Sample tracking error

Calculation contract

a_t = portfolio_t - benchmark_t; TE = sample_sd(a); TE_ann = TE*sqrt(K)

Method and interpretation

The default active returns are 0.01, 0.01, 0.01 and -0.005. Their sample standard deviation is 0.0075. At twelve periods per year the scaled number is approximately 0.0259808.

The sample convention divides the centered squared deviations by n-1. The card reports both the original-period and annualized quantities so that a user can see whether a comparison differs because of sampling frequency or the underlying dispersion.

Where the reference stops

Square-root time scaling is an assumption, not a calendar lookup. Autocorrelated or changing active returns can invalidate its interpretation. Tracking error says nothing by itself about whether average active performance is positive, and it is not the standard deviation of compounded relative wealth.

Published synthetic inputs

{
  "portfolio": [
    0.02,
    -0.01,
    0.03,
    0.01
  ],
  "benchmark": [
    0.01,
    -0.02,
    0.02,
    0.015
  ],
  "periods_per_year": 12
}

Frozen calculation output

{
  "active_returns": [
    0.01,
    0.01,
    0.009999999999999998,
    -0.004999999999999999
  ],
  "mean_active_return": 0.00625,
  "sample_tracking_error": 0.0075,
  "annualized_tracking_error": 0.025980762113533156
}

Arithmetic period differences with sample n-1 dispersion. Square-root annualization assumes stationary, uncorrelated active returns; it is not a compounded wealth difference.

Verification coverage

• Sample denominator, annualization and identical-series zero tested.
• Mismatched observations and unbounded scaling factors rejected.

Unit and independent-reference checks test arithmetic and rejection rules, not future investment performance.

Method references

• William F. Sharpe: differential-return ratios and time aggregation
• Simple Statistics: reference implementation and API

Read the assumptions in context