Keep the paired observations
The hypothetical portfolio returns are 0.02, -0.01, 0.03 and 0.01. The corresponding reference returns are 0.01, -0.02, 0.02 and 0.015. Treat them as four aligned monthly periods for this calculation only. There is no actual fund, benchmark instrument or historical observation date. Their differences are 0.01, 0.01, 0.01 and -0.005.
The arithmetic mean difference is 0.00625. This is not the difference between the two compounded terminal wealth values, and it is not a money-weighted investor return. The card uses an additive active-return definition, which is preserved in the export together with both original arrays.
Inspect the sample dispersion
The three positive differences each lie 0.00375 above the mean; the final difference lies 0.01125 below it. The sum of squared deviations is 0.00016875. Dividing by three, not four, produces sample variance 0.00005625 and sample standard deviation 0.0075. A population denominator would produce a different statistic, not a more accurate version of the same stated calculation.
Annualization adds an assumption
Multiplying period tracking error by the square root of twelve gives approximately 0.0259808. Dividing the mean active return by period tracking error gives 0.8333333; multiplying that ratio by the same square root gives about 2.8867513. The large numerical ratio is unsurprising in a deliberately constructed tiny sample. It does not imply demonstrated investment skill.
The square-root scaling needs a suitable dependence and stationarity interpretation. Four observations do not become twelve observed periods because the output is labelled annualized. A serious evaluation would examine window selection, benchmark relevance, serial dependence, costs, uncertainty and the number of alternatives considered. None of those questions is resolved by these two summary numbers.
The zero-denominator control
A second arithmetic control sets portfolio and benchmark returns equal. Tracking error becomes zero and the information ratio is withheld as null. The reference does not manufacture an infinite score or convert the undefined ratio to zero. This treatment is visible in the model card and executable tests. Both the valid example and the control are software-correctness checks; neither is a new independent sample of strategy performance.
| Statistic | Synthetic value | Convention |
|---|---|---|
| Mean active return | 0.00625 | Arithmetic mean |
| Tracking error | 0.0075 | Sample n-1 standard deviation |
| Annualized tracking error | 0.0259807621 | Square-root twelve scaling |
| Annualized information ratio | 2.8867513459 | Mean / sample dispersion times sqrt(12) |
Reproduce the referenced calculations
Primary references
These sources support methods and conventions. The example values are original constructions, not observations extracted from these documents.
Related investigations
Risk measurementTracking error needs an explicit convention
Distinguish centered active-return volatility from tracking difference, root-mean-square deviation and forward risk estimates, with explicit sample and annualization choices.
Validation examplesThe denominator defines the information ratio
Keep active-return means and tracking error on compatible scales, and separate a large historical ratio from evidence about repeatable performance.
Validation examplesOut-of-sample evidence and multiple testing
Separate chronological evaluation from repeated model selection, and quantify how a hypothetical search across many null tests changes the chance of a false discovery.