The Mathematics of Financial Modelingand Investment Management

17-Capital Asset Pricing Model Page 524 Wednesday, February 4, 2004 1:10 PM

524 The Mathematics of Financial Modeling and Investment Management

A difficulty with C(CAPM) is to identify the conditioning relation-

ships as well as the market portfolio. Jagannathan and Wang show that

the unconditional returns generated by a C(CAPM) can be thought of as

being generated by a two-factor model where one factor is the uncondi-

tional beta and the other represents the fluctuations of beta. This con-

clusion can be generalized. A C(CAPM) model is equivalent to a

nonlinear factor model.^16

Jagannathan and Wang show that the C(CAPM) is able to represent

the cross section of stock returns with a greater accuracy than the con-

ventional unconditional CAPM. They also show that the empirical accu-

racy of the unconditional CAPM is greatly improved by adding human

capital to the market portfolio. Human capital is not a tradable asset, at

least not in the same sense as financial assets.

BETA, BETA EVERYWHERE

In the development of both modern portfolio theory and CAPM, the

Greek letter beta appears. Certainly to the mathematically trained, this

presents no problem. However, it caused confusion in the investment

management community. The use of the term “beta” in the two theories

was as follows. First, because of the difficulty of working with the cova-

riance matrix at the time, Markowitz suggested using as a proxy mea-

sure of the full covariance matrix a covariance of a security’s return

with some index.^17 Sharpe picked up on this suggestion and proposed

the following model for doing so which he referred to as the market

model:^18

rit = αi + βi rmt + uit

Note that the index need not be a market portfolio—hence the use of m

rather than M in the above equation. When Sharpe estimated the market

model, he used a stock market index.

Then beta appeared in the CAPM where it is estimated from the

characteristic line which we discussed earlier. The market model and the

characteristic line look almost identical. The difference is simply that

(^16) For more on this subject see, for instance, Adrian Pagan, “The Econometrics of
Financial Markets,” Journal of Empirical Finance 3 (1996), pp. 15–102.
(^17) Harry M. Markowitz, Portfolio Selection: Second Edition (Cambridge, MA: Basil
Blackwell Ltd., 1991), p. 100.
(^18) William F. Sharpe, “A Simplified Model for Portfolio Analysis,” Management Sci-
ence (January 1963), pp. 277–293.