EDHEC-Risk Institute October 2016

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

4. Conclusion

analysis. We let the alpha and the market exposure depend on both the sector attribute and the three observable attributes that define the Fama-French-Carhart factors: market capitalisation, the book-to-market ratio and past one-year return. We show that the fundamental beta approach is more convenient when the multi-factor analysis is extended to additional dimensions (e.g. sector and regions). Finally, the fundamental beta provides an alternative measure of the conditional beta, which is a function of observable variables and is not subject to the lag issue that potentially affects betas estimated by a rolling-window regression. It immediately responds to changes in a stock's attributes, which allows us to assess the impact of a change in the portfolio composition on the factor exposure. We illustrate these benefits by constructing market-neutral portfolios based on the fundamental and the rolling-window methods, and we show that the former achieves better out-of-sample neutrality. We do not claim that a one-factor model with time-varying beta is the key to explaining any difference in expected returns. The two approaches – multi-factor model and conditional single-factor model - are not exclusive, and the true (still unknown) asset pricing model is likely to be a multi- factor one with betas depending on state variables. Our work can be extended in several dimensions. First, one may try to use attributes to decompose exposures to other risk factors with a conditional multi-factor model such as the Fama-French factor model. This requires the identification of the meaningful attributes for each factor. Another possible avenue for further research would consist in extending the empirical analysis by using macroeconomic variables (as in Ferson and Schadt, 1996) in addition to characteristics.

Multi-factor models are standard tools for analysing the performance and the risk of equity portfolios. In the Fama-French and Carhart models, the size, value and momentum factors are constructed by first sorting stocks on an attribute (market capitalisation, the book-to-market ratio or past short-term return), then by taking the excess return of a long leg over a short leg. These models do a much better job than the standard capital asset pricing model (CAPM) at explaining the differences across expected returns. However, numerous patterns have been identified in stock returns, raising concerns about a potential inflation in the number of long-short factors and their overlap. As noted by Cochrane (2001, p. 1060), one must ask “which characteristics really provide independent information about average returns” and “which are subsumed by others”. Our work suggests another meaningful approach for explaining the cross-section of expected returns, which consists in treating attributes of stocks as instrumental variables to estimate the beta with respect to the market factor. We stay with a limited number of risk factors by considering a one-factor model, and we estimate a conditional beta that depends on the same three characteristics that define the Fama-French and Carhart factors. We show that a conditional CAPM based on this “fundamental“ beta can capture the size, value and momentum effects as well as the Carhart model, but without the help of additional factors. The pricing errors are further reduced by introducing a time-varying market premium, which introduces the cyclical covariation between fundamental beta and the market risk premium as a driver of expected returns. Moreover, we use the fundamental beta approach to embed the sector dimension in our multi-factor risk and performance

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