The Mathematics of Financial Modelingand Investment Management

18-MultiFactorModels Page 530 Wednesday, February 4, 2004 1:10 PM

530 The Mathematics of Financial Modeling and Investment Management

It is likely that in the long run all price processes follow one single

common trend with the exception of disruptive events such as bankrupt-

cies or mergers and acquisitions. This trend-following behavior, how-

ever, might exhibit a complex dynamical structure. Within the time

horizons that are empirically available, multiple trends, mean reversion,

and structural breaks are at work. We will first analyze classical multi-

factor models of returns and how they are constructed and used in

investment management. Subsequently, we will discuss dynamic factor

models.

MULTIFACTOR MODELS

Let’s introduce multifactor models of returns. The general form of a lin-

ear multifactor market model of returns can be written in one of the fol-

lowing ways:

E []r = αααα+ ββββ' E [] f

Er[ it ft]=αi + ββββift

p

rit = αi + ∑βisfst + εt

s = 1

where:

rit = the return of the i-th security at time t

ai = constants specific for the i-th security

βis = the sensitivity of the i-th security to the s-th factor

ft = the s-th factor at time t and εt is a noise process

In this linear regression model, assuming that factors are orthogonal

(that is, uncorrelated), the sensitivities βis, referred to as betas, can be

written as:

cov(ritfst)

βis = --------------------------

var () fst

As both returns and factors are assumed to be stationary stochastic

processes, unconditional means and covariances are time-independent