EDHEC-Risk Institute October 2016

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

1. Literature and Practice Reviews

used by academics and practitioners to attribute the performance of a fund to systematic risk factors or to the manager’s skills. But other models have been developed by market practitioners to include other factors which are deemed important to explain the risk and the performance of a fund. The most famous example is the family of Barra models, which we present in Section 1.3.1. In Section 1.3.2 and Section 1.3.3, we take a closer look at the problem of multi-collinearity in models with a large number of factors and we present two possible methods to address this concern. 1.3.1 Barra Model There are in fact several classes of Barra models, which differ through the set of factors employed. One important difference between these models and the empirical asset pricing models of Fama and French or Carhart is that they treat factor exposures as observed quantities, as opposed to parameters to be estimated from a regression. It is the factors that are regarded as unobservable variables. The factor values at each date t are estimated through a cross-section regression of stock returns on the predetermined betas. Hence, the Barra method is an implicit factor method which involves extracting the factors directly from the historical returns. Mathematically, the excess return of a stock i between dates t – 1 and t is expressed as:

with

,

, ,

,

Here, the set of factors has been split into three subsets: C country factors, S sector factors and R risk factors. B i,t,c ( C ) denotes a dummy variable and takes the value one if asset i belongs to country c or 0 otherwise, and B i,t,s ( s ) is also a dummy variable and takes the value one if asset i belongs to sector s or 0 if not. The risk index exposures B i,t,r ( R ) are defined as continuous variables and are normalised within the country by ranking the company in each factor relatively to other local companies. The factor values at date t , , and , are estimated by performing the cross-section regression (1.10) at date t . This methodology is employed in the BARRA Global Equity Model (see the Risk Model Handbook (1998)). The Global Equity Model is a multi-factor model, partitioned into specific return and common factor return, whose main purpose is to assess the relative contributions of industry versus country factors. The common factors are industries, countries and risk indices. The equation for the Global Equity Model is Equation (1.10), with 90 factors in the MSCI version, and 93 factors in the FT version (1998).

(1.10)

35

An EDHEC-Risk Institute Publication