Passive/ActiveStatistical1 min read
◆ The PatternTracking error measures deviation from the benchmark
Tracking error (TE) is the standard deviation of the difference between portfolio and benchmark returns:
TE = σ(Rp − Rb)
Passive index funds target TE < 10 bps. Active managers accept TE of 2–8 % depending on mandate. Active share measures the fraction of holdings that differ from the benchmark — high active share combined with low TE signals closet indexing.
// Interactive — active share and tracking error
Active share %
Closet indexing. A fund with high fees but low active share is a bad deal — pay passive fees for passive exposure.
Pattern bridge: Standard deviation and variance are core in Indicators — Standard Deviation.
Worked example
Twelve months of portfolio and benchmark returns. The portfolio is ahead by 1.9% a year with a tracking error of 0.94%: an information ratio of 2.0. That looks superb, but it is one year — twelve numbers — and an IR estimated from twelve months has a standard error of about 1.
Where it misleads
- Low tracking error can mean a closet indexer charging active fees; active share (Cremers & Petajisto 2009) shows how different the holdings really are.
- Tracking error from the past understates future risk if the portfolio’s bets have changed.
- Minimizing tracking error alone can push a portfolio away from the mean-variance frontier (Roll 1992).
In code
port = np.array([1.2, -0.5, 2.1, 0.8, -1.9, 1.5, 0.3, 2.4, -0.7, 1.0, 0.6, -0.2]) / 100 # 12 months, made up bench = np.array([1.0, -0.8, 1.8, 1.1, -2.3, 1.2, 0.5, 2.0, -0.4, 0.9, 0.2, -0.5]) / 100 active = port - bench te = active.std(ddof=1) * np.sqrt(12) # annualized tracking error ir = active.mean() * 12 / te # information ratio
Assumes import numpy as np and import pandas as pd. Each snippet carries its own example numbers; the comments say which are made up or simulated.
Sources
- K. J. M. Cremers & A. Petajisto, “How Active Is Your Fund Manager? A New Measure That Predicts Performance”, Review of Financial Studies 22(9), 2009
- R. Roll, “A Mean/Variance Analysis of Tracking Error”, Journal of Portfolio Management 18(4), 1992
- R. C. Grinold, “The Fundamental Law of Active Management”, Journal of Portfolio Management 15(3), 1989
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