One of our clients asked us to calculate a variety of risk statistics for them this week and in the course of the assignment we were asked some intriguing questions, a couple of which I'll share with you today.
Annualizing Beta: can we annualize beta? I contacted a colleague who advised me that "yes, we can." We simply multiply each monthly return by 12 (because an arithmetic approach applies to risk, vs. a geometric for returns; translation: returns compound, risks don't). But what happens when we do this? We get the exact same result? And why? Because since we are uniformly altering both the portfolio and benchmark returns, the covariance and variance (which are used in the formula) are adjusted uniformly, resulting in the same answer as the "non-annualized" value. Furthermore, as another colleague put it, "Beta is the slope of the regression equation – it makes absolutely no sense to annualize it." Consequently, beta is beta ... annualization does not apply.
Deriving beta from CAPM: I found this suggestion somewhat intriguing. Rather than use the standard covariance/variance formula, why not derive beta directly from CAPM? The formula, as our client provide us, was:
Translation: the portfolio's expected return equals the risk free rate, plus the portfolio's beta times the difference between the benchmark (or market) return and the risk free rate. Simple algebra suggests that if one knows the risk free rate, market return and portfolio return, one can easily derive beta, yes? Simple, yes? Unfortunately, this doesn't work, for at least a couple reasons.
First, Fama/French (as well as others) showed that the expected return isn't derived just from beta: they posited a 3-factor model which has been deemed superior to the single-factor CAPM approach. Second, the formula as provided is missing an "error term," which is needed because, as just mentioned, the formula is flawed ... beta doesn't cover everything. The "=" sign should be replaced by an "≠" (i.e., not equal) sign, thus negating the simple algebraic approach.
More was discussed and this month's newsletter will be replete with some of this material.