The Mathematics of Financial Modelingand Investment Management

17-Capital Asset Pricing Model Page 523 Wednesday, February 4, 2004 1:10 PM

Capital Asset Pricing Model 523

though not necessarily in equal proportion. This eigenvector approxi-

mately corresponds to the entire market. The other largest eigenval-

ues correspond to eigenvectors that identify market sectors. The

eigenvectors corresponding to the largest eigenvalues exhibit some

degree of stability in time, the most stable being those corresponding

to the largest eigenvalues. Stability is measured by computing eigen-

values and eigenvectors on a moving window and counting the per-

centage of assets forming each eigenvector that remain unchanged.

Based on a remarkably large data set, work by Plerou et al. identi-

fies a number of different meaningful eigenvectors. The multiplicity of

eigenvectors corresponding to large eigenvalues suggests a structure of

multiple factors as portfolios. Note that the largest eigenvector is not

the market portfolio. In fact, the market portfolio includes all investable

assets. Therefore, it includes assets that are not in the largest eigenvec-

tor. This fact leaves open the door to a possible coexistence of CAPM

and multifactor models. In order to explore this point, we need first to

discuss the Conditional CAPM, Asset Pricing Theory, and multifactor

models. We discuss the Conditional CAPM in this chapter and the last

two models in the next chapter.

THE CONDITIONAL CAPM

As we have seen, the CAPM is embodied in a static linear regression of

asset returns over the market portfolio whose explanatory power has

been questioned by, among others, Fama and French.^14 Ravi Jagan-

nathan and Zheniu Wang^15 suggested a solution: They made the CAPM

regression coefficients conditional on some global information set,

thereby generalizing the model. Called the Conditional CAPM or

C(CAPM), this model represents each expected return rit given the

information set at time t by the conditional linear regression:

E[rt ⁄It – 1 ] = αααα+ ββββE[ft ⁄It – 1 ]

cov(ritfst ⁄It – 1 )

βis = -----------------------------------------

var(fst ⁄It – 1 )

(^14) Fama and French, “The Cross-Section of Expected Stock Returns.”
(^15) Ravi Jagannathan and Zhenyu Wang, “The Conditional CAPM and the Cross-
Section of Expected Returns,” Journal of Finance 51, no. 1, pp. 3–53.