Factor Models of Asset Returns

Factor models of security returns decompose the random return on each of a cross-section of assets into factor-related and asset-speci\u85c returns. Letting r denote the vector of random returns on n assets, and assuming k factors, a factor decomposition has the form: r = a+Bf + " (1) where B is a n k-matrix of factor betas, f is a random kvector of factor returns, and " is an nvector of asset-speci\u85c returns. The nvector of coe ¢-cients a is set so that E["] = 0: By de\u85ning B as the least squares projection B = cov(r; f)C1f; it follows that cov(f; ") = 0 kn: The factor decomposition (1) puts no empirical restrictions on returns be-yond 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-speci\u85c returns " are cross-sectionally uncorrelated, E[""0] = 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: cov(r; r0) = Bcov(f; f 0)B0 +D: (2) This is called a strict factor model. Without loss of generality one can assume that cov(f; f 0) has rank k; since otherwise one of the factors can be removed (giving a k 1 factor model) without a¤ecting the \u85t of the model. An important (and often troublesome) feature of factor models is their rota-tional indeterminacy. Let L denote any nonsingular k k-matrix and conside