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Okay, in general it may be true, but not often enough to make the blanket statement that indices are less risky than stocks.
The beta of a market is 1, it is not true that no stock's beta is <1.
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In the way that you used it, Beta represents correlation to the market.
Thus, while it can be used to measure the level of market risk; it can not be used to measure individual stock risk.
I suggest you spend some time on
riskgrades
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I know beta is not a measure of an individual stock's risk. I didn't mean to imply that. It was a seperate but somewhat related statement, just in case people reading this thread weren't aware that betas can be <1.