Build one active-return series
For each aligned row, active return aᵢ equals portfolio return minus benchmark return. Let μa be its arithmetic mean and sa its sample standard deviation, using N − 1 in the variance denominator. With P observations per year, annualized tracking error equals sa × sqrt(P). The arithmetic information ratio equals (μa/sa) × sqrt(P). Both use the same active-return series, so a risk-free rate does not enter either calculation.
Check the annualization with three observations
Suppose monthly active returns are 0%, 1%, and 2%. Their mean is 1 percentage point and sample standard deviation is also 1 percentage point. With P = 12, tracking error is approximately 3.464% annually and the information ratio is approximately 3.464. These intentionally simple values demonstrate the arithmetic. Three monthly observations would be far too little history to regard this unusually high sample ratio as stable.
Distinguish no variation from no risk
If every active return is identical, sample tracking error is zero and the information ratio has no finite denominator. The portfolio may still be volatile in absolute terms because the benchmark can move substantially. Square-root annualization assumes a time-scaling structure that autocorrelated returns may violate. The calculation also treats unusually strong outperformance as variation. Review the observation count, actual active-return path, and benchmark suitability before comparing ratios produced from different samples or reporting frequencies.