Why do we just assume that the β is symmetric for a stock? Could it not very well be the case that the β has a larger leverage (covariance with the market) for example 1.2 in a bear (baisse) market and a lower leverage (covariance) e.g. 0.8 in a bull (hausse) market for an individual stock, or the other way around? This would mean that the stock drops more than the index when the index drops and gains less than the index when the index gains. Of course this is neither a rational investment compared to index nor is it by definition a covariance, but it looks from numbers and graphs that several stocks or investments do not have the same β during a bull/ gaining market compared to a bear dropping market, it will drop more than the market when the market drops and not even outperform the market when the market gains.
Or is that something that would be explained by the alpha value of the CAPM, meaning that the alpha is some individual characteristic of the company?