The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 336 Wednesday, February 4, 2004 12:59 PM

336 The Mathematics of Financial Modeling and Investment Management

Maximize ωωωωP TΩΩΩΩωωωωP

subject to the normalization condition

Tωωωω

ωωωωP P = 1

where the product is a scalar product. It can be demonstrated that the

solution of this problem is the eigenvector ωωωω 1 corresponding to the larg-

est eigenvalue λ 1 of the variance-covariance matrix ΩΩΩΩ. As ΩΩΩΩis a vari-

ance-covariance matrix, the eigenvalues are all real.

Consider next the set of all normalized portfolios orthogonal to ωωωω 1 ,

that is, portfolios completely uncorrelated with ωωωω 1. These portfolios are

identified by the following relationship:

T

ωωωω 1 ωωωωP = ωωωωP

T

ωωωω 1 = 0

We can repeat the previous reasoning. Among this set, the portfolio of

maximum variance is given by the eigenvector ωωωω 2 corresponding to the

second largest eigenvalue λ 2 of the variance-covariance matrix ΩΩΩΩ. If

there are n distinct eigenvalues, we can repeat this process n times. In

this way, we determine the n portfolios Pi of maximum variance. The

weights of these portfolios are the ortho-normal eigenvectors of the

variance-covariance matrix ΩΩΩΩ. Note that each portfolio is a time series

which is a linear combination of the original time series Xi. The coeffi-

cients are the portfolios’ weights.

These portfolios of maximum variance are all mutually uncorre-

lated. It can be demonstrated that we can recover all the original return

time series as linear combinations of these portfolios:

n

Xi = ∑αiPi

i = 1

Thus far we have succeeded in replacing the original n correlated time

series Xi with n uncorrelated time series Pi with the additional insight

that each Xi is a linear combination of the Pi. Suppose now that only p

of the portfolios Pi have a significant variance, while the remaining n-p

have very small variances. We can then implement a dimensionality

reduction by choosing only those portfolios whose variance is signifi-

cantly different from zero. Let’s call these portfolios factors F.