CAPM and beta for individual stocks

Question

Why do we just assume that the beta is constant for a stock? Could it not very well be the case that the beta is 1.2 in a bear (baisse) market and 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 beta value 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?

Answer 1

I would say that we don't generally speaking assume that stock betas are constant. For example the paper Explanations for the Instability of Equity Beta: Risk-Free Rate Changes and Leverage Effects, from 1985 (!), cites at least six other earlier papers that the market risk of securities is not stable.

Many papers use a rolling window CAPM estimation approach (a popular way of allowing betas to vary) exactly because CAPM and more generally factor model betas do not seem to be constant. See for example the JFE paper CAPM for estimating the cost of equity capital: Interpreting the empirical evidence.

Answer 2

A beta is, by definition, constant (at least, over the time period it's calculated). You could divide a time period into subperiods based on what direction the index is going, and calculate two different measures for each direction, but then what you're calculating isn't the beta in general, it's a beta for those subperiods.