The Mathematics of Financial Modelingand Investment Management

17-Capital Asset Pricing Model Page 522 Wednesday, February 4, 2004 1:10 PM

522 The Mathematics of Financial Modeling and Investment Management

free rate is necessary for the theory to hold. Black’s argument was as fol-

lows. The beta of a risk-free asset is zero. Suppose that a portfolio can

be created such that it is uncorrelated with the market. That portfolio

would then have a beta of zero, and Black labeled that portfolio a

“zero-beta portfolio.” He set forth the conditions for constructing a

zero-beta portfolio and then showed how the CAPM can be modified

accordingly. Specifically, the return on the zero-beta portfolio is substi-

tuted for the risk-free rate.

Now let’s look at the assumption that the only relevant risk is the

variance of asset returns (Assumption 1). That is, it is assumed that the

only risk factor that an investor is concerned with is the uncertainty

about the future price of a security. Investors, however, usually are con-

cerned with other risks that will affect their ability to consume goods

and services in the future. Three examples would be the risks associated

with future labor income, the future relative prices of consumer goods,

and future investment opportunities. Consequently, using the variance

of expected returns as the sole measure of risk would be inappropriate

in the presence of these other risk factors. Recognizing these other risks

that investors face, Robert Merton modified the CAPM based on con-

sumers deriving their optimal lifetime consumption when they face such

non-market risk factors.^11

CAPM and Random Matrices

Let’s take a look at CAPM from a different angle. Under the assumption

of IID returns, the CAPM is the statement that the entire market is

driven by only one factor represented by the market portfolio. Plerou et

al^12 analyzed the distribution and stability of the eigenvalues of the vari-

ance-covariance matrix of large portfolios. Their conclusion can be

summarized as follows:

■ The majority of the eigenvalues fall within the bounds of Random

Matrix Theory (RMT). This means that the majority of eigenvalues do

not carry genuine correlation information. This confirms results

already described in the literature.^13

■ A number of eigenvalues are definitely outside the RMT bounds. The

eigenvector corresponding to the largest eigenvalue includes all assets,

(^11) Robert C. Merton, “An Intertemporal Capital Asset Pricing Model,” Econometri-
ca (September 1973), pp. 867–888.
(^12) Vasiliki Plerou, Parameswaran Gopikrishnan, Bernd Rosenow, Luis A. Nunes
Amaral, Thomas Guhr, and H. Eugene Stanley, “Random matrix approach to cross
correlations in financial data,” Physical Review 65 (June 2002).
(^13) Random matrices are covered in Chapter 12.