Regression Betas are betas that are basically a Return vs Return of the stock vs the index. There are multiple questions here: which index? and why?
Remember, R squared is a measure of how the portfolio is affected by price changes in the index. Higher the number, higher this relation. We always think that a higher R squared will give us the right answer.
You know what is the right answer? Both are screwed up. They are all crappy.
Reasons to hate BETAs:
To measure relative risk, you can use the standard deviation of returns or the downside risk. The downside risk is the probability-weighted average of returns below a certain threshold.