When 13 Minus 4 Equals 12, and 12 Equals 13
When would 13 minus 4 ever equal 12? How can 12 equal 13? When you’re talking about trading (whoops! “investing”) returns!
Imagine that, hypothetically, we KNEW with absolute certainty that two managers, trading different strategies, both had long-term cumulative annual growth rates (CAGR) of exactly 13%. Yes, there was variation from year to year, but we just KNOW they’re equivalent managers trading different styles/systems.
Now, by happenstance, imagine that after 47 months the managers are neck and neck, with the same returns through that timeframe. Along comes month 48, wherein the second manager has a return that is 4% lower than that of the first manager, who has the exact return needed to hit 13% annualized. Hmm, 13% annualized by four years, take 4% off the top, take the fourth root, golly gee, the second manager now has an 11.85% (roughly 12%) four-year annualized return! 13 minus 4 equals 12.
Now let’s forget that we know anything about the returns given by the two managers’ different styles. We are faced with two managers and their four-year return numbers, one of whom returned 12% annualized and one of whom returned 13% annualized, with similar volatility.
Is there really a difference in their returns?
Don’t confuse precision of measurement with accuracy; accept the fact that, statistically speaking, we can never know with precision, we can only become progressively more confident within a fog of uncertainty.
April 13th, 2008 at 3:16 pm
This analysis, even discussing the ‘pre-known’ relative returns, suffers from a variant of the ‘gambler fallacy’. http://en.wikipedia.org/wiki/Gambler’s_fallacy
A better approach would be to recognize that, after 48 months, when investor B under-performs investor A, you have to mark down your estimate of his returns, since you didn’t have -knowledge- of his future returns, you had -estimate- of same.
April 13th, 2008 at 3:35 pm
The first analysis is fallacious, since we can never know what the return stream distribution of a trader or manager or strategy is - and that is exactly my point, that we can never know what the distribution is, we only measure a sample from it.
I prefer to view the second case as an exercise in the uncertainty of sampling distributions - we never have full credibility of results, we only have a confidence interval about the results, and that interval is smaller (but never zero) for a given probability as sample size increases. In a sample of just a few years, I would not mark down either estimate; rather, if they were relatively close (12% vs. 13%) and had similar volatility, I would consider the results to be equivalent, and would grade them on other things (transaction expense, simplicity of model, etc.).
Now a 1% difference in annualized returns over a TEN year period? That would probably get my attention.