Investment Theory and Risk Management, + Website

Summary

I use Ross's theory of arbitrage pricing (APT) to motivate factor models and show that the CAPM—the single index model—is a special case of APT. Factor models can be specified as either cross-sectional (BARRA) or time series (BIRR); in either case, they attribute the variation in asset returns to risk-and-return factors whose contribution to return is measured by their factor sensitivities (the β_i). In cross-sectional models for which the number of assets exceeds the number of degrees of freedom available in the time series of returns, then factor models are essential to estimating the covariance matrix. The BARRA model is one such remedy to this problem. As we saw, factor selection appears to be somewhat arbitrary. While there is some comfort in appealing to the theory of fundamentals offered up by, say, dividend discount models, factor selection remains a challenging task. Principal components address this issue from two perspectives: first, principal components analysis helps locate the number of independent sources of variation, or number of factors, that help explain the cross-sectional variability in a collection of asset returns and, second, principal components are useful in finding a subset of independent factors to use in the factor model and in decomposing the variation in returns across this set of independent factors. However, since principal components are linear combinations of the original factors, it is often difficult to clearly discriminate among principal ...