlook the correlated component.

In this case, long-short and long-only factors should be differentiated. Long-only factors have a higher correlation than long-short factors, which implies a higher correlated component. Hence this approach is similar to the previous one for the risk attribution if long-short indices are used as investment vehicles for the factors. As for the first approach we need to compute factors’ covariance matrix estimate and estimate the factor loading vector . Decomposition of R-squared The last approach deals with assessing the relative explanation of regressors in linear regressions based on the portfolio variance decomposition. The key difficulty is to decompose the total variance or R-squared, when regressors (here, risk factors) are correlated. This problem has been discussed in the statistical literature on the relative importance of correlated regressors in multivariate regression models (Chevan and Sutherland, 1991).The idea is to take the average over all possible permutations of the marginal increase in R-squared related to the introduction of a new regressor starting from a given set of existing regressors. R2(1:K) denotes the coefficient of determination of the linear regression with the K factors as regressors and R2(k) denotes the coefficient of determination of the linear regression with only the k th factor. When the factors are pairwise uncorrelated, i.e. ρ ( F k , F l )=0 for k ≠ l , we have:

6 -This approach is similar to the approach developed in Deguest, Martellini and Milhau (2012), where the focus was on the contribution of various assets to investors' welfare, and where the correlated components were grouped together in a term that was called "diversification component".

Hence, we do not attribute the correlated components and leave them together as a separate contributor to portfolio risk. 6

This is the decomposition of R-squared in a multivariate model with orthogonal regressors.

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