Statistical Factor Analysis
The three types of factor models are statistical, macroeconomic and characteristic based. This chapter considers statistical factor models, which are the most technically difficult of the three classic types but also the most fundamental.
Section 4.1 describes the basic types of statistical factor models. Section 4.2 looks at approximate factor models, which impose an approximate structure on the covariance matrix as the number of assets becomes large. Section 4.3 discusses the arbitrage pricing theory and its applications to portfolio risk management. Section 4.4 considers “smalln” factor model estimation techniques, in which the number of assets is small relative to the number of time periods. Section 4.5 considers “large-n” techniques, in which the number of assets is large. Section 4.6 discusses techniques to determine the number of pervasive factors in returns.
Before defining a factor model it is useful to simply divide the returns on all assets into two parts: the part of the return correlated to a set of factors and the remaining (uncorrelated) part. The linear factor decomposi- tion expresses the vector of asset excess returns as a linear combination of factor returns and a vector of asset-specific returns:
where B is an n × k matrix of factor betas, f is a random vector of factor returns, and ε is an n-vector of asset-specific returns. The vector of coefficients a is set so that E[ε] = 0n. By defining B appropriately, in particular, it follows that cov(f, ε) = 0k×n.