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A full development of the CAPM can be found in any introductory finance textbook (for example, Brealey and Myers 1988), and so is omitted here. There are, however, several econometric issues involved in the more general model

Two straightforward tests of this model are

- a test of the null against the alternative that it is not equal to 0. A rejection would cast some doubt on the theoretical model as is assumed to be 0 in the first equation.
- a test of the null against the alternative that it is not equal to 1. Failure to reject would imply that the movement of asset prices of a particular company is the same as that of the market as a whole.

This example uses stock returns from the Tandy Corporation to estimate a CAPM and perform some general tests of the model.

The data for this example consist of monthly observations from January 1978 through December 1987 on the market return, the risk-free rate, and the return on the Tandy Corporation's common stock. Using a SAS DATA step, enter the data and create two new variables, R_TANDY = TANDY - R_F and R_MKT = R_M - R_F, that correspond to the risk premiums for the Tandy Corporation and the Market. A risk premium is the excess return of a security over the risk-free rate or, rather, the extra return that investors require for bearing risk.

data tandy; input r_m r_f tandy @@; r_tandy = tandy - r_f; r_mkt = r_m -r_f; label r_m='Market Rate of Return' r_f='Risk-Free Rate of Return' tandy='Rate of Return for Tandy Corporation' r_tandy='Risk Premium for Tandy Corporation' r_mkt='Risk Premium for Market'; datalines; -.045 .00487 -.075 .063 .00491 .055 .071 .00528 .194 ... more data ... ; run;

It is always a good idea to look at the data to be sure that it was input as expected. A simple plot allows for a visual assessment of the appropriateness of the proposed model. The following code uses the GPLOT procedure to plot the risk premium of the Tandy Corporation against the risk premium of the Market.

proc gplot data=tandy; plot r_tandy * r_mkt / haxis=axis1 hminor=4 cframe=ligr vaxis=axis2 vminor=4; symbol1 c=blue v=star; axis1 order=(-0.3 to 0.2 by 0.1); axis2 label=(angle=90 'Tandy Corp. Risk Premium') order=(-0.4 to 0.6 by 0.2); title 'TANDY Corporation CAPM Example'; title2'Plot of Risk Premiums'; title3'Tandy Corporation versus the Market'; run;

The AUTOREG procedure specifies a linear regression of R_TANDY on R_MKT. Note that a constant term is automatically included in the model unless the NOINT option is specified. The DWPROB option requests the Durbin-Watson test for autocorrelation of the residuals. The TEST statement enables you to perform hypothesis tests on parameter estimates; the following TEST statement is requesting a test that the coefficient of R_MKT is equal to 1. In addition, the OUTPUT data set TANDYOUT contains the predicted values of R_TANDY, the residuals of the regression, and the upper and lower bounds of the 90% confidence interval.

proc autoreg data=tandy; model r_tandy = r_mkt / dwprob; test r_mkt = 1; output out=tandyout p=p r=r ucl=u lcl=l alphacli=.10; title2; title3; run;

The *F*-statistic for the test that the coefficient of R_MKT is equal to one is 0.1254
with an associated *p*-value of 0.724, implying that there is little evidence that this
coefficient is different from 1.0. Thus there is little evidence that the returns of the Tandy stock were any more or less
volatile than the returns of the market.

R2 measures the proportion of the total variation of Y explained by the regression of Y on the independent variable, X. The
R^{2} value of 0.319 means that about 32% of the variation in the risk premium of
Tandy Corporation stock can be explained by the risk premium of the market. Or, put another way, 68% of the risk premium of
Tandy stock is firm specific.

The Durbin-Watson *d* statistic tests for autocorrelation and lack of independence of
residuals, which is a common problem in time series data. The *d* statistic ranges from 0
to 4. A value close to 2.0 indicates that you cannot reject the null hypothesis of no autocorrelation. The calculated value
of the *d* statistic in the CAPM for the Tandy Corporation is 1.933, and it has a *p*-value of 0.359. This value indicates that, for this model, you cannot reject the null
hypothesis of independent (nonautocorrelated) residuals at typically chosen levels of significance.

Using PROC GPLOT, you can investigate the residuals for patterns that would indicate a violation of the underlying assumptions of the model.

proc gplot data=tandyout; plot r * r_mkt / haxis=axis1 hminor=4 cframe=ligr vaxis=axis2 vminor=4 vref=0.0; symbol1 c=green v=star; axis1 order=(-0.3 to 0.2 by 0.1); axis2 label=(angle=90 'Tandy Corp. Risk Premium') order=(-0.3 to 0.4 by 0.1); title 'TANDY Corporation CAPM Example'; title2'OLS Residuals versus Market Risk Premium'; run;

Berndt, Ernst R. (1991), *The Practice of Econometrics: Classic and Contemporary*, New York: Addison-Wesley.

Brealey, Richard A. and Myers, Stewart C. (1988), *Principles of Corporate Finance*, Third Edition, New York:
McGraw-Hill.

SAS Institute Inc. (1993), *SAS/ETS Software: Applications Guide 2, Version 6, First Edition: Econometric Modeling,
Simulation, and Forecasting*, Cary, NC: SAS Institute Inc., 15-30.