The Mathematics of Financial Modelingand Investment Management

18-MultiFactorModels Page 532 Wednesday, February 4, 2004 1:10 PM

532 The Mathematics of Financial Modeling and Investment Management

Determination of Factors

Let’s now see how factors can be determined. Exogenous factors are

determined through considerations of macroeconomic theory and fun-

damentals of each firm. Abstract factors are determined through a pro-

cess of statistical analysis. We begin by describing the determination of

exogenous factors and then of abstract factors.

Exogenous Factors^2

There are several commercially available fundamental multifactor risk

models. Investment management companies often develop their own

proprietary models. Brokerage firms have developed models that they

make available to institutional clients. In this section, we will focus on a

commercially available model from Barra. The basic relationship to be

estimated in a multifactor risk model is

Ri − Rf = βi,F 1 RF 1 + βi,F 2 RF 2 + ... + βi,FH RFH + ei

where:

Ri = rate of return on stock i

Rf = risk-free rate of return

βi,Fj = sensitivity of stock i to risk factor j

RFj = rate of return on risk factor j

ei = nonfactor (specific) return on security i

The above function is referred to as a return generating function.

Fundamental factor models use company and industry attributes

and market data as “descriptors.” Examples are price/earnings ratios,

book/price ratios, estimated earnings growth, and trading activity. The

estimation of a fundamental factor model begins with an analysis of his-

torical stock returns and descriptors about a company. In the Barra

model, for example, the process of identifying the risk factors begins

with monthly returns for 1,900 companies that the descriptors must

explain. Descriptors are not the “risk factors” but instead they are the

candidates for risk factors. The descriptors are selected in terms of their

ability to explain stock returns. That is, all of the descriptors are poten-

tial risk factors but only those that appear to be important in explaining

stock returns are used in constructing risk factors.

(^2) The discussion in this section draws from Frank J. Fabozzi, Frank J. Jones, and Ra-
man Vardharaj, “Multi-Factor Equity Risk Models,” Chapter 13 in Frank J. Fabozzi
and Harry M. Markowitz (eds.), The Theory and Practice of Investment Manage-
ment (Hoboken, NJ: John Wiley & Sons, 2002).