EDHEC-Risk Institute October 2016

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

1. Literature and Practice Reviews

exposures to the extended model by linear regression. Introducing new explicit factors, however, creates concerns in terms of multi-collinearity: if some combination of the added factors is too correlated with a combination of the original ones, then exposures cannot be reliably estimated. As a solution to this problem, Menchero and Poduri (2008) propose to make the new factors orthogonal to the existing ones. The Blended Model We use the same notation as Menchero and Poduri (2008), albeit with a slightly different meaning. The core factor model consists of a set of K factors F (X) , which we interpret here as common sources of risk and performance. Thus, we think of F (X) as the Fama-French or Carhart factors for example, while the new factors F (Y) are L sector or country factors. In Menchero and Poduri’s work, F (X) represents the data-generating process for stock returns (a Barra model) and the “custom factors” F (Y) are the Fama-French factors. These differences in interpretation of the notation have no effect on the methodology. The blended model contains all factors, which can be numerous. For instance, standard sector classifications such as MSCI’s Global Industry Classification Standard, FTSE’s Industry Classification Benchmark and Thomson Reuter Business Classification contain ten sectors at the broadest level, and this number grows as one moves to finer decomposition levels. With so many regressors, a direct linear regression of portfolio returns may be unable to disentangle the respective effects of the various factors. Menchero and Poduri’s approach is to regress the factors F (Y) on the factors F (X) , and take the residuals F (Y) ⊥ so as to obtain the effects of factors Y net of factors X . Next, portfolio returns R p,t are regressed against the whole set of factors:

Note that this approach is return-based, meaning that it requires neither knowledge of the portfolio holdings nor of any fundamental or market information at the stock level. As such, it is less demanding in terms of inputs than a Barra-type model. Sector and Carhart Orthogonalised Factors We illustrate this method by taking the market as the single core factor because it is the first-order source of risk in returns, and consider the “custom” factors to be the sector and the Carhart factors. We consider the US equally-weighted broad index over the period 2002-2015. We first regress equally-weighted sector returns R s on the market factor, and we collect the residual plus the intercept term to isolate the “pure sector effect” ν s .

We do the same to extract the net effect of Carhart factors. Let R k be the return on one of the four Fama-French-Carhart portfolios and ν k be the corresponding factor purged of the market influence:

Finally, we regress US equally-weighted broad index returns R p on market factor augmented by the pure sector effect and the pure pricing factor effect:

37

An EDHEC-Risk Institute Publication