Rolls shows, that conventional tests of the CAPM are tautological, as, due to its mathmatical properties, any security or portfolio satifies ex-post the CAPM-Equation resp. lies on the SML, as long as the benchmark-portfolio, which is used to calculate the securities or portfolios beta, is efficient.
(A short description and proof is found, for instance, here
http://en.wikipedia.org/wiki/Rolls_critique)
However, this has very important implications for CAPM-based performance measures. The Jensen-Alpha, for instance, measures performance ex-post as the distance between the performance of a portfolio, that is predicted by its beta, and its actual performance.
Now, if ex-post any security or portfolio satifies the CAPM-Equation resp. lies on the SML, if the benchmark-portfolio used is efficient, how can the Jesen-Alpha significantly deviate from 0? If course, one reason my be that the benchmark-portfolio is not efficient. That, however, would question the validity of the Jensen-Measure, at all.
Has anyone an idea, how to solve this puzzle?
There are two aspects to this. First, it is both necessary and sufficient for at least one alpha to deviate from zero that the benchmark-portfolio ex-post does not lie on the SML. However, second, you are right that a non-zero alpha cannot strictly be interpreted in the Jensen sense as a CAPM risk-adjusted return because the CAPM does not hold (the benchmark portfolio cannot lie on the SML) and therefore any risk correction based on it is incorrect.
The Jensen-Alpha is an empirical matter... It is related to the variables proxy's choices, the equilibrium reference model and the market efficiency...
Maybe an easier way of looking at it is the following: there is always an equity premium and a vector of betas such that alpha is zero for every financial asset. Thus, you can always say that the reason you find alphas different from zero is not because CAPM is wrong, but rather because you don't use the right measure for the return of the market portfolio. There is no way to disentangle the two explanations.
We can have at least two main explanations for finding alphas different from zero: 1) we do not use the right proxy's choices when applying the CAPM or 2) we can ask about the market efficiency. Maybe one reason might be associated with the behavioural finance. Literature in behavioural finance suggests the presence of noise traders in the stock market with correlated behaviour and limits on arbitrage are conditions that can lead sentiment to influence asset prices (e.g. Shleifer and Summers, 1990; Baker and Wurgler, 2006). Shefrin (2001) conclude that stock price changes could be broken down into two stochastic processes, one related to fundamentals and the other to sentiment.
Your puzzle is related to the joint-hypothesis problem inherent in asset-pricing tests. Roll (1977) has indeed demonstrated that any CAPM assessment is in fact a joint assessment of (1) the pricing model and (2) the quality of the market index used. If a deviation from the pricing model (i.e., if a non-zero alpha is found) this can be due to failure of the CAPM. However, this can also be due to the inefficiency of the market index used. In an earlier JFQA paper, Roll (1968) demonstrate that the errors in the market index used induce a bias in the abnormal return estimated. This puzzle is a logjam inherent in all asset-pricing tests. Similar puzzles have also been identified. For example, Shanken (1982) pointed out that the APT is not testable because the factor model can be arbitrarily manipulated by repackaging a given set of securities. Hansen and Richard (1987) have argued that conditional models are not testable because it is impossible to observe the full information set of investors. This is also similar to the joint hypothesis problem inherent in the tests of market efficiency (see Fama (1970)).
This problem does not have a solution. It has been documented as a joint hypothesis problem.
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