R = the month’s return of the portfolio (KSE, NYSE, etc) of stocks
E = the month’s regression residual for a particular stock
(Grinblatt and Titman, 160)
The slope of the line of best fit is the beta. The second step obtains estimates of the intercept and slope coefficient of a single cross-sectional regression, in which each data observation corresponds to a stock. This is represented by the following model
r = v0 + v1B + v2unrel + u
Where: r = average monthly historical returns of a stock
B = estimated slope coefficient
unrel = a characteristic of the firm unrelated to the CAPM e.g the firm size
u = stock’s regression residual
(Grinblatt and Titman, 161)
If the CAPM is true the latter (second step) regression should produce the intercept (v0) equal to risk-free rf, the slope (v1) equal to the market portfolio’s risk premium, v2 equal to zero (since variables other than beta represented by unrel should not explain the mean returns once beta is accounted for). If any deviation of returns from the securities market line(SML) is found and upon graphing the deviations of the mean returns from the SML against the firm size if a relationship is found, this serves as an evidence to reject CAPM.
The second set of CAPM test, the time-series test, examines the restrictions on the intercepts of time-series market model regressions. This is evaluated by the following regression model
r – rf = a + B(R-rf) + z
(Grinblatt and Titman, 163)
This model is nearly identical to the former (first-step of cross sectional) with the exception that excess returns are used in place of returns. CAPM can be tested via this approach, using the returns of portfolios formed from characteristics such as the stock’s prior beta, firm’s size, and the ratio of the market value of a stock to its book value to estimate the coefficients of the above model. Betas estimated from both regressions are almost always nearly identical. The CAPM predicts that the intercept from such regression should be zero, however if the regression produces a positive intercept, this implies that CAPM has underestimated the returns or alternatively this alludes to the failure of the CAPM.
In the empirical tests of the CAPM, the returns of the low-beta stocks are much too high relative to its predictions, and the returns of high-beta stocks are much too low. More importantly, a number of stock characteristics (market cap, market-to-book ratio, momentum etc) explain historical average returns much better than the CAPM beta does). These empirical findings imply that CAPM does not properly explain the risk and return relationship.
On major problem of empirically testing the CAPM is that market portfolio (which is central to the specification of the various tests) is unobservable. Thus proxies such as S&P 500 are widely used. If, the model cannot be tested effectively because of the unobservability of the market portfolio or biases in the data available to researchers, it cannot be of any use for practical applications and is therefore quite vain. Furthermore, the CAPM is based on anticipations and since the agents do not publish their beliefs about the future, tests can only be based on the assumption that the future will more or less reflect the past. This implies that any test performed on the CAPM can, at best, only be partially conclusive (http://www.affi.asso.fr/images/cnf_11_doc_219.pdf).
In addition, a relation does exist between returns and betas, but the results of the regression of the average returns against betas are far from convincing. In part, this is explained by the fact that the betas are estimated using a proxy for the market portfolio which is possibly not mean–variance efficient and almost certainly not the real market portfolio (http://www.affi.asso.fr/images/cnf_11_doc_219.pdf).
Furthermore, some researchers, Fama and French (1992, 1995, 1996), have shown in empirical tests that adding variables (the book to market equity ratio, the size of the stocks etc) to the beta leads to a better explanation of the variations of expected returns across the range of assets. Thus this underscores the weak explanatory power of the traditional CAPM model.
These shortcomings have dented the potency of the CAPM, yet it is a widely used model of asset pricing in the field of financial economics.
Bibliography
CAPM PROBLEMS AND RETURNS DISTRIBUTIONS. Universit´e de Paris Dauphine. 4 Dec. 2005 <http://http://www.affi.asso.fr/images/cnf_11_doc_219.pdf>.
"Fama and French Three Factor Model." 5 Dec. 2005 <http://www.moneychimp.com/articles/risk/multifactor.htm>.
Grinblatt, Mark, and Sheridan Titman. Financial Markets and Corporate Strategy. International Edition ed. Mc Graw Hill, 2002. 160-163.
Understanding Risk and Return, the CAPM,. Tuck School of Business at Dartmouth. 5 Dec. 2005 <http://www.ssrn.com>.
The sock’s return in the recent past, usually last 6 months.
