The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 334 Wednesday, February 4, 2004 12:59 PM

334 The Mathematics of Financial Modeling and Investment Management

space models of prices—have been introduced to capture additional eco-

nomic information contained in asset prices, especially equity prices.^15

CAPM

Let’s begin by discussing multifactor models of returns. There are many

different ways of writing such models depending on the nature of the

factors. The first, and most famous, factor model is the Capital Asset

Pricing Model (CAPM) developed by Sharpe-Lintner-Mossin. In the

CAPM there is only one factor given by the portfolio of all investable

assets. Each log price process can be written as follows:

xi = βi + αif + σ

In its original formulation, the CAPM was derived as a general equi-

librium theory; the actual asset price process is the fixed point where the

collective action of all agents trying to maximize their utility does not

produce any change in the price process, thus the situation of equilibrium.

CAPM assumes the joint normality of returns and the independence

of returns from one period to another; the single factor evolves as an

arithmetic random walk. This version of the CAPM is conceptually

restrictive and difficult to test given that the market portfolio, which is

the portfolio of all investable assets, is difficult to define and measure.

A later version of CAPM called Conditional CAPM or C(CAPM) was

proposed. Essentially, the Conditional CAPM assumes that there is only

one factor driving all prices, but does not impose the restriction that such

a factor is the market portfolio or that it evolves as a random walk.

(^15) The literature on dynamic factor models is ample. Here is a selection of widely
quoted papers: M. Forni, M. Hallin, M. Lippi, and L. Reichlin, “The Generalized
Dynamic Factor Model: Identification and Estimation,” Review of Economics and
Statistics 82, no. 4 (2000), pp. 540–554; J.F. Geweke, “The Dynamic Factor Analy-
sis of Economic Time-Series Models” in D.J. Aigner and A.S. Goldberger (eds.) La-
tent Variables in Socioeconomic Models (Amsterdam: North Holland, 1981); J.F.
Geweke and K.J. Singleton, “Maximum Likelihood ‘Confirmatory’ Factor Analysis
of Economic Time Series,” International Economic Review 22, no. 1, pp. 37–54; D.
Quah and T.J. Sargent, “A Dynamic Index Model for Large Cross Sections,” in J.H.
Stock and M.W. Watson (eds.), Business Cycles, Indicators and Forecasting (Chica-
go, IL: The University of Chicago Press, 1993), pp. 285–309; J.H. Stock and M.W.
Watson, “Diffusion Indexes,” NBER Working Paper W6702, 1998; J.H. Stock and
M.W. Watson, “New Indexes of Coincident and Leading Economic Indications,” in
O.J. Blanchard and S. Fischer (eds.), NBER Macroeconomics Annual 1989 (Cam-
bridge, MA: M.I.T. Press, 1989); M.W. Watson and R.F. Engle, “Alternative Algo-
rithms for Estimation of Dynamic MIMIC, Factor, and Time Varying Coefficient
Regression Models,” Journal of Econometrics 23 (1983), pp. 385–400.