One of our clients introduced me to alternative ways to calculate two commonly used risk-adjusted return measures: information ratio and Sharpe ratio. Their immediate derivation is from Zephyr, and I am attempting to identify their origin. I confirmed that Morningstar also uses them.
In both cases, they annualize and then do the math, rather than do the math and then annualize. This calls to mind the associative property of mathematics, which says the order does not matter. While in addition, this holds (2 + 4 = 4 + 2 = 6, for example), it does not in these methods, as we get different results.
This also reminded me of how some attempted to derive my favorite risk-adjusted measure, Modigliani-Modigliani (annualize first or last?).
The information ratio differences:
And, the differences with Sharpe ratio: