In the specialised literature there are two main methods of testing the validity of the CAPM - as first introduced by Sharpe (1964), Lintner (1965) and Mosin (1966) - using a data set. The first one, used by Fama and MacBeth in 1973, uses as a starting point a cross sectional regression where the dependent variable is the excess return of each stock and the independent variable is the systematic risk - β. The second one is based on estimating a series of time regressions and was applied by Gibbons, Ross and Shanken in 1989. The test proposed by GRS consists of regressing the excess return of the stock on the excess return of the market and examines whether the intercept of the regressions equals to 0 for each of the stocks. This means that the chosen market portfolio is efficient. Thus, GRS method is at the same time testing the efficiency of the market portfolio.
In this paper, we are testing the validity of the CAPM using the Fama & MacBeth methodology. In order to perform the analysis, we selected the variables required by the model: a proxy for the risk-free asset, a proxy for the market portfolio and a series of stocks with adequate characteristics.
The data that we used for the study consists of monthly returns (calculated using the simple formula) of each stock within and the S&P 100 index for a period of 10 years, starting in Oct 2000 and ending in Oct 2010.
The risk-free asset is defined as having an exact and certain return and no correlation with the market or any other financial or real asset with uncertain returns. We chose to use the monthly T-bill rates which, theoretically speaking, is the best approximation for the risk-free rate, the state being considered as a default-free entity.
The CAPM states that, in equilibrium, there is just one risky portfolio that will be held by investors together with the risk-free asset. Considering the fact that investors have homogenous expectations, they will all want to hold the same portfolio. In empirical studies about the CAPM, the market is mostly defined through a market index, which can reproduce the complex market structure. Thus, we have chosen to use for our analysis the S&P 100 index as a proxy for the market portfolio.
In order to select the stocks which we later used for the cross sectional regression, we chose from a database consisting of all the 100 stocks included in the index. We then excluded the stocks which did not have data for the period we needed, thus being left with 90 stocks. As all the stocks were highly liquid and had stationary returns, we decided to focus on further econometric criteria. Therefore, we regressed the excess return of each stock on the excess return of the market for each of the 90 stocks and analysed the residuals of the 90 regressions using the following formula.
Rit = ai + βiRm,t + ei
The estimation was performed using the OLS method. OLS is the best estimation procedure if the residuals are observed from an independently identically distributed process - no autocorrelation and no heteroscedasticity. We used the Jacque Berra test for normality and the Durbin Watson test for autocorrelation and we found 14 stocks that passed the tests at 5% level and 6 stocks that passed at 2.5%. We also checked the significance of β's coefficient by using the t-test and took that into consideration when choosing the stocks. Obviously, if β's coefficient was insignificant (did not reject the null), then it would have been a problem when using the estimated value in the cross sectional regression. Another aspect that we checked was the goodness of fit of each regression, ending up with an average of approximately 30%. All things considered, after applying all the mentioned criteria we ended up with 20 stocks and 20 estimated βs (see appendix) - the starting point for the cross sectional regression.
Analysis
After separating the twenty most suitable stocks for our analysis, we tested CAPM's validity by using the methodology proposed by Fama - MacBeth. Firstly, we calculated the excess returns of the market portfolio (S&P 100 index) and of each stock for the whole examining period (Oct '00 - Oct '10). After this, we ran time-series regressions with dependent variable each stock's excess returns at time t and independent variable the market's excess return at time t. Our aim was to estimate the intercept and the slope of the following linear equation and to see if each stock's excess returns can be explained only by using excess market return and betas.
Rit - Rft = αi + βi(RMt - Rft) + eit
We then used the estimated betas from the time series regressions to perform the cross sectional one. The model we used was the following:
i = λ0 + λ1bi + εi
i - average return of each stock
In order to reach a conclusion about CAPM's validity we had to test whether the intercept (λ0) of the previous regression is equal to the average of the risk - free rate and the coefficient (λ1) is equal to the average of the excess market return. After conducting the cross - sectional regression we end up with this equation;
i = 0.0009 + 0.085bi + ei
(0.1683) (1.3238)
R2 = 8.87%
F = 1.75
Following Fama - MacBeth procedure we observed that the λ0 was not equal to the average of the risk-free rate for the 10 years' period (), as well as the λ1 was not equal to the average risk premium for the same period (see appendix). This was the first step that made us to reject CAPM's validity and its ability to explain the excess stock returns only by using the market's premium and each stock's systematic risk.
In the second stage, we included the square of the beta and the idiosyncratic risk (σεi2) of each stock. Our aim was to see whether by introducing more explanatory variables the model would provide a better explanation of the dependent variable. Our results were as follows:
i = -0.0021 + 0.00087bi - 0.0035bi2 + 2.428σεi2 + ei
(-0.2308) (0.0372) (-0.2892) (3.6363)
R2 = 51.63%
F = 5.69
As the ANOVA table from the appendix shows, all p values are bigger than 0.05 except for the variance of residuals we can conclude that the coefficient of intercept, beta and beta square are insignificant (we accept the null hypothesis that the estimated coefficients equal to 0). However, we reject the null hypothesis for the variance of residuals, meaning that the coefficient is significant. As a result, the variance is the only significant explanatory variable and thus we can conclude that CAPM is not valid.
We can also observe that by adding the idiosyncratic risk and the square of beta, the goodness of fit (R2) of the model increases from 8.87% to 51.63%. Also, when we test to see whether the model provides a good fit for our data, we can conclude that in the first case, when the only explanatory variable is the systematic risk, the model does not provide a good fit, while in the second case, by adding the two explanatory variables, we reject the null hypothesis that the model does not provide a good fit.
