## Factor Models

Factor models of security returns decompose the random return on each of a cross section of assets into factor-related and asset-specific returns. Letting *r* denote the vector of random returns on *n* assets, and assuming *k* factors, a factor decomposition has the form

where *B* is an *n* × *k*-matrix of factor betas, *f* is a random *k*-vector of factor returns, and *ε* is an *n*-vector of asset-specific returns. The *n*-vector of coefficients *a* is set so that *E*[*ε*] = 0. By defining *B* as the least squares projection , it follows that *cov*(*f*, *ε*) = 0^{k×n}.

The factor decomposition (1) puts no empirical restrictions on returns beyond requiring that the means and variances of *r* and *f* exist. So in this sense it is empty of empirical content. To add empirical structure, it is commonly assumed that the asset-specific returns *ε* are cross-sectionally uncorrelated, *E*[*εε*′] = *D* where *D* is a diagonal matrix. This implies that the covariance matrix of returns can be written as the sum of a matrix of rank *k* and a diagonal matrix:

This is called a *strict factor model*. Without loss of generality, one can assume ...

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