The Mathematics of Financial Modelingand Investment Management

18-MultiFactorModels Page 534 Wednesday, February 4, 2004 1:10 PM

534 The Mathematics of Financial Modeling and Investment Management

ual stocks in the portfolio. The same analysis can be applied to a stock

market index because an index is nothing more than a portfolio of stocks.

Abstract Factors

Suppose now that factors are abstract static factors under the assump-

tion that returns are normally distributed IID variables. Under this

assumption, two basic techniques can be used: factor analysis and prin-

cipal components analysis. We’ll begin with factor analysis.

Suppose that there is a strict factor structure with a known number

of undetermined factors of the form:

r = αααα+ Bf + εεεε

N

fi = ∑αs s r

s = 1

where factors are linear combinations of returns. A strict factor struc-

ture means that factors explain all the covariance between the process

components. Under this assumption, factors are processes with a vari-

ance-covariance matrix ΩΩΩΩF while the innovations εεεεare assumed to be

uncorrelated and have a diagonal variance-covariance matrix D. Under

these assumptions, the variance-covariance matrix ΩΩΩΩ of the multivariate

process r of returns can be written as the sum of two contributions:

ΩΩΩΩ= BΩΩΩΩFB′ + D

This representation is not unique as factors are not uniquely deter-

mined. In fact, given any set of factors, one obtains another set of fac-

tors by multiplying the former by an orthonormal matrix GGG, ′ = I.

This indeterminacy allows one to choose orthogonal factors with unit

variance so that their variance-covariance matrix is a unitary matrix

and the return process variance-covariance matrix can be written as:

ΩΩΩΩ = BB′ + D

This relationship is a constraint on the return variance-covariance

matrix. The latter can be estimated with MLE techniques. The resulting

computations are numerically complex. However, many software pack-

ages efficiently perform factor analysis. After estimating the matrix of

factor sensitivities, the factors themselves can be estimated with MLE