In this case the only support that we can provide to CAPM is a partly qualitative one, as λ1=0 which means that there is no linear relationship between average stock returns and systematic risk. However, we cannot say in any way that CAPM is valid, as there is no quantitative support for this conclusion since λ0, which is assumed to replicate the risk-free rate, is negative and the estimated market risk premium 0.00087 is more than the observed excess market return (see appendix). Also, we can see that σεi2 seems to play an important role in the excess stock returns and it does command a risk premium for the stocks.
All in all, our analysis offer us some proofs and we are in a position of declaring CAPM as invalid.
Comparison with other studies
Across literature there is a plenty of other studies about CAPM's validity, with the first of them been conducted about the early 1970s. These were tests from Black, Jensen and Scholes (1972) and Fama - MacBeth (1973). Our results did not tie with the results of these studies, as they declared that CAPM is valid and we do not. The reason that our results are different are due to different testing horizons, as Fama - MacBeth (FM) tested the period from 1929 - 1954, or maybe due to the reason that FM had much more data for their stocks and because of that they formed different portfolios as it was impossible to conduct a test with each of about 400 stocks.
Another relevant study which questioned the accuracy of the model tested by FM was conducted by Banz in 1981. Banz revealed that there are some other factors that can explain stock returns better than the market risk premium. His work stipulated that the coefficient of firm size is better than CAPM's beta at explaining the cross - sectional variation in average returns on a particular data set. This effect is known as the size effect of returns. Our results are consistent with those found by Banz regarding CAPM's validity and the fact that the stock returns are not only explained by its relationship with systematic risk.
Miller and Scholes (1972) used a sectional model of excess returns on the American market to verify a series of new hypothesis. One of them is that the model proposed by Fama McBeth does not entirely support the relationship between risk and return because this is not a linear one. A wrong functional form of the model because of non-linearity would make a too big intercept and a very small slope compared to Sharpe's model. Thus, they estimated a regression which also included beta squared as an explanatory variable. Testing for non linearity implied testing for the significance of the coefficient of beta squared. Miller and Scholes have not discovered any proof of non linearity and most of the recent tests which use Fama Mcbeth methodology have not managed to reject the null that the coefficient of beta squared is equal to 0. This is in accordance with the results we found when including beta squared as an explanatory variable.
One of the reasons that we may be forced to take CAPM as not valid is that our excess market returns have a high variance (σm2). Guermat, Bulkley, Freeman and Harris mention in their study that the higher the variance (σm2), the higher the standard deviation of the standard error of FM and the lower the power of the FM test. In order to avoid reaching to invalid conclusions about the CAPM's efficiency, they introduce a new test to be implemented that offers a larger approximation to the reality and how stock excess returns are explained by the market's excess returns.
Elton and Gruber state that if the real beta is correlated with the variance of the stochastic errors from the regressions where each stock's beta is estimated, this variance - residual risk - will serve as an approximation for the real beta. There will also be a significant relationship between the stocks' excess returns and this residual risk. The model will thus have the stocks' excess returns as a dependent variable and the independent variables will be an intercept, beta, beta squared, and the variance of the errors from the cross sectional regression proposed by Fama and McBeth. When testing for the significance of the residual risk, we rejected the null of the t-test, finding the residual risk significant in explaining stocks' returns.
GBFH after identifying the weaknesses of FM test, they try to explain them by subtracting the realized excess market return each month from the estimated FM slope coefficient MFM,t = FM,t - Rm,t . Their results are that by doing this, the standard deviation of the test statistic is reduced at a significant level and this makes their model more powerful. They called this a modified FM test and it tends to be more specific than the traditional FM test (1973).
However, there are several criticisms to the robustness of the approach we used. To start with, there might be a high level of correlation between the explanatory variables, thus leading to multicollinearity. If this would be the case, the t and F test results may be biased due to the different values of the standard error estimates. Therefore we might find some explanatory variables as insignificant when they actually are, and exclude them from the model.
Also, better results could be achieved by adding dummy variables to test the effect of variables such as investment behaviour or equity analyst recommendations. Moreover, macroeconomic factors such as GDP, inflation or unemployment rate could prove to be significant explanatory variables. Because those factors also have important effect on market prices therefore returns of each stock.
Finally, we used S&P 100 as a proxy for the market portfolio. We might get a better result with using larger stock indexes as our market portfolio because market portfolio needs to be mean variance efficient but we don't know that our index is MV efficient or not. Best method of solving this problem is examining all stocks and create MV efficient portfolio. Of course this right market portfolio creation process might take very very long time therefore not so efficient.
Unfortunately there is no exact test to see multicollinearity exists. We can fix multicollinearity problem with exclude highly correlated explanatory variables in our equation, also, we can eliminate multicollinearity by including some omitted explanatory variables in the equation, if multicollinearity exists. As a result our t and F tests will be reliable so our model will be reliable.
References
Banz, R. 1981. "The relationship between returns and market value of common stock". Journal of Financial Economics 9: pg 3-18.
C. Guermat, G. Bulkley, M.C. Freeman and R.D.F. Harris "Testing CAPM: A Simple Alternative to Fama and MacBeth" Department of Economics - Xfi Centre for Finance and Investment, University of Exeter, paper number 04/06 pg. 1 -21.
Eugene F. Fama and Kenneth R. French, "The Capital Asset Pricing Model: Theory and Evidence (August 2003)". CRSP Working Paper No. 550; Tuck Business School Working Paper No. 03-26. http://ssrn.com/abstract=440920
G. Michailidis, S. Tsopoglou, D. Papanastasiou, E. Mariola "Testing the Capital Asset Pricing Model (CAPM): The Case of the Emerging Greek Securities Market" International Research Journal of Finance and Economics, Issue 4(2006) pg. 78-81
John H. Cochrane "Asset pricing" Princeton University Press 2001 pg.235-237, 243-250, 260-262
