Volatility, Beta and Return - Was there ever a meaningful relationship?

This paper deals with the relationship between volatility or, more correctly, b values and return within the equities asset class. The existence of such a relationship has been controversial in actuarial circles, but the assumption that a positive linear relationship exists is now taught as part of the curriculum. Most of the empirical studies of the relationship between b values and mean return have used arithmetic means of ‘discrete’ rates of return. By contrast, many of the apparently contradictory studies used continuous compounding, geometric means or their equivalent. A great deal of the controversy therefore hinges on the definition of mean return, which has not really been identified as an important issue. If the appropriate definition of ‘mean return’ for long term investment or asset modeling is mean continuously compounded return or its equivalent, then much of the empirical support for a positive relationship between b values and return may need to be re-evaluated.