The Mathematics of Financial Modelingand Investment Management

17-Capital Asset Pricing Model Page 518 Wednesday, February 4, 2004 1:10 PM

518 The Mathematics of Financial Modeling and Investment Management

Estimating the Characteristic Line

The characteristic line is estimated using regression analysis. In fact, all

the data required are the same except for the risk-free rate each period.

The coefficient of determination, denoted by R-squared, indicates the

strength of the relationship. Specifically, it measures the percentage of

the variation in the return on a stock explained by the return by the

market portfolio (proxied by the S&P 500 in our illustration). The value

ranges from 0 to 1. The higher the R-squared, the greater the propor-

tion of systematic risk relative to total risk. For individual stocks, the R-

squared is typically in the 0.3 area. That is, for individual stocks system-

atic risk is small relative to nonsystematic risk. For well-diversified port-

folio, the R-squared is typically greater than 0.9.

TESTING THE CAPM

Testing the CAPM has been a major endeavor of financial econometrics.

The number of articles found under the general heading “tests of the

CAPM” is impressive. One bibliographic compilation lists almost 1,000

papers on the topic. Consequently, only the basic results are given here.

In general, a methodology referred to as “two-pass regression” is

used to test the CAPM. The first pass involves the estimation of beta for

each security by means of a time series regression described by charac-

teristic line. The betas from the first-pass regression are then used to

form portfolios of securities ranked by portfolio beta. The portfolio

returns, the return on the risk-free asset, and the portfolio betas are then

used to estimate the second-pass, cross-sectional regression:

Rp – RF = bo + b 1 βp + ep

where the parameters to be estimated are bo and b 1 , and ep is the error

term for the regression. The return data are frequently aggregated into

five-year periods for this regression.

Deriving the Empirical Analogue of the CML

The above equation is the empirical analogue of a beta version of the

CML. To see this, subtract RF from both sides of the CML equation

[ER( M)– Rf]

ER[]i = Rf + ----------------------------------cov(RiR )

var(RM)

m

which can then can be rewritten as