“If the stock moves exactly as the market does, the beta of it will be 0.”

Corrected, the statement should be:

“If the stock moves exactly as the market does, the beta of it will be 1″

Can I point out that this is just plain wrong? A beta of less than 1.0 does not mean an asset has a negatively correlation with the market. If an asset has a beta of less than 1.0 but greater than 0.0, it means the asset goes up when the market goes up but not as much as the market. It also means that the asset does not fall as much when the market declines.

Stocks with negative beta are inversely related to the market, so a stock with a beta of -1.0 would be expected to go in the opposite direction of the market but in the same magnitude. E.g. if the market is up 10% you would expect that stock to be down 10%. It is not clear that stocks with negative betas exist in the real world, other than such things as short funds.

Beta is indeed, as you say, a measure of risk. However, it is not a complete measure of risk, only a measure of one aspect of it. Stocks with high betas are by definition relatively volatile, because the market is volatile, but stocks with low betas are not, as you state, necessarily low risk. The classic example is a small biotech which could be very volatile but have a beta of zero, as it will go up or down based on the results of drug trials, etc.

Generally, the term alpha is used to refer to a manager’s stock picking value added, and is not used to describe a particular stock. Strictly speaking, it is the intercept of the regression mentioned above, not the error term, so even if applied to a single stock would not, as you say, represent everything that affects a stock returns other than beta.