## Friday, October 23, 2009

### Getting a grip on risk

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:

RP=RF+BP(RM-RF)

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.

#### 1 comment:

1. Stephen Campisi, CFA, Intuitive Performance SolutionsOctober 24, 2009 at 2:52 PM

An interesting discussion. What is missing here is a challenge to the idea that we get the same view of risk and return when we use data at any frequency: daily, weekly, monthly or annual. Everyone seems to blindly accept the untested and unproven idea that the risk/return relationship is unchanged by the frequency of the data. Why hasn't anyone asked whether annualization affects the outcome? Why is everyone so quick to discuss the nuances of math without understanding the nature of the data?

Let's look at this using the Fama-French monthly data for calendar years 1928 - 2008 for the small growth sector. I calculated the compound returns and standard deviations using monthly data and again using annual data (i.e. calendar year returns.) Then I tested one simple statistic - the coefficient of variation. This answers the question: "How much volatility risk must I take to earn one unit of compound return?" I tested this using monthly data, then I annualized the monthly results, and finally I performed the calculations using annual returns. It is obvious that you get a different result when you use monthly data and annualize it, or when you use annual data for the analysis.

Here are the results using monthly data:

Monthly data: Return = 0.70%, Std Dev = 7.86%, CV = 11.27
Annualized: Return = 8.70%, Std Dev = 27.23%, CV = 3.13

Here are the results using annual data (calendar year returns:)

Return = 8.70%, Std Dev = 33.61%, CV = 3.86

So, what about the assumption that annualizing from frequent data (monthly, weekly, daily) is OK? Clearly, more frequent data contains a lot of noise. And it's also clear that annualizing is not the same as calculating statistics from annual data. We should not be highly confident that we understand the risk of an investment from a short sample of data. Simply stated, the sample may not be representative. And, we see that the annualization process does not provide equivalence of results.